Library Coq.Lists.SetoidPermutation
Permutations of list modulo a setoid equality.
Contribution by Robbert Krebbers (Nijmegen University).
Section Permutation.
Context {A : Type} (eqA : relation A) (e : Equivalence eqA).
Inductive PermutationA : list A -> list A -> Prop :=
| permA_nil: PermutationA nil nil
| permA_skip x₁ x₂ l₁ l₂ :
eqA x₁ x₂ -> PermutationA l₁ l₂ -> PermutationA (x₁ :: l₁) (x₂ :: l₂)
| permA_swap x y l : PermutationA (y :: x :: l) (x :: y :: l)
| permA_trans l₁ l₂ l₃ :
PermutationA l₁ l₂ -> PermutationA l₂ l₃ -> PermutationA l₁ l₃.
Local Hint Constructors PermutationA.
Global Instance: Equivalence PermutationA.
Proof.
constructor.
- intro l. induction l; intuition.
- intros l₁ l₂. induction 1; eauto. apply permA_skip; intuition.
- exact permA_trans.
Qed.
Global Instance PermutationA_cons :
Proper (eqA ==> PermutationA ==> PermutationA) (@cons A).
Proof.
repeat intro. now apply permA_skip.
Qed.
Lemma PermutationA_app_head l₁ l₂ l :
PermutationA l₁ l₂ -> PermutationA (l ++ l₁) (l ++ l₂).
Proof.
induction l; trivial; intros. apply permA_skip; intuition.
Qed.
Global Instance PermutationA_app :
Proper (PermutationA ==> PermutationA ==> PermutationA) (@app A).
Proof.
intros l₁ l₂ Pl k₁ k₂ Pk.
induction Pl.
- easy.
- now apply permA_skip.
- etransitivity.
* rewrite <-!app_comm_cons. now apply permA_swap.
* rewrite !app_comm_cons. now apply PermutationA_app_head.
- do 2 (etransitivity; try eassumption).
apply PermutationA_app_head. now symmetry.
Qed.
Lemma PermutationA_app_tail l₁ l₂ l :
PermutationA l₁ l₂ -> PermutationA (l₁ ++ l) (l₂ ++ l).
Proof.
intros E. now rewrite E.
Qed.
Lemma PermutationA_cons_append l x :
PermutationA (x :: l) (l ++ x :: nil).
Proof.
induction l.
- easy.
- simpl. rewrite <-IHl. intuition.
Qed.
Lemma PermutationA_app_comm l₁ l₂ :
PermutationA (l₁ ++ l₂) (l₂ ++ l₁).
Proof.
induction l₁.
- now rewrite app_nil_r.
- rewrite <-app_comm_cons, IHl₁, app_comm_cons.
now rewrite PermutationA_cons_append, <-app_assoc.
Qed.
Lemma PermutationA_cons_app l l₁ l₂ x :
PermutationA l (l₁ ++ l₂) -> PermutationA (x :: l) (l₁ ++ x :: l₂).
Proof.
intros E. rewrite E.
now rewrite app_comm_cons, (PermutationA_cons_append l₁ x), <- app_assoc.
Qed.
Lemma PermutationA_middle l₁ l₂ x :
PermutationA (x :: l₁ ++ l₂) (l₁ ++ x :: l₂).
Proof.
now apply PermutationA_cons_app.
Qed.
Lemma PermutationA_equivlistA l₁ l₂ :
PermutationA l₁ l₂ -> equivlistA eqA l₁ l₂.
Proof.
induction 1.
- reflexivity.
- now apply equivlistA_cons_proper.
- now apply equivlistA_permute_heads.
- etransitivity; eassumption.
Qed.
Lemma NoDupA_equivlistA_PermutationA l₁ l₂ :
NoDupA eqA l₁ -> NoDupA eqA l₂ ->
equivlistA eqA l₁ l₂ -> PermutationA l₁ l₂.
Proof.
intros Pl₁. revert l₂. induction Pl₁ as [|x l₁ E1].
- intros l₂ _ H₂. symmetry in H₂. now rewrite (equivlistA_nil_eq eqA).
- intros l₂ Pl₂ E2.
destruct (@InA_split _ eqA l₂ x) as [l₂h [y [l₂t [E3 ?]]]].
{ rewrite <-E2. intuition. }
subst. transitivity (y :: l₁); [intuition |].
apply PermutationA_cons_app, IHPl₁.
now apply NoDupA_split with y.
apply equivlistA_NoDupA_split with x y; intuition.
Qed.
Lemma Permutation_eqlistA_commute l₁ l₂ l₃ :
eqlistA eqA l₁ l₂ -> Permutation l₂ l₃ ->
exists l₂', Permutation l₁ l₂' /\ eqlistA eqA l₂' l₃.
Proof.
intros E P. revert l₁ E.
induction P; intros.
- inversion_clear E. now exists nil.
- inversion_clear E.
destruct (IHP l0) as (l0',(P',E')); trivial. clear IHP.
exists (x0::l0'). split; auto.
- inversion_clear E. inversion_clear H0.
exists (x1::x0::l1). now repeat constructor.
- clear P1 P2.
destruct (IHP1 _ E) as (l₁',(P₁,E₁)).
destruct (IHP2 _ E₁) as (l₂',(P₂,E₂)).
exists l₂'. split; trivial. econstructor; eauto.
Qed.
Lemma PermutationA_decompose l₁ l₂ :
PermutationA l₁ l₂ ->
exists l, Permutation l₁ l /\ eqlistA eqA l l₂.
Proof.
induction 1.
- now exists nil.
- destruct IHPermutationA as (l,(P,E)). exists (x₁::l); auto.
- exists (x::y::l). split. constructor. reflexivity.
- destruct IHPermutationA1 as (l₁',(P,E)).
destruct IHPermutationA2 as (l₂',(P',E')).
destruct (@Permutation_eqlistA_commute l₁' l₂ l₂') as (l₁'',(P'',E''));
trivial.
exists l₁''. split. now transitivity l₁'. now transitivity l₂'.
Qed.
Lemma Permutation_PermutationA l₁ l₂ :
Permutation l₁ l₂ -> PermutationA l₁ l₂.
Proof.
induction 1.
- constructor.
- now constructor.
- apply permA_swap.
- econstructor; eauto.
Qed.
Lemma eqlistA_PermutationA l₁ l₂ :
eqlistA eqA l₁ l₂ -> PermutationA l₁ l₂.
Proof.
induction 1; now constructor.
Qed.
Lemma NoDupA_equivlistA_decompose l1 l2 :
NoDupA eqA l1 -> NoDupA eqA l2 -> equivlistA eqA l1 l2 ->
exists l, Permutation l1 l /\ eqlistA eqA l l2.
Proof.
intros. apply PermutationA_decompose.
now apply NoDupA_equivlistA_PermutationA.
Qed.
Lemma PermutationA_preserves_NoDupA l₁ l₂ :
PermutationA l₁ l₂ -> NoDupA eqA l₁ -> NoDupA eqA l₂.
Proof.
induction 1; trivial.
- inversion_clear 1; constructor; auto.
apply PermutationA_equivlistA in H0. contradict H2.
now rewrite H, H0.
- inversion_clear 1. inversion_clear H1. constructor.
+ contradict H. inversion_clear H; trivial.
elim H0. now constructor.
+ constructor; trivial.
contradict H0. now apply InA_cons_tl.
- eauto.
Qed.
End Permutation.