Ltac try_rewrite :=
repeat match goal with
| [ H : ?P |- _ ] => rewrite H
end.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.(** * 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.(** * 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.(** * 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).(* 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).(** 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).(** 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).(** 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).(** 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).Program Instance Decidable_PosEq : forall (p q : positive), Decidable (p = q) :=
  { Decidable_witness := Pos.eqb p q }.Program Instance Decidable_PosLt : forall p q, Decidable (Pos.lt p q) :=
  { Decidable_witness := Pos.ltb p q }.Program Instance Decidable_PosLe : forall p q, Decidable (Pos.le p q) :=
  { Decidable_witness := Pos.leb p q }.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
}.Program Instance Decidable_null : forall p, Decidable (null p) := {
  Decidable_witness := match p with Cst false => true | _ => false end
}.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
}.(** 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.

Lemma eval_null_zero : forall p var, null p -> eval var p = false.