[STATEMENT]
lemma (in Corps) H_Plimit_f:"\<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K;
     t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t;
     PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y;
     f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>;
     h0 \<noteq> \<zero>\<^bsub>R\<^esub>;
     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);
     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);
     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;

     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);

     erH R (Vr K v) X S (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;

        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 \<in> carrier R; h \<in> carrier R;
        Plimit\<^bsub> R X K v \<^esub>(Hfst K v R X t S Y f g0 h0) g;
        Plimit\<^bsub> R X K v \<^esub>(Hsnd K v R X t S Y f g0 h0) h;
        Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub>
                            (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h)\<rbrakk>
       \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub>
                              (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f"
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); erH R (Vr K v) X S (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; 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h)\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply(frule Vr_integral[of v], frule vp_ideal[of v],
       frule Vr_ring, frule pj_Hom[of "Vr K v" "vp K v"], assumption+,
       simp add:PolynRg.erH_mult[THEN sym, 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)" g0 h0])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); erH R (Vr K v) X S (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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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))\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply(frule PolynRg.P_mod_diff[of R "Vr K v" X "Vr K v \<diamondsuit>\<^sub>p t" S Y
        f "g0 \<cdot>\<^sub>r\<^bsub>R\<^esub> h0"], assumption+,
       frule PolynRg.is_Ring[of R], rule Ring.ring_tOp_closed,
       assumption+, simp)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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)\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
(** P_mod done **)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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)\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply (simp add:PolynRg.erH_mult[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)" g0 h0])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply (frule PolynRg.is_Ring[of R])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply (frule Hfst_PCauchy[of v R X S t Y g0 h0 f], assumption+,
       frule Hsnd_PCauchy[of v R X S t Y g0 h0 f], assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) f
[PROOF STEP]
apply (subst pol_limit_def)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> (\<forall>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<in> carrier R) \<and> (\<forall>N. \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f))
[PROOF STEP]
apply (rule conjI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<in> carrier R
 2. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>N. \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule allI)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>n. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<in> carrier R
 2. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>N. \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule Ring.ring_tOp_closed, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>n. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n \<in> carrier R
 2. \<And>n. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n \<in> carrier R
 3. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>N. \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (simp add:pol_Cauchy_seq_def, simp add:pol_Cauchy_seq_def)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>N. \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule allI)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (subgoal_tac "\<forall>m>N. P_mod R (Vr K v) X (vp K v\<^bsup> (Vr K v) (an N)\<^esup>)
                   ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m)
                       \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; \<forall>m>N. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m>M. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>m>N. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply blast
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0\<rbrakk> \<Longrightarrow> \<forall>m>N. P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule allI, rule impI, frule Vr_integral[of v])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v)\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule_tac t = t and g = g0 and h = h0 and m = "m - Suc 0" in
       PolynRg.P_mod_diffxxx5_1[of R "Vr K v" X _ S Y], assumption+,
       simp add:vp_gen_nonzero[of v],
       frule vp_maximal[of v], simp, assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); Idomain (Vr K v); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0) \<in> carrier R \<times> carrier R \<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)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0))) = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (fst (g0, h0)) \<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)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0))) = erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (snd (g0, h0)) \<and> deg R (Vr K v) X (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0))) \<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)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0)))) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (m - Suc 0)\<^esup>)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0))) \<and> deg R (Vr K v) X (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 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)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0)))) \<le> deg R (Vr K v) X f \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (m - Suc 0)\<^esup>)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0))) \<and> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (Suc (m - Suc 0))\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0)) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> Suc (m - Suc 0)))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (vp K v\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply ((erule conjE)+, simp del:npow_suc Hpr_Suc)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> 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)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) = erH R (Vr K v) X S (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)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) = erH R (Vr K v) X S (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; deg R (Vr K v) X (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) \<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); P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup>)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); deg R (Vr K v) X (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) + 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; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup>)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule Ring.ring_is_ag[of "R"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; 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); 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); 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; 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); 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); 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 \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); erH R (Vr K v) X S (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; Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> 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)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) = erH R (Vr K v) X S (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)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) = erH R (Vr K v) X S (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; deg R (Vr K v) X (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) \<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); P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup>)) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); deg R (Vr K v) X (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)) + 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; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup>)) (snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (thin_tac "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)",
       thin_tac "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)",
       thin_tac "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)",
       thin_tac "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)",
       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 =
         (erH R (Vr K v) X S (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)",
       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)",
       thin_tac "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",
        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)) (fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) =
        erH R (Vr K v) X S (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 "erH R (Vr K v) X S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y
         (pj (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)) (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) =
        erH R (Vr K v) X S (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 "deg R (Vr K v) X (fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m))
        \<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)",
       thin_tac "P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>(Vr K v) m\<^esup>))
        ( fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub>
             -\<^sub>a\<^bsub>R\<^esub> (fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)))",
       thin_tac "deg R (Vr K v) X (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) +
        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",
       thin_tac "P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>(Vr K v) m\<^esup>))
         (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> (m - Suc 0)) \<plusminus>\<^bsub>R\<^esub>
             -\<^sub>a\<^bsub>R\<^esub> (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)))")
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (case_tac "N = 0", simp add:r_apow_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (carrier (Vr K v)) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule_tac p = "(Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m)
               \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f" in PolynRg.P_mod_whole[of R "Vr K v" X], assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0\<rbrakk> \<Longrightarrow> (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (simp add:Hfst_def Hsnd_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0\<rbrakk> \<Longrightarrow> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_fst[of _ "carrier R" "carrier R"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_snd[of _
       "carrier R" "carrier R"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule Ring.ring_is_ag[of R],
       rule aGroup.ag_pOp_closed, assumption)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; aGroup R\<rbrakk> \<Longrightarrow> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; aGroup R\<rbrakk> \<Longrightarrow> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule Ring.ring_tOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; 0 < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v m\<^esup> \<cdot>\<^sub>r\<^bsub>Vr K v\<^esub> t)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N = 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; aGroup R\<rbrakk> \<Longrightarrow> -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule aGroup.ag_mOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule_tac g = "f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> (fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) \<cdot>\<^sub>r\<^bsub>R\<^esub>
       (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m))" and m = "N - Suc 0" and n = m in
       PolynRg.P_mod_n_m[of "R" "Vr K v" "X"], assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> N - Suc 0 \<le> m
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 4. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_fst[of _ "carrier R" "carrier R"],
       frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_snd[of _ "carrier R" "carrier R"])
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> N - Suc 0 \<le> m
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 4. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+, rule aGroup.ag_mOp_closed,
       assumption)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> N - Suc 0 \<le> m
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 4. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (rule Ring.ring_tOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (3 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> N - Suc 0 \<le> m
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (subst Suc_le_mono[THEN sym], simp)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply assumption
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; N \<noteq> 0; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc (N - Suc 0)\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (simp del:npow_suc)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v Suc m\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p (t^\<^bsup>Vr K v N\<^esup>)) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (simp only:Ring.principal_ideal_n_pow1[THEN sym, of "Vr K v"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (cut_tac n = N in an_neq_inf)
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
[PROOF STEP]
apply (subgoal_tac "an N \<noteq> 0")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t\<^bsup> Vr K v an N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (subst r_apow_def, simp)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v na (an N)\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (simp add:na_an)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule Ring.principal_ideal[of "Vr K v" t], assumption)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t)\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule_tac I = "Vr K v \<diamondsuit>\<^sub>p t" and n = N in Ring.ideal_pow_ideal[of "Vr K v"], assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>)\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_fst[of _ "carrier R" "carrier R"],
       frule_tac x = "Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m" in cart_prod_snd[of _ "carrier R" "carrier R"])
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (thin_tac "PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0",
       thin_tac "PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0")
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) ((Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (simp add:Hfst_def Hsnd_def)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule_tac x = "fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)" and y = "snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)" in Ring.ring_tOp_closed[of "R"],  assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule_tac  x = "(fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) \<cdot>\<^sub>r\<^bsub>R\<^esub> (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m))" in aGroup.ag_mOp_closed[of "R"], assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule_tac I = "Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>(Vr K v) N\<^esup>" and
       p = "f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> (fst (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m)) \<cdot>\<^sub>r\<^bsub>R\<^esub>
        (snd (Hpr\<^bsub> R (Vr K v) X t S Y f g0 h0\<^esub> m))" in
       PolynRg.P_mod_minus[of R "Vr K v" X], assumption+)
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (-\<^sub>a\<^bsub>R\<^esub> (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 4. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (frule Ring.ring_is_ag[of "R"])
[PROOF STATE]
proof (prove)
goal (4 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; aGroup R\<rbrakk> \<Longrightarrow> f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m))
 3. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (-\<^sub>a\<^bsub>R\<^esub> (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 4. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (rule aGroup.ag_pOp_closed, assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (-\<^sub>a\<^bsub>R\<^esub> (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)))\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (simp add:aGroup.ag_p_inv, simp add:aGroup.ag_inv_inv,
       frule aGroup.ag_mOp_closed[of "R" "f"], assumption+)
[PROOF STATE]
proof (prove)
goal (2 subgoals):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); 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)); Ring R; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<diamondsuit>\<^sub>r\<^bsub>Vr K v\<^esub> Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>; an N \<noteq> 0; ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>); fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<in> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (-\<^sub>a\<^bsub>R\<^esub> f \<plusminus>\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); -\<^sub>a\<^bsub>R\<^esub> f \<in> carrier R\<rbrakk> \<Longrightarrow> P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> f)
 2. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (simp add:aGroup.ag_pOp_commute[of "R" "-\<^sub>a\<^bsub>R\<^esub> f"])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> 0
[PROOF STEP]
apply (subst an_0[THEN sym])
[PROOF STATE]
proof (prove)
goal (1 subgoal):
 1. \<And>N m. \<lbrakk>valuation K v; Complete\<^bsub>v\<^esub> K; t \<in> carrier (Vr K v); vp K v = Vr K v \<diamondsuit>\<^sub>p t; PolynRg R (Vr K v) X; PolynRg S (Vr K v /\<^sub>r (Vr K v \<diamondsuit>\<^sub>p t)) Y; f \<in> carrier R; f \<noteq> \<zero>\<^bsub>R\<^esub>; g0 \<in> carrier R; h0 \<in> carrier R; g0 \<noteq> \<zero>\<^bsub>R\<^esub>; h0 \<noteq> \<zero>\<^bsub>R\<^esub>; g \<in> carrier R; h \<in> carrier R; Plimit\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0 g; Plimit\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0 h; Plimit\<^bsub> R X K v \<^esub>(\<lambda>n. (Hfst\<^bsub> K v R X t S Y f g0 h0\<^esub> n) \<cdot>\<^sub>r\<^bsub>R\<^esub> (Hsnd\<^bsub> K v R X t S Y f g0 h0\<^esub> n)) (g \<cdot>\<^sub>r\<^bsub>R\<^esub> h); ideal (Vr K v) (Vr K v \<diamondsuit>\<^sub>p t); 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)); Ring R; PCauchy\<^bsub> R X K v \<^esub>Hfst K v R X t S Y f g0 h0; PCauchy\<^bsub> R X K v \<^esub>Hsnd K v R X t S Y f g0 h0; N < m; Idomain (Vr K v); Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m \<in> carrier R \<times> carrier R; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v Suc m\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); aGroup R; 0 < N; P_mod R (Vr K v) X (Vr K v \<diamondsuit>\<^sub>p t \<^bsup>\<diamondsuit>Vr K v N\<^esup>) (f \<plusminus>\<^bsub>R\<^esub> -\<^sub>a\<^bsub>R\<^esub> fst (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m) \<cdot>\<^sub>r\<^bsub>R\<^esub> snd (Hpr\<^bsub> R Vr K v X t S Y f g0 h0\<^esub> m)); an N \<noteq> \<infinity>\<rbrakk> \<Longrightarrow> an N \<noteq> an 0
[PROOF STEP]
apply (subst aneq_natneq[of _ "0"], thin_tac "an N \<noteq> \<infinity>", simp)
[PROOF STATE]
proof (prove)
goal:
No subgoals!
[PROOF STEP]
done