[STATEMENT]
lemma (in Corps) Hensel_starter:"\<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K;
    PolynRg R (Vr K v) X; PolynRg S ((Vr K v) /\<^sub>r (vp K v)) Y;
    t \<in> carrier (Vr K v); vp K v = (Vr K v) \<diamondsuit>\<^sub>p t;
    f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S;
    0 < deg S ((Vr K v) /\<^sub>r (vp K v)) Y g';
    0 < deg S ((Vr K v) /\<^sub>r (vp K v)) Y h';
    ((erH R (Vr K v) X S ((Vr K v) /\<^sub>r (vp K v)) Y
             (pj  (Vr K v) (vp K v))) f) =  g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h';
    rel_prime_pols S ((Vr K v) /\<^sub>r (vp K v)) Y g' h'\<rbrakk> \<Longrightarrow>
 \<exists>g h. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and>
  deg R (Vr K v) X g \<le> deg S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
   (erH R (Vr K v) X S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
   (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) g) \<and> (deg R (Vr K v) X h +
    deg S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S
     ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) g)
    \<le> deg R (Vr K v) X f) \<and>
   (erH R (Vr K v) X S ((Vr K v) /\<^sub>r (vp K v)) Y
                            (pj  (Vr K v) (vp K v))) g = g' \<and>
    (erH R (Vr K v) X S ((Vr K v) /\<^sub>r (vp K v)) Y
                            (pj  (Vr K v) (vp K v))) h = h' \<and>
  0 < deg S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
   (erH R (Vr K v) X S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
   (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) g) \<and>
  0 < deg S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
    (erH R (Vr K v) X S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
   (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) h) \<and>
  rel_prime_pols S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
    (erH R (Vr K v) X S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
    (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) g)
    (erH R (Vr K v) X S ((Vr K v) /\<^sub>r ((Vr K v) \<diamondsuit>\<^sub>p t)) Y
    (pj (Vr K v) ((Vr K v) \<diamondsuit>\<^sub>p t)) h) \<and>
 P_mod R (Vr K v) X ((Vr K v) \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r vp K v) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r vp K v) Y g'; 0 < deg S (Vr K v /\<^sub>r vp K v) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r vp K v) Y g' h'\<rbrakk> \<Longrightarrow> \<exists>g h. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h)
[PROOF STEP]
apply (frule Vr_ring[of v],
       frule PolynRg.subring[of R "Vr K v" X],
       frule vp_maximal [of v], frule PolynRg.is_Ring,
       frule Ring.subring_Ring[of R "Vr K v"], assumption+,
       frule Ring.residue_field_cd[of "Vr K v" "vp K v"], assumption+,
       frule Corps.field_is_ring[of "Vr K v /\<^sub>r vp K v"],
       frule pj_Hom[of "Vr K v" "vp K v"], frule vp_ideal[of "v"],
       simp add:Ring.maximal_ideal_ideal)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r vp K v) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r vp K v) Y g'; 0 < deg S (Vr K v /\<^sub>r vp K v) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r vp K v) Y g' h'; Ring (Vr K v); Subring R (Vr K v); maximal_ideal (Vr K v) (vp K v); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r vp K v); Ring (Vr K v /\<^sub>r vp K v); pj (Vr K v) (vp K v) \<in> rHom (Vr K v) (Vr K v /\<^sub>r vp K v)\<rbrakk> \<Longrightarrow> \<exists>g h. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r vp K v) Y (pj (Vr K v) (vp K v)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h)
[PROOF STEP]
apply (frule Corps.field_is_idom[of "(Vr K v) /\<^sub>r (vp K v)"],
       frule Vr_integral[of v], simp,
       frule Vr_mem_f_mem[of v t], assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
[PROOF STEP]
apply (frule PolynRg.erH_inv[of R "Vr K v" X "Vr K v \<diamondsuit>\<^sub>p t" S Y "g'"],
       assumption+, simp add:Ring.maximal_ideal_ideal,
       simp add:PolynRg.is_Ring, assumption+, erule bexE, erule conjE)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g \<in> carrier R; deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g'\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
[PROOF STEP]
apply (frule PolynRg.erH_inv[of R "Vr K v" X "Vr K v \<diamondsuit>\<^sub>p t" S Y "h'"],
       assumption+, simp add:Ring.maximal_ideal_ideal,
       simp add:PolynRg.is_Ring, assumption+, erule bexE, erule conjE)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g ga. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g \<in> carrier R; deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g'; ga \<in> carrier R; deg R (Vr K v) X ga \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) ga = h'\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
[PROOF STEP]
apply (rename_tac g0 h0)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
[PROOF STEP]
apply (subgoal_tac " g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and>
  deg R (Vr K v) X g0 \<le>
  deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S
      (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and>
  deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
      (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and>
    0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S
      (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and>
   0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S
       (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and>
   rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
      (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0)
      (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and>
   P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (thin_tac "g' \<in> carrier S",
        thin_tac "h' \<in> carrier S",
        thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'",
        thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'",
        thin_tac "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'",
        thin_tac "rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'",
        thin_tac "Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t))",
        thin_tac "Ring  (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t))",
        thin_tac "vp K v = Vr K v \<diamondsuit>\<^sub>p t")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)\<rbrakk> \<Longrightarrow> \<exists>g. g \<noteq> \<zero>\<^bsub>R\<^esub> \<and> (\<exists>h. h \<noteq> \<zero>\<^bsub>R\<^esub> \<and> g \<in> carrier R \<and> h \<in> carrier R \<and> deg R (Vr K v) X g \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> deg R (Vr K v) X h + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<le> deg R (Vr K v) X f \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g = g' \<and> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h = h' \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g \<cdot>\<^sub>r\<^bsub>R\<^esub> h))
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> g0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rule conjI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> g0 \<noteq> \<zero>\<^bsub>R\<^esub>
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'",
    thin_tac "rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'",
   thin_tac "deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R\<rbrakk> \<Longrightarrow> g0 \<noteq> \<zero>\<^bsub>R\<^esub>
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rule contrapos_pp, simp+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; \<zero>\<^bsub>R\<^esub> \<in> carrier R; deg R (Vr K v) X \<zero>\<^bsub>R\<^esub> \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<zero>\<^bsub>R\<^esub> = g'; h0 \<in> carrier R; g0 = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> False
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (simp add:PolynRg.erH_rHom_0[of R "Vr K v" X S
      "Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y "pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; \<zero>\<^bsub>R\<^esub> \<in> carrier R; deg R (Vr K v) X \<zero>\<^bsub>R\<^esub> \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; \<zero>\<^bsub>S\<^esub> = g'; h0 \<in> carrier R; g0 = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> False
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rotate_tac -3, drule sym, simp add:deg_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; g0 = \<zero>\<^bsub>R\<^esub>; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; \<zero>\<^bsub>S\<^esub> \<in> carrier S; h' \<in> carrier S; 0 < - \<infinity>; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; \<zero>\<^bsub>R\<^esub> \<in> carrier R; g' = \<zero>\<^bsub>S\<^esub>\<rbrakk> \<Longrightarrow> False
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (drule aless_imp_le[of "0" "-\<infinity>"],
        cut_tac minf_le_any[of "0"],
        frule ale_antisym[of "0" "-\<infinity>"], simp only:ant_0[THEN sym], simp)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub> \<and> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rule conjI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub>
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'",
    thin_tac "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'",
    thin_tac "rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'",
   thin_tac "deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; h0 \<in> carrier R; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> h0 \<noteq> \<zero>\<^bsub>R\<^esub>
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rule contrapos_pp, simp+,
        simp add:PolynRg.erH_rHom_0[of R "Vr K v" X S
      "Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y "pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; \<zero>\<^bsub>R\<^esub> \<in> carrier R; \<zero>\<^bsub>S\<^esub> = h'; h0 = \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> False
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rotate_tac -2, drule sym, simp add:deg_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>g0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; \<zero>\<^bsub>S\<^esub> \<in> carrier S; 0 < - \<infinity>; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; (if g0 = \<zero>\<^bsub>R\<^esub> then - \<infinity> else an (deg_n R (Vr K v) X g0)) \<le> (if g' = \<zero>\<^bsub>S\<^esub> then - \<infinity> else an (deg_n S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g')); \<zero>\<^bsub>R\<^esub> \<in> carrier R; h' = \<zero>\<^bsub>S\<^esub>\<rbrakk> \<Longrightarrow> False
 2. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule aless_imp_le[of "0" "-\<infinity>"], thin_tac "0 < - \<infinity>",
        cut_tac minf_le_any[of "0"],
        frule ale_antisym[of "0" "-\<infinity>"], simp only:ant_0[THEN sym], simp)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule_tac x = "deg R (Vr K v) X h0" and
       y = "deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'" and
       z = "deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'" in aadd_le_mono)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h' + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'\<rbrakk> \<Longrightarrow> deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<le> deg R (Vr K v) X f \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) \<and> 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0) (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (simp add:PolynRg.deg_mult_pols1[THEN sym, of S
 "Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y "h'" "g'"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (h' \<cdot>\<^sub>r\<^bsub>S\<^esub> g')\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule PolynRg.is_Ring[of S "Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y],
        simp add:Ring.ring_tOp_commute[of S "h'" "g'"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'); Ring S\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rotate_tac 11, drule sym)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h'); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule PolynRg.erH_rHom[of R "Vr K v" X S
      "(Vr K v) /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y "pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)"],
      assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule PolynRg.pHom_dec_deg[of R "Vr K v" X S "(Vr K v) /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)" Y "erH R (Vr K v) X S ((Vr K v) /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t))
 Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t))" "f"], assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f) \<le> deg R (Vr K v) X f\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule_tac i = "deg R (Vr K v) X h0 +
        deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'" in ale_trans[of _
        "deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
           (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f)" "deg R (Vr K v) X f"],
       assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f) \<le> deg R (Vr K v) X f; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f\<rbrakk> \<Longrightarrow> deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; 0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f) \<le> deg R (Vr K v) X f; deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' \<le> deg R (Vr K v) X f\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (thin_tac "deg R (Vr K v) X h0 +
        deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'  \<le> deg R (Vr K v) X f",
       thin_tac "deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
           (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f)  \<le> deg R (Vr K v) X f",
       thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'",
       thin_tac "0 < deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'",
      thin_tac "deg R (Vr K v) X h0 + deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'
       \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
           (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f)",
     thin_tac "rel_prime_pols S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g' h'")
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 = g'; h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rotate_tac 12, drule sym)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = h'; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (drule sym)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y h'; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g' \<in> carrier S; h' \<in> carrier S; g' \<cdot>\<^sub>r\<^bsub>S\<^esub> h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y g'; g' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0; h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply simp
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0); g' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0; h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (frule_tac x = g0 and y = h0 in Ring.ring_tOp_closed[of "R"],
       assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0); Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0); g' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0; h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (thin_tac "deg R (Vr K v) X h0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
  (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0)",
       thin_tac "h' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0",
       thin_tac "g' = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0",
       thin_tac "deg R (Vr K v) X g0 \<le> deg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
             (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0)")
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (subst PolynRg.P_mod_diff[THEN sym, of R "Vr K v" X "Vr K v \<diamondsuit>\<^sub>p t"
       S Y f], assumption+)
[PROOF STATE]
proof (prove)
goal (5 subgoals):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)
 2. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
 3. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> f \<in> carrier R
 4. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R
 5. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (simp add:Ring.maximal_ideal_ideal, assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R\<rbrakk> \<Longrightarrow> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (rotate_tac 12, drule sym)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>g0 h0. \<lbrakk>erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) \<in> pHom R Vr K v X, S Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t) Y; Subring R (Vr K v); maximal_ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); Ring R; Ring (Vr K v); Corps (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Ring (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t) \<in> rHom (Vr K v) (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)); Idomain (Vr K v); t \<in> carrier K; g0 \<in> carrier R; g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0 \<in> carrier R; h0 \<in> carrier R; Ring S; valuation K v; Complete\<^bsub>v\<^esub> K; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0 \<in> carrier S; erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) g0 \<cdot>\<^sub>r\<^bsub>S\<^esub> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) h0\<rbrakk> \<Longrightarrow> erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) f = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0)
[PROOF STEP]
apply (subst PolynRg.erH_mult[of R "Vr K v" X S "Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)"
        Y], assumption+)
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done