1(* Title: HOL/BNF_Least_Fixpoint.thy 2 Author: Dmitriy Traytel, TU Muenchen 3 Author: Lorenz Panny, TU Muenchen 4 Author: Jasmin Blanchette, TU Muenchen 5 Copyright 2012, 2013, 2014 6 7Least fixpoint (datatype) operation on bounded natural functors. 8*) 9 10section \<open>Least Fixpoint (Datatype) Operation on Bounded Natural Functors\<close> 11 12theory BNF_Least_Fixpoint 13imports BNF_Fixpoint_Base 14keywords 15 "datatype" :: thy_decl and 16 "datatype_compat" :: thy_decl 17begin 18 19lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}" 20 by blast 21 22lemma image_Collect_subsetI: "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B" 23 by blast 24 25lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X" 26 by auto 27 28lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x" 29 by auto 30 31lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j" 32 unfolding underS_def by simp 33 34lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R" 35 unfolding underS_def by simp 36 37lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R" 38 unfolding underS_def Field_def by auto 39 40lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B" 41 unfolding bij_betw_def by auto 42 43lemma f_the_inv_into_f_bij_betw: 44 "bij_betw f A B \<Longrightarrow> (bij_betw f A B \<Longrightarrow> x \<in> B) \<Longrightarrow> f (the_inv_into A f x) = x" 45 unfolding bij_betw_def by (blast intro: f_the_inv_into_f) 46 47lemma ex_bij_betw: "|A| \<le>o (r :: 'b rel) \<Longrightarrow> \<exists>f B :: 'b set. bij_betw f B A" 48 by (subst (asm) internalize_card_of_ordLeq) (auto dest!: iffD2[OF card_of_ordIso ordIso_symmetric]) 49 50lemma bij_betwI': 51 "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y); 52 \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y; 53 \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y" 54 unfolding bij_betw_def inj_on_def by blast 55 56lemma surj_fun_eq: 57 assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 \<circ> f) x = (g2 \<circ> f) x" 58 shows "g1 = g2" 59proof (rule ext) 60 fix y 61 from surj_on obtain x where "x \<in> X" and "y = f x" by blast 62 thus "g1 y = g2 y" using eq_on by simp 63qed 64 65lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r" 66 unfolding wo_rel_def card_order_on_def by blast 67 68lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow> \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r" 69 unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit) 70 71lemma Card_order_trans: 72 "\<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" 73 unfolding card_order_on_def well_order_on_def linear_order_on_def 74 partial_order_on_def preorder_on_def trans_def antisym_def by blast 75 76lemma Cinfinite_limit2: 77 assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r" 78 shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)" 79proof - 80 from r have trans: "trans r" and total: "Total r" and antisym: "antisym r" 81 unfolding card_order_on_def well_order_on_def linear_order_on_def 82 partial_order_on_def preorder_on_def by auto 83 obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r" 84 using Cinfinite_limit[OF x1 r] by blast 85 obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r" 86 using Cinfinite_limit[OF x2 r] by blast 87 show ?thesis 88 proof (cases "y1 = y2") 89 case True with y1 y2 show ?thesis by blast 90 next 91 case False 92 with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r" 93 unfolding total_on_def by auto 94 thus ?thesis 95 proof 96 assume *: "(y1, y2) \<in> r" 97 with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast 98 with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def) 99 next 100 assume *: "(y2, y1) \<in> r" 101 with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast 102 with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def) 103 qed 104 qed 105qed 106 107lemma Cinfinite_limit_finite: 108 "\<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)" 109proof (induct X rule: finite_induct) 110 case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto 111next 112 case (insert x X) 113 then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast 114 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" 115 using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast 116 show ?case 117 apply (intro bexI ballI) 118 apply (erule insertE) 119 apply hypsubst 120 apply (rule z(2)) 121 using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3) 122 apply blast 123 apply (rule z(1)) 124 done 125qed 126 127lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A" 128 by auto 129 130lemmas well_order_induct_imp = wo_rel.well_order_induct[of r "\<lambda>x. x \<in> Field r \<longrightarrow> P x" for r P] 131 132lemma meta_spec2: 133 assumes "(\<And>x y. PROP P x y)" 134 shows "PROP P x y" 135 by (rule assms) 136 137lemma nchotomy_relcomppE: 138 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" 139 shows P 140proof (rule relcompp.cases[OF assms(2)], hypsubst) 141 fix b assume "r a b" "s b c" 142 moreover from assms(1) obtain b' where "b = f b'" by blast 143 ultimately show P by (blast intro: assms(3)) 144qed 145 146lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)" 147 unfolding vimage2p_def by auto 148 149lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R" 150 by (rule ssubst) 151 152lemma all_mem_range1: 153 "(\<And>y. y \<in> range f \<Longrightarrow> P y) \<equiv> (\<And>x. P (f x)) " 154 by (rule equal_intr_rule) fast+ 155 156lemma all_mem_range2: 157 "(\<And>fa y. fa \<in> range f \<Longrightarrow> y \<in> range fa \<Longrightarrow> P y) \<equiv> (\<And>x xa. P (f x xa))" 158 by (rule equal_intr_rule) fast+ 159 160lemma all_mem_range3: 161 "(\<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))" 162 by (rule equal_intr_rule) fast+ 163 164lemma all_mem_range4: 165 "(\<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> 166 (\<And>x xa xb xc. P (f x xa xb xc))" 167 by (rule equal_intr_rule) fast+ 168 169lemma all_mem_range5: 170 "(\<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> 171 y \<in> range fd \<Longrightarrow> P y) \<equiv> 172 (\<And>x xa xb xc xd. P (f x xa xb xc xd))" 173 by (rule equal_intr_rule) fast+ 174 175lemma all_mem_range6: 176 "(\<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> 177 fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> y \<in> range ff \<Longrightarrow> P y) \<equiv> 178 (\<And>x xa xb xc xd xe xf. P (f x xa xb xc xd xe xf))" 179 by (rule equal_intr_rule) (fastforce, fast) 180 181lemma all_mem_range7: 182 "(\<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> 183 fe \<in> range fd \<Longrightarrow> ff \<in> range fe \<Longrightarrow> fg \<in> range ff \<Longrightarrow> y \<in> range fg \<Longrightarrow> P y) \<equiv> 184 (\<And>x xa xb xc xd xe xf xg. P (f x xa xb xc xd xe xf xg))" 185 by (rule equal_intr_rule) (fastforce, fast) 186 187lemma all_mem_range8: 188 "(\<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> 189 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> 190 (\<And>x xa xb xc xd xe xf xg xh. P (f x xa xb xc xd xe xf xg xh))" 191 by (rule equal_intr_rule) (fastforce, fast) 192 193lemmas all_mem_range = all_mem_range1 all_mem_range2 all_mem_range3 all_mem_range4 all_mem_range5 194 all_mem_range6 all_mem_range7 all_mem_range8 195 196lemma pred_fun_True_id: "NO_MATCH id p \<Longrightarrow> pred_fun (\<lambda>x. True) p f = pred_fun (\<lambda>x. True) id (p \<circ> f)" 197 unfolding fun.pred_map unfolding comp_def id_def .. 198 199ML_file "Tools/BNF/bnf_lfp_util.ML" 200ML_file "Tools/BNF/bnf_lfp_tactics.ML" 201ML_file "Tools/BNF/bnf_lfp.ML" 202ML_file "Tools/BNF/bnf_lfp_compat.ML" 203ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML" 204ML_file "Tools/BNF/bnf_lfp_size.ML" 205 206end 207