diff options
Diffstat (limited to 'theories')
| -rw-r--r-- | theories/Init/Notations.v | 1 | ||||
| -rw-r--r-- | theories/Init/Peano.v | 4 | ||||
| -rw-r--r-- | theories/Init/Specif.v | 1 | ||||
| -rw-r--r-- | theories/Logic/ChoiceFacts.v | 4 | ||||
| -rw-r--r-- | theories/Reals/Ranalysis5.v | 62 | ||||
| -rw-r--r-- | theories/Sets/Multiset.v | 4 | ||||
| -rw-r--r-- | theories/Sets/Uniset.v | 4 | ||||
| -rw-r--r-- | theories/Sorting/Heap.v | 2 |
8 files changed, 35 insertions, 47 deletions
diff --git a/theories/Init/Notations.v b/theories/Init/Notations.v index 683a442cb..72073bb4f 100644 --- a/theories/Init/Notations.v +++ b/theories/Init/Notations.v @@ -77,6 +77,7 @@ Reserved Notation "{ x | P & Q }" (at level 0, x at level 99). Reserved Notation "{ x : A | P }" (at level 0, x at level 99). Reserved Notation "{ x : A | P & Q }" (at level 0, x at level 99). +Reserved Notation "{ x & P }" (at level 0, x at level 99). Reserved Notation "{ x : A & P }" (at level 0, x at level 99). Reserved Notation "{ x : A & P & Q }" (at level 0, x at level 99). diff --git a/theories/Init/Peano.v b/theories/Init/Peano.v index 73c8c5ef4..d5322d094 100644 --- a/theories/Init/Peano.v +++ b/theories/Init/Peano.v @@ -92,7 +92,9 @@ Lemma plus_n_O : forall n:nat, n = n + 0. Proof. induction n; simpl; auto. Qed. -Hint Resolve plus_n_O: core. + +Remove Hints eq_refl : core. +Hint Resolve plus_n_O eq_refl: core. (* We want eq_refl to have higher priority than plus_n_O *) Lemma plus_O_n : forall n:nat, 0 + n = n. Proof. diff --git a/theories/Init/Specif.v b/theories/Init/Specif.v index 137bd3a0f..b6afba29a 100644 --- a/theories/Init/Specif.v +++ b/theories/Init/Specif.v @@ -51,6 +51,7 @@ Notation "{ x | P & Q }" := (sig2 (fun x => P) (fun x => Q)) : type_scope. Notation "{ x : A | P }" := (sig (A:=A) (fun x => P)) : type_scope. Notation "{ x : A | P & Q }" := (sig2 (A:=A) (fun x => P) (fun x => Q)) : type_scope. +Notation "{ x & P }" := (sigT (fun x => P)) : type_scope. Notation "{ x : A & P }" := (sigT (A:=A) (fun x => P)) : type_scope. Notation "{ x : A & P & Q }" := (sigT2 (A:=A) (fun x => P) (fun x => Q)) : type_scope. diff --git a/theories/Logic/ChoiceFacts.v b/theories/Logic/ChoiceFacts.v index 9fd52866e..238ac7df0 100644 --- a/theories/Logic/ChoiceFacts.v +++ b/theories/Logic/ChoiceFacts.v @@ -28,6 +28,8 @@ intentional type theory, Journal of Symbolic Logic 70(2):488-514, 2005. [[Werner97]] Benjamin Werner, Sets in Types, Types in Sets, TACS, 1997. *) +Require Import RelationClasses Logic. + Set Implicit Arguments. Local Unset Intuition Negation Unfolding. @@ -125,8 +127,6 @@ Definition DependentFunctionalRelReification_on (A:Type) (B:A -> Type) := formulation of choice); Note also a typo in its intended formulation in [[Werner97]]. *) -Require Import RelationClasses Logic. - Definition RepresentativeFunctionalChoice_on := forall R:A->A->Prop, (Equivalence R) -> diff --git a/theories/Reals/Ranalysis5.v b/theories/Reals/Ranalysis5.v index 88ab4d6e1..afb78e1c8 100644 --- a/theories/Reals/Ranalysis5.v +++ b/theories/Reals/Ranalysis5.v @@ -29,46 +29,34 @@ Lemma f_incr_implies_g_incr_interv : forall f g:R->R, forall lb ub, (forall x , f lb <= x -> x <= f ub -> lb <= g x <= ub) -> (forall x y, f lb <= x -> x < y -> y <= f ub -> g x < g y). Proof. -intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub. - assert (x_encad : f lb <= x <= f ub). - split ; [assumption | apply Rle_trans with (r2:=y) ; [apply Rlt_le|] ; assumption]. - assert (y_encad : f lb <= y <= f ub). - split ; [apply Rle_trans with (r2:=x) ; [|apply Rlt_le] ; assumption | assumption]. - assert (Temp1 : lb <= lb) by intuition ; assert (Temp2 : ub <= ub) by intuition. - assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)). - assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)). - clear Temp1 Temp2. - case (Rlt_dec (g x) (g y)). - intuition. + intros f g lb ub lb_lt_ub f_incr f_eq_g g_ok x y lb_le_x x_lt_y y_le_ub. + assert (x_encad : f lb <= x <= f ub) by lra. + assert (y_encad : f lb <= y <= f ub) by lra. + assert (gx_encad := g_ok _ (proj1 x_encad) (proj2 x_encad)). + assert (gy_encad := g_ok _ (proj1 y_encad) (proj2 y_encad)). + case (Rlt_dec (g x) (g y)); [ easy |]. intros Hfalse. - assert (Temp := Rnot_lt_le _ _ Hfalse). - assert (Hcontradiction : y <= x). - replace y with (id y) by intuition ; replace x with (id x) by intuition ; - rewrite <- f_eq_g. rewrite <- f_eq_g. - assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y). + assert (Temp := Rnot_lt_le _ _ Hfalse). + enough (y <= x) by lra. + replace y with (id y) by easy. + replace x with (id x) by easy. + rewrite <- f_eq_g by easy. + rewrite <- f_eq_g by easy. + assert (f_incr2 : forall x y, lb <= x -> x <= y -> y < ub -> f x <= f y). { intros m n lb_le_m m_le_n n_lt_ub. case (m_le_n). - intros ; apply Rlt_le ; apply f_incr ; [| | apply Rlt_le] ; assumption. - intros Hyp ; rewrite Hyp ; apply Req_le ; reflexivity. - apply f_incr2. - intuition. intuition. - Focus 3. intuition. - Focus 2. intuition. - Focus 2. intuition. Focus 2. intuition. - assert (Temp2 : g x <> ub). - intro Hf. - assert (Htemp : (comp f g) x = f ub). - unfold comp ; rewrite Hf ; reflexivity. - rewrite f_eq_g in Htemp ; unfold id in Htemp. - assert (Htemp2 : x < f ub). - apply Rlt_le_trans with (r2:=y) ; intuition. - clear -Htemp Htemp2. fourier. - intuition. intuition. - clear -Temp2 gx_encad. - case (proj2 gx_encad). - intuition. - intro Hfalse ; apply False_ind ; apply Temp2 ; assumption. - apply False_ind. clear - Hcontradiction x_lt_y. fourier. + - intros; apply Rlt_le, f_incr, Rlt_le; assumption. + - intros Hyp; rewrite Hyp; apply Req_le; reflexivity. + } + apply f_incr2; intuition. + enough (g x <> ub) by lra. + intro Hf. + assert (Htemp : (comp f g) x = f ub). { + unfold comp; rewrite Hf; reflexivity. + } + rewrite f_eq_g in Htemp by easy. + unfold id in Htemp. + fourier. Qed. Lemma derivable_pt_id_interv : forall (lb ub x:R), diff --git a/theories/Sets/Multiset.v b/theories/Sets/Multiset.v index a50140628..a79ddead2 100644 --- a/theories/Sets/Multiset.v +++ b/theories/Sets/Multiset.v @@ -11,6 +11,7 @@ (* G. Huet 1-9-95 *) Require Import Permut Setoid. +Require Plus. (* comm. and ass. of plus *) Set Implicit Arguments. @@ -69,9 +70,6 @@ Section multiset_defs. unfold meq; unfold munion; simpl; auto. Qed. - - Require Plus. (* comm. and ass. of plus *) - Lemma munion_comm : forall x y:multiset, meq (munion x y) (munion y x). Proof. unfold meq; unfold multiplicity; unfold munion. diff --git a/theories/Sets/Uniset.v b/theories/Sets/Uniset.v index 95ba93232..7940bda1a 100644 --- a/theories/Sets/Uniset.v +++ b/theories/Sets/Uniset.v @@ -13,7 +13,7 @@ (* G. Huet 1-9-95 *) (* Updated Papageno 12/98 *) -Require Import Bool. +Require Import Bool Permut. Set Implicit Arguments. @@ -140,8 +140,6 @@ Hint Resolve seq_right. (** Here we should make uniset an abstract datatype, by hiding [Charac], [union], [charac]; all further properties are proved abstractly *) -Require Import Permut. - Lemma union_rotate : forall x y z:uniset, seq (union x (union y z)) (union z (union x y)). Proof. diff --git a/theories/Sorting/Heap.v b/theories/Sorting/Heap.v index 8f583be97..d9e5ad676 100644 --- a/theories/Sorting/Heap.v +++ b/theories/Sorting/Heap.v @@ -136,7 +136,7 @@ Section defs. (munion (list_contents _ eqA_dec l1) (list_contents _ eqA_dec l2)) -> (forall a, HdRel leA a l1 -> HdRel leA a l2 -> HdRel leA a l) -> merge_lem l1 l2. - Require Import Morphisms. + Import Morphisms. Instance: Equivalence (@meq A). Proof. constructor; auto with datatypes. red. apply meq_trans. Defined. |