# Theory Wellorder_Constructions

theory Wellorder_Constructions
imports Wellorder_Embedding Order_Union Cardinal_Notations
```(*  Title:      HOL/Cardinals/Wellorder_Constructions.thy
Author:     Andrei Popescu, TU Muenchen

Constructions on wellorders.
*)

section ‹Constructions on Wellorders›

theory Wellorder_Constructions
imports
Wellorder_Embedding Order_Union
"HOL-Library.Cardinal_Notations"
begin

declare
ordLeq_Well_order_simp[simp]
not_ordLeq_iff_ordLess[simp]
not_ordLess_iff_ordLeq[simp]
Func_empty[simp]
Func_is_emp[simp]

lemma Func_emp2[simp]: "A ≠ {} ⟹ Func A {} = {}" by auto

subsection ‹Restriction to a set›

lemma Restr_incr2:
"r <= r' ⟹ Restr r A <= Restr r' A"
by blast

lemma Restr_incr:
"⟦r ≤ r'; A ≤ A'⟧ ⟹ Restr r A ≤ Restr r' A'"
by blast

lemma Restr_Int:
"Restr (Restr r A) B = Restr r (A Int B)"
by blast

lemma Restr_iff: "(a,b) ∈ Restr r A = (a ∈ A ∧ b ∈ A ∧ (a,b) ∈ r)"

lemma Restr_subset1: "Restr r A ≤ r"
by auto

lemma Restr_subset2: "Restr r A ≤ A × A"
by auto

lemma wf_Restr:
"wf r ⟹ wf(Restr r A)"
using Restr_subset by (elim wf_subset) simp

lemma Restr_incr1:
"A ≤ B ⟹ Restr r A ≤ Restr r B"
by blast

subsection ‹Order filters versus restrictions and embeddings›

lemma ofilter_Restr:
assumes WELL: "Well_order r" and
OFA: "ofilter r A" and OFB: "ofilter r B" and SUB: "A ≤ B"
shows "ofilter (Restr r B) A"
proof-
let ?rB = "Restr r B"
have Well: "wo_rel r" unfolding wo_rel_def using WELL .
hence Refl: "Refl r" by (auto simp add: wo_rel.REFL)
hence Field: "Field ?rB = Field r Int B"
using Refl_Field_Restr by blast
have WellB: "wo_rel ?rB ∧ Well_order ?rB" using WELL
by (auto simp add: Well_order_Restr wo_rel_def)
(* Main proof *)
show ?thesis
fix a assume "a ∈ A"
hence "a ∈ Field r ∧ a ∈ B" using assms Well
with Field show "a ∈ Field(Restr r B)" by auto
next
fix a b assume *: "a ∈ A"  and "b ∈ under (Restr r B) a"
hence "b ∈ under r a"
using WELL OFB SUB ofilter_Restr_under[of r B a] by auto
thus "b ∈ A" using * Well OFA by(auto simp add: wo_rel.ofilter_def)
qed
qed

lemma ofilter_subset_iso:
assumes WELL: "Well_order r" and
OFA: "ofilter r A" and OFB: "ofilter r B"
shows "(A = B) = iso (Restr r A) (Restr r B) id"
using assms

subsection ‹Ordering the well-orders by existence of embeddings›

corollary ordLeq_refl_on: "refl_on {r. Well_order r} ordLeq"
using ordLeq_reflexive unfolding ordLeq_def refl_on_def
by blast

corollary ordLeq_trans: "trans ordLeq"
using trans_def[of ordLeq] ordLeq_transitive by blast

corollary ordLeq_preorder_on: "preorder_on {r. Well_order r} ordLeq"
by(auto simp add: preorder_on_def ordLeq_refl_on ordLeq_trans)

corollary ordIso_refl_on: "refl_on {r. Well_order r} ordIso"
using ordIso_reflexive unfolding refl_on_def ordIso_def
by blast

corollary ordIso_trans: "trans ordIso"
using trans_def[of ordIso] ordIso_transitive by blast

corollary ordIso_sym: "sym ordIso"
by (auto simp add: sym_def ordIso_symmetric)

corollary ordIso_equiv: "equiv {r. Well_order r} ordIso"
by (auto simp add:  equiv_def ordIso_sym ordIso_refl_on ordIso_trans)

lemma ordLess_Well_order_simp[simp]:
assumes "r <o r'"
shows "Well_order r ∧ Well_order r'"
using assms unfolding ordLess_def by simp

lemma ordIso_Well_order_simp[simp]:
assumes "r =o r'"
shows "Well_order r ∧ Well_order r'"
using assms unfolding ordIso_def by simp

lemma ordLess_irrefl: "irrefl ordLess"
by(unfold irrefl_def, auto simp add: ordLess_irreflexive)

lemma ordLess_or_ordIso:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "r <o r' ∨ r' <o r ∨ r =o r'"
unfolding ordLess_def ordIso_def
using assms embedS_or_iso[of r r'] by auto

corollary ordLeq_ordLess_Un_ordIso:
"ordLeq = ordLess ∪ ordIso"

lemma not_ordLeq_ordLess:
"r ≤o r' ⟹ ¬ r' <o r"
using not_ordLess_ordLeq by blast

lemma ordIso_or_ordLess:
assumes WELL: "Well_order r" and WELL': "Well_order r'"
shows "r =o r' ∨ r <o r' ∨ r' <o r"
using assms ordLess_or_ordLeq ordLeq_iff_ordLess_or_ordIso by blast

lemmas ord_trans = ordIso_transitive ordLeq_transitive ordLess_transitive
ordIso_ordLeq_trans ordLeq_ordIso_trans
ordIso_ordLess_trans ordLess_ordIso_trans
ordLess_ordLeq_trans ordLeq_ordLess_trans

lemma ofilter_ordLeq:
assumes "Well_order r" and "ofilter r A"
shows "Restr r A ≤o r"
proof-
have "A ≤ Field r" using assms by (auto simp add: wo_rel_def wo_rel.ofilter_def)
thus ?thesis using assms
wo_rel_def Restr_Field)
qed

corollary under_Restr_ordLeq:
"Well_order r ⟹ Restr r (under r a) ≤o r"
by (auto simp add: ofilter_ordLeq wo_rel.under_ofilter wo_rel_def)

subsection ‹Copy via direct images›

lemma Id_dir_image: "dir_image Id f ≤ Id"
unfolding dir_image_def by auto

lemma Un_dir_image:
"dir_image (r1 ∪ r2) f = (dir_image r1 f) ∪ (dir_image r2 f)"
unfolding dir_image_def by auto

lemma Int_dir_image:
assumes "inj_on f (Field r1 ∪ Field r2)"
shows "dir_image (r1 Int r2) f = (dir_image r1 f) Int (dir_image r2 f)"
proof
show "dir_image (r1 Int r2) f ≤ (dir_image r1 f) Int (dir_image r2 f)"
using assms unfolding dir_image_def inj_on_def by auto
next
show "(dir_image r1 f) Int (dir_image r2 f) ≤ dir_image (r1 Int r2) f"
proof(clarify)
fix a' b'
assume "(a',b') ∈ dir_image r1 f" "(a',b') ∈ dir_image r2 f"
then obtain a1 b1 a2 b2
where 1: "a' = f a1 ∧ b' = f b1 ∧ a' = f a2 ∧ b' = f b2" and
2: "(a1,b1) ∈ r1 ∧ (a2,b2) ∈ r2" and
3: "{a1,b1} ≤ Field r1 ∧ {a2,b2} ≤ Field r2"
unfolding dir_image_def Field_def by blast
hence "a1 = a2 ∧ b1 = b2" using assms unfolding inj_on_def by auto
hence "a' = f a1 ∧ b' = f b1 ∧ (a1,b1) ∈ r1 Int r2 ∧ (a2,b2) ∈ r1 Int r2"
using 1 2 by auto
thus "(a',b') ∈ dir_image (r1 ∩ r2) f"
unfolding dir_image_def by blast
qed
qed

(* More facts on ordinal sum: *)

lemma Osum_embed:
assumes FLD: "Field r Int Field r' = {}" and
WELL: "Well_order r" and WELL': "Well_order r'"
shows "embed r (r Osum r') id"
proof-
have 1: "Well_order (r Osum r')"
using assms by (auto simp add: Osum_Well_order)
moreover
have "compat r (r Osum r') id"
unfolding compat_def Osum_def by auto
moreover
have "inj_on id (Field r)" by simp
moreover
have "ofilter (r Osum r') (Field r)"
using 1 proof(auto simp add: wo_rel_def wo_rel.ofilter_def
Field_Osum under_def)
fix a b assume 2: "a ∈ Field r" and 3: "(b,a) ∈ r Osum r'"
moreover
{assume "(b,a) ∈ r'"
hence "a ∈ Field r'" using Field_def[of r'] by blast
hence False using 2 FLD by blast
}
moreover
{assume "a ∈ Field r'"
hence False using 2 FLD by blast
}
ultimately
show "b ∈ Field r" by (auto simp add: Osum_def Field_def)
qed
ultimately show ?thesis
using assms by (auto simp add: embed_iff_compat_inj_on_ofilter)
qed

corollary Osum_ordLeq:
assumes FLD: "Field r Int Field r' = {}" and
WELL: "Well_order r" and WELL': "Well_order r'"
shows "r ≤o r Osum r'"
using assms Osum_embed Osum_Well_order
unfolding ordLeq_def by blast

lemma Well_order_embed_copy:
assumes WELL: "well_order_on A r" and
INJ: "inj_on f A" and SUB: "f ` A ≤ B"
shows "∃r'. well_order_on B r' ∧ r ≤o r'"
proof-
have "bij_betw f A (f ` A)"
using INJ inj_on_imp_bij_betw by blast
then obtain r'' where "well_order_on (f ` A) r''" and 1: "r =o r''"
using WELL  Well_order_iso_copy by blast
hence 2: "Well_order r'' ∧ Field r'' = (f ` A)"
using well_order_on_Well_order by blast
(*  *)
let ?C = "B - (f ` A)"
obtain r''' where "well_order_on ?C r'''"
using well_order_on by blast
hence 3: "Well_order r''' ∧ Field r''' = ?C"
using well_order_on_Well_order by blast
(*  *)
let ?r' = "r'' Osum r'''"
have "Field r'' Int Field r''' = {}"
using 2 3 by auto
hence "r'' ≤o ?r'" using Osum_ordLeq[of r'' r'''] 2 3 by blast
hence 4: "r ≤o ?r'" using 1 ordIso_ordLeq_trans by blast
(*  *)
hence "Well_order ?r'" unfolding ordLeq_def by auto
moreover
have "Field ?r' = B" using 2 3 SUB by (auto simp add: Field_Osum)
ultimately show ?thesis using 4 by blast
qed

subsection ‹The maxim among a finite set of ordinals›

text ‹The correct phrasing would be ``a maxim of ...", as ‹≤o› is only a preorder.›

definition isOmax :: "'a rel set ⇒ 'a rel ⇒ bool"
where
"isOmax  R r ≡ r ∈ R ∧ (∀r' ∈ R. r' ≤o r)"

definition omax :: "'a rel set ⇒ 'a rel"
where
"omax R == SOME r. isOmax R r"

lemma exists_isOmax:
assumes "finite R" and "R ≠ {}" and "∀ r ∈ R. Well_order r"
shows "∃ r. isOmax R r"
proof-
have "finite R ⟹ R ≠ {} ⟶ (∀ r ∈ R. Well_order r) ⟶ (∃ r. isOmax R r)"
proof(clarsimp)
fix r :: "('a × 'a) set" and R assume *: "finite R" and **: "r ∉ R"
and ***: "Well_order r" and ****: "∀r∈R. Well_order r"
and IH: "R ≠ {} ⟶ (∃p. isOmax R p)"
let ?R' = "insert r R"
show "∃p'. (isOmax ?R' p')"
proof(cases "R = {}")
assume Case1: "R = {}"
thus ?thesis unfolding isOmax_def using ***
next
assume Case2: "R ≠ {}"
then obtain p where p: "isOmax R p" using IH by auto
hence 1: "Well_order p" using **** unfolding isOmax_def by simp
{assume Case21: "r ≤o p"
hence "isOmax ?R' p" using p unfolding isOmax_def by simp
hence ?thesis by auto
}
moreover
{assume Case22: "p ≤o r"
{fix r' assume "r' ∈ ?R'"
moreover
{assume "r' ∈ R"
hence "r' ≤o p" using p unfolding isOmax_def by simp
hence "r' ≤o r" using Case22 by(rule ordLeq_transitive)
}
moreover have "r ≤o r" using *** by(rule ordLeq_reflexive)
ultimately have "r' ≤o r" by auto
}
hence "isOmax ?R' r" unfolding isOmax_def by simp
hence ?thesis by auto
}
moreover have "r ≤o p ∨ p ≤o r"
using 1 *** ordLeq_total by auto
ultimately show ?thesis by blast
qed
qed
thus ?thesis using assms by auto
qed

lemma omax_isOmax:
assumes "finite R" and "R ≠ {}" and "∀ r ∈ R. Well_order r"
shows "isOmax R (omax R)"
unfolding omax_def using assms

lemma omax_in:
assumes "finite R" and "R ≠ {}" and "∀ r ∈ R. Well_order r"
shows "omax R ∈ R"
using assms omax_isOmax unfolding isOmax_def by blast

lemma Well_order_omax:
assumes "finite R" and "R ≠ {}" and "∀r∈R. Well_order r"
shows "Well_order (omax R)"
using assms apply - apply(drule omax_in) by auto

lemma omax_maxim:
assumes "finite R" and "∀ r ∈ R. Well_order r" and "r ∈ R"
shows "r ≤o omax R"
using assms omax_isOmax unfolding isOmax_def by blast

lemma omax_ordLeq:
assumes "finite R" and "R ≠ {}" and *: "∀ r ∈ R. r ≤o p"
shows "omax R ≤o p"
proof-
have "∀ r ∈ R. Well_order r" using * unfolding ordLeq_def by simp
thus ?thesis using assms omax_in by auto
qed

lemma omax_ordLess:
assumes "finite R" and "R ≠ {}" and *: "∀ r ∈ R. r <o p"
shows "omax R <o p"
proof-
have "∀ r ∈ R. Well_order r" using * unfolding ordLess_def by simp
thus ?thesis using assms omax_in by auto
qed

lemma omax_ordLeq_elim:
assumes "finite R" and "∀ r ∈ R. Well_order r"
and "omax R ≤o p" and "r ∈ R"
shows "r ≤o p"
using assms omax_maxim[of R r] apply simp
using ordLeq_transitive by blast

lemma omax_ordLess_elim:
assumes "finite R" and "∀ r ∈ R. Well_order r"
and "omax R <o p" and "r ∈ R"
shows "r <o p"
using assms omax_maxim[of R r] apply simp
using ordLeq_ordLess_trans by blast

lemma ordLeq_omax:
assumes "finite R" and "∀ r ∈ R. Well_order r"
and "r ∈ R" and "p ≤o r"
shows "p ≤o omax R"
using assms omax_maxim[of R r] apply simp
using ordLeq_transitive by blast

lemma ordLess_omax:
assumes "finite R" and "∀ r ∈ R. Well_order r"
and "r ∈ R" and "p <o r"
shows "p <o omax R"
using assms omax_maxim[of R r] apply simp
using ordLess_ordLeq_trans by blast

lemma omax_ordLeq_mono:
assumes P: "finite P" and R: "finite R"
and NE_P: "P ≠ {}" and Well_R: "∀ r ∈ R. Well_order r"
and LEQ: "∀ p ∈ P. ∃ r ∈ R. p ≤o r"
shows "omax P ≤o omax R"
proof-
let ?mp = "omax P"  let ?mr = "omax R"
{fix p assume "p ∈ P"
then obtain r where r: "r ∈ R" and "p ≤o r"
using LEQ by blast
moreover have "r <=o ?mr"
using r R Well_R omax_maxim by blast
ultimately have "p <=o ?mr"
using ordLeq_transitive by blast
}
thus "?mp <=o ?mr"
using NE_P P using omax_ordLeq by blast
qed

lemma omax_ordLess_mono:
assumes P: "finite P" and R: "finite R"
and NE_P: "P ≠ {}" and Well_R: "∀ r ∈ R. Well_order r"
and LEQ: "∀ p ∈ P. ∃ r ∈ R. p <o r"
shows "omax P <o omax R"
proof-
let ?mp = "omax P"  let ?mr = "omax R"
{fix p assume "p ∈ P"
then obtain r where r: "r ∈ R" and "p <o r"
using LEQ by blast
moreover have "r <=o ?mr"
using r R Well_R omax_maxim by blast
ultimately have "p <o ?mr"
using ordLess_ordLeq_trans by blast
}
thus "?mp <o ?mr"
using NE_P P omax_ordLess by blast
qed

subsection ‹Limit and succesor ordinals›

lemma embed_underS2:
assumes r: "Well_order r" and s: "Well_order s"  and g: "embed r s g" and a: "a ∈ Field r"
shows "g ` underS r a = underS s (g a)"
using embed_underS[OF assms] unfolding bij_betw_def by auto

lemma bij_betw_insert:
assumes "b ∉ A" and "f b ∉ A'" and "bij_betw f A A'"
shows "bij_betw f (insert b A) (insert (f b) A')"
using notIn_Un_bij_betw[OF assms] by auto

context wo_rel
begin

lemma underS_induct:
assumes "⋀a. (⋀ a1. a1 ∈ underS a ⟹ P a1) ⟹ P a"
shows "P a"
by (induct rule: well_order_induct) (rule assms, simp add: underS_def)

lemma suc_underS:
assumes B: "B ⊆ Field r" and A: "AboveS B ≠ {}" and b: "b ∈ B"
shows "b ∈ underS (suc B)"
using suc_AboveS[OF B A] b unfolding underS_def AboveS_def by auto

lemma underS_supr:
assumes bA: "b ∈ underS (supr A)" and A: "A ⊆ Field r"
shows "∃ a ∈ A. b ∈ underS a"
proof(rule ccontr, auto)
have bb: "b ∈ Field r" using bA unfolding underS_def Field_def by auto
assume "∀a∈A.  b ∉ underS a"
hence 0: "∀a ∈ A. (a,b) ∈ r" using A bA unfolding underS_def
by simp (metis REFL in_mono max2_def max2_greater refl_on_domain)
have "(supr A, b) ∈ r" apply(rule supr_least[OF A bb]) using 0 by auto
thus False using bA unfolding underS_def by simp (metis ANTISYM antisymD)
qed

lemma underS_suc:
assumes bA: "b ∈ underS (suc A)" and A: "A ⊆ Field r"
shows "∃ a ∈ A. b ∈ under a"
proof(rule ccontr, auto)
have bb: "b ∈ Field r" using bA unfolding underS_def Field_def by auto
assume "∀a∈A.  b ∉ under a"
hence 0: "∀a ∈ A. a ∈ underS b" using A bA unfolding underS_def
by simp (metis (lifting) bb max2_def max2_greater mem_Collect_eq under_def set_rev_mp)
have "(suc A, b) ∈ r" apply(rule suc_least[OF A bb]) using 0 unfolding underS_def by auto
thus False using bA unfolding underS_def by simp (metis ANTISYM antisymD)
qed

lemma (in wo_rel) in_underS_supr:
assumes j: "j ∈ underS i" and i: "i ∈ A" and A: "A ⊆ Field r" and AA: "Above A ≠ {}"
shows "j ∈ underS (supr A)"
proof-
have "(i,supr A) ∈ r" using supr_greater[OF A AA i] .
thus ?thesis using j unfolding underS_def
by simp (metis REFL TRANS max2_def max2_equals1 refl_on_domain transD)
qed

lemma inj_on_Field:
assumes A: "A ⊆ Field r" and f: "⋀ a b. ⟦a ∈ A; b ∈ A; a ∈ underS b⟧ ⟹ f a ≠ f b"
shows "inj_on f A"
unfolding inj_on_def proof safe
fix a b assume a: "a ∈ A" and b: "b ∈ A" and fab: "f a = f b"
{assume "a ∈ underS b"
hence False using f[OF a b] fab by auto
}
moreover
{assume "b ∈ underS a"
hence False using f[OF b a] fab by auto
}
ultimately show "a = b" using TOTALS A a b unfolding underS_def by auto
qed

lemma in_notinI:
assumes "(j,i) ∉ r ∨ j = i" and "i ∈ Field r" and "j ∈ Field r"
shows "(i,j) ∈ r" by (metis assms max2_def max2_greater_among)

lemma ofilter_init_seg_of:
assumes "ofilter F"
shows "Restr r F initial_segment_of r"
using assms unfolding ofilter_def init_seg_of_def under_def by auto

lemma underS_init_seg_of_Collect:
assumes "⋀j1 j2. ⟦j2 ∈ underS i; (j1, j2) ∈ r⟧ ⟹ R j1 initial_segment_of R j2"
shows "{R j |j. j ∈ underS i} ∈ Chains init_seg_of"
unfolding Chains_def proof safe
fix j ja assume jS: "j ∈ underS i" and jaS: "ja ∈ underS i"
and init: "(R ja, R j) ∉ init_seg_of"
hence jja: "{j,ja} ⊆ Field r" and j: "j ∈ Field r" and ja: "ja ∈ Field r"
and jjai: "(j,i) ∈ r" "(ja,i) ∈ r"
and i: "i ∉ {j,ja}" unfolding Field_def underS_def by auto
have jj: "(j,j) ∈ r" and jaja: "(ja,ja) ∈ r" using j ja by (metis in_notinI)+
show "R j initial_segment_of R ja"
using jja init jjai i
by (elim cases_Total3 disjE) (auto elim: cases_Total3 intro!: assms simp: underS_def)
qed

lemma (in wo_rel) Field_init_seg_of_Collect:
assumes "⋀j1 j2. ⟦j2 ∈ Field r; (j1, j2) ∈ r⟧ ⟹ R j1 initial_segment_of R j2"
shows "{R j |j. j ∈ Field r} ∈ Chains init_seg_of"
unfolding Chains_def proof safe
fix j ja assume jS: "j ∈ Field r" and jaS: "ja ∈ Field r"
and init: "(R ja, R j) ∉ init_seg_of"
hence jja: "{j,ja} ⊆ Field r" and j: "j ∈ Field r" and ja: "ja ∈ Field r"
unfolding Field_def underS_def by auto
have jj: "(j,j) ∈ r" and jaja: "(ja,ja) ∈ r" using j ja by (metis in_notinI)+
show "R j initial_segment_of R ja"
using jja init
by (elim cases_Total3 disjE) (auto elim: cases_Total3 intro!: assms simp: Field_def)
qed

subsubsection ‹Successor and limit elements of an ordinal›

definition "succ i ≡ suc {i}"

definition "isSucc i ≡ ∃ j. aboveS j ≠ {} ∧ i = succ j"

definition "zero = minim (Field r)"

definition "isLim i ≡ ¬ isSucc i"

lemma zero_smallest[simp]:
assumes "j ∈ Field r" shows "(zero, j) ∈ r"
unfolding zero_def
by (metis AboveS_Field assms subset_AboveS_UnderS subset_antisym subset_refl suc_def suc_least_AboveS)

lemma zero_in_Field: assumes "Field r ≠ {}"  shows "zero ∈ Field r"
using assms unfolding zero_def by (metis Field_ofilter minim_in ofilter_def)

lemma leq_zero_imp[simp]:
"(x, zero) ∈ r ⟹ x = zero"
by (metis ANTISYM WELL antisymD well_order_on_domain zero_smallest)

lemma leq_zero[simp]:
assumes "Field r ≠ {}"  shows "(x, zero) ∈ r ⟷ x = zero"
using zero_in_Field[OF assms] in_notinI[of x zero] by auto

lemma under_zero[simp]:
assumes "Field r ≠ {}" shows "under zero = {zero}"
using assms unfolding under_def by auto

lemma underS_zero[simp,intro]: "underS zero = {}"
unfolding underS_def by auto

lemma isSucc_succ: "aboveS i ≠ {} ⟹ isSucc (succ i)"
unfolding isSucc_def succ_def by auto

lemma succ_in_diff:
assumes "aboveS i ≠ {}"  shows "(i,succ i) ∈ r ∧ succ i ≠ i"
using assms suc_greater[of "{i}"] unfolding succ_def AboveS_def aboveS_def Field_def by auto

lemmas succ_in[simp] = succ_in_diff[THEN conjunct1]
lemmas succ_diff[simp] = succ_in_diff[THEN conjunct2]

lemma succ_in_Field[simp]:
assumes "aboveS i ≠ {}"  shows "succ i ∈ Field r"
using succ_in[OF assms] unfolding Field_def by auto

lemma succ_not_in:
assumes "aboveS i ≠ {}" shows "(succ i, i) ∉ r"
proof
assume 1: "(succ i, i) ∈ r"
hence "succ i ∈ Field r ∧ i ∈ Field r" unfolding Field_def by auto
hence "(i, succ i) ∈ r ∧ succ i ≠ i" using assms by auto
thus False using 1 by (metis ANTISYM antisymD)
qed

lemma not_isSucc_zero: "¬ isSucc zero"
proof
assume *: "isSucc zero"
then obtain i where "aboveS i ≠ {}" and 1: "minim (Field r) = succ i"
unfolding isSucc_def zero_def by auto
hence "succ i ∈ Field r" by auto
with * show False
by (metis REFL isSucc_def minim_least refl_on_domain
subset_refl succ_in succ_not_in zero_def)
qed

lemma isLim_zero[simp]: "isLim zero"
by (metis isLim_def not_isSucc_zero)

lemma succ_smallest:
assumes "(i,j) ∈ r" and "i ≠ j"
shows "(succ i, j) ∈ r"
unfolding succ_def apply(rule suc_least)
using assms unfolding Field_def by auto

lemma isLim_supr:
assumes f: "i ∈ Field r" and l: "isLim i"
shows "i = supr (underS i)"
proof(rule equals_supr)
fix j assume j: "j ∈ Field r" and 1: "⋀ j'. j' ∈ underS i ⟹ (j',j) ∈ r"
show "(i,j) ∈ r" proof(intro in_notinI[OF _ f j], safe)
assume ji: "(j,i) ∈ r" "j ≠ i"
hence a: "aboveS j ≠ {}" unfolding aboveS_def by auto
hence "i ≠ succ j" using l unfolding isLim_def isSucc_def by auto
moreover have "(succ j, i) ∈ r" using succ_smallest[OF ji] by auto
ultimately have "succ j ∈ underS i" unfolding underS_def by auto
hence "(succ j, j) ∈ r" using 1 by auto
thus False using succ_not_in[OF a] by simp
qed
qed(insert f, unfold underS_def Field_def, auto)

definition "pred i ≡ SOME j. j ∈ Field r ∧ aboveS j ≠ {} ∧ succ j = i"

lemma pred_Field_succ:
assumes "isSucc i" shows "pred i ∈ Field r ∧ aboveS (pred i) ≠ {} ∧ succ (pred i) = i"
proof-
obtain j where a: "aboveS j ≠ {}" and i: "i = succ j" using assms unfolding isSucc_def by auto
have 1: "j ∈ Field r" "j ≠ i" unfolding Field_def i
using succ_diff[OF a] a unfolding aboveS_def by auto
show ?thesis unfolding pred_def apply(rule someI_ex) using 1 i a by auto
qed

lemmas pred_Field[simp] = pred_Field_succ[THEN conjunct1]
lemmas aboveS_pred[simp] = pred_Field_succ[THEN conjunct2, THEN conjunct1]
lemmas succ_pred[simp] = pred_Field_succ[THEN conjunct2, THEN conjunct2]

lemma isSucc_pred_in:
assumes "isSucc i"  shows "(pred i, i) ∈ r"
proof-
define j where "j = pred i"
have i: "i = succ j" using assms unfolding j_def by simp
have a: "aboveS j ≠ {}" unfolding j_def using assms by auto
show ?thesis unfolding j_def[symmetric] unfolding i using succ_in[OF a] .
qed

lemma isSucc_pred_diff:
assumes "isSucc i"  shows "pred i ≠ i"
by (metis aboveS_pred assms succ_diff succ_pred)

(* todo: pred maximal, pred injective? *)

lemma succ_inj[simp]:
assumes "aboveS i ≠ {}" and "aboveS j ≠ {}"
shows "succ i = succ j ⟷ i = j"
proof safe
assume s: "succ i = succ j"
{assume "i ≠ j" and "(i,j) ∈ r"
hence "(succ i, j) ∈ r" using assms by (metis succ_smallest)
hence False using s assms by (metis succ_not_in)
}
moreover
{assume "i ≠ j" and "(j,i) ∈ r"
hence "(succ j, i) ∈ r" using assms by (metis succ_smallest)
hence False using s assms by (metis succ_not_in)
}
ultimately show "i = j" by (metis TOTALS WELL assms(1) assms(2) succ_in_diff well_order_on_domain)
qed

lemma pred_succ[simp]:
assumes "aboveS j ≠ {}"  shows "pred (succ j) = j"
unfolding pred_def apply(rule some_equality)
using assms apply(force simp: Field_def aboveS_def)
by (metis assms succ_inj)

lemma less_succ[simp]:
assumes "aboveS i ≠ {}"
shows "(j, succ i) ∈ r ⟷ (j,i) ∈ r ∨ j = succ i"
apply safe
apply (metis WELL assms in_notinI well_order_on_domain suc_singl_pred succ_def succ_in_diff)
apply (metis (hide_lams, full_types) REFL TRANS assms max2_def max2_equals1 refl_on_domain succ_in_Field succ_not_in transD)
apply (metis assms in_notinI succ_in_Field)
done

lemma underS_succ[simp]:
assumes "aboveS i ≠ {}"
shows "underS (succ i) = under i"
unfolding underS_def under_def by (auto simp: assms succ_not_in)

lemma succ_mono:
assumes "aboveS j ≠ {}" and "(i,j) ∈ r"
shows "(succ i, succ j) ∈ r"
by (metis (full_types) assms less_succ succ_smallest)

lemma under_succ[simp]:
assumes "aboveS i ≠ {}"
shows "under (succ i) = insert (succ i) (under i)"
using less_succ[OF assms] unfolding under_def by auto

definition mergeSL :: "('a ⇒ 'b ⇒ 'b) ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'b"
where
"mergeSL S L f i ≡
if isSucc i then S (pred i) (f (pred i))
else L f i"

subsubsection ‹Well-order recursion with (zero), succesor, and limit›

definition worecSL :: "('a ⇒ 'b ⇒ 'b) ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'b"
where "worecSL S L ≡ worec (mergeSL S L)"

definition "adm_woL L ≡ ∀f g i. isLim i ∧ (∀j∈underS i. f j = g j) ⟶ L f i = L g i"

lemma mergeSL:
fix f g :: "'a => 'b" and i :: 'a
assume 1: "∀j∈underS i. f j = g j"
show "mergeSL S L f i = mergeSL S L g i"
proof(cases "isSucc i")
case True
hence "pred i ∈ underS i" unfolding underS_def using isSucc_pred_in isSucc_pred_diff by auto
thus ?thesis using True 1 unfolding mergeSL_def by auto
next
case False hence "isLim i" unfolding isLim_def by auto
thus ?thesis using assms False 1 unfolding mergeSL_def adm_woL_def by auto
qed
qed

lemma worec_fixpoint1: "adm_wo H ⟹ worec H i = H (worec H) i"
by (metis worec_fixpoint)

lemma worecSL_isSucc:
assumes a: "adm_woL L" and i: "isSucc i"
shows "worecSL S L i = S (pred i) (worecSL S L (pred i))"
proof-
let ?H = "mergeSL S L"
have "worecSL S L i = ?H (worec ?H) i"
unfolding worecSL_def using worec_fixpoint1[OF mergeSL[OF a]] .
also have "... = S (pred i) (worecSL S L (pred i))"
unfolding worecSL_def mergeSL_def using i by simp
finally show ?thesis .
qed

lemma worecSL_succ:
assumes a: "adm_woL L" and i: "aboveS j ≠ {}"
shows "worecSL S L (succ j) = S j (worecSL S L j)"
proof-
define i where "i = succ j"
have i: "isSucc i" by (metis i i_def isSucc_succ)
have ij: "j = pred i" unfolding i_def using assms by simp
have 0: "succ (pred i) = i" using i by simp
show ?thesis unfolding ij using worecSL_isSucc[OF a i] unfolding 0 .
qed

lemma worecSL_isLim:
assumes a: "adm_woL L" and i: "isLim i"
shows "worecSL S L i = L (worecSL S L) i"
proof-
let ?H = "mergeSL S L"
have "worecSL S L i = ?H (worec ?H) i"
unfolding worecSL_def using worec_fixpoint1[OF mergeSL[OF a]] .
also have "... = L (worecSL S L) i"
using i unfolding worecSL_def mergeSL_def isLim_def by simp
finally show ?thesis .
qed

definition worecZSL :: "'b ⇒ ('a ⇒ 'b ⇒ 'b) ⇒ (('a ⇒ 'b) ⇒ 'a ⇒ 'b) ⇒ 'a ⇒ 'b"
where "worecZSL Z S L ≡ worecSL S (λ f a. if a = zero then Z else L f a)"

lemma worecZSL_zero:
shows "worecZSL Z S L zero = Z"
proof-
let ?L = "λ f a. if a = zero then Z else L f a"
have "worecZSL Z S L zero = ?L (worecZSL Z S L) zero"
unfolding worecZSL_def apply(rule worecSL_isLim)
using assms unfolding adm_woL_def by auto
also have "... = Z" by simp
finally show ?thesis .
qed

lemma worecZSL_succ:
assumes a: "adm_woL L" and i: "aboveS j ≠ {}"
shows "worecZSL Z S L (succ j) = S j (worecZSL Z S L j)"
unfolding worecZSL_def apply(rule  worecSL_succ)
using assms unfolding adm_woL_def by auto

lemma worecZSL_isLim:
assumes a: "adm_woL L" and "isLim i" and "i ≠ zero"
shows "worecZSL Z S L i = L (worecZSL Z S L) i"
proof-
let ?L = "λ f a. if a = zero then Z else L f a"
have "worecZSL Z S L i = ?L (worecZSL Z S L) i"
unfolding worecZSL_def apply(rule worecSL_isLim)
using assms unfolding adm_woL_def by auto
also have "... = L (worecZSL Z S L) i" using assms by simp
finally show ?thesis .
qed

subsubsection ‹Well-order succ-lim induction›

lemma ord_cases:
obtains j where "i = succ j" and "aboveS j ≠ {}"  | "isLim i"
by (metis isLim_def isSucc_def)

lemma well_order_inductSL[case_names Suc Lim]:
assumes SUCC: "⋀i. ⟦aboveS i ≠ {}; P i⟧ ⟹ P (succ i)" and
LIM: "⋀i. ⟦isLim i; ⋀j. j ∈ underS i ⟹ P j⟧ ⟹ P i"
shows "P i"
proof(induction rule: well_order_induct)
fix i assume 0:  "∀j. j ≠ i ∧ (j, i) ∈ r ⟶ P j"
show "P i" proof(cases i rule: ord_cases)
fix j assume i: "i = succ j" and j: "aboveS j ≠ {}"
hence "j ≠ i ∧ (j, i) ∈ r" by (metis  succ_diff succ_in)
hence 1: "P j" using 0 by simp
show "P i" unfolding i apply(rule SUCC) using 1 j by auto
qed(insert 0 LIM, unfold underS_def, auto)
qed

lemma well_order_inductZSL[case_names Zero Suc Lim]:
assumes ZERO: "P zero"
and SUCC: "⋀i. ⟦aboveS i ≠ {}; P i⟧ ⟹ P (succ i)" and
LIM: "⋀i. ⟦isLim i; i ≠ zero; ⋀j. j ∈ underS i ⟹ P j⟧ ⟹ P i"
shows "P i"
apply(induction rule: well_order_inductSL) using assms by auto

(* Succesor and limit ordinals *)
definition "isSuccOrd ≡ ∃ j ∈ Field r. ∀ i ∈ Field r. (i,j) ∈ r"
definition "isLimOrd ≡ ¬ isSuccOrd"

lemma isLimOrd_succ:
assumes isLimOrd and "i ∈ Field r"
shows "succ i ∈ Field r"
using assms unfolding isLimOrd_def isSuccOrd_def
by (metis REFL in_notinI refl_on_domain succ_smallest)

lemma isLimOrd_aboveS:
assumes l: isLimOrd and i: "i ∈ Field r"
shows "aboveS i ≠ {}"
proof-
obtain j where "j ∈ Field r" and "(j,i) ∉ r"
using assms unfolding isLimOrd_def isSuccOrd_def by auto
hence "(i,j) ∈ r ∧ j ≠ i" by (metis i max2_def max2_greater)
thus ?thesis unfolding aboveS_def by auto
qed

lemma succ_aboveS_isLimOrd:
assumes "∀ i ∈ Field r. aboveS i ≠ {} ∧ succ i ∈ Field r"
shows isLimOrd
unfolding isLimOrd_def isSuccOrd_def proof safe
fix j assume j: "j ∈ Field r" "∀i∈Field r. (i, j) ∈ r"
hence "(succ j, j) ∈ r" using assms by auto
moreover have "aboveS j ≠ {}" using assms j unfolding aboveS_def by auto
ultimately show False by (metis succ_not_in)
qed

lemma isLim_iff:
assumes l: "isLim i" and j: "j ∈ underS i"
shows "∃ k. k ∈ underS i ∧ j ∈ underS k"
proof-
have a: "aboveS j ≠ {}" using j unfolding underS_def aboveS_def by auto
show ?thesis apply(rule exI[of _ "succ j"]) apply safe
using assms a unfolding underS_def isLim_def
apply (metis (lifting, full_types) isSucc_def mem_Collect_eq succ_smallest)
by (metis (lifting, full_types) a mem_Collect_eq succ_diff succ_in)
qed

end (* context wo_rel *)

abbreviation "zero ≡ wo_rel.zero"
abbreviation "succ ≡ wo_rel.succ"
abbreviation "pred ≡ wo_rel.pred"
abbreviation "isSucc ≡ wo_rel.isSucc"
abbreviation "isLim ≡ wo_rel.isLim"
abbreviation "isLimOrd ≡ wo_rel.isLimOrd"
abbreviation "isSuccOrd ≡ wo_rel.isSuccOrd"
abbreviation "worecSL ≡ wo_rel.worecSL"
abbreviation "worecZSL ≡ wo_rel.worecZSL"

subsection ‹Projections of wellorders›

definition "oproj r s f ≡ Field s ⊆ f ` (Field r) ∧ compat r s f"

lemma oproj_in:
assumes "oproj r s f" and "(a,a') ∈ r"
shows "(f a, f a') ∈ s"
using assms unfolding oproj_def compat_def by auto

lemma oproj_Field:
assumes f: "oproj r s f" and a: "a ∈ Field r"
shows "f a ∈ Field s"
using oproj_in[OF f] a unfolding Field_def by auto

lemma oproj_Field2:
assumes f: "oproj r s f" and a: "b ∈ Field s"
shows "∃ a ∈ Field r. f a = b"
using assms unfolding oproj_def by auto

lemma oproj_under:
assumes f:  "oproj r s f" and a: "a ∈ under r a'"
shows "f a ∈ under s (f a')"
using oproj_in[OF f] a unfolding under_def by auto

(* An ordinal is embedded in another whenever it is embedded as an order
(not necessarily as initial segment):*)
theorem embedI:
assumes r: "Well_order r" and s: "Well_order s"
and f: "⋀ a. a ∈ Field r ⟹ f a ∈ Field s ∧ f ` underS r a ⊆ underS s (f a)"
shows "∃ g. embed r s g"
proof-
interpret r: wo_rel r by unfold_locales (rule r)
interpret s: wo_rel s by unfold_locales (rule s)
let ?G = "λ g a. suc s (g ` underS r a)"
define g where "g = worec r ?G"
(*  *)
{fix a assume "a ∈ Field r"
hence "bij_betw g (under r a) (under s (g a)) ∧
g a ∈ under s (f a)"
proof(induction a rule: r.underS_induct)
case (1 a)
hence a: "a ∈ Field r"
and IH1a: "⋀ a1. a1 ∈ underS r a ⟹ inj_on g (under r a1)"
and IH1b: "⋀ a1. a1 ∈ underS r a ⟹ g ` under r a1 = under s (g a1)"
and IH2: "⋀ a1. a1 ∈ underS r a ⟹ g a1 ∈ under s (f a1)"
unfolding underS_def Field_def bij_betw_def by auto
have fa: "f a ∈ Field s" using f[OF a] by auto
have g: "g a = suc s (g ` underS r a)"
using r.worec_fixpoint[OF adm] unfolding g_def fun_eq_iff by simp
have A0: "g ` underS r a ⊆ Field s"
using IH1b by (metis IH2 image_subsetI in_mono under_Field)
{fix a1 assume a1: "a1 ∈ underS r a"
from IH2[OF this] have "g a1 ∈ under s (f a1)" .
moreover have "f a1 ∈ underS s (f a)" using f[OF a] a1 by auto
ultimately have "g a1 ∈ underS s (f a)" by (metis s.ANTISYM s.TRANS under_underS_trans)
}
hence "f a ∈ AboveS s (g ` underS r a)" unfolding AboveS_def
using fa by simp (metis (lifting, full_types) mem_Collect_eq underS_def)
hence A: "AboveS s (g ` underS r a) ≠ {}" by auto
have B: "⋀ a1. a1 ∈ underS r a ⟹ g a1 ∈ underS s (g a)"
unfolding g apply(rule s.suc_underS[OF A0 A]) by auto
{fix a1 a2 assume a2: "a2 ∈ underS r a" and 1: "a1 ∈ underS r a2"
hence a12: "{a1,a2} ⊆ under r a2" and "a1 ≠ a2" using r.REFL a
unfolding underS_def under_def refl_on_def Field_def by auto
hence "g a1 ≠ g a2" using IH1a[OF a2] unfolding inj_on_def by auto
hence "g a1 ∈ underS s (g a2)" using IH1b[OF a2] a12
unfolding underS_def under_def by auto
} note C = this
have ga: "g a ∈ Field s" unfolding g using s.suc_inField[OF A0 A] .
have aa: "a ∈ under r a"
using a r.REFL unfolding under_def underS_def refl_on_def by auto
show ?case proof safe
show "bij_betw g (under r a) (under s (g a))" unfolding bij_betw_def proof safe
show "inj_on g (under r a)" proof(rule r.inj_on_Field)
fix a1 a2 assume "a1 ∈ under r a" and a2: "a2 ∈ under r a" and a1: "a1 ∈ underS r a2"
hence a22: "a2 ∈ under r a2" and a12: "a1 ∈ under r a2" "a1 ≠ a2"
using a r.REFL unfolding under_def underS_def refl_on_def by auto
show "g a1 ≠ g a2"
proof(cases "a2 = a")
case False hence "a2 ∈ underS r a"
using a2 unfolding underS_def under_def by auto
from IH1a[OF this] show ?thesis using a12 a22 unfolding inj_on_def by auto
qed(insert B a1, unfold underS_def, auto)
qed(unfold under_def Field_def, auto)
next
fix a1 assume a1: "a1 ∈ under r a"
show "g a1 ∈ under s (g a)"
proof(cases "a1 = a")
case True thus ?thesis
using ga s.REFL unfolding refl_on_def under_def by auto
next
case False
hence a1: "a1 ∈ underS r a" using a1 unfolding underS_def under_def by auto
thus ?thesis using B unfolding underS_def under_def by auto
qed
next
fix b1 assume b1: "b1 ∈ under s (g a)"
show "b1 ∈ g ` under r a"
proof(cases "b1 = g a")
case True thus ?thesis using aa by auto
next
case False
hence "b1 ∈ underS s (g a)" using b1 unfolding underS_def under_def by auto
from s.underS_suc[OF this[unfolded g] A0]
obtain a1 where a1: "a1 ∈ underS r a" and b1: "b1 ∈ under s (g a1)" by auto
obtain a2 where "a2 ∈ under r a1" and b1: "b1 = g a2" using IH1b[OF a1] b1 by auto
hence "a2 ∈ under r a" using a1
by (metis r.ANTISYM r.TRANS in_mono underS_subset_under under_underS_trans)
thus ?thesis using b1 by auto
qed
qed
next
have "(g a, f a) ∈ s" unfolding g proof(rule s.suc_least[OF A0])
fix b1 assume "b1 ∈ g ` underS r a"
then obtain a1 where a1: "b1 = g a1" and a1: "a1 ∈ underS r a" by auto
hence "b1 ∈ underS s (f a)"
using a by (metis ‹⋀a1. a1 ∈ underS r a ⟹ g a1 ∈ underS s (f a)›)
thus "f a ≠ b1 ∧ (b1, f a) ∈ s" unfolding underS_def by auto
qed(insert fa, auto)
thus "g a ∈ under s (f a)" unfolding under_def by auto
qed
qed
}
thus ?thesis unfolding embed_def by auto
qed

corollary ordLeq_def2:
"r ≤o s ⟷ Well_order r ∧ Well_order s ∧
(∃ f. ∀ a ∈ Field r. f a ∈ Field s ∧ f ` underS r a ⊆ underS s (f a))"
using embed_in_Field[of r s] embed_underS2[of r s] embedI[of r s]
unfolding ordLeq_def by fast

lemma iso_oproj:
assumes r: "Well_order r" and s: "Well_order s" and f: "iso r s f"
shows "oproj r s f"
using assms unfolding oproj_def
by (subst (asm) iso_iff3) (auto simp: bij_betw_def)

theorem oproj_embed:
assumes r: "Well_order r" and s: "Well_order s" and f: "oproj r s f"
shows "∃ g. embed s r g"
proof (rule embedI[OF s r, of "inv_into (Field r) f"], unfold underS_def, safe)
fix b assume "b ∈ Field s"
thus "inv_into (Field r) f b ∈ Field r" using oproj_Field2[OF f] by (metis imageI inv_into_into)
next
fix a b assume "b ∈ Field s" "a ≠ b" "(a, b) ∈ s"
"inv_into (Field r) f a = inv_into (Field r) f b"
with f show False by (auto dest!: inv_into_injective simp: Field_def oproj_def)
next
fix a b assume *: "b ∈ Field s" "a ≠ b" "(a, b) ∈ s"
{ assume "(inv_into (Field r) f a, inv_into (Field r) f b) ∉ r"
moreover
from *(3) have "a ∈ Field s" unfolding Field_def by auto
with *(1,2) have
"inv_into (Field r) f a ∈ Field r" "inv_into (Field r) f b ∈ Field r"
"inv_into (Field r) f a ≠ inv_into (Field r) f b"
by (auto dest!: oproj_Field2[OF f] inv_into_injective intro!: inv_into_into)
ultimately have "(inv_into (Field r) f b, inv_into (Field r) f a) ∈ r"
using r by (auto simp: well_order_on_def linear_order_on_def total_on_def)
with f[unfolded oproj_def compat_def] *(1) ‹a ∈ Field s›
f_inv_into_f[of b f "Field r"] f_inv_into_f[of a f "Field r"]
have "(b, a) ∈ s" by (metis in_mono)
with *(2,3) s have False
by (auto simp: well_order_on_def linear_order_on_def partial_order_on_def antisym_def)
} thus "(inv_into (Field r) f a, inv_into (Field r) f b) ∈ r" by blast
qed

corollary oproj_ordLeq:
assumes r: "Well_order r" and s: "Well_order s" and f: "oproj r s f"
shows "s ≤o r"
using oproj_embed[OF assms] r s unfolding ordLeq_def by auto

end
```