Theory Nat2

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theory Nat2
imports FOL
uses [Nat2.ML]
begin
(*  Title:      FOL/ex/nat2.thy
    ID:         $Id: Nat2.thy,v 1.5 2005/09/03 15:15:52 wenzelm Exp $
    Author:     Tobias Nipkow
    Copyright   1994  University of Cambridge
*)

header {* Theory for examples of simplification and induction on the natural numbers *}

theory Nat2
imports FOL
begin

typedecl nat
arities nat :: "term"

consts
  succ :: "nat => nat"
  pred :: "nat => nat"
  0 :: nat    ("0")
  add :: "[nat,nat] => nat"    (infixr "+" 90)
  lt :: "[nat,nat] => o"    (infixr "<" 70)
  leq :: "[nat,nat] => o"    (infixr "<=" 70)

axioms
 pred_0:         "pred(0) = 0"
 pred_succ:      "pred(succ(m)) = m"

 plus_0:         "0+n = n"
 plus_succ:      "succ(m)+n = succ(m+n)"

 nat_distinct1:  "~ 0=succ(n)"
 nat_distinct2:  "~ succ(m)=0"
 succ_inject:    "succ(m)=succ(n) <-> m=n"

 leq_0:          "0 <= n"
 leq_succ_succ:  "succ(m)<=succ(n) <-> m<=n"
 leq_succ_0:     "~ succ(m) <= 0"

 lt_0_succ:      "0 < succ(n)"
 lt_succ_succ:   "succ(m)<succ(n) <-> m<n"
 lt_0:           "~ m < 0"

 nat_ind:        "[| P(0); ALL n. P(n)-->P(succ(n)) |] ==> All(P)"

ML {* use_legacy_bindings (the_context ()) *}

end

theorem nat_exh: 

  [| P(0); !!x. P(succ(x)) |] ==> All(P)

theorem plus_0_right: 

  m + 0 = m

theorem plus_succ_right: 

  m + succ(n) = succ(m + n)

theorem le_imp_le_succ: 

  m <= n ==> m <= succ(n)

theorem le_plus: 

  m <= n --> m <= n + k

theorem succ_le: 

  succ(m) <= n --> m <= n

theorem not_less: 

  ¬ m < n <-> n <= m

theorem le_imp_not_less: 

  n <= m --> ¬ m < n

theorem not_le: 

  m < n --> ¬ n <= m

theorem plus_le: 

  m + k <= n --> m <= n

theorem not0: 

  [| m ≠ 0; m <= n |] ==> n ≠ 0

theorem plus_le_plus: 

  a <= a' ∧ b <= b' --> a + b <= a' + b'

theorem le_trans: 

  i <= j --> j <= k --> i <= k

theorem less_le_trans: 

  i < j --> j <= k --> i < k

theorem succ_le2: 

  succ(i) <= j <-> i < j

theorem less_succ: 

  i < succ(j) <-> i = j ∨ i < j