--- a/src/HOL/Cardinals/Constructions_on_Wellorders.thy Fri Jan 10 16:18:18 2014 +0100
+++ b/src/HOL/Cardinals/Constructions_on_Wellorders.thy Fri Jan 10 17:24:52 2014 +0100
@@ -15,7 +15,10 @@
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 \<noteq> {} \<Longrightarrow> Func A {} = {}" by auto
subsection {* Restriction to a set *}
@@ -430,4 +433,648 @@
using NE_P P omax_ordLess by blast
qed
+
+
+section {* 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 \<in> 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 \<notin> A" and "f b \<notin> 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 "\<And>a. (\<And> a1. a1 \<in> underS a \<Longrightarrow> P a1) \<Longrightarrow> P a"
+ shows "P a"
+ by (induct rule: well_order_induct) (rule assms, simp add: underS_def)
+
+lemma suc_underS:
+assumes B: "B \<subseteq> Field r" and A: "AboveS B \<noteq> {}" and b: "b \<in> B"
+shows "b \<in> underS (suc B)"
+using suc_AboveS[OF B A] b unfolding underS_def AboveS_def by auto
+
+lemma underS_supr:
+assumes bA: "b \<in> underS (supr A)" and A: "A \<subseteq> Field r"
+shows "\<exists> a \<in> A. b \<in> underS a"
+proof(rule ccontr, auto)
+ have bb: "b \<in> Field r" using bA unfolding underS_def Field_def by auto
+ assume "\<forall>a\<in>A. b \<notin> underS a"
+ hence 0: "\<forall>a \<in> A. (a,b) \<in> r" using A bA unfolding underS_def
+ by simp (metis REFL in_mono max2_def max2_greater refl_on_domain)
+ have "(supr A, b) \<in> 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 \<in> underS (suc A)" and A: "A \<subseteq> Field r"
+shows "\<exists> a \<in> A. b \<in> under a"
+proof(rule ccontr, auto)
+ have bb: "b \<in> Field r" using bA unfolding underS_def Field_def by auto
+ assume "\<forall>a\<in>A. b \<notin> under a"
+ hence 0: "\<forall>a \<in> A. a \<in> underS b" using A bA unfolding underS_def
+ by simp (metis (lifting) bb max2_def max2_greater mem_Collect_eq rel.under_def set_rev_mp)
+ have "(suc A, b) \<in> r" apply(rule suc_least[OF A bb]) using 0 unfolding rel.underS_def by auto
+ thus False using bA unfolding underS_def by simp (metis ANTISYM antisymD)
+qed
+
+lemma (in rel) AboveS_underS:
+assumes "i \<in> Field r"
+shows "i \<in> AboveS (underS i)"
+using assms unfolding AboveS_def underS_def by auto
+
+lemma (in wo_rel) in_underS_supr:
+assumes j: "j \<in> underS i" and i: "i \<in> A" and A: "A \<subseteq> Field r" and AA: "Above A \<noteq> {}"
+shows "j \<in> underS (supr A)"
+proof-
+ have "(i,supr A) \<in> r" using supr_greater[OF A AA i] .
+ thus ?thesis using j unfolding underS_def
+ by simp (metis REFL TRANS max2_def max2_equals1 rel.refl_on_domain transD)
+qed
+
+lemma inj_on_Field:
+assumes A: "A \<subseteq> Field r" and f: "\<And> a b. \<lbrakk>a \<in> A; b \<in> A; a \<in> underS b\<rbrakk> \<Longrightarrow> f a \<noteq> f b"
+shows "inj_on f A"
+unfolding inj_on_def proof safe
+ fix a b assume a: "a \<in> A" and b: "b \<in> A" and fab: "f a = f b"
+ {assume "a \<in> underS b"
+ hence False using f[OF a b] fab by auto
+ }
+ moreover
+ {assume "b \<in> 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) \<notin> r \<or> j = i" and "i \<in> Field r" and "j \<in> Field r"
+shows "(i,j) \<in> 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 "\<And>j1 j2. \<lbrakk>j2 \<in> underS i; (j1, j2) \<in> r\<rbrakk> \<Longrightarrow> R j1 initial_segment_of R j2"
+shows "{R j |j. j \<in> underS i} \<in> Chains init_seg_of"
+unfolding Chains_def proof safe
+ fix j ja assume jS: "j \<in> underS i" and jaS: "ja \<in> underS i"
+ and init: "(R ja, R j) \<notin> init_seg_of"
+ hence jja: "{j,ja} \<subseteq> Field r" and j: "j \<in> Field r" and ja: "ja \<in> Field r"
+ and jjai: "(j,i) \<in> r" "(ja,i) \<in> r"
+ and i: "i \<notin> {j,ja}" unfolding Field_def rel.underS_def by auto
+ have jj: "(j,j) \<in> r" and jaja: "(ja,ja) \<in> 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 "\<And>j1 j2. \<lbrakk>j2 \<in> Field r; (j1, j2) \<in> r\<rbrakk> \<Longrightarrow> R j1 initial_segment_of R j2"
+shows "{R j |j. j \<in> Field r} \<in> Chains init_seg_of"
+unfolding Chains_def proof safe
+ fix j ja assume jS: "j \<in> Field r" and jaS: "ja \<in> Field r"
+ and init: "(R ja, R j) \<notin> init_seg_of"
+ hence jja: "{j,ja} \<subseteq> Field r" and j: "j \<in> Field r" and ja: "ja \<in> Field r"
+ unfolding Field_def rel.underS_def by auto
+ have jj: "(j,j) \<in> r" and jaja: "(ja,ja) \<in> 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
+
+subsection {* Successor and limit elements of an ordinal *}
+
+definition "succ i \<equiv> suc {i}"
+
+definition "isSucc i \<equiv> \<exists> j. aboveS j \<noteq> {} \<and> i = succ j"
+
+definition "zero = minim (Field r)"
+
+definition "isLim i \<equiv> \<not> isSucc i"
+
+lemma zero_smallest[simp]:
+assumes "j \<in> Field r" shows "(zero, j) \<in> 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 \<noteq> {}" shows "zero \<in> Field r"
+using assms unfolding zero_def by (metis Field_ofilter minim_in ofilter_def)
+
+lemma leq_zero_imp[simp]:
+"(x, zero) \<in> r \<Longrightarrow> x = zero"
+by (metis ANTISYM WELL antisymD well_order_on_domain zero_smallest)
+
+lemma leq_zero[simp]:
+assumes "Field r \<noteq> {}" shows "(x, zero) \<in> r \<longleftrightarrow> x = zero"
+using zero_in_Field[OF assms] in_notinI[of x zero] by auto
+
+lemma under_zero[simp]:
+assumes "Field r \<noteq> {}" 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 \<noteq> {} \<Longrightarrow> isSucc (succ i)"
+unfolding isSucc_def succ_def by auto
+
+lemma succ_in_diff:
+assumes "aboveS i \<noteq> {}" shows "(i,succ i) \<in> r \<and> succ i \<noteq> 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 \<noteq> {}" shows "succ i \<in> Field r"
+using succ_in[OF assms] unfolding Field_def by auto
+
+lemma succ_not_in:
+assumes "aboveS i \<noteq> {}" shows "(succ i, i) \<notin> r"
+proof
+ assume 1: "(succ i, i) \<in> r"
+ hence "succ i \<in> Field r \<and> i \<in> Field r" unfolding Field_def by auto
+ hence "(i, succ i) \<in> r \<and> succ i \<noteq> i" using assms by auto
+ thus False using 1 by (metis ANTISYM antisymD)
+qed
+
+lemma not_isSucc_zero: "\<not> isSucc zero"
+proof
+ assume "isSucc zero"
+ moreover
+ then obtain i where "aboveS i \<noteq> {}" and 1: "minim (Field r) = succ i"
+ unfolding isSucc_def zero_def by auto
+ hence "succ i \<in> Field r" by auto
+ ultimately 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) \<in> r" and "i \<noteq> j"
+shows "(succ i, j) \<in> r"
+unfolding succ_def apply(rule suc_least)
+using assms unfolding Field_def by auto
+
+lemma isLim_supr:
+assumes f: "i \<in> Field r" and l: "isLim i"
+shows "i = supr (underS i)"
+proof(rule equals_supr)
+ fix j assume j: "j \<in> Field r" and 1: "\<And> j'. j' \<in> underS i \<Longrightarrow> (j',j) \<in> r"
+ show "(i,j) \<in> r" proof(intro in_notinI[OF _ f j], safe)
+ assume ji: "(j,i) \<in> r" "j \<noteq> i"
+ hence a: "aboveS j \<noteq> {}" unfolding aboveS_def by auto
+ hence "i \<noteq> succ j" using l unfolding isLim_def isSucc_def by auto
+ moreover have "(succ j, i) \<in> r" using succ_smallest[OF ji] by auto
+ ultimately have "succ j \<in> underS i" unfolding underS_def by auto
+ hence "(succ j, j) \<in> 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 \<equiv> SOME j. j \<in> Field r \<and> aboveS j \<noteq> {} \<and> succ j = i"
+
+lemma pred_Field_succ:
+assumes "isSucc i" shows "pred i \<in> Field r \<and> aboveS (pred i) \<noteq> {} \<and> succ (pred i) = i"
+proof-
+ obtain j where a: "aboveS j \<noteq> {}" and i: "i = succ j" using assms unfolding isSucc_def by auto
+ have 1: "j \<in> Field r" "j \<noteq> 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) \<in> r"
+proof-
+ def j \<equiv> "pred i"
+ have i: "i = succ j" using assms unfolding j_def by simp
+ have a: "aboveS j \<noteq> {}" 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 \<noteq> i"
+by (metis aboveS_pred assms succ_diff succ_pred)
+
+(* todo: pred maximal, pred injective? *)
+
+lemma succ_inj[simp]:
+assumes "aboveS i \<noteq> {}" and "aboveS j \<noteq> {}"
+shows "succ i = succ j \<longleftrightarrow> i = j"
+proof safe
+ assume s: "succ i = succ j"
+ {assume "i \<noteq> j" and "(i,j) \<in> r"
+ hence "(succ i, j) \<in> r" using assms by (metis succ_smallest)
+ hence False using s assms by (metis succ_not_in)
+ }
+ moreover
+ {assume "i \<noteq> j" and "(j,i) \<in> r"
+ hence "(succ j, i) \<in> 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 \<noteq> {}" 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 \<noteq> {}"
+shows "(j, succ i) \<in> r \<longleftrightarrow> (j,i) \<in> r \<or> j = succ i"
+apply safe
+ apply (metis WELL assms in_notinI rel.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 rel.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 \<noteq> {}"
+shows "underS (succ i) = under i"
+unfolding underS_def under_def by (auto simp: assms succ_not_in)
+
+lemma succ_mono:
+assumes "aboveS j \<noteq> {}" and "(i,j) \<in> r"
+shows "(succ i, succ j) \<in> r"
+by (metis (full_types) assms less_succ succ_smallest)
+
+lemma under_succ[simp]:
+assumes "aboveS i \<noteq> {}"
+shows "under (succ i) = insert (succ i) (under i)"
+using less_succ[OF assms] unfolding under_def by auto
+
+definition mergeSL :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where
+"mergeSL S L f i \<equiv>
+ if isSucc i then S (pred i) (f (pred i))
+ else L f i"
+
+
+subsection {* Well-order recursion with (zero), succesor, and limit *}
+
+definition worecSL :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where "worecSL S L \<equiv> worec (mergeSL S L)"
+
+definition "adm_woL L \<equiv> \<forall>f g i. isLim i \<and> (\<forall>j\<in>underS i. f j = g j) \<longrightarrow> L f i = L g i"
+
+lemma mergeSL:
+assumes "adm_woL L" shows "adm_wo (mergeSL S L)"
+unfolding adm_wo_def proof safe
+ fix f g :: "'a => 'b" and i :: 'a
+ assume 1: "\<forall>j\<in>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 \<in> 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 \<Longrightarrow> 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 \<noteq> {}"
+shows "worecSL S L (succ j) = S j (worecSL S L j)"
+proof-
+ def i \<equiv> "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 \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
+where "worecZSL Z S L \<equiv> worecSL S (\<lambda> f a. if a = zero then Z else L f a)"
+
+lemma worecZSL_zero:
+assumes a: "adm_woL L"
+shows "worecZSL Z S L zero = Z"
+proof-
+ let ?L = "\<lambda> 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 \<noteq> {}"
+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 \<noteq> zero"
+shows "worecZSL Z S L i = L (worecZSL Z S L) i"
+proof-
+ let ?L = "\<lambda> 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
+
+
+subsection {* Well-order succ-lim induction: *}
+
+lemma ord_cases:
+obtains j where "i = succ j" and "aboveS j \<noteq> {}" | "isLim i"
+by (metis isLim_def isSucc_def)
+
+lemma well_order_inductSL[case_names Suc Lim]:
+assumes SUCC: "\<And>i. \<lbrakk>aboveS i \<noteq> {}; P i\<rbrakk> \<Longrightarrow> P (succ i)" and
+LIM: "\<And>i. \<lbrakk>isLim i; \<And>j. j \<in> underS i \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i"
+shows "P i"
+proof(induction rule: well_order_induct)
+ fix i assume 0: "\<forall>j. j \<noteq> i \<and> (j, i) \<in> r \<longrightarrow> P j"
+ show "P i" proof(cases i rule: ord_cases)
+ fix j assume i: "i = succ j" and j: "aboveS j \<noteq> {}"
+ hence "j \<noteq> i \<and> (j, i) \<in> 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: "\<And>i. \<lbrakk>aboveS i \<noteq> {}; P i\<rbrakk> \<Longrightarrow> P (succ i)" and
+LIM: "\<And>i. \<lbrakk>isLim i; i \<noteq> zero; \<And>j. j \<in> underS i \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i"
+shows "P i"
+apply(induction rule: well_order_inductSL) using assms by auto
+
+(* Succesor and limit ordinals *)
+definition "isSuccOrd \<equiv> \<exists> j \<in> Field r. \<forall> i \<in> Field r. (i,j) \<in> r"
+definition "isLimOrd \<equiv> \<not> isSuccOrd"
+
+lemma isLimOrd_succ:
+assumes isLimOrd and "i \<in> Field r"
+shows "succ i \<in> 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 \<in> Field r"
+shows "aboveS i \<noteq> {}"
+proof-
+ obtain j where "j \<in> Field r" and "(j,i) \<notin> r"
+ using assms unfolding isLimOrd_def isSuccOrd_def by auto
+ hence "(i,j) \<in> r \<and> j \<noteq> i" by (metis i max2_def max2_greater)
+ thus ?thesis unfolding aboveS_def by auto
+qed
+
+lemma succ_aboveS_isLimOrd:
+assumes "\<forall> i \<in> Field r. aboveS i \<noteq> {} \<and> succ i \<in> Field r"
+shows isLimOrd
+unfolding isLimOrd_def isSuccOrd_def proof safe
+ fix j assume j: "j \<in> Field r" "\<forall>i\<in>Field r. (i, j) \<in> r"
+ hence "(succ j, j) \<in> r" using assms by auto
+ moreover have "aboveS j \<noteq> {}" 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 \<in> underS i"
+shows "\<exists> k. k \<in> underS i \<and> j \<in> underS k"
+proof-
+ have a: "aboveS j \<noteq> {}" 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 \<equiv> wo_rel.zero"
+abbreviation "succ \<equiv> wo_rel.succ"
+abbreviation "pred \<equiv> wo_rel.pred"
+abbreviation "isSucc \<equiv> wo_rel.isSucc"
+abbreviation "isLim \<equiv> wo_rel.isLim"
+abbreviation "isLimOrd \<equiv> wo_rel.isLimOrd"
+abbreviation "isSuccOrd \<equiv> wo_rel.isSuccOrd"
+abbreviation "adm_woL \<equiv> wo_rel.adm_woL"
+abbreviation "worecSL \<equiv> wo_rel.worecSL"
+abbreviation "worecZSL \<equiv> wo_rel.worecZSL"
+
+section {* Projections of Wellorders *}
+
+definition "oproj r s f \<equiv> Field s \<subseteq> f ` (Field r) \<and> compat r s f"
+
+lemma oproj_in:
+assumes "oproj r s f" and "(a,a') \<in> r"
+shows "(f a, f a') \<in> s"
+using assms unfolding oproj_def compat_def by auto
+
+lemma oproj_Field:
+assumes f: "oproj r s f" and a: "a \<in> Field r"
+shows "f a \<in> 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 \<in> Field s"
+shows "\<exists> a \<in> Field r. f a = b"
+using assms unfolding oproj_def by auto
+
+lemma oproj_under:
+assumes f: "oproj r s f" and a: "a \<in> under r a'"
+shows "f a \<in> under s (f a')"
+using oproj_in[OF f] a unfolding rel.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: "\<And> a. a \<in> Field r \<Longrightarrow> f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)"
+shows "\<exists> 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 = "\<lambda> g a. suc s (g ` underS r a)"
+ def g \<equiv> "worec r ?G"
+ have adm: "adm_wo r ?G" unfolding r.adm_wo_def image_def by auto
+ (* *)
+ {fix a assume "a \<in> Field r"
+ hence "bij_betw g (under r a) (under s (g a)) \<and>
+ g a \<in> under s (f a)"
+ proof(induction a rule: r.underS_induct)
+ case (1 a)
+ hence a: "a \<in> Field r"
+ and IH1a: "\<And> a1. a1 \<in> underS r a \<Longrightarrow> inj_on g (under r a1)"
+ and IH1b: "\<And> a1. a1 \<in> underS r a \<Longrightarrow> g ` under r a1 = under s (g a1)"
+ and IH2: "\<And> a1. a1 \<in> underS r a \<Longrightarrow> g a1 \<in> under s (f a1)"
+ unfolding rel.underS_def Field_def bij_betw_def by auto
+ have fa: "f a \<in> 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 \<subseteq> Field s"
+ using IH1b by (metis IH2 image_subsetI in_mono rel.under_Field)
+ {fix a1 assume a1: "a1 \<in> underS r a"
+ from IH2[OF this] have "g a1 \<in> under s (f a1)" .
+ moreover have "f a1 \<in> underS s (f a)" using f[OF a] a1 by auto
+ ultimately have "g a1 \<in> underS s (f a)" by (metis s.ANTISYM s.TRANS rel.under_underS_trans)
+ }
+ hence "f a \<in> AboveS s (g ` underS r a)" unfolding rel.AboveS_def
+ using fa by simp (metis (lifting, full_types) mem_Collect_eq rel.underS_def)
+ hence A: "AboveS s (g ` underS r a) \<noteq> {}" by auto
+ have B: "\<And> a1. a1 \<in> underS r a \<Longrightarrow> g a1 \<in> underS s (g a)"
+ unfolding g apply(rule s.suc_underS[OF A0 A]) by auto
+ {fix a1 a2 assume a2: "a2 \<in> underS r a" and 1: "a1 \<in> underS r a2"
+ hence a12: "{a1,a2} \<subseteq> under r a2" and "a1 \<noteq> a2" using r.REFL a
+ unfolding rel.underS_def rel.under_def refl_on_def Field_def by auto
+ hence "g a1 \<noteq> g a2" using IH1a[OF a2] unfolding inj_on_def by auto
+ hence "g a1 \<in> underS s (g a2)" using IH1b[OF a2] a12
+ unfolding rel.underS_def rel.under_def by auto
+ } note C = this
+ have ga: "g a \<in> Field s" unfolding g using s.suc_inField[OF A0 A] .
+ have aa: "a \<in> under r a"
+ using a r.REFL unfolding rel.under_def rel.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 \<in> under r a" and a2: "a2 \<in> under r a" and a1: "a1 \<in> underS r a2"
+ hence a22: "a2 \<in> under r a2" and a12: "a1 \<in> under r a2" "a1 \<noteq> a2"
+ using a r.REFL unfolding rel.under_def rel.underS_def refl_on_def by auto
+ show "g a1 \<noteq> g a2"
+ proof(cases "a2 = a")
+ case False hence "a2 \<in> underS r a"
+ using a2 unfolding rel.underS_def rel.under_def by auto
+ from IH1a[OF this] show ?thesis using a12 a22 unfolding inj_on_def by auto
+ qed(insert B a1, unfold rel.underS_def, auto)
+ qed(unfold rel.under_def Field_def, auto)
+ next
+ fix a1 assume a1: "a1 \<in> under r a"
+ show "g a1 \<in> under s (g a)"
+ proof(cases "a1 = a")
+ case True thus ?thesis
+ using ga s.REFL unfolding refl_on_def rel.under_def by auto
+ next
+ case False
+ hence a1: "a1 \<in> underS r a" using a1 unfolding rel.underS_def rel.under_def by auto
+ thus ?thesis using B unfolding rel.underS_def rel.under_def by auto
+ qed
+ next
+ fix b1 assume b1: "b1 \<in> under s (g a)"
+ show "b1 \<in> g ` under r a"
+ proof(cases "b1 = g a")
+ case True thus ?thesis using aa by auto
+ next
+ case False
+ hence "b1 \<in> underS s (g a)" using b1 unfolding rel.underS_def rel.under_def by auto
+ from s.underS_suc[OF this[unfolded g] A0]
+ obtain a1 where a1: "a1 \<in> underS r a" and b1: "b1 \<in> under s (g a1)" by auto
+ obtain a2 where "a2 \<in> under r a1" and b1: "b1 = g a2" using IH1b[OF a1] b1 by auto
+ hence "a2 \<in> under r a" using a1
+ by (metis r.ANTISYM r.TRANS in_mono rel.underS_subset_under rel.under_underS_trans)
+ thus ?thesis using b1 by auto
+ qed
+ qed
+ next
+ have "(g a, f a) \<in> s" unfolding g proof(rule s.suc_least[OF A0])
+ fix b1 assume "b1 \<in> g ` underS r a"
+ then obtain a1 where a1: "b1 = g a1" and a1: "a1 \<in> underS r a" by auto
+ hence "b1 \<in> underS s (f a)"
+ using a by (metis `\<And>a1. a1 \<in> underS r a \<Longrightarrow> g a1 \<in> underS s (f a)`)
+ thus "f a \<noteq> b1 \<and> (b1, f a) \<in> s" unfolding rel.underS_def by auto
+ qed(insert fa, auto)
+ thus "g a \<in> under s (f a)" unfolding rel.under_def by auto
+ qed
+ qed
+ }
+ thus ?thesis unfolding embed_def by auto
+qed
+
+corollary ordLeq_def2:
+ "r \<le>o s \<longleftrightarrow> Well_order r \<and> Well_order s \<and>
+ (\<exists> f. \<forall> a \<in> Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> 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 "\<exists> g. embed s r g"
+proof (rule embedI[OF s r, of "inv_into (Field r) f"], unfold rel.underS_def, safe)
+ fix b assume "b \<in> Field s"
+ thus "inv_into (Field r) f b \<in> Field r" using oproj_Field2[OF f] by (metis imageI inv_into_into)
+next
+ fix a b assume "b \<in> Field s" "a \<noteq> b" "(a, b) \<in> 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 \<in> Field s" "a \<noteq> b" "(a, b) \<in> s"
+ { assume "(inv_into (Field r) f a, inv_into (Field r) f b) \<notin> r"
+ moreover
+ from *(3) have "a \<in> Field s" unfolding Field_def by auto
+ with *(1,2) have
+ "inv_into (Field r) f a \<in> Field r" "inv_into (Field r) f b \<in> Field r"
+ "inv_into (Field r) f a \<noteq> 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) \<in> 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 \<in> Field s`
+ f_inv_into_f[of b f "Field r"] f_inv_into_f[of a f "Field r"]
+ have "(b, a) \<in> 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) \<in> 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 \<le>o r"
+using oproj_embed[OF assms] r s unfolding ordLeq_def by auto
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Cardinals/Ordinal_Arithmetic.thy Fri Jan 10 17:24:52 2014 +0100
@@ -0,0 +1,1728 @@
+(* Title: HOL/Cardinals/Ordinal_Arithmetic.thy
+ Author: Dmitriy Traytel, TU Muenchen
+ Copyright 2014
+
+Ordinal arithmetic.
+*)
+
+header {* Ordinal Arithmetic *}
+
+theory Ordinal_Arithmetic
+imports Constructions_on_Wellorders
+begin
+
+definition osum :: "'a rel \<Rightarrow> 'b rel \<Rightarrow> ('a + 'b) rel" (infixr "+o" 70)
+where
+ "r +o r' = map_pair Inl Inl ` r \<union> map_pair Inr Inr ` r' \<union>
+ {(Inl a, Inr a') | a a' . a \<in> Field r \<and> a' \<in> Field r'}"
+
+lemma Field_osum: "Field(r +o r') = Inl ` Field r \<union> Inr ` Field r'"
+ unfolding osum_def Field_def by auto
+
+lemma osum_Refl:"\<lbrakk>Refl r; Refl r'\<rbrakk> \<Longrightarrow> Refl (r +o r')"
+ (*Need first unfold Field_osum, only then osum_def*)
+ unfolding refl_on_def Field_osum unfolding osum_def by blast
+
+lemma osum_trans:
+assumes TRANS: "trans r" and TRANS': "trans r'"
+shows "trans (r +o r')"
+proof(unfold trans_def, safe)
+ fix x y z assume *: "(x, y) \<in> r +o r'" "(y, z) \<in> r +o r'"
+ thus "(x, z) \<in> r +o r'"
+ proof (cases x y z rule: sum.exhaust[case_product sum.exhaust sum.exhaust])
+ case (Inl_Inl_Inl a b c)
+ with * have "(a,b) \<in> r" "(b,c) \<in> r" unfolding osum_def by auto
+ with TRANS have "(a,c) \<in> r" unfolding trans_def by blast
+ with Inl_Inl_Inl show ?thesis unfolding osum_def by auto
+ next
+ case (Inl_Inl_Inr a b c)
+ with * have "a \<in> Field r" "c \<in> Field r'" unfolding osum_def Field_def by auto
+ with Inl_Inl_Inr show ?thesis unfolding osum_def by auto
+ next
+ case (Inl_Inr_Inr a b c)
+ with * have "a \<in> Field r" "c \<in> Field r'" unfolding osum_def Field_def by auto
+ with Inl_Inr_Inr show ?thesis unfolding osum_def by auto
+ next
+ case (Inr_Inr_Inr a b c)
+ with * have "(a,b) \<in> r'" "(b,c) \<in> r'" unfolding osum_def by auto
+ with TRANS' have "(a,c) \<in> r'" unfolding trans_def by blast
+ with Inr_Inr_Inr show ?thesis unfolding osum_def by auto
+ qed (auto simp: osum_def)
+qed
+
+lemma osum_Preorder: "\<lbrakk>Preorder r; Preorder r'\<rbrakk> \<Longrightarrow> Preorder (r +o r')"
+ unfolding preorder_on_def using osum_Refl osum_trans by blast
+
+lemma osum_antisym: "\<lbrakk>antisym r; antisym r'\<rbrakk> \<Longrightarrow> antisym (r +o r')"
+ unfolding antisym_def osum_def by auto
+
+lemma osum_Partial_order: "\<lbrakk>Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow> Partial_order (r +o r')"
+ unfolding partial_order_on_def using osum_Preorder osum_antisym by blast
+
+lemma osum_Total: "\<lbrakk>Total r; Total r'\<rbrakk> \<Longrightarrow> Total (r +o r')"
+ unfolding total_on_def Field_osum unfolding osum_def by blast
+
+lemma osum_Linear_order: "\<lbrakk>Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow> Linear_order (r +o r')"
+ unfolding linear_order_on_def using osum_Partial_order osum_Total by blast
+
+lemma osum_wf:
+assumes WF: "wf r" and WF': "wf r'"
+shows "wf (r +o r')"
+unfolding wf_eq_minimal2 unfolding Field_osum
+proof(intro allI impI, elim conjE)
+ fix A assume *: "A \<subseteq> Inl ` Field r \<union> Inr ` Field r'" and **: "A \<noteq> {}"
+ obtain B where B_def: "B = A Int Inl ` Field r" by blast
+ show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r +o r'"
+ proof(cases "B = {}")
+ case False
+ hence "B \<noteq> {}" "B \<le> Inl ` Field r" using B_def by auto
+ hence "Inl -` B \<noteq> {}" "Inl -` B \<le> Field r" unfolding vimage_def by auto
+ then obtain a where 1: "a \<in> Inl -` B" and "\<forall>a1 \<in> Inl -` B. (a1, a) \<notin> r"
+ using WF unfolding wf_eq_minimal2 by metis
+ hence "\<forall>a1 \<in> A. (a1, Inl a) \<notin> r +o r'"
+ unfolding osum_def using B_def ** by (auto simp: vimage_def Field_def)
+ thus ?thesis using 1 unfolding B_def by auto
+ next
+ case True
+ hence 1: "A \<le> Inr ` Field r'" using * B_def by auto
+ with ** have "Inr -`A \<noteq> {}" "Inr -` A \<le> Field r'" unfolding vimage_def by auto
+ with ** obtain a' where 2: "a' \<in> Inr -` A" and "\<forall>a1' \<in> Inr -` A. (a1',a') \<notin> r'"
+ using WF' unfolding wf_eq_minimal2 by metis
+ hence "\<forall>a1' \<in> A. (a1', Inr a') \<notin> r +o r'"
+ unfolding osum_def using ** 1 by (auto simp: vimage_def Field_def)
+ thus ?thesis using 2 by blast
+ qed
+qed
+
+lemma osum_minus_Id:
+ assumes r: "Total r" "\<not> (r \<le> Id)" and r': "Total r'" "\<not> (r' \<le> Id)"
+ shows "(r +o r') - Id \<le> (r - Id) +o (r' - Id)"
+ unfolding osum_def Total_Id_Field[OF r] Total_Id_Field[OF r'] by auto
+
+lemma osum_minus_Id1:
+ "r \<le> Id \<Longrightarrow> (r +o r') - Id \<le> (Inl ` Field r \<times> Inr ` Field r') \<union> (map_pair Inr Inr ` (r' - Id))"
+ unfolding osum_def by auto
+
+lemma osum_minus_Id2:
+ "r' \<le> Id \<Longrightarrow> (r +o r') - Id \<le> (map_pair Inl Inl ` (r - Id)) \<union> (Inl ` Field r \<times> Inr ` Field r')"
+ unfolding osum_def by auto
+
+lemma osum_wf_Id:
+ assumes TOT: "Total r" and TOT': "Total r'" and WF: "wf(r - Id)" and WF': "wf(r' - Id)"
+ shows "wf ((r +o r') - Id)"
+proof(cases "r \<le> Id \<or> r' \<le> Id")
+ case False
+ thus ?thesis
+ using osum_minus_Id[of r r'] assms osum_wf[of "r - Id" "r' - Id"]
+ wf_subset[of "(r - Id) +o (r' - Id)" "(r +o r') - Id"] by auto
+next
+ have 1: "wf (Inl ` Field r \<times> Inr ` Field r')" by (rule wf_Int_Times) auto
+ case True
+ thus ?thesis
+ proof (elim disjE)
+ assume "r \<subseteq> Id"
+ thus "wf ((r +o r') - Id)"
+ by (rule wf_subset[rotated, OF osum_minus_Id1 wf_Un[OF 1 wf_map_pair_image[OF WF']]]) auto
+ next
+ assume "r' \<subseteq> Id"
+ thus "wf ((r +o r') - Id)"
+ by (rule wf_subset[rotated, OF osum_minus_Id2 wf_Un[OF wf_map_pair_image[OF WF] 1]]) auto
+ qed
+qed
+
+lemma osum_Well_order:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "Well_order (r +o r')"
+proof-
+ have "Total r \<and> Total r'" using WELL WELL' by (auto simp add: order_on_defs)
+ thus ?thesis using assms unfolding well_order_on_def
+ using osum_Linear_order osum_wf_Id by blast
+qed
+
+lemma osum_embedL:
+ assumes WELL: "Well_order r" and WELL': "Well_order r'"
+ shows "embed r (r +o r') Inl"
+proof -
+ have 1: "Well_order (r +o r')" using assms by (auto simp add: osum_Well_order)
+ moreover
+ have "compat r (r +o r') Inl" unfolding compat_def osum_def by auto
+ moreover
+ have "ofilter (r +o r') (Inl ` Field r)"
+ unfolding wo_rel.ofilter_def[unfolded wo_rel_def, OF 1] Field_osum rel.under_def
+ unfolding osum_def Field_def by auto
+ ultimately show ?thesis using assms by (auto simp add: embed_iff_compat_inj_on_ofilter)
+qed
+
+corollary osum_ordLeqL:
+ assumes WELL: "Well_order r" and WELL': "Well_order r'"
+ shows "r \<le>o r +o r'"
+ using assms osum_embedL osum_Well_order unfolding ordLeq_def by blast
+
+lemma dir_image_alt: "dir_image r f = map_pair f f ` r"
+ unfolding dir_image_def map_pair_def by auto
+
+lemma map_pair_ordIso: "\<lbrakk>Well_order r; inj_on f (Field r)\<rbrakk> \<Longrightarrow> map_pair f f ` r =o r"
+ unfolding dir_image_alt[symmetric] by (rule ordIso_symmetric[OF dir_image_ordIso])
+
+definition oprod :: "'a rel \<Rightarrow> 'b rel \<Rightarrow> ('a \<times> 'b) rel" (infixr "*o" 80)
+where "r *o r' = {((x1, y1), (x2, y2)).
+ (((y1, y2) \<in> r' - Id \<and> x1 \<in> Field r \<and> x2 \<in> Field r) \<or>
+ ((y1, y2) \<in> Restr Id (Field r') \<and> (x1, x2) \<in> r))}"
+
+lemma Field_oprod: "Field (r *o r') = Field r \<times> Field r'"
+ unfolding oprod_def Field_def by auto blast+
+
+lemma oprod_Refl:"\<lbrakk>Refl r; Refl r'\<rbrakk> \<Longrightarrow> Refl (r *o r')"
+ unfolding refl_on_def Field_oprod unfolding oprod_def by auto
+
+lemma oprod_trans:
+ assumes "trans r" "trans r'" "antisym r" "antisym r'"
+ shows "trans (r *o r')"
+proof(unfold trans_def, safe)
+ fix x y z assume *: "(x, y) \<in> r *o r'" "(y, z) \<in> r *o r'"
+ thus "(x, z) \<in> r *o r'"
+ unfolding oprod_def
+ apply safe
+ apply (metis assms(2) transE)
+ apply (metis assms(2) transE)
+ apply (metis assms(2) transE)
+ apply (metis assms(4) antisymD)
+ apply (metis assms(4) antisymD)
+ apply (metis assms(2) transE)
+ apply (metis assms(4) antisymD)
+ apply (metis Field_def Range_iff Un_iff)
+ apply (metis Field_def Range_iff Un_iff)
+ apply (metis Field_def Range_iff Un_iff)
+ apply (metis Field_def Domain_iff Un_iff)
+ apply (metis Field_def Domain_iff Un_iff)
+ apply (metis Field_def Domain_iff Un_iff)
+ apply (metis assms(1) transE)
+ apply (metis assms(1) transE)
+ apply (metis assms(1) transE)
+ apply (metis assms(1) transE)
+ done
+qed
+
+lemma oprod_Preorder: "\<lbrakk>Preorder r; Preorder r'; antisym r; antisym r'\<rbrakk> \<Longrightarrow> Preorder (r *o r')"
+ unfolding preorder_on_def using oprod_Refl oprod_trans by blast
+
+lemma oprod_antisym: "\<lbrakk>antisym r; antisym r'\<rbrakk> \<Longrightarrow> antisym (r *o r')"
+ unfolding antisym_def oprod_def by auto
+
+lemma oprod_Partial_order: "\<lbrakk>Partial_order r; Partial_order r'\<rbrakk> \<Longrightarrow> Partial_order (r *o r')"
+ unfolding partial_order_on_def using oprod_Preorder oprod_antisym by blast
+
+lemma oprod_Total: "\<lbrakk>Total r; Total r'\<rbrakk> \<Longrightarrow> Total (r *o r')"
+ unfolding total_on_def Field_oprod unfolding oprod_def by auto
+
+lemma oprod_Linear_order: "\<lbrakk>Linear_order r; Linear_order r'\<rbrakk> \<Longrightarrow> Linear_order (r *o r')"
+ unfolding linear_order_on_def using oprod_Partial_order oprod_Total by blast
+
+lemma oprod_wf:
+assumes WF: "wf r" and WF': "wf r'"
+shows "wf (r *o r')"
+unfolding wf_eq_minimal2 unfolding Field_oprod
+proof(intro allI impI, elim conjE)
+ fix A assume *: "A \<subseteq> Field r \<times> Field r'" and **: "A \<noteq> {}"
+ then obtain y where y: "y \<in> snd ` A" "\<forall>y'\<in>snd ` A. (y', y) \<notin> r'"
+ using spec[OF WF'[unfolded wf_eq_minimal2], of "snd ` A"] by auto
+ let ?A = "fst ` A \<inter> {x. (x, y) \<in> A}"
+ from * y have "?A \<noteq> {}" "?A \<subseteq> Field r" by auto
+ then obtain x where x: "x \<in> ?A" and "\<forall>x'\<in> ?A. (x', x) \<notin> r"
+ using spec[OF WF[unfolded wf_eq_minimal2], of "?A"] by auto
+ with y have "\<forall>a'\<in>A. (a', (x, y)) \<notin> r *o r'"
+ unfolding oprod_def mem_Collect_eq split_beta fst_conv snd_conv Id_def by auto
+ moreover from x have "(x, y) \<in> A" by auto
+ ultimately show "\<exists>a\<in>A. \<forall>a'\<in>A. (a', a) \<notin> r *o r'" by blast
+qed
+
+lemma oprod_minus_Id:
+ assumes r: "Total r" "\<not> (r \<le> Id)" and r': "Total r'" "\<not> (r' \<le> Id)"
+ shows "(r *o r') - Id \<le> (r - Id) *o (r' - Id)"
+ unfolding oprod_def Total_Id_Field[OF r] Total_Id_Field[OF r'] by auto
+
+lemma oprod_minus_Id1:
+ "r \<le> Id \<Longrightarrow> r *o r' - Id \<le> {((x,y1), (x,y2)). x \<in> Field r \<and> (y1, y2) \<in> (r' - Id)}"
+ unfolding oprod_def by auto
+
+lemma wf_extend_oprod1:
+ assumes "wf r"
+ shows "wf {((x,y1), (x,y2)) . x \<in> A \<and> (y1, y2) \<in> r}"
+proof (unfold wf_eq_minimal2, intro allI impI, elim conjE)
+ fix B
+ assume *: "B \<subseteq> Field {((x,y1), (x,y2)) . x \<in> A \<and> (y1, y2) \<in> r}" and "B \<noteq> {}"
+ from image_mono[OF *, of snd] have "snd ` B \<subseteq> Field r" unfolding Field_def by force
+ with `B \<noteq> {}` obtain x where x: "x \<in> snd ` B" "\<forall>x'\<in>snd ` B. (x', x) \<notin> r"
+ using spec[OF assms[unfolded wf_eq_minimal2], of "snd ` B"] by auto
+ then obtain a where "(a, x) \<in> B" by auto
+ moreover
+ from * x have "\<forall>a'\<in>B. (a', (a, x)) \<notin> {((x,y1), (x,y2)) . x \<in> A \<and> (y1, y2) \<in> r}" by auto
+ ultimately show "\<exists>ax\<in>B. \<forall>a'\<in>B. (a', ax) \<notin> {((x,y1), (x,y2)) . x \<in> A \<and> (y1, y2) \<in> r}" by blast
+qed
+
+lemma oprod_minus_Id2:
+ "r' \<le> Id \<Longrightarrow> r *o r' - Id \<le> {((x1,y), (x2,y)). (x1, x2) \<in> (r - Id) \<and> y \<in> Field r'}"
+ unfolding oprod_def by auto
+
+lemma wf_extend_oprod2:
+ assumes "wf r"
+ shows "wf {((x1,y), (x2,y)) . (x1, x2) \<in> r \<and> y \<in> A}"
+proof (unfold wf_eq_minimal2, intro allI impI, elim conjE)
+ fix B
+ assume *: "B \<subseteq> Field {((x1, y), (x2, y)). (x1, x2) \<in> r \<and> y \<in> A}" and "B \<noteq> {}"
+ from image_mono[OF *, of fst] have "fst ` B \<subseteq> Field r" unfolding Field_def by force
+ with `B \<noteq> {}` obtain x where x: "x \<in> fst ` B" "\<forall>x'\<in>fst ` B. (x', x) \<notin> r"
+ using spec[OF assms[unfolded wf_eq_minimal2], of "fst ` B"] by auto
+ then obtain a where "(x, a) \<in> B" by auto
+ moreover
+ from * x have "\<forall>a'\<in>B. (a', (x, a)) \<notin> {((x1, y), x2, y). (x1, x2) \<in> r \<and> y \<in> A}" by auto
+ ultimately show "\<exists>xa\<in>B. \<forall>a'\<in>B. (a', xa) \<notin> {((x1, y), x2, y). (x1, x2) \<in> r \<and> y \<in> A}" by blast
+qed
+
+lemma oprod_wf_Id:
+ assumes TOT: "Total r" and TOT': "Total r'" and WF: "wf(r - Id)" and WF': "wf(r' - Id)"
+ shows "wf ((r *o r') - Id)"
+proof(cases "r \<le> Id \<or> r' \<le> Id")
+ case False
+ thus ?thesis
+ using oprod_minus_Id[of r r'] assms oprod_wf[of "r - Id" "r' - Id"]
+ wf_subset[of "(r - Id) *o (r' - Id)" "(r *o r') - Id"] by auto
+next
+ case True
+ thus ?thesis using wf_subset[OF wf_extend_oprod1[OF WF'] oprod_minus_Id1]
+ wf_subset[OF wf_extend_oprod2[OF WF] oprod_minus_Id2] by auto
+qed
+
+lemma oprod_Well_order:
+assumes WELL: "Well_order r" and WELL': "Well_order r'"
+shows "Well_order (r *o r')"
+proof-
+ have "Total r \<and> Total r'" using WELL WELL' by (auto simp add: order_on_defs)
+ thus ?thesis using assms unfolding well_order_on_def
+ using oprod_Linear_order oprod_wf_Id by blast
+qed
+
+lemma oprod_embed:
+ assumes WELL: "Well_order r" and WELL': "Well_order r'" and "r' \<noteq> {}"
+ shows "embed r (r *o r') (\<lambda>x. (x, minim r' (Field r')))" (is "embed _ _ ?f")
+proof -
+ from assms(3) have r': "Field r' \<noteq> {}" unfolding Field_def by auto
+ have minim[simp]: "minim r' (Field r') \<in> Field r'"
+ using wo_rel.minim_inField[unfolded wo_rel_def, OF WELL' _ r'] by auto
+ { fix b
+ assume "(b, minim r' (Field r')) \<in> r'"
+ moreover hence "b \<in> Field r'" unfolding Field_def by auto
+ hence "(minim r' (Field r'), b) \<in> r'"
+ using wo_rel.minim_least[unfolded wo_rel_def, OF WELL' subset_refl] r' by auto
+ ultimately have "b = minim r' (Field r')"
+ by (metis WELL' antisym_def linear_order_on_def partial_order_on_def well_order_on_def)
+ } note * = this
+ have 1: "Well_order (r *o r')" using assms by (auto simp add: oprod_Well_order)
+ moreover
+ from r' have "compat r (r *o r') ?f" unfolding compat_def oprod_def by auto
+ moreover
+ from * have "ofilter (r *o r') (?f ` Field r)"
+ unfolding wo_rel.ofilter_def[unfolded wo_rel_def, OF 1] Field_oprod rel.under_def
+ unfolding oprod_def by auto (auto simp: image_iff Field_def)
+ moreover have "inj_on ?f (Field r)" unfolding inj_on_def by auto
+ ultimately show ?thesis using assms by (auto simp add: embed_iff_compat_inj_on_ofilter)
+qed
+
+corollary oprod_ordLeq: "\<lbrakk>Well_order r; Well_order r'; r' \<noteq> {}\<rbrakk> \<Longrightarrow> r \<le>o r *o r'"
+ using oprod_embed oprod_Well_order unfolding ordLeq_def by blast
+
+definition "support z A f = {x \<in> A. f x \<noteq> z}"
+
+lemma support_Un[simp]: "support z (A \<union> B) f = support z A f \<union> support z B f"
+ unfolding support_def by auto
+
+lemma support_upd[simp]: "support z A (f(x := z)) = support z A f - {x}"
+ unfolding support_def by auto
+
+lemma support_upd_subset[simp]: "support z A (f(x := y)) \<subseteq> support z A f \<union> {x}"
+ unfolding support_def by auto
+
+lemma fun_unequal_in_support:
+ assumes "f \<noteq> g" "f \<in> Func A B" "g \<in> Func A C"
+ shows "(support z A f \<union> support z A g) \<inter> {a. f a \<noteq> g a} \<noteq> {}" (is "?L \<inter> ?R \<noteq> {}")
+proof -
+ from assms(1) obtain x where x: "f x \<noteq> g x" by blast
+ hence "x \<in> ?R" by simp
+ moreover from assms(2-3) x have "x \<in> A" unfolding Func_def by fastforce
+ with x have "x \<in> ?L" unfolding support_def by auto
+ ultimately show ?thesis by auto
+qed
+
+definition fin_support where
+ "fin_support z A = {f. finite (support z A f)}"
+
+lemma finite_support: "f \<in> fin_support z A \<Longrightarrow> finite (support z A f)"
+ unfolding support_def fin_support_def by auto
+
+lemma fin_support_Field_osum:
+ "f \<in> fin_support z (Inl ` A \<union> Inr ` B) \<longleftrightarrow>
+ (f o Inl) \<in> fin_support z A \<and> (f o Inr) \<in> fin_support z B" (is "?L \<longleftrightarrow> ?R1 \<and> ?R2")
+proof safe
+ assume ?L
+ from `?L` show ?R1 unfolding fin_support_def support_def
+ by (fastforce simp: image_iff elim: finite_surj[of _ _ "sum_case id undefined"])
+ from `?L` show ?R2 unfolding fin_support_def support_def
+ by (fastforce simp: image_iff elim: finite_surj[of _ _ "sum_case undefined id"])
+next
+ assume ?R1 ?R2
+ thus ?L unfolding fin_support_def support_Un
+ by (auto simp: support_def elim: finite_surj[of _ _ Inl] finite_surj[of _ _ Inr])
+qed
+
+lemma Func_upd: "\<lbrakk>f \<in> Func A B; x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> f(x := y) \<in> Func A B"
+ unfolding Func_def by auto
+
+context wo_rel
+begin
+
+definition isMaxim :: "'a set \<Rightarrow> 'a \<Rightarrow> bool"
+where "isMaxim A b \<equiv> b \<in> A \<and> (\<forall>a \<in> A. (a,b) \<in> r)"
+
+definition maxim :: "'a set \<Rightarrow> 'a"
+where "maxim A \<equiv> THE b. isMaxim A b"
+
+lemma isMaxim_unique[intro]: "\<lbrakk>isMaxim A x; isMaxim A y\<rbrakk> \<Longrightarrow> x = y"
+ unfolding isMaxim_def using antisymD[OF ANTISYM, of x y] by auto
+
+lemma maxim_isMaxim: "\<lbrakk>finite A; A \<noteq> {}; A \<subseteq> Field r\<rbrakk> \<Longrightarrow> isMaxim A (maxim A)"
+unfolding maxim_def
+proof (rule theI', rule ex_ex1I[OF _ isMaxim_unique, rotated], assumption+,
+ induct A rule: finite_induct)
+ case (insert x A)
+ thus ?case
+ proof (cases "A = {}")
+ case True
+ moreover have "isMaxim {x} x" unfolding isMaxim_def using refl_onD[OF REFL] insert(5) by auto
+ ultimately show ?thesis by blast
+ next
+ case False
+ with insert(3,5) obtain y where "isMaxim A y" by blast
+ with insert(2,5) have "if (y, x) \<in> r then isMaxim (insert x A) x else isMaxim (insert x A) y"
+ unfolding isMaxim_def subset_eq by (metis insert_iff max2_def max2_equals1 max2_iff)
+ thus ?thesis by metis
+ qed
+qed simp
+
+lemma maxim_in: "\<lbrakk>finite A; A \<noteq> {}; A \<subseteq> Field r\<rbrakk> \<Longrightarrow> maxim A \<in> A"
+ using maxim_isMaxim unfolding isMaxim_def by auto
+
+lemma maxim_greatest: "\<lbrakk>finite A; x \<in> A; A \<subseteq> Field r\<rbrakk> \<Longrightarrow> (x, maxim A) \<in> r"
+ using maxim_isMaxim unfolding isMaxim_def by auto
+
+lemma isMaxim_zero: "isMaxim A zero \<Longrightarrow> A = {zero}"
+ unfolding isMaxim_def by auto
+
+lemma maxim_insert:
+ assumes "finite A" "A \<noteq> {}" "A \<subseteq> Field r" "x \<in> Field r"
+ shows "maxim (insert x A) = max2 x (maxim A)"
+proof -
+ from assms have *: "isMaxim (insert x A) (maxim (insert x A))" "isMaxim A (maxim A)"
+ using maxim_isMaxim by auto
+ show ?thesis
+ proof (cases "(x, maxim A) \<in> r")
+ case True
+ with *(2) have "isMaxim (insert x A) (maxim A)" unfolding isMaxim_def
+ using transD[OF TRANS, of _ x "maxim A"] by blast
+ with *(1) True show ?thesis unfolding max2_def by (metis isMaxim_unique)
+ next
+ case False
+ hence "(maxim A, x) \<in> r" by (metis *(2) assms(3,4) in_mono in_notinI isMaxim_def)
+ with *(2) assms(4) have "isMaxim (insert x A) x" unfolding isMaxim_def
+ using transD[OF TRANS, of _ "maxim A" x] refl_onD[OF REFL, of x] by blast
+ with *(1) False show ?thesis unfolding max2_def by (metis isMaxim_unique)
+ qed
+qed
+
+lemma maxim_Un:
+ assumes "finite A" "A \<noteq> {}" "A \<subseteq> Field r" "finite B" "B \<noteq> {}" "B \<subseteq> Field r"
+ shows "maxim (A \<union> B) = max2 (maxim A) (maxim B)"
+proof -
+ from assms have *: "isMaxim (A \<union> B) (maxim (A \<union> B))" "isMaxim A (maxim A)" "isMaxim B (maxim B)"
+ using maxim_isMaxim by auto
+ show ?thesis
+ proof (cases "(maxim A, maxim B) \<in> r")
+ case True
+ with *(2,3) have "isMaxim (A \<union> B) (maxim B)" unfolding isMaxim_def
+ using transD[OF TRANS, of _ "maxim A" "maxim B"] by blast
+ with *(1) True show ?thesis unfolding max2_def by (metis isMaxim_unique)
+ next
+ case False
+ hence "(maxim B, maxim A) \<in> r" by (metis *(2,3) assms(3,6) in_mono in_notinI isMaxim_def)
+ with *(2,3) have "isMaxim (A \<union> B) (maxim A)" unfolding isMaxim_def
+ using transD[OF TRANS, of _ "maxim B" "maxim A"] by blast
+ with *(1) False show ?thesis unfolding max2_def by (metis isMaxim_unique)
+ qed
+qed
+
+lemma maxim_insert_zero:
+ assumes "finite A" "A \<noteq> {}" "A \<subseteq> Field r"
+ shows "maxim (insert zero A) = maxim A"
+using assms zero_in_Field maxim_in[OF assms] by (subst maxim_insert[unfolded max2_def]) auto
+
+lemma maxim_equality: "isMaxim A x \<Longrightarrow> maxim A = x"
+ unfolding maxim_def by (rule the_equality) auto
+
+lemma maxim_singleton:
+ "x \<in> Field r \<Longrightarrow> maxim {x} = x"
+ using refl_onD[OF REFL] by (intro maxim_equality) (simp add: isMaxim_def)
+
+lemma maxim_Int: "\<lbrakk>finite A; A \<noteq> {}; A \<subseteq> Field r; maxim A \<in> B\<rbrakk> \<Longrightarrow> maxim (A \<inter> B) = maxim A"
+ by (rule maxim_equality) (auto simp: isMaxim_def intro: maxim_in maxim_greatest)
+
+lemma maxim_mono: "\<lbrakk>X \<subseteq> Y; finite Y; X \<noteq> {}; Y \<subseteq> Field r\<rbrakk> \<Longrightarrow> (maxim X, maxim Y) \<in> r"
+ using maxim_in[OF finite_subset, of X Y] by (auto intro: maxim_greatest)
+
+definition "max_fun_diff f g \<equiv> maxim ({a \<in> Field r. f a \<noteq> g a})"
+
+lemma max_fun_diff_commute: "max_fun_diff f g = max_fun_diff g f"
+ unfolding max_fun_diff_def by metis
+
+lemma zero_under: "x \<in> Field r \<Longrightarrow> zero \<in> under x"
+ unfolding under_def by (auto intro: zero_smallest)
+
+end
+
+definition "FinFunc r s = Func (Field s) (Field r) \<inter> fin_support (zero r) (Field s)"
+
+lemma FinFuncD: "\<lbrakk>f \<in> FinFunc r s; x \<in> Field s\<rbrakk> \<Longrightarrow> f x \<in> Field r"
+ unfolding FinFunc_def Func_def by (fastforce split: option.splits)
+
+locale wo_rel2 =
+ fixes r s
+ assumes rWELL: "Well_order r"
+ and sWELL: "Well_order s"
+begin
+
+interpretation r!: wo_rel r by unfold_locales (rule rWELL)
+interpretation s!: wo_rel s by unfold_locales (rule sWELL)
+
+abbreviation "SUPP \<equiv> support r.zero (Field s)"
+abbreviation "FINFUNC \<equiv> FinFunc r s"
+lemmas FINFUNCD = FinFuncD[of _ r s]
+
+lemma fun_diff_alt: "{a \<in> Field s. f a \<noteq> g a} = (SUPP f \<union> SUPP g) \<inter> {a. f a \<noteq> g a}"
+ by (auto simp: support_def)
+
+lemma max_fun_diff_alt:
+ "s.max_fun_diff f g = s.maxim ((SUPP f \<union> SUPP g) \<inter> {a. f a \<noteq> g a})"
+ unfolding s.max_fun_diff_def fun_diff_alt ..
+
+lemma isMaxim_max_fun_diff: "\<lbrakk>f \<noteq> g; f \<in> FINFUNC; g \<in> FINFUNC\<rbrakk> \<Longrightarrow>
+ s.isMaxim {a \<in> Field s. f a \<noteq> g a} (s.max_fun_diff f g)"
+ using fun_unequal_in_support[of f g] unfolding max_fun_diff_alt fun_diff_alt fun_eq_iff
+ by (intro s.maxim_isMaxim) (auto simp: FinFunc_def fin_support_def support_def)
+
+lemma max_fun_diff_in: "\<lbrakk>f \<noteq> g; f \<in> FINFUNC; g \<in> FINFUNC\<rbrakk> \<Longrightarrow>
+ s.max_fun_diff f g \<in> {a \<in> Field s. f a \<noteq> g a}"
+ using isMaxim_max_fun_diff unfolding s.isMaxim_def by blast
+
+lemma max_fun_diff_max: "\<lbrakk>f \<noteq> g; f \<in> FINFUNC; g \<in> FINFUNC; x \<in> {a \<in> Field s. f a \<noteq> g a}\<rbrakk> \<Longrightarrow>
+ (x, s.max_fun_diff f g) \<in> s"
+ using isMaxim_max_fun_diff unfolding s.isMaxim_def by blast
+
+lemma max_fun_diff:
+ "\<lbrakk>f \<noteq> g; f \<in> FINFUNC; g \<in> FINFUNC\<rbrakk> \<Longrightarrow>
+ (\<exists>a b. a \<noteq> b \<and> a \<in> Field r \<and> b \<in> Field r \<and>
+ f (s.max_fun_diff f g) = a \<and> g (s.max_fun_diff f g) = b)"
+ using isMaxim_max_fun_diff[of f g] unfolding s.isMaxim_def FinFunc_def Func_def by auto
+
+lemma max_fun_diff_le_eq:
+ "\<lbrakk>(s.max_fun_diff f g, x) \<in> s; f \<noteq> g; f \<in> FINFUNC; g \<in> FINFUNC; x \<noteq> s.max_fun_diff f g\<rbrakk> \<Longrightarrow>
+ f x = g x"
+ using max_fun_diff_max[of f g x] antisymD[OF s.ANTISYM, of "s.max_fun_diff f g" x]
+ by (auto simp: Field_def)
+
+lemma max_fun_diff_max2:
+ assumes ineq: "s.max_fun_diff f g = s.max_fun_diff g h \<longrightarrow>
+ f (s.max_fun_diff f g) \<noteq> h (s.max_fun_diff g h)" and
+ fg: "f \<noteq> g" and gh: "g \<noteq> h" and fh: "f \<noteq> h" and
+ f: "f \<in> FINFUNC" and g: "g \<in> FINFUNC" and h: "h \<in> FINFUNC"
+ shows "s.max_fun_diff f h = s.max2 (s.max_fun_diff f g) (s.max_fun_diff g h)"
+ (is "?fh = s.max2 ?fg ?gh")
+proof (cases "?fg = ?gh")
+ case True
+ with ineq have "f ?fg \<noteq> h ?fg" by simp
+ moreover
+ { fix x assume x: "x \<in> {a \<in> Field s. f a \<noteq> h a}"
+ hence "(x, ?fg) \<in> s"
+ proof (cases "x = ?fg")
+ case False show ?thesis
+ proof (rule ccontr)
+ assume "(x, ?fg) \<notin> s"
+ with max_fun_diff_in[OF fg f g] x False have *: "(?fg, x) \<in> s" by (blast intro: s.in_notinI)
+ hence "f x = g x" by (rule max_fun_diff_le_eq[OF _ fg f g False])
+ moreover have "g x = h x" using max_fun_diff_le_eq[OF _ gh g h] False True * by simp
+ ultimately show False using x by simp
+ qed
+ qed (simp add: refl_onD[OF s.REFL])
+ }
+ ultimately have "s.isMaxim {a \<in> Field s. f a \<noteq> h a} ?fg"
+ unfolding s.isMaxim_def using max_fun_diff_in[OF fg f g] by simp
+ hence "?fh = ?fg" using isMaxim_max_fun_diff[OF fh f h] by blast
+ thus ?thesis unfolding True s.max2_def by simp
+next
+ case False note * = this
+ show ?thesis
+ proof (cases "(?fg, ?gh) \<in> s")
+ case True
+ hence *: "f ?gh = g ?gh" by (rule max_fun_diff_le_eq[OF _ fg f g *[symmetric]])
+ hence "s.isMaxim {a \<in> Field s. f a \<noteq> h a} ?gh" using isMaxim_max_fun_diff[OF gh g h]
+ isMaxim_max_fun_diff[OF fg f g] transD[OF s.TRANS _ True]
+ unfolding s.isMaxim_def by auto
+ hence "?fh = ?gh" using isMaxim_max_fun_diff[OF fh f h] by blast
+ thus ?thesis using True unfolding s.max2_def by simp
+ next
+ case False
+ with max_fun_diff_in[OF fg f g] max_fun_diff_in[OF gh g h] have True: "(?gh, ?fg) \<in> s"
+ by (blast intro: s.in_notinI)
+ hence *: "g ?fg = h ?fg" by (rule max_fun_diff_le_eq[OF _ gh g h *])
+ hence "s.isMaxim {a \<in> Field s. f a \<noteq> h a} ?fg" using isMaxim_max_fun_diff[OF gh g h]
+ isMaxim_max_fun_diff[OF fg f g] True transD[OF s.TRANS, of _ _ ?fg]
+ unfolding s.isMaxim_def by auto
+ hence "?fh = ?fg" using isMaxim_max_fun_diff[OF fh f h] by blast
+ thus ?thesis using False unfolding s.max2_def by simp
+ qed
+qed
+
+
+definition oexp where
+ "oexp = {(f, g) . f \<in> FINFUNC \<and> g \<in> FINFUNC \<and>
+ ((let m = s.max_fun_diff f g in (f m, g m) \<in> r) \<or> f = g)}"
+
+lemma Field_oexp: "Field oexp = FINFUNC"
+ unfolding oexp_def FinFunc_def by (auto simp: Let_def Field_def)
+
+lemma oexp_Refl: "Refl oexp"
+ unfolding refl_on_def Field_oexp unfolding oexp_def by (auto simp: Let_def)
+
+lemma oexp_trans: "trans oexp"
+proof (unfold trans_def, safe)
+ fix f g h :: "'b \<Rightarrow> 'a"
+ let ?fg = "s.max_fun_diff f g"
+ and ?gh = "s.max_fun_diff g h"
+ and ?fh = "s.max_fun_diff f h"
+ assume oexp: "(f, g) \<in> oexp" "(g, h) \<in> oexp"
+ thus "(f, h) \<in> oexp"
+ proof (cases "f = g \<or> g = h")
+ case False
+ with oexp have "f \<in> FINFUNC" "g \<in> FINFUNC" "h \<in> FINFUNC"
+ "(f ?fg, g ?fg) \<in> r" "(g ?gh, h ?gh) \<in> r" unfolding oexp_def Let_def by auto
+ note * = this False
+ show ?thesis
+ proof (cases "f \<noteq> h")
+ case True
+ show ?thesis
+ proof (cases "?fg = ?gh \<longrightarrow> f ?fg \<noteq> h ?gh")
+ case True
+ show ?thesis using max_fun_diff_max2[of f g h, OF True] * `f \<noteq> h` max_fun_diff_in
+ r.max2_iff[OF FINFUNCD FINFUNCD] r.max2_equals1[OF FINFUNCD FINFUNCD] max_fun_diff_le_eq
+ s.in_notinI[OF disjI1] unfolding oexp_def Let_def s.max2_def mem_Collect_eq by safe metis
+ next
+ case False with * show ?thesis unfolding oexp_def Let_def
+ using antisymD[OF r.ANTISYM, of "g ?gh" "h ?gh"] max_fun_diff_in[of g h] by auto
+ qed
+ qed (auto simp: oexp_def *(3))
+ qed auto
+qed
+
+lemma oexp_Preorder: "Preorder oexp"
+ unfolding preorder_on_def using oexp_Refl oexp_trans by blast
+
+lemma oexp_antisym: "antisym oexp"
+proof (unfold antisym_def, safe, rule ccontr)
+ fix f g assume "(f, g) \<in> oexp" "(g, f) \<in> oexp" "g \<noteq> f"
+ thus False using refl_onD[OF r.REFL FINFUNCD] max_fun_diff_in unfolding oexp_def Let_def
+ by (auto dest!: antisymD[OF r.ANTISYM] simp: s.max_fun_diff_commute)
+qed
+
+lemma oexp_Partial_order: "Partial_order oexp"
+ unfolding partial_order_on_def using oexp_Preorder oexp_antisym by blast
+
+lemma oexp_Total: "Total oexp"
+ unfolding total_on_def Field_oexp unfolding oexp_def using FINFUNCD max_fun_diff_in
+ by (auto simp: Let_def s.max_fun_diff_commute intro!: r.in_notinI)
+
+lemma oexp_Linear_order: "Linear_order oexp"
+ unfolding linear_order_on_def using oexp_Partial_order oexp_Total by blast
+
+definition "const = (\<lambda>x. if x \<in> Field s then r.zero else undefined)"
+
+lemma const_in[simp]: "x \<in> Field s \<Longrightarrow> const x = r.zero"
+ unfolding const_def by auto
+
+lemma const_notin[simp]: "x \<notin> Field s \<Longrightarrow> const x = undefined"
+ unfolding const_def by auto
+
+lemma const_Int_Field[simp]: "Field s \<inter> - {x. const x = r.zero} = {}"
+ by auto
+
+lemma const_FINFUNC[simp]: "Field r \<noteq> {} \<Longrightarrow> const \<in> FINFUNC"
+ unfolding FinFunc_def Func_def fin_support_def support_def const_def Int_iff mem_Collect_eq
+ using r.zero_in_Field by (metis (lifting) Collect_empty_eq finite.emptyI)
+
+lemma const_least:
+ assumes "Field r \<noteq> {}" "f \<in> FINFUNC"
+ shows "(const, f) \<in> oexp"
+proof (cases "f = const")
+ case True thus ?thesis using refl_onD[OF oexp_Refl] assms(2) unfolding Field_oexp by auto
+next
+ case False
+ with assms show ?thesis using max_fun_diff_in[of f const]
+ unfolding oexp_def Let_def by (auto intro: r.zero_smallest FinFuncD simp: s.max_fun_diff_commute)
+qed
+
+lemma support_not_const:
+ assumes "F \<subseteq> FINFUNC" and "const \<notin> F"
+ shows "\<forall>f \<in> F. finite (SUPP f) \<and> SUPP f \<noteq> {} \<and> SUPP f \<subseteq> Field s"
+proof (intro ballI conjI)
+ fix f assume "f \<in> F"
+ thus "finite (SUPP f)" "SUPP f \<subseteq> Field s"
+ using assms(1) unfolding FinFunc_def fin_support_def support_def by auto
+ show "SUPP f \<noteq> {}"
+ proof (rule ccontr, unfold not_not)
+ assume "SUPP f = {}"
+ moreover from `f \<in> F` assms(1) have "f \<in> FINFUNC" by blast
+ ultimately have "f = const"
+ by (auto simp: fun_eq_iff support_def FinFunc_def Func_def const_def)
+ with assms(2) `f \<in> F` show False by blast
+ qed
+qed
+
+lemma maxim_isMaxim_support:
+ assumes f: "F \<subseteq> FINFUNC" and "const \<notin> F"
+ shows "\<forall>f \<in> F. s.isMaxim (SUPP f) (s.maxim (SUPP f))"
+ using support_not_const[OF assms] by (auto intro!: s.maxim_isMaxim)
+
+lemma oexp_empty2: "Field s = {} \<Longrightarrow> oexp = {(\<lambda>x. undefined, \<lambda>x. undefined)}"
+ unfolding oexp_def FinFunc_def fin_support_def support_def by auto
+
+lemma oexp_empty: "\<lbrakk>Field r = {}; Field s \<noteq> {}\<rbrakk> \<Longrightarrow> oexp = {}"
+ unfolding oexp_def FinFunc_def Let_def by auto
+
+lemma fun_upd_FINFUNC: "\<lbrakk>f \<in> FINFUNC; x \<in> Field s; y \<in> Field r\<rbrakk> \<Longrightarrow> f(x := y) \<in> FINFUNC"
+ unfolding FinFunc_def Func_def fin_support_def
+ by (auto intro: finite_subset[OF support_upd_subset])
+
+lemma fun_upd_same_oexp:
+ assumes "(f, g) \<in> oexp" "f x = g x" "x \<in> Field s" "y \<in> Field r"
+ shows "(f(x := y), g(x := y)) \<in> oexp"
+proof -
+ from assms(1) fun_upd_FINFUNC[OF _ assms(3,4)] have fg: "f(x := y) \<in> FINFUNC" "g(x := y) \<in> FINFUNC"
+ unfolding oexp_def by auto
+ moreover from assms(2) have "s.max_fun_diff (f(x := y)) (g(x := y)) = s.max_fun_diff f g"
+ unfolding s.max_fun_diff_def by auto metis
+ ultimately show ?thesis using assms refl_onD[OF r.REFL] unfolding oexp_def Let_def by auto
+qed
+
+lemma fun_upd_smaller_oexp:
+ assumes "f \<in> FINFUNC" "x \<in> Field s" "y \<in> Field r" "(y, f x) \<in> r"
+ shows "(f(x := y), f) \<in> oexp"
+ using assms fun_upd_FINFUNC[OF assms(1-3)] s.maxim_singleton[of "x"]
+ unfolding oexp_def FinFunc_def Let_def fin_support_def s.max_fun_diff_def by (auto simp: fun_eq_iff)
+
+lemma oexp_wf_Id: "wf (oexp - Id)"
+proof (cases "Field r = {} \<or> Field s = {}")
+ case True thus ?thesis using oexp_empty oexp_empty2 by fastforce
+next
+ case False
+ hence Fields: "Field s \<noteq> {}" "Field r \<noteq> {}" by simp_all
+ hence [simp]: "r.zero \<in> Field r" by (intro r.zero_in_Field)
+ have const[simp]: "\<And>F. \<lbrakk>const \<in> F; F \<subseteq> FINFUNC\<rbrakk> \<Longrightarrow> \<exists>f0\<in>F. \<forall>f\<in>F. (f0, f) \<in> oexp"
+ using const_least[OF Fields(2)] by auto
+ show ?thesis
+ unfolding Linear_order_wf_diff_Id[OF oexp_Linear_order] Field_oexp
+ proof (intro allI impI)
+ fix A assume A: "A \<subseteq> FINFUNC" "A \<noteq> {}"
+ { fix y F
+ have "F \<subseteq> FINFUNC \<and> (\<exists>f \<in> F. y = s.maxim (SUPP f)) \<longrightarrow>
+ (\<exists>f0 \<in> F. \<forall>f \<in> F. (f0, f) \<in> oexp)" (is "?P F y")
+ proof (induct y arbitrary: F rule: s.well_order_induct)
+ case (1 y)
+ show ?case
+ proof (intro impI, elim conjE bexE)
+ fix f assume F: "F \<subseteq> FINFUNC" "f \<in> F" "y = s.maxim (SUPP f)"
+ thus "\<exists>f0\<in>F. \<forall>f\<in>F. (f0, f) \<in> oexp"
+ proof (cases "const \<in> F")
+ case False
+ with F have maxF: "\<forall>f \<in> F. s.isMaxim (SUPP f) (s.maxim (SUPP f))"
+ and SUPPF: "\<forall>f \<in> F. finite (SUPP f) \<and> SUPP f \<noteq> {} \<and> SUPP f \<subseteq> Field s"
+ using maxim_isMaxim_support support_not_const by auto
+ def z \<equiv> "s.minim {s.maxim (SUPP f) | f. f \<in> F}"
+ from F SUPPF maxF have zmin: "s.isMinim {s.maxim (SUPP f) | f. f \<in> F} z"
+ unfolding z_def by (intro s.minim_isMinim) (auto simp: s.isMaxim_def)
+ with F have zy: "(z, y) \<in> s" unfolding s.isMinim_def by auto
+ hence zField: "z \<in> Field s" unfolding Field_def by auto
+ def x0 \<equiv> "r.minim {f z | f. f \<in> F \<and> z = s.maxim (SUPP f)}"
+ from F(1,2) maxF(1) SUPPF zmin
+ have x0min: "r.isMinim {f z | f. f \<in> F \<and> z = s.maxim (SUPP f)} x0"
+ unfolding x0_def s.isMaxim_def s.isMinim_def
+ by (blast intro!: r.minim_isMinim FinFuncD[of _ r s])
+ with maxF(1) SUPPF F(1) have x0Field: "x0 \<in> Field r"
+ unfolding r.isMinim_def s.isMaxim_def by (auto intro!: FINFUNCD)
+ from x0min maxF(1) SUPPF F(1) have x0notzero: "x0 \<noteq> r.zero"
+ unfolding r.isMinim_def s.isMaxim_def FinFunc_def Func_def support_def
+ by fastforce
+ def G \<equiv> "{f(z := r.zero) | f. f \<in> F \<and> z = s.maxim (SUPP f) \<and> f z = x0}"
+ from zmin x0min have "G \<noteq> {}" unfolding G_def z_def s.isMinim_def r.isMinim_def by blast
+ have GF: "G \<subseteq> (\<lambda>f. f(z := r.zero)) ` F" unfolding G_def by auto
+ have "G \<subseteq> fin_support r.zero (Field s)"
+ unfolding FinFunc_def fin_support_def proof safe
+ fix g assume "g \<in> G"
+ with GF obtain f where "f \<in> F" "g = f(z := r.zero)" by auto
+ moreover with SUPPF have "finite (SUPP f)" by blast
+ ultimately show "finite (SUPP g)"
+ by (elim finite_subset[rotated]) (auto simp: support_def)
+ qed
+ moreover from F GF zField have "G \<subseteq> Func (Field s) (Field r)"
+ using Func_upd[of _ "Field s" "Field r" z r.zero] unfolding FinFunc_def by auto
+ ultimately have G: "G \<subseteq> FINFUNC" unfolding FinFunc_def by blast
+ hence "\<exists>g0\<in>G. \<forall>g\<in>G. (g0, g) \<in> oexp"
+ proof (cases "const \<in> G")
+ case False
+ with G have maxG: "\<forall>g \<in> G. s.isMaxim (SUPP g) (s.maxim (SUPP g))"
+ and SUPPG: "\<forall>g \<in> G. finite (SUPP g) \<and> SUPP g \<noteq> {} \<and> SUPP g \<subseteq> Field s"
+ using maxim_isMaxim_support support_not_const by auto
+ def y' \<equiv> "s.minim {s.maxim (SUPP f) | f. f \<in> G}"
+ from G SUPPG maxG `G \<noteq> {}` have y'min: "s.isMinim {s.maxim (SUPP f) | f. f \<in> G} y'"
+ unfolding y'_def by (intro s.minim_isMinim) (auto simp: s.isMaxim_def)
+ moreover
+ have "\<forall>g \<in> G. z \<notin> SUPP g" unfolding support_def G_def by auto
+ moreover
+ { fix g assume g: "g \<in> G"
+ then obtain f where "f \<in> F" "g = f(z := r.zero)" and z: "z = s.maxim (SUPP f)"
+ unfolding G_def by auto
+ with SUPPF bspec[OF SUPPG g] have "(s.maxim (SUPP g), z) \<in> s"
+ unfolding z by (intro s.maxim_mono) auto
+ }
+ moreover from y'min have "\<And>g. g \<in> G \<Longrightarrow> (y', s.maxim (SUPP g)) \<in> s"
+ unfolding s.isMinim_def by auto
+ ultimately have "y' \<noteq> z" "(y', z) \<in> s" using maxG
+ unfolding s.isMinim_def s.isMaxim_def by auto
+ with zy have "y' \<noteq> y" "(y', y) \<in> s" using antisymD[OF s.ANTISYM] transD[OF s.TRANS]
+ by blast+
+ moreover from `G \<noteq> {}` have "\<exists>g \<in> G. y' = wo_rel.maxim s (SUPP g)" using y'min
+ by (auto simp: G_def s.isMinim_def)
+ ultimately show ?thesis using mp[OF spec[OF mp[OF spec[OF 1]]], of y' G] G by auto
+ qed simp
+ then obtain g0 where g0: "g0 \<in> G" "\<forall>g \<in> G. (g0, g) \<in> oexp" by blast
+ hence g0z: "g0 z = r.zero" unfolding G_def by auto
+ def f0 \<equiv> "g0(z := x0)"
+ with x0notzero zField have SUPP: "SUPP f0 = SUPP g0 \<union> {z}" unfolding support_def by auto
+ from g0z have f0z: "f0(z := r.zero) = g0" unfolding f0_def fun_upd_upd by auto
+ have f0: "f0 \<in> F" using x0min g0(1)
+ Func_elim[OF set_mp[OF subset_trans[OF F(1)[unfolded FinFunc_def] Int_lower1]] zField]
+ unfolding f0_def r.isMinim_def G_def by (force simp: fun_upd_idem)
+ from g0(1) maxF(1) have maxf0: "s.maxim (SUPP f0) = z" unfolding SUPP G_def
+ by (intro s.maxim_equality) (auto simp: s.isMaxim_def)
+ show ?thesis
+ proof (intro bexI[OF _ f0] ballI)
+ fix f assume f: "f \<in> F"
+ show "(f0, f) \<in> oexp"
+ proof (cases "f0 = f")
+ case True thus ?thesis by (metis F(1) Field_oexp f0 in_mono oexp_Refl refl_onD)
+ next
+ case False
+ thus ?thesis
+ proof (cases "s.maxim (SUPP f) = z \<and> f z = x0")
+ case True
+ with f have "f(z := r.zero) \<in> G" unfolding G_def by blast
+ with g0(2) f0z have "(f0(z := r.zero), f(z := r.zero)) \<in> oexp" by auto
+ hence "(f0(z := r.zero, z := x0), f(z := r.zero, z := x0)) \<in> oexp"
+ by (elim fun_upd_same_oexp[OF _ _ zField x0Field]) simp
+ moreover
+ with f F(1) x0min True
+ have "(f(z := x0), f) \<in> oexp" unfolding G_def r.isMinim_def
+ by (intro fun_upd_smaller_oexp[OF _ zField x0Field]) auto
+ ultimately show ?thesis using transD[OF oexp_trans, of f0 "f(z := x0)" f]
+ unfolding f0_def by auto
+ next
+ case False note notG = this
+ thus ?thesis
+ proof (cases "s.maxim (SUPP f) = z")
+ case True
+ with notG have "f0 z \<noteq> f z" unfolding f0_def by auto
+ hence "f0 z \<noteq> f z" by metis
+ with True maxf0 f0 f SUPPF have "s.max_fun_diff f0 f = z"
+ using s.maxim_Un[of "SUPP f0" "SUPP f", unfolded s.max2_def]
+ unfolding max_fun_diff_alt by (intro trans[OF s.maxim_Int]) auto
+ moreover
+ from x0min True f have "(x0, f z) \<in> r" unfolding r.isMinim_def by auto
+ ultimately show ?thesis using f f0 F(1) unfolding oexp_def f0_def by auto
+ next
+ case False
+ with notG have *: "(z, s.maxim (SUPP f)) \<in> s" "z \<noteq> s.maxim (SUPP f)"
+ using zmin f unfolding s.isMinim_def G_def by auto
+ have f0f: "f0 (s.maxim (SUPP f)) = r.zero"
+ proof (rule ccontr)
+ assume "f0 (s.maxim (SUPP f)) \<noteq> r.zero"
+ with f SUPPF maxF(1) have "s.maxim (SUPP f) \<in> SUPP f0"
+ unfolding support_def[of _ _ f0] s.isMaxim_def by auto
+ with SUPPF f0 have "(s.maxim (SUPP f), z) \<in> s" unfolding maxf0[symmetric]
+ by (auto intro: s.maxim_greatest)
+ with * antisymD[OF s.ANTISYM] show False by simp
+ qed
+ moreover
+ have "f (s.maxim (SUPP f)) \<noteq> r.zero"
+ using bspec[OF maxF(1) f, unfolded s.isMaxim_def] by (auto simp: support_def)
+ with f0f * f f0 maxf0 SUPPF
+ have "s.max_fun_diff f0 f = s.maxim (SUPP f0 \<union> SUPP f)"
+ unfolding max_fun_diff_alt using s.maxim_Un[of "SUPP f0" "SUPP f"]
+ by (intro s.maxim_Int) (auto simp: s.max2_def)
+ moreover have "s.maxim (SUPP f0 \<union> SUPP f) = s.maxim (SUPP f)"
+ using s.maxim_Un[of "SUPP f0" "SUPP f"] * maxf0 SUPPF f0 f
+ by (auto simp: s.max2_def)
+ ultimately show ?thesis using f f0 F(1) maxF(1) SUPPF unfolding oexp_def Let_def
+ by (fastforce simp: s.isMaxim_def intro!: r.zero_smallest FINFUNCD)
+ qed
+ qed
+ qed
+ qed
+ qed simp
+ qed
+ qed
+ } note * = mp[OF this]
+ from A(2) obtain f where f: "f \<in> A" by blast
+ with A(1) show "\<exists>a\<in>A. \<forall>a'\<in>A. (a, a') \<in> oexp"
+ proof (cases "f = const")
+ case False with f A(1) show ?thesis using maxim_isMaxim_support[of "{f}"]
+ by (intro *[of _ "s.maxim (SUPP f)"]) (auto simp: s.isMaxim_def support_def)
+ qed simp
+ qed
+qed
+
+lemma oexp_Well_order: "Well_order oexp"
+ unfolding well_order_on_def using oexp_Linear_order oexp_wf_Id by blast
+
+interpretation o: wo_rel oexp by unfold_locales (rule oexp_Well_order)
+
+lemma zero_oexp: "Field r \<noteq> {} \<Longrightarrow> o.zero = const"
+ by (rule sym[OF o.leq_zero_imp[OF const_least]])
+ (auto intro!: o.zero_in_Field[unfolded Field_oexp] dest!: const_FINFUNC)
+
+end
+
+notation wo_rel2.oexp (infixl "^o" 90)
+lemmas oexp_def = wo_rel2.oexp_def[unfolded wo_rel2_def, OF conjI]
+lemmas oexp_Well_order = wo_rel2.oexp_Well_order[unfolded wo_rel2_def, OF conjI]
+lemmas Field_oexp = wo_rel2.Field_oexp[unfolded wo_rel2_def, OF conjI]
+
+definition "ozero = {}"
+
+lemma ozero_Well_order[simp]: "Well_order ozero"
+ unfolding ozero_def by simp
+
+lemma ozero_ordIso[simp]: "ozero =o ozero"
+ unfolding ozero_def ordIso_def iso_def[abs_def] embed_def bij_betw_def by auto
+
+lemma Field_ozero[simp]: "Field ozero = {}"
+ unfolding ozero_def by simp
+
+lemma iso_ozero_empty[simp]: "r =o ozero = (r = {})"
+ unfolding ozero_def ordIso_def iso_def[abs_def] embed_def bij_betw_def
+ by (auto dest: rel.well_order_on_domain)
+
+lemma ozero_ordLeq:
+assumes "Well_order r" shows "ozero \<le>o r"
+using assms unfolding ozero_def ordLeq_def embed_def[abs_def] rel.under_def by auto
+
+definition "oone = {((),())}"
+
+lemma oone_Well_order[simp]: "Well_order oone"
+ unfolding oone_def unfolding well_order_on_def linear_order_on_def partial_order_on_def
+ preorder_on_def total_on_def refl_on_def trans_def antisym_def by auto
+
+lemma Field_oone[simp]: "Field oone = {()}"
+ unfolding oone_def by simp
+
+lemma oone_ordIso: "oone =o {(x,x)}"
+ unfolding ordIso_def oone_def well_order_on_def linear_order_on_def partial_order_on_def
+ preorder_on_def total_on_def refl_on_def trans_def antisym_def
+ by (auto simp: iso_def embed_def bij_betw_def rel.under_def inj_on_def intro!: exI[of _ "\<lambda>_. x"])
+
+lemma osum_ordLeqR: "Well_order r \<Longrightarrow> Well_order s \<Longrightarrow> s \<le>o r +o s"
+ unfolding ordLeq_def2 rel.underS_def
+ by (auto intro!: exI[of _ Inr] osum_Well_order) (auto simp add: osum_def Field_def)
+
+lemma osum_congL:
+ assumes "r =o s" and t: "Well_order t"
+ shows "r +o t =o s +o t" (is "?L =o ?R")
+proof -
+ from assms(1) obtain f where r: "Well_order r" and s: "Well_order s" and f: "iso r s f"
+ unfolding ordIso_def by blast
+ let ?f = "sum_map f id"
+ from f have "inj_on ?f (Field ?L)"
+ unfolding Field_osum iso_def bij_betw_def inj_on_def by fastforce
+ with f have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_osum iso_def bij_betw_def image_image image_Un by auto
+ moreover from f have "compat ?L ?R ?f"
+ unfolding osum_def iso_iff3[OF r s] compat_def bij_betw_def
+ by (auto simp: map_pair_imageI)
+ ultimately have "iso ?L ?R ?f" by (subst iso_iff3) (auto intro: osum_Well_order r s t)
+ thus ?thesis unfolding ordIso_def by (auto intro: osum_Well_order r s t)
+qed
+
+lemma osum_congR:
+ assumes "r =o s" and t: "Well_order t"
+ shows "t +o r =o t +o s" (is "?L =o ?R")
+proof -
+ from assms(1) obtain f where r: "Well_order r" and s: "Well_order s" and f: "iso r s f"
+ unfolding ordIso_def by blast
+ let ?f = "sum_map id f"
+ from f have "inj_on ?f (Field ?L)"
+ unfolding Field_osum iso_def bij_betw_def inj_on_def by fastforce
+ with f have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_osum iso_def bij_betw_def image_image image_Un by auto
+ moreover from f have "compat ?L ?R ?f"
+ unfolding osum_def iso_iff3[OF r s] compat_def bij_betw_def
+ by (auto simp: map_pair_imageI)
+ ultimately have "iso ?L ?R ?f" by (subst iso_iff3) (auto intro: osum_Well_order r s t)
+ thus ?thesis unfolding ordIso_def by (auto intro: osum_Well_order r s t)
+qed
+
+lemma osum_cong:
+ assumes "t =o u" and "r =o s"
+ shows "t +o r =o u +o s"
+using ordIso_transitive[OF osum_congL[OF assms(1)] osum_congR[OF assms(2)]]
+ assms[unfolded ordIso_def] by auto
+
+lemma Well_order_empty[simp]: "Well_order {}"
+ unfolding Field_empty by (rule well_order_on_empty)
+
+lemma well_order_on_singleton[simp]: "well_order_on {x} {(x, x)}"
+ unfolding well_order_on_def linear_order_on_def partial_order_on_def preorder_on_def total_on_def
+ Field_def refl_on_def trans_def antisym_def by auto
+
+lemma oexp_empty[simp]:
+ assumes "Well_order r"
+ shows "r ^o {} = {(\<lambda>x. undefined, \<lambda>x. undefined)}"
+ unfolding oexp_def[OF assms Well_order_empty] FinFunc_def fin_support_def support_def by auto
+
+lemma oexp_empty2[simp]:
+ assumes "Well_order r" "r \<noteq> {}"
+ shows "{} ^o r = {}"
+proof -
+ from assms(2) have "Field r \<noteq> {}" unfolding Field_def by auto
+ thus ?thesis unfolding oexp_def[OF Well_order_empty assms(1)] FinFunc_def fin_support_def support_def
+ by auto
+qed
+
+lemma oprod_zero[simp]: "{} *o r = {}" "r *o {} = {}"
+ unfolding oprod_def by simp_all
+
+lemma oprod_congL:
+ assumes "r =o s" and t: "Well_order t"
+ shows "r *o t =o s *o t" (is "?L =o ?R")
+proof -
+ from assms(1) obtain f where r: "Well_order r" and s: "Well_order s" and f: "iso r s f"
+ unfolding ordIso_def by blast
+ let ?f = "map_pair f id"
+ from f have "inj_on ?f (Field ?L)"
+ unfolding Field_oprod iso_def bij_betw_def inj_on_def by fastforce
+ with f have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_oprod iso_def bij_betw_def by (auto intro!: map_pair_surj_on)
+ moreover from f have "compat ?L ?R ?f"
+ unfolding iso_iff3[OF r s] compat_def oprod_def bij_betw_def
+ by (auto simp: map_pair_imageI)
+ ultimately have "iso ?L ?R ?f" by (subst iso_iff3) (auto intro: oprod_Well_order r s t)
+ thus ?thesis unfolding ordIso_def by (auto intro: oprod_Well_order r s t)
+qed
+
+lemma oprod_congR:
+ assumes "r =o s" and t: "Well_order t"
+ shows "t *o r =o t *o s" (is "?L =o ?R")
+proof -
+ from assms(1) obtain f where r: "Well_order r" and s: "Well_order s" and f: "iso r s f"
+ unfolding ordIso_def by blast
+ let ?f = "map_pair id f"
+ from f have "inj_on ?f (Field ?L)"
+ unfolding Field_oprod iso_def bij_betw_def inj_on_def by fastforce
+ with f have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_oprod iso_def bij_betw_def by (auto intro!: map_pair_surj_on)
+ moreover from f rel.well_order_on_domain[OF r] have "compat ?L ?R ?f"
+ unfolding iso_iff3[OF r s] compat_def oprod_def bij_betw_def
+ by (auto simp: map_pair_imageI dest: inj_onD)
+ ultimately have "iso ?L ?R ?f" by (subst iso_iff3) (auto intro: oprod_Well_order r s t)
+ thus ?thesis unfolding ordIso_def by (auto intro: oprod_Well_order r s t)
+qed
+
+lemma oprod_cong:
+ assumes "t =o u" and "r =o s"
+ shows "t *o r =o u *o s"
+using ordIso_transitive[OF oprod_congL[OF assms(1)] oprod_congR[OF assms(2)]]
+ assms[unfolded ordIso_def] by auto
+
+lemma Field_singleton[simp]: "Field {(z,z)} = {z}"
+ by (metis rel.well_order_on_Field well_order_on_singleton)
+
+lemma zero_singleton[simp]: "zero {(z,z)} = z"
+ using wo_rel.zero_in_Field[unfolded wo_rel_def, of "{(z, z)}"] well_order_on_singleton[of z]
+ by auto
+
+lemma FinFunc_singleton: "FinFunc {(z,z)} s = {\<lambda>x. if x \<in> Field s then z else undefined}"
+ unfolding FinFunc_def Func_def fin_support_def support_def
+ by (auto simp: fun_eq_iff split: split_if_asm intro!: finite_subset[of _ "{}"])
+
+lemma oone_ordIso_oexp:
+ assumes "r =o oone" and s: "Well_order s"
+ shows "r ^o s =o oone" (is "?L =o ?R")
+proof -
+ from assms obtain f where *: "\<forall>x\<in>Field r. \<forall>y\<in>Field r. x = y" and "f ` Field r = {()}"
+ and r: "Well_order r"
+ unfolding ordIso_def oone_def by (auto simp: iso_def bij_betw_def inj_on_def)
+ then obtain x where "x \<in> Field r" by auto
+ with * have Fr: "Field r = {x}" by auto
+ interpret r: wo_rel r by unfold_locales (rule r)
+ from Fr r.well_order_on_domain[OF r] refl_onD[OF r.REFL, of x] have r_def: "r = {(x, x)}" by fast
+ interpret wo_rel2 r s by unfold_locales (rule r, rule s)
+ have "bij_betw (\<lambda>x. ()) (Field ?L) (Field ?R)"
+ unfolding bij_betw_def Field_oexp by (auto simp: r_def FinFunc_singleton)
+ moreover have "compat ?L ?R (\<lambda>x. ())" unfolding compat_def oone_def by auto
+ ultimately have "iso ?L ?R (\<lambda>x. ())" using s oone_Well_order
+ by (subst iso_iff3) (auto intro: oexp_Well_order)
+ thus ?thesis using s oone_Well_order unfolding ordIso_def by (auto intro: oexp_Well_order)
+qed
+
+(*Lemma 1.4.3 from Holz et al.*)
+context
+ fixes r s t
+ assumes r: "Well_order r"
+ assumes s: "Well_order s"
+ assumes t: "Well_order t"
+begin
+
+lemma osum_ozeroL: "ozero +o r =o r"
+ using r unfolding osum_def ozero_def by (auto intro: map_pair_ordIso)
+
+lemma osum_ozeroR: "r +o ozero =o r"
+ using r unfolding osum_def ozero_def by (auto intro: map_pair_ordIso)
+
+lemma osum_assoc: "(r +o s) +o t =o r +o s +o t" (is "?L =o ?R")
+proof -
+ let ?f =
+ "\<lambda>rst. case rst of Inl (Inl r) \<Rightarrow> Inl r | Inl (Inr s) \<Rightarrow> Inr (Inl s) | Inr t \<Rightarrow> Inr (Inr t)"
+ have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_osum bij_betw_def inj_on_def by (auto simp: image_Un image_iff)
+ moreover
+ have "compat ?L ?R ?f"
+ proof (unfold compat_def, safe)
+ fix a b
+ assume "(a, b) \<in> ?L"
+ thus "(?f a, ?f b) \<in> ?R"
+ unfolding osum_def[of "r +o s" t] osum_def[of r "s +o t"] Field_osum
+ unfolding osum_def Field_osum image_iff image_Un map_pair_def
+ by fastforce
+ qed
+ ultimately have "iso ?L ?R ?f" using r s t by (subst iso_iff3) (auto intro: osum_Well_order)
+ thus ?thesis using r s t unfolding ordIso_def by (auto intro: osum_Well_order)
+qed
+
+lemma osum_monoR:
+ assumes "s <o t"
+ shows "r +o s <o r +o t" (is "?L <o ?R")
+proof -
+ from assms obtain f where s: "Well_order s" and t:" Well_order t" and "embedS s t f"
+ unfolding ordLess_def by blast
+ hence *: "inj_on f (Field s)" "compat s t f" "ofilter t (f ` Field s)" "f ` Field s \<subset> Field t"
+ using embed_iff_compat_inj_on_ofilter[OF s t, of f] embedS_iff[OF s, of t f]
+ unfolding embedS_def by auto
+ let ?f = "sum_map id f"
+ from *(1) have "inj_on ?f (Field ?L)" unfolding Field_osum inj_on_def by fastforce
+ moreover
+ from *(2,4) have "compat ?L ?R ?f" unfolding compat_def osum_def map_pair_def by fastforce
+ moreover
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ interpret rt!: wo_rel ?R by unfold_locales (rule osum_Well_order[OF r t])
+ from *(3) have "ofilter ?R (?f ` Field ?L)"
+ unfolding t.ofilter_def rt.ofilter_def Field_osum image_Un image_image rel.under_def
+ by (auto simp: osum_def intro!: imageI) (auto simp: Field_def)
+ ultimately have "embed ?L ?R ?f" using embed_iff_compat_inj_on_ofilter[of ?L ?R ?f]
+ by (auto intro: osum_Well_order r s t)
+ moreover
+ from *(4) have "?f ` Field ?L \<subset> Field ?R" unfolding Field_osum image_Un image_image by auto
+ ultimately have "embedS ?L ?R ?f" using embedS_iff[OF osum_Well_order[OF r s], of ?R ?f] by auto
+ thus ?thesis unfolding ordLess_def by (auto intro: osum_Well_order r s t)
+qed
+
+lemma osum_monoL:
+ assumes "r \<le>o s"
+ shows "r +o t \<le>o s +o t"
+proof -
+ from assms obtain f where f: "\<forall>a\<in>Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)"
+ unfolding ordLeq_def2 by blast
+ let ?f = "sum_map f id"
+ from f have "\<forall>a\<in>Field (r +o t).
+ ?f a \<in> Field (s +o t) \<and> ?f ` underS (r +o t) a \<subseteq> underS (s +o t) (?f a)"
+ unfolding Field_osum rel.underS_def by (fastforce simp: osum_def)
+ thus ?thesis unfolding ordLeq_def2 by (auto intro: osum_Well_order r s t)
+qed
+
+lemma oprod_ozeroL: "ozero *o r =o ozero"
+ using ozero_ordIso unfolding ozero_def by simp
+
+lemma oprod_ozeroR: "r *o ozero =o ozero"
+ using ozero_ordIso unfolding ozero_def by simp
+
+lemma oprod_ooneR: "r *o oone =o r" (is "?L =o ?R")
+proof -
+ have "bij_betw fst (Field ?L) (Field ?R)" unfolding Field_oprod bij_betw_def inj_on_def by simp
+ moreover have "compat ?L ?R fst" unfolding compat_def oprod_def by auto
+ ultimately have "iso ?L ?R fst" using r oone_Well_order
+ by (subst iso_iff3) (auto intro: oprod_Well_order)
+ thus ?thesis using r oone_Well_order unfolding ordIso_def by (auto intro: oprod_Well_order)
+qed
+
+lemma oprod_ooneL: "oone *o r =o r" (is "?L =o ?R")
+proof -
+ have "bij_betw snd (Field ?L) (Field ?R)" unfolding Field_oprod bij_betw_def inj_on_def by simp
+ moreover have "Refl r" by (rule wo_rel.REFL[unfolded wo_rel_def, OF r])
+ hence "compat ?L ?R snd" unfolding compat_def oprod_def refl_on_def by auto
+ ultimately have "iso ?L ?R snd" using r oone_Well_order
+ by (subst iso_iff3) (auto intro: oprod_Well_order)
+ thus ?thesis using r oone_Well_order unfolding ordIso_def by (auto intro: oprod_Well_order)
+qed
+
+lemma oprod_monoR:
+ assumes "ozero <o r" "s <o t"
+ shows "r *o s <o r *o t" (is "?L <o ?R")
+proof -
+ from assms obtain f where s: "Well_order s" and t:" Well_order t" and "embedS s t f"
+ unfolding ordLess_def by blast
+ hence *: "inj_on f (Field s)" "compat s t f" "ofilter t (f ` Field s)" "f ` Field s \<subset> Field t"
+ using embed_iff_compat_inj_on_ofilter[OF s t, of f] embedS_iff[OF s, of t f]
+ unfolding embedS_def by auto
+ let ?f = "map_pair id f"
+ from *(1) have "inj_on ?f (Field ?L)" unfolding Field_oprod inj_on_def by fastforce
+ moreover
+ from *(2,4) the_inv_into_f_f[OF *(1)] have "compat ?L ?R ?f" unfolding compat_def oprod_def
+ by auto (metis rel.well_order_on_domain t, metis rel.well_order_on_domain s)
+ moreover
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ interpret rt!: wo_rel ?R by unfold_locales (rule oprod_Well_order[OF r t])
+ from *(3) have "ofilter ?R (?f ` Field ?L)"
+ unfolding t.ofilter_def rt.ofilter_def Field_oprod rel.under_def
+ by (auto simp: oprod_def image_iff) (fast | metis r rel.well_order_on_domain)+
+ ultimately have "embed ?L ?R ?f" using embed_iff_compat_inj_on_ofilter[of ?L ?R ?f]
+ by (auto intro: oprod_Well_order r s t)
+ moreover
+ from not_ordLess_ordIso[OF assms(1)] have "r \<noteq> {}" by (metis ozero_def ozero_ordIso)
+ hence "Field r \<noteq> {}" unfolding Field_def by auto
+ with *(4) have "?f ` Field ?L \<subset> Field ?R" unfolding Field_oprod
+ by auto (metis SigmaD2 SigmaI map_pair_surj_on)
+ ultimately have "embedS ?L ?R ?f" using embedS_iff[OF oprod_Well_order[OF r s], of ?R ?f] by auto
+ thus ?thesis unfolding ordLess_def by (auto intro: oprod_Well_order r s t)
+qed
+
+lemma oprod_monoL:
+ assumes "r \<le>o s"
+ shows "r *o t \<le>o s *o t"
+proof -
+ from assms obtain f where f: "\<forall>a\<in>Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)"
+ unfolding ordLeq_def2 by blast
+ let ?f = "map_pair f id"
+ from f have "\<forall>a\<in>Field (r *o t).
+ ?f a \<in> Field (s *o t) \<and> ?f ` underS (r *o t) a \<subseteq> underS (s *o t) (?f a)"
+ unfolding Field_oprod rel.underS_def unfolding map_pair_def oprod_def by auto
+ thus ?thesis unfolding ordLeq_def2 by (auto intro: oprod_Well_order r s t)
+qed
+
+lemma oprod_assoc: "(r *o s) *o t =o r *o s *o t" (is "?L =o ?R")
+proof -
+ let ?f = "\<lambda>((a,b),c). (a,b,c)"
+ have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding Field_oprod bij_betw_def inj_on_def by (auto simp: image_Un image_iff)
+ moreover
+ have "compat ?L ?R ?f"
+ proof (unfold compat_def, safe)
+ fix a1 a2 a3 b1 b2 b3
+ assume "(((a1, a2), a3), ((b1, b2), b3)) \<in> ?L"
+ thus "((a1, a2, a3), (b1, b2, b3)) \<in> ?R"
+ unfolding oprod_def[of "r *o s" t] oprod_def[of r "s *o t"] Field_oprod
+ unfolding oprod_def Field_oprod image_iff image_Un by fast
+ qed
+ ultimately have "iso ?L ?R ?f" using r s t by (subst iso_iff3) (auto intro: oprod_Well_order)
+ thus ?thesis using r s t unfolding ordIso_def by (auto intro: oprod_Well_order)
+qed
+
+lemma oprod_osum: "r *o (s +o t) =o r *o s +o r *o t" (is "?L =o ?R")
+proof -
+ let ?f = "\<lambda>(a,bc). case bc of Inl b \<Rightarrow> Inl (a, b) | Inr c \<Rightarrow> Inr (a, c)"
+ have "bij_betw ?f (Field ?L) (Field ?R)" unfolding Field_oprod Field_osum bij_betw_def inj_on_def
+ by (fastforce simp: image_Un image_iff split: sum.splits)
+ moreover
+ have "compat ?L ?R ?f"
+ proof (unfold compat_def, intro allI impI)
+ fix a b
+ assume "(a, b) \<in> ?L"
+ thus "(?f a, ?f b) \<in> ?R"
+ unfolding oprod_def[of r "s +o t"] osum_def[of "r *o s" "r *o t"] Field_oprod Field_osum
+ unfolding oprod_def osum_def Field_oprod Field_osum image_iff image_Un by auto
+ qed
+ ultimately have "iso ?L ?R ?f" using r s t
+ by (subst iso_iff3) (auto intro: oprod_Well_order osum_Well_order)
+ thus ?thesis using r s t unfolding ordIso_def by (auto intro: oprod_Well_order osum_Well_order)
+qed
+
+lemma ozero_oexp: "\<not> (s =o ozero) \<Longrightarrow> ozero ^o s =o ozero"
+ unfolding oexp_def[OF ozero_Well_order s] FinFunc_def
+ by simp (metis Func_emp2 bot.extremum_uniqueI emptyE rel.well_order_on_domain s subrelI)
+
+lemma oone_oexp: "oone ^o s =o oone" (is "?L =o ?R")
+ by (rule oone_ordIso_oexp[OF ordIso_reflexive[OF oone_Well_order] s])
+
+lemma oexp_monoR:
+ assumes "oone <o r" "s <o t"
+ shows "r ^o s <o r ^o t" (is "?L <o ?R")
+proof -
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rt!: wo_rel2 r t by unfold_locales (rule r, rule t)
+ interpret rexpt!: wo_rel "r ^o t" by unfold_locales (rule rt.oexp_Well_order)
+ interpret r!: wo_rel r by unfold_locales (rule r)
+ interpret s!: wo_rel s by unfold_locales (rule s)
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ have "Field r \<noteq> {}" by (metis assms(1) internalize_ordLess not_psubset_empty)
+ moreover
+ { assume "Field r = {r.zero}"
+ hence "r = {(r.zero, r.zero)}" using refl_onD[OF r.REFL, of r.zero] unfolding Field_def by auto
+ hence "r =o oone" by (metis oone_ordIso ordIso_symmetric)
+ with not_ordLess_ordIso[OF assms(1)] have False by (metis ordIso_symmetric)
+ }
+ ultimately obtain x where x: "x \<in> Field r" "r.zero \<in> Field r" "x \<noteq> r.zero"
+ by (metis insert_iff r.zero_in_Field subsetI subset_singletonD)
+ moreover from assms(2) obtain f where "embedS s t f" unfolding ordLess_def by blast
+ hence *: "inj_on f (Field s)" "compat s t f" "ofilter t (f ` Field s)" "f ` Field s \<subset> Field t"
+ using embed_iff_compat_inj_on_ofilter[OF s t, of f] embedS_iff[OF s, of t f]
+ unfolding embedS_def by auto
+ note invff = the_inv_into_f_f[OF *(1)] and injfD = inj_onD[OF *(1)]
+ def F \<equiv> "\<lambda>g z. if z \<in> f ` Field s then g (the_inv_into (Field s) f z) else
+ if z \<in> Field t then r.zero else undefined"
+ from *(4) x(2) the_inv_into_f_eq[OF *(1)] have FLR: "F ` Field ?L \<subseteq> Field ?R"
+ unfolding rt.Field_oexp rs.Field_oexp FinFunc_def Func_def fin_support_def support_def F_def
+ by (fastforce split: option.splits split_if_asm elim!: finite_surj[of _ _ f])
+ have "inj_on F (Field ?L)" unfolding rs.Field_oexp inj_on_def fun_eq_iff
+ proof safe
+ fix g h x assume "g \<in> FinFunc r s" "h \<in> FinFunc r s" "\<forall>y. F g y = F h y"
+ with invff show "g x = h x" unfolding F_def fun_eq_iff FinFunc_def Func_def
+ by auto (metis image_eqI)
+ qed
+ moreover
+ have "compat ?L ?R F" unfolding compat_def rs.oexp_def rt.oexp_def
+ proof (safe elim!: bspec[OF iffD1[OF image_subset_iff FLR[unfolded rs.Field_oexp rt.Field_oexp]]])
+ fix g h assume gh: "g \<in> FinFunc r s" "h \<in> FinFunc r s" "F g \<noteq> F h"
+ "let m = s.max_fun_diff g h in (g m, h m) \<in> r"
+ hence "g \<noteq> h" by auto
+ note max_fun_diff_in = rs.max_fun_diff_in[OF `g \<noteq> h` gh(1,2)]
+ and max_fun_diff_max = rs.max_fun_diff_max[OF `g \<noteq> h` gh(1,2)]
+ with *(4) invff *(2) have "t.max_fun_diff (F g) (F h) = f (s.max_fun_diff g h)"
+ unfolding t.max_fun_diff_def compat_def
+ by (intro t.maxim_equality) (auto simp: t.isMaxim_def F_def dest: injfD)
+ with gh invff max_fun_diff_in
+ show "let m = t.max_fun_diff (F g) (F h) in (F g m, F h m) \<in> r"
+ unfolding F_def Let_def by (auto simp: dest: injfD)
+ qed
+ moreover
+ from FLR have "ofilter ?R (F ` Field ?L)"
+ unfolding rexpt.ofilter_def rel.under_def rs.Field_oexp rt.Field_oexp unfolding rt.oexp_def
+ proof (safe elim!: imageI)
+ fix g h assume gh: "g \<in> FinFunc r s" "h \<in> FinFunc r t" "F g \<in> FinFunc r t"
+ "let m = t.max_fun_diff h (F g) in (h m, F g m) \<in> r"
+ thus "h \<in> F ` FinFunc r s"
+ proof (cases "h = F g")
+ case False
+ hence max_Field: "t.max_fun_diff h (F g) \<in> {a \<in> Field t. h a \<noteq> F g a}"
+ by (rule rt.max_fun_diff_in[OF _ gh(2,3)])
+ { assume *: "t.max_fun_diff h (F g) \<notin> f ` Field s"
+ with max_Field have "F g (t.max_fun_diff h (F g)) = r.zero" unfolding F_def by auto
+ moreover
+ with * gh(4) have "h (t.max_fun_diff h (F g)) = r.zero" unfolding Let_def by auto
+ ultimately have False using max_Field gh(2,3) unfolding FinFunc_def Func_def by auto
+ }
+ hence max_f_Field: "t.max_fun_diff h (F g) \<in> f ` Field s" by blast
+ { fix z assume z: "z \<in> Field t - f ` Field s"
+ have "(t.max_fun_diff h (F g), z) \<in> t"
+ proof (rule ccontr)
+ assume "(t.max_fun_diff h (F g), z) \<notin> t"
+ hence "(z, t.max_fun_diff h (F g)) \<in> t" using t.in_notinI[of "t.max_fun_diff h (F g)" z]
+ z max_Field by auto
+ hence "z \<in> f ` Field s" using *(3) max_f_Field unfolding t.ofilter_def rel.under_def
+ by fastforce
+ with z show False by blast
+ qed
+ hence "h z = r.zero" using rt.max_fun_diff_le_eq[OF _ False gh(2,3), of z]
+ z max_f_Field unfolding F_def by auto
+ } note ** = this
+ with *(3) gh(2) have "h = F (\<lambda>x. if x \<in> Field s then h (f x) else undefined)" using invff
+ unfolding F_def fun_eq_iff FinFunc_def Func_def Let_def t.ofilter_def rel.under_def by auto
+ moreover from gh(2) *(1,3) have "(\<lambda>x. if x \<in> Field s then h (f x) else undefined) \<in> FinFunc r s"
+ unfolding FinFunc_def Func_def fin_support_def support_def t.ofilter_def rel.under_def
+ by (auto intro: subset_inj_on elim!: finite_imageD[OF finite_subset[rotated]])
+ ultimately show "?thesis" by (rule image_eqI)
+ qed simp
+ qed
+ ultimately have "embed ?L ?R F" using embed_iff_compat_inj_on_ofilter[of ?L ?R F]
+ by (auto intro: oexp_Well_order r s t)
+ moreover
+ from FLR have "F ` Field ?L \<subset> Field ?R"
+ proof (intro psubsetI)
+ from *(4) obtain z where z: "z \<in> Field t" "z \<notin> f ` Field s" by auto
+ def h \<equiv> "\<lambda>z'. if z' \<in> Field t then if z' = z then x else r.zero else undefined"
+ from z x(3) have "rt.SUPP h = {z}" unfolding support_def h_def by simp
+ with x have "h \<in> Field ?R" unfolding h_def rt.Field_oexp FinFunc_def Func_def fin_support_def
+ by auto
+ moreover
+ { fix g
+ from z have "F g z = r.zero" "h z = x" unfolding support_def h_def F_def by auto
+ with x(3) have "F g \<noteq> h" unfolding fun_eq_iff by fastforce
+ }
+ hence "h \<notin> F ` Field ?L" by blast
+ ultimately show "F ` Field ?L \<noteq> Field ?R" by blast
+ qed
+ ultimately have "embedS ?L ?R F" using embedS_iff[OF rs.oexp_Well_order, of ?R F] by auto
+ thus ?thesis unfolding ordLess_def using r s t by (auto intro: oexp_Well_order)
+qed
+
+lemma oexp_monoL:
+ assumes "r \<le>o s"
+ shows "r ^o t \<le>o s ^o t"
+proof -
+ interpret rt!: wo_rel2 r t by unfold_locales (rule r, rule t)
+ interpret st!: wo_rel2 s t by unfold_locales (rule s, rule t)
+ interpret r!: wo_rel r by unfold_locales (rule r)
+ interpret s!: wo_rel s by unfold_locales (rule s)
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ show ?thesis
+ proof (cases "t = {}")
+ case True thus ?thesis using r s unfolding ordLeq_def2 rel.underS_def by auto
+ next
+ case False thus ?thesis
+ proof (cases "r = {}")
+ case True thus ?thesis using t `t \<noteq> {}` st.oexp_Well_order ozero_ordLeq[unfolded ozero_def]
+ by auto
+ next
+ case False
+ from assms obtain f where f: "embed r s f" unfolding ordLeq_def by blast
+ hence f_underS: "\<forall>a\<in>Field r. f a \<in> Field s \<and> f ` underS r a \<subseteq> underS s (f a)"
+ using embed_in_Field[OF rt.rWELL f] embed_underS2[OF rt.rWELL st.rWELL f] by auto
+ from f `t \<noteq> {}` False have *: "Field r \<noteq> {}" "Field s \<noteq> {}" "Field t \<noteq> {}"
+ unfolding Field_def embed_def rel.under_def bij_betw_def by auto
+ with f obtain x where "s.zero = f x" "x \<in> Field r" unfolding embed_def bij_betw_def
+ using embed_in_Field[OF r.WELL f] s.zero_under set_mp[OF r.under_Field] by blast
+ with f have fz: "f r.zero = s.zero" and inj: "inj_on f (Field r)" and compat: "compat r s f"
+ unfolding embed_iff_compat_inj_on_ofilter[OF r s] compat_def
+ by (fastforce intro: s.leq_zero_imp)+
+ let ?f = "\<lambda>g x. if x \<in> Field t then f (g x) else undefined"
+ { fix g assume g: "g \<in> Field (r ^o t)"
+ with fz f_underS have Field_fg: "?f g \<in> Field (s ^o t)"
+ unfolding st.Field_oexp rt.Field_oexp FinFunc_def Func_def fin_support_def support_def
+ by (auto elim!: finite_subset[rotated])
+ moreover
+ have "?f ` underS (r ^o t) g \<subseteq> underS (s ^o t) (?f g)"
+ proof safe
+ fix h
+ assume h_underS: "h \<in> underS (r ^o t) g"
+ hence "h \<in> Field (r ^o t)" unfolding rel.underS_def Field_def by auto
+ with fz f_underS have Field_fh: "?f h \<in> Field (s ^o t)"
+ unfolding st.Field_oexp rt.Field_oexp FinFunc_def Func_def fin_support_def support_def
+ by (auto elim!: finite_subset[rotated])
+ from h_underS have "h \<noteq> g" and hg: "(h, g) \<in> rt.oexp" unfolding rel.underS_def by auto
+ with f inj have neq: "?f h \<noteq> ?f g"
+ unfolding fun_eq_iff inj_on_def rt.oexp_def Option.map_def FinFunc_def Func_def Let_def
+ by simp metis
+ moreover
+ with hg have "t.max_fun_diff (?f h) (?f g) = t.max_fun_diff h g" unfolding rt.oexp_def
+ using rt.max_fun_diff[OF `h \<noteq> g`] rt.max_fun_diff_in[OF `h \<noteq> g`]
+ by (subst t.max_fun_diff_def, intro t.maxim_equality)
+ (auto simp: t.isMaxim_def intro: inj_onD[OF inj] intro!: rt.max_fun_diff_max)
+ with Field_fg Field_fh hg fz f_underS compat neq have "(?f h, ?f g) \<in> st.oexp"
+ using rt.max_fun_diff[OF `h \<noteq> g`] rt.max_fun_diff_in[OF `h \<noteq> g`] unfolding st.Field_oexp
+ unfolding rt.oexp_def st.oexp_def Let_def compat_def by auto
+ ultimately show "?f h \<in> underS (s ^o t) (?f g)" unfolding rel.underS_def by auto
+ qed
+ ultimately have "?f g \<in> Field (s ^o t) \<and> ?f ` underS (r ^o t) g \<subseteq> underS (s ^o t) (?f g)"
+ by blast
+ }
+ thus ?thesis unfolding ordLeq_def2 by (fastforce intro: oexp_Well_order r s t)
+ qed
+ qed
+qed
+
+lemma ordLeq_oexp2:
+ assumes "oone <o r"
+ shows "s \<le>o r ^o s"
+proof -
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret r!: wo_rel r by unfold_locales (rule r)
+ interpret s!: wo_rel s by unfold_locales (rule s)
+ from assms rel.well_order_on_domain[OF r] obtain x where
+ x: "x \<in> Field r" "r.zero \<in> Field r" "x \<noteq> r.zero"
+ unfolding ordLess_def oone_def embedS_def[abs_def] bij_betw_def embed_def rel.under_def
+ by (auto simp: image_def)
+ (metis equals0D insert_not_empty r.under_def r.zero_in_Field rel.under_empty)
+ let ?f = "\<lambda>a b. if b \<in> Field s then if b = a then x else r.zero else undefined"
+ from x(3) have SUPP: "\<And>y. y \<in> Field s \<Longrightarrow> rs.SUPP (?f y) = {y}" unfolding support_def by auto
+ { fix y assume y: "y \<in> Field s"
+ with x(1,2) SUPP have "?f y \<in> Field (r ^o s)" unfolding rs.Field_oexp
+ by (auto simp: FinFunc_def Func_def fin_support_def)
+ moreover
+ have "?f ` underS s y \<subseteq> underS (r ^o s) (?f y)"
+ proof safe
+ fix z
+ assume "z \<in> underS s y"
+ hence z: "z \<noteq> y" "(z, y) \<in> s" "z \<in> Field s" unfolding rel.underS_def Field_def by auto
+ from x(3) y z(1,3) have "?f z \<noteq> ?f y" unfolding fun_eq_iff by auto
+ moreover
+ { from x(1,2) have "?f z \<in> FinFunc r s" "?f y \<in> FinFunc r s"
+ unfolding FinFunc_def Func_def fin_support_def by (auto simp: SUPP[OF z(3)] SUPP[OF y])
+ moreover
+ from x(3) y z(1,2) refl_onD[OF s.REFL] have "s.max_fun_diff (?f z) (?f y) = y"
+ unfolding rs.max_fun_diff_alt SUPP[OF z(3)] SUPP[OF y]
+ by (intro s.maxim_equality) (auto simp: s.isMaxim_def)
+ ultimately have "(?f z, ?f y) \<in> rs.oexp" using y x(1)
+ unfolding rs.oexp_def Let_def by auto
+ }
+ ultimately show "?f z \<in> underS (r ^o s) (?f y)" unfolding rel.underS_def by blast
+ qed
+ ultimately have "?f y \<in> Field (r ^o s) \<and> ?f ` underS s y \<subseteq> underS (r ^o s) (?f y)" by blast
+ }
+ thus ?thesis unfolding ordLeq_def2 by (fast intro: oexp_Well_order r s)
+qed
+
+lemma FinFunc_osum:
+ "fg \<in> FinFunc r (s +o t) = (fg o Inl \<in> FinFunc r s \<and> fg o Inr \<in> FinFunc r t)"
+ (is "?L = (?R1 \<and> ?R2)")
+proof safe
+ assume ?L
+ from `?L` show ?R1 unfolding FinFunc_def Field_osum Func_def Int_iff fin_support_Field_osum o_def
+ by (auto split: sum.splits)
+ from `?L` show ?R2 unfolding FinFunc_def Field_osum Func_def Int_iff fin_support_Field_osum o_def
+ by (auto split: sum.splits)
+next
+ assume ?R1 ?R2
+ thus "?L" unfolding FinFunc_def Field_osum Func_def
+ by (auto simp: fin_support_Field_osum o_def image_iff split: sum.splits) (metis sumE)
+qed
+
+lemma max_fun_diff_eq_Inl:
+ assumes "wo_rel.max_fun_diff (s +o t) (sum_case f1 g1) (sum_case f2 g2) = Inl x"
+ "sum_case f1 g1 \<noteq> sum_case f2 g2"
+ "sum_case f1 g1 \<in> FinFunc r (s +o t)" "sum_case f2 g2 \<in> FinFunc r (s +o t)"
+ shows "wo_rel.max_fun_diff s f1 f2 = x" (is ?P) "g1 = g2" (is ?Q)
+proof -
+ interpret st!: wo_rel "s +o t" by unfold_locales (rule osum_Well_order[OF s t])
+ interpret s!: wo_rel s by unfold_locales (rule s)
+ interpret rst!: wo_rel2 r "s +o t" by unfold_locales (rule r, rule osum_Well_order[OF s t])
+ from assms(1) have *: "st.isMaxim {a \<in> Field (s +o t). sum_case f1 g1 a \<noteq> sum_case f2 g2 a} (Inl x)"
+ using rst.isMaxim_max_fun_diff[OF assms(2-4)] by simp
+ hence "s.isMaxim {a \<in> Field s. f1 a \<noteq> f2 a} x"
+ unfolding st.isMaxim_def s.isMaxim_def Field_osum by (auto simp: osum_def)
+ thus ?P unfolding s.max_fun_diff_def by (rule s.maxim_equality)
+ from assms(3,4) have **: "g1 \<in> FinFunc r t" "g2 \<in> FinFunc r t" unfolding FinFunc_osum
+ by (auto simp: o_def)
+ show ?Q
+ proof
+ fix x
+ from * ** show "g1 x = g2 x" unfolding st.isMaxim_def Field_osum FinFunc_def Func_def fun_eq_iff
+ unfolding osum_def by (case_tac "x \<in> Field t") auto
+ qed
+qed
+
+lemma max_fun_diff_eq_Inr:
+ assumes "wo_rel.max_fun_diff (s +o t) (sum_case f1 g1) (sum_case f2 g2) = Inr x"
+ "sum_case f1 g1 \<noteq> sum_case f2 g2"
+ "sum_case f1 g1 \<in> FinFunc r (s +o t)" "sum_case f2 g2 \<in> FinFunc r (s +o t)"
+ shows "wo_rel.max_fun_diff t g1 g2 = x" (is ?P) "g1 \<noteq> g2" (is ?Q)
+proof -
+ interpret st!: wo_rel "s +o t" by unfold_locales (rule osum_Well_order[OF s t])
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ interpret rst!: wo_rel2 r "s +o t" by unfold_locales (rule r, rule osum_Well_order[OF s t])
+ from assms(1) have *: "st.isMaxim {a \<in> Field (s +o t). sum_case f1 g1 a \<noteq> sum_case f2 g2 a} (Inr x)"
+ using rst.isMaxim_max_fun_diff[OF assms(2-4)] by simp
+ hence "t.isMaxim {a \<in> Field t. g1 a \<noteq> g2 a} x"
+ unfolding st.isMaxim_def t.isMaxim_def Field_osum by (auto simp: osum_def)
+ thus ?P ?Q unfolding t.max_fun_diff_def fun_eq_iff
+ by (auto intro: t.maxim_equality simp: t.isMaxim_def)
+qed
+
+lemma oexp_osum: "r ^o (s +o t) =o (r ^o s) *o (r ^o t)" (is "?R =o ?L")
+proof (rule ordIso_symmetric)
+ interpret rst!: wo_rel2 r "s +o t" by unfold_locales (rule r, rule osum_Well_order[OF s t])
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rt!: wo_rel2 r t by unfold_locales (rule r, rule t)
+ let ?f = "\<lambda>(f, g). sum_case f g"
+ have "bij_betw ?f (Field ?L) (Field ?R)"
+ unfolding bij_betw_def rst.Field_oexp rs.Field_oexp rt.Field_oexp Field_oprod proof (intro conjI)
+ show "inj_on ?f (FinFunc r s \<times> FinFunc r t)" unfolding inj_on_def
+ by (auto simp: fun_eq_iff split: sum.splits)
+ show "?f ` (FinFunc r s \<times> FinFunc r t) = FinFunc r (s +o t)"
+ proof safe
+ fix fg assume "fg \<in> FinFunc r (s +o t)"
+ thus "fg \<in> ?f ` (FinFunc r s \<times> FinFunc r t)"
+ by (intro image_eqI[of _ _ "(fg o Inl, fg o Inr)"])
+ (auto simp: FinFunc_osum fun_eq_iff split: sum.splits)
+ qed (auto simp: FinFunc_osum o_def)
+ qed
+ moreover have "compat ?L ?R ?f"
+ unfolding compat_def rst.Field_oexp rs.Field_oexp rt.Field_oexp oprod_def
+ unfolding rst.oexp_def Let_def rs.oexp_def rt.oexp_def
+ by (fastforce simp: Field_osum FinFunc_osum o_def split: sum.splits
+ dest: max_fun_diff_eq_Inl max_fun_diff_eq_Inr)
+ ultimately have "iso ?L ?R ?f" using r s t
+ by (subst iso_iff3) (auto intro: oexp_Well_order oprod_Well_order osum_Well_order)
+ thus "?L =o ?R" using r s t unfolding ordIso_def
+ by (auto intro: oexp_Well_order oprod_Well_order osum_Well_order)
+qed
+
+definition "rev_curr f b = (if b \<in> Field t then \<lambda>a. f (a, b) else undefined)"
+
+lemma rev_curr_FinFunc:
+ assumes Field: "Field r \<noteq> {}"
+ shows "rev_curr ` (FinFunc r (s *o t)) = FinFunc (r ^o s) t"
+proof safe
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rst!: wo_rel2 "r ^o s" t by unfold_locales (rule oexp_Well_order[OF r s], rule t)
+ fix g assume g: "g \<in> FinFunc r (s *o t)"
+ hence "finite (rst.SUPP (rev_curr g))" "\<forall>x \<in> Field t. finite (rs.SUPP (rev_curr g x))"
+ unfolding FinFunc_def Field_oprod rs.Field_oexp Func_def fin_support_def support_def
+ rs.zero_oexp[OF Field] rev_curr_def by (auto simp: fun_eq_iff rs.const_def elim!: finite_surj)
+ with g show "rev_curr g \<in> FinFunc (r ^o s) t"
+ unfolding FinFunc_def Field_oprod rs.Field_oexp Func_def
+ by (auto simp: rev_curr_def fin_support_def)
+next
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rst!: wo_rel2 "r ^o s" t by unfold_locales (rule oexp_Well_order[OF r s], rule t)
+ fix fg assume *: "fg \<in> FinFunc (r ^o s) t"
+ let ?g = "\<lambda>(a, b). if (a, b) \<in> Field (s *o t) then fg b a else undefined"
+ show "fg \<in> rev_curr ` FinFunc r (s *o t)"
+ proof (rule image_eqI[of _ _ ?g])
+ show "fg = rev_curr ?g"
+ proof
+ fix x
+ from * show "fg x = rev_curr ?g x"
+ unfolding FinFunc_def rs.Field_oexp Func_def rev_curr_def Field_oprod by auto
+ qed
+ next
+ have **: "(\<Union>g \<in> fg ` Field t. rs.SUPP g) =
+ (\<Union>g \<in> fg ` Field t - {rs.const}. rs.SUPP g)"
+ unfolding support_def by auto
+ from * have "\<forall>g \<in> fg ` Field t. finite (rs.SUPP g)" "finite (rst.SUPP fg)"
+ unfolding rs.Field_oexp FinFunc_def Func_def fin_support_def these_def by force+
+ moreover hence "finite (fg ` Field t - {rs.const})" using *
+ unfolding support_def rs.zero_oexp[OF Field] FinFunc_def Func_def
+ by (elim finite_surj[of _ _ fg]) (fastforce simp: image_iff these_def)
+ ultimately have "finite ((\<Union>g \<in> fg ` Field t. rs.SUPP g) \<times> rst.SUPP fg)"
+ by (subst **) (auto intro!: finite_cartesian_product)
+ with * show "?g \<in> FinFunc r (s *o t)"
+ unfolding Field_oprod rs.Field_oexp FinFunc_def Func_def fin_support_def these_def
+ support_def rs.zero_oexp[OF Field] by (auto elim!: finite_subset[rotated])
+ qed
+qed
+
+lemma rev_curr_app_FinFunc[elim!]:
+ "\<lbrakk>f \<in> FinFunc r (s *o t); z \<in> Field t\<rbrakk> \<Longrightarrow> rev_curr f z \<in> FinFunc r s"
+ unfolding rev_curr_def FinFunc_def Func_def Field_oprod fin_support_def support_def
+ by (auto elim: finite_surj)
+
+lemma max_fun_diff_oprod:
+ assumes Field: "Field r \<noteq> {}" and "f \<noteq> g" "f \<in> FinFunc r (s *o t)" "g \<in> FinFunc r (s *o t)"
+ defines "m \<equiv> wo_rel.max_fun_diff t (rev_curr f) (rev_curr g)"
+ shows "wo_rel.max_fun_diff (s *o t) f g =
+ (wo_rel.max_fun_diff s (rev_curr f m) (rev_curr g m), m)"
+proof -
+ interpret st!: wo_rel "s *o t" by unfold_locales (rule oprod_Well_order[OF s t])
+ interpret s!: wo_rel s by unfold_locales (rule s)
+ interpret t!: wo_rel t by unfold_locales (rule t)
+ interpret r_st!: wo_rel2 r "s *o t" by unfold_locales (rule r, rule oprod_Well_order[OF s t])
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rst!: wo_rel2 "r ^o s" t by unfold_locales (rule oexp_Well_order[OF r s], rule t)
+ from fun_unequal_in_support[OF assms(2), of "Field (s *o t)" "Field r" "Field r"] assms(3,4)
+ have diff1: "rev_curr f \<noteq> rev_curr g"
+ "rev_curr f \<in> FinFunc (r ^o s) t" "rev_curr g \<in> FinFunc (r ^o s) t" using rev_curr_FinFunc[OF Field]
+ unfolding fun_eq_iff rev_curr_def[abs_def] FinFunc_def support_def Field_oprod
+ by auto fast
+ hence diff2: "rev_curr f m \<noteq> rev_curr g m" "rev_curr f m \<in> FinFunc r s" "rev_curr g m \<in> FinFunc r s"
+ using rst.max_fun_diff[OF diff1] assms(3,4) rst.max_fun_diff_in unfolding m_def by auto
+ show ?thesis unfolding st.max_fun_diff_def
+ proof (intro st.maxim_equality, unfold st.isMaxim_def Field_oprod, safe)
+ show "s.max_fun_diff (rev_curr f m) (rev_curr g m) \<in> Field s"
+ using rs.max_fun_diff_in[OF diff2] by auto
+ next
+ show "m \<in> Field t" using rst.max_fun_diff_in[OF diff1] unfolding m_def by auto
+ next
+ assume "f (s.max_fun_diff (rev_curr f m) (rev_curr g m), m) =
+ g (s.max_fun_diff (rev_curr f m) (rev_curr g m), m)"
+ (is "f (?x, m) = g (?x, m)")
+ hence "rev_curr f m ?x = rev_curr g m ?x" unfolding rev_curr_def by auto
+ with rs.max_fun_diff[OF diff2] show False by auto
+ next
+ fix x y assume "f (x, y) \<noteq> g (x, y)" "x \<in> Field s" "y \<in> Field t"
+ thus "((x, y), (s.max_fun_diff (rev_curr f m) (rev_curr g m), m)) \<in> s *o t"
+ using rst.max_fun_diff_in[OF diff1] rs.max_fun_diff_in[OF diff2] diff1 diff2
+ rst.max_fun_diff_max[OF diff1, of y] rs.max_fun_diff_le_eq[OF _ diff2, of x]
+ unfolding oprod_def m_def rev_curr_def fun_eq_iff by auto (metis s.in_notinI)
+ qed
+qed
+
+lemma oexp_oexp: "(r ^o s) ^o t =o r ^o (s *o t)" (is "?R =o ?L")
+proof (cases "r = {}")
+ case True
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rst!: wo_rel2 "r ^o s" t by unfold_locales (rule oexp_Well_order[OF r s], rule t)
+ show ?thesis
+ proof (cases "s = {} \<or> t = {}")
+ case True with `r = {}` show ?thesis
+ by (auto simp: oexp_empty[OF oexp_Well_order[OF Well_order_empty s]]
+ intro!: ordIso_transitive[OF ordIso_symmetric[OF oone_ordIso] oone_ordIso]
+ ordIso_transitive[OF oone_ordIso_oexp[OF ordIso_symmetric[OF oone_ordIso] t] oone_ordIso])
+ next
+ case False
+ moreover hence "s *o t \<noteq> {}" unfolding oprod_def Field_def by fastforce
+ ultimately show ?thesis using `r = {}` ozero_ordIso
+ by (auto simp add: s t oprod_Well_order ozero_def)
+ qed
+next
+ case False
+ hence Field: "Field r \<noteq> {}" by (metis Field_def Range_empty_iff Un_empty)
+ show ?thesis
+ proof (rule ordIso_symmetric)
+ interpret r_st!: wo_rel2 r "s *o t" by unfold_locales (rule r, rule oprod_Well_order[OF s t])
+ interpret rs!: wo_rel2 r s by unfold_locales (rule r, rule s)
+ interpret rst!: wo_rel2 "r ^o s" t by unfold_locales (rule oexp_Well_order[OF r s], rule t)
+ have bij: "bij_betw rev_curr (Field ?L) (Field ?R)"
+ unfolding bij_betw_def r_st.Field_oexp rst.Field_oexp Field_oprod proof (intro conjI)
+ show "inj_on rev_curr (FinFunc r (s *o t))"
+ unfolding inj_on_def FinFunc_def Func_def Field_oprod rs.Field_oexp rev_curr_def[abs_def]
+ by (auto simp: fun_eq_iff) metis
+ show "rev_curr ` (FinFunc r (s *o t)) = FinFunc (r ^o s) t" by (rule rev_curr_FinFunc[OF Field])
+ qed
+ moreover
+ have "compat ?L ?R rev_curr"
+ unfolding compat_def proof safe
+ fix fg1 fg2 assume fg: "(fg1, fg2) \<in> r ^o (s *o t)"
+ show "(rev_curr fg1, rev_curr fg2) \<in> r ^o s ^o t"
+ proof (cases "fg1 = fg2")
+ assume "fg1 \<noteq> fg2"
+ with fg show ?thesis
+ using rst.max_fun_diff_in[of "rev_curr fg1" "rev_curr fg2"]
+ max_fun_diff_oprod[OF Field, of fg1 fg2] rev_curr_FinFunc[OF Field, symmetric]
+ unfolding r_st.Field_oexp rs.Field_oexp rst.Field_oexp unfolding r_st.oexp_def rst.oexp_def
+ by (auto simp: rs.oexp_def Let_def) (auto simp: rev_curr_def[abs_def])
+ next
+ assume "fg1 = fg2"
+ with fg bij show ?thesis unfolding r_st.Field_oexp rs.Field_oexp rst.Field_oexp bij_betw_def
+ by (auto simp: r_st.oexp_def rst.oexp_def)
+ qed
+ qed
+ ultimately have "iso ?L ?R rev_curr" using r s t
+ by (subst iso_iff3) (auto intro: oexp_Well_order oprod_Well_order)
+ thus "?L =o ?R" using r s t unfolding ordIso_def
+ by (auto intro: oexp_Well_order oprod_Well_order)
+ qed
+qed
+
+end (* context with 3 wellorders *)
+
+end