Library AAC_tactics.AAC


Theory for AAC Tactics

We define several base classes to package associative and possibly commutative/idempotent operators, and define a data-type for reified (or quoted) expressions (with morphisms).
We then define a reflexive decision procedure to decide the equality of reified terms: first normalise reified terms, then compare them. This allows us to close transitivity steps automatically, in the aac_rewrite tactic.
We restrict ourselves to the case where all symbols operate on a single fixed type. In particular, this means that we cannot handle situations like
H: ∀ x y, nat_of_pos (pos_of_nat (x) + y) + x = ...
where one occurrence of + operates on nat while the other one operates on positive.

From Coq Require Import Arith NArith List.
From Coq Require Import FMapPositive Relations RelationClasses.
From Coq Require Export Morphisms.
From AAC_tactics Require Import Utils Constants.

Set Implicit Arguments.
Set Asymmetric Patterns.

Local Open Scope signature_scope.

Environments for the reification process

Positive maps are used to index elements
Section sigma.
  Definition sigma := PositiveMap.t.
  Definition sigma_get A (null : A) (map : sigma A) (n : positive) : A :=
    match PositiveMap.find n map with
      | None ⇒ null
      | Some x ⇒ x
    end.
  Definition sigma_add := @PositiveMap.add.
  Definition sigma_empty := @PositiveMap.empty.

  Register sigma_get as aac_tactics.sigma.get.
  Register sigma_add as aac_tactics.sigma.add.
  Register sigma_empty as aac_tactics.sigma.empty.
End sigma.

Classes for properties of operators


Class Associative (X : Type) (R : relation X) (dot : X → X → X) :=
  law_assoc : ∀ x y z, R (dot x (dot y z)) (dot (dot x y) z).
Class Commutative (X : Type) (R : relation X) (plus : X → X → X) :=
  law_comm: ∀ x y, R (plus x y) (plus y x).
Class Idempotent (X : Type) (R : relation X) (plus : X → X → X) :=
  law_idem: ∀ x, R (plus x x) x.
Class Unit (X : Type) (R : relation X) (op : X → X → X) (unit : X) := {
  law_neutral_left: ∀ x, R (op unit x) x;
  law_neutral_right: ∀ x, R (op x unit) x
}.

Register Associative as aac_tactics.classes.Associative.
Register Commutative as aac_tactics.classes.Commutative.
Register Idempotent as aac_tactics.classes.Idempotent.
Register Unit as aac_tactics.classes.Unit.

Class used to find the equivalence relation on which operations are A or AC, starting from the relation appearing in the goal
Class AAC_lift (X : Type) (R : relation X) (E : relation X) := {
  aac_lift_equivalence : Equivalence E;
  aac_list_proper : Proper (E ==> E ==> iff) R
}.

Register AAC_lift as aac_tactics.internal.AAC_lift.
Register aac_lift_equivalence as aac_tactics.internal.aac_lift_equivalence.

Simple instances for when we have a subrelation or an equivalence
#[export] Instance aac_lift_subrelation {X} {R} {E} {HE: Equivalence E}
 {HR: @Transitive X R} {HER: subrelation E R} : AAC_lift R E | 3.
Proof.
  constructor; trivial.
  intros ? ? H ? ? H'. split; intro G.
   rewrite <- H, G. apply HER, H'.
   rewrite H, G. apply HER. symmetry. apply H'.
Qed.
#[export] Instance aac_lift_proper {X} {R : relation X} {E}
 {HE: Equivalence E} {HR: Proper (E==>E==>iff) R} : AAC_lift R E | 4 := {}.

Utilities for the evaluation function


Module Internal.

Section copy.

  Context {X} {R} {HR: @Equivalence X R} {plus}
   (op: Associative R plus) (op': Commutative R plus)
   (po: Proper (R ==> R ==> R) plus).

copy n x = x+...+x (n times)
  Fixpoint copy' n x :=
    match n with
      | xH ⇒ x
      | xI n ⇒ let xn := copy' n x in plus (plus xn xn) x
      | xO n ⇒ let xn := copy' n x in (plus xn xn)
    end.
  Definition copy n x := Prect (fun _ ⇒ X) x (fun _ xn ⇒ plus x xn) n.

  Lemma copy_plus : ∀ n m x, R (copy (n+m) x) (plus (copy n x) (copy m x)).
  Proof.
    unfold copy.
    induction n using Pind; intros m x.
    rewrite Prect_base. rewrite <- Pplus_one_succ_l.
    rewrite Prect_succ. reflexivity.
    rewrite Pplus_succ_permute_l. rewrite 2Prect_succ.
    rewrite IHn. apply op.
  Qed.

  Lemma copy_xH : ∀ x, R (copy 1 x) x.
  Proof. intros; unfold copy; rewrite Prect_base. reflexivity. Qed.

  Lemma copy_Psucc : ∀ n x, R (copy (Pos.succ n) x) (plus x (copy n x)).
  Proof. intros; unfold copy; rewrite Prect_succ. reflexivity. Qed.

  #[export] Instance copy_compat n : Proper (R ==> R) (copy n).
  Proof.
    unfold copy.
    induction n using Pind; intros x y H.
    rewrite 2Prect_base. assumption.
    rewrite 2Prect_succ. apply po; auto.
  Qed.

End copy.

Packaging structures

Free symbols


Module Sym.
  Section t.
    Context {X} {R : relation X} .

Type of an arity
    Fixpoint type_of (n: nat) :=
      match n with
        | O ⇒ X
        | S n ⇒ X → type_of n
      end.

Relation to be preserved at an arity
    Fixpoint rel_of n : relation (type_of n) :=
      match n with
        | O ⇒ R
      | S n ⇒ respectful R (rel_of n)
      end.

    Register type_of as aac_tactics.internal.sym.type_of.
    Register rel_of as aac_tactics.internal.sym.rel_of.

A symbol package contains:
  • an arity,
  • a value of the corresponding type, and
  • a proof that the value is a proper morphism
  Record pack : Type := mkPack {
    ar : nat;
    value :> type_of ar;
    morph : Proper (rel_of ar) value
   }.

  Register pack as aac_tactics.sym.pack.
  Register mkPack as aac_tactics.sym.mkPack.

Helper to build default values, when filling reification environments
  Definition null: pack := mkPack 1 (fun x ⇒ x) (fun _ _ H ⇒ H).

  Register null as aac_tactics.sym.null.

  End t.

End Sym.

Binary operations


Module Bin.
  Section t.
    Context {X} {R: relation X}.

    Record pack := mk_pack {
      value:> X → X → X;
      compat: Proper (R ==> R ==> R) value;
      assoc: Associative R value;
      comm: option (Commutative R value);
      idem: option (Idempotent R value)
    }.

    Register pack as aac_tactics.bin.pack.
    Register mk_pack as aac_tactics.bin.mkPack.
  End t.
See the Instances module for concrete instances of these classes
End Bin.

Reification, normalisation, and decision


Section s.
  Context {X} {R: relation X} {E: @Equivalence X R}.
  Infix "==" := R (at level 80).

We use environments to store the various operators and the morphisms

  Variable e_sym: idx → @Sym.pack X R.
  Variable e_bin: idx → @Bin.pack X R.

Packaging units (depends on e_bin)

  Record unit_of u := mk_unit_for {
    uf_idx: idx;
    uf_desc: Unit R (Bin.value (e_bin uf_idx)) u
  }.

  Record unit_pack := mk_unit_pack {
    u_value:> X;
    u_desc: list (unit_of u_value)
  }.

  Register unit_of as aac_tactics.internal.unit_of.
  Register mk_unit_for as aac_tactics.internal.mk_unit_for.
  Register unit_pack as aac_tactics.internal.unit_pack.
  Register mk_unit_pack as aac_tactics.internal.mk_unit_pack.

  Variable e_unit: positive → unit_pack.

  #[local] Hint Resolve e_bin e_unit: typeclass_instances.

Almost normalised syntax

A term in T is in normal form if:
  • sums do not contain sums
  • products do not contain products
  • there are no unary sums or products
  • lists and msets are lexicographically sorted according to the order we define below
vT n denotes the set of term vectors of size n (the mutual dependency could be removed).
Note that T and vT depend on the e_sym environment (which contains, among other things, the arity of symbols).

  Inductive T: Type :=
  | sum: idx → mset T → T
  | prd: idx → nelist T → T
  | sym: ∀ i, vT (Sym.ar (e_sym i)) → T
  | unit : idx → T
  with vT: nat → Type :=
  | vnil: vT O
  | vcons: ∀ n, T → vT n → vT (S n).

  Register T as aac_tactics.internal.T.
  Register sum as aac_tactics.internal.sum.
  Register prd as aac_tactics.internal.prd.
  Register sym as aac_tactics.internal.sym.
  Register unit as aac_tactics.internal.unit.

  Register vnil as aac_tactics.internal.vnil.
  Register vcons as aac_tactics.internal.vcons.

Lexicographic rpo over the normalised syntax
  Fixpoint compare (u v: T) :=
    match u,v with
      | sum i l, sum j vs ⇒ lex (idx_compare i j) (mset_compare compare l vs)
      | prd i l, prd j vs ⇒ lex (idx_compare i j) (list_compare compare l vs)
      | sym i l, sym j vs ⇒ lex (idx_compare i j) (vcompare l vs)
      | unit i , unit j ⇒ idx_compare i j
      | unit _ , _ ⇒ Lt
      | _ , unit _ ⇒ Gt
      | sum _ _, _ ⇒ Lt
      | _ , sum _ _ ⇒ Gt
      | prd _ _, _ ⇒ Lt
      | _ , prd _ _ ⇒ Gt

    end
  with vcompare i j (us: vT i) (vs: vT j) :=
    match us,vs with
      | vnil, vnil ⇒ Eq
      | vnil, _ ⇒ Lt
      | _, vnil ⇒ Gt
      | vcons _ u us, vcons _ v vs ⇒ lex (compare u v) (vcompare us vs)
    end.

Evaluation from syntax to the abstract domain


  Fixpoint eval u: X :=
    match u with
      | sum i l ⇒ let o := Bin.value (e_bin i) in
        fold_map (fun un ⇒ let '(u,n):=un in @copy _ o n (eval u)) o l
      | prd i l ⇒ fold_map eval (Bin.value (e_bin i)) l
      | sym i v ⇒ eval_aux v (Sym.value (e_sym i))
      | unit i ⇒ e_unit i
    end
  with eval_aux i (v: vT i): Sym.type_of i → X :=
    match v with
      | vnil ⇒ fun f ⇒ f
      | vcons _ u v ⇒ fun f ⇒ eval_aux v (f (eval u))
    end.

  Register eval as aac_tactics.internal.eval.

We need to show that compare reflects equality (this is because we work with msets rather than with lists with arities)
  Fixpoint tcompare_weak_spec u : ∀ (v : T), compare_weak_spec u v (compare u v)
  with vcompare_reflect_eqdep i us : ∀ j vs (H: i=j),
    vcompare us vs = Eq → cast vT H us = vs.
  Proof.
    induction u.
    - destruct v; simpl; try constructor.
      case (pos_compare_weak_spec p p0); intros; try constructor.
      case (mset_compare_weak_spec compare tcompare_weak_spec m m0); intros; try constructor.
    - destruct v; simpl; try constructor.
      case (pos_compare_weak_spec p p0); intros; try constructor.
      case (list_compare_weak_spec compare tcompare_weak_spec n n0); intros; try constructor.
    - destruct v0; simpl; try constructor.
      case_eq (idx_compare i i0); intro Hi; try constructor.
      apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst.
      case_eq (vcompare v v0); intro Hv; try constructor.
      rewrite <- (vcompare_reflect_eqdep _ _ _ _ eq_refl Hv). constructor.
    - destruct v; simpl; try constructor.
      case_eq (idx_compare p p0); intro Hi; try constructor.
      apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. constructor.
    - induction us; destruct vs; simpl; intros H Huv; try discriminate.
      apply cast_eq, eq_nat_dec.
      injection H; intro Hn.
      revert Huv; case (tcompare_weak_spec t t0); intros; try discriminate.
      symmetry in Hn. subst.
      rewrite <- (IHus _ _ eq_refl Huv).
      apply cast_eq, eq_nat_dec.
  Qed.

  Instance eval_aux_compat i (l: vT i): Proper (@Sym.rel_of X R i ==> R) (eval_aux l).
  Proof.
    induction l; simpl; repeat intro.
    assumption.
    apply IHl, H. reflexivity.
  Qed.

Is i a unit for j?
  Definition is_unit_of j i :=
    List.existsb (fun p ⇒ eq_idx_bool j (uf_idx p)) (u_desc (e_unit i)).

Is i commutative?
  Definition is_commutative i :=
    match Bin.comm (e_bin i) with Some _ ⇒ true | None ⇒ false end.

Is i idempotent?
  Definition is_idempotent i :=
    match Bin.idem (e_bin i) with Some _ ⇒ true | None ⇒ false end.

Normalisation


  #[universes(template)]
  Inductive discr {A} : Type :=
  | Is_op : A → discr
  | Is_unit : idx → discr
  | Is_nothing : discr.

This is called Datatypes.sum in the stdlib
  #[universes(template)]
  Inductive m {A} {B} :=
  | left : A → m
  | right : B → m.

  Definition comp A B (merge : B → B → B) (l : B) (l' : @m A B) : @m A B :=
    match l' with
      | left _ ⇒ right l
      | right l' ⇒ right (merge l l')
    end.

Auxiliary functions, to clean up sums
  Section sums.
    Variable i : idx.
    Variable is_unit : idx → bool.

    Definition sum' (u: mset T): T :=
      match u with
        | nil (u,xH) ⇒ u
        | _ ⇒ sum i u
      end.

    Definition is_sum (u: T) : @discr (mset T) :=
    match u with
      | sum j l ⇒ if eq_idx_bool j i then Is_op l else Is_nothing
      | unit j ⇒ if is_unit j then Is_unit j else Is_nothing
      | _ ⇒ Is_nothing
    end.

    Definition copy_mset n (l: mset T): mset T :=
      match n with
        | xH ⇒ l
        | _ ⇒ nelist_map (fun vm ⇒ let '(v,m):=vm in (v,Pmult n m)) l
      end.

    Definition return_sum u n :=
      match is_sum u with
        | Is_nothing ⇒ right (nil (u,n))
        | Is_op l' ⇒ right (copy_mset n l')
        | Is_unit j ⇒ left j
      end.

    Definition add_to_sum u n (l : @m idx (mset T)) :=
      match is_sum u with
        | Is_nothing ⇒ comp (merge_msets compare) (nil (u,n)) l
        | Is_op l' ⇒ comp (merge_msets compare) (copy_mset n l') l
        | Is_unit _ ⇒ l
    end.

    Definition norm_msets_ norm (l: mset T) :=
      fold_map'
        (fun un ⇒ let '(u,n) := un in return_sum (norm u) n)
        (fun un l ⇒ let '(u,n) := un in add_to_sum (norm u) n l) l.

  End sums.

Similar functions for products
  Section prds.

    Variable i : idx.
    Variable is_unit : idx → bool.

    Definition prd' (u: nelist T): T :=
    match u with
      | nil u ⇒ u
      | _ ⇒ prd i u
    end.

    Definition is_prd (u: T) : @discr (nelist T) :=
    match u with
      | prd j l ⇒ if eq_idx_bool j i then Is_op l else Is_nothing
      | unit j ⇒ if is_unit j then Is_unit j else Is_nothing
      | _ ⇒ Is_nothing
    end.

    Definition return_prd u :=
      match is_prd u with
        | Is_nothing ⇒ right (nil (u))
        | Is_op l' ⇒ right (l')
        | Is_unit j ⇒ left j
      end.

    Definition add_to_prd u (l : @m idx (nelist T)) :=
      match is_prd u with
        | Is_nothing ⇒ comp (@appne T) (nil (u)) l
        | Is_op l' ⇒ comp (@appne T) (l') l
        | Is_unit _ ⇒ l
      end.

    Definition norm_lists_ norm (l : nelist T) :=
      fold_map'
        (fun u ⇒ return_prd (norm u))
        (fun u l ⇒ add_to_prd (norm u) l) l.

  End prds.

  Definition run_list x :=
    match x with
      | left n ⇒ nil (unit n)
      | right l ⇒ l
    end.

  Definition norm_lists norm i l :=
    let is_unit := is_unit_of i in
    run_list (norm_lists_ i is_unit norm l).

  Definition run_msets x :=
    match x with
      | left n ⇒ nil (unit n, xH)
      | right l ⇒ l
    end.

  Definition norm_msets norm i l :=
    let is_unit := is_unit_of i in
      run_msets (norm_msets_ i is_unit norm l).

  Fixpoint norm u {struct u}:=
    match u with
      | sum i l ⇒ if is_commutative i then
                     if is_idempotent i then
                        sum' i (reduce_mset (norm_msets norm i l))
                     else sum' i (norm_msets norm i l)
                   else u
      | prd i l ⇒ prd' i (norm_lists norm i l)
      | sym i l ⇒ sym i (vnorm l)
      | unit i ⇒ unit i
    end
  with vnorm i (l: vT i): vT i :=
    match l with
      | vnil ⇒ vnil
      | vcons _ u l ⇒ vcons (norm u) (vnorm l)
    end.

Correctness


  Lemma is_unit_of_Unit : ∀ i j : idx,
   is_unit_of i j = true → Unit R (Bin.value (e_bin i)) (eval (unit j)).
  Proof.
    intros. unfold is_unit_of in H.
    rewrite existsb_exists in H.
    destruct H as [x [H H']].
    revert H' ; case (eq_idx_spec); [intros H' _ ; subst| intros _ H'; discriminate].
    simpl. destruct x. simpl. auto.
  Qed.

  Instance Binvalue_Commutative i (H : is_commutative i = true) :
    Commutative R (@Bin.value _ _ (e_bin i) ).
  Proof.
    unfold is_commutative in H.
    destruct (Bin.comm (e_bin i)); auto.
    discriminate.
  Qed.

  Instance Binvalue_Idempotent i (H : is_idempotent i = true) :
    Idempotent R (@Bin.value _ _ (e_bin i)).
  Proof.
    unfold is_idempotent in H.
    destruct (Bin.idem (e_bin i)); auto.
    discriminate.
  Qed.

  Instance Binvalue_Associative i : Associative R (@Bin.value _ _ (e_bin i)).
  Proof. destruct ((e_bin i)); auto. Qed.

  Instance Binvalue_Proper i : Proper (R ==> R ==> R) (@Bin.value _ _ (e_bin i) ).
  Proof. destruct ((e_bin i)); auto. Qed.

  #[local] Hint Resolve Binvalue_Proper Binvalue_Associative Binvalue_Commutative : core.

  #[local] Hint Resolve is_unit_of_Unit : core.

Auxiliary lemmas about sums
  Section sum_correctness.
    Variable i : idx.
    Variable is_unit : idx → bool.
    Hypothesis is_unit_sum_Unit : ∀ j, is_unit j = true →
      @Unit X R (Bin.value (e_bin i)) (eval (unit j)).

    Inductive is_sum_spec_ind : T → @discr (mset T) → Prop :=
    | is_sum_spec_op : ∀ j l, j = i → is_sum_spec_ind (sum j l) (Is_op l)
    | is_sum_spec_unit : ∀ j, is_unit j = true → is_sum_spec_ind (unit j) (Is_unit j)
    | is_sum_spec_nothing : ∀ u, is_sum_spec_ind u (Is_nothing).

    Lemma is_sum_spec u : is_sum_spec_ind u (is_sum i is_unit u).
    Proof.
      unfold is_sum; case u; intros; try constructor.
      case_eq (eq_idx_bool p i); intros; subst; try constructor; auto.
      revert H. case eq_idx_spec; try discriminate. auto.
      case_eq (is_unit p); intros; try constructor. auto.
    Qed.

    Instance assoc : @Associative X R (Bin.value (e_bin i)).
    Proof. destruct (e_bin i). simpl. assumption. Qed.

    Instance proper : Proper (R ==> R ==> R)(Bin.value (e_bin i)).
    Proof. destruct (e_bin i). simpl. assumption. Qed.

    Hypothesis comm : @Commutative X R (Bin.value (e_bin i)).

    Lemma sum'_sum : ∀ (l: mset T), eval (sum' i l) == eval (sum i l).
    Proof.
      intros [[a n] | [a n] l]; destruct n; simpl; reflexivity.
    Qed.

    Lemma eval_sum_nil x : eval (sum i (nil (x,xH))) == (eval x).
    Proof. rewrite <- sum'_sum. reflexivity. Qed.

    Lemma eval_sum_cons : ∀ n a (l: mset T),
      (eval (sum i ((a,n)::l))) == (@Bin.value _ _ (e_bin i)
        (@copy _ (@Bin.value _ _ (e_bin i)) n (eval a)) (eval (sum i l))).
    Proof. intros n a [[? ? ]|[b m] l]; simpl; reflexivity. Qed.

    Inductive compat_sum_unit : @m idx (mset T) → Prop :=
    | csu_left : ∀ x, is_unit x = true→ compat_sum_unit (left x)
    | csu_right : ∀ m, compat_sum_unit (right m).

    Lemma compat_sum_unit_return x n : compat_sum_unit (return_sum i is_unit x n).
    Proof.
      unfold return_sum.
      case is_sum_spec; intros; try constructor; auto.
    Qed.

    Lemma compat_sum_unit_add : ∀ x n h,
     compat_sum_unit h →
     compat_sum_unit (add_to_sum i (is_unit_of i) x n h).
    Proof.
      unfold add_to_sum;intros; inversion H;
        case_eq (is_sum i (is_unit_of i) x);
        intros; simpl; try constructor || eauto. apply H0.
    Qed.

    #[local] Hint Extern 5 (copy (?n + ?m) (eval ?a) ==
      Bin.value (copy ?n (eval ?a)) (copy ?m (eval ?a))) ⇒ apply copy_plus : core.
    #[local] Hint Extern 5 (?x == ?x) ⇒ reflexivity : core.
    #[local] Hint Extern 5 ( Bin.value ?x ?y == Bin.value ?y ?x) ⇒ apply Bin.comm : core.

    Lemma eval_merge_bin : ∀ (h k: mset T),
      eval (sum i (merge_msets compare h k)) ==
      @Bin.value _ _ (e_bin i) (eval (sum i h)) (eval (sum i k)).
    Proof.
      induction h as [[a n]|[a n] h IHh]; intro k.
      - simpl; induction k as [[b m]|[b m] k IHk]; simpl.
        × destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto.
          apply copy_plus; auto.
        × destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl; auto.
          rewrite copy_plus,law_assoc; auto.
          rewrite IHk; clear IHk. rewrite 2 law_assoc.
          apply proper; [apply law_comm|reflexivity].
      - induction k as [[b m]|[b m] k IHk]; simpl; simpl in IHh.
        × destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl.
          rewrite (law_comm _ (copy m (eval a))).
          rewrite law_assoc, <- copy_plus, Pplus_comm; auto.
          rewrite <- law_assoc, IHh. reflexivity.
          rewrite law_comm. reflexivity.
        × simpl in IHk.
          destruct (tcompare_weak_spec a b) as [a|a b|a b]; simpl.
          rewrite IHh; clear IHh. rewrite 2 law_assoc.
          rewrite (law_comm _ (copy m (eval a))).
          rewrite law_assoc, <- copy_plus, Pplus_comm; auto.
          rewrite IHh; clear IHh. simpl. rewrite law_assoc. reflexivity.
          rewrite 2 (law_comm (copy m (eval b))).
          rewrite law_assoc. apply proper; [ | reflexivity].
          rewrite <- IHk. reflexivity.
    Qed.

    Lemma copy_mset' n (l: mset T) :
      copy_mset n l = nelist_map (fun vm ⇒ let '(v,m):=vm in (v,Pmult n m)) l.
    Proof.
      unfold copy_mset. destruct n; try reflexivity.
      simpl. induction l as [|[a l] IHl]; simpl; try congruence.
      destruct a; reflexivity.
    Qed.

    Lemma copy_mset_succ n (l: mset T) :
      eval (sum i (copy_mset (Pos.succ n) l)) ==
      @Bin.value _ _ (e_bin i) (eval (sum i l)) (eval (sum i (copy_mset n l))).
    Proof.
      rewrite 2 copy_mset'.
      induction l as [[a m]|[a m] l IHl].
      simpl eval. rewrite <- copy_plus; auto.
      rewrite Pmult_Sn_m. reflexivity.
      simpl nelist_map. rewrite ! eval_sum_cons. rewrite IHl. clear IHl.
      rewrite Pmult_Sn_m. rewrite copy_plus; auto. rewrite <- !law_assoc.
      apply Binvalue_Proper; try reflexivity.
      rewrite law_comm . rewrite <- !law_assoc. apply proper; try reflexivity.
      apply law_comm.
    Qed.

    Lemma copy_mset_copy : ∀ n (m : mset T), eval (sum i (copy_mset n m)) ==
     @copy _ (@Bin.value _ _ (e_bin i)) n (eval (sum i m)).
    Proof.
      induction n using Pind; intros.
      - unfold copy_mset. rewrite copy_xH. reflexivity.
      - rewrite copy_mset_succ. rewrite copy_Psucc. rewrite IHn. reflexivity.
    Qed.

    Instance compat_sum_unit_Unit : ∀ p, compat_sum_unit (left p) →
      @Unit X R (Bin.value (e_bin i)) (eval (unit p)).
    Proof. intros; inversion H; subst; auto. Qed.

    Lemma copy_n_unit : ∀ j n, is_unit j = true →
      eval (unit j) == @copy _ (Bin.value (e_bin i)) n (eval (unit j)).
    Proof.
      intros; induction n using Prect.
      rewrite copy_xH. reflexivity.
      rewrite copy_Psucc. rewrite <- IHn.
      apply is_unit_sum_Unit in H. rewrite law_neutral_left. reflexivity.
    Qed.

    Lemma z0 l r (H : compat_sum_unit r) :
      eval (sum i (run_msets (comp (merge_msets compare) l r))) ==
      eval (sum i ((merge_msets compare) (l) (run_msets r))).
    Proof.
      unfold comp. unfold run_msets.
      case_eq r; intros; subst; [|reflexivity].
      rewrite eval_merge_bin; auto.
      rewrite eval_sum_nil.
      apply compat_sum_unit_Unit in H.
      rewrite law_neutral_right. reflexivity.
    Qed.

    Lemma z1 : ∀ n x,
      eval (sum i (run_msets (return_sum i (is_unit) x n ))) ==
      @copy _ (@Bin.value _ _ (e_bin i)) n (eval x).
    Proof.
      intros; unfold return_sum, run_msets.
      case (is_sum_spec); intros; subst.
      - rewrite copy_mset_copy; reflexivity.
      - rewrite eval_sum_nil. apply copy_n_unit. auto.
      - reflexivity.
    Qed.

    Lemma z2 : ∀ u n x, compat_sum_unit x →
      eval (sum i (run_msets (add_to_sum i (is_unit) u n x))) ==
      @Bin.value _ _ (e_bin i)
        (@copy _ (@Bin.value _ _ (e_bin i)) n (eval u)) (eval (sum i (run_msets x))).
    Proof.
      intros u n x Hix.
      unfold add_to_sum.
      case is_sum_spec; intros; subst.
      - rewrite z0 by auto. rewrite eval_merge_bin, copy_mset_copy. reflexivity.
      - rewrite <- copy_n_unit by assumption. apply is_unit_sum_Unit in H.
        rewrite law_neutral_left. reflexivity.
      - rewrite z0 by auto. rewrite eval_merge_bin. reflexivity.
    Qed.

  End sum_correctness.

  Lemma eval_norm_msets i norm
    (Comm : Commutative R (Bin.value (e_bin i)))
    (Hnorm: ∀ u, eval (norm u) == eval u) :
      ∀ h, eval (sum i (norm_msets norm i h)) == eval (sum i h).
  Proof.
    unfold norm_msets.
    assert (H : ∀ h : mset T,
     eval (sum i (run_msets (norm_msets_ i (is_unit_of i) norm h))) ==
     eval (sum i h) ∧ compat_sum_unit (is_unit_of i) (norm_msets_ i (is_unit_of i) norm h)).
    induction h as [[a n] | [a n] h [IHh IHh']]; simpl norm_msets_; split.
    - rewrite z1 by auto. rewrite Hnorm. reflexivity.
    - apply compat_sum_unit_return.
    - rewrite z2 by auto. rewrite IHh, eval_sum_cons, Hnorm. reflexivity.
    - apply compat_sum_unit_add, IHh'.
    - apply H.
  Defined.

  Lemma copy_idem i (Idem : Idempotent R (Bin.value (e_bin i))) n x :
    copy (plus:=(Bin.value (e_bin i))) n x == x.
  Proof.
    induction n using Pos.peano_ind; simpl.
    - apply copy_xH.
    - rewrite copy_Psucc, IHn; apply law_idem.
  Qed.

  Lemma eval_reduce_msets i (Idem : Idempotent R (Bin.value (e_bin i))) m :
    eval (sum i (reduce_mset m)) == eval (sum i m).
  Proof.
    induction m as [[a n]|[a n] m IH].
    - simpl. now rewrite 2copy_idem.
    - simpl. rewrite IH. now rewrite 2copy_idem.
  Qed.

Auxiliary lemmas about products
  Section prd_correctness.

    Variable i : idx.
    Variable is_unit : idx → bool.
    Hypothesis is_unit_prd_Unit : ∀ j, is_unit j = true →
     @Unit X R (Bin.value (e_bin i)) (eval (unit j)).

    Inductive is_prd_spec_ind : T → @discr (nelist T) → Prop :=
    | is_prd_spec_op :
      ∀ j l, j = i → is_prd_spec_ind (prd j l) (Is_op l)
    | is_prd_spec_unit :
      ∀ j, is_unit j = true → is_prd_spec_ind (unit j) (Is_unit j)
    | is_prd_spec_nothing :
      ∀ u, is_prd_spec_ind u (Is_nothing).

    Lemma is_prd_spec u : is_prd_spec_ind u (is_prd i is_unit u).
    Proof.
      unfold is_prd; case u; intros; try constructor.
      case (eq_idx_spec); intros; subst; try constructor; auto.
      case_eq (is_unit p); intros; try constructor; auto.
    Qed.

    Lemma prd'_prd : ∀ (l: nelist T), eval (prd' i l) == eval (prd i l).
    Proof.
      intros [?|? [|? ?]]; simpl; reflexivity.
    Qed.

    Lemma eval_prd_nil x: eval (prd i (nil x)) == eval x.
    Proof.
      rewrite <- prd'_prd. simpl. reflexivity.
    Qed.

    Lemma eval_prd_cons a : ∀ (l: nelist T),
      eval (prd i (a::l)) == @Bin.value _ _ (e_bin i) (eval a) (eval (prd i l)).
    Proof. intros [|b l]; simpl; reflexivity. Qed.

    Lemma eval_prd_app : ∀ (h k: nelist T),
      eval (prd i (h++k)) == @Bin.value _ _ (e_bin i) (eval (prd i h)) (eval (prd i k)).
    Proof.
      induction h; intro k. simpl; try reflexivity.
      simpl appne. rewrite 2 eval_prd_cons, IHh, law_assoc. reflexivity.
    Qed.

    Inductive compat_prd_unit : @m idx (nelist T) → Prop :=
    | cpu_left : ∀ x, is_unit x = true → compat_prd_unit (left x)
    | cpu_right : ∀ m, compat_prd_unit (right m).

    Lemma compat_prd_unit_return x : compat_prd_unit (return_prd i is_unit x).
    Proof.
      unfold return_prd.
      case (is_prd_spec); intros; try constructor; auto.
    Qed.

    Lemma compat_prd_unit_add : ∀ x h, compat_prd_unit h →
      compat_prd_unit (add_to_prd i is_unit x h).
    Proof.
      intros; unfold add_to_prd, comp.
      case (is_prd_spec); intros; try constructor; auto.
      - unfold comp; case h; try constructor.
      - unfold comp; case h; try constructor.
    Qed.

    Instance compat_prd_Unit : ∀ p, compat_prd_unit (left p) →
      @Unit X R (Bin.value (e_bin i)) (eval (unit p)).
    Proof.
      intros.
      inversion H; subst. apply is_unit_prd_Unit. assumption.
    Qed.

    Lemma z0' : ∀ l (r: @m idx (nelist T)), compat_prd_unit r →
      eval (prd i (run_list (comp (@appne T) l r))) ==
      eval (prd i ((appne (l) (run_list r)))).
    Proof.
      intros.
      unfold comp. unfold run_list. case_eq r; intros; auto; subst.
      rewrite eval_prd_app.
      rewrite eval_prd_nil.
      apply compat_prd_Unit in H. rewrite law_neutral_right. reflexivity.
      reflexivity.
    Qed.

    Lemma z1' a : eval (prd i (run_list (return_prd i is_unit a))) == eval (prd i (nil a)).
    Proof.
      intros. unfold return_prd. unfold run_list.
      case (is_prd_spec); intros; subst; reflexivity.
    Qed.

    Lemma z2' : ∀ u x, compat_prd_unit x →
      eval (prd i (run_list (add_to_prd i is_unit u x))) ==
      @Bin.value _ _ (e_bin i) (eval u) (eval (prd i (run_list x))).
    Proof.
      intros u x Hix.
      unfold add_to_prd.
      case (is_prd_spec); intros; subst.
      rewrite z0' by auto. rewrite eval_prd_app. reflexivity.
      apply is_unit_prd_Unit in H. rewrite law_neutral_left. reflexivity.
      rewrite z0' by auto. rewrite eval_prd_app. reflexivity.
    Qed.

  End prd_correctness.

  Lemma eval_norm_lists i (Hnorm: ∀ u, eval (norm u) == eval u) :
   ∀ h, eval (prd i (norm_lists norm i h)) == eval (prd i h).
  Proof.
    unfold norm_lists.
    assert (H : ∀ h : nelist T,
      eval (prd i (run_list (norm_lists_ i (is_unit_of i) norm h))) ==
      eval (prd i h)
      ∧ compat_prd_unit (is_unit_of i) (norm_lists_ i (is_unit_of i) norm h)). {
      induction h as [a | a h [IHh IHh']]; simpl norm_lists_; split.
      rewrite z1'. simpl. apply Hnorm.
      apply compat_prd_unit_return.
      rewrite z2'. rewrite IHh. rewrite eval_prd_cons.
      rewrite Hnorm. reflexivity. apply is_unit_of_Unit.
      auto.
      apply compat_prd_unit_add. auto.
    }
    apply H.
  Defined.

Correctness of the normalisation function

  Fixpoint eval_norm u: eval (norm u) == eval u
    with eval_norm_aux i l : ∀ (f: Sym.type_of i),
     Proper (@Sym.rel_of X R i) f → eval_aux (vnorm l) f == eval_aux l f.
  Proof.
    induction u as [ p m | p l | ? | ?]; simpl norm.
    - case_eq (is_commutative p); intros.
      case_eq (is_idempotent p); intros.
      rewrite sum'_sum.
      rewrite eval_reduce_msets. 2: eauto with typeclass_instances.
      apply eval_norm_msets; auto.
      rewrite sum'_sum.
      apply eval_norm_msets; auto.
      reflexivity.
    - rewrite prd'_prd.
      apply eval_norm_lists; auto.
    - apply eval_norm_aux, Sym.morph.
    - reflexivity.
    - induction l; simpl; intros f Hf. reflexivity.
      rewrite eval_norm. apply IHl, Hf; reflexivity.
  Qed.

Corollaries, for goal normalisation or decision

  Lemma normalise : ∀ (u v: T), eval (norm u) == eval (norm v) → eval u == eval v.
  Proof. intros u v. rewrite 2 eval_norm. trivial. Qed.

  Lemma compare_reflect_eq: ∀ u v, compare u v = Eq → eval u == eval v.
  Proof.
    intros u v. case (tcompare_weak_spec u v); intros; try congruence.
    reflexivity.
  Qed.

  Lemma decide: ∀ (u v: T), compare (norm u) (norm v) = Eq → eval u == eval v.
  Proof. intros u v H. apply normalise. apply compare_reflect_eq. apply H. Qed.

  Register decide as aac_tactics.internal.decide.

  Lemma lift_normalise {S} {H : AAC_lift S R} :
    ∀ (u v: T), (let x := norm u in let y := norm v in
      S (eval x) (eval y)) → S (eval u) (eval v).
  Proof. destruct H. intros u v; simpl; rewrite 2 eval_norm. trivial. Qed.

  Register lift_normalise as aac_tactics.internal.lift_normalise.

End s.

End Internal.

Local Ltac internal_normalize :=
  let x := fresh in let y := fresh in
  intro x; intro y; vm_compute in x; vm_compute in y; unfold x; unfold y;
  compute [Internal.eval Utils.fold_map Internal.copy Prect]; simpl.

Lemmas for performing transitivity steps given an AAC_lift instance


Section t.

  Context `{AAC_lift}.

  Lemma lift_transitivity_left (y x z : X): E x y → R y z → R x z.
  Proof. destruct H as [Hequiv Hproper]; intros G;rewrite G. trivial. Qed.

  Lemma lift_transitivity_right (y x z : X): E y z → R x y → R x z.
  Proof. destruct H as [Hequiv Hproper]; intros G. rewrite G. trivial. Qed.

  Lemma lift_reflexivity {HR :Reflexive R}: ∀ x y, E x y → R x y.
  Proof. destruct H. intros ? ? G. rewrite G. reflexivity. Qed.

  Register lift_transitivity_left as aac_tactics.internal.lift_transitivity_left.
  Register lift_transitivity_right as aac_tactics.internal.lift_transitivity_right.
  Register lift_reflexivity as aac_tactics.internal.lift_reflexivity.

End t.

Declare ML Module "aac_plugin:coq-aac-tactics.plugin".