src/HOL/BNF_Fixpoint_Base.thy
author wenzelm
Tue Sep 01 22:32:58 2015 +0200 (2015-09-01)
changeset 61076 bdc1e2f0a86a
parent 60918 4ceef1592e8c
child 61169 4de9ff3ea29a
permissions -rw-r--r--
eliminated \<Colon>;
     1 (*  Title:      HOL/BNF_Fixpoint_Base.thy
     2     Author:     Lorenz Panny, TU Muenchen
     3     Author:     Dmitriy Traytel, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Author:     Martin Desharnais, TU Muenchen
     6     Copyright   2012, 2013, 2014
     7 
     8 Shared fixpoint operations on bounded natural functors.
     9 *)
    10 
    11 section \<open>Shared Fixpoint Operations on Bounded Natural Functors\<close>
    12 
    13 theory BNF_Fixpoint_Base
    14 imports BNF_Composition Basic_BNFs
    15 begin
    16 
    17 lemma False_imp_eq_True: "(False \<Longrightarrow> Q) \<equiv> Trueprop True"
    18   by default simp_all
    19 
    20 lemma conj_imp_eq_imp_imp: "(P \<and> Q \<Longrightarrow> PROP R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> PROP R)"
    21   by default simp_all
    22 
    23 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
    24   by auto
    25 
    26 lemma predicate2D_conj: "P \<le> Q \<and> R \<Longrightarrow> P x y \<Longrightarrow> R \<and> Q x y"
    27   by auto
    28 
    29 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
    30   by blast
    31 
    32 lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
    33   by (cases u) (hypsubst, rule unit.case)
    34 
    35 lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
    36   by simp
    37 
    38 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    39   by simp
    40 
    41 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
    42   unfolding comp_def fun_eq_iff by simp
    43 
    44 lemma o_bij:
    45   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
    46   shows "bij f"
    47 unfolding bij_def inj_on_def surj_def proof safe
    48   fix a1 a2 assume "f a1 = f a2"
    49   hence "g ( f a1) = g (f a2)" by simp
    50   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
    51 next
    52   fix b
    53   have "b = f (g b)"
    54   using fg unfolding fun_eq_iff by simp
    55   thus "EX a. b = f a" by blast
    56 qed
    57 
    58 lemma case_sum_step:
    59   "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
    60   "case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
    61   by auto
    62 
    63 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
    64   by blast
    65 
    66 lemma type_copy_obj_one_point_absE:
    67   assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
    68   using type_definition.Rep_inverse[OF assms(1)]
    69   by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
    70 
    71 lemma obj_sumE_f:
    72   assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
    73   shows "\<forall>x. s = f x \<longrightarrow> P"
    74 proof
    75   fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
    76 qed
    77 
    78 lemma case_sum_if:
    79   "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    80   by simp
    81 
    82 lemma prod_set_simps:
    83   "fsts (x, y) = {x}"
    84   "snds (x, y) = {y}"
    85   unfolding prod_set_defs by simp+
    86 
    87 lemma sum_set_simps:
    88   "setl (Inl x) = {x}"
    89   "setl (Inr x) = {}"
    90   "setr (Inl x) = {}"
    91   "setr (Inr x) = {x}"
    92   unfolding sum_set_defs by simp+
    93 
    94 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
    95   by simp
    96 
    97 lemma Inr_Inl_False: "(Inr x = Inl y) = False"
    98   by simp
    99 
   100 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
   101   by blast
   102 
   103 lemma rewriteR_comp_comp: "\<lbrakk>g \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = f \<circ> r"
   104   unfolding comp_def fun_eq_iff by auto
   105 
   106 lemma rewriteR_comp_comp2: "\<lbrakk>g \<circ> h = r1 \<circ> r2; f \<circ> r1 = l\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = l \<circ> r2"
   107   unfolding comp_def fun_eq_iff by auto
   108 
   109 lemma rewriteL_comp_comp: "\<lbrakk>f \<circ> g = l\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l \<circ> h"
   110   unfolding comp_def fun_eq_iff by auto
   111 
   112 lemma rewriteL_comp_comp2: "\<lbrakk>f \<circ> g = l1 \<circ> l2; l2 \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l1 \<circ> r"
   113   unfolding comp_def fun_eq_iff by auto
   114 
   115 lemma convol_o: "\<langle>f, g\<rangle> \<circ> h = \<langle>f \<circ> h, g \<circ> h\<rangle>"
   116   unfolding convol_def by auto
   117 
   118 lemma map_prod_o_convol: "map_prod h1 h2 \<circ> \<langle>f, g\<rangle> = \<langle>h1 \<circ> f, h2 \<circ> g\<rangle>"
   119   unfolding convol_def by auto
   120 
   121 lemma map_prod_o_convol_id: "(map_prod f id \<circ> \<langle>id, g\<rangle>) x = \<langle>id \<circ> f, g\<rangle> x"
   122   unfolding map_prod_o_convol id_comp comp_id ..
   123 
   124 lemma o_case_sum: "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
   125   unfolding comp_def by (auto split: sum.splits)
   126 
   127 lemma case_sum_o_map_sum: "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
   128   unfolding comp_def by (auto split: sum.splits)
   129 
   130 lemma case_sum_o_map_sum_id: "(case_sum id g \<circ> map_sum f id) x = case_sum (f \<circ> id) g x"
   131   unfolding case_sum_o_map_sum id_comp comp_id ..
   132 
   133 lemma rel_fun_def_butlast:
   134   "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> (rel_fun S T) (f x) (g y))"
   135   unfolding rel_fun_def ..
   136 
   137 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
   138   by auto
   139 
   140 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
   141   by auto
   142 
   143 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   144   unfolding Grp_def id_apply by blast
   145 
   146 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   147    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
   148   unfolding Grp_def by rule auto
   149 
   150 lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
   151   unfolding vimage2p_def by blast
   152 
   153 lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
   154   unfolding vimage2p_def by auto
   155 
   156 lemma
   157   assumes "type_definition Rep Abs UNIV"
   158   shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs \<circ> Rep = id"
   159   unfolding fun_eq_iff comp_apply id_apply
   160     type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
   161 
   162 lemma type_copy_map_comp0_undo:
   163   assumes "type_definition Rep Abs UNIV"
   164           "type_definition Rep' Abs' UNIV"
   165           "type_definition Rep'' Abs'' UNIV"
   166   shows "Abs' \<circ> M \<circ> Rep'' = (Abs' \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> Rep'') \<Longrightarrow> M1 \<circ> M2 = M"
   167   by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
   168     type_definition.Abs_inverse[OF assms(1) UNIV_I]
   169     type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
   170 
   171 lemma vimage2p_id: "vimage2p id id R = R"
   172   unfolding vimage2p_def by auto
   173 
   174 lemma vimage2p_comp: "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
   175   unfolding fun_eq_iff vimage2p_def o_apply by simp
   176 
   177 lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
   178   unfolding rel_fun_def vimage2p_def by auto
   179 
   180 lemma fun_cong_unused_0: "f = (\<lambda>x. g) \<Longrightarrow> f (\<lambda>x. 0) = g"
   181   by (erule arg_cong)
   182 
   183 lemma inj_on_convol_ident: "inj_on (\<lambda>x. (x, f x)) X"
   184   unfolding inj_on_def by simp
   185 
   186 lemma map_sum_if_distrib_then:
   187   "\<And>f g e x y. map_sum f g (if e then Inl x else y) = (if e then Inl (f x) else map_sum f g y)"
   188   "\<And>f g e x y. map_sum f g (if e then Inr x else y) = (if e then Inr (g x) else map_sum f g y)"
   189   by simp_all
   190 
   191 lemma map_sum_if_distrib_else:
   192   "\<And>f g e x y. map_sum f g (if e then x else Inl y) = (if e then map_sum f g x else Inl (f y))"
   193   "\<And>f g e x y. map_sum f g (if e then x else Inr y) = (if e then map_sum f g x else Inr (g y))"
   194   by simp_all
   195 
   196 lemma case_prod_app: "case_prod f x y = case_prod (\<lambda>l r. f l r y) x"
   197   by (case_tac x) simp
   198 
   199 lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \<circ> f) (r \<circ> g) x"
   200   by (case_tac x) simp+
   201 
   202 lemma case_sum_transfer:
   203   "rel_fun (rel_fun R T) (rel_fun (rel_fun S T) (rel_fun (rel_sum R S) T)) case_sum case_sum"
   204   unfolding rel_fun_def by (auto split: sum.splits)
   205 
   206 lemma case_prod_map_prod: "case_prod h (map_prod f g x) = case_prod (\<lambda>l r. h (f l) (g r)) x"
   207   by (case_tac x) simp+
   208 
   209 lemma case_prod_o_map_prod: "case_prod f \<circ> map_prod g1 g2 = case_prod (\<lambda>l r. f (g1 l) (g2 r))"
   210   unfolding comp_def by auto
   211 
   212 lemma case_prod_transfer:
   213   "(rel_fun (rel_fun A (rel_fun B C)) (rel_fun (rel_prod A B) C)) case_prod case_prod"
   214   unfolding rel_fun_def by simp
   215 
   216 lemma eq_ifI: "(P \<longrightarrow> t = u1) \<Longrightarrow> (\<not> P \<longrightarrow> t = u2) \<Longrightarrow> t = (if P then u1 else u2)"
   217   by simp
   218 
   219 lemma comp_transfer:
   220   "rel_fun (rel_fun B C) (rel_fun (rel_fun A B) (rel_fun A C)) (op \<circ>) (op \<circ>)"
   221   unfolding rel_fun_def by simp
   222 
   223 lemma If_transfer: "rel_fun (op =) (rel_fun A (rel_fun A A)) If If"
   224   unfolding rel_fun_def by simp
   225 
   226 lemma Abs_transfer:
   227   assumes type_copy1: "type_definition Rep1 Abs1 UNIV"
   228   assumes type_copy2: "type_definition Rep2 Abs2 UNIV"
   229   shows "rel_fun R (vimage2p Rep1 Rep2 R) Abs1 Abs2"
   230   unfolding vimage2p_def rel_fun_def
   231     type_definition.Abs_inverse[OF type_copy1 UNIV_I]
   232     type_definition.Abs_inverse[OF type_copy2 UNIV_I] by simp
   233 
   234 lemma Inl_transfer:
   235   "rel_fun S (rel_sum S T) Inl Inl"
   236   by auto
   237 
   238 lemma Inr_transfer:
   239   "rel_fun T (rel_sum S T) Inr Inr"
   240   by auto
   241 
   242 lemma Pair_transfer: "rel_fun A (rel_fun B (rel_prod A B)) Pair Pair"
   243   unfolding rel_fun_def by simp
   244 
   245 ML_file "Tools/BNF/bnf_fp_util.ML"
   246 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
   247 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
   248 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
   249 ML_file "Tools/BNF/bnf_fp_n2m.ML"
   250 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
   251 
   252 end