diff options
Diffstat (limited to 'theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v')
| -rw-r--r-- | theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v | 16 |
1 files changed, 8 insertions, 8 deletions
diff --git a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v index 115f78be0..54d7aec52 100644 --- a/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v +++ b/theories/Numbers/Natural/SpecViaZ/NSigNAxioms.v @@ -35,9 +35,9 @@ Definition NZeq := N.eq. Definition NZ0 := N.zero. Definition NZsucc := N.succ. Definition NZpred := N.pred. -Definition NZplus := N.add. +Definition NZadd := N.add. Definition NZminus := N.sub. -Definition NZtimes := N.mul. +Definition NZmul := N.mul. Theorem NZeq_equiv : equiv N.t N.eq. Proof. @@ -64,7 +64,7 @@ rewrite 2 N.spec_pred0; congruence. rewrite 2 N.spec_pred; f_equal; auto; try omega. Qed. -Add Morphism NZplus with signature N.eq ==> N.eq ==> N.eq as NZplus_wd. +Add Morphism NZadd with signature N.eq ==> N.eq ==> N.eq as NZadd_wd. Proof. unfold N.eq; intros; rewrite 2 N.spec_add; f_equal; auto. Qed. @@ -77,7 +77,7 @@ rewrite 2 N.spec_sub0; f_equal; congruence. rewrite 2 N.spec_sub; f_equal; congruence. Qed. -Add Morphism NZtimes with signature N.eq ==> N.eq ==> N.eq as NZtimes_wd. +Add Morphism NZmul with signature N.eq ==> N.eq ==> N.eq as NZmul_wd. Proof. unfold N.eq; intros; rewrite 2 N.spec_mul; f_equal; auto. Qed. @@ -137,12 +137,12 @@ Qed. End Induction. -Theorem NZplus_0_l : forall n, 0 + n == n. +Theorem NZadd_0_l : forall n, 0 + n == n. Proof. intros; red; rewrite N.spec_add, N.spec_0; auto with zarith. Qed. -Theorem NZplus_succ_l : forall n m, (N.succ n) + m == N.succ (n + m). +Theorem NZadd_succ_l : forall n m, (N.succ n) + m == N.succ (n + m). Proof. intros; red; rewrite N.spec_add, 2 N.spec_succ, N.spec_add; auto with zarith. Qed. @@ -169,13 +169,13 @@ rewrite N.spec_pred, N.spec_sub; auto with zarith. rewrite N.spec_sub; auto with zarith. Qed. -Theorem NZtimes_0_l : forall n, 0 * n == 0. +Theorem NZmul_0_l : forall n, 0 * n == 0. Proof. intros; red. rewrite N.spec_mul, N.spec_0; auto with zarith. Qed. -Theorem NZtimes_succ_l : forall n m, (N.succ n) * m == n * m + m. +Theorem NZmul_succ_l : forall n m, (N.succ n) * m == n * m + m. Proof. intros; red. rewrite N.spec_add, 2 N.spec_mul, N.spec_succ; ring. |