src/HOL/Cardinals/Wellorder_Relation.thy

+(*  Title:      HOL/Cardinals/Wellorder_Relation.thy
+    Author:     Andrei Popescu, TU Muenchen
+    Copyright   2012
+
+Well-order relations.
+*)
+
+header {* Well-Order Relations *}
+
+theory Wellorder_Relation
+imports Wellorder_Relation_Base Wellfounded_More
+begin
+
+
+subsection {* Auxiliaries *}
+
+lemma (in wo_rel) PREORD: "Preorder r"
+using WELL order_on_defs[of _ r] by auto
+
+lemma (in wo_rel) PARORD: "Partial_order r"
+using WELL order_on_defs[of _ r] by auto
+
+lemma (in wo_rel) cases_Total2:
+"\<And> phi a b. \<lbrakk>{a,b} \<le> Field r; ((a,b) \<in> r - Id \<Longrightarrow> phi a b);
+              ((b,a) \<in> r - Id \<Longrightarrow> phi a b); (a = b \<Longrightarrow> phi a b)\<rbrakk>
+             \<Longrightarrow> phi a b"
+using TOTALS by auto
+
+
+subsection {* Well-founded induction and recursion adapted to non-strict well-order relations  *}
+
+lemma (in wo_rel) worec_unique_fixpoint:
+assumes ADM: "adm_wo H" and fp: "f = H f"
+shows "f = worec H"
+proof-
+  have "adm_wf (r - Id) H"
+  unfolding adm_wf_def
+  using ADM adm_wo_def[of H] underS_def by auto
+  hence "f = wfrec (r - Id) H"
+  using fp WF wfrec_unique_fixpoint[of "r - Id" H] by simp
+  thus ?thesis unfolding worec_def .
+qed
+
+
+subsubsection {* Properties of max2 *}
+
+lemma (in wo_rel) max2_iff:
+assumes "a \<in> Field r" and "b \<in> Field r"
+shows "((max2 a b, c) \<in> r) = ((a,c) \<in> r \<and> (b,c) \<in> r)"
+proof
+  assume "(max2 a b, c) \<in> r"
+  thus "(a,c) \<in> r \<and> (b,c) \<in> r"
+  using assms max2_greater[of a b] TRANS trans_def[of r] by blast
+next
+  assume "(a,c) \<in> r \<and> (b,c) \<in> r"
+  thus "(max2 a b, c) \<in> r"
+  using assms max2_among[of a b] by auto
+qed
+
+
+subsubsection{* Properties of minim *}
+
+lemma (in wo_rel) minim_Under:
+"\<lbrakk>B \<le> Field r; B \<noteq> {}\<rbrakk> \<Longrightarrow> minim B \<in> Under B"
+by(auto simp add: Under_def
+minim_in
+minim_inField
+minim_least
+under_ofilter
+underS_ofilter
+Field_ofilter
+ofilter_Under
+ofilter_UnderS
+ofilter_Un
+)
+
+lemma (in wo_rel) equals_minim_Under:
+"\<lbrakk>B \<le> Field r; a \<in> B; a \<in> Under B\<rbrakk>
+ \<Longrightarrow> a = minim B"
+by(auto simp add: Under_def equals_minim)
+
+lemma (in wo_rel) minim_iff_In_Under:
+assumes SUB: "B \<le> Field r" and NE: "B \<noteq> {}"
+shows "(a = minim B) = (a \<in> B \<and> a \<in> Under B)"
+proof
+  assume "a = minim B"
+  thus "a \<in> B \<and> a \<in> Under B"
+  using assms minim_in minim_Under by simp
+next
+  assume "a \<in> B \<and> a \<in> Under B"
+  thus "a = minim B"
+  using assms equals_minim_Under by simp
+qed
+
+lemma (in wo_rel) minim_Under_under:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "Under A = under (minim A)"
+proof-
+  (* Preliminary facts *)
+  have 1: "minim A \<in> A"
+  using assms minim_in by auto
+  have 2: "\<forall>x \<in> A. (minim A, x) \<in> r"
+  using assms minim_least by auto
+  (* Main proof *)
+  have "Under A \<le> under (minim A)"
+  proof
+    fix x assume "x \<in> Under A"
+    with 1 Under_def have "(x,minim A) \<in> r" by auto
+    thus "x \<in> under(minim A)" unfolding under_def by simp
+  qed
+  (*  *)
+  moreover
+  (*  *)
+  have "under (minim A) \<le> Under A"
+  proof
+    fix x assume "x \<in> under(minim A)"
+    hence 11: "(x,minim A) \<in> r" unfolding under_def by simp
+    hence "x \<in> Field r" unfolding Field_def by auto
+    moreover
+    {fix a assume "a \<in> A"
+     with 2 have "(minim A, a) \<in> r" by simp
+     with 11 have "(x,a) \<in> r"
+     using TRANS trans_def[of r] by blast
+    }
+    ultimately show "x \<in> Under A" by (unfold Under_def, auto)
+  qed
+  (*  *)
+  ultimately show ?thesis by blast
+qed
+
+lemma (in wo_rel) minim_UnderS_underS:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "UnderS A = underS (minim A)"
+proof-
+  (* Preliminary facts *)
+  have 1: "minim A \<in> A"
+  using assms minim_in by auto
+  have 2: "\<forall>x \<in> A. (minim A, x) \<in> r"
+  using assms minim_least by auto
+  (* Main proof *)
+  have "UnderS A \<le> underS (minim A)"
+  proof
+    fix x assume "x \<in> UnderS A"
+    with 1 UnderS_def have "x \<noteq> minim A \<and> (x,minim A) \<in> r" by auto
+    thus "x \<in> underS(minim A)" unfolding underS_def by simp
+  qed
+  (*  *)
+  moreover
+  (*  *)
+  have "underS (minim A) \<le> UnderS A"
+  proof
+    fix x assume "x \<in> underS(minim A)"
+    hence 11: "x \<noteq> minim A \<and> (x,minim A) \<in> r" unfolding underS_def by simp
+    hence "x \<in> Field r" unfolding Field_def by auto
+    moreover
+    {fix a assume "a \<in> A"
+     with 2 have 3: "(minim A, a) \<in> r" by simp
+     with 11 have "(x,a) \<in> r"
+     using TRANS trans_def[of r] by blast
+     moreover
+     have "x \<noteq> a"
+     proof
+       assume "x = a"
+       with 11 3 ANTISYM antisym_def[of r]
+       show False by auto
+     qed
+     ultimately
+     have "x \<noteq> a \<and> (x,a) \<in> r" by simp
+    }
+    ultimately show "x \<in> UnderS A" by (unfold UnderS_def, auto)
+  qed
+  (*  *)
+  ultimately show ?thesis by blast
+qed
+
+
+subsubsection{* Properties of supr *}
+
+lemma (in wo_rel) supr_Above:
+assumes SUB: "B \<le> Field r" and ABOVE: "Above B \<noteq> {}"
+shows "supr B \<in> Above B"
+proof(unfold supr_def)
+  have "Above B \<le> Field r"
+  using Above_Field by auto
+  thus "minim (Above B) \<in> Above B"
+  using assms by (simp add: minim_in)
+qed
+
+lemma (in wo_rel) supr_greater:
+assumes SUB: "B \<le> Field r" and ABOVE: "Above B \<noteq> {}" and
+        IN: "b \<in> B"
+shows "(b, supr B) \<in> r"
+proof-
+  from assms supr_Above
+  have "supr B \<in> Above B" by simp
+  with IN Above_def show ?thesis by simp
+qed
+
+lemma (in wo_rel) supr_least_Above:
+assumes SUB: "B \<le> Field r" and
+        ABOVE: "a \<in> Above B"
+shows "(supr B, a) \<in> r"
+proof(unfold supr_def)
+  have "Above B \<le> Field r"
+  using Above_Field by auto
+  thus "(minim (Above B), a) \<in> r"
+  using assms minim_least
+  by simp
+qed
+
+lemma (in wo_rel) supr_least:
+"\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> (b,a) \<in> r)\<rbrakk>
+ \<Longrightarrow> (supr B, a) \<in> r"
+by(auto simp add: supr_least_Above Above_def)
+
+lemma (in wo_rel) equals_supr_Above:
+assumes SUB: "B \<le> Field r" and ABV: "a \<in> Above B" and
+        MINIM: "\<And> a'. a' \<in> Above B \<Longrightarrow> (a,a') \<in> r"
+shows "a = supr B"
+proof(unfold supr_def)
+  have "Above B \<le> Field r"
+  using Above_Field by auto
+  thus "a = minim (Above B)"
+  using assms equals_minim by simp
+qed
+
+lemma (in wo_rel) equals_supr:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> Field r" and
+        ABV: "\<And> b. b \<in> B \<Longrightarrow> (b,a) \<in> r" and
+        MINIM: "\<And> a'. \<lbrakk> a' \<in> Field r; \<And> b. b \<in> B \<Longrightarrow> (b,a') \<in> r\<rbrakk> \<Longrightarrow> (a,a') \<in> r"
+shows "a = supr B"
+proof-
+  have "a \<in> Above B"
+  unfolding Above_def using ABV IN by simp
+  moreover
+  have "\<And> a'. a' \<in> Above B \<Longrightarrow> (a,a') \<in> r"
+  unfolding Above_def using MINIM by simp
+  ultimately show ?thesis
+  using equals_supr_Above SUB by auto
+qed
+
+lemma (in wo_rel) supr_inField:
+assumes "B \<le> Field r" and  "Above B \<noteq> {}"
+shows "supr B \<in> Field r"
+proof-
+  have "supr B \<in> Above B" using supr_Above assms by simp
+  thus ?thesis using assms Above_Field by auto
+qed
+
+lemma (in wo_rel) supr_above_Above:
+assumes SUB: "B \<le> Field r" and  ABOVE: "Above B \<noteq> {}"
+shows "Above B = above (supr B)"
+proof(unfold Above_def above_def, auto)
+  fix a assume "a \<in> Field r" "\<forall>b \<in> B. (b,a) \<in> r"
+  with supr_least assms
+  show "(supr B, a) \<in> r" by auto
+next
+  fix b assume "(supr B, b) \<in> r"
+  thus "b \<in> Field r"
+  using REFL refl_on_def[of _ r] by auto
+next
+  fix a b
+  assume 1: "(supr B, b) \<in> r" and 2: "a \<in> B"
+  with assms supr_greater
+  have "(a,supr B) \<in> r" by auto
+  thus "(a,b) \<in> r"
+  using 1 TRANS trans_def[of r] by blast
+qed
+
+lemma (in wo_rel) supr_under:
+assumes IN: "a \<in> Field r"
+shows "a = supr (under a)"
+proof-
+  have "under a \<le> Field r"
+  using under_Field by auto
+  moreover
+  have "under a \<noteq> {}"
+  using IN Refl_under_in REFL by auto
+  moreover
+  have "a \<in> Above (under a)"
+  using in_Above_under IN by auto
+  moreover
+  have "\<forall>a' \<in> Above (under a). (a,a') \<in> r"
+  proof(unfold Above_def under_def, auto)
+    fix a'
+    assume "\<forall>aa. (aa, a) \<in> r \<longrightarrow> (aa, a') \<in> r"
+    hence "(a,a) \<in> r \<longrightarrow> (a,a') \<in> r" by blast
+    moreover have "(a,a) \<in> r"
+    using REFL IN by (auto simp add: refl_on_def)
+    ultimately
+    show  "(a, a') \<in> r" by (rule mp)
+  qed
+  ultimately show ?thesis
+  using equals_supr_Above by auto
+qed
+
+
+subsubsection{* Properties of successor *}
+
+lemma (in wo_rel) suc_least:
+"\<lbrakk>B \<le> Field r; a \<in> Field r; (\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r)\<rbrakk>
+ \<Longrightarrow> (suc B, a) \<in> r"
+by(auto simp add: suc_least_AboveS AboveS_def)
+
+lemma (in wo_rel) equals_suc:
+assumes SUB: "B \<le> Field r" and IN: "a \<in> Field r" and
+ ABVS: "\<And> b. b \<in> B \<Longrightarrow> a \<noteq> b \<and> (b,a) \<in> r" and
+ MINIM: "\<And> a'. \<lbrakk>a' \<in> Field r; \<And> b. b \<in> B \<Longrightarrow> a' \<noteq> b \<and> (b,a') \<in> r\<rbrakk> \<Longrightarrow> (a,a') \<in> r"
+shows "a = suc B"
+proof-
+  have "a \<in> AboveS B"
+  unfolding AboveS_def using ABVS IN by simp
+  moreover
+  have "\<And> a'. a' \<in> AboveS B \<Longrightarrow> (a,a') \<in> r"
+  unfolding AboveS_def using MINIM by simp
+  ultimately show ?thesis
+  using equals_suc_AboveS SUB by auto
+qed
+
+lemma (in wo_rel) suc_above_AboveS:
+assumes SUB: "B \<le> Field r" and
+        ABOVE: "AboveS B \<noteq> {}"
+shows "AboveS B = above (suc B)"
+proof(unfold AboveS_def above_def, auto)
+  fix a assume "a \<in> Field r" "\<forall>b \<in> B. a \<noteq> b \<and> (b,a) \<in> r"
+  with suc_least assms
+  show "(suc B,a) \<in> r" by auto
+next
+  fix b assume "(suc B, b) \<in> r"
+  thus "b \<in> Field r"
+  using REFL refl_on_def[of _ r] by auto
+next
+  fix a b
+  assume 1: "(suc B, b) \<in> r" and 2: "a \<in> B"
+  with assms suc_greater[of B a]
+  have "(a,suc B) \<in> r" by auto
+  thus "(a,b) \<in> r"
+  using 1 TRANS trans_def[of r] by blast
+next
+  fix a
+  assume 1: "(suc B, a) \<in> r" and 2: "a \<in> B"
+  with assms suc_greater[of B a]
+  have "(a,suc B) \<in> r" by auto
+  moreover have "suc B \<in> Field r"
+  using assms suc_inField by simp
+  ultimately have "a = suc B"
+  using 1 2 SUB ANTISYM antisym_def[of r] by auto
+  thus False
+  using assms suc_greater[of B a] 2 by auto
+qed
+
+lemma (in wo_rel) suc_singl_pred:
+assumes IN: "a \<in> Field r" and ABOVE_NE: "aboveS a \<noteq> {}" and
+        REL: "(a',suc {a}) \<in> r" and DIFF: "a' \<noteq> suc {a}"
+shows "a' = a \<or> (a',a) \<in> r"
+proof-
+  have *: "suc {a} \<in> Field r \<and> a' \<in> Field r"
+  using WELL REL well_order_on_domain by auto
+  {assume **: "a' \<noteq> a"
+   hence "(a,a') \<in> r \<or> (a',a) \<in> r"
+   using TOTAL IN * by (auto simp add: total_on_def)
+   moreover
+   {assume "(a,a') \<in> r"
+    with ** * assms WELL suc_least[of "{a}" a']
+    have "(suc {a},a') \<in> r" by auto
+    with REL DIFF * ANTISYM antisym_def[of r]
+    have False by simp
+   }
+   ultimately have "(a',a) \<in> r"
+   by blast
+  }
+  thus ?thesis by blast
+qed
+
+lemma (in wo_rel) under_underS_suc:
+assumes IN: "a \<in> Field r" and ABV: "aboveS a \<noteq> {}"
+shows "underS (suc {a}) = under a"
+proof-
+  have 1: "AboveS {a} \<noteq> {}"
+  using ABV aboveS_AboveS_singl by auto
+  have 2: "a \<noteq> suc {a} \<and> (a,suc {a}) \<in> r"
+  using suc_greater[of "{a}" a] IN 1 by auto
+  (*   *)
+  have "underS (suc {a}) \<le> under a"
+  proof(unfold underS_def under_def, auto)
+    fix x assume *: "x \<noteq> suc {a}" and **: "(x,suc {a}) \<in> r"
+    with suc_singl_pred[of a x] IN ABV
+    have "x = a \<or> (x,a) \<in> r" by auto
+    with REFL refl_on_def[of _ r] IN
+    show "(x,a) \<in> r" by auto
+  qed
+  (*  *)
+  moreover
+  (*   *)
+  have "under a \<le> underS (suc {a})"
+  proof(unfold underS_def under_def, auto)
+    assume "(suc {a}, a) \<in> r"
+    with 2 ANTISYM antisym_def[of r]
+    show False by auto
+  next
+    fix x assume *: "(x,a) \<in> r"
+    with 2 TRANS trans_def[of r]
+    show "(x,suc {a}) \<in> r" by blast
+  (*  blast is often better than auto/auto for transitivity-like properties *)
+  qed
+  (*  *)
+  ultimately show ?thesis by blast
+qed
+
+
+subsubsection {* Properties of order filters  *}
+
+lemma (in wo_rel) ofilter_INTER:
+"\<lbrakk>I \<noteq> {}; \<And> i. i \<in> I \<Longrightarrow> ofilter(A i)\<rbrakk> \<Longrightarrow> ofilter (\<Inter> i \<in> I. A i)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_Inter:
+"\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> ofilter A\<rbrakk> \<Longrightarrow> ofilter (Inter S)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_Union:
+"(\<And> A. A \<in> S \<Longrightarrow> ofilter A) \<Longrightarrow> ofilter (Union S)"
+unfolding ofilter_def by blast
+
+lemma (in wo_rel) ofilter_under_Union:
+"ofilter A \<Longrightarrow> A = Union {under a| a. a \<in> A}"
+using ofilter_under_UNION[of A]
+by(unfold Union_eq, auto)
+
+
+subsubsection{* Other properties *}
+
+lemma (in wo_rel) Trans_Under_regressive:
+assumes NE: "A \<noteq> {}" and SUB: "A \<le> Field r"
+shows "Under(Under A) \<le> Under A"
+proof
+  let ?a = "minim A"
+  (*  Preliminary facts *)
+  have 1: "minim A \<in> Under A"
+  using assms minim_Under by auto
+  have 2: "\<forall>y \<in> A. (minim A, y) \<in> r"
+  using assms minim_least by auto
+  (* Main proof *)
+  fix x assume "x \<in> Under(Under A)"
+  with 1 have 1: "(x,minim A) \<in> r"
+  using Under_def by auto
+  with Field_def have "x \<in> Field r" by fastforce
+  moreover
+  {fix y assume *: "y \<in> A"
+   hence "(x,y) \<in> r"
+   using 1 2 TRANS trans_def[of r] by blast
+   with Field_def have "(x,y) \<in> r" by auto
+  }
+  ultimately
+  show "x \<in> Under A" unfolding Under_def by auto
+qed
+
+lemma (in wo_rel) ofilter_suc_Field:
+assumes OF: "ofilter A" and NE: "A \<noteq> Field r"
+shows "ofilter (A \<union> {suc A})"
+proof-
+  (* Preliminary facts *)
+  have 1: "A \<le> Field r" using OF ofilter_def by auto
+  hence 2: "AboveS A \<noteq> {}"
+  using ofilter_AboveS_Field NE OF by blast
+  from 1 2 suc_inField
+  have 3: "suc A \<in> Field r" by auto
+  (* Main proof *)
+  show ?thesis
+  proof(unfold ofilter_def, auto simp add: 1 3)
+    fix a x
+    assume "a \<in> A" "x \<in> under a" "x \<notin> A"
+    with OF ofilter_def have False by auto
+    thus "x = suc A" by simp
+  next
+    fix x assume *: "x \<in> under (suc A)" and **: "x \<notin> A"
+    hence "x \<in> Field r" using under_def Field_def by fastforce
+    with ** have "x \<in> AboveS A"
+    using ofilter_AboveS_Field[of A] OF by auto
+    hence "(suc A,x) \<in> r"
+    using suc_least_AboveS by auto
+    moreover
+    have "(x,suc A) \<in> r" using * under_def by auto
+    ultimately show "x = suc A"
+    using ANTISYM antisym_def[of r] by auto
+  qed
+qed
+
+(* FIXME: needed? *)
+declare (in wo_rel)
+  minim_in[simp]
+  minim_inField[simp]
+  minim_least[simp]
+  under_ofilter[simp]
+  underS_ofilter[simp]
+  Field_ofilter[simp]
+  ofilter_Under[simp]
+  ofilter_UnderS[simp]
+  ofilter_Int[simp]
+  ofilter_Un[simp]
+
+abbreviation "worec \<equiv> wo_rel.worec"
+abbreviation "adm_wo \<equiv> wo_rel.adm_wo"
+abbreviation "isMinim \<equiv> wo_rel.isMinim"
+abbreviation "minim \<equiv> wo_rel.minim"
+abbreviation "max2 \<equiv> wo_rel.max2"
+abbreviation "supr \<equiv> wo_rel.supr"
+abbreviation "suc \<equiv> wo_rel.suc"
+abbreviation "ofilter \<equiv> wo_rel.ofilter"
+
+end