src/HOL/Import/HOL4Compat.thy
author wenzelm
Wed Dec 29 17:34:41 2010 +0100 (2010-12-29)
changeset 41413 64cd30d6b0b8
parent 40607 30d512bf47a7
child 41550 efa734d9b221
permissions -rw-r--r--
explicit file specifications -- avoid secondary load path;
     1 (*  Title:      HOL/Import/HOL4Compat.thy
     2     Author:     Sebastian Skalberg (TU Muenchen)
     3 *)
     4 
     5 theory HOL4Compat
     6 imports
     7   HOL4Setup
     8   Complex_Main
     9   "~~/src/HOL/Old_Number_Theory/Primes"
    10   "~~/src/HOL/Library/ContNotDenum"
    11 begin
    12 
    13 abbreviation (input) mem (infixl "mem" 55) where "x mem xs \<equiv> List.member xs x"
    14 no_notation differentiable (infixl "differentiable" 60)
    15 no_notation sums (infixr "sums" 80)
    16 
    17 lemma EXISTS_UNIQUE_DEF: "(Ex1 P) = (Ex P & (ALL x y. P x & P y --> (x = y)))"
    18   by auto
    19 
    20 lemma COND_DEF:"(If b t f) = (@x. ((b = True) --> (x = t)) & ((b = False) --> (x = f)))"
    21   by auto
    22 
    23 definition LET :: "['a \<Rightarrow> 'b,'a] \<Rightarrow> 'b" where
    24   "LET f s == f s"
    25 
    26 lemma [hol4rew]: "LET f s = Let s f"
    27   by (simp add: LET_def Let_def)
    28 
    29 lemmas [hol4rew] = ONE_ONE_rew
    30 
    31 lemma bool_case_DEF: "(bool_case x y b) = (if b then x else y)"
    32   by simp
    33 
    34 lemma INR_INL_11: "(ALL y x. (Inl x = Inl y) = (x = y)) & (ALL y x. (Inr x = Inr y) = (x = y))"
    35   by safe
    36 
    37 (*lemma INL_neq_INR: "ALL v1 v2. Sum_Type.Inr v2 ~= Sum_Type.Inl v1"
    38   by simp*)
    39 
    40 primrec ISL :: "'a + 'b => bool" where
    41   "ISL (Inl x) = True"
    42 | "ISL (Inr x) = False"
    43 
    44 primrec ISR :: "'a + 'b => bool" where
    45   "ISR (Inl x) = False"
    46 | "ISR (Inr x) = True"
    47 
    48 lemma ISL: "(ALL x. ISL (Inl x)) & (ALL y. ~ISL (Inr y))"
    49   by simp
    50 
    51 lemma ISR: "(ALL x. ISR (Inr x)) & (ALL y. ~ISR (Inl y))"
    52   by simp
    53 
    54 primrec OUTL :: "'a + 'b => 'a" where
    55   "OUTL (Inl x) = x"
    56 
    57 primrec OUTR :: "'a + 'b => 'b" where
    58   "OUTR (Inr x) = x"
    59 
    60 lemma OUTL: "OUTL (Inl x) = x"
    61   by simp
    62 
    63 lemma OUTR: "OUTR (Inr x) = x"
    64   by simp
    65 
    66 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)"
    67   by simp;
    68 
    69 lemma one: "ALL v. v = ()"
    70   by simp;
    71 
    72 lemma option_case_def: "(!u f. option_case u f None = u) & (!u f x. option_case u f (Some x) = f x)"
    73   by simp
    74 
    75 lemma OPTION_MAP_DEF: "(!f x. Option.map f (Some x) = Some (f x)) & (!f. Option.map f None = None)"
    76   by simp
    77 
    78 primrec IS_SOME :: "'a option => bool" where
    79   "IS_SOME (Some x) = True"
    80 | "IS_SOME None = False"
    81 
    82 primrec IS_NONE :: "'a option => bool" where
    83   "IS_NONE (Some x) = False"
    84 | "IS_NONE None = True"
    85 
    86 lemma IS_NONE_DEF: "(!x. IS_NONE (Some x) = False) & (IS_NONE None = True)"
    87   by simp
    88 
    89 lemma IS_SOME_DEF: "(!x. IS_SOME (Some x) = True) & (IS_SOME None = False)"
    90   by simp
    91 
    92 primrec OPTION_JOIN :: "'a option option => 'a option" where
    93   "OPTION_JOIN None = None"
    94 | "OPTION_JOIN (Some x) = x"
    95 
    96 lemma OPTION_JOIN_DEF: "(OPTION_JOIN None = None) & (ALL x. OPTION_JOIN (Some x) = x)"
    97   by simp
    98 
    99 lemma PAIR: "(fst x,snd x) = x"
   100   by simp
   101 
   102 lemma PAIR_MAP: "map_pair f g p = (f (fst p),g (snd p))"
   103   by (simp add: map_pair_def split_def)
   104 
   105 lemma pair_case_def: "split = split"
   106   ..;
   107 
   108 lemma LESS_OR_EQ: "m <= (n::nat) = (m < n | m = n)"
   109   by auto
   110 
   111 definition nat_gt :: "nat => nat => bool" where
   112   "nat_gt == %m n. n < m"
   113 
   114 definition nat_ge :: "nat => nat => bool" where
   115   "nat_ge == %m n. nat_gt m n | m = n"
   116 
   117 lemma [hol4rew]: "nat_gt m n = (n < m)"
   118   by (simp add: nat_gt_def)
   119 
   120 lemma [hol4rew]: "nat_ge m n = (n <= m)"
   121   by (auto simp add: nat_ge_def nat_gt_def)
   122 
   123 lemma GREATER_DEF: "ALL m n. (n < m) = (n < m)"
   124   by simp
   125 
   126 lemma GREATER_OR_EQ: "ALL m n. n <= (m::nat) = (n < m | m = n)"
   127   by auto
   128 
   129 lemma LESS_DEF: "m < n = (? P. (!n. P (Suc n) --> P n) & P m & ~P n)"
   130 proof safe
   131   assume "m < n"
   132   def P == "%n. n <= m"
   133   have "(!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   134   proof (auto simp add: P_def)
   135     assume "n <= m"
   136     from prems
   137     show False
   138       by auto
   139   qed
   140   thus "? P. (!n. P (Suc n) \<longrightarrow> P n) & P m & ~P n"
   141     by auto
   142 next
   143   fix P
   144   assume alln: "!n. P (Suc n) \<longrightarrow> P n"
   145   assume pm: "P m"
   146   assume npn: "~P n"
   147   have "!k q. q + k = m \<longrightarrow> P q"
   148   proof
   149     fix k
   150     show "!q. q + k = m \<longrightarrow> P q"
   151     proof (induct k,simp_all)
   152       show "P m" by fact
   153     next
   154       fix k
   155       assume ind: "!q. q + k = m \<longrightarrow> P q"
   156       show "!q. Suc (q + k) = m \<longrightarrow> P q"
   157       proof (rule+)
   158         fix q
   159         assume "Suc (q + k) = m"
   160         hence "(Suc q) + k = m"
   161           by simp
   162         with ind
   163         have psq: "P (Suc q)"
   164           by simp
   165         from alln
   166         have "P (Suc q) --> P q"
   167           ..
   168         with psq
   169         show "P q"
   170           by simp
   171       qed
   172     qed
   173   qed
   174   hence "!q. q + (m - n) = m \<longrightarrow> P q"
   175     ..
   176   hence hehe: "n + (m - n) = m \<longrightarrow> P n"
   177     ..
   178   show "m < n"
   179   proof (rule classical)
   180     assume "~(m<n)"
   181     hence "n <= m"
   182       by simp
   183     with hehe
   184     have "P n"
   185       by simp
   186     with npn
   187     show "m < n"
   188       ..
   189   qed
   190 qed;
   191 
   192 definition FUNPOW :: "('a => 'a) => nat => 'a => 'a" where
   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   by (simp add: funpow_swap1)
   198 
   199 lemma [hol4rew]: "FUNPOW f n = f ^^ n"
   200   by (simp add: FUNPOW_def)
   201 
   202 lemma ADD: "(!n. (0::nat) + n = n) & (!m n. Suc m + n = Suc (m + n))"
   203   by simp
   204 
   205 lemma MULT: "(!n. (0::nat) * n = 0) & (!m n. Suc m * n = m * n + n)"
   206   by simp
   207 
   208 lemma SUB: "(!m. (0::nat) - m = 0) & (!m n. (Suc m) - n = (if m < n then 0 else Suc (m - n)))"
   209   by (simp) arith
   210 
   211 lemma MAX_DEF: "max (m::nat) n = (if m < n then n else m)"
   212   by (simp add: max_def)
   213 
   214 lemma MIN_DEF: "min (m::nat) n = (if m < n then m else n)"
   215   by (simp add: min_def)
   216 
   217 lemma DIVISION: "(0::nat) < n --> (!k. (k = k div n * n + k mod n) & k mod n < n)"
   218   by simp
   219 
   220 definition ALT_ZERO :: nat where 
   221   "ALT_ZERO == 0"
   222 
   223 definition NUMERAL_BIT1 :: "nat \<Rightarrow> nat" where 
   224   "NUMERAL_BIT1 n == n + (n + Suc 0)"
   225 
   226 definition NUMERAL_BIT2 :: "nat \<Rightarrow> nat" where 
   227   "NUMERAL_BIT2 n == n + (n + Suc (Suc 0))"
   228 
   229 definition NUMERAL :: "nat \<Rightarrow> nat" where 
   230   "NUMERAL x == x"
   231 
   232 lemma [hol4rew]: "NUMERAL ALT_ZERO = 0"
   233   by (simp add: ALT_ZERO_def NUMERAL_def)
   234 
   235 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT1 ALT_ZERO) = 1"
   236   by (simp add: ALT_ZERO_def NUMERAL_BIT1_def NUMERAL_def)
   237 
   238 lemma [hol4rew]: "NUMERAL (NUMERAL_BIT2 ALT_ZERO) = 2"
   239   by (simp add: ALT_ZERO_def NUMERAL_BIT2_def NUMERAL_def)
   240 
   241 lemma EXP: "(!m. m ^ 0 = (1::nat)) & (!m n. m ^ Suc n = m * (m::nat) ^ n)"
   242   by auto
   243 
   244 lemma num_case_def: "(!b f. nat_case b f 0 = b) & (!b f n. nat_case b f (Suc n) = f n)"
   245   by simp;
   246 
   247 lemma divides_def: "(a::nat) dvd b = (? q. b = q * a)"
   248   by (auto simp add: dvd_def);
   249 
   250 lemma list_case_def: "(!v f. list_case v f [] = v) & (!v f a0 a1. list_case v f (a0#a1) = f a0 a1)"
   251   by simp
   252 
   253 primrec list_size :: "('a \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> nat" where
   254   "list_size f [] = 0"
   255 | "list_size f (a0#a1) = 1 + (f a0 + list_size f a1)"
   256 
   257 lemma list_size_def': "(!f. list_size f [] = 0) &
   258          (!f a0 a1. list_size f (a0#a1) = 1 + (f a0 + list_size f a1))"
   259   by simp
   260 
   261 lemma list_case_cong: "! M M' v f. M = M' & (M' = [] \<longrightarrow>  v = v') &
   262            (!a0 a1. (M' = a0#a1) \<longrightarrow> (f a0 a1 = f' a0 a1)) -->
   263            (list_case v f M = list_case v' f' M')"
   264 proof clarify
   265   fix M M' v f
   266   assume "M' = [] \<longrightarrow> v = v'"
   267     and "!a0 a1. M' = a0 # a1 \<longrightarrow> f a0 a1 = f' a0 a1"
   268   show "list_case v f M' = list_case v' f' M'"
   269   proof (rule List.list.case_cong)
   270     show "M' = M'"
   271       ..
   272   next
   273     assume "M' = []"
   274     with prems
   275     show "v = v'"
   276       by auto
   277   next
   278     fix a0 a1
   279     assume "M' = a0 # a1"
   280     with prems
   281     show "f a0 a1 = f' a0 a1"
   282       by auto
   283   qed
   284 qed
   285 
   286 lemma list_Axiom: "ALL f0 f1. EX fn. (fn [] = f0) & (ALL a0 a1. fn (a0#a1) = f1 a0 a1 (fn a1))"
   287 proof safe
   288   fix f0 f1
   289   def fn == "list_rec f0 f1"
   290   have "fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   291     by (simp add: fn_def)
   292   thus "EX fn. fn [] = f0 & (ALL a0 a1. fn (a0 # a1) = f1 a0 a1 (fn a1))"
   293     by auto
   294 qed
   295 
   296 lemma list_Axiom_old: "EX! fn. (fn [] = x) & (ALL h t. fn (h#t) = f (fn t) h t)"
   297 proof safe
   298   def fn == "list_rec x (%h t r. f r h t)"
   299   have "fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   300     by (simp add: fn_def)
   301   thus "EX fn. fn [] = x & (ALL h t. fn (h # t) = f (fn t) h t)"
   302     by auto
   303 next
   304   fix fn1 fn2
   305   assume "ALL h t. fn1 (h # t) = f (fn1 t) h t"
   306   assume "ALL h t. fn2 (h # t) = f (fn2 t) h t"
   307   assume "fn2 [] = fn1 []"
   308   show "fn1 = fn2"
   309   proof
   310     fix xs
   311     show "fn1 xs = fn2 xs"
   312       by (induct xs,simp_all add: prems) 
   313   qed
   314 qed
   315 
   316 lemma NULL_DEF: "(List.null [] = True) & (!h t. List.null (h # t) = False)"
   317   by (simp add: null_def)
   318 
   319 definition sum :: "nat list \<Rightarrow> nat" where
   320   "sum l == foldr (op +) l 0"
   321 
   322 lemma SUM: "(sum [] = 0) & (!h t. sum (h#t) = h + sum t)"
   323   by (simp add: sum_def)
   324 
   325 lemma APPEND: "(!l. [] @ l = l) & (!l1 l2 h. (h#l1) @ l2 = h# l1 @ l2)"
   326   by simp
   327 
   328 lemma FLAT: "(concat [] = []) & (!h t. concat (h#t) = h @ (concat t))"
   329   by simp
   330 
   331 lemma LENGTH: "(length [] = 0) & (!h t. length (h#t) = Suc (length t))"
   332   by simp
   333 
   334 lemma MAP: "(!f. map f [] = []) & (!f h t. map f (h#t) = f h#map f t)"
   335   by simp
   336 
   337 lemma MEM: "(!x. List.member [] x = False) & (!x h t. List.member (h#t) x = ((x = h) | List.member t x))"
   338   by (simp add: member_def)
   339 
   340 lemma FILTER: "(!P. filter P [] = []) & (!P h t.
   341            filter P (h#t) = (if P h then h#filter P t else filter P t))"
   342   by simp
   343 
   344 lemma REPLICATE: "(ALL x. replicate 0 x = []) &
   345   (ALL n x. replicate (Suc n) x = x # replicate n x)"
   346   by simp
   347 
   348 definition FOLDR :: "[['a,'b]\<Rightarrow>'b,'b,'a list] \<Rightarrow> 'b" where
   349   "FOLDR f e l == foldr f l e"
   350 
   351 lemma [hol4rew]: "FOLDR f e l = foldr f l e"
   352   by (simp add: FOLDR_def)
   353 
   354 lemma FOLDR: "(!f e. foldr f [] e = e) & (!f e x l. foldr f (x#l) e = f x (foldr f l e))"
   355   by simp
   356 
   357 lemma FOLDL: "(!f e. foldl f e [] = e) & (!f e x l. foldl f e (x#l) = foldl f (f e x) l)"
   358   by simp
   359 
   360 lemma EVERY_DEF: "(!P. list_all P [] = True) & (!P h t. list_all P (h#t) = (P h & list_all P t))"
   361   by simp
   362 
   363 lemma list_exists_DEF: "(!P. list_ex P [] = False) & (!P h t. list_ex P (h#t) = (P h | list_ex P t))"
   364   by simp
   365 
   366 primrec map2 :: "[['a,'b]\<Rightarrow>'c,'a list,'b list] \<Rightarrow> 'c list" where
   367   map2_Nil: "map2 f [] l2 = []"
   368 | map2_Cons: "map2 f (x#xs) l2 = f x (hd l2) # map2 f xs (tl l2)"
   369 
   370 lemma MAP2: "(!f. map2 f [] [] = []) & (!f h1 t1 h2 t2. map2 f (h1#t1) (h2#t2) = f h1 h2#map2 f t1 t2)"
   371   by simp
   372 
   373 lemma list_INDUCT: "\<lbrakk> P [] ; !t. P t \<longrightarrow> (!h. P (h#t)) \<rbrakk> \<Longrightarrow> !l. P l"
   374 proof
   375   fix l
   376   assume "P []"
   377   assume allt: "!t. P t \<longrightarrow> (!h. P (h # t))"
   378   show "P l"
   379   proof (induct l)
   380     show "P []" by fact
   381   next
   382     fix h t
   383     assume "P t"
   384     with allt
   385     have "!h. P (h # t)"
   386       by auto
   387     thus "P (h # t)"
   388       ..
   389   qed
   390 qed
   391 
   392 lemma list_CASES: "(l = []) | (? t h. l = h#t)"
   393   by (induct l,auto)
   394 
   395 definition ZIP :: "'a list * 'b list \<Rightarrow> ('a * 'b) list" where
   396   "ZIP == %(a,b). zip a b"
   397 
   398 lemma ZIP: "(zip [] [] = []) &
   399   (!x1 l1 x2 l2. zip (x1#l1) (x2#l2) = (x1,x2)#zip l1 l2)"
   400   by simp
   401 
   402 lemma [hol4rew]: "ZIP (a,b) = zip a b"
   403   by (simp add: ZIP_def)
   404 
   405 primrec unzip :: "('a * 'b) list \<Rightarrow> 'a list * 'b list" where
   406   unzip_Nil: "unzip [] = ([],[])"
   407 | unzip_Cons: "unzip (xy#xys) = (let zs = unzip xys in (fst xy # fst zs,snd xy # snd zs))"
   408 
   409 lemma UNZIP: "(unzip [] = ([],[])) &
   410          (!x l. unzip (x#l) = (fst x#fst (unzip l),snd x#snd (unzip l)))"
   411   by (simp add: Let_def)
   412 
   413 lemma REVERSE: "(rev [] = []) & (!h t. rev (h#t) = (rev t) @ [h])"
   414   by simp;
   415 
   416 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)"
   417 proof safe
   418   fix x z
   419   assume allx: "ALL x. P x \<longrightarrow> 0 < x"
   420   assume px: "P x"
   421   assume allx': "ALL x. P x \<longrightarrow> x < z"
   422   have "EX s. ALL y. (EX x : Collect P. y < x) = (y < s)"
   423   proof (rule posreal_complete)
   424     show "ALL x : Collect P. 0 < x"
   425     proof safe
   426       fix x
   427       assume "P x"
   428       from allx
   429       have "P x \<longrightarrow> 0 < x"
   430         ..
   431       thus "0 < x"
   432         by (simp add: prems)
   433     qed
   434   next
   435     from px
   436     show "EX x. x : Collect P"
   437       by auto
   438   next
   439     from allx'
   440     show "EX y. ALL x : Collect P. x < y"
   441       apply simp
   442       ..
   443   qed
   444   thus "EX s. ALL y. (EX x. P x & y < x) = (y < s)"
   445     by simp
   446 qed
   447 
   448 lemma REAL_10: "~((1::real) = 0)"
   449   by simp
   450 
   451 lemma REAL_ADD_ASSOC: "(x::real) + (y + z) = x + y + z"
   452   by simp
   453 
   454 lemma REAL_MUL_ASSOC: "(x::real) * (y * z) = x * y * z"
   455   by simp
   456 
   457 lemma REAL_ADD_LINV:  "-x + x = (0::real)"
   458   by simp
   459 
   460 lemma REAL_MUL_LINV: "x ~= (0::real) ==> inverse x * x = 1"
   461   by simp
   462 
   463 lemma REAL_LT_TOTAL: "((x::real) = y) | x < y | y < x"
   464   by auto;
   465 
   466 lemma [hol4rew]: "real (0::nat) = 0"
   467   by simp
   468 
   469 lemma [hol4rew]: "real (1::nat) = 1"
   470   by simp
   471 
   472 lemma [hol4rew]: "real (2::nat) = 2"
   473   by simp
   474 
   475 lemma real_lte: "((x::real) <= y) = (~(y < x))"
   476   by auto
   477 
   478 lemma real_of_num: "((0::real) = 0) & (!n. real (Suc n) = real n + 1)"
   479   by (simp add: real_of_nat_Suc)
   480 
   481 lemma abs: "abs (x::real) = (if 0 <= x then x else -x)"
   482   by (simp add: abs_if)
   483 
   484 lemma pow: "(!x::real. x ^ 0 = 1) & (!x::real. ALL n. x ^ (Suc n) = x * x ^ n)"
   485   by simp
   486 
   487 definition real_gt :: "real => real => bool" where 
   488   "real_gt == %x y. y < x"
   489 
   490 lemma [hol4rew]: "real_gt x y = (y < x)"
   491   by (simp add: real_gt_def)
   492 
   493 lemma real_gt: "ALL x (y::real). (y < x) = (y < x)"
   494   by simp
   495 
   496 definition real_ge :: "real => real => bool" where
   497   "real_ge x y == y <= x"
   498 
   499 lemma [hol4rew]: "real_ge x y = (y <= x)"
   500   by (simp add: real_ge_def)
   501 
   502 lemma real_ge: "ALL x y. (y <= x) = (y <= x)"
   503   by simp
   504 
   505 end