diff options
Diffstat (limited to 'theories/Numbers/Integer')
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZBase.v | 6 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZDomain.v | 2 | ||||
| -rw-r--r-- | theories/Numbers/Integer/Abstract/ZMulOrder.v | 4 | ||||
| -rw-r--r-- | theories/Numbers/Integer/NatPairs/ZNatPairs.v | 4 |
4 files changed, 8 insertions, 8 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZBase.v b/theories/Numbers/Integer/Abstract/ZBase.v index d175c358c..648cde197 100644 --- a/theories/Numbers/Integer/Abstract/ZBase.v +++ b/theories/Numbers/Integer/Abstract/ZBase.v @@ -36,14 +36,14 @@ Proof NZpred_succ. Theorem Zeq_refl : forall n : Z, n == n. Proof (proj1 NZeq_equiv). -Theorem Zeq_symm : forall n m : Z, n == m -> m == n. +Theorem Zeq_sym : forall n m : Z, n == m -> m == n. Proof (proj2 (proj2 NZeq_equiv)). Theorem Zeq_trans : forall n m p : Z, n == m -> m == p -> n == p. Proof (proj1 (proj2 NZeq_equiv)). -Theorem Zneq_symm : forall n m : Z, n ~= m -> m ~= n. -Proof NZneq_symm. +Theorem Zneq_sym : forall n m : Z, n ~= m -> m ~= n. +Proof NZneq_sym. Theorem Zsucc_inj : forall n1 n2 : Z, S n1 == S n2 -> n1 == n2. Proof NZsucc_inj. diff --git a/theories/Numbers/Integer/Abstract/ZDomain.v b/theories/Numbers/Integer/Abstract/ZDomain.v index ce3ca21c2..4d927cb3b 100644 --- a/theories/Numbers/Integer/Abstract/ZDomain.v +++ b/theories/Numbers/Integer/Abstract/ZDomain.v @@ -49,7 +49,7 @@ assert (x == y); [rewrite Exx'; now rewrite Eyy' | rewrite <- H2; assert (H3 : e x y); [now apply -> eq_equiv_e | now inversion H3]]]. Qed. -Theorem neq_symm : forall n m, n # m -> m # n. +Theorem neq_sym : forall n m, n # m -> m # n. Proof. intros n m H1 H2; symmetry in H2; false_hyp H2 H1. Qed. diff --git a/theories/Numbers/Integer/Abstract/ZMulOrder.v b/theories/Numbers/Integer/Abstract/ZMulOrder.v index 46a8a38af..ee4ea3c72 100644 --- a/theories/Numbers/Integer/Abstract/ZMulOrder.v +++ b/theories/Numbers/Integer/Abstract/ZMulOrder.v @@ -173,7 +173,7 @@ Notation Zmul_neg := Zlt_mul_0 (only parsing). Theorem Zle_0_mul : forall n m : Z, 0 <= n * m -> 0 <= n /\ 0 <= m \/ n <= 0 /\ m <= 0. Proof. -assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_symm). +assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_sym). intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. rewrite Zlt_0_mul, Zeq_mul_0. pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. @@ -184,7 +184,7 @@ Notation Zmul_nonneg := Zle_0_mul (only parsing). Theorem Zle_mul_0 : forall n m : Z, n * m <= 0 -> 0 <= n /\ m <= 0 \/ n <= 0 /\ 0 <= m. Proof. -assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_symm). +assert (R : forall n : Z, 0 == n <-> n == 0) by (intros; split; apply Zeq_sym). intros n m. repeat rewrite Zlt_eq_cases. repeat rewrite R. rewrite Zlt_mul_0, Zeq_mul_0. pose proof (Zlt_trichotomy n 0); pose proof (Zlt_trichotomy m 0). tauto. diff --git a/theories/Numbers/Integer/NatPairs/ZNatPairs.v b/theories/Numbers/Integer/NatPairs/ZNatPairs.v index aa027103f..381b9baf6 100644 --- a/theories/Numbers/Integer/NatPairs/ZNatPairs.v +++ b/theories/Numbers/Integer/NatPairs/ZNatPairs.v @@ -110,7 +110,7 @@ Proof. unfold reflexive, Zeq. reflexivity. Qed. -Theorem ZE_symm : symmetric Z Zeq. +Theorem ZE_sym : symmetric Z Zeq. Proof. unfold symmetric, Zeq; now symmetry. Qed. @@ -127,7 +127,7 @@ Qed. Theorem NZeq_equiv : equiv Z Zeq. Proof. -unfold equiv; repeat split; [apply ZE_refl | apply ZE_trans | apply ZE_symm]. +unfold equiv; repeat split; [apply ZE_refl | apply ZE_trans | apply ZE_sym]. Qed. Add Relation Z Zeq |