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Commit 685148ab authored by Robbert Krebbers's avatar Robbert Krebbers
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Use scheme - then + then * for bullets.

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algebra/agree.v
View file @ 685148ab
......@@ -33,17 +33,17 @@ Qed.
Definition agree_cofe_mixin : CofeMixin (agree A).
Proof.
split.
* intros x y; split.
- intros x y; split.
+ by intros Hxy n; split; intros; apply Hxy.
+ by intros Hxy; split; intros; apply Hxy with n.
* split.
- split.
+ by split.
+ by intros x y Hxy; split; intros; symmetry; apply Hxy; auto; apply Hxy.
+ intros x y z Hxy Hyz; split; intros n'; intros.
- transitivity (agree_is_valid y n'). by apply Hxy. by apply Hyz.
- transitivity (y n'). by apply Hxy. by apply Hyz, Hxy.
* intros n x y Hxy; split; intros; apply Hxy; auto.
* intros c n; apply and_wlog_r; intros;
* transitivity (agree_is_valid y n'). by apply Hxy. by apply Hyz.
* transitivity (y n'). by apply Hxy. by apply Hyz, Hxy.
- intros n x y Hxy; split; intros; apply Hxy; auto.
- intros c n; apply and_wlog_r; intros;
symmetry; apply (chain_cauchy c); naive_solver.
Qed.
Canonical Structure agreeC := CofeT agree_cofe_mixin.
......@@ -74,8 +74,8 @@ Proof.
intros n x y1 y2 [Hy' Hy]; split; [|done].
split; intros (?&?&Hxy); repeat (intro || split);
try apply Hy'; eauto using agree_valid_le.
* etransitivity; [apply Hxy|apply Hy]; eauto using agree_valid_le.
* etransitivity; [apply Hxy|symmetry; apply Hy, Hy'];
- etransitivity; [apply Hxy|apply Hy]; eauto using agree_valid_le.
- etransitivity; [apply Hxy|symmetry; apply Hy, Hy'];
eauto using agree_valid_le.
Qed.
Instance: Proper (dist n ==> dist n ==> dist n) (@op (agree A) _).
......@@ -95,13 +95,13 @@ Qed.
Definition agree_cmra_mixin : CMRAMixin (agree A).
Proof.
split; try (apply _ || done).
* by intros n x1 x2 Hx y1 y2 Hy.
* intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?].
- by intros n x1 x2 Hx y1 y2 Hy.
- intros n x [? Hx]; split; [by apply agree_valid_S|intros n' ?].
rewrite (Hx n'); last auto.
symmetry; apply dist_le with n; try apply Hx; auto.
* intros x; apply agree_idemp.
* by intros x y n [(?&?&?) ?].
* by intros x y n; rewrite agree_includedN.
- intros x; apply agree_idemp.
- by intros x y n [(?&?&?) ?].
- by intros x y n; rewrite agree_includedN.
Qed.
Lemma agree_op_inv (x1 x2 : agree A) n : ✓{n} (x1 ⋅ x2) → x1 ≡{n}≡ x2.
Proof. intros Hxy; apply Hxy. Qed.
......@@ -113,8 +113,8 @@ Qed.
Definition agree_cmra_extend_mixin : CMRAExtendMixin (agree A).
Proof.
intros n x y1 y2 Hval Hx; exists (x,x); simpl; split.
* by rewrite agree_idemp.
* by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp.
- by rewrite agree_idemp.
- by move: Hval; rewrite Hx; move=> /agree_op_inv->; rewrite agree_idemp.
Qed.
Canonical Structure agreeRA : cmraT :=
CMRAT agree_cofe_mixin agree_cmra_mixin agree_cmra_extend_mixin.
......
algebra/auth.v
View file @ 685148ab
......@@ -39,14 +39,14 @@ Instance auth_compl : Compl (auth A) := λ c,
Definition auth_cofe_mixin : CofeMixin (auth A).
Proof.
split.
* intros x y; unfold dist, auth_dist, equiv, auth_equiv.
- intros x y; unfold dist, auth_dist, equiv, auth_equiv.
rewrite !equiv_dist; naive_solver.
* intros n; split.
- intros n; split.
+ by intros ?; split.
+ by intros ?? [??]; split; symmetry.
+ intros ??? [??] [??]; split; etransitivity; eauto.
* by intros ? [??] [??] [??]; split; apply dist_S.
* intros c n; split. apply (conv_compl (chain_map authoritative c) n).
- by intros ? [??] [??] [??]; split; apply dist_S.
- intros c n; split. apply (conv_compl (chain_map authoritative c) n).
apply (conv_compl (chain_map own c) n).
Qed.
Canonical Structure authC := CofeT auth_cofe_mixin.
......@@ -99,24 +99,24 @@ Proof. destruct x as [[]]; naive_solver eauto using cmra_validN_includedN. Qed.
Definition auth_cmra_mixin : CMRAMixin (auth A).
Proof.
split.
* by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
* by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
* intros n [x a] [y b] [Hx Ha]; simpl in *;
- by intros n x y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
- by intros n y1 y2 [Hy Hy']; split; simpl; rewrite ?Hy ?Hy'.
- intros n [x a] [y b] [Hx Ha]; simpl in *;
destruct Hx; intros ?; cofe_subst; auto.
* by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
- by intros n x1 x2 [Hx Hx'] y1 y2 [Hy Hy'];
split; simpl; rewrite ?Hy ?Hy' ?Hx ?Hx'.
* intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
* by split; simpl; rewrite assoc.
* by split; simpl; rewrite comm.
* by split; simpl; rewrite ?cmra_unit_l.
* by split; simpl; rewrite ?cmra_unit_idemp.
* intros n ??; rewrite! auth_includedN; intros [??].
- intros n [[] ?] ?; naive_solver eauto using cmra_includedN_S, cmra_validN_S.
- by split; simpl; rewrite assoc.
- by split; simpl; rewrite comm.
- by split; simpl; rewrite ?cmra_unit_l.
- by split; simpl; rewrite ?cmra_unit_idemp.
- intros n ??; rewrite! auth_includedN; intros [??].
by split; simpl; apply cmra_unit_preservingN.
* assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a).
- assert (∀ n (a b1 b2 : A), b1 ⋅ b2 ≼{n} a → b1 ≼{n} a).
{ intros n a b1 b2 <-; apply cmra_includedN_l. }
intros n [[a1| |] b1] [[a2| |] b2];
naive_solver eauto using cmra_validN_op_l, cmra_validN_includedN.
* by intros n ??; rewrite auth_includedN;
- by intros n ??; rewrite auth_includedN;
intros [??]; split; simpl; apply cmra_op_minus.
Qed.
Definition auth_cmra_extend_mixin : CMRAExtendMixin (auth A).
......@@ -150,9 +150,9 @@ Context `{Empty A, !CMRAIdentity A}.
Global Instance auth_cmra_identity : CMRAIdentity authRA.
Proof.
split; simpl.
* by apply (@cmra_empty_valid A _).
* by intros x; constructor; rewrite /= left_id.
* apply _.
- by apply (@cmra_empty_valid A _).
- by intros x; constructor; rewrite /= left_id.
- apply _.
Qed.
Lemma auth_frag_op a b : ◯ (a ⋅ b) ≡ ◯ a ⋅ ◯ b.
Proof. done. Qed.
......@@ -221,9 +221,9 @@ Instance auth_map_cmra_monotone {A B : cmraT} (f : A → B) :
CMRAMonotone f → CMRAMonotone (auth_map f).
Proof.
split.
* by intros n [x a] [y b]; rewrite !auth_includedN /=;
- by intros n [x a] [y b]; rewrite !auth_includedN /=;
intros [??]; split; simpl; apply: includedN_preserving.
* intros n [[a| |] b]; rewrite /= /cmra_validN;
- intros n [[a| |] b]; rewrite /= /cmra_validN;
naive_solver eauto using @includedN_preserving, @validN_preserving.
Qed.
Definition authC_map {A B} (f : A -n> B) : authC A -n> authC B :=
......
algebra/cmra.v
View file @ 685148ab
......@@ -243,8 +243,8 @@ Qed.
Global Instance cmra_includedN_preorder n : PreOrder (@includedN A _ _ n).
Proof.
split.
* by intros x; exists (unit x); rewrite cmra_unit_r.
* intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
- by intros x; exists (unit x); rewrite cmra_unit_r.
- intros x y z [z1 Hy] [z2 Hz]; exists (z1 ⋅ z2).
by rewrite assoc -Hy -Hz.
Qed.
Global Instance cmra_included_preorder: PreOrder (@included A _ _).
......@@ -349,8 +349,8 @@ Proof. split. by intros x y. intros x y y' ?? z ?; naive_solver. Qed.
Lemma cmra_update_updateP x y : x ~~> y ↔ x ~~>: (y =).
Proof.
split.
* by intros Hx z ?; exists y; split; [done|apply (Hx z)].
* by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
- by intros Hx z ?; exists y; split; [done|apply (Hx z)].
- by intros Hx z n ?; destruct (Hx z n) as (?&<-&?).
Qed.
Lemma cmra_updateP_id (P : A → Prop) x : P x → x ~~>: P.
Proof. by intros ? z n ?; exists x. Qed.
......@@ -402,8 +402,8 @@ Instance cmra_monotone_compose {A B C : cmraT} (f : A → B) (g : B → C) :
CMRAMonotone f → CMRAMonotone g → CMRAMonotone (g ∘ f).
Proof.
split.
* move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
* move=> n x Hx /=. by apply validN_preserving, validN_preserving.
- move=> n x y Hxy /=. by apply includedN_preserving, includedN_preserving.
- move=> n x Hx /=. by apply validN_preserving, validN_preserving.
Qed.
Section cmra_monotone.
......@@ -527,20 +527,20 @@ Section prod.
Definition prod_cmra_mixin : CMRAMixin (A * B).
Proof.
split; try apply _.
* by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
* by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
* by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
* by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
- by intros n x y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
- by intros n y1 y2 [Hy1 Hy2]; split; rewrite /= ?Hy1 ?Hy2.
- by intros n y1 y2 [Hy1 Hy2] [??]; split; rewrite /= -?Hy1 -?Hy2.
- by intros n x1 x2 [Hx1 Hx2] y1 y2 [Hy1 Hy2];
split; rewrite /= ?Hx1 ?Hx2 ?Hy1 ?Hy2.
* by intros n x [??]; split; apply cmra_validN_S.
* by split; rewrite /= assoc.
* by split; rewrite /= comm.
* by split; rewrite /= cmra_unit_l.
* by split; rewrite /= cmra_unit_idemp.
* intros n x y; rewrite !prod_includedN.
- by intros n x [??]; split; apply cmra_validN_S.
- by split; rewrite /= assoc.
- by split; rewrite /= comm.
- by split; rewrite /= cmra_unit_l.
- by split; rewrite /= cmra_unit_idemp.
- intros n x y; rewrite !prod_includedN.
by intros [??]; split; apply cmra_unit_preservingN.
* intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
* intros x y n; rewrite prod_includedN; intros [??].
- intros n x y [??]; split; simpl in *; eauto using cmra_validN_op_l.
- intros x y n; rewrite prod_includedN; intros [??].
by split; apply cmra_op_minus.
Qed.
Definition prod_cmra_extend_mixin : CMRAExtendMixin (A * B).
......@@ -556,9 +556,9 @@ Section prod.
CMRAIdentity A → CMRAIdentity B → CMRAIdentity prodRA.
Proof.
split.
* split; apply cmra_empty_valid.
* by split; rewrite /=left_id.
* by intros ? [??]; split; apply (timeless _).
- split; apply cmra_empty_valid.
- by split; rewrite /=left_id.
- by intros ? [??]; split; apply (timeless _).
Qed.
Lemma prod_update x y : x.1 ~~> y.1 → x.2 ~~> y.2 → x ~~> y.
Proof. intros ?? z n [??]; split; simpl in *; auto. Qed.
......@@ -579,7 +579,7 @@ Instance prod_map_cmra_monotone {A A' B B' : cmraT} (f : A → A') (g : B → B'
CMRAMonotone f → CMRAMonotone g → CMRAMonotone (prod_map f g).
Proof.
split.
* intros n x y; rewrite !prod_includedN; intros [??]; simpl.
- intros n x y; rewrite !prod_includedN; intros [??]; simpl.
by split; apply includedN_preserving.
* by intros n x [??]; split; simpl; apply validN_preserving.
- by intros n x [??]; split; simpl; apply validN_preserving.
Qed.
algebra/cmra_big_op.v
View file @ 685148ab
......@@ -21,9 +21,9 @@ Proof. done. Qed.
Global Instance big_op_permutation : Proper ((≡ₚ) ==> (≡)) big_op.
Proof.
induction 1 as [|x xs1 xs2 ? IH|x y xs|xs1 xs2 xs3]; simpl; auto.
* by rewrite IH.
* by rewrite !assoc (comm _ x).
* by transitivity (big_op xs2).
- by rewrite IH.
- by rewrite !assoc (comm _ x).
- by transitivity (big_op xs2).
Qed.
Global Instance big_op_proper : Proper ((≡) ==> (≡)) big_op.
Proof. by induction 1; simpl; repeat apply (_ : Proper (_ ==> _ ==> _) op). Qed.
......@@ -35,10 +35,10 @@ Qed.
Lemma big_op_contains xs ys : xs `contains` ys → big_op xs ≼ big_op ys.
Proof.
induction 1 as [|x xs ys|x y xs|x xs ys|xs ys zs]; rewrite //=.
* by apply cmra_preserving_l.
* by rewrite !assoc (comm _ y).
* by transitivity (big_op ys); last apply cmra_included_r.
* by transitivity (big_op ys).
- by apply cmra_preserving_l.
- by rewrite !assoc (comm _ y).
- by transitivity (big_op ys); last apply cmra_included_r.
- by transitivity (big_op ys).
Qed.
Lemma big_op_delete xs i x :
xs !! i = Some x → x ⋅ big_op (delete i xs) ≡ big_op xs.
......
algebra/cofe.v
View file @ 685148ab
......@@ -97,15 +97,15 @@ Section cofe.
Global Instance cofe_equivalence : Equivalence ((≡) : relation A).
Proof.
split.
* by intros x; rewrite equiv_dist.
* by intros x y; rewrite !equiv_dist.
* by intros x y z; rewrite !equiv_dist; intros; transitivity y.
- by intros x; rewrite equiv_dist.
- by intros x y; rewrite !equiv_dist.
- by intros x y z; rewrite !equiv_dist; intros; transitivity y.
Qed.
Global Instance dist_ne n : Proper (dist n ==> dist n ==> iff) (@dist A _ n).
Proof.
intros x1 x2 ? y1 y2 ?; split; intros.
* by transitivity x1; [|transitivity y1].
* by transitivity x2; [|transitivity y2].
- by transitivity x1; [|transitivity y1].
- by transitivity x2; [|transitivity y2].
Qed.
Global Instance dist_proper n : Proper ((≡) ==> (≡) ==> iff) (@dist A _ n).
Proof.
......@@ -158,8 +158,8 @@ Program Definition fixpoint_chain {A : cofeT} `{Inhabited A} (f : A → A)
`{!Contractive f} : chain A := {| chain_car i := Nat.iter (S i) f inhabitant |}.
Next Obligation.
intros A ? f ? n. induction n as [|n IH]; intros [|i] ?; simpl; try omega.
* apply (contractive_0 f).
* apply (contractive_S f), IH; auto with omega.
- apply (contractive_0 f).
- apply (contractive_S f), IH; auto with omega.
Qed.
Program Definition fixpoint {A : cofeT} `{Inhabited A} (f : A → A)
`{!Contractive f} : A := compl (fixpoint_chain f).
......@@ -212,14 +212,14 @@ Section cofe_mor.
Definition cofe_mor_cofe_mixin : CofeMixin (cofeMor A B).
Proof.
split.
* intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
- intros f g; split; [intros Hfg n k; apply equiv_dist, Hfg|].
intros Hfg k; apply equiv_dist; intros n; apply Hfg.
* intros n; split.
- intros n; split.
+ by intros f x.
+ by intros f g ? x.
+ by intros f g h ?? x; transitivity (g x).
* by intros n f g ? x; apply dist_S.
* intros c n x; simpl.
- by intros n f g ? x; apply dist_S.
- intros c n x; simpl.
by rewrite (conv_compl (fun_chain c x) n) /=.
Qed.
Canonical Structure cofe_mor : cofeT := CofeT cofe_mor_cofe_mixin.
......@@ -274,11 +274,11 @@ Section product.
Definition prod_cofe_mixin : CofeMixin (A * B).
Proof.
split.
* intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
- intros x y; unfold dist, prod_dist, equiv, prod_equiv, prod_relation.
rewrite !equiv_dist; naive_solver.
* apply _.
* by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
* intros c n; split. apply (conv_compl (chain_map fst c) n).
- apply _.
- by intros n [x1 y1] [x2 y2] [??]; split; apply dist_S.
- intros c n; split. apply (conv_compl (chain_map fst c) n).
apply (conv_compl (chain_map snd c) n).
Qed.
Canonical Structure prodC : cofeT := CofeT prod_cofe_mixin.
......@@ -308,10 +308,10 @@ Section discrete_cofe.
Definition discrete_cofe_mixin : CofeMixin A.
Proof.
split.
* intros x y; split; [done|intros Hn; apply (Hn 0)].
* done.
* done.
* intros c n. rewrite /compl /discrete_compl /=.
- intros x y; split; [done|intros Hn; apply (Hn 0)].
- done.
- done.
- intros c n. rewrite /compl /discrete_compl /=.
symmetry; apply (chain_cauchy c 0 (S n)); omega.
Qed.
Definition discreteC : cofeT := CofeT discrete_cofe_mixin.
......@@ -347,14 +347,14 @@ Section later.
Definition later_cofe_mixin : CofeMixin (later A).
Proof.
split.
* intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
- intros x y; unfold equiv, later_equiv; rewrite !equiv_dist.
split. intros Hxy [|n]; [done|apply Hxy]. intros Hxy n; apply (Hxy (S n)).
* intros [|n]; [by split|split]; unfold dist, later_dist.
- intros [|n]; [by split|split]; unfold dist, later_dist.
+ by intros [x].
+ by intros [x] [y].
+ by intros [x] [y] [z] ??; transitivity y.
* intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
* intros c [|n]; [done|by apply (conv_compl (later_chain c) n)].
- intros [|n] [x] [y] ?; [done|]; unfold dist, later_dist; by apply dist_S.
- intros c [|n]; [done|by apply (conv_compl (later_chain c) n)].
Qed.
Canonical Structure laterC : cofeT := CofeT later_cofe_mixin.
Global Instance Next_contractive : Contractive (@Next A).
......
algebra/cofe_solver.v
View file @ 685148ab
......@@ -43,8 +43,8 @@ Qed.
Lemma fg {k} (x : A (S (S k))) : f (S k) (g (S k) x) ≡{k}≡ x.
Proof.
induction k as [|k IH]; simpl.
* rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. apply (contractive_0 map).
* rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. by apply (contractive_S map).
- rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. apply (contractive_0 map).
- rewrite f_S g_S -{2}(map_id _ _ x) -map_comp. by apply (contractive_S map).
Qed.
Record tower := {
......@@ -66,15 +66,15 @@ Qed.
Definition tower_cofe_mixin : CofeMixin tower.
Proof.
split.
* intros X Y; split; [by intros HXY n k; apply equiv_dist|].
- intros X Y; split; [by intros HXY n k; apply equiv_dist|].
intros HXY k; apply equiv_dist; intros n; apply HXY.
* intros k; split.
- intros k; split.
+ by intros X n.
+ by intros X Y ? n.
+ by intros X Y Z ?? n; transitivity (Y n).
* intros k X Y HXY n; apply dist_S.
- intros k X Y HXY n; apply dist_S.
by rewrite -(g_tower X) (HXY (S n)) g_tower.
* intros c n k; rewrite /= (conv_compl (tower_chain c k) n).
- intros c n k; rewrite /= (conv_compl (tower_chain c k) n).
apply (chain_cauchy c); lia.
Qed.
Definition T : cofeT := CofeT tower_cofe_mixin.
......@@ -136,12 +136,12 @@ Lemma g_embed_coerce {k i} (x : A k) :
g i (embed_coerce (S i) x) ≡ embed_coerce i x.
Proof.
unfold embed_coerce; destruct (le_lt_dec (S i) k), (le_lt_dec i k); simpl.
* symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
* exfalso; lia.
* assert (i = k) by lia; subst.
- symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
- exfalso; lia.
- assert (i = k) by lia; subst.
rewrite (ff_ff _ (eq_refl (1 + (0 + k)))) /= gf.
by rewrite (gg_gg _ (eq_refl (0 + (0 + k)))).
* assert (H : 1 + ((i - k) + k) = S i) by lia.
- assert (H : 1 + ((i - k) + k) = S i) by lia.
rewrite (ff_ff _ H) /= -{2}(gf (ff (i - k) x)) g_coerce.
by erewrite coerce_proper by done.
Qed.
......@@ -156,22 +156,22 @@ Lemma embed_f k (x : A k) : embed (S k) (f k x) ≡ embed k x.
Proof.
rewrite equiv_dist=> n i; rewrite /embed /= /embed_coerce.
destruct (le_lt_dec i (S k)), (le_lt_dec i k); simpl.
* assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=.
- assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H) /=.
by erewrite g_coerce, gf, coerce_proper by done.
* assert (S k = 0 + (0 + i)) as H by lia.
- assert (S k = 0 + (0 + i)) as H by lia.
rewrite (gg_gg _ H); simplify_equality'.
by rewrite (ff_ff _ (eq_refl (1 + (0 + k)))).
* exfalso; lia.
* assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
- exfalso; lia.
- assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H) /=.
by erewrite coerce_proper by done.
Qed.
Lemma embed_tower k (X : T) : embed (S k) (X (S k)) ≡{k}≡ X.
Proof.
intros i; rewrite /= /embed_coerce.
destruct (le_lt_dec i (S k)) as [H|H]; simpl.
* rewrite -(gg_tower i (S k - i) X).
- rewrite -(gg_tower i (S k - i) X).
apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _).
* rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
- rewrite (ff_tower k (i - S k) X). by destruct (Nat.sub_add _ _ _).
Qed.
Program Definition unfold_chain (X : T) : chain (F T T) :=
......@@ -206,7 +206,7 @@ Proof. by intros n X Y HXY k; rewrite /fold /= HXY. Qed.
Theorem result : solution F.
Proof.
apply (Solution F T (CofeMor unfold) (CofeMor fold)).
* move=> X /=.
- move=> X /=.
rewrite equiv_dist; intros n k; unfold unfold, fold; simpl.
rewrite -g_tower -(gg_tower _ n); apply (_ : Proper (_ ==> _) (g _)).
transitivity (map (ff n, gg n) (X (S (n + k)))).
......@@ -228,7 +228,7 @@ Proof.
assert (H: S n + k = n + S k) by lia.
rewrite (map_ff_gg _ _ _ H).
apply (_ : Proper (_ ==> _) (gg _)); by destruct H.
* intros X; rewrite equiv_dist=> n /=.
- intros X; rewrite equiv_dist=> n /=.
rewrite /unfold /= (conv_compl (unfold_chain (fold X)) n) /=.
rewrite g_S -!map_comp -{2}(map_id _ _ X).
apply (contractive_ne map); split => Y /=.
......
algebra/dra.v
View file @ 685148ab
......@@ -57,9 +57,9 @@ Instance validity_equiv : Equiv T := λ x y,
Instance validity_equivalence : Equivalence ((≡) : relation T).
Proof.
split; unfold equiv, validity_equiv.
* by intros [x px ?]; simpl.
* intros [x px ?] [y py ?]; naive_solver.
* intros [x px ?] [y py ?] [z pz ?] [? Hxy] [? Hyz]; simpl in *.
- by intros [x px ?]; simpl.
- intros [x px ?] [y py ?]; naive_solver.
- intros [x px ?] [y py ?] [z pz ?] [? Hxy] [? Hyz]; simpl in *.
split; [|intros; transitivity y]; tauto.
Qed.
Instance dra_valid_proper' : Proper ((≡) ==> iff) (valid : A → Prop).
......@@ -69,8 +69,8 @@ Proof. by intros x1 x2 Hx; split; rewrite /= Hx. Qed.
Instance: Proper ((≡) ==> (≡) ==> iff) (⊥).
Proof.
intros x1 x2 Hx y1 y2 Hy; split.
* by rewrite Hy (symmetry_iff (⊥) x1) (symmetry_iff (⊥) x2) Hx.
* by rewrite -Hy (symmetry_iff (⊥) x2) (symmetry_iff (⊥) x1) -Hx.
- by rewrite Hy (symmetry_iff (⊥) x1) (symmetry_iff (⊥) x2) Hx.
- by rewrite -Hy (symmetry_iff (⊥) x2) (symmetry_iff (⊥) x1) -Hx.
Qed.
Lemma dra_disjoint_rl x y z : ✓ x → ✓ y → ✓ z → y ⊥ z → x ⊥ y ⋅ z → x ⊥ y.
Proof. intros ???. rewrite !(symmetry_iff _ x). by apply dra_disjoint_ll. Qed.
......@@ -103,31 +103,31 @@ Solve Obligations with naive_solver auto using dra_minus_valid.
Definition validity_ra : RA (discreteC T).
Proof.
split.
* intros ??? [? Heq]; split; simpl; [|by intros (?&?&?); rewrite Heq].
- intros ??? [? Heq]; split; simpl; [|by intros (?&?&?); rewrite Heq].
split; intros (?&?&?); split_ands';
first [rewrite ?Heq; tauto|rewrite -?Heq; tauto|tauto].
* by intros ?? [? Heq]; split; [done|]; simpl; intros ?; rewrite Heq.
* intros ?? [??]; naive_solver.
* intros x1 x2 [? Hx] y1 y2 [? Hy];
- by intros ?? [? Heq]; split; [done|]; simpl; intros ?; rewrite Heq.
- intros ?? [??]; naive_solver.
- intros x1 x2 [? Hx] y1 y2 [? Hy];
split; simpl; [|by intros (?&?&?); rewrite Hx // Hy].
split; intros (?&?&z&?&?); split_ands'; try tauto.
+ exists z. by rewrite -Hy // -Hx.
+ exists z. by rewrite Hx ?Hy; tauto.
* intros [x px ?] [y py ?] [z pz ?]; split; simpl;
- intros [x px ?] [y py ?] [z pz ?]; split; simpl;
[intuition eauto 2 using dra_disjoint_lr, dra_disjoint_rl
|by intros; rewrite assoc].
* intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
* intros [x px ?]; split;
- intros [x px ?] [y py ?]; split; naive_solver eauto using dra_comm.
- intros [x px ?]; split;
naive_solver eauto using dra_unit_l, dra_unit_disjoint_l.
* intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.
* intros x y Hxy; exists (unit y ⩪ unit x).
- intros [x px ?]; split; naive_solver eauto using dra_unit_idemp.