diff options
| author |  Russell O'Connor <roconnor@blockstream.io> | 2017-06-29 15:17:06 -0400 |
|---|---|---|
| committer | Russell O'Connor <roconnor@blockstream.io> | 2017-06-29 15:17:06 -0400 |
| commit | f4f609f598fb4804acf272209782b60d4647cad4 (patch) | |
| tree | 9c5b298fd555708716cccca99dec6982d3f2a9f8 /theories | |
| parent | 6d7c392b73eaa021083ab03c9042d271fb4c28c0 (diff) | |
Add Z.mod_div lemma to standard library.
Diffstat (limited to 'theories')
| -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 | 6 |
4 files changed, 30 insertions, 0 deletions
diff --git a/theories/Numbers/Integer/Abstract/ZDivEucl.v b/theories/Numbers/Integer/Abstract/ZDivEucl.v index c2fa69e53..c5b8d5ebe 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 310748ddb..eeb803eea 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 043010779..a9b72d815 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 2ba865bd0..509384104 100644 --- a/theories/ZArith/Zdiv.v +++ b/theories/ZArith/Zdiv.v @@ -508,6 +508,12 @@ 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 (a b : Z) : a mod b / b = 0. +Proof. + zero_or_not b. + auto using Z.mod_div. +Qed. + (** A last inequality: *) Theorem Zdiv_mult_le: |