# Library AAC_tactics.Instances

# Instances for AAC Tactics

From Stdlib Require PeanoNat ZArith Zminmax NArith List Permutation.
From Stdlib Require QArith Qminmax Relations.
From AAC_tactics Require Export AAC.
This one is not declared as an instance; this would interfere badly with setoid_rewrite
Lemma eq_subr {X} {R} `{@Reflexive X R} : subrelation eq R.
Proof. intros x y →. reflexivity. Qed.
Module Peano.
Import PeanoNat.

## Peano instances

#[export] Instance aac_Nat_add_Assoc : Associative eq Nat.add := Nat.add_assoc.
#[export] Instance aac_Nat_add_Comm : Commutative eq Nat.add := Nat.add_comm.
#[export] Instance aac_Nat_mul_Comm : Commutative eq Nat.mul := Nat.mul_comm.
#[export] Instance aac_Nat_mul_Assoc : Associative eq Nat.mul := Nat.mul_assoc.
#[export] Instance aac_Nat_min_Comm : Commutative eq Nat.min := Nat.min_comm.
#[export] Instance aac_Nat_min_Assoc : Associative eq Nat.min := Nat.min_assoc.
#[export] Instance aac_Nat_min_Idem : Idempotent eq Nat.min := Nat.min_idempotent.
#[export] Instance aac_Nat_max_Comm : Commutative eq Nat.max := Nat.max_comm.
#[export] Instance aac_Nat_max_Assoc : Associative eq Nat.max := Nat.max_assoc.
#[export] Instance aac_Nat_max_Idem : Idempotent eq Nat.max := Nat.max_idempotent.
#[export] Instance aac_Nat_gcd_Comm : Commutative eq Nat.gcd := Nat.gcd_comm.
#[export] Instance aac_Nat_gcd_Assoc : Associative eq Nat.gcd := Nat.gcd_assoc.
#[export] Instance aac_Nat_gcd_Idem : Idempotent eq Nat.gcd := Nat.gcd_diag.
#[export] Instance aac_Nat_lcm_Comm : Commutative eq Nat.lcm := Nat.lcm_comm.
#[export] Instance aac_Nat_lcm_Assoc : Associative eq Nat.lcm := Nat.lcm_assoc.
#[export] Instance aac_Nat_lcm_Idem : Idempotent eq Nat.lcm := Nat.lcm_diag.
#[export] Instance aac_Nat_mul_1_Unit : Unit eq Nat.mul 1 :=
Build_Unit eq Nat.mul 1 Nat.mul_1_l Nat.mul_1_r.
#[export] Instance aac_Nat_lcm_1_Unit : Unit eq Nat.lcm 1 :=
Build_Unit eq Nat.lcm 1 Nat.lcm_1_l Nat.lcm_1_r.
#[export] Instance aac_Nat_add_0_Unit : Unit eq Nat.add 0 :=
Build_Unit eq Nat.add (0) Nat.add_0_l Nat.add_0_r.
#[export] Instance aac_Nat_max_0_Unit : Unit eq Nat.max 0 :=
Build_Unit eq Nat.max 0 Nat.max_0_l Nat.max_0_r.
#[export] Instance aac_Nat_gcd_0_Unit : Unit eq Nat.gcd 0 :=
Build_Unit eq Nat.gcd 0 Nat.gcd_0_l Nat.gcd_0_r.
We also provide liftings from Nat.le to eq
#[export] Instance Nat_le_PreOrder : PreOrder Nat.le :=
Build_PreOrder _ Nat.le_refl Nat.le_trans.
#[export] Instance aac_Nat_le_eq_lift : AAC_lift Nat.le eq :=
Build_AAC_lift eq_equivalence _.
End Peano.
Module Z.
Import ZArith Zminmax.
Open Scope Z_scope.

## Z instances

#[export] Instance aac_Z_add_Assoc : Associative eq Z.add := Z.add_assoc.
#[export] Instance aac_Z_add_Comm : Commutative eq Z.add := Z.add_comm.
#[export] Instance aac_Z_mul_Comm : Commutative eq Z.mul := Z.mul_comm.
#[export] Instance aac_Z_mul_Assoc : Associative eq Z.mul := Z.mul_assoc.
#[export] Instance aac_Z_min_Comm : Commutative eq Z.min := Z.min_comm.
#[export] Instance aac_Z_min_Assoc : Associative eq Z.min := Z.min_assoc.
#[export] Instance aac_Z_min_Idem : Idempotent eq Z.min := Z.min_idempotent.
#[export] Instance aac_Z_max_Comm : Commutative eq Z.max := Z.max_comm.
#[export] Instance aac_Z_max_Assoc : Associative eq Z.max := Z.max_assoc.
#[export] Instance aac_Z_max_Idem : Idempotent eq Z.max := Z.max_idempotent.
#[export] Instance aac_Z_gcd_Comm : Commutative eq Z.gcd := Z.gcd_comm.
#[export] Instance aac_Z_gcd_Assoc : Associative eq Z.gcd := Z.gcd_assoc.
#[export] Instance aac_Z_lcm_Comm : Commutative eq Z.lcm := Z.lcm_comm.
#[export] Instance aac_Z_lcm_Assoc : Associative eq Z.lcm := Z.lcm_assoc.
#[export] Instance aac_Z_mul_1_Unit : Unit eq Z.mul 1 :=
Build_Unit eq Z.mul 1 Z.mul_1_l Z.mul_1_r.
#[export] Instance aac_Z_add_0_Unit : Unit eq Z.add 0 :=
Build_Unit eq Z.add 0 Z.add_0_l Z.add_0_r.
We also provide liftings from Z.le to eq
#[export] Instance Z_le_PreOrder : PreOrder Z.le :=
Build_PreOrder _ Z.le_refl Z.le_trans.
#[export] Instance aac_Z_le_eq_lift : AAC_lift Z.le eq :=
Build_AAC_lift eq_equivalence _.
End Z.
Module Lists.
Import List Permutation.

## List instances

#[export] Instance aac_List_app_Assoc {A} : Associative eq (@app A) :=
@app_assoc A.
#[export] Instance aac_List_app_nil_Unit {A} : Unit eq (@app A) (@nil A) :=
Build_Unit _ (@app A) (@nil A) (@app_nil_l A) (@app_nil_r A).
Exported Morphisms module provides a Proper instance
#[export] Instance aac_List_app_Permutation_Assoc {A} :
Associative (@Permutation A) (@app A).
Proof. repeat intro; rewrite app_assoc; apply Permutation_refl. Qed.
#[export] Instance aac_List_app_Permutation_Comm {A} :
Commutative (@Permutation A) (@app A) := @Permutation_app_comm A.
#[export] Instance aac_List_app_nil_Permutation_Unit {A} :
Unit (@Permutation A) (@app A) (@nil A) :=
Build_Unit (@Permutation A) (@app A) (@nil A) (fun x ⇒ Permutation_refl x)
(fun x ⇒ eq_ind_r (fun l ⇒ Permutation l _) (Permutation_refl x) (app_nil_r x)).
Permutation_app' in the Stdlib provides a Proper instance
End Lists.
Module N.
Import NArith.
Open Scope N_scope.

## N instances

#[export] Instance aac_N_add_Assoc : Associative eq N.add := N.add_assoc.
#[export] Instance aac_N_add_Comm : Commutative eq N.add := N.add_comm.
#[export] Instance aac_N_mul_Comm : Commutative eq N.mul := N.mul_comm.
#[export] Instance aac_N_mul_Assoc : Associative eq N.mul := N.mul_assoc.
#[export] Instance aac_N_min_Comm : Commutative eq N.min := N.min_comm.
#[export] Instance aac_N_min_Assoc : Associative eq N.min := N.min_assoc.
#[export] Instance aac_N_min_Idem : Idempotent eq N.min := N.min_idempotent.
#[export] Instance aac_N_max_Comm : Commutative eq N.max := N.max_comm.
#[export] Instance aac_N_max_Assoc : Associative eq N.max := N.max_assoc.
#[export] Instance aac_N_max_Idem : Idempotent eq N.max := N.max_idempotent.
#[export] Instance aac_N_gcd_Comm : Commutative eq N.gcd := N.gcd_comm.
#[export] Instance aac_N_gcd_Assoc : Associative eq N.gcd := N.gcd_assoc.
#[export] Instance aac_N_gcd_Idem : Idempotent eq N.gcd := N.gcd_diag.
#[export] Instance aac_N_lcm_Comm : Commutative eq N.lcm := N.lcm_comm.
#[export] Instance aac_N_lcm_Assoc : Associative eq N.lcm := N.lcm_assoc.
#[export] Instance aac_N_lcm_Idem : Idempotent eq N.lcm := N.lcm_diag.
#[export] Instance aac_N_mul_1_Unit : Unit eq N.mul (1)%N :=
Build_Unit eq N.mul 1 N.mul_1_l N.mul_1_r.
#[export] Instance aac_N_lcm_1_Unit : Unit eq N.lcm (1)%N :=
Build_Unit eq N.lcm 1 N.lcm_1_l N.lcm_1_r.
#[export] Instance aac_N_add_0_Unit : Unit eq N.add (0)%N :=
Build_Unit eq N.add 0 N.add_0_l N.add_0_r.
#[export] Instance aac_N_max_0_Unit : Unit eq N.max 0 :=
Build_Unit eq N.max 0 N.max_0_l N.max_0_r.
#[export] Instance aac_N_gcd_0_Unit : Unit eq N.gcd 0 :=
Build_Unit eq N.gcd 0 N.gcd_0_l N.gcd_0_r.
#[export] Instance N_le_PreOrder : PreOrder N.le :=
Build_PreOrder N.le N.le_refl N.le_trans.
#[export] Instance aac_N_le_eq_lift : AAC_lift N.le eq :=
Build_AAC_lift eq_equivalence _.
End N.
Module P.
Import NArith.
Open Scope positive_scope.

## Positive instances

#[export] Instance aac_Pos_add_Assoc : Associative eq Pos.add := Pos.add_assoc.
#[export] Instance aac_Pos_add_Comm : Commutative eq Pos.add := Pos.add_comm.
#[export] Instance aac_Pos_mul_Comm : Commutative eq Pos.mul := Pos.mul_comm.
#[export] Instance aac_Pos_mul_Assoc : Associative eq Pos.mul := Pos.mul_assoc.
#[export] Instance aac_Pos_min_Comm : Commutative eq Pos.min := Pos.min_comm.
#[export] Instance aac_Pos_min_Assoc : Associative eq Pos.min := Pos.min_assoc.
#[export] Instance aac_Pos_min_Idem : Idempotent eq Pos.min := Pos.min_idempotent.
#[export] Instance aac_Pos_max_Comm : Commutative eq Pos.max := Pos.max_comm.
#[export] Instance aac_Pos_max_Assoc : Associative eq Pos.max := Pos.max_assoc.
#[export] Instance aac_Pos_max_Idem : Idempotent eq Pos.max := Pos.max_idempotent.
#[export] Instance aac_Pos_mul_1_Unit : Unit eq Pos.mul 1 :=
Build_Unit eq Pos.mul 1 Pos.mul_1_l Pos.mul_1_r.
#[export] Instance aac_Pos_max_1_Unit : Unit eq Pos.max 1 :=
Build_Unit eq Pos.max 1 Pos.max_1_l Pos.max_1_r.
We also provide liftings from Pos.le to eq
#[export] Instance Pos_le_PreOrder : PreOrder Pos.le :=
Build_PreOrder Pos.le Pos.le_refl Pos.le_trans.
#[export] Instance aac_Pos_le_eq_lift : AAC_lift Pos.le eq :=
Build_AAC_lift eq_equivalence _.
End P.
Module Q.
Import QArith Qminmax.

## Q instances

#[export] Instance aac_Q_Qplus_Qeq_Assoc : Associative Qeq Qplus := Qplus_assoc.
#[export] Instance aac_Q_Qplus_Qeq_Comm : Commutative Qeq Qplus := Qplus_comm.
#[export] Instance aac_Q_Qmult_Qeq_Comm : Commutative Qeq Qmult := Qmult_comm.
#[export] Instance aac_Q_Qmult_Qeq_Assoc : Associative Qeq Qmult := Qmult_assoc.
#[export] Instance aac_Q_Qmin_Qeq_Comm : Commutative Qeq Qmin := Q.min_comm.
#[export] Instance aac_Q_Qmin_Qeq_Assoc : Associative Qeq Qmin := Q.min_assoc.
#[export] Instance aac_Q_Qmin_Qeq_Idem : Idempotent Qeq Qmin := Q.min_idempotent.
#[export] Instance aac_Q_Qmax_Qeq_Comm : Commutative Qeq Qmax := Q.max_comm.
#[export] Instance aac_Q_Qmax_Qeq_Assoc : Associative Qeq Qmax := Q.max_assoc.
#[export] Instance aac_Q_Qmax_Qeq_Idem : Idempotent Qeq Qmax := Q.max_idempotent.
#[export] Instance aac_Q_Qmult_1_Qeq_Unit : Unit Qeq Qmult 1 :=
Build_Unit Qeq Qmult 1 Qmult_1_l Qmult_1_r.
#[export] Instance aac_Q_Qplus_0_Qeq_Unit : Unit Qeq Qplus 0 :=
Build_Unit Qeq Qplus 0 Qplus_0_l Qplus_0_r.
we also provide liftings from le to eq
#[export] Instance Q_Qle_PreOrder : PreOrder Qle :=
Build_PreOrder Qle Qle_refl Qle_trans.
#[export] Instance aac_Q_Qle_eq_lift : AAC_lift Qle Qeq :=
Build_AAC_lift QOrderedType.QOrder.TO.eq_equiv _.
End Q.
Module Prop_ops.

## Prop instances

#[export] Instance aac_Prop_or_iff_Assoc : Associative iff or.
Proof. unfold Associative; tauto. Qed.
#[export] Instance aac_Prop_or_iff_Comm : Commutative iff or.
Proof. unfold Commutative; tauto. Qed.
#[export] Instance aac_Prop_or_iff_Idem : Idempotent iff or.
Proof. unfold Idempotent; tauto. Qed.
#[export] Instance aac_Prop_and_iff_Assoc : Associative iff and.
Proof. unfold Associative; tauto. Qed.
#[export] Instance aac_Prop_and_iff_Comm : Commutative iff and.
Proof. unfold Commutative; tauto. Qed.
#[export] Instance aac_Prop_and_iff_Idem : Idempotent iff and.
Proof. unfold Idempotent; tauto. Qed.
#[export] Instance aac_Prop_or_False_iff_Unit : Unit iff or False.
Proof. constructor; firstorder. Qed.
#[export] Instance aac_Prop_and_True_iff_Unit : Unit iff and True.
Proof. constructor; firstorder. Qed.
#[export] Program Instance not_iff_compat : Proper (iff ==> iff) not.
Next Obligation. unfold iff; split; intros; tauto. Qed.
Solve All Obligations with firstorder.
#[export] Instance aac_Prop_impl_iff_lift : AAC_lift Basics.impl iff :=
Build_AAC_lift _ _.
End Prop_ops.
Module Bool.

## Boolean instances

#[export] Instance aac_Bool_orb_Assoc : Associative eq orb.
Proof. unfold Associative; firstorder with bool. Qed.
#[export] Instance aac_Bool_orb_Comm : Commutative eq orb.
Proof. unfold Commutative; firstorder with bool. Qed.
#[export] Instance aac_Bool_orb_Idem : Idempotent eq orb.
Proof. intro; apply Bool.orb_diag. Qed.
#[export] Instance aac_Bool_andb_Assoc : Associative eq andb.
Proof. unfold Associative; firstorder with bool. Qed.
#[export] Instance aac_Bool_andb_Comm : Commutative eq andb.
Proof. unfold Commutative; firstorder with bool. Qed.
#[export] Instance aac_Bool_andb_Idem : Idempotent eq andb.
Proof. intro; apply Bool.andb_diag. Qed.
#[export] Instance aac_Bool_orb_false_Unit : Unit eq orb false.
Proof. constructor; firstorder with bool. Qed.
#[export] Instance aac_Bool_andb_true_Unit : Unit eq andb true.
Proof. constructor; intros [|]; firstorder. Qed.
#[export] Instance negb_compat : Proper (eq ==> eq) negb.
Proof. intros [|] [|]; auto. Qed.
End Bool.
Module Relations.
Import Relations.

## Relation instances

Section defs.
Variable T : Type.
Variables R S: relation T.
Definition inter : relation T := fun x y ⇒ R x y ∧ S x y.
Definition compo : relation T := fun x y ⇒ ∃ z : T, R x z ∧ S z y.
Definition negr : relation T := fun x y ⇒ ¬ R x y.
union and converse are already defined in the standard library
Definition bot : relation T := fun _ _ ⇒ False.
Definition top : relation T := fun _ _ ⇒ True.
End defs.
#[export] Instance same_relation_Equivalence T :
Equivalence (same_relation T).
Proof. firstorder. Qed.
#[export] Instance aac_union_same_relation_Comm T :
Commutative (same_relation T) (union T).
Proof. unfold Commutative; compute; intuition. Qed.
#[export] Instance aac_union_same_relation_Assoc T :
Associative (same_relation T) (union T).
Proof. unfold Associative; compute; intuition. Qed.
#[export] Instance aac_union_same_relation_Idem T :
Idempotent (same_relation T) (union T).
Proof. unfold Idempotent; compute; intuition. Qed.
#[export] Instance aac_bot_union_same_relation_Unit T :
Unit (same_relation T) (union T) (bot T).
Proof. constructor; compute; intuition. Qed.
#[export] Instance aac_inter_same_relation_Comm T :
Commutative (same_relation T) (inter T).
Proof. unfold Commutative; compute; intuition. Qed.
#[export] Instance aac_inter_same_relation_Assoc T :
Associative (same_relation T) (inter T).
Proof. unfold Associative; compute; intuition. Qed.
#[export] Instance aac_inter_same_relation_Idem T :
Idempotent (same_relation T) (inter T).
Proof. unfold Idempotent; compute; intuition. Qed.
#[export] Instance aac_inter_top_same_relation_Unit T :
Unit (same_relation T) (inter T) (top T).
Proof. constructor; compute; intuition. Qed.
Note that we use eq directly as a neutral element for composition
#[export] Instance aac_compo_same_relation_Assoc T :
Associative (same_relation T) (compo T).
Proof. unfold Associative; compute; firstorder. Qed.
#[export] Instance aac_compo_eq_same_relation_Unit T :
Unit (same_relation T) (compo T) (eq).
Proof. compute; firstorder subst; trivial. Qed.
#[export] Instance negr_same_relation_compat T :
Proper (same_relation T ==> same_relation T) (negr T).
Proof. compute. firstorder. Qed.
#[export] Instance transp_same_relation_compat T :
Proper (same_relation T ==> same_relation T) (transp T).
Proof. compute. firstorder. Qed.
#[export] Instance clos_trans_incr T :
Proper (inclusion T ==> inclusion T) (clos_trans T).
Proof.
intros R S H x y Hxy. induction Hxy.
constructor 1. apply H. assumption.
econstructor 2; eauto 3.
Qed.
#[export] Instance clos_trans_same_relation_compat T :
Proper (same_relation T ==> same_relation T) (clos_trans T).
Proof. intros R S H; split; apply clos_trans_incr, H. Qed.
#[export] Instance clos_refl_trans_incr T :
Proper (inclusion T ==> inclusion T) (clos_refl_trans T).
Proof.
intros R S H x y Hxy. induction Hxy.
constructor 1. apply H. assumption.
constructor 2.
econstructor 3; eauto 3.
Qed.
#[export] Instance clos_refl_trans_same_relation_compat T :
Proper (same_relation T ==> same_relation T) (clos_refl_trans T).
Proof. intros R S H; split; apply clos_refl_trans_incr, H. Qed.
#[export] Instance inclusion_PreOrder T : PreOrder (inclusion T).
Proof. constructor; unfold Reflexive, Transitive, inclusion; intuition. Qed.
#[export] Program Instance aac_inclusion_same_relation_lift T :
AAC_lift (inclusion T) (same_relation T) :=
Build_AAC_lift (same_relation_Equivalence T) _.
Next Obligation. firstorder. Qed.
End Relations.
Module All.
Export Peano.
Export Z.
Export Lists.
Export P.
Export N.
Export Prop_ops.
Export Bool.
Export Relations.
End All.
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