diff --git a/theories/algebra/cmra.v b/theories/algebra/cmra.v index b1304e73985b39146323e0cb0e767e3823702bb2..4a7325f8bbf842e860357885f6af65158bfc39cc 100644 --- a/theories/algebra/cmra.v +++ b/theories/algebra/cmra.v @@ -1425,7 +1425,7 @@ Section option. Instance option_unit : Unit (option A) := None. Lemma option_ucmra_mixin : UcmraMixin optionR. - Proof. split; [ done | by intros [] | done ]. Qed. + Proof. split; [done| |done]. by intros []. Qed. Canonical Structure optionUR := UcmraT (option A) option_ucmra_mixin. (** Misc *) @@ -1456,13 +1456,13 @@ Section option. Lemma exclusiveN_Some_l n a `{!Exclusive a} mb : ✓{n} (Some a ⋅ mb) → mb = None. - Proof. destruct mb; [|done]. move=> /(exclusiveN_l _ a) []. Qed. + Proof. destruct mb; last done. move=> /(exclusiveN_l _ a) []. Qed. Lemma exclusiveN_Some_r n a `{!Exclusive a} mb : ✓{n} (mb ⋅ Some a) → mb = None. Proof. rewrite comm. by apply exclusiveN_Some_l. Qed. Lemma exclusive_Some_l a `{!Exclusive a} mb : ✓ (Some a ⋅ mb) → mb = None. - Proof. destruct mb; [|done]. move=> /(exclusive_l a) []. Qed. + Proof. destruct mb; last done. move=> /(exclusive_l a) []. Qed. Lemma exclusive_Some_r a `{!Exclusive a} mb : ✓ (mb ⋅ Some a) → mb = None. Proof. rewrite comm. by apply exclusive_Some_l. Qed. @@ -1474,9 +1474,9 @@ Section option. Proof. rewrite Some_included; eauto. Qed. Lemma Some_includedN_total `{CmraTotal A} n a b : Some a ≼{n} Some b ↔ a ≼{n} b. - Proof. rewrite Some_includedN. split; [|eauto]. by intros [->|?]. Qed. + Proof. rewrite Some_includedN. split; [|by eauto]. by intros [->|?]. Qed. Lemma Some_included_total `{CmraTotal A} a b : Some a ≼ Some b ↔ a ≼ b. - Proof. rewrite Some_included. split; [|eauto]. by intros [->|?]. Qed. + Proof. rewrite Some_included. split; [|by eauto]. by intros [->|?]. Qed. Lemma Some_includedN_exclusive n a `{!Exclusive a} b : Some a ≼{n} Some b → ✓{n} b → a ≡{n}≡ b. diff --git a/theories/algebra/coPset.v b/theories/algebra/coPset.v index 0250c6922dd282329fdd19bbe1eb38808869d8de..dcd07527b2f44ed3006152d757a4ee7d56e42a71 100644 --- a/theories/algebra/coPset.v +++ b/theories/algebra/coPset.v @@ -56,7 +56,7 @@ Section coPset. Proof. intros (Z&->&?)%subseteq_disjoint_union_L. rewrite local_update_unital_discrete=> Z' _ /leibniz_equiv_iff->. - split; [done|]. rewrite coPset_op_union. set_solver. + split; first done. rewrite coPset_op_union. set_solver. Qed. End coPset. diff --git a/theories/algebra/gmultiset.v b/theories/algebra/gmultiset.v index 27f1eab6753378149c3cd1ab98a48dc822f82e33..822a952e3156f2659266ee6fc354f9213db9bebc 100644 --- a/theories/algebra/gmultiset.v +++ b/theories/algebra/gmultiset.v @@ -47,7 +47,10 @@ Section gmultiset. Proof. apply discrete_cmra_discrete. Qed. Lemma gmultiset_ucmra_mixin : UcmraMixin (gmultiset K). - Proof. split; [done | | done]. intros X. by rewrite gmultiset_op_disj_union left_id_L. Qed. + Proof. + split; [done | | done]. intros X. + by rewrite gmultiset_op_disj_union left_id_L. + Qed. Canonical Structure gmultisetUR := UcmraT (gmultiset K) gmultiset_ucmra_mixin. Global Instance gmultiset_cancelable X : Cancelable X. diff --git a/theories/algebra/ofe.v b/theories/algebra/ofe.v index 8a7f5cb70720e48409668969acd96bd2bb5d9549..0b0963f54a6172a35c875a23c9fb08336323d53f 100644 --- a/theories/algebra/ofe.v +++ b/theories/algebra/ofe.v @@ -281,7 +281,9 @@ Section limit_preserving. LimitPreserving (λ x, P1 x ∧ P2 x). Proof. intros Hlim1 Hlim2 c Hc. - split; [ apply Hlim1, Hc | apply Hlim2, Hc ]. + split. + - apply Hlim1, Hc. + - apply Hlim2, Hc. Qed. Lemma limit_preserving_impl (P1 P2 : A → Prop) : diff --git a/theories/bi/derived_laws.v b/theories/bi/derived_laws.v index f8bc17d885b01866f85b28dd2ed4beb9dca3c92e..13926381f41499485cb2fe164f6adba79d64ff88 100644 --- a/theories/bi/derived_laws.v +++ b/theories/bi/derived_laws.v @@ -1565,19 +1565,19 @@ Qed. Global Instance bi_persistently_sep_weak_homomorphism `{BiPositive PROP} : WeakMonoidHomomorphism bi_sep bi_sep (≡) (@bi_persistently PROP). -Proof. split; try apply _. apply persistently_sep. Qed. +Proof. split; [by apply _ ..|]. apply persistently_sep. Qed. Global Instance bi_persistently_sep_homomorphism `{BiAffine PROP} : MonoidHomomorphism bi_sep bi_sep (≡) (@bi_persistently PROP). -Proof. split; try apply _. apply persistently_emp. Qed. +Proof. split; [by apply _ ..|]. apply persistently_emp. Qed. Global Instance bi_persistently_sep_entails_weak_homomorphism : WeakMonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@bi_persistently PROP). -Proof. split; try apply _. intros P Q; by rewrite persistently_sep_2. Qed. +Proof. split; [by apply _ ..|]. intros P Q; by rewrite persistently_sep_2. Qed. Global Instance bi_persistently_sep_entails_homomorphism : MonoidHomomorphism bi_sep bi_sep (flip (⊢)) (@bi_persistently PROP). -Proof. split; try apply _. simpl. apply persistently_emp_intro. Qed. +Proof. split; [by apply _ ..|]. simpl. apply persistently_emp_intro. Qed. (* Limits *) Lemma limit_preserving_entails {A : ofeT} `{Cofe A} (Φ Ψ : A → PROP) : diff --git a/theories/bi/derived_laws_later.v b/theories/bi/derived_laws_later.v index d693e22e0ad860d8047b2d2a9d667c52611ec2e9..1af9dd81e8fb0dbcf9edb8b867791d29cbb8f195 100644 --- a/theories/bi/derived_laws_later.v +++ b/theories/bi/derived_laws_later.v @@ -118,7 +118,7 @@ Proof. Qed. Lemma löb_alt_strong : BiLöb PROP ↔ ∀ P, (▷ P → P) ⊢ P. -Proof. split; intros HLöb P; [apply löb|]. by intros ->%entails_impl_True. Qed. +Proof. split; intros HLöb P; [by apply löb|]. by intros ->%entails_impl_True. Qed. (** Proof following https://en.wikipedia.org/wiki/L%C3%B6b's_theorem#Proof_of_L%C3%B6b's_theorem. Their [Ψ] is called [Q] in our proof. *) diff --git a/theories/bi/monpred.v b/theories/bi/monpred.v index bb30eb594a5f0d60b2240624e1e09f7292911611..f2bb21e9a3f0d255a31d0ab44790c21675dbce3d 100644 --- a/theories/bi/monpred.v +++ b/theories/bi/monpred.v @@ -520,14 +520,20 @@ Qed. Lemma monPred_objectively_and P Q : <obj> (P ∧ Q) ⊣⊢ <obj> P ∧ <obj> Q. Proof. unseal. split=>i. apply bi.equiv_spec; split=>/=. - - apply bi.and_intro; do 2 f_equiv; [apply bi.and_elim_l | apply bi.and_elim_r ]. + - apply bi.and_intro; do 2 f_equiv. + + apply bi.and_elim_l. + + apply bi.and_elim_r. - apply bi.forall_intro=>?. by rewrite !bi.forall_elim. Qed. Lemma monPred_objectively_exist {A} (Φ : A → monPred) : (∃ x, <obj> (Φ x)) ⊢ <obj> (∃ x, (Φ x)). Proof. apply bi.exist_elim=>?. f_equiv. apply bi.exist_intro. Qed. Lemma monPred_objectively_or P Q : <obj> P ∨ <obj> Q ⊢ <obj> (P ∨ Q). -Proof. apply bi.or_elim; f_equiv; [apply bi.or_intro_l | apply bi.or_intro_r]. Qed. +Proof. + apply bi.or_elim; f_equiv. + - apply bi.or_intro_l. + - apply bi.or_intro_r. +Qed. Lemma monPred_objectively_sep_2 P Q : <obj> P ∗ <obj> Q ⊢ <obj> (P ∗ Q). Proof. unseal. split=>i /=. apply bi.forall_intro=>?. by rewrite !bi.forall_elim. Qed. @@ -554,7 +560,11 @@ Lemma monPred_subjectively_forall {A} (Φ : A → monPred) : (<subj> (∀ x, Φ x)) ⊢ ∀ x, <subj> (Φ x). Proof. apply bi.forall_intro=>?. f_equiv. apply bi.forall_elim. Qed. Lemma monPred_subjectively_and P Q : <subj> (P ∧ Q) ⊢ <subj> P ∧ <subj> Q. -Proof. apply bi.and_intro; f_equiv; [apply bi.and_elim_l | apply bi.and_elim_r]. Qed. +Proof. + apply bi.and_intro; f_equiv. + - apply bi.and_elim_l. + - apply bi.and_elim_r. +Qed. Lemma monPred_subjectively_exist {A} (Φ : A → monPred) : <subj> (∃ x, Φ x) ⊣⊢ ∃ x, <subj> (Φ x). Proof. unseal. split=>i. apply bi.equiv_spec; split=>/=; @@ -564,7 +574,9 @@ Lemma monPred_subjectively_or P Q : <subj> (P ∨ Q) ⊣⊢ <subj> P ∨ <subj> Proof. unseal. split=>i. apply bi.equiv_spec; split=>/=. - apply bi.exist_elim=>?. by rewrite -!bi.exist_intro. - - apply bi.or_elim; do 2 f_equiv; [apply bi.or_intro_l | apply bi.or_intro_r]. + - apply bi.or_elim; do 2 f_equiv. + + apply bi.or_intro_l. + + apply bi.or_intro_r. Qed. Lemma monPred_subjectively_sep P Q : <subj> (P ∗ Q) ⊢ <subj> P ∗ <subj> Q.