Theory Wellorder_Embedding

theory Wellorder_Embedding
imports Fun_More Wellorder_Relation
```(*  Title:      HOL/Cardinals/Wellorder_Embedding.thy
Author:     Andrei Popescu, TU Muenchen

Well-order embeddings.
*)

section ‹Well-Order Embeddings›

theory Wellorder_Embedding
imports Fun_More Wellorder_Relation
begin

subsection ‹Auxiliaries›

lemma UNION_bij_betw_ofilter:
assumes WELL: "Well_order r" and
OF: "⋀ i. i ∈ I ⟹ ofilter r (A i)" and
BIJ: "⋀ i. i ∈ I ⟹ bij_betw f (A i) (A' i)"
shows "bij_betw f (⋃i ∈ I. A i) (⋃i ∈ I. A' i)"
proof-
have "wo_rel r" using WELL by (simp add: wo_rel_def)
hence "⋀ i j. ⟦i ∈ I; j ∈ I⟧ ⟹ A i ≤ A j ∨ A j ≤ A i"
using wo_rel.ofilter_linord[of r] OF by blast
with WELL BIJ show ?thesis
by (auto simp add: bij_betw_UNION_chain)
qed

subsection ‹(Well-order) embeddings, strict embeddings, isomorphisms and order-compatible
functions›

lemma embed_halfcong:
assumes EQ: "⋀ a. a ∈ Field r ⟹ f a = g a" and
EMB: "embed r r' f"
shows "embed r r' g"
proof(unfold embed_def, auto)
fix a assume *: "a ∈ Field r"
hence "bij_betw f (under r a) (under r' (f a))"
using EMB unfolding embed_def by simp
moreover
{have "under r a ≤ Field r"
by (auto simp add: under_Field)
hence "⋀ b. b ∈ under r a ⟹ f b = g b"
using EQ by blast
}
moreover have "f a = g a" using * EQ by auto
ultimately show "bij_betw g (under r a) (under r' (g a))"
using bij_betw_cong[of "under r a" f g "under r' (f a)"] by auto
qed

lemma embed_cong[fundef_cong]:
assumes "⋀ a. a ∈ Field r ⟹ f a = g a"
shows "embed r r' f = embed r r' g"
using assms embed_halfcong[of r f g r']
embed_halfcong[of r g f r'] by auto

lemma embedS_cong[fundef_cong]:
assumes "⋀ a. a ∈ Field r ⟹ f a = g a"
shows "embedS r r' f = embedS r r' g"
unfolding embedS_def using assms
embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast

lemma iso_cong[fundef_cong]:
assumes "⋀ a. a ∈ Field r ⟹ f a = g a"
shows "iso r r' f = iso r r' g"
unfolding iso_def using assms
embed_cong[of r f g r'] bij_betw_cong[of "Field r" f g "Field r'"] by blast

lemma id_compat: "compat r r id"
by(auto simp add: id_def compat_def)

lemma comp_compat:
"⟦compat r r' f; compat r' r'' f'⟧ ⟹ compat r r'' (f' o f)"
by(auto simp add: comp_def compat_def)

corollary one_set_greater:
"(∃f::'a ⇒ 'a'. f ` A ≤ A' ∧ inj_on f A) ∨ (∃g::'a' ⇒ 'a. g ` A' ≤ A ∧ inj_on g A')"
proof-
obtain r where "well_order_on A r" by (fastforce simp add: well_order_on)
hence 1: "A = Field r ∧ Well_order r"
using well_order_on_Well_order by auto
obtain r' where 2: "well_order_on A' r'" by (fastforce simp add: well_order_on)
hence 2: "A' = Field r' ∧ Well_order r'"
using well_order_on_Well_order by auto
hence "(∃f. embed r r' f) ∨ (∃g. embed r' r g)"
using 1 2 by (auto simp add: wellorders_totally_ordered)
moreover
{fix f assume "embed r r' f"
hence "f`A ≤ A' ∧ inj_on f A"
using 1 2 by (auto simp add: embed_Field embed_inj_on)
}
moreover
{fix g assume "embed r' r g"
hence "g`A' ≤ A ∧ inj_on g A'"
using 1 2 by (auto simp add: embed_Field embed_inj_on)
}
ultimately show ?thesis by blast
qed

corollary one_type_greater:
"(∃f::'a ⇒ 'a'. inj f) ∨ (∃g::'a' ⇒ 'a. inj g)"
using one_set_greater[of UNIV UNIV] by auto

subsection ‹Uniqueness of embeddings›

lemma comp_embedS:
assumes WELL: "Well_order r" and WELL': "Well_order r'" and WELL'': "Well_order r''"
and  EMB: "embedS r r' f" and EMB': "embedS r' r'' f'"
shows "embedS r r'' (f' o f)"
proof-
have "embed r' r'' f'" using EMB' unfolding embedS_def by simp
thus ?thesis using assms by (auto simp add: embedS_comp_embed)
qed

lemma iso_iff4:
assumes WELL: "Well_order r"  and WELL': "Well_order r'"
shows "iso r r' f = (embed r r' f ∧ embed r' r (inv_into (Field r) f))"
using assms embed_bothWays_iso
by(unfold iso_def, auto simp add: inv_into_Field_embed_bij_betw)

lemma embed_embedS_iso:
"embed r r' f = (embedS r r' f ∨ iso r r' f)"
unfolding embedS_def iso_def by blast

lemma not_embedS_iso:
"¬ (embedS r r' f ∧ iso r r' f)"
unfolding embedS_def iso_def by blast

lemma embed_embedS_iff_not_iso:
assumes "embed r r' f"
shows "embedS r r' f = (¬ iso r r' f)"
using assms unfolding embedS_def iso_def by blast

lemma iso_inv_into:
assumes WELL: "Well_order r" and ISO: "iso r r' f"
shows "iso r' r (inv_into (Field r) f)"
using assms unfolding iso_def
using bij_betw_inv_into inv_into_Field_embed_bij_betw by blast

lemma embedS_or_iso:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "(∃g. embedS r r' g) ∨ (∃h. embedS r' r h) ∨ (∃f. iso r r' f)"
proof-
{fix f assume *: "embed r r' f"
{assume "bij_betw f (Field r) (Field r')"
hence ?thesis using * by (auto simp add: iso_def)
}
moreover
{assume "¬ bij_betw f (Field r) (Field r')"
hence ?thesis using * by (auto simp add: embedS_def)
}
ultimately have ?thesis by auto
}
moreover
{fix f assume *: "embed r' r f"
{assume "bij_betw f (Field r') (Field r)"
hence "iso r' r f" using * by (auto simp add: iso_def)
hence "iso r r' (inv_into (Field r') f)"
using WELL' by (auto simp add: iso_inv_into)
hence ?thesis by blast
}
moreover
{assume "¬ bij_betw f (Field r') (Field r)"
hence ?thesis using * by (auto simp add: embedS_def)
}
ultimately have ?thesis by auto
}
ultimately show ?thesis using WELL WELL'
wellorders_totally_ordered[of r r'] by blast
qed

end
```