src/HOL/Cardinals/Wellorder_Relation.thy

tuned;

+
−(*  Title:      HOL/Cardinals/Wellorder_Relation.thy
+
−    Author:     Andrei Popescu, TU Muenchen
+
−    Copyright   2012
+
−
+
−Well-order relations.
+
−*)
+
−
+
−section \<open>Well-Order Relations\<close>
+
−
+
−theory Wellorder_Relation
+
−imports BNF_Wellorder_Relation Wellfounded_More
+
−begin
+
−
+
−context wo_rel
+
−begin
+
−
+
−subsection \<open>Auxiliaries\<close>
+
−
+
−lemma PREORD: "Preorder r"
+
−using WELL order_on_defs[of _ r] by auto
+
−
+
−lemma PARORD: "Partial_order r"
+
−using WELL order_on_defs[of _ r] by auto
+
−
+
−lemma 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 \<open>Well-founded induction and recursion adapted to non-strict well-order relations\<close>
+
−
+
−lemma 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[of r] 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 \<open>Properties of max2\<close>
+
−
+
−lemma 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 \<open>Properties of minim\<close>
+
−
+
−lemma minim_Under:
+
−"\<lbrakk>B \<le> Field r; B \<noteq> {}\<rbrakk> \<Longrightarrow> minim B \<in> Under B"
+
−by(auto simp add: Under_def minim_inField minim_least)
+
−
+
−lemma 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 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 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[of r] 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 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[of r] 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 \<open>Properties of supr\<close>
+
−
+
−lemma 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[of r] by auto
+
−  thus "minim (Above B) \<in> Above B"
+
−  using assms by (simp add: minim_in)
+
−qed
+
−
+
−lemma 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[of r] show ?thesis by simp
+
−qed
+
−
+
−lemma 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[of r] by auto
+
−  thus "(minim (Above B), a) \<in> r"
+
−  using assms minim_least
+
−  by simp
+
−qed
+
−
+
−lemma 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 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[of r] by auto
+
−  thus "a = minim (Above B)"
+
−  using assms equals_minim by simp
+
−qed
+
−
+
−lemma 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 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[of r] by auto
+
−qed
+
−
+
−lemma 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 supr_under:
+
−assumes IN: "a \<in> Field r"
+
−shows "a = supr (under a)"
+
−proof-
+
−  have "under a \<le> Field r"
+
−  using under_Field[of r] by auto
+
−  moreover
+
−  have "under a \<noteq> {}"
+
−  using IN Refl_under_in[of r] REFL by auto
+
−  moreover
+
−  have "a \<in> Above (under a)"
+
−  using in_Above_under[of _ r] 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 \<open>Properties of successor\<close>
+
−
+
−lemma 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 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 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 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 metis
+
−  {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 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[of r] 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 \<open>Properties of order filters\<close>
+
−
+
−lemma ofilter_Under[simp]:
+
−assumes "A \<le> Field r"
+
−shows "ofilter(Under A)"
+
−proof(unfold ofilter_def, auto)
+
−  fix x assume "x \<in> Under A"
+
−  thus "x \<in> Field r"
+
−  using Under_Field[of r] assms by auto
+
−next
+
−  fix a x
+
−  assume "a \<in> Under A" and "x \<in> under a"
+
−  thus "x \<in> Under A"
+
−  using TRANS under_Under_trans[of r] by auto
+
−qed
+
−
+
−lemma ofilter_UnderS[simp]:
+
−assumes "A \<le> Field r"
+
−shows "ofilter(UnderS A)"
+
−proof(unfold ofilter_def, auto)
+
−  fix x assume "x \<in> UnderS A"
+
−  thus "x \<in> Field r"
+
−  using UnderS_Field[of r] assms by auto
+
−next
+
−  fix a x
+
−  assume "a \<in> UnderS A" and "x \<in> under a"
+
−  thus "x \<in> UnderS A"
+
−  using TRANS ANTISYM under_UnderS_trans[of r] by auto
+
−qed
+
−
+
−lemma ofilter_Int[simp]: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A Int B)"
+
−unfolding ofilter_def by blast
+
−
+
−lemma ofilter_Un[simp]: "\<lbrakk>ofilter A; ofilter B\<rbrakk> \<Longrightarrow> ofilter(A \<union> B)"
+
−unfolding ofilter_def by blast
+
−
+
−lemma 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 ofilter_Inter:
+
−"\<lbrakk>S \<noteq> {}; \<And> A. A \<in> S \<Longrightarrow> ofilter A\<rbrakk> \<Longrightarrow> ofilter (\<Inter>S)"
+
−unfolding ofilter_def by blast
+
−
+
−lemma ofilter_Union:
+
−"(\<And> A. A \<in> S \<Longrightarrow> ofilter A) \<Longrightarrow> ofilter (\<Union>S)"
+
−unfolding ofilter_def by blast
+
−
+
−lemma ofilter_under_Union:
+
−"ofilter A \<Longrightarrow> A = \<Union>{under a| a. a \<in> A}"
+
−using ofilter_under_UNION [of A] by auto
+
−
+
−
+
−subsubsection \<open>Other properties\<close>
+
−
+
−lemma 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[of r] 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 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[of r] by auto
+
−    ultimately show "x = suc A"
+
−    using ANTISYM antisym_def[of r] by auto
+
−  qed
+
−qed
+
−
+
−(* FIXME: needed? *)
+
−declare
+
−  minim_in[simp]
+
−  minim_inField[simp]
+
−  minim_least[simp]
+
−  under_ofilter[simp]
+
−  underS_ofilter[simp]
+
−  Field_ofilter[simp]
+
−
+
−end
+
−
+
−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"
+
−
+
−end