src/HOL/Import/HOLLightCompat.thy
author haftmann
Mon Mar 01 13:40:23 2010 +0100 (2010-03-01)
changeset 35416 d8d7d1b785af
parent 19064 bf19cc5a7899
child 41589 bbd861837ebc
permissions -rw-r--r--
replaced a couple of constsdefs by definitions (also some old primrecs by modern ones)
     1 (*  Title:      HOL/Import/HOLLightCompat.thy
     2     ID:         $Id$
     3     Author:     Steven Obua and Sebastian Skalberg (TU Muenchen)
     4 *)
     5 
     6 theory HOLLightCompat imports HOL4Setup HOL4Compat Divides Primes Real begin
     7 
     8 lemma light_imp_def: "(t1 --> t2) = ((t1 & t2) = t1)"
     9   by auto;
    10 
    11 lemma comb_rule: "[| P1 = P2 ; Q1 = Q2 |] ==> P1 Q1 = P2 Q2"
    12   by simp
    13 
    14 lemma light_and_def: "(t1 & t2) = ((%f. f t1 t2::bool) = (%f. f True True))"
    15 proof auto
    16   assume a: "(%f. f t1 t2::bool) = (%f. f True True)"
    17   have b: "(%(x::bool) (y::bool). x) = (%x y. x)" ..
    18   with a
    19   have "t1 = True"
    20     by (rule comb_rule)
    21   thus t1
    22     by simp
    23 next
    24   assume a: "(%f. f t1 t2::bool) = (%f. f True True)"
    25   have b: "(%(x::bool) (y::bool). y) = (%x y. y)" ..
    26   with a
    27   have "t2 = True"
    28     by (rule comb_rule)
    29   thus t2
    30     by simp
    31 qed
    32 
    33 definition Pred :: "nat \<Rightarrow> nat" where
    34    "Pred n \<equiv> n - (Suc 0)"
    35 
    36 lemma Pred_altdef: "Pred = (SOME PRE. PRE 0 = 0 & (ALL n. PRE (Suc n) = n))"
    37   apply (rule some_equality[symmetric])
    38   apply (simp add: Pred_def)
    39   apply (rule ext)
    40   apply (induct_tac x)
    41   apply (auto simp add: Pred_def)
    42   done
    43 
    44 lemma NUMERAL_rew[hol4rew]: "NUMERAL x = x" by (simp add: NUMERAL_def)
    45 
    46 lemma REP_ABS_PAIR: "\<forall> x y. Rep_Prod (Abs_Prod (Pair_Rep x y)) = Pair_Rep x y"
    47   apply (subst Abs_Prod_inverse)
    48   apply (auto simp add: Prod_def)
    49   done
    50 
    51 lemma fst_altdef: "fst = (%p. SOME x. EX y. p = (x, y))"
    52   apply (rule ext, rule someI2)
    53   apply (auto intro: fst_conv[symmetric])
    54   done
    55 
    56 lemma snd_altdef: "snd = (%p. SOME x. EX y. p = (y, x))"
    57   apply (rule ext, rule someI2)
    58   apply (auto intro: snd_conv[symmetric])
    59   done
    60 
    61 lemma add_altdef: "op + = (SOME add. (ALL n. add 0 n = n) & (ALL m n. add (Suc m) n = Suc (add m n)))"
    62   apply (rule some_equality[symmetric])
    63   apply auto
    64   apply (rule ext)+
    65   apply (induct_tac x)
    66   apply auto
    67   done
    68 
    69 lemma mult_altdef: "op * = (SOME mult. (ALL n. mult 0 n = 0) & (ALL m n. mult (Suc m) n = mult m n + n))"
    70   apply (rule some_equality[symmetric])
    71   apply auto
    72   apply (rule ext)+
    73   apply (induct_tac x)
    74   apply auto
    75   done
    76 
    77 lemma sub_altdef: "op - = (SOME sub. (ALL m. sub m 0 = m) & (ALL m n. sub m (Suc n) = Pred (sub m n)))"
    78   apply (simp add: Pred_def)
    79   apply (rule some_equality[symmetric])
    80   apply auto
    81   apply (rule ext)+
    82   apply (induct_tac xa)
    83   apply auto
    84   done
    85 
    86 definition NUMERAL_BIT0 :: "nat \<Rightarrow> nat" where
    87   "NUMERAL_BIT0 n \<equiv> n + n"
    88 
    89 lemma NUMERAL_BIT1_altdef: "NUMERAL_BIT1 n = Suc (n + n)"
    90   by (simp add: NUMERAL_BIT1_def)
    91 
    92 consts
    93   sumlift :: "('a \<Rightarrow> 'c) \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> (('a + 'b) \<Rightarrow> 'c)"
    94 
    95 primrec
    96   "sumlift f g (Inl a) = f a"
    97   "sumlift f g (Inr b) = g b"
    98   
    99 lemma sum_Recursion: "\<exists> f. (\<forall> a. f (Inl a) = Inl' a) \<and> (\<forall> b. f (Inr b) = Inr' b)"
   100   apply (rule exI[where x="sumlift Inl' Inr'"])
   101   apply auto
   102   done
   103 
   104 end