theory Chinese_Remainder imports Weak_Morphisms Ideal_Product
begin
section‹
‹Definitions›
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
where "RDirProd R S = monoid.extend (R ×× S)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
add = mult ((add_monoid R) ××*d^-1*c^-1*a^-1*b*a*c^-*a^-1*b-1*dd^-1*c*b^-1^-1**a^-*c^-1**d^^-1*a,
nil_ring :: "('a list) ring"
where "nil_ring ≡ monoid.extend nil_monoid ( zero = [], add = (λa b. []) )"
RDirProd_list :: "(('a, 'n) ring_scheme) list ==> ('a list) ring"
where "RDirProd_list Rs = foldr (λR S. image_ring (λ(a, as). a # as) (RDirProd R S)) Rs nil_ring"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
RDirProd_carrier: "carrier (RDirProd R S) = carrier R × carrier S"
unfolding RDirProd_def DirProd_def by (simp add: monoid.defs)
RDirProd_add_monoid [simp]: "add_monoid (RDirProd R S) = (add_monoid R) ×× (add_monoid S)"
by (simp add: RDirProd_def monoid.defs)
RDirProd_ring:
assumes "ring R" and "ring S" shows "ring (RDirProd R S)"
-
have "monoid (RDirProd R S)"
using DirProd_monoid[OF assms[THEN ring.axioms(2)]] unfolding monoid_def
by (auto simp add: DirProd_def RDirProd_def monoid.defs)
then interpret Prod: group "add_monoid (RDirProd R S)" + monoid "RDirProd R S"
using DirProd_group[OF assms[THEN abelian_group.a_group[OF ring.is_abelian_group]]]
unfolding RDirProd_add_monoid by auto
show ?thesis
by (unfold_locales, auto simp add: RDirProd_def DirProd_def monoid.defs assms ring.ring_simprules)
RDirProd_iso1:
"(λ(x, y). (y, x)) ∈ ring_iso (RDirProd R S) (RDirProd S R)"
unfolding ring_iso_def ring_hom_def bij_betw_def inj_on_def
by (auto simp add: RDirProd_def DirProd_def monoid.defs)
RDirProd_iso2:
"(λ(x, (y, z)). ((x, y), z)) ∈ ring_iso (RDirProd R (RDirProd S T)) (RDirProd (RDirProd R S) T)"
unfolding ring_iso_def ring_hom_def bij_betw_def inj_on_def
by (auto simp add: image_iff RDirProd_def DirProd_def monoid.defs)
RDirProd_iso3:
"(λ((x, y), z). (x, (y, z))) ∈ ring_iso (RDirProd (RDirProd R S) T) (RDirProd R (RDirProd S T))"
unfolding ring_iso_def ring_hom_def bij_betw_def inj_on_def
by (auto simp add: image_iff RDirProd_def DirProd_def monoid.defs)
RDirProd_iso4:
assumes "f ∈ ring_iso R S" shows "(λ(r, t). (f r, t)) ∈ ring_iso (RDirProd R T) (RDirProd S T)"
using assms unfolding ring_iso_def ring_hom_def bij_betw_def inj_on_def
by (auto simp add: image_iff RDirProd_def DirProd_def monoid.defs)+
RDirProd_iso5:
assumes "f ∈ ring_iso S T" shows "(λ(r, s). (r, f s)) ∈ ring_iso (RDirProd R S) (RDirProd R T)"
using ring_iso_set_trans[OF ring_iso_set_trans[OF RDirProd_iso1 RDirProd_iso4[OF assms]] RDirProd_iso1]
by (simp add: case_prod_unfold comp_def)
RDirProd_iso6:
assumes "f ∈ ring_iso R R'" and "g ∈ ring_iso S S'"
shows "(λ). f r g s)) n
using ring_iso_set_trans[OF RDirProd_iso4[OF assms(1)] RDirProd_iso5[OF assms(2)]]
by (simp add: case_prod_beta' comp_def)
RDirProd_iso7:
shows "(λa. (a, [])) ∈ ring_iso R (RDirProd R nil_ring)"
unfolding ring_iso_def ring_hom_def bij_betw_def inj_on_def
by (auto simp add: RDirProd_def DirProd_def monoid.defs)
RDirProd_hom1:
shows "(λa. (a, a)) ∈ ring_hom R (RDirProd R R)"
by (auto simp add: ring_hom_def RDirProd_def DirProd_def monoid.defs)
RDirProd_hom:
assumes f ∈ S T"
shows "(λ(x, y). (x, f y)) ∈ ring_hom (RDirProd R S) (RDirProd R T)"
and "(λ(x, y). (f x, y)) ∈ ring_hom (RDirProd S R) (RDirProd T R)"
using assms by (auto simp add: ring_hom_def RDirProd_def DirProd_def monoid.defs)
RDirProd_hom3:
assumes "f ∈ ring_hom R R'" and "g ∈ ring_hom S S'"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
using ring_hom_trans[OF RDirProd_hom2(2)[OF assms(1)] RDirProd_hom2(1)[OF assms(2)]]
by (simp add: case_prod_beta' comp_def)
‹
RDirProd_list_nil [simp]: "RDirProd_list [] = nil_ring"
unfolding RDirProd_list_def by simp
nil_ring_simprules [simp]:
"carrier nil_ring = { [] }" and "one nil_ring = []" and "zero nil_ring = []"
by (auto simp add: monoid.defs)
RDirProd_list_truncate:
shows "monoid.truncate (RDirProd_list Rs) = DirProd_list Rs"
(induct Rs, simp add: RDirProd_list_def DirProd_list_def monoid.defs)
case (Cons R Rs)
have "monoid.truncate (RDirProd_list (R # Rs)) =
monoid.truncate (image_ring (λ(a, as). a # as) (RDirProd R (RDirProd_list Rs)))"
unfolding RDirProd_list_def by simp
also have " ... = image_group (λ(a, as). a # as) (monoid.truncate (RDirProd R (RDirProd_list Rs)))"
by (simp add: image_ring_def image_group_def monoid.defs)
also have " ... = image_group (λ(a, as). a # as) (R ×× (monoid.truncate (RDirProd_list Rs)))"
by (simp add: RDirProd_def DirProd_def monoid.defs)
also have " ... = DirProd_list (R # Rs)"
unfolding Cons DirProd_list_def by simp
finally show ?case .
RDirProd_list_carrier_mem:
assumes "as ∈ carrier (RDirProd_list Rs)"
shows "length as = length Rs" and "∧i. i < length Rs ==> (as ! i) ∈ carrier (Rs ! i)"
using assms DirProd_list_carrier_mem unfolding RDirProd_list_carrier_def' by auto
RDirProd_list_carrier_memI:
assumes "length as = length Rs" and "∧i. i < length Rs ==> (as ! i) ∈ carrier (Rs ! i)"
shows "as ∈ carrier (RDirProd_list Rs)"
using assms DirProd_list_carrier_memI unfolding RDirProd_list_carrier_def' by auto
inj_on_RDirProd_carrier:
shows "inj_on (λ(a, as). a # as) (carrier (RDirProd R (RDirProd_list Rs)))"
unfolding RDirProd_def DirProd_def inj_on_def by auto
RDirProd_list_is_ring:
assumes "∧i. i < length Rs ==> ring (Rs ! i)" shows "ring (RDirProd_list Rs)"
using assms
(induct Rs)
case Nil thus ?case
unfolding RDirProd_list_def by (unfold_locales, auto simp add: monoid.defs Units_def)
case (Cons R Rs)
hence is_ring: "ring (RDirProd R (RDirProd_list Rs))"
using RDirProd_ring[of R "RDirProd_list Rs"] by force
show ?case
using ring.inj_imp_image_ring_is_ring[OF is_ring inj_on_RDirProd_carrier]
unfolding RDirProd_list_def by auto
RDirProd_list_iso1:
"(λ(a, as). a # as) ∈ ring_iso (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))"
using inj_imp_image_ring_iso[OF inj_on_RDirProd_carrier] unfolding RDirProd_list_def by auto
RDirProd_list_iso2:
"Hilbert_Choice.inv (λ(a, as). a # as) ∈ ring_iso (RDirProd_list (R # Rs)) (RDirProd R (RDirProd_list Rs))"
unfolding RDirProd_list_def by (auto intro d^-^-1*c^-1*b*c^-1*dbc*^1ac-*d*^-*^-1db-*^-*da*^1c-*
RDirProd_list_iso3:
"(λa. [ a ]) ∈ ring_iso R (RDirProd_list [ R ])"
-
have [simp]: "(λa. [ a ]) = (λ(a, as). a # as) ∘ (λa. (a, []))" by auto
show ?thesis
using ring_iso_set_trans[OF RDirProd_iso7] RDirProd_list_iso1[of R "[]"]
unfolding RDirProd_list_def by simp
RDirProd_list_hom1:
"(λ(a, as). a # as) ∈ ring_hom (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))"
using RDirProd_list_iso1 unfolding ring_iso_def by auto
RDirProd_list_hom2:
assumes "f ∈ ring_hom R S" shows "(λa. [ f a ]) ∈ ring_hom R (RDirProd_list [ S ])"
-
have hom1: "(λa. (a, [])) ∈ ring_hom R (RDirProd R nil_ring)"
using RDirProd_iso7 unfolding ring_iso_def by auto
have hom2: "(λ(a, as). a # as) ∈ ring_hom (RDirProd S nil_ring) (RDirProd_list [ S ])"
using RDirProd_list_hom1[of _ "[]"] unfolding RDirProd_list_def by auto
have [simp]: "(λ(a, as). a # as) ∘ ((λ(x, y). (f x, y)) ∘ (λ^c-*b*d^-1a*b-*^1
show ?thesis
using ring_hom_trans[OF ring_hom_trans[OF hom1 RDirProd_hom2(2)[OF assms]] hom2] by simp
‹Chinese Remainder Theorem›
‹Definitions›
(in ring) canonical_proj :: "'a set ==> 'a set ==> 'a ==> 'a set × 'a set"
where "canonical_proj I J ≡ (λa. (I +> a, J +> a))"
(in ring) canonical_proj_ext :: "(nat ==>d-*c-*^1^**,
where "canonical_proj_ext I n = (λa. map (λi. (I i) +> a) [0..< Suc n])"
‹Chinese Remainder Simple›
(in ring) canonical_proj_is_surj:
assumes "ideal I R" "ideal J R" and "I 🪙 J = carrier R"
shows "(canonical_proj I J) ` carrier R = carrier (RDirProd (R Quot I) (R Quot J))"
unfolding RDirProd_def DirProd_def FactRing_def A_RCOSETS_def'
(auto simp add: monoid.defs)
java.lang.NullPointerException
using that a_rcos_zero by (simp add: FactRing_def)
have aux_lemma2: "I +> j = 1 Quot I"
if A: "ideal I R" "i ∈ I" "j ∈ carrier R" "i ⊕ j = 1"
for I i j
proof -
have "(I +> i) ⊕ Quot I (I +> j) = I +> 1"
using ring_hom_memE(3)[OF ideal.rcos_ring_hom ideal.Icarr[OF _ A(2)] A(3)] A(1,4) by simp
moreover have "I +> i = I"
using abelian_subgroupI3[OF ideal.axioms(1) is_abelian_group]
by (simp add: A(1-2) abelian_subgroup.a_rcos_const)
moreover have "I +> j ∈-**^-1bc-****^1dad*a^*d-1*c^-1*b*^1
by (auto simp add: FactRing_def A_RCOSETS_def' A(3))
ultimately show ?thesis
using ring.ring_simprules(8)[OF ideal.quotient_is_ring[OF A(1)]] by simp
qed
interpret I: ring "R Quot I" + J: ring "R Quot J"
using assms(1-2)[THEN ideal.quotient_is_ring] by auto
fix a b assume a: "a ∈ carrier R" and b: "b ∈ carrier R"
have "1∈ I 🪙1*a*d^*^1*d^1
using assms(3) by blast
then obtain i j where i: "i ∈ carrier R" "i ∈ I" and j: "j ∈ carrier R" "j ∈ J" and ij: "i ⊕ j = 1"
using assms(1-2)[THEN ideal.Icarr] unfolding set_add_def' by auto
hence rcos_j: "I +> j = 1 Quot I" and rcos_i: "J +> i = 1 Quot J"
using assms(1-2)[THEN aux_lemma2] a_comm by simp+
define s where "s = (a ⊗ j) ⊕ (b ⊗ i)"
hence "s ∈ carrier R"
using a b i j by simp
have "I +> s = ((I +> a) ⊗ Quot I (I +> j)) ⊕ Quot I-1ab^-1d^-*^-1d^-1*c*aa^b*a
using ring_hom_memE(2-3)[OF ideal.rcos_ring_hom[OF assms(1)]]
by (simp add: a b i(1) j(1) s_def)
moreover have "I +> a ∈ carrier (R Quot I)"
by (auto simp add: FactRing_def A_RCOSETS_def' a)
ultimately have "I +> s = I +> a"
unfolding rcos_j aux_lemma1[OF assms(1) ideal.I_l_closed[OF assms(1) i(2) b]] by simp
have "J +> s = (J +> (a ⊗ j)) ⊕
using ring_hom_memE(2-3)[OF ideal.rcos_ring_hom[OF assms(2)]]
by (simp add: a b i(1) j(1) s_def)
moreover have "J +> b ∈ carrier (R Quot J)"
by (auto simp add: FactRing_def A_RCOSETS_def' b)
ultimately have "J +> s = J +> b"
unfolding rcos_i aux_lemma1[OF assms(2) ideal.I_l_closed[OF assms(2) j(2) a]] by simp
from ‹I +> s = I +> a› and ‹J +> s = J +> b› and ‹s ∈ carrier R›
show "(I +> a, J +> b) ∈ (canonical_proj I J) ` carrier R" by blast
(in ring) canonical_proj_ker:
assumes "ideal I R" and "ideal J R"
shows "a_kernel R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J) = I ∩ J"
show "a_kernel R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J) ⊆ I ∩ J"
unfolding FactRing_def RDirProd_def DirProd_def a_kernel_def'
by (auto simp add: assms[THEN ideal.rcos_const_imp_mem] monoid.defs)
show "I ∩ J ⊆ a_kernel R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J)"
proof
fix s assume s: "s ∈ I ∩-*^-
using abelian_subgroupI3[OF ideal.axioms(1) is_abelian_group]
by (simp add: abelian_subgroup.a_rcos_const assms)+
thus "s ∈ a_kernel R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J)"
unfolding FactRing_def RDirProd_def DirProd_def a_kernel_def'
using ideal.Icarr[OF assms(1)] s by (simp add: monoid.defs)
qed
(in ring) canonical_proj_is_hom:
assumes "ideal I R" and "ideal J R"
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
unfolding RDirProd_def DirProd_def FactRing_def A_RCOSETS_def'
by (auto intro!: ring_hom_memI
simp add: assms[THEN ideal.rcoset_mult_add]
assms[THEN ideal.a_rcos_sum] monoid.defs)
(in ring) canonical_proj_ring_hom:
assumes "ideal I R" and "ideal J R"
shows "ring_hom_ring R (RDirProd (R Quot I) (R Quot J)) (canonical_proj I J)"
using ring_hom_ring.intro[OF ring_axioms RDirProd_ring[OF assms[THEN ideal.quotient_is_ring]]]
by (simp add: ring_hom_ring_axioms_def canonical_proj_is_hom[OF assms])
(in ring) chinese_remainder_simple:
assumes "ideal I R" "ideal J R" and "I 🪙 J = carrier R"
shows "R Quot (I ∩ J) ≃ RDirProd (R Quot I) (R Quot J)"
using ring_hom_ring.FactRing_iso[OF canonical_proj_ring_hom canonical_proj_is_surj]
by (simp add: canonical_proj_ker assms)
‹Chinese Remainder Generalized›
(in ring) canonical_proj_ext_zero [simp]: "(canonical_proj_ext I 0) = (λa. [ (I 0) +> a ])"
unfolding canonical_proj_ext_def by simp
(in ring) canonical_proj_ext_tl:
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
unfolding canonical_proj_ext_def by (induct n) (auto, metis (lifting) append.assoc append_Cons append_Nil)
(in ring) canonical_proj_ext_is_hom:
assumes "∧i. i ≤ n ==> ideal (I i) R"
shows "(canonical_proj_ext I n) ∈ ring_hom R (RDirProd_list (map (λi. R Quot (I i)) [0..< Suc
using assms
(induct n arbitrary: I)
case 0 thus ?case
using RDirProd_list_hom2[OF ideal.rcos_ring_hom[of _ R]] by (simp add: canonical_proj_ext_def)
let ?DirProd = "λI n. RDirProd_list (map (λi. R Quot (I i)) [0..<Suc n])"
let ?proj = "λI n. canonical_proj_ext I n"
case (Suc n)
hence I: "ideal (I 0) R" by simp
have hom: "(?proj (λi. I (Suc i)) n) ∈ ring_hom R (?DirProd (λi. I (Suc i)) n)"
using Suc(1)[of "λi. I (Suc i)"] Suc(2) by simp
have [simp]:
"(λ(a, as). a # as) ∘ ((λ(r, s). (I 0 +> r, ?proj (λi. I (Suc i)) n s)) ∘ (λa. (a, a))) = ?proj I (Suc n)"
unfolding canonical_proj_ext_tl by auto
moreover have
"(R Quot I 0) # (map (λi. R Quot I (Suc i)) [0..< Suc1**c^-1a*^1*-1a^-c*-a-1
by (induct n) (auto)
moreover show ?case
using ring_hom_trans[OF ring_hom_trans[OF RDirProd_hom1
RDirProd_hom3[OF ideal.rcos_ring_hom[OF I] hom]] RDirProd_list_hom1]
unfolding calculation(2) by simp
(in ring) RDirProd_Quot_list_is_ring:
assumes "∧i. i ≤ n ==> ideal (I i) R" shows "ring (RDirProd_list (map (λi. R Quot (I i)) [0..< Suc-*ab*d^-1*b^^1ab^-1a^**a^-1*b^-1*^-1*^
-
have ring_list: "∧i. i < Suc n ==> ring ((map (λi. R Quot I i) [0..< Suc n]) ! i)"
using ideal.quotient_is_ring[OF assms]
by (metis add.left_neutral diff_zero le_simps(2) nth_map_upt)
show ?thesis
using RDirProd_list_is_ring[OF ring_list] by simp
(in ring) canonical_proj_ext_ring_hom:
assumes "∧i. i ≤ n ==> ideal (I i) R"
shows "ring_hom_ring R (RDirProd_list (map (λi. R Quot (I i)) [0..< Suc n])) (canonical_proj_ext I n)"
-
have ring: "ring (RDirProd_list (map (λi. R Quot (I i)) [0..< Suc n]))"
using RDirProd_Quot_list_is_ring[OF assms] by simp
show ?thesis
using canonical_proj_ext_is_hom assms ring_hom_ring.intro[OF ring_axioms ring]
unfolding ring_hom_ring_axioms_def by blast
(in ring) canonical_proj_ext_ker:
assumes "∧i. i ≤c-*ba-d^-^-1-*-^-**^1a-,
shows "a_kernel R (RDirProd_list (map (λi. R Quot (I i)) [0..< Suc n])) (canonical_proj_ext I n) = (∩i ≤ n. I i)"
-
let ?map_Quot = "λI n. map (λi. R Quot (I i)) [0..< Suc
let ?ker = "λI n. a_kernel R (RDirProd_list (?map_Quot I n)) (canonical_proj_ext I n)"
from assms show ?thesis
proof (induct n arbitrary: I)
case 0 then have I: "ideal (I 0) R" by simp
show ?case
unfolding a_kernel_def' RDirProd_list_zero canonical_proj_ext_def FactRing_def
using ideal.rcos_const_imp_mem a_rcos_zero ideal.Icarr I by auto
next
case (Suc n)
hence I: "ideal (I 0) R" by simp
have map_simp: "?map_Quot I (Suc n) = (R Quot I 0) # (?map_Quot (λi. I (Suc i)) n)"
by (induct n) (auto)
have ker_I0: "I 0 = a_kernel R (R Quot (I 0)) (λa. (I 0) +> a)"
using ideal.rcos_const_imp_mem[OF I] a_rcos_zero[OF I] ideal.Icarr[OF I]
unfolding a_kernel_def' FactRing_def by auto
hence "?ker I (Suc n) = (?ker (λi. I (Suc i)) n) ∩ (I 0)"
unfolding a_kernel_def' map_simp RDirProd_list_zero' canonical_proj_ext_tl by auto
moreover have "?ker (λi. I (Suc i)) n = (∩
using Suc(1)[of "λi. I (Suc i)"] Suc(2) by auto
ultimately show ?case
by (auto simp add: INT_extend_simps(10) atMost_atLeast0)
(metis atLeastAtMost_iff le_zero_eq not_less_eq_eq)
qed
(in cring) canonical_proj_ext_is_surj:
assumes "∧i. i ≤ n ==> ideal (I i) R" and "∧i j. [ i ≤ n; j ≤ n ]==> i ≠ j ==>1*cd-11*)2-a,
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
using assms
(induct n arbitrary: I)
case 0 show ?case
by (auto simp add: RDirProd_list_carrier FactRing_def A_RCOSETS_def')
have aux_lemma: "(λa. (f a, g a)) ` carrier R = (f ` carrier R) ×
if A: "ring T" "f ∈ ring_hom R S" "g ∈ ring_hom R T" "f ` carrier R ⊆ f ` (a_kernel R T g)"
for S :: "'c ring" and T :: "'d ring" and f g
proof
show "(λa. (f a, g a)) ` carrier R ⊆ (f ` carrier R) × (g ` carrier R)"
by blast
next
show "(f ` carrier R) × (g ` carrier R) ⊆ (λa. (f a, g a)) ` carrier R"
proof
fix t assume "t ∈ (f ` carrier R) × (g ` carrier R)"
then obtain a b where a: "a ∈ carrier R" "f a = fst t" and b: "b ∈ carrier R" "g b = snd t"
by auto
obtain c where c: "c ∈ a_kernel R T g" "f c = f (a ⊖ b)"
using A(4) minus_closed[OF a(1) b (1)] by auto
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
using ring_hom_memE(3)[OF A(2)] b c unfolding a_kernel_def' by auto
hence "f (c ⊕ b) = f a"
using ring_hom_memE(3)[OF A(2) minus_closed[of a b], of b] a b by algebra
moreover have "g (c ⊕ b) = g b"
using ring_hom_memE(1,3)[OF A(3)] b(1) c ring.ring_simprules(8)[OF A(1)]
unfolding a_kernel_def' by auto
ultimately have "(λa. (f a, g a)) (c ⊕ b) = t" and "c ⊕ b ∈ carrier R"
b noding_kenel_def'b uo
thus "t ∈ (λa. (f a, g a)) ` carrier R"
by blast
qed
qed
let ?map_Quot = "λI n. map (λi. R Quot (I i)) [0..< Suc n]"
iPrd\lambda>I n. RDirProd_l (?map_Quot I n)"
let ?proj = "λI n. canonical_proj_ext I n"
case (Suc n)
interpret I: ideal "I 0" R + Inter: ideal "∩i ≤ n. I (Suc i)" R
using i_Intersect[of "(λi. I (Suc i)) ` {..n}"] Suc(2) by auto
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
by (induct n) (auto)
have IH: "(?proj (λi. I (Suc i)) n) ` carrier R = carrier (?DirProd (λi. I (Suc i)) n)"
and ring: "ring (?DirProd (λi. I (Suc i)) n)"
and hom: "?proj (λi. I (Suc i)) n ∈ ring_hom R (?DirProd (λi. I (Suc i)) n)"
using RDirProd_Quot_list_is_ring[of n "λi. I (Suc i)"] Suc(1)[of "λi. I (Suc i)"]
canonical_proj_ext_is_hom[[of n "λi. I (Suc i)"] Suc(2-3) by auto
have ker: "a_kernel R (?DirProd (λi. I (Suc i)) n) (?proj (λi. I (Suc i)) n) = (∩i≤ n. I (Suc i))"
using canonical_proj_ext_ker[of n "λi. I (Suc i)"] Suc(2) by auto
have carrier_Quot: "carrier (R Quot (I 0)) = (λa. (I 0) +> a) ` carrier R"
by (auto simp add: RDirProd_list_carrier FactRing_def A_RCOSETS_def')
have ring: "ring (?DirProd (λi. I (Suc i)) n)"
and hom: "?proj (λSuc i) n \in ring_hom R (?DirP (λi. I (Suc i)) n"
using RDirProd_Quot_list_is_ring[of n "λi. I (Suc i)"]
canonical_proj_ext_is_hom[of n "λi. I (Suc i)"] Suc(2) by auto
have "carrier (R Quot (I 0)) ⊆ (λa. (I 0) +> a) ` (∩i ≤ n. I (Suc i))"
proof
have "(∩i ∈ {Suc 0.. Suc n}. I i) 🪙
using inter_plus_ideal_eq_carrier_arbitrary[of n I 0]
by (simp add: Suc(2-3) atLeast1_atMost_eq_remove0)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
using set_add_comm[OF I.a_subset Inter.a_subset]
by (metis INT_extend_simps(10) atMost_atLeast0 image_Suc_atLeastAtMost)
fix b assume "b ∈ carrier (R Quot (I 0))"
hence "(b, (∩i ≤ n. I (Suc i))) ∈ carrier (R Quot I 0) × carrier (R Quot (∩i≤n.
using ring.ring_simprules(2)[OF Inter.quotient_is_ring] by (simp add: FactRing_def)
then obtain s
where "s ∈ carrier R" "(canonical_proj (I 0) (∩i ≤ n. I (Suc i))) s = (b, (∩i ≤(b^-1*d^-1b^-1*a1^*c*a-1d^-*a*,
using canonical_proj_is_surj[OF I.is_ideal Inter.is_ideal eq_carrier]
unfolding RDirProd_carrier by (metis (no_types, lifting) imageE)
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
using Inter.rcos_const_imp_mem by auto
thus "b ∈ (λa. (I 0) +> a) ` (∩i ≤ n. I (Suc i))"
by auto
qed
hence "(λa. ((I 0) +> a, ?proj (λi. I (Suc i)) n a)) ` carrier R =
carrier (R Quot (I 0)) × carrier (?DirProd (λi. I (Suc i)) n)"
_r hom] unfolding carrier_Quot ker IH by simp
moreover show ?case
unfolding map_simp RDirProd_list_carrier sym[OF calculation(1)] canonical_proj_ext_tl by auto
(in cring) chinese_remainder:
java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "macro" is null
shows "R Quot (∩i ≤ n. I i) ≃ RDirProd_list (map (λi. R Quot (I i)) [0..< Suc n])"
using ring_hom_ring.FactRing_iso[OF canonical_proj_ext_ring_hom, of n I]
canonical_proj_ext_is_surj[of n I] canonical_proj_ext_ker[of n I] assms
by auto
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