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Diffstat (limited to 'src')
| -rw-r--r-- | src/Util/ZUtil/Morphisms.v | 21 |
1 files changed, 21 insertions, 0 deletions
diff --git a/src/Util/ZUtil/Morphisms.v b/src/Util/ZUtil/Morphisms.v index d58b89c91..731219a6a 100644 --- a/src/Util/ZUtil/Morphisms.v +++ b/src/Util/ZUtil/Morphisms.v @@ -14,45 +14,66 @@ Module Z. [Local Existing Instances]. *) Lemma succ_le_Proper : Proper (Z.le ==> Z.le) Z.succ. Proof. repeat (omega || intro). Qed. + Hint Resolve succ_le_Proper : zarith. Lemma add_le_Proper : Proper (Z.le ==> Z.le ==> Z.le) Z.add. Proof. repeat (omega || intro). Qed. + Hint Resolve add_le_Proper : zarith. Lemma add_le_Proper' x : Proper (Z.le ==> Z.le) (Z.add x). Proof. repeat (omega || intro). Qed. + Hint Resolve add_le_Proper' : zarith. Lemma sub_le_ge_Proper : Proper (Z.le ==> Z.ge ==> Z.le) Z.sub. Proof. repeat (omega || intro). Qed. + Hint Resolve sub_le_ge_Proper : zarith. Lemma sub_le_flip_le_Proper : Proper (Z.le ==> Basics.flip Z.le ==> Z.le) Z.sub. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve sub_le_flip_le_Proper : zarith. Lemma sub_le_eq_Proper : Proper (Z.le ==> Logic.eq ==> Z.le) Z.sub. Proof. repeat (omega || intro). Qed. + Hint Resolve sub_le_eq_Proper : zarith. Lemma mul_Zpos_le_Proper p : Proper (Z.le ==> Z.le) (Z.mul (Z.pos p)). Proof. repeat (nia || intro). Qed. + Hint Resolve mul_Zpos_le_Proper : zarith. Lemma log2_up_le_Proper : Proper (Z.le ==> Z.le) Z.log2_up. Proof. intros ???; apply Z.log2_up_le_mono; assumption. Qed. + Hint Resolve log2_up_le_Proper : zarith. Lemma log2_le_Proper : Proper (Z.le ==> Z.le) Z.log2. Proof. intros ???; apply Z.log2_le_mono; assumption. Qed. + Hint Resolve log2_le_Proper : zarith. Lemma pow_Zpos_le_Proper x : Proper (Z.le ==> Z.le) (Z.pow (Z.pos x)). Proof. intros ???; apply Z.pow_le_mono_r; try reflexivity; try assumption. Qed. + Hint Resolve pow_Zpos_le_Proper : zarith. Lemma lt_le_flip_Proper_flip_impl : Proper (Z.le ==> Basics.flip Z.le ==> Basics.flip Basics.impl) Z.lt. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve lt_le_flip_Proper_flip_impl : zarith. Lemma le_Proper_ge_le_flip_impl : Proper (Z.le ==> Z.ge ==> Basics.flip Basics.impl) Z.le. Proof. intros ???????; omega. Qed. + Hint Resolve le_Proper_ge_le_flip_impl : zarith. Lemma add_le_Proper_flip : Proper (Basics.flip Z.le ==> Basics.flip Z.le ==> Basics.flip Z.le) Z.add. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve add_le_Proper_flip : zarith. Lemma sub_le_ge_Proper_flip : Proper (Basics.flip Z.le ==> Basics.flip Z.ge ==> Basics.flip Z.le) Z.sub. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve sub_le_ge_Proper_flip : zarith. Lemma sub_flip_le_le_Proper_flip : Proper (Basics.flip Z.le ==> Z.le ==> Basics.flip Z.le) Z.sub. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve sub_flip_le_le_Proper_flip : zarith. Lemma sub_le_eq_Proper_flip : Proper (Basics.flip Z.le ==> Logic.eq ==> Basics.flip Z.le) Z.sub. Proof. unfold Basics.flip; repeat (omega || intro). Qed. + Hint Resolve sub_le_eq_Proper_flip : zarith. Lemma log2_up_le_Proper_flip : Proper (Basics.flip Z.le ==> Basics.flip Z.le) Z.log2_up. Proof. intros ???; apply Z.log2_up_le_mono; assumption. Qed. + Hint Resolve log2_up_le_Proper_flip : zarith. Lemma log2_le_Proper_flip : Proper (Basics.flip Z.le ==> Basics.flip Z.le) Z.log2. Proof. intros ???; apply Z.log2_le_mono; assumption. Qed. + Hint Resolve log2_le_Proper_flip : zarith. Lemma pow_Zpos_le_Proper_flip x : Proper (Basics.flip Z.le ==> Basics.flip Z.le) (Z.pow (Z.pos x)). Proof. intros ???; apply Z.pow_le_mono_r; try reflexivity; try assumption. Qed. + Hint Resolve pow_Zpos_le_Proper_flip : zarith. Lemma add_with_carry_le_Proper : Proper (Z.le ==> Z.le ==> Z.le ==> Z.le) Z.add_with_carry. Proof. unfold Z.add_with_carry; repeat (omega || intro). Qed. + Hint Resolve add_with_carry_le_Proper : zarith. Lemma sub_with_borrow_le_Proper : Proper (Basics.flip Z.le ==> Z.le ==> Basics.flip Z.le ==> Z.le) Z.sub_with_borrow. Proof. unfold Z.sub_with_borrow, Z.add_with_carry, Basics.flip; repeat (omega || intro). Qed. + Hint Resolve sub_with_borrow_le_Proper : zarith. End Z. |