src/HOL/Import/HOL4Compat.thy
author krauss
Mon Jul 31 18:05:40 2006 +0200 (2006-07-31)
changeset 20269 c40070317ab8
parent 19064 bf19cc5a7899
child 20432 07ec57376051
permissions -rw-r--r--
Removed an "apply arith" where there are already "No Subgoals"
     1 (*  Title:      HOL/Import/HOL4Compat.thy
     2     ID:         $Id$
     3     Author:     Sebastian Skalberg (TU Muenchen)
     4 *)
     5 
     6 theory HOL4Compat imports HOL4Setup Divides Primes Real 
     7 begin
     8 
     9 lemma EXISTS_UNIQUE_DEF: "(Ex1 P) = (Ex P & (ALL x y. P x & P y --> (x = y)))"
    10   by auto
    11 
    12 lemma COND_DEF:"(If b t f) = (@x. ((b = True) --> (x = t)) & ((b = False) --> (x = f)))"
    13   by auto
    14 
    15 constdefs
    16   LET :: "['a \<Rightarrow> 'b,'a] \<Rightarrow> 'b"
    17   "LET f s == f s"
    18 
    19 lemma [hol4rew]: "LET f s = Let s f"
    20   by (simp add: LET_def Let_def)
    21 
    22 lemmas [hol4rew] = ONE_ONE_rew
    23 
    24 lemma bool_case_DEF: "(bool_case x y b) = (if b then x else y)"
    25   by simp;
    26 
    27 lemma INR_INL_11: "(ALL y x. (Inl x = Inl y) = (x = y)) & (ALL y x. (Inr x = Inr y) = (x = y))"
    28   by safe
    29 
    30 (*lemma INL_neq_INR: "ALL v1 v2. Sum_Type.Inr v2 ~= Sum_Type.Inl v1"
    31   by simp*)
    32 
    33 consts
    34   ISL :: "'a + 'b => bool"
    35   ISR :: "'a + 'b => bool"
    36 
    37 primrec ISL_def:
    38   "ISL (Inl x) = True"
    39   "ISL (Inr x) = False"
    40 
    41 primrec ISR_def:
    42   "ISR (Inl x) = False"
    43   "ISR (Inr x) = True"
    44 
    45 lemma ISL: "(ALL x. ISL (Inl x)) & (ALL y. ~ISL (Inr y))"
    46   by simp
    47 
    48 lemma ISR: "(ALL x. ISR (Inr x)) & (ALL y. ~ISR (Inl y))"
    49   by simp
    50 
    51 consts
    52   OUTL :: "'a + 'b => 'a"
    53   OUTR :: "'a + 'b => 'b"
    54 
    55 primrec OUTL_def:
    56   "OUTL (Inl x) = x"
    57 
    58 primrec OUTR_def:
    59   "OUTR (Inr x) = x"
    60 
    61 lemma OUTL: "OUTL (Inl x) = x"
    62   by simp
    63 
    64 lemma OUTR: "OUTR (Inr x) = x"
    65   by simp
    66 
    67 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)"
    68   by simp;
    69 
    70 lemma one: "ALL v. v = ()"
    71   by simp;
    72 
    73 lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)"
    74   by simp
    75 
    76 lemma OPTION_MAP_DEF: "(!f x. option_map f (Some x) = Some (f x)) & (!f. option_map f None = None)"
    77   by simp
    78 
    79 consts
    80   IS_SOME :: "'a option => bool"
    81   IS_NONE :: "'a option => bool"
    82 
    83 primrec IS_SOME_def:
    84   "IS_SOME (Some x) = True"
    85   "IS_SOME None = False"
    86 
    87 primrec IS_NONE_def:
    88   "IS_NONE (Some x) = False"
    89   "IS_NONE None = True"
    90 
    91 lemma IS_NONE_DEF: "(!x. IS_NONE (Some x) = False) & (IS_NONE None = True)"
    92   by simp
    93 
    94 lemma IS_SOME_DEF: "(!x. IS_SOME (Some x) = True) & (IS_SOME None = False)"
    95   by simp
    96 
    97 consts
    98   OPTION_JOIN :: "'a option option => 'a option"
    99 
   100 primrec OPTION_JOIN_def:
   101   "OPTION_JOIN None = None"
   102   "OPTION_JOIN (Some x) = x"
   103 
   104 lemma OPTION_JOIN_DEF: "(OPTION_JOIN None = None) & (ALL x. OPTION_JOIN (Some x) = x)"
   105   by simp;
   106 
   107 lemma PAIR: "(fst x,snd x) = x"
   108   by simp
   109 
   110 lemma PAIR_MAP: "prod_fun f g p = (f (fst p),g (snd p))"
   111   by (simp add: prod_fun_def split_def)
   112 
   113 lemma pair_case_def: "split = split"
   114   ..;
   115 
   116 lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)"
   117   by auto
   118 
   119 constdefs
   120   nat_gt :: "nat => nat => bool"
   121   "nat_gt == %m n. n < m"
   122   nat_ge :: "nat => nat => bool"
   123   "nat_ge == %m n. nat_gt m n | m = n"
   124 
   125 lemma [hol4rew]: "nat_gt m n = (n < m)"
   126   by (simp add: nat_gt_def)
   127 
   128 lemma [hol4rew]: "nat_ge m n = (n <= m)"
   129   by (auto simp add: nat_ge_def nat_gt_def)
   130 
   131 lemma GREATER_DEF: "ALL m n. (n < m) = (n < m)"
   132   by simp
   133 
   134 lemma GREATER_OR_EQ: "ALL m n. n <= (m::nat) = (n < m | m = n)"
   135   by auto
   136 
   137 lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)"
   138 proof safe
   139   assume "m < n"
   140   def P == "%n. n <= m"
   141   have "(!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   142   proof (auto simp add: P_def)
   143     assume "n <= m"
   144     from prems
   145     show False
   146       by auto
   147   qed
   148   thus "? P. (!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   149     by auto
   150 next
   151   fix P
   152   assume alln: "!n. P (Suc n) \<longrightarrow> P n"
   153   assume pm: "P m"
   154   assume npn: "~P n"
   155   have "!k q. q + k = m \<longrightarrow> P q"
   156   proof
   157     fix k
   158     show "!q. q + k = m \<longrightarrow> P q"
   159     proof (induct k,simp_all)
   160       show "P m" .
   161     next
   162       fix k
   163       assume ind: "!q. q + k = m \<longrightarrow> P q"
   164       show "!q. Suc (q + k) = m \<longrightarrow> P q"
   165       proof (rule+)
   166 	fix q
   167 	assume "Suc (q + k) = m"
   168 	hence "(Suc q) + k = m"
   169 	  by simp
   170 	with ind
   171 	have psq: "P (Suc q)"
   172 	  by simp
   173 	from alln
   174 	have "P (Suc q) --> P q"
   175 	  ..
   176 	with psq
   177 	show "P q"
   178 	  by simp
   179       qed
   180     qed
   181   qed
   182   hence "!q. q + (m - n) = m \<longrightarrow> P q"
   183     ..
   184   hence hehe: "n + (m - n) = m \<longrightarrow> P n"
   185     ..
   186   show "m < n"
   187   proof (rule classical)
   188     assume "~(m<n)"
   189     hence "n <= m"
   190       by simp
   191     with hehe
   192     have "P n"
   193       by simp
   194     with npn
   195     show "m < n"
   196       ..
   197   qed
   198 qed;
   199 
   200 constdefs
   201   FUNPOW :: "('a => 'a) => nat => 'a => 'a"
   202   "FUNPOW f n == f ^ n"
   203 
   204 lemma FUNPOW: "(ALL f x. (f ^ 0) x = x) &
   205   (ALL f n x. (f ^ Suc n) x = (f ^ n) (f x))"
   206 proof auto
   207   fix f n x
   208   have "ALL x. f ((f ^ n) x) = (f ^ n) (f x)"
   209     by (induct n,auto)
   210   thus "f ((f ^ n) x) = (f ^ n) (f x)"
   211     ..
   212 qed
   213 
   214 lemma [hol4rew]: "FUNPOW f n = f ^ n"
   215   by (simp add: FUNPOW_def)
   216 
   217 lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))"
   218   by simp
   219 
   220 lemma MULT: "(!n. (0::nat) * n = 0) & (!m n. Suc m * n = m * n + n)"
   221   by simp
   222 
   223 lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))"
   224   by simp
   225 
   226 lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)"
   227   by (simp add: max_def)
   228 
   229 lemma MIN_DEF: "min (m::nat) n = (if m < n then m else n)"
   230   by (simp add: min_def)
   231 
   232 lemma DIVISION: "(0::nat) < n --> (!k. (k = k div n * n + k mod n) & k mod n < n)"
   233   by simp
   234 
   235 constdefs
   236   ALT_ZERO :: nat
   237   "ALT_ZERO == 0"
   238   NUMERAL_BIT1 :: "nat \<Rightarrow> nat"
   239   "NUMERAL_BIT1 n == n + (n + Suc 0)"
   240   NUMERAL_BIT2 :: "nat \<Rightarrow> nat"
   241   "NUMERAL_BIT2 n == n + (n + Suc (Suc 0))"
   242   NUMERAL :: "nat \<Rightarrow> nat"
   243   "NUMERAL x == x"
   244 
   245 lemma [hol4rew]: "NUMERAL ALT_ZERO = 0"
   246   by (simp add: ALT_ZERO_def NUMERAL_def)
   247 
   248 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT1 ALT_ZERO) = 1"
   249   by (simp add: ALT_ZERO_def NUMERAL_BIT1_def NUMERAL_def)
   250 
   251 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT2 ALT_ZERO) = 2"
   252   by (simp add: ALT_ZERO_def NUMERAL_BIT2_def NUMERAL_def)
   253 
   254 lemma EXP: "(!m. m ^ 0 = (1::nat)) & (!m n. m ^ Suc n = m * (m::nat) ^ n)"
   255   by auto
   256 
   257 lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)"
   258   by simp;
   259 
   260 lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)"
   261   by (auto simp add: dvd_def);
   262 
   263 lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)"
   264   by simp
   265 
   266 consts
   267   list_size :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat"
   268 
   269 primrec
   270   "list_size f [] = 0"
   271   "list_size f (a0#a1) = 1 + (f a0 + list_size f a1)"
   272 
   273 lemma list_size_def: "(!f. list_size f [] = 0) &
   274          (!f a0 a1. list_size f (a0#a1) = 1 + (f a0 + list_size f a1))"
   275   by simp
   276 
   277 lemma list_case_cong: "! M M' v f. M = M' & (M' = [] \<longrightarrow>  v = v') &
   278            (!a0 a1. (M' = a0#a1) \<longrightarrow> (f a0 a1 = f' a0 a1)) -->
   279            (list_case v f M = list_case v' f' M')"
   280 proof clarify
   281   fix M M' v f
   282   assume "M' = [] \<longrightarrow> v = v'"
   283     and "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
   284   show "list_case v f M' = list_case v' f' M'"
   285   proof (rule List.list.case_cong)
   286     show "M' = M'"
   287       ..
   288   next
   289     assume "M' = []"
   290     with prems
   291     show "v = v'"
   292       by auto
   293   next
   294     fix a0 a1
   295     assume "M' = a0 # a1"
   296     with prems
   297     show "f a0 a1 = f' a0 a1"
   298       by auto
   299   qed
   300 qed
   301 
   302 lemma list_Axiom: "ALL f0 f1. EX fn. (fn [] = f0) & (ALL a0 a1. fn (a0#a1) = f1 a0 a1 (fn a1))"
   303 proof safe
   304   fix f0 f1
   305   def fn == "list_rec f0 f1"
   306   have "fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   307     by (simp add: fn_def)
   308   thus "EX fn. fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   309     by auto
   310 qed
   311 
   312 lemma list_Axiom_old: "EX! fn. (fn [] = x) & (ALL h t. fn (h#t) = f (fn t) h t)"
   313 proof safe
   314   def fn == "list_rec x (%h t r. f r h t)"
   315   have "fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   316     by (simp add: fn_def)
   317   thus "EX fn. fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   318     by auto
   319 next
   320   fix fn1 fn2
   321   assume "ALL h t. fn1 (h # t) = f (fn1 t) h t"
   322   assume "ALL h t. fn2 (h # t) = f (fn2 t) h t"
   323   assume "fn2 [] = fn1 []"
   324   show "fn1 = fn2"
   325   proof
   326     fix xs
   327     show "fn1 xs = fn2 xs"
   328       by (induct xs,simp_all add: prems) 
   329   qed
   330 qed
   331 
   332 lemma NULL_DEF: "(null [] = True) & (!h t. null (h # t) = False)"
   333   by simp
   334 
   335 constdefs
   336   sum :: "nat list \<Rightarrow> nat"
   337   "sum l == foldr (op +) l 0"
   338 
   339 lemma SUM: "(sum [] = 0) & (!h t. sum (h#t) = h + sum t)"
   340   by (simp add: sum_def)
   341 
   342 lemma APPEND: "(!l. [] @ l = l) & (!l1 l2 h. (h#l1) @ l2 = h# l1 @ l2)"
   343   by simp
   344 
   345 lemma FLAT: "(concat [] = []) & (!h t. concat (h#t) = h @ (concat t))"
   346   by simp
   347 
   348 lemma LENGTH: "(length [] = 0) & (!h t. length (h#t) = Suc (length t))"
   349   by simp
   350 
   351 lemma MAP: "(!f. map f [] = []) & (!f h t. map f (h#t) = f h#map f t)"
   352   by simp
   353 
   354 lemma MEM: "(!x. x mem [] = False) & (!x h t. x mem (h#t) = ((x = h) | x mem t))"
   355   by auto
   356 
   357 lemma FILTER: "(!P. filter P [] = []) & (!P h t.
   358            filter P (h#t) = (if P h then h#filter P t else filter P t))"
   359   by simp
   360 
   361 lemma REPLICATE: "(ALL x. replicate 0 x = []) &
   362   (ALL n x. replicate (Suc n) x = x # replicate n x)"
   363   by simp
   364 
   365 constdefs
   366   FOLDR :: "[['a,'b]\<Rightarrow>'b,'b,'a list] \<Rightarrow> 'b"
   367   "FOLDR f e l == foldr f l e"
   368 
   369 lemma [hol4rew]: "FOLDR f e l = foldr f l e"
   370   by (simp add: FOLDR_def)
   371 
   372 lemma FOLDR: "(!f e. foldr f [] e = e) & (!f e x l. foldr f (x#l) e = f x (foldr f l e))"
   373   by simp
   374 
   375 lemma FOLDL: "(!f e. foldl f e [] = e) & (!f e x l. foldl f e (x#l) = foldl f (f e x) l)"
   376   by simp
   377 
   378 lemma EVERY_DEF: "(!P. list_all P [] = True) & (!P h t. list_all P (h#t) = (P h & list_all P t))"
   379   by simp
   380 
   381 consts
   382   list_exists :: "['a \<Rightarrow> bool,'a list] \<Rightarrow> bool"
   383 
   384 primrec
   385   list_exists_Nil: "list_exists P Nil = False"
   386   list_exists_Cons: "list_exists P (x#xs) = (if P x then True else list_exists P xs)"
   387 
   388 lemma list_exists_DEF: "(!P. list_exists P [] = False) &
   389          (!P h t. list_exists P (h#t) = (P h | list_exists P t))"
   390   by simp
   391 
   392 consts
   393   map2 :: "[['a,'b]\<Rightarrow>'c,'a list,'b list] \<Rightarrow> 'c list"
   394 
   395 primrec
   396   map2_Nil: "map2 f [] l2 = []"
   397   map2_Cons: "map2 f (x#xs) l2 = f x (hd l2) # map2 f xs (tl l2)"
   398 
   399 lemma MAP2: "(!f. map2 f [] [] = []) & (!f h1 t1 h2 t2. map2 f (h1#t1) (h2#t2) = f h1 h2#map2 f t1 t2)"
   400   by simp
   401 
   402 lemma list_INDUCT: "\<lbrakk> P [] ; !t. P t \<longrightarrow> (!h. P (h#t)) \<rbrakk> \<Longrightarrow> !l. P l"
   403 proof
   404   fix l
   405   assume "P []"
   406   assume allt: "!t. P t \<longrightarrow> (!h. P (h # t))"
   407   show "P l"
   408   proof (induct l)
   409     show "P []" .
   410   next
   411     fix h t
   412     assume "P t"
   413     with allt
   414     have "!h. P (h # t)"
   415       by auto
   416     thus "P (h # t)"
   417       ..
   418   qed
   419 qed
   420 
   421 lemma list_CASES: "(l = []) | (? t h. l = h#t)"
   422   by (induct l,auto)
   423 
   424 constdefs
   425   ZIP :: "'a list * 'b list \<Rightarrow> ('a * 'b) list"
   426   "ZIP == %(a,b). zip a b"
   427 
   428 lemma ZIP: "(zip [] [] = []) &
   429   (!x1 l1 x2 l2. zip (x1#l1) (x2#l2) = (x1,x2)#zip l1 l2)"
   430   by simp
   431 
   432 lemma [hol4rew]: "ZIP (a,b) = zip a b"
   433   by (simp add: ZIP_def)
   434 
   435 consts
   436   unzip :: "('a * 'b) list \<Rightarrow> 'a list * 'b list"
   437 
   438 primrec
   439   unzip_Nil: "unzip [] = ([],[])"
   440   unzip_Cons: "unzip (xy#xys) = (let zs = unzip xys in (fst xy # fst zs,snd xy # snd zs))"
   441 
   442 lemma UNZIP: "(unzip [] = ([],[])) &
   443          (!x l. unzip (x#l) = (fst x#fst (unzip l),snd x#snd (unzip l)))"
   444   by (simp add: Let_def)
   445 
   446 lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])"
   447   by simp;
   448 
   449 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)"
   450 proof safe
   451   fix x z
   452   assume allx: "ALL x. P x \<longrightarrow> 0 < x"
   453   assume px: "P x"
   454   assume allx': "ALL x. P x \<longrightarrow> x < z"
   455   have "EX s. ALL y. (EX x : Collect P. y < x) = (y < s)"
   456   proof (rule posreal_complete)
   457     show "ALL x : Collect P. 0 < x"
   458     proof safe
   459       fix x
   460       assume "P x"
   461       from allx
   462       have "P x \<longrightarrow> 0 < x"
   463 	..
   464       thus "0 < x"
   465 	by (simp add: prems)
   466     qed
   467   next
   468     from px
   469     show "EX x. x : Collect P"
   470       by auto
   471   next
   472     from allx'
   473     show "EX y. ALL x : Collect P. x < y"
   474       apply simp
   475       ..
   476   qed
   477   thus "EX s. ALL y. (EX x. P x & y < x) = (y < s)"
   478     by simp
   479 qed
   480 
   481 lemma REAL_10: "~((1::real) = 0)"
   482   by simp
   483 
   484 lemma REAL_ADD_ASSOC: "(x::real) + (y + z) = x + y + z"
   485   by simp
   486 
   487 lemma REAL_MUL_ASSOC: "(x::real) * (y * z) = x * y * z"
   488   by simp
   489 
   490 lemma REAL_ADD_LINV:  "-x + x = (0::real)"
   491   by simp
   492 
   493 lemma REAL_MUL_LINV: "x ~= (0::real) ==> inverse x * x = 1"
   494   by simp
   495 
   496 lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x"
   497   by auto;
   498 
   499 lemma [hol4rew]: "real (0::nat) = 0"
   500   by simp
   501 
   502 lemma [hol4rew]: "real (1::nat) = 1"
   503   by simp
   504 
   505 lemma [hol4rew]: "real (2::nat) = 2"
   506   by simp
   507 
   508 lemma real_lte: "((x::real) <= y) = (~(y < x))"
   509   by auto
   510 
   511 lemma real_of_num: "((0::real) = 0) & (!n. real (Suc n) = real n + 1)"
   512   by (simp add: real_of_nat_Suc)
   513 
   514 lemma abs: "abs (x::real) = (if 0 <= x then x else -x)"
   515   by (simp add: abs_if)
   516 
   517 lemma pow: "(!x::real. x ^ 0 = 1) & (!x::real. ALL n. x ^ (Suc n) = x * x ^ n)"
   518   by simp
   519 
   520 constdefs
   521   real_gt :: "real => real => bool" 
   522   "real_gt == %x y. y < x"
   523 
   524 lemma [hol4rew]: "real_gt x y = (y < x)"
   525   by (simp add: real_gt_def)
   526 
   527 lemma real_gt: "ALL x (y::real). (y < x) = (y < x)"
   528   by simp
   529 
   530 constdefs
   531   real_ge :: "real => real => bool"
   532   "real_ge x y == y <= x"
   533 
   534 lemma [hol4rew]: "real_ge x y = (y <= x)"
   535   by (simp add: real_ge_def)
   536 
   537 lemma real_ge: "ALL x y. (y <= x) = (y <= x)"
   538   by simp
   539 
   540 end