src/HOL/BNF_Least_Fixpoint.thy
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(*  Title:      HOL/BNF_Least_Fixpoint.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Lorenz Panny, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012, 2013, 2014
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Least fixed point operation on bounded natural functors.
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*)
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header {* Least Fixed Point Operation on Bounded Natural Functors *}
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theory BNF_Least_Fixpoint
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imports BNF_Fixpoint_Base
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keywords
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  "datatype_new" :: thy_decl and
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  "datatype_compat" :: thy_decl
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begin
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lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
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  by blast
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lemma image_Collect_subsetI: "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
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  by blast
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lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
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  by auto
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lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
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  by auto
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lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j"
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  unfolding underS_def by simp
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lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
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  unfolding underS_def by simp
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lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R"
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  unfolding underS_def Field_def by auto
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lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
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  unfolding Field_def by auto
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lemma fst_convol': "fst (\<langle>f, g\<rangle> x) = f x"
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  using fst_convol unfolding convol_def by simp
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lemma snd_convol': "snd (\<langle>f, g\<rangle> x) = g x"
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  using snd_convol unfolding convol_def by simp
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lemma convol_expand_snd: "fst o f = g \<Longrightarrow> \<langle>g, snd o f\<rangle> = f"
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  unfolding convol_def by auto
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lemma convol_expand_snd':
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  assumes "(fst o f = g)"
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  shows "h = snd o f \<longleftrightarrow> \<langle>g, h\<rangle> = f"
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proof -
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  from assms have *: "\<langle>g, snd o f\<rangle> = f" by (rule convol_expand_snd)
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  then have "h = snd o f \<longleftrightarrow> h = snd o \<langle>g, snd o f\<rangle>" by simp
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  moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol)
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  moreover have "\<dots> \<longleftrightarrow> \<langle>g, h\<rangle> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
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  ultimately show ?thesis by simp
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qed
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lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
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  unfolding bij_betw_def by auto
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lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
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  unfolding bij_betw_def by auto
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lemma f_the_inv_into_f_bij_betw:
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  "bij_betw f A B \<Longrightarrow> (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x"
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  unfolding bij_betw_def by (blast intro: f_the_inv_into_f)
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lemma ex_bij_betw: "|A| \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A"
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  by (subst (asm) internalize_card_of_ordLeq) (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric])
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lemma bij_betwI':
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  "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
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    \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
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    \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
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  unfolding bij_betw_def inj_on_def by blast
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lemma surj_fun_eq:
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  assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
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  shows "g1 = g2"
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proof (rule ext)
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  fix y
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  from surj_on obtain x where "x \<in> X" and "y = f x" by blast
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  thus "g1 y = g2 y" using eq_on by simp
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qed
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lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
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  unfolding wo_rel_def card_order_on_def by blast
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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"
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  unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
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lemma Card_order_trans:
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  "\<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"
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  unfolding card_order_on_def well_order_on_def linear_order_on_def
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    partial_order_on_def preorder_on_def trans_def antisym_def by blast
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lemma Cinfinite_limit2:
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  assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
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  shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
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proof -
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  from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
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    unfolding card_order_on_def well_order_on_def linear_order_on_def
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      partial_order_on_def preorder_on_def by auto
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  obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
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    using Cinfinite_limit[OF x1 r] by blast
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  obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
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    using Cinfinite_limit[OF x2 r] by blast
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  show ?thesis
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  proof (cases "y1 = y2")
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    case True with y1 y2 show ?thesis by blast
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  next
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    case False
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    with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
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      unfolding total_on_def by auto
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    thus ?thesis
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    proof
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      assume *: "(y1, y2) \<in> r"
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      with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
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      with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
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    next
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      assume *: "(y2, y1) \<in> r"
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      with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
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      with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
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    qed
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  qed
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qed
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lemma Cinfinite_limit_finite:
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  "\<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)"
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proof (induct X rule: finite_induct)
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  case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
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next
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  case (insert x X)
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  then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
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  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"
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    using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
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  show ?case
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    apply (intro bexI ballI)
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    apply (erule insertE)
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    apply hypsubst
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    apply (rule z(2))
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    using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
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    apply blast
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    apply (rule z(1))
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    done
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qed
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lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
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  by auto
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lemmas well_order_induct_imp = wo_rel.well_order_induct[of r "\<lambda>x. x \<in> Field r \<longrightarrow> P x" for r P]
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lemma meta_spec2:
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  assumes "(\<And>x y. PROP P x y)"
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  shows "PROP P x y"
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  by (rule assms)
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lemma nchotomy_relcomppE:
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  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"
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  shows P
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proof (rule relcompp.cases[OF assms(2)], hypsubst)
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  fix b assume "r a b" "s b c"
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  moreover from assms(1) obtain b' where "b = f b'" by blast
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  ultimately show P by (blast intro: assms(3))
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qed
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lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
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  unfolding rel_fun_def vimage2p_def by auto
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lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
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  unfolding vimage2p_def by auto
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lemma id_transfer: "rel_fun A A id id"
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  unfolding rel_fun_def by simp
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lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R"
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  by (rule ssubst)
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ML_file "Tools/BNF/bnf_lfp_util.ML"
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ML_file "Tools/BNF/bnf_lfp_tactics.ML"
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ML_file "Tools/BNF/bnf_lfp.ML"
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ML_file "Tools/BNF/bnf_lfp_compat.ML"
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ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML"
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ML_file "Tools/BNF/bnf_lfp_size.ML"
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ML_file "Tools/Function/old_size.ML"
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lemma size_bool[code]: "size (b\<Colon>bool) = 0"
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  by (cases b) auto
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lemma size_nat[simp, code]: "size (n\<Colon>nat) = n"
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  by (induct n) simp_all
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declare prod.size[no_atp]
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lemma size_sum_o_map: "size_sum g1 g2 \<circ> map_sum f1 f2 = size_sum (g1 \<circ> f1) (g2 \<circ> f2)"
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  by (rule ext) (case_tac x, auto)
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lemma size_prod_o_map: "size_prod g1 g2 \<circ> map_prod f1 f2 = size_prod (g1 \<circ> f1) (g2 \<circ> f2)"
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  by (rule ext) auto
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setup {*
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BNF_LFP_Size.register_size_global @{type_name sum} @{const_name size_sum} @{thms sum.size}
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  @{thms size_sum_o_map}
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#> BNF_LFP_Size.register_size_global @{type_name prod} @{const_name size_prod} @{thms prod.size}
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  @{thms size_prod_o_map}
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*}
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hide_fact (open) id_transfer
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end