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+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+
+ +
+

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: forall 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 : forall 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: forall x y, R (plus x y) (plus y x).
+Class Idempotent (X : Type) (R : relation X) (plus : X -> X -> X) :=
+  law_idem: forall x, R (plus x x) x.
+Class Unit (X : Type) (R : relation X) (op : X -> X -> X) (unit : X) := {
+  law_neutral_left: forall x, R (op unit x) x;
+  law_neutral_right: forall 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 : forall 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 : forall x, R (copy 1 x) x.
+  Proof. intros; unfold copy; rewrite Prect_base. reflexivity. Qed.
+ +
+  Lemma copy_Psucc : forall 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: forall i, vT (Sym.ar (e_sym i)) -> T
+  | unit : idx -> T
+  with vT: nat -> Type :=
+  | vnil: vT O
+  | vcons: forall 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 (Pos.compare i j) (mset_compare compare l vs)
+      | prd i l, prd j vs => lex (Pos.compare i j) (list_compare compare l vs)
+      | sym i l, sym j vs => lex (Pos.compare i j) (vcompare l vs)
+      | unit i , unit j => Pos.compare i j
+      | unit _ , _ => Lt
+      | _ , unit _ => Gt
+      | sym _ _, _ => Lt
+      | _ , sym _ _ => 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 : forall (v : T), compare_weak_spec u v (compare u v)
+  with vcompare_reflect_eqdep i us : forall 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 (Pos.compare i i0); intro Hi; try constructor.
+      (* the symmetry is required *)
+      apply pos_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 (Pos.compare p p0); intro Hi; try constructor.
+      apply pos_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.
+      (* the symmetry is required *)
+      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 : forall 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 : forall 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 : forall j l, j = i -> is_sum_spec_ind (sum j l) (Is_op l)
+    | is_sum_spec_unit : forall j, is_unit j = true -> is_sum_spec_ind (unit j) (Is_unit j)
+    | is_sum_spec_nothing : forall 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 : forall (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 : forall 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 : forall x, is_unit x = true-> compat_sum_unit (left x)
+    | csu_right : forall 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 : forall 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.
+ +
+    (* Hint Resolve copy_plus. : this lags because of the inference of the implicit arguments *)
+    #[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 : forall (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 : forall 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 : forall 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 : forall 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 : forall 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 : forall 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: forall u, eval (norm u) == eval u) :
+      forall h, eval (sum i (norm_msets norm i h)) == eval (sum i h).
+  Proof.
+    unfold norm_msets.
+    assert (H : forall 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 : forall 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 :
+      forall j l, j = i -> is_prd_spec_ind (prd j l) (Is_op l)
+    | is_prd_spec_unit :
+      forall j, is_unit j = true -> is_prd_spec_ind (unit j) (Is_unit j)
+    | is_prd_spec_nothing :
+      forall 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 : forall (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 : forall (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 : forall (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 : forall x, is_unit x = true -> compat_prd_unit (left x)
+    | cpu_right : forall 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 : forall 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 : forall 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' : forall 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' : forall 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: forall u, eval (norm u) == eval u) :
+   forall h, eval (prd i (norm_lists norm i h)) == eval (prd i h).
+  Proof.
+    unfold norm_lists.
+    assert (H : forall 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 : forall (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 : forall (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: forall 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: forall (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} :
+    forall (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}: forall 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 "coq-aac-tactics.plugin".
+ +
+Lemma transitivity4 {A R} {H: @Equivalence A R} a b a' b': R a a' -> R b b' -> R a b -> R a' b'.
+Proof. now intros -> ->. Qed.
+Tactic Notation "aac_normalise" "in" hyp(H) :=
+  eapply transitivity4 in H; [| aac_normalise; reflexivity | aac_normalise; reflexivity].
+ +
+Ltac aac_normalise_all :=
+    aac_normalise;
+    repeat match goal with
+      | H: _ |- _ => aac_normalise in H
+      end.
+Tactic Notation "aac_normalise" "in" "*" := aac_normalise_all.
+
+
+ +
+ + + diff --git a/docs/coqdoc/AAC_tactics.Caveats.html b/docs/coqdoc/AAC_tactics.Caveats.html new file mode 100644 index 0000000..339164f --- /dev/null +++ b/docs/coqdoc/AAC_tactics.Caveats.html @@ -0,0 +1,760 @@ + + + + + + + + + + + + + +
+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+
+ +
+

Currently known limitations and caveats of AAC Tactics

+ +
+
+ +
+From Coq Require NArith PeanoNat.
+From AAC_tactics Require Import AAC.
+From AAC_tactics Require Instances.
+ +
+
+ +
+

Limitations

+ +
+ +

Dependent parameters

+ + +
+ + The type of the rewriting hypothesis must be of the form + +
+ + forall (x_1: T_1) ... (x_n: T_n), R l r + +
+ + where R is a relation over some type T and such that for all + variable x_i appearing in the left-hand side (l), we actually + have T_i = T. The goal should be of the form S g d, where + S is a relation on T. + +
+ + In other words, we cannot instantiate arguments of an exogeneous type. + +
+
+ +
+Section parameters.
+ +
+  Context {X} {R} {E: @Equivalence X R} {plus}
+   {plus_A: Associative R plus} {plus_C: Commutative R plus}
+   {plus_Proper: Proper (R ==> R ==> R) plus}
+   {zero} {Zero: Unit R plus zero}.
+ +
+  Notation "x == y" := (R x y) (at level 70).
+  Notation "x + y" := (plus x y) (at level 50, left associativity).
+  Notation "0" := (zero).
+ +
+  Variable f : nat -> X -> X.
+ +
+
+ +
+ In Hf, since the parameter n has type nat, + it cannot be instantiated automatically + +
+
+  Hypothesis Hf: forall n x, f n x + x == x.
+  Hypothesis Hf': forall n, Proper (R ==> R) (f n).
+ +
+  Goal forall a b k, a + f k (b+a) + b == a+b.
+    intros.
+
+ +
+aac_rewrite does not instantiate n automatically +
+
+    Fail aac_rewrite Hf.
+
+ +
+of course, this argument can be given explicitly +
+
+    aac_rewrite (Hf k).
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+For the same reason, we cannot handle higher-order parameters (here, g) +
+
+  Hypothesis H : forall g x y, g x + g y == g (x + y).
+  Variable g : X -> X.
+  Hypothesis Hg : Proper (R ==> R) g.
+ +
+  Goal forall a b c, g a + g b + g c == g (a + b + c).
+    intros.
+    Fail aac_rewrite H.
+    do 2 aac_rewrite (H g). aac_reflexivity.
+  Qed.
+ +
+End parameters.
+ +
+
+ +
+

Exogeneous morphisms

+ + +
+ + We do not handle "exogeneous" morphisms: morphisms that move from + type T to some other type T'. + +
+
+ +
+Section morphism.
+  Import NArith PeanoNat.
+  Import Instances.N Instances.Peano.
+ +
+  Open Scope nat_scope.
+ +
+
+ +
+ Typically, although N_of_nat is a proper morphism from + @eq nat to @eq N, we cannot rewrite under N_of_nat + +
+
+  Goal forall a b: nat, N_of_nat (a+b-(b+a)) = 0%N.
+    intros.
+    Fail aac_rewrite Nat.sub_diag.
+  Abort.
+ +
+
+ +
+ More generally, this prevents us from rewriting under + propositional contexts + +
+
+  Context {P} {HP : Proper (@eq nat ==> iff) P}.
+  Hypothesis H : P 0.
+ +
+  Goal forall a b, P (a + b - (b + a)).
+    intros a b.
+    Fail aac_rewrite Nat.sub_diag.
+
+ +
+a solution is to introduce an evar to replace the part to be + rewritten - this tiresome process should be improved in the + future; here, it can be done using eapply and the morphism +
+
+    eapply HP.
+    aac_rewrite Nat.sub_diag.
+    reflexivity.
+    exact H.
+  Qed.
+ +
+  Goal forall a b, a+b-(b+a) = 0 /\ b-b = 0.
+    intros.
+
+ +
+similarly, we need to bring equations to the toplevel before + being able to rewrite +
+
+    Fail aac_rewrite Nat.sub_diag.
+    split; aac_rewrite Nat.sub_diag; reflexivity.
+  Qed.
+ +
+End morphism.
+ +
+
+ +
+

Treatment of variance with inequations

+ + +
+ + We do not take variance into account when we compute the set of + solutions to a matching problem modulo AC. As a consequence, + aac_instances may propose solutions for which aac_rewrite will + fail, due to the lack of adequate morphisms. + +
+
+ +
+Section ineq.
+  Import ZArith.
+  Import Instances.Z.
+ +
+  Open Scope Z_scope.
+ +
+  #[local] Instance Z_add_incr: Proper (Z.le ==> Z.le ==> Z.le) Z.add.
+  Proof. intros ? ? H ? ? H'. apply Zplus_le_compat; assumption. Qed.
+ +
+  Hypothesis H : forall x, x+x <= x.
+ +
+  Goal forall a b c, c + - (a + a) + b + b <= c.
+    intros.
+
+ +
+this fails because the first solution is not valid (Z.opp is not increasing) +
+
+    Fail aac_rewrite H.
+    aac_instances H.
+
+ +
+on the contrary, the second solution is valid: +
+
+    aac_rewrite H at 1.
+
+ +
+currently, we cannot filter out such invalid solutions in an easy way; + this should be fixed in the future +
+
+  Abort.
+ +
+End ineq.
+ +
+
+ +
+

Caveats

+ +
+ +

Special treatment for units

+ + +
+ + S 0 is considered as a unit for multiplication whenever a Nat.mul + appears in the goal. The downside is that S x does not match 1, + and 1 does not match S (0 + 0) whenever Nat.mul appears in + the goal. + +
+
+ +
+Section Peano.
+  Import Instances.Peano.
+ +
+  Hypothesis H : forall x, x + S x = S (x+x).
+ +
+  Goal 1 = 1.
+
+ +
+OK (no multiplication around), x is instantiated with O +
+
+    aacu_rewrite H.
+  Abort.
+ +
+  Goal 1 * 1 = 1.
+
+ +
+fails since 1 is seen as a unit, not the application of the + morphism S to the constant O +
+
+    Fail aacu_rewrite H.
+  Abort.
+ +
+  Hypothesis H': forall x, x + 1 = 1 + x.
+ +
+  Goal forall a, a + S (0+0) = 1 + a.
+
+ +
+OK (no multiplication around), x is instantiated with a +
+
+    intro. aac_rewrite H'.
+  Abort.
+ +
+  Goal forall a, a * a + S (0+0) = 1 + a * a.
+
+ +
+fails: although S (0+0) is understood as the application of + the morphism S to the constant O, it is not recognised + as the unit S O of multiplication +
+
+    intro. Fail aac_rewrite H'.
+  Abort.
+ +
+
+ +
+ More generally, similar counter-intuitive behaviours can appear + when declaring an applied morphism as a unit + +
+
+End Peano.
+ +
+
+ +
+

Existential variables

+ + +
+ + We implemented an algorithm for matching modulo AC, not for + unifying modulo AC. As a consequence, existential variables + appearing in a goal are considered as constants and will not be + instantiated. +
+
+ +
+Section evars.
+  Import ZArith.
+  Import Instances.Z.
+ +
+  Variable P: Prop.
+  Hypothesis H: forall x y, x + y + x = x -> P.
+  Hypothesis idem: forall x, x + x = x.
+ +
+  Goal P.
+    eapply H.
+
+ +
+this works: x is instantiated with an evar +
+
+    aac_rewrite idem.
+    instantiate (2 := 0).
+
+ +
+this does work but there are remaining evars in the end +
+
+    symmetry. aac_reflexivity.
+  Abort.
+ +
+  Hypothesis H': forall x, 3+x = x -> P.
+ +
+  Goal P.
+    eapply H'.
+
+ +
+this fails since we do not instantiate evars +
+
+    Fail aac_rewrite idem.
+  Abort.
+ +
+End evars.
+ +
+
+ +
+

Distinction between aac_rewrite and aacu_rewrite

+ +
+
+ +
+Section U.
+  Context {X} {R} {E: @Equivalence X R}
+   {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot}
+   {one} {One: Unit R dot one}.
+ +
+  Infix "==" := R (at level 70).
+  Infix "*" := dot.
+  Notation "1" := one.
+ +
+
+ +
+in some situations, the aac_rewrite tactic allows + instantiations of a variable with a unit, when the variable occurs + directly under a function symbol: +
+
+  Variable f : X -> X.
+  Hypothesis Hf : Proper (R ==> R) f.
+  Hypothesis dot_inv_left : forall x, f x*x == x.
+ +
+  Goal f 1 == 1.
+    aac_rewrite dot_inv_left. reflexivity.
+  Qed.
+ +
+
+ +
+this behaviour seems desirable in most situations: these + solutions with units are less peculiar than the other ones, since + the unit comes from the goal; however, this policy is not properly + enforced for now (hard to do with the current algorithm): +
+
+  Hypothesis dot_inv_right : forall x, x*f x == x.
+ +
+  Goal f 1 == 1.
+    Fail aac_rewrite dot_inv_right.
+    aacu_rewrite dot_inv_right. reflexivity.
+  Qed.
+ +
+End U.
+ +
+
+ +
+

Rewriting units

+ +
+
+Section V.
+  Context {X} {R} {E: @Equivalence X R}
+  {dot} {dot_A: Associative R dot} {dot_Proper: Proper (R ==> R ==> R) dot}
+  {one} {One: Unit R dot one}.
+ +
+  Infix "==" := R (at level 70).
+  Infix "*" := dot.
+  Notation "1" := one.
+ +
+
+ +
+ aac_rewrite uses the symbols appearing in the goal and the + hypothesis to infer the AC and A operations. In the following + example, dot appears neither in the left-hand-side of the goal, + nor in the right-hand side of the hypothesis. Hence, 1 is not + recognised as a unit. To circumvent this problem, we can force + aac_rewrite to take into account a given operation, by giving + it an extra argument. This extra argument seems useful only in + this peculiar case. + +
+
+ +
+  Lemma inv_unique: forall x y y', x*y == 1 -> y'*x == 1 -> y==y'.
+  Proof.
+    intros x y y' Hxy Hy'x.
+    aac_instances <- Hy'x [dot].
+    aac_rewrite <- Hy'x at 1 [dot].
+    aac_rewrite Hxy. aac_reflexivity.
+  Qed.
+ +
+End V.
+ +
+
+ +
+

Rewriting too many things

+ +
+
+ +
+Section W.
+  Import Instances.Peano.
+ +
+  Variables a b c: nat.
+  Hypothesis H: 0 = c.
+ +
+  Goal b*(a+a) <= b*(c+a+a+1).
+
+ +
+aac_rewrite finds a pattern modulo AC that matches a given + hypothesis, and then makes a call to setoid_rewrite. This + setoid_rewrite can unfortunately make several rewrites (in a + non-intuitive way: below, the 1 in the right-hand side is + rewritten into S c) +
+
+    aac_rewrite H.
+
+ +
+to this end, we provide a companion tactic to aac_rewrite + and aacu_rewrite, that makes the transitivity step, but not the + setoid_rewrite; this allows the user to select the relevant + occurrences in which to rewrite: +
+
+    aac_pattern H at 2. setoid_rewrite H at 1.
+  Abort.
+ +
+End W.
+ +
+
+ +
+

Rewriting nullifiable patterns

+ +
+
+ +
+Section Z.
+  Import Instances.Peano.
+ +
+
+ +
+ If the pattern of the rewritten hypothesis does not contain "hard" + symbols (like constants, function symbols, AC or A symbols without + units), there can be infinitely many subterms such that the pattern + matches: it is possible to build "subterms" modulo ACU that make the + size of the term increase (by making neutral elements appear in a + layered fashion). Hence, we settled with heuristics to propose only + "some" of these solutions. In such cases, the tactic displays a + (conservative) warning. + +
+
+ +
+  Variables a b c: nat.
+  Variable f: nat -> nat.
+  Hypothesis H0: forall x, 0 = x - x.
+  Hypothesis H1: forall x, 1 = x * x.
+ +
+  Goal a + b * c = c.
+    aac_instances H0.
+
+ +
+in this case, only three solutions are proposed, while there are + infinitely many solutions, for example: +
    +
  • a+b*c*(1+☐) + +
    • +
  • a+b*c*(1+0*(1+ ☐)) + +
    • +
  • ... + +
    • +
+
+
+  Abort.
+ +
+
+ +
+

If the pattern is a unit or can be instantiated to be equal to a unit

+ + +
+ + The heuristic is to make the unit appear at each possible position + in the term, e.g. transforming a into 1*a and a*1, but this + process is not recursive (we will not transform 1*a) into + (1+0*1)*a. + +
+
+ +
+  Goal a+b+c = c.
+    aac_instances H0 [mult].
+
+ +
+1 solution, we miss solutions like (a+b+c*(1+0*(1+[]))) and so on +
+
+    aac_instances H1 [mult].
+
+ +
+7 solutions, we miss solutions like (a+b+c+0*(1+0*[])) +
+
+  Abort.
+ +
+
+ +
+

Another example of the former case

+ + +
+ + In the following, the hypothesis can be instantiated + to be equal to 1. + +
+
+ +
+  Hypothesis H : forall x y, (x+y)*x = x*x + y*x.
+ +
+  Goal a*a+b*a + c = c.
+
+ +
+here, only one solution if we use the aac_instance tactic +
+
+    aac_instances <- H.
+
+ +
+there are 8 solutions using aacu_instances (but, here, + there are infinitely many different solutions); we miss, e.g., a*a +b*a + + (x*x + y*x)*c, which seems to be more peculiar +
+
+    aacu_instances <- H.
+
+ +
+the 7 last solutions are the same as if we were matching 1 +
+
+    aacu_instances H1.
+  Abort.
+ +
+
+ +
+ The behavior of the tactic is not satisfying in this case. It is + still unclear how to handle properly this kind of situation; we plan + to investigate this in the future. + +
+
+End Z.
+
+
+ +
+ + + diff --git a/docs/coqdoc/AAC_tactics.Constants.html b/docs/coqdoc/AAC_tactics.Constants.html new file mode 100644 index 0000000..ca89e9e --- /dev/null +++ b/docs/coqdoc/AAC_tactics.Constants.html @@ -0,0 +1,80 @@ + + + + + + + + + + + + + +
+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+Register Init.Datatypes.O as aac_tactics.nat.O.
+Register Init.Datatypes.S as aac_tactics.nat.S.
+Register Init.Datatypes.nat as aac_tactics.nat.type.
+ +
+Register Init.Datatypes.pair as aac_tactics.pair.pair.
+Register Init.Datatypes.prod as aac_tactics.pair.prod.
+ +
+Register Init.Datatypes.option as aac_tactics.option.typ.
+Register Init.Datatypes.Some as aac_tactics.option.Some.
+Register Init.Datatypes.None as aac_tactics.option.None.
+ +
+Register Init.Datatypes.list as aac_tactics.list.typ.
+Register Init.Datatypes.nil as aac_tactics.list.nil.
+Register Init.Datatypes.cons as aac_tactics.list.cons.
+ +
+From Coq Require BinNums.
+Register BinNums.positive as aac_tactics.pos.typ.
+Register BinNums.xI as aac_tactics.pos.xI.
+Register BinNums.xO as aac_tactics.pos.xO.
+Register BinNums.xH as aac_tactics.pos.xH.
+ +
+From Coq Require Classes.Morphisms.
+Register Morphisms.Proper as aac_tactics.coq.classes.morphisms.Proper.
+From Coq Require Classes.RelationClasses.
+ +
+Register RelationClasses.Equivalence as aac_tactics.coq.RelationClasses.Equivalence.
+Register RelationClasses.Reflexive as aac_tactics.coq.RelationClasses.Reflexive.
+Register RelationClasses.Transitive as aac_tactics.coq.RelationClasses.Transitive.
+
+
+ +
+ + + diff --git a/docs/coqdoc/AAC_tactics.Instances.html b/docs/coqdoc/AAC_tactics.Instances.html new file mode 100644 index 0000000..b7326c9 --- /dev/null +++ b/docs/coqdoc/AAC_tactics.Instances.html @@ -0,0 +1,629 @@ + + + + + + + + + + + + + +
+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+
+ +
+

Instances for AAC Tactics

+ +
+
+ +
+From Coq Require PeanoNat ZArith Zminmax NArith List Permutation.
+From Coq 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.
+ +
+  (* We also provide liftings from N.le to eq *)
+  #[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.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 => exists 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.
+ +
+
+ +
+ Here, we should not see any instance of our classes: + Print HintDb typeclass_instances. + +
+
+
+
+ +
+ + + diff --git a/docs/coqdoc/AAC_tactics.Tutorial.html b/docs/coqdoc/AAC_tactics.Tutorial.html new file mode 100644 index 0000000..c6f860d --- /dev/null +++ b/docs/coqdoc/AAC_tactics.Tutorial.html @@ -0,0 +1,820 @@ + + + + + + + + + + + + + +
+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+
+ +
+

Tutorial for using AAC Tactics

+ +
+
+ +
+From Coq Require PeanoNat ZArith List Permutation Lia.
+From AAC_tactics Require Import AAC.
+From AAC_tactics Require Instances.
+ +
+
+ +
+

Introductory example

+ + +
+ + Here is a first example with relative numbers (Z): one can + rewrite an universally quantified hypothesis modulo the + associativity and commutativity of Z.add. +
+
+ +
+Section introduction.
+  Import ZArith.
+  Import Instances.Z.
+ +
+  Variables a b c : Z.
+  Hypothesis H: forall x, x + Z.opp x = 0.
+ +
+  Goal a + b + c + Z.opp (c + a) = b.
+    aac_rewrite H.
+    aac_reflexivity.
+  Qed.
+ +
+  Goal a + c + Z.opp (b + a + Z.opp b) = c.
+    do 2 aac_rewrite H.
+    reflexivity.
+  Qed.
+ +
+
+ +
+note: +
    +
  • the tactic handles arbitrary function symbols like Z.opp (as + long as they are proper morphisms w.r.t. the considered + equivalence relation) + +
    • +
  • here, the ring tactic would have done the job + +
    • +
+ +
+ + several associative/commutative operations can be used at the same time, + here, Z.mul and Z.add, which are both associative and commutative (AC) + +
+
+  Goal (b + c) * (c + b) + a + Z.opp ((c + b) * (b + c)) = a.
+    aac_rewrite H.
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+some commutative operations can be declared as idempotent, here Z.max + which is taken into account by the aac_normalise and aac_reflexivity tactics; + however, aac_rewrite does not match modulo idempotency + +
+
+  Goal Z.max (b + c) (c + b) + a + Z.opp (c + b) = a.
+    aac_normalise.
+    aac_rewrite H.
+    aac_reflexivity.
+  Qed.
+ +
+  Goal Z.max c (Z.max b c) + a + Z.opp (Z.max c b) = a.
+    aac_normalise.
+    aac_rewrite H.
+    aac_reflexivity.
+  Qed.
+ +
+End introduction.
+ +
+
+ +
+

Usage

+ + +
+ + One can also work in an abstract context, with arbitrary + associative and commutative operators. (Note that one can declare + several operations of each kind.) + +
+
+ +
+Section base.
+  Context {X} {R} {E: Equivalence R}
+  {plus} {dot} {zero} {one}
+  {dot_A: @Associative X R dot }
+  {plus_A: @Associative X R plus }
+  {plus_C: @Commutative X R plus }
+  {dot_Proper :Proper (R ==> R ==> R) dot}
+  {plus_Proper :Proper (R ==> R ==> R) plus}
+  {Zero : Unit R plus zero}
+  {One : Unit R dot one}.
+ +
+  Notation "x == y" := (R x y) (at level 70).
+  Notation "x * y" := (dot x y) (at level 40, left associativity).
+  Notation "1" := (one).
+  Notation "x + y" := (plus x y) (at level 50, left associativity).
+  Notation "0" := (zero).
+ +
+
+ +
+in the very first example, ring would have solved the + goal; here, since dot does not necessarily distribute over plus, + it is not possible to rely on it +
+
+  Section reminder.
+    Hypothesis H : forall x, x * x == x.
+    Variables a b c : X.
+ +
+    Goal (a+b+c)*(c+a+b) == a+b+c.
+      aac_rewrite H.
+      aac_reflexivity.
+    Qed.
+ +
+
+ +
+the tactic starts by normalising terms, so that trailing units + are always eliminated +
+
+    Goal ((a+b)+0+c)*((c+a)+b*1) == a+b+c.
+      aac_rewrite H.
+      aac_reflexivity.
+    Qed.
+  End reminder.
+ +
+
+ +
+the tactic can deal with "proper" morphisms of arbitrary arity + (here f and g, or Z.opp earlier): it rewrites under such + morphisms (g), and, more importantly, it is able to reorder + terms modulo AC under these morphisms (f) +
+
+  Section morphisms.
+    Variable f : X -> X -> X.
+    Hypothesis Hf : Proper (R ==> R ==> R) f.
+    Variable g : X -> X.
+    Hypothesis Hg : Proper (R ==> R) g.
+    Variable a b: X.
+    Hypothesis H : forall x y, x+f (b+y) x == y+x.
+ +
+    Goal g ((f (a+b) a) + a) == g (a+a).
+      aac_rewrite H.
+      reflexivity.
+    Qed.
+  End morphisms.
+ +
+
+ +
+

Selecting what and where to rewrite

+ + +
+ + There are sometimes several solutions to the matching problem. We + now show how to interact with the tactic to select the desired one. + +
+
+ +
+  Section occurrence.
+    Variable f : X -> X.
+    Variable a : X.
+    Hypothesis Hf : Proper (R ==> R) f.
+    Hypothesis H : forall x, x + x == x.
+ +
+    Goal f(a+a)+f(a+a) == f a.
+      (* in case there are several possible solutions, one can print
+        the different solutions using the aac_instances tactic (in
+        ProofGeneral, look at the *coq* buffer): *)
+      aac_instances H.
+      (* the default choice is the occurrence with the smallest
+        possible context (number 0), but one can choose the desired one *)
+      aac_rewrite H at 1.
+      (* now the goal is f a + f a == f a, there is only one solution *)
+      aac_rewrite H.
+      reflexivity.
+    Qed.
+ +
+  End occurrence.
+ +
+  Section subst.
+    Variables a b c d : X.
+    Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b.
+    Hypothesis H': forall x, x + x == x.
+ +
+    Goal a*c*d*c*d*b == a*c*d*b.
+      (* here, there is only one possible occurrence, but several substitutions: *)
+      aac_instances H.
+      (* one can select them with the proper keyword: *)
+      aac_rewrite H subst 1.
+      aac_rewrite H'.
+      aac_reflexivity.
+    Qed.
+  End subst.
+ +
+
+ +
+As expected, one can use both keywords together to select the + occurrence and the substitution. We also provide a keyword to + specify that the rewrite should be done in the right-hand side of + the equation. + +
+
+ +
+  Section both.
+    Variables a b c d : X.
+    Hypothesis H: forall x y, a*x*y*b == a*(x+y)*b.
+    Hypothesis H': forall x, x + x == x.
+ +
+    Goal a*c*d*c*d*b*b == a*(c*d*b)*b.
+      aac_instances H.
+      aac_rewrite H at 1 subst 1.
+      aac_instances H.
+      aac_rewrite H.
+      aac_rewrite H'.
+      aac_rewrite H at 0 subst 1 in_right.
+      aac_reflexivity.
+    Qed.
+ +
+  End both.
+ +
+
+ +
+

Distinction between aac_rewrite and aacu_rewrite:

+ + +
+ + aac_rewrite rejects solutions in which variables are instantiated + by units, while the companion tactic, aacu_rewrite allows such + solutions. + +
+
+ +
+  Section dealing_with_units.
+    Variables a b c: X.
+    Hypothesis H: forall x, a*x*a == x.
+ +
+    Goal a*a == 1.
+      (* here, x must be instantiated with 1, so that the aac_*
+        tactics give no solutions: *)
+      try aac_instances H.
+      (* while we get solutions with the aacu_* tactics: *)
+      aacu_instances H.
+      aacu_rewrite H.
+      reflexivity.
+    Qed.
+ +
+
+ +
+we introduced this distinction because it allows us to rule + out dummy cases in common situations: +
+
+    Hypothesis H': forall x y z, x*y + x*z == x*(y+z).
+    Goal a*b*c + a*c + a*b == a*(c+b*(1+c)).
+      (* 6 solutions without units  *)
+      aac_instances H'.
+      aac_rewrite H' at 0.
+      (* more than 52 with units *)
+      aacu_instances H'.
+    Abort.
+ +
+  End dealing_with_units.
+End base.
+ +
+
+ +
+

Declaring instances

+ + +
+ + To use one's own operations: it suffices to declare them as + instances of our classes. (Note that the following instances are + already declared in the Instances module.) + +
+
+ +
+Section Peano.
+  Import PeanoNat.
+ +
+  #[local] Instance aac_Nat_add_Assoc : Associative eq Nat.add := Nat.add_assoc.
+  #[local] Instance aac_Nat_add_Comm : Commutative eq Nat.add := Nat.add_comm.
+ +
+  #[local] Instance aac_Nat_mul_Comm : Commutative eq Nat.mul := Nat.mul_comm.
+  #[local] Instance aac_Nat_mul_Assoc : Associative eq Nat.mul := Nat.mul_assoc.
+ +
+  #[local] 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.
+  #[local] Instance aac_Nat_add_0_Unit : Unit eq Nat.add 0 :=
+    Build_Unit eq Nat.add (O) Nat.add_0_l Nat.add_0_r.
+ +
+
+ +
+Two (or more) operations may share the same units: in the + following example, 0 is understood as the unit of Nat.max as well as + the unit of Nat.add. +
+
+ +
+  #[local] Instance aac_Nat_max_Comm : Commutative eq Nat.max := Nat.max_comm.
+  #[local] Instance aac_Nat_max_Assoc : Associative eq Nat.max := Nat.max_assoc.
+ +
+
+ +
+Commutative operations may additionally be declared as idempotent. + This does not change the behaviour of aac_rewrite, but this enables + more simplifications in aac_normalise and aac_reflexivity. + +
+
+ +
+  #[local] Instance aac_Nat_max_Idem : Idempotent eq Nat.max := Nat.max_idempotent.
+ +
+  #[local] 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.
+ +
+  Variable a b c : nat.
+ +
+  Goal Nat.max (a + 0) 0 = a.
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+here, we use idempotency: +
+
+  Goal Nat.max (a + 0) a = a.
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+furthermore, several operators can be mixed: +
+
+  Hypothesis H : forall x y z, Nat.max (x + y) (x + z) = x + Nat.max y z.
+ +
+  Goal Nat.max (a + b) (c + (a * 1)) = Nat.max c b + a.
+    aac_instances H. aac_rewrite H. aac_reflexivity.
+  Qed.
+ +
+  Goal Nat.max (a + b) (c + Nat.max (a*1+0) 0) = a + Nat.max b c.
+    aac_instances H. aac_rewrite H. aac_reflexivity.
+  Qed.
+ +
+
+ +
+

Working with inequations

+ + +
+ + To be able to use the tactics, the goal must be a relation R + applied to two arguments, and the rewritten hypothesis must end + with a relation Q applied to two arguments. These relations are + not necessarily equivalences, but they should be related + according to the occurrence where the rewrite takes place; we + leave this check to the underlying call to setoid_rewrite. + +
+ + one can rewrite equations in the left member of inequations: +
+
+  Goal (forall x, x + x = x) -> a + b + b + a <= a + b.
+    intro Hx.
+    aac_rewrite Hx.
+    reflexivity.
+  Qed.
+ +
+
+ +
+or in the right member of inequations, using the in_right keyword: +
+
+  Goal (forall x, x + x = x) -> a + b <= a + b + b + a.
+    intro Hx.
+    aac_rewrite Hx in_right.
+    reflexivity.
+  Qed.
+ +
+
+ +
+similarly, one can rewrite inequations in inequations: +
+
+  Goal (forall x, x + x <= x) -> a + b + b + a <= a + b.
+    intro Hx.
+    aac_rewrite Hx.
+    reflexivity.
+  Qed.
+ +
+
+ +
+possibly in the right-hand side: +
+
+  Goal (forall x, x <= x + x) -> a + b <= a + b + b + a.
+    intro Hx.
+    aac_rewrite <- Hx in_right.
+    reflexivity.
+  Qed.
+ +
+
+ +
+aac_reflexivity deals with "trivial" inequations too: +
+
+  Goal Nat.max (a + b) (c + a) <= Nat.max (b + a) (c + 1*a).
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+In the last three examples, there were no equivalence relations + involved in the goal. However, we actually had to guess the + equivalence relation with respect to which the operators + (add,max,0) were AC. In this case, it was Leibniz equality + eq so that it was automatically inferred; more generally, one + can specify which equivalence relation to use by declaring + instances of the AAC_lift type class: + +
+
+ +
+  #[local] Instance aac_le_eq_lift : AAC_lift le eq := {}.
+
+ +
+(This instance is automatically inferred because eq is always a + valid candidate, here for le.) +
+
+ +
+End Peano.
+ +
+
+ +
+

Normalising goals

+ + +
+ + We also provide a tactic to normalise terms modulo AC. This + normalisation is the one we use internally. + +
+
+ +
+Section AAC_normalise.
+  Import Instances.Z.
+  Import ZArith Lia.
+  Open Scope Z_scope.
+ +
+  Variable a b c d : Z.
+  Goal a + (b + c*c*d) + a + 0 + d*1 = a.
+    aac_normalise.
+  Abort.
+ +
+  Goal b + 0 + a = c*1 -> a+b = c.
+    intro H.
+    aac_normalise in H.
+    assumption.
+  Qed.
+ +
+  Goal b + 0 + a = c*1+d -> a+b = d*(1+0)+c.
+    intro.
+    aac_normalise in *.
+    assumption.
+  Qed.
+ +
+  Goal Z.max (a+b) (b+a) = a+b.
+    aac_reflexivity.
+  Abort.
+ +
+
+ +
+Example by Abhishek Anand extracted from verification of a C++ gcd function +
+
+  Goal forall a b a' b' : Z,
+    0 < b' -> Z.gcd a' b' = Z.gcd a b -> Z.gcd b' (a' mod b') = Z.gcd a b.
+  Proof.
+    intros.
+    aac_rewrite Z.gcd_mod; try lia.
+    aac_normalise_all.
+    lia.
+  Qed.
+End AAC_normalise.
+ +
+
+ +
+

Examples from previous website

+ +
+
+ +
+Section Examples.
+ +
+  Import Instances.Z.
+  Import ZArith Lia.
+  Open Scope Z_scope.
+ +
+
+ +
+

Reverse triangle inequality

+ +
+
+ +
+  Lemma Z_abs_triangle : forall x y, Z.abs (x + y) <= Z.abs x + Z.abs y.
+  Proof Z.abs_triangle.
+ +
+  Lemma Z_add_opp_diag_r : forall x, x + -x = 0.
+  Proof Z.add_opp_diag_r.
+ +
+
+ +
+the following morphisms are required to perform the required rewrites: +
+
+  #[local] Instance Z_opp_ge_le_compat : Proper (Z.ge ==> Z.le) Z.opp.
+  Proof. intros x y. lia. Qed.
+ +
+  #[local] Instance Z_add_le_compat : Proper (Z.le ==> Z.le ==> Z.le) Z.add.
+  Proof. intros ? ? ? ? ? ?; lia. Qed.
+ +
+  Goal forall a b, Z.abs a - Z.abs b <= Z.abs (a - b).
+    intros. unfold Z.sub.
+    aac_instances <- (Z.sub_diag b).
+    aac_rewrite <- (Z.sub_diag b) at 3.
+    unfold Z.sub.
+    aac_rewrite Z_abs_triangle.
+    aac_rewrite Z_add_opp_diag_r.
+    aac_reflexivity.
+  Qed.
+ +
+
+ +
+

Pythagorean triples

+ +
+
+ +
+  Notation "x ^2" := (x*x) (at level 40).
+  Notation "2 ⋅ x" := (x+x) (at level 41).
+ +
+  Lemma Hbin1: forall x y, (x+y)^2 = x^2 + y^2 + 2⋅x*y.
+  Proof. intros; ring. Qed.
+  Lemma Hbin2: forall x y, x^2 + y^2 = (x+y)^2 + -(2⋅x*y).
+  Proof. intros; ring. Qed.
+  Lemma Hopp : forall x, x + -x = 0.
+  Proof. apply Zplus_opp_r. Qed.
+ +
+  Variables a b c : Z.
+  Hypothesis H : c^2 + 2⋅(a+1)*b = (a+1+b)^2.
+ +
+  Goal a^2 + b^2 + 2⋅a + 1 = c^2.
+    aacu_rewrite <- Hbin1.
+    rewrite Hbin2.
+    aac_rewrite <- H.
+    aac_rewrite Hopp.
+    aac_reflexivity.
+  Qed.
+
+ +
+note: after the aac_rewrite <- H, one could use ring to close the proof +
+
+ +
+End Examples.
+ +
+
+ +
+

List examples

+ +
+
+ +
+Section Lists.
+  Import List Permutation.
+  Import Instances.Lists.
+ +
+  Variables (X : Type) (l1 l2 l3 : list X).
+ +
+  Goal l1 ++ (l2 ++ l3) = (l1 ++ l2) ++ l3.
+    aac_reflexivity.
+  Qed.
+ +
+  Goal Permutation (l1 ++ l2) (l2 ++ l1).
+    aac_reflexivity.
+  Qed.
+ +
+  Hypothesis H : Permutation l1 l2.
+ +
+  Goal Permutation (l1 ++ l3) (l3 ++ l2).
+    aac_rewrite H.
+    aac_reflexivity.
+  Qed.
+End Lists.
+ +
+
+ +
+

Prop examples

+ +
+
+ +
+Section Props.
+  Import Instances.Prop_ops.
+ +
+  Variables (P Q : Prop).
+ +
+  Goal (Q /\ P) <-> (P /\ (Q /\ True)).
+    aac_reflexivity.
+  Qed.
+ +
+  Goal (Q \/ P) <-> (P \/ (Q \/ False)).
+    aac_reflexivity.
+  Qed.
+End Props.
+
+
+ +
+ + + diff --git a/docs/coqdoc/AAC_tactics.Utils.html b/docs/coqdoc/AAC_tactics.Utils.html new file mode 100644 index 0000000..65327c2 --- /dev/null +++ b/docs/coqdoc/AAC_tactics.Utils.html @@ -0,0 +1,430 @@ + + + + + + + + + + + + + +
+
+
+(* *********************************************************************** *)
+(*  This is part of aac_tactics, it is distributed under the terms of the  *)
+(*         GNU Lesser General Public License version 3                     *)
+(*              (see file LICENSE for more details)                        *)
+(*                                                                         *)
+(*       Copyright 2009-2010: Thomas Braibant, Damien Pous.                *)
+(* *********************************************************************** *)
+ +
+
+ +
+

Utility functions and results for AAC Tactics

+ +
+
+ +
+From Coq Require Import Arith NArith List RelationClasses.
+ +
+Set Implicit Arguments.
+Set Asymmetric Patterns.
+ +
+
+ +
+

Utilities for positive numbers

+ + +
+ + We use positive numbers as: +
    +
  • indices for morphisms and symbols + +
    • +
  • multiplicity of terms in sums + +
    • +
+ +
+
+ +
+Notation idx := positive.
+ +
+Fixpoint eq_idx_bool i j :=
+  match i,j with
+    | xH, xH => true
+    | xO i, xO j => eq_idx_bool i j
+    | xI i, xI j => eq_idx_bool i j
+    | _, _ => false
+  end.
+ +
+
+ +
+Specification predicate for boolean binary functions +
+
+Inductive decide_spec {A} {B} (R : A -> B -> Prop) (x : A) (y : B) : bool -> Prop :=
+| decide_true : R x y -> decide_spec R x y true
+| decide_false : ~(R x y) -> decide_spec R x y false.
+ +
+Lemma eq_idx_spec : forall i j, decide_spec (@eq _) i j (eq_idx_bool i j).
+Proof.
+  induction i; destruct j; simpl; try (constructor; congruence).
+   case (IHi j); constructor; congruence.
+   case (IHi j); constructor; congruence.
+Qed.
+ +
+
+ +
+ Weak specification predicate for comparison functions: + only the Eq case is specified + +
+
+Inductive compare_weak_spec A: A -> A -> comparison -> Prop :=
+| pcws_eq: forall i, compare_weak_spec i i Eq
+| pcws_lt: forall i j, compare_weak_spec i j Lt
+| pcws_gt: forall i j, compare_weak_spec i j Gt.
+ +
+Lemma pos_compare_weak_spec: forall i j, compare_weak_spec i j (Pos.compare i j).
+Proof.
+  intros. case Pos.compare_spec; try constructor.
+  intros <-; constructor.
+Qed.
+ +
+Lemma pos_compare_reflect_eq: forall i j, Pos.compare i j = Eq -> i=j.
+Proof.
+  intros ??.
+  case pos_compare_weak_spec; intros; congruence.
+Qed.
+ +
+
+ +
+

Dependent types utilities

+ +
+
+ +
+Notation cast T H u := (eq_rect _ T u _ H).
+ +
+Section dep.
+  Variable U: Type.
+  Variable T: U -> Type.
+ +
+  Lemma cast_eq: (forall u v: U, {u=v}+{u<>v}) ->
+    forall A (H: A=A) (u: T A), cast T H u = u.
+  Proof.
+    intros. rewrite <- Eqdep_dec.eq_rect_eq_dec; trivial.
+  Qed.
+ +
+  Variable f: forall A B, T A -> T B -> comparison.
+  Definition reflect_eqdep := forall A u B v (H: A=B),
+   @f A B u v = Eq -> cast T H u = v.
+ +
+
+ +
+ These lemmas have to remain transparent to get structural recursion + in the lemma tcompare_weak_spec below + +
+
+  Lemma reflect_eqdep_eq: reflect_eqdep ->
+    forall A u v, @f A A u v = Eq -> u = v.
+  Proof. intros H A u v He. apply (H _ _ _ _ eq_refl He). Defined.
+ +
+  Lemma reflect_eqdep_weak_spec: reflect_eqdep ->
+    forall A u v, compare_weak_spec u v (@f A A u v).
+  Proof.
+    intros. case_eq (f u v); try constructor.
+    intro H'. apply reflect_eqdep_eq in H'. subst. constructor. assumption.
+  Defined.
+End dep.
+ +
+
+ +
+

Utilities about (non-empty) lists and multisets

+ +
+
+ +
+#[universes(template)]
+Inductive nelist (A : Type) : Type :=
+| nil : A -> nelist A
+| cons : A -> nelist A -> nelist A.
+ +
+Register nil as aac_tactics.nelist.nil.
+Register cons as aac_tactics.nelist.cons.
+ +
+Notation "x :: y" := (cons x y).
+ +
+Fixpoint nelist_map (A B: Type) (f: A -> B) l :=
+  match l with
+    | nil x => nil ( f x)
+    | cons x l => cons ( f x) (nelist_map f l)
+  end.
+ +
+Fixpoint appne A l l' : nelist A :=
+  match l with
+    nil x => cons x l'
+    | cons t q => cons t (appne A q l')
+  end.
+ +
+Notation "x ++ y" := (appne x y).
+ +
+
+ +
+Finite multisets are represented with ordered lists with multiplicities +
+
+Definition mset A := nelist (A*positive).
+ +
+
+ +
+Lexicographic composition of comparisons (this is a notation to keep it lazy) +
+
+Notation lex e f := (match e with Eq => f | _ => e end).
+ +
+Section lists.
+ +
+
+ +
+Comparison functions +
+
+  Section c.
+    Variables A B: Type.
+    Variable compare: A -> B -> comparison.
+    Fixpoint list_compare h k :=
+      match h,k with
+        | nil x, nil y => compare x y
+        | nil x, _ => Lt
+        | _, nil x => Gt
+        | u::h, v::k => lex (compare u v) (list_compare h k)
+      end.
+  End c.
+ +
+  Definition mset_compare A B compare: mset A -> mset B -> comparison :=
+    list_compare (fun un vm =>
+      let '(u,n) := un in
+      let '(v,m) := vm in
+      lex (compare u v) (Pos.compare n m)).
+ +
+  Section list_compare_weak_spec.
+    Variable A: Type.
+    Variable compare: A -> A -> comparison.
+    Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v).
+
+ +
+ +
+ + This lemma has to remain transparent to get structural recursion + in the lemma tcompare_weak_spec below + +
+
+    Lemma list_compare_weak_spec: forall h k,
+      compare_weak_spec h k (list_compare compare h k).
+    Proof.
+      induction h as [|u h IHh]; destruct k as [|v k]; simpl; try constructor.
+      case (Hcompare a a0 ); try constructor.
+      case (Hcompare u v ); try constructor.
+      case (IHh k); intros; constructor.
+    Defined.
+  End list_compare_weak_spec.
+ +
+  Section mset_compare_weak_spec.
+    Variable A: Type.
+    Variable compare: A -> A -> comparison.
+    Hypothesis Hcompare: forall u v, compare_weak_spec u v (compare u v).
+
+ +
+ This lemma has to remain transparent to get structural recursion + in the lemma tcompare_weak_spec below + +
+
+    Lemma mset_compare_weak_spec: forall h k,
+      compare_weak_spec h k (mset_compare compare h k).
+    Proof.
+      apply list_compare_weak_spec.
+      intros [u n] [v m].
+      case (Hcompare u v); try constructor.
+      case (pos_compare_weak_spec n m); try constructor.
+    Defined.
+  End mset_compare_weak_spec.
+ +
+
+ +
+Merging functions (sorted) +
+
+  Section m.
+    Variable A: Type.
+    Variable compare: A -> A -> comparison.
+    Definition insert n1 h1 :=
+      let fix insert_aux l2 :=
+      match l2 with
+        | nil (h2,n2) =>
+          match compare h1 h2 with
+            | Eq => nil (h1,Pplus n1 n2)
+            | Lt => (h1,n1):: nil (h2,n2)
+            | Gt => (h2,n2):: nil (h1,n1)
+          end
+        | (h2,n2)::t2 =>
+          match compare h1 h2 with
+            | Eq => (h1,Pplus n1 n2):: t2
+            | Lt => (h1,n1)::l2
+            | Gt => (h2,n2)::insert_aux t2
+          end
+      end
+      in insert_aux.
+ +
+    Fixpoint merge_msets l1 :=
+      match l1 with
+        | nil (h1,n1) => fun l2 => insert n1 h1 l2
+        | (h1,n1)::t1 =>
+          let fix merge_aux l2 :=
+            match l2 with
+               | nil (h2,n2) =>
+                match compare h1 h2 with
+                  | Eq => (h1,Pplus n1 n2) :: t1
+                  | Lt => (h1,n1):: merge_msets t1 l2
+                  | Gt => (h2,n2):: l1
+                end
+              | (h2,n2)::t2 =>
+                match compare h1 h2 with
+                  | Eq => (h1,Pplus n1 n2)::merge_msets t1 t2
+                  | Lt => (h1,n1)::merge_msets t1 l2
+                  | Gt => (h2,n2)::merge_aux t2
+                end
+            end
+            in merge_aux
+      end.
+ +
+
+ +
+Setting all multiplicities to one, in order to implement idempotency +
+
+    Definition reduce_mset: mset A -> mset A := nelist_map (fun x => (fst x,xH)).
+ +
+
+ +
+Interpretation of a list with a constant and a binary operation +
+
+    Variable B: Type.
+    Variable map: A -> B.
+    Variable b2: B -> B -> B.
+    Fixpoint fold_map l :=
+      match l with
+        | nil x => map x
+        | u::l => b2 (map u) (fold_map l)
+      end.
+ +
+
+ +
+Mapping and merging +
+
+    Variable merge: A -> nelist B -> nelist B.
+    Fixpoint merge_map (l: nelist A): nelist B :=
+      match l with
+        | nil x => nil (map x)
+        | u::l => merge u (merge_map l)
+      end.
+ +
+    Variable ret : A -> B.
+    Variable bind : A -> B -> B.
+    Fixpoint fold_map' (l : nelist A) : B :=
+      match l with
+        | nil x => ret x
+        | u::l => bind u (fold_map' l)
+      end.
+ +
+  End m.
+End lists.
+
+
+ +
+ + + diff --git a/docs/coqdoc/config.js b/docs/coqdoc/config.js new file mode 100644 index 0000000..1902b36 --- /dev/null +++ b/docs/coqdoc/config.js @@ -0,0 +1,79 @@ +var coqdocjs = coqdocjs || {}; + +coqdocjs.repl = { + "forall": "∀", + "exists": "∃", + "~": "¬", + "/\\": "∧", + "\\/": "∨", + "->": "→", + "<-": "←", + "<->": "↔", + "=>": "⇒", + "<>": "≠", + "<=": "≤", + ">=": "≥", + "el": "∈", + "nel": "∉", + "<<=": "⊆", + "|-": "⊢", + ">>": "»", + "<<": "⊆", + "++": "⧺", + "===": "≡", + "=/=": "≢", + "=~=": "≅", + "==>": "⟹", + "<==": "⟸", + "False": "⊥", + "True": "⊤", + ":=": "≔", + "-|": "⊣", + "*": "×", + "::": "∷", + "lhd": "⊲", + "rhd": "⊳", + "nat": "ℕ", + "alpha": "α", + "beta": "β", + "gamma": "γ", + "delta": "δ", + "epsilon": "ε", + "eta": "η", + "iota": "ι", + "kappa": "κ", + "lambda": "λ", + "mu": "μ", + "nu": "ν", + "omega": "ω", + "phi": "ϕ", + "pi": "π", + "psi": "ψ", + "rho": "ρ", + "sigma": "σ", + "tau": "τ", + "theta": "θ", + "xi": "ξ", + "zeta": "ζ", + "Delta": "Δ", + "Gamma": "Γ", + "Pi": "Π", + "Sigma": "Σ", + "Omega": "Ω", + "Xi": "Ξ" +}; + +coqdocjs.subscr = { + "0" : "₀", + "1" : "₁", + "2" : "₂", + "3" : "₃", + "4" : "₄", + "5" : "₅", + "6" : "₆", + "7" : "₇", + "8" : "₈", + "9" : "₉", +}; + +coqdocjs.replInText = ["==>","<=>", "=>", "->", "<-", ":="]; diff --git a/docs/coqdoc/coqdoc.css b/docs/coqdoc/coqdoc.css new file mode 100644 index 0000000..18dad89 --- /dev/null +++ b/docs/coqdoc/coqdoc.css @@ -0,0 +1,197 @@ +@import url(https://fonts.googleapis.com/css?family=Open+Sans:400,700); + +body{ + font-family: 'Open Sans', sans-serif; + font-size: 14px; + color: #2D2D2D +} + +a { + text-decoration: none; + border-radius: 3px; + padding-left: 3px; + padding-right: 3px; + margin-left: -3px; + margin-right: -3px; + color: inherit; + font-weight: bold; +} + +#main .code a, #main .inlinecode a, #toc a { + font-weight: inherit; +} + +a[href]:hover, [clickable]:hover{ + background-color: rgba(0,0,0,0.1); + cursor: pointer; +} + +h, h1, h2, h3, h4, h5 { + line-height: 1; + color: black; + text-rendering: optimizeLegibility; + font-weight: normal; + letter-spacing: 0.1em; + text-align: left; +} + +div + br { + display: none; +} + +div:empty{ display: none;} + +#main h1 { + font-size: 2em; +} + +#main h2 { + font-size: 1.667rem; +} + +#main h3 { + font-size: 1.333em; +} + +#main h4, #main h5, #main h6 { + font-size: 1em; +} + +#toc h2 { + padding-bottom: 0; +} + +#main .doc { + margin: 0; + text-align: justify; +} + +.inlinecode, .code, #main pre { + font-family: monospace; +} + +.code > br:first-child { + display: none; +} + +.doc + .code{ + margin-top:0.5em; +} + +.block{ + display: block; + margin-top: 5px; + margin-bottom: 5px; + padding: 10px; + text-align: center; +} + +.block img{ + margin: 15px; +} + +table.infrule { + border: 0px; + margin-left: 50px; + margin-top: 10px; + margin-bottom: 10px; +} + +td.infrule { + font-family: "Droid Sans Mono", "DejaVu Sans Mono", monospace; + text-align: center; + padding: 0; + line-height: 1; +} + +tr.infrulemiddle hr { + margin: 1px 0 1px 0; +} + +.infrulenamecol { + color: rgb(60%,60%,60%); + padding-left: 1em; + padding-bottom: 0.1em +} + +.id[type="constructor"], .id[type="projection"], .id[type="method"], +.id[title="constructor"], .id[title="projection"], .id[title="method"] { + color: #A30E16; +} + +.id[type="var"], .id[type="variable"], +.id[title="var"], .id[title="variable"] { + color: inherit; +} + +.id[type="definition"], .id[type="record"], .id[type="class"], .id[type="instance"], .id[type="inductive"], .id[type="library"], +.id[title="definition"], .id[title="record"], .id[title="class"], .id[title="instance"], .id[title="inductive"], .id[title="library"] { + color: #A6650F; +} + +.id[type="lemma"], +.id[title="lemma"]{ + color: #188B0C; +} + +.id[type="keyword"], .id[type="notation"], .id[type="abbreviation"], +.id[title="keyword"], .id[title="notation"], .id[title="abbreviation"]{ + color : #2874AE; +} + +.comment { + color: #808080; +} + +/* TOC */ + +#toc h2{ + letter-spacing: 0; + font-size: 1.333em; +} + +/* Index */ + +#index { + margin: 0; + padding: 0; + width: 100%; +} + +#index #frontispiece { + margin: 1em auto; + padding: 1em; + width: 60%; +} + +.booktitle { font-size : 140% } +.authors { font-size : 90%; + line-height: 115%; } +.moreauthors { font-size : 60% } + +#index #entrance { + text-align: center; +} + +#index #entrance .spacer { + margin: 0 30px 0 30px; +} + +ul.doclist { + margin-top: 0em; + margin-bottom: 0em; +} + +#toc > * { + clear: both; +} + +#toc > a { + display: block; + float: left; + margin-top: 1em; +} + +#toc a h2{ + display: inline; +} diff --git a/docs/coqdoc/coqdocjs.css b/docs/coqdoc/coqdocjs.css new file mode 100644 index 0000000..959b42e --- /dev/null +++ b/docs/coqdoc/coqdocjs.css @@ -0,0 +1,239 @@ +/* replace unicode */ + +.id[repl] .hidden { + font-size: 0; +} + +.id[repl]:before{ + content: attr(repl); +} + +/* folding proofs */ + +@keyframes show-proof { + 0% { + max-height: 1.2em; + opacity: 1; + } + 99% { + max-height: 1000em; + } + 100%{ + } +} + +@keyframes hide-proof { + from { + visibility: visible; + max-height: 10em; + opacity: 1; + } + to { + max-height: 1.2em; + } +} + +.proof { + cursor: pointer; +} +.proof * { + cursor: pointer; +} + +.proof { + overflow: hidden; + position: relative; + transition: opacity 1s; + display: inline-block; +} + +.proof[show="false"] { + max-height: 1.2em; + visibility: visible; + opacity: 0.3; +} + +.proof[show="false"][animate] { + animation-name: hide-proof; + animation-duration: 0.25s; +} + +.proof[show="true"] { + animation-name: show-proof; + animation-duration: 10s; +} + +.proof[show="true"]:before { + content: "\25BC"; /* arrow down */ +} +.proof[show="false"]:before { + content: "\25B6"; /* arrow right */ +} + +.proof[show="false"]:hover { + visibility: visible; + opacity: 0.5; +} + +#toggle-proofs[proof-status="no-proofs"] { + display: none; +} + +#toggle-proofs[proof-status="some-hidden"]:before { + content: "Show Proofs"; +} + +#toggle-proofs[proof-status="all-shown"]:before { + content: "Hide Proofs"; +} + + +/* page layout */ + +html, body { + height: 100%; + margin:0; + padding:0; +} + +@media only screen { /* no div with internal scrolling to allow printing of whole content */ + body { + display: flex; + flex-direction: column + } + + #content { + flex: 1; + overflow: auto; + display: flex; + flex-direction: column; + } +} + +#content:focus { + outline: none; /* prevent glow in OS X */ +} + +#main { + display: block; + padding: 16px; + padding-top: 1em; + padding-bottom: 2em; + margin-left: auto; + margin-right: auto; + max-width: 60em; + flex: 1 0 auto; +} + +.libtitle { + display: none; +} + +/* header */ +#header { + width:100%; + padding: 0; + margin: 0; + display: flex; + align-items: center; + background-color: rgb(21,57,105); + color: white; + font-weight: bold; + overflow: hidden; +} + + +.button { + cursor: pointer; +} + +#header * { + text-decoration: none; + vertical-align: middle; + margin-left: 15px; + margin-right: 15px; +} + +#header > .right, #header > .left { + display: flex; + flex: 1; + align-items: center; +} +#header > .left { + text-align: left; +} +#header > .right { + flex-direction: row-reverse; +} + +#header a, #header .button { + color: white; + box-sizing: border-box; +} + +#header a { + border-radius: 0; + padding: 0.2em; +} + +#header .button { + background-color: rgb(63, 103, 156); + border-radius: 1em; + padding-left: 0.5em; + padding-right: 0.5em; + margin: 0.2em; +} + +#header a:hover, #header .button:hover { + background-color: rgb(181, 213, 255); + color: black; +} + +#header h1 { padding: 0; + margin: 0;} + +/* footer */ +#footer { + text-align: center; + opacity: 0.5; + font-size: 75%; +} + +/* hyperlinks */ + +@keyframes highlight { + 50%{ + background-color: black; + } +} + +:target * { + animation-name: highlight; + animation-duration: 1s; +} + +a[name]:empty { + float: right; +} + +/* Proviola */ + +div.code { + width: auto; + float: none; +} + +div.goal { + position: fixed; + left: 75%; + width: 25%; + top: 3em; +} + +div.doc { + clear: both; +} + +span.command:hover { + background-color: inherit; +} diff --git a/docs/coqdoc/coqdocjs.js b/docs/coqdoc/coqdocjs.js new file mode 100644 index 0000000..727da8c --- /dev/null +++ b/docs/coqdoc/coqdocjs.js @@ -0,0 +1,197 @@ +var coqdocjs = coqdocjs || {}; +(function(){ + +function replace(s){ + var m; + if (m = s.match(/^(.+)'/)) { + return replace(m[1])+"'"; + } else if (m = s.match(/^([A-Za-z]+)_?(\d+)$/)) { + return replace(m[1])+m[2].replace(/\d/g, function(d){ + if (coqdocjs.subscr.hasOwnProperty(d)) { + return coqdocjs.subscr[d]; + } else { + return d; + } + }); + } else if (coqdocjs.repl.hasOwnProperty(s)){ + return coqdocjs.repl[s] + } else { + return s; + } +} + +function toArray(nl){ + return Array.prototype.slice.call(nl); +} + +function replInTextNodes() { + // Get all the nodes up front. + var nodes = Array.from(document.querySelectorAll(".code, .inlinecode")) + .flatMap(elem => Array.from(elem.childNodes) + .filter(e => e.nodeType == Node.TEXT_NODE) + ); + + // Create a replacement template node to clone from. + var replacementTemplate = document.createElement("span"); + replacementTemplate.setAttribute("class", "id"); + replacementTemplate.setAttribute("type", "keyword"); + + // Do the replacements. + coqdocjs.replInText.forEach(function(toReplace){ + var replacement = replacementTemplate.cloneNode(true); + replacement.appendChild(document.createTextNode(toReplace)); + + nodes.forEach(node => { + var fragments = node.textContent.split(toReplace); + node.textContent = fragments[fragments.length-1]; + for (var k = 0; k < fragments.length - 1; ++k) { + fragments[k] && node.parentNode.insertBefore(document.createTextNode(fragments[k]),node); + node.parentNode.insertBefore(replacement.cloneNode(true), node); + } + }); + }); +} + +function replNodes() { + toArray(document.getElementsByClassName("id")).forEach(function(node){ + if (["var", "variable", "keyword", "notation", "definition", "inductive"].indexOf(node.getAttribute("type"))>=0){ + var text = node.textContent; + var replText = replace(text); + if(text != replText) { + node.setAttribute("repl", replText); + node.setAttribute("title", text); + var hidden = document.createElement("span"); + hidden.setAttribute("class", "hidden"); + while (node.firstChild) { + hidden.appendChild(node.firstChild); + } + node.appendChild(hidden); + } + } + }); +} + +function isVernacStart(l, t){ + t = t.trim(); + for(var s of l){ + if (t == s || t.startsWith(s+" ") || t.startsWith(s+".")){ + return true; + } + } + return false; +} + +function isProofStart(n){ + return isVernacStart(["Proof"], n.textContent) || + (isVernacStart(["Next"], n.textContent) && isVernacStart(["Obligation"], n.nextSibling.nextSibling.textContent)); +} + +function isProofEnd(s){ + return isVernacStart(["Qed", "Admitted", "Defined", "Abort"], s); +} + +function proofStatus(){ + var proofs = toArray(document.getElementsByClassName("proof")); + if(proofs.length) { + for(var proof of proofs) { + if (proof.getAttribute("show") === "false") { + return "some-hidden"; + } + } + return "all-shown"; + } + else { + return "no-proofs"; + } +} + +function updateView(){ + document.getElementById("toggle-proofs").setAttribute("proof-status", proofStatus()); +} + +function foldProofs() { + var hasCommands = true; + var nodes = document.getElementsByClassName("command"); + if(nodes.length == 0) { + hasCommands = false; + console.log("no command tags found") + nodes = document.getElementsByClassName("id"); + } + toArray(nodes).forEach(function(node){ + if(isProofStart(node)) { + var proof = document.createElement("span"); + proof.setAttribute("class", "proof"); + + node.parentNode.insertBefore(proof, node); + if(proof.previousSibling.nodeType === Node.TEXT_NODE) + proof.appendChild(proof.previousSibling); + while(node && !isProofEnd(node.textContent)) { + proof.appendChild(node); + node = proof.nextSibling; + } + if (proof.nextSibling) proof.appendChild(proof.nextSibling); // the Qed + if (!hasCommands && proof.nextSibling) proof.appendChild(proof.nextSibling); // the dot after the Qed + + proof.addEventListener("click", function(proof){return function(e){ + if (e.target.parentNode.tagName.toLowerCase() === "a") + return; + proof.setAttribute("show", proof.getAttribute("show") === "true" ? "false" : "true"); + proof.setAttribute("animate", ""); + updateView(); + };}(proof)); + proof.setAttribute("show", "false"); + } + }); +} + +function toggleProofs(){ + var someProofsHidden = proofStatus() === "some-hidden"; + toArray(document.getElementsByClassName("proof")).forEach(function(proof){ + proof.setAttribute("show", someProofsHidden); + proof.setAttribute("animate", ""); + }); + updateView(); +} + +function repairDom(){ + // pull whitespace out of command + toArray(document.getElementsByClassName("command")).forEach(function(node){ + while(node.firstChild && node.firstChild.textContent.trim() == ""){ + console.log("try move"); + node.parentNode.insertBefore(node.firstChild, node); + } + }); + toArray(document.getElementsByClassName("id")).forEach(function(node){ + node.setAttribute("type", node.getAttribute("title")); + }); + toArray(document.getElementsByClassName("idref")).forEach(function(ref){ + toArray(ref.childNodes).forEach(function(child){ + if (["var", "variable"].indexOf(child.getAttribute("type")) > -1) + ref.removeAttribute("href"); + }); + }); + +} + +function fixTitle(){ + var url = "/" + window.location.pathname; + var basename = url.substring(url.lastIndexOf('/')+1, url.lastIndexOf('.')); + if (basename === "toc") {document.title = "Table of Contents";} + else if (basename === "indexpage") {document.title = "Index";} + else {document.title = basename;} +} + +function postprocess(){ + repairDom(); + replInTextNodes() + replNodes(); + foldProofs(); + document.getElementById("toggle-proofs").addEventListener("click", toggleProofs); + updateView(); +} + +fixTitle(); +document.addEventListener('DOMContentLoaded', postprocess); + +coqdocjs.toggleProofs = toggleProofs; +})(); diff --git a/docs/coqdoc/indexpage.html b/docs/coqdoc/indexpage.html new file mode 100644 index 0000000..2d3492d --- /dev/null +++ b/docs/coqdoc/indexpage.html @@ -0,0 +1,2455 @@ + + + + + + + + + + + + + +
+
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Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(685 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(13 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(176 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(6 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(55 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(31 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(14 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(39 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(161 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(140 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
+
+

Global Index

+

A

+AAC [library]
+aac_lift_proper [instance, in AAC_tactics.AAC]
+aac_lift_subrelation [instance, in AAC_tactics.AAC]
+aac_list_proper [projection, in AAC_tactics.AAC]
+aac_lift_equivalence [projection, in AAC_tactics.AAC]
+AAC_lift [record, in AAC_tactics.AAC]
+AAC_normalise.d [variable, in AAC_tactics.Tutorial]
+AAC_normalise.c [variable, in AAC_tactics.Tutorial]
+AAC_normalise.b [variable, in AAC_tactics.Tutorial]
+AAC_normalise.a [variable, in AAC_tactics.Tutorial]
+AAC_normalise [section, in AAC_tactics.Tutorial]
+aac_le_eq_lift [instance, in AAC_tactics.Tutorial]
+aac_Nat_max_0_Unit [instance, in AAC_tactics.Tutorial]
+aac_Nat_max_Idem [instance, in AAC_tactics.Tutorial]
+aac_Nat_max_Assoc [instance, in AAC_tactics.Tutorial]
+aac_Nat_max_Comm [instance, in AAC_tactics.Tutorial]
+aac_Nat_add_0_Unit [instance, in AAC_tactics.Tutorial]
+aac_Nat_mul_1_Unit [instance, in AAC_tactics.Tutorial]
+aac_Nat_mul_Assoc [instance, in AAC_tactics.Tutorial]
+aac_Nat_mul_Comm [instance, in AAC_tactics.Tutorial]
+aac_Nat_add_Comm [instance, in AAC_tactics.Tutorial]
+aac_Nat_add_Assoc [instance, in AAC_tactics.Tutorial]
+All [module, in AAC_tactics.Instances]
+appne [definition, in AAC_tactics.Utils]
+Associative [record, in AAC_tactics.AAC]
+Associative [inductive, in AAC_tactics.AAC]
+

B

+base [section, in AAC_tactics.Tutorial]
+base.both [section, in AAC_tactics.Tutorial]
+base.both.a [variable, in AAC_tactics.Tutorial]
+base.both.b [variable, in AAC_tactics.Tutorial]
+base.both.c [variable, in AAC_tactics.Tutorial]
+base.both.d [variable, in AAC_tactics.Tutorial]
+base.both.H [variable, in AAC_tactics.Tutorial]
+base.both.H' [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units.H' [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units.H [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units.c [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units.b [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units.a [variable, in AAC_tactics.Tutorial]
+base.dealing_with_units [section, in AAC_tactics.Tutorial]
+base.dot [variable, in AAC_tactics.Tutorial]
+base.dot_Proper [variable, in AAC_tactics.Tutorial]
+base.dot_A [variable, in AAC_tactics.Tutorial]
+base.E [variable, in AAC_tactics.Tutorial]
+base.morphisms [section, in AAC_tactics.Tutorial]
+base.morphisms.a [variable, in AAC_tactics.Tutorial]
+base.morphisms.b [variable, in AAC_tactics.Tutorial]
+base.morphisms.f [variable, in AAC_tactics.Tutorial]
+base.morphisms.g [variable, in AAC_tactics.Tutorial]
+base.morphisms.H [variable, in AAC_tactics.Tutorial]
+base.morphisms.Hf [variable, in AAC_tactics.Tutorial]
+base.morphisms.Hg [variable, in AAC_tactics.Tutorial]
+base.occurrence [section, in AAC_tactics.Tutorial]
+base.occurrence.a [variable, in AAC_tactics.Tutorial]
+base.occurrence.f [variable, in AAC_tactics.Tutorial]
+base.occurrence.H [variable, in AAC_tactics.Tutorial]
+base.occurrence.Hf [variable, in AAC_tactics.Tutorial]
+base.One [variable, in AAC_tactics.Tutorial]
+base.one [variable, in AAC_tactics.Tutorial]
+base.plus [variable, in AAC_tactics.Tutorial]
+base.plus_Proper [variable, in AAC_tactics.Tutorial]
+base.plus_C [variable, in AAC_tactics.Tutorial]
+base.plus_A [variable, in AAC_tactics.Tutorial]
+base.R [variable, in AAC_tactics.Tutorial]
+base.reminder [section, in AAC_tactics.Tutorial]
+base.reminder.a [variable, in AAC_tactics.Tutorial]
+base.reminder.b [variable, in AAC_tactics.Tutorial]
+base.reminder.c [variable, in AAC_tactics.Tutorial]
+base.reminder.H [variable, in AAC_tactics.Tutorial]
+base.subst [section, in AAC_tactics.Tutorial]
+base.subst.a [variable, in AAC_tactics.Tutorial]
+base.subst.b [variable, in AAC_tactics.Tutorial]
+base.subst.c [variable, in AAC_tactics.Tutorial]
+base.subst.d [variable, in AAC_tactics.Tutorial]
+base.subst.H [variable, in AAC_tactics.Tutorial]
+base.subst.H' [variable, in AAC_tactics.Tutorial]
+base.X [variable, in AAC_tactics.Tutorial]
+base.Zero [variable, in AAC_tactics.Tutorial]
+base.zero [variable, in AAC_tactics.Tutorial]
+_ + _ [notation, in AAC_tactics.Tutorial]
+_ * _ [notation, in AAC_tactics.Tutorial]
+_ == _ [notation, in AAC_tactics.Tutorial]
+0 [notation, in AAC_tactics.Tutorial]
+1 [notation, in AAC_tactics.Tutorial]
+Bool [module, in AAC_tactics.Instances]
+Bool.aac_Bool_andb_true_Unit [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_orb_false_Unit [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Idem [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Comm [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Assoc [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Idem [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Comm [instance, in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Assoc [instance, in AAC_tactics.Instances]
+Bool.negb_compat [instance, in AAC_tactics.Instances]
+

C

+cast [abbreviation, in AAC_tactics.Utils]
+cast_eq [lemma, in AAC_tactics.Utils]
+Caveats [library]
+Commutative [record, in AAC_tactics.AAC]
+Commutative [inductive, in AAC_tactics.AAC]
+compare_weak_spec_sind [definition, in AAC_tactics.Utils]
+compare_weak_spec_ind [definition, in AAC_tactics.Utils]
+compare_weak_spec [inductive, in AAC_tactics.Utils]
+cons [constructor, in AAC_tactics.Utils]
+Constants [library]
+

D

+decide_spec_sind [definition, in AAC_tactics.Utils]
+decide_spec_ind [definition, in AAC_tactics.Utils]
+decide_false [constructor, in AAC_tactics.Utils]
+decide_true [constructor, in AAC_tactics.Utils]
+decide_spec [inductive, in AAC_tactics.Utils]
+dep [section, in AAC_tactics.Utils]
+dep.f [variable, in AAC_tactics.Utils]
+dep.T [variable, in AAC_tactics.Utils]
+dep.U [variable, in AAC_tactics.Utils]
+

E

+eq_idx_spec [lemma, in AAC_tactics.Utils]
+eq_idx_bool [definition, in AAC_tactics.Utils]
+eq_subr [lemma, in AAC_tactics.Instances]
+evars [section, in AAC_tactics.Caveats]
+evars.H [variable, in AAC_tactics.Caveats]
+evars.H' [variable, in AAC_tactics.Caveats]
+evars.idem [variable, in AAC_tactics.Caveats]
+evars.P [variable, in AAC_tactics.Caveats]
+Examples [section, in AAC_tactics.Tutorial]
+Examples.a [variable, in AAC_tactics.Tutorial]
+Examples.b [variable, in AAC_tactics.Tutorial]
+Examples.c [variable, in AAC_tactics.Tutorial]
+Examples.H [variable, in AAC_tactics.Tutorial]
+_ ^2 [notation, in AAC_tactics.Tutorial]
+2 ⋅ _ [notation, in AAC_tactics.Tutorial]
+

F

+fold_map' [definition, in AAC_tactics.Utils]
+fold_map [definition, in AAC_tactics.Utils]
+

H

+Hbin1 [lemma, in AAC_tactics.Tutorial]
+Hbin2 [lemma, in AAC_tactics.Tutorial]
+Hopp [lemma, in AAC_tactics.Tutorial]
+

I

+Idempotent [record, in AAC_tactics.AAC]
+Idempotent [inductive, in AAC_tactics.AAC]
+idx [abbreviation, in AAC_tactics.Utils]
+ineq [section, in AAC_tactics.Caveats]
+ineq.H [variable, in AAC_tactics.Caveats]
+insert [definition, in AAC_tactics.Utils]
+Instances [library]
+Internal [module, in AAC_tactics.AAC]
+Internal.add_to_prd [definition, in AAC_tactics.AAC]
+Internal.add_to_sum [definition, in AAC_tactics.AAC]
+Internal.assoc [instance, in AAC_tactics.AAC]
+Internal.Bin [module, in AAC_tactics.AAC]
+Internal.Binvalue_Proper [instance, in AAC_tactics.AAC]
+Internal.Binvalue_Associative [instance, in AAC_tactics.AAC]
+Internal.Binvalue_Idempotent [instance, in AAC_tactics.AAC]
+Internal.Binvalue_Commutative [instance, in AAC_tactics.AAC]
+Internal.Bin.assoc [projection, in AAC_tactics.AAC]
+Internal.Bin.comm [projection, in AAC_tactics.AAC]
+Internal.Bin.compat [projection, in AAC_tactics.AAC]
+Internal.Bin.idem [projection, in AAC_tactics.AAC]
+Internal.Bin.pack [record, in AAC_tactics.AAC]
+Internal.Bin.t [section, in AAC_tactics.AAC]
+Internal.Bin.t.R [variable, in AAC_tactics.AAC]
+Internal.Bin.t.X [variable, in AAC_tactics.AAC]
+Internal.Bin.value [projection, in AAC_tactics.AAC]
+Internal.comp [definition, in AAC_tactics.AAC]
+Internal.compare [definition, in AAC_tactics.AAC]
+Internal.compare_reflect_eq [lemma, in AAC_tactics.AAC]
+Internal.compat_prd_Unit [instance, in AAC_tactics.AAC]
+Internal.compat_prd_unit_add [lemma, in AAC_tactics.AAC]
+Internal.compat_prd_unit_return [lemma, in AAC_tactics.AAC]
+Internal.compat_prd_unit_sind [definition, in AAC_tactics.AAC]
+Internal.compat_prd_unit_ind [definition, in AAC_tactics.AAC]
+Internal.compat_prd_unit [inductive, in AAC_tactics.AAC]
+Internal.compat_sum_unit_Unit [instance, in AAC_tactics.AAC]
+Internal.compat_sum_unit_add [lemma, in AAC_tactics.AAC]
+Internal.compat_sum_unit_return [lemma, in AAC_tactics.AAC]
+Internal.compat_sum_unit_sind [definition, in AAC_tactics.AAC]
+Internal.compat_sum_unit_ind [definition, in AAC_tactics.AAC]
+Internal.compat_sum_unit [inductive, in AAC_tactics.AAC]
+Internal.copy [definition, in AAC_tactics.AAC]
+Internal.copy [section, in AAC_tactics.AAC]
+Internal.copy_idem [lemma, in AAC_tactics.AAC]
+Internal.copy_n_unit [lemma, in AAC_tactics.AAC]
+Internal.copy_mset_copy [lemma, in AAC_tactics.AAC]
+Internal.copy_mset_succ [lemma, in AAC_tactics.AAC]
+Internal.copy_mset' [lemma, in AAC_tactics.AAC]
+Internal.copy_mset [definition, in AAC_tactics.AAC]
+Internal.copy_compat [instance, in AAC_tactics.AAC]
+Internal.copy_Psucc [lemma, in AAC_tactics.AAC]
+Internal.copy_xH [lemma, in AAC_tactics.AAC]
+Internal.copy_plus [lemma, in AAC_tactics.AAC]
+Internal.copy' [definition, in AAC_tactics.AAC]
+Internal.copy.HR [variable, in AAC_tactics.AAC]
+Internal.copy.op [variable, in AAC_tactics.AAC]
+Internal.copy.op' [variable, in AAC_tactics.AAC]
+Internal.copy.plus [variable, in AAC_tactics.AAC]
+Internal.copy.po [variable, in AAC_tactics.AAC]
+Internal.copy.R [variable, in AAC_tactics.AAC]
+Internal.copy.X [variable, in AAC_tactics.AAC]
+Internal.cpu_right [constructor, in AAC_tactics.AAC]
+Internal.cpu_left [constructor, in AAC_tactics.AAC]
+Internal.csu_right [constructor, in AAC_tactics.AAC]
+Internal.csu_left [constructor, in AAC_tactics.AAC]
+Internal.decide [lemma, in AAC_tactics.AAC]
+Internal.discr [inductive, in AAC_tactics.AAC]
+Internal.discr_sind [definition, in AAC_tactics.AAC]
+Internal.discr_rec [definition, in AAC_tactics.AAC]
+Internal.discr_ind [definition, in AAC_tactics.AAC]
+Internal.discr_rect [definition, in AAC_tactics.AAC]
+Internal.eval [definition, in AAC_tactics.AAC]
+Internal.eval_norm_aux [definition, in AAC_tactics.AAC]
+Internal.eval_norm [definition, in AAC_tactics.AAC]
+Internal.eval_norm_lists [lemma, in AAC_tactics.AAC]
+Internal.eval_prd_app [lemma, in AAC_tactics.AAC]
+Internal.eval_prd_cons [lemma, in AAC_tactics.AAC]
+Internal.eval_prd_nil [lemma, in AAC_tactics.AAC]
+Internal.eval_reduce_msets [lemma, in AAC_tactics.AAC]
+Internal.eval_norm_msets [lemma, in AAC_tactics.AAC]
+Internal.eval_merge_bin [lemma, in AAC_tactics.AAC]
+Internal.eval_sum_cons [lemma, in AAC_tactics.AAC]
+Internal.eval_sum_nil [lemma, in AAC_tactics.AAC]
+Internal.eval_aux_compat [instance, in AAC_tactics.AAC]
+Internal.eval_aux [definition, in AAC_tactics.AAC]
+Internal.is_prd_spec [lemma, in AAC_tactics.AAC]
+Internal.is_prd_spec_ind_sind [definition, in AAC_tactics.AAC]
+Internal.is_prd_spec_ind_ind [definition, in AAC_tactics.AAC]
+Internal.is_prd_spec_nothing [constructor, in AAC_tactics.AAC]
+Internal.is_prd_spec_unit [constructor, in AAC_tactics.AAC]
+Internal.is_prd_spec_op [constructor, in AAC_tactics.AAC]
+Internal.is_prd_spec_ind [inductive, in AAC_tactics.AAC]
+Internal.is_sum_spec [lemma, in AAC_tactics.AAC]
+Internal.is_sum_spec_ind_sind [definition, in AAC_tactics.AAC]
+Internal.is_sum_spec_ind_ind [definition, in AAC_tactics.AAC]
+Internal.is_sum_spec_nothing [constructor, in AAC_tactics.AAC]
+Internal.is_sum_spec_unit [constructor, in AAC_tactics.AAC]
+Internal.is_sum_spec_op [constructor, in AAC_tactics.AAC]
+Internal.is_sum_spec_ind [inductive, in AAC_tactics.AAC]
+Internal.is_unit_of_Unit [lemma, in AAC_tactics.AAC]
+Internal.is_prd [definition, in AAC_tactics.AAC]
+Internal.is_sum [definition, in AAC_tactics.AAC]
+Internal.Is_nothing [constructor, in AAC_tactics.AAC]
+Internal.Is_unit [constructor, in AAC_tactics.AAC]
+Internal.Is_op [constructor, in AAC_tactics.AAC]
+Internal.is_idempotent [definition, in AAC_tactics.AAC]
+Internal.is_commutative [definition, in AAC_tactics.AAC]
+Internal.is_unit_of [definition, in AAC_tactics.AAC]
+Internal.left [constructor, in AAC_tactics.AAC]
+Internal.lift_normalise [lemma, in AAC_tactics.AAC]
+Internal.m [inductive, in AAC_tactics.AAC]
+Internal.m_sind [definition, in AAC_tactics.AAC]
+Internal.m_rec [definition, in AAC_tactics.AAC]
+Internal.m_ind [definition, in AAC_tactics.AAC]
+Internal.m_rect [definition, in AAC_tactics.AAC]
+Internal.norm [definition, in AAC_tactics.AAC]
+Internal.normalise [lemma, in AAC_tactics.AAC]
+Internal.norm_msets [definition, in AAC_tactics.AAC]
+Internal.norm_lists [definition, in AAC_tactics.AAC]
+Internal.norm_lists_ [definition, in AAC_tactics.AAC]
+Internal.norm_msets_ [definition, in AAC_tactics.AAC]
+Internal.prd [constructor, in AAC_tactics.AAC]
+Internal.prd' [definition, in AAC_tactics.AAC]
+Internal.prd'_prd [lemma, in AAC_tactics.AAC]
+Internal.proper [instance, in AAC_tactics.AAC]
+Internal.return_prd [definition, in AAC_tactics.AAC]
+Internal.return_sum [definition, in AAC_tactics.AAC]
+Internal.right [constructor, in AAC_tactics.AAC]
+Internal.run_msets [definition, in AAC_tactics.AAC]
+Internal.run_list [definition, in AAC_tactics.AAC]
+Internal.s [section, in AAC_tactics.AAC]
+Internal.sum [constructor, in AAC_tactics.AAC]
+Internal.sum' [definition, in AAC_tactics.AAC]
+Internal.sum'_sum [lemma, in AAC_tactics.AAC]
+Internal.sym [constructor, in AAC_tactics.AAC]
+Internal.Sym [module, in AAC_tactics.AAC]
+Internal.Sym.ar [projection, in AAC_tactics.AAC]
+Internal.Sym.morph [projection, in AAC_tactics.AAC]
+Internal.Sym.null [definition, in AAC_tactics.AAC]
+Internal.Sym.pack [record, in AAC_tactics.AAC]
+Internal.Sym.rel_of [definition, in AAC_tactics.AAC]
+Internal.Sym.t [section, in AAC_tactics.AAC]
+Internal.Sym.type_of [definition, in AAC_tactics.AAC]
+Internal.Sym.t.R [variable, in AAC_tactics.AAC]
+Internal.Sym.t.X [variable, in AAC_tactics.AAC]
+Internal.Sym.value [projection, in AAC_tactics.AAC]
+Internal.s.E [variable, in AAC_tactics.AAC]
+Internal.s.e_unit [variable, in AAC_tactics.AAC]
+Internal.s.e_bin [variable, in AAC_tactics.AAC]
+Internal.s.e_sym [variable, in AAC_tactics.AAC]
+Internal.s.prds [section, in AAC_tactics.AAC]
+Internal.s.prds.i [variable, in AAC_tactics.AAC]
+Internal.s.prds.is_unit [variable, in AAC_tactics.AAC]
+Internal.s.prd_correctness.is_unit_prd_Unit [variable, in AAC_tactics.AAC]
+Internal.s.prd_correctness.is_unit [variable, in AAC_tactics.AAC]
+Internal.s.prd_correctness.i [variable, in AAC_tactics.AAC]
+Internal.s.prd_correctness [section, in AAC_tactics.AAC]
+Internal.s.R [variable, in AAC_tactics.AAC]
+Internal.s.sums [section, in AAC_tactics.AAC]
+Internal.s.sums.i [variable, in AAC_tactics.AAC]
+Internal.s.sums.is_unit [variable, in AAC_tactics.AAC]
+Internal.s.sum_correctness.comm [variable, in AAC_tactics.AAC]
+Internal.s.sum_correctness.is_unit_sum_Unit [variable, in AAC_tactics.AAC]
+Internal.s.sum_correctness.is_unit [variable, in AAC_tactics.AAC]
+Internal.s.sum_correctness.i [variable, in AAC_tactics.AAC]
+Internal.s.sum_correctness [section, in AAC_tactics.AAC]
+Internal.s.X [variable, in AAC_tactics.AAC]
+_ == _ [notation, in AAC_tactics.AAC]
+Internal.T [inductive, in AAC_tactics.AAC]
+Internal.tcompare_weak_spec [definition, in AAC_tactics.AAC]
+Internal.T_sind [definition, in AAC_tactics.AAC]
+Internal.T_rec [definition, in AAC_tactics.AAC]
+Internal.T_ind [definition, in AAC_tactics.AAC]
+Internal.T_rect [definition, in AAC_tactics.AAC]
+Internal.uf_desc [projection, in AAC_tactics.AAC]
+Internal.uf_idx [projection, in AAC_tactics.AAC]
+Internal.unit [constructor, in AAC_tactics.AAC]
+Internal.unit_pack [record, in AAC_tactics.AAC]
+Internal.unit_of [record, in AAC_tactics.AAC]
+Internal.u_desc [projection, in AAC_tactics.AAC]
+Internal.u_value [projection, in AAC_tactics.AAC]
+Internal.vcompare [definition, in AAC_tactics.AAC]
+Internal.vcompare_reflect_eqdep [definition, in AAC_tactics.AAC]
+Internal.vcons [constructor, in AAC_tactics.AAC]
+Internal.vnil [constructor, in AAC_tactics.AAC]
+Internal.vnorm [definition, in AAC_tactics.AAC]
+Internal.vT [inductive, in AAC_tactics.AAC]
+Internal.vT_sind [definition, in AAC_tactics.AAC]
+Internal.vT_rec [definition, in AAC_tactics.AAC]
+Internal.vT_ind [definition, in AAC_tactics.AAC]
+Internal.vT_rect [definition, in AAC_tactics.AAC]
+Internal.z0 [lemma, in AAC_tactics.AAC]
+Internal.z0' [lemma, in AAC_tactics.AAC]
+Internal.z1 [lemma, in AAC_tactics.AAC]
+Internal.z1' [lemma, in AAC_tactics.AAC]
+Internal.z2 [lemma, in AAC_tactics.AAC]
+Internal.z2' [lemma, in AAC_tactics.AAC]
+introduction [section, in AAC_tactics.Tutorial]
+introduction.a [variable, in AAC_tactics.Tutorial]
+introduction.b [variable, in AAC_tactics.Tutorial]
+introduction.c [variable, in AAC_tactics.Tutorial]
+introduction.H [variable, in AAC_tactics.Tutorial]
+inv_unique [lemma, in AAC_tactics.Caveats]
+

L

+law_neutral_right [projection, in AAC_tactics.AAC]
+law_neutral_left [projection, in AAC_tactics.AAC]
+law_idem [projection, in AAC_tactics.AAC]
+law_idem [constructor, in AAC_tactics.AAC]
+law_comm [projection, in AAC_tactics.AAC]
+law_comm [constructor, in AAC_tactics.AAC]
+law_assoc [projection, in AAC_tactics.AAC]
+law_assoc [constructor, in AAC_tactics.AAC]
+lex [abbreviation, in AAC_tactics.Utils]
+lift_reflexivity [lemma, in AAC_tactics.AAC]
+lift_transitivity_right [lemma, in AAC_tactics.AAC]
+lift_transitivity_left [lemma, in AAC_tactics.AAC]
+lists [section, in AAC_tactics.Utils]
+Lists [module, in AAC_tactics.Instances]
+Lists [section, in AAC_tactics.Tutorial]
+Lists.aac_List_app_nil_Permutation_Unit [instance, in AAC_tactics.Instances]
+Lists.aac_List_app_Permutation_Comm [instance, in AAC_tactics.Instances]
+Lists.aac_List_app_Permutation_Assoc [instance, in AAC_tactics.Instances]
+Lists.aac_List_app_nil_Unit [instance, in AAC_tactics.Instances]
+Lists.aac_List_app_Assoc [instance, in AAC_tactics.Instances]
+lists.c [section, in AAC_tactics.Utils]
+lists.c.A [variable, in AAC_tactics.Utils]
+lists.c.B [variable, in AAC_tactics.Utils]
+lists.c.compare [variable, in AAC_tactics.Utils]
+Lists.H [variable, in AAC_tactics.Tutorial]
+lists.list_compare_weak_spec.Hcompare [variable, in AAC_tactics.Utils]
+lists.list_compare_weak_spec.compare [variable, in AAC_tactics.Utils]
+lists.list_compare_weak_spec.A [variable, in AAC_tactics.Utils]
+lists.list_compare_weak_spec [section, in AAC_tactics.Utils]
+Lists.l1 [variable, in AAC_tactics.Tutorial]
+Lists.l2 [variable, in AAC_tactics.Tutorial]
+Lists.l3 [variable, in AAC_tactics.Tutorial]
+lists.m [section, in AAC_tactics.Utils]
+lists.mset_compare_weak_spec.Hcompare [variable, in AAC_tactics.Utils]
+lists.mset_compare_weak_spec.compare [variable, in AAC_tactics.Utils]
+lists.mset_compare_weak_spec.A [variable, in AAC_tactics.Utils]
+lists.mset_compare_weak_spec [section, in AAC_tactics.Utils]
+lists.m.A [variable, in AAC_tactics.Utils]
+lists.m.B [variable, in AAC_tactics.Utils]
+lists.m.bind [variable, in AAC_tactics.Utils]
+lists.m.b2 [variable, in AAC_tactics.Utils]
+lists.m.compare [variable, in AAC_tactics.Utils]
+lists.m.map [variable, in AAC_tactics.Utils]
+lists.m.merge [variable, in AAC_tactics.Utils]
+lists.m.ret [variable, in AAC_tactics.Utils]
+Lists.X [variable, in AAC_tactics.Tutorial]
+list_compare_weak_spec [lemma, in AAC_tactics.Utils]
+list_compare [definition, in AAC_tactics.Utils]
+

M

+merge_map [definition, in AAC_tactics.Utils]
+merge_msets [definition, in AAC_tactics.Utils]
+morphism [section, in AAC_tactics.Caveats]
+morphism.H [variable, in AAC_tactics.Caveats]
+morphism.HP [variable, in AAC_tactics.Caveats]
+morphism.P [variable, in AAC_tactics.Caveats]
+mset [definition, in AAC_tactics.Utils]
+mset_compare_weak_spec [lemma, in AAC_tactics.Utils]
+mset_compare [definition, in AAC_tactics.Utils]
+

N

+N [module, in AAC_tactics.Instances]
+nelist [inductive, in AAC_tactics.Utils]
+nelist_map [definition, in AAC_tactics.Utils]
+nelist_sind [definition, in AAC_tactics.Utils]
+nelist_rec [definition, in AAC_tactics.Utils]
+nelist_ind [definition, in AAC_tactics.Utils]
+nelist_rect [definition, in AAC_tactics.Utils]
+nil [constructor, in AAC_tactics.Utils]
+N.aac_N_le_eq_lift [instance, in AAC_tactics.Instances]
+N.aac_N_gcd_0_Unit [instance, in AAC_tactics.Instances]
+N.aac_N_max_0_Unit [instance, in AAC_tactics.Instances]
+N.aac_N_add_0_Unit [instance, in AAC_tactics.Instances]
+N.aac_N_lcm_1_Unit [instance, in AAC_tactics.Instances]
+N.aac_N_mul_1_Unit [instance, in AAC_tactics.Instances]
+N.aac_N_lcm_Idem [instance, in AAC_tactics.Instances]
+N.aac_N_lcm_Assoc [instance, in AAC_tactics.Instances]
+N.aac_N_lcm_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_gcd_Idem [instance, in AAC_tactics.Instances]
+N.aac_N_gcd_Assoc [instance, in AAC_tactics.Instances]
+N.aac_N_gcd_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_max_Idem [instance, in AAC_tactics.Instances]
+N.aac_N_max_Assoc [instance, in AAC_tactics.Instances]
+N.aac_N_max_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_min_Idem [instance, in AAC_tactics.Instances]
+N.aac_N_min_Assoc [instance, in AAC_tactics.Instances]
+N.aac_N_min_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_mul_Assoc [instance, in AAC_tactics.Instances]
+N.aac_N_mul_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_add_Comm [instance, in AAC_tactics.Instances]
+N.aac_N_add_Assoc [instance, in AAC_tactics.Instances]
+N.N_le_PreOrder [instance, in AAC_tactics.Instances]
+

P

+P [module, in AAC_tactics.Instances]
+parameters [section, in AAC_tactics.Caveats]
+parameters.E [variable, in AAC_tactics.Caveats]
+parameters.f [variable, in AAC_tactics.Caveats]
+parameters.g [variable, in AAC_tactics.Caveats]
+parameters.H [variable, in AAC_tactics.Caveats]
+parameters.Hf [variable, in AAC_tactics.Caveats]
+parameters.Hf' [variable, in AAC_tactics.Caveats]
+parameters.Hg [variable, in AAC_tactics.Caveats]
+parameters.plus [variable, in AAC_tactics.Caveats]
+parameters.plus_Proper [variable, in AAC_tactics.Caveats]
+parameters.plus_C [variable, in AAC_tactics.Caveats]
+parameters.plus_A [variable, in AAC_tactics.Caveats]
+parameters.R [variable, in AAC_tactics.Caveats]
+parameters.X [variable, in AAC_tactics.Caveats]
+parameters.Zero [variable, in AAC_tactics.Caveats]
+parameters.zero [variable, in AAC_tactics.Caveats]
+_ + _ [notation, in AAC_tactics.Caveats]
+_ == _ [notation, in AAC_tactics.Caveats]
+0 [notation, in AAC_tactics.Caveats]
+pcws_gt [constructor, in AAC_tactics.Utils]
+pcws_lt [constructor, in AAC_tactics.Utils]
+pcws_eq [constructor, in AAC_tactics.Utils]
+Peano [section, in AAC_tactics.Caveats]
+Peano [module, in AAC_tactics.Instances]
+Peano [section, in AAC_tactics.Tutorial]
+Peano.a [variable, in AAC_tactics.Tutorial]
+Peano.aac_Nat_le_eq_lift [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_0_Unit [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_max_0_Unit [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_add_0_Unit [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_1_Unit [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_mul_1_Unit [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Idem [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Assoc [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Idem [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Assoc [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_max_Idem [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_max_Assoc [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_max_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_min_Idem [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_min_Assoc [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_min_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_mul_Assoc [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_mul_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_add_Comm [instance, in AAC_tactics.Instances]
+Peano.aac_Nat_add_Assoc [instance, in AAC_tactics.Instances]
+Peano.b [variable, in AAC_tactics.Tutorial]
+Peano.c [variable, in AAC_tactics.Tutorial]
+Peano.H [variable, in AAC_tactics.Caveats]
+Peano.H [variable, in AAC_tactics.Tutorial]
+Peano.H' [variable, in AAC_tactics.Caveats]
+Peano.Nat_le_PreOrder [instance, in AAC_tactics.Instances]
+pos_compare_reflect_eq [lemma, in AAC_tactics.Utils]
+pos_compare_weak_spec [lemma, in AAC_tactics.Utils]
+Props [section, in AAC_tactics.Tutorial]
+Props.P [variable, in AAC_tactics.Tutorial]
+Props.Q [variable, in AAC_tactics.Tutorial]
+Prop_ops.aac_Prop_impl_iff_lift [instance, in AAC_tactics.Instances]
+Prop_ops.not_iff_compat [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_True_iff_Unit [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_False_iff_Unit [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Idem [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Comm [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Assoc [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Idem [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Comm [instance, in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Assoc [instance, in AAC_tactics.Instances]
+Prop_ops [module, in AAC_tactics.Instances]
+P.aac_Pos_le_eq_lift [instance, in AAC_tactics.Instances]
+P.aac_Pos_max_1_Unit [instance, in AAC_tactics.Instances]
+P.aac_Pos_mul_1_Unit [instance, in AAC_tactics.Instances]
+P.aac_Pos_max_Idem [instance, in AAC_tactics.Instances]
+P.aac_Pos_max_Assoc [instance, in AAC_tactics.Instances]
+P.aac_Pos_max_Comm [instance, in AAC_tactics.Instances]
+P.aac_Pos_min_Idem [instance, in AAC_tactics.Instances]
+P.aac_Pos_min_Assoc [instance, in AAC_tactics.Instances]
+P.aac_Pos_min_Comm [instance, in AAC_tactics.Instances]
+P.aac_Pos_mul_Assoc [instance, in AAC_tactics.Instances]
+P.aac_Pos_mul_Comm [instance, in AAC_tactics.Instances]
+P.aac_Pos_add_Comm [instance, in AAC_tactics.Instances]
+P.aac_Pos_add_Assoc [instance, in AAC_tactics.Instances]
+P.Pos_le_PreOrder [instance, in AAC_tactics.Instances]
+

Q

+Q [module, in AAC_tactics.Instances]
+Q.aac_Q_Qle_eq_lift [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qplus_0_Qeq_Unit [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmult_1_Qeq_Unit [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Idem [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Assoc [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Comm [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Idem [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Assoc [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Comm [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmult_Qeq_Assoc [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qmult_Qeq_Comm [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qplus_Qeq_Comm [instance, in AAC_tactics.Instances]
+Q.aac_Q_Qplus_Qeq_Assoc [instance, in AAC_tactics.Instances]
+Q.Q_Qle_PreOrder [instance, in AAC_tactics.Instances]
+

R

+reduce_mset [definition, in AAC_tactics.Utils]
+reflect_eqdep_weak_spec [lemma, in AAC_tactics.Utils]
+reflect_eqdep_eq [lemma, in AAC_tactics.Utils]
+reflect_eqdep [definition, in AAC_tactics.Utils]
+Relations [module, in AAC_tactics.Instances]
+Relations.aac_inclusion_same_relation_lift [instance, in AAC_tactics.Instances]
+Relations.aac_compo_eq_same_relation_Unit [instance, in AAC_tactics.Instances]
+Relations.aac_compo_same_relation_Assoc [instance, in AAC_tactics.Instances]
+Relations.aac_inter_top_same_relation_Unit [instance, in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Idem [instance, in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Assoc [instance, in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Comm [instance, in AAC_tactics.Instances]
+Relations.aac_bot_union_same_relation_Unit [instance, in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Idem [instance, in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Assoc [instance, in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Comm [instance, in AAC_tactics.Instances]
+Relations.bot [definition, in AAC_tactics.Instances]
+Relations.clos_refl_trans_same_relation_compat [instance, in AAC_tactics.Instances]
+Relations.clos_refl_trans_incr [instance, in AAC_tactics.Instances]
+Relations.clos_trans_same_relation_compat [instance, in AAC_tactics.Instances]
+Relations.clos_trans_incr [instance, in AAC_tactics.Instances]
+Relations.compo [definition, in AAC_tactics.Instances]
+Relations.defs [section, in AAC_tactics.Instances]
+Relations.defs.R [variable, in AAC_tactics.Instances]
+Relations.defs.S [variable, in AAC_tactics.Instances]
+Relations.defs.T [variable, in AAC_tactics.Instances]
+Relations.inclusion_PreOrder [instance, in AAC_tactics.Instances]
+Relations.inter [definition, in AAC_tactics.Instances]
+Relations.negr [definition, in AAC_tactics.Instances]
+Relations.negr_same_relation_compat [instance, in AAC_tactics.Instances]
+Relations.same_relation_Equivalence [instance, in AAC_tactics.Instances]
+Relations.top [definition, in AAC_tactics.Instances]
+Relations.transp_same_relation_compat [instance, in AAC_tactics.Instances]
+

S

+sigma [definition, in AAC_tactics.AAC]
+sigma [section, in AAC_tactics.AAC]
+sigma_empty [definition, in AAC_tactics.AAC]
+sigma_add [definition, in AAC_tactics.AAC]
+sigma_get [definition, in AAC_tactics.AAC]
+

T

+t [section, in AAC_tactics.AAC]
+transitivity4 [lemma, in AAC_tactics.AAC]
+Tutorial [library]
+

U

+U [section, in AAC_tactics.Caveats]
+Unit [record, in AAC_tactics.AAC]
+Unnamed_thm5 [definition, in AAC_tactics.Caveats]
+Unnamed_thm5 [definition, in AAC_tactics.Caveats]
+Unnamed_thm5 [definition, in AAC_tactics.Caveats]
+Unnamed_thm5 [definition, in AAC_tactics.Caveats]
+Unnamed_thm4 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm3 [definition, in AAC_tactics.Caveats]
+Unnamed_thm2 [definition, in AAC_tactics.Caveats]
+Unnamed_thm1 [definition, in AAC_tactics.Caveats]
+Unnamed_thm1 [definition, in AAC_tactics.Caveats]
+Unnamed_thm0 [definition, in AAC_tactics.Caveats]
+Unnamed_thm [definition, in AAC_tactics.Caveats]
+Unnamed_thm29 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm28 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm27 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm26 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm25 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm24 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm23 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm22 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm22 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm21 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm20 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm20 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm19 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm18 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm17 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm16 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm15 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm14 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm13 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm12 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm11 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm11 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm10 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm9 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm8 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm7 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm6 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm5 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm4 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm3 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm2 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm1 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm0 [definition, in AAC_tactics.Tutorial]
+Unnamed_thm [definition, in AAC_tactics.Tutorial]
+Utils [library]
+U.dot [variable, in AAC_tactics.Caveats]
+U.dot_inv_right [variable, in AAC_tactics.Caveats]
+U.dot_inv_left [variable, in AAC_tactics.Caveats]
+U.dot_Proper [variable, in AAC_tactics.Caveats]
+U.dot_A [variable, in AAC_tactics.Caveats]
+U.E [variable, in AAC_tactics.Caveats]
+U.f [variable, in AAC_tactics.Caveats]
+U.Hf [variable, in AAC_tactics.Caveats]
+U.One [variable, in AAC_tactics.Caveats]
+U.one [variable, in AAC_tactics.Caveats]
+U.R [variable, in AAC_tactics.Caveats]
+U.X [variable, in AAC_tactics.Caveats]
+_ * _ [notation, in AAC_tactics.Caveats]
+_ == _ [notation, in AAC_tactics.Caveats]
+1 [notation, in AAC_tactics.Caveats]
+

V

+V [section, in AAC_tactics.Caveats]
+V.dot [variable, in AAC_tactics.Caveats]
+V.dot_Proper [variable, in AAC_tactics.Caveats]
+V.dot_A [variable, in AAC_tactics.Caveats]
+V.E [variable, in AAC_tactics.Caveats]
+V.One [variable, in AAC_tactics.Caveats]
+V.one [variable, in AAC_tactics.Caveats]
+V.R [variable, in AAC_tactics.Caveats]
+V.X [variable, in AAC_tactics.Caveats]
+_ * _ [notation, in AAC_tactics.Caveats]
+_ == _ [notation, in AAC_tactics.Caveats]
+1 [notation, in AAC_tactics.Caveats]
+

W

+W [section, in AAC_tactics.Caveats]
+W.a [variable, in AAC_tactics.Caveats]
+W.b [variable, in AAC_tactics.Caveats]
+W.c [variable, in AAC_tactics.Caveats]
+W.H [variable, in AAC_tactics.Caveats]
+

Z

+Z [section, in AAC_tactics.Caveats]
+Z [module, in AAC_tactics.Instances]
+Z_add_incr [instance, in AAC_tactics.Caveats]
+Z_add_le_compat [instance, in AAC_tactics.Tutorial]
+Z_opp_ge_le_compat [instance, in AAC_tactics.Tutorial]
+Z_add_opp_diag_r [lemma, in AAC_tactics.Tutorial]
+Z_abs_triangle [lemma, in AAC_tactics.Tutorial]
+Z.a [variable, in AAC_tactics.Caveats]
+Z.aac_Z_le_eq_lift [instance, in AAC_tactics.Instances]
+Z.aac_Z_add_0_Unit [instance, in AAC_tactics.Instances]
+Z.aac_Z_mul_1_Unit [instance, in AAC_tactics.Instances]
+Z.aac_Z_lcm_Assoc [instance, in AAC_tactics.Instances]
+Z.aac_Z_lcm_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_gcd_Assoc [instance, in AAC_tactics.Instances]
+Z.aac_Z_gcd_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_max_Idem [instance, in AAC_tactics.Instances]
+Z.aac_Z_max_Assoc [instance, in AAC_tactics.Instances]
+Z.aac_Z_max_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_min_Idem [instance, in AAC_tactics.Instances]
+Z.aac_Z_min_Assoc [instance, in AAC_tactics.Instances]
+Z.aac_Z_min_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_mul_Assoc [instance, in AAC_tactics.Instances]
+Z.aac_Z_mul_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_add_Comm [instance, in AAC_tactics.Instances]
+Z.aac_Z_add_Assoc [instance, in AAC_tactics.Instances]
+Z.b [variable, in AAC_tactics.Caveats]
+Z.c [variable, in AAC_tactics.Caveats]
+Z.f [variable, in AAC_tactics.Caveats]
+Z.H [variable, in AAC_tactics.Caveats]
+Z.H0 [variable, in AAC_tactics.Caveats]
+Z.H1 [variable, in AAC_tactics.Caveats]
+Z.Z_le_PreOrder [instance, in AAC_tactics.Instances]
+

other

+_ ++ _ [notation, in AAC_tactics.Utils]
+_ :: _ [notation, in AAC_tactics.Utils]
+

+

Notation Index

+

B

+_ + _ [in AAC_tactics.Tutorial]
+_ * _ [in AAC_tactics.Tutorial]
+_ == _ [in AAC_tactics.Tutorial]
+0 [in AAC_tactics.Tutorial]
+1 [in AAC_tactics.Tutorial]
+

E

+_ ^2 [in AAC_tactics.Tutorial]
+2 ⋅ _ [in AAC_tactics.Tutorial]
+

I

+_ == _ [in AAC_tactics.AAC]
+

P

+_ + _ [in AAC_tactics.Caveats]
+_ == _ [in AAC_tactics.Caveats]
+0 [in AAC_tactics.Caveats]
+

U

+_ * _ [in AAC_tactics.Caveats]
+_ == _ [in AAC_tactics.Caveats]
+1 [in AAC_tactics.Caveats]
+

V

+_ * _ [in AAC_tactics.Caveats]
+_ == _ [in AAC_tactics.Caveats]
+1 [in AAC_tactics.Caveats]
+

other

+_ ++ _ [in AAC_tactics.Utils]
+_ :: _ [in AAC_tactics.Utils]
+

+

Module Index

+

A

+All [in AAC_tactics.Instances]
+

B

+Bool [in AAC_tactics.Instances]
+

I

+Internal [in AAC_tactics.AAC]
+Internal.Bin [in AAC_tactics.AAC]
+Internal.Sym [in AAC_tactics.AAC]
+

L

+Lists [in AAC_tactics.Instances]
+

N

+N [in AAC_tactics.Instances]
+

P

+P [in AAC_tactics.Instances]
+Peano [in AAC_tactics.Instances]
+Prop_ops [in AAC_tactics.Instances]
+

Q

+Q [in AAC_tactics.Instances]
+

R

+Relations [in AAC_tactics.Instances]
+

Z

+Z [in AAC_tactics.Instances]
+

+

Variable Index

+

A

+AAC_normalise.d [in AAC_tactics.Tutorial]
+AAC_normalise.c [in AAC_tactics.Tutorial]
+AAC_normalise.b [in AAC_tactics.Tutorial]
+AAC_normalise.a [in AAC_tactics.Tutorial]
+

B

+base.both.a [in AAC_tactics.Tutorial]
+base.both.b [in AAC_tactics.Tutorial]
+base.both.c [in AAC_tactics.Tutorial]
+base.both.d [in AAC_tactics.Tutorial]
+base.both.H [in AAC_tactics.Tutorial]
+base.both.H' [in AAC_tactics.Tutorial]
+base.dealing_with_units.H' [in AAC_tactics.Tutorial]
+base.dealing_with_units.H [in AAC_tactics.Tutorial]
+base.dealing_with_units.c [in AAC_tactics.Tutorial]
+base.dealing_with_units.b [in AAC_tactics.Tutorial]
+base.dealing_with_units.a [in AAC_tactics.Tutorial]
+base.dot [in AAC_tactics.Tutorial]
+base.dot_Proper [in AAC_tactics.Tutorial]
+base.dot_A [in AAC_tactics.Tutorial]
+base.E [in AAC_tactics.Tutorial]
+base.morphisms.a [in AAC_tactics.Tutorial]
+base.morphisms.b [in AAC_tactics.Tutorial]
+base.morphisms.f [in AAC_tactics.Tutorial]
+base.morphisms.g [in AAC_tactics.Tutorial]
+base.morphisms.H [in AAC_tactics.Tutorial]
+base.morphisms.Hf [in AAC_tactics.Tutorial]
+base.morphisms.Hg [in AAC_tactics.Tutorial]
+base.occurrence.a [in AAC_tactics.Tutorial]
+base.occurrence.f [in AAC_tactics.Tutorial]
+base.occurrence.H [in AAC_tactics.Tutorial]
+base.occurrence.Hf [in AAC_tactics.Tutorial]
+base.One [in AAC_tactics.Tutorial]
+base.one [in AAC_tactics.Tutorial]
+base.plus [in AAC_tactics.Tutorial]
+base.plus_Proper [in AAC_tactics.Tutorial]
+base.plus_C [in AAC_tactics.Tutorial]
+base.plus_A [in AAC_tactics.Tutorial]
+base.R [in AAC_tactics.Tutorial]
+base.reminder.a [in AAC_tactics.Tutorial]
+base.reminder.b [in AAC_tactics.Tutorial]
+base.reminder.c [in AAC_tactics.Tutorial]
+base.reminder.H [in AAC_tactics.Tutorial]
+base.subst.a [in AAC_tactics.Tutorial]
+base.subst.b [in AAC_tactics.Tutorial]
+base.subst.c [in AAC_tactics.Tutorial]
+base.subst.d [in AAC_tactics.Tutorial]
+base.subst.H [in AAC_tactics.Tutorial]
+base.subst.H' [in AAC_tactics.Tutorial]
+base.X [in AAC_tactics.Tutorial]
+base.Zero [in AAC_tactics.Tutorial]
+base.zero [in AAC_tactics.Tutorial]
+

D

+dep.f [in AAC_tactics.Utils]
+dep.T [in AAC_tactics.Utils]
+dep.U [in AAC_tactics.Utils]
+

E

+evars.H [in AAC_tactics.Caveats]
+evars.H' [in AAC_tactics.Caveats]
+evars.idem [in AAC_tactics.Caveats]
+evars.P [in AAC_tactics.Caveats]
+Examples.a [in AAC_tactics.Tutorial]
+Examples.b [in AAC_tactics.Tutorial]
+Examples.c [in AAC_tactics.Tutorial]
+Examples.H [in AAC_tactics.Tutorial]
+

I

+ineq.H [in AAC_tactics.Caveats]
+Internal.Bin.t.R [in AAC_tactics.AAC]
+Internal.Bin.t.X [in AAC_tactics.AAC]
+Internal.copy.HR [in AAC_tactics.AAC]
+Internal.copy.op [in AAC_tactics.AAC]
+Internal.copy.op' [in AAC_tactics.AAC]
+Internal.copy.plus [in AAC_tactics.AAC]
+Internal.copy.po [in AAC_tactics.AAC]
+Internal.copy.R [in AAC_tactics.AAC]
+Internal.copy.X [in AAC_tactics.AAC]
+Internal.Sym.t.R [in AAC_tactics.AAC]
+Internal.Sym.t.X [in AAC_tactics.AAC]
+Internal.s.E [in AAC_tactics.AAC]
+Internal.s.e_unit [in AAC_tactics.AAC]
+Internal.s.e_bin [in AAC_tactics.AAC]
+Internal.s.e_sym [in AAC_tactics.AAC]
+Internal.s.prds.i [in AAC_tactics.AAC]
+Internal.s.prds.is_unit [in AAC_tactics.AAC]
+Internal.s.prd_correctness.is_unit_prd_Unit [in AAC_tactics.AAC]
+Internal.s.prd_correctness.is_unit [in AAC_tactics.AAC]
+Internal.s.prd_correctness.i [in AAC_tactics.AAC]
+Internal.s.R [in AAC_tactics.AAC]
+Internal.s.sums.i [in AAC_tactics.AAC]
+Internal.s.sums.is_unit [in AAC_tactics.AAC]
+Internal.s.sum_correctness.comm [in AAC_tactics.AAC]
+Internal.s.sum_correctness.is_unit_sum_Unit [in AAC_tactics.AAC]
+Internal.s.sum_correctness.is_unit [in AAC_tactics.AAC]
+Internal.s.sum_correctness.i [in AAC_tactics.AAC]
+Internal.s.X [in AAC_tactics.AAC]
+introduction.a [in AAC_tactics.Tutorial]
+introduction.b [in AAC_tactics.Tutorial]
+introduction.c [in AAC_tactics.Tutorial]
+introduction.H [in AAC_tactics.Tutorial]
+

L

+lists.c.A [in AAC_tactics.Utils]
+lists.c.B [in AAC_tactics.Utils]
+lists.c.compare [in AAC_tactics.Utils]
+Lists.H [in AAC_tactics.Tutorial]
+lists.list_compare_weak_spec.Hcompare [in AAC_tactics.Utils]
+lists.list_compare_weak_spec.compare [in AAC_tactics.Utils]
+lists.list_compare_weak_spec.A [in AAC_tactics.Utils]
+Lists.l1 [in AAC_tactics.Tutorial]
+Lists.l2 [in AAC_tactics.Tutorial]
+Lists.l3 [in AAC_tactics.Tutorial]
+lists.mset_compare_weak_spec.Hcompare [in AAC_tactics.Utils]
+lists.mset_compare_weak_spec.compare [in AAC_tactics.Utils]
+lists.mset_compare_weak_spec.A [in AAC_tactics.Utils]
+lists.m.A [in AAC_tactics.Utils]
+lists.m.B [in AAC_tactics.Utils]
+lists.m.bind [in AAC_tactics.Utils]
+lists.m.b2 [in AAC_tactics.Utils]
+lists.m.compare [in AAC_tactics.Utils]
+lists.m.map [in AAC_tactics.Utils]
+lists.m.merge [in AAC_tactics.Utils]
+lists.m.ret [in AAC_tactics.Utils]
+Lists.X [in AAC_tactics.Tutorial]
+

M

+morphism.H [in AAC_tactics.Caveats]
+morphism.HP [in AAC_tactics.Caveats]
+morphism.P [in AAC_tactics.Caveats]
+

P

+parameters.E [in AAC_tactics.Caveats]
+parameters.f [in AAC_tactics.Caveats]
+parameters.g [in AAC_tactics.Caveats]
+parameters.H [in AAC_tactics.Caveats]
+parameters.Hf [in AAC_tactics.Caveats]
+parameters.Hf' [in AAC_tactics.Caveats]
+parameters.Hg [in AAC_tactics.Caveats]
+parameters.plus [in AAC_tactics.Caveats]
+parameters.plus_Proper [in AAC_tactics.Caveats]
+parameters.plus_C [in AAC_tactics.Caveats]
+parameters.plus_A [in AAC_tactics.Caveats]
+parameters.R [in AAC_tactics.Caveats]
+parameters.X [in AAC_tactics.Caveats]
+parameters.Zero [in AAC_tactics.Caveats]
+parameters.zero [in AAC_tactics.Caveats]
+Peano.a [in AAC_tactics.Tutorial]
+Peano.b [in AAC_tactics.Tutorial]
+Peano.c [in AAC_tactics.Tutorial]
+Peano.H [in AAC_tactics.Caveats]
+Peano.H [in AAC_tactics.Tutorial]
+Peano.H' [in AAC_tactics.Caveats]
+Props.P [in AAC_tactics.Tutorial]
+Props.Q [in AAC_tactics.Tutorial]
+

R

+Relations.defs.R [in AAC_tactics.Instances]
+Relations.defs.S [in AAC_tactics.Instances]
+Relations.defs.T [in AAC_tactics.Instances]
+

U

+U.dot [in AAC_tactics.Caveats]
+U.dot_inv_right [in AAC_tactics.Caveats]
+U.dot_inv_left [in AAC_tactics.Caveats]
+U.dot_Proper [in AAC_tactics.Caveats]
+U.dot_A [in AAC_tactics.Caveats]
+U.E [in AAC_tactics.Caveats]
+U.f [in AAC_tactics.Caveats]
+U.Hf [in AAC_tactics.Caveats]
+U.One [in AAC_tactics.Caveats]
+U.one [in AAC_tactics.Caveats]
+U.R [in AAC_tactics.Caveats]
+U.X [in AAC_tactics.Caveats]
+

V

+V.dot [in AAC_tactics.Caveats]
+V.dot_Proper [in AAC_tactics.Caveats]
+V.dot_A [in AAC_tactics.Caveats]
+V.E [in AAC_tactics.Caveats]
+V.One [in AAC_tactics.Caveats]
+V.one [in AAC_tactics.Caveats]
+V.R [in AAC_tactics.Caveats]
+V.X [in AAC_tactics.Caveats]
+

W

+W.a [in AAC_tactics.Caveats]
+W.b [in AAC_tactics.Caveats]
+W.c [in AAC_tactics.Caveats]
+W.H [in AAC_tactics.Caveats]
+

Z

+Z.a [in AAC_tactics.Caveats]
+Z.b [in AAC_tactics.Caveats]
+Z.c [in AAC_tactics.Caveats]
+Z.f [in AAC_tactics.Caveats]
+Z.H [in AAC_tactics.Caveats]
+Z.H0 [in AAC_tactics.Caveats]
+Z.H1 [in AAC_tactics.Caveats]
+

+

Library Index

+

A

+AAC
+

C

+Caveats
+Constants
+

I

+Instances
+

T

+Tutorial
+

U

+Utils
+

+

Lemma Index

+

C

+cast_eq [in AAC_tactics.Utils]
+

E

+eq_idx_spec [in AAC_tactics.Utils]
+eq_subr [in AAC_tactics.Instances]
+

H

+Hbin1 [in AAC_tactics.Tutorial]
+Hbin2 [in AAC_tactics.Tutorial]
+Hopp [in AAC_tactics.Tutorial]
+

I

+Internal.compare_reflect_eq [in AAC_tactics.AAC]
+Internal.compat_prd_unit_add [in AAC_tactics.AAC]
+Internal.compat_prd_unit_return [in AAC_tactics.AAC]
+Internal.compat_sum_unit_add [in AAC_tactics.AAC]
+Internal.compat_sum_unit_return [in AAC_tactics.AAC]
+Internal.copy_idem [in AAC_tactics.AAC]
+Internal.copy_n_unit [in AAC_tactics.AAC]
+Internal.copy_mset_copy [in AAC_tactics.AAC]
+Internal.copy_mset_succ [in AAC_tactics.AAC]
+Internal.copy_mset' [in AAC_tactics.AAC]
+Internal.copy_Psucc [in AAC_tactics.AAC]
+Internal.copy_xH [in AAC_tactics.AAC]
+Internal.copy_plus [in AAC_tactics.AAC]
+Internal.decide [in AAC_tactics.AAC]
+Internal.eval_norm_lists [in AAC_tactics.AAC]
+Internal.eval_prd_app [in AAC_tactics.AAC]
+Internal.eval_prd_cons [in AAC_tactics.AAC]
+Internal.eval_prd_nil [in AAC_tactics.AAC]
+Internal.eval_reduce_msets [in AAC_tactics.AAC]
+Internal.eval_norm_msets [in AAC_tactics.AAC]
+Internal.eval_merge_bin [in AAC_tactics.AAC]
+Internal.eval_sum_cons [in AAC_tactics.AAC]
+Internal.eval_sum_nil [in AAC_tactics.AAC]
+Internal.is_prd_spec [in AAC_tactics.AAC]
+Internal.is_sum_spec [in AAC_tactics.AAC]
+Internal.is_unit_of_Unit [in AAC_tactics.AAC]
+Internal.lift_normalise [in AAC_tactics.AAC]
+Internal.normalise [in AAC_tactics.AAC]
+Internal.prd'_prd [in AAC_tactics.AAC]
+Internal.sum'_sum [in AAC_tactics.AAC]
+Internal.z0 [in AAC_tactics.AAC]
+Internal.z0' [in AAC_tactics.AAC]
+Internal.z1 [in AAC_tactics.AAC]
+Internal.z1' [in AAC_tactics.AAC]
+Internal.z2 [in AAC_tactics.AAC]
+Internal.z2' [in AAC_tactics.AAC]
+inv_unique [in AAC_tactics.Caveats]
+

L

+lift_reflexivity [in AAC_tactics.AAC]
+lift_transitivity_right [in AAC_tactics.AAC]
+lift_transitivity_left [in AAC_tactics.AAC]
+list_compare_weak_spec [in AAC_tactics.Utils]
+

M

+mset_compare_weak_spec [in AAC_tactics.Utils]
+

P

+pos_compare_reflect_eq [in AAC_tactics.Utils]
+pos_compare_weak_spec [in AAC_tactics.Utils]
+

R

+reflect_eqdep_weak_spec [in AAC_tactics.Utils]
+reflect_eqdep_eq [in AAC_tactics.Utils]
+

T

+transitivity4 [in AAC_tactics.AAC]
+

Z

+Z_add_opp_diag_r [in AAC_tactics.Tutorial]
+Z_abs_triangle [in AAC_tactics.Tutorial]
+

+

Constructor Index

+

C

+cons [in AAC_tactics.Utils]
+

D

+decide_false [in AAC_tactics.Utils]
+decide_true [in AAC_tactics.Utils]
+

I

+Internal.cpu_right [in AAC_tactics.AAC]
+Internal.cpu_left [in AAC_tactics.AAC]
+Internal.csu_right [in AAC_tactics.AAC]
+Internal.csu_left [in AAC_tactics.AAC]
+Internal.is_prd_spec_nothing [in AAC_tactics.AAC]
+Internal.is_prd_spec_unit [in AAC_tactics.AAC]
+Internal.is_prd_spec_op [in AAC_tactics.AAC]
+Internal.is_sum_spec_nothing [in AAC_tactics.AAC]
+Internal.is_sum_spec_unit [in AAC_tactics.AAC]
+Internal.is_sum_spec_op [in AAC_tactics.AAC]
+Internal.Is_nothing [in AAC_tactics.AAC]
+Internal.Is_unit [in AAC_tactics.AAC]
+Internal.Is_op [in AAC_tactics.AAC]
+Internal.left [in AAC_tactics.AAC]
+Internal.prd [in AAC_tactics.AAC]
+Internal.right [in AAC_tactics.AAC]
+Internal.sum [in AAC_tactics.AAC]
+Internal.sym [in AAC_tactics.AAC]
+Internal.unit [in AAC_tactics.AAC]
+Internal.vcons [in AAC_tactics.AAC]
+Internal.vnil [in AAC_tactics.AAC]
+

L

+law_idem [in AAC_tactics.AAC]
+law_comm [in AAC_tactics.AAC]
+law_assoc [in AAC_tactics.AAC]
+

N

+nil [in AAC_tactics.Utils]
+

P

+pcws_gt [in AAC_tactics.Utils]
+pcws_lt [in AAC_tactics.Utils]
+pcws_eq [in AAC_tactics.Utils]
+

+

Inductive Index

+

A

+Associative [in AAC_tactics.AAC]
+

C

+Commutative [in AAC_tactics.AAC]
+compare_weak_spec [in AAC_tactics.Utils]
+

D

+decide_spec [in AAC_tactics.Utils]
+

I

+Idempotent [in AAC_tactics.AAC]
+Internal.compat_prd_unit [in AAC_tactics.AAC]
+Internal.compat_sum_unit [in AAC_tactics.AAC]
+Internal.discr [in AAC_tactics.AAC]
+Internal.is_prd_spec_ind [in AAC_tactics.AAC]
+Internal.is_sum_spec_ind [in AAC_tactics.AAC]
+Internal.m [in AAC_tactics.AAC]
+Internal.T [in AAC_tactics.AAC]
+Internal.vT [in AAC_tactics.AAC]
+

N

+nelist [in AAC_tactics.Utils]
+

+

Projection Index

+

A

+aac_list_proper [in AAC_tactics.AAC]
+aac_lift_equivalence [in AAC_tactics.AAC]
+

I

+Internal.Bin.assoc [in AAC_tactics.AAC]
+Internal.Bin.comm [in AAC_tactics.AAC]
+Internal.Bin.compat [in AAC_tactics.AAC]
+Internal.Bin.idem [in AAC_tactics.AAC]
+Internal.Bin.value [in AAC_tactics.AAC]
+Internal.Sym.ar [in AAC_tactics.AAC]
+Internal.Sym.morph [in AAC_tactics.AAC]
+Internal.Sym.value [in AAC_tactics.AAC]
+Internal.uf_desc [in AAC_tactics.AAC]
+Internal.uf_idx [in AAC_tactics.AAC]
+Internal.u_desc [in AAC_tactics.AAC]
+Internal.u_value [in AAC_tactics.AAC]
+

L

+law_neutral_right [in AAC_tactics.AAC]
+law_neutral_left [in AAC_tactics.AAC]
+law_idem [in AAC_tactics.AAC]
+law_comm [in AAC_tactics.AAC]
+law_assoc [in AAC_tactics.AAC]
+

+

Section Index

+

A

+AAC_normalise [in AAC_tactics.Tutorial]
+

B

+base [in AAC_tactics.Tutorial]
+base.both [in AAC_tactics.Tutorial]
+base.dealing_with_units [in AAC_tactics.Tutorial]
+base.morphisms [in AAC_tactics.Tutorial]
+base.occurrence [in AAC_tactics.Tutorial]
+base.reminder [in AAC_tactics.Tutorial]
+base.subst [in AAC_tactics.Tutorial]
+

D

+dep [in AAC_tactics.Utils]
+

E

+evars [in AAC_tactics.Caveats]
+Examples [in AAC_tactics.Tutorial]
+

I

+ineq [in AAC_tactics.Caveats]
+Internal.Bin.t [in AAC_tactics.AAC]
+Internal.copy [in AAC_tactics.AAC]
+Internal.s [in AAC_tactics.AAC]
+Internal.Sym.t [in AAC_tactics.AAC]
+Internal.s.prds [in AAC_tactics.AAC]
+Internal.s.prd_correctness [in AAC_tactics.AAC]
+Internal.s.sums [in AAC_tactics.AAC]
+Internal.s.sum_correctness [in AAC_tactics.AAC]
+introduction [in AAC_tactics.Tutorial]
+

L

+lists [in AAC_tactics.Utils]
+Lists [in AAC_tactics.Tutorial]
+lists.c [in AAC_tactics.Utils]
+lists.list_compare_weak_spec [in AAC_tactics.Utils]
+lists.m [in AAC_tactics.Utils]
+lists.mset_compare_weak_spec [in AAC_tactics.Utils]
+

M

+morphism [in AAC_tactics.Caveats]
+

P

+parameters [in AAC_tactics.Caveats]
+Peano [in AAC_tactics.Caveats]
+Peano [in AAC_tactics.Tutorial]
+Props [in AAC_tactics.Tutorial]
+

R

+Relations.defs [in AAC_tactics.Instances]
+

S

+sigma [in AAC_tactics.AAC]
+

T

+t [in AAC_tactics.AAC]
+

U

+U [in AAC_tactics.Caveats]
+

V

+V [in AAC_tactics.Caveats]
+

W

+W [in AAC_tactics.Caveats]
+

Z

+Z [in AAC_tactics.Caveats]
+

+

Instance Index

+

A

+aac_lift_proper [in AAC_tactics.AAC]
+aac_lift_subrelation [in AAC_tactics.AAC]
+aac_le_eq_lift [in AAC_tactics.Tutorial]
+aac_Nat_max_0_Unit [in AAC_tactics.Tutorial]
+aac_Nat_max_Idem [in AAC_tactics.Tutorial]
+aac_Nat_max_Assoc [in AAC_tactics.Tutorial]
+aac_Nat_max_Comm [in AAC_tactics.Tutorial]
+aac_Nat_add_0_Unit [in AAC_tactics.Tutorial]
+aac_Nat_mul_1_Unit [in AAC_tactics.Tutorial]
+aac_Nat_mul_Assoc [in AAC_tactics.Tutorial]
+aac_Nat_mul_Comm [in AAC_tactics.Tutorial]
+aac_Nat_add_Comm [in AAC_tactics.Tutorial]
+aac_Nat_add_Assoc [in AAC_tactics.Tutorial]
+

B

+Bool.aac_Bool_andb_true_Unit [in AAC_tactics.Instances]
+Bool.aac_Bool_orb_false_Unit [in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Idem [in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Comm [in AAC_tactics.Instances]
+Bool.aac_Bool_andb_Assoc [in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Idem [in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Comm [in AAC_tactics.Instances]
+Bool.aac_Bool_orb_Assoc [in AAC_tactics.Instances]
+Bool.negb_compat [in AAC_tactics.Instances]
+

I

+Internal.assoc [in AAC_tactics.AAC]
+Internal.Binvalue_Proper [in AAC_tactics.AAC]
+Internal.Binvalue_Associative [in AAC_tactics.AAC]
+Internal.Binvalue_Idempotent [in AAC_tactics.AAC]
+Internal.Binvalue_Commutative [in AAC_tactics.AAC]
+Internal.compat_prd_Unit [in AAC_tactics.AAC]
+Internal.compat_sum_unit_Unit [in AAC_tactics.AAC]
+Internal.copy_compat [in AAC_tactics.AAC]
+Internal.eval_aux_compat [in AAC_tactics.AAC]
+Internal.proper [in AAC_tactics.AAC]
+

L

+Lists.aac_List_app_nil_Permutation_Unit [in AAC_tactics.Instances]
+Lists.aac_List_app_Permutation_Comm [in AAC_tactics.Instances]
+Lists.aac_List_app_Permutation_Assoc [in AAC_tactics.Instances]
+Lists.aac_List_app_nil_Unit [in AAC_tactics.Instances]
+Lists.aac_List_app_Assoc [in AAC_tactics.Instances]
+

N

+N.aac_N_le_eq_lift [in AAC_tactics.Instances]
+N.aac_N_gcd_0_Unit [in AAC_tactics.Instances]
+N.aac_N_max_0_Unit [in AAC_tactics.Instances]
+N.aac_N_add_0_Unit [in AAC_tactics.Instances]
+N.aac_N_lcm_1_Unit [in AAC_tactics.Instances]
+N.aac_N_mul_1_Unit [in AAC_tactics.Instances]
+N.aac_N_lcm_Idem [in AAC_tactics.Instances]
+N.aac_N_lcm_Assoc [in AAC_tactics.Instances]
+N.aac_N_lcm_Comm [in AAC_tactics.Instances]
+N.aac_N_gcd_Idem [in AAC_tactics.Instances]
+N.aac_N_gcd_Assoc [in AAC_tactics.Instances]
+N.aac_N_gcd_Comm [in AAC_tactics.Instances]
+N.aac_N_max_Idem [in AAC_tactics.Instances]
+N.aac_N_max_Assoc [in AAC_tactics.Instances]
+N.aac_N_max_Comm [in AAC_tactics.Instances]
+N.aac_N_min_Idem [in AAC_tactics.Instances]
+N.aac_N_min_Assoc [in AAC_tactics.Instances]
+N.aac_N_min_Comm [in AAC_tactics.Instances]
+N.aac_N_mul_Assoc [in AAC_tactics.Instances]
+N.aac_N_mul_Comm [in AAC_tactics.Instances]
+N.aac_N_add_Comm [in AAC_tactics.Instances]
+N.aac_N_add_Assoc [in AAC_tactics.Instances]
+N.N_le_PreOrder [in AAC_tactics.Instances]
+

P

+Peano.aac_Nat_le_eq_lift [in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_0_Unit [in AAC_tactics.Instances]
+Peano.aac_Nat_max_0_Unit [in AAC_tactics.Instances]
+Peano.aac_Nat_add_0_Unit [in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_1_Unit [in AAC_tactics.Instances]
+Peano.aac_Nat_mul_1_Unit [in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Idem [in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Assoc [in AAC_tactics.Instances]
+Peano.aac_Nat_lcm_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Idem [in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Assoc [in AAC_tactics.Instances]
+Peano.aac_Nat_gcd_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_max_Idem [in AAC_tactics.Instances]
+Peano.aac_Nat_max_Assoc [in AAC_tactics.Instances]
+Peano.aac_Nat_max_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_min_Idem [in AAC_tactics.Instances]
+Peano.aac_Nat_min_Assoc [in AAC_tactics.Instances]
+Peano.aac_Nat_min_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_mul_Assoc [in AAC_tactics.Instances]
+Peano.aac_Nat_mul_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_add_Comm [in AAC_tactics.Instances]
+Peano.aac_Nat_add_Assoc [in AAC_tactics.Instances]
+Peano.Nat_le_PreOrder [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_impl_iff_lift [in AAC_tactics.Instances]
+Prop_ops.not_iff_compat [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_True_iff_Unit [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_False_iff_Unit [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Idem [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Comm [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_and_iff_Assoc [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Idem [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Comm [in AAC_tactics.Instances]
+Prop_ops.aac_Prop_or_iff_Assoc [in AAC_tactics.Instances]
+P.aac_Pos_le_eq_lift [in AAC_tactics.Instances]
+P.aac_Pos_max_1_Unit [in AAC_tactics.Instances]
+P.aac_Pos_mul_1_Unit [in AAC_tactics.Instances]
+P.aac_Pos_max_Idem [in AAC_tactics.Instances]
+P.aac_Pos_max_Assoc [in AAC_tactics.Instances]
+P.aac_Pos_max_Comm [in AAC_tactics.Instances]
+P.aac_Pos_min_Idem [in AAC_tactics.Instances]
+P.aac_Pos_min_Assoc [in AAC_tactics.Instances]
+P.aac_Pos_min_Comm [in AAC_tactics.Instances]
+P.aac_Pos_mul_Assoc [in AAC_tactics.Instances]
+P.aac_Pos_mul_Comm [in AAC_tactics.Instances]
+P.aac_Pos_add_Comm [in AAC_tactics.Instances]
+P.aac_Pos_add_Assoc [in AAC_tactics.Instances]
+P.Pos_le_PreOrder [in AAC_tactics.Instances]
+

Q

+Q.aac_Q_Qle_eq_lift [in AAC_tactics.Instances]
+Q.aac_Q_Qplus_0_Qeq_Unit [in AAC_tactics.Instances]
+Q.aac_Q_Qmult_1_Qeq_Unit [in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Idem [in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Assoc [in AAC_tactics.Instances]
+Q.aac_Q_Qmax_Qeq_Comm [in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Idem [in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Assoc [in AAC_tactics.Instances]
+Q.aac_Q_Qmin_Qeq_Comm [in AAC_tactics.Instances]
+Q.aac_Q_Qmult_Qeq_Assoc [in AAC_tactics.Instances]
+Q.aac_Q_Qmult_Qeq_Comm [in AAC_tactics.Instances]
+Q.aac_Q_Qplus_Qeq_Comm [in AAC_tactics.Instances]
+Q.aac_Q_Qplus_Qeq_Assoc [in AAC_tactics.Instances]
+Q.Q_Qle_PreOrder [in AAC_tactics.Instances]
+

R

+Relations.aac_inclusion_same_relation_lift [in AAC_tactics.Instances]
+Relations.aac_compo_eq_same_relation_Unit [in AAC_tactics.Instances]
+Relations.aac_compo_same_relation_Assoc [in AAC_tactics.Instances]
+Relations.aac_inter_top_same_relation_Unit [in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Idem [in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Assoc [in AAC_tactics.Instances]
+Relations.aac_inter_same_relation_Comm [in AAC_tactics.Instances]
+Relations.aac_bot_union_same_relation_Unit [in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Idem [in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Assoc [in AAC_tactics.Instances]
+Relations.aac_union_same_relation_Comm [in AAC_tactics.Instances]
+Relations.clos_refl_trans_same_relation_compat [in AAC_tactics.Instances]
+Relations.clos_refl_trans_incr [in AAC_tactics.Instances]
+Relations.clos_trans_same_relation_compat [in AAC_tactics.Instances]
+Relations.clos_trans_incr [in AAC_tactics.Instances]
+Relations.inclusion_PreOrder [in AAC_tactics.Instances]
+Relations.negr_same_relation_compat [in AAC_tactics.Instances]
+Relations.same_relation_Equivalence [in AAC_tactics.Instances]
+Relations.transp_same_relation_compat [in AAC_tactics.Instances]
+

Z

+Z_add_incr [in AAC_tactics.Caveats]
+Z_add_le_compat [in AAC_tactics.Tutorial]
+Z_opp_ge_le_compat [in AAC_tactics.Tutorial]
+Z.aac_Z_le_eq_lift [in AAC_tactics.Instances]
+Z.aac_Z_add_0_Unit [in AAC_tactics.Instances]
+Z.aac_Z_mul_1_Unit [in AAC_tactics.Instances]
+Z.aac_Z_lcm_Assoc [in AAC_tactics.Instances]
+Z.aac_Z_lcm_Comm [in AAC_tactics.Instances]
+Z.aac_Z_gcd_Assoc [in AAC_tactics.Instances]
+Z.aac_Z_gcd_Comm [in AAC_tactics.Instances]
+Z.aac_Z_max_Idem [in AAC_tactics.Instances]
+Z.aac_Z_max_Assoc [in AAC_tactics.Instances]
+Z.aac_Z_max_Comm [in AAC_tactics.Instances]
+Z.aac_Z_min_Idem [in AAC_tactics.Instances]
+Z.aac_Z_min_Assoc [in AAC_tactics.Instances]
+Z.aac_Z_min_Comm [in AAC_tactics.Instances]
+Z.aac_Z_mul_Assoc [in AAC_tactics.Instances]
+Z.aac_Z_mul_Comm [in AAC_tactics.Instances]
+Z.aac_Z_add_Comm [in AAC_tactics.Instances]
+Z.aac_Z_add_Assoc [in AAC_tactics.Instances]
+Z.Z_le_PreOrder [in AAC_tactics.Instances]
+

+

Abbreviation Index

+

C

+cast [in AAC_tactics.Utils]
+

I

+idx [in AAC_tactics.Utils]
+

L

+lex [in AAC_tactics.Utils]
+

+

Definition Index

+

A

+appne [in AAC_tactics.Utils]
+

C

+compare_weak_spec_sind [in AAC_tactics.Utils]
+compare_weak_spec_ind [in AAC_tactics.Utils]
+

D

+decide_spec_sind [in AAC_tactics.Utils]
+decide_spec_ind [in AAC_tactics.Utils]
+

E

+eq_idx_bool [in AAC_tactics.Utils]
+

F

+fold_map' [in AAC_tactics.Utils]
+fold_map [in AAC_tactics.Utils]
+

I

+insert [in AAC_tactics.Utils]
+Internal.add_to_prd [in AAC_tactics.AAC]
+Internal.add_to_sum [in AAC_tactics.AAC]
+Internal.comp [in AAC_tactics.AAC]
+Internal.compare [in AAC_tactics.AAC]
+Internal.compat_prd_unit_sind [in AAC_tactics.AAC]
+Internal.compat_prd_unit_ind [in AAC_tactics.AAC]
+Internal.compat_sum_unit_sind [in AAC_tactics.AAC]
+Internal.compat_sum_unit_ind [in AAC_tactics.AAC]
+Internal.copy [in AAC_tactics.AAC]
+Internal.copy_mset [in AAC_tactics.AAC]
+Internal.copy' [in AAC_tactics.AAC]
+Internal.discr_sind [in AAC_tactics.AAC]
+Internal.discr_rec [in AAC_tactics.AAC]
+Internal.discr_ind [in AAC_tactics.AAC]
+Internal.discr_rect [in AAC_tactics.AAC]
+Internal.eval [in AAC_tactics.AAC]
+Internal.eval_norm_aux [in AAC_tactics.AAC]
+Internal.eval_norm [in AAC_tactics.AAC]
+Internal.eval_aux [in AAC_tactics.AAC]
+Internal.is_prd_spec_ind_sind [in AAC_tactics.AAC]
+Internal.is_prd_spec_ind_ind [in AAC_tactics.AAC]
+Internal.is_sum_spec_ind_sind [in AAC_tactics.AAC]
+Internal.is_sum_spec_ind_ind [in AAC_tactics.AAC]
+Internal.is_prd [in AAC_tactics.AAC]
+Internal.is_sum [in AAC_tactics.AAC]
+Internal.is_idempotent [in AAC_tactics.AAC]
+Internal.is_commutative [in AAC_tactics.AAC]
+Internal.is_unit_of [in AAC_tactics.AAC]
+Internal.m_sind [in AAC_tactics.AAC]
+Internal.m_rec [in AAC_tactics.AAC]
+Internal.m_ind [in AAC_tactics.AAC]
+Internal.m_rect [in AAC_tactics.AAC]
+Internal.norm [in AAC_tactics.AAC]
+Internal.norm_msets [in AAC_tactics.AAC]
+Internal.norm_lists [in AAC_tactics.AAC]
+Internal.norm_lists_ [in AAC_tactics.AAC]
+Internal.norm_msets_ [in AAC_tactics.AAC]
+Internal.prd' [in AAC_tactics.AAC]
+Internal.return_prd [in AAC_tactics.AAC]
+Internal.return_sum [in AAC_tactics.AAC]
+Internal.run_msets [in AAC_tactics.AAC]
+Internal.run_list [in AAC_tactics.AAC]
+Internal.sum' [in AAC_tactics.AAC]
+Internal.Sym.null [in AAC_tactics.AAC]
+Internal.Sym.rel_of [in AAC_tactics.AAC]
+Internal.Sym.type_of [in AAC_tactics.AAC]
+Internal.tcompare_weak_spec [in AAC_tactics.AAC]
+Internal.T_sind [in AAC_tactics.AAC]
+Internal.T_rec [in AAC_tactics.AAC]
+Internal.T_ind [in AAC_tactics.AAC]
+Internal.T_rect [in AAC_tactics.AAC]
+Internal.vcompare [in AAC_tactics.AAC]
+Internal.vcompare_reflect_eqdep [in AAC_tactics.AAC]
+Internal.vnorm [in AAC_tactics.AAC]
+Internal.vT_sind [in AAC_tactics.AAC]
+Internal.vT_rec [in AAC_tactics.AAC]
+Internal.vT_ind [in AAC_tactics.AAC]
+Internal.vT_rect [in AAC_tactics.AAC]
+

L

+list_compare [in AAC_tactics.Utils]
+

M

+merge_map [in AAC_tactics.Utils]
+merge_msets [in AAC_tactics.Utils]
+mset [in AAC_tactics.Utils]
+mset_compare [in AAC_tactics.Utils]
+

N

+nelist_map [in AAC_tactics.Utils]
+nelist_sind [in AAC_tactics.Utils]
+nelist_rec [in AAC_tactics.Utils]
+nelist_ind [in AAC_tactics.Utils]
+nelist_rect [in AAC_tactics.Utils]
+

R

+reduce_mset [in AAC_tactics.Utils]
+reflect_eqdep [in AAC_tactics.Utils]
+Relations.bot [in AAC_tactics.Instances]
+Relations.compo [in AAC_tactics.Instances]
+Relations.inter [in AAC_tactics.Instances]
+Relations.negr [in AAC_tactics.Instances]
+Relations.top [in AAC_tactics.Instances]
+

S

+sigma [in AAC_tactics.AAC]
+sigma_empty [in AAC_tactics.AAC]
+sigma_add [in AAC_tactics.AAC]
+sigma_get [in AAC_tactics.AAC]
+

U

+Unnamed_thm5 [in AAC_tactics.Caveats]
+Unnamed_thm5 [in AAC_tactics.Caveats]
+Unnamed_thm5 [in AAC_tactics.Caveats]
+Unnamed_thm5 [in AAC_tactics.Caveats]
+Unnamed_thm4 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm3 [in AAC_tactics.Caveats]
+Unnamed_thm2 [in AAC_tactics.Caveats]
+Unnamed_thm1 [in AAC_tactics.Caveats]
+Unnamed_thm1 [in AAC_tactics.Caveats]
+Unnamed_thm0 [in AAC_tactics.Caveats]
+Unnamed_thm [in AAC_tactics.Caveats]
+Unnamed_thm29 [in AAC_tactics.Tutorial]
+Unnamed_thm28 [in AAC_tactics.Tutorial]
+Unnamed_thm27 [in AAC_tactics.Tutorial]
+Unnamed_thm26 [in AAC_tactics.Tutorial]
+Unnamed_thm25 [in AAC_tactics.Tutorial]
+Unnamed_thm24 [in AAC_tactics.Tutorial]
+Unnamed_thm23 [in AAC_tactics.Tutorial]
+Unnamed_thm22 [in AAC_tactics.Tutorial]
+Unnamed_thm22 [in AAC_tactics.Tutorial]
+Unnamed_thm21 [in AAC_tactics.Tutorial]
+Unnamed_thm20 [in AAC_tactics.Tutorial]
+Unnamed_thm20 [in AAC_tactics.Tutorial]
+Unnamed_thm19 [in AAC_tactics.Tutorial]
+Unnamed_thm18 [in AAC_tactics.Tutorial]
+Unnamed_thm17 [in AAC_tactics.Tutorial]
+Unnamed_thm16 [in AAC_tactics.Tutorial]
+Unnamed_thm15 [in AAC_tactics.Tutorial]
+Unnamed_thm14 [in AAC_tactics.Tutorial]
+Unnamed_thm13 [in AAC_tactics.Tutorial]
+Unnamed_thm12 [in AAC_tactics.Tutorial]
+Unnamed_thm11 [in AAC_tactics.Tutorial]
+Unnamed_thm11 [in AAC_tactics.Tutorial]
+Unnamed_thm10 [in AAC_tactics.Tutorial]
+Unnamed_thm9 [in AAC_tactics.Tutorial]
+Unnamed_thm8 [in AAC_tactics.Tutorial]
+Unnamed_thm7 [in AAC_tactics.Tutorial]
+Unnamed_thm6 [in AAC_tactics.Tutorial]
+Unnamed_thm5 [in AAC_tactics.Tutorial]
+Unnamed_thm4 [in AAC_tactics.Tutorial]
+Unnamed_thm3 [in AAC_tactics.Tutorial]
+Unnamed_thm2 [in AAC_tactics.Tutorial]
+Unnamed_thm1 [in AAC_tactics.Tutorial]
+Unnamed_thm0 [in AAC_tactics.Tutorial]
+Unnamed_thm [in AAC_tactics.Tutorial]
+

+

Record Index

+

A

+AAC_lift [in AAC_tactics.AAC]
+Associative [in AAC_tactics.AAC]
+

C

+Commutative [in AAC_tactics.AAC]
+

I

+Idempotent [in AAC_tactics.AAC]
+Internal.Bin.pack [in AAC_tactics.AAC]
+Internal.Sym.pack [in AAC_tactics.AAC]
+Internal.unit_pack [in AAC_tactics.AAC]
+Internal.unit_of [in AAC_tactics.AAC]
+

U

+Unit [in AAC_tactics.AAC]
+

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
Global IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(685 entries)
Notation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Module IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(13 entries)
Variable IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(176 entries)
Library IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(6 entries)
Lemma IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(55 entries)
Constructor IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(31 entries)
Inductive IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(14 entries)
Projection IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(19 entries)
Section IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(39 entries)
Instance IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(161 entries)
Abbreviation IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(3 entries)
Definition IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(140 entries)
Record IndexABCDEFGHIJKLMNOPQRSTUVWXYZ_other(9 entries)
+
+ +
+ + + diff --git a/docs/coqdoc/toc.html b/docs/coqdoc/toc.html new file mode 100644 index 0000000..85db8c6 --- /dev/null +++ b/docs/coqdoc/toc.html @@ -0,0 +1,223 @@ + + + + + + + + + + + + + +
+
+
+ +
+
+ +
+ + + diff --git a/docs/ocamldoc/Aac_rewrite.html b/docs/ocamldoc/Aac_rewrite.html new file mode 100644 index 0000000..b15c33f --- /dev/null +++ b/docs/ocamldoc/Aac_rewrite.html @@ -0,0 +1,39 @@ + + + + + + + + + + + + + + + + + + + +Aac_rewrite + + +
Previous + Up + 
+

Module Aac_rewrite

+ +
module Aac_rewrite: sig .. end
+
+

aac_rewrite -- rewriting modulo A or AC

+
+
+
+ +
val aac_reflexivity : unit Proofview.tactic
+
val aac_normalise : unit Proofview.tactic
+
val aac_rewrite : args:(string * int) list ->
       ?abort:bool ->
       ?l2r:bool ->
       ?show:bool ->
       ?strict:bool -> ?extra:EConstr.t -> EConstr.constr -> unit Proofview.tactic
+
val add : string -> 'a -> (string * 'a) list -> (string * 'a) list
+
val pr_aac_args : 'a -> 'b -> 'c -> (string * int) list -> Pp.t
diff --git a/docs/ocamldoc/Coq.Classes.html b/docs/ocamldoc/Coq.Classes.html new file mode 100644 index 0000000..7b9fe16 --- /dev/null +++ b/docs/ocamldoc/Coq.Classes.html @@ -0,0 +1,40 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Classes + + +
Previous + Up + Next +
+

Module Coq.Classes

+ +
module Classes: sig .. end
+
+

Coq typeclasses

+
+
+
+ +
val mk_morphism : EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+
val mk_equivalence : EConstr.constr -> EConstr.constr -> EConstr.constr
+
val mk_reflexive : EConstr.constr -> EConstr.constr -> EConstr.constr
+
val mk_transitive : EConstr.constr -> EConstr.constr -> EConstr.constr
diff --git a/docs/ocamldoc/Coq.Equivalence.html b/docs/ocamldoc/Coq.Equivalence.html new file mode 100644 index 0000000..84900ce --- /dev/null +++ b/docs/ocamldoc/Coq.Equivalence.html @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Equivalence + + +
Previous + Up + Next +
+

Module Coq.Equivalence

+ +
module Equivalence: sig .. end
+ +
type t = {
+ + + + + + + + + + + + + + +
+   +carrier : EConstr.constr;
+   +eq : EConstr.constr;
+   +equivalence : EConstr.constr;
+} + + + +
val make : EConstr.constr -> EConstr.constr -> EConstr.constr -> t
+
val infer : Environ.env ->
       Evd.evar_map ->
       EConstr.constr -> EConstr.constr -> t * Evd.evar_map
+
val from_relation : Environ.env ->
       Evd.evar_map -> Coq.Relation.t -> t * Evd.evar_map
+
val to_relation : t -> Coq.Relation.t
+
val split : t -> EConstr.constr * EConstr.constr * EConstr.constr
diff --git a/docs/ocamldoc/Coq.List.html b/docs/ocamldoc/Coq.List.html new file mode 100644 index 0000000..ce6fab9 --- /dev/null +++ b/docs/ocamldoc/Coq.List.html @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + + + + + + +Coq.List + + +
 Up + Next +
+

Module Coq.List

+ +
module List: sig .. end
+
+

Coq lists

+
+
+
+ +
val of_list : EConstr.constr -> EConstr.constr list -> EConstr.constr
+
+

of_list ty l

+
+
+ +
val type_of_list : EConstr.constr -> EConstr.constr
+
+

type_of_list ty

+
+
+ diff --git a/docs/ocamldoc/Coq.Nat.html b/docs/ocamldoc/Coq.Nat.html new file mode 100644 index 0000000..04b6ff9 --- /dev/null +++ b/docs/ocamldoc/Coq.Nat.html @@ -0,0 +1,38 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Nat + + +
Previous + Up + Next +
+

Module Coq.Nat

+ +
module Nat: sig .. end
+
+

Coq unary numbers (peano)

+
+
+
+ +
val typ : Coq.lazy_ref
+
val of_int : int -> Constr.constr
diff --git a/docs/ocamldoc/Coq.Option.html b/docs/ocamldoc/Coq.Option.html new file mode 100644 index 0000000..293ec43 --- /dev/null +++ b/docs/ocamldoc/Coq.Option.html @@ -0,0 +1,35 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Option + + +
Previous + Up + Next +
+

Module Coq.Option

+ +
module Option: sig .. end
+ +
val typ : Coq.lazy_ref
+
val some : EConstr.constr -> EConstr.constr -> EConstr.constr
+
val none : EConstr.constr -> EConstr.constr
+
val of_option : EConstr.constr -> EConstr.constr option -> EConstr.constr
diff --git a/docs/ocamldoc/Coq.Pair.html b/docs/ocamldoc/Coq.Pair.html new file mode 100644 index 0000000..4c02a96 --- /dev/null +++ b/docs/ocamldoc/Coq.Pair.html @@ -0,0 +1,39 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Pair + + +
Previous + Up + Next +
+

Module Coq.Pair

+ +
module Pair: sig .. end
+
+

Coq pairs

+
+
+
+ +
val typ : Coq.lazy_ref
+
val pair : Coq.lazy_ref
+
val of_pair : EConstr.constr ->
       EConstr.constr -> EConstr.constr * EConstr.constr -> EConstr.constr
diff --git a/docs/ocamldoc/Coq.Pos.html b/docs/ocamldoc/Coq.Pos.html new file mode 100644 index 0000000..635f0b1 --- /dev/null +++ b/docs/ocamldoc/Coq.Pos.html @@ -0,0 +1,38 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Pos + + +
Previous + Up + Next +
+

Module Coq.Pos

+ +
module Pos: sig .. end
+
+

Coq positive numbers (pos)

+
+
+
+ +
val typ : Coq.lazy_ref
+
val of_int : int -> EConstr.constr
diff --git a/docs/ocamldoc/Coq.Relation.html b/docs/ocamldoc/Coq.Relation.html new file mode 100644 index 0000000..bd18049 --- /dev/null +++ b/docs/ocamldoc/Coq.Relation.html @@ -0,0 +1,52 @@ + + + + + + + + + + + + + + + + + + + + +Coq.Relation + + +
Previous + Up + Next +
+

Module Coq.Relation

+ +
module Relation: sig .. end
+ +
type t = {
+ + + + + + + + + +
+   +carrier : EConstr.constr;
+   +r : EConstr.constr;
+} + + + +
val make : EConstr.constr -> EConstr.constr -> t
+
val split : t -> EConstr.constr * EConstr.constr
diff --git a/docs/ocamldoc/Coq.Rewrite.html b/docs/ocamldoc/Coq.Rewrite.html new file mode 100644 index 0000000..b37d387 --- /dev/null +++ b/docs/ocamldoc/Coq.Rewrite.html @@ -0,0 +1,181 @@ + + + + + + + + + + + + + + + + + + + + + +Coq.Rewrite + + +
Previous + Up + 
+

Module Coq.Rewrite

+ +
module Rewrite: sig .. end
+

The rewriting tactics used in aac_rewrite, build as handlers of the usual setoid_rewrite

+

Datatypes

+
type hypinfo = {
+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
+   +hyp : EConstr.constr;(*
+
+

the actual constr corresponding to the hypothese

+
+
+		*)
+   +hyptype : EConstr.constr;(*
+
+

the type of the hypothesis

+
+
+		*)
+   +context : EConstr.rel_context;(*
+
+

the quantifications of the hypothese

+
+
+		*)
+   +body : EConstr.constr;(*
+
+

the body of the hypothese

+
+
+		*)
+   +rel : Coq.Relation.t;(*
+
+

the relation

+
+
+		*)
+   +left : EConstr.constr;(*
+
+

left hand side

+
+
+		*)
+   +right : EConstr.constr;(*
+
+

right hand side

+
+
+		*)
+   +l2r : bool;(*
+
+

rewriting from left to right

+
+
+		*)
+} + +
+
+

We keep some informations about the hypothesis, with an (informal) + invariant:

+ +
+
+ + +
val get_hypinfo : Environ.env ->
       Evd.evar_map ->
       ?check_type:(EConstr.types -> bool) ->
       EConstr.constr -> l2r:bool -> hypinfo
+
+

get_hypinfo ?check_type H l2r analyse the hypothesis H, and + build the related hypinfo. Moreover, an optionnal + function can be provided to check the type of every free + variable of the body of the hypothesis.

+
+
+

Rewriting with bindings

+

The problem : Given a term to rewrite of type H :forall xn ... x1, + t, we have to instanciate the subset of xi of type + carrier. subst : (int * constr) is the mapping the De Bruijn + indices in t to the constrs. We need to handle the rest of the + indexes. Two ways :

+ + +

Both these terms have the same type.

+ +
val build : hypinfo -> (int * EConstr.constr) list -> EConstr.constr
+
+

build the constr to rewrite, with lambda abstractions

+
+
+

build the constr to rewrite, in CPS style, with evars

+ +
val rewrite : ?abort:bool ->
       hypinfo -> (int * EConstr.constr) list -> unit Proofview.tactic
+
+

rewrite ?abort hypinfo subst builds the rewriting tactic + associated with the given subst and hypinfo. +If abort is set to true, we build + tclIDTAC instead.

+
+
+ diff --git a/docs/ocamldoc/Coq.html b/docs/ocamldoc/Coq.html new file mode 100644 index 0000000..80c4913 --- /dev/null +++ b/docs/ocamldoc/Coq.html @@ -0,0 +1,135 @@ + + + + + + + + + + + + + + + + + + + + + + + + + +Coq + + +
 Up + Next +
+

Module Coq

+ +
module Coq: sig .. end
+
+

Interface with Coq where we define some handlers for Coq's API, + and we import several definitions from Coq's standard library.

+ +

This general purpose library could be reused by other plugins.

+ +

Some salient points:

+ +
+
+
+ +
type tactic = unit Proofview.tactic
+ + +
type lazy_ref = Names.GlobRef.t Stdlib.Lazy.t
+
+
+

Getting Coq terms from the environment

+
+ + +
val get_fresh : lazy_ref -> Constr.constr
+
val get_efresh : lazy_ref -> EConstr.constr
+
val find_global : string -> lazy_ref

General purpose functions

+
val cps_resolve_one_typeclass : ?error:Pp.t -> EConstr.types -> (EConstr.constr -> tactic) -> tactic
+
val evar_binary : Environ.env ->
       Evd.evar_map -> EConstr.constr -> Evd.evar_map * EConstr.constr
+
val evar_relation : Environ.env ->
       Evd.evar_map -> EConstr.constr -> Evd.evar_map * EConstr.constr
+
val mk_letin : string -> EConstr.constr -> EConstr.constr Proofview.tactic
+
val tclRETYPE : EConstr.constr -> unit Proofview.tactic
+
val decomp_term : Evd.evar_map ->
       EConstr.constr ->
       (EConstr.constr, EConstr.types, EConstr.ESorts.t, EConstr.EInstance.t,
        EConstr.ERelevance.t)
       Constr.kind_of_term

Bindings with Coq' Standard Library

+
module List: sig .. end
+

Coq lists

+ +
+ +
module Pair: sig .. end
+

Coq pairs

+ +
+ +
module Option: sig .. end
+
module Pos: sig .. end
+

Coq positive numbers (pos)

+ +
+ +
module Nat: sig .. end
+

Coq unary numbers (peano)

+ +
+ +
module Classes: sig .. end
+

Coq typeclasses

+ +
+ +
module Relation: sig .. end
+
module Equivalence: sig .. end
+
val match_as_equation : ?context:EConstr.rel_context ->
       Environ.env ->
       Evd.evar_map ->
       EConstr.constr -> (EConstr.constr * EConstr.constr * Relation.t) option
+
+

match_as_equation ?context env sigma c try to decompose c as a + relation applied to two terms.

+
+
+

Some tacticials

+
val tclTIME : string -> tactic -> tactic
+
+

time the execution of a tactic

+
+
+ +
val tclDEBUG : string -> unit Proofview.tactic
+
+

emit debug messages to see which tactics are failing

+
+
+ +
val tclPRINT : unit Proofview.tactic
+
+

print the current goal

+
+
+

Error related mechanisms

+
val anomaly : string -> 'a
+
val user_error : Pp.t -> 'a
+
val warning : string -> unit

Helpers

+
val show_proof : Declare.Proof.t -> unit
+
+

print the current proof term

+
+
+

Rewriting tactics used in aac_rewrite

+
module Rewrite: sig .. end
diff --git a/docs/ocamldoc/Helper.CONTROL.html b/docs/ocamldoc/Helper.CONTROL.html new file mode 100644 index 0000000..55fdf17 --- /dev/null +++ b/docs/ocamldoc/Helper.CONTROL.html @@ -0,0 +1,30 @@ + + + + + + + + + + + + + + + + + + +Helper.CONTROL + + +
 Up + 
+

Module type Helper.CONTROL

+ +
module type CONTROL = sig .. end
+ +
val debug : bool
+
val time : bool
+
val printing : bool
diff --git a/docs/ocamldoc/Helper.Debug.html b/docs/ocamldoc/Helper.Debug.html new file mode 100644 index 0000000..921e9c5 --- /dev/null +++ b/docs/ocamldoc/Helper.Debug.html @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + +Helper.Debug + + +
 Up + 
+

Functor Helper.Debug

+ +
module Debug: 
functor (X : CONTROL) -> sig .. end
+ + + + +
Parameters: + + + + +
+X:CONTROL +
+
+
+ +
val debug : string -> unit
+
+

debug prints the string and end it with a newline

+
+
+ +
val debug_exception : string -> exn -> unit
+
val time : ('a -> 'b) ->
       'a -> (float -> unit, Stdlib.out_channel, unit) Stdlib.format -> 'b
+
+

time computes the time spent in a function, and then + print it using the given format

+
+
+ +
val pr_constr : Environ.env -> Evd.evar_map -> string -> EConstr.constr -> unit
+
+

pr_constr print a Coq constructor, that can be labelled + by a string

+
+
+ diff --git a/docs/ocamldoc/Helper.html b/docs/ocamldoc/Helper.html new file mode 100644 index 0000000..4c4ac84 --- /dev/null +++ b/docs/ocamldoc/Helper.html @@ -0,0 +1,38 @@ + + + + + + + + + + + + + + + + + + + + +Helper + + +
Previous + Up + Next +
+

Module Helper

+ +
module Helper: sig .. end
+
+

Debugging functions, that can be triggered on a per-file base.

+
+
+
+ +
module type CONTROL = sig .. end
+
module Debug: 
functor (X : CONTROL) -> sig .. end
diff --git a/docs/ocamldoc/Matcher.Subst.html b/docs/ocamldoc/Matcher.Subst.html new file mode 100644 index 0000000..d780c1a --- /dev/null +++ b/docs/ocamldoc/Matcher.Subst.html @@ -0,0 +1,47 @@ + + + + + + + + + + + + + + + + + + + +Matcher.Subst + + +
Previous + Up + 
+

Module Matcher.Subst

+ +
module Subst: sig .. end
+
+

Substitutions (or environments)

+ +

The module Matcher.Subst contains infrastructure to deal with + substitutions, i.e., functions from variables to terms. Only a + restricted subsets of these functions need to be exported.

+ +

As expected, a particular substitution can be used to + instantiate a pattern.

+
+
+
+ +
type t
+ + +
val sprint : t -> string
+
val instantiate : t -> Matcher.Terms.t -> Matcher.Terms.t
+
val to_list : t -> (Matcher.var * Matcher.Terms.t) list
diff --git a/docs/ocamldoc/Matcher.Terms.html b/docs/ocamldoc/Matcher.Terms.html new file mode 100644 index 0000000..3d6e003 --- /dev/null +++ b/docs/ocamldoc/Matcher.Terms.html @@ -0,0 +1,135 @@ + + + + + + + + + + + + + + + + + + + + + + + + +Matcher.Terms + + +
 Up + Next +
+

Module Matcher.Terms

+ +
module Terms: sig .. end
+
+

Representations of expressions

+ +

The module Matcher.Terms defines two different types for expressions.

+ +
+
+
+

Abstract syntax tree of terms and patterns

+

We represent both terms and patterns using the following datatype.

+ +

Values of type symbol are used to index symbols. Typically, + given two associative operations (^) and ( * ), and two + morphisms f and g, the term f (a^b) (a*g b) is represented + by the following value + Sym(0,[| Dot(1, Sym(2,[||]), Sym(3,[||])); + Dot(4, Sym(2,[||]), Sym(5,[|Sym(3,[||])|])) |]) + where the implicit symbol environment associates + f to 0, (^) to 1, a to 2, b to 3, ( * ) to 4, and g to 5,

+ +

Accordingly, the following value, that contains "variables" + Sym(0,[| Dot(1, Var x, Unit (1); Dot(4, Var x, + Sym(5,[|Sym(3,[||])|])) |]) represents the pattern forall x, f + (x^1) (x*g b). The relationship between 1 and ( * ) is only + mentionned in the units table.

+ +
type t =
+ + + + + + + + + + + + + + + + + + + + + + + + +
+| +Dot of (Matcher.symbol * t * t)
+| +Plus of (Matcher.symbol * t * t)
+| +Sym of (Matcher.symbol * t array)
+| +Var of Matcher.var
+| +Unit of Matcher.symbol
+ + + +
val equal_aac : Matcher.units -> t -> t -> bool
+
+

Test for equality of terms modulo AACU (relies on the following + canonical representation of terms). Note than two different + units of a same operator are not considered equal.

+
+
+ +
val map_syms : (Matcher.symbol -> Matcher.symbol) -> t -> t

Normalised terms (canonical representation)

+

A term in normal form is the canonical representative of the + equivalence class of all the terms that are equal modulo AACU. + This representation is only used internally; it is exported here + for the sake of completeness

+ +
type nf_term
+ +

Comparisons

+
val nf_term_compare : nf_term -> nf_term -> int
+
val nf_equal : nf_term -> nf_term -> bool

Printing function

+
val sprint_nf_term : nf_term -> string

Conversion functions

+
val term_of_t : Matcher.units -> t -> nf_term
+
+

we have the following property: a and b are equal modulo AACU + iif nf_equal (term_of_t a) (term_of_t b) = true

+
+
+ +
val t_of_term : nf_term -> t
diff --git a/docs/ocamldoc/Matcher.html b/docs/ocamldoc/Matcher.html new file mode 100644 index 0000000..839af8d --- /dev/null +++ b/docs/ocamldoc/Matcher.html @@ -0,0 +1,158 @@ + + + + + + + + + + + + + + + + + + + + + + +Matcher + + +
Previous + Up + Next +
+

Module Matcher

+ +
module Matcher: sig .. end
+
+

Standalone module containing the algorithm for matching modulo + associativity and associativity and commutativity + (AAC). Additionnaly, some A or AC operators can have units (U).

+ +

This module could be reused outside of the Coq plugin.

+ +

Matching a pattern p against a term t modulo AACU boils down + to finding a substitution env such that the pattern p + instantiated with env is equal to t modulo AACU.

+ +

We proceed by structural decomposition of the pattern, trying all + possible non-deterministic splittings of the subject, when needed. The + function Matcher.matcher is limited to top-level matching, that is, the + subject must make a perfect match against the pattern (x+x does + not match a+a+b ).

+ +

We use a search monad Search_monad to perform non-deterministic + choices in an almost transparent way.

+ +

We also provide a function Matcher.subterm for finding a match that is + a subterm of the subject modulo AACU. In particular, this function + gives a solution to the aforementioned case (x+x against + a+b+a).

+ +

On a slightly more involved level :

+ + +
+
+
+

Utility functions

+
type symbol = int
+ + +
type var = int
+ +

Relationship between units and operators. This is a sparse + representation of a matrix of couples (op,unit) where op is + the index of the operation, and unit the index of the relevant + unit. We make the assumption that any operation has 0 or 1 unit, + and that operations can share a unit).

+ +
type units = (symbol * symbol) list
+ + +
type ext_units = {
+ + + + + + + + + +
+   +unit_for : units;
+   +is_ac : (symbol * bool) list;
+} + + + +
type 'a mset = ('a * int) list
+
+
+

The arguments of sums (or AC operators) are represented using finite multisets. + (Typically, a+b+a corresponds to 2.a+b, i.e. Sum[a,2;b,1])

+
+
+ + +
val linear : 'a mset -> 'a list
+
+

linear expands a multiset into a simple list

+
+
+ +
module Terms: sig .. end
+

Representations of expressions

+ +
+ +
module Subst: sig .. end
+

Substitutions (or environments)

+ +
+

Main functions exported by this module

+
val matcher : ?strict:bool ->
       ext_units ->
       Terms.t -> Terms.t -> Subst.t Search_monad.m
+
+

matcher p t computes the set of solutions to the given top-level + matching problem (p is the pattern, t is the term). If the + strict flag is set, solutions where units are used to + instantiate some variables are excluded, unless this unit appears + directly under a function symbol (e.g., f(x) still matches f(1), + while x+x+y does not match a+b+c, since this would require to + assign 1 to x).

+
+
+ +
val subterm : ?strict:bool ->
       ext_units ->
       Terms.t ->
       Terms.t ->
       (int * Terms.t * Subst.t Search_monad.m) Search_monad.m
+
+

subterm p t computes a set of solutions to the given + subterm-matching problem.

+ +

Return a collection of possible solutions (each with the + associated depth, the context, and the solutions of the matching + problem). The context is actually a Matcher.Terms.t where the variables + are yet to be instantiated by one of the associated substitutions

+
+
+ diff --git a/docs/ocamldoc/Print.html b/docs/ocamldoc/Print.html new file mode 100644 index 0000000..407263a --- /dev/null +++ b/docs/ocamldoc/Print.html @@ -0,0 +1,46 @@ + + + + + + + + + + + + + + + + + + + + +Print + + +
Previous + Up + Next +
+

Module Print

+ +
module Print: sig .. end
+
+

Pretty printing functions we use for the aac_instances + tactic.

+
+
+
+ +
val print : Coq.Relation.t ->
       Theory.Trans.ir ->
       (int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m ->
       EConstr.rel_context -> unit Proofview.tactic
+
+

The main printing function. Print.print uses the rel-context + to rename the variables, and rebuilds raw Coq terms (for the given + context, and the terms in the environment). In order to do so, it + requires the information gathered by the Theory.Trans module.

+
+
+ diff --git a/docs/ocamldoc/Search_monad.html b/docs/ocamldoc/Search_monad.html new file mode 100644 index 0000000..501938f --- /dev/null +++ b/docs/ocamldoc/Search_monad.html @@ -0,0 +1,83 @@ + + + + + + + + + + + + + + + + + + + + + + +Search_monad + + +
Previous + Up + Next +
+

Module Search_monad

+ +
module Search_monad: sig .. end
+
+

Search monad that allows to express non-deterministic algorithms + in a legible maner, or programs that solve combinatorial problems.

+
+ +
+
+ +
type 'a m
+
+
+

A data type that represent a collection of 'a

+
+
+ +

Monadic operations

+
val (>>) : 'a m -> ('a -> 'b m) -> 'b m
+
+

bind and return

+
+
+ +
val return : 'a -> 'a m
+
val (>>|) : 'a m -> 'a m -> 'a m
+
+

non-deterministic choice

+
+
+ +
val fail : unit -> 'a m
+
+

failure

+
+
+ +
val fold : ('a -> 'b -> 'b) -> 'a m -> 'b -> 'b
+
+

folding through the collection

+
+
+

Derived facilities

+
val sprint : ('a -> string) -> 'a m -> string
+
val count : 'a m -> int
+
val choose : 'a m -> 'a option
+
val to_list : 'a m -> 'a list
+
val sort : ('a -> 'a -> int) -> 'a m -> 'a m
+
val is_empty : 'a m -> bool
+
val filter : ('a -> bool) -> 'a m -> 'a m
diff --git a/docs/ocamldoc/Theory.Sigma.html b/docs/ocamldoc/Theory.Sigma.html new file mode 100644 index 0000000..713b0e4 --- /dev/null +++ b/docs/ocamldoc/Theory.Sigma.html @@ -0,0 +1,58 @@ + + + + + + + + + + + + + + + + + + + +Theory.Sigma + + +
 Up + Next +
+

Module Theory.Sigma

+ +
module Sigma: sig .. end
+
+

Environments

+
+
+
+ +
val add : EConstr.constr ->
       EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+
+

add ty n x map adds the value x of type ty with key n in map

+
+
+ +
val empty : EConstr.constr -> EConstr.constr
+
+

empty ty create an empty map of type ty

+
+
+ +
val of_list : EConstr.constr ->
       EConstr.constr -> (int * EConstr.constr) list -> EConstr.constr
+
+

of_list ty null l translates an OCaml association list into a Coq one

+
+
+ +
val to_fun : EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+
+

to_fun ty null map converts a Coq association list into a Coq function (with default value null)

+
+
+ diff --git a/docs/ocamldoc/Theory.Stubs.html b/docs/ocamldoc/Theory.Stubs.html new file mode 100644 index 0000000..b716d16 --- /dev/null +++ b/docs/ocamldoc/Theory.Stubs.html @@ -0,0 +1,57 @@ + + + + + + + + + + + + + + + + + + + + +Theory.Stubs + + +
Previous + Up + Next +
+

Module Theory.Stubs

+ +
module Stubs: sig .. end
+
+

We need to export some Coq stubs out of this module. They are used + by the main tactic, see Rewrite

+
+
+
+ +
val lift : Coq.lazy_ref
+
val lift_proj_equivalence : Coq.lazy_ref
+
val lift_transitivity_left : Coq.lazy_ref
+
val lift_transitivity_right : Coq.lazy_ref
+
val lift_reflexivity : Coq.lazy_ref
+
val eval : Coq.lazy_ref
+
+

The evaluation fonction, used to convert a reified coq term to a + raw coq term

+
+
+ +
val decide_thm : Coq.lazy_ref
+
+

The main lemma of our theory, that is + compare (norm u) (norm v) = Eq -> eval u == eval v

+
+
+ +
val lift_normalise_thm : Coq.lazy_ref
diff --git a/docs/ocamldoc/Theory.Sym.html b/docs/ocamldoc/Theory.Sym.html new file mode 100644 index 0000000..eb2f541 --- /dev/null +++ b/docs/ocamldoc/Theory.Sym.html @@ -0,0 +1,80 @@ + + + + + + + + + + + + + + + + + + + + +Theory.Sym + + +
Previous + Up + Next +
+

Module Theory.Sym

+ +
module Sym: sig .. end
+
+

Dynamically typed morphisms

+
+
+
+ +
type pack = {
+ + + + + + + + + + + + + + +
+   +ar : Constr.t;
+   +value : Constr.t;
+   +morph : Constr.t;
+} + +
+
+

mimics the Coq record Sym.pack

+
+
+ + +
val typ : Coq.lazy_ref
+
val mk_pack : Coq.Relation.t -> pack -> EConstr.constr
+
+

mk_pack rlt (ar,value,morph)

+
+
+ +
val null : Coq.Relation.t -> EConstr.constr
+
+

null builds a dummy (identity) symbol, given an Coq.Relation.t

+
+
+ diff --git a/docs/ocamldoc/Theory.Trans.html b/docs/ocamldoc/Theory.Trans.html new file mode 100644 index 0000000..165a5ec --- /dev/null +++ b/docs/ocamldoc/Theory.Trans.html @@ -0,0 +1,157 @@ + + + + + + + + + + + + + + + + + + + + + + + +Theory.Trans + + +
Previous + Up + 
+

Module Theory.Trans

+ +
module Trans: sig .. end
+
+

Tranlations between Coq and OCaml

+
+
+
+

This module provides facilities to interpret a term with + arbitrary operators as an instance of an abstract syntax tree + Matcher.Terms.t.

+ +

For each Coq application f x_1 ... x_n, this amounts to + deciding whether one of the partial applications f x_1 + ... x_i, i<=n is a proper morphism, whether the partial + application with i=n-2 yields an A or AC binary operator, and + whether the whole term is the unit for some A or AC operator. We + use typeclass resolution to test each of these possibilities.

+ +

Note that there are ambiguous terms:

+ +

To achieve this reification, one need to record informations + about the collected operators (symbols, binary operators, + units). We use the following imperative internal data-structure to + this end.

+ +
type envs
+ + +
val empty_envs : unit -> envs

Reification: from Coq terms to AST Matcher.Terms.t

t_of_constr goal rlt envs (left,right) builds the abstract + syntax tree of the terms left and right. We rely on the goal + to perform typeclasses resolutions to find morphisms compatible + with the relation rlt. Doing so, it modifies the reification + environment envs. Moreover, we need to create fresh + evars; this is why we give back the goal, accordingly + updated.

+ +
val t_of_constr : Environ.env ->
       Evd.evar_map ->
       Coq.Relation.t ->
       envs ->
       EConstr.constr * EConstr.constr ->
       Matcher.Terms.t * Matcher.Terms.t * Evd.evar_map
+
val add_symbol : Environ.env ->
       Evd.evar_map ->
       Coq.Relation.t -> envs -> Constr.t -> Evd.evar_map
+
+

add_symbol adds a given binary symbol to the environment of + known stuff.

+
+
+

Reconstruction: from AST back to Coq terms

+

The next functions allow one to map OCaml abstract syntax trees + to Coq terms. We need two functions to rebuild different kind of + terms: first, raw terms, like the one encountered by + Theory.Trans.t_of_constr; second, reified Coq terms, that are required for + the reflexive decision procedure.

+ +
type ir
+ + +
val ir_of_envs : Environ.env ->
       Evd.evar_map ->
       Coq.Relation.t -> envs -> Evd.evar_map * ir
+
val ir_to_units : ir -> Matcher.ext_units

Building raw, natural, terms

+
val raw_constr_of_t : ir ->
       Coq.Relation.t -> EConstr.rel_context -> Matcher.Terms.t -> EConstr.constr
+
+

raw_constr_of_t rebuilds a term in the raw representation, and + reconstruct the named products on top of it. In particular, this + allow us to print the context put around the left (or right) + hand side of a pattern.

+
+
+

Building reified terms

The reification environments, as Coq constrs

+ +
type sigmas = {
+ + + + + + + + + + + + + + +
+   +env_sym : EConstr.constr;
+   +env_bin : EConstr.constr;
+   +env_units : EConstr.constr;
+} + + + +
type reifier
+
+
+

We need to reify two terms (left and right members of a goal) + that share the same reification envirnoment. Therefore, we need + to add letins to the proof context in order to ensure some + sharing in the proof terms we produce.

+ +

Moreover, in order to have as much sharing as possible, we also + add letins for various partial applications that are used + throughout the terms.

+ +

To achieve this, we decompose the reconstruction function into + two steps: first, we build the reification environment and then + reify each term successively.

+
+
+ + +
val mk_reifier : Coq.Relation.t ->
       EConstr.constr ->
       ir ->
       (sigmas * reifier) Proofview.tactic
+
val reif_constr_of_t : reifier -> Matcher.Terms.t -> EConstr.constr
+
+

reif_constr_of_t reifier t rebuilds the term t in the + reified form.

+
+
+ diff --git a/docs/ocamldoc/Theory.html b/docs/ocamldoc/Theory.html new file mode 100644 index 0000000..3f3761c --- /dev/null +++ b/docs/ocamldoc/Theory.html @@ -0,0 +1,78 @@ + + + + + + + + + + + + + + + + + + + + + + +Theory + + +
Previous + Up + Next +
+

Module Theory

+ +
module Theory: sig .. end
+
+

Bindings for Coq constants that are specific to the plugin; + reification and translation functions.

+ +

Note: this module is highly correlated with the definitions of AAC_rewrite.v.

+ +

This module interfaces with the above Coq module; it provides + facilities to interpret a term with arbitrary operators as an + abstract syntax tree, and to convert an AST into a Coq term + (either using the Coq "raw" terms, as written in the starting + goal, or using the reified Coq datatypes we define in AAC_rewrite.v).

+
+
+
+

Both in OCaml and Coq, we represent finite multisets using + weighted lists (('a*int) list), see Matcher.mset.

+ +

mk_mset ty l constructs a Coq multiset from an OCaml multiset + l of Coq terms of type ty

+ +
val mk_mset : EConstr.constr -> (EConstr.constr * int) list -> EConstr.constr

Packaging modules

+
module Sigma: sig .. end
+

Environments

+ +
+ +
module Sym: sig .. end
+

Dynamically typed morphisms

+ +
+ +
module Stubs: sig .. end
+

We need to export some Coq stubs out of this module.

+ +
+

Building reified terms

+

We define a bundle of functions to build reified versions of the + terms (those that will be given to the reflexive decision + procedure). In particular, each field takes as first argument the + index of the symbol rather than the symbol itself.

+ +
module Trans: sig .. end
+

Tranlations between Coq and OCaml

+ +
+ diff --git a/docs/ocamldoc/index.html b/docs/ocamldoc/index.html new file mode 100644 index 0000000..50d4567 --- /dev/null +++ b/docs/ocamldoc/index.html @@ -0,0 +1,74 @@ + + + + + + + + + + + + + + + + + + + + +
+ +
+ + + + + + + + + +
Coq
+

Interface with Coq where we define some handlers for Coq's API, + and we import several definitions from Coq's standard library.

+ +
+
Helper
+

Debugging functions, that can be triggered on a per-file base.

+ +
+
Search_monad
+

Search monad that allows to express non-deterministic algorithms + in a legible maner, or programs that solve combinatorial problems.

+ +
+
Matcher
+

Standalone module containing the algorithm for matching modulo + associativity and associativity and commutativity + (AAC).

+ +
+
Theory
+

Bindings for Coq constants that are specific to the plugin; + reification and translation functions.

+ +
+
Print
+

Pretty printing functions we use for the aac_instances + tactic.

+ +
+
Aac_rewrite
+

aac_rewrite -- rewriting modulo A or AC

+ +
+
+ + diff --git a/docs/ocamldoc/index_attributes.html b/docs/ocamldoc/index_attributes.html new file mode 100644 index 0000000..4923044 --- /dev/null +++ b/docs/ocamldoc/index_attributes.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of class attributes + + +
 Up + 
+

Index of class attributes

+ +
+ + diff --git a/docs/ocamldoc/index_class_types.html b/docs/ocamldoc/index_class_types.html new file mode 100644 index 0000000..b6c33cb --- /dev/null +++ b/docs/ocamldoc/index_class_types.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of class types + + +
 Up + 
+

Index of class types

+ +
+ + diff --git a/docs/ocamldoc/index_classes.html b/docs/ocamldoc/index_classes.html new file mode 100644 index 0000000..8666e16 --- /dev/null +++ b/docs/ocamldoc/index_classes.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of classes + + +
 Up + 
+

Index of classes

+ +
+ + diff --git a/docs/ocamldoc/index_exceptions.html b/docs/ocamldoc/index_exceptions.html new file mode 100644 index 0000000..3852218 --- /dev/null +++ b/docs/ocamldoc/index_exceptions.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of exceptions + + +
 Up + 
+

Index of exceptions

+ +
+ + diff --git a/docs/ocamldoc/index_extensions.html b/docs/ocamldoc/index_extensions.html new file mode 100644 index 0000000..03ada85 --- /dev/null +++ b/docs/ocamldoc/index_extensions.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of extensions + + +
 Up + 
+

Index of extensions

+ +
+ + diff --git a/docs/ocamldoc/index_methods.html b/docs/ocamldoc/index_methods.html new file mode 100644 index 0000000..21014d9 --- /dev/null +++ b/docs/ocamldoc/index_methods.html @@ -0,0 +1,26 @@ + + + + + + + + + + + + + + + + +Index of class methods + + +
 Up + 
+

Index of class methods

+ +
+ + diff --git a/docs/ocamldoc/index_module_types.html b/docs/ocamldoc/index_module_types.html new file mode 100644 index 0000000..f62a16b --- /dev/null +++ b/docs/ocamldoc/index_module_types.html @@ -0,0 +1,29 @@ + + + + + + + + + + + + + + + + +Index of module types + + +
 Up + 
+

Index of module types

+ + + + +
C
CONTROL [Helper]
+ + diff --git a/docs/ocamldoc/index_modules.html b/docs/ocamldoc/index_modules.html new file mode 100644 index 0000000..4c3f3e5 --- /dev/null +++ b/docs/ocamldoc/index_modules.html @@ -0,0 +1,163 @@ + + + + + + + + + + + + + + + + +Index of modules + + +
 Up + 
+

Index of modules

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
A
Aac_rewrite
+

aac_rewrite -- rewriting modulo A or AC

+ +
+
C
Classes [Coq]
+

Coq typeclasses

+ +
+
Coq
+

Interface with Coq where we define some handlers for Coq's API, + and we import several definitions from Coq's standard library.

+ +
+
D
Debug [Helper]
E
Equivalence [Coq]
H
Helper
+

Debugging functions, that can be triggered on a per-file base.

+ +
+
L
List [Coq]
+

Coq lists

+ +
+
M
Matcher
+

Standalone module containing the algorithm for matching modulo + associativity and associativity and commutativity + (AAC).

+ +
+
N
Nat [Coq]
+

Coq unary numbers (peano)

+ +
+
O
Option [Coq]
P
Pair [Coq]
+

Coq pairs

+ +
+
Pos [Coq]
+

Coq positive numbers (pos)

+ +
+
Print
+

Pretty printing functions we use for the aac_instances + tactic.

+ +
+
R
Relation [Coq]
Rewrite [Coq]
S
Search_monad
+

Search monad that allows to express non-deterministic algorithms + in a legible maner, or programs that solve combinatorial problems.

+ +
+
Sigma [Theory]
+

Environments

+ +
+
Stubs [Theory]
+

We need to export some Coq stubs out of this module.

+ +
+
Subst [Matcher]
+

Substitutions (or environments)

+ +
+
Sym [Theory]
+

Dynamically typed morphisms

+ +
+
T
Terms [Matcher]
+

Representations of expressions

+ +
+
Theory
+

Bindings for Coq constants that are specific to the plugin; + reification and translation functions.

+ +
+
Trans [Theory]
+

Tranlations between Coq and OCaml

+ +
+
+ + diff --git a/docs/ocamldoc/index_types.html b/docs/ocamldoc/index_types.html new file mode 100644 index 0000000..e994c1b --- /dev/null +++ b/docs/ocamldoc/index_types.html @@ -0,0 +1,102 @@ + + + + + + + + + + + + + + + + +Index of types + + +
 Up + 
+

Index of types

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
E
envs [Theory.Trans]
ext_units [Matcher]
H
hypinfo [Coq.Rewrite]
+

We keep some informations about the hypothesis, with an (informal) + invariant: typeof hyp = typ, typ = forall context, body, body = rel left right

+ +
+
I
ir [Theory.Trans]
L
lazy_ref [Coq]
+

Getting Coq terms from the environment

+ +
+
M
m [Search_monad]
+

A data type that represent a collection of 'a

+ +
+
mset [Matcher]
+

The arguments of sums (or AC operators) are represented using finite multisets.

+ +
+
N
nf_term [Matcher.Terms]
P
pack [Theory.Sym]
+

mimics the Coq record Sym.pack

+ +
+
R
reifier [Theory.Trans]
+

We need to reify two terms (left and right members of a goal) + that share the same reification envirnoment.

+ +
+
S
sigmas [Theory.Trans]
symbol [Matcher]
T
t [Matcher.Subst]
t [Matcher.Terms]
t [Coq.Equivalence]
t [Coq.Relation]
tactic [Coq]
U
units [Matcher]
V
var [Matcher]
+ + diff --git a/docs/ocamldoc/index_values.html b/docs/ocamldoc/index_values.html new file mode 100644 index 0000000..7512944 --- /dev/null +++ b/docs/ocamldoc/index_values.html @@ -0,0 +1,407 @@ + + + + + + + + + + + + + + + + +Index of values + + +
 Up + 
+

Index of values

+ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
(>>) [Search_monad]
+

bind and return

+ +
+
(>>|) [Search_monad]
+

non-deterministic choice

+ +
+
A
aac_normalise [Aac_rewrite]
aac_reflexivity [Aac_rewrite]
aac_rewrite [Aac_rewrite]
add [Aac_rewrite]
add [Theory.Sigma]
+

add ty n x map adds the value x of type ty with key n in map

+ +
+
add_symbol [Theory.Trans]
+

add_symbol adds a given binary symbol to the environment of + known stuff.

+ +
+
anomaly [Coq]
B
build [Coq.Rewrite]
+

build the constr to rewrite, with lambda abstractions

+ +
+
C
choose [Search_monad]
count [Search_monad]
cps_resolve_one_typeclass [Coq]
D
debug [Helper.CONTROL]
debug [Helper.Debug]
+

debug prints the string and end it with a newline

+ +
+
debug_exception [Helper.Debug]
decide_thm [Theory.Stubs]
+

The main lemma of our theory, that is + compare (norm u) (norm v) = Eq -> eval u == eval v

+ +
+
decomp_term [Coq]
E
empty [Theory.Sigma]
+

empty ty create an empty map of type ty

+ +
+
empty_envs [Theory.Trans]
equal_aac [Matcher.Terms]
+

Test for equality of terms modulo AACU (relies on the following + canonical representation of terms).

+ +
+
eval [Theory.Stubs]
+

The evaluation fonction, used to convert a reified coq term to a + raw coq term

+ +
+
evar_binary [Coq]
evar_relation [Coq]
F
fail [Search_monad]
+

failure

+ +
+
filter [Search_monad]
find_global [Coq]
fold [Search_monad]
+

folding through the collection

+ +
+
from_relation [Coq.Equivalence]
G
get_efresh [Coq]
get_fresh [Coq]
get_hypinfo [Coq.Rewrite]
+

get_hypinfo ?check_type H l2r analyse the hypothesis H, and + build the related hypinfo.

+ +
+
I
infer [Coq.Equivalence]
instantiate [Matcher.Subst]
ir_of_envs [Theory.Trans]
ir_to_units [Theory.Trans]
is_empty [Search_monad]
L
lift [Theory.Stubs]
lift_normalise_thm [Theory.Stubs]
lift_proj_equivalence [Theory.Stubs]
lift_reflexivity [Theory.Stubs]
lift_transitivity_left [Theory.Stubs]
lift_transitivity_right [Theory.Stubs]
linear [Matcher]
+

linear expands a multiset into a simple list

+ +
+
M
make [Coq.Equivalence]
make [Coq.Relation]
map_syms [Matcher.Terms]
match_as_equation [Coq]
+

match_as_equation ?context env sigma c try to decompose c as a + relation applied to two terms.

+ +
+
matcher [Matcher]
+

matcher p t computes the set of solutions to the given top-level + matching problem (p is the pattern, t is the term).

+ +
+
mk_equivalence [Coq.Classes]
mk_letin [Coq]
mk_morphism [Coq.Classes]
mk_mset [Theory]
mk_pack [Theory.Sym]
+

mk_pack rlt (ar,value,morph)

+ +
+
mk_reflexive [Coq.Classes]
mk_reifier [Theory.Trans]
mk_transitive [Coq.Classes]
N
nf_equal [Matcher.Terms]
nf_term_compare [Matcher.Terms]
none [Coq.Option]
null [Theory.Sym]
+

null builds a dummy (identity) symbol, given an Coq.Relation.t

+ +
+
O
of_int [Coq.Nat]
of_int [Coq.Pos]
of_list [Theory.Sigma]
+

of_list ty null l translates an OCaml association list into a Coq one

+ +
+
of_list [Coq.List]
+

of_list ty l

+ +
+
of_option [Coq.Option]
of_pair [Coq.Pair]
P
pair [Coq.Pair]
pr_aac_args [Aac_rewrite]
pr_constr [Helper.Debug]
+

pr_constr print a Coq constructor, that can be labelled + by a string

+ +
+
print [Print]
+

The main printing function.

+ +
+
printing [Helper.CONTROL]
R
raw_constr_of_t [Theory.Trans]
+

raw_constr_of_t rebuilds a term in the raw representation, and + reconstruct the named products on top of it.

+ +
+
reif_constr_of_t [Theory.Trans]
+

reif_constr_of_t reifier t rebuilds the term t in the + reified form.

+ +
+
return [Search_monad]
rewrite [Coq.Rewrite]
+

rewrite ?abort hypinfo subst builds the rewriting tactic + associated with the given subst and hypinfo.

+ +
+
S
show_proof [Coq]
+

print the current proof term

+ +
+
some [Coq.Option]
sort [Search_monad]
split [Coq.Equivalence]
split [Coq.Relation]
sprint [Matcher.Subst]
sprint [Search_monad]
sprint_nf_term [Matcher.Terms]
subterm [Matcher]
+

subterm p t computes a set of solutions to the given + subterm-matching problem.

+ +
+
T
t_of_constr [Theory.Trans]
t_of_term [Matcher.Terms]
tclDEBUG [Coq]
+

emit debug messages to see which tactics are failing

+ +
+
tclPRINT [Coq]
+

print the current goal

+ +
+
tclRETYPE [Coq]
tclTIME [Coq]
+

time the execution of a tactic

+ +
+
term_of_t [Matcher.Terms]
+

we have the following property: a and b are equal modulo AACU + iif nf_equal (term_of_t a) (term_of_t b) = true

+ +
+
time [Helper.CONTROL]
time [Helper.Debug]
+

time computes the time spent in a function, and then + print it using the given format

+ +
+
to_fun [Theory.Sigma]
+

to_fun ty null map converts a Coq association list into a Coq function (with default value null)

+ +
+
to_list [Matcher.Subst]
to_list [Search_monad]
to_relation [Coq.Equivalence]
typ [Theory.Sym]
typ [Coq.Nat]
typ [Coq.Pos]
typ [Coq.Option]
typ [Coq.Pair]
type_of_list [Coq.List]
+

type_of_list ty

+ +
+
U
user_error [Coq]
W
warning [Coq]
+ + diff --git a/docs/ocamldoc/style.css b/docs/ocamldoc/style.css new file mode 100644 index 0000000..259bd49 --- /dev/null +++ b/docs/ocamldoc/style.css @@ -0,0 +1,43 @@ +.keyword { font-weight : bold ; color : Red } +.keywordsign { color : #C04600 } +.comment { color : Green } +.constructor { color : Blue } +.type { color : #5C6585 } +.string { color : Maroon } +.warning { color : Red ; font-weight : bold } +.info { margin-left : 3em; margin-right: 3em } +.param_info { margin-top: 4px; margin-left : 3em; margin-right : 3em } +.code { color : #465F91 ; } +.typetable { border-style : hidden } +.paramstable { border-style : hidden ; padding: 5pt 5pt} +tr { background-color : White } +td.typefieldcomment { background-color : #FFFFFF ; font-size: smaller ;} +div.sig_block {margin-left: 2em} +*:target { background: yellow; } +body {font: 13px sans-serif; color: black; text-align: left; padding: 5px; margin: 0} +h1 { font-size : 20pt ; text-align: center; } +h2 { font-size : 20pt ; text-align: center; } +h3 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #90BDFF ;padding: 2px; } +h4 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #90DDFF ;padding: 2px; } +h5 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #90EDFF ;padding: 2px; } +h6 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #90FDFF ;padding: 2px; } +div.h7 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #90BDFF ; padding: 2px; } +div.h8 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #E0FFFF ; padding: 2px; } +div.h9 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #F0FFFF ; padding: 2px; } +div.h10 { font-size : 20pt ; border: 1px solid #000000; margin-top: 5px; margin-bottom: 2px;text-align: center; background-color: #FFFFFF ; padding: 2px; } +a {color: #416DFF; text-decoration: none} +a:hover {background-color: #ddd; text-decoration: underline} +pre { margin-bottom: 4px; font-family: monospace; } +pre.verbatim, pre.codepre { } +.indextable {border: 1px #ddd solid; border-collapse: collapse} +.indextable td, .indextable th {border: 1px #ddd solid; min-width: 80px} +.indextable td.module {background-color: #eee ; padding-left: 2px; padding-right: 2px} +.indextable td.module a {color: #4E6272; text-decoration: none; display: block; width: 100%} +.indextable td.module a:hover {text-decoration: underline; background-color: transparent} +.deprecated {color: #888; font-style: italic} +.indextable tr td div.info { margin-left: 2px; margin-right: 2px } +ul.indexlist { margin-left: 0; padding-left: 0;} +ul.indexlist li { list-style-type: none ; margin-left: 0; padding-left: 0; } +ul.info-attributes {list-style: none; margin: 0; padding: 0; } +div.info > p:first-child { margin-top:0; } +div.info-desc > p:first-child { margin-top:0; margin-bottom:0; } \ No newline at end of file diff --git a/docs/ocamldoc/type_Aac_rewrite.html b/docs/ocamldoc/type_Aac_rewrite.html new file mode 100644 index 0000000..8f159a5 --- /dev/null +++ b/docs/ocamldoc/type_Aac_rewrite.html @@ -0,0 +1,31 @@ + + + + + + + + + + + + + + + +Aac_rewrite + + +sig
+  val aac_reflexivity : unit Proofview.tactic
+  val aac_normalise : unit Proofview.tactic
+  val aac_rewrite :
+    args:(string * int) list ->
+    ?abort:bool ->
+    ?l2r:bool ->
+    ?show:bool ->
+    ?strict:bool ->
+    ?extra:EConstr.t -> EConstr.constr -> unit Proofview.tactic
+  val add : string -> 'a -> (string * 'a) list -> (string * 'a) list
+  val pr_aac_args : 'a -> 'b -> 'c -> (string * int) list -> Pp.t
+end diff --git a/docs/ocamldoc/type_Coq.Classes.html b/docs/ocamldoc/type_Coq.Classes.html new file mode 100644 index 0000000..b2c6c05 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Classes.html @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +Coq.Classes + + +sig
+  val mk_morphism :
+    EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+  val mk_equivalence : EConstr.constr -> EConstr.constr -> EConstr.constr
+  val mk_reflexive : EConstr.constr -> EConstr.constr -> EConstr.constr
+  val mk_transitive : EConstr.constr -> EConstr.constr -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.Equivalence.html b/docs/ocamldoc/type_Coq.Equivalence.html new file mode 100644 index 0000000..1bcdda9 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Equivalence.html @@ -0,0 +1,37 @@ + + + + + + + + + + + + + + + +Coq.Equivalence + + +sig
+  type t = {
+    carrier : EConstr.constr;
+    eq : EConstr.constr;
+    equivalence : EConstr.constr;
+  }
+  val make :
+    EConstr.constr -> EConstr.constr -> EConstr.constr -> Coq.Equivalence.t
+  val infer :
+    Environ.env ->
+    Evd.evar_map ->
+    EConstr.constr -> EConstr.constr -> Coq.Equivalence.t * Evd.evar_map
+  val from_relation :
+    Environ.env ->
+    Evd.evar_map -> Coq.Relation.t -> Coq.Equivalence.t * Evd.evar_map
+  val to_relation : Coq.Equivalence.t -> Coq.Relation.t
+  val split :
+    Coq.Equivalence.t -> EConstr.constr * EConstr.constr * EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.List.html b/docs/ocamldoc/type_Coq.List.html new file mode 100644 index 0000000..2e913af --- /dev/null +++ b/docs/ocamldoc/type_Coq.List.html @@ -0,0 +1,22 @@ + + + + + + + + + + + + + + + +Coq.List + + +sig
+  val of_list : EConstr.constr -> EConstr.constr list -> EConstr.constr
+  val type_of_list : EConstr.constr -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.Nat.html b/docs/ocamldoc/type_Coq.Nat.html new file mode 100644 index 0000000..243ed36 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Nat.html @@ -0,0 +1,19 @@ + + + + + + + + + + + + + + + +Coq.Nat + + +sig val typ : Coq.lazy_ref val of_int : int -> Constr.constr end diff --git a/docs/ocamldoc/type_Coq.Option.html b/docs/ocamldoc/type_Coq.Option.html new file mode 100644 index 0000000..ca10254 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Option.html @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +Coq.Option + + +sig
+  val typ : Coq.lazy_ref
+  val some : EConstr.constr -> EConstr.constr -> EConstr.constr
+  val none : EConstr.constr -> EConstr.constr
+  val of_option : EConstr.constr -> EConstr.constr option -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.Pair.html b/docs/ocamldoc/type_Coq.Pair.html new file mode 100644 index 0000000..c2f8f74 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Pair.html @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +Coq.Pair + + +sig
+  val typ : Coq.lazy_ref
+  val pair : Coq.lazy_ref
+  val of_pair :
+    EConstr.constr ->
+    EConstr.constr -> EConstr.constr * EConstr.constr -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.Pos.html b/docs/ocamldoc/type_Coq.Pos.html new file mode 100644 index 0000000..ea67180 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Pos.html @@ -0,0 +1,19 @@ + + + + + + + + + + + + + + + +Coq.Pos + + +sig val typ : Coq.lazy_ref val of_int : int -> EConstr.constr end diff --git a/docs/ocamldoc/type_Coq.Relation.html b/docs/ocamldoc/type_Coq.Relation.html new file mode 100644 index 0000000..dd2db05 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Relation.html @@ -0,0 +1,23 @@ + + + + + + + + + + + + + + + +Coq.Relation + + +sig
+  type t = { carrier : EConstr.constr; r : EConstr.constr; }
+  val make : EConstr.constr -> EConstr.constr -> Coq.Relation.t
+  val split : Coq.Relation.t -> EConstr.constr * EConstr.constr
+end diff --git a/docs/ocamldoc/type_Coq.Rewrite.html b/docs/ocamldoc/type_Coq.Rewrite.html new file mode 100644 index 0000000..a414198 --- /dev/null +++ b/docs/ocamldoc/type_Coq.Rewrite.html @@ -0,0 +1,41 @@ + + + + + + + + + + + + + + + +Coq.Rewrite + + +sig
+  type hypinfo = {
+    hyp : EConstr.constr;
+    hyptype : EConstr.constr;
+    context : EConstr.rel_context;
+    body : EConstr.constr;
+    rel : Coq.Relation.t;
+    left : EConstr.constr;
+    right : EConstr.constr;
+    l2r : bool;
+  }
+  val get_hypinfo :
+    Environ.env ->
+    Evd.evar_map ->
+    ?check_type:(EConstr.types -> bool) ->
+    EConstr.constr -> l2r:bool -> Coq.Rewrite.hypinfo
+  val build :
+    Coq.Rewrite.hypinfo -> (int * EConstr.constr) list -> EConstr.constr
+  val rewrite :
+    ?abort:bool ->
+    Coq.Rewrite.hypinfo ->
+    (int * EConstr.constr) list -> unit Proofview.tactic
+end diff --git a/docs/ocamldoc/type_Coq.html b/docs/ocamldoc/type_Coq.html new file mode 100644 index 0000000..8aef783 --- /dev/null +++ b/docs/ocamldoc/type_Coq.html @@ -0,0 +1,139 @@ + + + + + + + + + + + + + + + +Coq + + +sig
+  type tactic = unit Proofview.tactic
+  type lazy_ref = Names.GlobRef.t Stdlib.Lazy.t
+  val get_fresh : Coq.lazy_ref -> Constr.constr
+  val get_efresh : Coq.lazy_ref -> EConstr.constr
+  val find_global : string -> Coq.lazy_ref
+  val cps_resolve_one_typeclass :
+    ?error:Pp.t ->
+    EConstr.types -> (EConstr.constr -> Coq.tactic) -> Coq.tactic
+  val evar_binary :
+    Environ.env ->
+    Evd.evar_map -> EConstr.constr -> Evd.evar_map * EConstr.constr
+  val evar_relation :
+    Environ.env ->
+    Evd.evar_map -> EConstr.constr -> Evd.evar_map * EConstr.constr
+  val mk_letin : string -> EConstr.constr -> EConstr.constr Proofview.tactic
+  val tclRETYPE : EConstr.constr -> unit Proofview.tactic
+  val decomp_term :
+    Evd.evar_map ->
+    EConstr.constr ->
+    (EConstr.constr, EConstr.types, EConstr.ESorts.t, EConstr.EInstance.t,
+     EConstr.ERelevance.t)
+    Constr.kind_of_term
+  module List :
+    sig
+      val of_list : EConstr.constr -> EConstr.constr list -> EConstr.constr
+      val type_of_list : EConstr.constr -> EConstr.constr
+    end
+  module Pair :
+    sig
+      val typ : Coq.lazy_ref
+      val pair : Coq.lazy_ref
+      val of_pair :
+        EConstr.constr ->
+        EConstr.constr -> EConstr.constr * EConstr.constr -> EConstr.constr
+    end
+  module Option :
+    sig
+      val typ : Coq.lazy_ref
+      val some : EConstr.constr -> EConstr.constr -> EConstr.constr
+      val none : EConstr.constr -> EConstr.constr
+      val of_option :
+        EConstr.constr -> EConstr.constr option -> EConstr.constr
+    end
+  module Pos :
+    sig val typ : Coq.lazy_ref val of_int : int -> EConstr.constr end
+  module Nat :
+    sig val typ : Coq.lazy_ref val of_int : int -> Constr.constr end
+  module Classes :
+    sig
+      val mk_morphism :
+        EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+      val mk_equivalence : EConstr.constr -> EConstr.constr -> EConstr.constr
+      val mk_reflexive : EConstr.constr -> EConstr.constr -> EConstr.constr
+      val mk_transitive : EConstr.constr -> EConstr.constr -> EConstr.constr
+    end
+  module Relation :
+    sig
+      type t = { carrier : EConstr.constr; r : EConstr.constr; }
+      val make : EConstr.constr -> EConstr.constr -> Coq.Relation.t
+      val split : Coq.Relation.t -> EConstr.constr * EConstr.constr
+    end
+  module Equivalence :
+    sig
+      type t = {
+        carrier : EConstr.constr;
+        eq : EConstr.constr;
+        equivalence : EConstr.constr;
+      }
+      val make :
+        EConstr.constr ->
+        EConstr.constr -> EConstr.constr -> Coq.Equivalence.t
+      val infer :
+        Environ.env ->
+        Evd.evar_map ->
+        EConstr.constr -> EConstr.constr -> Coq.Equivalence.t * Evd.evar_map
+      val from_relation :
+        Environ.env ->
+        Evd.evar_map -> Coq.Relation.t -> Coq.Equivalence.t * Evd.evar_map
+      val to_relation : Coq.Equivalence.t -> Coq.Relation.t
+      val split :
+        Coq.Equivalence.t -> EConstr.constr * EConstr.constr * EConstr.constr
+    end
+  val match_as_equation :
+    ?context:EConstr.rel_context ->
+    Environ.env ->
+    Evd.evar_map ->
+    EConstr.constr ->
+    (EConstr.constr * EConstr.constr * Coq.Relation.t) option
+  val tclTIME : string -> Coq.tactic -> Coq.tactic
+  val tclDEBUG : string -> unit Proofview.tactic
+  val tclPRINT : unit Proofview.tactic
+  val anomaly : string -> 'a
+  val user_error : Pp.t -> 'a
+  val warning : string -> unit
+  val show_proof : Declare.Proof.t -> unit
+  module Rewrite :
+    sig
+      type hypinfo = {
+        hyp : EConstr.constr;
+        hyptype : EConstr.constr;
+        context : EConstr.rel_context;
+        body : EConstr.constr;
+        rel : Coq.Relation.t;
+        left : EConstr.constr;
+        right : EConstr.constr;
+        l2r : bool;
+      }
+      val get_hypinfo :
+        Environ.env ->
+        Evd.evar_map ->
+        ?check_type:(EConstr.types -> bool) ->
+        EConstr.constr -> l2r:bool -> Coq.Rewrite.hypinfo
+      val build :
+        Coq.Rewrite.hypinfo -> (int * EConstr.constr) list -> EConstr.constr
+      val rewrite :
+        ?abort:bool ->
+        Coq.Rewrite.hypinfo ->
+        (int * EConstr.constr) list -> unit Proofview.tactic
+    end
+end diff --git a/docs/ocamldoc/type_Helper.CONTROL.html b/docs/ocamldoc/type_Helper.CONTROL.html new file mode 100644 index 0000000..0bf847c --- /dev/null +++ b/docs/ocamldoc/type_Helper.CONTROL.html @@ -0,0 +1,19 @@ + + + + + + + + + + + + + + + +Helper.CONTROL + + +sig val debug : bool val time : bool val printing : bool end diff --git a/docs/ocamldoc/type_Helper.Debug.html b/docs/ocamldoc/type_Helper.Debug.html new file mode 100644 index 0000000..2520d76 --- /dev/null +++ b/docs/ocamldoc/type_Helper.Debug.html @@ -0,0 +1,28 @@ + + + + + + + + + + + + + + + +Helper.Debug + + +functor (X : CONTROL) ->
+  sig
+    val debug : string -> unit
+    val debug_exception : string -> exn -> unit
+    val time :
+      ('a -> 'b) ->
+      'a -> (float -> unit, Stdlib.out_channel, unit) Stdlib.format -> 'b
+    val pr_constr :
+      Environ.env -> Evd.evar_map -> string -> EConstr.constr -> unit
+  end diff --git a/docs/ocamldoc/type_Helper.html b/docs/ocamldoc/type_Helper.html new file mode 100644 index 0000000..c9485f3 --- /dev/null +++ b/docs/ocamldoc/type_Helper.html @@ -0,0 +1,33 @@ + + + + + + + + + + + + + + + +Helper + + +sig
+  module type CONTROL =
+    sig val debug : bool val time : bool val printing : bool end
+  module Debug :
+    functor (X : CONTROL) ->
+      sig
+        val debug : string -> unit
+        val debug_exception : string -> exn -> unit
+        val time :
+          ('a -> 'b) ->
+          'a -> (float -> unit, Stdlib.out_channel, unit) Stdlib.format -> 'b
+        val pr_constr :
+          Environ.env -> Evd.evar_map -> string -> EConstr.constr -> unit
+      end
+end diff --git a/docs/ocamldoc/type_Matcher.Subst.html b/docs/ocamldoc/type_Matcher.Subst.html new file mode 100644 index 0000000..afbf434 --- /dev/null +++ b/docs/ocamldoc/type_Matcher.Subst.html @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +Matcher.Subst + + +sig
+  type t
+  val sprint : Matcher.Subst.t -> string
+  val instantiate : Matcher.Subst.t -> Matcher.Terms.t -> Matcher.Terms.t
+  val to_list : Matcher.Subst.t -> (Matcher.var * Matcher.Terms.t) list
+end diff --git a/docs/ocamldoc/type_Matcher.Terms.html b/docs/ocamldoc/type_Matcher.Terms.html new file mode 100644 index 0000000..e091d41 --- /dev/null +++ b/docs/ocamldoc/type_Matcher.Terms.html @@ -0,0 +1,35 @@ + + + + + + + + + + + + + + + +Matcher.Terms + + +sig
+  type t =
+      Dot of (Matcher.symbol * Matcher.Terms.t * Matcher.Terms.t)
+    | Plus of (Matcher.symbol * Matcher.Terms.t * Matcher.Terms.t)
+    | Sym of (Matcher.symbol * Matcher.Terms.t array)
+    | Var of Matcher.var
+    | Unit of Matcher.symbol
+  val equal_aac : Matcher.units -> Matcher.Terms.t -> Matcher.Terms.t -> bool
+  val map_syms :
+    (Matcher.symbol -> Matcher.symbol) -> Matcher.Terms.t -> Matcher.Terms.t
+  type nf_term
+  val nf_term_compare : Matcher.Terms.nf_term -> Matcher.Terms.nf_term -> int
+  val nf_equal : Matcher.Terms.nf_term -> Matcher.Terms.nf_term -> bool
+  val sprint_nf_term : Matcher.Terms.nf_term -> string
+  val term_of_t : Matcher.units -> Matcher.Terms.t -> Matcher.Terms.nf_term
+  val t_of_term : Matcher.Terms.nf_term -> Matcher.Terms.t
+end diff --git a/docs/ocamldoc/type_Matcher.html b/docs/ocamldoc/type_Matcher.html new file mode 100644 index 0000000..ccb1a8e --- /dev/null +++ b/docs/ocamldoc/type_Matcher.html @@ -0,0 +1,68 @@ + + + + + + + + + + + + + + + +Matcher + + +sig
+  type symbol = int
+  type var = int
+  type units = (Matcher.symbol * Matcher.symbol) list
+  type ext_units = {
+    unit_for : Matcher.units;
+    is_ac : (Matcher.symbol * bool) list;
+  }
+  type 'a mset = ('a * int) list
+  val linear : 'a Matcher.mset -> 'a list
+  module Terms :
+    sig
+      type t =
+          Dot of (Matcher.symbol * Matcher.Terms.t * Matcher.Terms.t)
+        | Plus of (Matcher.symbol * Matcher.Terms.t * Matcher.Terms.t)
+        | Sym of (Matcher.symbol * Matcher.Terms.t array)
+        | Var of Matcher.var
+        | Unit of Matcher.symbol
+      val equal_aac :
+        Matcher.units -> Matcher.Terms.t -> Matcher.Terms.t -> bool
+      val map_syms :
+        (Matcher.symbol -> Matcher.symbol) ->
+        Matcher.Terms.t -> Matcher.Terms.t
+      type nf_term
+      val nf_term_compare :
+        Matcher.Terms.nf_term -> Matcher.Terms.nf_term -> int
+      val nf_equal : Matcher.Terms.nf_term -> Matcher.Terms.nf_term -> bool
+      val sprint_nf_term : Matcher.Terms.nf_term -> string
+      val term_of_t :
+        Matcher.units -> Matcher.Terms.t -> Matcher.Terms.nf_term
+      val t_of_term : Matcher.Terms.nf_term -> Matcher.Terms.t
+    end
+  module Subst :
+    sig
+      type t
+      val sprint : Matcher.Subst.t -> string
+      val instantiate : Matcher.Subst.t -> Matcher.Terms.t -> Matcher.Terms.t
+      val to_list : Matcher.Subst.t -> (Matcher.var * Matcher.Terms.t) list
+    end
+  val matcher :
+    ?strict:bool ->
+    Matcher.ext_units ->
+    Matcher.Terms.t -> Matcher.Terms.t -> Matcher.Subst.t Search_monad.m
+  val subterm :
+    ?strict:bool ->
+    Matcher.ext_units ->
+    Matcher.Terms.t ->
+    Matcher.Terms.t ->
+    (int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m
+end diff --git a/docs/ocamldoc/type_Print.html b/docs/ocamldoc/type_Print.html new file mode 100644 index 0000000..263e5e7 --- /dev/null +++ b/docs/ocamldoc/type_Print.html @@ -0,0 +1,25 @@ + + + + + + + + + + + + + + + +Print + + +sig
+  val print :
+    Coq.Relation.t ->
+    Theory.Trans.ir ->
+    (int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m ->
+    EConstr.rel_context -> unit Proofview.tactic
+end diff --git a/docs/ocamldoc/type_Search_monad.html b/docs/ocamldoc/type_Search_monad.html new file mode 100644 index 0000000..ea09c0e --- /dev/null +++ b/docs/ocamldoc/type_Search_monad.html @@ -0,0 +1,34 @@ + + + + + + + + + + + + + + + +Search_monad + + +sig
+  type 'a m
+  val ( >> ) :
+    'a Search_monad.m -> ('a -> 'b Search_monad.m) -> 'b Search_monad.m
+  val return : 'a -> 'a Search_monad.m
+  val ( >>| ) : 'a Search_monad.m -> 'a Search_monad.m -> 'a Search_monad.m
+  val fail : unit -> 'a Search_monad.m
+  val fold : ('a -> 'b -> 'b) -> 'a Search_monad.m -> 'b -> 'b
+  val sprint : ('a -> string) -> 'a Search_monad.m -> string
+  val count : 'a Search_monad.m -> int
+  val choose : 'a Search_monad.m -> 'a option
+  val to_list : 'a Search_monad.m -> 'a list
+  val sort : ('a -> 'a -> int) -> 'a Search_monad.m -> 'a Search_monad.m
+  val is_empty : 'a Search_monad.m -> bool
+  val filter : ('a -> bool) -> 'a Search_monad.m -> 'a Search_monad.m
+end diff --git a/docs/ocamldoc/type_Theory.Sigma.html b/docs/ocamldoc/type_Theory.Sigma.html new file mode 100644 index 0000000..f30c143 --- /dev/null +++ b/docs/ocamldoc/type_Theory.Sigma.html @@ -0,0 +1,29 @@ + + + + + + + + + + + + + + + +Theory.Sigma + + +sig
+  val add :
+    EConstr.constr ->
+    EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+  val empty : EConstr.constr -> EConstr.constr
+  val of_list :
+    EConstr.constr ->
+    EConstr.constr -> (int * EConstr.constr) list -> EConstr.constr
+  val to_fun :
+    EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Theory.Stubs.html b/docs/ocamldoc/type_Theory.Stubs.html new file mode 100644 index 0000000..0fbf9c5 --- /dev/null +++ b/docs/ocamldoc/type_Theory.Stubs.html @@ -0,0 +1,28 @@ + + + + + + + + + + + + + + + +Theory.Stubs + + +sig
+  val lift : Coq.lazy_ref
+  val lift_proj_equivalence : Coq.lazy_ref
+  val lift_transitivity_left : Coq.lazy_ref
+  val lift_transitivity_right : Coq.lazy_ref
+  val lift_reflexivity : Coq.lazy_ref
+  val eval : Coq.lazy_ref
+  val decide_thm : Coq.lazy_ref
+  val lift_normalise_thm : Coq.lazy_ref
+end diff --git a/docs/ocamldoc/type_Theory.Sym.html b/docs/ocamldoc/type_Theory.Sym.html new file mode 100644 index 0000000..e20ca16 --- /dev/null +++ b/docs/ocamldoc/type_Theory.Sym.html @@ -0,0 +1,24 @@ + + + + + + + + + + + + + + + +Theory.Sym + + +sig
+  type pack = { ar : Constr.t; value : Constr.t; morph : Constr.t; }
+  val typ : Coq.lazy_ref
+  val mk_pack : Coq.Relation.t -> Theory.Sym.pack -> EConstr.constr
+  val null : Coq.Relation.t -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Theory.Trans.html b/docs/ocamldoc/type_Theory.Trans.html new file mode 100644 index 0000000..cd10c81 --- /dev/null +++ b/docs/ocamldoc/type_Theory.Trans.html @@ -0,0 +1,56 @@ + + + + + + + + + + + + + + + +Theory.Trans + + +sig
+  type envs
+  val empty_envs : unit -> Theory.Trans.envs
+  val t_of_constr :
+    Environ.env ->
+    Evd.evar_map ->
+    Coq.Relation.t ->
+    Theory.Trans.envs ->
+    EConstr.constr * EConstr.constr ->
+    Matcher.Terms.t * Matcher.Terms.t * Evd.evar_map
+  val add_symbol :
+    Environ.env ->
+    Evd.evar_map ->
+    Coq.Relation.t -> Theory.Trans.envs -> Constr.t -> Evd.evar_map
+  type ir
+  val ir_of_envs :
+    Environ.env ->
+    Evd.evar_map ->
+    Coq.Relation.t -> Theory.Trans.envs -> Evd.evar_map * Theory.Trans.ir
+  val ir_to_units : Theory.Trans.ir -> Matcher.ext_units
+  val raw_constr_of_t :
+    Theory.Trans.ir ->
+    Coq.Relation.t ->
+    EConstr.rel_context -> Matcher.Terms.t -> EConstr.constr
+  type sigmas = {
+    env_sym : EConstr.constr;
+    env_bin : EConstr.constr;
+    env_units : EConstr.constr;
+  }
+  type reifier
+  val mk_reifier :
+    Coq.Relation.t ->
+    EConstr.constr ->
+    Theory.Trans.ir ->
+    (Theory.Trans.sigmas * Theory.Trans.reifier) Proofview.tactic
+  val reif_constr_of_t :
+    Theory.Trans.reifier -> Matcher.Terms.t -> EConstr.constr
+end diff --git a/docs/ocamldoc/type_Theory.html b/docs/ocamldoc/type_Theory.html new file mode 100644 index 0000000..7bf9ff1 --- /dev/null +++ b/docs/ocamldoc/type_Theory.html @@ -0,0 +1,91 @@ + + + + + + + + + + + + + + + +Theory + + +sig
+  val mk_mset :
+    EConstr.constr -> (EConstr.constr * int) list -> EConstr.constr
+  module Sigma :
+    sig
+      val add :
+        EConstr.constr ->
+        EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+      val empty : EConstr.constr -> EConstr.constr
+      val of_list :
+        EConstr.constr ->
+        EConstr.constr -> (int * EConstr.constr) list -> EConstr.constr
+      val to_fun :
+        EConstr.constr -> EConstr.constr -> EConstr.constr -> EConstr.constr
+    end
+  module Sym :
+    sig
+      type pack = { ar : Constr.t; value : Constr.t; morph : Constr.t; }
+      val typ : Coq.lazy_ref
+      val mk_pack : Coq.Relation.t -> Theory.Sym.pack -> EConstr.constr
+      val null : Coq.Relation.t -> EConstr.constr
+    end
+  module Stubs :
+    sig
+      val lift : Coq.lazy_ref
+      val lift_proj_equivalence : Coq.lazy_ref
+      val lift_transitivity_left : Coq.lazy_ref
+      val lift_transitivity_right : Coq.lazy_ref
+      val lift_reflexivity : Coq.lazy_ref
+      val eval : Coq.lazy_ref
+      val decide_thm : Coq.lazy_ref
+      val lift_normalise_thm : Coq.lazy_ref
+    end
+  module Trans :
+    sig
+      type envs
+      val empty_envs : unit -> Theory.Trans.envs
+      val t_of_constr :
+        Environ.env ->
+        Evd.evar_map ->
+        Coq.Relation.t ->
+        Theory.Trans.envs ->
+        EConstr.constr * EConstr.constr ->
+        Matcher.Terms.t * Matcher.Terms.t * Evd.evar_map
+      val add_symbol :
+        Environ.env ->
+        Evd.evar_map ->
+        Coq.Relation.t -> Theory.Trans.envs -> Constr.t -> Evd.evar_map
+      type ir
+      val ir_of_envs :
+        Environ.env ->
+        Evd.evar_map ->
+        Coq.Relation.t -> Theory.Trans.envs -> Evd.evar_map * Theory.Trans.ir
+      val ir_to_units : Theory.Trans.ir -> Matcher.ext_units
+      val raw_constr_of_t :
+        Theory.Trans.ir ->
+        Coq.Relation.t ->
+        EConstr.rel_context -> Matcher.Terms.t -> EConstr.constr
+      type sigmas = {
+        env_sym : EConstr.constr;
+        env_bin : EConstr.constr;
+        env_units : EConstr.constr;
+      }
+      type reifier
+      val mk_reifier :
+        Coq.Relation.t ->
+        EConstr.constr ->
+        Theory.Trans.ir ->
+        (Theory.Trans.sigmas * Theory.Trans.reifier) Proofview.tactic
+      val reif_constr_of_t :
+        Theory.Trans.reifier -> Matcher.Terms.t -> EConstr.constr
+    end
+end diff --git a/index.html b/index.html new file mode 100644 index 0000000..82ba2e8 --- /dev/null +++ b/index.html @@ -0,0 +1,54 @@ + + + + + + + AAC Tactics + + + + + + + + +
+

AAC Tactics

+
+
+ View the project on GitHub +
+

About

+

Welcome to the AAC Tactics project website! This project is part of coq-community.

+

This Coq plugin provides tactics for rewriting and proving universally quantified equations modulo associativity and commutativity of some operator, with idempotent commutative operators enabling additional simplifications. The tactics can be applied for custom operators by registering the operators and their properties as type class instances. Instances for many commonly used operators, such as for binary integer arithmetic and booleans, are provided with the plugin.

+

This is an open source project, licensed under the GNU Lesser General Public License v3.0 or later.

+

Get the code

+

The current stable release of AAC Tactics can be downloaded from GitHub.

+

Documentation

+ +

Help and contact

+ +

Authors

+ + +