src/HOL/BNF_FP_Base.thy
changeset 55058 4e700eb471d4
parent 54485 b61b8c9e4cf7
child 55059 ef2e0fb783c6
     1.1 --- /dev/null	Thu Jan 01 00:00:00 1970 +0000
     1.2 +++ b/src/HOL/BNF_FP_Base.thy	Mon Jan 20 18:24:56 2014 +0100
     1.3 @@ -0,0 +1,170 @@
     1.4 +(*  Title:      HOL/BNF/BNF_FP_Base.thy
     1.5 +    Author:     Lorenz Panny, TU Muenchen
     1.6 +    Author:     Dmitriy Traytel, TU Muenchen
     1.7 +    Author:     Jasmin Blanchette, TU Muenchen
     1.8 +    Copyright   2012, 2013
     1.9 +
    1.10 +Shared fixed point operations on bounded natural functors, including
    1.11 +*)
    1.12 +
    1.13 +header {* Shared Fixed Point Operations on Bounded Natural Functors *}
    1.14 +
    1.15 +theory BNF_FP_Base
    1.16 +imports BNF_Comp Ctr_Sugar
    1.17 +begin
    1.18 +
    1.19 +lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
    1.20 +by auto
    1.21 +
    1.22 +lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
    1.23 +by blast
    1.24 +
    1.25 +lemma unit_case_Unity: "(case u of () \<Rightarrow> f) = f"
    1.26 +by (cases u) (hypsubst, rule unit.cases)
    1.27 +
    1.28 +lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
    1.29 +by simp
    1.30 +
    1.31 +lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    1.32 +by simp
    1.33 +
    1.34 +lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    1.35 +by clarify
    1.36 +
    1.37 +lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    1.38 +by auto
    1.39 +
    1.40 +lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
    1.41 +unfolding o_def fun_eq_iff by simp
    1.42 +
    1.43 +lemma o_bij:
    1.44 +  assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
    1.45 +  shows "bij f"
    1.46 +unfolding bij_def inj_on_def surj_def proof safe
    1.47 +  fix a1 a2 assume "f a1 = f a2"
    1.48 +  hence "g ( f a1) = g (f a2)" by simp
    1.49 +  thus "a1 = a2" using gf unfolding fun_eq_iff by simp
    1.50 +next
    1.51 +  fix b
    1.52 +  have "b = f (g b)"
    1.53 +  using fg unfolding fun_eq_iff by simp
    1.54 +  thus "EX a. b = f a" by blast
    1.55 +qed
    1.56 +
    1.57 +lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
    1.58 +
    1.59 +lemma sum_case_step:
    1.60 +"sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
    1.61 +"sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
    1.62 +by auto
    1.63 +
    1.64 +lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
    1.65 +by simp
    1.66 +
    1.67 +lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
    1.68 +by blast
    1.69 +
    1.70 +lemma obj_sumE_f:
    1.71 +"\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f x \<longrightarrow> P"
    1.72 +by (rule allI) (metis sumE)
    1.73 +
    1.74 +lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
    1.75 +by (cases s) auto
    1.76 +
    1.77 +lemma sum_case_if:
    1.78 +"sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    1.79 +by simp
    1.80 +
    1.81 +lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
    1.82 +by blast
    1.83 +
    1.84 +lemma UN_compreh_eq_eq:
    1.85 +"\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
    1.86 +"\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
    1.87 +by blast+
    1.88 +
    1.89 +lemma Inl_Inr_False: "(Inl x = Inr y) = False"
    1.90 +by simp
    1.91 +
    1.92 +lemma prod_set_simps:
    1.93 +"fsts (x, y) = {x}"
    1.94 +"snds (x, y) = {y}"
    1.95 +unfolding fsts_def snds_def by simp+
    1.96 +
    1.97 +lemma sum_set_simps:
    1.98 +"setl (Inl x) = {x}"
    1.99 +"setl (Inr x) = {}"
   1.100 +"setr (Inl x) = {}"
   1.101 +"setr (Inr x) = {x}"
   1.102 +unfolding sum_set_defs by simp+
   1.103 +
   1.104 +lemma prod_rel_simp:
   1.105 +"prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"
   1.106 +unfolding prod_rel_def by simp
   1.107 +
   1.108 +lemma sum_rel_simps:
   1.109 +"sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"
   1.110 +"sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"
   1.111 +"sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"
   1.112 +"sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"
   1.113 +unfolding sum_rel_def by simp+
   1.114 +
   1.115 +lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
   1.116 +by blast
   1.117 +
   1.118 +lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
   1.119 +  unfolding o_def fun_eq_iff by auto
   1.120 +
   1.121 +lemma rewriteR_comp_comp2: "\<lbrakk>g o h = r1 o r2; f o r1 = l\<rbrakk> \<Longrightarrow> f o g o h = l o r2"
   1.122 +  unfolding o_def fun_eq_iff by auto
   1.123 +
   1.124 +lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
   1.125 +  unfolding o_def fun_eq_iff by auto
   1.126 +
   1.127 +lemma rewriteL_comp_comp2: "\<lbrakk>f o g = l1 o l2; l2 o h = r\<rbrakk> \<Longrightarrow> f o (g o h) = l1 o r"
   1.128 +  unfolding o_def fun_eq_iff by auto
   1.129 +
   1.130 +lemma convol_o: "<f, g> o h = <f o h, g o h>"
   1.131 +  unfolding convol_def by auto
   1.132 +
   1.133 +lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"
   1.134 +  unfolding convol_def by auto
   1.135 +
   1.136 +lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"
   1.137 +  unfolding map_pair_o_convol id_o o_id ..
   1.138 +
   1.139 +lemma o_sum_case: "h o sum_case f g = sum_case (h o f) (h o g)"
   1.140 +  unfolding o_def by (auto split: sum.splits)
   1.141 +
   1.142 +lemma sum_case_o_sum_map: "sum_case f g o sum_map h1 h2 = sum_case (f o h1) (g o h2)"
   1.143 +  unfolding o_def by (auto split: sum.splits)
   1.144 +
   1.145 +lemma sum_case_o_sum_map_id: "(sum_case id g o sum_map f id) x = sum_case (f o id) g x"
   1.146 +  unfolding sum_case_o_sum_map id_o o_id ..
   1.147 +
   1.148 +lemma fun_rel_def_butlast:
   1.149 +  "(fun_rel R (fun_rel S T)) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"
   1.150 +  unfolding fun_rel_def ..
   1.151 +
   1.152 +lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
   1.153 +  by auto
   1.154 +
   1.155 +lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
   1.156 +  by auto
   1.157 +
   1.158 +lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   1.159 +  unfolding Grp_def id_apply by blast
   1.160 +
   1.161 +lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   1.162 +   (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
   1.163 +  unfolding Grp_def by rule auto
   1.164 +
   1.165 +ML_file "Tools/bnf_fp_util.ML"
   1.166 +ML_file "Tools/bnf_fp_def_sugar_tactics.ML"
   1.167 +ML_file "Tools/bnf_fp_def_sugar.ML"
   1.168 +ML_file "Tools/bnf_fp_n2m_tactics.ML"
   1.169 +ML_file "Tools/bnf_fp_n2m.ML"
   1.170 +ML_file "Tools/bnf_fp_n2m_sugar.ML"
   1.171 +ML_file "Tools/bnf_fp_rec_sugar_util.ML"
   1.172 +
   1.173 +end