[STATEMENT]
lemma instant_index_increase:
  assumes \<open>\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
  shows   \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>)  \<hookrightarrow>\<^bsup>k\<^esup>  (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                         \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof (insert assms, induct \<Psi> arbitrary: \<Gamma> \<Phi>)
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>a \<Psi> \<Gamma> \<Phi>. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (Nil \<Gamma> \<Phi>)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>a \<Psi> \<Gamma> \<Phi>. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>1\<^esup> (\<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [])\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []
[PROOF STEP]
using instant_i intro_part
[PROOF STATE]
proof (prove)
using this:
?\<Gamma>, ?n \<turnstile> [] \<triangleright> ?\<Phi> \<hookrightarrow>\<^sub>i ?\<Gamma>, Suc ?n \<turnstile> ?\<Phi> \<triangleright> []
?\<Gamma>\<^sub>1, ?n\<^sub>1 \<turnstile> ?\<Psi>\<^sub>1 \<triangleright> ?\<Phi>\<^sub>1 \<hookrightarrow>\<^sub>i ?\<Gamma>\<^sub>2, ?n\<^sub>2 \<turnstile> ?\<Psi>\<^sub>2 \<triangleright> ?\<Phi>\<^sub>2 \<Longrightarrow> ?\<Gamma>\<^sub>1, ?n\<^sub>1 \<turnstile> ?\<Psi>\<^sub>1 \<triangleright> ?\<Phi>\<^sub>1 \<hookrightarrow> ?\<Gamma>\<^sub>2, ?n\<^sub>2 \<turnstile> ?\<Psi>\<^sub>2 \<triangleright> ?\<Phi>\<^sub>2

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []
[PROOF STEP]
by fastforce
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using assms Nil.prems calculation(2)
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []
\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>1\<^esup> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []
\<lbrakk> \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> [] \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>a \<Psi> \<Gamma> \<Phi>. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>a \<Psi> \<Gamma> \<Phi>. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (Cons \<psi> \<Psi>)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<psi> # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>a \<Psi> \<Gamma> \<Phi>. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> a # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<psi> # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<psi> # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<psi> # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof (induct \<psi>)
[PROOF STATE]
proof (state)
goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (SporadicOn K\<^sub>1 \<tau> K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches: \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                      = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                      \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_sporadicon_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 sporadic ?\<tau> on ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 sporadic ?\<tau> on ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> ?K\<^sub>1 \<Up> ?n # ?K\<^sub>2 \<Down> ?n @ ?\<tau> # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br1: \<open>\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                    \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
                      ((\<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                          \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                      \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h1: \<open>\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>))
                                      \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                                 \<and> (\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g)\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h1 SporadicOn.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
              fp:\<open>((\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>))
                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have
            \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part sporadic_on_e1)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with fp relpowp_Suc_I2
[PROOF STATE]
proof (chain)
picking this:
\<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>
[PROOF STEP]
have
            \<open>((\<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
thus ?thesis
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using fp
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                  \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                        \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                            \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi>)
                                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                             \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h2 SporadicOn.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
            where fp:\<open>((((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi>)
                            \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
              and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have pc:\<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi>)\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>
[PROOF STEP]
by (simp add: elims_part sporadic_on_e2)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi>)
                    \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi>
K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from branches SporadicOn.prems(2)
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have
          \<open>\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
             \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Down> n @ \<tau>) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (8 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 sporadic x2 on x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 8. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with br1 br2
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Down> n @ \<tau> # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 sporadic \<tau> on K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (7 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (7 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (TagRelation K\<^sub>1 K\<^sub>2 R)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (7 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches: \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          = \<lbrakk> ((\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>2, n)\<rfloor> \<in> R) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_tagrel_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (time-relation \<lfloor>?K\<^sub>1, ?K\<^sub>2\<rfloor> \<in> ?R) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (?K\<^sub>1, ?n), \<tau>\<^sub>v\<^sub>a\<^sub>r (?K\<^sub>2, ?n)\<rfloor> \<in> ?R # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (time-relation \<lfloor>?K\<^sub>1, ?K\<^sub>2\<rfloor> \<in> ?R) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (7 subgoals):
 1. \<And>x1 x2 x3. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>x1, x2\<rfloor> \<in> x3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 7. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
thus ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
            ((((\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>2, n)\<rfloor> \<in> R) # \<Gamma>), n
                \<turnstile> \<Psi> \<triangleright> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>))
              \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)) \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using TagRelation.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
          where fp:\<open>((((\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r(K\<^sub>2, n)\<rfloor> \<in> R) # \<Gamma>), n
                        \<turnstile> \<Psi> \<triangleright> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>))
                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
            and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have pc:\<open>(\<Gamma>, n \<turnstile> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi>) \<triangleright> \<Phi>)
            \<hookrightarrow> (((\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R) # \<Gamma>), n
                  \<turnstile> \<Psi> \<triangleright> ((time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>
[PROOF STEP]
by (simp add: elims_part tagrel_e)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi>)
                \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi>
\<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lfloor>\<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>1, n), \<tau>\<^sub>v\<^sub>a\<^sub>r (K\<^sub>2, n)\<rfloor> \<in> R # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (time-relation \<lfloor>K\<^sub>1, K\<^sub>2\<rfloor> \<in> R) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (Implies K\<^sub>1 K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches: \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 implies K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          = \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_implies_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 implies ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> ?K\<^sub>1 \<not>\<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 implies ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> ?K\<^sub>1 \<Up> ?n # ?K\<^sub>2 \<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 implies ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 implies K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
                    ((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))
                        \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h1 Implies.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
          fp:\<open>((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))
            \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
          and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have pc:\<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi>)
                  \<hookrightarrow> (((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part implies_e1)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n
                                \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                            \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 implies K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                              \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n
                          \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. (
                        (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))
                          \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)
                    ) \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h2 Implies.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
            fp:\<open>(((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
        and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 implies K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part implies_e2)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 implies K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>
K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (6 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 6. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using Implies.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (ImpliesNot K\<^sub>1 K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches: \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 implies not K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          = \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_implies_not_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 implies not ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> ?K\<^sub>1 \<not>\<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 implies not ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> ?K\<^sub>1 \<Up> ?n # ?K\<^sub>2 \<not>\<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 implies not ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
                            \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 implies not K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
                    ((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>))
                      \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h1 ImpliesNot.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
          fp:\<open>((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>))
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
          and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have pc:\<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi>)
                  \<hookrightarrow> (((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part implies_not_e1)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n
                            \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                            \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 implies not K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                              \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n
                          \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. (
                     (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n
                       \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>)) \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)
                    ) \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h2 ImpliesNot.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
            fp:\<open>(((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>))
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
        and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 implies not K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 implies not K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part implies_not_e2)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 implies not K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi>
K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (5 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 implies not x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 5. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using ImpliesNot.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies not K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 implies not K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (TimeDelayedBy K\<^sub>1 \<delta>\<tau> K\<^sub>2 K\<^sub>3)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches:
        \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          = \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_timedelayed_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 time-delayed by ?\<delta>\<tau> on ?K\<^sub>2 implies ?K\<^sub>3) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> ?K\<^sub>1 \<not>\<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 time-delayed by ?\<delta>\<tau> on ?K\<^sub>2 implies ?K\<^sub>3) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> ?K\<^sub>1 \<Up> ?n # ?K\<^sub>2 @ ?n \<oplus> ?\<delta>\<tau> \<Rightarrow> ?K\<^sub>3 # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 time-delayed by ?\<delta>\<tau> on ?K\<^sub>2 implies ?K\<^sub>3) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br1:
        \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
            ((\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
            \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
                         \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
          ((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))
            \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
          \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h1 TimeDelayedBy.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
          where fp:\<open>(((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
                      \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))
                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
            and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n
                    \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>
[PROOF STEP]
by (simp add: elims_part timedelayed_e1)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br2:
        \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
              ((\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
              \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h2: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
                    \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
                                 \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))
                                 \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                                \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h2 TimeDelayedBy.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
          where fp:\<open>(((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
                         \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))
                       \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
            and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<Up> n) # (K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3) # \<Gamma>), n
                  \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>
[PROOF STEP]
by (simp add: elims_part timedelayed_e2)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with fp relpowp_Suc_I2
[PROOF STATE]
proof (chain)
picking this:
K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>
[PROOF STEP]
have
          \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi>) \<triangleright> \<Phi>)
            \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (4 subgoals):
 1. \<And>x1 x2 x3 x4. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 time-delayed by x2 on x3 implies x4) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 4. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using TimeDelayedBy.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 @ n \<oplus> \<delta>\<tau> \<Rightarrow> K\<^sub>3 # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 time-delayed by \<delta>\<tau> on K\<^sub>2 implies K\<^sub>3) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (WeaklyPrecedes K\<^sub>1 K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g =
        \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
            \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_weakly_precedes_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 weakly precedes ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> ?K\<^sub>2 ?n, #\<^sup>\<le> ?K\<^sub>1 ?n\<rceil> \<in> \<lambda>(x, y). x \<le> y # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 weakly precedes ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<rho> \<in> \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                            \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
            \<Longrightarrow> (\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                  \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                \<and> (\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume \<open>\<rho> \<in> \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                        \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                                  \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>))
                             \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                          \<and> (\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g)\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using  WeaklyPrecedes.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
          where fp:\<open>(((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                                  \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>))
                             \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
            and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow> (((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part weakly_precedes_e)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with fp relpowp_Suc_I2
[PROOF STATE]
proof (chain)
picking this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                      \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (3 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 weakly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 3. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using WeaklyPrecedes.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>\<le> K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 weakly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (StrictlyPrecedes K\<^sub>1 K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g =
        \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
          \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_strictly_precedes_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 strictly precedes ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> ?K\<^sub>2 ?n, #\<^sup>< ?K\<^sub>1 ?n\<rceil> \<in> \<lambda>(x, y). x \<le> y # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 strictly precedes ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<rho> \<in> \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                            \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
            \<Longrightarrow> (\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                  \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                \<and> (\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume \<open>\<rho> \<in> \<lbrakk> ((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                        \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                                  \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>))
                             \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                            \<and> (\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g)\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using  StrictlyPrecedes.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k
          where fp:\<open>(((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
                                  \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>))
                             \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
            and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                \<hookrightarrow> (((\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> (\<lambda>(x, y). x \<le> y)) # \<Gamma>), n
              \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part strictly_precedes_e)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with fp relpowp_Suc_I2
[PROOF STATE]
proof (chain)
picking this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                      \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (2 subgoals):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 strictly precedes x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
 2. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using StrictlyPrecedes.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
next
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
case (Kills K\<^sub>1 K\<^sub>2)
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have branches: \<open>\<lbrakk> \<Gamma>, n \<turnstile> ((K\<^sub>1 kills K\<^sub>2) # \<Psi>) \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          = \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<union> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> \<ge> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using HeronConf_interp_stepwise_kills_cases
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> ?\<Gamma>, ?n \<turnstile> (?K\<^sub>1 kills ?K\<^sub>2) # ?\<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> ?K\<^sub>1 \<not>\<Up> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 kills ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> ?K\<^sub>1 \<Up> ?n # ?K\<^sub>2 \<not>\<Up> \<ge> ?n # ?\<Gamma>, ?n \<turnstile> ?\<Psi> \<triangleright> (?K\<^sub>1 kills ?K\<^sub>2) # ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
                \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 kills K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h1: \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k.
                    ((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))
                    \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                  \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h1 Kills.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
          fp:\<open>((((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))\<close>
          and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have pc:\<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi>)
                  \<hookrightarrow> (((K\<^sub>1 \<not>\<Up> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part kills_e1)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>
K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
moreover
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have br2:
        \<open>\<rho> \<in> \<lbrakk> ((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> \<ge> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>) \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
          \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ((\<Gamma>, n \<turnstile> ((K\<^sub>1 kills K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
                                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k))
                          \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
proof -
[PROOF STATE]
proof (state)
goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
assume h2: \<open>\<rho> \<in> \<lbrakk>((K\<^sub>1 \<Up> n)#(K\<^sub>2 \<not>\<Up> \<ge> n)#\<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2)#\<Phi>)\<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
then
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. (
                    (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> \<ge> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))
                      \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)
                    ) \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using h2 Kills.prems
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. ?\<Gamma>, n \<turnstile> \<Psi> \<triangleright> ?\<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by simp
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
from this
[PROOF STATE]
proof (chain)
picking this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
obtain \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k where
            fp:\<open>(((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> \<ge> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))
                \<hookrightarrow>\<^bsup>k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
        and rc:\<open>\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<close>
[PROOF STATE]
proof (prove)
using this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. (\<And>k \<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k. \<lbrakk>K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k; \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
have \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 kills K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>)
              \<hookrightarrow> (((K\<^sub>1 \<Up> n) # (K\<^sub>2 \<not>\<Up> \<ge> n) # \<Gamma>), n \<turnstile> \<Psi> \<triangleright> ((K\<^sub>1 kills K\<^sub>2) # \<Phi>))\<close>
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>
[PROOF STEP]
by (simp add: elims_part kills_e2)
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
hence \<open>(\<Gamma>, n \<turnstile> ((K\<^sub>1 kills K\<^sub>2) # \<Psi>) \<triangleright> \<Phi>) \<hookrightarrow>\<^bsup>Suc k\<^esup> (\<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k)\<close>
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
using fp relpowp_Suc_I2
[PROOF STATE]
proof (prove)
using this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi>
K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
\<lbrakk>?P ?x ?y; (?P ^^ ?n) ?y ?z\<rbrakk> \<Longrightarrow> (?P ^^ Suc ?n) ?x ?z

goal (1 subgoal):
 1. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
by auto
[PROOF STATE]
proof (state)
this:
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
with rc
[PROOF STATE]
proof (chain)
picking this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k
[PROOF STEP]
show ?thesis
[PROOF STATE]
proof (prove)
using this:
\<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>Suc k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<And>x1 x2. \<lbrakk>\<And>\<Gamma> \<Phi>. \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g; \<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g\<rbrakk> \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (x1 kills x2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
ultimately
[PROOF STATE]
proof (chain)
picking this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
show ?case
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
using Kills.prems(2)
[PROOF STATE]
proof (prove)
using this:
\<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g = \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<union> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<not>\<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> K\<^sub>1 \<Up> n # K\<^sub>2 \<not>\<Up> \<ge> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 kills K\<^sub>2) # \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g \<Longrightarrow> \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
\<rho> \<in> \<lbrakk> \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal (1 subgoal):
 1. \<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g
[PROOF STEP]
by blast
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> (K\<^sub>1 kills K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed
[PROOF STATE]
proof (state)
this:
\<exists>\<Gamma>\<^sub>k \<Psi>\<^sub>k \<Phi>\<^sub>k k. \<Gamma>, n \<turnstile> \<psi> # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^bsup>k\<^esup> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<and> \<rho> \<in> \<lbrakk> \<Gamma>\<^sub>k, Suc n \<turnstile> \<Psi>\<^sub>k \<triangleright> \<Phi>\<^sub>k \<rbrakk>\<^sub>c\<^sub>o\<^sub>n\<^sub>f\<^sub>i\<^sub>g

goal:
No subgoals!
[PROOF STEP]
qed