Local Open Scope logic.
Hint Rewrite <- prop_and : gather_prop.
Hint Rewrite gather_prop_left gather_prop_right : gather_prop.
Definition not_a_prop {A} (P: A) := True.
Ltac not_a_prop := match goal with
| |- not_a_prop  (prop _) => fail 1
| |- _ => apply Coq.Init.Logic.I
end.
Hint Rewrite @gather_prop3 using not_a_prop : gather_prop.
Hint Rewrite @gather_prop4 using not_a_prop : gather_prop.
Hint Rewrite @gather_prop5 using not_a_prop : gather_prop.
Hint Rewrite @sepcon_andp_prop @sepcon_andp_prop' : gather_prop.
(*Hint Rewrite <- sepcon_assoc : gather_prop.*)
(* These versions can sometimes take minutes, when A and B can't be unified Hint Extern 1 (_ |-- _) => (simple apply (@derives_refl mpred _) ) : cancel. Hint Extern 1 (_ |-- |> _) => (simple apply (@now_later mpred _ _) ) : cancel. *)
Hint Extern 2 (?A |-- ?B) => (constr_eq A B; simple apply derives_refl) : cancel.
Hint Extern 2 (?A |-- |> ?B) => (constr_eq A B; simple apply now_later) : cancel.
Ltac cancel1 :=
first [
simple apply cancel1_here; [
try match goal with H := _ : list mpred |- _ => clear H end; (*
this line is to work around Coq 8.4 bug,
Anomaly: undefined_evars_of_term *)
solve [eauto with nocore cancel]
| ]
| simple apply cancel1_next; cancel1
| simple apply cancel1_last; [
try match goal with H := _ : list mpred |- _ => clear H end; (*
this line is to work around Coq 8.4 bug,
Anomaly: undefined_evars_of_term *)
solve [eauto with nocore cancel] | ]
].
Ltac cancel2 :=
simple apply cancel1_start;
cancel1;
repeat simple apply cancel1_finish1;
try simple apply cancel1_finish2.
Ltac lift1 a e1 rho  :=
match e1 with
| lift0 rho => constr:(a)
| lift0 ?a1 => constr: (lift0 (a a1))
| _ => constr:(lift1 a e1)
end.
Ltac lift2 a e1 e2 rho :=
match e1 with
|  lift0 ?a1 => lift1 (a a1) e2 rho
| _ => constr:(lift2 a e1 e2)
end.
Ltac lift3 a e1 e2 e3 rho :=
match e1 with
|  lift0 ?a1 => lift2 (a a1) e2 e3 rho
| _ => constr:(lift3 a e1 e2 e3)
end.
Ltac lift4 a e1 e2 e3 e4 rho :=
match e1 with
|  lift0 ?a1 => lift3 (a a1) e2 e3 e4 rho
| _ => constr:(lift4 a e1 e2 e3 e4)
end.
Ltac abstract_env rho P :=
match P with
| @emp mpred _ _ => constr:(@emp (environ->mpred) _ _)
| @sepcon mpred _ _ ?e1 ?e2 =>
let e1' := abstract_env rho e1 in let e2' := abstract_env rho e2
in constr:(@sepcon (environ->mpred) _ _ e1' e2')
| ?a0 ?a1 ?a2 ?e1 ?e2 ?e3 ?e4 =>
let e1' := abstract_env rho e1  in let e2' := abstract_env rho e2 in let e3' := abstract_env rho e3 in let e4' := abstract_env rho e4
in lift3 (a0 a1 a2) e1' e2' e3' e4' rho
| ?a0 ?a1 ?e1 ?e2 ?e3 ?e4 =>
let e1' := abstract_env rho e1 in let e2' := abstract_env rho e2 in let e3' := abstract_env rho e3 in let e4' := abstract_env rho e4
in lift3 (a0 a1) e1' e2' e3' e4' rho
| ?a0 ?e1 ?e2 ?e3 ?e4 =>
let e1' := abstract_env rho e1 in let e2' := abstract_env rho e2 in let e3' := abstract_env rho e3 in let e4' := abstract_env rho e4
in lift4 a0 e1' e2' e3' e4' rho
| ?a0 ?e1 ?e2 ?e3 =>
let e1' := abstract_env rho e1 in let e2' := abstract_env rho e2 in let e3' := abstract_env rho e3
in lift3 a0 e1' e2' e3' rho
| ?a0 ?e1 ?e2 =>
let e1' := abstract_env rho e1 in let e2' := abstract_env rho e2
in lift2 a0 e1' e2' rho
| ?a0 ?e1 => let e1' := abstract_env rho e1 in lift1 a0 e1' rho
| rho => constr: (lift1)
| ?a => constr:(lift0 a)
end.
Ltac fixup_lifts :=
repeat
match goal with
| |- context[@lift0 mpred] => change (@lift0 mpred) with (@liftx (LiftEnviron mpred))
| |- context[@lift1 ?A] => change (@lift1 A mpred) with (@liftx (Tarrow A (LiftEnviron mpred)))
| |- context[@lift2 ?A ?B] =>  change (@lift2 A B mpred) with (@liftx (Tarrow A (Tarrow B (LiftEnviron mpred))))
| |- context[@lift3 ?A ?B ?C] => change (@lift3 A B C mpred) with (@liftx (Tarrow A (Tarrow B (Tarrow C (LiftEnviron mpred)))))
| |- context[@lift4 ?A ?B ?C ?D] => change (@lift4 A B C D mpred) with (@liftx (Tarrow A (Tarrow B (Tarrow C (Tarrow D (LiftEnviron mpred))))))
end.
Fixpoint fold_right_sepconx (l: list mpred) : mpred :=
match l with
| nil => emp
| a::nil => a
| a::b => a * fold_right_sepconx b
end.
Definition fold_left_sepconx (l: list mpred) : mpred :=
match l with
| nil => emp
| a::l => (fix fold_left_sepconx (a: mpred) (l: list mpred) {struct l}: mpred :=
match l with
| nil => a
| b :: l => fold_left_sepconx (sepcon a b) l
end) a l
end.
Ltac unfold_right_sepcon A :=
lazymatch A with
| (?B * ?C)%logic => let x := unfold_right_sepcon C
in let y := constr:(B :: x)
in y
| ?D => let y := constr:(D::nil) in y
end.
Ltac cancel_frame :=
match goal with
| |- _ |-- fold_right_sepcon _  => (* setup *)
rewrite !sepcon_assoc; cancel_frame
| F := ?v |- ?A |-- fold_right_sepcon ?F  => (* fast way *)
is_evar v;
apply fold_right_sepconx_eqx;
let w := unfold_right_sepcon A
in instantiate (1:=w) in (value of F);
unfold F;
unfold fold_right_sepconx;
simple apply derives_refl
(* | |- _ |-- fold_right_sepcon ?F  =>  (* slow way *)
repeat apply cancel_frame2_low;
try (unfold F; apply cancel_frame0_low);
try (unfold F; apply cancel_frame1_low)
*)
| |- ?P |-- fold_right _ _ ?F ?rho  =>
let P' := abstract_env rho P in
change ( P' rho |-- fold_right sepcon emp F rho);
fixup_lifts; cbv beta;
repeat rewrite sepcon_assoc;
repeat match goal with |- (_ * _) _ |-- _ =>
apply cancel_frame2
end;
try (unfold F; apply cancel_frame1);
try (instantiate (1:=nil) in (value of F); unfold F; apply cancel_frame0)
end.
Ltac pull_left A :=
(* For some reason in Coq 8.4pl3 and perhaps other versions, this works better than the version in log_normalize which is simply, repeat rewrite <- (pull_right A) || rewrite <- (pull_right0 A) and which sometimes fails when the terms get complicated. *)
repeat match goal with
| |- context [?Q * ?R * A] => rewrite <- (pull_right A Q R)
| |- context [?Q * A] => rewrite <- (pull_right0 A Q)
end.

Lemma cancel_left: forall P Q R: mpred,
Q |-- R -> P * Q |-- P * R.