author | traytel |
Thu, 07 Apr 2016 17:56:26 +0200 | |
changeset 62906 | 75ca185db27f |
parent 62905 | 52c5a25e0c96 |
child 63561 | fba08009ff3e |
permissions | -rw-r--r-- |
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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 fixpoint (datatype) operation on bounded natural functors. |
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*) |
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section \<open>Least Fixpoint (Datatype) Operation on Bounded Natural Functors\<close> |
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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" :: 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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|
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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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|
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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 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 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 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 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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lemma all_mem_range1: |
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"(\<And>y. y \<in> range f \<Longrightarrow> P y) \<equiv> (\<And>x. P (f x)) " |
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by (rule equal_intr_rule) fast+ |
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lemma all_mem_range2: |
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"(\<And>fa y. fa \<in> range f \<Longrightarrow> y \<in> range fa \<Longrightarrow> P y) \<equiv> (\<And>x xa. P (f x xa))" |
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by (rule equal_intr_rule) fast+ |
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lemma all_mem_range3: |
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"(\<And>fa fb y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> y \<in> range fb \<Longrightarrow> P y) \<equiv> (\<And>x xa xb. P (f x xa xb))" |
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by (rule equal_intr_rule) fast+ |
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lemma all_mem_range4: |
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"(\<And>fa fb fc y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> y \<in> range fc \<Longrightarrow> P y) \<equiv> |
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(\<And>x xa xb xc. P (f x xa xb xc))" |
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by (rule equal_intr_rule) fast+ |
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lemma all_mem_range5: |
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"(\<And>fa fb fc fd y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow> |
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y \<in> range fd \<Longrightarrow> P y) \<equiv> |
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(\<And>x xa xb xc xd. P (f x xa xb xc xd))" |
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by (rule equal_intr_rule) fast+ |
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lemma all_mem_range6: |
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"(\<And>fa fb fc fd fe ff y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow> |
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fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> y \<in> range ff \<Longrightarrow> P y) \<equiv> |
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(\<And>x xa xb xc xd xe xf. P (f x xa xb xc xd xe xf))" |
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by (rule equal_intr_rule) (fastforce, fast) |
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lemma all_mem_range7: |
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"(\<And>fa fb fc fd fe ff fg y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow> |
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fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> fg \<in> range ff \<Longrightarrow> y \<in> range fg \<Longrightarrow> P y) \<equiv> |
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(\<And>x xa xb xc xd xe xf xg. P (f x xa xb xc xd xe xf xg))" |
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by (rule equal_intr_rule) (fastforce, fast) |
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lemma all_mem_range8: |
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191 |
"(\<And>fa fb fc fd fe ff fg fh y. fa \<in> range f \<Longrightarrow> fb \<in> range fa \<Longrightarrow> fc \<in> range fb \<Longrightarrow> fd \<in> range fc \<Longrightarrow> |
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fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> fg \<in> range ff \<Longrightarrow> fh \<in> range fg \<Longrightarrow> y \<in> range fh \<Longrightarrow> P y) \<equiv> |
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193 |
(\<And>x xa xb xc xd xe xf xg xh. P (f x xa xb xc xd xe xf xg xh))" |
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194 |
by (rule equal_intr_rule) (fastforce, fast) |
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195 |
|
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lemmas all_mem_range = all_mem_range1 all_mem_range2 all_mem_range3 all_mem_range4 all_mem_range5 |
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197 |
all_mem_range6 all_mem_range7 all_mem_range8 |
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198 |
|
55062 | 199 |
ML_file "Tools/BNF/bnf_lfp_util.ML" |
200 |
ML_file "Tools/BNF/bnf_lfp_tactics.ML" |
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201 |
ML_file "Tools/BNF/bnf_lfp.ML" |
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202 |
ML_file "Tools/BNF/bnf_lfp_compat.ML" |
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55571 | 203 |
ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML" |
58179 | 204 |
ML_file "Tools/BNF/bnf_lfp_size.ML" |
205 |
||
48975
7f79f94a432c
added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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206 |
end |