diff options
Diffstat (limited to 'theories/ZArith')
| -rw-r--r-- | theories/ZArith/Wf_Z.v | 8 | ||||
| -rw-r--r-- | theories/ZArith/Zcomplements.v | 9 |
2 files changed, 8 insertions, 9 deletions
diff --git a/theories/ZArith/Wf_Z.v b/theories/ZArith/Wf_Z.v index 3935e1248..f1bfb027f 100644 --- a/theories/ZArith/Wf_Z.v +++ b/theories/ZArith/Wf_Z.v @@ -151,9 +151,7 @@ Section Efficient_Rec. forall P:Z -> Prop, (forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> 0 <= x -> P x) -> forall x:Z, 0 <= x -> P x. - Proof. - exact Zlt_0_rec. - Qed. + Proof. intros; now apply Zlt_0_rec. Qed. (** Obsolete version of [Z.lt] induction principle on non-negative numbers *) @@ -170,7 +168,7 @@ Section Efficient_Rec. (forall x:Z, (forall y:Z, 0 <= y < x -> P y) -> P x) -> forall x:Z, 0 <= x -> P x. Proof. - exact Z_lt_rec. + intros; now apply Z_lt_rec. Qed. (** An even more general induction principle using [Z.lt]. *) @@ -196,7 +194,7 @@ Section Efficient_Rec. (forall x:Z, (forall y:Z, z <= y < x -> P y) -> z <= x -> P x) -> forall x:Z, z <= x -> P x. Proof. - exact Zlt_lower_bound_rec. + intros; now apply Zlt_lower_bound_rec with z. Qed. End Efficient_Rec. diff --git a/theories/ZArith/Zcomplements.v b/theories/ZArith/Zcomplements.v index b4163ef99..a5e710504 100644 --- a/theories/ZArith/Zcomplements.v +++ b/theories/ZArith/Zcomplements.v @@ -53,10 +53,11 @@ Theorem Z_lt_abs_rec : forall n:Z, P n. Proof. intros P HP p. - set (Q := fun z => 0 <= z -> P z * P (- z)) in *. - cut (Q (Z.abs p)); [ intros | apply (Z_lt_rec Q); auto with zarith ]. - elim (Zabs_dec p); intro eq; rewrite eq; elim H; auto with zarith. - unfold Q; clear Q; intros. + set (Q := fun z => 0 <= z -> P z * P (- z)). + cut (Q (Z.abs p)); [ intros H | apply (Z_lt_rec Q); auto with zarith ]. + elim (Zabs_dec p); intro eq; rewrite eq; + elim H; auto with zarith. + intros x H; subst Q. split; apply HP. rewrite Z.abs_eq; auto; intros. elim (H (Z.abs m)); intros; auto with zarith. |