# Theory Valuation3

theory Valuation3
imports Valuation2
```(**        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⇘K v⇙ f b ⟷ (∀N. ∃M. (∀n. M < n ⟶
((f n) ±⇘K⇙ (-⇩a⇘K⇙ 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 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 (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 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+,
frule_tac h = x in Ring.ideal_subset[of "Vr K v" "vp K v"],
apply (cut_tac x = x in val_pos_mem_Vr[of v], assumption) apply(
apply (frule_tac x = x in val_pos_n_val_pos[of v],
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 (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⇘K 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+,

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 (subst val_pos_n_val_pos[of  "v"], assumption+)
apply (rule aGroup.ag_pOp_closed, assumption+) apply simp
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⇘K 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"],
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 (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)
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⇘K v⇙ f b; lim⇘K 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"],
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)) +⇩K -⇩K b) +⇩K (-⇩K (f (Suc (max xb xc)) +⇩K -⇩K c)) ∈ vpr K G a i v ⇗♢Vr K G a i v xa⇖" *)
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+)
cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
(** addition of  -⇩K f (Suc (max xb xc)) +⇩K b  and f (Suc (max xb xc)) +⇩K -⇩K c
is included in vpr K G a i v ⇗♢(Vr K G a i v) xa⇖ **)
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,
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_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"],
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⇘K v⇙ f b⟧ ⟹
∃N. (∀m. N < m ⟶ (n_val K v) (f m) = (n_val K v) b)"
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 ≤ (n_val K v ( f n +⇩K -⇩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 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+
done

lemma (in Corps) limit_val:"⟦b ∈ carrier K; b ≠ 𝟬; ∀j. f j ∈ carrier K;
valuation K v; lim⇘K 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⇘K v⇙ f 𝟬⟧ ⟹ ∀N.(∃M. (∀m. M < m ⟶ (an N) ≤ (n_val K v (f m))))"
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)
done

lemma (in Corps) not_limit_zero:"⟦valuation K v; ∀j. f j ∈ carrier K;
¬ (lim⇘K v⇙ f 𝟬)⟧ ⟹ ∃N.(∀M. (∃m. (M < m) ∧
((n_val K v) (f m)) < (an N)))"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
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)
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 (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⇘K v⇙ f b; lim⇘K v⇙ g c⟧
⟹ lim⇘K v⇙ (λj. (f j) ± (g j)) (b ± c)"
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 (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
done

lemma (in Corps) Abs_ant_abs[simp]:"Abs (ant z) = ant (abs z)"
apply (simp only:ant_0[THEN sym], simp only:aless_zless)
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⇘K v⇙ f b; lim⇘K v⇙ g c⟧ ⟹ lim⇘K 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+,
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+,
frule aGroup.ag_mOp_closed[of "K" "b"], assumption+,
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 (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+)
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,

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⇘K v⇙ f 𝟬; lim⇘K v⇙ g c⟧ ⟹ lim⇘K v⇙ (λj. (f j) ⋅⇩r (g j)) 𝟬"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
apply (subgoal_tac "∀j. (f j) ⋅⇩r (g j) ∈ carrier K",
prefer 2 apply (rule allI, simp add:Ring.ring_tOp_closed)
apply (case_tac "c = 𝟬⇘K⇙")
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+,

(** case c ≠ 0⇩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+,
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 (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"])
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 (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+)
done

lemma (in Corps) limit_minus:"⟦valuation K v; ∀j. f j ∈ carrier K;
b ∈ carrier K; lim⇘K v⇙ f b⟧ ⟹ lim⇘K v⇙ (λj. (-⇩a (f j))) (-⇩a b)"
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 (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
frule aGroup.ag_mOp_closed[of "K" "b"], assumption+)
done

lemma (in Corps) inv_diff:"⟦x ∈ carrier K; x ≠ 𝟬; y ∈ carrier K; y ≠ 𝟬⟧ ⟹
(x⇗‐K⇖) ± (-⇩a (y⇗‐K⇖)) = (x⇗‐K⇖) ⋅⇩r ( y⇗‐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⇗‐K⇖" "y⇗‐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⇗‐K⇖"], assumption+,
frule aGroup.ag_mOp_closed[of "K" "y⇗‐K⇖"], assumption+,
frule aGroup.ag_pOp_closed[of "K" "x" "-⇩a y"], assumption+)

simp only:Ring.ring_distrib1[of "K" "(x⇗‐K⇖) ⋅⇩r (y⇗‐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+,
cut_tac linvf[of "y"], simp+,
simp only:Ring.ring_r_one)
rule aGroup.ag_pOp_commute[of "K" "x⇗‐K⇖" "-⇩a y⇗‐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⇘K v⇙ f b⟧ ⟹
lim⇘K v⇙ (λj. if (f j) = 𝟬 then 𝟬 else (f j)⇗‐K⇖) (b⇗‐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⇗‐K⇖) ± -⇩a b⇗‐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⇗‐K⇖" "b⇗‐K⇖"], 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⇗‐ K⇖ ⋅⇩r b⇗‐ 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 (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 (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 (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 (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⇘K v⇙ f ⟷ (∀n. (f n) ∈ carrier K) ∧ (
∀N. ∃M. (∀n m. M < n ∧ M < m ⟶
((f n) ±⇘K⇙ (-⇩a⇘K⇙ (f m))) ∈ (vp K v)⇗(Vr K v) (an N)⇖))"

definition
v_complete :: "['b ⇒ ant, _] ⇒ bool"
("(2Complete⇘_⇙ _)"  [90,91]90) where
"Complete⇘v⇙ K ⟷ (∀f. (Cauchy⇘K v⇙ f) ⟶
(∃b. b ∈ (carrier K) ∧ lim⇘K v⇙ f b))"

lemma (in Corps) has_limit_Cauchy:"⟦valuation K v; ∀j. f j ∈ carrier K;
b ∈ carrier K; lim⇘K v⇙ f b⟧ ⟹ Cauchy⇘K v⇙ f"
apply (rule allI)
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⇘Vr 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 = "-⇩a⇘(Vr K v)⇙ (f m ± -⇩a b)" in Ring.ideal_pOp_closed[of "Vr K v"],
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 (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_commute[of "K" "-⇩a b"])
apply (frule_tac x = "f m" in aGroup.ag_mOp_closed[of "K"], assumption+)
apply (simp add:aGroup.ag_pOp_assoc[of "K" _ "b" "-⇩a b"])
done

lemma (in Corps) no_limit_zero_Cauchy:"⟦valuation K v; Cauchy⇘K v⇙ f;
¬ (lim⇘K 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)
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 ± -⇩a f m ∈ vpr K G a i v ⇗♢Vr K G a i v L⇖ 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"],
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+,
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 ± (f ma ± -⇩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 (frule_tac x = "f m" and y = "f ma ± -⇩a (f m)" in
aGroup.ag_pOp_commute[of "K"], assumption+)
done

lemma (in Corps) no_limit_zero_Cauchy1:"⟦valuation K v; ∀j. f j ∈ carrier K;
Cauchy⇘K v⇙ f; ¬ (lim⇘K 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 v'⇙ K K' ⟷ subfield K K' ∧
Complete⇘v'⇙ K' ∧ (∀x ∈ carrier K. v x = v' x) ∧
(∀x ∈ carrier K'. (∃f. Cauchy⇘K v⇙ f ∧ lim⇘K' v'⇙ f x))"

lemma (in Corps) subfield_zero:"⟦Corps K'; subfield K K'⟧ ⟹ 𝟬⇘K⇙ = 𝟬⇘K'⇙"
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⇘K⇙"], 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 ±⇘K'⇙ 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'"])
frule conjunct1)
apply (thin_tac "I⇘K⇙ ∈ aHom K K' ∧
(∀x∈carrier K. ∀y∈carrier K. I⇘K⇙ (x ⋅⇩r y) = I⇘K⇙ x ⋅⇩r⇘K'⇙ I⇘K⇙ y) ∧
I⇘K⇙ 1⇩r = 1⇩r⇘K'⇙")
apply (frule aHom_add[of "K" "K'" "I⇘K⇙" "x" "y"], assumption+,
frule aGroup.ag_pOp_closed[of "K" "x" "y"], assumption+,
done

lemma (in Corps) subfield_mOp:"⟦Corps K'; subfield K K'; x ∈ carrier K⟧ ⟹
-⇩a x = -⇩a⇘K'⇙ 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'"])
frule conjunct1)
apply (thin_tac "I⇘K⇙ ∈ aHom K K' ∧
(∀x∈carrier K. ∀y∈carrier K. I⇘K⇙ (x ⋅⇩r y) = I⇘K⇙ x ⋅⇩r⇘K'⇙ I⇘K⇙ y) ∧
I⇘K⇙ 1⇩r = 1⇩r⇘K'⇙")
apply (frule aHom_inv_inv[of "K" "K'" "I⇘K⇙" "x"], assumption+,
frule aGroup.ag_mOp_closed[of "K" "x"], assumption+)
done

lemma (in Corps) completion_val_eq:"⟦Corps K'; valuation K v; valuation K' v';
x ∈ carrier K;  Completion⇘v 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 v'⇙ K K'⟧ ⟹  carrier K ⊆ carrier K'"
apply (unfold v_completion_def, (erule conjE)+)
done

lemma (in Corps) completion_subfield:"⟦Corps K'; valuation K v;
valuation K' v'; Completion⇘v v'⇙ K K'⟧ ⟹  subfield K K'"
done

lemma (in Corps) subfield_sub:"subfield K K' ⟹ carrier K ⊆ carrier K'"
done

lemma (in Corps) completion_Vring_sub:"⟦Corps K'; valuation K v;
valuation K' v'; Completion⇘v 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+,
done

lemma (in Corps) completion_idmap_rHom:"⟦Corps K'; valuation K v;
valuation K' v';  Completion⇘v 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 (rule conjI)
rule conjI,
apply (rule conjI)
apply ((rule ballI)+) apply (
frule_tac x = a and y = b in aGroup.ag_pOp_closed, assumption+,
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+,
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 (rule conjI)
apply ((rule ballI)+,
frule_tac x = x and y = y in Ring.ring_tOp_closed, assumption+,
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 (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⇘K⇙"],
apply (frule Ring.ring_one[of "Vr K v"], simp add:idmap_def)
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⇘K⇙"], assumption+, simp add:idmap_def)
done

lemma (in Corps) completion_vpr_sub:"⟦Corps K'; valuation K v; valuation K' v';
Completion⇘v 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+,
done

lemma (in Corps) val_v_completion:"⟦Corps K'; valuation K v; valuation K' v';
x ∈ carrier K'; x ≠ 𝟬⇘K'⇙; Completion⇘v v'⇙ K K'⟧ ⟹
∃f. (Cauchy⇘K 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,
done

lemma (in Corps) v_completion_v_limit:"⟦Corps K'; valuation K v;
x ∈ carrier K; subfield K K'; Complete⇘v'⇙ K'; ∀j. f j ∈ carrier K;
valuation K' v'; ∀x∈carrier K. v x = v' x; lim⇘K' v' ⇙f x⟧ ⟹ lim⇘K 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+,
subgoal_tac "∀n. f n ±⇘K'⇙ -⇩a⇘K'⇙ x = f n ± -⇩a x")
apply (rule allI)
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 (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 (subgoal_tac "I⇘(Vr K v)⇙ ∈ carrier (Vr K v) → carrier (Vr K' v')")
apply simp
apply (rule conjI)
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+,
thin_tac "I⇘K⇙ ∈ aHom K K' ∧
(∀x∈carrier K. ∀y∈carrier K. I⇘K⇙ (x ⋅⇩r y) = I⇘K⇙ x ⋅⇩r⇘K'⇙ I⇘K⇙ y) ∧
I⇘K⇙ 1⇩r = 1⇩r⇘K'⇙")
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⇘K⇙"], assumption+,
frule_tac x = a and y = b in aGroup.ag_pOp_closed[of K], assumption+,
apply (rule Pi_I,
drule_tac x = x in bspec, simp add:Vr_mem_f_mem)
apply (frule_tac x1 = x in val_pos_mem_Vr[THEN sym, of v],
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+,
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 (cut_tac amult_pos1[of "tna a" "an N"])
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⇘K' v' ⇙f⟧
⟹  Cauchy⇘K 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 ((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)
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 (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
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 v'⇙ K K'; Cauchy⇘ K v ⇙f⟧ ⟹ Cauchy⇘ K' v' ⇙f"
erule conjE,
rule conjI, unfold v_completion_def, frule conjunct1,
fold v_completion_def, rule allI, frule subfield_sub[of K'])

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 ±⇘K'⇙ (-⇩a⇘K'⇙ (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+)
frule_tac x = "f n  ±⇘K'⇙ -⇩a⇘K'⇙ f m" and n = "(Lv K' v') * (an N)"
in n_value_x_1[of v]) (*apply (
thin_tac "f n ±' (-⇩a' (f m)) ∈ vp K v⇗(Vr K v) ( (Lv K' v') * (an N))⇖",
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
apply assumption
apply (frule_tac x = "f n ±⇘K'⇙ -⇩a⇘K'⇙ 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
apply (frule_tac j = "ant za * an N" and k = "n_val K v (f n ±⇘K'⇙ -⇩a⇘K'⇙ (f m))"
in ale_trans[of "0"], assumption+)
apply (frule_tac w1 = z and x1 = 0 and y1 = "n_val K v ( f n ±⇘K'⇙ -⇩a⇘K'⇙ (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 ±⇘K'⇙ -⇩a⇘K'⇙ (f m))"
and k = "v ( f n ±⇘K'⇙ -⇩a⇘K'⇙ (f m))" in ale_trans, assumption+)
apply (thin_tac "f n ±⇘K'⇙ -⇩a⇘K'⇙ (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 ±⇘K'⇙ -⇩a⇘K'⇙ (f m))",
thin_tac "n_val K v ( f n ±⇘K'⇙ -⇩a⇘K'⇙ (f m)) ≤ v ( f n ±⇘K'⇙ -⇩a⇘K'⇙ (f m))")

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 ±⇘K'⇙ -⇩a⇘K'⇙ (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 (rule_tac x = "f n ±⇘K'⇙ -⇩a⇘K'⇙ (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 ±⇘K'⇙ -⇩a⇘K'⇙ (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)"

lemma (in Corps) completion_approx:"⟦Corps K'; valuation K v; valuation K' v';
Completion⇘v v'⇙ K K'; x ∈ carrier (Vr K' v')⟧ ⟹
∃y∈carrier (Vr K v). (y ±⇘K'⇙ -⇩a⇘K'⇙ 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+,
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⇘K'⇙",
simp, frule Corps.field_is_ring[of "K'"],
frule Ring.ring_is_ag[of "K'"],
subgoal_tac " 𝟬⇘K'⇙ ∈ carrier (Vr K v)",
subgoal_tac " (𝟬⇘K'⇙ ±⇘K'⇙ -⇩a⇘K'⇙ 𝟬⇘K'⇙)∈ vp K' v'", blast,
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+,
cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"],
frule aGroup.ag_inc_zero[of "K"],
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)
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 ±⇘K'⇙ -⇩a⇘K'⇙ 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 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 (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,
frule pj_Hom[of "Vr K' v'" "vp K' v'"], assumption+) apply (
apply (rule surj_to_test)
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 (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 (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+)
done

lemma (in Corps) res_v_completion_ker:"⟦Corps K'; valuation K v;
valuation K' v'; Completion⇘v 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 (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 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 (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 (frule_tac x = x in Corps.vp_mem_val_poss[of "K'" "v'"],
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 (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+,
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 (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 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 (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+)
done

text‹expansion of x in a complete field K, with normal valuation v. Here
we 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⇙ k}"
(**  x *⇩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⇘K⇙ t⇘K⇙⇗-(tna (v x))⇖))) ⋅⇩r⇘K⇙ (t⇘K⇙⇗(tna (v x))⇖)"
| psum_Suc: "psum⇘ K x v t⇙ (Suc n) = (psum⇘ K x v t⇙ n) ±⇘K⇙
((csrp_fn (Vr K v) (vp K v) (pj (Vr K v) (vp K v)
((x ±⇘K⇙ -⇩a⇘K⇙ (psum⇘ K x v t⇙ n)) ⋅⇩r⇘K⇙ (t⇘K⇙⇗- (tna (v x) + int (Suc n))⇖)))) ⋅⇩r⇘K⇙
(t⇘K⇙⇗(tna (v x) + int (Suc n))⇖))"

definition
expand_coeff :: "[_ , 'b ⇒ ant, 'b, 'b]
⇒ nat ⇒ 'b"
("(5ecf⇘_ _ _ _⇙ _)" [96,96,96,96,97]96) where
"ecf⇘K 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⇘K⇙ t⇘K⇙⇗(-(tna (v x)))⇖))
else csrp_fn (Vr K v) (vp K v) (pj (Vr K v)
(vp K v) ((x ±⇘K⇙ -⇩a⇘K⇙ (psum⇘ K x v t⇙ (n - 1))) ⋅⇩r⇘K⇙ (t⇘K⇙⇗(- (tna (v x) + int n))⇖))))"

definition
expand_term :: "[_, 'b ⇒ ant, 'b, 'b]
⇒ nat ⇒ 'b"
("(5etm⇘ _ _ _ _⇙ _)" [96,96,96,96,97]96) where

"etm⇘K v t x⇙ n = (ecf⇘K v t x⇙ n)⋅⇩r⇘K⇙ (t⇘K⇙⇗(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 (∈ O), and  a⇩0 be an
element of the complete set of representatives such that a⇩0 = x t⇧-⇧j mod P.
We see that (a⇩0 - x t⇧-⇧j)/t is an element of O, and then we choose a⇩1 an
element of the complete set of representatives which is equal to (a⇩0 - x t⇧-⇧j)/tmodulo P. We see x - (a⇩0t⇧j + a⇩1t⇗(j+1)⇖ + … + a⇩st⇗(j+s)⇖) ∈ (t⇗(j+s+1)⇖).
"psum G a i K v t x s" is the partial sum a⇩0t⇧j + a⇩1t⇗(j+1)⇖ + … + a⇩st⇗(j+s)⇖ ***)

lemma (in Corps) Kxa_val_ge:"⟦valuation K v; t ∈ carrier K; v t = 1⟧
⟹  Kxa K v (t⇘K⇙⇗j⇖) = {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,
frule_tac x = k in Vr_mem_f_mem[of "v"], assumption+,
frule npowf_mem[of "t" "j"], simp,
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 (rule subsetI, simp, erule conjE)
apply (case_tac "x = 𝟬⇘K⇙")
apply (frule Vr_ring[of "v"],
frule Ring.ring_zero[of "Vr K v"],
frule_tac x1 = "t⇘K⇙⇗j⇖" in Ring.ring_times_x_0[THEN sym, of "K"],
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⇘K⇙⇗j⇖)"],
apply (cut_tac npowf_mem[of t j])
defer
apply assumption apply simp
apply (frule value_of_inv[THEN sym, of "v" "t⇘K⇙⇗j⇖"], assumption+)

apply (cut_tac invf_closed1[of "t⇘K⇙⇗j⇖"], simp, erule conjE)
apply (frule_tac x1 = x in val_t2p[THEN sym, of "v" _ "(t⇘K⇙⇗j⇖)⇗‐K⇖"],
assumption+, simp)
apply (frule_tac x = "(t⇘K⇙⇗j⇖)⇗‐K⇖" and y = x in Ring.ring_tOp_closed[of "K"],
assumption+)
apply (simp add:Ring.ring_tOp_commute[of "K" _ "(t⇘K⇙⇗j⇖)⇗‐K⇖"])
apply (frule_tac x = "((t⇘K⇙⇗j⇖)⇗‐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⇘K⇙⇗j⇖" "(t⇘K⇙⇗j⇖)⇗‐K⇖"],
assumption+)
apply (simp add:Ring.ring_tOp_commute[of K "t⇘K⇙⇗j⇖" "(t⇘K⇙⇗j⇖)⇗‐ K⇖"] linvf,
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⇘K⇙⇗j⇖) = (vp K v)⇗(Vr K v) (ant j)⇖"
apply (frule val_surj_n_val[of v], blast)
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 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 (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⇗‐K⇖) ± (-⇩a b)) ⋅⇩r x"
apply (cut_tac field_is_ring,
cut_tac invf_closed1[of x], simp, erule conjE) apply (
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⇗‐K⇖"], assumption+,
apply simp
done

lemma a0_le_1[simp]:"(0::ant) ≤ 1"

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"],
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 (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⇘K⇙⇗((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⇘K⇙⇗- 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⇘K⇙⇗- tna (v x)⇖))"],
assumption+,
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⇘K⇙⇗(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⇘K⇙⇗- tna (v x)⇖))"],
assumption+,
apply (frule Ring.csrp_diff_in_vpr[of "Vr K v" "vp K v"
"x ⋅⇩r ((t⇘K⇙⇗(tna (v x))⇖)⇗‐K⇖)"], assumption+)

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⇘K⇙⇗- 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⇘K⇙⇗- 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⇘K⇙⇗- tna (v x)⇖)))" in aGroup.ag_mOp_closed[of "Vr K v"], assumption+)
apply (frule_tac x = "x ⋅⇩r (t⇘K⇙⇗- tna (v x)⇖) ± -⇩a (csrp_fn (Vr K v) (vp K v)
(pj (Vr K v) (vp K v) (x ⋅⇩r (t⇘K⇙⇗- 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 (frule npowf_mem[of "t" "1 + tna (v x)"], assumption+)
apply (simp add:Ring.ring_tOp_commute[of "K" _ "t⇘K⇙⇗(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⇘K⇙⇗- tna (v x)⇖"], assumption+)
apply (subst val_pos_mem_Vr[THEN sym, of v], assumption+)
apply (frule value_in_aug_inf[of "v" "x"], assumption+,
apply (frule val_nonzero_noninf[of "v" "x"], assumption+,

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 (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 (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⇘K⇙⇗(tna (v x) + (1 + int n))⇖" and y = k in
Ring.ring_tOp_commute[of "K"], assumption+) apply simp
apply (thin_tac "(t⇘K⇙⇗(tna (v x) + (1 + int n))⇖) ⋅⇩r k =
k ⋅⇩r (t⇘K⇙⇗(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⇘K⇙⇗(tna (v x) + (1 + int n))⇖" in Ring.ring_tOp_closed[of "K"], assumption+)
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⇘K⇙⇗(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 (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 (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 (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⇘K 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 (rule Vr_mem_f_mem[of v], assumption+)
apply (rule Ring.csrp_fn_mem, assumption+)
apply (subgoal_tac "x ⋅⇩r (t⇘K⇙⇗- tna (v x)⇖) ∈ carrier (Vr K v)")
apply (frule npowf_mem[of  "t" "- tna (v x)"], assumption+,
subst val_pos_mem_Vr[THEN sym, of v "x ⋅⇩r (t⇘K⇙⇗(- tna(v x))⇖)"],
assumption+,
rule Ring.ring_tOp_closed, assumption+)
simp add:val_exp[THEN sym, of  "v" "t" "- tna (v x)"])
apply (frule value_in_aug_inf[of  "v" "x"], assumption+,
apply (frule val_nonzero_noninf[of  "v" "x"], assumption+)

apply (frule psum_diff_mem_Kxa[of  "v" "t" "x" "n - 1"],
assumption+, erule conjE)
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 (thin_tac "x ± -⇩a psum⇘ K x v t⇙ (n - Suc 0) =
(t⇘K⇙⇗(tna (v x) + (1 + int (n - Suc 0)))⇖) ⋅⇩r k")
apply(simp add:Ring.ring_tOp_commute[of "K" "t⇘K⇙⇗(tna (v x) + (1 + int (n - Suc 0)))⇖"])
apply (thin_tac "t⇘K⇙⇗(tna (v x) + (1 + int (n - Suc 0)))⇖ ∈ carrier K",
thin_tac "t⇘K⇙⇗(- tna (v x) - int n)⇖ ∈ carrier K")
apply (rule Vr_mem_f_mem, assumption+)
apply (rule Ring.csrp_fn_mem, assumption+)
done

lemma (in Corps) etm_mem:"⟦valuation K v; t ∈ carrier K; v t = 1;
x ∈ carrier K; x ≠ 𝟬⟧ ⟹ etm⇘K v t x⇙ n ∈ carrier K"
apply (frule val1_neq_0[of v t], assumption+)
apply (cut_tac field_is_ring,
rule Ring.ring_tOp_closed[of "K"], assumption)
done

lemma (in Corps) psum_sum_etm:"⟦valuation K v; t ∈ carrier K; v t = 1;
x ∈ carrier K; x ≠ 𝟬⟧ ⟹
psum⇘K x v t⇙ n = nsum K (λj. (ecf⇘K v t x⇙ j)⋅⇩r (t⇘K⇙⇗(tna (v x) + (int j))⇖)) n"
apply (frule val1_neq_0[of v t], assumption+)
apply (induct_tac "n")
apply (rotate_tac -1, drule sym)
apply simp
apply (thin_tac "Σ⇩e K (λj. ecf⇘K v t x⇙ j ⋅⇩r t⇘K⇙⇗(tna (v x) + int j)⇖) n =
psum⇘ K x v t⇙ n")
done

lemma zabs_pos:"0 ≤ (abs (z::int))"

lemma abs_p_self_pos:"0 ≤ z + (abs (z::int))"

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⇘K v ⇙(partial_sum K x v t) x"
apply (cut_tac field_is_ring, frule Ring.ring_is_ag[of "K"])
apply (rule allI)
apply (subgoal_tac "∀n. (N + na (Abs (v x))) < n ⟶
psum⇘K 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⇘K⇙⇗(tna (v x) + (1 + int n))⇖) =
{xa ∈ carrier K. ant (tna (v x) + (1 + int n)) ≤ v xa}")
apply (erule conjE)

apply (subgoal_tac "(an N) ≤ v (psum⇘ K x v t⇙ n ± -⇩a x)")

apply (frule value_in_aug_inf[of v x], assumption+,
apply (frule val_nonzero_noninf[of v x], assumption+)
apply (frule val_surj_n_val[of v], blast)
apply (rule_tac x = "psum⇘K 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 (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⇘K 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"],
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 (cut_tac aneg_less[THEN sym, of "0" "Abs (v x)"], simp)
apply (frule val_nonzero_noninf[of v x], assumption+)
apply (simp only:ant_1[THEN sym], simp del:ant_1 add:a_zpz)
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
apply (simp only:add.assoc[THEN sym, of "1"])
apply (cut_tac ?a1 = z and ?b1 = "abs z" and ?c1 = "1 + int N" in
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
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 (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⇘R 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 ±⇘R⇙ -⇩a⇘R⇙ (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⇘R 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) ±⇘R⇙ -⇩a⇘R⇙ 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⇘R X K v d⇙ F = (λn. (ldeg_p R (Vr K v) X d (F n)))"

(** deg R (Vr K v) X (F n) ≤ 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⇘R 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) ≤ 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⇘R X K v d⇙ F) n ∈ carrier R"
apply (frule PolynRg.is_Ring)
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⇘R 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 (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 (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+,
apply (subst pol_coeff_def, rule allI, rule impI,
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⇘R⇙ (X^⇗R (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) ±⇘R⇙ (ldeg_p R (Vr K v) X d q) =
ldeg_p R (Vr K v) X d (p ±⇘R⇙ 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) ±⇘R⇙
(hdeg_p R (Vr K v) X (Suc d) q) = hdeg_p R (Vr K v) X (Suc d) (p ±⇘R⇙ 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⇘R⇙ (ldeg_p R (Vr K v) X d p) = ldeg_p R (Vr K v) X d (-⇩a⇘R⇙ p)"

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⇘R⇙ (hdeg_p R (Vr K v) X (Suc d) p) = hdeg_p R (Vr K v) X (Suc d) (-⇩a⇘R⇙ p)"

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 ±⇘R⇙ -⇩a⇘R⇙ (F m))⟧
⟹ P_mod R (Vr K v) X (vp K v⇗(Vr K v) (an N)⇖)
(((Pseql⇘R X K v d⇙ F) n) ±⇘R⇙ -⇩a⇘R⇙ ((Pseql⇘R 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 (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⇘R⇙ (F m)"], assumption+,
simp, rule aGroup.ag_mOp_closed, assumption, simp, simp,
apply (rule PCauchy_lTr[of v R X "F n ±⇘R⇙ -⇩a⇘R⇙ (F m)" "d" "N"],
assumption+,
rule aGroup.ag_pOp_closed[of "R" "F n" "-⇩a⇘R⇙ (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⇘R⇙ (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 ±⇘R⇙ -⇩a⇘R⇙ (F m))⟧
⟹ P_mod R (Vr K v) X (vp K v⇗(Vr K v) (an N)⇖)
(((Pseqh⇘R X K v d⇙ F) n) ±⇘R⇙ -⇩a⇘R⇙ ((Pseqh⇘R 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)

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⇘R⇙ (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 ±⇘R⇙ -⇩a⇘R⇙ (F m)" "d" "N"],
assumption+,
rule aGroup.ag_pOp_closed[of "R" "F n" "-⇩a⇘R⇙ (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⇘R⇙ (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⇘R X K v d⇙ F) n) ±⇘R⇙ ((Pseqh⇘R X K v d⇙ F) n)"
apply (frule PolynRg.is_Ring)
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⇘R⇙", simp,
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⇘v⇙ K;
PolynRg R (Vr K v) X; ∀n. f n ∈ carrier (Vr K v);
∀n. F n = (f n) ⋅⇩r⇘R⇙ (X^⇗R 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 ±⇘R⇙ -⇩a⇘R⇙ (F m))⟧ ⟹
∃b∈carrier (Vr K v). Plimit⇘ R X K v ⇙F (b ⋅⇩r⇘R⇙ (X^⇗R 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 (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⇘R⇙ (X^⇗R d⇖) ±⇘R⇙ -⇩a⇘R⇙ (f m) ⋅⇩r⇘R⇙ (X^⇗R 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⇘R⇙ (X^⇗R d⇖))", blast)

(******* b ∈ carrier (Vr K v) ***********)
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 (thin_tac "∀n. F n = f n ⋅⇩r⇘R⇙ X^⇗R d⇖")
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⇘R⇙ (X^⇗R d⇖) ±⇘R⇙ -⇩a⇘R⇙ (b ⋅⇩r⇘R⇙ (X^⇗R 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 (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^⇗R 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 ±⇘R⇙ -⇩a⇘R⇙ b" and
p = "(f n ±⇘R⇙ -⇩a⇘R⇙ b) ⋅⇩r⇘R⇙ (X^⇗R d⇖)" in
PolynRg.monomial_P_mod_mod[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_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 ±⇘R⇙ -⇩a⇘R⇙ b" and
p1 = "(f n ±⇘R⇙ -⇩a⇘R⇙ b) ⋅⇩r⇘R⇙ (X^⇗R 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 (case_tac "b = 𝟬⇘K⇙", simp add:Vr_0_f_0[THEN sym],
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⇘R⇙ X^⇗R 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⇘R⇙ X^⇗R d⇖ ±⇘R⇙ -⇩a⇘R⇙ f m ⋅⇩r⇘R⇙ X^⇗R 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],

apply (rule allI)
apply (rotate_tac -3, frule_tac x = N in spec)

apply (thin_tac "∀n. F n = f n ⋅⇩r⇘R⇙ X^⇗R 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⇘R⇙ (X^⇗R 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"],
frule_tac x = "f m" in aGroup.ag_mOp_closed[of "Vr K v"], assumption+,
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⇘R⇙ X^⇗R 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)
frule_tac y1 = "f n" and z1 = "-⇩a⇘R⇙ f m" in Ring.ring_distrib2[
THEN sym, of R "X^⇗R d⇖"], assumption+, simp,
thin_tac "f n ⋅⇩r⇘R⇙ X^⇗R d⇖ ±⇘R⇙ (-⇩a⇘R⇙ f m) ⋅⇩r⇘R⇙ X^⇗R d⇖ =
(f n ±⇘R⇙ -⇩a⇘R⇙ f m) ⋅⇩r⇘R⇙ X^⇗R 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"],
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⇘R⇙ (X^⇗R d⇖); c ∈ carrier (Vr K v);
Plimit⇘ R X K v ⇙F (c ⋅⇩r⇘R⇙ (X^⇗R d⇖))⟧  ⟹ lim⇘ K v ⇙f c"
apply (frule PolynRg.is_Ring,
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⇘R⇙ X^⇗R 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+)
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^⇗R d⇖" _ "-⇩a⇘R⇙ c"],
frule Ring.ring_is_ag[of "Vr K v"],
frule aGroup.ag_mOp_closed[of "Vr K v" "c"], assumption+)
frule_tac x = "f n" in aGroup.ag_pOp_closed[of "Vr K v" _
"-⇩a⇘(Vr 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)⇙ -⇩a⇘(Vr K v)⇙ c" and p1 = "(f n ±⇘(Vr K v)⇙ -⇩a⇘(Vr K v)⇙ c)
⋅⇩r⇘R⇙ (X^⇗R d⇖)" in PolynRg.monomial_P_mod_mod[THEN sym, of R "Vr K v"
X _ _ _ d], assumption+, simp, simp)
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⇘R⇙ (X^⇗R d⇖); b ∈ carrier (Vr K v); c ∈ carrier (Vr K v);
Plimit⇘ R X K v ⇙F (b ⋅⇩r⇘R⇙ (X^⇗R d⇖)); Plimit⇘ R X K v ⇙F (c ⋅⇩r⇘R⇙ (X^⇗R d⇖))⟧ ⟹
b  ⋅⇩r⇘R⇙ (X^⇗R d⇖) = c ⋅⇩r⇘R⇙ (X^⇗R 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⇘R⇙", simp add:deg_def)
apply (rule contrapos_pp, simp+,
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,
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 (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))
(fst (s_cf R (Vr K v) X p),
λj. -⇩a⇘Vr 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⇘Vr K v⇙ snd (s_cf R (Vr K v) X p) j)"],
assumption+, simp,
thin_tac "-⇩a⇘R⇙ 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⇘Vr 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 ±⇘R⇙
polyn_expr R X (fst (s_cf R (Vr K v) X p))
(fst (s_cf R (Vr K v) X p), λj. -⇩a⇘Vr 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⇘Vr 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⇘Vr 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⇘Vr 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+,
rule PolynRg.add_cf_pol_coeff[of R "Vr K v" X], assumption+,
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⇘Vr K v⇙ snd (s_cf R (Vr K v) X p) j)))",
(**** we obtained snd (add_cf (Vr K v) f (fst (s_cf R (Vr K v) X p),
λj. -⇩a⇘Vr 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+,
apply (frule_tac I = "vp K v⇗ (Vr K v) (an (Suc n))⇖" and
x = "-⇩a⇘Vr K v⇙ snd (s_cf R (Vr K v) X p) (fst f)" in
Ring.ideal_inv1_closed[of "Vr K v"], assumption+)
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
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+)
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⇘R⇙")
apply (frule Plimit_deg[of v R X F 0 p], assumption+,
apply (case_tac "p = 𝟬⇘R⇙", 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+,
apply (subgoal_tac "p = 𝟬⇘R⇙", simp)
apply (thin_tac "p ≠ 𝟬⇘R⇙")
apply (rule contrapos_pp, simp+)

apply (frule n_val_valuation[of v])
apply (frule val_nonzero_z[of "n_val K v" "p"])
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 (frule val_pos_mem_Vr[THEN sym, of "v" "p"])
apply (frule val_pos_n_val_pos[of "v" "p"])
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 ±⇘R⇙ -⇩a⇘R⇙ 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 ±⇘R⇙ -⇩a⇘R⇙ 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⇘R⇙ ±⇘R⇙ -⇩a⇘R⇙ p)")
apply blast
apply simp
apply (frule Ring.ring_is_ag[of "R"])
apply (frule aGroup.ag_mOp_closed[of "R" "p"], assumption)
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" _ "-⇩a⇘(Vr 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" _ "-⇩a⇘(Vr K v)⇙ p"], assumption+)
apply (frule_tac n = "an (Suc n)" in n_value_x_1[of "v" _ "p"], simp)
apply assumption
apply simp

apply (fold pol_limit_def)
apply (case_tac "ad = ∞", simp)
apply simp apply (erule exE)
apply (frule Plimit_deg[of "v" "R" "X" "F" "na ad" "p"], assumption+)
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 (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) ±⇘R⇙ -⇩a⇘R⇙ (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 ±⇘R⇙ -⇩a⇘R⇙ 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⇘R⇙ 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 ±⇘R⇙ -⇩a⇘R⇙ 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⇘R⇙ p"],
assumption+)
apply (simp, rule aGroup.ag_mOp_closed, assumption+)
apply (frule PolynRg.deg_minus_eq1 [of ```