Library HeapSolver

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.
Goal phcDisj PHCfree PHCfree.
Goal forall x, phcValid x -> phcDisj x PHCfree.
Goal forall x, phcValid x -> phcDisj PHCfree x.
Goal forall x y, phcDisj x y -> phcDisj x y.
Goal forall x y, phcDisj x y -> phcDisj y x.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj y z.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj x y.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion x y) z.
Goal forall x y, phcDisj x y -> phcValid (phcUnion x y).
Goal forall x y, phcDisj y x -> phcValid (phcUnion x y).
Goal forall x y, phcDisj x y -> phcValid x.
Goal forall x y, phcDisj x y -> phcValid y.
Goal forall x y, phcDisj x y -> phcValid x /\ phcValid y.
Goal forall x y, phcDisj y x -> phcValid x /\ phcValid y.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion x y) z -> phcDisj y z.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion y x) z -> phcDisj y z.
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj z y.
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion y z) -> phcDisj x y.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion z y) -> phcDisj x y.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj y x.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion y x) z -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj x y -> phcDisj (phcUnion x y) z -> phcDisj x (phcUnion z y).
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion x y) z.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion x y) z.
Goal forall x y z, phcDisj y z -> phcDisj x (phcUnion y z) -> phcDisj (phcUnion y x) z.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj y z.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj y z.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion y x) -> phcDisj y z.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj z y.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion y z) x -> phcDisj x y.
Goal forall x y z, phcDisj z y -> phcDisj (phcUnion y z) x -> phcDisj x y.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion z y) x -> phcDisj x y.
Goal forall x y z, phcDisj y z -> phcDisj (phcUnion y z) x -> phcDisj y x.
Goal forall x y z, phcDisj y x -> phcDisj (phcUnion y x) z -> phcDisj x (phcUnion z y).
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion y x) z.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj z y -> phcDisj x (phcUnion z y) -> phcDisj (phcUnion y x) z.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj x (phcUnion z y).
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion y z).
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj x (phcUnion z y).
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion y z) x.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion y z) x.
Goal forall x y z, phcDisj x y -> phcDisj z (phcUnion x y) -> phcDisj (phcUnion z y) x.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj (phcUnion y z) x.
Goal forall x y z, phcDisj y x -> phcDisj z (phcUnion y x) -> phcDisj (phcUnion z y) x.
Goal forall x y z, phcUnion x (phcUnion y z) = phcUnion (phcUnion x y) z.
Goal forall x y z, phcUnion (phcUnion x y) z = phcUnion x (phcUnion y z).
Goal forall x y, phcUnion x y = phcUnion y x.
Goal forall x, phcUnion x PHCfree = x.
Goal forall x, phcUnion PHCfree x = x.
Goal forall x, x = phcUnion x PHCfree.
Goal forall x, x = phcUnion PHCfree x.
Goal forall x v, phcUnion x (phIden v) = x.
Goal forall x v, phcUnion (phIden v) x = x.
Goal forall x y z, phcUnion x (phcUnion y z) = phcUnion y (phcUnion x z).
Goal forall x y z, phcUnion (phcUnion x y) z = phcUnion (phcUnion x z) y.
Goal forall x y z w, phcUnion (phcUnion x z) (phcUnion y w) = phcUnion (phcUnion x y) (phcUnion z w).
Goal forall x y z w, phcUnion (phcUnion x y) (phcUnion z w) = phcUnion (phcUnion x z) (phcUnion y w).
Goal forall x y, phcDisj x y -> phcEntire x -> phcEntire (phcUnion x y).
Goal forall x y, phcDisj y x -> phcEntire x -> phcEntire (phcUnion x y).
Goal forall x y, phcDisj x y -> phcEntire y -> phcEntire (phcUnion x y).
Goal forall x y, phcDisj y x -> phcEntire y -> phcEntire (phcUnion x y).
Goal forall x y v, phDisj x y -> phcEntire (x v) -> phcEntire (phUnion x y v).
Goal forall x y v, phDisj y x -> phcEntire (x v) -> phcEntire (phUnion x y v).
Goal forall x y v, phDisj x y -> phcEntire (y v) -> phcEntire (phUnion x y v).
Goal forall x y v, phDisj y x -> phcEntire (y v) -> phcEntire (phUnion x y v).

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.
Goal phDisj phIden phIden.
Goal forall x, phValid x -> phDisj x phIden.
Goal forall x, phValid x -> phDisj phIden x.
Goal forall x y, phDisj x y -> phDisj x y.
Goal forall x y, phDisj x y -> phDisj y x.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj y z.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj x y.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj x (phUnion y z).
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj (phUnion x y) z.
Goal forall x y, phDisj x y -> phValid (phUnion x y).
Goal forall x y, phDisj y x -> phValid (phUnion x y).
Goal forall x y, phDisj x y -> phValid x.
Goal forall x y, phDisj x y -> phValid y.
Goal forall x y, phDisj x y -> phValid x /\ phValid y.
Goal forall x y, phDisj y x -> phValid x /\ phValid y.
Goal forall x y z, phDisj y x -> phDisj (phUnion x y) z -> phDisj y z.
Goal forall x y z, phDisj x y -> phDisj (phUnion y x) z -> phDisj y z.
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj z y.
Goal forall x y z, phDisj z y -> phDisj x (phUnion y z) -> phDisj x y.
Goal forall x y z, phDisj y z -> phDisj x (phUnion z y) -> phDisj x y.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj y x.
Goal forall x y z, phDisj y x -> phDisj (phUnion x y) z -> phDisj x (phUnion y z).
Goal forall x y z, phDisj x y -> phDisj (phUnion y x) z -> phDisj x (phUnion y z).
Goal forall x y z, phDisj x y -> phDisj (phUnion x y) z -> phDisj x (phUnion z y).
Goal forall x y z, phDisj z y -> phDisj x (phUnion y z) -> phDisj (phUnion x y) z.
Goal forall x y z, phDisj y z -> phDisj x (phUnion z y) -> phDisj (phUnion x y) z.
Goal forall x y z, phDisj y z -> phDisj x (phUnion y z) -> phDisj (phUnion y x) z.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj y z.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj y z.
Goal forall x y z, phDisj x y -> phDisj z (phUnion y x) -> phDisj y z.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj z y.
Goal forall x y z, phDisj y z -> phDisj (phUnion y z) x -> phDisj x y.
Goal forall x y z, phDisj z y -> phDisj (phUnion y z) x -> phDisj x y.
Goal forall x y z, phDisj y z -> phDisj (phUnion z y) x -> phDisj x y.
Goal forall x y z, phDisj y z -> phDisj (phUnion y z) x -> phDisj y x.
Goal forall x y z, phDisj y x -> phDisj (phUnion y x) z -> phDisj x (phUnion z y).
Goal forall x y z, phDisj z y -> phDisj x (phUnion z y) -> phDisj (phUnion y x) z.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj x (phUnion y z).
Goal forall x y z, phDisj z y -> phDisj x (phUnion z y) -> phDisj (phUnion y x) z.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj x (phUnion y z).
Goal forall x y z, phDisj x y -> phDisj z (phUnion y x) -> phDisj x (phUnion y z).
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj x (phUnion z y).
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj x (phUnion y z).
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj x (phUnion z y).
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj (phUnion y z) x.
Goal forall x y z, phDisj y x -> phDisj z (phUnion x y) -> phDisj (phUnion y z) x.
Goal forall x y z, phDisj x y -> phDisj z (phUnion x y) -> phDisj (phUnion z y) x.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj (phUnion y z) x.
Goal forall x y z, phDisj y x -> phDisj z (phUnion y x) -> phDisj (phUnion z y) x.
Goal forall x y v, phDisj x y -> phcDisj (x v) (y v).
Goal forall x y v, phDisj y x -> phcDisj (x v) (y v).
Goal forall x y l v w, phDisj x y -> phcDisj v w -> phDisj (phUpdate x l v) (phUpdate y l w).
Goal forall x y z, phUnion x (phUnion y z) = phUnion (phUnion x y) z.
Goal forall x y z, phUnion (phUnion x y) z = phUnion x (phUnion y z).
Goal forall x y, phUnion x y = phUnion y x.
Goal forall x, phUnion x phIden = x.
Goal forall x, phUnion phIden x = x.
Goal forall x y z, phUnion x (phUnion y z) = phUnion y (phUnion x z).
Goal forall x y z, phUnion (phUnion x y) z = phUnion (phUnion x z) y.
Goal forall x y z w, phUnion (phUnion x z) (phUnion y w) = phUnion (phUnion x y) (phUnion z w).
Goal forall x y z w, phUnion (phUnion x y) (phUnion z w) = phUnion (phUnion x z) (phUnion y w).
Goal forall x y v, phUnion x y v = phcUnion (x v) (y v).
Goal forall x y v, phcUnion (x v) (y v) = phUnion x y v.
Goal forall x y v, phcDisj (x v) (y v) -> phcEntire (x v) -> phcEntire (phUnion x y v).

End HeapSolver.