src/HOL/BNF_Least_Fixpoint.thy
changeset 58128 43a1ba26a8cb
parent 58123 62765d39539f
child 58136 10f92532f128
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/BNF_Least_Fixpoint.thy	Mon Sep 01 16:34:40 2014 +0200
@@ -0,0 +1,198 @@
+(*  Title:      HOL/BNF_Least_Fixpoint.thy
+    Author:     Dmitriy Traytel, TU Muenchen
+    Author:     Lorenz Panny, TU Muenchen
+    Author:     Jasmin Blanchette, TU Muenchen
+    Copyright   2012, 2013, 2014
+
+Least fixed point operation on bounded natural functors.
+*)
+
+header {* Least Fixed Point Operation on Bounded Natural Functors *}
+
+theory BNF_Least_Fixpoint
+imports BNF_Fixpoint_Base
+keywords
+  "datatype_new" :: thy_decl and
+  "datatype_compat" :: thy_decl
+begin
+
+lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
+  by blast
+
+lemma image_Collect_subsetI: "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
+  by blast
+
+lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
+  by auto
+
+lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
+  by auto
+
+lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j"
+  unfolding underS_def by simp
+
+lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
+  unfolding underS_def by simp
+
+lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R"
+  unfolding underS_def Field_def by auto
+
+lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
+  unfolding Field_def by auto
+
+lemma fst_convol': "fst (\<langle>f, g\<rangle> x) = f x"
+  using fst_convol unfolding convol_def by simp
+
+lemma snd_convol': "snd (\<langle>f, g\<rangle> x) = g x"
+  using snd_convol unfolding convol_def by simp
+
+lemma convol_expand_snd: "fst o f = g \<Longrightarrow> \<langle>g, snd o f\<rangle> = f"
+  unfolding convol_def by auto
+
+lemma convol_expand_snd':
+  assumes "(fst o f = g)"
+  shows "h = snd o f \<longleftrightarrow> \<langle>g, h\<rangle> = f"
+proof -
+  from assms have *: "\<langle>g, snd o f\<rangle> = f" by (rule convol_expand_snd)
+  then have "h = snd o f \<longleftrightarrow> h = snd o \<langle>g, snd o f\<rangle>" by simp
+  moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol)
+  moreover have "\<dots> \<longleftrightarrow> \<langle>g, h\<rangle> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
+  ultimately show ?thesis by simp
+qed
+
+lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
+  unfolding bij_betw_def by auto
+
+lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
+  unfolding bij_betw_def by auto
+
+lemma f_the_inv_into_f_bij_betw: "bij_betw f A B \<Longrightarrow>
+  (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x"
+  unfolding bij_betw_def by (blast intro: f_the_inv_into_f)
+
+lemma ex_bij_betw: "|A| \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A"
+  by (subst (asm) internalize_card_of_ordLeq)
+    (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])
+
+lemma bij_betwI':
+  "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
+    \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
+    \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
+  unfolding bij_betw_def inj_on_def by blast
+
+lemma surj_fun_eq:
+  assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
+  shows "g1 = g2"
+proof (rule ext)
+  fix y
+  from surj_on obtain x where "x \<in> X" and "y = f x" by blast
+  thus "g1 y = g2 y" using eq_on by simp
+qed
+
+lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
+unfolding wo_rel_def card_order_on_def by blast
+
+lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
+  \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
+unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
+
+lemma Card_order_trans:
+  "\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
+unfolding card_order_on_def well_order_on_def linear_order_on_def
+  partial_order_on_def preorder_on_def trans_def antisym_def by blast
+
+lemma Cinfinite_limit2:
+ assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
+ shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
+proof -
+  from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
+    unfolding card_order_on_def well_order_on_def linear_order_on_def
+      partial_order_on_def preorder_on_def by auto
+  obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
+    using Cinfinite_limit[OF x1 r] by blast
+  obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
+    using Cinfinite_limit[OF x2 r] by blast
+  show ?thesis
+  proof (cases "y1 = y2")
+    case True with y1 y2 show ?thesis by blast
+  next
+    case False
+    with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
+      unfolding total_on_def by auto
+    thus ?thesis
+    proof
+      assume *: "(y1, y2) \<in> r"
+      with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
+      with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
+    next
+      assume *: "(y2, y1) \<in> r"
+      with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
+      with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
+    qed
+  qed
+qed
+
+lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
+ \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
+proof (induct X rule: finite_induct)
+  case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
+next
+  case (insert x X)
+  then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
+  then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
+    using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
+  show ?case
+    apply (intro bexI ballI)
+    apply (erule insertE)
+    apply hypsubst
+    apply (rule z(2))
+    using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
+    apply blast
+    apply (rule z(1))
+    done
+qed
+
+lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
+by auto
+
+(*helps resolution*)
+lemma well_order_induct_imp:
+  "wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
+     x \<in> Field r \<longrightarrow> P x"
+by (erule wo_rel.well_order_induct)
+
+lemma meta_spec2:
+  assumes "(\<And>x y. PROP P x y)"
+  shows "PROP P x y"
+by (rule assms)
+
+lemma nchotomy_relcomppE:
+  assumes "\<And>y. \<exists>x. y = f x" "(r OO s) a c" "\<And>b. r a (f b) \<Longrightarrow> s (f b) c \<Longrightarrow> P"
+  shows P
+proof (rule relcompp.cases[OF assms(2)], hypsubst)
+  fix b assume "r a b" "s b c"
+  moreover from assms(1) obtain b' where "b = f b'" by blast
+  ultimately show P by (blast intro: assms(3))
+qed
+
+lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
+  unfolding rel_fun_def vimage2p_def by auto
+
+lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
+  unfolding vimage2p_def by auto
+
+lemma id_transfer: "rel_fun A A id id"
+  unfolding rel_fun_def by simp
+
+lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R"
+  by (rule ssubst)
+
+ML_file "Tools/BNF/bnf_lfp_util.ML"
+ML_file "Tools/BNF/bnf_lfp_tactics.ML"
+ML_file "Tools/BNF/bnf_lfp.ML"
+ML_file "Tools/BNF/bnf_lfp_compat.ML"
+ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML"
+
+hide_fact (open) id_transfer
+
+end