diff options
| -rw-r--r-- | theories/Init/Datatypes.v | 5 | ||||
| -rw-r--r-- | theories/PArith/BinPos.v | 28 |
2 files changed, 33 insertions, 0 deletions
diff --git a/theories/Init/Datatypes.v b/theories/Init/Datatypes.v index 4e3f2e80f3..11d80dbc33 100644 --- a/theories/Init/Datatypes.v +++ b/theories/Init/Datatypes.v @@ -262,6 +262,11 @@ Inductive comparison : Set := | Lt : comparison | Gt : comparison. +Lemma comparison_eq_stable : forall c c' : comparison, ~~ c = c' -> c = c'. +Proof. + destruct c, c'; intro H; reflexivity || destruct H; discriminate. +Qed. + Definition CompOpp (r:comparison) := match r with | Eq => Eq diff --git a/theories/PArith/BinPos.v b/theories/PArith/BinPos.v index 7baf102aaa..d6385ee314 100644 --- a/theories/PArith/BinPos.v +++ b/theories/PArith/BinPos.v @@ -813,6 +813,34 @@ Proof. rewrite compare_cont_spec. unfold ge. destruct (p ?= q); easy'. Qed. +Lemma compare_cont_Lt_not_Lt p q : + compare_cont Lt p q <> Lt <-> p > q. +Proof. + rewrite compare_cont_Lt_Lt. + unfold gt, le, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + +Lemma compare_cont_Lt_not_Gt p q : + compare_cont Lt p q <> Gt <-> p <= q. +Proof. + apply not_iff_compat, compare_cont_Lt_Gt. +Qed. + +Lemma compare_cont_Gt_not_Lt p q : + compare_cont Gt p q <> Lt <-> p >= q. +Proof. + apply not_iff_compat, compare_cont_Gt_Lt. +Qed. + +Lemma compare_cont_Gt_not_Gt p q : + compare_cont Gt p q <> Gt <-> p < q. +Proof. + rewrite compare_cont_Gt_Gt. + unfold ge, lt, compare. + now destruct compare_cont; split; try apply comparison_eq_stable. +Qed. + (** We can express recursive equations for [compare] *) Lemma compare_xO_xO p q : (p~0 ?= q~0) = (p ?= q). |