src/HOL/BNF_FP_Base.thy
 author desharna Tue Jul 01 17:01:28 2014 +0200 (2014-07-01) changeset 57471 11cd462e31ec parent 57303 498a62e65f5f child 57489 8f0ba9f2d10f permissions -rw-r--r--
generate 'rel_induct' theorem for 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 mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
```
```    17 by auto
```
```    18
```
```    19 lemma predicate2D_conj: "P \<le> Q \<and> R \<Longrightarrow> P x y \<Longrightarrow> R \<and> Q x y"
```
```    20   by auto
```
```    21
```
```    22 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
```
```    23 by blast
```
```    24
```
```    25 lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
```
```    26 by (cases u) (hypsubst, rule unit.case)
```
```    27
```
```    28 lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
```
```    29 by simp
```
```    30
```
```    31 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
```
```    32 by simp
```
```    33
```
```    34 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
```
```    35 unfolding comp_def fun_eq_iff by simp
```
```    36
```
```    37 lemma o_bij:
```
```    38   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
```
```    39   shows "bij f"
```
```    40 unfolding bij_def inj_on_def surj_def proof safe
```
```    41   fix a1 a2 assume "f a1 = f a2"
```
```    42   hence "g ( f a1) = g (f a2)" by simp
```
```    43   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
```
```    44 next
```
```    45   fix b
```
```    46   have "b = f (g b)"
```
```    47   using fg unfolding fun_eq_iff by simp
```
```    48   thus "EX a. b = f a" by blast
```
```    49 qed
```
```    50
```
```    51 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
```
```    52
```
```    53 lemma case_sum_step:
```
```    54 "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
```
```    55 "case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
```
```    56 by auto
```
```    57
```
```    58 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
```
```    59 by blast
```
```    60
```
```    61 lemma type_copy_obj_one_point_absE:
```
```    62   assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
```
```    63   using type_definition.Rep_inverse[OF assms(1)]
```
```    64   by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
```
```    65
```
```    66 lemma obj_sumE_f:
```
```    67   assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
```
```    68   shows "\<forall>x. s = f x \<longrightarrow> P"
```
```    69 proof
```
```    70   fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
```
```    71 qed
```
```    72
```
```    73 lemma case_sum_if:
```
```    74 "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
```
```    75 by simp
```
```    76
```
```    77 lemma prod_set_simps:
```
```    78 "fsts (x, y) = {x}"
```
```    79 "snds (x, y) = {y}"
```
```    80 unfolding fsts_def snds_def by simp+
```
```    81
```
```    82 lemma sum_set_simps:
```
```    83 "setl (Inl x) = {x}"
```
```    84 "setl (Inr x) = {}"
```
```    85 "setr (Inl x) = {}"
```
```    86 "setr (Inr x) = {x}"
```
```    87 unfolding sum_set_defs by simp+
```
```    88
```
```    89 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
```
```    90   by simp
```
```    91
```
```    92 lemma Inr_Inl_False: "(Inr x = Inl y) = False"
```
```    93   by simp
```
```    94
```
```    95 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
```
```    96 by blast
```
```    97
```
```    98 lemma rewriteR_comp_comp: "\<lbrakk>g \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = f \<circ> r"
```
```    99   unfolding comp_def fun_eq_iff by auto
```
```   100
```
```   101 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"
```
```   102   unfolding comp_def fun_eq_iff by auto
```
```   103
```
```   104 lemma rewriteL_comp_comp: "\<lbrakk>f \<circ> g = l\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l \<circ> h"
```
```   105   unfolding comp_def fun_eq_iff by auto
```
```   106
```
```   107 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"
```
```   108   unfolding comp_def fun_eq_iff by auto
```
```   109
```
```   110 lemma convol_o: "<f, g> \<circ> h = <f \<circ> h, g \<circ> h>"
```
```   111   unfolding convol_def by auto
```
```   112
```
```   113 lemma map_prod_o_convol: "map_prod h1 h2 \<circ> <f, g> = <h1 \<circ> f, h2 \<circ> g>"
```
```   114   unfolding convol_def by auto
```
```   115
```
```   116 lemma map_prod_o_convol_id: "(map_prod f id \<circ> <id , g>) x = <id \<circ> f , g> x"
```
```   117   unfolding map_prod_o_convol id_comp comp_id ..
```
```   118
```
```   119 lemma o_case_sum: "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
```
```   120   unfolding comp_def by (auto split: sum.splits)
```
```   121
```
```   122 lemma case_sum_o_map_sum: "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
```
```   123   unfolding comp_def by (auto split: sum.splits)
```
```   124
```
```   125 lemma case_sum_o_map_sum_id: "(case_sum id g \<circ> map_sum f id) x = case_sum (f \<circ> id) g x"
```
```   126   unfolding case_sum_o_map_sum id_comp comp_id ..
```
```   127
```
```   128 lemma rel_fun_def_butlast:
```
```   129   "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> (rel_fun S T) (f x) (g y))"
```
```   130   unfolding rel_fun_def ..
```
```   131
```
```   132 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
```
```   133   by auto
```
```   134
```
```   135 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
```
```   136   by auto
```
```   137
```
```   138 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
```
```   139   unfolding Grp_def id_apply by blast
```
```   140
```
```   141 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
```
```   142    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
```
```   143   unfolding Grp_def by rule auto
```
```   144
```
```   145 lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
```
```   146   unfolding vimage2p_def by blast
```
```   147
```
```   148 lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
```
```   149   unfolding vimage2p_def by auto
```
```   150
```
```   151 lemma
```
```   152   assumes "type_definition Rep Abs UNIV"
```
```   153   shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs \<circ> Rep = id"
```
```   154   unfolding fun_eq_iff comp_apply id_apply
```
```   155     type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
```
```   156
```
```   157 lemma type_copy_map_comp0_undo:
```
```   158   assumes "type_definition Rep Abs UNIV"
```
```   159           "type_definition Rep' Abs' UNIV"
```
```   160           "type_definition Rep'' Abs'' UNIV"
```
```   161   shows "Abs' \<circ> M \<circ> Rep'' = (Abs' \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> Rep'') \<Longrightarrow> M1 \<circ> M2 = M"
```
```   162   by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
```
```   163     type_definition.Abs_inverse[OF assms(1) UNIV_I]
```
```   164     type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
```
```   165
```
```   166 lemma vimage2p_id: "vimage2p id id R = R"
```
```   167   unfolding vimage2p_def by auto
```
```   168
```
```   169 lemma vimage2p_comp: "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
```
```   170   unfolding fun_eq_iff vimage2p_def o_apply by simp
```
```   171
```
```   172 lemma fun_cong_unused_0: "f = (\<lambda>x. g) \<Longrightarrow> f (\<lambda>x. 0) = g"
```
```   173   by (erule arg_cong)
```
```   174
```
```   175 lemma inj_on_convol_ident: "inj_on (\<lambda>x. (x, f x)) X"
```
```   176   unfolding inj_on_def by simp
```
```   177
```
```   178 lemma case_prod_app: "case_prod f x y = case_prod (\<lambda>l r. f l r y) x"
```
```   179   by (case_tac x) simp
```
```   180
```
```   181 lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \<circ> f) (r \<circ> g) x"
```
```   182   by (case_tac x) simp+
```
```   183
```
```   184 lemma case_prod_map_prod: "case_prod h (map_prod f g x) = case_prod (\<lambda>l r. h (f l) (g r)) x"
```
```   185   by (case_tac x) simp+
```
```   186
```
```   187 lemma prod_inj_map: "inj f \<Longrightarrow> inj g \<Longrightarrow> inj (map_prod f g)"
```
```   188   by (simp add: inj_on_def)
```
```   189
```
```   190 ML_file "Tools/BNF/bnf_fp_util.ML"
```
```   191 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
```
```   192 ML_file "Tools/BNF/bnf_lfp_size.ML"
```
```   193 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
```
```   194 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
```
```   195 ML_file "Tools/BNF/bnf_fp_n2m.ML"
```
```   196 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
```
```   197
```
```   198 ML_file "Tools/Function/size.ML"
```
```   199 setup Size.setup
```
```   200
```
```   201 lemma size_bool[code]: "size (b\<Colon>bool) = 0"
```
```   202   by (cases b) auto
```
```   203
```
```   204 lemma size_nat[simp, code]: "size (n\<Colon>nat) = n"
```
```   205   by (induct n) simp_all
```
```   206
```
```   207 declare prod.size[no_atp]
```
```   208
```
```   209 lemma size_sum_o_map: "size_sum g1 g2 \<circ> map_sum f1 f2 = size_sum (g1 \<circ> f1) (g2 \<circ> f2)"
```
```   210   by (rule ext) (case_tac x, auto)
```
```   211
```
```   212 lemma size_prod_o_map: "size_prod g1 g2 \<circ> map_prod f1 f2 = size_prod (g1 \<circ> f1) (g2 \<circ> f2)"
```
```   213   by (rule ext) auto
```
```   214
```
```   215 setup {*
```
```   216 BNF_LFP_Size.register_size_global @{type_name sum} @{const_name size_sum} @{thms sum.size}
```
```   217   @{thms size_sum_o_map}
```
```   218 #> BNF_LFP_Size.register_size_global @{type_name prod} @{const_name size_prod} @{thms prod.size}
```
```   219   @{thms size_prod_o_map}
```
```   220 *}
```
```   221
```
```   222 end
```