SSL Valuation3.thy

(**        Valuation3  
                            author Hidetsune Kobayashi
                            Group You Santo
                            Department of Mathematics
                            Nihon University
                            h_coba@math.cst.nihon-u.ac.jp
                            June 24, 2005
                            July 20, 2007

   chapter 1. elementary properties of a valuation
     section 9. completion 
      subsection Hensel's theorem
   **)

theory Valuation3 
imports Valuation2
begin

section "Completion"

text‹In this section we formalize "completion" of the ground field K›

definition
  limit :: "[_, 'b ==> ant, nat ==> 'b, 'b]
           ==> bool" (‹(4lim_{_ _} _ _)› [90,90,90,91]90) where
  "lim _v f b ⟷ (∀N. ∃M. (∀n. M < n ⟶
                ((f n) ± (-_a b)) ∈ (vp K v)^{(Vr K v) (an N)}))"

(* In this definition, f represents a sequence of elements of K, which is
   converging to b. Key lemmas of this section are n_value_x_1 and
   n_value_x_2 *) 


lemma not_in_singleton_noneq:"x ∉ {a} ==> x ≠ a"
apply simp
done   (* later move this lemma to Algebra1 *) 

lemma noneq_not_in_singleton:"x ≠ a ==> x ∉ {a}"
apply simp
done

lemma inf_neq_1[simp]:"∞ ≠ 1"
by (simp only:ant_1[THEN sym], rule z_neq_inf[THEN not_sym, of 1])

lemma a1_neq_0[simp]:"(1::ant) ≠ 0"
by (simp only:an_1[THEN sym], simp only:an_0[THEN sym],
    subst aneq_natneq[of 1 0], simp)

lemma a1_poss[simp]:"(0::ant) < 1"
by (cut_tac zposs_aposss[of 1], simp)

lemma a_p1_gt[simp]:"[a ≠ ∞; a ≠ -∞] ==> a < a + 1"
apply (cut_tac aadd_poss_less[of a 1],
       simp add:aadd_commute, assumption+) 
apply simp
done

lemma (in Corps) vpr_pow_inter_zero:"valuation K v ==>
      (∩ {I. ∃n. I = (vp K v)^{Vr K v) (an n)}}) = {0}"
apply (frule Vr_ring[of v], frule vp_ideal[of v])
apply (rule equalityI) 
defer
 apply (rule subsetI)
 apply simp
 apply (rule allI, rule impI, erule exE, simp)
 apply (cut_tac n = "an n" in vp_apow_ideal[of v], assumption+)
 apply simp
 apply (cut_tac I = "(vp K v)^{(Vr K v) (an n)}" in  Ring.ideal_zero[of "Vr K v"],
        assumption+)
 apply (simp add:Vr_0_f_0)

apply (rule subsetI, simp) 
 apply (rule contrapos_pp, simp+)
 apply (subgoal_tac "x ∈ vp K v")
 prefer 2 
 apply (drule_tac x = "vp K v^{(Vr K v) (an 1)}" in spec)
 apply (subgoal_tac "∃n. vp K v^{(Vr K v) (an (Suc 0))} = vp K v^{(Vr K v) (an n)}", 
        simp,
        thin_tac " ∃n. vp K v^{(Vr K v) (an (Suc 0))} = vp K v^{(Vr K v) (an n)}")
 apply (simp add:r_apow_def an_def) 
 apply (simp only:na_1)
  apply (simp only:Ring.idealpow_1_self[of "Vr K v" "vp K v"])

  apply blast

apply (frule n_val_valuation[of v])

apply (frule_tac x = x in val_nonzero_z[of "n_val K v"],
       frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"],
       assumption+,
       simp add:Vr_mem_f_mem, assumption+) apply (
       frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"],
       simp add:Vr_mem_f_mem, assumption+)
apply (cut_tac x = x in val_pos_mem_Vr[of v], assumption) apply(
       simp add:Vr_mem_f_mem, simp) 
       apply (frule_tac x = x in val_pos_n_val_pos[of v],
       simp add:Vr_mem_f_mem, simp)
apply (cut_tac x = "n_val K v x" and y = 1 in aadd_pos_poss, assumption+,
       simp) apply (frule_tac y = "n_val K v x + 1" in aless_imp_le[of 0])
apply (cut_tac x1 = x and n1 = "(n_val K v x) + 1" in n_val_n_pow[THEN sym, 
       of v], assumption+) 
apply (drule_tac a = "vp K v^{(Vr K v) (n_val K v x + 1)}" in forall_spec)
apply (erule exE, simp)
apply (simp only:ant_1[THEN sym] a_zpz,
       cut_tac z = "z + 1" in z_neq_inf)
 apply (subst an_na[THEN sym], assumption+, blast)
 apply simp
 apply (cut_tac a = "n_val K v x" in a_p1_gt)
 apply (erule exE, simp only:ant_1[THEN sym], simp only:a_zpz z_neq_inf)
 apply (cut_tac i = "z + 1" and j = z in ale_zle, simp)
 apply (erule exE, simp add:z_neq_minf)
 apply (cut_tac y1 = "n_val K v x" and x1 = "n_val K v x + 1" in 
        aneg_le[THEN sym], simp)
done 

lemma (in Corps) limit_diff_n_val:"[b ∈ carrier K; ∀j. f j ∈ carrier K;
      valuation K v] ==> (lim _v f b) = (∀N. ∃M. ∀n. M < n ⟶
       (an N) ≤ (n_val K v ((f n) ± (-_a b))))" 
apply (rule iffI)
apply (rule allI)
apply (simp add:limit_def) apply (rotate_tac -1)
 apply (drule_tac x = N in spec)
 apply (erule exE)
apply (subgoal_tac "∀n>M. (an N) ≤ (n_val K v (f n ± (-_a b)))")
apply blast
 apply (rule allI, rule impI) apply (rotate_tac -2)
 apply (drule_tac x = n in spec, simp) 
 apply (rule n_value_x_1[of v], assumption+,
        simp add:an_nat_pos, assumption)

apply (simp add:limit_def)
 apply (rule allI, rotate_tac -1, drule_tac x = N in spec)

 apply (erule exE)
 apply (subgoal_tac "∀n>M. f n ± (-_a b) ∈ vp K v ^{Vr K v) (an N)}")
 apply blast
apply (rule allI, rule impI)
apply (rotate_tac -2, drule_tac x = n in spec, simp)
 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
 apply (rule_tac x = "f n ± -_a b" and n = "an N" in n_value_x_2[of "v"],
        assumption+) 
 apply (subst val_pos_mem_Vr[THEN sym, of "v"], assumption+)
 apply (drule_tac x = n in spec)
 apply (rule aGroup.ag_pOp_closed[of "K"], assumption+)
 apply (simp add:aGroup.ag_mOp_closed)
 apply (subst val_pos_n_val_pos[of  "v"], assumption+)
 apply (rule aGroup.ag_pOp_closed, assumption+) apply simp
 apply (simp add:aGroup.ag_mOp_closed)
 apply (rule_tac j = "an N" and k = "n_val K v ( f n ± -_a b)" in 
        ale_trans[of "0"], simp, assumption+)   
        apply simp 
done

lemma (in Corps) an_na_Lv:"valuation K v ==> an (na (Lv K v)) = Lv K v" 
apply (frule Lv_pos[of "v"])
apply (frule aless_imp_le[of "0" "Lv K v"])
apply (frule apos_neq_minf[of "Lv K v"])
apply (frule Lv_z[of "v"], erule exE)
apply (rule an_na)
 apply (rule contrapos_pp, simp+)
done

lemma (in Corps) limit_diff_val:"[b ∈ carrier K; ∀j. f j ∈ carrier K;
      valuation K v] ==> (lim _v f b) = (∀N. ∃M. ∀n. M < n ⟶
             (an N) ≤ (v ((f n) ± (-_a b))))" 
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       simp add:limit_diff_n_val[of "b" "f" "v"])
apply (rule iffI)
 apply (rule allI,
   rotate_tac -1, drule_tac x = N in spec, erule exE) 
 apply (subgoal_tac "∀n > M. an N ≤ v( f n ± -_a b)", blast)
 apply (rule allI, rule impI)
 apply (drule_tac x = n in spec,
        drule_tac x = n in spec, simp) 
 apply (frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
     frule_tac x = "f n" and y = "-_a b" in aGroup.ag_pOp_closed, assumption+)
 apply (frule_tac x = "f n ± -_a b" in n_val_le_val[of "v"], 
         assumption+)
 apply (cut_tac n = N in an_nat_pos)
 apply (frule_tac j = "an N" and k = "n_val K v ( f n ± -_a b)" in 
                   ale_trans[of "0"], assumption+)
 apply (subst val_pos_n_val_pos, assumption+)
 apply (rule_tac i = "an N" and j = "n_val K v ( f n ± -_a b)" and 
        k = "v ( f n ± -_a b)" in ale_trans, assumption+)
 apply (rule allI)
 apply (rotate_tac 3, drule_tac x = "N * (na (Lv K v))" in spec)
 apply (erule exE)
 apply (subgoal_tac "∀n. M < n ⟶ (an N) ≤ n_val K v (f n ± -_a b)", blast)
 apply (rule allI, rule impI)
 apply (rotate_tac -2, drule_tac x = n in spec, simp)

  apply (drule_tac x = n in spec)
  apply (frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
      frule_tac x = "f n" and y = "-_a b" in aGroup.ag_pOp_closed, assumption+)
 apply (cut_tac n = "N * na (Lv K v)" in an_nat_pos)
 apply (frule_tac i = 0 and j = "an (N * na (Lv K v))" and 
                  k = "v ( f n ± -_a b)" in ale_trans, assumption+)
 apply (simp add:amult_an_an, simp add:an_na_Lv)
 apply (frule Lv_pos[of "v"])
 apply (frule_tac x1 = "f n ± -_a b" in n_val[THEN sym, of v], 
        assumption+, simp)
 apply (frule Lv_z[of v], erule exE, simp)
 apply (simp add:amult_pos_mono_r)
done

text‹uniqueness of the limit is derived from ‹vp_pow_inter_zero››
lemma (in Corps) limit_unique:"[b ∈ carrier K; ∀j. f j ∈ carrier K;
      valuation K v; c ∈ carrier K; lim _v f b; lim _v f c] ==> b = c" 
apply (rule contrapos_pp, simp+, simp add:limit_def,
       cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       simp add:aGroup.ag_neq_diffnonzero[of "K" "b" "c"], 
       frule vpr_pow_inter_zero[THEN sym, of v], 
       frule noneq_not_in_singleton[of "b ± -_a c" "0"])
apply simp
apply (rotate_tac -1, erule exE, erule conjE)
apply (erule exE, simp, thin_tac "x = (vp K v)^{Vr K v) (an n)}")
apply (rotate_tac 3, drule_tac x = n in spec,
       erule exE,
       drule_tac x = n in spec,
       erule exE)
apply (rename_tac x N M1 M2)
apply (subgoal_tac "M1 < Suc (max M1 M2)",
       subgoal_tac "M2 < Suc (max M1 M2)",
       drule_tac x = "Suc (max M1 M2)" in spec,
       drule_tac x = "Suc (max M1 M2)" in spec,
       drule_tac x = "Suc (max M1 M2)" in spec)
 apply simp

(* We see (f (Suc (max xb xc)) +\<^sub>K -\<^sub>K b) +\<^sub>K (-\<^sub>K (f (Suc (max xb xc)) +\<^sub>K -\<^sub>K c)) \<in> vpr K G a i v \<^bsup>\<diamondsuit>Vr K G a i v xa\<^esup>" *) 
 apply (frule_tac n = "an N" in vp_apow_ideal[of "v"], 
       frule_tac x = "f (Suc (max M1 M2)) ± -_a b" and N = "an N" in 
       mem_vp_apow_mem_Vr[of "v"], simp,
       frule Vr_ring[of "v"],  simp, simp)
      
apply (frule Vr_ring[of "v"], 
       frule_tac I = "vp K v^{Vr K v) (an N)}" and x = "f (Suc (max M1 M2)) ± -_a b"
       in Ring.ideal_inv1_closed[of "Vr K v"], assumption+)
 (** mOp of Vring and that of K is the same **)
apply (frule_tac I = "vp K v^{Vr K v) (an N)}" and h = "f (Suc (max M1 M2)) ± -_a b"
 in Ring.ideal_subset[of "Vr K v"], assumption+)
 apply (simp add:Vr_mOp_f_mOp,
        cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
        frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
        simp add:aGroup.ag_p_inv, simp add:aGroup.ag_inv_inv)
 (** addition of  -\<^sub>K f (Suc (max xb xc)) +\<^sub>K b  and f (Suc (max xb xc)) +\<^sub>K -\<^sub>K c
    is included in vpr K G a i v \<^bsup>\<diamondsuit>(Vr K G a i v) xa\<^esup> **)
 apply (frule_tac  I = "vp K v^{Vr K v) (an N)}" and x = 
 "-_a (f (Suc (max M1 M2))) ± b " and y = "f (Suc (max M1 M2)) ± (-_a c)" in 
  Ring.ideal_pOp_closed[of "Vr K v"], assumption+)
 apply (frule_tac x = "f (Suc (max M1 M2)) ± -_a c" and N = "an N" in 
        mem_vp_apow_mem_Vr[of v], simp, assumption,
        frule_tac x = "-_a (f (Suc (max M1 M2))) ± b" and N = "an N" in 
        mem_vp_apow_mem_Vr[of "v"], simp,
        simp add:Vr_pOp_f_pOp, simp add:Vr_pOp_f_pOp,
   frule aGroup.ag_mOp_closed[of "K" "c"], assumption+,
   frule_tac x = "f (Suc (max M1 M2))" in aGroup.ag_mOp_closed[of "K"], 
      assumption+,
   frule_tac x = "f (Suc (max M1 M2))" and y = "-_a c" in 
      aGroup.ag_pOp_closed[of "K"], assumption+) 

apply (simp add:aGroup.ag_pOp_assoc[of "K" _ "b" _],
       simp add:aGroup.ag_pOp_assoc[THEN sym, of "K" "b" _ "-_a c"],
       simp add:aGroup.ag_pOp_commute[of "K" "b"],
       simp add:aGroup.ag_pOp_assoc[of "K" _ "b" "-_a c"],
       frule aGroup.ag_pOp_closed[of "K" "b" "-_a c"], assumption+,
       simp add:aGroup.ag_pOp_assoc[THEN sym, of "K" _ _ "b ± -_a c"],
       simp add:aGroup.ag_l_inv1, simp add:aGroup.ag_l_zero)
apply (simp add:aGroup.ag_pOp_commute[of "K" _ "b"])
apply arith apply arith
done


(** The following lemma will be used to prove lemma limit_t. This lemma and
 them next lemma show that the valuation v is continuous (see lemma 
 n_val **) 
lemma (in Corps) limit_n_val:"[b ∈ carrier K; b ≠ 0; valuation K v;
       ∀j. f j ∈ carrier K; lim _v f b] ==>
       ∃N. (∀m. N < m ⟶ (n_val K v) (f m) = (n_val K v) b)"
apply (simp add:limit_def)
apply (frule n_val_valuation[of "v"])
apply (frule val_nonzero_z[of "n_val K v" "b"], assumption+, erule exE,
       rename_tac L)
apply (rotate_tac -3, drule_tac x = "nat (abs L + 1)" in spec)
apply (erule exE, rename_tac M)

    (* |L| + 1 \<le> (n_val K v ( f n +\<^sub>K -\<^sub>K b)). *) 
 apply (subgoal_tac "∀n. M < n ⟶ n_val K v (f n) = n_val K v b", blast)
 apply (rule allI, rule impI) 
 apply (rotate_tac -2, 
        drule_tac x = n in spec,
        simp)
apply (frule_tac x = "f n ± -_a b" and n = "an (nat (∣L∣ + 1))" in 
       n_value_x_1[of "v"],
       thin_tac "f n ± -_a b ∈ vp K v^{Vr K v) (an (nat (∣L∣ + 1)))}")
apply (simp add:an_def) 
 apply (simp add: zpos_apos [symmetric])
 apply assumption

 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
 apply (frule aGroup.ag_mOp_closed[of "K" "b"], assumption+)
 
 apply (drule_tac x = n in spec, 
        frule_tac x = "f n" in aGroup.ag_pOp_closed[of "K" _ "-_a b"], 
        assumption+,  
        frule_tac x = "b" and y = "(f n) ± (-_a b)" in value_less_eq[of 
        "n_val K v"], assumption+)
 apply simp 
 apply (rule_tac x = "ant L" and y = "an (nat (∣L∣ + 1))" and 
        z = "n_val K v ( f n ± -_a b)" in aless_le_trans)
 apply (subst an_def)
 apply (subst aless_zless) apply arith apply assumption+
 apply (simp add:aGroup.ag_pOp_commute[of "K" "b"])
 apply (simp add:aGroup.ag_pOp_assoc) apply (simp add:aGroup.ag_l_inv1)
 apply (simp add:aGroup.ag_r_zero)
done 

lemma (in Corps) limit_val:"[b ∈ carrier K; b ≠ 0; ∀j. f j ∈ carrier K;
      valuation K v; lim _v f b] ==> ∃N. (∀n. N < n ⟶ v (f n) = v b)"
apply (frule limit_n_val[of "b" "v" "f"], assumption+)
 apply (erule exE)
 apply (subgoal_tac "∀m. N < m ⟶ v (f m) = v b")
 apply blast
apply (rule allI, rule impI)
 apply (drule_tac x = m in spec)
 apply (drule_tac x = m in spec)
  apply simp
 apply (frule Lv_pos[of "v"])
 apply (simp add:n_val[THEN sym, of "v"])
done

lemma (in Corps) limit_val_infinity:"[valuation K v; ∀j. f j ∈ carrier K;
      lim _v f 0] ==> ∀N.(∃M. (∀m. M < m ⟶ (an N) ≤ (n_val K v (f m))))"
apply (simp add:limit_def)
 apply (rule allI)
 apply (drule_tac x = N in spec, 
        erule exE)
       
 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
        simp only:aGroup.ag_inv_zero[of "K"], simp only:aGroup.ag_r_zero)
apply (subgoal_tac "∀n. M < n ⟶ an N ≤ n_val K v (f n)", blast)

 apply (rule allI, rule impI)
 apply (drule_tac x = n in spec,
        drule_tac x = n in spec, simp)
 apply (simp add:n_value_x_1)
done

lemma (in Corps) not_limit_zero:"[valuation K v; ∀j. f j ∈ carrier K;
      ¬ (lim _v f 0)] ==> ∃N.(∀M. (∃m. (M < m) ∧
                                    ((n_val K v) (f m)) < (an N)))"
apply (simp add:limit_def)
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
apply (simp add:aGroup.ag_inv_zero aGroup.ag_r_zero)
apply (erule exE)
 apply (case_tac "N = 0", simp add:r_apow_def) 
 apply (subgoal_tac "∀M. ∃n. M < n ∧ n_val K v (f n) < (an 0)") apply blast
 apply (rule allI, 
        rotate_tac 4, frule_tac x = M in spec,
        erule exE, erule conjE)
 apply (frule_tac x1 = "f n" in val_pos_mem_Vr[THEN sym, of "v"]) apply simp 
 apply simp
 apply (frule_tac x = "f n" in val_pos_n_val_pos[of "v"])
 apply simp apply simp apply (thin_tac "¬ 0 ≤ v (f n)")
 apply (simp add:aneg_le) apply blast

apply (simp)
 apply (subgoal_tac "∀n. ((f n) ∈ vp K v^{(Vr K v) (an N)}) =
                                ((an N) ≤ n_val K v (f n))")
 apply simp
 apply (thin_tac "∀n. (f n ∈ vp K v^{(Vr K v) (an N)}) = (an N ≤ n_val K v (f n))")
 apply (simp add:aneg_le) apply blast

apply (rule allI)
 apply (thin_tac "∀M. ∃n. M < n ∧ f n ∉ vp K v^{(Vr K v) (an N)}")
 apply (rule iffI)
 apply (frule_tac x1 = "f n" and n1 = "an N" in n_val_n_pow[THEN sym, of v])
 apply (rule_tac N = "an N" and x = "f n" in mem_vp_apow_mem_Vr[of v],
        assumption+, simp, assumption, simp, simp)
 apply (frule_tac x = "f n" and n = "an N" in n_val_n_pow[of "v"])
 apply (frule_tac x1 = "f n" in val_pos_mem_Vr[THEN sym, of "v"])
 apply simp
 apply (cut_tac n = N in an_nat_pos) 
 apply (frule_tac j = "an N" and k = "n_val K v (f n)" in ale_trans[of "0"],
         assumption+)
 apply (frule_tac x1 = "f n" in val_pos_n_val_pos[THEN sym, of "v"]) 
 apply simp+ 
done

lemma (in Corps) limit_p:"[valuation K v; ∀j. f j ∈ carrier K;
   ∀j. g j ∈ carrier K; b ∈ carrier K; c ∈ carrier K; lim _v f b; lim _v g c]
   ==> lim _v (λj. (f j) ± (g j)) (b ± c)"
apply (simp add:limit_def)
 apply (rule allI) apply (rotate_tac 3,
       drule_tac x = N in spec,
       drule_tac x = N in spec,
       (erule exE)+) 
 apply (frule_tac x = M and y = Ma in two_inequalities, simp,
        subgoal_tac "∀n > (max M Ma). (f n) ± (g n) ± -_a (b ± c)
        ∈ (vp K v)^{Vr K v) (an N)}")
 apply (thin_tac "∀n. Ma < n ⟶
                     g n ± -_a c ∈ (vp K v)^{Vr K v) (an N)}",
        thin_tac "∀n. M < n ⟶
                      f n ± -_a b ∈(vp K v)^{Vr K v) (an N)}", blast)
 apply (frule Vr_ring[of v], 
        frule_tac n = "an N" in vp_apow_ideal[of v])
apply simp 
 apply (rule allI, rule impI)
 apply (thin_tac "∀n>M. f n ± -_a b ∈ vp K v^{(Vr K v) (an N)}",
        thin_tac "∀n>Ma. g n ± -_a c ∈ vp K v^{(Vr K v) (an N)}",
        frule_tac I = "vp K v^{Vr K v) (an N)}" and x = "f n ± -_a b" 
        and y = "g n ± -_a c" in Ring.ideal_pOp_closed[of "Vr K v"], 
        assumption+, simp, simp) 
apply (frule_tac I = "vp K v^{Vr K v) (an N)}" and h = "f n ± -_a b" 
        in Ring.ideal_subset[of "Vr K v"], assumption+, simp,
        frule_tac I = "vp K v^{Vr K v) (an N)}" and h = "g n ± -_a c" in
                   Ring.ideal_subset[of "Vr K v"], assumption+, simp)
 apply (simp add:Vr_pOp_f_pOp) 
 apply (thin_tac "f n ± -_a b ∈ carrier (Vr K v)",
        thin_tac "g n ± -_a c ∈ carrier (Vr K v)")

 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
        frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
        frule aGroup.ag_mOp_closed[of "K" "c"], assumption+,
        frule_tac a = "f n" and b = "-_a b" and c = "g n" and d = "-_a c" in 
                  aGroup.ag_add4_rel[of "K"], simp+)
 apply (simp add:aGroup.ag_p_inv[THEN sym]) 
done

lemma (in Corps) Abs_ant_abs[simp]:"Abs (ant z) = ant (abs z)"
apply (simp add:Abs_def, simp add:aminus)
apply (simp only:ant_0[THEN sym], simp only:aless_zless)
apply (simp add:zabs_def)
done

lemma (in Corps) limit_t_nonzero:"[valuation K v; ∀(j::nat). (f j) ∈ carrier K; ∀(j::nat). g j ∈ carrier K; b ∈ carrier K; c ∈ carrier K; b ≠ 0; c ≠ 0;
 lim _v f b; lim _v g c] ==> lim _v (λj. (f j) ⋅_r (g j)) (b ⋅_r c)"
apply (cut_tac field_is_ring, simp add:limit_def, rule allI) 
 apply (subgoal_tac "∀j. (f j) ⋅_r (g j) ± -_a (b ⋅_r c) =
 ((f j) ⋅_r (g j) ± (f j) ⋅_r (-_a c)) ± ((f j) ⋅_r c ± -_a (b ⋅_r c))", simp,
 thin_tac "∀j. f j ⋅_r g j ± -_a b ⋅_r c =
             f j ⋅_r g j ± f j ⋅_r (-_a c) ± (f j ⋅_r c ± -_a b ⋅_r c)")
 apply (frule limit_n_val[of  "b" "v" "f"], assumption+,
        simp add:limit_def)
 apply (frule n_val_valuation[of "v"])
 apply (frule val_nonzero_z[of "n_val K v" "b"], assumption+, 
        rotate_tac -1, erule exE,
        frule val_nonzero_z[of "n_val K v" "c"], assumption+,
        rotate_tac -1, erule exE, rename_tac N bz cz)
 apply (rotate_tac 5, 
  drule_tac x = "N + nat (abs cz)" in spec, 
  drule_tac x = "N + nat (abs bz)" in spec)
  apply (erule exE)+ 
  apply (rename_tac N bz cz M M1 M2)
(** three inequalities together **)
apply (subgoal_tac "∀n. (max (max M1 M2) M) < n ⟶
 (f n) ⋅_r (g n) ± (f n) ⋅_r (-_a c) ± ((f n) ⋅_r c ± (-_a (b ⋅_r c)))
                  ∈ vp K v ^{Vr K v) (an N)}",  blast)
apply (rule allI, rule impI) apply (simp, (erule conjE)+) 
 apply (rotate_tac 11, drule_tac x = n in spec,
      drule_tac x = n in spec, simp,
      drule_tac x = n in spec, simp)
 apply (frule_tac b = "g n ± -_a c" and n = "an N" and x = "f n" in 
        convergenceTr1[of v])
 apply simp apply simp apply (simp add:an_def a_zpz[THEN sym]) apply simp
 apply (frule_tac b = "f n ± -_a b" and n = "an N" in convergenceTr1[of  
  "v" "c"], assumption+, simp) apply (simp add:an_def)
   apply (simp add:a_zpz[THEN sym]) apply simp

 apply (drule_tac x = n in spec, 
        drule_tac x = n in spec)
 apply (simp add:Ring.ring_inv1_1[of "K" "b" "c"],
        cut_tac Ring.ring_is_ag, frule aGroup.ag_mOp_closed[of "K" "c"], 
        assumption+,
        simp add:Ring.ring_distrib1[THEN sym],
        frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
        simp add:Ring.ring_distrib2[THEN sym],
        subst Ring.ring_tOp_commute[of "K" _ "c"], assumption+,
        rule aGroup.ag_pOp_closed, assumption+) 
apply (cut_tac n = N in an_nat_pos)
apply (frule_tac n = "an N" in vp_apow_ideal[of "v"], assumption+)
apply (frule Vr_ring[of "v"]) 

apply (frule_tac x = "(f n) ⋅_r (g n ± -_a c)" and y = "c ⋅_r (f n ± -_a b)"
   and I = "vp K v^{(Vr K v) (an N)}" in Ring.ideal_pOp_closed[of "Vr K v"], 
   assumption+)
apply (frule_tac R = "Vr K v" and I = "vp K v^{(Vr K v) (an N)}" and 
       h = "(f n) ⋅_r (g n ± -_a c)" in Ring.ideal_subset, assumption+,
       frule_tac R = "Vr K v" and I = "vp K v^{(Vr K v) (an N)}" and 
       h = "c ⋅_r (f n ± -_a b)" in Ring.ideal_subset, assumption+) 
apply (simp add:Vr_pOp_f_pOp, assumption+)
apply (rule allI)
 apply (thin_tac "∀N. ∃M. ∀n>M. f n ± -_a b ∈ vp K v^{(Vr K v) (an N)}",
        thin_tac "∀N. ∃M. ∀n>M. g n ± -_a c ∈ vp K v^{(Vr K v) (an N)}")
 apply (drule_tac x = j in spec,
        drule_tac x = j in spec,
        cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule_tac x = "f j" and y = "g j" in Ring.ring_tOp_closed, assumption+,
       frule_tac x = "b" and y = "c" in Ring.ring_tOp_closed, assumption+,
       frule_tac x = "f j" and y = "c" in Ring.ring_tOp_closed, assumption+,
       frule_tac x = c in aGroup.ag_mOp_closed[of "K"], assumption+,
       frule_tac x = "f j" and y = "-_a c" in Ring.ring_tOp_closed, assumption+,
       frule_tac x = "b ⋅_r c" in aGroup.ag_mOp_closed[of "K"], assumption+)
 apply (subst aGroup.pOp_assocTr41[THEN sym, of "K"], assumption+,
        subst aGroup.pOp_assocTr42[of "K"], assumption+,
        subst Ring.ring_distrib1[THEN sym, of "K"], assumption+)
 apply (simp add:aGroup.ag_l_inv1, simp add:Ring.ring_times_x_0, 
        simp add:aGroup.ag_r_zero)
done

lemma an_npn[simp]:"an (n + m) = an n + an m"
by (simp add:an_def a_zpz) (** move **) 

lemma Abs_noninf:"a ≠ -∞ ∧ a ≠ ∞ ==> Abs a ≠ ∞"
by (cut_tac mem_ant[of "a"], simp, erule exE, simp add:Abs_def,
       simp add:aminus)

lemma (in Corps) limit_t_zero:"[c ∈ carrier K; valuation K v;
      ∀(j::nat). f j ∈ carrier K; ∀(j::nat). g j ∈ carrier K;
      lim _v f 0; lim _v g c] ==> lim _v (λj. (f j) ⋅_r (g j)) 0"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"], 
       simp add:limit_def, simp add:aGroup.ag_inv_zero, simp add:aGroup.ag_r_zero)
apply (subgoal_tac "∀j. (f j) ⋅_r (g j) ∈ carrier K",
       simp add:aGroup.ag_r_zero)
 prefer 2 apply (rule allI, simp add:Ring.ring_tOp_closed)
apply (case_tac "c = 0")
 apply (simp add:aGroup.ag_inv_zero, simp add:aGroup.ag_r_zero)
 apply (rule allI,
        rotate_tac 4,
        drule_tac x = N in spec,
        drule_tac x = N in spec, (erule exE)+,
        rename_tac N M1 M2)
 apply (subgoal_tac "∀n>(max M1 M2). (f n) ⋅_r (g n) ∈ (vp K v)^{Vr K v) (an N)}", 
        blast) 
 apply (rule allI, rule impI)
 apply (drule_tac x = M1 and y = M2 in two_inequalities, assumption+,
        thin_tac "∀n>M2. g n ∈ vp K v^{(Vr K v) (an N)}")
apply (thin_tac "∀j. f j ⋅_r g j ∈ carrier K",
       thin_tac "∀j. f j ∈ carrier K",
       thin_tac "∀j. g j ∈ carrier K",
       drule_tac x = n in spec, simp, erule conjE,
       erule conjE,
       frule Vr_ring[of v])
 apply (cut_tac n = N in an_nat_pos) 
 apply (frule_tac x = "f n" in mem_vp_apow_mem_Vr[of  "v"], assumption+,
        frule_tac x = "g n" in mem_vp_apow_mem_Vr[of  "v"], assumption+,
       frule_tac n = "an N" in vp_apow_ideal[of  "v"], assumption+)
 apply (frule_tac I = "vp K v^{Vr K v) (an N)}" and x = "g n" and 
        r = "f n" in Ring.ideal_ring_multiple[of "Vr K v"], assumption+,
        simp add:Vr_tOp_f_tOp)


 (** case c \<noteq> 0\<^sub>K **)
 apply (rule allI)
 apply (subgoal_tac "∀j. (f j) ⋅_r (g j) =
        (f j) ⋅_r ((g j) ± (-_a c)) ± (f j) ⋅_r c", simp,
        thin_tac "∀j. (f j) ⋅_r (g j) =
        (f j) ⋅_r ((g j) ± (-_a c)) ± (f j) ⋅_r c",
        thin_tac "∀j. (f j) ⋅_r ( g j ± -_a c) ± (f j) ⋅_r c ∈ carrier K")
apply (rotate_tac 4,
       drule_tac x = "N + na (Abs (n_val K v c))" in  spec,
       drule_tac x = N in spec)
 apply (erule exE)+ apply (rename_tac N M1 M2)
apply (subgoal_tac "∀n. (max M1 M2) < n ⟶ (f n) ⋅_r (g n ± -_a c) ±
                     (f n) ⋅_r c ∈ vp K v^{Vr K v) (an N)}", blast)
apply (rule allI, rule impI, simp, erule conjE,
      drule_tac x = n in spec,
      drule_tac x = n in spec,
      drule_tac x = n in spec) 
apply (frule n_val_valuation[of "v"])
apply (frule value_in_aug_inf[of "n_val K v" "c"], assumption+, 
       simp add:aug_inf_def)
apply (frule val_nonzero_noninf[of "n_val K v" "c"], assumption+)
apply (cut_tac Abs_noninf[of "n_val K v c"])
apply (cut_tac Abs_pos[of "n_val K v c"]) apply (simp add:an_na)

apply (drule_tac x = n in spec, simp)
 apply (frule_tac b = "f n" and n = "an N" in convergenceTr1[of 
                                   "v" "c"], assumption+)
 apply simp

apply (frule_tac x = "f n" and N = "an N + Abs (n_val K v c)" in 
       mem_vp_apow_mem_Vr[of "v"], 
       frule_tac n = "an N" in vp_apow_ideal[of "v"])
      apply simp
apply (rule_tac x = "an N" and y = "Abs (n_val K v c)" in aadd_two_pos)
    apply simp apply (simp add:Abs_pos) apply assumption
   
apply (frule_tac x = "g n ± (-_a c)" and N = "an N" in mem_vp_apow_mem_Vr[of 
       "v"], simp, assumption+) apply (
       frule_tac x = "c ⋅_r (f n)" and N = "an N" in mem_vp_apow_mem_Vr[of 
       "v"], simp)  apply simp  
apply (simp add:Ring.ring_tOp_commute[of "K" "c"]) apply (
       frule Vr_ring[of  "v"], 
       frule_tac I = "(vp K v)^{Vr K v) (an N)}" and x = "g n ± (-_a c)" 
       and r = "f n" in Ring.ideal_ring_multiple[of "Vr K v"]) 
apply (simp add:vp_apow_ideal) apply assumption+
apply (frule_tac I = "vp K v^{Vr K v) (an N)}" and 
       x = "(f n) ⋅_r (g n ± -_a c)" and y = "(f n) ⋅_r c" in 
                      Ring.ideal_pOp_closed[of "Vr K v"])
    apply (simp add:vp_apow_ideal)
 apply (simp add:Vr_tOp_f_tOp,  assumption)
 apply (simp add:Vr_tOp_f_tOp Vr_pOp_f_pOp,
       frule_tac x = "(f n) ⋅_r (g n ± -_a c)" and N = "an N" in mem_vp_apow_mem_Vr[of "v"], simp add:Vr_pOp_f_pOp, assumption+)
 apply (frule_tac N = "an N" and x = "(f n) ⋅_r c" in mem_vp_apow_mem_Vr[of 
        "v"]) apply simp apply assumption
 apply (simp add:Vr_pOp_f_pOp) apply simp

 apply (thin_tac "∀N. ∃M. ∀n>M. f n ∈ vp K v^{(Vr K v) (an N)}",
        thin_tac "∀N. ∃M. ∀n>M. g n ± -_a c ∈ vp K v^{(Vr K v) (an N)}",
        rule allI)
 apply (drule_tac x = j in spec, 
    drule_tac x = j in spec,
    drule_tac x = j in spec,
    frule  aGroup.ag_mOp_closed[of "K" "c"], assumption+,
    frule_tac x = "f j" in Ring.ring_tOp_closed[of "K" _ "c"], assumption+,
    frule_tac x = "f j" in Ring.ring_tOp_closed[of "K" _ "-_a c"], assumption+)
 apply (simp add:Ring.ring_distrib1, simp add:aGroup.ag_pOp_assoc, 
        simp add:Ring.ring_distrib1[THEN sym],
        simp add:aGroup.ag_l_inv1, simp add:Ring.ring_times_x_0, 
        simp add:aGroup.ag_r_zero)
done

lemma (in Corps) limit_minus:"[valuation K v; ∀j. f j ∈ carrier K;
      b ∈ carrier K; lim _v f b] ==> lim _v (λj. (-_a (f j))) (-_a b)"
apply (simp add:limit_def)
 apply (rule allI,
       rotate_tac -1, frule_tac x = N in spec,
        thin_tac "∀N. ∃M. ∀n. M < n ⟶
                   f n ± -_a b ∈ (vp K v)^{Vr K v) (an N)}",
       erule exE,
       subgoal_tac "∀n. M < n ⟶
           (-_a (f n)) ± (-_a (-_a b)) ∈ (vp K v)^{Vr K v) (an N)}",
      blast) 
 apply (rule allI, rule impI,
        frule_tac x = n in spec,
        frule_tac x = n in spec, simp) 

apply (frule Vr_ring[of "v"],
       frule_tac n = "an N" in vp_apow_ideal[of "v"], simp)
apply (frule_tac x = "f n ± -_a b" and N = "an N" in 
       mem_vp_apow_mem_Vr[of "v"], simp+,
       frule_tac I = "vp K v^{Vr K v) (an N)}" and x = "f n ± (-_a b)" in 
       Ring.ideal_inv1_closed[of "Vr K v"], assumption+, simp) 
 apply (simp add:Vr_mOp_f_mOp) 
 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
        frule aGroup.ag_mOp_closed[of "K" "b"], assumption+)
 apply (simp add:aGroup.ag_p_inv[of "K"])
done

lemma (in Corps) inv_diff:"[x ∈ carrier K; x ≠ 0; y ∈ carrier K; y ≠ 0] ==>
                (x^<hyphen>K) ± (-_a (y^<hyphen>K)) = (x^<hyphen>K) ⋅_r ( y^<hyphen>K) ⋅_r (-_a (x ± (-_a y)))"
apply (cut_tac invf_closed1[of "x"], simp, erule conjE,
       cut_tac invf_closed1[of y], simp, erule conjE,
       cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule Ring.ring_tOp_closed[of "K" "x^<hyphen>K" "y^<hyphen>K"], assumption+,
       frule aGroup.ag_mOp_closed[of "K" "x"], assumption+,
       frule aGroup.ag_mOp_closed[of "K" "y"], assumption+,
       frule aGroup.ag_mOp_closed[of "K" "x^<hyphen>K"], assumption+,
       frule aGroup.ag_mOp_closed[of "K" "y^<hyphen>K"], assumption+,
       frule aGroup.ag_pOp_closed[of "K" "x" "-_a y"], assumption+) 
                                    
apply (simp add:Ring.ring_inv1_2[THEN sym],
       simp only:Ring.ring_distrib1[of "K" "(x^<hyphen>K) ⋅_r (y^<hyphen>K)" "x" "-_a y"],
       simp only:Ring.ring_tOp_commute[of "K" _ x],
       simp only:Ring.ring_inv1_2[THEN sym, of "K"], 
       simp only:Ring.ring_tOp_assoc[THEN sym],
       simp only:Ring.ring_tOp_commute[of "K" "x"],
       cut_tac linvf[of  "x"], simp+,
       simp add:Ring.ring_l_one, simp only:Ring.ring_tOp_assoc,
       cut_tac linvf[of "y"], simp+, 
       simp only:Ring.ring_r_one) 
apply (simp add:aGroup.ag_p_inv[of "K"], simp add:aGroup.ag_inv_inv,
       rule aGroup.ag_pOp_commute[of "K" "x^<hyphen>K" "-_a y^<hyphen>K"], assumption+)
apply simp+
done

lemma times2plus:"(2::nat)*n = n + n"
by simp

lemma (in Corps) limit_inv:"[valuation K v; ∀j. f j ∈ carrier K;
      b ∈ carrier K; b ≠ 0; lim _v f b] ==>
           lim _v (λj. if (f j) = 0 then 0 else (f j)^<hyphen>K) (b^<hyphen>K)"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
apply (frule limit_n_val[of "b" "v" "f"], assumption+) 
apply (subst limit_def)
 apply (rule allI, erule exE)
 apply (subgoal_tac "∀m>Na. f m ≠ 0") 
 prefer 2
 apply (rule allI, rule impI, rotate_tac -2,
        drule_tac x = m in spec, simp)
 apply (frule n_val_valuation[of v]) 
 apply (frule val_nonzero_noninf[of "n_val K v" b], assumption+)
 apply (rule contrapos_pp, simp+, simp add:value_of_zero)
 apply (unfold limit_def)
  apply (rotate_tac 2,
         frule_tac x = "N + 2*(na (Abs (n_val K v b)))" in 
         spec)
  apply (erule exE)
  apply (subgoal_tac "∀n>(max Na M).
          (if f n = 0 then 0 else f n^<hyphen>K) ± -_a b^<hyphen>K ∈ vp K v^{Vr K v) (an N)}", 
         blast) 
  apply (rule allI, rule impI)
  apply (cut_tac x = "Na" and y = "max Na M" and z = n
         in le_less_trans)
  apply simp+
  apply (thin_tac "∀N. ∃M. ∀n>M. f n ± -_a b ∈ vp K v^{(Vr K v) (an N)}")
 apply (drule_tac x = n in spec,
        drule_tac x = n in spec,
        drule_tac x = n in spec,
        drule_tac x = n in spec, simp)
 apply (subst inv_diff, assumption+)
 apply (cut_tac x = "f n" in invf_closed1, simp,
        cut_tac x = b in invf_closed1, simp, simp, (erule conjE)+)
(* apply (frule field_is_idom[of "K"], frule field_iOp_closed[of "K" "b"], 
                                                 simp, simp, erule conjE,
        frule idom_tOp_nonzeros [of "K" "b\<^bsup>\<hyphen>K\<^esup>" "b\<^bsup>\<hyphen>K\<^esup>"], assumption+) *)
 apply (frule_tac n = "an N + an (2 * na (Abs (n_val K v b)))" and 
        x = "f n ± -_a b" in n_value_x_1[of v])
 apply (simp only:an_npn[THEN sym], rule an_nat_pos) 
 apply assumption
 apply (rule_tac x = "f n^{<hyphen> K} ⋅_r b^{<hyphen> K} ⋅_r (-_a (f n ± -_a b))" and v = v and 
        n = "an N" in n_value_x_2, assumption+)
 apply (frule n_val_valuation[of v])
 apply (subst val_pos_mem_Vr[THEN sym, of "v"], assumption+)
  apply (rule Ring.ring_tOp_closed, assumption+)+
  apply (rule aGroup.ag_mOp_closed, assumption)
  apply (rule aGroup.ag_pOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+)
  apply (subst val_pos_n_val_pos[of v], assumption+,
         rule Ring.ring_tOp_closed, assumption+,
         rule Ring.ring_tOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+,
         rule aGroup.ag_pOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+)
  apply (subst val_t2p[of "n_val K v"], assumption+,
         rule Ring.ring_tOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+,
         rule aGroup.ag_pOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+,
         subst val_minus_eq[of "n_val K v"], assumption+,
          (rule aGroup.ag_pOp_closed, assumption+),
          (rule aGroup.ag_mOp_closed, assumption+))
  apply (subst val_t2p[of "n_val K v"], assumption+)
  apply (simp add:value_of_inv)
  apply (frule_tac x = "an N + an (2 * na (Abs (n_val K v b)))" and y = "n_val K v (f n ± -_a b)" and z = "- n_val K v b + - n_val K v b" in aadd_le_mono)
  apply (cut_tac z = "n_val K v b" in Abs_pos)
  apply (frule val_nonzero_z[of "n_val K v" b], assumption+, erule exE)
  apply (rotate_tac -1, drule sym, cut_tac z = z in z_neq_minf,
         cut_tac z = z in z_neq_inf, simp,
         cut_tac a = "(n_val K v b)" in Abs_noninf, simp)
  apply (simp only:times2plus an_npn, simp add:an_na)
  apply (rotate_tac -4, drule sym, simp)
  apply (thin_tac "f n ± -_a b ∈ vp K v^{(Vr K v) (an N + (ant ∣z∣ + ant ∣z∣))}")
  apply (simp add:an_def, simp add:aminus, (simp add:a_zpz)+)
  apply (subst aadd_commute)
  apply (rule_tac i = 0 and j = "ant (int N + 2 * ∣z∣ - 2 * z)" and 
         k = "n_val K v (f n ± -_a b) + ant (- (2 * z))" in ale_trans)
  apply (subst ant_0[THEN sym])
  apply (subst ale_zle, simp, assumption)

  apply (frule n_val_valuation[of v])
  apply (subst val_t2p[of "n_val K v"], assumption+)
  apply (rule Ring.ring_tOp_closed, assumption+)+
  apply (rule aGroup.ag_mOp_closed, assumption)
  apply (rule aGroup.ag_pOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+)
  apply (subst val_t2p[of "n_val K v"], assumption+) 
  apply (subst val_minus_eq[of "n_val K v"], assumption+,
         rule aGroup.ag_pOp_closed, assumption+,
         rule aGroup.ag_mOp_closed, assumption+)

  apply (simp add:value_of_inv)
  apply (frule_tac x = "an N + an (2 * na (Abs (n_val K v b)))" and y = "n_val K v (f n ± -_a b)" and z = "- n_val K v b + - n_val K v b" in aadd_le_mono)
  apply (cut_tac z = "n_val K v b" in Abs_pos)
  apply (frule val_nonzero_z[of "n_val K v" b], assumption+, erule exE)
  apply (rotate_tac -1, drule sym, cut_tac z = z in z_neq_minf,
         cut_tac z = z in z_neq_inf, simp,
         cut_tac a = "(n_val K v b)" in Abs_noninf, simp)
  apply (simp only:times2plus an_npn, simp add:an_na)
  apply (rotate_tac -4, drule sym, simp)
  apply (thin_tac "f n ± -_a b ∈ vp K v^{(Vr K v) (an N + (ant ∣z∣ + ant ∣z∣))}")
  apply (simp add:an_def, simp add:aminus, (simp add:a_zpz)+)
  apply (subst aadd_commute)
  apply (rule_tac i = "ant (int N)" and j = "ant (int N + 2 * ∣z∣ - 2 * z)" 
        and k = "n_val K v (f n ± -_a b) + ant (- (2 * z))" in ale_trans)
  apply (subst ale_zle, simp, assumption)
 
  apply simp
done

definition
  Cauchy_seq :: "[_ , 'b ==> ant, nat ==> 'b]
           ==> bool" (‹(3Cauchy_{_ _} _)› [90,90,91]90) where
  "Cauchy _v f ⟷ (∀n. (f n) ∈ carrier K) ∧ (
  ∀N. ∃M. (∀n m. M < n ∧ M < m ⟶
                ((f n) ± (-_a (f m))) ∈ (vp K v)^{Vr K v) (an N)}))"

definition
  v_complete :: "['b ==> ant, _] ==> bool"
                    (‹(2Complete _)›  [90,91]90) where
  "Complete K ⟷ (∀f. (Cauchy _v f) ⟶
                           (∃b. b ∈ (carrier K) ∧ lim _v f b))"

lemma (in Corps) has_limit_Cauchy:"[valuation K v; ∀j. f j ∈ carrier K;
      b ∈ carrier K; lim _v f b] ==> Cauchy _v f" 
apply (simp add:Cauchy_seq_def)
apply (rule allI)
apply (simp add:limit_def)
 apply (rotate_tac -1)
 apply (drule_tac x = N in spec)
 apply (erule exE)
 apply (subgoal_tac "∀n m. M < n ∧ M < m ⟶
                        f n ± -_a (f m) ∈ vp K v^{Vr K v) (an N)}")
 apply blast
 apply ((rule allI)+, rule impI, erule conjE)
 apply (frule_tac x = n in spec,
        frule_tac x = m in spec,
        thin_tac "∀j. f j ∈ carrier K",
        frule_tac x = n in spec,
        frule_tac x = m in spec,
        thin_tac "∀n. M < n ⟶ f n ± -_a b ∈ vp K v^{Vr K v) (an N)}",
        simp)
 apply (frule_tac n = "an N" in vp_apow_ideal[of v], simp)
 apply (frule Vr_ring[of "v"])
 apply (frule_tac x = "f m ± -_a b" and I = "vp K v^{Vr K v) (an N)}" in 
           Ring.ideal_inv1_closed[of "Vr K v"], assumption+)
 apply (frule_tac h = "f m ± -_a b" and I = "vp K v^{Vr K v) (an N)}" in 
           Ring.ideal_subset[of "Vr K v"], assumption+,
        frule_tac h = "f n ± -_a b" and I = "vp K v^{Vr K v) (an N)}" in 
           Ring.ideal_subset[of "Vr K v"], assumption+)  
apply (frule_tac h = "-_{a K v} (f m ± -_a b)" and I = "vp K v^{Vr K v) (an N)}" in         Ring.ideal_subset[of "Vr K v"], assumption+,
       frule_tac h = "f n ± -_a b" and I = "vp K v^{Vr K v) (an N)}" in 
        Ring.ideal_subset[of "Vr K v"], assumption+) 
apply (frule_tac I = "(vp K v)^{(Vr K v) (an N)}" and x = "f n ± -_a b" and
       y = "-_{aVr K v)} (f m ± -_a b)" in Ring.ideal_pOp_closed[of "Vr K v"],
       assumption+)
apply (simp add:Vr_pOp_f_pOp) apply (simp add:Vr_mOp_f_mOp)
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
 apply (frule aGroup.ag_mOp_closed[of "K" "b"], assumption+)
 apply (frule_tac x = "f m ± -_a b" in Vr_mem_f_mem[of "v"], assumption+)
 apply (frule_tac x = "f m ± -_a b" in aGroup.ag_mOp_closed[of "K"], 
         assumption+)
 apply (simp add:aGroup.ag_pOp_assoc)
 apply (simp add:aGroup.ag_pOp_commute[of "K" "-_a b"])
 apply (simp add:aGroup.ag_p_inv[of "K"])
 apply (frule_tac x = "f m" in aGroup.ag_mOp_closed[of "K"], assumption+)
 apply (simp add:aGroup.ag_inv_inv)
 apply (simp add:aGroup.ag_pOp_assoc[of "K" _ "b" "-_a b"])
 apply (simp add:aGroup.ag_r_inv1[of "K"], simp add:aGroup.ag_r_zero)
done

lemma (in Corps) no_limit_zero_Cauchy:"[valuation K v; Cauchy _v f;
    ¬ (lim _v f 0)] ==>
 ∃N M. (∀m. N < m ⟶ ((n_val K v) (f M)) = ((n_val K v) (f m)))"
apply (frule not_limit_zero[of "v" "f"], thin_tac "¬ lim_{K v} f 0")
apply (simp add:Cauchy_seq_def, assumption) apply (erule exE)
apply (rename_tac L)
apply (simp add:Cauchy_seq_def, erule conjE,
       rotate_tac -1,
       frule_tac x = L in spec, thin_tac "∀N. ∃M. ∀n m.
       M < n ∧ M < m ⟶ f n ± -_a (f m) ∈ vp K v^{Vr K v) (an N)}")
apply (erule exE)
apply (drule_tac x = M in spec)
apply (erule exE, erule conjE)
apply (rotate_tac -3,
       frule_tac x = m in spec)
    apply (thin_tac "∀n m. M < n ∧ M < m ⟶
               f n ± -_a (f m) ∈ (vp K v)^{Vr K v) (an L)}")
   apply (subgoal_tac "M < m ∧ (∀ma. M < ma ⟶
                             n_val K v (f m) = n_val K v (f ma))")
   apply blast
  apply simp
 
 apply (rule allI, rule impI)
 apply (rotate_tac -2)
 apply (drule_tac x = ma in spec)
 apply simp
 (** we have f ma \<plusminus> -\<^sub>a f m \<in> vpr K G a i v \<^bsup>\<diamondsuit>Vr K G a i v L\<^esup> as **) 
 apply (frule Vr_ring[of "v"], 
        frule_tac n = "an L" in vp_apow_ideal[of "v"], simp)
apply (frule_tac I = "vp K v^{Vr K v) (an L)}" and x = "f m ± -_a (f ma)"
        in Ring.ideal_inv1_closed[of "Vr K v"], assumption+) apply (
        frule_tac I = "vp K v^{Vr K v) (an L)}" and 
        h = "f m ± -_a (f ma)" in Ring.ideal_subset[of "Vr K v"], 
        assumption+, simp add:Vr_mOp_f_mOp)
   apply (frule_tac x = m in spec,
          drule_tac x = ma in spec)  apply (
         
          cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
          frule_tac x = "f ma" in aGroup.ag_mOp_closed[of "K"], assumption+,
          frule_tac x = "f m" and y = "-_a (f ma)" in aGroup.ag_p_inv[of "K"],
          assumption+, simp only:aGroup.ag_inv_inv,
          frule_tac x = "f m" in aGroup.ag_mOp_closed[of "K"], assumption+,
          simp add:aGroup.ag_pOp_commute,
          thin_tac "-_a ( f m ± -_a (f ma)) = f ma ± -_a (f m)",
          thin_tac "f m ± -_a (f ma) ∈ vp K v^{Vr K v) (an L)}")
  (** finally, by f ma = f m \<plusminus> (f ma \<plusminus> -\<^sub>a (f m)) and value_less_eq 
      we have the conclusion **)
   apply (frule_tac x = "f ma ± -_a (f m)" and n = "an L" in n_value_x_1[of 
         "v" ], simp) apply assumption apply (
         frule n_val_valuation[of "v"], 
         frule_tac x = "f m" and y = "f ma ± -_a (f m)" in value_less_eq[of 
         "n_val K v"], assumption+) apply (simp add:aGroup.ag_pOp_closed) 
  apply (
         rule_tac x = "n_val K v (f m)" and y = "an L" and
         z = "n_val K v ( f ma ± -_a (f m))" in  
         aless_le_trans, assumption+) 
  apply (frule_tac x = "f ma ± -_a (f m)" in Vr_mem_f_mem[of "v"])
  apply (simp add:Ring.ideal_subset)
  apply (frule_tac x = "f m" and y = "f ma ± -_a (f m)" in 
         aGroup.ag_pOp_commute[of "K"], assumption+) 
  apply (simp add:aGroup.ag_pOp_assoc,
         simp add:aGroup.ag_l_inv1, simp add:aGroup.ag_r_zero)
done 

lemma (in Corps) no_limit_zero_Cauchy1:"[valuation K v; ∀j. f j ∈ carrier K;
  Cauchy _v f; ¬ (lim _v f 0)] ==> ∃N M. (∀m. N < m ⟶ v (f M) = v (f m))"
apply (frule no_limit_zero_Cauchy[of "v" "f"], assumption+)
 apply (erule exE)+
 apply (subgoal_tac "∀m. N < m ⟶ v (f M) = v (f m)") apply blast
 apply (rule allI, rule impI)
 apply (frule_tac x = M in spec,
        drule_tac x = m in spec,
        drule_tac x = m in spec, simp)
 apply (simp add:n_val[THEN sym, of "v"])
done
 
definition
  subfield :: "[_, ('b, 'm1) Ring_scheme] ==> bool" where
  "subfield K K' ⟷ Corps K' ∧ carrier K ⊆ carrier K' ∧
                   idmap (carrier K) ∈ rHom K K'"

definition
  v_completion :: "['b ==> ant, 'b ==> ant, _, ('b, 'm) Ring_scheme] ==> bool" 
               (‹(4Completion _{_} _ _)› [90,90,90,91]90) where
  "Completion _v' K K' ⟷ subfield K K' ∧
      Complete_' K' ∧ (∀x ∈ carrier K. v x = v' x) ∧
      (∀x ∈ carrier K'. (∃f. Cauchy _v f ∧ lim_{' v'} f x))"

lemma (in Corps) subfield_zero:"[Corps K'; subfield K K'] ==> 0 = 0_'"
apply (simp add:subfield_def, (erule conjE)+)
apply (simp add:rHom_def, (erule conjE)+)
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule Corps.field_is_ring[of "K'"], frule Ring.ring_is_ag[of "K'"])
apply (frule aHom_0_0[of "K" "K'" "I"], assumption+)
apply (frule aGroup.ag_inc_zero[of "K"], simp add:idmap_def)
done

lemma (in Corps) subfield_pOp:"[Corps K'; subfield K K'; x ∈ carrier K;
      y ∈ carrier K] ==> x ± y = x ±_' y"
apply (cut_tac field_is_ring, frule Corps.field_is_ring[of "K'"],
       frule Ring.ring_is_ag[of "K"], frule Ring.ring_is_ag[of "K'"])
apply (simp add:subfield_def, erule conjE, simp add:rHom_def,
       frule conjunct1)
apply (thin_tac "I ∈ aHom K K' ∧
     (∀x∈carrier K. ∀y∈carrier K. I (x ⋅_r y) = I x ⋅_r' I y) ∧
     I 1_r = 1_r'")
apply (frule aHom_add[of "K" "K'" "I" "x" "y"], assumption+,
       frule aGroup.ag_pOp_closed[of "K" "x" "y"], assumption+, 
       simp add:idmap_def)
done

lemma (in Corps) subfield_mOp:"[Corps K'; subfield K K'; x ∈ carrier K] ==>
                     -_a x = -_a' x"
apply (cut_tac field_is_ring, frule Corps.field_is_ring[of "K'"],
       frule Ring.ring_is_ag[of "K"], frule Ring.ring_is_ag[of "K'"])
apply (simp add:subfield_def, erule conjE, simp add:rHom_def,
       frule conjunct1)
apply (thin_tac "I ∈ aHom K K' ∧
     (∀x∈carrier K. ∀y∈carrier K. I (x ⋅_r y) = I x ⋅_r' I y) ∧
     I 1_r = 1_r'")
apply (frule aHom_inv_inv[of "K" "K'" "I" "x"], assumption+,
       frule aGroup.ag_mOp_closed[of "K" "x"], assumption+)
apply (simp add:idmap_def)
done

lemma (in Corps) completion_val_eq:"[Corps K'; valuation K v; valuation K' v';
 x ∈ carrier K; Completion _v' K K'] ==> v x = v' x"
apply (unfold v_completion_def, (erule conjE)+)
apply simp
done

lemma (in Corps) completion_subset:"[Corps K'; valuation K v; valuation K' v';
 Completion _v' K K'] ==> carrier K ⊆ carrier K'"
apply (unfold v_completion_def, (erule conjE)+)
apply (simp add:subfield_def)
done

lemma (in Corps) completion_subfield:"[Corps K'; valuation K v;
       valuation K' v'; Completion _v' K K'] ==> subfield K K'"
apply (simp add:v_completion_def)
done

lemma (in Corps) subfield_sub:"subfield K K' ==> carrier K ⊆ carrier K'"
apply (simp add:subfield_def)
done

lemma (in Corps) completion_Vring_sub:"[Corps K'; valuation K v;
  valuation K' v'; Completion _v' K K'] ==>
                     carrier (Vr K v) ⊆ carrier (Vr K' v')"
apply (rule subsetI,
      frule completion_subset[of  "K'" "v" "v'"], assumption+,
      frule_tac x = x in Vr_mem_f_mem[of "v"], assumption+,
      frule_tac x = x in completion_val_eq[of "K'" "v" "v'"],
        assumption+)
apply (frule_tac x1 = x in val_pos_mem_Vr[THEN sym, of  "v"],
       assumption+, simp,
       frule_tac c = x in subsetD[of "carrier K" "carrier K'"], assumption+,
      simp add:Corps.val_pos_mem_Vr[of "K'" "v'"])
done

lemma (in Corps) completion_idmap_rHom:"[Corps K'; valuation K v;
  valuation K' v'; Completion _v' K K'] ==>
             I_{Vr K v)} ∈ rHom (Vr K v) (Vr K' v')"
apply (frule completion_Vring_sub[of  "K'" "v" "v'"],
         assumption+,
       frule completion_subfield[of "K'" "v" "v'"], 
         assumption+,
       frule Vr_ring[of "v"],
        frule Ring.ring_is_ag[of "Vr K v"],
       frule Corps.Vr_ring[of "K'" "v'"], assumption+,
       frule Ring.ring_is_ag[of "Vr K' v'"])
apply (simp add:rHom_def)
apply (rule conjI) 
 apply (simp add:aHom_def,
        rule conjI,
        simp add:idmap_def, simp add:subsetD)
apply (rule conjI)
 apply (simp add:idmap_def extensional_def)
 apply ((rule ballI)+) apply (
        frule_tac x = a and y = b in aGroup.ag_pOp_closed, assumption+,
        simp add:idmap_def, 
   frule_tac c = a in subsetD[of "carrier (Vr K v)" 
                         "carrier (Vr K' v')"], assumption+,
   frule_tac c = b in subsetD[of "carrier (Vr K v)" 
                         "carrier (Vr K' v')"], assumption+,
   simp add:Vr_pOp_f_pOp,
   frule_tac x = a in Vr_mem_f_mem[of v], assumption,
   frule_tac x = b in Vr_mem_f_mem[of v], assumption,
   simp add:Corps.Vr_pOp_f_pOp,
   simp add:subfield_pOp)
apply (rule conjI)
 apply ((rule ballI)+, 
        frule_tac x = x and y = y in Ring.ring_tOp_closed, assumption+,
        simp add:idmap_def, simp add:subfield_def, erule conjE)
 apply (frule_tac c = x in subsetD[of "carrier (Vr K v)" 
            "carrier (Vr K' v')"], assumption+,
        frule_tac c = y in subsetD[of "carrier (Vr K v)" 
            "carrier (Vr K' v')"], assumption+)
 apply (simp add:Vr_tOp_f_tOp Corps.Vr_tOp_f_tOp)
apply (frule_tac x = x in Vr_mem_f_mem[of "v"], assumption+,
       frule_tac x = y in Vr_mem_f_mem[of "v"], assumption+,
     frule_tac x = x in Corps.Vr_mem_f_mem[of "K'" "v'"], assumption+,
     frule_tac x = y in Corps.Vr_mem_f_mem[of "K'" "v'"], assumption+,
       cut_tac field_is_ring, frule Corps.field_is_ring[of "K'"],
       frule_tac x = x and y = y in Ring.ring_tOp_closed[of "K"], assumption+)
apply (frule_tac x = x and y = y in rHom_tOp[of "K" "K'" _ _ "I"], 
                 assumption+, simp add:idmap_def)
apply (frule Ring.ring_one[of "Vr K v"], simp add:idmap_def)
apply (simp add:Vr_1_f_1 Corps.Vr_1_f_1)
apply (simp add:subfield_def, (erule conjE)+)
apply (cut_tac field_is_ring, frule Corps.field_is_ring[of "K'"],
       frule Ring.ring_one[of "K"],
       frule rHom_one[of "K" "K'" "I"], assumption+, simp add:idmap_def)
done

lemma (in Corps) completion_vpr_sub:"[Corps K'; valuation K v; valuation K' v';
      Completion _v' K K'] ==> vp K v ⊆ vp K' v'"
apply (rule subsetI,
      frule completion_subset[of "K'" "v" "v'"], assumption+,
      frule Vr_ring[of "v"], frule vp_ideal[of "v"],
        frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"],
        assumption+,
      frule_tac x = x in Vr_mem_f_mem[of  "v"], assumption+,
      frule_tac x = x in completion_val_eq[of "K'" "v" "v'"],
        assumption+)
apply (frule completion_subset[of "K'" "v" "v'"],
        assumption+,
       frule_tac c = x in subsetD[of "carrier K" "carrier K'"], assumption+,
       simp add:Corps.vp_mem_val_poss vp_mem_val_poss)
done

lemma (in Corps) val_v_completion:"[Corps K'; valuation K v; valuation K' v';
      x ∈ carrier K'; x ≠ 0_'; Completion _v' K K'] ==>
   ∃f. (Cauchy _v f) ∧ (∃N. (∀m. N < m ⟶ v (f m) = v' x))"
apply (simp add:v_completion_def, erule conjE, (erule conjE)+)
 apply (rotate_tac -1, drule_tac x = x in bspec, assumption+,
        erule exE, erule conjE,
        subgoal_tac "∃N. ∀m. N < m ⟶ v (f m) = v' x", blast) 
    thm Corps.limit_val
 apply (frule_tac f = f and v = v' in  Corps.limit_val[of "K'" "x"],
        assumption+,
        unfold Cauchy_seq_def, frule conjunct1, fold Cauchy_seq_def) 
apply (rule allI, drule_tac x = j in spec,
       simp add:subfield_def, erule conjE, simp add:subsetD, assumption+)
 apply (simp add:Cauchy_seq_def)
done

lemma (in Corps) v_completion_v_limit:"[Corps K'; valuation K v;
      x ∈ carrier K; subfield K K'; Complete_' K'; ∀j. f j ∈ carrier K;
      valuation K' v'; ∀x∈carrier K. v x = v' x; lim_{' v'} f x] ==> lim _v f x"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule Corps.field_is_ring[of K'], frule Ring.ring_is_ag[of "K'"],
       subgoal_tac "∀j. f j ∈ carrier K'",
       unfold subfield_def, frule conjunct2, fold subfield_def, erule conjE)
apply (frule subsetD[of "carrier K" "carrier K'" "x"], assumption+,
       simp add:limit_diff_val[of "x" "f"],
       subgoal_tac "∀n. f n ±_' -_a' x = f n ± -_a x")
 apply (rule allI)
 apply (simp add:limit_def)
 apply (rotate_tac 6, drule_tac x = N in spec,
        erule exE)
 apply (subgoal_tac "∀n>M. an N ≤ v'(f n ± -_a x)",
        subgoal_tac "∀n. (v' (f n ± -_a x) = v (f n ± -_a x))", simp, blast)
 apply (rule allI)
 apply (frule_tac x = "f n ± -_a x" in bspec,
        rule aGroup.ag_pOp_closed, assumption+, simp,
        rule aGroup.ag_mOp_closed, assumption+) apply simp
 apply (rule allI, rule impI)
 apply (frule_tac v = v' and n = "an N" and x = "f n ± -_a x" in 
         Corps.n_value_x_1[of K'], assumption+, simp, simp)
 apply (frule_tac v = v' and x = "f n ± -_a x" in Corps.n_val_le_val[of K'],
         assumption+) 
 apply (cut_tac x = "f n" and y = "-_a x" in aGroup.ag_pOp_closed, assumption,
        simp, rule aGroup.ag_mOp_closed, assumption+, simp add:subsetD)
 apply (subst Corps.val_pos_n_val_pos[of K' v'], assumption+)
   apply (cut_tac x = "f n" and y = "-_a x" in aGroup.ag_pOp_closed, assumption,
        simp, rule aGroup.ag_mOp_closed, assumption+, simp add:subsetD)
apply (rule_tac i = 0 and j = "an N" and k = "n_val K' v' (f n ± -_a x)" in 
       ale_trans, simp+, rule allI)
 apply (subst subfield_pOp[of K'], assumption+, simp+,
        rule aGroup.ag_mOp_closed, assumption+)
 apply (simp add:subfield_mOp[of K'])
 apply (cut_tac subfield_sub[of K'], simp add:subsetD, assumption+)
done
    
lemma (in Corps) Vr_idmap_aHom:"[Corps K'; valuation K v; valuation K' v';
      subfield K K'; ∀x∈carrier K. v x = v' x] ==>
                              I_{Vr K v)} ∈ aHom (Vr K v) (Vr K' v')"
apply (simp add:aHom_def)
apply (subgoal_tac "I_{Vr K v)} ∈ carrier (Vr K v) → carrier (Vr K' v')")
apply simp
apply (rule conjI)
 apply (simp add:idmap_def)
apply (rule ballI)+
 apply (frule Vr_ring[of "v"],
        frule Ring.ring_is_ag[of "Vr K v"],
        frule Corps.Vr_ring[of "K'" "v'"], assumption+,
        frule Ring.ring_is_ag[of "Vr K' v'"]) 
 apply (frule_tac x = a and y = b in aGroup.ag_pOp_closed[of "Vr K v"],
               assumption+,
        frule_tac x = a in funcset_mem[of "I_{Vr K v)}" 
        "carrier (Vr K v)" "carrier (Vr K' v')"], assumption+,
        frule_tac x = b in funcset_mem[of "I_{Vr K v)}" 
        "carrier (Vr K v)" "carrier (Vr K' v')"], assumption+,
        frule_tac x = "(I_{Vr K v)}) a" and y = "(I_{Vr K v)}) b" in 
        aGroup.ag_pOp_closed[of "Vr K' v'"], assumption+, 
        simp add:Vr_pOp_f_pOp)
 apply (simp add:idmap_def, simp add:subfield_def, erule conjE,
        simp add:rHom_def, frule conjunct1,
        thin_tac "I ∈ aHom K K' ∧
           (∀x∈carrier K. ∀y∈carrier K. I (x ⋅_r y) = I x ⋅_r' I y) ∧
           I 1_r = 1_r'")
  apply (simp add:Corps.Vr_pOp_f_pOp[of K' v'])
  apply (frule_tac x = a in Vr_mem_f_mem[of v], assumption+,
         frule_tac x = b in Vr_mem_f_mem[of v], assumption+)
  apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of K],
         frule Corps.field_is_ring[of K'], frule Ring.ring_is_ag[of K']) 
  apply (frule_tac a = a and b = b in aHom_add[of K K' "I"], assumption+,
         frule_tac x = a and y = b in aGroup.ag_pOp_closed[of K], assumption+,
         simp add:idmap_def)
apply (rule Pi_I,
       drule_tac x = x in bspec, simp add:Vr_mem_f_mem)
apply (simp add:idmap_def)
 apply (frule_tac x1 = x in val_pos_mem_Vr[THEN sym, of v],
         simp add:Vr_mem_f_mem, simp)
 apply (subst Corps.val_pos_mem_Vr[THEN sym, of K' v'], assumption+,
        frule_tac x = x in Vr_mem_f_mem[of v], assumption+,
        frule subfield_sub[of K'], simp add:subsetD)
 apply assumption
done

lemma amult_pos_pos:"0 ≤ a ==> 0 ≤ a * an N"
apply (case_tac "N = 0", simp add:an_0)
 apply (case_tac "a = ∞", simp) 
 apply (frule apos_neq_minf[of a])
 apply (subst ant_tna[THEN sym, of a], simp)
 apply (subst amult_0_r, simp)
apply (case_tac "a = ∞", simp add:an_def) 
 apply (frule apos_neq_minf[of a])
 apply (subst ant_tna[THEN sym, of a], simp)
 apply (case_tac "a = 0", simp)
 apply (simp add:ant_0 an_def amult_0_l)
apply (cut_tac amult_pos1[of "tna a" "an N"]) 
 apply (simp add:ant_tna)
 apply (rule_tac ale_trans[of 0 "an N" "a * an N"], simp+)
 apply (frule ale_neq_less[of 0 a], rule not_sym, assumption)
 apply (subst aless_zless[THEN sym, of 0 "tna a"], simp add:ant_tna ant_0)
 apply simp
done

lemma (in Corps) Cauchy_down:"[Corps K'; valuation K v; valuation K' v';
  subfield K K'; ∀x∈carrier K. v x = v' x; ∀j. f j ∈ carrier K; Cauchy_{' v'} f]
  ==> Cauchy _v f" 
apply (simp add:Cauchy_seq_def, rule allI, erule conjE) 
apply (rotate_tac -1, drule_tac 
  x = "na (Lv K v) * N" in spec,
  erule exE,
  subgoal_tac "∀n m. M < n ∧ M < m ⟶
      f n ± (-_a (f m)) ∈ vp K v^{Vr K v) (an N)}", blast)
 apply (simp add:amult_an_an) apply (simp add:an_na_Lv)
 apply ((rule allI)+, rule impI, erule conjE) apply (
      rotate_tac -3, drule_tac x = n in spec,
      rotate_tac -1, drule_tac x = m in spec,
      simp)
 apply (rotate_tac 7,
        frule_tac x = n in spec,
        drule_tac x = m in spec)
 apply (simp add:subfield_mOp[THEN sym],
        cut_tac field_is_ring, frule Ring.ring_is_ag[of K],
        frule_tac x = "f m" in aGroup.ag_mOp_closed[of K], assumption+)
 apply (simp add:subfield_pOp[THEN sym])
 apply (frule_tac x = "f n" and y = "-_a f m" in aGroup.ag_pOp_closed, 
        assumption+,
        frule subfield_sub[of K'],
        frule_tac c = "f n ± -_a f m" in subsetD[of "carrier K" "carrier K'"], 
        assumption+)
 apply (frule Lv_pos[of v],
        frule aless_imp_le[of 0 "Lv K v"])
 apply (frule_tac N = N in amult_pos_pos[of "Lv K v"])
 apply (frule_tac n = "(Lv K v) * an N" and x = "f n ± -_a f m" in 
        Corps.n_value_x_1[of K' v'], assumption+)
 apply (frule_tac x = "f n ± -_a f m" in Corps.n_val_le_val[of K' v'],
        assumption+,
        frule_tac j = "Lv K v * an N" and k = "n_val K' v' (f n ± -_a f m)" in
         ale_trans[of 0], assumption+, simp add:Corps.val_pos_n_val_pos)
 apply (frule_tac i = "Lv K v * an N" and j = "n_val K' v' (f n ± -_a f m)" 
        and k = "v' (f n ± -_a f m)" in ale_trans, assumption+,
        thin_tac "n_val K' v' (f n ± -_a f m) ≤ v' (f n ± -_a f m)",
        thin_tac "Lv K v * an N ≤ n_val K' v' (f n ± -_a f m)")
 apply (rotate_tac 1,
        drule_tac x = "f n ± -_a f m" in bspec, assumption,
        rotate_tac -1, drule sym, simp)
 apply (frule_tac v1 = v and x1 = "f n ± -_a f m" in n_val[THEN sym],
        assumption)
 apply simp
 apply (simp only:amult_commute[of _ "Lv K v"])
apply (frule Lv_z[of v], erule exE)
 
apply (cut_tac w = z and x = "an N" and y = "n_val K v (f n ± -_a f m)" in
       amult_pos_mono_l,
       cut_tac m = 0 and n = z in aless_zless, simp add:ant_0)
 apply simp
 apply (rule_tac x="f n ± -_a (f m)" and n = "an N" in n_value_x_2[of v], 
        assumption+) 
 apply (subst val_pos_mem_Vr[THEN sym, of v], assumption+)
 apply (subst val_pos_n_val_pos[of v], assumption+)
 apply (rule_tac j = "an N" and k = "n_val K v (f n ± -_a f m)" in 
        ale_trans[of 0], simp, assumption+)
 apply simp
done

lemma (in Corps) Cauchy_up:"[Corps K'; valuation K v; valuation K' v';
      Completion _v' K K'; Cauchy_{K v} f] ==> Cauchy_{K' v'} f" 
apply (simp add:Cauchy_seq_def,
       erule conjE,
       rule conjI, unfold v_completion_def, frule conjunct1,
       fold v_completion_def, rule allI, frule subfield_sub[of K'])
apply (simp add:subsetD) 

apply (rule allI)
apply (rotate_tac -1, drule_tac x = "na (Lv K' v') * N" 
      in spec, erule exE) 
apply (subgoal_tac "∀n m. M < n ∧ M < m ⟶
         f n ±_' (-_a' (f m)) ∈ vp K' v'^{Vr K' v') (an N)}", blast,
       (rule allI)+, rule impI, erule conjE)
apply (rotate_tac -3, drule_tac x = n in spec,
       rotate_tac -1, 
       drule_tac x = m in spec, simp,
       frule_tac x = n in spec,
       drule_tac x = m in spec)
 apply(unfold v_completion_def, frule conjunct1, fold v_completion_def,
       cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule_tac x = "f m" in aGroup.ag_mOp_closed[of "K"], assumption+,
       frule_tac x = "f n" and y = "-_a (f m)" in aGroup.ag_pOp_closed[of "K"],
       assumption+)
 apply (simp add:amult_an_an) apply (simp add:Corps.an_na_Lv)
apply (simp add:subfield_mOp, simp add:subfield_pOp) apply (
       frule_tac x = "f n ±_' -_a' f m" and n = "(Lv K' v') * (an N)"
       in n_value_x_1[of v]) (*apply (
  thin_tac "f n \<plusminus>' (-\<^sub>a' (f m)) \<in> vp K v\<^bsup>(Vr K v) ( (Lv K' v') * (an N))\<^esup>",
   simp add:v_completion_def, (erule conjE)+) *) 
apply (frule Corps.Lv_pos[of "K'" "v'"], assumption+, 
       frule Corps.Lv_z[of "K'" "v'"],
       assumption, erule exE, simp, 
       cut_tac n = N in an_nat_pos,
       frule_tac w1 = z and x1 = 0 and y1 = "an N" in 
             amult_pos_mono_l[THEN sym], simp, simp add:amult_0_r)
apply assumption
apply (frule_tac x = "f n ±_' -_a' f m " in n_val_le_val[of v],
        assumption+)
apply (subst n_val[THEN sym, of "v"], assumption+)
apply (frule Lv_pos[of "v"], frule Lv_z[of v], erule exE, simp)
apply (frule Corps.Lv_pos[of "K'" "v'"], assumption+, 
       frule Corps.Lv_z[of "K'" "v'"],   assumption, erule exE, simp, 
       cut_tac n = N in an_nat_pos,
       frule_tac w1 = za and x1 = 0 and y1 = "an N" in 
             amult_pos_mono_l[THEN sym], simp, simp add:amult_0_r)
apply (frule_tac j = "ant za * an N" and k = "n_val K v (f n ±_' -_a' (f m))"
       in ale_trans[of "0"], assumption+)
apply (frule_tac w1 = z and x1 = 0 and y1 = "n_val K v ( f n ±_' -_a' (f m))"
       in amult_pos_mono_r[THEN sym], simp, simp add:amult_0_l)
apply (frule_tac i = "Lv K' v' * an N" and j ="n_val K v ( f n ±_' -_a' (f m))"
       and k = "v ( f n ±_' -_a' (f m))" in ale_trans, assumption+)
apply (thin_tac "f n ±_' -_a' (f m) ∈ vp K v^{(Vr K v) (Lv K' v') * (an N)}",
       thin_tac "Lv K' v' * an N ≤ n_val K v ( f n ±_' -_a' (f m))",
       thin_tac "n_val K v ( f n ±_' -_a' (f m)) ≤ v ( f n ±_' -_a' (f m))")

apply (simp add:v_completion_def, (erule conjE)+)
 apply (thin_tac "∀x∈carrier K. v x = v' x",
        thin_tac "∀x∈carrier K'. ∃f. Cauchy_{K v} f ∧ lim_{K' v'} f x")
 apply (frule subfield_sub[of K'],
        frule_tac c = "f n ±_' -_a' (f m)" in 
                  subsetD[of "carrier K" "carrier K'"], assumption+)
apply (simp add:Corps.n_val[THEN sym, of "K'" "v'"])
apply (simp add:amult_commute[of _ "Lv K' v'"])
apply (frule Corps.Lv_pos[of "K'" "v'"], assumption, 
       frule Corps.Lv_z[of "K'" "v'"], assumption+, erule exE, simp)
apply (simp add:amult_pos_mono_l)

apply (rule_tac x = "f n ±_' -_a' (f m)" and n = "an N" in 
       Corps.n_value_x_2[of "K'" "v'"], assumption+)
apply (cut_tac n = N in an_nat_pos)
 apply (frule_tac j = "an N" and k = "n_val K' v' (f n ±_' -_a' (f m))" in 
        ale_trans[of "0"], assumption+)
 apply (simp add:Corps.val_pos_n_val_pos[THEN sym, of "K'" "v'"])
 apply (simp add:Corps.val_pos_mem_Vr) apply assumption apply simp
done

lemma max_gtTr:"(n::nat) < max (Suc n) (Suc m) ∧ m < max (Suc n) (Suc m)"
by (simp add:max_def)

lemma (in Corps) completion_approx:"[Corps K'; valuation K v; valuation K' v';
      Completion _v' K K'; x ∈ carrier (Vr K' v')] ==>
               ∃y∈carrier (Vr K v). (y ±_' -_a' x) ∈ (vp K' v')"
(** show an element y near by x is included in (Vr K v) **) 
apply (frule Corps.Vr_mem_f_mem [of "K'" "v'" "x"], assumption+,
       frule Corps.val_pos_mem_Vr[THEN sym, of "K'" "v'" "x"], assumption+,
       simp add:v_completion_def, (erule conjE)+,
       rotate_tac -1, drule_tac x = x in bspec, assumption+, 
       erule exE, erule conjE)
apply (unfold Cauchy_seq_def, frule conjunct1, fold Cauchy_seq_def)
apply (case_tac "x = 0_'", 
       simp, frule Corps.field_is_ring[of "K'"], 
       frule Ring.ring_is_ag[of "K'"],
       subgoal_tac " 0_' ∈ carrier (Vr K v)",
       subgoal_tac " (0_' ±_' -_a' 0_')∈ vp K' v'", blast,
       frule aGroup.ag_inc_zero[of "K'"], simp add:aGroup.ag_r_inv1,
       simp add:Corps.Vr_0_f_0[THEN sym, of "K'" "v'"],
       frule Corps.Vr_ring[of "K'" "v'"], assumption+,
       frule Corps.vp_ideal[of "K'" "v'"], assumption+,
       simp add:Ring.ideal_zero,
       simp add:subfield_zero[THEN sym, of  "K'"],
       cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule aGroup.ag_inc_zero[of "K"],
       simp add:Vr_0_f_0[THEN sym, of "v"],
       frule Vr_ring[of "v"],simp add:Ring.ring_zero)
      apply (frule_tac f = f in Corps.limit_val[of "K'" "x" _ "v'"], 
                         assumption+)
      apply (rule allI, rotate_tac -2, frule_tac x = j in spec,
             frule subfield_sub[of K'], simp add:subsetD, assumption+)
apply (erule exE)
 apply (simp add:limit_def,
        frule Corps.Vr_ring[of K' v'], assumption+,
        rotate_tac 10,
              drule_tac x = "Suc 0" in spec, erule exE,
       rotate_tac 1,
              frule_tac x = N and y = M in two_inequalities, assumption+,
              thin_tac "∀n>N. v' (f n) = v' x",
              thin_tac "∀n>M. f n ±_' -_a' x ∈ vp K' v'^{(Vr K' v') (an (Suc 0))}")
       apply (frule Corps.vp_ideal[of K' v'], assumption+,
              simp add:Ring.r_apow_Suc[of "Vr K' v'" "vp K' v'"])
       apply (drule_tac x = "N + M + 1" in spec, simp,
              drule_tac x = "N + M + 1" in spec, simp,
              erule conjE)
 apply (drule_tac x = "f (Suc (N + M))" in bspec, assumption+)
 apply simp 
 apply (cut_tac x = "f (Suc (N + M))" in val_pos_mem_Vr[of v], assumption+)
 apply simp apply blast
done

lemma (in Corps) res_v_completion_surj:"[ Corps K'; valuation K v;
      valuation K' v'; Completion _v' K K'] ==>
      surjec_{Vr K v),(qring (Vr K' v') (vp K' v'))}
            (compos (Vr K v) (pj (Vr K' v') (vp K' v')) (I_{Vr K v)}))"
apply (frule Vr_ring[of "v"], 
       frule Ring.ring_is_ag[of "Vr K v"],
       frule Corps.Vr_ring[of "K'" "v'"], assumption+,
       frule Ring.ring_is_ag[of "Vr K' v'"],
       frule Ring.ring_is_ag[of "Vr K v"])
apply (frule Corps.vp_ideal[of "K'" "v'"], assumption+, 
       frule Ring.qring_ring[of "Vr K' v'" "vp K' v'"], assumption+)
apply (simp add:surjec_def)
apply (frule aHom_compos[of "Vr K v" "Vr K' v'" 
      "qring (Vr K' v') (vp K' v')" "I_{Vr K v)}" 
      "pj (Vr K' v') (vp K' v')"], assumption+, simp add:Ring.ring_is_ag)
apply (rule Vr_idmap_aHom, assumption+) apply (simp add:completion_subfield,
      simp add:v_completion_def) apply (
      frule pj_Hom[of "Vr K' v'" "vp K' v'"], assumption+) apply ( 
      simp add:rHom_def, simp)
apply (rule surj_to_test)
 apply (simp add:aHom_def)
apply (rule ballI) 
 apply (thin_tac "Ring (Vr K' v' /_r vp K' v')",
  thin_tac "compos (Vr K v) (pj (Vr K' v') (vp K' v')) (I_{Vr K v)}) ∈
               aHom (Vr K v) (Vr K' v' /_r vp K' v')")
 apply (simp add:Ring.qring_carrier)
 apply (erule bexE)
 apply (frule_tac x = a in completion_approx[of "K'" "v" "v'"], 
       assumption+, erule bexE)
 apply (subgoal_tac "compos (Vr K v) (pj (Vr K' v')
                     (vp K' v')) ((I_{Vr K v)})) y = b", blast)
 apply (simp add:compos_def compose_def idmap_def)
 apply (frule completion_Vring_sub[of "K'" "v" "v'"], assumption+)
 apply (frule_tac c = y in subsetD[of "carrier (Vr K v)" "carrier (Vr K' v')"],
         assumption+)
 apply (frule_tac x = y in pj_mem[of "Vr K' v'" "vp K' v'"], assumption+, simp,
        thin_tac "pj (Vr K' v') (vp K' v') y = y ⊎_{Vr K' v')} (vp K' v')")
 apply (rotate_tac -5, frule sym, thin_tac "a ⊎_{Vr K' v')} (vp K' v') = b", 
       simp)
 apply (rule_tac b1 = y and a1 = a in Ring.ar_coset_same1[THEN sym, 
        of "Vr K' v'" "vp K' v'"], assumption+)
 apply (frule Ring.ring_is_ag[of "Vr K' v'"], 
        frule_tac x = a in aGroup.ag_mOp_closed[of "Vr K' v'"],
        assumption+)
 apply (simp add:Corps.Vr_mOp_f_mOp, simp add:Corps.Vr_pOp_f_pOp)
done

lemma (in Corps) res_v_completion_ker:"[Corps K'; valuation K v;
      valuation K' v'; Completion _v' K K'] ==>
  ker_{Vr K v), (qring (Vr K' v') (vp K' v'))}
  (compos (Vr K v) (pj (Vr K' v') (vp K' v')) (I_{Vr K v)})) = vp K v"
apply (rule equalityI)
 apply (rule subsetI)
 apply (simp add:ker_def, (erule conjE)+)
 apply (frule Corps.Vr_ring[of "K'" "v'"], assumption+,
        frule Corps.vp_ideal[of "K'" "v'"], assumption+,
        frule Ring.qring_ring[of "Vr K' v'" "vp K' v'"], assumption+,
        simp add:Ring.qring_zero)
 apply (simp add:compos_def, simp add:compose_def, simp add:idmap_def)
 apply (frule completion_Vring_sub[of "K'" "v" "v'"], assumption+)
 apply (frule_tac c = x in subsetD[of "carrier (Vr K v)" "carrier (Vr K' v')"],
        assumption+)
 apply (simp add:pj_mem)
 apply (frule_tac a = x in Ring.qring_zero_1[of "Vr K' v'" _  "vp K' v'"], 
        assumption+)
 apply (subst vp_mem_val_poss[of v], assumption+) 
 apply (simp add:Vr_mem_f_mem)
 apply (frule_tac x = x in Corps.vp_mem_val_poss[of "K'" "v'"],
        assumption+, simp add:Corps.Vr_mem_f_mem, simp)
apply (frule_tac x = x in Vr_mem_f_mem[of v], assumption+)
 apply (frule_tac x = x in completion_val_eq[of "K'" "v" "v'"],
         assumption+, simp)
apply (rule subsetI)
 apply (simp add:ker_def)
 apply (frule Vr_ring[of  "v"])
 apply (frule vp_ideal[of "v"])
 apply (frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"],
         assumption+, simp)
 apply (frule Corps.Vr_ring[of "K'" "v'"], assumption+,
        frule Corps.vp_ideal[of "K'" "v'"], assumption+, 
        simp add:Ring.qring_zero)
 apply (simp add:compos_def compose_def idmap_def)
 apply (frule completion_Vring_sub[of "K'" "v" "v'"],
        assumption+, frule_tac c = x in subsetD[of "carrier (Vr K v)"
        "carrier (Vr K' v')"], assumption+)
 apply (simp add:pj_mem)
 apply (frule completion_vpr_sub[of "K'" "v" "v'"], assumption+,
        frule_tac c = x in subsetD[of "vp K v" "vp K' v'"], assumption+)
 apply (simp add:Ring.ar_coset_same4[of "Vr K' v'" "vp K' v'"])
done 

lemma (in Corps) completion_res_qring_isom:"[Corps K'; valuation K v;
  valuation K' v'; Completion _v' K K'] ==>
   r_isom ((Vr K v) /_r (vp K v)) ((Vr K' v') /_r (vp K' v'))"
apply (subst r_isom_def)
apply (frule res_v_completion_surj[of "K'" "v" "v'"], assumption+)
apply (frule Vr_ring[of "v"],
       frule Corps.Vr_ring[of "K'" "v'"], assumption+,
       frule Corps.vp_ideal[of "K'" "v'"], assumption+,
       frule Ring.qring_ring[of "Vr K' v'" "vp K' v'"], assumption+)
apply (frule rHom_compos[of "Vr K v" "Vr K' v'" "(Vr K' v' /_r vp K' v')" 
       "(I_{Vr K v)})" "pj (Vr K' v') (vp K' v')"], assumption+)
apply (simp add:completion_idmap_rHom)
apply (simp add:pj_Hom) 
apply (frule surjec_ind_bijec [of "Vr K v" "(Vr K' v' /_r vp K' v')" 
      "compos (Vr K v) (pj (Vr K' v') (vp K' v')) (I_{Vr K v)})"], assumption+)
apply (frule ind_hom_rhom[of "Vr K v" "(Vr K' v' /_r vp K' v')" 
       "compos (Vr K v) (pj (Vr K' v') (vp K' v')) (I_{Vr K v)})"], assumption+)
apply (simp add:res_v_completion_ker) apply blast
done

text‹expansion of x in a complete field K, with normal valuation v. Here
  suppose t is an element of K satisfying the equation ‹v t = 1›.›

definition
  Kxa :: "[_, 'b ==> ant, 'b] ==> 'b set" where
  "Kxa K v x = {y. ∃k∈carrier (Vr K v). y = x ⋅_r k}"
    (**  x *\<^sub>r carrier (Vr K v) **)

primrec 
  partial_sum :: "[('b, 'm) Ring_scheme, 'b, 'b ==> ant, 'b]
                                   ==> nat ==> 'b"
        (‹(5psum_{_ _ _ _} _)› [96,96,96,96,97]96)
where
  psum_0: "psum_{K x v t} 0 = (csrp_fn (Vr K v) (vp K v)
     (pj (Vr K v) (vp K v) (x ⋅_r t^{(tna (v x))}))) ⋅_r (t^{tna (v x))})"
| psum_Suc: "psum_{K x v t} (Suc n) = (psum_{K x v t} n) ±
  ((csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v)
   ((x ± -_a (psum_{K x v t} n)) ⋅_r (t ^{(tna (v x) + int (Suc n))})))) ⋅_r
         (t^{tna (v x) + int (Suc n))}))" 

definition
  expand_coeff :: "[_ , 'b ==> ant, 'b, 'b]
                      ==> nat ==> 'b"
                   (‹(5ecf _{_ _ _} _)› [96,96,96,96,97]96) where
  "ecf _{v t x} n = (if n = 0 then csrp_fn (Vr K v) (vp K v)
     (pj (Vr K v) (vp K v) (x ⋅_r t^{-(tna (v x)))}))
     else csrp_fn (Vr K v) (vp K v) (pj (Vr K v)
  (vp K v) ((x ± -_a (psum_{K x v t} (n - 1))) ⋅_r (t^{- (tna (v x) + int n))}))))" 

definition
  expand_term :: "[_, 'b ==> ant, 'b, 'b]
                      ==> nat ==> 'b"
                   (‹(5etm_{_ _ _ _} _)› [96,96,96,96,97]96) where
        
  "etm _{v t x} n = (ecf _{v t x} n)⋅_r (t^{tna (v x) + int n)})"



(*** Let O be the valuation ring with respect to the valuation v and let P
be the maximal ideal of O. Let j be the value of x (\<in> O), and  a\<^sub>0 be an 
element of the complete set of representatives such that a\<^sub>0 = x t\<^sup>-\<^sup>j mod P.
 We see that (a\<^sub>0 - x t\<^sup>-\<^sup>j)/t is an element of O, and then we choose a\<^sub>1 an 
element of the complete set of representatives which is equal to (a\<^sub>0 - x t\<^sup>-\<^sup>j)/tmodulo P. We see x - (a\<^sub>0t\<^sup>j + a\<^sub>1t\<^bsup>(j+1)\<^esup> + \<dots> + a\<^sub>st\<^bsup>(j+s)\<^esup>) \<in> (t\<^bsup>(j+s+1)\<^esup>). 
"psum G a i K v t x s" is the partial sum a\<^sub>0t\<^sup>j + a\<^sub>1t\<^bsup>(j+1)\<^esup> + \<dots> + a\<^sub>st\<^bsup>(j+s)\<^esup> ***)  

lemma (in Corps) Kxa_val_ge:"[valuation K v; t ∈ carrier K; v t = 1]
 ==> Kxa K v (t) = {x. x ∈ carrier K ∧ (ant j) ≤ (v x)}"
apply (frule val1_neq_0[of v t], assumption+)
apply (cut_tac field_is_ring) 
apply (rule equalityI)
 apply (rule subsetI,
        simp add:Kxa_def, erule bexE,
        frule_tac x = k in Vr_mem_f_mem[of "v"], assumption+,
        frule npowf_mem[of "t" "j"], simp, 
        simp add:Ring.ring_tOp_closed)
 apply (simp add:val_t2p) apply (simp add:val_exp[THEN sym]) 
 apply (simp add:val_pos_mem_Vr[THEN sym, of "v"])
 apply (frule_tac x = 0 and y = "v k" in aadd_le_mono[of _ _ "ant j"])
 apply (simp add:aadd_0_l aadd_commute)
apply (rule subsetI, simp, erule conjE)
 apply (simp add:Kxa_def)
 apply (case_tac "x = 0")
 apply (frule Vr_ring[of "v"], 
        frule Ring.ring_zero[of "Vr K v"],
        simp add:Vr_0_f_0,
        frule_tac x1 = "t" in Ring.ring_times_x_0[THEN sym, of "K"],
        simp add:npowf_mem, blast)
 apply (frule val_exp[of "v" "t" "j"], assumption+, simp) 
 apply (frule field_potent_nonzero1[of "t" "j"], 
        frule npowf_mem[of "t" "j"], assumption+)
 apply (frule_tac y = "v x" in ale_diff_pos[of "v (t)"], 
        simp add:diff_ant_def)
apply (cut_tac npowf_mem[of t j]) 
 defer 
 apply assumption apply simp
apply (frule value_of_inv[THEN sym, of "v" "t"], assumption+) 
 
 apply (cut_tac invf_closed1[of "t"], simp, erule conjE)
 apply (frule_tac x1 = x in val_t2p[THEN sym, of "v" _ "(t)^<hyphen>K"], 
        assumption+, simp)
 apply (frule_tac x = "(t)^<hyphen>K" and y = x in Ring.ring_tOp_closed[of "K"], 
          assumption+)
 apply (simp add:Ring.ring_tOp_commute[of "K" _ "(t)^<hyphen>K"])
 apply (frule_tac x = "((t)^<hyphen>K) ⋅_r x" in val_pos_mem_Vr[of v], assumption+, 
        simp)
 apply (frule_tac z = x in Ring.ring_tOp_assoc[of "K" "t" "(t)^<hyphen>K"], 
         assumption+) 
 apply (simp add:Ring.ring_tOp_commute[of K "t" "(t)^{<hyphen> K}"] linvf,
        simp add:Ring.ring_l_one, blast) 
 apply simp
done

lemma (in Corps) Kxa_pow_vpr:"[ valuation K v; t ∈ carrier K; v t = 1;
    (0::int) ≤ j] ==> Kxa K v (t) = (vp K v)^{Vr K v) (ant j)}"
apply (frule val_surj_n_val[of v], blast) 
apply (simp add:Kxa_val_ge) 
apply (rule equalityI)
 apply (rule subsetI, simp, erule conjE)
 apply (rule_tac x = x in n_value_x_2[of "v" _ "(ant j)"],
            assumption+)
 apply (cut_tac ale_zle[THEN sym, of "0" "j"]) 
 apply (frule_tac a = "0 ≤ j" and b = "ant 0 ≤ ant j" in a_b_exchange,
         assumption+) 
 apply (thin_tac "(0 ≤ j) = (ant 0 ≤ ant j)", simp add:ant_0)
 apply (frule_tac k = "v x" in ale_trans[of "0" "ant j"], assumption+)
 apply (simp add:val_pos_mem_Vr) apply simp
 apply (simp only:ale_zle[THEN sym, of "0" "j"], simp add:ant_0)
apply (rule subsetI)
 apply simp 
 apply (frule_tac x = x in mem_vp_apow_mem_Vr[of "v" "ant j"]) 
 apply (simp only:ale_zle[THEN sym, of "0" "j"], simp add:ant_0) 
 apply assumption
apply (simp add:Vr_mem_f_mem[of "v"])
apply (frule_tac x = x in n_value_x_1[of "v" "ant j" _ ])
 apply (simp only:ale_zle[THEN sym, of "0" "j"], simp add:ant_0) 
 apply assumption apply simp
done  

lemma (in Corps) field_distribTr:"[a ∈ carrier K; b ∈ carrier K;
      x ∈ carrier K; x ≠ 0] ==> a ± (-_a (b ⋅_r x)) = (a ⋅_r (x^<hyphen>K) ± (-_a b)) ⋅_r x"
apply (cut_tac field_is_ring,
       cut_tac invf_closed1[of x], simp, erule conjE) apply (
       simp add:Ring.ring_inv1_1[of "K" "b" "x"],
       frule Ring.ring_is_ag[of "K"],
       frule aGroup.ag_mOp_closed[of "K" "b"], assumption+)
apply (frule Ring.ring_tOp_closed[of "K" "a" "x^<hyphen>K"], assumption+,
       simp add:Ring.ring_distrib2, simp add:Ring.ring_tOp_assoc, 
       simp add:linvf,
       simp add:Ring.ring_r_one)
apply simp
done

lemma a0_le_1[simp]:"(0::ant) ≤ 1"
by (cut_tac a0_less_1, simp add:aless_imp_le)

lemma (in Corps) vp_mem_times_t:"[valuation K v; t ∈ carrier K; t ≠ 0;
       v t = 1; x ∈ vp K v] ==> ∃a∈carrier (Vr K v). x = a ⋅_r t" 
apply (frule Vr_ring[of v],
       frule vp_ideal[of v]) 
 apply (frule val_surj_n_val[of v], blast)
 apply (frule vp_mem_val_poss[of v x],
        frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"], 
        assumption+, simp add:Vr_mem_f_mem, simp) 
 apply (frule gt_a0_ge_1[of "v x"])
 apply (subgoal_tac "v t ≤ v x") 
 apply (thin_tac "1 ≤ v x")
 apply (frule val_Rxa_gt_a_1[of "v" "t" "x"])
 apply (subst val_pos_mem_Vr[THEN sym, of "v" "t"], assumption+)
 apply simp 
 apply (simp add:vp_mem_Vr_mem) apply assumption+
 apply (simp add:Rxa_def, erule bexE) 
 apply (cut_tac a0_less_1) 
 apply (subgoal_tac "0 ≤ v t")
 apply (frule val_pos_mem_Vr[of "v" "t"], assumption+)
 apply (simp, simp add:Vr_tOp_f_tOp, blast, simp+)
done

lemma (in Corps) psum_diff_mem_Kxa:"[valuation K v; t ∈ carrier K;
      v t = 1; x ∈ carrier K; x ≠ 0] ==>
     (psum_{K x v t} n) ∈ carrier K ∧
     ( x ± (-_a (psum_{K x v t} n))) ∈ Kxa K v (t^{(tna (v x)) + (1 + int n))})"
apply (frule val1_neq_0[of v t], assumption+)
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
       frule Vr_ring[of v], frule vp_ideal[of v])
apply (induct_tac n)
apply (subgoal_tac "x ⋅_r (t ^{tna (v x)}) ∈ carrier (Vr K v)",
       rule conjI, simp, rule Ring.ring_tOp_closed[of "K"], assumption+,
       frule Ring.csrp_fn_mem[of "Vr K v" "vp K v" 
       "pj (Vr K v) (vp K v) (x ⋅_r (t ^{tna (v x)}))"], 
       assumption+,
       simp add:pj_mem Ring.qring_carrier, blast,
       simp add:Vr_mem_f_mem, simp add:npowf_mem)
apply (simp, 
     frule npowf_mem[of "t" "tna (v x)"], assumption+,
     frule field_potent_nonzero1[of "t" "tna (v x)"], assumption+,
     subst field_distribTr[of "x" _ "t^{tna (v x))}"], assumption+,
     frule Ring.csrp_fn_mem[of "Vr K v" "vp K v" 
     "pj (Vr K v) (vp K v) (x ⋅_r (t ^{tna (v x)}))"], 
     assumption+,
     simp add:pj_mem Ring.qring_carrier, blast, simp add:Vr_mem_f_mem,
     simp add:npowf_mem, assumption) 
apply (frule Ring.csrp_diff_in_vpr[of "Vr K v" "vp K v" 
       "x ⋅_r ((t^{tna (v x))})^<hyphen>K)"], assumption+) 
      apply (simp add:npowf_minus)
 
 apply (simp add:npowf_minus)
 apply (frule pj_Hom[of "Vr K v" "vp K v"], assumption+)
 apply (frule rHom_mem[of "pj (Vr K v) (vp K v)" "Vr K v" "Vr K v /_r vp K v" 
        "x ⋅_r (t ^{tna (v x)})"], assumption+)
 apply (frule Ring.csrp_fn_mem[of "Vr K v" "vp K v" 
         "pj (Vr K v) (vp K v) (x ⋅_r (t ^{tna (v x)}))"], assumption+)
 apply (frule Ring.ring_is_ag[of "Vr K v"], 
        frule_tac x = "csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v)
       (x ⋅_r (t ^{tna (v x)})))" in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp add:Vr_pOp_f_pOp) apply (simp add:Vr_mOp_f_mOp)
apply (frule_tac x = "x ⋅_r (t ^{tna (v x)}) ± -_a (csrp_fn (Vr K v) (vp K v)
      (pj (Vr K v) (vp K v) (x ⋅_r (t ^{tna (v x)}))))" in 
      vp_mem_times_t[of "v" "t"], assumption+, erule bexE, simp)
apply (frule_tac x = a in Vr_mem_f_mem[of  "v"], assumption+)
apply (simp add:Ring.ring_tOp_assoc[of "K"])
 apply (simp add:npowf_exp_1_add[THEN sym, of "t" "tna (v x)"]) 
 apply (simp add:add.commute)
apply (simp add:Kxa_def) 
apply (frule npowf_mem[of "t" "1 + tna (v x)"], assumption+)
apply (simp add:Ring.ring_tOp_commute[of "K" _ "t^{1 + tna (v x))}"])
 apply blast
 apply (frule npowf_mem[of  "t" "- tna (v x)"], assumption+)
 apply (frule Ring.ring_tOp_closed[of "K" "x" "t ^{tna (v x)}"], assumption+)
apply (subst val_pos_mem_Vr[THEN sym, of v], assumption+)
 apply (simp add:val_t2p) apply (simp add:val_exp[THEN sym, of "v" "t"])
 apply (simp add:aminus[THEN sym])
 apply (frule value_in_aug_inf[of "v" "x"], assumption+, 
        simp add:aug_inf_def)
 apply (frule val_nonzero_noninf[of "v" "x"], assumption+,
         simp add:ant_tna)
 apply (simp add:aadd_minus_r)

apply (erule conjE)
 apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
 apply (rule conjI)
 apply simp
 apply (rule aGroup.ag_pOp_closed, assumption+)
 apply (rule Ring.ring_tOp_closed[of "K"], assumption+)
 apply (simp add:Kxa_def, erule bexE, simp)
 apply (subst Ring.ring_tOp_commute, assumption+)
 apply (rule npowf_mem, assumption+) apply (simp add:Vr_mem_f_mem)
 apply (frule_tac n = "tna (v x) + (1 + int n)" in npowf_mem[of t ], 
        assumption,
        frule_tac n = "- tna (v x) + (-1 - int n)" in npowf_mem[of t ], 
        assumption,
        frule_tac x = k in Vr_mem_f_mem[of v], assumption+)
 apply (simp add:Ring.ring_tOp_assoc npowf_exp_add[THEN sym, of t])
  apply (simp add:field_npowf_exp_zero) 
  apply (simp add:Ring.ring_r_one)

 apply (frule pj_Hom[of "Vr K v" "vp K v"], assumption+)
 apply (frule_tac a = k in rHom_mem[of "pj (Vr K v) (vp K v)" "Vr K v" 
        "Vr K v /_r vp K v"], assumption+)
 apply (frule_tac x = "pj (Vr K v) (vp K v) k" in Ring.csrp_fn_mem[of "Vr K v" 
        "vp K v"], assumption+)
 apply (simp add:Vr_mem_f_mem)
 apply (rule npowf_mem, assumption+)

apply (simp add:Kxa_def) apply (erule bexE, simp)
apply (frule_tac x = k in Vr_mem_f_mem[of "v"], assumption+)
apply (frule_tac n = "tna (v x) + (1 + int n)" in npowf_mem[of "t"], 
       assumption+)
apply (frule_tac n = "- tna (v x) + (- 1 - int n)" in npowf_mem[of "t"], 
       assumption+)
apply (frule_tac x = "t^{tna (v x) + (1 + int n))}" and y = k in 
      Ring.ring_tOp_commute[of "K"], assumption+) apply simp
apply (simp add:Ring.ring_tOp_assoc, simp add:npowf_exp_add[THEN sym])
   apply (simp add:field_npowf_exp_zero) 
   apply (simp add:Ring.ring_r_one)
   apply (thin_tac "(t^{tna (v x) + (1 + int n))}) ⋅_r k =
            k ⋅_r (t^{tna (v x) + (1 + int n))})")
apply (frule pj_Hom[of "Vr K v" "vp K v"], assumption+)
apply (frule_tac a = k in rHom_mem[of "pj (Vr K v) (vp K v)" "Vr K v" 
        "Vr K v /_r vp K v"], assumption+)
 apply (frule_tac x = "pj (Vr K v) (vp K v) k" in Ring.csrp_fn_mem[of "Vr K v" 
        "vp K v"], assumption+)
 apply (frule_tac x = "csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v) k)" in 
        Vr_mem_f_mem[of v], assumption+)
 apply (frule_tac x = "csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v) k)" and
 y = "t^{tna (v x) + (1 + int n))}" in Ring.ring_tOp_closed[of "K"], assumption+)
 apply (simp add:aGroup.ag_p_inv[of "K"])
 apply (frule_tac x = "psum_{K x v t} n" in aGroup.ag_mOp_closed[of "K"], 
          assumption+)
 apply (frule_tac x = "(csrp_fn (Vr K v) (vp K v)(pj (Vr K v) (vp K v) k)) ⋅_r
       (t^{tna (v x) + (1 + int n))})" in aGroup.ag_mOp_closed[of "K"], assumption+)
 apply (subst aGroup.ag_pOp_assoc[THEN sym], assumption+)
 apply simp
 apply (simp add:Ring.ring_inv1_1)
 apply (subst Ring.ring_distrib2[THEN sym, of "K"], assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+)
 apply (frule_tac x = k in  Ring.csrp_diff_in_vpr[of "Vr K v" "vp K v"]
     , assumption+)
 apply (frule Ring.ring_is_ag[of "Vr K v"])
 apply (frule_tac x = "csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v) k)" in 
        aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp add:Vr_pOp_f_pOp) apply (simp add:Vr_mOp_f_mOp)
 apply (frule_tac x = "k ± -_a (csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v)
 k))" in vp_mem_times_t[of "v" "t"], assumption+, erule bexE, simp)
 apply (frule_tac x = a in Vr_mem_f_mem[of v], assumption+)
 apply (simp add:Ring.ring_tOp_assoc[of "K"])
 apply (simp add:npowf_exp_1_add[THEN sym, of "t"])
 apply (simp add:add.commute[of "2"])
 apply (simp add:add.assoc)
 apply (subst Ring.ring_tOp_commute, assumption+)
 apply (rule npowf_mem, assumption+) apply blast
done

lemma Suc_diff_int:"0 < n ==> int (n - Suc 0) = int n - 1" 
  by (cut_tac of_nat_Suc[of "n - Suc 0"], simp)

lemma (in Corps) ecf_mem:"[valuation K v; t ∈ carrier K; v t = 1;
      x ∈ carrier K; x ≠ 0 ] ==> ecf _{v t x} n ∈ carrier K"
apply (frule val1_neq_0[of v t], assumption+)
apply (cut_tac field_is_ring,
       frule Vr_ring[of v], frule vp_ideal[of v])
apply (case_tac "n = 0")
 apply (simp add:expand_coeff_def)
 apply (rule Vr_mem_f_mem[of v], assumption+)
 apply (rule Ring.csrp_fn_mem, assumption+)
 apply (subgoal_tac "x ⋅_r (t ^{tna (v x)}) ∈ carrier (Vr K v)")
 apply (simp add:pj_mem Ring.qring_carrier, blast)
apply (frule npowf_mem[of  "t" "- tna (v x)"], assumption+,
       subst val_pos_mem_Vr[THEN sym, of v "x ⋅_r (t^{- tna(v x))})"], 
       assumption+,
       rule Ring.ring_tOp_closed, assumption+) 
apply (simp add:val_t2p,
       simp add:val_exp[THEN sym, of  "v" "t" "- tna (v x)"]) 
apply (frule value_in_aug_inf[of  "v" "x"], assumption+,
       simp add:aug_inf_def)
      apply (frule val_nonzero_noninf[of  "v" "x"], assumption+)
  apply (simp add:aminus[THEN sym], simp add:ant_tna)
  apply (simp add:aadd_minus_r)

apply (simp add:expand_coeff_def) 
apply (frule psum_diff_mem_Kxa[of  "v" "t" "x" "n - 1"],
                   assumption+, erule conjE)
apply (simp add:Kxa_def, erule bexE,
       frule_tac x = k in Vr_mem_f_mem[of v], assumption+,
     frule npowf_mem[of  "t" "tna (v x) + (1 + int (n - Suc 0))"], 
     assumption+,
   frule npowf_mem[of  "t" "-tna (v x) - int n"], assumption+)
  apply simp
apply(simp add:Ring.ring_tOp_commute[of "K" "t^{tna (v x) + (int n))}"])
 apply (simp add:Ring.ring_tOp_assoc, simp add:npowf_exp_add[THEN sym])

 apply (simp add:npowf_def, simp add:Ring.ring_r_one)
apply (rule Vr_mem_f_mem, assumption+)
 apply (rule Ring.csrp_fn_mem, assumption+)
 apply (simp add:pj_mem Ring.qring_carrier, blast)
done

lemma (in Corps) etm_mem:"[valuation K v; t ∈ carrier K; v t = 1;
     x ∈ carrier K; x ≠ 0] ==> etm _{v t x} n ∈ carrier K"
apply (frule val1_neq_0[of v t], assumption+)
apply (simp add:expand_term_def)
apply (cut_tac field_is_ring,
       rule Ring.ring_tOp_closed[of "K"], assumption)
apply (simp add:ecf_mem)
apply (simp add:npowf_mem)
done

lemma (in Corps) psum_sum_etm:"[valuation K v; t ∈ carrier K; v t = 1;
      x ∈ carrier K; x ≠ 0] ==>
 psum _{x v t} n = nsum K (λj. (ecf _{v t x} j)⋅_r (t^{tna (v x) + (int j))})) n" 
apply (frule val1_neq_0[of v t], assumption+)
apply (induct_tac "n")
 apply (simp add:expand_coeff_def)
apply (rotate_tac -1, drule sym)
apply simp
 apply (thin_tac "Σ_e K (λj. ecf _{v t x} j ⋅_r t^{tna (v x) + int j)}) n =
         psum_{K x v t} n")
 apply (simp add:expand_coeff_def)
done

lemma zabs_pos:"0 ≤ (abs (z::int))"
by (simp add:zabs_def)

lemma abs_p_self_pos:"0 ≤ z + (abs (z::int))"
by (simp add:zabs_def)

lemma zadd_right_mono:"(i::int) ≤ j ==> i + k ≤ j + k"
by simp

theorem (in Corps) expansion_thm:"[valuation K v; t ∈ carrier K;
       v t = 1; x ∈ carrier K; x ≠ 0] ==> lim _v (partial_sum K x v t) x" 
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
apply (simp add:limit_def)
 apply (rule allI)
 apply (subgoal_tac "∀n. (N + na (Abs (v x))) < n ⟶
                     psum _{x v t} n ± -_a x ∈ vp K v^{Vr K v) (an N)}")
 apply blast
apply (rule allI, rule impI)
apply (frule_tac n = n in psum_diff_mem_Kxa[of "v" "t" "x"],
          assumption+, erule conjE) 
apply (frule_tac j = "tna (v x) + (1 + int n)" in  Kxa_val_ge[of "v" "t"], 
       assumption+)
 apply simp
 apply (thin_tac "Kxa K v (t^{tna (v x) + (1 + int n))}) =
           {xa ∈ carrier K. ant (tna (v x) + (1 + int n)) ≤ v xa}")
 apply (erule conjE)
 apply (simp add:a_zpz[THEN sym])

apply (subgoal_tac "(an N) ≤ v (psum_{K x v t} n ± -_a x)")
 
 apply (frule value_in_aug_inf[of v x], assumption+,
       simp add:aug_inf_def)
      apply (frule val_nonzero_noninf[of v x], assumption+)
  apply (simp add:ant_tna)
 apply (frule val_surj_n_val[of v], blast)
apply (rule_tac x = "psum _{x v t} n ± -_a x" and n = "an N" in 
                     n_value_x_2[of  "v"], assumption+) 
apply (subst val_pos_mem_Vr[THEN sym, of v], assumption+)
apply (rule aGroup.ag_pOp_closed[of "K"], assumption+)
apply (simp add:aGroup.ag_mOp_closed)

apply (cut_tac n = N in an_nat_pos)
apply (rule_tac i = 0 and j = "an N" and k = "v (psum_{K x v t} n ± -_a x)" in 
       ale_trans, assumption+)
apply simp
apply simp

apply (frule_tac x1 = "x ± (-_a psum _{x v t} n)" in val_minus_eq[THEN sym,
      of v], assumption+, simp,
      thin_tac "v ( x ± -_a psum_{K x v t} n) = v (-_a ( x ± -_a psum_{K x v t} n))")
 apply (frule_tac x = "psum_{K x v t} n" in aGroup.ag_mOp_closed[of "K"],
        assumption+, simp add:aGroup.ag_p_inv, simp add:aGroup.ag_inv_inv)
 apply (frule aGroup.ag_mOp_closed[of "K" "x"], assumption+)
 apply (simp add:aGroup.ag_pOp_commute[of "K" "-_a x"])

apply (cut_tac Abs_pos[of "v x"])
apply (frule val_nonzero_z[of v x], assumption+, erule exE, simp) 
apply (simp add:na_def) 
apply (cut_tac aneg_less[THEN sym, of "0" "Abs (v x)"], simp)
apply (frule val_nonzero_noninf[of v x], assumption+)
apply (simp add:tna_ant)
apply (simp only:ant_1[THEN sym], simp del:ant_1 add:a_zpz)
apply (simp add:add.assoc[THEN sym])
apply (cut_tac m1 = "N + nat ∣z∣" and n1 = n in of_nat_less_iff [where ?'a = int, THEN sym], simp)
apply (frule_tac a = "int N + abs z" and b = "int n" and c = "z + 1" in 
       add_strict_right_mono, simp only:add.commute)
apply (simp only:add.assoc[of _ "1"])
apply (simp only:add.assoc[THEN sym, of "1"])
apply (simp only:add.commute[of "1"])
apply (simp only:add.assoc[of _ "1"])
apply (cut_tac ?a1 = z and ?b1 = "abs z" and ?c1 = "1 + int N" in 
       add.assoc[THEN sym])
apply (thin_tac "∣z∣ + int N < int n")
apply (frule_tac a = "z + (∣z∣ + (1 + int N))" and b = "z + ∣z∣ + (1 + int N)" and c = "int n + (z + 1)" in ineq_conv1, assumption+)
apply (thin_tac "z + (∣z∣ + (1 + int N)) < int n + (z + 1)",
       thin_tac "z + (∣z∣ + (1 + int N)) = z + ∣z∣ + (1 + int N)",
       thin_tac "N + nat ∣z∣ < n")
apply (cut_tac z = z in abs_p_self_pos)
apply (frule_tac i = 0 and j = "z + abs z" and k = "1 + int N" in 
        zadd_right_mono) 
apply (simp only:add_0)
apply (frule_tac i = "1 + int N" and j = "z + ∣z∣ + (1 + int N)" and 
       k = "int n + (z + 1)" in zle_zless_trans, assumption+) 
apply (thin_tac "z + ∣z∣ + (1 + int N) < int n + (z + 1)",
       thin_tac "0 ≤ z + ∣z∣",
       thin_tac "1 + int N ≤ z + ∣z∣ + (1 + int N)")
apply (cut_tac m1 = "1 + int N" and n1 = "int n + (z + 1)" in 
       aless_zless[THEN sym], simp)

apply (frule_tac x = "ant (1 + int N)" and y = "ant (int n + (z + 1))"
       and z = "v ( psum_{K x v t} n ± -_a x)" in aless_le_trans, assumption+)
apply (thin_tac "ant (1 + int N) < ant (int n + (z + 1))")

apply (simp add:a_zpz[THEN sym])
apply (frule_tac x = "1 + ant (int N)" and y = "v ( psum_{K x v t} n ± -_a x)" in 
       aless_imp_le, thin_tac "1 + ant (int N) < v ( psum_{K x v t} n ± -_a x)")
apply (cut_tac a0_less_1, frule aless_imp_le[of "0" "1"])
apply (frule_tac b = "ant (int N)" in aadd_pos_le[of "1"])
apply (subst an_def)
apply (rule_tac i = "ant (int N)" and j = "1 + ant (int N)" and 
       k = "v ( psum_{K x v t} n ± -_a x)" in ale_trans, assumption+)
done

subsection "Hensel's theorem"

definition
(*** Cauchy sequence of polynomials in (Vr K v)[X] ***)
  pol_Cauchy_seq :: "[('b, 'm) Ring_scheme, 'b, _, 'b ==> ant,
         nat ==> 'b] ==> bool" (‹(5PCauchy_{_ _ _ _} _)› [90,90,90,90,91]90) where
  "PCauchy _{X K v} F ⟷ (∀n. (F n) ∈ carrier R) ∧
    (∃d. (∀n. deg R (Vr K v) X (F n) ≤ (an d))) ∧
    (∀N. ∃M. (∀n m. M < n ∧ M < m ⟶
        P_mod R (Vr K v) X ((vp K v)^{Vr K v) (an N)}) (F n ± -_a (F m))))"

definition
  pol_limit :: "[('b, 'm) Ring_scheme, 'b, _, 'b ==> ant,
             nat ==> 'b, 'b] ==> bool" 
             (‹(6Plimit_{_ _ _ _} _ _)› [90,90,90,90,90,91]90) where
  "Plimit _{X K v} F p ⟷ (∀n. (F n) ∈ carrier R) ∧
    (∀N. ∃M. (∀m. M < m ⟶
        P_mod R (Vr K v) X ((vp K v)^{Vr K v) (an N)}) ((F m) ± -_a p)))"

definition
  Pseql :: "[('b, 'm) Ring_scheme, 'b, _, 'b ==> ant, nat,
             nat ==> 'b] ==> nat ==> 'b" 
            (‹(6Pseql _{_ _ _ _} _)› [90,90,90,90,90,91]90) where
  "Pseql _{X K v d} F = (λn. (ldeg_p R (Vr K v) X d (F n)))"

   (** deg R (Vr K v) X (F n) \<le> an (Suc d) **)

definition
  Pseqh :: "[('b, 'm) Ring_scheme, 'b, _, 'b ==> ant, nat, nat ==> 'b] ==>
        nat ==> 'b" 
           (‹(6Pseqh_{_ _ _ _ _} _)› [90,90,90,90,90,91]90) where
  "Pseqh _{X K v d} F = (λn. (hdeg_p R (Vr K v) X (Suc d) (F n)))"

    (** deg R (Vr K v) X (F n) \<le> an (Suc d) **)

lemma an_neq_minf[simp]:"∀n. -∞ ≠ an n"
apply (rule allI)
apply (simp add:an_def) apply (rule not_sym) apply simp
done

lemma an_neq_minf1[simp]:"∀n. an n ≠ -∞" 
apply (rule allI) apply (simp add:an_def)
done

lemma (in Corps) Pseql_mem:"[valuation K v; PolynRg R (Vr K v) X;
      F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d)] ==>
                     (Pseql _{X K v d} F) n ∈ carrier R"
apply (frule PolynRg.is_Ring)
apply (simp add:Pseql_def)
apply (frule Vr_ring[of "v"], 
       rule PolynRg.ldeg_p_mem, assumption+, simp)
done

lemma (in Corps) Pseqh_mem:"[valuation K v; PolynRg R (Vr K v) X;
      F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d)] ==>
                     (Pseqh _{X K v d} F) n ∈ carrier R"
apply (frule PolynRg.is_Ring)
apply (frule Vr_ring[of "v"]) 
apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"])
apply (simp del:npow_suc add:Pseqh_def)
apply (rule PolynRg.hdeg_p_mem, assumption+, simp)
done

lemma (in Corps) PCauchy_lTr:"[valuation K v; PolynRg R (Vr K v) X;
    p ∈ carrier R; deg R (Vr K v) X p ≤ an (Suc d);
    P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) p] ==>
    P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) (ldeg_p R (Vr K v) X d p)"
apply (frule PolynRg.is_Ring)
apply (simp add:ldeg_p_def)
apply (frule Vr_ring[of v])
apply (frule PolynRg.scf_d_pol[of "R" "Vr K v" "X" "p" "Suc d"], assumption+,
       (erule conjE)+)
apply (frule_tac n = "an N" in vp_apow_ideal[of v], simp)
apply (frule PolynRg.P_mod_mod[THEN sym, of R "Vr K v" X "vp K v^{(Vr K v) (an N)}"
       p "scf_d R (Vr K v) X p (Suc d)"], assumption+, simp)
apply (subst PolynRg.polyn_expr_short[of R "Vr K v" X 
             "scf_d R (Vr K v) X p (Suc d)" d], assumption+, simp)
apply (subst PolynRg.P_mod_mod[THEN sym, of R "Vr K v" X "vp K v^{(Vr K v) (an N)}"
       "polyn_expr R X d (d, snd (scf_d R (Vr K v) X p (Suc d)))" 
       "(d, snd (scf_d R (Vr K v) X p (Suc d)))"], assumption+)
apply (subst PolynRg.polyn_expr_short[THEN sym], simp+,
       simp add:PolynRg.polyn_mem)
apply (subst pol_coeff_def, rule allI, rule impI, 
       simp add:PolynRg.pol_coeff_mem)
apply simp+
done

lemma (in Corps) PCauchy_hTr:"[valuation K v; PolynRg R (Vr K v) X;
      p ∈ carrier R; deg R (Vr K v) X p ≤ an (Suc d);
      P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) p]
  ==> P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) (hdeg_p R (Vr K v) X (Suc d) p)"
apply (frule PolynRg.is_Ring)
apply (cut_tac Vr_ring[of v])
apply (frule PolynRg.scf_d_pol[of R "Vr K v" X p "Suc d"], assumption+)
apply (frule_tac n = "an N" in vp_apow_ideal[of v], simp)
apply (frule PolynRg.P_mod_mod[THEN sym, of "R" "Vr K v" "X" 
       "vp K v^{(Vr K v) (an N)}" p "scf_d R (Vr K v) X p (Suc d)"], assumption+,
       simp+)
apply (subst hdeg_p_def) 
apply (subst PolynRg.monomial_P_mod_mod[THEN sym, of "R" "Vr K v" "X" 
       "vp K v^{(Vr K v) (an N)}" "snd (scf_d R (Vr K v) X p (Suc d)) (Suc d)" 
       "(snd (scf_d R (Vr K v) X p (Suc d)) (Suc d)) ⋅_r (X^ ^{(Suc d)})" 
       "Suc d"], 
       assumption+)
apply (rule PolynRg.pol_coeff_mem[of R "Vr K v" X 
       "scf_d R (Vr K v) X p (Suc d)" "Suc d"], assumption+, simp+)
done
 
lemma (in Corps) v_ldeg_p_pOp:"[valuation K v; PolynRg R (Vr K v) X;
      p ∈ carrier R; q ∈ carrier R; deg R (Vr K v) X p ≤ an (Suc d);
      deg R (Vr K v) X q ≤ an (Suc d)] ==>
      (ldeg_p R (Vr K v) X d p) ± (ldeg_p R (Vr K v) X d q) =
                              ldeg_p R (Vr K v) X d (p ± q)"
by (simp add:PolynRg.ldeg_p_pOp[of R "Vr K v" X p q d]) 

lemma (in Corps) v_hdeg_p_pOp:"[valuation K v; PolynRg R (Vr K v) X;
      p ∈ carrier R; q ∈ carrier R; deg R (Vr K v) X p ≤ an (Suc d);
      deg R (Vr K v) X q ≤ an (Suc d)] ==> (hdeg_p R (Vr K v) X (Suc d) p) ±
      (hdeg_p R (Vr K v) X (Suc d) q) = hdeg_p R (Vr K v) X (Suc d) (p ± q)"
by (simp add:PolynRg.hdeg_p_pOp[of R "Vr K v" X p q d])

lemma (in Corps) v_ldeg_p_mOp:"[valuation K v; PolynRg R (Vr K v) X;
       p ∈ carrier R;deg R (Vr K v) X p ≤ an (Suc d)] ==>
       -_a (ldeg_p R (Vr K v) X d p) = ldeg_p R (Vr K v) X d (-_a p)"
by (simp add:PolynRg.ldeg_p_mOp)


lemma (in Corps) v_hdeg_p_mOp:"[valuation K v; PolynRg R (Vr K v) X;
    p ∈ carrier R;deg R (Vr K v) X p ≤ an (Suc d)] ==>
   -_a (hdeg_p R (Vr K v) X (Suc d) p) = hdeg_p R (Vr K v) X (Suc d) (-_a p)"
by (simp add:PolynRg.hdeg_p_mOp)

lemma (in Corps) PCauchy_lPCauchy:"[valuation K v; PolynRg R (Vr K v) X;
      ∀n. F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d);
      P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) (F n ± -_a (F m))]
      ==> P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)})
                     (((Pseql _{X K v d} F) n) ± -_a ((Pseql _{X K v d} F) m))"
apply (frule PolynRg.is_Ring)
apply (cut_tac Vr_ring[of v],
       frule Ring.ring_is_ag[of "R"],
       frule PolynRg.subring[of "R" "Vr K v" "X"])
apply (simp add:Pseql_def)
apply (subst v_ldeg_p_mOp[of "v" "R" "X" _ "d"], assumption+,
       simp, simp)
apply (subst v_ldeg_p_pOp[of v R X "F n" "-_a (F m)"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption, simp, simp,
       simp add:PolynRg.deg_minus_eq1)
apply (rule PCauchy_lTr[of v R X "F n ± -_a (F m)" "d" "N"],
       assumption+,
       rule aGroup.ag_pOp_closed[of "R" "F n" "-_a (F m)"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption+, simp)
apply (frule PolynRg.deg_minus_eq1 [of "R" "Vr K v" "X" "F m"], simp)
apply (rule PolynRg.polyn_deg_add4[of "R" "Vr K v" "X" "F n" 
       "-_a (F m)" "Suc d"], assumption+, simp,
       rule aGroup.ag_mOp_closed, assumption, simp+)
done

lemma (in Corps) PCauchy_hPCauchy:"[valuation K v; PolynRg R (Vr K v) X;
      ∀n. F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d);
      P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) (F n ± -_a (F m))]
      ==> P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)})
                     (((Pseqh _{X K v d} F) n) ± -_a ((Pseqh _{X K v d} F) m))"
apply (frule PolynRg.is_Ring)
apply (frule Vr_ring[of v], frule Ring.ring_is_ag[of "R"],
       frule PolynRg.subring[of "R" "Vr K v" "X"], 
       frule vp_apow_ideal[of v "an N"], simp) 

apply (simp add:Pseqh_def,
       subst v_hdeg_p_mOp[of v R X "F m" "d"], assumption+,
       simp+)

apply (subst v_hdeg_p_pOp[of v R X "F n" "-_a (F m)"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption, simp, simp,
       frule PolynRg.deg_minus_eq1 [of "R" "Vr K v" "X" "F m"],
       simp+ )
apply (frule PCauchy_hTr[of "v" "R" "X" "F n ± -_a (F m)" "d" "N"],
    assumption+,
    rule aGroup.ag_pOp_closed[of "R" "F n" "-_a (F m)"], assumption+,
    simp, rule aGroup.ag_mOp_closed, assumption+, simp)
apply (frule PolynRg.deg_minus_eq1 [of "R" "Vr K v" "X" "F m"], simp+)
apply (rule PolynRg.polyn_deg_add4[of "R" "Vr K v" "X" "F n" "-_a (F m)" 
       "Suc d"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption, simp+)
done

(** Don't forget  t_vp_apow  ***)
 
lemma (in Corps) Pseq_decompos:"[valuation K v; PolynRg R (Vr K v) X;
      F n ∈ carrier R; deg R (Vr K v) X (F n) ≤ an (Suc d)]
      ==> F n = ((Pseql _{X K v d} F) n) ± ((Pseqh _{X K v d} F) n)"
apply (frule PolynRg.is_Ring)
apply (simp del:npow_suc add:Pseql_def Pseqh_def)
apply (frule Vr_ring[of v])
apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
apply (rule PolynRg.decompos_p[of "R" "Vr K v" "X" "F n" "d"], assumption+)
done

lemma (in Corps) deg_0_const:"[valuation K v; PolynRg R (Vr K v) X;
      p ∈ carrier R; deg R (Vr K v) X p ≤ 0] ==> p ∈ carrier (Vr K v)"
apply (frule Vr_ring[of v])
apply (frule PolynRg.subring)
apply (frule PolynRg.is_Ring)
apply (case_tac "p = 0", simp,
       simp add:Ring.Subring_zero_ring_zero[THEN sym],
       simp add:Ring.ring_zero)
apply (subst PolynRg.pol_of_deg0[THEN sym, of "R" "Vr K v" "X" "p"], 
        assumption+)
apply (simp add:PolynRg.deg_an, simp only:an_0[THEN sym])
apply (simp only:ale_nat_le[of "deg_n R (Vr K v) X p" "0"])
done

lemma (in Corps) monomial_P_limt:"[valuation K v; Complete K;
     PolynRg R (Vr K v) X; ∀n. f n ∈ carrier (Vr K v);
     ∀n. F n = (f n) ⋅_r (X^ ^d); ∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
     P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)}) (F n ± -_a (F m))] ==>
        ∃b∈carrier (Vr K v). Plimit_{R X K v} F (b ⋅_r (X^ ^d))"
apply (frule PolynRg.is_Ring)
apply (frule Vr_ring[of v])
apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
apply simp

apply (subgoal_tac "Cauchy_{K v} f")
 apply (simp add:v_complete_def)
 apply (drule_tac a = f in forall_spec)
 apply (thin_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
        P_mod R (Vr K v) X (vp K v^{Vr K v) (an N)})
        ((f n) ⋅_r (X^ ^d) ± -_a (f m) ⋅_r (X^ ^d))", assumption)
 apply (erule exE, erule conjE)
 apply (subgoal_tac "b ∈ carrier (Vr K v)")
 apply (subgoal_tac "Plimit_{R X K v} F (b ⋅_r (X^ ^d))", blast)
 
(******* b \<in> carrier (Vr K v) ***********)
apply (simp add:pol_limit_def)
apply (rule conjI)
 apply (rule allI)
 apply (rule Ring.ring_tOp_closed[of "R"], assumption)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
 apply (rule Ring.mem_subring_mem_ring[of "R" "Vr K v"], assumption+) 
 apply simp
 
 apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"])
 apply (simp add:Ring.npClose)
apply (thin_tac "∀n. F n = f n ⋅_r X^ ^d") 
apply (simp add:limit_def)
 apply (rule allI)
(* apply (simp add:t_vp_apow[of "K" "v" "t"]) *) 
 apply (rotate_tac -2, drule_tac x = N in spec) 
 apply (erule exE)
 apply (subgoal_tac "∀n> M. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                   ((f n)⋅_r (X^ ^d) ± -_a (b ⋅_r (X^ ^d)))", blast) 
 apply (rule allI, rule impI)
 apply (rotate_tac -2, drule_tac x = n in spec, simp)
 apply (drule_tac x = n in spec)

apply (frule_tac x = "f n" in Ring.mem_subring_mem_ring[of "R" "Vr K v"], 
       assumption+,
       frule_tac x = b in Ring.mem_subring_mem_ring[of "R" "Vr K v"], 
       assumption+)
apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"])
 apply (frule Ring.npClose[of "R" "X" "d"], assumption+)
 apply (simp add:Ring.ring_inv1_1)
 apply (frule Ring.ring_is_ag[of "R"],
        frule_tac x = b in aGroup.ag_mOp_closed[of "R"], assumption+)
 apply (subst Ring.ring_distrib2[THEN sym, of "R" "X^ ^d"], assumption+)

 apply (frule_tac n = "an N" in vp_apow_ideal[of v], simp) 
 apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and c = "f n ± -_a b" and 
        p = "(f n ± -_a b) ⋅_r (X^ ^d)" in  
        PolynRg.monomial_P_mod_mod[of "R" "Vr K v" "X" _ _ _ "d"], assumption+)
 apply (simp add:Ring.Subring_minus_ring_minus[THEN sym])
 apply (frule Ring.ring_is_ag[of "Vr K v"])
 apply (frule_tac x = b in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp only:Ring.Subring_pOp_ring_pOp[THEN sym])
 apply (rule aGroup.ag_pOp_closed, assumption+) apply simp
 apply (frule_tac I1 = "vp K v^{(Vr K v) (an N)}" and c1 = "f n ± -_a b" and 
     p1 = "(f n ± -_a b) ⋅_r (X^ ^d)" in PolynRg.monomial_P_mod_mod[THEN sym, 
     of "R" "Vr K v" "X" _ _ _ "d"], assumption+)
 apply (frule Ring.ring_is_ag[of "Vr K v"])
 apply (frule_tac x = b in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp only:Ring.Subring_minus_ring_minus[THEN sym])
 apply (simp only:Ring.Subring_pOp_ring_pOp[THEN sym])
 apply (rule aGroup.ag_pOp_closed, assumption+) apply simp apply simp
 apply (simp only:Vr_mOp_f_mOp[THEN sym])
 apply (frule Ring.ring_is_ag[of "Vr K v"])
 apply (frule_tac x = b in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp add:Vr_pOp_f_pOp[THEN sym])
 apply (simp add:Ring.Subring_pOp_ring_pOp)
 apply (simp add:Ring.Subring_minus_ring_minus)

 apply (case_tac "b = 0", simp add:Vr_0_f_0[THEN sym], 
        simp add:Ring.ring_zero)
 apply (frule_tac b = b in limit_val[of  _ "f" "v"], assumption+,
        rule allI,
        frule_tac x = j in spec, simp add:Vr_mem_f_mem,
        assumption+, erule exE)
 apply (thin_tac "∀n. F n = f n ⋅_r X^ ^d",
        thin_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
                         P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                          (f n ⋅_r X^ ^d ± -_a f m ⋅_r X^ ^d)")  
 apply (rotate_tac -1, drule_tac x = "Suc N" in spec, simp)
 apply (drule_tac x = "Suc N" in spec)
 apply (frule_tac x1 = "f (Suc N)" in val_pos_mem_Vr[THEN sym, of v],
        simp add:Vr_mem_f_mem, simp, simp add:val_pos_mem_Vr[of v]) 

apply (simp add:Cauchy_seq_def)
 apply (simp add:Vr_mem_f_mem)
 apply (rule allI)
 apply (rotate_tac -3, frule_tac x = N in spec)

 apply (thin_tac "∀n. F n = f n ⋅_r X^ ^d")
 (*apply (simp add:t_vp_apow[of "K" "v" "t"]) *) 
 apply (frule_tac n = "an N" in vp_apow_ideal[of "v"], simp)
 apply (drule_tac x = N in spec, erule exE) 
 apply (subgoal_tac "∀n m. M < n ∧ M < m ⟶
                      f n ± -_a (f m) ∈ vp K v^{(Vr K v) (an N)}", blast)
 apply (rule allI)+
 apply (rule impI, erule conjE)
 apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and c = "f n ± -_a (f m)" and 
   p = "(f n ± -_a (f m)) ⋅_r (X^ ^d)" in
   PolynRg.monomial_P_mod_mod[of "R" "Vr K v" "X" _ _ _ "d"], assumption+)
 
 apply (frule_tac x = n in spec,
        drule_tac x = m in spec)
 apply (frule Ring.ring_is_ag[of "Vr K v"],
        simp add:Vr_mOp_f_mOp[THEN sym],
        frule_tac x = "f m" in aGroup.ag_mOp_closed[of "Vr K v"], assumption+,
        simp add:Vr_pOp_f_pOp[THEN sym])
 apply (rule aGroup.ag_pOp_closed, assumption+, simp)
 apply simp 
 apply (thin_tac "(f n ± -_a f m ∈ vp K v^{(Vr K v) (an N)}) =
        P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) ((f n ± -_a f m) ⋅_r X^ ^d)")
 apply (rotate_tac -3, drule_tac x = n in spec,
        rotate_tac -1, drule_tac x = m in spec, simp)
 apply (frule_tac x = n in spec,
        drule_tac x = m in spec)
 apply (frule_tac x = "f n" in Ring.mem_subring_mem_ring[of R "Vr K v"],
         assumption+,
        frule_tac x = "f m" in Ring.mem_subring_mem_ring[of R "Vr K v"],
         assumption+,
        frule Ring.ring_is_ag[of R],
        frule_tac x = "f m" in aGroup.ag_mOp_closed[of R], assumption+,
        frule PolynRg.X_mem_R[of R "Vr K v" X],
        frule Ring.npClose[of R X d], assumption)
 apply (simp add:Ring.ring_inv1_1[of R],
        frule_tac y1 = "f n" and z1 = "-_a f m" in Ring.ring_distrib2[
        THEN sym, of R "X^ ^d"], assumption+, simp,
        thin_tac "f n ⋅_r X^ ^d ± (-_a f m) ⋅_r X^ ^d =
                                           (f n ± -_a f m) ⋅_r X^ ^d")
 apply (simp only:Ring.Subring_minus_ring_minus[THEN sym,of R "Vr K v"])
 apply (frule Ring.subring_Ring[of R "Vr K v"], assumption,
        frule Ring.ring_is_ag[of "Vr K v"],
        frule_tac x = "f m" in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
 apply (simp add:Ring.Subring_pOp_ring_pOp[THEN sym, of R "Vr K v"],
        simp add:Vr_pOp_f_pOp, simp add:Vr_mOp_f_mOp)
done

lemma (in Corps) mPlimit_uniqueTr:"[valuation K v;
     PolynRg R (Vr K v) X; ∀n. f n ∈ carrier (Vr K v);
     ∀n. F n = (f n) ⋅_r (X^ ^d); c ∈ carrier (Vr K v);
     Plimit_{R X K v} F (c ⋅_r (X^ ^d))] ==> lim_{K v} f c"
apply (frule PolynRg.is_Ring,
       simp add:pol_limit_def limit_def,
       rule allI,
       erule conjE,
       rotate_tac -1, drule_tac x = N in spec,
       erule exE)
apply (subgoal_tac "∀n. M < n ⟶ f n ± -_a c ∈ vp K v^{(Vr K v) (an N)}", blast)
apply (rule allI, rule impI,
       rotate_tac -2, drule_tac x = n in spec, simp,
       drule_tac x = n in spec,
       drule_tac x = n in spec,
       thin_tac "∀n. f n ⋅_r X^ ^d ∈ carrier R")
apply (frule Vr_ring[of v],
       frule PolynRg.X_mem_R[of "R" "Vr K v" "X"],
       frule Ring.npClose[of "R" "X" "d"], assumption+,
       frule PolynRg.subring[of "R" "Vr K v" "X"])
apply (frule_tac x = c in Ring.mem_subring_mem_ring[of "R" "Vr K v"],
       assumption+,
      frule_tac x = "f n" in Ring.mem_subring_mem_ring[of "R" "Vr K v"],
       assumption+)
apply (simp add:Ring.ring_inv1_1,
       frule Ring.ring_is_ag[of "R"],
       frule aGroup.ag_mOp_closed[of "R" "c"], assumption+)
apply (simp add:Ring.ring_distrib2[THEN sym, of "R" "X^ ^d" _ "-_a c"],
       simp add:Ring.Subring_minus_ring_minus[THEN sym],
       frule Ring.ring_is_ag[of "Vr K v"],
       frule aGroup.ag_mOp_closed[of "Vr K v" "c"], assumption+)
apply (simp add:Ring.Subring_pOp_ring_pOp[THEN sym],
       frule_tac x = "f n" in aGroup.ag_pOp_closed[of "Vr K v" _
        "-_{aVr K v)} c"], assumption+,
       frule_tac n = "an N" in vp_apow_ideal[of "v"], simp,
       frule_tac I1 = "vp K v^{(Vr K v) (an N)}" and
        c1 = "f n ±_{Vr K v)} -_{aVr K v)} c" and p1 = "(f n ±_{Vr K v)} -_{aVr K v)} c)
       ⋅_r (X^ ^d)" in PolynRg.monomial_P_mod_mod[THEN sym, of R "Vr K v"
        X _ _ _ d], assumption+, simp, simp)
 apply (simp add:Vr_pOp_f_pOp, simp add:Vr_mOp_f_mOp)
done

lemma (in Corps) mono_P_limt_unique:"[valuation K v;
  PolynRg R (Vr K v) X; ∀n. f n ∈ carrier (Vr K v);
  ∀n. F n = (f n) ⋅_r (X^ ^d); b ∈ carrier (Vr K v); c ∈ carrier (Vr K v);
  Plimit_{R X K v} F (b ⋅_r (X^ ^d)); Plimit_{R X K v} F (c ⋅_r (X^ ^d))] ==>
        b ⋅_r (X^ ^d) = c ⋅_r (X^ ^d)"
apply (frule PolynRg.is_Ring)
apply (frule_tac mPlimit_uniqueTr[of v R X f F d b], assumption+,
       frule_tac mPlimit_uniqueTr[of v R X f F d c], assumption+)
apply (frule Vr_ring[of v],
       frule PolynRg.subring[of "R" "Vr K v" "X"],
       frule Vr_mem_f_mem[of v b], assumption+,
       frule Vr_mem_f_mem[of v c], assumption+,
       frule limit_unique[of b f v c])
apply (rule allI, simp add:Vr_mem_f_mem, assumption+, simp)
done

lemma (in Corps) Plimit_deg:"[valuation K v; PolynRg R (Vr K v) X;
  ∀n. F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ (an d);
  p ∈ carrier R; Plimit_{R X K v} F p] ==> deg R (Vr K v) X p ≤ (an d)"
apply (frule PolynRg.is_Ring, frule Vr_ring[of v])
apply (case_tac "p = 0", simp add:deg_def)
apply (rule contrapos_pp, simp+,
       simp add:aneg_le,
       frule PolynRg.s_cf_expr[of R "Vr K v" X p], assumption+, (erule conjE)+,
       frule PolynRg.s_cf_deg[of R "Vr K v" X p], assumption+,
       frule PolynRg.pol_coeff_mem[of R "Vr K v" X "s_cf R (Vr K v) X p"
        "fst (s_cf R (Vr K v) X p)"], assumption+, simp,
       frule Vr_mem_f_mem[of v "snd (s_cf R (Vr K v) X p)
            (fst (s_cf R (Vr K v) X p))"], assumption+)
(* show the value of the leading coefficient is noninf *)
apply (frule val_nonzero_noninf[of "v"
       "snd (s_cf R (Vr K v) X p) (fst (s_cf R (Vr K v) X p))"], assumption,
       simp add:Vr_0_f_0,
       frule val_pos_mem_Vr[THEN sym, of v "snd (s_cf R (Vr K v) X p)
         (fst (s_cf R (Vr K v) X p))"], assumption+, simp,
       frule value_in_aug_inf[of "v" "snd (s_cf R (Vr K v) X p)
          (fst (s_cf R (Vr K v) X p))"], assumption+,
       cut_tac mem_ant[of "v (snd (s_cf R (Vr K v) X p)
       (fst (s_cf R (Vr K v) X p)))"], simp add:aug_inf_def,
       erule exE, simp, simp only:ant_0[THEN sym], simp only:ale_zle,
       frule_tac z = z in zpos_nat, erule exE, simp,
       thin_tac "z = int n")

(* show that the leading coefficient of p shoule be 0. contradiction *)
apply (simp add:pol_limit_def)
apply (rotate_tac 5, drule_tac x = "Suc n" in spec)
apply (erule exE)
apply (rotate_tac -1,
       drule_tac x = "Suc M" in spec, simp del:npow_suc,
       drule_tac x = "Suc M" in spec,
       drule_tac x = "Suc M" in spec)

  (**** Only formula manipulation to obtain
        P_mod R (Vr K v) X (vp K v^{Vr K v an (Suc n)})
         (polyn_expr R X (fst (s_cf R (Vr K v) X p))
           (add_cf (Vr K v) f
             (fst (s_cf R (Vr K v) X p),
              λj. -_{a K v} snd (s_cf R (Vr K v) X p) j))) *****)
apply (frule PolynRg.polyn_minus[of R "Vr K v" X "s_cf R (Vr K v) X p"
       "fst (s_cf R (Vr K v) X p)"], assumption+, simp,
   frule PolynRg.minus_pol_coeff[of R "Vr K v" X "s_cf R (Vr K v) X p"],
        assumption+, drule sym,
   frule_tac x = "deg R (Vr K v) X (F (Suc M))" in ale_less_trans[of _
       "an d" "deg R (Vr K v) X p"], assumption+,
   frule_tac p = "F (Suc M)" and d = "deg_n R (Vr K v) X p" in
            PolynRg.pol_expr_edeg[of "R" "Vr K v" "X"], assumption+,
   frule_tac x = "deg R (Vr K v) X (F (Suc M))" and
         y = "deg R (Vr K v) X p" in aless_imp_le,
      subst PolynRg.deg_an[THEN sym, of "R" "Vr K v" "X" "p"], assumption+,
      erule exE, (erule conjE)+,
   frule_tac c = f in PolynRg.polyn_add1[of "R" "Vr K v" "X" _
    "(fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j)"],
    assumption+, simp,
   thin_tac "-_a p = polyn_expr R X (fst (s_cf R (Vr K v) X p))
    (fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j)",
   thin_tac "polyn_expr R X (fst (s_cf R (Vr K v) X p))
             (s_cf R (Vr K v) X p) = p",
   thin_tac "F (Suc M) = polyn_expr R X (fst (s_cf R (Vr K v) X p)) f",
   thin_tac "polyn_expr R X (fst (s_cf R (Vr K v) X p)) f ±
       polyn_expr R X (fst (s_cf R (Vr K v) X p))
        (fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j) =
       polyn_expr R X (fst (s_cf R (Vr K v) X p)) (add_cf (Vr K v) f
       (fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j))")

(** apply P_mod_mod to obtain a condition of coefficients **) 
 apply (frule_tac n = "an (Suc n)" in vp_apow_ideal[of "v"], simp,
   frule_tac p1 = "polyn_expr R X (fst (s_cf R (Vr K v) X p))(add_cf (Vr K v) f
   (fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j))" and
  I1= "vp K v^{(Vr K v) (an (Suc n))}" and c1 = "add_cf (Vr K v) f (fst
   (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j)" in
  PolynRg.P_mod_mod[THEN sym, of R "Vr K v" X], assumption+,
  rule PolynRg.polyn_mem[of R "Vr K v" X], assumption+,
  rule PolynRg.add_cf_pol_coeff[of R "Vr K v" X], assumption+,
  simp add:PolynRg.add_cf_len,
  rule PolynRg.add_cf_pol_coeff[of R "Vr K v" X], assumption+,
  simp add:PolynRg.add_cf_len,
  simp add:PolynRg.add_cf_len)
 apply (drule_tac x = "fst (s_cf R (Vr K v) X p)" in spec, simp,
        thin_tac "P_mod R (Vr K v) X (vp K v^{(Vr K v) (an (Suc n))})
         (polyn_expr R X (fst (s_cf R (Vr K v) X p)) (add_cf (Vr K v) f
      (fst (s_cf R (Vr K v) X p), λj. -_{a K v} snd (s_cf R (Vr K v) X p) j)))",
      simp add:add_cf_def)
 (**** we obtained snd (add_cf (Vr K v) f (fst (s_cf R (Vr K v) X p),
     λj. -_{a K v} snd (s_cf R (Vr K v) X p) j)) (fst (s_cf R (Vr K v) X p))
           ∈ vp K v^{Vr K v an (Suc n)} ***)
 apply (frule_tac p = "polyn_expr R X (fst (s_cf R (Vr K v) X p)) f" and
       c = f and j = "fst f" in PolynRg.pol_len_gt_deg[of R "Vr K v" X],
       assumption+, simp, drule sym, simp add:PolynRg.deg_an) apply simp
 apply (rotate_tac -4, drule sym, simp)
 apply (frule Ring.ring_is_ag[of "Vr K v"],
        frule_tac x = "snd (s_cf R (Vr K v) X p) (fst f)" in
        aGroup.ag_mOp_closed[of "Vr K v"], assumption+,
        simp add:aGroup.ag_l_zero)
 apply (frule_tac I = "vp K v^{(Vr K v) (an (Suc n))}" and
        x = "-_{a K v} snd (s_cf R (Vr K v) X p) (fst f)" in
        Ring.ideal_inv1_closed[of "Vr K v"], assumption+)
 apply (simp add:aGroup.ag_inv_inv)
 apply (frule_tac n = "an (Suc n)" and x = "snd (s_cf R (Vr K v) X p) (fst f)"
         in n_value_x_1[of v], simp+)
 apply (frule_tac x = "snd (s_cf R (Vr K v) X p) (fst f)" in
         n_val_le_val[of v], assumption+, simp add:ant_int)
 apply (drule_tac i = "an (Suc n)" and
        j = "n_val K v (snd (s_cf R (Vr K v) X p) (fst f))" and
        k = "v (snd (s_cf R (Vr K v) X p) (fst f))" in ale_trans,
        assumption+)
 apply (simp add:ant_int ale_natle)
done

lemma (in Corps) Plimit_deg1:"[valuation K v; Ring R; PolynRg R (Vr K v) X;
  ∀n. F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ ad;
  p ∈ carrier R; Plimit_{R X K v} F p] ==> deg R (Vr K v) X p ≤ ad"
apply (frule Vr_ring[of v])
apply (case_tac "∀n. F n = 0")
apply (frule Plimit_deg[of v R X F 0 p], assumption+,
       rule allI, simp add:deg_def, assumption+)
apply (case_tac "p = 0", simp add:deg_def,
       frule PolynRg.nonzero_deg_pos[of R "Vr K v" X p], assumption+,
       simp,
       frule PolynRg.pols_const[of "R" "Vr K v" "X" "p"], assumption+,
       simp,
       frule PolynRg.pols_const[of "R" "Vr K v" "X" "p"], assumption+,
       simp add:ale_refl)
 apply (subgoal_tac "p = 0", simp)
 apply (thin_tac "p ≠ 0")
 apply (rule contrapos_pp, simp+)

apply (frule n_val_valuation[of v])
apply (frule val_nonzero_z[of "n_val K v" "p"])
 apply (simp add:Vr_mem_f_mem)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
 apply (simp only:Ring.Subring_zero_ring_zero[THEN sym, of "R" "Vr K v"])
 apply (simp add:Vr_0_f_0, erule exE)
 apply (frule val_pos_mem_Vr[THEN sym, of "v" "p"])
 apply (simp add:Vr_mem_f_mem, simp)
 apply (frule val_pos_n_val_pos[of "v" "p"])
 apply (simp add:Vr_mem_f_mem, simp)
 apply (simp add:ant_0[THEN sym])
apply (frule_tac z = z in zpos_nat, erule exE)
apply (unfold pol_limit_def, erule conjE)
 apply (rotate_tac -1, drule_tac x = "Suc n" in spec)
 apply (subgoal_tac "¬ (∃M. ∀m. M < m ⟶
             P_mod R (Vr K v) X (vp K v^{(Vr K v) (an (Suc n))}) ( F m ± -_a p))")
 apply blast
 apply (thin_tac "∃M. ∀m. M < m ⟶
           P_mod R (Vr K v) X (vp K v^{(Vr K v) (an (Suc n))}) (F m ± -_a p)")
 apply simp
 apply (subgoal_tac "M < (Suc M) ∧
       ¬ P_mod R (Vr K v) X (vp K v^{(Vr K v) (an (Suc n))}) (0 ± -_a p)")
 apply blast
 apply simp
 apply (frule Ring.ring_is_ag[of "R"])
 apply (frule aGroup.ag_mOp_closed[of "R" "p"], assumption)
 apply (simp add:aGroup.ag_l_zero)
 apply (frule Ring.ring_is_ag[of "Vr K v"])
 apply (frule aGroup.ag_mOp_closed[of "Vr K v" "p"], assumption)
 apply (frule_tac n = "an (Suc n)" in vp_apow_ideal[of v], simp)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
 apply (simp add:Ring.Subring_minus_ring_minus[THEN sym, of "R" "Vr K v"])
 apply (simp add:PolynRg.P_mod_coeffTr[of "R" "Vr K v" "X" _ "-_{aVr K v)} p"])
 apply (rule contrapos_pp, simp+)

 apply (frule_tac I = "vp K v^{(Vr K v) (an (Suc n))}" in
        Ring.ideal_inv1_closed[of "Vr K v" _ "-_{aVr K v)} p"], assumption+)
 apply (simp add:aGroup.ag_inv_inv)
 apply (frule_tac n = "an (Suc n)" in n_value_x_1[of "v" _ "p"], simp)
 apply assumption
 apply simp
 apply (simp add:ant_int, simp add:ale_natle)

apply (fold pol_limit_def)
apply (case_tac "ad = ∞", simp)
apply simp apply (erule exE)
apply (subgoal_tac "0 ≤ ad")
apply (frule Plimit_deg[of "v" "R" "X" "F" "na ad" "p"], assumption+)
 apply (simp add:an_na)+
apply (drule_tac x = n in spec,
       drule_tac x = n in spec)

apply (frule_tac p = "F n" in PolynRg.nonzero_deg_pos[of "R" "Vr K v" "X"],
           assumption+)
apply (rule_tac j = "deg R (Vr K v) X (F n)" in ale_trans[of "0" _ "ad"],
           assumption+)
done

lemma (in Corps) Plimit_ldeg:"[valuation K v; PolynRg R (Vr K v) X;
     ∀n. F n ∈ carrier R; p ∈ carrier R;
     ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d);
      Plimit_{R X K v} F p] ==> Plimit_{R X K v} (Pseql_{R X K v d} F)
                                             (ldeg_p R (Vr K v) X d p)"
apply (frule Vr_ring[of v], frule PolynRg.is_Ring,
       frule Ring.ring_is_ag[of "R"])
apply (frule Plimit_deg[of v R X F "Suc d" p], assumption+)
apply (simp add:Pseql_def, simp add:pol_limit_def)
apply (rule conjI, rule allI)
 apply (rule PolynRg.ldeg_p_mem, assumption+, simp+)
 apply (rule allI)
 apply (rotate_tac -5, drule_tac x = N in spec, erule exE)
 apply (subgoal_tac "∀m > M. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
              (ldeg_p R (Vr K v) X d (F m) ± -_a (ldeg_p R (Vr K v) X d p))",
        blast)
 apply (rule allI, rule impI)
 apply (rotate_tac -2,
        frule_tac x = m in spec,
        thin_tac "∀m. M < m ⟶
            P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) ( F m ± -_a p)",
        simp)
 apply (subst v_ldeg_p_mOp[of v R X _ d], assumption+)
 apply (subst v_ldeg_p_pOp[of v R X _ "-_a p"], assumption+)
 apply (simp, rule aGroup.ag_mOp_closed, assumption, simp, simp)
 apply (frule PolynRg.deg_minus_eq1 [THEN sym, of "R" "Vr K v" "X" "p"],
        assumption+)
 apply simp
apply (rule_tac p = "F m ± -_a p" and N = N in PCauchy_lTr[of "v"
       "R" "X" _ "d" ], assumption+)
 apply (rule_tac x = "F m" in aGroup.ag_pOp_closed[of "R" _ "-_a p"],
       assumption+)
 apply (simp, rule aGroup.ag_mOp_closed, assumption+)
apply (frule PolynRg.deg_minus_eq1 [of "R" "Vr K v" "X" "p"], assumption+)
apply (rule PolynRg.polyn_deg_add4[of "R" "Vr K v" "X" _ "-_a p" "Suc d"],
         assumption+)
  apply (simp, rule aGroup.ag_mOp_closed, assumption, simp+)
done

lemma (in Corps) Plimit_hdeg:"[valuation K v; PolynRg R (Vr K v) X;
     ∀n. F n ∈ carrier R; ∀n. deg R (Vr K v) X (F n) ≤ an (Suc d);
      p ∈ carrier R; Plimit_{R X K v} F p] ==>
      Plimit_{R X K v} (Pseqh_{R X K v d} F) (hdeg_p R (Vr K v) X (Suc d) p)"
apply (frule Vr_ring[of "v"], frule PolynRg.is_Ring,
       frule Ring.ring_is_ag[of "R"])
apply (frule Plimit_deg[of v R X F "Suc d" p], assumption+)
apply (simp add:Pseqh_def, simp add:pol_limit_def)
apply (rule conjI, rule allI)
 apply (rule PolynRg.hdeg_p_mem, assumption+, simp+)
 apply (rule allI)
 apply (rotate_tac -5, drule_tac x = N in spec)
 apply (erule exE)
 apply (subgoal_tac "∀m>M. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
  (hdeg_p R (Vr K v) X (Suc d) (F m) ± -_a (hdeg_p R (Vr K v) X (Suc d) p))",
    blast)
 apply (rule allI, rule impI)
 apply (rotate_tac -2,
        drule_tac x = m in spec, simp)
 apply (subst v_hdeg_p_mOp[of v R X _ d], assumption+)
 apply (subst v_hdeg_p_pOp[of v R X _ "-_a p"], assumption+)
 apply (simp, rule aGroup.ag_mOp_closed, assumption, simp, simp)
 apply (frule PolynRg.deg_minus_eq1 [THEN sym, of R "Vr K v" X p], assumption+)
 apply simp
apply (rule_tac p = "F m ± -_a p" and N = N in PCauchy_hTr[of v R X _ d ],
       assumption+)
 apply (rule_tac x = "F m" in aGroup.ag_pOp_closed[of R _ "-_a p"],
        assumption+)
 apply (simp, rule aGroup.ag_mOp_closed, assumption+)
apply (frule PolynRg.deg_minus_eq1 [of "R" "Vr K v" "X" "p"], assumption+)
apply (rule PolynRg.polyn_deg_add4[of "R" "Vr K v" "X" _ "-_a p" "Suc d"],
         assumption+)
  apply (simp, rule aGroup.ag_mOp_closed, assumption, simp+)
done

lemma (in Corps) P_limit_uniqueTr:"[valuation K v; PolynRg R (Vr K v) X] ==>
 ∀F. ((∀n. F n ∈ carrier R) ∧ (∀n. deg R (Vr K v) X (F n) ≤ (an d)) ⟶
      (∀p1 p2. p1 ∈ carrier R ∧ p2 ∈ carrier R ∧ Plimit_{R X K v} F p1 ∧
           Plimit_{R X K v} F p2 ⟶ p1 = p2))"
apply (frule PolynRg.is_Ring)
apply (induct_tac d)
apply (rule allI, rule impI, (rule allI)+, rule impI)
apply (erule conjE)+
apply (subgoal_tac "∀n. F n ∈ carrier (Vr K v)")
apply (frule Vr_ring[of "v"])
apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"])
apply (frule_tac f = F and F = F and d = 0 and b = p1 and c = p2 in
        mono_P_limt_unique[of v R X], assumption+)
apply (rule allI, drule_tac x = n in spec,
       simp add:Ring.ring_r_one)
apply (frule_tac F = F and p = p1 in Plimit_deg[of v R X _ 0],
       assumption+, simp add:deg_0_const,
       frule_tac F = F and p = p2 in Plimit_deg[of v R X _ 0],
       assumption+, simp add:deg_0_const)
apply (simp add:Ring.ring_r_one)+
apply (simp add:deg_0_const)

(******** case Suc d **********)
apply (rename_tac d)
apply (rule allI, rule impI)
apply (erule conjE)
apply ((rule allI)+, rule impI, (erule conjE)+)
apply (frule_tac F = F and p = p1 and d = d in Plimit_ldeg[of v R X],
                 assumption+,
       frule_tac F = F and p = p2 and d = d in Plimit_ldeg[of v R X],
                 assumption+,
       frule_tac F = F and p = p1 and d = d in Plimit_hdeg[of v R X],
                 assumption+,
       frule_tac F = F and p = p2 and d = d in Plimit_hdeg[of v R X],
                 assumption+)
apply (frule_tac a = "Pseql_{R X K v d} F" in forall_spec)
 apply (rule conjI)
 apply (rule allI)
 apply (rule Pseql_mem, assumption+, simp)
 apply (rule allI, simp)
 apply (rule allI)
apply (subst Pseql_def)
 apply (rule_tac p = "F n" and d = d in PolynRg.deg_ldeg_p[of R "Vr K v" X],
                      assumption+) apply (simp add:Vr_ring)
 apply simp
 apply (thin_tac "∀F. (∀n. F n ∈ carrier R) ∧
       (∀n. deg R (Vr K v) X (F n) ≤ an d) ⟶
         (∀p1 p2.
                p1 ∈ carrier R ∧
                p2 ∈ carrier R ∧
                Plimit_{R X K v} F p1 ∧ Plimit_{R X K v} F p2 ⟶
                p1 = p2)")
apply (frule Vr_ring[of v])
apply (frule_tac F = F and d = "Suc d" and p = p1 in
                           Plimit_deg[of v R X], assumption+,
       frule_tac F = F and d = "Suc d" and p = p2 in
                           Plimit_deg[of v R X], assumption+)
apply (subgoal_tac "(ldeg_p R (Vr K v) X d p1) = (ldeg_p R (Vr K v) X d p2)")
apply (subgoal_tac "hdeg_p R (Vr K v) X (Suc d) p1 =
                                        hdeg_p R (Vr K v) X (Suc d) p2")

apply (frule_tac p = p1 and d = d in PolynRg.decompos_p[of "R" "Vr K v" "X"],
                        assumption+,
       frule_tac p = p2 and d = d in PolynRg.decompos_p[of "R" "Vr K v" "X"],
                        assumption+)
 apply simp
 apply (thin_tac "Plimit_{R X K v} Pseql_{R X K v d} F (ldeg_p R (Vr K v) X d p1)",
        thin_tac "Plimit_{R X K v} Pseql_{R X K v d} F (ldeg_p R (Vr K v) X d p2)",
        thin_tac "∀p1 p2. p1 ∈ carrier R ∧ p2 ∈ carrier R ∧
           Plimit_{R X K v} Pseql_{R X K v d} F p1 ∧
           Plimit_{R X K v} Pseql_{R X K v d} F p2 ⟶ p1 = p2")
 apply (simp only:hdeg_p_def)
 apply (rule_tac f = "λj. snd (scf_d R (Vr K v) X (F j) (Suc d)) (Suc d)"
    and F = "Pseqh_{R X K v d} F"
    and b = "(snd (scf_d R (Vr K v) X p1 (Suc d))) (Suc d)"
    and d = "Suc d" and c = "snd (scf_d R (Vr K v) X p2 (Suc d)) (Suc d)" in
        mono_P_limt_unique[of v R X], assumption+)
 apply (rule allI)
apply (frule_tac p = "F n" and d = "Suc d" in
                 PolynRg.scf_d_pol[of "R" "Vr K v" "X"])
 apply (simp del:npow_suc)+
 apply (frule_tac c = "scf_d R (Vr K v) X (F n) (Suc d)" and
        j = "Suc d" in PolynRg.pol_coeff_mem[of R "Vr K v" X])
 apply simp apply (simp del:npow_suc)+
 apply (rule allI)
apply (frule_tac c = "scf_d R (Vr K v) X (F n) (Suc d)" and
        j = "Suc d" in PolynRg.pol_coeff_mem[of R "Vr K v" X])
 apply (frule_tac p = "F n" and d = "Suc d" in PolynRg.scf_d_pol[of R
           "Vr K v" X], (simp del:npow_suc)+)
 apply (cut_tac p = "F n" and d = "Suc d" in PolynRg.scf_d_pol[of R
           "Vr K v" X], (simp del:npow_suc)+)
          
 apply (subst Pseqh_def) apply (simp only:hdeg_p_def)
apply (frule_tac p = p1 and d = "Suc d" in PolynRg.scf_d_pol[of R "Vr K v" X],
       assumption+)
 apply (rule_tac c = "scf_d R (Vr K v) X p1 (Suc d)" and
        j = "Suc d" in PolynRg.pol_coeff_mem[of R "Vr K v" X], assumption+,
   simp, simp)
apply (frule_tac p = p2 and d = "Suc d" in PolynRg.scf_d_pol[of R "Vr K v" X],
       assumption+)
 apply (rule_tac c = "scf_d R (Vr K v) X p2 (Suc d)" and
        j = "Suc d" in PolynRg.pol_coeff_mem[of R "Vr K v" X], assumption+,
   simp, simp) apply simp apply simp
 apply (rotate_tac -4,
        drule_tac x = "ldeg_p R (Vr K v) X d p1" in spec,
        rotate_tac -1,
        drule_tac x = "ldeg_p R (Vr K v) X d p2" in spec)
 apply (simp add:PolynRg.ldeg_p_mem)
done

lemma (in Corps) P_limit_unique:"[valuation K v; Complete K;
  PolynRg R (Vr K v) X; ∀n. F n ∈ carrier R;
  ∀n. deg R (Vr K v) X (F n) ≤ (an d); p1 ∈ carrier R; p2 ∈ carrier R;
  Plimit_{R X K v} F p1; Plimit_{R X K v} F p2] ==> p1 = p2"
apply (frule P_limit_uniqueTr[of "v" "R" "X" "d"], assumption+)
apply blast
done

lemma (in Corps) P_limitTr:"[valuation K v; Complete K; PolynRg R (Vr K v) X]
 ==> ∀F. ((∀n. F n ∈ carrier R) ∧ (∀n. deg R (Vr K v) X (F n) ≤ (an d)) ∧
        (∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
        P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) (F n ± -_a (F m))) ⟶
        (∃p∈carrier R. Plimit_{R X K v} F p))"
apply (frule PolynRg.is_Ring)
apply (frule Vr_ring[of v])
 apply (induct_tac d)

 apply simp
 apply (rule allI, rule impI, (erule conjE)+)
 apply (frule_tac F = F and f = F in monomial_P_limt[of v R X _ _ 0],
        assumption+)
 apply (rule allI)
 apply (rotate_tac 5, drule_tac x = n in spec)
 apply (simp add:deg_0_const)
 apply (rule allI)
  apply (drule_tac x = n in spec,
         thin_tac "∀n. deg R (Vr K v) X (F n) ≤ 0")
  apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"],
         frule PolynRg.is_Ring,
         simp add:Ring.ring_r_one, assumption)

 apply (erule bexE)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
 apply (cut_tac x = b in Ring.mem_subring_mem_ring[of "R" "Vr K v"])
  apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"], assumption+)
  apply (simp add:Ring.ring_r_one, blast)

(*********** case n ***************)
apply (rule allI, rule impI) apply (rename_tac d F)
apply (erule conjE)+
 apply (subgoal_tac "(∀n.(Pseql_{R X K v d} F) n ∈ carrier R) ∧
    (∀n. deg R (Vr K v) X ((Pseql_{R X K v d} F) n) ≤ an d) ∧
    (∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
     P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
        ((Pseql_{R X K v d} F) n ± -_a ((Pseql_{R X K v d} F) m)))")
 apply (frule_tac a = "Pseql_{R X K v d} F" in forall_spec, assumption)
 apply (erule bexE)
 apply (thin_tac "∀F. (∀n. F n ∈ carrier R) ∧
     (∀n. deg R (Vr K v) X (F n) ≤ an d) ∧
     (∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
        P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) (F n ± -_a (F m))) ⟶
        (∃p∈carrier R. Plimit_{R X K v} F p)",
     thin_tac "(∀n. (Pseql_{R X K v d} F) n ∈ carrier R) ∧
     (∀n. deg R (Vr K v) X ((Pseql_{R X K v d} F) n) ≤ an d) ∧
    (∀N. ∃M. ∀n m. M < n ∧ M < m ⟶ P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
    ((Pseql_{R X K v d} F) n ± -_a ((Pseql_{R X K v d} F) m)))")
 apply (subgoal_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
                      P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                       ((Pseqh _{X K v d} F) n ± -_a ((Pseqh _{X K v d} F) m))")
 apply (frule_tac f = "λj. snd (scf_d R (Vr K v) X (F j) (Suc d)) (Suc d)"
  and F = "Pseqh_{R X K v d} F" and d = "Suc d" in monomial_P_limt[of v R X],
       assumption+)
 apply (rule allI)
 apply (drule_tac x = n in spec,
        drule_tac x = n in spec,
        frule_tac p = "F n" and d = "Suc d" in PolynRg.scf_d_pol[of R "Vr K v"
       "X"], assumption+, (erule conjE)+)
 apply (rule_tac c = "scf_d R (Vr K v) X (F n) (Suc d)" and j = "Suc d" in
        PolynRg.pol_coeff_mem[of R "Vr K v" X], assumption+)
 apply simp
 apply (rule allI)
  apply (thin_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
         P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                       ( (Pseqh_{R X K v d} F) n ± -_a ((Pseqh_{R X K v d} F) m))")
  apply (simp only:Pseqh_def hdeg_p_def, assumption, erule bexE)

apply (thin_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
       P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) (F n ± -_a (F m))",
       thin_tac "∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
               P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
               ((Pseqh_{R X K v d} F) n ± -_a ((Pseqh_{R X K v d} F) m))")

apply (subgoal_tac "Plimit_{R X K v} F (p ± b ⋅_r (X^ ^{(Suc d)}))",
       subgoal_tac "p ± b ⋅_r(X^ ^{(Suc d)}) ∈ carrier R", blast)

apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"],
       frule_tac n = "Suc d" in Ring.npClose[of "R" "X"], assumption+,
       frule Ring.ring_is_ag[of "R"],
       rule aGroup.ag_pOp_closed[of "R"], assumption+,
       rule Ring.ring_tOp_closed, assumption+)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"],
        simp add:Ring.mem_subring_mem_ring, assumption)

apply (simp del:npow_suc add:pol_limit_def,
       rule allI,
       subgoal_tac "∀n. F n = (Pseql_{R X K v d} F) n ± ((Pseqh_{R X K v d} F) n)",
       simp del:npow_suc,
       subgoal_tac "∀m. (Pseql_{R X K v d} F) m ± ((Pseqh_{R X K v d} F) m) ±
        -_a (p ± (b ⋅_r (X^ ^{(Suc d)}))) = (Pseql_{R X K v d} F) m ± -_a p ±
         ((Pseqh_{R X K v d} F) m ± -_a (b ⋅_r (X^ ^{(Suc d)})))",
       simp del:npow_suc)
 
apply (thin_tac "∀m. (Pseql _{X K v d} F) m ± (Pseqh_{R X K v d} F) m ±
    -_a (p ± b ⋅_r X^ ^{(Suc d)}) = (Pseql _{X K v d} F) m ± -_a p ±
     ((Pseqh_{R X K v d} F) m ± -_a b ⋅_r X^ ^{(Suc d)})")
apply (erule conjE)+
apply (rotate_tac -3,
       drule_tac x = N in spec, erule exE)
apply (rotate_tac 1,
       drule_tac x = N in spec, erule exE)
apply (rename_tac d F p b N M1 M2)
apply (subgoal_tac "∀m. (max M1 M2) < m ⟶
           P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
           ((Pseql_{R X K v d} F) m ± -_a p ±
            ((Pseqh_{R X K v d} F) m ± -_a (b ⋅_r (X^ ^{(Suc d)}))))", blast)
apply (rule allI, rule impI)
apply (rotate_tac -2,
       drule_tac x = m in spec, simp del:npow_suc)
apply (erule conjE)
apply (rotate_tac -5,
       drule_tac x = m in spec, simp del:npow_suc)
 apply (frule_tac n = "an N" in vp_apow_ideal[of v], simp del:npow_suc)
 apply (frule Ring.ring_is_ag[of "R"])
 apply (rule_tac I = "vp K v^{(Vr K v) (an N)}" and
        p = "(Pseql_{R X K v d} F) m ± -_a p" and
        q = "(Pseqh_{R X K v d} F) m ± -_a (b ⋅_r (X^ ^{(Suc d)}))" in
        PolynRg.P_mod_add[of R "Vr K v" X], assumption+)

apply (rule aGroup.ag_pOp_closed, assumption+)
 apply (simp add:Pseql_mem, rule aGroup.ag_mOp_closed, assumption+)

 apply (rule aGroup.ag_pOp_closed[of "R"], assumption)
 apply (simp add:Pseqh_mem, rule aGroup.ag_mOp_closed, assumption)
 apply (rule Ring.ring_tOp_closed, assumption)
 apply (frule PolynRg.subring[of "R" "Vr K v" "X"])
 apply (simp add:Ring.mem_subring_mem_ring)
 apply (frule PolynRg.X_mem_R[of "R" "Vr K v" "X"])
 apply (rule Ring.npClose, assumption+)

apply (rule allI)
 apply (thin_tac "∀n.(Pseql_{R X K v d} F) n ±((Pseqh_{R X K v d} F) n) ∈ carrier R")
 apply (erule conjE)+
 apply (thin_tac "∀n. deg R (Vr K v) X
             ((Pseql _{X K v d} F) n ± (Pseqh_{R X K v d} F) n) ≤ an (Suc d)")
 apply (thin_tac "∀N. ∃M. ∀m>M. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                       ((Pseql _{X K v d} F) m ± -_a p)",
        thin_tac "∀N. ∃M. ∀m>M. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                       ((Pseqh_{R X K v d} F) m ± -_a b ⋅_r X^ ^{(Suc d)})",
        thin_tac "∀n. F n = (Pseql_{R X K v d} F) n ± ((Pseqh_{R X K v d} F) n)")
 apply (drule_tac x = m in spec,
        drule_tac x = m in spec)
 apply (subgoal_tac "b ⋅_r X^ ^{(Suc d)} ∈ carrier R")
 apply (frule Ring.ring_is_ag[of "R"])
 apply (frule_tac x = p in aGroup.ag_mOp_closed[of "R"], assumption+)
 apply (subst aGroup.ag_pOp_assoc[of "R"], assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+)
 apply (rule aGroup.ag_pOp_closed, assumption+)
 apply (frule_tac x1 = "-_a p" and y1 = "(Pseqh_{R X K v d} F) m" and z1 =
  "-_a (b ⋅_r (X^ ^{(Suc d)}))" in aGroup.ag_pOp_assoc[THEN sym, of "R"],
  assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+, simp del:npow_suc)
 apply (subst aGroup.ag_pOp_assoc[THEN sym], assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+,
        rule aGroup.ag_pOp_closed, assumption+)
 apply (subst aGroup.ag_add4_rel[of R], assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+)
 apply (subst aGroup.ag_p_inv[THEN sym, of R], assumption+, simp del:npow_suc)

 apply (rule Ring.ring_tOp_closed, assumption+,
        frule PolynRg.subring[of R "Vr K v" X],
        simp add:Ring.mem_subring_mem_ring,
        rule Ring.npClose, assumption+, simp add:PolynRg.X_mem_R)
apply (rule allI)
apply (rule_tac F = F and n = n and d = d in Pseq_decompos[of "v" "R" "X"],
       assumption+, simp, simp)
apply (rule allI)
apply (rotate_tac -3, drule_tac x = N in spec)
apply (erule exE)
 apply (subgoal_tac "∀n m. M < n ∧ M < m ⟶
                    P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                     ((Pseqh_{R X K v d} F) n ± -_a ((Pseqh_{R X K v d} F) m))")
 apply blast
 apply ((rule allI)+, rule impI)
 apply (rotate_tac -2,
        drule_tac x = n in spec,
        rotate_tac -1,
        drule_tac x = m in spec,
        simp)
 
apply (simp only: Pseqh_def)
apply (subst v_hdeg_p_mOp[of "v" "R" "X"], assumption+)
 apply simp+

apply (frule Ring.ring_is_ag[of "R"])
apply (subst v_hdeg_p_pOp[of "v" "R" "X"], assumption+)
apply (simp, rule aGroup.ag_mOp_closed, assumption, simp, simp,
       frule_tac p1 = "F m" in PolynRg.deg_minus_eq1 [THEN sym, of R "Vr K v"
       X])
apply simp
apply (rotate_tac -1, frule sym,
       thin_tac "deg R (Vr K v) X (F m) = deg R (Vr K v) X (-_a (F m))", simp)
apply (rule PCauchy_hTr[of v R X], assumption+)
apply (rule_tac x = "F n" and y = "-_a (F m)" in aGroup.ag_pOp_closed[of "R"],
       assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption+, simp)
apply (frule_tac p = "F m" in PolynRg.deg_minus_eq1 [of R "Vr K v" X],
       simp,
       rule_tac p = "F n" and q = "-_a (F m)" and n = "Suc d" in
       PolynRg.polyn_deg_add4[of "R" "Vr K v" "X"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption, simp+)

apply (rule conjI, rule allI, rule Pseql_mem, assumption+, simp)
 apply (rule allI, simp)

apply (thin_tac "∀F. (∀n. F n ∈ carrier R) ∧
      (∀n. deg R (Vr K v) X (F n) ≤ an d) ∧
       (∀N. ∃M. ∀n m. M < n ∧ M < m ⟶
          P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)}) (F n ± -_a (F m))) ⟶
               (∃p∈carrier R. Plimit_{R X K v} F p)")

apply (rule conjI, rule allI)
 apply (subst Pseql_def)
 apply (rule_tac p = "F n" and d = d in PolynRg.deg_ldeg_p[of R "Vr K v" X],
        assumption+)
 apply simp+

apply (simp only: Pseql_def)
 apply (rule allI)
 apply (rotate_tac -1, drule_tac x = N in spec)
 apply (erule exE)
 apply (subgoal_tac "∀n m. M < n ∧ M < m ⟶
        P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
          (ldeg_p R (Vr K v) X d (F n) ± -_a (ldeg_p R (Vr K v) X d (F m)))")
 apply blast
 apply ((rule allI)+, rule impI, erule conjE)
 apply (rotate_tac -3, drule_tac x = n in spec,
        rotate_tac -1, drule_tac x = m in spec, simp)

apply (subst v_ldeg_p_mOp[of v R X], assumption+, simp+)

apply (frule Ring.ring_is_ag[of "R"])
apply (subst v_ldeg_p_pOp[of v R X], assumption+, simp,
       rule aGroup.ag_mOp_closed, assumption, simp, simp,
       frule_tac p = "F m" in PolynRg.deg_minus_eq1[of R "Vr K v" X], simp)
apply simp
apply (rule PCauchy_lTr[of v R X], assumption+)
apply (rule_tac x = "F n" and y = "-_a (F m)" in aGroup.ag_pOp_closed[of R],
       assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption+, simp)

apply (frule_tac p = "F m" in PolynRg.deg_minus_eq1[of R "Vr K v" X], simp,
       rule_tac p = "F n" and q = "-_a (F m)" and n = "Suc d" in
       PolynRg.polyn_deg_add4[of "R" "Vr K v" "X"], assumption+,
       simp, rule aGroup.ag_mOp_closed, assumption, simp+)
done

lemma (in Corps) PCauchy_Plimit:"[valuation K v; Complete K;
      PolynRg R (Vr K v) X; PCauchy _{X K v} F] ==>
        ∃p∈carrier R. Plimit _{X K v} F p"
  by (metis P_limitTr pol_Cauchy_seq_def)

lemma (in Corps) P_limit_mult:"[valuation K v; PolynRg R (Vr K v) X;
  ∀n. F n ∈ carrier R; ∀n. G n ∈ carrier R; p1 ∈ carrier R; p2 ∈ carrier R;
  Plimit_{R X K v} F p1; Plimit_{R X K v} G p2] ==>
                  Plimit_{R X K v} (λn. (F n) ⋅_r (G n)) (p1 ⋅_r p2)"
apply (frule Vr_ring[of v],
       frule PolynRg.is_Ring,
       frule Ring.ring_is_ag[of "R"])
apply (simp add:pol_limit_def)
apply (rule conjI)
apply (rule allI)
 apply (drule_tac x = n in spec,
        drule_tac x = n in spec)

 apply (simp add:Ring.ring_tOp_closed[of "R"])

apply (rule allI)
 apply (rotate_tac 6,
        drule_tac x = N in spec,
        drule_tac x = N in spec)
 apply (erule exE, erule exE, rename_tac N M1 M2)
 apply (subgoal_tac "∀m. (max M1 M2) < m ⟶
          P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                     ((F m) ⋅_r (G m) ± -_a (p1 ⋅_r p2))")
 apply blast
 
apply (rule allI, rule impI, simp, erule conjE)
 apply (rotate_tac -4,
        drule_tac x = m in spec,
        drule_tac x = m in spec, simp)
 apply (subgoal_tac "(F m) ⋅_r (G m) ± -_a p1 ⋅_r p2 =
             ((F m) ± -_a p1) ⋅_r (G m) ± p1 ⋅_r ((G m) ± -_a p2)", simp)
 apply (frule_tac n = "an N" in vp_apow_ideal[of v])
 apply simp
 apply (rule_tac I = "vp K v^{(Vr K v) (an N)}" and
        p = "((F m) ± -_a p1) ⋅_r (G m)" and q = "p1 ⋅_r ((G m) ± -_a p2)"
        in PolynRg.P_mod_add[of R "Vr K v" "X"],
         assumption+)
 apply (rule Ring.ring_tOp_closed[of "R"], assumption+)
 apply (rule aGroup.ag_pOp_closed, assumption) apply simp
 apply (rule aGroup.ag_mOp_closed, assumption+) apply simp
 apply (rule Ring.ring_tOp_closed[of "R"], assumption+)
 apply (rule aGroup.ag_pOp_closed, assumption) apply simp
  apply (rule aGroup.ag_mOp_closed, assumption+)
apply (frule Ring.whole_ideal[of "Vr K v"])
apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and J = "carrier (Vr K v)" and
   p = "F m ± -_a p1" and q = "G m" in PolynRg.P_mod_mult1[of R "Vr K v" X],
   assumption+,
   rule aGroup.ag_pOp_closed, assumption+, simp, rule aGroup.ag_mOp_closed,
      assumption+) apply simp apply assumption
apply (rotate_tac 8,
       drule_tac x = m in spec)
apply (case_tac "G m = 0", simp add:P_mod_def)
apply (frule_tac p = "G m" in PolynRg.s_cf_expr[of R "Vr K v" X], assumption+,
       (erule conjE)+)
thm PolynRg.P_mod_mod
apply (frule_tac I1 = "carrier (Vr K v)" and p1 = "G m" and
       c1 = "s_cf R (Vr K v) X (G m)" in PolynRg.P_mod_mod[THEN sym,
       of R "Vr K v" X], assumption+)
apply (simp,
      thin_tac "P_mod R (Vr K v) X (carrier (Vr K v)) (G m) =
        (∀j≤fst (s_cf R (Vr K v) X (G m)).
            snd (s_cf R (Vr K v) X (G m)) j ∈ carrier (Vr K v))")
apply (rule allI, rule impI)
 apply (simp add:PolynRg.pol_coeff_mem)
apply (simp add:Ring.idealprod_whole_r[of "Vr K v"])

apply (cut_tac I = "carrier (Vr K v)" and J = "vp K v^{(Vr K v) (an N)}" and
      p = p1 and q = "G m ± -_a p2" in PolynRg.P_mod_mult1[of R "Vr K v" X],
      assumption+)
apply (simp only: Ring.whole_ideal, assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+, simp, rule aGroup.ag_mOp_closed,
      assumption+)
apply (frule PolynRg.s_cf_expr0[of R "Vr K v" X p1], assumption+)
  thm PolynRg.P_mod_mod
apply (cut_tac I1 = "carrier (Vr K v)" and p1 = p1 and
       c1 = "s_cf R (Vr K v) X p1" in PolynRg.P_mod_mod[THEN sym,
        of R "Vr K v" X], assumption+)
apply (simp add:Ring.whole_ideal, assumption+)
apply (simp, simp, simp, (erule conjE)+,
       thin_tac "P_mod R (Vr K v) X (carrier (Vr K v)) p1 =
        (∀j≤fst (s_cf R (Vr K v) X p1).
            snd (s_cf R (Vr K v) X p1) j ∈ carrier (Vr K v))")
 apply (rule allI, rule impI)
 apply (simp add:PolynRg.pol_coeff_mem, assumption)
apply (simp add:Ring.idealprod_whole_l[of "Vr K v"])
apply (drule_tac x = m in spec,
       drule_tac x = m in spec)
apply (frule aGroup.ag_mOp_closed[of R p1], assumption,
       frule aGroup.ag_mOp_closed[of R p2], assumption )
apply (simp add:Ring.ring_distrib1 Ring.ring_distrib2)
 apply (subst aGroup.pOp_assocTr43[of R], assumption+,
        (rule Ring.ring_tOp_closed, assumption+)+)
 apply (simp add:Ring.ring_inv1_1[THEN sym],
        simp add:Ring.ring_inv1_2[THEN sym])
apply (frule_tac x = p1 and y = "G m" in Ring.ring_tOp_closed, assumption+,
       frule_tac x = "F m" and y = "G m" in Ring.ring_tOp_closed, assumption+,
       simp add:aGroup.ag_l_inv1 aGroup.ag_r_zero)
done
       (** Hfst K v R X t S Y f g h m**)
definition
  Hfst :: "[_, 'b ==> ant, ('b, 'm1) Ring_scheme, 'b,'b, ('b set, 'm2) Ring_scheme, 'b set, 'b, 'b, 'b, nat] ==> 'b"
        (‹(11Hfst_{_ _ _ _ _ _ _ _ _ _} _)› [67,67,67,67,67,67,67,67,67,67,68]67) where
  "Hfst _{v R X t S Y f g h} m = fst (Hpr _{(Vr K v) X t S Y f g h} m)"

definition
  Hsnd :: "[_, 'b ==> ant, ('b, 'm1) Ring_scheme, 'b,'b, ('b set, 'm2) Ring_scheme, 'b set, 'b, 'b, 'b, nat] ==> 'b"
        (‹(11Hsnd_{_ _ _ _ _ _ _ _ _ _} _)› [67,67,67,67,67,67,67,67,67,67,68]67) where
 
  "Hsnd _{v R X t S Y f g h} m = snd (Hpr _{(Vr K v) X t S Y f g h} m)"

lemma (in Corps) Hensel_starter:"[valuation K v; Complete K;
    PolynRg R (Vr K v) X; PolynRg S ((Vr K v) /_r (vp K v)) Y;
    t ∈ carrier (Vr K v); vp K v = (Vr K v) ♢_p t;
    f ∈ carrier R; f ≠ 0; g' ∈ carrier S; h' ∈ carrier S;
    0 < deg S ((Vr K v) /_r (vp K v)) Y g';
    0 < deg S ((Vr K v) /_r (vp K v)) Y h';
    ((erH R (Vr K v) X S ((Vr K v) /_r (vp K v)) Y
             (pj (Vr K v) (vp K v))) f) = g' ⋅_r h';
    rel_prime_pols S ((Vr K v) /_r (vp K v)) Y g' h'] ==>
 ∃g h. g ≠ 0 ∧ h ≠ 0 ∧ g ∈ carrier R ∧ h ∈ carrier R ∧
  deg R (Vr K v) X g ≤ deg S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
   (erH R (Vr K v) X S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
   (pj (Vr K v) ((Vr K v) ♢_p t)) g) ∧ (deg R (Vr K v) X h +
    deg S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y (erH R (Vr K v) X S
     ((Vr K v) /_r ((Vr K v) ♢_p t)) Y (pj (Vr K v) ((Vr K v) ♢_p t)) g)
    ≤ deg R (Vr K v) X f) ∧
   (erH R (Vr K v) X S ((Vr K v) /_r (vp K v)) Y
                            (pj (Vr K v) (vp K v))) g = g' ∧
    (erH R (Vr K v) X S ((Vr K v) /_r (vp K v)) Y
                            (pj (Vr K v) (vp K v))) h = h' ∧
  0 < deg S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
   (erH R (Vr K v) X S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
   (pj (Vr K v) ((Vr K v) ♢_p t)) g) ∧
  0 < deg S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
    (erH R (Vr K v) X S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
   (pj (Vr K v) ((Vr K v) ♢_p t)) h) ∧
  rel_prime_pols S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
    (erH R (Vr K v) X S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
    (pj (Vr K v) ((Vr K v) ♢_p t)) g)
    (erH R (Vr K v) X S ((Vr K v) /_r ((Vr K v) ♢_p t)) Y
    (pj (Vr K v) ((Vr K v) ♢_p t)) h) ∧
 P_mod R (Vr K v) X ((Vr K v) ♢_p t) (f ± -_a (g ⋅_r h))"
apply (frule Vr_ring[of v],
       frule PolynRg.subring[of R "Vr K v" X],
       frule vp_maximal [of v], frule PolynRg.is_Ring,
       frule Ring.subring_Ring[of R "Vr K v"], assumption+,
       frule Ring.residue_field_cd[of "Vr K v" "vp K v"], assumption+,
       frule Corps.field_is_ring[of "Vr K v /_r vp K v"],
       frule pj_Hom[of "Vr K v" "vp K v"], frule vp_ideal[of "v"],
       simp add:Ring.maximal_ideal_ideal)
apply (frule Corps.field_is_idom[of "(Vr K v) /_r (vp K v)"],
       frule Vr_integral[of v], simp,
       frule Vr_mem_f_mem[of v t], assumption+)
apply (frule PolynRg.erH_inv[of R "Vr K v" X "Vr K v ♢_p t" S Y "g'"],
       assumption+, simp add:Ring.maximal_ideal_ideal,
       simp add:PolynRg.is_Ring, assumption+, erule bexE, erule conjE)
apply (frule PolynRg.erH_inv[of R "Vr K v" X "Vr K v ♢_p t" S Y "h'"],
       assumption+, simp add:Ring.maximal_ideal_ideal,
       simp add:PolynRg.is_Ring, assumption+, erule bexE, erule conjE)
apply (rename_tac g0 h0)
apply (subgoal_tac " g0 ≠ 0 ∧ h0 ≠ 0 ∧
  deg R (Vr K v) X g0 ≤
  deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
      (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0) ∧
  deg R (Vr K v) X h0 + deg S (Vr K v /_r (Vr K v ♢_p t)) Y
      (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) g0) ≤ deg R (Vr K v) X f ∧
    0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
      (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0) ∧
   0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) h0) ∧
   rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y
      (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) g0)
      (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) h0) ∧
   P_mod R (Vr K v) X (Vr K v ♢_p t) (f ± -_a g0 ⋅_r h0)")
 apply (thin_tac "g' ∈ carrier S",
        thin_tac "h' ∈ carrier S",
        thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'",
        thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'",
        thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) f = g' ⋅_r h'",
        thin_tac "rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y g' h'",
        thin_tac "Corps (Vr K v /_r (Vr K v ♢_p t))",
        thin_tac "Ring (Vr K v /_r (Vr K v ♢_p t))",
        thin_tac "vp K v = Vr K v ♢_p t")
apply blast
apply (rule conjI)
 apply (thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) f = g' ⋅_r h'",
    thin_tac "rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y g' h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) h0 = h'",
   thin_tac "deg R (Vr K v) X h0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'")
 apply (rule contrapos_pp, simp+)
 apply (simp add:PolynRg.erH_rHom_0[of R "Vr K v" X S
      "Vr K v /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)"])
 apply (rotate_tac -3, drule sym, simp add:deg_def)
  apply (drule aless_imp_le[of "0" "-∞"],
        cut_tac minf_le_any[of "0"],
        frule ale_antisym[of "0" "-∞"], simp only:ant_0[THEN sym], simp)

apply (rule conjI)
 apply (thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'",
    thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) f = g' ⋅_r h'",
    thin_tac "rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y g' h'",
    thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) g0 = g'",
   thin_tac "deg R (Vr K v) X h0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'")
 apply (rule contrapos_pp, simp+,
        simp add:PolynRg.erH_rHom_0[of R "Vr K v" X S
      "Vr K v /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)"])
 apply (rotate_tac -2, drule sym, simp add:deg_def)
 apply (frule aless_imp_le[of "0" "-∞"], thin_tac "0 < - ∞",
        cut_tac minf_le_any[of "0"],
        frule ale_antisym[of "0" "-∞"], simp only:ant_0[THEN sym], simp)

apply (frule_tac x = "deg R (Vr K v) X h0" and
       y = "deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'" and
       z = "deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'" in aadd_le_mono)
apply (simp add:PolynRg.deg_mult_pols1[THEN sym, of S
 "Vr K v /_r (Vr K v ♢_p t)" Y "h'" "g'"])
 apply (frule PolynRg.is_Ring[of S "Vr K v /_r (Vr K v ♢_p t)" Y],
        simp add:Ring.ring_tOp_commute[of S "h'" "g'"])
 apply (rotate_tac 11, drule sym)
 apply simp
apply (frule PolynRg.erH_rHom[of R "Vr K v" X S
      "(Vr K v) /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)"],
      assumption+)
apply (frule PolynRg.pHom_dec_deg[of R "Vr K v" X S "(Vr K v) /_r (Vr K v ♢_p t)" Y "erH R (Vr K v) X S ((Vr K v) /_r (Vr K v ♢_p t))
 Y (pj (Vr K v) (Vr K v ♢_p t))" "f"], assumption+)
apply (frule_tac i = "deg R (Vr K v) X h0 +
        deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'" in ale_trans[of _
        "deg S (Vr K v /_r (Vr K v ♢_p t)) Y
           (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) f)" "deg R (Vr K v) X f"],
       assumption+) apply simp
apply (thin_tac "deg R (Vr K v) X h0 +
        deg S (Vr K v /_r (Vr K v ♢_p t)) Y g' ≤ deg R (Vr K v) X f",
       thin_tac "deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) f) ≤ deg R (Vr K v) X f",
       thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'",
       thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y h'",
      thin_tac "deg R (Vr K v) X h0 + deg S (Vr K v /_r (Vr K v ♢_p t)) Y g'
       ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
           (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) f)",
     thin_tac "rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y g' h'")
 apply (rotate_tac 12, drule sym)
 apply (drule sym)
 apply simp
apply (frule_tac x = g0 and y = h0 in Ring.ring_tOp_closed[of "R"],
       assumption+)
 apply (thin_tac "deg R (Vr K v) X h0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
  (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) h0)",
       thin_tac "h' = erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) h0",
       thin_tac "g' = erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) g0",
       thin_tac "deg R (Vr K v) X g0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) g0)")
apply (subst PolynRg.P_mod_diff[THEN sym, of R "Vr K v" X "Vr K v ♢_p t"
       S Y f], assumption+) apply (simp add:Ring.maximal_ideal_ideal, assumption+)
apply (rotate_tac 12, drule sym)
apply (subst PolynRg.erH_mult[of R "Vr K v" X S "Vr K v /_r (Vr K v ♢_p t)"
        Y], assumption+)
done

lemma aadd_plus_le_plus:"[ a ≤ (a'::ant); b ≤ b'] ==> a + b ≤ a' + b'"
  by (metis aadd_commute aadd_le_mono ale_trans)

lemma (in Corps) Hfst_PCauchy:"[valuation K v; Complete K;
  PolynRg R (Vr K v) X; PolynRg S (Vr K v /_r (Vr K v ♢_p t)) Y; g0 ∈ carrier R;
  h0 ∈ carrier R; f ∈ carrier R; f ≠ 0; g0 ≠ 0; h0 ≠ 0;
  t ∈ carrier (Vr K v); vp K v = Vr K v ♢_p t;
  deg R (Vr K v) X g0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0);
  deg R (Vr K v) X h0 + deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0)
        ≤ deg R (Vr K v) X f;
  0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0);
  0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) h0);
  rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
         (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0)
    (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                                    (pj (Vr K v) (Vr K v ♢_p t)) h0);

   erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
      (pj (Vr K v) (Vr K v ♢_p t)) f =
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                        (pj (Vr K v) (Vr K v ♢_p t)) g0 ⋅_r
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                               (pj (Vr K v) (Vr K v ♢_p t)) h0] ==>
      PCauchy_{R X K v} Hfst K v R X t S Y f g0 h0"
(*  P_mod begin*)
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 /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)" g0 h0],
       frule PolynRg.P_mod_diff[THEN sym, of R "Vr K v" X "Vr K v ♢_p t" S Y
        f "g0 ⋅_r h0"], assumption+,
       frule PolynRg.is_Ring[of R], rule Ring.ring_tOp_closed,
       assumption+) (** P_mod done **)
apply (simp add:pol_Cauchy_seq_def, rule conjI)
 apply (rule allI)

apply (frule_tac t = t and g = g0 and h = h0 and m = n in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f],
       rule Vr_integral[of v], assumption+, simp add:vp_gen_nonzero,
       drule sym, simp add:vp_maximal, assumption+)

apply (subst Hfst_def)
apply (rule cart_prod_fst, assumption)
apply (rule conjI)
apply (subgoal_tac "∀n. deg R (Vr K v) X (Hfst_{K v R X t S Y f g0 h0} n) ≤
                     an (deg_n R (Vr K v) X f)")
apply blast
apply (rule allI)
apply (frule Vr_integral[of v],
       frule_tac t = t and g = g0 and h = h0 and m = n in
       PolynRg.P_mod_diffxxx5_4[of R "Vr K v" X _ S Y f], assumption+,
       simp add:vp_gen_nonzero,
       drule sym, simp add:vp_maximal, assumption+)
apply (subst PolynRg.deg_an[THEN sym], (erule conjE)+, assumption+)
apply (simp add:Hfst_def, (erule conjE)+)
apply (frule_tac i = "deg R (Vr K v) X (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" and
      j = "deg R (Vr K v) X g0" and k = "deg S (Vr K v /_r (Vr K v ♢_p t)) Y
            (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
              (pj (Vr K v) (Vr K v ♢_p t)) g0)" in ale_trans, assumption+,
       frule PolynRg.nonzero_deg_pos[of R "Vr K v" X h0], assumption+,
       frule_tac x = 0 and y = "deg R (Vr K v) X h0" and z = "deg S
  (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
      (pj (Vr K v) (Vr K v ♢_p t)) g0)" in aadd_le_mono, simp add:aadd_0_l)
apply (rule allI)
apply (subgoal_tac "∀n m. N < n ∧ N < m ⟶
  P_mod R (Vr K v) X (Vr K v ♢_p t^{(Vr K v) (an N)})
     ( Hfst_{K v R X t S Y f g0 h0} n ± -_a (Hfst_{K v R X t S Y f g0 h0} m))")
apply blast
apply ((rule allI)+, rule impI, (erule conjE)+,
       frule Vr_integral[of v], frule vp_gen_nonzero[of v t], assumption+,
       frule vp_maximal[of v])
apply (frule_tac t = t and g = g0 and h = h0 and m = n in
      PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
      assumption+,
      frule_tac t = t and g = g0 and h = h0 and m = m in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
       assumption+,
      frule_tac t = t and g = g0 and h = h0 and m = N in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
       assumption+,
      frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_fst[of _
       "carrier R" "carrier R"],
       frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} N" in cart_prod_fst[of _
       "carrier R" "carrier R"],
      frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} n" in cart_prod_fst[of _
       "carrier R" "carrier R"],
       thin_tac "Hpr_{R (Vr K v) X t S Y f g0 h0} n ∈ carrier R × carrier R",
       thin_tac "Hpr_{R (Vr K v) X t S Y f g0 h0} m ∈ carrier R × carrier R")
apply (frule PolynRg.is_Ring)
apply (case_tac "N = 0", simp add:r_apow_def)
apply (rule_tac p = "Hfst_{K v R X t S Y f g0 h0} n ±
               -_a (Hfst_{K v R X t S Y f g0 h0} m)" in
       PolynRg.P_mod_whole[of "R" "Vr K v" "X"], assumption+,
       frule Ring.ring_is_ag[of "R"], simp add:Hfst_def,
       rule aGroup.ag_pOp_closed, assumption+, rule aGroup.ag_mOp_closed,
       assumption+)

apply (frule_tac t = t and g = g0 and h = h0 and m = N and n = "n - N" in
       PolynRg.P_mod_diffxxx5_3[of R "Vr K v" X _ S Y f], assumption+,
       simp+, (erule conjE)+,
       frule_tac t = t and g = g0 and h = h0 and m = N and n = "m - N" in
       PolynRg.P_mod_diffxxx5_3[of R "Vr K v" X _ S Y f], assumption+,
       simp, (erule conjE)+,
       thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) N})) (snd
     (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ± -_a (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n)))",
      thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) N})) (snd
    (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ± -_a (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)))")
apply (frule Vr_ring[of v],
       simp only:Ring.principal_ideal_n_pow1[THEN sym],
       drule sym, simp, frule vp_ideal[of v],
       simp add:Ring.ring_pow_apow,
       frule_tac n = "an N" in vp_apow_ideal[of v], simp,
       frule Ring.ring_is_ag[of R],
       frule_tac x = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n)" in
        aGroup.ag_mOp_closed[of R], assumption)
apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and
      p = "(fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N)) ±
      -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" in PolynRg.P_mod_minus[of R
     "Vr K v" X], assumption+)

apply (rule aGroup.ag_pOp_closed, assumption+,
       simp add:aGroup.ag_p_inv aGroup.ag_inv_inv)
 apply (frule Ring.ring_is_ag,
       frule_tac x = "-_a fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" and
       y = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n)" in aGroup.ag_pOp_commute)
 apply(rule aGroup.ag_mOp_closed, assumption+, simp,
       thin_tac "-_a fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ±
       fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n) = fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n) ±
       -_a fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N)")
apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and
       p = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n) ±
          -_a fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" and
       q = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ±
          -_a fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" in PolynRg.P_mod_add[of
          R "Vr K v" X], assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+,
       rule aGroup.ag_mOp_closed, assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+,
       rule aGroup.ag_mOp_closed, assumption+)
apply (frule_tac x = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" in
       aGroup.ag_mOp_closed, assumption+,
       frule_tac x = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" in
       aGroup.ag_mOp_closed, assumption+,
       simp add:aGroup.pOp_assocTr43[of R] aGroup.ag_l_inv1 aGroup.ag_r_zero)
apply (simp add:Hfst_def)
done

lemma (in Corps) Hsnd_PCauchy:"[valuation K v; Complete K;
  PolynRg R (Vr K v) X; PolynRg S (Vr K v /_r (Vr K v ♢_p t)) Y; g0 ∈ carrier R;
  h0 ∈ carrier R; f ∈ carrier R; f ≠ 0; g0 ≠ 0; h0 ≠ 0;
  t ∈ carrier (Vr K v); vp K v = Vr K v ♢_p t;
  deg R (Vr K v) X g0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0);
  deg R (Vr K v) X h0 + deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0)
        ≤ deg R (Vr K v) X f;
  0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0);
  0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
       (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) h0);
  rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S
         (Vr K v /_r (Vr K v ♢_p t)) Y (pj (Vr K v) (Vr K v ♢_p t)) g0)
    (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                                    (pj (Vr K v) (Vr K v ♢_p t)) h0);
   erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
      (pj (Vr K v) (Vr K v ♢_p t)) f =
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                        (pj (Vr K v) (Vr K v ♢_p t)) g0 ⋅_r
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                               (pj (Vr K v) (Vr K v ♢_p t)) h0] ==>
      PCauchy_{R X K v} Hsnd K v R X t S Y f g0 h0"
(*  P_mod begin*)
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 /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)" g0 h0],
       frule PolynRg.P_mod_diff[THEN sym, of R "Vr K v" X "Vr K v ♢_p t" S Y
        f "g0 ⋅_r h0"], assumption+,
       frule PolynRg.is_Ring[of R], rule Ring.ring_tOp_closed,
       assumption+) (** P_mod done **)
apply (simp add:pol_Cauchy_seq_def, rule conjI)
 apply (rule allI)

apply (frule_tac t = t and g = g0 and h = h0 and m = n in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f],
       rule Vr_integral[of v], assumption+, simp add:vp_gen_nonzero,
       drule sym, simp add:vp_maximal, assumption+)

apply (subst Hsnd_def)
apply (rule cart_prod_snd, assumption)
apply (rule conjI)
apply (subgoal_tac "∀n. deg R (Vr K v) X (Hsnd_{K v R X t S Y f g0 h0} n) ≤
                     an (deg_n R (Vr K v) X f)")
apply blast
apply (rule allI)
apply (frule Vr_integral[of v],
       frule_tac t = t and g = g0 and h = h0 and m = n in
       PolynRg.P_mod_diffxxx5_4[of R "Vr K v" X _ S Y f], assumption+,
       simp add:vp_gen_nonzero,
       drule sym, simp add:vp_maximal, assumption+)
apply (subst PolynRg.deg_an[THEN sym], (erule conjE)+, assumption+)

apply (simp add:Hsnd_def)
apply (rule allI)
apply (subgoal_tac "∀n m. N < n ∧ N < m ⟶
  P_mod R (Vr K v) X (Vr K v ♢_p t^{(Vr K v) (an N)})
     ( Hsnd_{K v R X t S Y f g0 h0} n ± -_a (Hsnd_{K v R X t S Y f g0 h0} m))")
apply blast
apply ((rule allI)+, rule impI, (erule conjE)+)
apply (frule Vr_integral[of v], frule vp_gen_nonzero[of v t], assumption+)
apply (frule vp_maximal[of v])
apply (frule_tac t = t and g = g0 and h = h0 and m = n in
      PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
      assumption+)
apply (frule_tac t = t and g = g0 and h = h0 and m = m in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
       assumption+,
      frule_tac t = t and g = g0 and h = h0 and m = N in
       PolynRg.P_mod_diffxxx5_2[of R "Vr K v" X _ S Y f], assumption+, simp,
       assumption+,
      frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_snd[of _
       "carrier R" "carrier R"],
       frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} N" in cart_prod_snd[of _
       "carrier R" "carrier R"],
      frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} n" in cart_prod_snd[of _
       "carrier R" "carrier R"],
       thin_tac "Hpr_{R (Vr K v) X t S Y f g0 h0} n ∈ carrier R × carrier R",
       thin_tac "Hpr_{R (Vr K v) X t S Y f g0 h0} m ∈ carrier R × carrier R")
apply (frule PolynRg.is_Ring)
apply (case_tac "N = 0", simp add:r_apow_def)
apply (rule_tac p = "Hsnd_{K v R X t S Y f g0 h0} n ±
               -_a (Hsnd_{K v R X t S Y f g0 h0} m)" in
       PolynRg.P_mod_whole[of "R" "Vr K v" "X"], assumption+,
       frule Ring.ring_is_ag[of "R"], simp add:Hsnd_def,
       rule aGroup.ag_pOp_closed, assumption+, rule aGroup.ag_mOp_closed,
       assumption+)

apply (frule_tac t = t and g = g0 and h = h0 and m = N and n = "n - N" in
       PolynRg.P_mod_diffxxx5_3[of R "Vr K v" X _ S Y f], assumption+)
apply simp+
apply (erule conjE)+
apply (frule_tac t = t and g = g0 and h = h0 and m = N and n = "m - N" in
       PolynRg.P_mod_diffxxx5_3[of R "Vr K v" X _ S Y f], assumption+)
apply simp apply (erule conjE)+
apply (thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) N})) (fst
    (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ± -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n)))",
      thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) N})) (fst
    (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ± -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)))")
apply (frule Vr_ring[of v])
apply (simp only:Ring.principal_ideal_n_pow1[THEN sym])
apply (drule sym, simp, frule vp_ideal[of v])
apply (simp add:Ring.ring_pow_apow,
      frule_tac n = "an N" in vp_apow_ideal[of v], simp,
      frule Ring.ring_is_ag[of R],
      frule_tac x = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n)" in
        aGroup.ag_mOp_closed[of R], assumption)
apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and
      p = "(snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N)) ±
      -_a (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" in PolynRg.P_mod_minus[of R
     "Vr K v" X], assumption+)

apply (rule aGroup.ag_pOp_closed, assumption+,
       simp add:aGroup.ag_p_inv aGroup.ag_inv_inv)
 apply (frule Ring.ring_is_ag,
       frule_tac x = "-_a snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" and
       y = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n)" in aGroup.ag_pOp_commute)
 apply(rule aGroup.ag_mOp_closed, assumption+, simp,
       thin_tac "-_a snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ±
       snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n) = snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n) ±
       -_a snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N)")
apply (frule_tac I = "vp K v^{(Vr K v) (an N)}" and
       p = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n) ±
          -_a snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" and
       q = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N) ±
          -_a snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" in PolynRg.P_mod_add[of
          R "Vr K v" X], assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+,
       rule aGroup.ag_mOp_closed, assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+,
       rule aGroup.ag_mOp_closed, assumption+)
apply (frule_tac x = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} N)" in
       aGroup.ag_mOp_closed, assumption+,
       frule_tac x = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" in
       aGroup.ag_mOp_closed, assumption+,
       simp add:aGroup.pOp_assocTr43[of R] aGroup.ag_l_inv1 aGroup.ag_r_zero)
apply (simp add:Hsnd_def)
done

lemma (in Corps) H_Plimit_f:"[valuation K v; Complete K;
     t ∈ carrier (Vr K v); vp K v = Vr K v ♢_p t;
     PolynRg R (Vr K v) X; PolynRg S (Vr K v /_r (Vr K v ♢_p t)) Y;
     f ∈ carrier R; f ≠ 0; g0 ∈ carrier R; h0 ∈ carrier R; g0 ≠ 0;
     h0 ≠ 0;
     0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y
             (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
               (pj (Vr K v) (Vr K v ♢_p t)) g0);
     0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y
             (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
               (pj (Vr K v) (Vr K v ♢_p t)) h0);
     deg R (Vr K v) X h0 +
        deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0) ≤ deg R (Vr K v) X f;
       
     rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                      (pj (Vr K v) (Vr K v ♢_p t)) g0)
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
                       (pj (Vr K v) (Vr K v ♢_p t)) h0);
      
     erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) f =
      erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0 ⋅_r
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) h0;
      
        deg R (Vr K v) X g0
        ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
           (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) g0);
      
      g ∈ carrier R; h ∈ carrier R;
        Plimit_{R X K v} (Hfst K v R X t S Y f g0 h0) g;
        Plimit_{R X K v} (Hsnd K v R X t S Y f g0 h0) h;
        Plimit_{R X K v} (λn. (Hfst_{K v R X t S Y f g0 h0} n) ⋅_r
                            (Hsnd_{K v R X t S Y f g0 h0} n)) (g ⋅_r h)]
       ==> Plimit_{R X K v} (λn. (Hfst_{K v R X t S Y f g0 h0} n) ⋅_r
                              (Hsnd_{K v R X t S Y f g0 h0} n)) f"
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 /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)" g0 h0])
apply(frule PolynRg.P_mod_diff[of R "Vr K v" X "Vr K v ♢_p t" S Y
        f "g0 ⋅_r h0"], assumption+,
       frule PolynRg.is_Ring[of R], rule Ring.ring_tOp_closed,
       assumption+, simp) (** P_mod done **)
apply (simp add:PolynRg.erH_mult[of R "Vr K v" X S
          "Vr K v /_r (Vr K v ♢_p t)" Y "pj (Vr K v) (Vr K v ♢_p t)" g0 h0])
apply (frule PolynRg.is_Ring[of R])
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+)
apply (subst pol_limit_def)
apply (rule conjI)
 apply (rule allI)
 apply (rule Ring.ring_tOp_closed, assumption)
 apply (simp add:pol_Cauchy_seq_def, simp add:pol_Cauchy_seq_def)
 apply (rule allI)
 apply (subgoal_tac "∀m>N. P_mod R (Vr K v) X (vp K v^{(Vr K v) (an N)})
                   ((Hfst_{K v R X t S Y f g0 h0} m) ⋅_r (Hsnd_{K v R X t S Y f g0 h0} m)
                       ± -_a f)")
apply blast

apply (rule allI, rule impI, frule Vr_integral[of v])
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+)
apply ((erule conjE)+, simp del:npow_suc Hpr_Suc)
apply (frule Ring.ring_is_ag[of "R"])
apply (thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y
       (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
               (pj (Vr K v) (Vr K v ♢_p t)) g0)",
       thin_tac "0 < deg S (Vr K v /_r (Vr K v ♢_p t)) Y
             (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
               (pj (Vr K v) (Vr K v ♢_p t)) h0)",
       thin_tac "rel_prime_pols S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0)
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) h0)",
       thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p t) ( f ± -_a g0 ⋅_r h0)",
       thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) f =
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0) ⋅_r
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) h0)",
       thin_tac "deg R (Vr K v) X g0 ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) g0)",
       thin_tac "deg R (Vr K v) X h0 + deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0) ≤ deg R (Vr K v) X f",
        thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) =
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) g0",
       thin_tac "erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) =
        erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) h0",
       thin_tac "deg R (Vr K v) X (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m))
        ≤ deg S (Vr K v /_r (Vr K v ♢_p t)) Y
           (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
             (pj (Vr K v) (Vr K v ♢_p t)) g0)",
       thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) m}))
        ( fst (Hpr_{R (Vr K v) X t S Y f g0 h0} (m - Suc 0)) ±
             -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)))",
       thin_tac "deg R (Vr K v) X (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) +
        deg S (Vr K v /_r (Vr K v ♢_p t)) Y
         (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
           (pj (Vr K v) (Vr K v ♢_p t)) g0) ≤ deg R (Vr K v) X f",
       thin_tac "P_mod R (Vr K v) X (Vr K v ♢_p (t^^{Vr K v) m}))
         (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} (m - Suc 0)) ±
             -_a (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)))")
apply (case_tac "N = 0", simp add:r_apow_def)
apply (rule_tac p = "(Hfst_{K v R X t S Y f g0 h0} m) ⋅_r (Hsnd_{K v R X t S Y f g0 h0} m)
               ± -_a f" in PolynRg.P_mod_whole[of R "Vr K v" X], assumption+)
apply (simp add:Hfst_def Hsnd_def)
apply (frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_fst[of _ "carrier R" "carrier R"])
apply (frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_snd[of _
       "carrier R" "carrier R"])
apply (frule Ring.ring_is_ag[of R],
       rule aGroup.ag_pOp_closed, assumption)
 apply (rule Ring.ring_tOp_closed, assumption+)
 apply (rule aGroup.ag_mOp_closed, assumption+)

apply (frule_tac g = "f ± -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) ⋅_r
       (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m))" and m = "N - Suc 0" and n = m in
       PolynRg.P_mod_n_m[of "R" "Vr K v" "X"], assumption+)
apply (frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_fst[of _ "carrier R" "carrier R"],
       frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_snd[of _ "carrier R" "carrier R"])

apply (rule aGroup.ag_pOp_closed, assumption+, rule aGroup.ag_mOp_closed,
       assumption)
apply (rule Ring.ring_tOp_closed, assumption+)
apply (subst Suc_le_mono[THEN sym], simp)
apply assumption
apply (simp del:npow_suc)
apply (simp only:Ring.principal_ideal_n_pow1[THEN sym, of "Vr K v"])
apply (cut_tac n = N in an_neq_inf)
apply (subgoal_tac "an N ≠ 0")
apply (subst r_apow_def, simp) apply (simp add:na_an)
apply (frule Ring.principal_ideal[of "Vr K v" t], assumption)
apply (frule_tac I = "Vr K v ♢_p t" and n = N in Ring.ideal_pow_ideal[of "Vr K v"], assumption+)
apply (frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_fst[of _ "carrier R" "carrier R"],
       frule_tac x = "Hpr_{R (Vr K v) X t S Y f g0 h0} m" in cart_prod_snd[of _ "carrier R" "carrier R"])

apply (thin_tac "PCauchy_{R X K v} Hfst K v R X t S Y f g0 h0",
       thin_tac "PCauchy_{R X K v} Hsnd K v R X t S Y f g0 h0")
apply (simp add:Hfst_def Hsnd_def)
apply (frule_tac x = "fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" and y = "snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m)" in Ring.ring_tOp_closed[of "R"], assumption+)

apply (frule_tac x = "(fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) ⋅_r (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m))" in aGroup.ag_mOp_closed[of "R"], assumption+)
apply (frule_tac I = "Vr K v ♢_p t ^{<diamondsuit>(Vr K v) N}" and
       p = "f ± -_a (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} m)) ⋅_r
        (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} m))" in
       PolynRg.P_mod_minus[of R "Vr K v" X], assumption+)
apply (frule Ring.ring_is_ag[of "R"])
apply (rule aGroup.ag_pOp_closed, assumption+)
apply (simp add:aGroup.ag_p_inv, simp add:aGroup.ag_inv_inv,
       frule aGroup.ag_mOp_closed[of "R" "f"], assumption+)
apply (simp add:aGroup.ag_pOp_commute[of "R" "-_a f"])
apply (subst an_0[THEN sym])
apply (subst aneq_natneq[of _ "0"], thin_tac "an N ≠ ∞", simp)
done

theorem (in Corps) Hensel:"[valuation K v; Complete K;
    PolynRg R (Vr K v) X; PolynRg S ((Vr K v) /_r (vp K v)) Y;
    f ∈ carrier R; f ≠ 0; g' ∈ carrier S; h' ∈ carrier S;
    0 < deg S ((Vr K v) /_r (vp K v)) Y g';
    0 < deg S ((Vr K v) /_r (vp K v)) Y h';
    ((erH R (Vr K v) X S ((Vr K v) /_r (vp K v)) Y
             (pj (Vr K v) (vp K v))) f) = g' ⋅_r h';
    rel_prime_pols S ((Vr K v) /_r (vp K v)) Y g' h'] ==>
  ∃g h. g ∈ carrier R ∧ h ∈ carrier R ∧
        deg R (Vr K v) X g ≤ deg S ((Vr K v) /_r (vp K v)) Y g' ∧
                  f = g ⋅_r h"
apply (frule PolynRg.is_Ring[of R "Vr K v" X],
       frule PolynRg.is_Ring[of S "Vr K v /_r vp K v" Y],
       frule vp_gen_t[of v], erule bexE,
       frule_tac t = t in vp_gen_nonzero[of v], assumption)
apply (frule_tac t = t in Hensel_starter[of v R X S Y _ f g' h'], assumption+)
apply ((erule exE)+, (erule conjE)+, rename_tac g0 h0)
apply (frule Vr_ring[of v], frule Vr_integral[of v])
apply (rotate_tac 22, drule sym, drule sym, simp)
apply (frule vp_maximal[of v], simp)
apply (frule_tac mx = "Vr K v ♢_p t" in Ring.residue_field_cd[of "Vr K v"],
                 assumption)
apply (frule_tac mx = "Vr K v ♢_p t" in Ring.maximal_ideal_ideal[of "Vr K v"],
          assumption)
apply (frule_tac I = "Vr K v ♢_p t" in Ring.qring_ring[of "Vr K v"],
       assumption+)
apply (frule_tac B = "Vr K v /_r (Vr K v ♢_p t)" and h = "pj (Vr K v) (Vr K v ♢_p t)" in PolynRg.erH_rHom[of R "Vr K v" X S _ Y], assumption+)
 apply (simp add:pj_Hom)
apply (frule_tac ?g0.0 = g0 and ?h0.0 = h0 in Hfst_PCauchy[of v R X S _ Y _ _
       f], assumption+)
apply (frule_tac ?g0.0 = g0 and ?h0.0 = h0 in Hsnd_PCauchy[of v R X S _ Y _ _
       f], assumption+)
apply (frule_tac F = "λj. Hfst _{v R X t S Y f g0 h0} j" in PCauchy_Plimit[of v R X]
      , assumption+)
apply (frule_tac F = "λj. Hsnd _{v R X t S Y f g0 h0} j" in PCauchy_Plimit[of v R X]
, assumption+)
apply ((erule bexE)+, rename_tac g0 h0 g h)

apply (frule_tac F = "λj. Hfst _{v R X t S Y f g0 h0} j" and
        G = "λj. Hsnd _{v R X t S Y f g0 h0} j" and ?p1.0 = g and ?p2.0 = h
       in P_limit_mult[of v R X], assumption+, rule allI)

apply (simp add:pol_Cauchy_seq_def)
apply (simp add:pol_Cauchy_seq_def, assumption+)
apply (frule_tac t = t and ?g0.0 = g0 and ?h0.0 = h0 and g = g and h = h in
       H_Plimit_f[of v _ R X S Y f], assumption+)
apply (frule_tac F = "λn. (Hfst_{K v R X t S Y f g0 h0} n) ⋅_r (Hsnd_{K v R X t S Y f g0 h0} n)" and ?p1.0 = "g ⋅_r h" and d = "na (deg R (Vr K v) X g0 + deg R (Vr K v) X f)" in P_limit_unique[of v R X _ _ _ f], assumption+)
apply (rule allI)
 apply (frule PolynRg.is_Ring[of R],
        rule Ring.ring_tOp_closed, assumption)
 apply (simp add:pol_Cauchy_seq_def)
 apply (simp add:pol_Cauchy_seq_def)
 apply (rule allI)
apply (thin_tac "Plimit_{R X K v} (Hfst K v R X t S Y f g0 h0) g",
       thin_tac "Plimit_{R X K v} (Hsnd K v R X t S Y f g0 h0) h",
       thin_tac "Plimit_{R X K v} (λn. (Hfst_{K v R X t S Y f g0 h0} n) ⋅_r
                             (Hsnd_{K v R X t S Y f g0 h0} n)) (g ⋅_r h)",
       thin_tac "Plimit_{R X K v} (λn. (Hfst_{K v R X t S Y f g0 h0} n) ⋅_r
                             (Hsnd_{K v R X t S Y f g0 h0} n)) f")
apply (subst PolynRg.deg_mult_pols1[of R "Vr K v" X], assumption+)

 apply (simp add:pol_Cauchy_seq_def, simp add:pol_Cauchy_seq_def,
        thin_tac "g' = erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) g0",
        thin_tac "h' = erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
         (pj (Vr K v) (Vr K v ♢_p t)) h0")
apply (subst PolynRg.deg_mult_pols1[THEN sym, of R "Vr K v" X], assumption+)
apply (simp add:pol_Cauchy_seq_def, simp add:pol_Cauchy_seq_def)

apply (frule_tac p1 = g0 in PolynRg.deg_mult_pols1[THEN sym, of R "Vr K v" X _ f], assumption+, simp)
apply (frule_tac x = g0 in Ring.ring_tOp_closed[of R _ f], assumption+)
apply (frule_tac p = "g0 ⋅_r f" in PolynRg.nonzero_deg_pos[of R "Vr K v" X],
          assumption+)
apply (frule_tac p = "g0 ⋅_r f" in PolynRg.deg_in_aug_minf[of R "Vr K v" X],
       assumption+, simp add:aug_minf_def,
       simp add:PolynRg.polyn_ring_integral[of R "Vr K v" X],
       simp add:Idomain.idom_tOp_nonzeros[of R _ f],
       frule_tac p = "g0 ⋅_r f" in PolynRg.deg_noninf[of R "Vr K v" X],
       assumption+)
apply (simp add:an_na)
apply (subst PolynRg.deg_mult_pols1[of "R" "Vr K v" "X"], assumption+,
       simp add:pol_Cauchy_seq_def, simp add:pol_Cauchy_seq_def)
apply (frule_tac t = t and g = g0 and h = h0 and m = n in
       PolynRg.P_mod_diffxxx5_4[of R "Vr K v" X _ S Y f], assumption+)
apply (erule conjE)

apply (frule_tac a = "deg R (Vr K v) X (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" and a' = "deg R (Vr K v) X g0" and b = "deg R (Vr K v) X (snd (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" and b' = "deg R (Vr K v) X f" in aadd_plus_le_plus, assumption+)
 apply simp
apply (simp add:Hfst_def Hsnd_def)

apply (rule Ring.ring_tOp_closed, assumption+)
 apply (rotate_tac -1, drule sym)
apply (frule_tac F = "λj. Hfst _{v R X t S Y f g0 h0} j" and p = g and ad = "deg S
 (Vr K v /_r (Vr K v ♢_p t)) Y (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
  (pj (Vr K v) (Vr K v ♢_p t)) g0)" in Plimit_deg1[of v R X], assumption+,
  simp add:pol_Cauchy_seq_def)
apply (rule allI)
apply (frule_tac t = t and g = g0 and h = h0 and m = n in
      PolynRg.P_mod_diffxxx5_4[of R "Vr K v" X _ S Y f], assumption+,
      erule conjE)
apply (frule_tac i = "deg R (Vr K v) X (fst (Hpr_{R (Vr K v) X t S Y f g0 h0} n))" and
 j = "deg R (Vr K v) X g0" and k = "deg S (Vr K v /_r (Vr K v ♢_p t)) Y
      (erH R (Vr K v) X S (Vr K v /_r (Vr K v ♢_p t)) Y
               (pj (Vr K v) (Vr K v ♢_p t)) g0)" in ale_trans, assumption+)
 apply (subst Hfst_def, assumption+, blast)
done

end