src/HOL/Import/HOLLightList.thy
author nipkow
Mon Jan 30 21:49:41 2012 +0100 (2012-01-30)
changeset 46372 6fa9cdb8b850
parent 45827 66c68453455c
permissions -rw-r--r--
added "'a rel"
     1 (*  Title:      HOL/Import/HOLLightList.thy
     2     Author:     Cezary Kaliszyk
     3 *)
     4 
     5 header {* Compatibility theorems for HOL Light lists *}
     6 
     7 theory HOLLightList
     8 imports List
     9 begin
    10 
    11 lemma FINITE_SET_OF_LIST:
    12   "finite (set l)"
    13   by simp
    14 
    15 lemma AND_ALL2:
    16   "(list_all2 P l m \<and> list_all2 Q l m) = list_all2 (\<lambda>x y. P x y \<and> Q x y) l m"
    17   by (induct l m rule: list_induct2') auto
    18 
    19 lemma MEM_EXISTS_EL:
    20   "(x \<in> set l) = (\<exists>i<length l. x = l ! i)"
    21   by (auto simp add: in_set_conv_nth)
    22 
    23 lemma INJECTIVE_MAP:
    24   "(\<forall>l m. map f l = map f m --> l = m) = (\<forall>x y. f x = f y --> x = y)"
    25 proof (intro iffI allI impI)
    26   fix x y
    27   assume "\<forall>l m. map f l = map f m \<longrightarrow> l = m" "f x = f y"
    28   then show "x = y"
    29     by (drule_tac x="[x]" in spec) (drule_tac x="[y]" in spec, simp)
    30 next
    31   fix l m
    32   assume a: "\<forall>x y. f x = f y \<longrightarrow> x = y"
    33   assume "map f l = map f m"
    34   then show "l = m"
    35     by (induct l m rule: list_induct2') (simp_all add: a)
    36 qed
    37 
    38 lemma SURJECTIVE_MAP:
    39   "(\<forall>m. EX l. map f l = m) = (\<forall>y. EX x. f x = y)"
    40   apply (intro iffI allI)
    41   apply (drule_tac x="[y]" in spec)
    42   apply (elim exE)
    43   apply (case_tac l)
    44   apply simp
    45   apply (rule_tac x="a" in exI)
    46   apply simp
    47   apply (induct_tac m)
    48   apply simp
    49   apply (drule_tac x="a" in spec)
    50   apply (elim exE)
    51   apply (rule_tac x="x # l" in exI)
    52   apply simp
    53   done
    54 
    55 lemma LENGTH_TL:
    56   "l \<noteq> [] \<longrightarrow> length (tl l) = length l - 1"
    57   by simp
    58 
    59 lemma DEF_APPEND:
    60   "op @ = (SOME APPEND. (\<forall>l. APPEND [] l = l) \<and> (\<forall>h t l. APPEND (h # t) l = h # APPEND t l))"
    61   apply (rule some_equality[symmetric])
    62   apply (auto simp add: fun_eq_iff)
    63   apply (induct_tac x)
    64   apply simp_all
    65   done
    66 
    67 lemma DEF_REVERSE:
    68   "rev = (SOME REVERSE. REVERSE [] = [] \<and> (\<forall>l x. REVERSE (x # l) = (REVERSE l) @ [x]))"
    69   apply (rule some_equality[symmetric])
    70   apply (auto simp add: fun_eq_iff)
    71   apply (induct_tac x)
    72   apply simp_all
    73   done
    74 
    75 lemma DEF_LENGTH:
    76   "length = (SOME LENGTH. LENGTH [] = 0 \<and> (\<forall>h t. LENGTH (h # t) = Suc (LENGTH t)))"
    77   apply (rule some_equality[symmetric])
    78   apply (auto simp add: fun_eq_iff)
    79   apply (induct_tac x)
    80   apply simp_all
    81   done
    82 
    83 lemma DEF_MAP:
    84   "map = (SOME MAP. (\<forall>f. MAP f [] = []) \<and> (\<forall>f h t. MAP f (h # t) = f h # MAP f t))"
    85   apply (rule some_equality[symmetric])
    86   apply (auto simp add: fun_eq_iff)
    87   apply (induct_tac xa)
    88   apply simp_all
    89   done
    90 
    91 lemma DEF_REPLICATE:
    92   "replicate =
    93     (SOME REPLICATE. (\<forall>x. REPLICATE 0 x = []) \<and> (\<forall>n x. REPLICATE (Suc n) x = x # REPLICATE n x))"
    94   apply (rule some_equality[symmetric])
    95   apply (auto simp add: fun_eq_iff)
    96   apply (induct_tac x)
    97   apply simp_all
    98   done
    99 
   100 lemma DEF_ITLIST:
   101   "foldr = (SOME ITLIST. (\<forall>f b. ITLIST f [] b = b) \<and> (\<forall>h f t b. ITLIST f (h # t) b = f h (ITLIST f t b)))"
   102   apply (rule some_equality[symmetric])
   103   apply (auto simp add: fun_eq_iff)
   104   apply (induct_tac xa)
   105   apply simp_all
   106   done
   107 
   108 lemma DEF_ALL2: "list_all2 =
   109   (SOME ALL2.
   110     (\<forall>P l2. ALL2 P [] l2 = (l2 = [])) \<and>
   111     (\<forall>h1 P t1 l2.
   112       ALL2 P (h1 # t1) l2 = (if l2 = [] then False else P h1 (hd l2) \<and> ALL2 P t1 (tl l2))))"
   113   apply (rule some_equality[symmetric])
   114   apply (auto)
   115   apply (case_tac l2, simp_all)
   116   apply (case_tac l2, simp_all)
   117   apply (case_tac l2, simp_all)
   118   apply (simp add: fun_eq_iff)
   119   apply (intro allI)
   120   apply (induct_tac xa xb rule: list_induct2')
   121   apply simp_all
   122   done
   123 
   124 lemma ALL2:
   125  "list_all2 P [] [] = True \<and>
   126   list_all2 P (h1 # t1) [] = False \<and>
   127   list_all2 P [] (h2 # t2) = False \<and>
   128   list_all2 P (h1 # t1) (h2 # t2) = (P h1 h2 \<and> list_all2 P t1 t2)"
   129   by simp
   130 
   131 lemma DEF_FILTER:
   132   "filter = (SOME FILTER. (\<forall>P. FILTER P [] = []) \<and>
   133      (\<forall>h P t. FILTER P (h # t) = (if P h then h # FILTER P t else FILTER P t)))"
   134   apply (rule some_equality[symmetric])
   135   apply (auto simp add: fun_eq_iff)
   136   apply (induct_tac xa)
   137   apply simp_all
   138   done
   139 
   140 fun map2 where
   141   "map2 f [] [] = []"
   142 | "map2 f (h1 # t1) (h2 # t2) = (f h1 h2) # (map2 f t1 t2)"
   143 
   144 lemma MAP2:
   145   "map2 f [] [] = [] \<and> map2 f (h1 # t1) (h2 # t2) = f h1 h2 # map2 f t1 t2"
   146   by simp
   147 
   148 fun fold2 where
   149   "fold2 f [] [] b = b"
   150 | "fold2 f (h1 # t1) (h2 # t2) b = f h1 h2 (fold2 f t1 t2 b)"
   151 
   152 lemma ITLIST2:
   153   "fold2 f [] [] b = b \<and> fold2 f (h1 # t1) (h2 # t2) b = f h1 h2 (fold2 f t1 t2 b)"
   154   by simp
   155 
   156 definition [simp]: "list_el x xs = nth xs x"
   157 
   158 lemma ZIP:
   159   "zip [] [] = [] \<and> zip (h1 # t1) (h2 # t2) = (h1, h2) # (zip t1 t2)"
   160   by simp
   161 
   162 lemma LAST_CLAUSES:
   163   "last [h] = h \<and> last (h # k # t) = last (k # t)"
   164   by simp
   165 
   166 lemma DEF_NULL:
   167   "List.null = (SOME NULL. NULL [] = True \<and> (\<forall>h t. NULL (h # t) = False))"
   168   apply (rule some_equality[symmetric])
   169   apply (auto simp add: fun_eq_iff null_def)
   170   apply (case_tac x)
   171   apply simp_all
   172   done
   173 
   174 lemma DEF_ALL:
   175   "list_all = (SOME u. (\<forall>P. u P [] = True) \<and> (\<forall>h P t. u P (h # t) = (P h \<and> u P t)))"
   176   apply (rule some_equality[symmetric])
   177   apply auto[1]
   178   apply (simp add: fun_eq_iff)
   179   apply (intro allI)
   180   apply (induct_tac xa)
   181   apply simp_all
   182   done
   183 
   184 lemma MAP_EQ:
   185   "list_all (\<lambda>x. f x = g x) l \<longrightarrow> map f l = map g l"
   186   by (induct l) auto
   187 
   188 definition [simp]: "list_mem x xs = List.member xs x"
   189 
   190 lemma DEF_MEM:
   191   "list_mem = (SOME MEM. (\<forall>x. MEM x [] = False) \<and> (\<forall>h x t. MEM x (h # t) = (x = h \<or> MEM x t)))"
   192   apply (rule some_equality[symmetric])
   193   apply (auto simp add: member_def)[1]
   194   apply (simp add: fun_eq_iff)
   195   apply (intro allI)
   196   apply (induct_tac xa)
   197   apply (simp_all add: member_def)
   198   done
   199 
   200 lemma DEF_EX:
   201   "list_ex = (SOME u. (\<forall>P. u P [] = False) \<and> (\<forall>h P t. u P (h # t) = (P h \<or> u P t)))"
   202   apply (rule some_equality[symmetric])
   203   apply (auto)
   204   apply (simp add: fun_eq_iff)
   205   apply (intro allI)
   206   apply (induct_tac xa)
   207   apply (simp_all)
   208   done
   209 
   210 lemma ALL_IMP:
   211   "(\<forall>x. x \<in> set l \<and> P x \<longrightarrow> Q x) \<and> list_all P l \<longrightarrow> list_all Q l"
   212   by (simp add: list_all_iff)
   213 
   214 lemma NOT_EX: "(\<not> list_ex P l) = list_all (\<lambda>x. \<not> P x) l"
   215   by (simp add: list_all_iff list_ex_iff)
   216 
   217 lemma NOT_ALL: "(\<not> list_all P l) = list_ex (\<lambda>x. \<not> P x) l"
   218   by (simp add: list_all_iff list_ex_iff)
   219 
   220 lemma ALL_MAP: "list_all P (map f l) = list_all (P o f) l"
   221   by (simp add: list_all_iff)
   222 
   223 lemma ALL_T: "list_all (\<lambda>x. True) l"
   224   by (simp add: list_all_iff)
   225 
   226 lemma MAP_EQ_ALL2: "list_all2 (\<lambda>x y. f x = f y) l m \<longrightarrow> map f l = map f m"
   227   by (induct l m rule: list_induct2') simp_all
   228 
   229 lemma ALL2_MAP: "list_all2 P (map f l) l = list_all (\<lambda>a. P (f a) a) l"
   230   by (induct l) simp_all
   231 
   232 lemma MAP_EQ_DEGEN: "list_all (\<lambda>x. f x = x) l --> map f l = l"
   233   by (induct l) simp_all
   234 
   235 lemma ALL2_AND_RIGHT:
   236    "list_all2 (\<lambda>x y. P x \<and> Q x y) l m = (list_all P l \<and> list_all2 Q l m)"
   237   by (induct l m rule: list_induct2') auto
   238 
   239 lemma ITLIST_EXTRA:
   240   "foldr f (l @ [a]) b = foldr f l (f a b)"
   241   by simp
   242 
   243 lemma ALL_MP:
   244   "list_all (\<lambda>x. P x \<longrightarrow> Q x) l \<and> list_all P l \<longrightarrow> list_all Q l"
   245   by (simp add: list_all_iff)
   246 
   247 lemma AND_ALL:
   248   "(list_all P l \<and> list_all Q l) = list_all (\<lambda>x. P x \<and> Q x) l"
   249   by (auto simp add: list_all_iff)
   250 
   251 lemma EX_IMP:
   252   "(\<forall>x. x\<in>set l \<and> P x \<longrightarrow> Q x) \<and> list_ex P l \<longrightarrow> list_ex Q l"
   253   by (auto simp add: list_ex_iff)
   254 
   255 lemma ALL_MEM:
   256   "(\<forall>x. x\<in>set l \<longrightarrow> P x) = list_all P l"
   257   by (auto simp add: list_all_iff)
   258 
   259 lemma EX_MAP:
   260   "ALL P f l. list_ex P (map f l) = list_ex (P o f) l"
   261   by (simp add: list_ex_iff)
   262 
   263 lemma EXISTS_EX:
   264   "\<forall>P l. (EX x. list_ex (P x) l) = list_ex (\<lambda>s. EX x. P x s) l"
   265   by (auto simp add: list_ex_iff)
   266 
   267 lemma FORALL_ALL:
   268   "\<forall>P l. (\<forall>x. list_all (P x) l) = list_all (\<lambda>s. \<forall>x. P x s) l"
   269   by (auto simp add: list_all_iff)
   270 
   271 lemma MEM_APPEND: "\<forall>x l1 l2. (x\<in>set (l1 @ l2)) = (x\<in>set l1 \<or> x\<in>set l2)"
   272   by simp
   273 
   274 lemma MEM_MAP: "\<forall>f y l. (y\<in>set (map f l)) = (EX x. x\<in>set l \<and> y = f x)"
   275   by auto
   276 
   277 lemma MEM_FILTER: "\<forall>P l x. (x\<in>set (filter P l)) = (P x \<and> x\<in>set l)"
   278   by auto
   279 
   280 lemma EX_MEM: "(EX x. P x \<and> x\<in>set l) = list_ex P l"
   281   by (auto simp add: list_ex_iff)
   282 
   283 lemma ALL2_MAP2:
   284   "list_all2 P (map f l) (map g m) = list_all2 (\<lambda>x y. P (f x) (g y)) l m"
   285   by (simp add: list_all2_map1 list_all2_map2)
   286 
   287 lemma ALL2_ALL:
   288   "list_all2 P l l = list_all (\<lambda>x. P x x) l"
   289   by (induct l) simp_all
   290 
   291 lemma LENGTH_MAP2:
   292   "length l = length m \<longrightarrow> length (map2 f l m) = length m"
   293   by (induct l m rule: list_induct2') simp_all
   294 
   295 lemma DEF_set_of_list:
   296   "set = (SOME sol. sol [] = {} \<and> (\<forall>h t. sol (h # t) = insert h (sol t)))"
   297   apply (rule some_equality[symmetric])
   298   apply (simp_all)
   299   apply (rule ext)
   300   apply (induct_tac x)
   301   apply simp_all
   302   done
   303 
   304 lemma IN_SET_OF_LIST:
   305   "(x : set l) = (x : set l)"
   306   by simp
   307 
   308 lemma DEF_BUTLAST:
   309   "butlast = (SOME B. B [] = [] \<and> (\<forall>h t. B (h # t) = (if t = [] then [] else h # B t)))"
   310   apply (rule some_equality[symmetric])
   311   apply auto
   312   apply (rule ext)
   313   apply (induct_tac x)
   314   apply auto
   315   done
   316 
   317 lemma MONO_ALL:
   318   "(ALL x. P x \<longrightarrow> Q x) \<longrightarrow> list_all P l \<longrightarrow> list_all Q l"
   319   by (simp add: list_all_iff)
   320 
   321 lemma EL_TL: "l \<noteq> [] \<Longrightarrow> tl l ! x = l ! (x + 1)"
   322   by (induct l) (simp_all)
   323 
   324 (* Assume the same behaviour outside of the usual domain.
   325    For HD, LAST, EL it follows from "undefined = SOME _. False".
   326 
   327    The definitions of TL and ZIP are different for empty lists.
   328  *)
   329 axiomatization where
   330   DEF_HD: "hd = (SOME HD. \<forall>t h. HD (h # t) = h)"
   331 
   332 axiomatization where
   333   DEF_LAST: "last =
   334     (SOME LAST. \<forall>h t. LAST (h # t) = (if t = [] then h else LAST t))"
   335 
   336 axiomatization where
   337   DEF_EL: "list_el =
   338     (SOME EL. (\<forall>l. EL 0 l = hd l) \<and> (\<forall>n l. EL (Suc n) l = EL n (tl l)))"
   339 
   340 end