aac_tactics/ 0000775 0000000 0000000 00000000000 12211622232 0013305 5 ustar 00root root 0000000 0000000 aac_tactics/.gitignore 0000664 0000000 0000000 00000000011 12211622232 0015265 0 ustar 00root root 0000000 0000000 doc
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aac_tactics/AAC.v 0000664 0000000 0000000 00000111040 12211622232 0014055 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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 file for the aac_rewrite tactic
We define several base classes to package associative and possibly
commutative 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. *)
Require Import Arith NArith.
Require Import List.
Require Import FMapPositive FMapFacts.
Require Import RelationClasses Equality.
Require Export Morphisms.
Set Implicit Arguments.
Local Open Scope signature_scope.
(** * Environments for the reification process: we use positive maps 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.
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 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
}.
(** 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 (R: relation X) (E : relation X) := {
aac_lift_equivalence : Equivalence E;
aac_list_proper : Proper (E ==> E ==> iff) R
}.
(** simple instances, when we have a subrelation, or an equivalence *)
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.
Instance aac_lift_proper {X} {R : relation X} {E} {HE: Equivalence E}
{HR: Proper (E==>E==>iff) R}: AAC_lift R E | 4 := {}.
Module Internal.
(** * Utilities for the evaluation function *)
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 (Psucc n) x) (plus x (copy n x)).
Proof. intros; unfold copy; rewrite Prect_succ. reflexivity. Qed.
Global 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.
(** * Utilities for positive numbers
which we use as:
- indices for morphisms and symbols
- multiplicity of terms in sums *)
Local 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.
Fixpoint idx_compare i j :=
match i,j with
| xH, xH => Eq
| xH, _ => Lt
| _, xH => Gt
| xO i, xO j => idx_compare i j
| xI i, xI j => idx_compare i j
| xI _, xO _ => Gt
| xO _, xI _ => Lt
end.
Local Notation pos_compare := idx_compare (only parsing).
(** 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. induction i; destruct j; simpl; try constructor; case (IHi j); intros; constructor. Qed.
Lemma idx_compare_reflect_eq: forall i j, idx_compare i j = Eq -> i=j.
Proof. intros i j. case (pos_compare_weak_spec i j); intros; congruence. Qed.
(** * Dependent types utilities *)
Local 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 *)
Inductive nelist (A : Type) : Type :=
| nil : A -> nelist A
| cons : A -> nelist A -> nelist A.
Local 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.
Local 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) *)
Local 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.
(** (sorted) merging functions *)
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.
(** 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.
(** * 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.
(** 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
}.
(** helper to build default values, when filling reification environments *)
Definition null: pack := mkPack 1 (fun x => x) (fun _ _ H => H).
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)
}.
End t.
(* See #Instances.v# 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)
}.
Variable e_unit: positive -> unit_pack.
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).
(** lexicographic rpo over the normalised syntax *)
Fixpoint compare (u v: T) :=
match u,v with
| sum i l, sum j vs => lex (idx_compare i j) (mset_compare compare l vs)
| prd i l, prd j vs => lex (idx_compare i j) (list_compare compare l vs)
| sym i l, sym j vs => lex (idx_compare i j) (vcompare l vs)
| unit i , unit j => idx_compare i j
| unit _ , _ => Lt
| _ , unit _ => Gt
| sum _ _, _ => Lt
| _ , sum _ _ => Gt
| prd _ _, _ => Lt
| _ , prd _ _ => Gt
end
with vcompare i j (us: vT i) (vs: vT j) :=
match us,vs with
| vnil, vnil => Eq
| vnil, _ => Lt
| _, vnil => Gt
| vcons _ u us, vcons _ v vs => lex (compare u v) (vcompare us vs)
end.
(** ** Evaluation from syntax to the abstract domain *)
Fixpoint eval u: X :=
match u with
| sum i l => let o := Bin.value (e_bin i) in
fold_map (fun un => let '(u,n):=un in @copy _ o n (eval u)) o l
| prd i l => fold_map eval (Bin.value (e_bin i)) l
| sym i v => eval_aux v (Sym.value (e_sym i))
| unit i => e_unit i
end
with eval_aux i (v: vT i): Sym.type_of i -> X :=
match v with
| vnil => fun f => f
| vcons _ u v => fun f => eval_aux v (f (eval u))
end.
(** we need to show that compare reflects equality (this is because
we work with msets rather than lists with arities) *)
Lemma tcompare_weak_spec: forall (u v : T), compare_weak_spec u v (compare u v)
with vcompare_reflect_eqdep: forall i us j vs (H: i=j), vcompare us vs = Eq -> cast vT H us = vs.
Proof.
induction u.
destruct v; simpl; try constructor.
case (pos_compare_weak_spec p p0); intros; try constructor.
case (mset_compare_weak_spec compare tcompare_weak_spec m m0); intros; try constructor.
destruct v; simpl; try constructor.
case (pos_compare_weak_spec p p0); intros; try constructor.
case (list_compare_weak_spec compare tcompare_weak_spec n n0); intros; try constructor.
destruct v0; simpl; try constructor.
case_eq (idx_compare i i0); intro Hi; try constructor.
apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. (* the [symmetry] is required ! *)
case_eq (vcompare v v0); intro Hv; try constructor.
rewrite <- (vcompare_reflect_eqdep _ _ _ _ eq_refl Hv). constructor.
destruct v; simpl; try constructor.
case_eq (idx_compare p p0); intro Hi; try constructor.
apply idx_compare_reflect_eq in Hi. symmetry in Hi. subst. constructor.
induction us; destruct vs; simpl; intros H Huv; try discriminate.
apply cast_eq, eq_nat_dec.
injection H; intro Hn.
revert Huv; case (tcompare_weak_spec t t0); intros; try discriminate.
symmetry in Hn. subst. (* symmetry required *)
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.
(** ** Normalisation *)
Inductive discr {A} : Type :=
| Is_op : A -> discr
| Is_unit : idx -> discr
| Is_nothing : discr .
(* This is called sum in the std lib *)
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
| u => 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
| u => 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 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_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.
Hint Resolve Binvalue_Proper Binvalue_Associative Binvalue_Commutative.
(** auxiliary lemmas about sums *)
Hint Resolve is_unit_of_Unit.
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 *)
Hint Extern 5 (copy (?n + ?m) (eval ?a) == Bin.value (copy ?n (eval ?a)) (copy ?m (eval ?a))) => apply copy_plus.
Hint Extern 5 (?x == ?x) => reflexivity.
Hint Extern 5 ( Bin.value ?x ?y == Bin.value ?y ?x) => apply Bin.comm.
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))), 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 (Psucc 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.
rewrite eval_merge_bin; auto.
rewrite eval_sum_nil.
apply compat_sum_unit_Unit in H. rewrite law_neutral_right. reflexivity.
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. unfold 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. rewrite 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. auto.
apply compat_sum_unit_return.
rewrite z2 by auto. rewrite IHh.
rewrite eval_sum_cons. rewrite Hnorm. reflexivity. apply compat_sum_unit_add, IHh'.
apply H.
Defined.
(** 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.
unfold 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 *)
Theorem eval_norm: forall u, eval (norm u) == eval u
with eval_norm_aux: forall i (l: vT i) (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.
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.
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.
End s.
End Internal.
(** * Lemmas for performing transitivity steps
given an instance of AAC_lift *)
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.
End t.
Declare ML Module "aac".
aac_tactics/CHANGELOG 0000664 0000000 0000000 00000002073 12211622232 0014521 0 ustar 00root root 0000000 0000000
AAC_tactics 0.3 :
-----------------
- New release for Coq 8.4
AAC_tactics 0.2-pl2 :
-----------------
- Improved the handling of nullifiable patterns.
AAC_tactics 0.2.1 :
-----------------
- backport of some debian patches (thanks to S. Glondu)
AAC_tactics 0.2 :
-----------------
- Several operators can share a given unit (like max and plus sharing zero)
- Added some support to rewrite in inequations (using inequations)
- Better priting functions for aac_instances
- Overhauled inference of morphisms and operators :
* Lift the previous requirement to have at leat one AC and one A operator
* Binary operations are infered before morphisms (hence List.assoc can be recognized as being Associative)
- Should now be able to handle goal with evars (but this is not unification modulo AC)
- Added several new instances of Associative and Commutative operators
- The old syntax to declare AC and A operators is no longer supported
- The tactics do not abstract the proof they built (was troublesome if evars appeared)
AAC_tactics 0.1 :
-----------------
Initial release
aac_tactics/COPYING 0000664 0000000 0000000 00000104513 12211622232 0014344 0 ustar 00root root 0000000 0000000 GNU GENERAL PUBLIC LICENSE
Version 3, 29 June 2007
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aac_tactics/Caveats.v 0000664 0000000 0000000 00000027241 12211622232 0015070 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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 caveats and limitations of the [aac_tactics] library.
Depending on your installation, either uncomment the following two
lines, or add them to your .coqrc files, replacing "."
with the path to the [aac_tactics] library
*)
Add Rec LoadPath "." as AAC_tactics.
Require Import AAC.
Require Instances.
(** ** Limitations *)
(** *** 1. 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], 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.
Fail aac_rewrite Hf. (** [aac_rewrite] does not instantiate [n] automatically *)
aac_rewrite (Hf k). (** of course, this argument can be given explicitly *)
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.
(** *** 2. Exogeneous morphisms
We do not handle `exogeneous' morphisms: morphisms that move from
type [T] to some other type [T']. *)
Section morphism.
Require Import NArith Minus.
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 minus_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 minus_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 minus_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 minus_diag.
split; aac_rewrite minus_diag; reflexivity.
Qed.
End morphism.
(** *** 3. 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.
Require Import ZArith.
Import Instances.Z.
Open Scope Z_scope.
Instance Zplus_incr: Proper (Zle ==> Zle ==> Zle) Zplus.
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 ([Zopp] 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 *)
(** *** 1. Special treatment for units.
[S O] is considered as a unit for multiplication whenever a [Peano.mult]
appears in the goal. The downside is that [S x] does not match [1],
and [1] does not match [S(0+0)] whenever [Peano.mult] 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 an unit. *)
End Peano.
(** *** 2. 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, they will not be
instantiated. *)
Section evars.
Require 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.
aac_rewrite idem. (** this works: [x] is instantiated with an evar *)
instantiate (2 := 0).
symmetry. aac_reflexivity. (** this does work but there are remaining evars in the end *)
Abort.
Hypothesis H': forall x, 3+x = x -> P.
Goal P.
eapply H'.
Fail aac_rewrite idem. (** this fails since we do not instantiate evars *)
Abort.
End evars.
(** *** 3. 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.
(** *** 4. 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.
(** *** 5. Rewriting too much things. *)
Section W.
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.
(** *** 6. Rewriting nullifiable patterns. *)
Section Z.
(** 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. E.g.
- a+b*c*(1+[])
- a+b*c*(1+0*(1+ []))
- ...
*)
Abort.
(** **** If the pattern is a unit or can be instanciated 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 is the following, where the hypothesis can be instanciated 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 on this in the future *)
End Z.
aac_tactics/Instances.v 0000664 0000000 0000000 00000025426 12211622232 0015434 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
Require Export AAC.
(** Instances for aac_rewrite.*)
(* This one is not declared as an instance: this interferes badly with setoid_rewrite *)
Lemma eq_subr {X} {R} `{@Reflexive X R}: subrelation eq R.
Proof. intros x y ->. reflexivity. Qed.
(* At the moment, all the instances are exported even if they are packaged into modules. Even using LocalInstances in fact*)
Module Peano.
Require Import Arith NArith Max Min.
Instance aac_plus_Assoc : Associative eq plus := plus_assoc.
Instance aac_plus_Comm : Commutative eq plus := plus_comm.
Instance aac_mult_Comm : Commutative eq mult := mult_comm.
Instance aac_mult_Assoc : Associative eq mult := mult_assoc.
Instance aac_min_Comm : Commutative eq min := min_comm.
Instance aac_min_Assoc : Associative eq min := min_assoc.
Instance aac_max_Comm : Commutative eq max := max_comm.
Instance aac_max_Assoc : Associative eq max := max_assoc.
Instance aac_one : Unit eq mult 1 := Build_Unit eq mult 1 mult_1_l mult_1_r.
Instance aac_zero_plus : Unit eq plus O := Build_Unit eq plus (O) plus_0_l plus_0_r.
Instance aac_zero_max : Unit eq max O := Build_Unit eq max 0 max_0_l max_0_r.
(* We also provide liftings from le to eq *)
Instance preorder_le : PreOrder le := Build_PreOrder _ _ le_refl le_trans.
Instance lift_le_eq : AAC_lift le eq := Build_AAC_lift eq_equivalence _.
End Peano.
Module Z.
Require Import ZArith Zminmax.
Open Scope Z_scope.
Instance aac_Zplus_Assoc : Associative eq Zplus := Zplus_assoc.
Instance aac_Zplus_Comm : Commutative eq Zplus := Zplus_comm.
Instance aac_Zmult_Comm : Commutative eq Zmult := Zmult_comm.
Instance aac_Zmult_Assoc : Associative eq Zmult := Zmult_assoc.
Instance aac_Zmin_Comm : Commutative eq Zmin := Zmin_comm.
Instance aac_Zmin_Assoc : Associative eq Zmin := Zmin_assoc.
Instance aac_Zmax_Comm : Commutative eq Zmax := Zmax_comm.
Instance aac_Zmax_Assoc : Associative eq Zmax := Zmax_assoc.
Instance aac_one : Unit eq Zmult 1 := Build_Unit eq Zmult 1 Zmult_1_l Zmult_1_r.
Instance aac_zero_Zplus : Unit eq Zplus 0 := Build_Unit eq Zplus 0 Zplus_0_l Zplus_0_r.
(* We also provide liftings from le to eq *)
Instance preorder_Zle : PreOrder Zle := Build_PreOrder _ _ Zle_refl Zle_trans.
Instance lift_le_eq : AAC_lift Zle eq := Build_AAC_lift eq_equivalence _.
End Z.
Module Lists.
Require Import List.
Instance aac_append_Assoc {A} : Associative eq (@app A) := @app_assoc A.
Instance aac_nil_append {A} : @Unit (list A) eq (@app A) (@nil A) := Build_Unit _ (@app A) (@nil A) (@app_nil_l A) (@app_nil_r A).
Instance aac_append_Proper {A} : Proper (eq ==> eq ==> eq) (@app A).
Proof.
repeat intro.
subst.
reflexivity.
Qed.
End Lists.
Module N.
Require Import NArith.
Open Scope N_scope.
Instance aac_Nplus_Assoc : Associative eq Nplus := Nplus_assoc.
Instance aac_Nplus_Comm : Commutative eq Nplus := Nplus_comm.
Instance aac_Nmult_Comm : Commutative eq Nmult := Nmult_comm.
Instance aac_Nmult_Assoc : Associative eq Nmult := Nmult_assoc.
Instance aac_Nmin_Comm : Commutative eq Nmin := N.min_comm.
Instance aac_Nmin_Assoc : Associative eq Nmin := N.min_assoc.
Instance aac_Nmax_Comm : Commutative eq Nmax := N.max_comm.
Instance aac_Nmax_Assoc : Associative eq Nmax := N.max_assoc.
Instance aac_one : Unit eq Nmult (1)%N := Build_Unit eq Nmult (1)%N Nmult_1_l Nmult_1_r.
Instance aac_zero : Unit eq Nplus (0)%N := Build_Unit eq Nplus (0)%N Nplus_0_l Nplus_0_r.
Instance aac_zero_max : Unit eq Nmax 0 := Build_Unit eq Nmax 0 N.max_0_l N.max_0_r.
(* We also provide liftings from le to eq *)
Instance preorder_le : PreOrder Nle := Build_PreOrder _ Nle N.le_refl N.le_trans.
Instance lift_le_eq : AAC_lift Nle eq := Build_AAC_lift eq_equivalence _.
End N.
Module P.
Require Import NArith.
Open Scope positive_scope.
Instance aac_Pplus_Assoc : Associative eq Pplus := Pplus_assoc.
Instance aac_Pplus_Comm : Commutative eq Pplus := Pplus_comm.
Instance aac_Pmult_Comm : Commutative eq Pmult := Pmult_comm.
Instance aac_Pmult_Assoc : Associative eq Pmult := Pmult_assoc.
Instance aac_Pmin_Comm : Commutative eq Pmin := Pos.min_comm.
Instance aac_Pmin_Assoc : Associative eq Pmin := Pos.min_assoc.
Instance aac_Pmax_Comm : Commutative eq Pmax := Pos.max_comm.
Instance aac_Pmax_Assoc : Associative eq Pmax := Pos.max_assoc.
Instance aac_one : Unit eq Pmult 1 := Build_Unit eq Pmult 1 _ Pmult_1_r.
intros; reflexivity. Qed. (* TODO : add this lemma in the stdlib *)
Instance aac_one_max : Unit eq Pmax 1 := Build_Unit eq Pmax 1 Pos.max_1_l Pos.max_1_r.
(* We also provide liftings from le to eq *)
Instance preorder_le : PreOrder Ple := Build_PreOrder _ Ple Pos.le_refl Pos.le_trans.
Instance lift_le_eq : AAC_lift Ple eq := Build_AAC_lift eq_equivalence _.
End P.
Module Q.
Require Import QArith Qminmax.
Instance aac_Qplus_Assoc : Associative Qeq Qplus := Qplus_assoc.
Instance aac_Qplus_Comm : Commutative Qeq Qplus := Qplus_comm.
Instance aac_Qmult_Comm : Commutative Qeq Qmult := Qmult_comm.
Instance aac_Qmult_Assoc : Associative Qeq Qmult := Qmult_assoc.
Instance aac_Qmin_Comm : Commutative Qeq Qmin := Q.min_comm.
Instance aac_Qmin_Assoc : Associative Qeq Qmin := Q.min_assoc.
Instance aac_Qmax_Comm : Commutative Qeq Qmax := Q.max_comm.
Instance aac_Qmax_Assoc : Associative Qeq Qmax := Q.max_assoc.
Instance aac_one : Unit Qeq Qmult 1 := Build_Unit Qeq Qmult 1 Qmult_1_l Qmult_1_r.
Instance aac_zero_Qplus : Unit Qeq Qplus 0 := Build_Unit Qeq Qplus 0 Qplus_0_l Qplus_0_r.
(* We also provide liftings from le to eq *)
Instance preorder_le : PreOrder Qle := Build_PreOrder _ Qle Qle_refl Qle_trans.
Instance lift_le_eq : AAC_lift Qle Qeq := Build_AAC_lift QOrderedType.QOrder.TO.eq_equiv _.
End Q.
Module Prop_ops.
Instance aac_or_Assoc : Associative iff or. Proof. unfold Associative; tauto. Qed.
Instance aac_or_Comm : Commutative iff or. Proof. unfold Commutative; tauto. Qed.
Instance aac_and_Assoc : Associative iff and. Proof. unfold Associative; tauto. Qed.
Instance aac_and_Comm : Commutative iff and. Proof. unfold Commutative; tauto. Qed.
Instance aac_True : Unit iff or False. Proof. constructor; firstorder. Qed.
Instance aac_False : Unit iff and True. Proof. constructor; firstorder. Qed.
Program Instance aac_not_compat : Proper (iff ==> iff) not.
Solve All Obligations using firstorder.
Instance lift_impl_iff : AAC_lift Basics.impl iff := Build_AAC_lift _ _.
End Prop_ops.
Module Bool.
Instance aac_orb_Assoc : Associative eq orb. Proof. unfold Associative; firstorder. Qed.
Instance aac_orb_Comm : Commutative eq orb. Proof. unfold Commutative; firstorder. Qed.
Instance aac_andb_Assoc : Associative eq andb. Proof. unfold Associative; firstorder. Qed.
Instance aac_andb_Comm : Commutative eq andb. Proof. unfold Commutative; firstorder. Qed.
Instance aac_true : Unit eq orb false. Proof. constructor; firstorder. Qed.
Instance aac_false : Unit eq andb true. Proof. constructor; intros [|];firstorder. Qed.
Instance negb_compat : Proper (eq ==> eq) negb. Proof. intros [|] [|]; auto. Qed.
End Bool.
Module Relations.
Require Import Relations.
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.
Instance eq_same_relation T : Equivalence (same_relation T). Proof. firstorder. Qed.
Instance aac_union_Comm T : Commutative (same_relation T) (union T). Proof. unfold Commutative; compute; intuition. Qed.
Instance aac_union_Assoc T : Associative (same_relation T) (union T). Proof. unfold Associative; compute; intuition. Qed.
Instance aac_bot T : Unit (same_relation T) (union T) (bot T). Proof. constructor; compute; intuition. Qed.
Instance aac_inter_Comm T : Commutative (same_relation T) (inter T). Proof. unfold Commutative; compute; intuition. Qed.
Instance aac_inter_Assoc T : Associative (same_relation T) (inter T). Proof. unfold Associative; compute; intuition. Qed.
Instance aac_top 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 *)
Instance aac_compo T : Associative (same_relation T) (compo T). Proof. unfold Associative; compute; firstorder. Qed.
Instance aac_eq T : Unit (same_relation T) (compo T) (eq). Proof. compute; firstorder subst; trivial. Qed.
Instance negr_compat T : Proper (same_relation T ==> same_relation T) (negr T).
Proof. compute. firstorder. Qed.
Instance transp_compat T : Proper (same_relation T ==> same_relation T) (transp T).
Proof. compute. firstorder. Qed.
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.
Instance clos_trans_compat T: Proper (same_relation T ==> same_relation T) (clos_trans T).
Proof. intros R S H; split; apply clos_trans_incr, H. Qed.
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.
Instance clos_refl_trans_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.
Instance preorder_inclusion T : PreOrder (inclusion T).
Proof. constructor; unfold Reflexive, Transitive, inclusion; intuition. Qed.
Instance lift_inclusion_same_relation T: AAC_lift (inclusion T) (same_relation T) :=
Build_AAC_lift (eq_same_relation T) _.
Proof. firstorder. Qed.
End Relations.
Module All.
Export Peano.
Export Z.
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.
*)
aac_tactics/LICENSE 0000664 0000000 0000000 00000001213 12211622232 0014307 0 ustar 00root root 0000000 0000000 The aac_tactics plugin library is free software: you can redistribute
it and/or modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation, either version 3
of the License, or (at your option) any later version.
This library is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with this library. If not, see
.
aac_tactics/Makefile 0000664 0000000 0000000 00000001303 12211622232 0014742 0 ustar 00root root 0000000 0000000
FILES := coq.mli helper.mli search_monad.mli matcher.mli theory.mli print.mli \
evm_compute.mli evm_compute.ml \
coq.ml helper.ml search_monad.ml matcher.ml theory.ml print.ml rewrite.ml4 \
aac.mlpack \
AAC.v Instances.v Tutorial.v Caveats.v
ARGS := -R . AAC_tactics
.PHONY: coq clean doc
world: all doc
all: Makefile.coq
$(MAKE) -f Makefile.coq all
install: Makefile.coq
$(MAKE) -f Makefile.coq install
coq: Makefile.coq
$(MAKE) -f Makefile.coq
doc: Makefile.coq
$(MAKE) -f Makefile.coq html
$(MAKE) -f Makefile.coq mlihtml
Makefile.coq: Makefile $(VS)
coq_makefile $(ARGS) $(FILES) -o Makefile.coq
clean:: Makefile.coq
$(MAKE) -f Makefile.coq clean
rm -f Makefile.coq .depend
aac_tactics/README.txt 0000664 0000000 0000000 00000004275 12211622232 0015013 0 ustar 00root root 0000000 0000000
aac_tactics
===========
Thomas Braibant & Damien Pous
Laboratoire d'Informatique de Grenoble (UMR 5217), INRIA, CNRS, France
Webpage of the project: http://sardes.inrialpes.fr/~braibant/aac_tactics/
FOREWORD
========
This plugin provides tactics for rewriting universally quantified
equations, modulo associativity and commutativity of some operators.
INSTALLATION
============
This plugin should work with Coq v8.4
- running [make] in the top-level directory will generate a Makefile
(using coq_makefile), and will build the plugin and its documentation
Option 1
--------
To install the plugin in Coq's directory, as, e.g., omega or ring.
- run [sudo make install CMXSFILES='aac_tactics.cmxs aac_tactics.cma']
to copy the relevant files of the plugin
Option 2
--------
If you chose not to use the previous option, you will need to add the
following lines to (all) your .v files to be able to use the plugin:
Add Rec LoadPath "absolute_path_to_aac_tactics".
Add ML Path "absolute_path_to_aac_tactics".
DOCUMENTATION
=============
Please refer to Tutorial.v for a succinct introduction on how to use
this plugin.
To understand the inner-working of the tactic, please refer to the
.mli files as the main source of information on each .ml
file. Alternatively, [make world] generates ocamldoc/coqdoc
documentation in directories doc/ and html/, respectively.
File Instances.v defines several instances for frequent use-cases of
this plugin, that should allow you to use it out-of-the-shelf. Namely,
we have instances for:
- Peano naturals (Import Instances.Peano)
- Z binary numbers (Import Instances.Z)
- N binary numbers (Import Instances.N)
- P binary numbers (Import Instances.P)
- Rationnal numbers (Import Instances.Q)
- Prop (Import Instances.Prop_ops)
- Booleans (Import Instances.Bool)
- Relations (Import Instances.Relations)
- All of the above (Import Instances.All)
ACKNOWLEDGEMENTS
================
We are grateful to Evelyne Contejean, Hugo Herbelin, Assia Mahboubi
and Matthieu Sozeau for highly instructive discussions.
This plugin took inspiration from the plugin tutorial "constructors",
distributed under the LGPL 2.1, copyrighted by Matthieu Sozeau
aac_tactics/Tutorial.v 0000664 0000000 0000000 00000031057 12211622232 0015305 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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 the [aac_tactics] library.
Depending on your installation, either modify the following two
lines, or add them to your .coqrc files, replacing "." with the
path to the [aac_tactics] library. *)
Add Rec LoadPath "." as AAC_tactics.
Add ML Path ".".
Require Import AAC.
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 [Zplus]. *)
Section introduction.
Import ZArith.
Import Instances.Z.
Variables a b c : Z.
Hypothesis H: forall x, x + Zopp x = 0.
Goal a + b + c + Zopp (c + a) = b.
aac_rewrite H.
aac_reflexivity.
Qed.
Goal a + c + Zopp (b + a + Zopp b) = c.
do 2 aac_rewrite H.
reflexivity.
Qed.
(** Notes:
- the tactic handles arbitrary function symbols like [Zopp] (as
long as they are proper morphisms w.r.t. the considered
equivalence relation);
- here, ring would have done the job.
*)
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 [Zopp] 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
proof-general, look at buffer *coq* ): *)
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 file [Instances.v].) *)
Section Peano.
Require Import Arith.
Instance aac_plus_Assoc : Associative eq plus := plus_assoc.
Instance aac_plus_Comm : Commutative eq plus := plus_comm.
Instance aac_one : Unit eq mult 1 :=
Build_Unit eq mult 1 mult_1_l mult_1_r.
Instance aac_zero_plus : Unit eq plus O :=
Build_Unit eq plus (O) plus_0_l plus_0_r.
(** Two (or more) operations may share the same units: in the
following example, [0] is understood as the unit of [max] as well as
the unit of [plus]. *)
Instance aac_max_Comm : Commutative eq Max.max := Max.max_comm.
Instance aac_max_Assoc : Associative eq Max.max := Max.max_assoc.
Instance aac_zero_max : Unit eq Max.max O :=
Build_Unit eq Max.max 0 Max.max_0_l Max.max_0_r.
Variable a b c : nat.
Goal Max.max (a + 0) 0 = a.
aac_reflexivity.
Qed.
(** Furthermore, several operators can be mixed: *)
Hypothesis H : forall x y z, Max.max (x + y) (x + z) = x+ Max.max y z.
Goal Max.max (a + b) (c + (a * 1)) = Max.max c b + a.
aac_instances H. aac_rewrite H. aac_reflexivity.
Qed.
Goal Max.max (a + b) (c + Max.max (a*1+0) 0) = a + Max.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 Max.max (a + b) (c + a) <= Max.max (b + a) (c + 1*a).
aac_reflexivity.
Qed.
(** In the last three examples, there were no equivalence relation
involved in the goal. However, we actually had to guess the
equivalence relation with respect to which the operators
([plus,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: *)
Instance lift_le_eq : 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.
Require Import ZArith.
Open Scope Z_scope.
Variable a b c d : Z.
Goal a + (b + c*c*d) + a + 0 + d*1 = a.
aac_normalise.
Abort.
End AAC_normalise.
(** ** Examples from the web page *)
Section Examples.
Import Instances.Z.
Require Import ZArith.
Open Scope Z_scope.
(** *** Reverse triangle inequality *)
Lemma Zabs_triangle : forall x y, Zabs (x + y) <= Zabs x + Zabs y .
Proof Zabs_triangle.
Lemma Zplus_opp_r : forall x, x + -x = 0.
Proof Zplus_opp_r.
(** The following morphisms are required to perform the required rewrites *)
Instance Zminus_compat : Proper (Zge ==> Zle) Zopp.
Proof. intros x y. omega. Qed.
Instance Proper_Zplus : Proper (Zle ==> Zle ==> Zle) Zplus.
Proof. firstorder. Qed.
Goal forall a b, Zabs a - Zabs b <= Zabs (a - b).
intros. unfold Zminus.
aac_instances <- (Zminus_diag b).
aac_rewrite <- (Zminus_diag b) at 3.
unfold Zminus.
aac_rewrite Zabs_triangle.
aac_rewrite Zplus_opp_r.
aac_reflexivity.
Qed.
(** *** Pythagorean triples *)
Notation "x ^2" := (x*x) (at level 40).
Notation "2 â‹… x" := (x+x) (at level 41).
Lemma Hbin1: forall x y, (x+y)^2 = x^2 + y^2 + 2â‹…x*y. Proof. intros; ring. Qed.
Lemma Hbin2: forall x y, x^2 + y^2 = (x+y)^2 + -(2â‹…x*y). Proof. intros; ring. Qed.
Lemma Hopp : forall x, x + -x = 0. Proof Zplus_opp_r.
Variables a b c : Z.
Hypothesis H : c^2 + 2â‹…(a+1)*b = (a+1+b)^2.
Goal a^2 + b^2 + 2â‹…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.
Section Match.
(* The following example is inspired by a question by Francois
Pottier, wokring in the context of separation logic. *)
Variable T : Type.
Variable star : T -> T -> T.
Hypothesis P : T -> Prop.
Context (starA : Associative eq star).
Context (starC : Commutative eq star).
Hypothesis P_hereditary_left : forall a b, P (star a b) -> P a.
Hypothesis P_hereditary_right : forall a b, P (star a b) -> P b.
Notation "a * b" := (star a b).
Ltac crush :=
match goal with
H: P ?R' |- P ?R =>
let h := fresh in
aac_match (fun x => x * R) R' h;
rewrite <- h in H;
try eauto using P_hereditary_right
end.
Goal forall a b c, P (c * a * c * b) -> P (b * a).
Proof. intros. crush. Qed.
Goal forall a b c, exists d, P (d * a * c * b) -> P (b * d) /\ b = d.
Proof. intros. eexists. intros. split. crush. reflexivity. Qed.
End Match.
aac_tactics/aac.mlpack 0000664 0000000 0000000 00000000101 12211622232 0015212 0 ustar 00root root 0000000 0000000 Coq
Helper
Search_monad
Matcher
Theory
Print
Evm_compute
Rewrite
aac_tactics/coq.ml 0000664 0000000 0000000 00000042652 12211622232 0014432 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Interface with Coq *)
type constr = Term.constr
open Term
open Names
open Coqlib
(* The contrib name is used to locate errors when loading constrs *)
let contrib_name = "aac_tactics"
(* Getting constrs (primitive Coq terms) from existing Coq
libraries. *)
let find_constant contrib dir s =
Libnames.constr_of_global (Coqlib.find_reference contrib dir s)
let init_constant dir s = find_constant contrib_name dir s
(* A clause specifying that the [let] should not try to fold anything
in the goal *)
let nowhere =
{ Tacexpr.onhyps = Some [];
Tacexpr.concl_occs = Glob_term.no_occurrences_expr
}
let cps_mk_letin
(name:string)
(c: constr)
(k : constr -> Proof_type.tactic)
: Proof_type.tactic =
fun goal ->
let name = (Names.id_of_string name) in
let name = Tactics.fresh_id [] name goal in
let letin = (Tactics.letin_tac None (Name name) c None nowhere) in
Tacticals.tclTHEN letin (k (mkVar name)) goal
(** {2 General functions} *)
type goal_sigma = Proof_type.goal Tacmach.sigma
let goal_update (goal : goal_sigma) evar_map : goal_sigma=
let it = Tacmach.sig_it goal in
Tacmach.re_sig it evar_map
let fresh_evar goal ty : constr * goal_sigma =
let env = Tacmach.pf_env goal in
let evar_map = Tacmach.project goal in
let (em,x) = Evarutil.new_evar evar_map env ty in
x,( goal_update goal em)
let resolve_one_typeclass goal ty : constr*goal_sigma=
let env = Tacmach.pf_env goal in
let evar_map = Tacmach.project goal in
let em,c = Typeclasses.resolve_one_typeclass env evar_map ty in
c, (goal_update goal em)
let general_error =
"Cannot resolve a typeclass : please report"
let cps_resolve_one_typeclass ?error : Term.types -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic = fun t k goal ->
Tacmach.pf_apply
(fun env em -> let em ,c =
try Typeclasses.resolve_one_typeclass env em t
with Not_found ->
begin match error with
| None -> Util.anomaly "Cannot resolve a typeclass : please report"
| Some x -> Util.error x
end
in
Tacticals.tclTHENLIST [Refiner.tclEVARS em; k c] goal
) goal
let nf_evar goal c : Term.constr=
let evar_map = Tacmach.project goal in
Evarutil.nf_evar evar_map c
let evar_unit (gl : goal_sigma) (x : constr) : constr * goal_sigma =
let env = Tacmach.pf_env gl in
let evar_map = Tacmach.project gl in
let (em,x) = Evarutil.new_evar evar_map env x in
x,(goal_update gl em)
let evar_binary (gl: goal_sigma) (x : constr) =
let env = Tacmach.pf_env gl in
let evar_map = Tacmach.project gl in
let ty = mkArrow x (mkArrow x x) in
let (em,x) = Evarutil.new_evar evar_map env ty in
x,( goal_update gl em)
let evar_relation (gl: goal_sigma) (x: constr) =
let env = Tacmach.pf_env gl in
let evar_map = Tacmach.project gl in
let ty = mkArrow x (mkArrow x (mkSort prop_sort)) in
let (em,r) = Evarutil.new_evar evar_map env ty in
r,( goal_update gl em)
let cps_evar_relation (x: constr) k = fun goal ->
Tacmach.pf_apply
(fun env em ->
let ty = mkArrow x (mkArrow x (mkSort prop_sort)) in
let (em,r) = Evarutil.new_evar em env ty in
Tacticals.tclTHENLIST [Refiner.tclEVARS em; k r] goal
) goal
(* decomp_term : constr -> (constr, types) kind_of_term *)
let decomp_term c = kind_of_term (strip_outer_cast c)
let lapp c v = mkApp (Lazy.force c, v)
(** {2 Bindings with Coq' Standard Library} *)
module Std = struct
(* Here we package the module to be able to use List, later *)
module Pair = struct
let path = ["Coq"; "Init"; "Datatypes"]
let typ = lazy (init_constant path "prod")
let pair = lazy (init_constant path "pair")
let of_pair t1 t2 (x,y) =
mkApp (Lazy.force pair, [| t1; t2; x ; y|] )
end
module Bool = struct
let path = ["Coq"; "Init"; "Datatypes"]
let typ = lazy (init_constant path "bool")
let ctrue = lazy (init_constant path "true")
let cfalse = lazy (init_constant path "false")
let of_bool b =
if b then Lazy.force ctrue else Lazy.force cfalse
end
module Comparison = struct
let path = ["Coq"; "Init"; "Datatypes"]
let typ = lazy (init_constant path "comparison")
let eq = lazy (init_constant path "Eq")
let lt = lazy (init_constant path "Lt")
let gt = lazy (init_constant path "Gt")
end
module Leibniz = struct
let path = ["Coq"; "Init"; "Logic"]
let eq_refl = lazy (init_constant path "eq_refl")
let eq_refl ty x = lapp eq_refl [| ty;x|]
let eq ty = Term.mkApp (init_constant path "eq", [| ty |])
end
module Option = struct
let path = ["Coq"; "Init"; "Datatypes"]
let typ = lazy (init_constant path "option")
let some = lazy (init_constant path "Some")
let none = lazy (init_constant path "None")
let some t x = mkApp (Lazy.force some, [| t ; x|])
let none t = mkApp (Lazy.force none, [| t |])
let of_option t x = match x with
| Some x -> some t x
| None -> none t
end
module Pos = struct
let path = ["Coq" ; "PArith"; "BinPos"]
let typ = lazy (init_constant path "positive")
let xI = lazy (init_constant path "xI")
let xO = lazy (init_constant path "xO")
let xH = lazy (init_constant path "xH")
(* A coq positive from an int *)
let of_int n =
let rec aux n =
begin match n with
| n when n < 1 -> assert false
| 1 -> Lazy.force xH
| n -> mkApp
(
(if n mod 2 = 0
then Lazy.force xO
else Lazy.force xI
), [| aux (n/2)|]
)
end
in
aux n
end
module Nat = struct
let path = ["Coq" ; "Init"; "Datatypes"]
let typ = lazy (init_constant path "nat")
let _S = lazy (init_constant path "S")
let _O = lazy (init_constant path "O")
(* A coq nat from an int *)
let of_int n =
let rec aux n =
begin match n with
| n when n < 0 -> assert false
| 0 -> Lazy.force _O
| n -> mkApp
(
(Lazy.force _S
), [| aux (n-1)|]
)
end
in
aux n
end
(** Lists from the standard library*)
module List = struct
let path = ["Coq"; "Lists"; "List"]
let typ = lazy (init_constant path "list")
let nil = lazy (init_constant path "nil")
let cons = lazy (init_constant path "cons")
let cons ty h t =
mkApp (Lazy.force cons, [| ty; h ; t |])
let nil ty =
(mkApp (Lazy.force nil, [| ty |]))
let rec of_list ty = function
| [] -> nil ty
| t::q -> cons ty t (of_list ty q)
let type_of_list ty =
mkApp (Lazy.force typ, [|ty|])
end
(** Morphisms *)
module Classes =
struct
let classes_path = ["Coq";"Classes"; ]
let morphism = lazy (init_constant (classes_path@["Morphisms"]) "Proper")
let equivalence = lazy (init_constant (classes_path@ ["RelationClasses"]) "Equivalence")
let reflexive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Reflexive")
let transitive = lazy (init_constant (classes_path@ ["RelationClasses"]) "Transitive")
(** build the type [Equivalence ty rel] *)
let mk_equivalence ty rel =
mkApp (Lazy.force equivalence, [| ty; rel|])
(** build the type [Reflexive ty rel] *)
let mk_reflexive ty rel =
mkApp (Lazy.force reflexive, [| ty; rel|])
(** build the type [Proper rel t] *)
let mk_morphism ty rel t =
mkApp (Lazy.force morphism, [| ty; rel; t|])
(** build the type [Transitive ty rel] *)
let mk_transitive ty rel =
mkApp (Lazy.force transitive, [| ty; rel|])
end
module Relation = struct
type t =
{
carrier : constr;
r : constr;
}
let make ty r = {carrier = ty; r = r }
let split t = t.carrier, t.r
end
module Transitive = struct
type t =
{
carrier : constr;
leq : constr;
transitive : constr;
}
let make ty leq transitive = {carrier = ty; leq = leq; transitive = transitive}
let infer goal ty leq =
let ask = Classes.mk_transitive ty leq in
let transitive , goal = resolve_one_typeclass goal ask in
make ty leq transitive, goal
let from_relation goal rlt =
infer goal (rlt.Relation.carrier) (rlt.Relation.r)
let cps_from_relation rlt k =
let ty =rlt.Relation.carrier in
let r = rlt.Relation.r in
let ask = Classes.mk_transitive ty r in
cps_resolve_one_typeclass ask
(fun trans -> k (make ty r trans) )
let to_relation t =
{Relation.carrier = t.carrier; Relation.r = t.leq}
end
module Equivalence = struct
type t =
{
carrier : constr;
eq : constr;
equivalence : constr;
}
let make ty eq equivalence = {carrier = ty; eq = eq; equivalence = equivalence}
let infer goal ty eq =
let ask = Classes.mk_equivalence ty eq in
let equivalence , goal = resolve_one_typeclass goal ask in
make ty eq equivalence, goal
let from_relation goal rlt =
infer goal (rlt.Relation.carrier) (rlt.Relation.r)
let cps_from_relation rlt k =
let ty =rlt.Relation.carrier in
let r = rlt.Relation.r in
let ask = Classes.mk_equivalence ty r in
cps_resolve_one_typeclass ask (fun equiv -> k (make ty r equiv) )
let to_relation t =
{Relation.carrier = t.carrier; Relation.r = t.eq}
let split t =
t.carrier, t.eq, t.equivalence
end
end
(**[ match_as_equation goal eqt] see [eqt] as an equation. An
optionnal rel_context can be provided to ensure taht the term
remains typable*)
let match_as_equation ?(context = Term.empty_rel_context) goal equation : (constr*constr* Std.Relation.t) option =
let env = Tacmach.pf_env goal in
let env = Environ.push_rel_context context env in
let evar_map = Tacmach.project goal in
begin
match decomp_term equation with
| App(c,ca) when Array.length ca >= 2 ->
let n = Array.length ca in
let left = ca.(n-2) in
let right = ca.(n-1) in
let r = (mkApp (c, Array.sub ca 0 (n - 2))) in
let carrier = Typing.type_of env evar_map left in
let rlt =Std.Relation.make carrier r
in
Some (left, right, rlt )
| _ -> None
end
(** {1 Tacticals} *)
let tclTIME msg tac = fun gl ->
let u = Sys.time () in
let r = tac gl in
let _ = Pp.msgnl (Pp.str (Printf.sprintf "%s:%fs" msg (Sys.time ()-. u))) in
r
let tclDEBUG msg tac = fun gl ->
let _ = Pp.msgnl (Pp.str msg) in
tac gl
let tclPRINT tac = fun gl ->
let _ = Pp.msgnl (Printer.pr_constr (Tacmach.pf_concl gl)) in
tac gl
(** {1 Error related mechanisms} *)
(* functions to handle the failures of our tactic. Some should be
reported [anomaly], some are on behalf of the user [user_error]*)
let anomaly msg =
Util.anomaly ("[aac_tactics] " ^ msg)
let user_error msg =
Util.error ("[aac_tactics] " ^ msg)
let warning msg =
Pp.msg_warning (Pp.str ("[aac_tactics]" ^ msg))
(** {1 Rewriting tactics used in aac_rewrite} *)
module Rewrite = struct
(** Some informations about the hypothesis, with an (informal)
invariant:
- [typeof hyp = hyptype]
- [hyptype = forall context, body]
- [body = rel left right]
*)
type hypinfo =
{
hyp : Term.constr; (** the actual constr corresponding to the hypothese *)
hyptype : Term.constr; (** the type of the hypothesis *)
context : Term.rel_context; (** the quantifications of the hypothese *)
body : Term.constr; (** the body of the type of the hypothesis*)
rel : Std.Relation.t; (** the relation *)
left : Term.constr; (** left hand side *)
right : Term.constr; (** right hand side *)
l2r : bool; (** rewriting from left to right *)
}
let get_hypinfo c ~l2r ?check_type (k : hypinfo -> Proof_type.tactic) : Proof_type.tactic = fun goal ->
let ctype = Tacmach.pf_type_of goal c in
let (rel_context, body_type) = Term.decompose_prod_assum ctype in
let rec check f e =
match decomp_term e with
| Term.Rel i ->
let name, constr_option, types = Term.lookup_rel i rel_context
in f types
| _ -> Term.fold_constr (fun acc x -> acc && check f x) true e
in
begin match check_type with
| None -> ()
| Some f ->
if not (check f body_type)
then user_error "Unable to deal with higher-order or heterogeneous patterns";
end;
begin
match match_as_equation ~context:rel_context goal body_type with
| None ->
user_error "The hypothesis is not an applied relation"
| Some (hleft,hright,hrlt) ->
k {
hyp = c;
hyptype = ctype;
body = body_type;
l2r = l2r;
context = rel_context;
rel = hrlt ;
left =hleft;
right = hright;
}
goal
end
(* 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 debruijn
indices in [t] to the [constrs]. We need to handle the rest of the
indexes. Two ways :
- either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ]
- either building a term like [fun 1 2 => H tn 1 tn-2 2]
Both these terms have the same type.
*)
(* Fresh evars for everyone (should be the good way to do this
recompose in Coq v8.4) *)
let recompose_prod
(context : Term.rel_context)
(subst : (int * Term.constr) list)
env
em
: Evd.evar_map * Term.constr list =
(* the last index of rel relevant for the rewriting *)
let min_n = List.fold_left
(fun acc (x,_) -> min acc x)
(List.length context) subst in
let rec aux context acc em n =
let _ = Printf.printf "%i\n%!" n in
match context with
| [] ->
env, em, acc
| ((name,c,ty) as t)::q ->
let env, em, acc = aux q acc em (n+1) in
if n >= min_n
then
let em,x =
try em, List.assoc n subst
with | Not_found ->
Evarutil.new_evar em env (Term.substl acc ty)
in
(Environ.push_rel t env), em,x::acc
else
env,em,acc
in
let _,em,acc = aux context [] em 1 in
em, acc
(* no fresh evars : instead, use a lambda abstraction around an
application to handle non-instanciated variables. *)
let recompose_prod'
(context : Term.rel_context)
(subst : (int *Term.constr) list)
c
=
let rec popn pop n l =
if n <= 0 then l
else match l with
| [] -> []
| t::q -> pop t :: (popn pop (n-1) q)
in
let pop_rel_decl (name,c,ty) = (name,c,Termops.pop ty) in
let rec aux sign n next app ctxt =
match sign with
| [] -> List.rev app, List.rev ctxt
| decl::sign ->
try let term = (List.assoc n subst) in
aux sign (n+1) next (term::app) (None :: ctxt)
with
| Not_found ->
let term = Term.mkRel next in
aux sign (n+1) (next+1) (term::app) (Some decl :: ctxt)
in
let app,ctxt = aux context 1 1 [] [] in
(* substitutes in the context *)
let rec update ctxt app =
match ctxt,app with
| [],_ -> []
| _,[] -> []
| None :: sign, _ :: app ->
None :: update sign (List.map (Termops.pop) app)
| Some decl :: sign, _ :: app ->
Some (Term.substl_decl app decl)::update sign (List.map (Termops.pop) app)
in
let ctxt = update ctxt app in
(* updates the rel accordingly, taking some care not to go to far
beyond: it is important not to lift indexes homogeneously, we
have to update *)
let rec update ctxt res n =
match ctxt with
| [] -> List.rev res
| None :: sign ->
(update (sign) (popn pop_rel_decl n res) 0)
| Some decl :: sign ->
update sign (decl :: res) (n+1)
in
let ctxt = update ctxt [] 0 in
let c = Term.applistc c (List.rev app) in
let c = Term.it_mkLambda_or_LetIn c (ctxt) in
c
(* Version de Matthieu
let subst_rel_context k cstr ctx =
let (_, ctx') =
List.fold_left (fun (k, ctx') (id, b, t) -> (succ k, (id, Option.map
(Term.substnl [cstr] k) b, Term.substnl [cstr] k t) :: ctx')) (k, [])
ctx in List.rev ctx'
let recompose_prod' context subst c =
let len = List.length context in
let rec aux sign n next app ctxt =
match sign with
| [] -> List.rev app, List.rev ctxt
| decl::sign ->
try let term = (List.assoc n subst) in
aux (subst_rel_context 0 term sign) (pred n) (succ next)
(term::List.map (Term.lift (-1)) app) ctxt
with Not_found ->
let term = Term.mkRel (len - next) in
aux sign (pred n) (succ next) (term::app) (decl :: ctxt)
in
let app,ctxt = aux (List.rev context) len 0 [] [] in
Term.it_mkLambda_or_LetIn (Term.applistc c(app)) (List.rev ctxt)
*)
let build
(h : hypinfo)
(subst : (int *Term.constr) list)
(k :Term.constr -> Proof_type.tactic)
: Proof_type.tactic = fun goal ->
let c = recompose_prod' h.context subst h.hyp in
Tacticals.tclTHENLIST [k c] goal
let build_with_evar
(h : hypinfo)
(subst : (int *Term.constr) list)
(k :Term.constr -> Proof_type.tactic)
: Proof_type.tactic
= fun goal ->
Tacmach.pf_apply
(fun env em ->
let evar_map, acc = recompose_prod h.context subst env em in
let c = Term.applistc h.hyp (List.rev acc) in
Tacticals.tclTHENLIST [Refiner.tclEVARS evar_map; k c] goal
) goal
let rewrite ?(abort=false)hypinfo subst k =
build hypinfo subst
(fun rew ->
let tac =
if not abort then
Equality.general_rewrite_bindings
hypinfo.l2r
Termops.all_occurrences
true (* tell if existing evars must be frozen for instantiation *)
false
(rew,Glob_term.NoBindings)
true
else
Tacticals.tclIDTAC
in k tac
)
end
include Std
aac_tactics/coq.mli 0000664 0000000 0000000 00000020177 12211622232 0014601 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** 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:
- we use Caml's module system to mimic Coq's one, and avoid cluttering
the namespace;
- we also provide several handlers for standard coq tactics, in
order to provide a unified setting (we replace functions that
modify the evar_map by functions that operate on the whole
goal, and repack the modified evar_map with it).
*)
(** {2 Getting Coq terms from the environment} *)
val init_constant : string list -> string -> Term.constr
(** {2 General purpose functions} *)
type goal_sigma = Proof_type.goal Tacmach.sigma
val goal_update : goal_sigma -> Evd.evar_map -> goal_sigma
val resolve_one_typeclass : Proof_type.goal Tacmach.sigma -> Term.types -> Term.constr * goal_sigma
val cps_resolve_one_typeclass: ?error:string -> Term.types -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic
val nf_evar : goal_sigma -> Term.constr -> Term.constr
val fresh_evar :goal_sigma -> Term.types -> Term.constr* goal_sigma
val evar_unit :goal_sigma ->Term.constr -> Term.constr* goal_sigma
val evar_binary: goal_sigma -> Term.constr -> Term.constr* goal_sigma
val evar_relation: goal_sigma -> Term.constr -> Term.constr* goal_sigma
val cps_evar_relation : Term.constr -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic
(** [cps_mk_letin name v] binds the constr [v] using a letin tactic *)
val cps_mk_letin : string -> Term.constr -> ( Term.constr -> Proof_type.tactic) -> Proof_type.tactic
val decomp_term : Term.constr -> (Term.constr , Term.types) Term.kind_of_term
val lapp : Term.constr lazy_t -> Term.constr array -> Term.constr
(** {2 Bindings with Coq' Standard Library} *)
(** Coq lists *)
module List:
sig
(** [of_list ty l] *)
val of_list:Term.constr ->Term.constr list ->Term.constr
(** [type_of_list ty] *)
val type_of_list:Term.constr ->Term.constr
end
(** Coq pairs *)
module Pair:
sig
val typ:Term.constr lazy_t
val pair:Term.constr lazy_t
val of_pair : Term.constr -> Term.constr -> Term.constr * Term.constr -> Term.constr
end
module Bool : sig
val typ : Term.constr lazy_t
val of_bool : bool -> Term.constr
end
module Comparison : sig
val typ : Term.constr lazy_t
val eq : Term.constr lazy_t
val lt : Term.constr lazy_t
val gt : Term.constr lazy_t
end
module Leibniz : sig
val eq_refl : Term.types -> Term.constr -> Term.constr
val eq : Term.types -> Term.constr
end
module Option : sig
val some : Term.constr -> Term.constr -> Term.constr
val none : Term.constr -> Term.constr
val of_option : Term.constr -> Term.constr option -> Term.constr
end
(** Coq positive numbers (pos) *)
module Pos:
sig
val typ:Term.constr lazy_t
val of_int: int ->Term.constr
end
(** Coq unary numbers (peano) *)
module Nat:
sig
val typ:Term.constr lazy_t
val of_int: int ->Term.constr
end
(** Coq typeclasses *)
module Classes:
sig
val mk_morphism: Term.constr -> Term.constr -> Term.constr -> Term.constr
val mk_equivalence: Term.constr -> Term.constr -> Term.constr
val mk_reflexive: Term.constr -> Term.constr -> Term.constr
val mk_transitive: Term.constr -> Term.constr -> Term.constr
end
module Relation : sig
type t = {carrier : Term.constr; r : Term.constr;}
val make : Term.constr -> Term.constr -> t
val split : t -> Term.constr * Term.constr
end
module Transitive : sig
type t =
{
carrier : Term.constr;
leq : Term.constr;
transitive : Term.constr;
}
val make : Term.constr -> Term.constr -> Term.constr -> t
val infer : goal_sigma -> Term.constr -> Term.constr -> t * goal_sigma
val from_relation : goal_sigma -> Relation.t -> t * goal_sigma
val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic
val to_relation : t -> Relation.t
end
module Equivalence : sig
type t =
{
carrier : Term.constr;
eq : Term.constr;
equivalence : Term.constr;
}
val make : Term.constr -> Term.constr -> Term.constr -> t
val infer : goal_sigma -> Term.constr -> Term.constr -> t * goal_sigma
val from_relation : goal_sigma -> Relation.t -> t * goal_sigma
val cps_from_relation : Relation.t -> (t -> Proof_type.tactic) -> Proof_type.tactic
val to_relation : t -> Relation.t
val split : t -> Term.constr * Term.constr * Term.constr
end
(** [match_as_equation ?context goal c] try to decompose c as a
relation applied to two terms. An optionnal rel_context can be
provided to ensure that the term remains typable *)
val match_as_equation : ?context:Term.rel_context -> goal_sigma -> Term.constr -> (Term.constr * Term.constr * Relation.t) option
(** {2 Some tacticials} *)
(** time the execution of a tactic *)
val tclTIME : string -> Proof_type.tactic -> Proof_type.tactic
(** emit debug messages to see which tactics are failing *)
val tclDEBUG : string -> Proof_type.tactic -> Proof_type.tactic
(** print the current goal *)
val tclPRINT : Proof_type.tactic -> Proof_type.tactic
(** {2 Error related mechanisms} *)
val anomaly : string -> 'a
val user_error : string -> 'a
val warning : string -> unit
(** {2 Rewriting tactics used in aac_rewrite} *)
module Rewrite : sig
(** The rewriting tactics used in aac_rewrite, build as handlers of the usual [setoid_rewrite]*)
(** {2 Datatypes} *)
(** We keep some informations about the hypothesis, with an (informal)
invariant:
- [typeof hyp = typ]
- [typ = forall context, body]
- [body = rel left right]
*)
type hypinfo =
{
hyp : Term.constr; (** the actual constr corresponding to the hypothese *)
hyptype : Term.constr; (** the type of the hypothesis *)
context : Term.rel_context; (** the quantifications of the hypothese *)
body : Term.constr; (** the body of the hypothese*)
rel : Relation.t; (** the relation *)
left : Term.constr; (** left hand side *)
right : Term.constr; (** right hand side *)
l2r : bool; (** rewriting from left to right *)
}
(** [get_hypinfo H l2r ?check_type k] analyse the hypothesis H, and
build the related hypinfo, in CPS style. Moreover, an optionnal
function can be provided to check the type of every free
variable of the body of the hypothesis. *)
val get_hypinfo :Term.constr -> l2r:bool -> ?check_type:(Term.types -> bool) -> (hypinfo -> Proof_type.tactic) -> Proof_type.tactic
(** {2 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 :
- either using fresh evars and rewriting [H tn ?1 tn-2 ?2 ]
- either building a term like [fun 1 2 => H tn 1 tn-2 2]
Both these terms have the same type.
*)
(** build the constr to rewrite, in CPS style, with lambda abstractions *)
val build : hypinfo -> (int * Term.constr) list -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic
(** build the constr to rewrite, in CPS style, with evars *)
val build_with_evar : hypinfo -> (int * Term.constr) list -> (Term.constr -> Proof_type.tactic) -> Proof_type.tactic
(** [rewrite ?abort hypinfo subst] builds the rewriting tactic
associated with the given [subst] and [hypinfo], and feeds it to
the given continuation. If [abort] is set to [true], we build
[tclIDTAC] instead. *)
val rewrite : ?abort:bool -> hypinfo -> (int * Term.constr) list -> (Proof_type.tactic -> Proof_type.tactic) -> Proof_type.tactic
end
aac_tactics/evm_compute.ml 0000664 0000000 0000000 00000016120 12211622232 0016162 0 ustar 00root root 0000000 0000000 (*i camlp4deps: "parsing/grammar.cma" i*)
(*i camlp4use: "pa_extend.cmp" i*)
let pp_constr fmt x = Pp.pp_with fmt (Printer.pr_constr x)
let pp_list pp fmt l = List.iter (fun x -> Format.fprintf fmt "%a; " pp x) l
let pp_list_nl pp fmt l = List.iter (fun x -> Format.fprintf fmt "%a;\n" pp x) l
let pp_constrs fmt l = (pp_list pp_constr) fmt l
type constr = Term.constr
module Replace (X : sig val eq: Term.constr -> Term.constr -> bool
val subst : (Term.constr * Term.constr) list end) =
struct
(* assumes that [c] and [t] have no outer casts, and all
applications have been flattened *)
let rec find l (t: constr) =
match l with
| [] -> None
| (c,c') :: q ->
begin
match Term.kind_of_term c, Term.kind_of_term t with
| Term.App (f1,args1), Term.App (f2, args2) ->
let l1 = Array.length args1 in
let l2 = Array.length args2 in
if l1 <= l2 && X.eq c (Term.mkApp (f2, Array.sub args2 0 l1))
then
(* we must return the array of arguments, to make the
substitution in them too *)
Some (c',Array.sub args2 l1 (l2 - l1))
else
find q t
| _, _ ->
if X.eq c t
then Some (c', [| |])
else find q t
end
let replace_terms t =
let rec aux (k:int) t =
match find X.subst t with
| Some (t',v) ->
let v' = Array.map (Term.map_constr_with_binders (succ) aux k) v in
Term.mkApp (Term.lift k t', v')
| None ->
Term.map_constr_with_binders succ aux k t
in aux 0 t
end
let nowhere =
{ Tacexpr.onhyps = Some [];
Tacexpr.concl_occs = Glob_term.no_occurrences_expr
}
let mk_letin
(name:string)
(c: constr)
(k : Names .identifier -> Proof_type.tactic)
: Proof_type.tactic =
fun goal ->
let name = (Names.id_of_string name) in
let name = Tactics.fresh_id [] name goal in
let letin = (Tactics.letin_tac None (Names.Name name) c None nowhere) in
Tacticals.tclTHEN letin (k name) goal
let assert_tac
(name:string)
(c: constr)
(by:Proof_type.tactic)
(k : Names.identifier -> Proof_type.tactic)
: Proof_type.tactic =
fun goal ->
let name = (Names.id_of_string name) in
let name = Tactics.fresh_id [] name goal in
let t = (Tactics.assert_tac (Names.Name name) c) in
Tacticals.tclTHENS t [by; (k name)] goal
(* The contrib name is used to locate errors when loading constrs *)
let contrib_name = "evm_compute"
(* Getting constrs (primitive Coq terms) from existing Coq
libraries. *)
let find_constant contrib dir s =
Libnames.constr_of_global (Coqlib.find_reference contrib dir s)
let init_constant dir s = find_constant contrib_name dir s
module Leibniz = struct
let path = ["Coq"; "Init"; "Logic"]
let eq_refl t x=
Term.mkApp (init_constant path "eq_refl", [| t; x|])
let eq t x y =
Term.mkApp (init_constant path "eq", [| t; x ; y|])
let eq_ind_r ty x p px y yx =
Term.mkApp (init_constant path "eq_ind_r", [|ty;x;p;px;y;yx|])
end
let mk_vm_cast t c = Term.mkCast (c,Term.VMcast,t)
let mk_let
(name:Names.identifier)
(c: constr)
(t: constr)
(k : Names.identifier -> constr) =
Term.mkNamedLetIn name c t (Term.subst_vars [name] (k name))
let mk_fun
(name:Names.identifier)
(t: constr)
(k : Names.identifier -> constr) =
Term.mkNamedLambda name t (Term.subst_vars [name] (k name))
let rec has_evar x =
match Term.kind_of_term x with
| Term.Evar _ -> true
| Term.Rel _ | Term.Var _ | Term.Meta _ | Term.Sort _ | Term.Const _ | Term.Ind _ | Term.Construct _ ->
false
| Term.Cast (t1, _, t2) | Term.Prod (_, t1, t2) | Term.Lambda (_, t1, t2) ->
has_evar t1 || has_evar t2
| Term.LetIn (_, t1, t2, t3) ->
has_evar t1 || has_evar t2 || has_evar t3
| Term.App (t1, ts) ->
has_evar t1 || has_evar_array ts
| Term.Case (_, t1, t2, ts) ->
has_evar t1 || has_evar t2 || has_evar_array ts
| Term.Fix ((_, tr)) | Term.CoFix ((_, tr)) ->
has_evar_prec tr
and has_evar_array x =
Util.array_exists has_evar x
and has_evar_prec (_, ts1, ts2) =
Util.array_exists has_evar ts1 || Util.array_exists has_evar ts2
let evm_compute eq blacklist = fun gl ->
(* the type of the conclusion of the goal is [concl] *)
let concl = Tacmach.pf_concl gl in
let extra =
List.fold_left (fun acc (id,body,ty) ->
match body with
| None -> acc
| Some body -> if has_evar body then (Term.mkVar id :: acc) else acc)
[] (Tacmach.pf_hyps gl) in
(* the set of evars that appear in the goal *)
let evars = Evd.evar_list (Tacmach.project gl) concl in
(* the arguments of the function are: the constr that are blacklisted, then the evars *)
let args = extra @ blacklist @ (List.map (fun x -> Term.mkEvar x) evars) in
let argsv = Array.of_list args in
let context = (Termops.rel_list 0 (List.length args)) in
(* we associate to each argument the proper de bruijn index *)
let (subst: (Term.constr * Term.constr) list) = List.combine args context in
let module R = Replace(struct let eq = eq let subst = subst end) in
let t = R.replace_terms concl in
(* we have to retype both the blacklist and the evars to know how to build the final product *)
let rel_context = List.map (fun x -> Names.Anonymous, None, Tacmach.pf_type_of gl x) args in
(* the abstraction *)
let t = Term.it_mkLambda_or_LetIn t (List.rev rel_context) in
let typeof_t = (Tacmach.pf_type_of gl t) in
(* the normal form of the head function *)
let nft = Vnorm.cbv_vm (Tacmach.pf_env gl) t typeof_t in
let (!!) x = Tactics.fresh_id [] ((Names.id_of_string x)) gl in
(* p = [fun x => x a_i] which corresponds to the type of the goal when applied to [t] *)
let p = mk_fun (!! "x") typeof_t (fun x -> Term.mkApp (Term.mkVar x,argsv)) in
let proof_term pnft = begin
mk_let (!! "nft") nft typeof_t (fun nft -> let nft' = Term.mkVar nft in
mk_let (!! "t") t typeof_t (fun t -> let t' = Term.mkVar t in
mk_let (!! "H") (mk_vm_cast (Leibniz.eq typeof_t t' nft') (Leibniz.eq_refl typeof_t nft')) (Leibniz.eq typeof_t t' nft')
(fun h ->
(* typeof_t -> Prop *)
let body = Leibniz.eq_ind_r typeof_t
nft' p pnft t' (Term.mkVar h)
in
Term.mkCast (body, Term.DEFAULTcast, Term.mkApp (t', argsv))
)))
end in
try
assert_tac "subgoal" (Term.mkApp (p,[| nft |]))
Tacticals.tclIDTAC
(fun subgoal ->
(* We use the tactic [exact_no_check]. It has two consequences:
- first, it means that in theory we could produce ill typed proof terms, that fail to type check at Qed;
- second, it means that we are able to use vm_compute and vm_compute casts, that will be checkable at Qed time when all evars have been instantiated.
*)
Tactics.exact_no_check (proof_term (Term.mkVar subgoal))
) gl
with
| e ->
Tacticals.tclFAIL 0 (Pp.str (Printf.sprintf "evm_compute failed with an exception %s" (Printexc.to_string e))) gl
;;
let evm_compute_in eq blacklist h = fun gl ->
let concl = Tacmach.pf_concl gl in
Tacticals.tclTHENLIST
[Tactics.revert [h];
evm_compute eq (concl :: blacklist);
Tactics.introduction h
]
gl
aac_tactics/evm_compute.mli 0000664 0000000 0000000 00000001016 12211622232 0016331 0 ustar 00root root 0000000 0000000
(** [evm_compute eq blacklist] performs a vm_compute step with the
following provisos: evars can appear in the goal; terms that are
equal (modulo eq) to terms in the blacklist are abstracted
before-hand. *)
val evm_compute : (Term.constr -> Term.constr -> bool) -> Term.constr list -> Proof_type.tactic
(** [evm_compute eq blacklist h] performs an evm_compute step in the hypothesis h *)
val evm_compute_in : (Term.constr -> Term.constr -> bool) -> Term.constr list -> Names.identifier -> Proof_type.tactic
aac_tactics/files.txt 0000664 0000000 0000000 00000000215 12211622232 0015146 0 ustar 00root root 0000000 0000000 AAC_coq.ml
AAC_helper.ml
AAC_search_monad.ml
AAC_matcher.ml
AAC_theory.ml
AAC_print.ml
AAC_rewrite.ml
AAC.v
Instances.v
Tutorial.v
Caveats.v
aac_tactics/helper.ml 0000664 0000000 0000000 00000002162 12211622232 0015117 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
module type CONTROL = sig
val debug : bool
val time : bool
val printing : bool
end
module Debug (X : CONTROL) = struct
open X
let debug x =
if debug then
Printf.printf "%s\n%!" x
let time f x fmt =
if time then
let t = Sys.time () in
let r = f x in
Printf.printf fmt (Sys.time () -. t);
r
else f x
let pr_constr msg constr =
if printing then
( Pp.msgnl (Pp.str (Printf.sprintf "=====%s====" msg));
Pp.msgnl (Printer.pr_constr constr);
)
let debug_exception msg e =
debug (msg ^ (Printexc.to_string e))
end
aac_tactics/helper.mli 0000664 0000000 0000000 00000002355 12211622232 0015274 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Debugging functions, that can be triggered on a per-file base. *)
module type CONTROL =
sig
val debug : bool
val time : bool
val printing : bool
end
module Debug :
functor (X : CONTROL) ->
sig
(** {!debug} prints the string and end it with a newline *)
val debug : string -> unit
val debug_exception : string -> exn -> unit
(** {!time} computes the time spent in a function, and then
print it using the given format *)
val time :
('a -> 'b) -> 'a -> (float -> unit, out_channel, unit) format -> 'b
(** {!pr_constr} print a Coq constructor, that can be labelled
by a string *)
val pr_constr : string -> Term.constr -> unit
end
aac_tactics/matcher.ml 0000664 0000000 0000000 00000106002 12211622232 0015261 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** This module defines our matching functions, modulo associativity
and commutativity (AAC).
The basic idea is to find a substitution [env] such that the
pattern [p] instantiated by [env] is equal to [t] modulo AAC.
We proceed by structural decomposition of the pattern, and try all
possible non-deterministic split of the subject when needed. The
function {!matcher} is limited to top-level matching, that is, the
subject must make a perfect match against the pattern ([x+x] do
not match [a+a+b] ). We use a search monad {!Search} to perform
non-deterministic splits in an almost transparent way. We also
provide a function {!subterm} for finding a match that is a
subterm modulo AAC of the subject. Therefore, we are able to solve
the aforementioned case [x+x] against [a+b+a].
This file is structured as follows. First, we define the search
monad. Then,we define the two representations of terms (one
representing the AST, and one in normal form ), and environments
from variables to terms. Then, we use these parts to solve
matching problem. Finally, we wrap this function {!matcher} into
{!subterm}
*)
module Control =
struct
let debug = false
let time = false
let printing = false
end
module Debug = Helper.Debug (Control)
open Debug
module Search = Search_monad (* a handle *)
type symbol = int
type var = int
type units = (symbol * symbol) list (* from AC/A symbols to the unit *)
type ext_units =
{
unit_for : units;
is_ac : (symbol * bool) list
}
let print_units units=
List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units;
Printf.printf "\n%!"
exception NoUnit
let get_unit units op =
try List.assoc op units
with
| Not_found -> raise NoUnit
let is_unit units op unit = List.mem (op,unit) units
open Search
type 'a mset = ('a * int) list
let linear p =
let rec ncons t l = function
| 0 -> l
| n -> t::ncons t l (n-1)
in
let rec aux = function
[ ] -> []
| (t,n)::q -> let q = aux q in
ncons t q n
in aux p
(** The module {!Terms} defines two different types for expressions.
- a public type {!Terms.t} that represent abstract syntax trees of
expressions with binary associative (and commutative) operators
- a private type {!Terms.nf_term} that represent an equivalence
class for terms that are equal modulo AAC. The constructions
functions on this type ensure the property that the term is in
normal form (that is, no sum can appear as a subterm of the same
sum, no trailing units, etc...).
*)
module Terms : sig
(** {1 Abstract syntax tree of terms}
Terms represented using this datatype are representation of the
AST of an expression. *)
type t =
Dot of (symbol * t * t)
| Plus of (symbol * t * t)
| Sym of (symbol * t array)
| Var of var
| Unit of symbol
val equal_aac : units -> t -> t -> bool
val size: t -> int
(** {1 Terms in normal form}
A term in normal form is the canonical representative of the
equivalence class of all the terms that are equal modulo
Associativity and Commutativity. Outside the {!Matcher} module,
one does not need to access the actual representation of this
type. *)
type nf_term = private
| TAC of symbol * nf_term mset
| TA of symbol * nf_term list
| TSym of symbol * nf_term list
| TUnit of symbol
| TVar of var
(** {2 Fresh variables: we provide some functions to pick some fresh variables with respect to a term} *)
val fresh_var_term : t -> int
val fresh_var_nfterm : nf_term -> int
(** {2 Constructors: we ensure that the terms are always
normalised
braibant - Fri 27 Aug 2010, 15:11
Moreover, we assure that we will not build empty sums or empty
products, leaving this task to the caller function.
}
*)
val mk_TAC : units -> symbol -> nf_term mset -> nf_term
val mk_TA : units -> symbol -> nf_term list -> nf_term
val mk_TSym : symbol -> nf_term list -> nf_term
val mk_TVar : var -> nf_term
val mk_TUnit : symbol -> nf_term
(** {2 Comparisons} *)
val nf_term_compare : nf_term -> nf_term -> int
val nf_equal : nf_term -> nf_term -> bool
(** {2 Printing function} *)
val sprint_nf_term : nf_term -> string
(** {2 Conversion functions} *)
val term_of_t : units -> t -> nf_term
val t_of_term : nf_term -> t (* we could return the units here *)
end
= struct
type t =
Dot of (symbol * t * t)
| Plus of (symbol * t * t)
| Sym of (symbol * t array)
| Var of var
| Unit of symbol
let rec size = function
| Dot (_,x,y)
| Plus (_,x,y) -> size x+ size y + 1
| Sym (_,v)-> Array.fold_left (fun acc x -> size x + acc) 1 v
| _ -> 1
type nf_term =
| TAC of symbol * nf_term mset
| TA of symbol * nf_term list
| TSym of symbol * nf_term list
| TUnit of symbol
| TVar of var
(** {2 Picking fresh variables} *)
(** [fresh_var_term] picks a fresh variable with respect to a term *)
let fresh_var_term t =
let rec aux = function
| Dot (_,t1,t2)
| Plus (_,t1,t2) -> max (aux t1) (aux t2)
| Sym (_,v) -> Array.fold_left (fun acc x -> max acc (aux x)) 0 v
| Var v -> assert (v >= 0); v
| Unit _ -> 0
in
aux t
(** [fresh_var_nfterm] picks a fresh_variable with respect to a term *)
let fresh_var_nfterm t =
let rec aux = function
| TAC (_,l) -> List.fold_left (fun acc (x,_) -> max acc (aux x)) 0 l
| TA (_,l)
| TSym (_,l) -> List.fold_left (fun acc x -> max acc (aux x)) 0 l
| TVar v -> assert (v >= 0); v
| TUnit _ -> 0
in
aux t
(** {2 Constructors: we ensure that the terms are always
normalised} *)
(** {3 Pre constructors : These constructors ensure that sums and
products are not degenerated (no trailing units)} *)
let mk_TAC' units (s : symbol) l = match l with
| [] -> TUnit (get_unit units s )
| [_,0] -> assert false
| [t,1] -> t
| _ -> TAC (s,l)
let mk_TA' units (s : symbol) l = match l with
| [] -> TUnit (get_unit units s )
| [t] -> t
| _ -> TA (s,l)
(** {2 Comparison} *)
let nf_term_compare = Pervasives.compare
let nf_equal a b =
a = b
(** [merge_ac comp l1 l2] merges two lists of terms with coefficients
into one. Terms that are equal modulo the comparison function
[comp] will see their arities added. *)
(* This function could be improved by the use of sorted msets *)
let merge_ac (compare : 'a -> 'a -> int) (l : 'a mset) (l' : 'a mset) : 'a mset =
let rec aux l l'=
match l,l' with
| [], _ -> l'
| _, [] -> l
| (t,tar)::q, (t',tar')::q' ->
begin match compare t t' with
| 0 -> ( t,tar+tar'):: aux q q'
| -1 -> (t, tar):: aux q l'
| _ -> (t', tar'):: aux l q'
end
in aux l l'
(** [merge_map f l] uses the combinator [f] to combine the head of the
list [l] with the merge_maped tail of [l] *)
let rec merge_map (f : 'a -> 'b list -> 'b list) (l : 'a list) : 'b list =
match l with
| [] -> []
| t::q -> f t (merge_map f q)
(** This function has to deal with the arities *)
let rec merge (l : nf_term mset) (l' : nf_term mset) : nf_term mset=
merge_ac nf_term_compare l l'
let extract_A units s t =
match t with
| TA (s',l) when s' = s -> l
| TUnit u when is_unit units s u -> []
| _ -> [t]
let extract_AC units s (t,ar) : nf_term mset =
match t with
| TAC (s',l) when s' = s -> List.map (fun (x,y) -> (x,y*ar)) l
| TUnit u when is_unit units s u -> []
| _ -> [t,ar]
(** {3 Constructors of {!nf_term}}*)
let mk_TAC units (s : symbol) (l : (nf_term *int) list) =
mk_TAC' units s
(merge_map (fun u l -> merge (extract_AC units s u) l) l)
let mk_TA units s l =
mk_TA' units s
(merge_map (fun u l -> (extract_A units s u) @ l) l)
let mk_TSym s l = TSym (s,l)
let mk_TVar v = TVar v
let mk_TUnit s = TUnit s
(** {2 Printing function} *)
let print_binary_list (single : 'a -> string)
(unit : string)
(binary : string -> string -> string) (l : 'a list) =
let rec aux l =
match l with
[] -> unit
| [t] -> single t
| t::q ->
let r = aux q in
Printf.sprintf "%s" (binary (single t) r)
in
aux l
let print_symbol s =
match s with
| s, None -> Printf.sprintf "%i" s
| s , Some u -> Printf.sprintf "%i(unit %i)" s u
let sprint_ac (single : 'a -> string) (l : 'a mset) =
(print_binary_list
(fun (x,t) ->
if t = 1
then single x
else Printf.sprintf "%i*%s" t (single x)
)
"0"
(fun x y -> x ^ " , " ^ y)
l
)
let print_symbol single s l =
match l with
[] -> Printf.sprintf "%i" s
| _ ->
Printf.sprintf "%i(%s)"
s
(print_binary_list single "" (fun x y -> x ^ "," ^ y) l)
let print_ac single s l =
Printf.sprintf "[%s:AC]{%s}"
(string_of_int s )
(sprint_ac
single
l
)
let print_a single s l =
Printf.sprintf "[%s:A]{%s}"
(string_of_int s)
(print_binary_list single "1" (fun x y -> x ^ " , " ^ y) l)
let rec sprint_nf_term = function
| TSym (s,l) -> print_symbol sprint_nf_term s l
| TAC (s,l) -> print_ac sprint_nf_term s l
| TA (s,l) -> print_a sprint_nf_term s l
| TVar v -> Printf.sprintf "x%i" v
| TUnit s -> Printf.sprintf "unit%i" s
(** {2 Conversion functions} *)
(* rebuilds a tree out of a list, under the assumption that the list is not empty *)
let rec binary_of_list f comb l =
let l = List.rev l in
let rec aux = function
| [] -> assert false
| [t] -> f t
| t::q -> comb (aux q) (f t)
in
aux l
let term_of_t units : t -> nf_term =
let rec term_of_t = function
| Dot (s,l,r) ->
let l = term_of_t l in
let r = term_of_t r in
mk_TA units s [l;r]
| Plus (s,l,r) ->
let l = term_of_t l in
let r = term_of_t r in
mk_TAC units s [l,1;r,1]
| Unit x ->
mk_TUnit x
| Sym (s,t) ->
let t = Array.to_list t in
let t = List.map term_of_t t in
mk_TSym s t
| Var i ->
mk_TVar ( i)
in
term_of_t
let rec t_of_term : nf_term -> t = function
| TAC (s,l) ->
(binary_of_list
t_of_term
(fun l r -> Plus ( s,l,r))
(linear l)
)
| TA (s,l) ->
(binary_of_list
t_of_term
(fun l r -> Dot ( s,l,r))
l
)
| TSym (s,l) ->
let v = Array.of_list l in
let v = Array.map (t_of_term) v in
Sym ( s,v)
| TVar x -> Var x
| TUnit s -> Unit s
let equal_aac units x y =
nf_equal (term_of_t units x) (term_of_t units y)
end
(** Terms environments defined as association lists from variables to
terms in normal form {! Terms.nf_term} *)
module Subst : sig
type t
val find : t -> var -> Terms.nf_term option
val add : t -> var -> Terms.nf_term -> t
val empty : t
val instantiate : t -> Terms.t -> Terms.t
val sprint : t -> string
val to_list : t -> (var*Terms.t) list
end
=
struct
open Terms
(** Terms environments, with nf_terms, to avoid costly conversions
of {!Terms.nf_terms} to {!Terms.t}, that will be mostly discarded*)
type t = (var * nf_term) list
let find : t -> var -> nf_term option = fun t x ->
try Some (List.assoc x t) with | _ -> None
let add t x v = (x,v) :: t
let empty = []
let sprint (l : t) =
match l with
| [] -> Printf.sprintf "Empty environment\n"
| _ ->
let s = List.fold_left
(fun acc (x,y) ->
Printf.sprintf "%sX%i -> %s\n"
acc
x
(sprint_nf_term y)
)
""
(List.rev l) in
Printf.sprintf "%s\n%!" s
(** [instantiate] is an homomorphism except for the variables*)
let instantiate (t: t) (x:Terms.t) : Terms.t =
let rec aux = function
| Unit _ as x -> x
| Sym (s,t) -> Sym (s,Array.map aux t)
| Plus (s,l,r) -> Plus (s, aux l, aux r)
| Dot (s,l,r) -> Dot (s, aux l, aux r)
| Var i ->
begin match find t i with
| None -> Util.error "aac_tactics: instantiate failure"
| Some x -> t_of_term x
end
in aux x
let to_list t = List.map (fun (x,y) -> x,Terms.t_of_term y) t
end
(******************)
(* MATCHING UTILS *)
(******************)
open Terms
(** Since most of the folowing functions require an extra parameter,
the units, we package them in a module. This functor will then be
applied to a module containing the units, in the exported
functions. *)
module M (X : sig
val units : units
val is_ac : (symbol * bool) list
val strict : bool (* variables cannot be instantiated with units *)
end) = struct
open X
let print_units ()=
List.iter (fun (op,unit) -> Printf.printf "%i %i\t" op unit) units;
Printf.printf "\n%!"
let mk_TAC s l = mk_TAC units s l
let mk_TA s l = mk_TA units s l
let mk_TAC' s l =
try return( mk_TAC s l)
with _ -> fail ()
let mk_TA' s l =
try return( mk_TA s l)
with _ -> fail ()
(** First, we need to be able to perform non-deterministic choice of
term splitting to satisfy a pattern. Indeed, we want to show that:
(x+a*b)*c <= a*b*c
*)
let a_nondet_split_raw t : ('a list * 'a list) m =
let rec aux l l' =
match l' with
| [] ->
return ( l,[])
| t::q ->
return ( l,l' )
>>| aux (l @ [t]) q
in
aux [] t
(** Same as the previous [a_nondet_split], but split the list in 3
parts *)
let a_nondet_middle t : ('a list * 'a list * 'a list) m =
a_nondet_split_raw t >>
(fun (left, right) ->
a_nondet_split_raw left >>
(fun (left, middle) -> return (left, middle, right))
)
(** Non deterministic splits of ac lists *)
let dispatch f n =
let rec aux k =
if k = 0 then return (f n 0)
else return (f (n-k) k) >>| aux (k-1)
in
aux (n )
let add_with_arith x ar l =
if ar = 0 then l else (x,ar) ::l
let ac_nondet_split_raw (l : 'a mset) : ('a mset * 'a mset) m =
let rec aux = function
| [] -> return ([],[])
| (t,tar)::q ->
aux q
>>
(fun (left,right) ->
dispatch (fun arl arr ->
add_with_arith t arl left,
add_with_arith t arr right
)
tar
)
in
aux l
let a_nondet_split current t : (nf_term * nf_term list) m=
a_nondet_split_raw t
>>
(fun (l,r) ->
if strict && (l=[])
then fail()
else
mk_TA' current l >>
fun t -> return (t, r)
)
let ac_nondet_split current t : (nf_term * nf_term mset) m=
ac_nondet_split_raw t
>>
(fun (l,r) ->
if strict && (l=[])
then fail()
else
mk_TAC' current l >>
fun t -> return (t, r)
)
(** Try to affect the variable [x] to each left factor of [t]*)
let var_a_nondet_split env current x t =
a_nondet_split current t
>>
(fun (t,res) ->
return ((Subst.add env x t), res)
)
(** Try to affect variable [x] to _each_ subset of t. *)
let var_ac_nondet_split (current: symbol) env (x : var) (t : nf_term mset) : (Subst.t * (nf_term mset)) m =
ac_nondet_split current t
>>
(fun (t,res) ->
return ((Subst.add env x t), res)
)
(** See the term t as a given AC symbol. Unwrap the first constructor
if necessary *)
let get_AC (s : symbol) (t : nf_term) : (nf_term *int) list =
match t with
| TAC (s',l) when s' = s -> l
| TUnit u when is_unit units s u -> []
| _ -> [t,1]
(** See the term t as a given A symbol. Unwrap the first constructor
if necessary *)
let get_A (s : symbol) (t : nf_term) : nf_term list =
match t with
| TA (s',l) when s' = s -> l
| TUnit u when is_unit units s u -> []
| _ -> [t]
(** See the term [t] as an symbol [s]. Fail if it is not such
symbol. *)
let get_Sym s t =
match t with
| TSym (s',l) when s' = s -> return l
| _ -> fail ()
(*************)
(* A Removal *)
(*************)
(** We remove the left factor v in a term list. This function runs
linearly with respect to the size of the first pattern symbol *)
let left_factor current (v : nf_term) (t : nf_term list) =
let rec aux a b =
match a,b with
| t::q , t' :: q' when nf_equal t t' -> aux q q'
| [], q -> return q
| _, _ -> fail ()
in
match v with
| TA (s,l) when s = current -> aux l t
| TUnit u when is_unit units current u -> return t
| _ ->
begin match t with
| [] -> fail ()
| t::q ->
if nf_equal v t
then return q
else fail ()
end
(**************)
(* AC Removal *)
(**************)
(** {!pick_sym} gather all elements of a list that satisfies a
predicate, and combine them with the residual of the list. That
is, each element of the residual contains exactly one element less
than the original term.
TODO : This function not as efficient as it could be, using a
proper data-structure *)
let pick_sym (s : symbol) (t : nf_term mset ) =
let rec aux front back =
match back with
| [] -> fail ()
| (t,tar)::q ->
begin match t with
| TSym (s',v') when s = s' ->
let back =
if tar > 1
then (t,tar -1) ::q
else q
in
return (v' , List.rev_append front back )
>>| aux ((t,tar)::front) q
| _ -> aux ((t,tar)::front) q
end
in
aux [] t
(** We have to check if we are trying to remove a unit from a term. Could also be located in Terms*)
let is_unit_AC s t =
try nf_equal t (mk_TAC s [])
with | NoUnit -> false
let is_same_AC s t : nf_term mset option=
match t with
TAC (s',l) when s = s' -> Some l
| _ -> None
(** We want to remove the term [v] from the term list [t] under an AC
symbol *)
let single_AC_factor (s : symbol) (v : nf_term) v_ar (t : nf_term mset) : (nf_term mset) m =
let rec aux front back =
match back with
| [] -> fail ()
| (t,tar)::q ->
begin
if nf_equal v t
then
match () with
| _ when tar < v_ar -> fail ()
| _ when tar = v_ar -> return (List.rev_append front q)
| _ -> return (List.rev_append front ((t,tar-v_ar)::q))
else
aux ((t,tar) :: front) q
end
in
if is_unit_AC s v
then
return t
else
aux [] t
(* Remove a constant from a mset. If this constant is also a mset for
the same symbol, we remove every elements, one at a time (and we do
care of the arities) *)
let factor_AC (s : symbol) (v: nf_term) (t : nf_term mset) : ( nf_term mset ) m =
match is_same_AC s v with
| None -> single_AC_factor s v 1 t
| Some l ->
(* We are trying to remove an AC factor *)
List.fold_left (fun acc (v,v_ar) ->
acc >> (single_AC_factor s v v_ar)
)
(return t)
l
(** [tri_fold f l acc] folds on the list [l] and give to f the
beginning of the list in reverse order, the considered element, and
the last part of the list
as an exemple, on the list [1;2;3;4], we get the trace
f () [] 1 [2; 3; 4]
f () [1] 2 [3; 4]
f () [2;1] 3 [ 4]
f () [3;2;1] 4 []
it is the duty of the user to reverse the front if needed
*)
let tri_fold f (l : 'a list) (acc : 'b)= match l with
[] -> acc
| _ ->
let _,_,acc = List.fold_left (fun acc (t : 'a) ->
let l,r,acc = acc in
let r = List.tl r in
t::l,r,f acc l t r
) ([], l,acc) l
in acc
(* let test = tri_fold (fun acc l t r -> (l,t,r) :: acc) [1;2;3;4] [] *)
(*****************************)
(* ENUMERATION DES POSITIONS *)
(*****************************)
(** The pattern is a unit: we need to try to make it appear at each
position. We do not need to go further with a real matching, since
the match should be trivial. Hence, we proceed recursively to
enumerate all the positions. *)
module Positions = struct
let ac (l: 'a mset) : ('a mset * 'a )m =
let rec aux = function
| [] -> assert false
| [t,1] -> return ([],t)
| [t,tar] -> return ([t,tar -1],t)
| (t,tar) as h :: q ->
(aux q >> (fun (c,x) -> return (h :: c,x)))
>>| (if tar > 1 then return ((t,tar-1) :: q,t) else return (q,t))
in
aux l
let ac' current (l: 'a mset) : ('a mset * 'a )m =
ac_nondet_split_raw l >>
(fun (l,r) ->
if l = [] || r = []
then fail ()
else
mk_TAC' current r >>
fun t -> return (l, t)
)
let a (l : 'a list) : ('a list * 'a * 'a list) m =
let rec aux left right : ('a list * 'a * 'a list) m =
match right with
| [] -> assert false
| [t] -> return (left,t,[])
| t::q ->
aux (left@[t]) q
>>| return (left,t,q)
in
aux [] l
end
let build_ac (current : symbol) (context : nf_term mset) (p : t) : t m=
if context = []
then return p
else
mk_TAC' current context >>
fun t -> return (Plus (current,t_of_term t,p))
let build_a (current : symbol)
(left : nf_term list) (right : nf_term list) (p : t) : t m=
let right_pat p =
if right = []
then return p
else
mk_TA' current right >>
fun t -> return (Dot (current,p,t_of_term t))
in
let left_pat p=
if left = []
then return p
else
mk_TA' current left >>
fun t -> return (Dot (current,t_of_term t,p))
in
right_pat p >> left_pat >> (fun p -> return p)
let conts (hole : t) (l : symbol list) p : t m =
let p = t_of_term p in
(* - aller chercher les symboles ac et les symboles a
- construire pour chaque
* * +
/ \ / \ / \
1 p p 1 p 1
*)
let ac,a = List.partition (fun s -> List.assoc s is_ac) l in
let acc = fail () in
let acc = List.fold_left
(fun acc s ->
acc >>| return (Plus (s,p,hole))
) acc ac in
let acc =
List.fold_left
(fun acc s ->
acc >>| return (Dot (s,p,hole)) >>| return (Dot (s,hole,p))
) acc a
in acc
(**
Return the same thing as subterm :
- The size of the context
- The context
- A collection of substitutions (here == return Subst.empty)
*)
let unit_subterm (t : nf_term) (unit : symbol) (hole : t): (int * t * Subst.t m) m =
let symbols = List.fold_left
(fun acc (ac,u) -> if u = unit then ac :: acc else acc ) [] units
in
(* make a unit appear at each strict sub-position of the term*)
let rec positions (t : nf_term) : t m =
match t with
(* with a final unit at the end *)
| TAC (s,l) ->
let symbols' = List.filter (fun x -> x <> s) symbols in
(
ac_nondet_split_raw l >>
(fun (l,r) -> if l = [] || r = [] then fail () else
(
match r with
| [p,1] ->
positions p >>| conts hole symbols' p
| _ ->
mk_TAC' s r >> conts hole symbols'
) >> build_ac s l ))
| TA (s,l) ->
let symbols' = List.filter (fun x -> x <> s) symbols in
(
(* first the other symbols, and then the more simple case of
this particular symbol *)
a_nondet_middle l >>
(fun (l,m,r) ->
(* meant to break the symmetry *)
if (l = [] && r = [])
then fail ()
else
(
match m with
| [p] ->
positions p >>| conts hole symbols' p
| _ ->
mk_TA' s m >> conts hole symbols'
) >> build_a s l r ))
>>|
(
if List.mem s symbols then
begin match l with
| [a] -> assert false
| [a;b] -> build_a s [a] [b] (hole)
| _ ->
(* on ne construit que les elements interieurs,
d'ou la disymetrie *)
(Positions.a l >>
(fun (left,p,right) ->
if left = [] then fail () else
(build_a s left right (Dot (s,hole,t_of_term p)))))
end
else fail ()
)
| TSym (s,l) ->
tri_fold (fun acc l t r ->
((positions t) >>
(fun (p) ->
let l = List.map t_of_term l in
let r = List.map t_of_term r in
return (Sym (s, Array.of_list (List.rev_append l (p::r)))) ))
>>|
(
conts hole symbols t >>
(fun (p) ->
let l = List.map t_of_term l in
let r = List.map t_of_term r in
return (Sym (s, Array.of_list (List.rev_append l (p::r)))) )
)
>>| acc
) l (fail())
| TVar x -> assert false
| TUnit x when x = unit -> return (hole)
| TUnit x as t -> conts hole symbols t
in
(positions t
>>|
(match t with
| TSym _ -> conts hole symbols t
| TAC (s,l) -> conts hole symbols t
| TA (s,l) -> conts hole symbols t
| _ -> fail())
)
>> fun (p) -> return (Terms.size p,p,return Subst.empty)
(************)
(* Matching *)
(************)
(** {!matching} is the generic matching judgement. Each time a
non-deterministic split is made, we have to come back to this one.
{!matchingSym} is used to match two applications that have the
same (free) head-symbol.
{!matchingAC} is used to match two sums (with the subtlety that
[x+y] matches [f a] which is a function application or [a*b] which
is a product).
{!matchingA} is used to match two products (with the subtlety that
[x*y] matches [f a] which is a function application, or [a+b]
which is a sum).
*)
let matching : Subst.t -> nf_term -> nf_term -> Subst.t Search.m=
let rec matching env (p : nf_term) (t: nf_term) : Subst.t Search.m=
match p with
| TAC (s,l) ->
let l = linear l in
matchingAC env s l (get_AC s t)
| TA (s,l) ->
matchingA env s l (get_A s t)
| TSym (s,l) ->
(get_Sym s t)
>> (fun t -> matchingSym env l t)
| TVar x ->
begin match Subst.find env x with
| None -> return (Subst.add env x t)
| Some v -> if nf_equal v t then return env else fail ()
end
| TUnit s ->
if nf_equal p t then return env else fail ()
and
matchingAC (env : Subst.t) (current : symbol) (l : nf_term list) (t : (nf_term *int) list) =
match l with
| TSym (s,v):: q ->
pick_sym s t
>> (fun (v',t') ->
matchingSym env v v'
>> (fun env -> matchingAC env current q t'))
| TAC (s,v)::q when s = current ->
assert false
| TVar x:: q -> (* This is an optimization *)
begin match Subst.find env x with
| None ->
(var_ac_nondet_split current env x t)
>> (fun (env,residual) -> matchingAC env current q residual)
| Some v ->
(factor_AC current v t)
>> (fun residual -> matchingAC env current q residual)
end
| TUnit s as v :: q -> (* this is an optimization *)
(factor_AC current v t) >>
(fun residual -> matchingAC env current q residual)
| h :: q ->(* PAC =/= curent or PA or unit for this symbol*)
(ac_nondet_split current t)
>>
(fun (t,right) ->
matching env h t
>>
(
fun env ->
matchingAC env current q right
)
)
| [] -> if t = [] then return env else fail ()
and
matchingA (env : Subst.t) (current : symbol) (l : nf_term list) (t : nf_term list) =
match l with
| TSym (s,v) :: l ->
begin match t with
| TSym (s',v') :: r when s = s' ->
(matchingSym env v v')
>> (fun env -> matchingA env current l r)
| _ -> fail ()
end
| TA (s,v) :: l when s = current ->
assert false
| TVar x :: l ->
begin match Subst.find env x with
| None ->
debug (Printf.sprintf "var %i (%s)" x
(let b = Buffer.create 21 in List.iter (fun t -> Buffer.add_string b ( Terms.sprint_nf_term t)) t; Buffer.contents b ));
var_a_nondet_split env current x t
>> (fun (env,residual)-> debug (Printf.sprintf "pl %i %i" x(List.length residual)); matchingA env current l residual)
| Some v ->
(left_factor current v t)
>> (fun residual -> matchingA env current l residual)
end
| TUnit s as v :: q -> (* this is an optimization *)
(left_factor current v t) >>
(fun residual -> matchingA env current q residual)
| h :: l ->
a_nondet_split current t
>> (fun (t,r) ->
matching env h t
>> (fun env -> matchingA env current l r)
)
| [] -> if t = [] then return env else fail ()
and
matchingSym (env : Subst.t) (l : nf_term list) (t : nf_term list) =
List.fold_left2
(fun acc p t -> acc >> (fun env -> matching env p t))
(return env)
l
t
in
fun env l r ->
let _ = debug (Printf.sprintf "pattern :%s\nterm :%s\n%!" (Terms.sprint_nf_term l) (Terms.sprint_nf_term r)) in
let m = matching env l r in
let _ = debug (Printf.sprintf "count %i" (Search.count m)) in
m
(** [unitifiable p : Subst.t m] *)
let unitifiable p : (symbol * Subst.t m) m =
let m = List.fold_left
(fun acc (_,unit) -> acc >>|
let m = matching Subst.empty p (mk_TUnit unit) in
if Search.is_empty m then
fail ()
else
begin
return (unit,m)
end
) (fail ()) units
in
m
;;
let nullifiable p =
let nullable = not strict in
let has_unit s =
try let _ = get_unit units s in true
with NoUnit -> false
in
let rec aux = function
| TA (s,l) -> has_unit s && List.for_all (aux) l
| TAC (s,l) -> has_unit s && List.for_all (fun (x,n) -> aux x) l
| TSym _ -> false
| TVar _ -> nullable
| TUnit _ -> true
in aux p
let unit_warning p ~nullif ~unitif =
assert ((Search.is_empty unitif) || nullif);
if not (Search.is_empty unitif)
then
begin
Pp.msg_warning
(Pp.str
"[aac_tactics] This pattern can be instanciated to match units, some solutions can be missing");
end
;;
(***********)
(* Subterm *)
(***********)
(** [subterm] solves a sub-term pattern matching.
This function is more high-level than {!matcher}, thus takes {!t}
as arguments rather than terms in normal form {!nf_term}.
We use three mutually recursive functions {!subterm},
{!subterm_AC}, {!subterm_A} to find the matching subterm, making
non-deterministic choices to split the term into a context and an
intersting sub-term. Intuitively, the only case in which we have to
go in depth is when we are left with a sub-term that is atomic.
Indeed, rewriting [H: b = c |- a+b+a = a+a+c], we do not want to
find recursively the sub-terms of [a+b] and [b+a], since they will
overlap with the sub-terms of [a+b+a].
We rebuild the context on the fly, leaving the variables in the
pattern uninstantiated. We do so in order to allow interaction
with the user, to choose the env.
Strange patterms like x*y*x can be instanciated by nothing, inside
a product. Therefore, we need to check that all the term is not
going into the context. With proper support for interaction with
the user, we should lift these tests. However, at the moment, they
serve as heuristics to return "interesting" matchings
*)
let return_non_empty raw_p m =
if is_empty m
then
fail ()
else
return (raw_p ,m)
let subterm (raw_p:t) (raw_t:t): (int * t * Subst.t m) m=
let _ = debug (String.make 40 '=') in
let p = term_of_t units raw_p in
let t = term_of_t units raw_t in
let nullif = nullifiable p in
let unitif = unitifiable p in
let _ = unit_warning p ~nullif ~unitif in
let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term p)) in
let _ = debug (Printf.sprintf "%s" (Terms.sprint_nf_term t)) in
let filter_right current right (p,m) =
if right = []
then return (p,m)
else
mk_TAC' current right >>
fun t -> return (Plus (current,p,t_of_term t),m)
in
let filter_middle current left right (p,m) =
let right_pat p =
if right = []
then return p
else
mk_TA' current right >>
fun t -> return (Dot (current,p,t_of_term t))
in
let left_pat p=
if left = []
then return p
else
mk_TA' current left >>
fun t -> return (Dot (current,t_of_term t,p))
in
right_pat p >> left_pat >> (fun p -> return (p,m))
in
let rec subterm (t:nf_term) : (t * Subst.t m) m=
match t with
| TAC (s,l) ->
((ac_nondet_split_raw l) >>
(fun (left,right) ->
(subterm_AC s left) >> (filter_right s right)
))
| TA (s,l) ->
(a_nondet_middle l) >>
(fun (left, middle, right) ->
(subterm_A s middle) >>
(filter_middle s left right)
)
| TSym (s, l) ->
let init =
return_non_empty raw_p (matching Subst.empty p t)
in
tri_fold (fun acc l t r ->
((subterm t) >>
(fun (p,m) ->
let l = List.map t_of_term l in
let r = List.map t_of_term r in
let p = Sym (s, Array.of_list (List.rev_append l (p::r))) in
return (p,m)
)) >>| acc
) l init
| TVar x -> assert false
(* this case is superseded by the later disjunction *)
| TUnit s -> fail ()
and subterm_AC s tl =
match tl with
[x,1] -> subterm x
| _ ->
mk_TAC' s tl >> fun t ->
return_non_empty raw_p (matching Subst.empty p t)
and subterm_A s tl =
match tl with
[x] -> subterm x
| _ ->
mk_TA' s tl >>
fun t ->
return_non_empty raw_p (matching Subst.empty p t)
in
match p with
| TUnit unit -> unit_subterm t unit raw_p
| _ when not (Search.is_empty unitif) ->
let unit_matches =
Search.fold
(fun (unit,inst) acc ->
Search.fold
(fun subst acc' ->
let m = unit_subterm t unit (Subst.instantiate subst raw_p)
in
m>>| acc'
)
inst
acc
)
unitif
(fail ())
in
let nullifies (t : Subst.t) =
List.for_all (fun (_,x) ->
List.exists (fun (_,y) -> Unit y = x ) units
) (Subst.to_list t)
in
let nonunit_matches =
subterm t >>
(
fun (p,m) ->
let m = Search.filter (fun subst -> not( nullifies subst)) m in
if Search.is_empty m
then fail ()
else return (Terms.size p,p,m)
)
in
unit_matches >>| nonunit_matches
| _ -> (subterm t >> fun (p,m) -> return (Terms.size p,p,m))
end
(* The functions we export, handlers for the previous ones. Some debug
information also *)
let subterm ?(strict = false) units raw t =
let module M = M (struct
let is_ac = units.is_ac
let units = units.unit_for
let strict = strict
end) in
let sols = time (M.subterm raw) t "%fs spent in subterm (including matching)\n" in
debug
(Printf.sprintf "%i possible solution(s)\n"
(Search.fold (fun (_,_,envm) acc -> count envm + acc) sols 0));
sols
let matcher ?(strict = false) units p t =
let module M = M (struct
let is_ac = units.is_ac
let units = units.unit_for
let strict = false
end) in
let units = units.unit_for in
let sols = time
(fun (p,t) ->
let p = (Terms.term_of_t units p) in
let t = (Terms.term_of_t units t) in
M.matching Subst.empty p t) (p,t)
"%fs spent in the matcher\n"
in
debug (Printf.sprintf "%i solutions\n" (count sols));
sols
aac_tactics/matcher.mli 0000664 0000000 0000000 00000016271 12211622232 0015442 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** 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} 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 {!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 :
- it must be noted that we allow several AC/A operators to share
the same units, but that a given AC/A operator can have at most
one unit.
- if the pattern 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 AAC and U that make the size
of the term increase (by making neutral elements appear in a
layered fashion). Hence, in this case, a warning is issued, and
some solutions are omitted.
*)
(** {2 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 (* from AC/A symbols to the unit *)
type ext_units =
{
unit_for : units; (* from AC/A symbols to the unit *)
is_ac : (symbol * bool) 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]]) *)
type 'a mset = ('a * int) list
(** [linear] expands a multiset into a simple list *)
val linear : 'a mset -> 'a list
(** Representations of expressions
The module {!Terms} defines two different types for expressions.
- a public type {!Terms.t} that represents abstract syntax trees
of expressions with binary associative and commutative operators
- a private type {!Terms.nf_term}, corresponding to a canonical
representation for the above terms modulo AACU. The construction
functions on this type ensure that these terms are in normal form
(that is, no sum can appear as a subterm of the same sum, no
trailing units, lists corresponding to multisets are sorted,
etc...).
*)
module Terms :
sig
(** {2 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 (symbol * t * t)
| Plus of (symbol * t * t)
| Sym of (symbol * t array)
| Var of var
| Unit of symbol
(** 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 equal_aac : units -> t -> t -> bool
(** {2 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
(** {3 Comparisons} *)
val nf_term_compare : nf_term -> nf_term -> int
val nf_equal : nf_term -> nf_term -> bool
(** {3 Printing function} *)
val sprint_nf_term : nf_term -> string
(** {3 Conversion functions} *)
(** 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 term_of_t : units -> t -> nf_term
val t_of_term : nf_term -> t
end
(** Substitutions (or environments)
The module {!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.
*)
module Subst :
sig
type t
val sprint : t -> string
val instantiate : t -> Terms.t-> Terms.t
val to_list : t -> (var*Terms.t) list
end
(** {2 Main functions exported by this module} *)
(** [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 matcher : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> Subst.t 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 {!Terms.t} where the variables
are yet to be instantiated by one of the associated substitutions
*)
val subterm : ?strict:bool -> ext_units -> Terms.t -> Terms.t -> (int * Terms.t * Subst.t Search_monad.m) Search_monad.m
aac_tactics/print.ml 0000664 0000000 0000000 00000007034 12211622232 0014777 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(* A very basic way to interact with the envs, to choose a possible
solution *)
open Pp
open Matcher
type named_env = (Names.name * Terms.t) list
(** {pp_env} prints a substitution mapping names to terms, using the
provided printer *)
let pp_env pt : named_env -> Pp.std_ppcmds = fun env ->
List.fold_left
(fun acc (v,t) ->
begin match v with
| Names.Name s -> str (Printf.sprintf "%s: " (Names.string_of_id s))
| Names.Anonymous -> str ("_")
end
++ pt t ++ str "; " ++ acc
)
(str "")
env
(** {pp_envm} prints a collection of possible environments, and number
them. This number must remain compatible with the parameters given to
{!aac_rewrite} *)
let pp_envm pt : named_env Search_monad.m -> Pp.std_ppcmds = fun m ->
let _,s =
Search_monad.fold
(fun env (n,acc) ->
n+1, h 0 (str (Printf.sprintf "%i:\t[" n) ++pp_env pt env ++ str "]") ++ fnl () :: acc
) m (0,[])
in
List.fold_left (fun acc s -> s ++ acc) (str "") (s)
let trivial_substitution envm =
match Search_monad.choose envm with
| None -> true (* assert false *)
| Some l -> l=[]
(** {pp_all} prints a collection of possible contexts and related
possibles substitutions, giving a number to each. This number must
remain compatible with the parameters of {!aac_rewrite} *)
let pp_all pt : (int * Terms.t * named_env Search_monad.m) Search_monad.m -> Pp.std_ppcmds = fun m ->
let _,s = Search_monad.fold
(fun (size,context,envm) (n,acc) ->
let s = str (Printf.sprintf "occurence %i: transitivity through " n) in
let s = s ++ pt context ++ str "\n" in
let s = if trivial_substitution envm then s else
s ++ str (Printf.sprintf "%i possible(s) substitution(s)" (Search_monad.count envm) )
++ fnl () ++ pp_envm pt envm
in
n+1, s::acc
) m (0,[]) in
List.fold_left (fun acc s -> s ++ str "\n" ++ acc) (str "") (s)
(** The main printing function. {!print} uses the debruijn_env the
rename the variables, and rebuilds raw Coq terms (for the context, and
the terms in the environment). In order to do so, it requires the
information gathered by the {!Theory.Trans} module.*)
let print rlt ir m (context : Term.rel_context) goal =
if Search_monad.count m = 0
then
(
Tacticals.tclFAIL 0 (Pp.str "No subterm modulo AC") goal
)
else
let _ = Pp.msgnl (Pp.str "All solutions:") in
let m = Search_monad.(>>) m
(fun (i,t,envm) ->
let envm = Search_monad.(>>) envm ( fun env ->
let l = Matcher.Subst.to_list env in
let l = List.sort (fun (n,_) (n',_) -> Pervasives.compare n n') l in
let l =
List.map (fun (v,t) ->
let (name,body,types) = Term.lookup_rel v context in
(name,t)
) l
in
Search_monad.return l
)
in
Search_monad.return (i,t,envm)
)
in
let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in
let _ = Pp.msgnl
(pp_all
(fun t -> Printer.pr_constr (Theory.Trans.raw_constr_of_t ir rlt context t) ) m
)
in
Tacticals.tclIDTAC goal
aac_tactics/print.mli 0000664 0000000 0000000 00000002066 12211622232 0015150 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Pretty printing functions we use for the aac_instances
tactic. *)
(** The main printing function. {!print} uses the [Term.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.*)
val print :
Coq.Relation.t ->
Theory.Trans.ir ->
(int * Matcher.Terms.t * Matcher.Subst.t Search_monad.m) Search_monad.m ->
Term.rel_context ->
Proof_type.tactic
aac_tactics/rewrite.ml4 0000664 0000000 0000000 00000043225 12211622232 0015412 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** aac_rewrite -- rewriting modulo *)
module Control = struct
let debug = false
let printing = false
let time = false
end
module Debug = Helper.Debug (Control)
open Debug
let time_tac msg tac =
if Control.time then Coq.tclTIME msg tac else tac
let tac_or_exn tac exn msg = fun gl ->
try tac gl with e ->
pr_constr "last goal" (Tacmach.pf_concl gl);
exn msg e
(* helper to be used with the previous function: raise a new anomaly
except if a another one was previously raised *)
let push_anomaly msg = function
| Util.Anomaly _ as e -> raise e
| _ -> Coq.anomaly msg
module M = Matcher
open Term
open Names
open Coqlib
open Proof_type
(** The various kind of relation we can encounter, as a hierarchy *)
type rew_relation =
| Bare of Coq.Relation.t
| Transitive of Coq.Transitive.t
| Equivalence of Coq.Equivalence.t
(** {!promote try to go higher in the aforementionned hierarchy} *)
let promote (rlt : Coq.Relation.t) (k : rew_relation -> Proof_type.tactic) =
try Coq.Equivalence.cps_from_relation rlt
(fun e -> k (Equivalence e))
with
| Not_found ->
begin
try Coq.Transitive.cps_from_relation rlt
(fun trans -> k (Transitive trans))
with
|Not_found -> k (Bare rlt)
end
(*
Various situations:
p == q |- left == right : rewrite <- ->
p <= q |- left <= right : rewrite ->
p <= q |- left == right : failure
p == q |- left <= right : rewrite <- ->
Not handled
p <= q |- left >= right : failure
*)
(** aac_lift : the ideal type beyond AAC.v/Lift
A base relation r, together with an equivalence relation, and the
proof that the former lifts to the later. Howver, we have to
ensure manually the invariant : r.carrier == e.carrier, and that
lift connects the two things *)
type aac_lift =
{
r : Coq.Relation.t;
e : Coq.Equivalence.t;
lift : Term.constr
}
type rewinfo =
{
hypinfo : Coq.Rewrite.hypinfo;
in_left : bool; (** are we rewriting in the left hand-sie of the goal *)
pattern : constr;
subject : constr;
morph_rlt : Coq.Relation.t; (** the relation we look for in morphism *)
eqt : Coq.Equivalence.t; (** the equivalence we use as workbase *)
rlt : Coq.Relation.t; (** the relation in the goal *)
lifting: aac_lift
}
let infer_lifting (rlt: Coq.Relation.t) (k : lift:aac_lift -> Proof_type.tactic) : Proof_type.tactic =
Coq.cps_evar_relation rlt.Coq.Relation.carrier (fun e ->
let lift_ty =
mkApp (Lazy.force Theory.Stubs.lift,
[|
rlt.Coq.Relation.carrier;
rlt.Coq.Relation.r;
e
|]
) in
Coq.cps_resolve_one_typeclass ~error:"Cannot infer a lifting"
lift_ty (fun lift goal ->
let x = rlt.Coq.Relation.carrier in
let r = rlt.Coq.Relation.r in
let eq = (Coq.nf_evar goal e) in
let equiv = Coq.lapp Theory.Stubs.lift_proj_equivalence [| x;r;eq; lift |] in
let lift =
{
r = rlt;
e = Coq.Equivalence.make x eq equiv;
lift = lift;
}
in
k ~lift:lift goal
))
(** Builds a rewinfo, once and for all *)
let dispatch in_left (left,right,rlt) hypinfo (k: rewinfo -> Proof_type.tactic ) : Proof_type.tactic=
let l2r = hypinfo.Coq.Rewrite.l2r in
infer_lifting rlt
(fun ~lift ->
let eq = lift.e in
k {
hypinfo = hypinfo;
in_left = in_left;
pattern = if l2r then hypinfo.Coq.Rewrite.left else hypinfo.Coq.Rewrite.right;
subject = if in_left then left else right;
morph_rlt = Coq.Equivalence.to_relation eq;
eqt = eq;
lifting = lift;
rlt = rlt
}
)
(** {1 Tactics} *)
(** Build the reifiers, the reified terms, and the evaluation fonction *)
let handle eqt zero envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) k =
let (x,r,_) = Coq.Equivalence.split eqt in
Theory.Trans.mk_reifier (Coq.Equivalence.to_relation eqt) zero envs
(fun (maps, reifier) ->
(* fold through a term and reify *)
let t = Theory.Trans.reif_constr_of_t reifier t in
let t' = Theory.Trans.reif_constr_of_t reifier t' in
(* Some letins *)
let eval = (mkApp (Lazy.force Theory.Stubs.eval, [|x;r; maps.Theory.Trans.env_sym; maps.Theory.Trans.env_bin; maps.Theory.Trans.env_units|])) in
Coq.cps_mk_letin "eval" eval (fun eval ->
Coq.cps_mk_letin "left" t (fun t ->
Coq.cps_mk_letin "right" t' (fun t' ->
k maps eval t t'))))
(** [by_aac_reflexivity] is a sub-tactic that closes a sub-goal that
is merely a proof of equality of two terms modulo AAC *)
let by_aac_reflexivity zero
eqt envs (t : Matcher.Terms.t) (t' : Matcher.Terms.t) : Proof_type.tactic =
handle eqt zero envs t t'
(fun maps eval t t' ->
let (x,r,e) = Coq.Equivalence.split eqt in
let decision_thm = Coq.lapp Theory.Stubs.decide_thm
[|x;r;e;
maps.Theory.Trans.env_sym;
maps.Theory.Trans.env_bin;
maps.Theory.Trans.env_units;
t;t';
|]
in
(* This convert is required to deal with evars in a proper
way *)
let convert_to = mkApp (r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in
let convert = Tactics.convert_concl convert_to Term.VMcast in
let apply_tac = Tactics.apply decision_thm in
(Tacticals.tclTHENLIST
[
convert ;
tac_or_exn apply_tac Coq.user_error "unification failure";
tac_or_exn (time_tac "vm_norm" (Tactics.normalise_in_concl)) Coq.anomaly "vm_compute failure";
Tacticals.tclORELSE Tactics.reflexivity
(Tacticals.tclFAIL 0 (Pp.str "Not an equality modulo A/AC"))
])
)
let by_aac_normalise zero lift ir t t' =
let eqt = lift.e in
let rlt = lift.r in
handle eqt zero ir t t'
(fun maps eval t t' ->
let (x,r,e) = Coq.Equivalence.split eqt in
let normalise_thm = Coq.lapp Theory.Stubs.lift_normalise_thm
[|x;r;e;
maps.Theory.Trans.env_sym;
maps.Theory.Trans.env_bin;
maps.Theory.Trans.env_units;
rlt.Coq.Relation.r;
lift.lift;
t;t';
|]
in
(* This convert is required to deal with evars in a proper
way *)
let convert_to = mkApp (rlt.Coq.Relation.r, [| mkApp (eval,[| t |]); mkApp (eval, [|t'|])|]) in
let convert = Tactics.convert_concl convert_to Term.VMcast in
let apply_tac = Tactics.apply normalise_thm in
(Tacticals.tclTHENLIST
[
convert ;
apply_tac;
])
)
(** A handler tactic, that reifies the goal, and infer the liftings,
and then call its continuation *)
let aac_conclude
(k : Term.constr -> aac_lift -> Theory.Trans.ir -> Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic) = fun goal ->
let (equation : Term.types) = Tacmach.pf_concl goal in
let envs = Theory.Trans.empty_envs () in
let left, right,r =
match Coq.match_as_equation goal equation with
| None -> Coq.user_error "The goal is not an applied relation"
| Some x -> x in
try infer_lifting r
(fun ~lift goal ->
let eq = Coq.Equivalence.to_relation lift.e in
let tleft,tright, goal = Theory.Trans.t_of_constr goal eq envs (left,right) in
let goal, ir = Theory.Trans.ir_of_envs goal eq envs in
let concl = Tacmach.pf_concl goal in
let _ = pr_constr "concl "concl in
let evar_map = Tacmach.project goal in
Tacticals.tclTHEN
(Refiner.tclEVARS evar_map)
(k left lift ir tleft tright)
goal
)goal
with
| Not_found -> Coq.user_error "No lifting from the goal's relation to an equivalence"
open Libnames
open Tacinterp
let aac_normalise = fun goal ->
let ids = Tacmach.pf_ids_of_hyps goal in
Tacticals.tclTHENLIST
[
aac_conclude by_aac_normalise;
Tacinterp.interp (
<:tactic<
intro x;
intro y;
vm_compute in x;
vm_compute in y;
unfold x;
unfold y;
compute [Internal.eval Internal.fold_map Internal.copy Prect]; simpl
>>
);
Tactics.keep ids
] goal
let aac_reflexivity = fun goal ->
aac_conclude
(fun zero lift ir t t' ->
let x,r = Coq.Relation.split (lift.r) in
let r_reflexive = (Coq.Classes.mk_reflexive x r) in
Coq.cps_resolve_one_typeclass ~error:"The goal's relation is not reflexive"
r_reflexive
(fun reflexive ->
let lift_reflexivity =
mkApp (Lazy.force (Theory.Stubs.lift_reflexivity),
[|
x;
r;
lift.e.Coq.Equivalence.eq;
lift.lift;
reflexive
|])
in
Tacticals.tclTHEN
(Tactics.apply lift_reflexivity)
(fun goal ->
let concl = Tacmach.pf_concl goal in
let _ = pr_constr "concl "concl in
by_aac_reflexivity zero lift.e ir t t' goal)
)
) goal
(** A sub-tactic to lift the rewriting using Lift *)
let lift_transitivity in_left (step:constr) preorder lifting (using_eq : Coq.Equivalence.t): tactic =
fun goal ->
(* catch the equation and the two members*)
let concl = Tacmach.pf_concl goal in
let (left, right, _ ) = match Coq.match_as_equation goal concl with
| Some x -> x
| None -> Coq.user_error "The goal is not an equation"
in
let lift_transitivity =
let thm =
if in_left
then
Lazy.force Theory.Stubs.lift_transitivity_left
else
Lazy.force Theory.Stubs.lift_transitivity_right
in
mkApp (thm,
[|
preorder.Coq.Relation.carrier;
preorder.Coq.Relation.r;
using_eq.Coq.Equivalence.eq;
lifting;
step;
left;
right;
|])
in
Tacticals.tclTHENLIST
[
Tactics.apply lift_transitivity
] goal
(** The core tactic for aac_rewrite *)
let core_aac_rewrite ?abort
rewinfo
subst
(by_aac_reflexivity : Matcher.Terms.t -> Matcher.Terms.t -> Proof_type.tactic)
(tr : constr) t left : tactic =
pr_constr "transitivity through" tr;
let tran_tac =
lift_transitivity rewinfo.in_left tr rewinfo.rlt rewinfo.lifting.lift rewinfo.eqt
in
Coq.Rewrite.rewrite ?abort rewinfo.hypinfo subst (fun rew ->
Tacticals.tclTHENSV
(tac_or_exn (tran_tac) Coq.anomaly "Unable to make the transitivity step")
(
if rewinfo.in_left
then [| by_aac_reflexivity left t ; rew |]
else [| by_aac_reflexivity t left ; rew |]
)
)
exception NoSolutions
(** Choose a substitution from a
[(int * Terms.t * Env.env Search_monad.m) Search_monad.m ] *)
(* WARNING: Beware, since the printing function can change the order of the
printed monad, this function has to be updated accordingly *)
let choose_subst subterm sol m=
try
let (depth,pat,envm) = match subterm with
| None -> (* first solution *)
List.nth ( List.rev (Search_monad.to_list m)) 0
| Some x ->
List.nth ( List.rev (Search_monad.to_list m)) x
in
let env = match sol with
None ->
List.nth ( (Search_monad.to_list envm)) 0
| Some x -> List.nth ( (Search_monad.to_list envm)) x
in
pat, env
with
| _ -> raise NoSolutions
(** rewrite the constr modulo AC from left to right in the left member
of the goal *)
let aac_rewrite ?abort rew ?(l2r=true) ?(show = false) ?(in_left=true) ?strict ~occ_subterm ~occ_sol ?extra : Proof_type.tactic = fun goal ->
let envs = Theory.Trans.empty_envs () in
let (concl : Term.types) = Tacmach.pf_concl goal in
let (_,_,rlt) as concl =
match Coq.match_as_equation goal concl with
| None -> Coq.user_error "The goal is not an applied relation"
| Some (left, right, rlt) -> left,right,rlt
in
let check_type x =
Tacmach.pf_conv_x goal x rlt.Coq.Relation.carrier
in
Coq.Rewrite.get_hypinfo rew ~l2r ?check_type:(Some check_type)
(fun hypinfo ->
dispatch in_left concl hypinfo
(
fun rewinfo ->
let goal =
match extra with
| Some t -> Theory.Trans.add_symbol goal rewinfo.morph_rlt envs t
| None -> goal
in
let pattern, subject, goal =
Theory.Trans.t_of_constr goal rewinfo.morph_rlt envs
(rewinfo.pattern , rewinfo.subject)
in
let goal, ir = Theory.Trans.ir_of_envs goal rewinfo.morph_rlt envs in
let units = Theory.Trans.ir_to_units ir in
let m = Matcher.subterm ?strict units pattern subject in
(* We sort the monad in increasing size of contet *)
let m = Search_monad.sort (fun (x,_,_) (y,_,_) -> x - y) m in
if show
then
Print.print rewinfo.morph_rlt ir m (hypinfo.Coq.Rewrite.context)
else
try
let pat,subst = choose_subst occ_subterm occ_sol m in
let tr_step = Matcher.Subst.instantiate subst pat in
let tr_step_raw = Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt [] tr_step in
let conv = (Theory.Trans.raw_constr_of_t ir rewinfo.morph_rlt (hypinfo.Coq.Rewrite.context)) in
let subst = Matcher.Subst.to_list subst in
let subst = List.map (fun (x,y) -> x, conv y) subst in
let by_aac_reflexivity = (by_aac_reflexivity rewinfo.subject rewinfo.eqt ir) in
core_aac_rewrite ?abort rewinfo subst by_aac_reflexivity tr_step_raw tr_step subject
with
| NoSolutions ->
Tacticals.tclFAIL 0
(Pp.str (if occ_subterm = None && occ_sol = None
then "No matching occurence modulo AC found"
else "No such solution"))
)
) goal
(** [aac_match eq (fun x1 ... xn => p) t H] match the term [t] against
the pattern, and introduces an hypothesis H of type p = t. (Note
that we have performed the substitution in p).
*)
let aac_match ~eq pattern term h = fun gl ->
let env = Tacmach.pf_env gl in
let evar_map = Tacmach.project gl in
let carrier = Typing.type_of env evar_map term in
let rel = Coq.Relation.make carrier eq in
let equiv,gl = Coq.Equivalence.from_relation gl rel in
(* first, we decompose the pattern as an arity. *)
let x_args, pat = Term.decompose_lam pattern in
(* then, we reify the pattern and the term, using the provided equality *)
let envs = Theory.Trans.empty_envs () in
let left, right,gl = Theory.Trans.t_of_constr gl rel envs (pat,term) in
let gl,ir = Theory.Trans.ir_of_envs gl rel envs in
let solutions = Matcher.matcher (Theory.Trans.ir_to_units ir) left right in
(* then, we pick the first solution to the matching problem *)
let sigma = match Search_monad.choose solutions with
| None -> Coq.user_error "no solution to the matching problem"
| Some sigma -> sigma
in
let p_sigma = Matcher.Subst.instantiate sigma left in
let args = List.map (fun (x,t) -> x,Theory.Trans.raw_constr_of_t ir rel [] t) (Matcher.Subst.to_list sigma) in
(* Then, we have to assert the fact that p_sigma is equal to term *)
let equation = mkApp (eq, [| Theory.Trans.raw_constr_of_t ir rel [] p_sigma ; term |]) in
let evar_map = Tacmach.project gl in
Tacticals.tclTHENLIST
[
Refiner.tclEVARS evar_map;
Tactics.assert_by h equation (by_aac_reflexivity term equiv ir p_sigma right);
] gl
open Coq.Rewrite
open Tacmach
open Tacticals
open Tacexpr
open Tacinterp
open Extraargs
open Genarg
let rec add k x = function
| [] -> [k,x]
| k',_ as ky::q ->
if k'=k then Coq.user_error ("redondant argument ("^k^")")
else ky::add k x q
let get k l = try Some (List.assoc k l) with Not_found -> None
let get_lhs l = try List.assoc "in_right" l; false with Not_found -> true
let aac_rewrite ~args =
aac_rewrite ~occ_subterm:(get "at" args) ~occ_sol:(get "subst" args) ~in_left:(get_lhs args)
let pr_aac_args _ _ _ l =
List.fold_left
(fun acc -> function
| ("in_right" as s,_) -> Pp.(++) (Pp.str s) acc
| (k,i) -> Pp.(++) (Pp.(++) (Pp.str k) (Pp.int i)) acc
) (Pp.str "") l
ARGUMENT EXTEND aac_args
TYPED AS ((string * int) list )
PRINTED BY pr_aac_args
| [ "at" integer(n) aac_args(q) ] -> [ add "at" n q ]
| [ "subst" integer(n) aac_args(q) ] -> [ add "subst" n q ]
| [ "in_right" aac_args(q) ] -> [ add "in_right" 0 q ]
| [ ] -> [ [] ]
END
let pr_constro _ _ _ = fun b ->
match b with
| None -> Pp.str ""
| Some o -> Pp.str ""
ARGUMENT EXTEND constro
TYPED AS (constr option)
PRINTED BY pr_constro
| [ "[" constr(c) "]" ] -> [ Some c ]
| [ ] -> [ None ]
END
TACTIC EXTEND _aac_reflexivity_
| [ "aac_reflexivity" ] -> [ aac_reflexivity ]
END
TACTIC EXTEND _aac_normalise_
| [ "aac_normalise" ] -> [ aac_normalise ]
END
TACTIC EXTEND _aac_rewrite_
| [ "aac_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true c gl ]
END
TACTIC EXTEND _aac_pattern_
| [ "aac_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~abort:true c gl ]
END
TACTIC EXTEND _aac_instances_
| [ "aac_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:true ~show:true c gl ]
END
TACTIC EXTEND _aacu_rewrite_
| [ "aacu_rewrite" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false c gl ]
END
TACTIC EXTEND _aacu_pattern_
| [ "aacu_pattern" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~abort:true c gl ]
END
TACTIC EXTEND _aacu_instances_
| [ "aacu_instances" orient(l2r) constr(c) aac_args(args) constro(extra)] ->
[ fun gl -> aac_rewrite ?extra ~args ~l2r ~strict:false ~show:true c gl ]
END
TACTIC EXTEND _aac_match_
| [ "aac_match" constr(p) constr(t) ident(h)] ->
[ fun gl ->
let ty = Tacmach.pf_type_of gl t in
let eq = Coq.Leibniz.eq ty in
aac_match ~eq p t (Names.Name h) gl]
END
aac_tactics/search_monad.ml 0000664 0000000 0000000 00000003764 12211622232 0016274 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
type 'a m = | F of 'a
| N of 'a m list
let fold (f : 'a -> 'b -> 'b) (m : 'a m) (acc : 'b) =
let rec aux acc = function
F x -> f x acc
| N l ->
(List.fold_left (fun acc x ->
match x with
| (N []) -> acc
| x -> aux acc x
) acc l)
in
aux acc m
let rec (>>) : 'a m -> ('a -> 'b m) -> 'b m =
fun m f ->
match m with
| F x -> f x
| N l ->
N (List.fold_left (fun acc x ->
match x with
| (N []) -> acc
| x -> (x >> f)::acc
) [] l)
let (>>|) (m : 'a m) (n :'a m) : 'a m = match (m,n) with
| N [],_ -> n
| _,N [] -> m
| F x, N l -> N (F x::l)
| N l, F x -> N (F x::l)
| x,y -> N [x;y]
let return : 'a -> 'a m = fun x -> F x
let fail : unit -> 'a m = fun () -> N []
let sprint f m =
fold (fun x acc -> Printf.sprintf "%s\n%s" acc (f x)) m ""
let rec count = function
| F _ -> 1
| N l -> List.fold_left (fun acc x -> acc+count x) 0 l
let opt_comb f x y = match x with None -> f y | _ -> x
let rec choose = function
| F x -> Some x
| N l -> List.fold_left (fun acc x ->
opt_comb choose acc x
) None l
let is_empty = fun x -> choose x = None
let to_list m = (fold (fun x acc -> x::acc) m [])
let sort f m =
N (List.map (fun x -> F x) (List.sort f (to_list m)))
(* preserve the structure of the heap *)
let filter f m =
fold (fun x acc -> (if f x then return x else fail ()) >>| acc) m (N [])
aac_tactics/search_monad.mli 0000664 0000000 0000000 00000002654 12211622232 0016442 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Search monad that allows to express non-deterministic algorithms
in a legible maner, or programs that solve combinatorial problems.
@see the
inspiration of this module
*)
(** A data type that represent a collection of ['a] *)
type 'a m
(** {2 Monadic operations} *)
(** bind and return *)
val ( >> ) : 'a m -> ('a -> 'b m) -> 'b m
val return : 'a -> 'a m
(** non-deterministic choice *)
val ( >>| ) : 'a m -> 'a m -> 'a m
(** failure *)
val fail : unit -> 'a m
(** folding through the collection *)
val fold : ('a -> 'b -> 'b) -> 'a m -> 'b -> 'b
(** {2 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
aac_tactics/theory.ml 0000664 0000000 0000000 00000102401 12211622232 0015147 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Constr from the theory of this particular development
The inner-working of this module is highly correlated with the
particular structure of {b AAC.v}, therefore, it should
be of little interest to most readers.
*)
open Term
module Control = struct
let printing = true
let debug = false
let time = false
end
module Debug = Helper.Debug (Control)
open Debug
(** {1 HMap : Specialized Hashtables based on constr} *)
module Hashed_constr =
struct
type t = constr
let equal = (=)
let hash = Hashtbl.hash
end
(* TODO module HMap = Hashtbl, du coup ? *)
module HMap = Hashtbl.Make(Hashed_constr)
let ac_theory_path = ["AAC_tactics"; "AAC"]
module Stubs = struct
let path = ac_theory_path@["Internal"]
(** The constants from the inductive type *)
let _Tty = lazy (Coq.init_constant path "T")
let vTty = lazy (Coq.init_constant path "vT")
let rsum = lazy (Coq.init_constant path "sum")
let rprd = lazy (Coq.init_constant path "prd")
let rsym = lazy (Coq.init_constant path "sym")
let runit = lazy (Coq.init_constant path "unit")
let vnil = lazy (Coq.init_constant path "vnil")
let vcons = lazy (Coq.init_constant path "vcons")
let eval = lazy (Coq.init_constant path "eval")
let decide_thm = lazy (Coq.init_constant path "decide")
let lift_normalise_thm = lazy (Coq.init_constant path "lift_normalise")
let lift =
lazy (Coq.init_constant ac_theory_path "AAC_lift")
let lift_proj_equivalence=
lazy (Coq.init_constant ac_theory_path "aac_lift_equivalence")
let lift_transitivity_left =
lazy(Coq.init_constant ac_theory_path "lift_transitivity_left")
let lift_transitivity_right =
lazy(Coq.init_constant ac_theory_path "lift_transitivity_right")
let lift_reflexivity =
lazy(Coq.init_constant ac_theory_path "lift_reflexivity")
end
module Classes = struct
module Associative = struct
let path = ac_theory_path
let typ = lazy (Coq.init_constant path "Associative")
let ty (rlt : Coq.Relation.t) (value : Term.constr) =
mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier;
rlt.Coq.Relation.r;
value
|] )
let infer goal rlt value =
let ty = ty rlt value in
Coq.resolve_one_typeclass goal ty
end
module Commutative = struct
let path = ac_theory_path
let typ = lazy (Coq.init_constant path "Commutative")
let ty (rlt : Coq.Relation.t) (value : Term.constr) =
mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier;
rlt.Coq.Relation.r;
value
|] )
end
module Proper = struct
let path = ac_theory_path @ ["Internal";"Sym"]
let typeof = lazy (Coq.init_constant path "type_of")
let relof = lazy (Coq.init_constant path "rel_of")
let mk_typeof : Coq.Relation.t -> int -> constr = fun rlt n ->
let x = rlt.Coq.Relation.carrier in
mkApp (Lazy.force typeof, [| x ; Coq.Nat.of_int n |])
let mk_relof : Coq.Relation.t -> int -> constr = fun rlt n ->
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force relof, [| x;r ; Coq.Nat.of_int n |])
let ty rlt op ar =
let typeof = mk_typeof rlt ar in
let relof = mk_relof rlt ar in
Coq.Classes.mk_morphism typeof relof op
let infer goal rlt op ar =
let ty = ty rlt op ar in
Coq.resolve_one_typeclass goal ty
end
module Unit = struct
let path = ac_theory_path
let typ = lazy (Coq.init_constant path "Unit")
let ty (rlt : Coq.Relation.t) (value : Term.constr) (unit : Term.constr)=
mkApp (Lazy.force typ, [| rlt.Coq.Relation.carrier;
rlt.Coq.Relation.r;
value;
unit
|] )
end
end
(* Non empty lists *)
module NEList = struct
let path = ac_theory_path @ ["Internal"]
let typ = lazy (Coq.init_constant path "list")
let nil = lazy (Coq.init_constant path "nil")
let cons = lazy (Coq.init_constant path "cons")
let cons ty h t =
mkApp (Lazy.force cons, [| ty; h ; t |])
let nil ty x =
(mkApp (Lazy.force nil, [| ty ; x|]))
let rec of_list ty = function
| [] -> invalid_arg "NELIST"
| [x] -> nil ty x
| t::q -> cons ty t (of_list ty q)
let type_of_list ty =
mkApp (Lazy.force typ, [|ty|])
end
(** a [mset] is a ('a * pos) list *)
let mk_mset ty (l : (constr * int) list) =
let pos = Lazy.force Coq.Pos.typ in
let pair (x : constr) (ar : int) =
Coq.Pair.of_pair ty pos (x, Coq.Pos.of_int ar)
in
let pair_ty = Coq.lapp Coq.Pair.typ [| ty ; pos|] in
let rec aux = function
| [ ] -> assert false
| [x,ar] -> NEList.nil pair_ty (pair x ar)
| (t,ar)::q -> NEList.cons pair_ty (pair t ar) (aux q)
in
aux l
module Sigma = struct
let sigma = lazy (Coq.init_constant ac_theory_path "sigma")
let sigma_empty = lazy (Coq.init_constant ac_theory_path "sigma_empty")
let sigma_add = lazy (Coq.init_constant ac_theory_path "sigma_add")
let sigma_get = lazy (Coq.init_constant ac_theory_path "sigma_get")
let add ty n x map =
mkApp (Lazy.force sigma_add,[|ty; n; x ; map|])
let empty ty =
mkApp (Lazy.force sigma_empty,[|ty |])
let typ ty =
mkApp (Lazy.force sigma, [|ty|])
let to_fun ty null map =
mkApp (Lazy.force sigma_get, [|ty;null;map|])
let of_list ty null l =
match l with
| (_,t)::q ->
let map =
List.fold_left
(fun acc (i,t) ->
assert (i > 0);
add ty (Coq.Pos.of_int i) ( t) acc)
(empty ty)
q
in to_fun ty (t) map
| [] -> debug "of_list empty" ; to_fun ty (null) (empty ty)
end
module Sym = struct
type pack = {ar: Term.constr; value: Term.constr ; morph: Term.constr}
let path = ac_theory_path @ ["Internal";"Sym"]
let typ = lazy (Coq.init_constant path "pack")
let mkPack = lazy (Coq.init_constant path "mkPack")
let value = lazy (Coq.init_constant path "value")
let null = lazy (Coq.init_constant path "null")
let mk_pack (rlt: Coq.Relation.t) s =
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force mkPack, [|x;r; s.ar;s.value;s.morph|])
let null rlt =
let x = rlt.Coq.Relation.carrier in
let r = rlt.Coq.Relation.r in
mkApp (Lazy.force null, [| x;r;|])
let mk_ty : Coq.Relation.t -> constr = fun rlt ->
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force typ, [| x; r|] )
end
module Bin =struct
type pack = {value : Term.constr;
compat : Term.constr;
assoc : Term.constr;
comm : Term.constr option;
}
let path = ac_theory_path @ ["Internal";"Bin"]
let typ = lazy (Coq.init_constant path "pack")
let mkPack = lazy (Coq.init_constant path "mk_pack")
let mk_pack: Coq.Relation.t -> pack -> Term.constr = fun (rlt) s ->
let (x,r) = Coq.Relation.split rlt in
let comm_ty = Classes.Commutative.ty rlt s.value in
mkApp (Lazy.force mkPack , [| x ; r;
s.value;
s.compat ;
s.assoc;
Coq.Option.of_option comm_ty s.comm
|])
let mk_ty : Coq.Relation.t -> constr = fun rlt ->
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force typ, [| x; r|] )
end
module Unit = struct
let path = ac_theory_path @ ["Internal"]
let unit_of_ty = lazy (Coq.init_constant path "unit_of")
let unit_pack_ty = lazy (Coq.init_constant path "unit_pack")
let mk_unit_of = lazy (Coq.init_constant path "mk_unit_for")
let mk_unit_pack = lazy (Coq.init_constant path "mk_unit_pack")
type unit_of =
{
uf_u : Term.constr; (* u *)
uf_idx : Term.constr;
uf_desc : Term.constr; (* Unit R (e_bin uf_idx) u *)
}
type pack = {
u_value : Term.constr; (* X *)
u_desc : (unit_of) list (* unit_of u_value *)
}
let ty_unit_of rlt e_bin u =
let (x,r) = Coq.Relation.split rlt in
mkApp ( Lazy.force unit_of_ty, [| x; r; e_bin; u |])
let ty_unit_pack rlt e_bin =
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force unit_pack_ty, [| x; r; e_bin |])
let mk_unit_of rlt e_bin u unit_of =
let (x,r) = Coq.Relation.split rlt in
mkApp (Lazy.force mk_unit_of , [| x;
r;
e_bin ;
u;
unit_of.uf_idx;
unit_of.uf_desc
|])
let mk_pack rlt e_bin pack : Term.constr =
let (x,r) = Coq.Relation.split rlt in
let ty = ty_unit_of rlt e_bin pack.u_value in
let mk_unit_of = mk_unit_of rlt e_bin pack.u_value in
let u_desc =Coq.List.of_list ( ty ) (List.map mk_unit_of pack.u_desc) in
mkApp (Lazy.force mk_unit_pack, [|x;r;
e_bin ;
pack.u_value;
u_desc
|])
let default x : pack =
{ u_value = x;
u_desc = []
}
end
let anomaly msg =
Util.anomaly ("aac_tactics: " ^ msg)
let user_error msg =
Util.error ("aac_tactics: " ^ msg)
module Trans = struct
(** {1 From Coq to Abstract Syntax Trees (AST)}
This module provides facilities to interpret a Coq term with
arbitrary operators as an abstract syntax tree. Considering an
application, we try to infer instances of our classes.
We consider that [A] operators are coarser than [AC] operators,
which in turn are coarser than raw morphisms. That means that
[List.append], of type [(A : Type) -> list A -> list A -> list
A] can be understood as an [A] operator.
During this reification, we gather some informations that will
be used to rebuild Coq terms from AST ( type {!envs})
We use a main hash-table from [constr] to [pack], in order to
discriminate the various constructors. All these are mixed in
order to improve on the number of comparisons in the tables *)
(* 'a * (unit, op_unit) option *)
type 'a with_unit = 'a * (Unit.unit_of) option
type pack =
(* used to infer the AC/A structure in the first pass {!Gather} *)
| Bin of Bin.pack with_unit
(* will only be used in the second pass : {!Parse}*)
| Sym of Sym.pack
| Unit of constr (* to avoid confusion in bloom *)
(** discr is a map from {!Term.constr} to {!pack}.
[None] means that it is not possible to instantiate this partial
application to an interesting class.
[Some x] means that we found something in the past. This means
in particular that a single [constr] cannot be two things at the
same time.
The field [bloom] allows to give a unique number to each class we
found. *)
type envs =
{
discr : (pack option) HMap.t ;
bloom : (pack, int ) Hashtbl.t;
bloom_back : (int, pack ) Hashtbl.t;
bloom_next : int ref;
}
let empty_envs () =
{
discr = HMap.create 17;
bloom = Hashtbl.create 17;
bloom_back = Hashtbl.create 17;
bloom_next = ref 1;
}
let add_bloom envs pack =
if Hashtbl.mem envs.bloom pack
then ()
else
let x = ! (envs.bloom_next) in
Hashtbl.add envs.bloom pack x;
Hashtbl.add envs.bloom_back x pack;
incr (envs.bloom_next)
let find_bloom envs pack =
try Hashtbl.find envs.bloom pack
with Not_found -> assert false
(********************************************)
(* (\* Gather the occuring AC/A symbols *\) *)
(********************************************)
(** This modules exhibit a function that memoize in the environment
all the AC/A operators as well as the morphism that occur. This
staging process allows us to prefer AC/A operators over raw
morphisms. Moreover, for each AC/A operators, we need to try to
infer units. Otherwise, we do not have [x * y * x <= a * a] since 1
does not occur.
But, do we also need to check whether constants are
units. Otherwise, we do not have the ability to rewrite [0 = a +
a] in [a = ...]*)
module Gather : sig
val gather : Coq.goal_sigma -> Coq.Relation.t -> envs -> Term.constr -> Coq.goal_sigma
end
= struct
let memoize envs t pack : unit =
begin
HMap.add envs.discr t (Some pack);
add_bloom envs pack;
match pack with
| Bin (_, None) | Sym _ -> ()
| Bin (_, Some (unit_of)) ->
let unit = unit_of.Unit.uf_u in
HMap.add envs.discr unit (Some (Unit unit));
add_bloom envs (Unit unit);
| Unit _ -> assert false
end
let get_unit (rlt : Coq.Relation.t) op goal :
(Coq.goal_sigma * constr * constr ) option=
let x = (rlt.Coq.Relation.carrier) in
let unit, goal = Coq.evar_unit goal x in
let ty =Classes.Unit.ty rlt op unit in
let result =
try
let t,goal = Coq.resolve_one_typeclass goal ty in
Some (goal,t,unit)
with Not_found -> None
in
match result with
| None -> None
| Some (goal,s,unit) ->
let unit = Coq.nf_evar goal unit in
Some (goal, unit, s)
(** gives back the class and the operator *)
let is_bin (rlt: Coq.Relation.t) (op: constr) ( goal: Coq.goal_sigma)
: (Coq.goal_sigma * Bin.pack) option =
let assoc_ty = Classes.Associative.ty rlt op in
let comm_ty = Classes.Commutative.ty rlt op in
let proper_ty = Classes.Proper.ty rlt op 2 in
try
let proper , goal = Coq.resolve_one_typeclass goal proper_ty in
let assoc, goal = Coq.resolve_one_typeclass goal assoc_ty in
let comm , goal =
try
let comm, goal = Coq.resolve_one_typeclass goal comm_ty in
Some comm, goal
with Not_found ->
None, goal
in
let bin =
{Bin.value = op;
Bin.compat = proper;
Bin.assoc = assoc;
Bin.comm = comm }
in
Some (goal,bin)
with |Not_found -> None
let is_bin (rlt : Coq.Relation.t) (op : constr) (goal : Coq.goal_sigma)=
match is_bin rlt op goal with
| None -> None
| Some (goal, bin_pack) ->
match get_unit rlt op goal with
| None -> Some (goal, Bin (bin_pack, None))
| Some (gl, unit,s) ->
let unit_of =
{
Unit.uf_u = unit;
(* TRICK : this term is not well-typed by itself,
but it is okay the way we use it in the other
functions *)
Unit.uf_idx = op;
Unit.uf_desc = s;
}
in Some (gl,Bin (bin_pack, Some (unit_of)))
(** {is_morphism} try to infer the kind of operator we are
dealing with *)
let is_morphism goal (rlt : Coq.Relation.t) (papp : Term.constr) (ar : int) : (Coq.goal_sigma * pack ) option =
let typeof = Classes.Proper.mk_typeof rlt ar in
let relof = Classes.Proper.mk_relof rlt ar in
let m = Coq.Classes.mk_morphism typeof relof papp in
try
let m,goal = Coq.resolve_one_typeclass goal m in
let pack = {Sym.ar = (Coq.Nat.of_int ar); Sym.value= papp; Sym.morph= m} in
Some (goal, Sym pack)
with
| Not_found -> None
(* [crop_app f [| a_0 ; ... ; a_n |]]
returns Some (f a_0 ... a_(n-2), [|a_(n-1); a_n |] )
*)
let crop_app t ca : (constr * constr array) option=
let n = Array.length ca in
if n <= 1
then None
else
let papp = Term.mkApp (t, Array.sub ca 0 (n-2) ) in
let args = Array.sub ca (n-2) 2 in
Some (papp, args )
let fold goal (rlt : Coq.Relation.t) envs t ca cont =
let fold_morphism t ca =
let nb_params = Array.length ca in
let rec aux n =
assert (n < nb_params && 0 < nb_params );
let papp = Term.mkApp (t, Array.sub ca 0 n) in
match is_morphism goal rlt papp (nb_params - n) with
| None ->
(* here we have to memoize the failures *)
HMap.add envs.discr papp None;
if n < nb_params - 1 then aux (n+1) else goal
| Some (goal, pack) ->
let args = Array.sub ca (n) (nb_params -n)in
let goal = Array.fold_left cont goal args in
memoize envs papp pack;
goal
in
if nb_params = 0 then goal else aux 0
in
let is_aac t = is_bin rlt t in
match crop_app t ca with
| None ->
fold_morphism t ca
| Some (papp, args) ->
begin match is_aac papp goal with
| None -> fold_morphism t ca
| Some (goal, pack) ->
memoize envs papp pack;
Array.fold_left cont goal args
end
(* update in place the envs of known stuff, using memoization. We
have to memoize failures, here. *)
let gather goal (rlt : Coq.Relation.t ) envs t : Coq.goal_sigma =
let rec aux goal x =
match Coq.decomp_term x with
| App (t,ca) ->
fold goal rlt envs t ca (aux )
| _ -> goal
in
aux goal t
end
(** We build a term out of a constr, updating in place the
environment if needed (the only kind of such updates are the
constants). *)
module Parse :
sig
val t_of_constr : Coq.goal_sigma -> Coq.Relation.t -> envs -> constr -> Matcher.Terms.t * Coq.goal_sigma
end
= struct
(** [discriminates goal envs rlt t ca] infer :
- its {! pack } (is it an AC operator, or an A operator, or a
Symbol ?),
- the relevant partial application,
- the vector of the remaining arguments
We use an expansion to handle the special case of units,
before going back to the general discrimination
procedure. That means that a unit is more coarse than a raw
morphism
This functions is prevented to go through [ar < 0] by the fact
that a constant is a morphism. But not an eva. *)
let is_morphism goal (rlt : Coq.Relation.t) (papp : Term.constr) (ar : int) : (Coq.goal_sigma * pack ) option =
let typeof = Classes.Proper.mk_typeof rlt ar in
let relof = Classes.Proper.mk_relof rlt ar in
let m = Coq.Classes.mk_morphism typeof relof papp in
try
let m,goal = Coq.resolve_one_typeclass goal m in
let pack = {Sym.ar = (Coq.Nat.of_int ar); Sym.value= papp; Sym.morph= m} in
Some (goal, Sym pack)
with
| e -> None
exception NotReflexive
let discriminate goal envs (rlt : Coq.Relation.t) t ca : Coq.goal_sigma * pack * constr * constr array =
let nb_params = Array.length ca in
let rec fold goal ar :Coq.goal_sigma * pack * constr * constr array =
begin
assert (0 <= ar && ar <= nb_params);
let p_app = mkApp (t, Array.sub ca 0 (nb_params - ar)) in
begin
try
begin match HMap.find envs.discr p_app with
| None ->
fold goal (ar-1)
| Some pack ->
(goal, pack, p_app, Array.sub ca (nb_params -ar ) ar)
end
with
Not_found -> (* Could not find this constr *)
memoize (is_morphism goal rlt p_app ar) p_app ar
end
end
and memoize (x) p_app ar =
assert (0 <= ar && ar <= nb_params);
match x with
| Some (goal, pack) ->
HMap.add envs.discr p_app (Some pack);
add_bloom envs pack;
(goal, pack, p_app, Array.sub ca (nb_params-ar) ar)
| None ->
if ar = 0 then raise NotReflexive;
begin
(* to memoise the failures *)
HMap.add envs.discr p_app None;
(* will terminate, since [const] is capped, and it is
easy to find an instance of a constant *)
fold goal (ar-1)
end
in
try match HMap.find envs.discr (mkApp (t,ca))with
| None -> fold goal (nb_params)
| Some pack -> goal, pack, (mkApp (t,ca)), [| |]
with Not_found -> fold goal (nb_params)
let discriminate goal envs rlt x =
try
match Coq.decomp_term x with
| App (t,ca) ->
discriminate goal envs rlt t ca
| _ -> discriminate goal envs rlt x [| |]
with
| NotReflexive -> user_error "The relation to which the goal was lifted is not Reflexive"
(* TODO: is it the only source of invalid use that fall
into this catch_all ? *)
| e ->
user_error "Cannot handle this kind of hypotheses (variables occuring under a function symbol which is not a proper morphism)."
(** [t_of_constr goal rlt envs cstr] builds the abstract syntax tree
of the term [cstr]. Doing so, it modifies the environment of
known stuff [envs], and eventually creates fresh
evars. Therefore, we give back the goal updated accordingly *)
let t_of_constr goal (rlt: Coq.Relation.t ) envs : constr -> Matcher.Terms.t * Coq.goal_sigma =
let r_goal = ref (goal) in
let rec aux x =
match Coq.decomp_term x with
| Rel i -> Matcher.Terms.Var i
| _ ->
let goal, pack , p_app, ca = discriminate (!r_goal) envs rlt x in
r_goal := goal;
let k = find_bloom envs pack
in
match pack with
| Bin (pack,_) ->
begin match pack.Bin.comm with
| Some _ ->
assert (Array.length ca = 2);
Matcher.Terms.Plus ( k, aux ca.(0), aux ca.(1))
| None ->
assert (Array.length ca = 2);
Matcher.Terms.Dot ( k, aux ca.(0), aux ca.(1))
end
| Unit _ ->
assert (Array.length ca = 0);
Matcher.Terms.Unit ( k)
| Sym _ ->
Matcher.Terms.Sym ( k, Array.map aux ca)
in
(
fun x -> let r = aux x in r, !r_goal
)
end (* Parse *)
let add_symbol goal rlt envs l =
let goal = Gather.gather goal rlt envs (Term.mkApp (l, [| Term.mkRel 0;Term.mkRel 0|])) in
goal
(* [t_of_constr] buils a the abstract syntax tree of a constr,
updating in place the environment. Doing so, we infer all the
morphisms and the AC/A operators. It is mandatory to do so both
for the pattern and the term, since AC symbols can occur in one
and not the other *)
let t_of_constr goal rlt envs (l,r) =
let goal = Gather.gather goal rlt envs l in
let goal = Gather.gather goal rlt envs r in
let l,goal = Parse.t_of_constr goal rlt envs l in
let r, goal = Parse.t_of_constr goal rlt envs r in
l, r, goal
(* An intermediate representation of the environment, with association lists for AC/A operators, and also the raw [packer] information *)
type ir =
{
packer : (int, pack) Hashtbl.t; (* = bloom_back *)
bin : (int * Bin.pack) list ;
units : (int * Unit.pack) list;
sym : (int * Term.constr) list ;
matcher_units : Matcher.ext_units
}
let ir_to_units ir = ir.matcher_units
let ir_of_envs goal (rlt : Coq.Relation.t) envs =
let add x y l = (x,y)::l in
let nil = [] in
let sym ,
(bin : (int * Bin.pack with_unit) list),
(units : (int * constr) list) =
Hashtbl.fold
(fun int pack (sym,bin,units) ->
match pack with
| Bin s ->
sym, add (int) s bin, units
| Sym s ->
add (int) s sym, bin, units
| Unit s ->
sym, bin, add int s units
)
envs.bloom_back
(nil,nil,nil)
in
let matcher_units =
let unit_for , is_ac =
List.fold_left
(fun ((unit_for,is_ac) as acc) (n,(bp,wu)) ->
match wu with
| None -> acc
| Some (unit_of) ->
let unit_n = Hashtbl.find envs.bloom
(Unit unit_of.Unit.uf_u)
in
( n, unit_n)::unit_for,
(n, bp.Bin.comm <> None )::is_ac
)
([],[]) bin
in
{Matcher.unit_for = unit_for; Matcher.is_ac = is_ac}
in
let units : (int * Unit.pack) list =
List.fold_left
(fun acc (n,u) ->
(* first, gather all bins with this unit *)
let all_bin =
List.fold_left
( fun acc (nop,(bp,wu)) ->
match wu with
| None -> acc
| Some unit_of ->
if unit_of.Unit.uf_u = u
then
{unit_of with
Unit.uf_idx = (Coq.Pos.of_int nop)} :: acc
else
acc
)
[] bin
in
(n,{
Unit.u_value = u;
Unit.u_desc = all_bin
})::acc
)
[] units
in
goal, {
packer = envs.bloom_back;
bin = (List.map (fun (n,(s,_)) -> n, s) bin);
units = units;
sym = (List.map (fun (n,s) -> n,(Sym.mk_pack rlt s)) sym);
matcher_units = matcher_units
}
(** {1 From AST back to Coq }
The next functions allow one to map OCaml abstract syntax trees
to Coq terms *)
(** {2 Building raw, natural, terms} *)
(** [raw_constr_of_t_debruijn] rebuilds a term in the raw
representation, without products on top, and maybe escaping free
debruijn indices (in the case of a pattern for example). *)
let raw_constr_of_t_debruijn ir (t : Matcher.Terms.t) : Term.constr * int list =
let add_set,get =
let r = ref [] in
let rec add x = function
[ ] -> [x]
| t::q when t = x -> t::q
| t::q -> t:: (add x q)
in
(fun x -> r := add x !r),(fun () -> !r)
in
(* Here, we rely on the invariant that the maps are well formed:
it is meanigless to fail to find a symbol in the maps, or to
find the wrong kind of pack in the maps *)
let rec aux t =
match t with
| Matcher.Terms.Plus (s,l,r) ->
begin match Hashtbl.find ir.packer s with
| Bin (s,_) ->
mkApp (s.Bin.value , [|(aux l);(aux r)|])
| _ -> Printf.printf "erreur:%i\n%!"s;
assert false
end
| Matcher.Terms.Dot (s,l,r) ->
begin match Hashtbl.find ir.packer s with
| Bin (s,_) ->
mkApp (s.Bin.value, [|(aux l);(aux r)|])
| _ -> assert false
end
| Matcher.Terms.Sym (s,t) ->
begin match Hashtbl.find ir.packer s with
| Sym s ->
mkApp (s.Sym.value, Array.map aux t)
| _ -> assert false
end
| Matcher.Terms.Unit x ->
begin match Hashtbl.find ir.packer x with
| Unit s -> s
| _ -> assert false
end
| Matcher.Terms.Var i -> add_set i;
mkRel (i)
in
let t = aux t in
t , get ( )
(** [raw_constr_of_t] rebuilds a term in the raw representation *)
let raw_constr_of_t ir rlt (context:rel_context) t =
(** cap rebuilds the products in front of the constr *)
let rec cap c = function [] -> c
| t::q ->
let i = Term.lookup_rel t context in
cap (mkProd_or_LetIn i c) q
in
let t,indices = raw_constr_of_t_debruijn ir t in
cap t (List.sort (Pervasives.compare) indices)
(** {2 Building reified terms} *)
(* Some informations to be able to build the maps *)
type reif_params =
{
bin_null : Bin.pack; (* the default A operator *)
sym_null : constr;
unit_null : Unit.pack;
sym_ty : constr; (* the type, as it appears in e_sym *)
bin_ty : constr
}
(** A record containing the environments that will be needed by the
decision procedure, as a Coq constr. Contains the functions
from the symbols (as ints) to indexes (as constr) *)
type sigmas = {
env_sym : Term.constr;
env_bin : Term.constr;
env_units : Term.constr; (* the [idx -> X:constr] *)
}
type sigma_maps = {
sym_to_pos : int -> Term.constr;
bin_to_pos : int -> Term.constr;
units_to_pos : int -> Term.constr;
}
(** infers some stuff that will be required when we will build
environments (our environments need a default case, so we need
an Op_AC, an Op_A, and a symbol) *)
(* Note : this function can fail if the user is using the wrong
relation, like proving a = b, while the classes are defined with
another relation (==) *)
let build_reif_params goal (rlt : Coq.Relation.t) (zero) :
reif_params * Coq.goal_sigma =
let carrier = rlt.Coq.Relation.carrier in
let bin_null =
try
let op,goal = Coq.evar_binary goal carrier in
let assoc,goal = Classes.Associative.infer goal rlt op in
let compat,goal = Classes.Proper.infer goal rlt op 2 in
let op = Coq.nf_evar goal op in
{
Bin.value = op;
Bin.compat = compat;
Bin.assoc = assoc;
Bin.comm = None
}
with Not_found -> user_error "Cannot infer a default A operator (required at least to be Proper and Associative)"
in
let zero, goal =
try
let evar_op,goal = Coq.evar_binary goal carrier in
let evar_unit, goal = Coq.evar_unit goal carrier in
let query = Classes.Unit.ty rlt evar_op evar_unit in
let _, goal = Coq.resolve_one_typeclass goal query in
Coq.nf_evar goal evar_unit, goal
with _ -> zero, goal in
let sym_null = Sym.null rlt in
let unit_null = Unit.default zero in
let record =
{
bin_null = bin_null;
sym_null = sym_null;
unit_null = unit_null;
sym_ty = Sym.mk_ty rlt ;
bin_ty = Bin.mk_ty rlt
}
in
record, goal
(* We want to lift down the indexes of symbols. *)
let renumber (l: (int * 'a) list ) =
let _, l = List.fold_left (fun (next,acc) (glob,x) ->
(next+1, (next,glob,x)::acc)
) (1,[]) l
in
let rec to_global loc = function
| [] -> assert false
| (local, global,x)::q when local = loc -> global
| _::q -> to_global loc q
in
let rec to_local glob = function
| [] -> assert false
| (local, global,x)::q when global = glob -> local
| _::q -> to_local glob q
in
let locals = List.map (fun (local,global,x) -> local,x) l in
locals, (fun x -> to_local x l) , (fun x -> to_global x l)
(** [build_sigma_maps] given a envs and some reif_params, we are
able to build the sigmas *)
let build_sigma_maps (rlt : Coq.Relation.t) zero ir (k : sigmas * sigma_maps -> Proof_type.tactic ) : Proof_type.tactic = fun goal ->
let rp,goal = build_reif_params goal rlt zero in
let renumbered_sym, to_local, to_global = renumber ir.sym in
let env_sym = Sigma.of_list
rp.sym_ty
(rp.sym_null)
renumbered_sym
in
Coq.cps_mk_letin "env_sym" env_sym
(fun env_sym ->
let bin = (List.map ( fun (n,s) -> n, Bin.mk_pack rlt s) ir.bin) in
let env_bin =
Sigma.of_list
rp.bin_ty
(Bin.mk_pack rlt rp.bin_null)
bin
in
Coq.cps_mk_letin "env_bin" env_bin
(fun env_bin ->
let units = (List.map (fun (n,s) -> n, Unit.mk_pack rlt env_bin s)ir.units) in
let env_units =
Sigma.of_list
(Unit.ty_unit_pack rlt env_bin)
(Unit.mk_pack rlt env_bin rp.unit_null )
units
in
Coq.cps_mk_letin "env_units" env_units
(fun env_units ->
let sigmas =
{
env_sym = env_sym ;
env_bin = env_bin ;
env_units = env_units;
} in
let f = List.map (fun (x,_) -> (x,Coq.Pos.of_int x)) in
let sigma_maps =
{
sym_to_pos = (let sym = f renumbered_sym in fun x -> (List.assoc (to_local x) sym));
bin_to_pos = (let bin = f bin in fun x -> (List.assoc x bin));
units_to_pos = (let units = f units in fun x -> (List.assoc x units));
}
in
k (sigmas, sigma_maps )
)
)
) goal
(** builders for the reification *)
type reif_builders =
{
rsum: constr -> constr -> constr -> constr;
rprd: constr -> constr -> constr -> constr;
rsym: constr -> constr array -> constr;
runit : constr -> constr
}
(* donne moi une tactique, je rajoute ma part. Potentiellement, il
est possible d'utiliser la notation 'do' a la Haskell:
http://www.cas.mcmaster.ca/~carette/pa_monad/ *)
let (>>) : 'a * Proof_type.tactic -> ('a -> 'b * Proof_type.tactic) -> 'b * Proof_type.tactic =
fun (x,t) f ->
let b,t' = f x in
b, Tacticals.tclTHEN t t'
let return x = x, Tacticals.tclIDTAC
let mk_vect vnil vcons v =
let ar = Array.length v in
let rec aux = function
| 0 -> vnil
| n -> let n = n-1 in
mkApp( vcons,
[|
(Coq.Nat.of_int n);
v.(ar - 1 - n);
(aux (n))
|]
)
in aux ar
(* TODO: use a do notation *)
let mk_reif_builders (rlt: Coq.Relation.t) (env_sym:constr) (k: reif_builders -> Proof_type.tactic) =
let x = (rlt.Coq.Relation.carrier) in
let r = (rlt.Coq.Relation.r) in
let x_r_env = [|x;r;env_sym|] in
let tty = mkApp (Lazy.force Stubs._Tty, x_r_env) in
let rsum = mkApp (Lazy.force Stubs.rsum, x_r_env) in
let rprd = mkApp (Lazy.force Stubs.rprd, x_r_env) in
let rsym = mkApp (Lazy.force Stubs.rsym, x_r_env) in
let vnil = mkApp (Lazy.force Stubs.vnil, x_r_env) in
let vcons = mkApp (Lazy.force Stubs.vcons, x_r_env) in
Coq.cps_mk_letin "tty" tty
(fun tty ->
Coq.cps_mk_letin "rsum" rsum
(fun rsum ->
Coq.cps_mk_letin "rprd" rprd
(fun rprd ->
Coq.cps_mk_letin "rsym" rsym
(fun rsym ->
Coq.cps_mk_letin "vnil" vnil
(fun vnil ->
Coq.cps_mk_letin "vcons" vcons
(fun vcons ->
let r ={
rsum =
begin fun idx l r ->
mkApp (rsum, [| idx ; mk_mset tty [l,1 ; r,1]|])
end;
rprd =
begin fun idx l r ->
let lst = NEList.of_list tty [l;r] in
mkApp (rprd, [| idx; lst|])
end;
rsym =
begin fun idx v ->
let vect = mk_vect vnil vcons v in
mkApp (rsym, [| idx; vect|])
end;
runit = fun idx -> (* could benefit of a letin *)
mkApp (Lazy.force Stubs.runit , [|x;r;env_sym;idx; |])
}
in k r
))))))
type reifier = sigma_maps * reif_builders
let mk_reifier rlt zero envs (k : sigmas *reifier -> Proof_type.tactic) =
build_sigma_maps rlt zero envs
(fun (s,sm) ->
mk_reif_builders rlt s.env_sym
(fun rb ->k (s,(sm,rb)) )
)
(** [reif_constr_of_t reifier t] rebuilds the term [t] in the
reified form. We use the [reifier] to minimise the size of the
terms (we make as much lets as possible)*)
let reif_constr_of_t (sm,rb) (t:Matcher.Terms.t) : constr =
let rec aux = function
| Matcher.Terms.Plus (s,l,r) ->
let idx = sm.bin_to_pos s in
rb.rsum idx (aux l) (aux r)
| Matcher.Terms.Dot (s,l,r) ->
let idx = sm.bin_to_pos s in
rb.rprd idx (aux l) (aux r)
| Matcher.Terms.Sym (s,t) ->
let idx = sm.sym_to_pos s in
rb.rsym idx (Array.map aux t)
| Matcher.Terms.Unit s ->
let idx = sm.units_to_pos s in
rb.runit idx
| Matcher.Terms.Var i ->
anomaly "call to reif_constr_of_t on a term with variables."
in aux t
end
aac_tactics/theory.mli 0000664 0000000 0000000 00000017101 12211622232 0015322 0 ustar 00root root 0000000 0000000 (***************************************************************************)
(* 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. *)
(***************************************************************************)
(** Bindings for Coq constants that are specific to the plugin;
reification and translation functions.
Note: this module is highly correlated with the definitions of {i
AAC.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 {i
AAC.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:Term.constr -> (Term.constr * int) list ->Term.constr
(** {2 Packaging modules} *)
(** Environments *)
module Sigma:
sig
(** [add ty n x map] adds the value [x] of type [ty] with key [n] in [map] *)
val add: Term.constr ->Term.constr ->Term.constr ->Term.constr ->Term.constr
(** [empty ty] create an empty map of type [ty] *)
val empty: Term.constr ->Term.constr
(** [of_list ty null l] translates an OCaml association list into a Coq one *)
val of_list: Term.constr -> Term.constr -> (int * Term.constr ) list -> Term.constr
(** [to_fun ty null map] converts a Coq association list into a Coq function (with default value [null]) *)
val to_fun: Term.constr ->Term.constr ->Term.constr ->Term.constr
end
(** Dynamically typed morphisms *)
module Sym:
sig
(** mimics the Coq record [Sym.pack] *)
type pack = {ar: Term.constr; value: Term.constr ; morph: Term.constr}
val typ: Term.constr lazy_t
(** [mk_pack rlt (ar,value,morph)] *)
val mk_pack: Coq.Relation.t -> pack -> Term.constr
(** [null] builds a dummy (identity) symbol, given an {!Coq.Relation.t} *)
val null: Coq.Relation.t -> Term.constr
end
(** We need to export some Coq stubs out of this module. They are used
by the main tactic, see {!Rewrite} *)
module Stubs : sig
val lift : Term.constr Lazy.t
val lift_proj_equivalence : Term.constr Lazy.t
val lift_transitivity_left : Term.constr Lazy.t
val lift_transitivity_right : Term.constr Lazy.t
val lift_reflexivity : Term.constr Lazy.t
(** The evaluation fonction, used to convert a reified coq term to a
raw coq term *)
val eval: Term.constr lazy_t
(** The main lemma of our theory, that is
[compare (norm u) (norm v) = Eq -> eval u == eval v] *)
val decide_thm:Term.constr lazy_t
val lift_normalise_thm : Term.constr lazy_t
end
(** {2 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. *)
(** Tranlations between Coq and OCaml *)
module Trans : sig
(** 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:
- a term like [f x y] might yield a unary morphism ([f x]) and a
binary one ([f]); we select the latter one (unless [f] is A or
AC, in which case we declare it accordingly);
- a term like [S O] can be considered as a morphism ([S])
applied to a unit for [(+)], or as a unit for [( * )]; we
chose to give priority to units, so that the latter
interpretation is selected in this case;
- an element might be the unit for several operations
*)
(** 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
(** {2 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 : Coq.goal_sigma -> Coq.Relation.t -> envs -> (Term.constr * Term.constr) -> Matcher.Terms.t * Matcher.Terms.t * Coq.goal_sigma
(** [add_symbol] adds a given binary symbol to the environment of
known stuff. *)
val add_symbol : Coq.goal_sigma -> Coq.Relation.t -> envs -> Term.constr -> Coq.goal_sigma
(** {2 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
{!t_of_constr}; second, reified Coq terms, that are required for
the reflexive decision procedure. *)
type ir
val ir_of_envs : Coq.goal_sigma -> Coq.Relation.t -> envs -> Coq.goal_sigma * ir
val ir_to_units : ir -> Matcher.ext_units
(** {2 Building raw, natural, terms} *)
(** [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. *)
val raw_constr_of_t : ir -> Coq.Relation.t -> (Term.rel_context) ->Matcher.Terms.t -> Term.constr
(** {2 Building reified terms} *)
(** The reification environments, as Coq constrs *)
type sigmas = {
env_sym : Term.constr;
env_bin : Term.constr;
env_units : Term.constr; (* the [idx -> X:constr] *)
}
(** 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.*)
type reifier
val mk_reifier : Coq.Relation.t -> Term.constr -> ir -> (sigmas * reifier -> Proof_type.tactic) -> Proof_type.tactic
(** [reif_constr_of_t reifier t] rebuilds the term [t] in the
reified form. *)
val reif_constr_of_t : reifier -> Matcher.Terms.t -> Term.constr
end