diff options
| author | glondu | 2009-09-17 15:58:14 +0000 |
|---|---|---|
| committer | glondu | 2009-09-17 15:58:14 +0000 |
| commit | 61ccbc81a2f3b4662ed4a2bad9d07d2003dda3a2 (patch) | |
| tree | 961cc88c714aa91a0276ea9fbf8bc53b2b9d5c28 /theories/Numbers/Integer/SpecViaZ | |
| parent | 6d3fbdf36c6a47b49c2a4b16f498972c93c07574 (diff) | |
Delete trailing whitespaces in all *.{v,ml*} files
git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@12337 85f007b7-540e-0410-9357-904b9bb8a0f7
Diffstat (limited to 'theories/Numbers/Integer/SpecViaZ')
| -rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSig.v | 8 | ||||
| -rw-r--r-- | theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v | 6 |
2 files changed, 7 insertions, 7 deletions
diff --git a/theories/Numbers/Integer/SpecViaZ/ZSig.v b/theories/Numbers/Integer/SpecViaZ/ZSig.v index 4e45939831..00e292db0f 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSig.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSig.v @@ -58,7 +58,7 @@ Module Type ZType. Parameter spec_eq_bool: forall x y, if eq_bool x y then [x] = [y] else [x] <> [y]. - + Parameter succ : t -> t. Parameter spec_succ: forall n, [succ n] = [n] + 1. @@ -93,21 +93,21 @@ Module Type ZType. Parameter sqrt : t -> t. - Parameter spec_sqrt: forall x, 0 <= [x] -> + Parameter spec_sqrt: forall x, 0 <= [x] -> [sqrt x] ^ 2 <= [x] < ([sqrt x] + 1) ^ 2. Parameter div_eucl : t -> t -> t * t. Parameter spec_div_eucl: forall x y, [y] <> 0 -> let (q,r) := div_eucl x y in ([q], [r]) = Zdiv_eucl [x] [y]. - + Parameter div : t -> t -> t. Parameter spec_div: forall x y, [y] <> 0 -> [div x y] = [x] / [y]. Parameter modulo : t -> t -> t. - Parameter spec_modulo: forall x y, [y] <> 0 -> + Parameter spec_modulo: forall x y, [y] <> 0 -> [modulo x y] = [x] mod [y]. Parameter gcd : t -> t -> t. diff --git a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v index 4d1054553f..030c589ff9 100644 --- a/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v +++ b/theories/Numbers/Integer/SpecViaZ/ZSigZAxioms.v @@ -27,7 +27,7 @@ Infix "-" := Z.sub : IntScope. Infix "*" := Z.mul : IntScope. Notation "- x" := (Z.opp x) : IntScope. -Hint Rewrite +Hint Rewrite Z.spec_0 Z.spec_1 Z.spec_add Z.spec_sub Z.spec_pred Z.spec_succ Z.spec_mul Z.spec_opp Z.spec_of_Z : Zspec. @@ -91,7 +91,7 @@ Section Induction. Variable A : Z.t -> Prop. Hypothesis A_wd : predicate_wd Z.eq A. Hypothesis A0 : A 0. -Hypothesis AS : forall n, A n <-> A (Z.succ n). +Hypothesis AS : forall n, A n <-> A (Z.succ n). Add Morphism A with signature Z.eq ==> iff as A_morph. Proof. apply A_wd. Qed. @@ -214,7 +214,7 @@ Proof. Qed. Add Morphism Z.compare with signature Z.eq ==> Z.eq ==> (@eq comparison) as compare_wd. -Proof. +Proof. intros x x' Hx y y' Hy. rewrite 2 spec_compare_alt; unfold Z.eq in *; rewrite Hx, Hy; intuition. Qed. |
