Ltac try_rewrite :=
repeat match goal with
| [ H : ?P |- _ ] => rewrite H
end.
(* We opacify here decide for proofs, and will make it transparent for reflexive tactics later on. *)
Global Opaque decide.
Ltac tac_decide :=
match goal with
| [ H : @decide ?P ?D = true |- _ ] => apply (@Decidable_sound P D) in H
| [ H : @decide ?P ?D = false |- _ ] => apply (@Decidable_complete_alt P D) in H
| [ |- @decide ?P ?D = true ] => apply (@Decidable_complete P D)
| [ |- @decide ?P ?D = false ] => apply (@Decidable_sound_alt P D)
| [ |- negb ?b = true ] => apply negb_true_iff
| [ |- negb ?b = false ] => apply negb_false_iff
| [ H : negb ?b = true |- _ ] => apply negb_true_iff in H
| [ H : negb ?b = false |- _ ] => apply negb_false_iff in H
end.
Ltac try_decide := repeat tac_decide.
Ltac make_decide P := match goal with
| [ |- context [@decide P ?D] ] =>
let b := fresh "b" in
let H := fresh "H" in
define (@decide P D) b H; destruct b; try_decide
| [ X : context [@decide P ?D] |- _ ] =>
let b := fresh "b" in
let H := fresh "H" in
define (@decide P D) b H; destruct b; try_decide
end.
Ltac case_decide := match goal with
| [ |- context [@decide ?P ?D] ] =>
let b := fresh "b" in
let H := fresh "H" in
define (@decide P D) b H; destruct b; try_decide
| [ X : context [@decide ?P ?D] |- _ ] =>
let b := fresh "b" in
let H := fresh "H" in
define (@decide P D) b H; destruct b; try_decide
| [ |- context [Pos.compare ?x ?y] ] =>
destruct (Pos.compare_spec x y); try lia
| [ X : context [Pos.compare ?x ?y] |- _ ] =>
destruct (Pos.compare_spec x y); try lia
end.
Section Definitions.
(** * Global, inductive definitions. *)
(** A Horner polynomial is either a constant, or a product P × (i + Q), where i is a variable. *)
Inductive poly :=
| Cst : bool -> poly
| Poly : poly -> positive -> poly -> poly.
(* TODO: We should use [positive] instead of [nat] to encode variables, for efficiency purpose. *)
Inductive null : poly -> Prop :=
| null_intro : null (Cst false).
(** Polynomials satisfy a uniqueness condition whenever they are valid. A polynomial [p] satisfies [valid n p] whenever it is well-formed and each of its variable indices is < [n]. *)
Inductive valid : positive -> poly -> Prop :=
| valid_cst : forall k c, valid k (Cst c)
| valid_poly : forall k p i q,
Pos.lt i k -> ~ null q -> valid i p -> valid (Pos.succ i) q -> valid k (Poly p i q).
(** Linear polynomials are valid polynomials in which every variable appears at most once. *)
Inductive linear : positive -> poly -> Prop :=
| linear_cst : forall k c, linear k (Cst c)
| linear_poly : forall k p i q, Pos.lt i k -> ~ null q ->
linear i p -> linear i q -> linear k (Poly p i q).
End Definitions.
Section Computational.
Program Instance Decidable_PosEq : forall (p q : positive), Decidable (p = q) :=
{ Decidable_witness := Pos.eqb p q }.
Next Obligation.
apply Pos.eqb_eq.
Qed.
Program Instance Decidable_PosLt : forall p q, Decidable (Pos.lt p q) :=
{ Decidable_witness := Pos.ltb p q }.
Next Obligation.
apply Pos.ltb_lt.
Qed.
Program Instance Decidable_PosLe : forall p q, Decidable (Pos.le p q) :=
{ Decidable_witness := Pos.leb p q }.
Next Obligation.
apply Pos.leb_le.
Qed.
(** * The core reflexive part. *)
#[local]
Hint Constructors valid : core.
Fixpoint beq_poly pl pr :=
match pl with
| Cst cl =>
match pr with
| Cst cr => decide (cl = cr)
| Poly _ _ _ => false
end
| Poly pl il ql =>
match pr with
| Cst _ => false
| Poly pr ir qr =>
decide (il = ir) && beq_poly pl pr && beq_poly ql qr
end
end.
(* We could do that with [decide equality] but dependency in proofs is heavy *)
Program Instance Decidable_eq_poly : forall (p q : poly), Decidable (eq p q) := {
Decidable_witness := beq_poly p q
}.
Next Obligation.
split.
- revert q; induction p; intros [] ?; simpl in *; bool; try_decide;
f_equal; first [intuition congruence|auto].
- revert q; induction p; intros [] Heq; simpl in *; bool; try_decide; intuition;
try injection Heq; first[congruence|intuition].
Qed.
Program Instance Decidable_null : forall p, Decidable (null p) := {
Decidable_witness := match p with Cst false => true | _ => false end
}.
Next Obligation.
split.
- destruct p as [[]|]; first [discriminate|constructor].
- inversion 1; trivial.
Qed.
Definition list_nth {A} p (l : list A) def :=
Pos.peano_rect (fun _ => list A -> A)
(fun l => match l with nil => def | cons t l => t end)
(fun _ F l => match l with nil => def | cons t l => F l end) p l.
Fixpoint eval var (p : poly) :=
match p with
| Cst c => c
| Poly p i q =>
let vi := list_nth i var false in
xorb (eval var p) (andb vi (eval var q))
end.
Fixpoint valid_dec k p :=
match p with
| Cst c => true
| Poly p i q =>
negb (decide (null q)) && decide (i < k)%positive &&
valid_dec i p && valid_dec (Pos.succ i) q
end.
Program Instance Decidable_valid : forall n p, Decidable (valid n p) := {
Decidable_witness := valid_dec n p
}.
Next Obligation.
split.
- revert n; induction p; unfold valid_dec in *; intuition; bool; try_decide; auto.
- intros H; induction H; unfold valid_dec in *; bool; try_decide; auto.
Qed.
(** Basic algebra *)
(* Addition of polynomials *)
Fixpoint poly_add pl {struct pl} :=
match pl with
| Cst cl =>
fix F pr := match pr with
| Cst cr => Cst (xorb cl cr)
| Poly pr ir qr => Poly (F pr) ir qr
end
| Poly pl il ql =>
fix F pr {struct pr} := match pr with
| Cst cr => Poly (poly_add pl pr) il ql
| Poly pr ir qr =>
match Pos.compare il ir with
| Eq =>
let qs := poly_add ql qr in
(* Ensure validity *)
if decide (null qs) then poly_add pl pr
else Poly (poly_add pl pr) il qs
| Gt => Poly (poly_add pl (Poly pr ir qr)) il ql
| Lt => Poly (F pr) ir qr
end
end
end.
(* Multiply a polynomial by a constant *)
Fixpoint poly_mul_cst v p :=
match p with
| Cst c => Cst (andb c v)
| Poly p i q =>
let r := poly_mul_cst v q in
(* Ensure validity *)
if decide (null r) then poly_mul_cst v p
else Poly (poly_mul_cst v p) i r
end.
(* Multiply a polynomial by a monomial *)
Fixpoint poly_mul_mon k p :=
match p with
| Cst c =>
if decide (null p) then p
else Poly (Cst false) k p
| Poly p i q =>
if decide (i <= k)%positive then Poly (Cst false) k (Poly p i q)
else Poly (poly_mul_mon k p) i (poly_mul_mon k q)
end.
(* Multiplication of polynomials *)
Fixpoint poly_mul pl {struct pl} :=
match pl with
| Cst cl => poly_mul_cst cl
| Poly pl il ql =>
fun pr =>
(* Multiply by a factor *)
let qs := poly_mul ql pr in
(* Ensure validity *)
if decide (null qs) then poly_mul pl pr
else poly_add (poly_mul pl pr) (poly_mul_mon il qs)
end.
(** Quotienting a polynomial by the relation X_i^2 ~ X_i *)
(* Remove the multiple occurrences of monomials x_k *)
Fixpoint reduce_aux k p :=
match p with
| Cst c => Cst c
| Poly p i q =>
if decide (i = k) then poly_add (reduce_aux k p) (reduce_aux k q)
else
let qs := reduce_aux i q in
(* Ensure validity *)
if decide (null qs) then (reduce_aux k p)
else Poly (reduce_aux k p) i qs
end.
(* Rewrite any x_k ^ {n + 1} to x_k *)
Fixpoint reduce p :=
match p with
| Cst c => Cst c
| Poly p i q =>
let qs := reduce_aux i q in
(* Ensure validity *)
if decide (null qs) then reduce p
else Poly (reduce p) i qs
end.
End Computational.
Section Validity.
(* Decision procedure of validity *)
#[local]
Hint Constructors valid linear : core.

Lemma valid_le_compat : forall k l p, valid k p -> (k <= l)%positive -> valid l p.