(*  Title:      HOL/BNF/BNF_LFP.thy

     Author:     Dmitriy Traytel, TU Muenchen

     Copyright   2012

 Least fixed point operation on bounded natural functors.

 *)

 header {* Least Fixed Point Operation on Bounded Natural Functors *}

 theory BNF_LFP

 imports BNF_FP

 keywords

   "data" :: 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> rel.underS R j"

 unfolding rel.underS_def by simp

 lemma underS_E: "i \<in> rel.underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"

 unfolding rel.underS_def by simp

 lemma underS_Field: "i \<in> rel.underS R j \<Longrightarrow> i \<in> Field R"

 unfolding rel.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 (<f, g> x) = f x"

 using fst_convol unfolding convol_def by simp

 lemma snd_convol': "snd (<f, g> x) = g x"

 using snd_convol unfolding convol_def by simp

 lemma convol_expand_snd: "fst o f = g \<Longrightarrow>  <g, snd o f> = f"

 unfolding convol_def by auto

 definition inver where

   "inver g f A = (ALL a : A. g (f a) = a)"

 lemma bij_betw_iff_ex:

   "bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")

 proof (rule iffI)

   assume ?L

   hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto

   let ?phi = "% b a. a : A \<and> f a = b"

   have "ALL b : B. EX a. ?phi b a" using f by blast

   then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"

     using bchoice[of B ?phi] by blast

   hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast

   have gf: "inver g f A" unfolding inver_def

     by (metis (no_types) gg imageI[of _ A f] the_inv_into_f_f[OF inj_f])

   moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast

   moreover have "A \<le> g ` B"

   proof safe

     fix a assume a: "a : A"

     hence "f a : B" using f by auto

     moreover have "a = g (f a)" using a gf unfolding inver_def by auto

     ultimately show "a : g ` B" by blast

   qed

   ultimately show ?R by blast

 next

   assume ?R

   then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast

   show ?L unfolding bij_betw_def

   proof safe

     show "inj_on f A" unfolding inj_on_def

     proof safe

       fix a1 a2 assume a: "a1 : A"  "a2 : A" and "f a1 = f a2"

       hence "g (f a1) = g (f a2)" by simp

       thus "a1 = a2" using a g unfolding inver_def by simp

     qed

   next

     fix a assume "a : A"

     then obtain b where b: "b : B" and a: "a = g b" using g by blast

     hence "b = f (g b)" using g unfolding inver_def by auto

     thus "f a : B" unfolding a using b by simp

   next

     fix b assume "b : B"

     hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto

     thus "b : f ` A" by auto

   qed

 qed

 lemma bij_betw_ex_weakE:

   "\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"

 by (auto simp only: bij_betw_iff_ex)

 lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"

 unfolding inver_def by auto (rule rev_image_eqI, auto)

 lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"

 unfolding inver_def by auto

 lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"

 unfolding inver_def by simp

 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 inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"

 unfolding inver_def by auto

 lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"

 unfolding bij_betw_def inver_def by auto

 lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"

 unfolding bij_betw_def inver_def by auto

 lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"

 by (drule bij_betw_imageE, unfold bij_betw_iff_ex) blast

 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

 apply (rule conjI)

  apply blast

 by (erule thin_rl) 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 `(\<And>x y. PROP P x y)`)

 ML_file "Tools/bnf_lfp_util.ML"

 ML_file "Tools/bnf_lfp_tactics.ML"

 ML_file "Tools/bnf_lfp.ML"

 end