[STATEMENT]
lemma SA_plus:
"(\<oplus>\<^bsub>SA n\<^esub>) = (\<oplus>\<^bsub>Fun\<^bsub>n\<^esub> Q\<^sub>p\<^esub>)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<oplus>\<^bsub>SA n\<^esub>) = (\<oplus>\<^bsub>function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p\<^esub>)
[PROOF STEP]
unfolding SA_def
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. (\<oplus>\<^bsub>function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p\<lparr>carrier := semialg_functions n\<rparr>\<^esub>) = (\<oplus>\<^bsub>function_ring (carrier (Q\<^sub>p\<^bsup>n\<^esup>)) Q\<^sub>p\<^esub>)
[PROOF STEP]
by simp


[STATEMENT]
lemma "(13::real) < 18 - 4"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. 13 < 18 - 4
[PROOF STEP]
by simp


[STATEMENT]
lemma surjective_linorder_matrix:
  fixes f :: "('a::finite,'b::linorder_stone_relation_algebra_expansion) square"
  shows "matrix_bounded_idempotent_semiring.surjective f \<longleftrightarrow> (\<forall>j . \<exists>i . f (i,j) = top)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. matrix_bounded_idempotent_semiring.surjective f = (\<forall>j. \<exists>i. f (i, j) = top)
[PROOF STEP]
by (unfold matrix_stone_relation_algebra.surjective_conv_total total_linorder_matrix) (simp add: conv_matrix_def)


[STATEMENT]
lemma euler_mascheroni_pos: "euler_mascheroni > (0::real)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. 0 < euler_mascheroni
[PROOF STEP]
using euler_mascheroni_lower[of 0] ln_2_less_1
[PROOF STATE]
proof (prove)
using this:
harm (Suc 0) - ln (real (0 + 2)) + 1 / real (2 * (0 + 2)) \<le> euler_mascheroni
ln 2 < 1

goal (1 subgoal):
 1. 0 < euler_mascheroni
[PROOF STEP]
by (simp add: harm_def)


[STATEMENT]
lemma le_Suc_eq_insert: "{i. i <Suc n} = insert n {i. i< n}"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. {i. i < Suc n} = insert n {i. i < n}
[PROOF STEP]
by auto


[STATEMENT]
lemma "dist x y \<ge> v \<Longrightarrow> dist x y + dist (a::'a) b \<ge> v" for x::"('a::metric_space)"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. v \<le> dist x y \<Longrightarrow> v \<le> dist x y + dist a b
[PROOF STEP]
by metric_eisbach