# Library Coq.setoid_ring.Ncring_polynom

Set Implicit Arguments.
Require Import Setoid.
Require Import BinList.
Require Import BinPos.
Require Import BinNat.
Require Import BinInt.
Require Export Ring_polynom. Require Export Ncring.

#[local] Create HintDb rsimpl.

Section MakeRingPol.

Context (C R:Type) `{Rh:Ring_morphism C R}.

Variable phiCR_comm: forall (c:C)(x:R), x * [c] == [c] * x.

Ltac rsimpl := repeat (gen_rewrite || rewrite phiCR_comm).

#[local] Hint Rewrite
ring_mul_1_l ring_mul_1_r
ring_mul_0_l ring_mul_0_r
ring_distr_l ring_distr_r
: rsimpl.

Inductive Pol : Type :=
| Pc : C -> Pol
| PX : Pol -> positive -> positive -> Pol -> Pol.
Definition cO:C .
Definition cI:C .

Definition P0 := Pc 0.
Definition P1 := Pc 1.

Variable Ceqb:C->C->bool.
#[universes(template)]
Class Equalityb (A : Type):= {equalityb : A -> A -> bool}.
Notation "x =? y" := (equalityb x y) (at level 70, no associativity).
Variable Ceqb_eq: forall x y:C, Ceqb x y = true -> (x == y).

Instance equalityb_coef : Equalityb C :=
{equalityb x y := Ceqb x y}.

Fixpoint Peq (P P' : Pol) {struct P'} : bool :=
match P, P' with
| Pc c, Pc c' => c =? c'
| PX P i n Q, PX P' i' n' Q' =>
match Pos.compare i i', Pos.compare n n' with
| Eq, Eq => if Peq P P' then Peq Q Q' else false
| _,_ => false
end
| _, _ => false
end.

Instance equalityb_pol : Equalityb Pol :=
{equalityb x y := Peq x y}.

Definition mkPX P i n Q :=
match P with
| Pc c => if c =? 0 then Q else PX P i n Q
| PX P' i' n' Q' =>
match Pos.compare i i' with
| Eq => if Q' =? P0 then PX P' i (n + n') Q else PX P i n Q
| _ => PX P i n Q
end
end.

Definition mkXi i n := PX P1 i n P0.

Definition mkX i := mkXi i 1.

Fixpoint Popp (P:Pol) : Pol :=
match P with
| Pc c => Pc (- c)
| PX P i n Q => PX (Popp P) i n (Popp Q)
end.

Notation "-- P" := (Popp P)(at level 30).

Fixpoint PaddCl (c:C)(P:Pol) {struct P} : Pol :=
match P with
| Pc c1 => Pc (c + c1)
| PX P i n Q => PX P i n (PaddCl c Q)
end.

Variable P:Pol.

match Q with
| Pc c => mkPX P i n Q
| PX P' i' n' Q' =>
match Pos.compare i i' with
|
Gt => mkPX P i n Q
|
Lt => mkPX P' i' n' (PaddX i n Q')
|
Eq => match Z.pos_sub n n' with
|
Zpos k => mkPX (PaddX i k P') i' n' Q'
|
Z0 => mkPX (Padd P P') i n Q'
|
Zneg k => mkPX (Padd P (mkPX P' i k P0)) i n Q'
end
end
end.

Fixpoint Padd (P1 P2: Pol) {struct P1} : Pol :=
match P1 with
| Pc c => PaddCl c P2
| PX P' i' n' Q' =>
end.

Notation "P ++ P'" := (Padd P P').

Definition Psub(P P':Pol):= P ++ (--P').

Notation "P -- P'" := (Psub P P')(at level 50).

Multiplication

Fixpoint PmulC_aux (P:Pol) (c:C) {struct P} : Pol :=
match P with
| Pc c' => Pc (c' * c)
| PX P i n Q => mkPX (PmulC_aux P c) i n (PmulC_aux Q c)
end.

Definition PmulC P c :=
if c =? 0 then P0 else
if c =? 1 then P else PmulC_aux P c.

Fixpoint Pmul (P1 P2 : Pol) {struct P2} : Pol :=
match P2 with
| Pc c => PmulC P1 c
| PX P i n Q =>
end.

Notation "P ** P'" := (Pmul P P')(at level 40).

Definition Psquare (P:Pol) : Pol := P ** P.

Evaluation of a polynomial towards R

Fixpoint Pphi(l:list R) (P:Pol) {struct P} : R :=
match P with
| Pc c => [c]
| PX P i n Q =>
let x := nth 0 i l in
let xn := pow_pos x n in
(Pphi l P) * xn + (Pphi l Q)
end.

Reserved Notation "P @ l " (at level 10, no associativity).
Notation "P @ l " := (Pphi l P).

Proofs

Ltac destr_pos_sub H :=
match goal with |- context [Z.pos_sub ?x ?y] =>
assert (H := Z.pos_sub_discr x y); destruct (Z.pos_sub x y)
end.

Lemma Peq_ok : forall P P',
(P =? P') = true -> forall l, P@l == P'@ l.

Lemma Pphi0 : forall l, P0@l == 0.

Lemma Pphi1 : forall l, P1@l == 1.

Lemma mkPX_ok : forall l P i n Q,
(mkPX P i n Q)@l == P@l * (pow_pos (nth 0 i l) n) + Q@l.

Ltac Esimpl :=
repeat (progress (
match goal with
| |- context [?P@?l] =>
match P with
| P0 => rewrite (Pphi0 l)
| P1 => rewrite (Pphi1 l)
| (mkPX ?P ?i ?n ?Q) => rewrite (mkPX_ok l P i n Q)
end
| |- context [[?c]] =>
match c with
| 0 => rewrite ring_morphism0
| 1 => rewrite ring_morphism1
| ?x + ?y => rewrite ring_morphism_add
| ?x * ?y => rewrite ring_morphism_mul
| ?x - ?y => rewrite ring_morphism_sub
| - ?x => rewrite ring_morphism_opp
end
end));
simpl; rsimpl.

Lemma PaddCl_ok : forall c P l, (PaddCl c P)@l == [c] + P@l .

Lemma PmulC_aux_ok : forall c P l, (PmulC_aux P c)@l == P@l * [c].

Lemma PmulC_ok : forall c P l, (PmulC P c)@l == P@l * [c].

Lemma Popp_ok : forall P l, (--P)@l == - P@l.

Ltac Esimpl2 :=
Esimpl;
repeat (progress (
match goal with
| |- context [(PaddCl ?c ?P)@?l] => rewrite (PaddCl_ok c P l)
| |- context [(PmulC ?P ?c)@?l] => rewrite (PmulC_ok c P l)
| |- context [(--?P)@?l] => rewrite (Popp_ok P l)
end)); Esimpl.

Lemma PaddXPX: forall P i n Q,
match Q with
| Pc c => mkPX P i n Q
| PX P' i' n' Q' =>
match Pos.compare i i' with
|
Gt => mkPX P i n Q
|
Lt => mkPX P' i' n' (PaddX Padd P i n Q')
|
Eq => match Z.pos_sub n n' with
|
Zpos k => mkPX (PaddX Padd P i k P') i' n' Q'
|
Z0 => mkPX (Padd P P') i n Q'
|
Zneg k => mkPX (Padd P (mkPX P' i k P0)) i n Q'
end
end
end.

(forall P l, (P2 ++ P) @ l == P2 @ l + P @ l)
/\
(forall P k n l,
P2 @ l * pow_pos (nth 0 k l) n + P @ l).

Lemma Padd_ok : forall P Q l, (P ++ Q) @ l == P @ l + Q @ l.

Lemma PaddX_ok : forall P2 P k n l,
(PaddX Padd P2 k n P) @ l == P2 @ l * pow_pos (nth 0 k l) n + P @ l.

Lemma Psub_ok : forall P' P l, (P -- P')@l == P@l - P'@l.

Lemma Pmul_ok : forall P P' l, (P**P')@l == P@l * P'@l.

Lemma Psquare_ok : forall P l, (Psquare P)@l == P@l * P@l.

Definition of polynomial expressions

Specification of the power function
Section POWER.
Variable Cpow : Set.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.

Record power_theory : Prop := mkpow_th {
rpow_pow_N : forall r n, (rpow r (Cp_phi n))== (pow_N r n)
}.

End POWER.
Variable Cpow : Set.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.
Variable pow_th : power_theory Cp_phi rpow.

evaluation of polynomial expressions towards R

Fixpoint PEeval (l:list R) (pe:PExpr C) {struct pe} : R :=
match pe with
| PEO => 0
| PEI => 1
| PEc c => [c]
| PEX _ j => nth 0 j l
| PEadd pe1 pe2 => (PEeval l pe1) + (PEeval l pe2)
| PEsub pe1 pe2 => (PEeval l pe1) - (PEeval l pe2)
| PEmul pe1 pe2 => (PEeval l pe1) * (PEeval l pe2)
| PEopp pe1 => - (PEeval l pe1)
| PEpow pe1 n => rpow (PEeval l pe1) (Cp_phi n)
end.

Strategy expand [PEeval].

Definition mk_X j := mkX j.

Correctness proofs

Lemma mkX_ok : forall p l, nth 0 p l == (mk_X p) @ l.

Ltac Esimpl3 :=
repeat match goal with
| |- context [(?P1 ++ ?P2)@?l] => rewrite (Padd_ok P1 P2 l)
| |- context [(?P1 -- ?P2)@?l] => rewrite (Psub_ok P1 P2 l)

Section POWER2.
Variable subst_l : Pol -> Pol.
Fixpoint Ppow_pos (res P:Pol) (p:positive){struct p} : Pol :=
match p with
| xH => subst_l (Pmul P res)
| xO p => Ppow_pos (Ppow_pos res P p) P p
| xI p => subst_l (Pmul P (Ppow_pos (Ppow_pos res P p) P p))
end.

Definition Ppow_N P n :=
match n with
| N0 => P1
| Npos p => Ppow_pos P1 P p
end.

Fixpoint pow_pos_gen (R:Type)(m:R->R->R)(x:R) (i:positive) {struct i}: R :=
match i with
| xH => x
| xO i => let p := pow_pos_gen m x i in m p p
| xI i => let p := pow_pos_gen m x i in m x (m p p)
end.

Lemma Ppow_pos_ok : forall l, (forall P, subst_l P@l == P@l) ->
forall res P p, (Ppow_pos res P p)@l == (pow_pos_gen Pmul P p)@l * res@l.

Definition pow_N_gen (R:Type)(x1:R)(m:R->R->R)(x:R) (p:N) :=
match p with
| N0 => x1
| Npos p => pow_pos_gen m x p
end.

Lemma Ppow_N_ok : forall l, (forall P, subst_l P@l == P@l) ->
forall P n, (Ppow_N P n)@l == (pow_N_gen P1 Pmul P n)@l.

End POWER2.

Normalization and rewriting

Section NORM_SUBST_REC.
Let subst_l (P:Pol) := P.
Let Pmul_subst P1 P2 := subst_l (Pmul P1 P2).
Let Ppow_subst := Ppow_N subst_l.

Fixpoint norm_aux (pe:PExpr C) : Pol :=
match pe with
| PEO => Pc cO
| PEI => Pc cI
| PEc c => Pc c
| PEX _ j => mk_X j
| PEadd pe1 (PEopp pe2) =>
Psub (norm_aux pe1) (norm_aux pe2)
| PEsub pe1 pe2 => Psub (norm_aux pe1) (norm_aux pe2)
| PEmul pe1 pe2 => Pmul (norm_aux pe1) (norm_aux pe2)
| PEopp pe1 => Popp (norm_aux pe1)
| PEpow pe1 n => Ppow_N (fun p => p) (norm_aux pe1) n
end.

Definition norm_subst pe := subst_l (norm_aux pe).

Lemma norm_aux_spec : forall l pe, PEeval l pe == (norm_aux pe)@l.

Lemma norm_subst_spec : forall l pe, PEeval l pe == (norm_subst pe)@l.

End NORM_SUBST_REC.

Fixpoint interp_PElist (l:list R) (lpe:list (PExpr C * PExpr C)) {struct lpe} : Prop :=
match lpe with
| nil => True
| (me,pe)::lpe =>
match lpe with
| nil => PEeval l me == PEeval l pe
| _ => PEeval l me == PEeval l pe /\ interp_PElist l lpe
end
end.

Lemma norm_subst_ok : forall l pe, PEeval l pe == (norm_subst pe)@l.

Lemma ring_correct : forall l pe1 pe2,
(norm_subst pe1 =? norm_subst pe2) = true ->
PEeval l pe1 == PEeval l pe2.

End MakeRingPol.