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author | coq <coq@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-08-02 17:17:42 +0000 |
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committer | coq <coq@85f007b7-540e-0410-9357-904b9bb8a0f7> | 2002-08-02 17:17:42 +0000 |
commit | 12965209478bd99dfbe57f07d5b525e51b903f22 (patch) | |
tree | 36a7f5e4802cd321caf02fed0be8349100be09fb /test-suite/modules/Nat.v | |
parent | 8b26fd6ba739d4f49fae99ed764b086022e44b50 (diff) |
Modules dans COQ\!\!\!\!
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@2957 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'test-suite/modules/Nat.v')
-rw-r--r-- | test-suite/modules/Nat.v | 68 |
1 files changed, 68 insertions, 0 deletions
diff --git a/test-suite/modules/Nat.v b/test-suite/modules/Nat.v new file mode 100644 index 000000000..d0ad6a5e2 --- /dev/null +++ b/test-suite/modules/Nat.v @@ -0,0 +1,68 @@ +Definition T:=nat. + +Definition le:=Peano.le. + +Hints Unfold le. + +Lemma le_refl:(n:nat)(le n n). + Auto. +Save. + +Lemma le_trans:(n,m,k:nat)(le n m) -> (le m k) -> (le n k). + Unfold le. + Intros. + Generalize H. + Clear H. + Elim H0. + Auto. + Auto. +Save. + +Lemma le_mono_S : (n,m:nat)(le n m) -> (le (S n) (S m)). + Unfold le. + NewInduction 1. + Auto. + Auto. +Save. + +Hints Resolve le_mono_S. + +Lemma le_pred_mono : (n,m:nat) (le n m) -> (le (pred n) (pred m)). + Unfold le. + Induction 1. + Auto. + Intro. + Case m0. + Auto. + Intro. + Simpl. + Auto. +Save. + +Hints Resolve le_pred_mono. + +Lemma le_S_mono : (m,n:nat)(le (S n) (S m)) -> (le n m). + Intros. + Change (le (pred (S n)) (pred (S m))). + Auto. +Save. + +Hints Resolve le_S_mono. + +Lemma le_antis:(n,m:nat)(le n m) -> (le m n) -> n=m. + NewInduction n. + Intros. + Inversion H0. + Reflexivity. + + Intros. + Inversion H. + Auto. + Rewrite (IHn m0). + Auto. + Rewrite <- H2 in H. + Auto. + Rewrite <- H2 in H0. + Auto. +Save. + |