# Theory Free_Abelian_Groups

```section‹Free Abelian Groups›

theory Free_Abelian_Groups
imports
Product_Groups FiniteProduct "HOL-Cardinals.Cardinal_Arithmetic"
"HOL-Library.Countable_Set" "HOL-Library.Poly_Mapping" "HOL-Library.Equipollence"

begin

(*Move? But where?*)
lemma eqpoll_Fpow:
assumes "infinite A" shows "Fpow A ≈ A"
unfolding eqpoll_iff_card_of_ordIso
by (metis assms card_of_Fpow_infinite)

lemma infinite_iff_card_of_countable: "⟦countable B; infinite B⟧ ⟹ infinite A ⟷ ( |B| ≤o |A| )"
unfolding infinite_iff_countable_subset card_of_ordLeq countable_def
by (force intro: card_of_ordLeqI ordLeq_transitive)

lemma iso_imp_eqpoll_carrier: "G ≅ H ⟹ carrier G ≈ carrier H"
by (auto simp: is_iso_def iso_def eqpoll_def)

subsection‹Generalised finite product›

definition
gfinprod :: "[('b, 'm) monoid_scheme, 'a ⇒ 'b, 'a set] ⇒ 'b"
where "gfinprod G f A =
(if finite {x ∈ A. f x ≠ 𝟭⇘G⇙} then finprod G f {x ∈ A. f x ≠ 𝟭⇘G⇙} else 𝟭⇘G⇙)"

context comm_monoid begin

lemma gfinprod_closed [simp]:
"f ∈ A → carrier G ⟹ gfinprod G f A ∈ carrier G"
unfolding gfinprod_def
by (auto simp: image_subset_iff_funcset intro: finprod_closed)

lemma gfinprod_cong:
"⟦A = B; f ∈ B → carrier G;
⋀i. i ∈ B =simp=> f i = g i⟧ ⟹ gfinprod G f A = gfinprod G g B"
unfolding gfinprod_def
by (auto simp: simp_implies_def cong: conj_cong intro: finprod_cong)

lemma gfinprod_eq_finprod [simp]: "⟦finite A; f ∈ A → carrier G⟧ ⟹ gfinprod G f A = finprod G f A"
by (auto simp: gfinprod_def intro: finprod_mono_neutral_cong_left)

lemma gfinprod_insert [simp]:
assumes "finite {x ∈ A. f x ≠ 𝟭⇘G⇙}" "f ∈ A → carrier G" "f i ∈ carrier G"
shows "gfinprod G f (insert i A) = (if i ∈ A then gfinprod G f A else f i ⊗ gfinprod G f A)"
proof -
have f: "f ∈ {x ∈ A. f x ≠ 𝟭} → carrier G"
using assms by (auto simp: image_subset_iff_funcset)
have "{x. x = i ∧ f x ≠ 𝟭 ∨ x ∈ A ∧ f x ≠ 𝟭} = (if f i = 𝟭 then {x ∈ A. f x ≠ 𝟭} else insert i {x ∈ A. f x ≠ 𝟭})"
by auto
then show ?thesis
using assms
unfolding gfinprod_def by (simp add: conj_disj_distribR insert_absorb f split: if_split_asm)
qed

lemma gfinprod_distrib:
assumes fin: "finite {x ∈ A. f x ≠ 𝟭⇘G⇙}" "finite {x ∈ A. g x ≠ 𝟭⇘G⇙}"
and "f ∈ A → carrier G" "g ∈ A → carrier G"
shows "gfinprod G (λi. f i ⊗ g i) A = gfinprod G f A ⊗ gfinprod G g A"
proof -
have "finite {x ∈ A. f x ⊗ g x ≠ 𝟭}"
by (auto intro: finite_subset [OF _ finite_UnI [OF fin]])
then have "gfinprod G (λi. f i ⊗ g i) A = gfinprod G (λi. f i ⊗ g i) ({i ∈ A. f i ≠ 𝟭⇘G⇙} ∪ {i ∈ A. g i ≠ 𝟭⇘G⇙})"
unfolding gfinprod_def
using assms by (force intro: finprod_mono_neutral_cong)
also have "… = gfinprod G f A ⊗ gfinprod G g A"
proof -
have "finprod G f ({i ∈ A. f i ≠ 𝟭⇘G⇙} ∪ {i ∈ A. g i ≠ 𝟭⇘G⇙}) = gfinprod G f A"
"finprod G g ({i ∈ A. f i ≠ 𝟭⇘G⇙} ∪ {i ∈ A. g i ≠ 𝟭⇘G⇙}) = gfinprod G g A"
using assms by (auto simp: gfinprod_def intro: finprod_mono_neutral_cong_right)
moreover have "(λi. f i ⊗ g i) ∈ {i ∈ A. f i ≠ 𝟭} ∪ {i ∈ A. g i ≠ 𝟭} → carrier G"
using assms by (force simp: image_subset_iff_funcset)
ultimately show ?thesis
using assms
apply simp
apply (subst finprod_multf, auto)
done
qed
finally show ?thesis .
qed

lemma gfinprod_mono_neutral_cong_left:
assumes "A ⊆ B"
and 1: "⋀i. i ∈ B - A ⟹ h i = 𝟭"
and gh: "⋀x. x ∈ A ⟹ g x = h x"
and h: "h ∈ B → carrier G"
shows "gfinprod G g A = gfinprod G h B"
proof (cases "finite {x ∈ B. h x ≠ 𝟭}")
case True
then have "finite {x ∈ A. h x ≠ 𝟭}"
apply (rule rev_finite_subset)
using ‹A ⊆ B› by auto
with True assms show ?thesis
apply (simp add: gfinprod_def cong: conj_cong)
apply (auto intro!: finprod_mono_neutral_cong_left)
done
next
case False
have "{x ∈ B. h x ≠ 𝟭} ⊆ {x ∈ A. h x ≠ 𝟭}"
using 1 by auto
with False have "infinite {x ∈ A. h x ≠ 𝟭}"
using infinite_super by blast
with False assms show ?thesis
by (simp add: gfinprod_def cong: conj_cong)
qed

lemma gfinprod_mono_neutral_cong_right:
assumes "A ⊆ B" "⋀i. i ∈ B - A ⟹ g i = 𝟭" "⋀x. x ∈ A ⟹ g x = h x" "g ∈ B → carrier G"
shows "gfinprod G g B = gfinprod G h A"
using assms  by (auto intro!: gfinprod_mono_neutral_cong_left [symmetric])

lemma gfinprod_mono_neutral_cong:
assumes [simp]: "finite B" "finite A"
and *: "⋀i. i ∈ B - A ⟹ h i = 𝟭" "⋀i. i ∈ A - B ⟹ g i = 𝟭"
and gh: "⋀x. x ∈ A ∩ B ⟹ g x = h x"
and g: "g ∈ A → carrier G"
and h: "h ∈ B → carrier G"
shows "gfinprod G g A = gfinprod G h B"
proof-
have "gfinprod G g A = gfinprod G g (A ∩ B)"
by (rule gfinprod_mono_neutral_cong_right) (use assms in auto)
also have "… = gfinprod G h (A ∩ B)"
by (rule gfinprod_cong) (use assms in auto)
also have "… = gfinprod G h B"
by (rule gfinprod_mono_neutral_cong_left) (use assms in auto)
finally show ?thesis .
qed

end

lemma (in comm_group) hom_group_sum:
assumes hom: "⋀i. i ∈ I ⟹ f i ∈ hom (A i) G" and grp: "⋀i. i ∈ I ⟹ group (A i)"
shows "(λx. gfinprod G (λi. (f i) (x i)) I) ∈ hom (sum_group I A) G"
unfolding hom_def
proof (intro CollectI conjI ballI)
show "(λx. gfinprod G (λi. f i (x i)) I) ∈ carrier (sum_group I A) → carrier G"
using assms
by (force simp: hom_def carrier_sum_group intro: gfinprod_closed simp flip: image_subset_iff_funcset)
next
fix x y
assume x: "x ∈ carrier (sum_group I A)" and y: "y ∈ carrier (sum_group I A)"
then have finx: "finite {i ∈ I. x i ≠ 𝟭⇘A i⇙}" and finy: "finite {i ∈ I. y i ≠ 𝟭⇘A i⇙}"
using assms by (auto simp: carrier_sum_group)
have finfx: "finite {i ∈ I. f i (x i) ≠ 𝟭}"
using assms by (auto simp: is_group hom_one [OF hom] intro: finite_subset [OF _ finx])
have finfy: "finite {i ∈ I. f i (y i) ≠ 𝟭}"
using assms by (auto simp: is_group hom_one [OF hom] intro: finite_subset [OF _ finy])
have carr: "f i (x i) ∈ carrier G"  "f i (y i) ∈ carrier G" if "i ∈ I" for i
using hom_carrier [OF hom] that x y assms
have lam: "(λi. f i ( x i ⊗⇘A i⇙ y i)) ∈ I → carrier G"
using x y assms by (auto simp: hom_def carrier_sum_group PiE_def Pi_def)
have lam': "(λi. f i (if i ∈ I then x i ⊗⇘A i⇙ y i else undefined)) ∈ I → carrier G"
with lam x y assms
show "gfinprod G (λi. f i ((x ⊗⇘sum_group I A⇙ y) i)) I
= gfinprod G (λi. f i (x i)) I ⊗ gfinprod G (λi. f i (y i)) I"
by (simp add: carrier_sum_group PiE_def Pi_def hom_mult [OF hom]
gfinprod_distrib finfx finfy carr cong: gfinprod_cong)
qed

subsection‹Free Abelian groups on a set, using the "frag" type constructor.          ›

definition free_Abelian_group :: "'a set ⇒ ('a ⇒⇩0 int) monoid"
where "free_Abelian_group S = ⦇carrier = {c. Poly_Mapping.keys c ⊆ S}, monoid.mult = (+), one  = 0⦈"

lemma group_free_Abelian_group [simp]: "group (free_Abelian_group S)"
proof -
have "⋀x. Poly_Mapping.keys x ⊆ S ⟹ x ∈ Units (free_Abelian_group S)"
unfolding free_Abelian_group_def Units_def
then show ?thesis
unfolding free_Abelian_group_def
by unfold_locales (auto simp: dest: subsetD [OF keys_add])
qed

lemma carrier_free_Abelian_group_iff [simp]:
shows "x ∈ carrier (free_Abelian_group S) ⟷ Poly_Mapping.keys x ⊆ S"
by (auto simp: free_Abelian_group_def)

lemma one_free_Abelian_group [simp]: "𝟭⇘free_Abelian_group S⇙ = 0"
by (auto simp: free_Abelian_group_def)

lemma mult_free_Abelian_group [simp]: "x ⊗⇘free_Abelian_group S⇙ y = x + y"
by (auto simp: free_Abelian_group_def)

lemma inv_free_Abelian_group [simp]: "Poly_Mapping.keys x ⊆ S ⟹ inv⇘free_Abelian_group S⇙ x = -x"
by (rule group.inv_equality [OF group_free_Abelian_group]) auto

lemma abelian_free_Abelian_group: "comm_group(free_Abelian_group S)"
apply (rule group.group_comm_groupI [OF group_free_Abelian_group])

lemma pow_free_Abelian_group [simp]:
fixes n::nat
shows "Group.pow (free_Abelian_group S) x n = frag_cmul (int n) x"
by (induction n) (auto simp: nat_pow_def free_Abelian_group_def frag_cmul_distrib)

lemma int_pow_free_Abelian_group [simp]:
fixes n::int
assumes "Poly_Mapping.keys x ⊆ S"
shows "Group.pow (free_Abelian_group S) x n = frag_cmul n x"
proof (induction n)
case (nonneg n)
then show ?case
next
case (neg n)
have "x [^]⇘free_Abelian_group S⇙ - int (Suc n)
= inv⇘free_Abelian_group S⇙ (x [^]⇘free_Abelian_group S⇙ int (Suc n))"
by (rule group.int_pow_neg [OF group_free_Abelian_group]) (use assms in ‹simp add: free_Abelian_group_def›)
also have "… = frag_cmul (- int (Suc n)) x"
by (metis assms inv_free_Abelian_group pow_free_Abelian_group int_pow_int minus_frag_cmul
order_trans keys_cmul)
finally show ?case .
qed

lemma frag_of_in_free_Abelian_group [simp]:
"frag_of x ∈ carrier(free_Abelian_group S) ⟷ x ∈ S"
by simp

lemma free_Abelian_group_induct:
assumes major: "Poly_Mapping.keys x ⊆ S"
and minor: "P(0)"
"⋀x y. ⟦Poly_Mapping.keys x ⊆ S; Poly_Mapping.keys y ⊆ S; P x; P y⟧ ⟹ P(x-y)"
"⋀a. a ∈ S ⟹ P(frag_of a)"
shows "P x"
proof -
have "Poly_Mapping.keys x ⊆ S ∧ P x"
using major
proof (induction x rule: frag_induction)
case (diff a b)
then show ?case
by (meson Un_least minor(2) order.trans keys_diff)
qed (auto intro: minor)
then show ?thesis ..
qed

lemma sum_closed_free_Abelian_group:
"(⋀i. i ∈ I ⟹ x i ∈ carrier (free_Abelian_group S)) ⟹ sum x I ∈ carrier (free_Abelian_group S)"
apply (induction I rule: infinite_finite_induct, auto)
by (metis (no_types, opaque_lifting) UnE subsetCE keys_add)

lemma (in comm_group) free_Abelian_group_universal:
fixes f :: "'c ⇒ 'a"
assumes "f ` S ⊆ carrier G"
obtains h where "h ∈ hom (free_Abelian_group S) G" "⋀x. x ∈ S ⟹ h(frag_of x) = f x"
proof
have fin: "Poly_Mapping.keys u ⊆ S ⟹ finite {x ∈ S. f x [^] poly_mapping.lookup u x ≠ 𝟭}" for u :: "'c ⇒⇩0 int"
apply (rule finite_subset [OF _ finite_keys [of u]])
unfolding keys.rep_eq by force
define h :: "('c ⇒⇩0 int) ⇒ 'a"
where "h ≡ λx. gfinprod G (λa. f a [^] poly_mapping.lookup x a) S"
show "h ∈ hom (free_Abelian_group S) G"
proof (rule homI)
fix x y
assume xy: "x ∈ carrier (free_Abelian_group S)" "y ∈ carrier (free_Abelian_group S)"
then show "h (x ⊗⇘free_Abelian_group S⇙ y) = h x ⊗ h y"
using assms unfolding h_def free_Abelian_group_def
qed (use assms in ‹force simp: free_Abelian_group_def h_def intro: gfinprod_closed›)
show "h(frag_of x) = f x" if "x ∈ S" for x
proof -
have fin: "(λa. f x [^] (1::int)) ∈ {x} → carrier G" "f x [^] (1::int) ∈ carrier G"
using assms that by force+
show ?thesis
by (cases " f x [^] (1::int) = 𝟭") (use assms that in ‹auto simp: h_def gfinprod_def finprod_singleton›)
qed
qed

lemma eqpoll_free_Abelian_group_infinite:
assumes "infinite A" shows "carrier(free_Abelian_group A) ≈ A"
proof (rule lepoll_antisym)
have "carrier (free_Abelian_group A) ≲ {f::'a⇒int. f ` A ⊆ UNIV ∧ {x. f x ≠ 0} ⊆ A ∧ finite {x. f x ≠ 0}}"
unfolding lepoll_def
by (rule_tac x="Poly_Mapping.lookup" in exI) (auto simp: poly_mapping_eqI lookup_not_eq_zero_eq_in_keys inj_onI)
also have "… ≲ Fpow (A × (UNIV::int set))"
by (rule lepoll_restricted_funspace)
also have "… ≈ A × (UNIV::int set)"
proof (rule eqpoll_Fpow)
show "infinite (A × (UNIV::int set))"
using assms finite_cartesian_productD1 by fastforce
qed
also have "… ≈ A"
unfolding eqpoll_iff_card_of_ordIso
proof -
have "|A × (UNIV::int set)| <=o |A|"
by (simp add: assms card_of_Times_ordLeq_infinite flip: infinite_iff_card_of_countable)
moreover have "|A| ≤o |A × (UNIV::int set)|"
by simp
ultimately have "|A| *c |(UNIV::int set)| =o |A|"
then show "|A × (UNIV::int set)| =o |A|"
by (metis Times_cprod ordIso_transitive)
qed
finally show "carrier (free_Abelian_group A) ≲ A" .
have "inj_on frag_of A"
moreover have "frag_of ` A ⊆ carrier (free_Abelian_group A)"
ultimately show "A ≲ carrier (free_Abelian_group A)"
by (force simp: lepoll_def)
qed

proposition (in comm_group) eqpoll_homomorphisms_from_free_Abelian_group:
"{f. f ∈ extensional (carrier(free_Abelian_group S)) ∧ f ∈ hom (free_Abelian_group S) G}
≈ (S →⇩E carrier G)"  (is "?lhs ≈ ?rhs")
unfolding eqpoll_def bij_betw_def
proof (intro exI conjI)
let ?f = "λf. restrict (f ∘ frag_of) S"
show "inj_on ?f ?lhs"
proof (clarsimp simp: inj_on_def)
fix g h
assume
g: "g ∈ extensional (carrier (free_Abelian_group S))" "g ∈ hom (free_Abelian_group S) G"
and h: "h ∈ extensional (carrier (free_Abelian_group S))" "h ∈ hom (free_Abelian_group S) G"
and eq: "restrict (g ∘ frag_of) S = restrict (h ∘ frag_of) S"
have 0: "0 ∈ carrier (free_Abelian_group S)"
by simp
interpret hom_g: group_hom "free_Abelian_group S" G g
using g by (auto simp: group_hom_def group_hom_axioms_def is_group)
interpret hom_h: group_hom "free_Abelian_group S" G h
using h by (auto simp: group_hom_def group_hom_axioms_def is_group)
have "Poly_Mapping.keys c ⊆ S ⟹ Poly_Mapping.keys c ⊆ S ∧ g c = h c" for c
proof (induction c rule: frag_induction)
case zero
show ?case
using hom_g.hom_one hom_h.hom_one by auto
next
case (one x)
then show ?case
using eq by (simp add: fun_eq_iff) (metis comp_def)
next
case (diff a b)
then show ?case
using hom_g.hom_mult hom_h.hom_mult hom_g.hom_inv hom_h.hom_inv
apply (auto simp: dest: subsetD [OF keys_diff])
qed
then show "g = h"
by (meson g h carrier_free_Abelian_group_iff extensionalityI)
qed
have "f ∈ (λf. restrict (f ∘ frag_of) S) `
{f ∈ extensional (carrier (free_Abelian_group S)). f ∈ hom (free_Abelian_group S) G}"
if f: "f ∈ S →⇩E carrier G"
for f :: "'c ⇒ 'a"
proof -
obtain h where h: "h ∈ hom (free_Abelian_group S) G" "⋀x. x ∈ S ⟹ h(frag_of x) = f x"
proof (rule free_Abelian_group_universal)
show "f ` S ⊆ carrier G"
using f by blast
qed auto
let ?h = "restrict h (carrier (free_Abelian_group S))"
show ?thesis
proof
show "f = restrict (?h ∘ frag_of) S"
using f by (force simp: h)
show "?h ∈ {f ∈ extensional (carrier (free_Abelian_group S)). f ∈ hom (free_Abelian_group S) G}"
using h by (auto simp: hom_def dest!: subsetD [OF keys_add])
qed
qed
then show "?f ` ?lhs = S →⇩E carrier G"
by (auto simp: hom_def Ball_def Pi_def)
qed

lemma hom_frag_minus:
assumes "h ∈ hom (free_Abelian_group S) (free_Abelian_group T)" "Poly_Mapping.keys a ⊆ S"
shows "h (-a) = - (h a)"
proof -
have "Poly_Mapping.keys (h a) ⊆ T"
by (meson assms carrier_free_Abelian_group_iff hom_in_carrier)
then show ?thesis
by (metis (no_types) assms carrier_free_Abelian_group_iff group_free_Abelian_group group_hom.hom_inv group_hom_axioms_def group_hom_def inv_free_Abelian_group)
qed

assumes "h ∈ hom (free_Abelian_group S) (free_Abelian_group T)" "Poly_Mapping.keys a ⊆ S" "Poly_Mapping.keys b ⊆ S"
shows "h (a+b) = h a + h b"
proof -
have "Poly_Mapping.keys (h a) ⊆ T"
by (meson assms carrier_free_Abelian_group_iff hom_in_carrier)
moreover
have "Poly_Mapping.keys (h b) ⊆ T"
by (meson assms carrier_free_Abelian_group_iff hom_in_carrier)
ultimately show ?thesis
using assms hom_mult by fastforce
qed

lemma hom_frag_diff:
assumes "h ∈ hom (free_Abelian_group S) (free_Abelian_group T)" "Poly_Mapping.keys a ⊆ S" "Poly_Mapping.keys b ⊆ S"
shows "h (a-b) = h a - h b"

proposition isomorphic_free_Abelian_groups:
"free_Abelian_group S ≅ free_Abelian_group T ⟷ S ≈ T"  (is "(?FS ≅ ?FT) = ?rhs")
proof
interpret S: group "?FS"
by simp
interpret T: group "?FT"
by simp
interpret G2: comm_group "integer_mod_group 2"
by (rule abelian_integer_mod_group)
let ?Two = "{0..<2::int}"
have [simp]: "¬ ?Two ⊆ {a}" for a
assume L: "?FS ≅ ?FT"
let ?HS = "{h ∈ extensional (carrier ?FS). h ∈ hom ?FS (integer_mod_group 2)}"
let ?HT = "{h ∈ extensional (carrier ?FT). h ∈ hom ?FT (integer_mod_group 2)}"
have "S →⇩E ?Two ≈ ?HS"
apply (rule eqpoll_sym)
using G2.eqpoll_homomorphisms_from_free_Abelian_group by (simp add: carrier_integer_mod_group)
also have "… ≈ ?HT"
proof -
obtain f g where "group_isomorphisms ?FS ?FT f g"
using L S.iso_iff_group_isomorphisms by (force simp: is_iso_def)
then have f: "f ∈ hom ?FS ?FT"
and g: "g ∈ hom ?FT ?FS"
and gf: "∀x ∈ carrier ?FS. g(f x) = x"
and fg: "∀y ∈ carrier ?FT. f(g y) = y"
by (auto simp: group_isomorphisms_def)
let ?f = "λh. restrict (h ∘ g) (carrier ?FT)"
let ?g = "λh. restrict (h ∘ f) (carrier ?FS)"
show ?thesis
proof (rule lepoll_antisym)
show "?HS ≲ ?HT"
unfolding lepoll_def
proof (intro exI conjI)
show "inj_on ?f ?HS"
apply (rule inj_on_inverseI [where g = ?g])
using hom_in_carrier [OF f]
by (auto simp: gf fun_eq_iff carrier_integer_mod_group Ball_def Pi_def extensional_def)
show "?f ` ?HS ⊆ ?HT"
proof clarsimp
fix h
assume h: "h ∈ hom ?FS (integer_mod_group 2)"
have "h ∘ g ∈ hom ?FT (integer_mod_group 2)"
by (rule hom_compose [OF g h])
moreover have "restrict (h ∘ g) (carrier ?FT) x = (h ∘ g) x" if "x ∈ carrier ?FT" for x
using g that by (simp add: hom_def)
ultimately show "restrict (h ∘ g) (carrier ?FT) ∈ hom ?FT (integer_mod_group 2)"
using T.hom_restrict by fastforce
qed
qed
next
show "?HT ≲ ?HS"
unfolding lepoll_def
proof (intro exI conjI)
show "inj_on ?g ?HT"
apply (rule inj_on_inverseI [where g = ?f])
using hom_in_carrier [OF g]
by (auto simp: fg fun_eq_iff carrier_integer_mod_group Ball_def Pi_def extensional_def)
show "?g ` ?HT ⊆ ?HS"
proof clarsimp
fix k
assume k: "k ∈ hom ?FT (integer_mod_group 2)"
have "k ∘ f ∈ hom ?FS (integer_mod_group 2)"
by (rule hom_compose [OF f k])
moreover have "restrict (k ∘ f) (carrier ?FS) x = (k ∘ f) x" if "x ∈ carrier ?FS" for x
using f that by (simp add: hom_def)
ultimately show "restrict (k ∘ f) (carrier ?FS) ∈ hom ?FS (integer_mod_group 2)"
using S.hom_restrict by fastforce
qed
qed
qed
qed
also have "… ≈ T →⇩E ?Two"
using G2.eqpoll_homomorphisms_from_free_Abelian_group by (simp add: carrier_integer_mod_group)
finally have *: "S →⇩E ?Two ≈ T →⇩E ?Two" .
then have "finite (S →⇩E ?Two) ⟷ finite (T →⇩E ?Two)"
by (rule eqpoll_finite_iff)
then have "finite S ⟷ finite T"
by (auto simp: finite_funcset_iff)
then consider "finite S" "finite T" | "~ finite S" "~ finite T"
by blast
then show ?rhs
proof cases
case 1
with * have "2 ^ card S = (2::nat) ^ card T"
by (simp add: card_PiE finite_PiE eqpoll_iff_card)
then have "card S = card T"
by auto
then show ?thesis
using eqpoll_iff_card 1 by blast
next
case 2
have "carrier (free_Abelian_group S) ≈ carrier (free_Abelian_group T)"
using L by (simp add: iso_imp_eqpoll_carrier)
then show ?thesis
using 2 eqpoll_free_Abelian_group_infinite eqpoll_sym eqpoll_trans by metis
qed
next
assume ?rhs
then obtain f g where f: "⋀x. x ∈ S ⟹ f x ∈ T ∧ g(f x) = x"
and g: "⋀y. y ∈ T ⟹ g y ∈ S ∧ f(g y) = y"
using eqpoll_iff_bijections by metis
interpret S: comm_group "?FS"
interpret T: comm_group "?FT"
have "(frag_of ∘ f) ` S ⊆ carrier (free_Abelian_group T)"
using f by auto
then obtain h where h: "h ∈ hom (free_Abelian_group S) (free_Abelian_group T)"
and h_frag: "⋀x. x ∈ S ⟹ h (frag_of x) = (frag_of ∘ f) x"
using T.free_Abelian_group_universal [of "frag_of ∘ f" S] by blast
interpret hhom: group_hom "free_Abelian_group S" "free_Abelian_group T" h
by (simp add: h group_hom_axioms_def group_hom_def)
have "(frag_of ∘ g) ` T ⊆ carrier (free_Abelian_group S)"
using g by auto
then obtain k where k: "k ∈ hom (free_Abelian_group T) (free_Abelian_group S)"
and k_frag: "⋀x. x ∈ T ⟹ k (frag_of x) = (frag_of ∘ g) x"
using S.free_Abelian_group_universal [of "frag_of ∘ g" T] by blast
interpret khom: group_hom "free_Abelian_group T" "free_Abelian_group S" k
by (simp add: k group_hom_axioms_def group_hom_def)
have kh: "Poly_Mapping.keys x ⊆ S ⟹ Poly_Mapping.keys x ⊆ S ∧ k (h x) = x" for x
proof (induction rule: frag_induction)
case zero
then show ?case
apply auto
by (metis group_free_Abelian_group h hom_one k one_free_Abelian_group)
next
case (one x)
then show ?case
by (auto simp: h_frag k_frag f)
next
case (diff a b)
with keys_diff have "Poly_Mapping.keys (a - b) ⊆ S"
by (metis Un_least order_trans)
with diff hhom.hom_closed show ?case
by (simp add: hom_frag_diff [OF h] hom_frag_diff [OF k])
qed
have hk: "Poly_Mapping.keys y ⊆ T ⟹ Poly_Mapping.keys y ⊆ T ∧ h (k y) = y" for y
proof (induction rule: frag_induction)
case zero
then show ?case
apply auto
by (metis group_free_Abelian_group h hom_one k one_free_Abelian_group)
next
case (one y)
then show ?case
by (auto simp: h_frag k_frag g)
next
case (diff a b)
with keys_diff have "Poly_Mapping.keys (a - b) ⊆ T"
by (metis Un_least order_trans)
with diff khom.hom_closed show ?case
by (simp add: hom_frag_diff [OF h] hom_frag_diff [OF k])
qed
have "h ∈ iso ?FS ?FT"
unfolding iso_def bij_betw_iff_bijections mem_Collect_eq
proof (intro conjI exI ballI h)
fix x
assume x: "x ∈ carrier (free_Abelian_group S)"
show "h x ∈ carrier (free_Abelian_group T)"
by (meson x h hom_in_carrier)
show "k (h x) = x"
using x by (simp add: kh)
next
fix y
assume y: "y ∈ carrier (free_Abelian_group T)"
show "k y ∈ carrier (free_Abelian_group S)"
by (meson y k hom_in_carrier)
show "h (k y) = y"
using y by (simp add: hk)
qed
then show "?FS ≅ ?FT"
by (auto simp: is_iso_def)
qed

lemma isomorphic_group_integer_free_Abelian_group_singleton:
"integer_group ≅ free_Abelian_group {x}"
proof -
have "(λn. frag_cmul n (frag_of x)) ∈ iso integer_group (free_Abelian_group {x})"
proof (rule isoI [OF homI])
show "bij_betw (λn. frag_cmul n (frag_of x)) (carrier integer_group) (carrier (free_Abelian_group {x}))"
apply (rule bij_betwI [where g = "λy. Poly_Mapping.lookup y x"])
by (auto simp: integer_group_def in_keys_iff intro!: poly_mapping_eqI)
qed (auto simp: frag_cmul_distrib)
then show ?thesis
unfolding is_iso_def
by blast
qed

lemma group_hom_free_Abelian_groups_id:
"id ∈ hom (free_Abelian_group S) (free_Abelian_group T) ⟷ S ⊆ T"
proof -
have "x ∈ T" if ST: "⋀c:: 'a ⇒⇩0 int. Poly_Mapping.keys c ⊆ S ⟶ Poly_Mapping.keys c ⊆ T" and "x ∈ S" for x
using ST [of "frag_of x"] ‹x ∈ S› by simp
then show ?thesis
by (auto simp: hom_def free_Abelian_group_def Pi_def)
qed

proposition iso_free_Abelian_group_sum:
assumes "pairwise (λi j. disjnt (S i) (S j)) I"
shows "(λf. sum' f I) ∈ iso (sum_group I (λi. free_Abelian_group(S i))) (free_Abelian_group (⋃(S ` I)))"
(is "?h ∈ iso ?G ?H")
proof (rule isoI)
show hom: "?h ∈ hom ?G ?H"
proof (rule homI)
show "?h c ∈ carrier ?H" if "c ∈ carrier ?G" for c
using that
apply (rule order_trans [OF keys_sum])
apply (auto simp: free_Abelian_group_def)
done
show "?h (x ⊗⇘?G⇙ y) = ?h x ⊗⇘?H⇙ ?h y"
if "x ∈ carrier ?G" "y ∈ carrier ?G"  for x y
using that by (simp add: sum.finite_Collect_op carrier_sum_group sum.distrib')
qed
interpret GH: group_hom "?G" "?H" "?h"
using hom by (simp add: group_hom_def group_hom_axioms_def)
show "bij_betw ?h (carrier ?G) (carrier ?H)"
unfolding bij_betw_def
proof (intro conjI subset_antisym)
show "?h ` carrier ?G ⊆ carrier ?H"
apply (clarsimp simp: sum.G_def carrier_sum_group simp del: carrier_free_Abelian_group_iff)
by (force simp: PiE_def Pi_iff intro!: sum_closed_free_Abelian_group)
have *: "poly_mapping.lookup (Abs_poly_mapping (λj. if j ∈ S i then poly_mapping.lookup x j else 0)) k
= (if k ∈ S i then poly_mapping.lookup x k else 0)" if "i ∈ I" for i k and x :: "'b ⇒⇩0 int"
using that by (auto simp: conj_commute cong: conj_cong)
have eq: "Abs_poly_mapping (λj. if j ∈ S i then poly_mapping.lookup x j else 0) = 0
⟷ (∀c ∈ S i. poly_mapping.lookup x c = 0)" if "i ∈ I" for i and x :: "'b ⇒⇩0 int"
apply (auto simp: poly_mapping_eq_iff fun_eq_iff)
apply (simp add: * Abs_poly_mapping_inverse conj_commute cong: conj_cong)
apply (force dest!: spec split: if_split_asm)
done
have "x ∈ ?h ` {x ∈ Π⇩E i∈I. {c. Poly_Mapping.keys c ⊆ S i}. finite {i ∈ I. x i ≠ 0}}"
if x: "Poly_Mapping.keys x ⊆ ⋃ (S ` I)" for x :: "'b ⇒⇩0 int"
proof -
let ?f = "(λi c. if c ∈ S i then Poly_Mapping.lookup x c else 0)"
define J where "J ≡ {i ∈ I. ∃c∈S i. c ∈ Poly_Mapping.keys x}"
have "J ⊆ (λc. THE i. i ∈ I ∧ c ∈ S i) ` Poly_Mapping.keys x"
proof (clarsimp simp: J_def)
show "i ∈ (λc. THE i. i ∈ I ∧ c ∈ S i) ` Poly_Mapping.keys x"
if "i ∈ I" "c ∈ S i" "c ∈ Poly_Mapping.keys x" for i c
proof
show "i = (THE i. i ∈ I ∧ c ∈ S i)"
using assms that by (auto simp: pairwise_def disjnt_def intro: the_equality [symmetric])
qed
then have fin: "finite J"
using finite_subset finite_keys by blast
have [simp]: "Poly_Mapping.keys (Abs_poly_mapping (?f i)) = {k. ?f i k ≠ 0}" if "i ∈ I" for i
by (simp add: eq_onp_def keys.abs_eq conj_commute cong: conj_cong)
have [simp]: "Poly_Mapping.lookup (Abs_poly_mapping (?f i)) c = ?f i c" if "i ∈ I" for i c
by (auto simp: Abs_poly_mapping_inverse conj_commute cong: conj_cong)
show ?thesis
proof
have "poly_mapping.lookup x c = poly_mapping.lookup (?h (λi∈I. Abs_poly_mapping (?f i))) c"
for c
proof (cases "c ∈ Poly_Mapping.keys x")
case True
then obtain i where "i ∈ I" "c ∈ S i" "?f i c ≠ 0"
using x by (auto simp: in_keys_iff)
then have 1: "poly_mapping.lookup (sum' (λj. Abs_poly_mapping (?f j)) (I - {i})) c = 0"
using assms
apply (simp add: sum.G_def Poly_Mapping.lookup_sum pairwise_def disjnt_def)
apply (force simp: eq split: if_split_asm intro!: comm_monoid_add_class.sum.neutral)
done
have 2: "poly_mapping.lookup x c = poly_mapping.lookup (Abs_poly_mapping (?f i)) c"
by (auto simp: ‹c ∈ S i› Abs_poly_mapping_inverse conj_commute cong: conj_cong)
have "finite {i ∈ I. Abs_poly_mapping (?f i) ≠ 0}"
by (rule finite_subset [OF _ fin]) (use ‹i ∈ I› J_def eq in ‹auto simp: in_keys_iff›)
with ‹i ∈ I› have "?h (λj∈I. Abs_poly_mapping (?f j)) = Abs_poly_mapping (?f i) + sum' (λj. Abs_poly_mapping (?f j)) (I - {i})"
then show ?thesis
next
case False
then have "poly_mapping.lookup x c = 0"
using keys.rep_eq by force
then show ?thesis
qed
then show "x = ?h (λi∈I. Abs_poly_mapping (?f i))"
by (rule poly_mapping_eqI)
have "(λi. Abs_poly_mapping (?f i)) ∈ (Π i∈I. {c. Poly_Mapping.keys c ⊆ S i})"
by (auto simp: PiE_def Pi_def in_keys_iff)
then show "(λi∈I. Abs_poly_mapping (?f i))
∈ {x ∈ Π⇩E i∈I. {c. Poly_Mapping.keys c ⊆ S i}. finite {i ∈ I. x i ≠ 0}}"
using fin unfolding J_def by (force simp add: eq in_keys_iff cong: conj_cong)
qed
qed
then show "carrier ?H ⊆ ?h ` carrier ?G"
by (simp add: carrier_sum_group) (auto simp: free_Abelian_group_def)
show "inj_on ?h (carrier (sum_group I (λi. free_Abelian_group (S i))))"
unfolding GH.inj_on_one_iff
proof clarify
fix x
assume "x ∈ carrier ?G" "?h x = 𝟭⇘?H⇙"
then have eq0: "sum' x I = 0"
and xs: "⋀i. i ∈ I ⟹ Poly_Mapping.keys (x i) ⊆ S i" and xext: "x ∈ extensional I"
and fin: "finite {i ∈ I. x i ≠ 0}"
by (simp_all add: carrier_sum_group PiE_def Pi_def)
have "x i = 0" if "i ∈ I" for i
proof -
have "sum' x (insert i (I - {i})) = 0"
using eq0 that by (simp add: insert_absorb)
moreover have "Poly_Mapping.keys (sum' x (I - {i})) = {}"
proof -
have "x i = - sum' x (I - {i})"
by (metis (mono_tags, lifting) diff_zero eq0 fin sum_diff1' minus_diff_eq that)
then have "Poly_Mapping.keys (x i) = Poly_Mapping.keys (sum' x (I - {i}))"
by simp
then have "Poly_Mapping.keys (sum' x (I - {i})) ⊆ S i"
using that xs by metis
moreover
have "Poly_Mapping.keys (sum' x (I - {i})) ⊆ (⋃j ∈ I - {i}. S j)"
proof -
have "Poly_Mapping.keys (sum' x (I - {i})) ⊆ (⋃i∈{j ∈ I. j ≠ i ∧ x j ≠ 0}. Poly_Mapping.keys (x i))"
using keys_sum [of x "{j ∈ I. j ≠ i ∧ x j ≠ 0}"] by (simp add: sum.G_def)
also have "… ⊆ ⋃ (S ` (I - {i}))"
using xs by force
finally show ?thesis .
qed
moreover have "A = {}" if "A ⊆ S i" "A ⊆ ⋃ (S ` (I - {i}))" for A
using assms that ‹i ∈ I›
by (force simp: pairwise_def disjnt_def image_def subset_iff)
ultimately show ?thesis
by metis
qed
then have [simp]: "sum' x (I - {i}) = 0"
by (auto simp: sum.G_def)
have "sum' x (insert i (I - {i})) = x i"
by (subst sum.insert' [OF finite_subset [OF _ fin]]) auto
ultimately show ?thesis
by metis
qed
with xext [unfolded extensional_def]
show "x = 𝟭⇘sum_group I (λi. free_Abelian_group (S i))⇙"
by (force simp: free_Abelian_group_def)
qed
qed
qed

lemma isomorphic_free_Abelian_group_Union:
"pairwise disjnt I ⟹ free_Abelian_group(⋃ I) ≅ sum_group I free_Abelian_group"
using iso_free_Abelian_group_sum [of "λX. X" I]
by (metis SUP_identity_eq empty_iff group.iso_sym group_free_Abelian_group is_iso_def sum_group)

lemma isomorphic_sum_integer_group:
"sum_group I (λi. integer_group) ≅ free_Abelian_group I"
proof -
have "sum_group I (λi. integer_group) ≅ sum_group I (λi. free_Abelian_group {i})"
by (rule iso_sum_groupI) (auto simp: isomorphic_group_integer_free_Abelian_group_singleton)
also have "… ≅ free_Abelian_group I"
using iso_free_Abelian_group_sum [of "λx. {x}" I] by (auto simp: is_iso_def)
finally show ?thesis .
qed

end
```