src/HOL/BNF_FP_Base.thy
author desharna
Mon Jul 07 16:06:46 2014 +0200 (2014-07-07)
changeset 57525 f9dd8a33f820
parent 57489 8f0ba9f2d10f
child 57641 dc59f147b27d
permissions -rw-r--r--
generate 'rel_cases' theorem for (co)datatypes
     1 (*  Title:      HOL/BNF_FP_Base.thy
     2     Author:     Lorenz Panny, TU Muenchen
     3     Author:     Dmitriy Traytel, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Copyright   2012, 2013
     6 
     7 Shared fixed point operations on bounded natural functors.
     8 *)
     9 
    10 header {* Shared Fixed Point Operations on Bounded Natural Functors *}
    11 
    12 theory BNF_FP_Base
    13 imports BNF_Comp Basic_BNFs
    14 begin
    15 
    16 lemma False_imp_eq_True: "(False \<Longrightarrow> Q) \<equiv> Trueprop True"
    17   by default simp_all
    18 
    19 lemma conj_imp_eq_imp_imp: "(P \<and> Q \<Longrightarrow> PROP R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> PROP R)"
    20   by default simp_all
    21 
    22 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
    23 by auto
    24 
    25 lemma predicate2D_conj: "P \<le> Q \<and> R \<Longrightarrow> P x y \<Longrightarrow> R \<and> Q x y"
    26   by auto
    27 
    28 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
    29 by blast
    30 
    31 lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
    32 by (cases u) (hypsubst, rule unit.case)
    33 
    34 lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
    35 by simp
    36 
    37 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    38 by simp
    39 
    40 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
    41 unfolding comp_def fun_eq_iff by simp
    42 
    43 lemma o_bij:
    44   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
    45   shows "bij f"
    46 unfolding bij_def inj_on_def surj_def proof safe
    47   fix a1 a2 assume "f a1 = f a2"
    48   hence "g ( f a1) = g (f a2)" by simp
    49   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
    50 next
    51   fix b
    52   have "b = f (g b)"
    53   using fg unfolding fun_eq_iff by simp
    54   thus "EX a. b = f a" by blast
    55 qed
    56 
    57 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
    58 
    59 lemma case_sum_step:
    60 "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
    61 "case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
    62 by auto
    63 
    64 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
    65 by blast
    66 
    67 lemma type_copy_obj_one_point_absE:
    68   assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
    69   using type_definition.Rep_inverse[OF assms(1)]
    70   by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
    71 
    72 lemma obj_sumE_f:
    73   assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
    74   shows "\<forall>x. s = f x \<longrightarrow> P"
    75 proof
    76   fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
    77 qed
    78 
    79 lemma case_sum_if:
    80 "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    81 by simp
    82 
    83 lemma prod_set_simps:
    84 "fsts (x, y) = {x}"
    85 "snds (x, y) = {y}"
    86 unfolding fsts_def snds_def by simp+
    87 
    88 lemma sum_set_simps:
    89 "setl (Inl x) = {x}"
    90 "setl (Inr x) = {}"
    91 "setr (Inl x) = {}"
    92 "setr (Inr x) = {x}"
    93 unfolding sum_set_defs by simp+
    94 
    95 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
    96   by simp
    97 
    98 lemma Inr_Inl_False: "(Inr x = Inl y) = False"
    99   by simp
   100 
   101 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
   102 by blast
   103 
   104 lemma rewriteR_comp_comp: "\<lbrakk>g \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = f \<circ> r"
   105   unfolding comp_def fun_eq_iff by auto
   106 
   107 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"
   108   unfolding comp_def fun_eq_iff by auto
   109 
   110 lemma rewriteL_comp_comp: "\<lbrakk>f \<circ> g = l\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l \<circ> h"
   111   unfolding comp_def fun_eq_iff by auto
   112 
   113 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"
   114   unfolding comp_def fun_eq_iff by auto
   115 
   116 lemma convol_o: "<f, g> \<circ> h = <f \<circ> h, g \<circ> h>"
   117   unfolding convol_def by auto
   118 
   119 lemma map_prod_o_convol: "map_prod h1 h2 \<circ> <f, g> = <h1 \<circ> f, h2 \<circ> g>"
   120   unfolding convol_def by auto
   121 
   122 lemma map_prod_o_convol_id: "(map_prod f id \<circ> <id , g>) x = <id \<circ> f , g> x"
   123   unfolding map_prod_o_convol id_comp comp_id ..
   124 
   125 lemma o_case_sum: "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
   126   unfolding comp_def by (auto split: sum.splits)
   127 
   128 lemma case_sum_o_map_sum: "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
   129   unfolding comp_def by (auto split: sum.splits)
   130 
   131 lemma case_sum_o_map_sum_id: "(case_sum id g \<circ> map_sum f id) x = case_sum (f \<circ> id) g x"
   132   unfolding case_sum_o_map_sum id_comp comp_id ..
   133 
   134 lemma rel_fun_def_butlast:
   135   "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> (rel_fun S T) (f x) (g y))"
   136   unfolding rel_fun_def ..
   137 
   138 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
   139   by auto
   140 
   141 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
   142   by auto
   143 
   144 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   145   unfolding Grp_def id_apply by blast
   146 
   147 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   148    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
   149   unfolding Grp_def by rule auto
   150 
   151 lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
   152   unfolding vimage2p_def by blast
   153 
   154 lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
   155   unfolding vimage2p_def by auto
   156 
   157 lemma
   158   assumes "type_definition Rep Abs UNIV"
   159   shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs \<circ> Rep = id"
   160   unfolding fun_eq_iff comp_apply id_apply
   161     type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
   162 
   163 lemma type_copy_map_comp0_undo:
   164   assumes "type_definition Rep Abs UNIV"
   165           "type_definition Rep' Abs' UNIV"
   166           "type_definition Rep'' Abs'' UNIV"
   167   shows "Abs' \<circ> M \<circ> Rep'' = (Abs' \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> Rep'') \<Longrightarrow> M1 \<circ> M2 = M"
   168   by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
   169     type_definition.Abs_inverse[OF assms(1) UNIV_I]
   170     type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
   171 
   172 lemma vimage2p_id: "vimage2p id id R = R"
   173   unfolding vimage2p_def by auto
   174 
   175 lemma vimage2p_comp: "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
   176   unfolding fun_eq_iff vimage2p_def o_apply by simp
   177 
   178 lemma fun_cong_unused_0: "f = (\<lambda>x. g) \<Longrightarrow> f (\<lambda>x. 0) = g"
   179   by (erule arg_cong)
   180 
   181 lemma inj_on_convol_ident: "inj_on (\<lambda>x. (x, f x)) X"
   182   unfolding inj_on_def by simp
   183 
   184 lemma case_prod_app: "case_prod f x y = case_prod (\<lambda>l r. f l r y) x"
   185   by (case_tac x) simp
   186 
   187 lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \<circ> f) (r \<circ> g) x"
   188   by (case_tac x) simp+
   189 
   190 lemma case_prod_map_prod: "case_prod h (map_prod f g x) = case_prod (\<lambda>l r. h (f l) (g r)) x"
   191   by (case_tac x) simp+
   192 
   193 lemma prod_inj_map: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (map_prod f g)"
   194   by (simp add: inj_on_def)
   195 
   196 lemma eq_ifI: "(P \<longrightarrow> t = u1) \<Longrightarrow> (\<not> P \<longrightarrow> t = u2) \<Longrightarrow> t = (if P then u1 else u2)"
   197   by simp
   198 
   199 ML_file "Tools/BNF/bnf_fp_util.ML"
   200 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
   201 ML_file "Tools/BNF/bnf_lfp_size.ML"
   202 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
   203 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
   204 ML_file "Tools/BNF/bnf_fp_n2m.ML"
   205 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
   206 
   207 ML_file "Tools/Function/size.ML"
   208 setup Size.setup
   209 
   210 lemma size_bool[code]: "size (b\<Colon>bool) = 0"
   211   by (cases b) auto
   212 
   213 lemma size_nat[simp, code]: "size (n\<Colon>nat) = n"
   214   by (induct n) simp_all
   215 
   216 declare prod.size[no_atp]
   217 
   218 lemma size_sum_o_map: "size_sum g1 g2 \<circ> map_sum f1 f2 = size_sum (g1 \<circ> f1) (g2 \<circ> f2)"
   219   by (rule ext) (case_tac x, auto)
   220 
   221 lemma size_prod_o_map: "size_prod g1 g2 \<circ> map_prod f1 f2 = size_prod (g1 \<circ> f1) (g2 \<circ> f2)"
   222   by (rule ext) auto
   223 
   224 setup {*
   225 BNF_LFP_Size.register_size_global @{type_name sum} @{const_name size_sum} @{thms sum.size}
   226   @{thms size_sum_o_map}
   227 #> BNF_LFP_Size.register_size_global @{type_name prod} @{const_name size_prod} @{thms prod.size}
   228   @{thms size_prod_o_map}
   229 *}
   230 
   231 end