src/Sequents/LK/Nat.thy
author wenzelm
Sat Feb 01 18:00:28 2014 +0100 (2014-02-01)
changeset 55229 08f2ebb65078
parent 55228 901a6696cdd8
child 55230 cb5ef74b32f9
permissions -rw-r--r--
simplified sessions;
     1 (*  Title:      Sequents/LK/Nat.thy
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Copyright   1999  University of Cambridge
     4 *)
     5 
     6 header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}
     7 
     8 theory Nat
     9 imports "../LK"
    10 begin
    11 
    12 typedecl nat
    13 arities nat :: "term"
    14 
    15 axiomatization
    16   Zero :: nat      ("0") and
    17   Suc :: "nat=>nat" and
    18   rec :: "[nat, 'a, [nat,'a]=>'a] => 'a"
    19 where
    20   induct:  "[| $H |- $E, P(0), $F;
    21               !!x. $H, P(x) |- $E, P(Suc(x)), $F |] ==> $H |- $E, P(n), $F" and
    22 
    23   Suc_inject:  "|- Suc(m)=Suc(n) --> m=n" and
    24   Suc_neq_0:   "|- Suc(m) ~= 0" and
    25   rec_0:       "|- rec(0,a,f) = a" and
    26   rec_Suc:     "|- rec(Suc(m), a, f) = f(m, rec(m,a,f))"
    27 
    28 definition add :: "[nat, nat] => nat"  (infixl "+" 60)
    29   where "m + n == rec(m, n, %x y. Suc(y))"
    30 
    31 
    32 declare Suc_neq_0 [simp]
    33 
    34 lemma Suc_inject_rule: "$H, $G, m = n |- $E \<Longrightarrow> $H, Suc(m) = Suc(n), $G |- $E"
    35   by (rule L_of_imp [OF Suc_inject])
    36 
    37 lemma Suc_n_not_n: "|- Suc(k) ~= k"
    38   apply (rule_tac n = k in induct)
    39   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms Suc_neq_0}) 1 *})
    40   apply (fast add!: Suc_inject_rule)
    41   done
    42 
    43 lemma add_0: "|- 0+n = n"
    44   apply (unfold add_def)
    45   apply (rule rec_0)
    46   done
    47 
    48 lemma add_Suc: "|- Suc(m)+n = Suc(m+n)"
    49   apply (unfold add_def)
    50   apply (rule rec_Suc)
    51   done
    52 
    53 declare add_0 [simp] add_Suc [simp]
    54 
    55 lemma add_assoc: "|- (k+m)+n = k+(m+n)"
    56   apply (rule_tac n = "k" in induct)
    57   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
    58   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
    59   done
    60 
    61 lemma add_0_right: "|- m+0 = m"
    62   apply (rule_tac n = "m" in induct)
    63   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
    64   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
    65   done
    66 
    67 lemma add_Suc_right: "|- m+Suc(n) = Suc(m+n)"
    68   apply (rule_tac n = "m" in induct)
    69   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
    70   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
    71   done
    72 
    73 lemma "(!!n. |- f(Suc(n)) = Suc(f(n))) ==> |- f(i+j) = i+f(j)"
    74   apply (rule_tac n = "i" in induct)
    75   apply (tactic {* simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_0}) 1 *})
    76   apply (tactic {* asm_simp_tac (put_simpset LK_ss @{context} addsimps @{thms add_Suc}) 1 *})
    77   done
    78 
    79 end