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(***************************************************************************)
(* This is part of aac_tactics, it is distributed under the terms of the *)
(* GNU Lesser General Public License version 3 *)
(* (see file LICENSE for more details) *)
(* *)
(* Copyright 2009-2010: Thomas Braibant, Damien Pous. *)
(***************************************************************************)
(** * Theory 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 #<a href="Instances.html">Instances.v</a># 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".
|