diff options
Diffstat (limited to 'theories')
-rw-r--r-- | theories/FSets/FMapPositive.v | 8 | ||||
-rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivEucl.v | 8 | ||||
-rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivFloor.v | 8 | ||||
-rw-r--r-- | theories/Numbers/Integer/Abstract/ZDivTrunc.v | 8 | ||||
-rw-r--r-- | theories/ZArith/Zdiv.v | 7 |
5 files changed, 35 insertions, 4 deletions
diff --git a/theories/FSets/FMapPositive.v b/theories/FSets/FMapPositive.v index b1c0fdaa2..3e7664929 100644 --- a/theories/FSets/FMapPositive.v +++ b/theories/FSets/FMapPositive.v @@ -654,19 +654,19 @@ Module PositiveMap <: S with Module E:=PositiveOrderedTypeBits. Lemma xelements_bits_lt_1 : forall p p0 q m v, List.In (p0,v) (xelements m (append p (xO q))) -> E.bits_lt p0 p. - Proof. + Proof using. intros. generalize (xelements_complete _ _ _ _ H); clear H; intros. - revert p0 q m v H. + revert p0 H. induction p; destruct p0; simpl; intros; eauto; try discriminate. Qed. Lemma xelements_bits_lt_2 : forall p p0 q m v, List.In (p0,v) (xelements m (append p (xI q))) -> E.bits_lt p p0. - Proof. + Proof using. intros. generalize (xelements_complete _ _ _ _ H); clear H; intros. - revert p0 q m v H. + revert p0 H. induction p; destruct p0; simpl; intros; eauto; try discriminate. Qed. diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v index ce49877e4..967b68d36 100644 --- a/theories/Numbers/Integer/Abstract/ZDivEucl.v +++ b/theories/Numbers/Integer/Abstract/ZDivEucl.v @@ -602,6 +602,14 @@ Proof. apply div_mod; order. Qed. +Lemma mod_div: forall a b, b~=0 -> + a mod b / b == 0. +Proof. + intros a b Hb. + rewrite div_small_iff by assumption. + auto using mod_always_pos. +Qed. + (** A last inequality: *) Theorem div_mul_le: diff --git a/theories/Numbers/Integer/Abstract/ZDivFloor.v b/theories/Numbers/Integer/Abstract/ZDivFloor.v index ca6197002..a9077127e 100644 --- a/theories/Numbers/Integer/Abstract/ZDivFloor.v +++ b/theories/Numbers/Integer/Abstract/ZDivFloor.v @@ -650,6 +650,14 @@ Proof. apply div_mod; order. Qed. +Lemma mod_div: forall a b, b~=0 -> + a mod b / b == 0. +Proof. + intros a b Hb. + rewrite div_small_iff by assumption. + auto using mod_bound_or. +Qed. + (** A last inequality: *) Theorem div_mul_le: diff --git a/theories/Numbers/Integer/Abstract/ZDivTrunc.v b/theories/Numbers/Integer/Abstract/ZDivTrunc.v index 46c36015d..bbb8ad5ae 100644 --- a/theories/Numbers/Integer/Abstract/ZDivTrunc.v +++ b/theories/Numbers/Integer/Abstract/ZDivTrunc.v @@ -622,6 +622,14 @@ Proof. apply quot_rem; order. Qed. +Lemma rem_quot: forall a b, b~=0 -> + a rem b รท b == 0. +Proof. + intros a b Hb. + rewrite quot_small_iff by assumption. + auto using rem_bound_abs. +Qed. + (** A last inequality: *) Theorem quot_mul_le: diff --git a/theories/ZArith/Zdiv.v b/theories/ZArith/Zdiv.v index 55fc90f21..fa1ddf56f 100644 --- a/theories/ZArith/Zdiv.v +++ b/theories/ZArith/Zdiv.v @@ -508,6 +508,13 @@ Qed. (** Unfortunately, the previous result isn't always true on negative numbers. For instance: 3/(-2)/(-2) = 1 <> 0 = 3 / (-2*-2) *) +Lemma Zmod_div : forall a b, a mod b / b = 0. +Proof. + intros a b. + zero_or_not b. + auto using Z.mod_div. +Qed. + (** A last inequality: *) Theorem Zdiv_mult_le: |