src/HOL/Import/HOL4Compat.thy
author skalberg
Fri Apr 02 17:37:45 2004 +0200 (2004-04-02)
changeset 14516 a183dec876ab
child 14620 1be590fd2422
permissions -rw-r--r--
Added HOL proof importer.
     1 theory HOL4Compat = HOL4Setup + Divides + Primes + Real:
     2 
     3 lemma EXISTS_UNIQUE_DEF: "(Ex1 P) = (Ex P & (ALL x y. P x & P y --> (x = y)))"
     4   by auto
     5 
     6 lemma COND_DEF:"(If b t f) = (@x. ((b = True) --> (x = t)) & ((b = False) --> (x = f)))"
     7   by auto
     8 
     9 constdefs
    10   LET :: "['a \<Rightarrow> 'b,'a] \<Rightarrow> 'b"
    11   "LET f s == f s"
    12 
    13 lemma [hol4rew]: "LET f s = Let s f"
    14   by (simp add: LET_def Let_def)
    15 
    16 lemmas [hol4rew] = ONE_ONE_rew
    17 
    18 lemma bool_case_DEF: "(bool_case x y b) = (if b then x else y)"
    19   by simp;
    20 
    21 lemma INR_INL_11: "(ALL y x. (Inl x = Inl y) = (x = y)) & (ALL y x. (Inr x = Inr y) = (x = y))"
    22   by safe
    23 
    24 consts
    25   ISL :: "'a + 'b => bool"
    26   ISR :: "'a + 'b => bool"
    27 
    28 primrec ISL_def:
    29   "ISL (Inl x) = True"
    30   "ISL (Inr x) = False"
    31 
    32 primrec ISR_def:
    33   "ISR (Inl x) = False"
    34   "ISR (Inr x) = True"
    35 
    36 lemma ISL: "(ALL x. ISL (Inl x)) & (ALL y. ~ISL (Inr y))"
    37   by simp
    38 
    39 lemma ISR: "(ALL x. ISR (Inr x)) & (ALL y. ~ISR (Inl y))"
    40   by simp
    41 
    42 consts
    43   OUTL :: "'a + 'b => 'a"
    44   OUTR :: "'a + 'b => 'b"
    45 
    46 primrec OUTL_def:
    47   "OUTL (Inl x) = x"
    48 
    49 primrec OUTR_def:
    50   "OUTR (Inr x) = x"
    51 
    52 lemma OUTL: "OUTL (Inl x) = x"
    53   by simp
    54 
    55 lemma OUTR: "OUTR (Inr x) = x"
    56   by simp
    57 
    58 lemma sum_case_def: "(ALL f g x. sum_case f g (Inl x) = f x) & (ALL f g y. sum_case f g (Inr y) = g y)"
    59   by simp;
    60 
    61 lemma one: "ALL v. v = ()"
    62   by simp;
    63 
    64 lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)"
    65   by simp
    66 
    67 lemma OPTION_MAP_DEF: "(!f x. option_map f (Some x) = Some (f x)) & (!f. option_map f None = None)"
    68   by simp
    69 
    70 consts
    71   IS_SOME :: "'a option => bool"
    72   IS_NONE :: "'a option => bool"
    73 
    74 primrec IS_SOME_def:
    75   "IS_SOME (Some x) = True"
    76   "IS_SOME None = False"
    77 
    78 primrec IS_NONE_def:
    79   "IS_NONE (Some x) = False"
    80   "IS_NONE None = True"
    81 
    82 lemma IS_NONE_DEF: "(!x. IS_NONE (Some x) = False) & (IS_NONE None = True)"
    83   by simp
    84 
    85 lemma IS_SOME_DEF: "(!x. IS_SOME (Some x) = True) & (IS_SOME None = False)"
    86   by simp
    87 
    88 consts
    89   OPTION_JOIN :: "'a option option => 'a option"
    90 
    91 primrec OPTION_JOIN_def:
    92   "OPTION_JOIN None = None"
    93   "OPTION_JOIN (Some x) = x"
    94 
    95 lemma OPTION_JOIN_DEF: "(OPTION_JOIN None = None) & (ALL x. OPTION_JOIN (Some x) = x)"
    96   by simp;
    97 
    98 lemma PAIR: "(fst x,snd x) = x"
    99   by simp
   100 
   101 lemma PAIR_MAP: "prod_fun f g p = (f (fst p),g (snd p))"
   102   by (simp add: prod_fun_def split_def)
   103 
   104 lemma pair_case_def: "split = split"
   105   ..;
   106 
   107 lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)"
   108   by auto
   109 
   110 constdefs
   111   nat_gt :: "nat => nat => bool"
   112   "nat_gt == %m n. n < m"
   113   nat_ge :: "nat => nat => bool"
   114   "nat_ge == %m n. nat_gt m n | m = n"
   115 
   116 lemma [hol4rew]: "nat_gt m n = (n < m)"
   117   by (simp add: nat_gt_def)
   118 
   119 lemma [hol4rew]: "nat_ge m n = (n <= m)"
   120   by (auto simp add: nat_ge_def nat_gt_def)
   121 
   122 lemma GREATER_DEF: "ALL m n. (n < m) = (n < m)"
   123   by simp
   124 
   125 lemma GREATER_OR_EQ: "ALL m n. n <= (m::nat) = (n < m | m = n)"
   126   by auto
   127 
   128 lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)"
   129 proof safe
   130   assume "m < n"
   131   def P == "%n. n <= m"
   132   have "(!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   133   proof (auto simp add: P_def)
   134     assume "n <= m"
   135     from prems
   136     show False
   137       by auto
   138   qed
   139   thus "? P. (!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   140     by auto
   141 next
   142   fix P
   143   assume alln: "!n. P (Suc n) \<longrightarrow> P n"
   144   assume pm: "P m"
   145   assume npn: "~P n"
   146   have "!k q. q + k = m \<longrightarrow> P q"
   147   proof
   148     fix k
   149     show "!q. q + k = m \<longrightarrow> P q"
   150     proof (induct k,simp_all)
   151       show "P m" .
   152     next
   153       fix k
   154       assume ind: "!q. q + k = m \<longrightarrow> P q"
   155       show "!q. Suc (q + k) = m \<longrightarrow> P q"
   156       proof (rule+)
   157 	fix q
   158 	assume "Suc (q + k) = m"
   159 	hence "(Suc q) + k = m"
   160 	  by simp
   161 	with ind
   162 	have psq: "P (Suc q)"
   163 	  by simp
   164 	from alln
   165 	have "P (Suc q) --> P q"
   166 	  ..
   167 	with psq
   168 	show "P q"
   169 	  by simp
   170       qed
   171     qed
   172   qed
   173   hence "!q. q + (m - n) = m \<longrightarrow> P q"
   174     ..
   175   hence hehe: "n + (m - n) = m \<longrightarrow> P n"
   176     ..
   177   show "m < n"
   178   proof (rule classical)
   179     assume "~(m<n)"
   180     hence "n <= m"
   181       by simp
   182     with hehe
   183     have "P n"
   184       by simp
   185     with npn
   186     show "m < n"
   187       ..
   188   qed
   189 qed;
   190 
   191 constdefs
   192   FUNPOW :: "('a => 'a) => nat => 'a => 'a"
   193   "FUNPOW f n == f ^ n"
   194 
   195 lemma FUNPOW: "(ALL f x. (f ^ 0) x = x) &
   196   (ALL f n x. (f ^ Suc n) x = (f ^ n) (f x))"
   197 proof auto
   198   fix f n x
   199   have "ALL x. f ((f ^ n) x) = (f ^ n) (f x)"
   200     by (induct n,auto)
   201   thus "f ((f ^ n) x) = (f ^ n) (f x)"
   202     ..
   203 qed
   204 
   205 lemma [hol4rew]: "FUNPOW f n = f ^ n"
   206   by (simp add: FUNPOW_def)
   207 
   208 lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))"
   209   by simp
   210 
   211 lemma MULT: "(!n. (0::nat) * n = 0) & (!m n. Suc m * n = m * n + n)"
   212   by simp
   213 
   214 lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))"
   215   apply simp
   216   apply arith
   217   done
   218 
   219 lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)"
   220   by (simp add: max_def)
   221 
   222 lemma MIN_DEF: "min (m::nat) n = (if m < n then m else n)"
   223   by (simp add: min_def)
   224 
   225 lemma DIVISION: "(0::nat) < n --> (!k. (k = k div n * n + k mod n) & k mod n < n)"
   226   by simp
   227 
   228 constdefs
   229   ALT_ZERO :: nat
   230   "ALT_ZERO == 0"
   231   NUMERAL_BIT1 :: "nat \<Rightarrow> nat"
   232   "NUMERAL_BIT1 n == n + (n + Suc 0)"
   233   NUMERAL_BIT2 :: "nat \<Rightarrow> nat"
   234   "NUMERAL_BIT2 n == n + (n + Suc (Suc 0))"
   235   NUMERAL :: "nat \<Rightarrow> nat"
   236   "NUMERAL x == x"
   237 
   238 lemma [hol4rew]: "NUMERAL ALT_ZERO = 0"
   239   by (simp add: ALT_ZERO_def NUMERAL_def)
   240 
   241 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT1 ALT_ZERO) = 1"
   242   by (simp add: ALT_ZERO_def NUMERAL_BIT1_def NUMERAL_def)
   243 
   244 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT2 ALT_ZERO) = 2"
   245   by (simp add: ALT_ZERO_def NUMERAL_BIT2_def NUMERAL_def)
   246 
   247 lemma EXP: "(!m. m ^ 0 = (1::nat)) & (!m n. m ^ Suc n = m * (m::nat) ^ n)"
   248   by auto
   249 
   250 lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)"
   251   by simp;
   252 
   253 lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)"
   254   by (auto simp add: dvd_def);
   255 
   256 lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)"
   257   by simp
   258 
   259 consts
   260   list_size :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat"
   261 
   262 primrec
   263   "list_size f [] = 0"
   264   "list_size f (a0#a1) = 1 + (f a0 + list_size f a1)"
   265 
   266 lemma list_size_def: "(!f. list_size f [] = 0) &
   267          (!f a0 a1. list_size f (a0#a1) = 1 + (f a0 + list_size f a1))"
   268   by simp
   269 
   270 lemma list_case_cong: "! M M' v f. M = M' & (M' = [] \<longrightarrow>  v = v') &
   271            (!a0 a1. (M' = a0#a1) \<longrightarrow> (f a0 a1 = f' a0 a1)) -->
   272            (list_case v f M = list_case v' f' M')"
   273 proof clarify
   274   fix M M' v f
   275   assume "M' = [] \<longrightarrow> v = v'"
   276     and "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
   277   show "list_case v f M' = list_case v' f' M'"
   278   proof (rule List.list.case_cong)
   279     show "M' = M'"
   280       ..
   281   next
   282     assume "M' = []"
   283     with prems
   284     show "v = v'"
   285       by auto
   286   next
   287     fix a0 a1
   288     assume "M' = a0 # a1"
   289     with prems
   290     show "f a0 a1 = f' a0 a1"
   291       by auto
   292   qed
   293 qed
   294 
   295 lemma list_Axiom: "ALL f0 f1. EX fn. (fn [] = f0) & (ALL a0 a1. fn (a0#a1) = f1 a0 a1 (fn a1))"
   296 proof safe
   297   fix f0 f1
   298   def fn == "list_rec f0 f1"
   299   have "fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   300     by (simp add: fn_def)
   301   thus "EX fn. fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   302     by auto
   303 qed
   304 
   305 lemma list_Axiom_old: "EX! fn. (fn [] = x) & (ALL h t. fn (h#t) = f (fn t) h t)"
   306 proof safe
   307   def fn == "list_rec x (%h t r. f r h t)"
   308   have "fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   309     by (simp add: fn_def)
   310   thus "EX fn. fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   311     by auto
   312 next
   313   fix fn1 fn2
   314   assume "ALL h t. fn1 (h # t) = f (fn1 t) h t"
   315   assume "ALL h t. fn2 (h # t) = f (fn2 t) h t"
   316   assume "fn2 [] = fn1 []"
   317   show "fn1 = fn2"
   318   proof
   319     fix xs
   320     show "fn1 xs = fn2 xs"
   321       by (induct xs,simp_all add: prems) 
   322   qed
   323 qed
   324 
   325 lemma NULL_DEF: "(null [] = True) & (!h t. null (h # t) = False)"
   326   by simp
   327 
   328 constdefs
   329   sum :: "nat list \<Rightarrow> nat"
   330   "sum l == foldr (op +) l 0"
   331 
   332 lemma SUM: "(sum [] = 0) & (!h t. sum (h#t) = h + sum t)"
   333   by (simp add: sum_def)
   334 
   335 lemma APPEND: "(!l. [] @ l = l) & (!l1 l2 h. (h#l1) @ l2 = h# l1 @ l2)"
   336   by simp
   337 
   338 lemma FLAT: "(concat [] = []) & (!h t. concat (h#t) = h @ (concat t))"
   339   by simp
   340 
   341 lemma LENGTH: "(length [] = 0) & (!h t. length (h#t) = Suc (length t))"
   342   by simp
   343 
   344 lemma MAP: "(!f. map f [] = []) & (!f h t. map f (h#t) = f h#map f t)"
   345   by simp
   346 
   347 lemma MEM: "(!x. x mem [] = False) & (!x h t. x mem (h#t) = ((x = h) | x mem t))"
   348   by auto
   349 
   350 lemma FILTER: "(!P. filter P [] = []) & (!P h t.
   351            filter P (h#t) = (if P h then h#filter P t else filter P t))"
   352   by simp
   353 
   354 lemma REPLICATE: "(ALL x. replicate 0 x = []) &
   355   (ALL n x. replicate (Suc n) x = x # replicate n x)"
   356   by simp
   357 
   358 constdefs
   359   FOLDR :: "[['a,'b]\<Rightarrow>'b,'b,'a list] \<Rightarrow> 'b"
   360   "FOLDR f e l == foldr f l e"
   361 
   362 lemma [hol4rew]: "FOLDR f e l = foldr f l e"
   363   by (simp add: FOLDR_def)
   364 
   365 lemma FOLDR: "(!f e. foldr f [] e = e) & (!f e x l. foldr f (x#l) e = f x (foldr f l e))"
   366   by simp
   367 
   368 lemma FOLDL: "(!f e. foldl f e [] = e) & (!f e x l. foldl f e (x#l) = foldl f (f e x) l)"
   369   by simp
   370 
   371 lemma EVERY_DEF: "(!P. list_all P [] = True) & (!P h t. list_all P (h#t) = (P h & list_all P t))"
   372   by simp
   373 
   374 consts
   375   list_exists :: "['a \<Rightarrow> bool,'a list] \<Rightarrow> bool"
   376 
   377 primrec
   378   list_exists_Nil: "list_exists P Nil = False"
   379   list_exists_Cons: "list_exists P (x#xs) = (if P x then True else list_exists P xs)"
   380 
   381 lemma list_exists_DEF: "(!P. list_exists P [] = False) &
   382          (!P h t. list_exists P (h#t) = (P h | list_exists P t))"
   383   by simp
   384 
   385 consts
   386   map2 :: "[['a,'b]\<Rightarrow>'c,'a list,'b list] \<Rightarrow> 'c list"
   387 
   388 primrec
   389   map2_Nil: "map2 f [] l2 = []"
   390   map2_Cons: "map2 f (x#xs) l2 = f x (hd l2) # map2 f xs (tl l2)"
   391 
   392 lemma MAP2: "(!f. map2 f [] [] = []) & (!f h1 t1 h2 t2. map2 f (h1#t1) (h2#t2) = f h1 h2#map2 f t1 t2)"
   393   by simp
   394 
   395 lemma list_INDUCT: "\<lbrakk> P [] ; !t. P t \<longrightarrow> (!h. P (h#t)) \<rbrakk> \<Longrightarrow> !l. P l"
   396 proof
   397   fix l
   398   assume "P []"
   399   assume allt: "!t. P t \<longrightarrow> (!h. P (h # t))"
   400   show "P l"
   401   proof (induct l)
   402     show "P []" .
   403   next
   404     fix h t
   405     assume "P t"
   406     with allt
   407     have "!h. P (h # t)"
   408       by auto
   409     thus "P (h # t)"
   410       ..
   411   qed
   412 qed
   413 
   414 lemma list_CASES: "(l = []) | (? t h. l = h#t)"
   415   by (induct l,auto)
   416 
   417 constdefs
   418   ZIP :: "'a list * 'b list \<Rightarrow> ('a * 'b) list"
   419   "ZIP == %(a,b). zip a b"
   420 
   421 lemma ZIP: "(zip [] [] = []) &
   422   (!x1 l1 x2 l2. zip (x1#l1) (x2#l2) = (x1,x2)#zip l1 l2)"
   423   by simp
   424 
   425 lemma [hol4rew]: "ZIP (a,b) = zip a b"
   426   by (simp add: ZIP_def)
   427 
   428 consts
   429   unzip :: "('a * 'b) list \<Rightarrow> 'a list * 'b list"
   430 
   431 primrec
   432   unzip_Nil: "unzip [] = ([],[])"
   433   unzip_Cons: "unzip (xy#xys) = (let zs = unzip xys in (fst xy # fst zs,snd xy # snd zs))"
   434 
   435 lemma UNZIP: "(unzip [] = ([],[])) &
   436          (!x l. unzip (x#l) = (fst x#fst (unzip l),snd x#snd (unzip l)))"
   437   by (simp add: Let_def)
   438 
   439 lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])"
   440   by simp;
   441 
   442 lemma REAL_SUP_ALLPOS: "\<lbrakk> ALL x. P (x::real) \<longrightarrow> 0 < x ; EX x. P x; EX z. ALL x. P x \<longrightarrow> x < z \<rbrakk> \<Longrightarrow> EX s. ALL y. (EX x. P x & y < x) = (y < s)"
   443 proof safe
   444   fix x z
   445   assume allx: "ALL x. P x \<longrightarrow> 0 < x"
   446   assume px: "P x"
   447   assume allx': "ALL x. P x \<longrightarrow> x < z"
   448   have "EX s. ALL y. (EX x : Collect P. y < x) = (y < s)"
   449   proof (rule posreal_complete)
   450     show "ALL x : Collect P. 0 < x"
   451     proof safe
   452       fix x
   453       assume "P x"
   454       from allx
   455       have "P x \<longrightarrow> 0 < x"
   456 	..
   457       thus "0 < x"
   458 	by (simp add: prems)
   459     qed
   460   next
   461     from px
   462     show "EX x. x : Collect P"
   463       by auto
   464   next
   465     from allx'
   466     show "EX y. ALL x : Collect P. x < y"
   467       apply simp
   468       ..
   469   qed
   470   thus "EX s. ALL y. (EX x. P x & y < x) = (y < s)"
   471     by simp
   472 qed
   473 
   474 lemma REAL_10: "~((1::real) = 0)"
   475   by simp
   476 
   477 lemma REAL_ADD_ASSOC: "(x::real) + (y + z) = x + y + z"
   478   by simp
   479 
   480 lemma REAL_MUL_ASSOC: "(x::real) * (y * z) = x * y * z"
   481   by simp
   482 
   483 lemma REAL_ADD_LINV:  "-x + x = (0::real)"
   484   by simp
   485 
   486 lemma REAL_MUL_LINV: "x ~= (0::real) ==> inverse x * x = 1"
   487   by simp
   488 
   489 lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x"
   490   by auto;
   491 
   492 lemma [hol4rew]: "real (0::nat) = 0"
   493   by simp
   494 
   495 lemma [hol4rew]: "real (1::nat) = 1"
   496   by simp
   497 
   498 lemma [hol4rew]: "real (2::nat) = 2"
   499   by simp
   500 
   501 lemma real_lte: "((x::real) <= y) = (~(y < x))"
   502   by auto
   503 
   504 lemma real_of_num: "((0::real) = 0) & (!n. real (Suc n) = real n + 1)"
   505   by (simp add: real_of_nat_Suc)
   506 
   507 lemma abs: "abs (x::real) = (if 0 <= x then x else -x)"
   508   by (simp add: real_abs_def)
   509 
   510 lemma pow: "(!x::real. x ^ 0 = 1) & (!x::real. ALL n. x ^ (Suc n) = x * x ^ n)"
   511   by simp;
   512 
   513 constdefs
   514   real_gt :: "real => real => bool" 
   515   "real_gt == %x y. y < x"
   516 
   517 lemma [hol4rew]: "real_gt x y = (y < x)"
   518   by (simp add: real_gt_def)
   519 
   520 lemma real_gt: "ALL x (y::real). (y < x) = (y < x)"
   521   by simp
   522 
   523 constdefs
   524   real_ge :: "real => real => bool"
   525   "real_ge x y == y <= x"
   526 
   527 lemma [hol4rew]: "real_ge x y = (y <= x)"
   528   by (simp add: real_ge_def)
   529 
   530 lemma real_ge: "ALL x y. (y <= x) = (y <= x)"
   531   by simp
   532 
   533 end