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Diffstat (limited to 'theories/Arith/Peano_dec.v')
| -rwxr-xr-x | theories/Arith/Peano_dec.v | 26 |
1 files changed, 12 insertions, 14 deletions
diff --git a/theories/Arith/Peano_dec.v b/theories/Arith/Peano_dec.v index 96a8523f9f..4d657d0602 100755 --- a/theories/Arith/Peano_dec.v +++ b/theories/Arith/Peano_dec.v @@ -8,29 +8,27 @@ (*i $Id$ i*) -Require Decidable. +Require Import Decidable. -V7only [Import nat_scope.]. Open Local Scope nat_scope. -Implicit Variables Type m,n,x,y:nat. +Implicit Types m n x y : nat. -Theorem O_or_S : (n:nat)({m:nat|(S m)=n})+{O=n}. +Theorem O_or_S : forall n, {m : nat | S m = n} + {0 = n}. Proof. -NewInduction n. -Auto. -Left; Exists n; Auto. +induction n. +auto. +left; exists n; auto. Defined. -Theorem eq_nat_dec : (n,m:nat){n=m}+{~(n=m)}. +Theorem eq_nat_dec : forall n m, {n = m} + {n <> m}. Proof. -NewInduction n; NewInduction m; Auto. -Elim (IHn m); Auto. +induction n; induction m; auto. +elim (IHn m); auto. Defined. -Hints Resolve O_or_S eq_nat_dec : arith. +Hint Resolve O_or_S eq_nat_dec: arith. -Theorem dec_eq_nat:(x,y:nat)(decidable (x=y)). -Intros x y; Unfold decidable; Elim (eq_nat_dec x y); Auto with arith. +Theorem dec_eq_nat : forall n m, decidable (n = m). +intros x y; unfold decidable in |- *; elim (eq_nat_dec x y); auto with arith. Defined. - |