Commit bd3a8a9f authored by Jacques-Henri Jourdan's avatar Jacques-Henri Jourdan

Improve proofmode for monpred.

parent ad06e03a
...@@ -16,8 +16,50 @@ Arguments MakeMonPredCar _ _%I _%I. ...@@ -16,8 +16,50 @@ Arguments MakeMonPredCar _ _%I _%I.
Global Instance make_monPred_car_pure φ i : MakeMonPredCar i ⌜φ⌝ ⌜φ⌝. Global Instance make_monPred_car_pure φ i : MakeMonPredCar i ⌜φ⌝ ⌜φ⌝.
Proof. rewrite /MakeMonPredCar. by unseal. Qed. Proof. rewrite /MakeMonPredCar. by unseal. Qed.
Global Instance make_monPred_car_internal_eq {A : ofeT} (x y : A) i :
MakeMonPredCar i (x y) (x y).
Proof. rewrite /MakeMonPredCar. by unseal. Qed.
Global Instance make_monPred_car_emp i : MakeMonPredCar i emp emp. Global Instance make_monPred_car_emp i : MakeMonPredCar i emp emp.
Proof. rewrite /MakeMonPredCar. by unseal. Qed. Proof. rewrite /MakeMonPredCar. by unseal. Qed.
Global Instance make_monPred_car_sep i P 𝓟 Q 𝓠 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i Q 𝓠
MakeMonPredCar i (P Q) (𝓟 𝓠).
Proof. rewrite /MakeMonPredCar=><-<-. by unseal. Qed.
Global Instance make_monPred_car_and i P 𝓟 Q 𝓠 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i Q 𝓠
MakeMonPredCar i (P Q) (𝓟 𝓠).
Proof. rewrite /MakeMonPredCar=><-<-. by unseal. Qed.
Global Instance make_monPred_car_or i P 𝓟 Q 𝓠 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i Q 𝓠
MakeMonPredCar i (P Q) (𝓟 𝓠).
Proof. rewrite /MakeMonPredCar=><-<-. by unseal. Qed.
Global Instance make_monPred_car_forall {A} i (Φ : A monPred) (Ψ : A PROP) :
( a, MakeMonPredCar i (Φ a) (Ψ a)) MakeMonPredCar i ( a, Φ a) ( a, Ψ a).
Proof. rewrite /MakeMonPredCar=>H. setoid_rewrite <- H. by unseal. Qed.
Global Instance make_monPred_car_exists {A} i (Φ : A monPred) (Ψ : A PROP) :
( a, MakeMonPredCar i (Φ a) (Ψ a)) MakeMonPredCar i ( a, Φ a) ( a, Ψ a).
Proof. rewrite /MakeMonPredCar=>H. setoid_rewrite <- H. by unseal. Qed.
Global Instance make_monPred_car_persistently i P 𝓟 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i (bi_persistently P) (bi_persistently 𝓟).
Proof. rewrite /MakeMonPredCar=><-. by unseal. Qed.
Global Instance make_monPred_car_affinely i P 𝓟 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i (bi_affinely P) (bi_affinely 𝓟).
Proof. rewrite /MakeMonPredCar=><-. by unseal. Qed.
Global Instance make_monPred_car_absorbingly i P 𝓟 :
MakeMonPredCar i P 𝓟 MakeMonPredCar i (bi_absorbingly P) (bi_absorbingly 𝓟).
Proof. rewrite /MakeMonPredCar=><-. by unseal. Qed.
Global Instance make_monPred_car_plainly i P Φ :
( j, MakeMonPredCar j P (Φ j))
MakeMonPredCar i (bi_plainly P) ( j, bi_plainly (Φ j)).
Proof. rewrite /MakeMonPredCar=>H. unseal. by do 3 f_equiv. Qed.
Global Instance make_monPred_car_persistently_if i P 𝓟 p :
MakeMonPredCar i P 𝓟
MakeMonPredCar i (bi_persistently_if p P) (bi_persistently_if p 𝓟).
Proof. destruct p; simpl; apply _. Qed.
Global Instance make_monPred_car_affinely_if i P 𝓟 p :
MakeMonPredCar i P 𝓟
MakeMonPredCar i (bi_affinely_if p P) (bi_affinely_if p 𝓟).
Proof. destruct p; simpl; apply _. Qed.
Global Instance make_monPred_car_embed i : MakeMonPredCar i P P. Global Instance make_monPred_car_embed i : MakeMonPredCar i P P.
Proof. rewrite /MakeMonPredCar. by unseal. Qed. Proof. rewrite /MakeMonPredCar. by unseal. Qed.
Global Instance make_monPred_car_in i j : MakeMonPredCar j (monPred_in i) i j. Global Instance make_monPred_car_in i j : MakeMonPredCar j (monPred_in i) i j.
...@@ -25,14 +67,47 @@ Proof. rewrite /MakeMonPredCar. by unseal. Qed. ...@@ -25,14 +67,47 @@ Proof. rewrite /MakeMonPredCar. by unseal. Qed.
Global Instance make_monPred_car_default i P : MakeMonPredCar i P (P i) | 100. Global Instance make_monPred_car_default i P : MakeMonPredCar i P (P i) | 100.
Proof. by rewrite /MakeMonPredCar. Qed. Proof. by rewrite /MakeMonPredCar. Qed.
Global Instance from_assumption_make_monPred_car_l p i P 𝓟 :
MakeMonPredCar i P 𝓟 FromAssumption p (P i) 𝓟.
Proof.
rewrite /MakeMonPredCar /FromAssumption=><-.
apply bi.affinely_persistently_if_elim.
Qed.
Global Instance from_assumption_make_monPred_car_r p i P 𝓟 :
MakeMonPredCar i P 𝓟 FromAssumption p 𝓟 (P i).
Proof.
rewrite /MakeMonPredCar /FromAssumption=><-.
apply bi.affinely_persistently_if_elim.
Qed.
Global Instance as_valid_monPred_car φ P (Φ : I PROP) : Global Instance as_valid_monPred_car φ P (Φ : I PROP) :
AsValid φ P ( i, MakeMonPredCar i P (Φ i)) AsValid φ ( i, Φ i). AsValid φ P ( i, MakeMonPredCar i P (Φ i)) AsValid φ ( i, Φ i) | 100.
Proof. Proof.
rewrite /MakeMonPredCar /AsValid /bi_valid=> -> EQ. setoid_rewrite <-EQ. rewrite /MakeMonPredCar /AsValid /bi_valid=> -> EQ. setoid_rewrite <-EQ.
unseal; split. unseal; split.
- move=>[/= /bi.forall_intro //]. - move=>[/= /bi.forall_intro //].
- move=>H. split=>i. rewrite /= H bi.forall_elim //. - move=>H. split=>i. rewrite /= H bi.forall_elim //.
Qed. Qed.
Global Instance as_valid_monPred_car_wand φ P Q (Φ Ψ : I PROP) :
AsValid φ (P - Q)
( i, MakeMonPredCar i P (Φ i)) ( i, MakeMonPredCar i Q (Ψ i))
AsValid φ ( i, Φ i - Ψ i).
Proof.
rewrite /AsValid /MakeMonPredCar. intros -> EQ1 EQ2.
setoid_rewrite <-EQ1. setoid_rewrite <-EQ2. split.
- move=>/bi.wand_entails H. setoid_rewrite H. by iIntros (i) "$".
- move=>H. apply bi.entails_wand. split=>i. iIntros "H". by iApply H.
Qed.
Global Instance as_valid_monPred_car_equiv φ P Q (Φ Ψ : I PROP) :
AsValid φ (P - Q)
( i, MakeMonPredCar i P (Φ i)) ( i, MakeMonPredCar i Q (Ψ i))
AsValid φ ( i, Φ i - Ψ i).
Proof.
rewrite /AsValid /MakeMonPredCar. intros -> EQ1 EQ2.
setoid_rewrite <-EQ1. setoid_rewrite <-EQ2. split.
- move=>/bi.wand_iff_equiv H. setoid_rewrite H. iIntros. iSplit; iIntros "$".
- move=>H. apply bi.equiv_wand_iff. split=>i. by iSplit; iIntros; iApply H.
Qed.
Global Instance into_pure_monPred_car P φ i : Global Instance into_pure_monPred_car P φ i :
IntoPure P φ IntoPure (P i) φ. IntoPure P φ IntoPure (P i) φ.
...@@ -66,14 +141,24 @@ Proof. ...@@ -66,14 +141,24 @@ Proof.
by destruct a, pe=><-; try unseal. by destruct a, pe=><-; try unseal.
Qed. Qed.
Global Instance into_wand_monPred_car p q R P 𝓟 Q 𝓠 i : (* FIXME : P and Q do not go through MakeMonPredCar, which prevent
IntoWand p q R P Q MakeMonPredCar i P 𝓟 MakeMonPredCar i Q 𝓠 lazilly unfolding of monPred connectives there. The reason we
IntoWand p q (R i) 𝓟 𝓠. cannot do that is that IntoWand is sometimes called with known
values, which we do not wnat to replace with existentials. *)
Global Instance into_wand_monPred_car p q R P Q i :
IntoWand p q R P Q IntoWand p q (R i) (P i) (Q i).
Proof. Proof.
rewrite /IntoWand /MakeMonPredCar /bi_affinely_if /bi_persistently_if. rewrite /IntoWand /MakeMonPredCar /bi_affinely_if /bi_persistently_if.
destruct p, q=> /bi.wand_elim_l' [/(_ i) H] <- <-; apply bi.wand_intro_r; destruct p, q=> /bi.wand_elim_l' [/(_ i) H]; apply bi.wand_intro_r;
revert H; unseal; done. revert H; unseal; done.
Qed. Qed.
Global Instance into_wand_wand'_monPred p q P Q 𝓟 𝓠 i :
IntoWand' p q ((P - Q) i) 𝓟 𝓠 IntoWand p q ((P - Q) i) 𝓟 𝓠 | 100.
Proof. done. Qed.
Global Instance into_wand_impl'_monPred p q P Q 𝓟 𝓠 i :
IntoWand' p q ((P Q) i) 𝓟 𝓠 IntoWand p q ((P Q) i) 𝓟 𝓠 | 100.
Proof. done. Qed.
Global Instance from_forall_monPred_car_wand P Q (Φ Ψ : I PROP) i : Global Instance from_forall_monPred_car_wand P Q (Φ Ψ : I PROP) i :
( j, MakeMonPredCar j P (Φ j)) ( j, MakeMonPredCar j Q (Ψ j)) ( j, MakeMonPredCar j P (Φ j)) ( j, MakeMonPredCar j Q (Ψ j))
FromForall ((P - Q) i)%I (λ j, i j Φ j - Ψ j)%I. FromForall ((P - Q) i)%I (λ j, i j Φ j - Ψ j)%I.
...@@ -201,6 +286,16 @@ Global Instance is_except_0_monPred_car i P : ...@@ -201,6 +286,16 @@ Global Instance is_except_0_monPred_car i P :
IsExcept0 P IsExcept0 (P i). IsExcept0 P IsExcept0 (P i).
Proof. rewrite /IsExcept0=>- [/(_ i)]. by unseal. Qed. Proof. rewrite /IsExcept0=>- [/(_ i)]. by unseal. Qed.
Global Instance make_monPred_car_except_0 i P 𝓠 :
MakeMonPredCar i P 𝓠 MakeMonPredCar i ( P)%I ( 𝓠)%I.
Proof. rewrite /MakeMonPredCar=><-. by unseal. Qed.
Global Instance make_monPred_car_later i P 𝓠 :
MakeMonPredCar i P 𝓠 MakeMonPredCar i ( P)%I ( 𝓠)%I.
Proof. rewrite /MakeMonPredCar=><-. by unseal. Qed.
Global Instance make_monPred_car_laterN i n P 𝓠 :
MakeMonPredCar i P 𝓠 MakeMonPredCar i (^n P)%I (^n 𝓠)%I.
Proof. rewrite /MakeMonPredCar=> <-. elim n=>//= ? <-. by unseal. Qed.
Global Instance into_except_0_monPred_car_fwd i P Q 𝓠 : Global Instance into_except_0_monPred_car_fwd i P Q 𝓠 :
IntoExcept0 P Q MakeMonPredCar i Q 𝓠 IntoExcept0 (P i) 𝓠. IntoExcept0 P Q MakeMonPredCar i Q 𝓠 IntoExcept0 (P i) 𝓠.
Proof. rewrite /IntoExcept0 /MakeMonPredCar=> -> <-. by unseal. Qed. Proof. rewrite /IntoExcept0 /MakeMonPredCar=> -> <-. by unseal. Qed.
...@@ -211,13 +306,9 @@ Proof. rewrite /IntoExcept0 /MakeMonPredCar=> H <-. rewrite H. by unseal. Qed. ...@@ -211,13 +306,9 @@ Proof. rewrite /IntoExcept0 /MakeMonPredCar=> H <-. rewrite H. by unseal. Qed.
Global Instance into_later_monPred_car i n P Q 𝓠 : Global Instance into_later_monPred_car i n P Q 𝓠 :
IntoLaterN n P Q MakeMonPredCar i Q 𝓠 IntoLaterN n (P i) 𝓠. IntoLaterN n P Q MakeMonPredCar i Q 𝓠 IntoLaterN n (P i) 𝓠.
Proof. Proof.
rewrite /IntoLaterN /MakeMonPredCar=> -> <-. rewrite /IntoLaterN /MakeMonPredCar=> -> <-. elim n=>//= ? <-. by unseal.
induction n as [|? IH]=>//. rewrite /= -IH. by unseal.
Qed. Qed.
Global Instance from_later_monPred_car i n P Q 𝓠 : Global Instance from_later_monPred_car i n P Q 𝓠 :
FromLaterN n P Q MakeMonPredCar i Q 𝓠 FromLaterN n (P i) 𝓠. FromLaterN n P Q MakeMonPredCar i Q 𝓠 FromLaterN n (P i) 𝓠.
Proof. Proof. rewrite /FromLaterN /MakeMonPredCar=> <- <-. elim n=>//= ? ->. by unseal. Qed.
rewrite /FromLaterN /MakeMonPredCar=> <- <-.
induction n as [|? IH]=>//. rewrite /= IH. by unseal.
Qed.
End sbi. End sbi.
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!
Please register or to comment