The Coq Proof Assistant

# Library Coq.btauto.Algebra

Require Import Bool PArith DecidableClass Ring Lia.
Ltac bool :=
repeat match goal with
| [ H : ?P && ?Q = true |- _ ] =>
apply andb_true_iff in H; destruct H
| |- ?P && ?Q = true =>
apply <- andb_true_iff; split
end.
Arguments decide P /H.
#[global]
Hint Extern 5 => progress bool : core.
Ltac define t x H :=
set (x := t) in *; assert (H : t = x) by reflexivity; clearbody x.
Lemma Decidable_sound : forall P (H : Decidable P),
decide P = true -> P.
Lemma Decidable_complete : forall P (H : Decidable P),
P -> decide P = true.
Lemma Decidable_sound_alt : forall P (H : Decidable P),
~ P -> decide P = false.
Lemma Decidable_complete_alt : forall P (H : Decidable P),
decide P = false -> ~ P.
Ltac try_rewrite :=
repeat match goal with
| [ H : ?P |- _ ] => rewrite H
end.
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.
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 }.
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 }.

# 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.
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
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
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.
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
if decide (null r) then poly_mul_cst v p
else Poly (poly_mul_cst v p) i r
end.
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.
Fixpoint poly_mul pl {struct pl} :=
match pl with
| Cst cl => poly_mul_cst cl
| Poly pl il ql =>
fun pr =>
let qs := poly_mul ql pr in
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
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
if decide (null qs) then (reduce_aux k p)
else Poly (reduce_aux k p) i qs
end.
Fixpoint reduce p :=
match p with
| Cst c => Cst c
| Poly p i q =>
let qs := reduce_aux i q in
if decide (null qs) then reduce p
else Poly (reduce p) i qs
end.
End Computational.
Section Validity.
#[local]
Hint Constructors valid linear : core.
Lemma valid_le_compat : forall k l p, valid k p -> (k <= l)%positive -> valid l p.
Lemma linear_le_compat : forall k l p, linear k p -> (k <= l)%positive -> linear l p.
Lemma linear_valid_incl : forall k p, linear k p -> valid k p.
End Validity.
Section Evaluation.
Lemma eval_null_zero : forall p var, null p -> eval var p = false.
Lemma eval_extensional_eq_compat : forall p var1 var2,
(forall x, list_nth x var1 false = list_nth x var2 false) -> eval var1 p = eval var2 p.
Lemma eval_suffix_compat : forall k p var1 var2,
(forall i, (i < k)%positive -> list_nth i var1 false = list_nth i var2 false) -> valid k p ->
eval var1 p = eval var2 p.
End Evaluation.
Section Algebra.
Lemma poly_add_compat : forall pl pr var, eval var (poly_add pl pr) = xorb (eval var pl) (eval var pr).
Lemma poly_mul_cst_compat : forall v p var,
eval var (poly_mul_cst v p) = andb v (eval var p).
Lemma poly_mul_mon_compat : forall i p var,
eval var (poly_mul_mon i p) = (list_nth i var false && eval var p).
Lemma poly_mul_compat : forall pl pr var, eval var (poly_mul pl pr) = andb (eval var pl) (eval var pr).
#[local]
Hint Extern 5 =>
match goal with
| [ |- (Pos.max ?x ?y <= ?z)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (?z <= Pos.max ?x ?y)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (Pos.max ?x ?y < ?z)%positive ] =>
apply Pos.max_case_strong; intros; lia
| [ |- (?z < Pos.max ?x ?y)%positive ] =>
apply Pos.max_case_strong; intros; lia
| _ => lia
end : core.
#[local]
Hint Resolve Pos.le_max_r Pos.le_max_l : core.
#[local]
Hint Constructors valid linear : core.
Lemma poly_add_valid_compat : forall kl kr pl pr, valid kl pl -> valid kr pr ->
valid (Pos.max kl kr) (poly_add pl pr).
Lemma poly_mul_cst_valid_compat : forall k v p, valid k p -> valid k (poly_mul_cst v p).
Lemma poly_mul_mon_null_compat : forall i p, null (poly_mul_mon i p) -> null p.
Lemma poly_mul_mon_valid_compat : forall k i p,
valid k p -> valid (Pos.max (Pos.succ i) k) (poly_mul_mon i p).
Lemma poly_mul_valid_compat : forall kl kr pl pr, valid kl pl -> valid kr pr ->
valid (Pos.max kl kr) (poly_mul pl pr).
Lemma poly_add_linear_compat : forall kl kr pl pr, linear kl pl -> linear kr pr ->
linear (Pos.max kl kr) (poly_add pl pr).
End Algebra.
Section Reduce.
Lemma reduce_aux_eval_compat : forall k p var, valid (Pos.succ k) p ->
(list_nth k var false && eval var (reduce_aux k p) = list_nth k var false && eval var p).
Lemma reduce_eval_compat : forall k p var, valid k p ->
eval var (reduce p) = eval var p.
Lemma reduce_aux_le_compat : forall k l p, valid k p -> (k <= l)%positive ->
reduce_aux l p = reduce_aux k p.
Lemma linear_reduce_aux : forall i p, valid (Pos.succ i) p -> linear i (reduce_aux i p).
Lemma linear_reduce : forall k p, valid k p -> linear k (reduce p).
End Reduce.