The Coq Proof Assistant

# Library Coq.setoid_ring.Ring_polynom

Set Implicit Arguments.
Require Import Setoid Morphisms.
Require Import BinList BinPos BinNat BinInt.
Require Export Ring_theory.
Local Open Scope positive_scope.
Import RingSyntax.
Section MakeRingPol.
Variable R:Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R->R).
Variable req : R -> R -> Prop.
Variable Rsth : Equivalence req.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Variable ARth : almost_ring_theory rO rI radd rmul rsub ropp req.
Variable C: Type.
Variable (cO cI: C) (cadd cmul csub : C->C->C) (copp : C->C).
Variable ceqb : C->C->bool.
Variable phi : C -> R.
Variable CRmorph : ring_morph rO rI radd rmul rsub ropp req
cO cI cadd cmul csub copp ceqb phi.
Variable Cpow : Type.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.
Variable pow_th : power_theory rI rmul req Cp_phi rpow.
Variable cdiv: C -> C -> C * C.
Variable div_th: div_theory req cadd cmul phi cdiv.
Notation "0" := rO. Notation "1" := rI.
Infix "+" := radd. Infix "*" := rmul.
Infix "-" := rsub. Notation "- x" := (ropp x).
Infix "==" := req.
Infix "^" := (pow_pos rmul).
Infix "+!" := cadd. Infix "*!" := cmul.
Infix "-! " := csub. Notation "-! x" := (copp x).
Infix "?=!" := ceqb. Notation "[ x ]" := (phi x).
Add Morphism radd with signature (req ==> req ==> req) as radd_ext.
Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext.
Add Morphism ropp with signature (req ==> req) as ropp_ext.
Add Morphism rsub with signature (req ==> req ==> req) as rsub_ext.
Ltac rsimpl := gen_srewrite Rsth Reqe ARth.
Ltac add_push := gen_add_push radd Rsth Reqe ARth.
Ltac mul_push := gen_mul_push rmul Rsth Reqe ARth.
Ltac add_permut_rec t :=
match t with
| ?x + ?y => add_permut_rec y || add_permut_rec x
| _ => add_push t; apply (Radd_ext Reqe); [|reflexivity]
end.
Ltac add_permut :=
repeat (reflexivity ||
match goal with |- ?t == _ => add_permut_rec t end).
Ltac mul_permut_rec t :=
match t with
| ?x * ?y => mul_permut_rec y || mul_permut_rec x
| _ => mul_push t; apply (Rmul_ext Reqe); [|reflexivity]
end.
Ltac mul_permut :=
repeat (reflexivity ||
match goal with |- ?t == _ => mul_permut_rec t end).
Inductive Pol : Type :=
| Pc : C -> Pol
| Pinj : positive -> Pol -> Pol
| PX : Pol -> positive -> Pol -> Pol.
Definition P0 := Pc cO.
Definition P1 := Pc cI.
Fixpoint Peq (P P' : Pol) {struct P'} : bool :=
match P, P' with
| Pc c, Pc c' => c ?=! c'
| Pinj j Q, Pinj j' Q' =>
match j ?= j' with
| Eq => Peq Q Q'
| _ => false
end
| PX P i Q, PX P' i' Q' =>
match i ?= i' with
| Eq => if Peq P P' then Peq Q Q' else false
| _ => false
end
| _, _ => false
end.
Infix "?==" := Peq.
Definition mkPinj j P :=
match P with
| Pc _ => P
| Pinj j' Q => Pinj (j + j') Q
| _ => Pinj j P
end.
Definition mkPinj_pred j P:=
match j with
| xH => P
| xO j => Pinj (Pos.pred_double j) P
| xI j => Pinj (xO j) P
end.
Definition mkPX P i Q :=
match P with
| Pc c => if c ?=! cO then mkPinj xH Q else PX P i Q
| Pinj _ _ => PX P i Q
| PX P' i' Q' => if Q' ?== P0 then PX P' (i' + i) Q else PX P i Q
end.
Definition mkXi i := PX P1 i P0.
Definition mkX := mkXi 1.
Opposite of addition
Fixpoint Popp (P:Pol) : Pol :=
match P with
| Pc c => Pc (-! c)
| Pinj j Q => Pinj j (Popp Q)
| PX P i Q => PX (Popp P) i (Popp Q)
end.
Notation "-- P" := (Popp P).
Addition et subtraction
Fixpoint PaddC (P:Pol) (c:C) : Pol :=
match P with
| Pc c1 => Pc (c1 +! c)
| Pinj j Q => Pinj j (PaddC Q c)
| PX P i Q => PX P i (PaddC Q c)
end.
Fixpoint PsubC (P:Pol) (c:C) : Pol :=
match P with
| Pc c1 => Pc (c1 -! c)
| Pinj j Q => Pinj j (PsubC Q c)
| PX P i Q => PX P i (PsubC Q c)
end.
Section PopI.
Variable Pop : Pol -> Pol -> Pol.
Variable Q : Pol.
Fixpoint PaddI (j:positive) (P:Pol) : Pol :=
match P with
| Pc c => mkPinj j (PaddC Q c)
| Pinj j' Q' =>
match Z.pos_sub j' j with
| Zpos k => mkPinj j (Pop (Pinj k Q') Q)
| Z0 => mkPinj j (Pop Q' Q)
| Zneg k => mkPinj j' (PaddI k Q')
end
| PX P i Q' =>
match j with
| xH => PX P i (Pop Q' Q)
| xO j => PX P i (PaddI (Pos.pred_double j) Q')
| xI j => PX P i (PaddI (xO j) Q')
end
end.
Fixpoint PsubI (j:positive) (P:Pol) : Pol :=
match P with
| Pc c => mkPinj j (PaddC (--Q) c)
| Pinj j' Q' =>
match Z.pos_sub j' j with
| Zpos k => mkPinj j (Pop (Pinj k Q') Q)
| Z0 => mkPinj j (Pop Q' Q)
| Zneg k => mkPinj j' (PsubI k Q')
end
| PX P i Q' =>
match j with
| xH => PX P i (Pop Q' Q)
| xO j => PX P i (PsubI (Pos.pred_double j) Q')
| xI j => PX P i (PsubI (xO j) Q')
end
end.
Variable P' : Pol.
Fixpoint PaddX (i':positive) (P:Pol) : Pol :=
match P with
| Pc c => PX P' i' P
| Pinj j Q' =>
match j with
| xH => PX P' i' Q'
| xO j => PX P' i' (Pinj (Pos.pred_double j) Q')
| xI j => PX P' i' (Pinj (xO j) Q')
end
| PX P i Q' =>
match Z.pos_sub i i' with
| Zpos k => mkPX (Pop (PX P k P0) P') i' Q'
| Z0 => mkPX (Pop P P') i Q'
| Zneg k => mkPX (PaddX k P) i Q'
end
end.
Fixpoint PsubX (i':positive) (P:Pol) : Pol :=
match P with
| Pc c => PX (--P') i' P
| Pinj j Q' =>
match j with
| xH => PX (--P') i' Q'
| xO j => PX (--P') i' (Pinj (Pos.pred_double j) Q')
| xI j => PX (--P') i' (Pinj (xO j) Q')
end
| PX P i Q' =>
match Z.pos_sub i i' with
| Zpos k => mkPX (Pop (PX P k P0) P') i' Q'
| Z0 => mkPX (Pop P P') i Q'
| Zneg k => mkPX (PsubX k P) i Q'
end
end.
End PopI.
Fixpoint Padd (P P': Pol) {struct P'} : Pol :=
match P' with
| Pc c' => PaddC P c'
| Pinj j' Q' => PaddI Padd Q' j' P
| PX P' i' Q' =>
match P with
| Pc c => PX P' i' (PaddC Q' c)
| Pinj j Q =>
match j with
| xH => PX P' i' (Padd Q Q')
| xO j => PX P' i' (Padd (Pinj (Pos.pred_double j) Q) Q')
| xI j => PX P' i' (Padd (Pinj (xO j) Q) Q')
end
| PX P i Q =>
match Z.pos_sub i i' with
| Zpos k => mkPX (Padd (PX P k P0) P') i' (Padd Q Q')
| Z0 => mkPX (Padd P P') i (Padd Q Q')
| Zneg k => mkPX (PaddX Padd P' k P) i (Padd Q Q')
end
end
end.
Infix "++" := Padd.
Fixpoint Psub (P P': Pol) {struct P'} : Pol :=
match P' with
| Pc c' => PsubC P c'
| Pinj j' Q' => PsubI Psub Q' j' P
| PX P' i' Q' =>
match P with
| Pc c => PX (--P') i' (PaddC (--Q') c)
| Pinj j Q =>
match j with
| xH => PX (--P') i' (Psub Q Q')
| xO j => PX (--P') i' (Psub (Pinj (Pos.pred_double j) Q) Q')
| xI j => PX (--P') i' (Psub (Pinj (xO j) Q) Q')
end
| PX P i Q =>
match Z.pos_sub i i' with
| Zpos k => mkPX (Psub (PX P k P0) P') i' (Psub Q Q')
| Z0 => mkPX (Psub P P') i (Psub Q Q')
| Zneg k => mkPX (PsubX Psub P' k P) i (Psub Q Q')
end
end
end.
Infix "--" := Psub.
Multiplication
Fixpoint PmulC_aux (P:Pol) (c:C) : Pol :=
match P with
| Pc c' => Pc (c' *! c)
| Pinj j Q => mkPinj j (PmulC_aux Q c)
| PX P i Q => mkPX (PmulC_aux P c) i (PmulC_aux Q c)
end.
Definition PmulC P c :=
if c ?=! cO then P0 else
if c ?=! cI then P else PmulC_aux P c.
Section PmulI.
Variable Pmul : Pol -> Pol -> Pol.
Variable Q : Pol.
Fixpoint PmulI (j:positive) (P:Pol) : Pol :=
match P with
| Pc c => mkPinj j (PmulC Q c)
| Pinj j' Q' =>
match Z.pos_sub j' j with
| Zpos k => mkPinj j (Pmul (Pinj k Q') Q)
| Z0 => mkPinj j (Pmul Q' Q)
| Zneg k => mkPinj j' (PmulI k Q')
end
| PX P' i' Q' =>
match j with
| xH => mkPX (PmulI xH P') i' (Pmul Q' Q)
| xO j' => mkPX (PmulI j P') i' (PmulI (Pos.pred_double j') Q')
| xI j' => mkPX (PmulI j P') i' (PmulI (xO j') Q')
end
end.
End PmulI.
Fixpoint Pmul (P P'' : Pol) {struct P''} : Pol :=
match P'' with
| Pc c => PmulC P c
| Pinj j' Q' => PmulI Pmul Q' j' P
| PX P' i' Q' =>
match P with
| Pc c => PmulC P'' c
| Pinj j Q =>
let QQ' :=
match j with
| xH => Pmul Q Q'
| xO j => Pmul (Pinj (Pos.pred_double j) Q) Q'
| xI j => Pmul (Pinj (xO j) Q) Q'
end in
mkPX (Pmul P P') i' QQ'
| PX P i Q=>
let QQ' := Pmul Q Q' in
let PQ' := PmulI Pmul Q' xH P in
let QP' := Pmul (mkPinj xH Q) P' in
let PP' := Pmul P P' in
(mkPX (mkPX PP' i P0 ++ QP') i' P0) ++ mkPX PQ' i QQ'
end
end.
Infix "**" := Pmul.
Monomial
A monomial is X1^k1...Xi^ki. Its representation is a simplified version of the polynomial representation:

- mon0 correspond to the polynom P1.
- (zmon j M) corresponds to (Pinj j ...), i.e. skip j variable indices.
- (vmon i M) is X^i*M with X the current variable, its corresponds to (PX P1 i ...)]

Inductive Mon: Set :=
| mon0: Mon
| zmon: positive -> Mon -> Mon
| vmon: positive -> Mon -> Mon.
Definition mkZmon j M :=
match M with mon0 => mon0 | _ => zmon j M end.
Definition zmon_pred j M :=
match j with xH => M | _ => mkZmon (Pos.pred j) M end.
Definition mkVmon i M :=
match M with
| mon0 => vmon i mon0
| zmon j m => vmon i (zmon_pred j m)
| vmon i' m => vmon (i+i') m
end.
Fixpoint CFactor (P: Pol) (c: C) {struct P}: Pol * Pol :=
match P with
| Pc c1 => let (q,r) := cdiv c1 c in (Pc r, Pc q)
| Pinj j1 P1 =>
let (R,S) := CFactor P1 c in
(mkPinj j1 R, mkPinj j1 S)
| PX P1 i Q1 =>
let (R1, S1) := CFactor P1 c in
let (R2, S2) := CFactor Q1 c in
(mkPX R1 i R2, mkPX S1 i S2)
end.
Fixpoint MFactor (P: Pol) (c: C) (M: Mon) {struct P}: Pol * Pol :=
match P, M with
_, mon0 => if (ceqb c cI) then (Pc cO, P) else CFactor P c
| Pc _, _ => (P, Pc cO)
| Pinj j1 P1, zmon j2 M1 =>
match j1 ?= j2 with
Eq => let (R,S) := MFactor P1 c M1 in
(mkPinj j1 R, mkPinj j1 S)
| Lt => let (R,S) := MFactor P1 c (zmon (j2 - j1) M1) in
(mkPinj j1 R, mkPinj j1 S)
| Gt => (P, Pc cO)
end
| Pinj _ _, vmon _ _ => (P, Pc cO)
| PX P1 i Q1, zmon j M1 =>
let M2 := zmon_pred j M1 in
let (R1, S1) := MFactor P1 c M in
let (R2, S2) := MFactor Q1 c M2 in
(mkPX R1 i R2, mkPX S1 i S2)
| PX P1 i Q1, vmon j M1 =>
match i ?= j with
Eq => let (R1,S1) := MFactor P1 c (mkZmon xH M1) in
(mkPX R1 i Q1, S1)
| Lt => let (R1,S1) := MFactor P1 c (vmon (j - i) M1) in
(mkPX R1 i Q1, S1)
| Gt => let (R1,S1) := MFactor P1 c (mkZmon xH M1) in
(mkPX R1 i Q1, mkPX S1 (i-j) (Pc cO))
end
end.
Definition POneSubst (P1: Pol) (cM1: C * Mon) (P2: Pol): option Pol :=
let (c,M1) := cM1 in
let (Q1,R1) := MFactor P1 c M1 in
match R1 with
(Pc c) => if c ?=! cO then None
else Some (Padd Q1 (Pmul P2 R1))
| _ => Some (Padd Q1 (Pmul P2 R1))
end.
Fixpoint PNSubst1 (P1: Pol) (cM1: C * Mon) (P2: Pol) (n: nat) : Pol :=
match POneSubst P1 cM1 P2 with
Some P3 => match n with S n1 => PNSubst1 P3 cM1 P2 n1 | _ => P3 end
| _ => P1
end.
Definition PNSubst (P1: Pol) (cM1: C * Mon) (P2: Pol) (n: nat): option Pol :=
match POneSubst P1 cM1 P2 with
Some P3 => match n with S n1 => Some (PNSubst1 P3 cM1 P2 n1) | _ => None end
| _ => None
end.
Fixpoint PSubstL1 (P1: Pol) (LM1: list ((C * Mon) * Pol)) (n: nat) : Pol :=
match LM1 with
cons (M1,P2) LM2 => PSubstL1 (PNSubst1 P1 M1 P2 n) LM2 n
| _ => P1
end.
Fixpoint PSubstL (P1: Pol) (LM1: list ((C * Mon) * Pol)) (n: nat) : option Pol :=
match LM1 with
cons (M1,P2) LM2 =>
match PNSubst P1 M1 P2 n with
Some P3 => Some (PSubstL1 P3 LM2 n)
| None => PSubstL P1 LM2 n
end
| _ => None
end.
Fixpoint PNSubstL (P1: Pol) (LM1: list ((C * Mon) * Pol)) (m n: nat) : Pol :=
match PSubstL P1 LM1 n with
Some P3 => match m with S m1 => PNSubstL P3 LM1 m1 n | _ => P3 end
| _ => P1
end.
Evaluation of a polynomial towards R
Fixpoint Pphi(l:list R) (P:Pol) : R :=
match P with
| Pc c => [c]
| Pinj j Q => Pphi (jump j l) Q
| PX P i Q => Pphi l P * (hd l) ^ i + Pphi (tail l) Q
end.
Reserved Notation "P @ l " (at level 10, no associativity).
Notation "P @ l " := (Pphi l P).
Definition Pequiv (P Q : Pol) := forall l, P@l == Q@l.
Infix "===" := Pequiv (at level 70, no associativity).
Instance Pequiv_eq : Equivalence Pequiv.
Instance Pphi_ext : Proper (eq ==> Pequiv ==> req) Pphi.
Instance Pinj_ext : Proper (eq ==> Pequiv ==> Pequiv) Pinj.
Instance PX_ext : Proper (Pequiv ==> eq ==> Pequiv ==> Pequiv) PX.
Evaluation of a monomial towards R
Fixpoint Mphi(l:list R) (M: Mon) : R :=
match M with
| mon0 => rI
| zmon j M1 => Mphi (jump j l) M1
| vmon i M1 => Mphi (tail l) M1 * (hd l) ^ i
end.
Notation "M @@ l" := (Mphi l M) (at level 10, no associativity).
Proofs
Ltac destr_pos_sub :=
match goal with |- context [Z.pos_sub ?x ?y] =>
generalize (Z.pos_sub_discr x y); destruct (Z.pos_sub x y)
end.
Lemma jump_add' i j (l:list R) : jump (i + j) l = jump j (jump i l).
Lemma Peq_ok P P' : (P ?== P') = true -> P === P'.
Lemma Peq_spec P P' : BoolSpec (P === P') True (P ?== P').
Lemma Pphi0 l : P0@l == 0.
Lemma Pphi1 l : P1@l == 1.
Lemma mkPinj_ok j l P : (mkPinj j P)@l == P@(jump j l).
Instance mkPinj_ext : Proper (eq ==> Pequiv ==> Pequiv) mkPinj.
Lemma pow_pos_add x i j : x^(j + i) == x^i * x^j.
Lemma ceqb_spec c c' : BoolSpec ([c] == [c']) True (c ?=! c').
Lemma mkPX_ok l P i Q :
(mkPX P i Q)@l == P@l * (hd l)^i + Q@(tail l).
Instance mkPX_ext : Proper (Pequiv ==> eq ==> Pequiv ==> Pequiv) mkPX.
Hint Rewrite
Pphi0
Pphi1
mkPinj_ok
mkPX_ok
(morph0 CRmorph)
(morph1 CRmorph)
(morph0 CRmorph)
(morph_add CRmorph)
(morph_mul CRmorph)
(morph_sub CRmorph)
(morph_opp CRmorph)
: Esimpl.
Ltac Esimpl := try rewrite_db Esimpl; rsimpl; simpl.
Lemma PaddC_ok c P l : (PaddC P c)@l == P@l + [c].
Lemma PsubC_ok c P l : (PsubC P c)@l == P@l - [c].
Lemma PmulC_aux_ok c P l : (PmulC_aux P c)@l == P@l * [c].
Lemma PmulC_ok c P l : (PmulC P c)@l == P@l * [c].
Lemma Popp_ok P l : (--P)@l == - P@l.
Hint Rewrite PaddC_ok PsubC_ok PmulC_ok Popp_ok : Esimpl.
Lemma PaddX_ok P' P k l :
(forall P l, (P++P')@l == P@l + P'@l) ->
(PaddX Padd P' k P) @ l == P@l + P'@l * (hd l)^k.
Lemma Padd_ok P' P l : (P ++ P')@l == P@l + P'@l.
Lemma Psub_opp P' P : P -- P' === P ++ (--P').
Lemma Psub_ok P' P l : (P -- P')@l == P@l - P'@l.
Lemma PmulI_ok P' :
(forall P l, (Pmul P P') @ l == P @ l * P' @ l) ->
forall P p l, (PmulI Pmul P' p P) @ l == P @ l * P' @ (jump p l).
Lemma Pmul_ok P P' l : (P**P')@l == P@l * P'@l.
Lemma mkZmon_ok M j l :
(mkZmon j M) @@ l == (zmon j M) @@ l.
Lemma zmon_pred_ok M j l :
(zmon_pred j M) @@ (tail l) == (zmon j M) @@ l.
Lemma mkVmon_ok M i l :
(mkVmon i M)@@l == M@@l * (hd l)^i.
Ltac destr_factor := match goal with
| H : context [CFactor ?P _] |- context [CFactor ?P ?c] =>
destruct (CFactor P c); destr_factor; rewrite H; clear H
| H : context [MFactor ?P _ _] |- context [MFactor ?P ?c ?M] =>
specialize (H M); destruct (MFactor P c M); destr_factor; rewrite H; clear H
| _ => idtac
end.
Lemma Mcphi_ok P c l :
let (Q,R) := CFactor P c in
P@l == Q@l + [c] * R@l.
Lemma Mphi_ok P (cM: C * Mon) l :
let (c,M) := cM in
let (Q,R) := MFactor P c M in
P@l == Q@l + [c] * M@@l * R@l.
Lemma POneSubst_ok P1 cM1 P2 P3 l :
POneSubst P1 cM1 P2 = Some P3 ->
[fst cM1] * (snd cM1)@@l == P2@l -> P1@l == P3@l.
Lemma PNSubst1_ok n P1 cM1 P2 l :
[fst cM1] * (snd cM1)@@l == P2@l ->
P1@l == (PNSubst1 P1 cM1 P2 n)@l.
Lemma PNSubst_ok n P1 cM1 P2 l P3 :
PNSubst P1 cM1 P2 n = Some P3 ->
[fst cM1] * (snd cM1)@@l == P2@l -> P1@l == P3@l.
Fixpoint MPcond (LM1: list (C * Mon * Pol)) (l: list R) : Prop :=
match LM1 with
| (M1,P2) :: LM2 => ([fst M1] * (snd M1)@@l == P2@l) /\ MPcond LM2 l
| _ => True
end.
Lemma PSubstL1_ok n LM1 P1 l :
MPcond LM1 l -> P1@l == (PSubstL1 P1 LM1 n)@l.
Lemma PSubstL_ok n LM1 P1 P2 l :
PSubstL P1 LM1 n = Some P2 -> MPcond LM1 l -> P1@l == P2@l.
Lemma PNSubstL_ok m n LM1 P1 l :
MPcond LM1 l -> P1@l == (PNSubstL P1 LM1 m n)@l.
Definition of polynomial expressions
Inductive PExpr : Type :=
| PEO : PExpr
| PEI : PExpr
| PEc : C -> PExpr
| PEX : positive -> PExpr
| PEadd : PExpr -> PExpr -> PExpr
| PEsub : PExpr -> PExpr -> PExpr
| PEmul : PExpr -> PExpr -> PExpr
| PEopp : PExpr -> PExpr
| PEpow : PExpr -> N -> PExpr.
evaluation of polynomial expressions towards R
Definition mk_X j := mkPinj_pred j mkX.
evaluation of polynomial expressions towards R
Fixpoint PEeval (l:list R) (pe:PExpr) {struct pe} : R :=
match pe with
| PEO => rO
| PEI => rI
| PEc c => phi 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.
Correctness proofs
Lemma mkX_ok p l : nth 0 p l == (mk_X p) @ l.
Hint Rewrite Padd_ok Psub_ok : Esimpl.
Section POWER.
Variable subst_l : Pol -> Pol.
Fixpoint Ppow_pos (res P:Pol) (p:positive) : Pol :=
match p with
| xH => subst_l (res ** P)
| xO p => Ppow_pos (Ppow_pos res P p) P p
| xI p => subst_l ((Ppow_pos (Ppow_pos res P p) P p) ** P)
end.
Definition Ppow_N P n :=
match n with
| N0 => P1
| Npos p => Ppow_pos P1 P p
end.
Lemma Ppow_pos_ok l :
(forall P, subst_l P@l == P@l) ->
forall res P p, (Ppow_pos res P p)@l == res@l * (pow_pos Pmul P p)@l.
Lemma Ppow_N_ok l :
(forall P, subst_l P@l == P@l) ->
forall P n, (Ppow_N P n)@l == (pow_N P1 Pmul P n)@l.
End POWER.
Normalization and rewriting
Section NORM_SUBST_REC.
Variable n : nat.
Variable lmp:list (C*Mon*Pol).
Let subst_l P := PNSubstL P lmp n n.
Let Pmul_subst P1 P2 := subst_l (P1 ** P2).
Let Ppow_subst := Ppow_N subst_l.
Fixpoint norm_aux (pe:PExpr) : Pol :=
match pe with
| PEO => Pc cO
| PEI => Pc cI
| PEc c => Pc c
| PEX j => mk_X j
| PEadd (PEopp pe1) pe2 => (norm_aux pe2) -- (norm_aux pe1)
| PEadd pe1 (PEopp pe2) => (norm_aux pe1) -- (norm_aux pe2)
| PEadd pe1 pe2 => (norm_aux pe1) ++ (norm_aux pe2)
| PEsub pe1 pe2 => (norm_aux pe1) -- (norm_aux pe2)
| PEmul pe1 pe2 => (norm_aux pe1) ** (norm_aux pe2)
| PEopp pe1 => -- (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).
Internally, norm_aux is expanded in a large number of cases. To speed-up proofs, we use an alternative definition.
Definition get_PEopp pe :=
match pe with
| PEopp pe' => Some pe'
| _ => None
end.
Lemma norm_aux_PEadd pe1 pe2 :
norm_aux (PEadd pe1 pe2) =
match get_PEopp pe1, get_PEopp pe2 with
| Some pe1', _ => (norm_aux pe2) -- (norm_aux pe1')
| None, Some pe2' => (norm_aux pe1) -- (norm_aux pe2')
| None, None => (norm_aux pe1) ++ (norm_aux pe2)
end.
Lemma norm_aux_PEopp pe :
match get_PEopp pe with
| Some pe' => norm_aux pe = -- (norm_aux pe')
| None => True
end.
Lemma norm_aux_spec l pe :
PEeval l pe == (norm_aux pe)@l.
Lemma norm_subst_spec :
forall l pe, MPcond lmp l ->
PEeval l pe == (norm_subst pe)@l.
End NORM_SUBST_REC.
Fixpoint interp_PElist (l:list R) (lpe:list (PExpr*PExpr)) {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.
Fixpoint mon_of_pol (P:Pol) : option (C * Mon) :=
match P with
| Pc c => if (c ?=! cO) then None else Some (c, mon0)
| Pinj j P =>
match mon_of_pol P with
| None => None
| Some (c,m) => Some (c, mkZmon j m)
end
| PX P i Q =>
if Peq Q P0 then
match mon_of_pol P with
| None => None
| Some (c,m) => Some (c, mkVmon i m)
end
else None
end.
Fixpoint mk_monpol_list (lpe:list (PExpr * PExpr)) : list (C*Mon*Pol) :=
match lpe with
| nil => nil
| (me,pe)::lpe =>
match mon_of_pol (norm_subst 0 nil me) with
| None => mk_monpol_list lpe
| Some m => (m,norm_subst 0 nil pe):: mk_monpol_list lpe
end
end.
Lemma mon_of_pol_ok : forall P m, mon_of_pol P = Some m ->
forall l, [fst m] * Mphi l (snd m) == P@l.
Lemma interp_PElist_ok : forall l lpe,
interp_PElist l lpe -> MPcond (mk_monpol_list lpe) l.
Lemma norm_subst_ok : forall n l lpe pe,
interp_PElist l lpe ->
PEeval l pe == (norm_subst n (mk_monpol_list lpe) pe)@l.
Lemma ring_correct : forall n l lpe pe1 pe2,
interp_PElist l lpe ->
(let lmp := mk_monpol_list lpe in
norm_subst n lmp pe1 ?== norm_subst n lmp pe2) = true ->
PEeval l pe1 == PEeval l pe2.
Generic evaluation of polynomial towards R avoiding parenthesis
Variable get_sign : C -> option C.
Variable get_sign_spec : sign_theory copp ceqb get_sign.
Section EVALUATION.
Variable mkpow : R -> positive -> R.
Variable mkopp_pow : R -> positive -> R.
Variable mkmult_pow : R -> R -> positive -> R.
Fixpoint mkmult_rec (r:R) (lm:list (R*positive)) {struct lm}: R :=
match lm with
| nil => r
| cons (x,p) t => mkmult_rec (mkmult_pow r x p) t
end.
Definition mkmult1 lm :=
match lm with
| nil => 1
| cons (x,p) t => mkmult_rec (mkpow x p) t
end.
Definition mkmultm1 lm :=
match lm with
| nil => ropp rI
| cons (x,p) t => mkmult_rec (mkopp_pow x p) t
end.
Definition mkmult_c_pos c lm :=
if c ?=! cI then mkmult1 (rev' lm)
else mkmult_rec [c] (rev' lm).
Definition mkmult_c c lm :=
match get_sign c with
| None => mkmult_c_pos c lm
| Some c' =>
if c' ?=! cI then mkmultm1 (rev' lm)
else mkmult_rec [c] (rev' lm)
end.
Definition mkadd_mult rP c lm :=
match get_sign c with
| None => rP + mkmult_c_pos c lm
| Some c' => rP - mkmult_c_pos c' lm
end.
Definition add_pow_list (r:R) n l :=
match n with
| N0 => l
| Npos p => (r,p)::l
end.
Fixpoint add_mult_dev
(rP:R) (P:Pol) (fv:list R) (n:N) (lm:list (R*positive)) {struct P} : R :=
match P with
| Pc c =>
let lm := add_pow_list (hd fv) n lm in
mkadd_mult rP c lm
| Pinj j Q =>
add_mult_dev rP Q (jump j fv) N0 (add_pow_list (hd fv) n lm)
| PX P i Q =>
let rP := add_mult_dev rP P fv (N.add (Npos i) n) lm in
if Q ?== P0 then rP
else add_mult_dev rP Q (tail fv) N0 (add_pow_list (hd fv) n lm)
end.
Fixpoint mult_dev (P:Pol) (fv : list R) (n:N)
(lm:list (R*positive)) {struct P} : R :=
match P with
| Pc c => mkmult_c c (add_pow_list (hd fv) n lm)
| Pinj j Q => mult_dev Q (jump j fv) N0 (add_pow_list (hd fv) n lm)
| PX P i Q =>
let rP := mult_dev P fv (N.add (Npos i) n) lm in
if Q ?== P0 then rP
else
let lmq := add_pow_list (hd fv) n lm in
add_mult_dev rP Q (tail fv) N0 lmq
end.
Definition Pphi_avoid fv P := mult_dev P fv N0 nil.
Fixpoint r_list_pow (l:list (R*positive)) : R :=
match l with
| nil => rI
| cons (r,p) l => pow_pos rmul r p * r_list_pow l
end.
Hypothesis mkpow_spec : forall r p, mkpow r p == pow_pos rmul r p.
Hypothesis mkopp_pow_spec : forall r p, mkopp_pow r p == - (pow_pos rmul r p).
Hypothesis mkmult_pow_spec : forall r x p, mkmult_pow r x p == r * pow_pos rmul x p.
Lemma mkmult_rec_ok : forall lm r, mkmult_rec r lm == r * r_list_pow lm.
Lemma mkmult1_ok : forall lm, mkmult1 lm == r_list_pow lm.
Lemma mkmultm1_ok : forall lm, mkmultm1 lm == - r_list_pow lm.
Lemma r_list_pow_rev : forall l, r_list_pow (rev' l) == r_list_pow l.
Lemma mkmult_c_pos_ok : forall c lm, mkmult_c_pos c lm == [c]* r_list_pow lm.
Lemma mkmult_c_ok : forall c lm, mkmult_c c lm == [c] * r_list_pow lm.
Lemma mkadd_mult_ok : forall rP c lm, mkadd_mult rP c lm == rP + [c]*r_list_pow lm.
Lemma add_pow_list_ok :
forall r n l, r_list_pow (add_pow_list r n l) == pow_N rI rmul r n * r_list_pow l.
Lemma add_mult_dev_ok : forall P rP fv n lm,
add_mult_dev rP P fv n lm == rP + P@fv*pow_N rI rmul (hd fv) n * r_list_pow lm.
Lemma mult_dev_ok : forall P fv n lm,
mult_dev P fv n lm == P@fv * pow_N rI rmul (hd fv) n * r_list_pow lm.
Lemma Pphi_avoid_ok : forall P fv, Pphi_avoid fv P == P@fv.
End EVALUATION.
Definition Pphi_pow :=
let mkpow x p :=
match p with xH => x | _ => rpow x (Cp_phi (Npos p)) end in
let mkopp_pow x p := ropp (mkpow x p) in
let mkmult_pow r x p := rmul r (mkpow x p) in
Pphi_avoid mkpow mkopp_pow mkmult_pow.
Lemma local_mkpow_ok r p :
match p with
| xI _ => rpow r (Cp_phi (Npos p))
| xO _ => rpow r (Cp_phi (Npos p))
| 1 => r
end == pow_pos rmul r p.
Lemma Pphi_pow_ok : forall P fv, Pphi_pow fv P == P@fv.
Lemma ring_rw_pow_correct : forall n lH l,
interp_PElist l lH ->
forall lmp, mk_monpol_list lH = lmp ->
forall pe npe, norm_subst n lmp pe = npe ->
PEeval l pe == Pphi_pow l npe.
Fixpoint mkmult_pow (r x:R) (p: positive) {struct p} : R :=
match p with
| xH => r*x
| xO p => mkmult_pow (mkmult_pow r x p) x p
| xI p => mkmult_pow (mkmult_pow (r*x) x p) x p
end.
Definition mkpow x p :=
match p with
| xH => x
| xO p => mkmult_pow x x (Pos.pred_double p)
| xI p => mkmult_pow x x (xO p)
end.
Definition mkopp_pow x p :=
match p with
| xH => -x
| xO p => mkmult_pow (-x) x (Pos.pred_double p)
| xI p => mkmult_pow (-x) x (xO p)
end.
Definition Pphi_dev := Pphi_avoid mkpow mkopp_pow mkmult_pow.
Lemma mkmult_pow_ok p r x : mkmult_pow r x p == r * x^p.
Lemma mkpow_ok p x : mkpow x p == x^p.
Lemma mkopp_pow_ok p x : mkopp_pow x p == - x^p.
Lemma Pphi_dev_ok : forall P fv, Pphi_dev fv P == P@fv.
Lemma ring_rw_correct : forall n lH l,
interp_PElist l lH ->
forall lmp, mk_monpol_list lH = lmp ->
forall pe npe, norm_subst n lmp pe = npe ->
PEeval l pe == Pphi_dev l npe.
End MakeRingPol.