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HeapSolver.v
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HeapSolver.v
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Require Import HahnBase.
Require Import Heaps.
Require Import Permissions.
Require Import Prelude.
(** * Heap solver *)
(** This section contains tactics that provide more
proof automation for working with heaps, in particular
for doing proofs that involve heap (cell) validity
and disjointness. *)
Module Type HeapSolver
(domains : Domains)
(heaps : Heaps domains).
Export domains heaps.
(** ** Permission heap cells *)
(** The following tactic, [phcsolve], gives some proof automation
for simple (but often occuring) properties on validity
and disjointness of permission heaps. *)
(** This tactic is defined simply as a table of patterns,
and matching proofs for these patterns. *)
Ltac phcsolve :=
clarify; match goal with
| [|- phcValid PHCfree] => apply phcValid_free
| [|- phcDisj PHCfree PHCfree] => by apply phcDisj_free_l, phcValid_free
| [|- phcDisj ?x PHCfree] => apply phcDisj_free_l; phcsolve
| [|- phcDisj PHCfree ?x] => apply phcDisj_free_r; phcsolve
| [H: phcDisj ?x ?y |- phcDisj ?x ?y] => exact H
| [H: phcDisj ?x ?y |- phcDisj ?y ?x] => symmetry; exact H
| [H: phDisj ?x ?y |- phcDisj (?x ?v) (?y ?v)] => by apply H
| [H: phDisj ?y ?x |- phcDisj (?x ?v) (?y ?v)] => symmetry; by apply H
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?y ?z] => apply phcDisj_union_l with x; [exact H1|exact H2]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj ?x ?y] => apply phcDisj_union_r with z; [exact H1|exact H2]
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?x (phcUnion ?y ?z)] => apply phcDisj_assoc_l; [exact H1|exact H2]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj (phcUnion ?x ?y) ?z] => apply phcDisj_assoc_r; [exact H1|exact H2]
| [H: phcDisj ?x ?y |- phcValid (phcUnion ?x ?y)] => by apply phcUnion_valid
| [H: phcDisj ?y ?x |- phcValid (phcUnion ?x ?y)] => symmetry in H; by apply phcUnion_valid
| [H: phcDisj ?x ?y |- phcValid ?x] => by apply phcDisj_valid_l with y
| [H: phcDisj ?x ?y |- phcValid ?y] => by apply phcDisj_valid_r with x
| [H: phcDisj ?x ?y |- phcValid ?x /\ phcValid ?y] => by apply phcDisj_valid
| [H: phcDisj ?y ?x |- phcValid ?x /\ phcValid ?y] => apply phcDisj_valid; by symmetry
| [H1: phcDisj ?y ?x, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?y ?z] => apply phcDisj_union_l with x; [symmetry; exact H1|exact H2]
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?y ?x) ?z |- phcDisj ?y ?z] => apply phcDisj_union_l with x; [exact H1|by rewrite phcUnion_comm]
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?z ?y] => symmetry; apply phcDisj_union_l with x; [exact H1|exact H2]
| [H1: phcDisj ?z ?y, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj ?x ?y] => apply phcDisj_union_r with z; [symmetry; exact H1|exact H2]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?z ?y) |- phcDisj ?x ?y] => apply phcDisj_union_r with z; [exact H1|by rewrite phcUnion_comm]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj ?y ?x] => symmetry; by apply phcDisj_union_r with z
| [H1: phcDisj ?y ?x, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?x (phcUnion ?y ?z)] => apply phcDisj_assoc_l; [symmetry; exact H1|exact H2]
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?y ?x) ?z |- phcDisj ?x (phcUnion ?y ?z)] => apply phcDisj_assoc_l; [exact H1|by rewrite phcUnion_comm]
| [H1: phcDisj ?x ?y, H2: phcDisj (phcUnion ?x ?y) ?z |- phcDisj ?x (phcUnion ?z ?y)] => rewrite phcUnion_comm; by apply phcDisj_assoc_l
| [H1: phcDisj ?z ?y, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj (phcUnion ?x ?y) ?z] => apply phcDisj_assoc_r; [symmetry; exact H1|exact H2]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?z ?y) |- phcDisj (phcUnion ?x ?y) ?z] => apply phcDisj_assoc_r; [exact H1|by rewrite phcUnion_comm]
| [H1: phcDisj ?y ?z, H2: phcDisj ?x (phcUnion ?y ?z) |- phcDisj (phcUnion ?y ?x) ?z] => rewrite phcUnion_comm; by apply phcDisj_assoc_r
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?y ?z] => symmetry in H2; by apply phcDisj_union_l with x
| [H1: phcDisj ?y ?x, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?y ?z] => symmetry in H1; symmetry in H2; by apply phcDisj_union_l with x
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?y ?x) |- phcDisj ?y ?z] => symmetry in H2; apply phcDisj_union_l with x; auto; rewrite phcUnion_comm; auto; fail
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?z ?y] => symmetry; symmetry in H2; apply phcDisj_union_l with x; auto; fail
| [H1: phcDisj ?y ?z, H2: phcDisj (phcUnion ?y ?z) ?x |- phcDisj ?x ?y] => symmetry in H2; by apply phcDisj_union_r with z
| [H1: phcDisj ?z ?y, H2: phcDisj (phcUnion ?y ?z) ?x |- phcDisj ?x ?y] => symmetry in H1; symmetry in H2; by apply phcDisj_union_r with z
| [H1: phcDisj ?y ?z, H2: phcDisj (phcUnion ?z ?y) ?x |- phcDisj ?x ?y] => symmetry in H2; apply phcDisj_union_r with z; auto; by rewrite phcUnion_comm
| [H1: phcDisj ?y ?z, H2: phcDisj (phcUnion ?y ?z) ?x |- phcDisj ?y ?x] => symmetry; symmetry in H2; by apply phcDisj_union_r with z
| [H1: phcDisj ?y ?x, H2: phcDisj (phcUnion ?y ?x) ?z |- phcDisj ?x (phcUnion ?z ?y)] => rewrite phcUnion_comm in H2; rewrite phcUnion_comm; apply phcDisj_assoc_l; [by symmetry|exact H2]
| [H1: phcDisj ?z ?y, H2: phcDisj ?x (phcUnion ?z ?y) |- phcDisj (phcUnion ?y ?x) ?z] => rewrite phcUnion_comm in H2; rewrite phcUnion_comm; apply phcDisj_assoc_r; [by symmetry|exact H2]
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?x (phcUnion ?y ?z)] => symmetry in H2; by apply phcDisj_assoc_l
| [H1: phcDisj ?y ?x, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?x (phcUnion ?y ?z)] => symmetry in H1; symmetry in H2; by apply phcDisj_assoc_l
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?y ?x) |- phcDisj ?x (phcUnion ?y ?z)] => apply phcDisj_assoc_l; [exact H1|symmetry; by rewrite phcUnion_comm]
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj ?x (phcUnion ?z ?y)] => rewrite phcUnion_comm; apply phcDisj_assoc_l; [exact H1|by symmetry]
| [H1: phcDisj ?y ?x, H2: phcDisj ?z (phcUnion ?y ?x) |- phcDisj ?x (phcUnion ?y ?z)] => apply phcDisj_assoc_l; [by symmetry|]; symmetry; rewrite phcUnion_comm; auto; by symmetry
| [H1: phcDisj ?y ?x, H2: phcDisj ?z (phcUnion ?y ?x) |- phcDisj ?x (phcUnion ?z ?y)] => rewrite phcUnion_comm in H2; rewrite phcUnion_comm; apply phcDisj_assoc_l; auto; by symmetry
| [H1: phcDisj ?y ?x, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj (phcUnion ?y ?z) ?x] => symmetry; apply phcDisj_assoc_l; by symmetry
| [H1: phcDisj ?x ?y, H2: phcDisj ?z (phcUnion ?x ?y) |- phcDisj (phcUnion ?z ?y) ?x] => apply phcDisj_assoc_r; [by symmetry|by rewrite phcUnion_comm]
| [|- phcUnion ?x (phcUnion ?y ?z) = phcUnion (phcUnion ?x ?y) ?z] => by apply phcUnion_assoc
| [|- phcUnion (phcUnion ?x ?y) ?z = phcUnion ?x (phcUnion ?y ?z)] => symmetry; by apply phcUnion_assoc
| [|- phcUnion ?x ?y = phcUnion ?y ?x] => by apply phcUnion_comm
| [|- phcUnion ?x PHCfree = ?x] => by apply phcUnion_free_l
| [|- phcUnion PHCfree ?x = ?x] => by apply phcUnion_free_r
| [|- ?x = phcUnion ?x PHCfree] => symmetry; by apply phcUnion_free_l
| [|- ?x = phcUnion PHCfree ?x] => symmetry; by apply phcUnion_free_r
| [|- phcUnion ?x (phIden _) = ?x] => unfold phIden; by apply phcUnion_free_l
| [|- phcUnion (phIden _) ?x = ?x] => unfold phIden; by apply phcUnion_free_r
| [|- phcUnion ?x (phcUnion ?y ?z) = phcUnion ?y (phcUnion ?x ?z)] => by apply phcUnion_swap_l
| [|- phcUnion (phcUnion ?x ?y) ?z = phcUnion (phcUnion ?x ?z) ?y] => by apply phcUnion_swap_r
| [|- phcUnion (phcUnion ?x ?z) (phcUnion ?y ?w) = phcUnion (phcUnion ?x ?y) (phcUnion ?z ?w)] => by apply phcUnion_compat
| [H: phcEntire ?x |- phcEntire (phcUnion ?x ?y)] => apply phcEntire_union; [clear H; phcsolve|by left]
| [H: phcEntire ?y |- phcEntire (phcUnion ?x ?y)] => apply phcEntire_union; [clear H; phcsolve|by right]
| [H: phcEntire (?x ?v) |- phcEntire (phUnion ?x ?y ?v)] => apply phcEntire_union; [clear H; phcsolve|by left]
| [H: phcEntire (?y ?v) |- phcEntire (phUnion ?x ?y ?v)] => apply phcEntire_union; [clear H; phcsolve|by right]
| _ => fail
end.
(** *** Unit tests *)
(** Below several unit tests are given for [phcsolve]. *)
Goal phcValid PHCfree.
Proof. phcsolve. Qed.
Goal phcDisj PHCfree PHCfree.
Proof. phcsolve. Qed.
Goal forall x, phcValid x -> phcDisj x PHCfree.
Proof. intros. phcsolve. Qed.
Goal forall x, phcValid x -> phcDisj PHCfree x.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcDisj y x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion x y) z.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcValid (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj y x -> phcValid (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcValid x.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcValid y.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcValid x /\ phcValid y.
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj y x -> phcValid x /\ phcValid y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion x y) z -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion y x) z -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj z y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion y z) -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion z y) -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj y x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion y x) z -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion z y).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion x y) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion x y) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion y x) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion y x) -> phcDisj y z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj z y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion y z) x -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj z y -> phcDisj (phcUnion y z) x -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion z y) x -> phcDisj x y.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion y z) x -> phcDisj y x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion y x) z -> phcDisj x (phcUnion z y).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion y x) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion y x) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion z y).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion z y).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion y z) x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion y z) x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion z y) x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj (phcUnion y z) x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj (phcUnion z y) x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcUnion x (phcUnion y z) = phcUnion (phcUnion x y) z.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcUnion (phcUnion x y) z = phcUnion x (phcUnion y z).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcUnion x y = phcUnion y x.
Proof. intros. phcsolve. Qed.
Goal forall x, phcUnion x PHCfree = x.
Proof. intros. phcsolve. Qed.
Goal forall x, phcUnion PHCfree x = x.
Proof. intros. phcsolve. Qed.
Goal forall x, x = phcUnion x PHCfree.
Proof. intros. phcsolve. Qed.
Goal forall x, x = phcUnion PHCfree x.
Proof. intros. phcsolve. Qed.
Goal forall x v, phcUnion x (phIden v) = x.
Proof. intros. phcsolve. Qed.
Goal forall x v, phcUnion (phIden v) x = x.
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcUnion x (phcUnion y z) = phcUnion y (phcUnion x z).
Proof. intros. phcsolve. Qed.
Goal forall x y z, phcUnion (phcUnion x y) z = phcUnion (phcUnion x z) y.
Proof. intros. phcsolve. Qed.
Goal forall x y z w, phcUnion (phcUnion x z) (phcUnion y w) = phcUnion (phcUnion x y) (phcUnion z w).
Proof. intros. phcsolve. Qed.
Goal forall x y z w, phcUnion (phcUnion x y) (phcUnion z w) = phcUnion (phcUnion x z) (phcUnion y w).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcEntire x -> phcEntire (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj y x -> phcEntire x -> phcEntire (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj x y -> phcEntire y -> phcEntire (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y, phcDisj y x -> phcEntire y -> phcEntire (phcUnion x y).
Proof. intros. phcsolve. Qed.
Goal forall x y v, phDisj x y -> phcEntire (x v) -> phcEntire (phUnion x y v).
Proof. intros. phcsolve. Qed.
Goal forall x y v, phDisj y x -> phcEntire (x v) -> phcEntire (phUnion x y v).
Proof. intros. phcsolve. Qed.
Goal forall x y v, phDisj x y -> phcEntire (y v) -> phcEntire (phUnion x y v).
Proof. intros. phcsolve. Qed.
Goal forall x y v, phDisj y x -> phcEntire (y v) -> phcEntire (phUnion x y v).
Proof. intros. phcsolve. Qed.
(** ** Permission heap *)
(** The following tactic, [phsolve], gives some proof automation
for simple (but often occuring) properties on validity
and disjointness (and more) of permission heaps. *)
(** This tactic is defined simply as a table of patterns,
and matching proofs for these patterns. *)
Ltac phsolve :=
clarify; match goal with
| [|- phValid phIden] => apply phValid_iden
| [|- phDisj phIden phIden] => by apply phDisj_iden_l, phValid_iden
| [|- phDisj ?x phIden] => apply phDisj_iden_l; phsolve
| [|- phDisj phIden ?x] => apply phDisj_iden_r; phsolve
| [H: phDisj ?x ?y |- phDisj ?x ?y] => exact H
| [H: phDisj ?x ?y |- phDisj ?y ?x] => symmetry; exact H
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?y ?z] => apply phDisj_union_l with x; [exact H1|exact H2]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?y ?z) |- phDisj ?x ?y] => apply phDisj_union_r with z; [exact H1|exact H2]
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?x (phUnion ?y ?z)] => apply phDisj_assoc_l; [exact H1|exact H2]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?y ?z) |- phDisj (phUnion ?x ?y) ?z] => apply phDisj_assoc_r; [exact H1|exact H2]
| [H: phDisj ?x ?y |- phValid (phUnion ?x ?y)] => by apply phUnion_valid
| [H: phDisj ?y ?x |- phValid (phUnion ?x ?y)] => symmetry in H; by apply phUnion_valid
| [H: phDisj ?x ?y |- phValid ?x] => by apply phDisj_valid_l with y
| [H: phDisj ?x ?y |- phValid ?y] => by apply phDisj_valid_r with x
| [H: phDisj ?x ?y |- phValid ?x /\ phValid ?y] => by apply phDisj_valid
| [H: phDisj ?y ?x |- phValid ?x /\ phValid ?y] => apply phDisj_valid; by symmetry
| [H1: phDisj ?y ?x, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?y ?z] => apply phDisj_union_l with x; [symmetry; exact H1|exact H2]
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?y ?x) ?z |- phDisj ?y ?z] => apply phDisj_union_l with x; [exact H1|by rewrite phUnion_comm]
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?z ?y] => symmetry; apply phDisj_union_l with x; [exact H1|exact H2]
| [H1: phDisj ?z ?y, H2: phDisj ?x (phUnion ?y ?z) |- phDisj ?x ?y] => apply phDisj_union_r with z; [symmetry; exact H1|exact H2]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?z ?y) |- phDisj ?x ?y] => apply phDisj_union_r with z; [exact H1|by rewrite phUnion_comm]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?y ?z) |- phDisj ?y ?x] => symmetry; by apply phDisj_union_r with z
| [H1: phDisj ?y ?x, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?x (phUnion ?y ?z)] => apply phDisj_assoc_l; [symmetry; exact H1|exact H2]
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?y ?x) ?z |- phDisj ?x (phUnion ?y ?z)] => apply phDisj_assoc_l; [exact H1|by rewrite phUnion_comm]
| [H1: phDisj ?x ?y, H2: phDisj (phUnion ?x ?y) ?z |- phDisj ?x (phUnion ?z ?y)] => rewrite phUnion_comm; by apply phDisj_assoc_l
| [H1: phDisj ?z ?y, H2: phDisj ?x (phUnion ?y ?z) |- phDisj (phUnion ?x ?y) ?z] => apply phDisj_assoc_r; [symmetry; exact H1|exact H2]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?z ?y) |- phDisj (phUnion ?x ?y) ?z] => apply phDisj_assoc_r; [exact H1|by rewrite phUnion_comm]
| [H1: phDisj ?y ?z, H2: phDisj ?x (phUnion ?y ?z) |- phDisj (phUnion ?y ?x) ?z] => rewrite phUnion_comm; by apply phDisj_assoc_r
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?y ?z] => symmetry in H2; by apply phDisj_union_l with x
| [H1: phDisj ?y ?x, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?y ?z] => symmetry in H1; symmetry in H2; by apply phDisj_union_l with x
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?y ?x) |- phDisj ?y ?z] => symmetry in H2; apply phDisj_union_l with x; auto; rewrite phUnion_comm; auto; fail
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?z ?y] => symmetry; symmetry in H2; apply phDisj_union_l with x; auto; fail
| [H1: phDisj ?y ?z, H2: phDisj (phUnion ?y ?z) ?x |- phDisj ?x ?y] => symmetry in H2; by apply phDisj_union_r with z
| [H1: phDisj ?z ?y, H2: phDisj (phUnion ?y ?z) ?x |- phDisj ?x ?y] => symmetry in H1; symmetry in H2; by apply phDisj_union_r with z
| [H1: phDisj ?y ?z, H2: phDisj (phUnion ?z ?y) ?x |- phDisj ?x ?y] => symmetry in H2; apply phDisj_union_r with z; auto; by rewrite phUnion_comm
| [H1: phDisj ?y ?z, H2: phDisj (phUnion ?y ?z) ?x |- phDisj ?y ?x] => symmetry; symmetry in H2; by apply phDisj_union_r with z
| [H1: phDisj ?y ?x, H2: phDisj (phUnion ?y ?x) ?z |- phDisj ?x (phUnion ?z ?y)] => rewrite phUnion_comm in H2; rewrite phUnion_comm; apply phDisj_assoc_l; [by symmetry|exact H2]
| [H1: phDisj ?z ?y, H2: phDisj ?x (phUnion ?z ?y) |- phDisj (phUnion ?y ?x) ?z] => rewrite phUnion_comm in H2; rewrite phUnion_comm; apply phDisj_assoc_r; [by symmetry|exact H2]
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?x (phUnion ?y ?z)] => symmetry in H2; by apply phDisj_assoc_l
| [H1: phDisj ?y ?x, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?x (phUnion ?y ?z)] => symmetry in H1; symmetry in H2; by apply phDisj_assoc_l
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?y ?x) |- phDisj ?x (phUnion ?y ?z)] => apply phDisj_assoc_l; [exact H1|symmetry; by rewrite phUnion_comm]
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?x ?y) |- phDisj ?x (phUnion ?z ?y)] => rewrite phUnion_comm; apply phDisj_assoc_l; [exact H1|by symmetry]
| [H1: phDisj ?y ?x, H2: phDisj ?z (phUnion ?y ?x) |- phDisj ?x (phUnion ?y ?z)] => apply phDisj_assoc_l; [by symmetry|]; symmetry; rewrite phUnion_comm; auto; by symmetry
| [H1: phDisj ?y ?x, H2: phDisj ?z (phUnion ?y ?x) |- phDisj ?x (phUnion ?z ?y)] => rewrite phUnion_comm in H2; rewrite phUnion_comm; apply phDisj_assoc_l; auto; by symmetry
| [H1: phDisj ?y ?x, H2: phDisj ?z (phUnion ?x ?y) |- phDisj (phUnion ?y ?z) ?x] => symmetry; apply phDisj_assoc_l; by symmetry
| [H1: phDisj ?x ?y, H2: phDisj ?z (phUnion ?x ?y) |- phDisj (phUnion ?z ?y) ?x] => apply phDisj_assoc_r; [by symmetry|by rewrite phUnion_comm]
| [H: phDisj ?x ?y |- phcDisj (?x ?v) (?y ?v)] => by apply H
| [H: phDisj ?y ?x |- phcDisj (?x ?v) (?y ?v)] => symmetry; by apply H
| [|- phDisj (phUpdate _ ?l _) (phUpdate _ ?l _)] => apply phDisj_upd; [phcsolve|phsolve]
| [|- phUnion (phUnion ?x ?y) ?z = phUnion ?x (phUnion ?y ?z)] => by apply phUnion_assoc
| [|- phUnion ?x (phUnion ?y ?z) = phUnion (phUnion ?x ?y) ?z] => symmetry; by apply phUnion_assoc
| [|- phUnion ?x ?y = phUnion ?y ?x] => by apply phUnion_comm
| [|- phUnion ?x phIden = ?x] => by apply phUnion_iden_l
| [|- phUnion phIden ?x = ?x] => by apply phUnion_iden_r
| [|- phUnion ?x (phUnion ?y ?z) = phUnion ?y (phUnion ?x ?z)] => by apply phUnion_swap_l
| [|- phUnion (phUnion ?x ?y) ?z = phUnion (phUnion ?x ?z) ?y] => by apply phUnion_swap_r
| [|- phUnion (phUnion ?x ?z) (phUnion ?y ?w) = phUnion (phUnion ?x ?y) (phUnion ?z ?w)] => by apply phUnion_compat
| [|- phUnion ?x ?y ?v = phcUnion (?x ?v) (?y ?v)] => by apply phUnion_cell
| [|- phcUnion (?x ?v) (?y ?v) = phUnion ?x ?y ?v] => by apply phUnion_cell
| [|- phcEntire (phUnion ?x ?y ?v)] => rewrite <- phUnion_cell; phcsolve
| _ => fail
end.
(** *** Unit tests *)
(** Below several unit tests are given for [pmesolve]. *)
Goal phValid phIden.
Proof. phsolve. Qed.
Goal phDisj phIden phIden.
Proof. phsolve. Qed.
Goal forall x, phValid x -> phDisj x phIden.
Proof. intros. phsolve. Qed.
Goal forall x, phValid x -> phDisj phIden x.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phDisj y x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj (phUnion x y) z.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phValid (phUnion x y).
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj y x -> phValid (phUnion x y).
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phValid x.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phValid y.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj x y -> phValid x /\ phValid y.
Proof. intros. phsolve. Qed.
Goal forall x y, phDisj y x -> phValid x /\ phValid y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj (phUnion x y) z -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion y x) z -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj z y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj z y -> phDisj x (phUnion y z) -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion z y) -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj y x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj (phUnion x y) z -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion y x) z -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj x (phUnion z y).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj z y -> phDisj x (phUnion y z) -> phDisj (phUnion x y) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion z y) -> phDisj (phUnion x y) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj (phUnion y x) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion y x) -> phDisj y z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj z y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj (phUnion y z) x -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj z y -> phDisj (phUnion y z) x -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj (phUnion z y) x -> phDisj x y.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y z -> phDisj (phUnion y z) x -> phDisj y x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj (phUnion y x) z -> phDisj x (phUnion z y).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj z y -> phDisj x (phUnion z y) -> phDisj (phUnion y x) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj z y -> phDisj x (phUnion z y) -> phDisj (phUnion y x) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion y x) -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj x (phUnion z y).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj x (phUnion z y).
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj (phUnion y z) x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj (phUnion y z) x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj (phUnion z y) x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj (phUnion y z) x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj (phUnion z y) x.
Proof. intros. phsolve. Qed.
Goal forall x y v, phDisj x y -> phcDisj (x v) (y v).
Proof. intros. phsolve. Qed.
Goal forall x y v, phDisj y x -> phcDisj (x v) (y v).
Proof. intros. phsolve. Qed.
Goal forall x y l v w, phDisj x y -> phcDisj v w -> phDisj (phUpdate x l v) (phUpdate y l w).
Proof. intros. phsolve. Qed.
Goal forall x y z, phUnion x (phUnion y z) = phUnion (phUnion x y) z.
Proof. intros. phsolve. Qed.
Goal forall x y z, phUnion (phUnion x y) z = phUnion x (phUnion y z).
Proof. intros. phsolve. Qed.
Goal forall x y, phUnion x y = phUnion y x.
Proof. intros. phsolve. Qed.
Goal forall x, phUnion x phIden = x.
Proof. intros. phsolve. Qed.
Goal forall x, phUnion phIden x = x.
Proof. intros. phsolve. Qed.
Goal forall x y z, phUnion x (phUnion y z) = phUnion y (phUnion x z).
Proof. intros. phsolve. Qed.
Goal forall x y z, phUnion (phUnion x y) z = phUnion (phUnion x z) y.
Proof. intros. phsolve. Qed.
Goal forall x y z w, phUnion (phUnion x z) (phUnion y w) = phUnion (phUnion x y) (phUnion z w).
Proof. intros. phsolve. Qed.
Goal forall x y z w, phUnion (phUnion x y) (phUnion z w) = phUnion (phUnion x z) (phUnion y w).
Proof. intros. phsolve. Qed.
Goal forall x y v, phUnion x y v = phcUnion (x v) (y v).
Proof. intros. phsolve. Qed.
Goal forall x y v, phcUnion (x v) (y v) = phUnion x y v.
Proof. intros. phsolve. Qed.
Goal forall x y v, phcDisj (x v) (y v) -> phcEntire (x v) -> phcEntire (phUnion x y v).
Proof. intros. phsolve. Qed.
End HeapSolver.