# Library Coq.nsatz.Nsatz

Require Import List.
Require Import Setoid.
Require Import BinPos.
Require Import BinList.
Require Import Znumtheory.
Require Import Ring_polynom Ring_tac InitialRing.
Require Export Morphisms Setoid Bool.
Class Zero (A : Type) := {zero : A}.
Notation "0" := zero.
Class One (A : Type) := {one : A}.
Notation "1" := one.
Class Addition (A : Type) := {addition : A -> A -> A}.
Notation "x + y" := (addition x y).
Class Multiplication (A : Type) := {multiplication : A -> A -> A}.
Notation "x * y" := (multiplication x y).
Class Subtraction (A : Type) := {subtraction : A -> A -> A}.
Notation "x - y" := (subtraction x y).
Class Opposite (A : Type) := {opposite : A -> A}.
Notation "- x" := (opposite x).
Class Ring (R:Type) := {
ring0: R; ring1: R;
ring_plus: R->R->R; ring_mult: R->R->R;
ring_sub: R->R->R; ring_opp: R->R;
ring_eq : R -> R -> Prop;
ring_ring:
ring_theory ring0 ring1 ring_plus ring_mult ring_sub
ring_opp ring_eq;
ring_setoid: Equivalence ring_eq;
ring_plus_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_plus;
ring_mult_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_mult;
ring_sub_comp: Proper (ring_eq==>ring_eq==>ring_eq) ring_sub;
ring_opp_comp: Proper (ring_eq==>ring_eq) ring_opp
}.
Class Domain (R : Type) := {
domain_ring:> Ring R;
domain_axiom_product:
forall x y, ring_eq (ring_mult x y) ring0 -> (ring_eq x ring0) \/ (ring_eq y ring0);
domain_axiom_one_zero: not (ring_eq ring1 ring0)}.
Section domain.
Variable R: Type.
Variable Rd: Domain R.
Add Ring Rr: (@ring_ring R (@domain_ring R Rd)).
Instance zero_ring : Zero R := {zero := ring0}.
Instance one_ring : One R := {one := ring1}.
Instance addition_ring : Addition R := {addition x y := ring_plus x y}.
Instance multiplication_ring : Multiplication R := {multiplication x y := ring_mult x y}.
Instance subtraction_ring : Subtraction R := {subtraction x y := ring_sub x y}.
Instance opposite_ring : Opposite R := {opposite x := ring_opp x}.
Infix "==" := ring_eq (at level 70, no associativity).
Lemma psos_r1b: forall x y:R, x - y == 0 -> x == y.
Lemma psos_r1: forall x y, x == y -> x - y == 0.
Lemma nsatzR_diff: forall x y:R, not (x == y) -> not (x - y == 0).
Require Import ZArith.
Definition PolZ := Pol Z.
Definition PEZ := PExpr Z.
Definition P0Z : PolZ := @P0 Z 0%Z.
Definition PolZadd : PolZ -> PolZ -> PolZ :=
@Padd Z 0%Z Zplus Zeq_bool.
Definition PolZmul : PolZ -> PolZ -> PolZ :=
@Pmul Z 0%Z 1%Z Zplus Zmult Zeq_bool.
Definition PolZeq := @Peq Z Zeq_bool.
Definition norm :=
@norm_aux Z 0%Z 1%Z Zplus Zmult Zminus Zopp Zeq_bool.
Fixpoint mult_l (la : list PEZ) (lp: list PolZ) : PolZ :=
match la, lp with
| a::la, p::lp => PolZadd (PolZmul (norm a) p) (mult_l la lp)
| _, _ => P0Z
end.
Fixpoint compute_list (lla: list (list PEZ)) (lp:list PolZ) :=
match lla with
| List.nil => lp
| la::lla => compute_list lla ((mult_l la lp)::lp)
end.
Definition check (lpe:list PEZ) (qe:PEZ) (certif: list (list PEZ) * list PEZ) :=
let (lla, lq) := certif in
let lp := List.map norm lpe in
PolZeq (norm qe) (mult_l lq (compute_list lla lp)).
Definition PhiR : list R -> PolZ -> R :=
(Pphi 0 ring_plus ring_mult (gen_phiZ 0 1 ring_plus ring_mult ring_opp)).
Definition pow (r : R) (n : nat) := pow_N 1 ring_mult r (Nnat.N_of_nat n).
Definition PEevalR : list R -> PEZ -> R :=
PEeval 0 ring_plus ring_mult ring_sub ring_opp
(gen_phiZ 0 1 ring_plus ring_mult ring_opp)
Nnat.nat_of_N pow.
Lemma P0Z_correct : forall l, PhiR l P0Z = 0.
Lemma Rext: ring_eq_ext ring_plus ring_mult ring_opp ring_eq.
Lemma Rset : Setoid_Theory R ring_eq.
Lemma PolZadd_correct : forall P' P l,
PhiR l (PolZadd P P') == ((PhiR l P) + (PhiR l P')).
Lemma PolZmul_correct : forall P P' l,
PhiR l (PolZmul P P') == ((PhiR l P) * (PhiR l P')).
Lemma R_power_theory
: power_theory 1 ring_mult ring_eq Nnat.nat_of_N pow.
Lemma norm_correct :
forall (l : list R) (pe : PEZ), PEevalR l pe == PhiR l (norm pe).
Lemma PolZeq_correct : forall P P' l,
PolZeq P P' = true ->
PhiR l P == PhiR l P'.
Fixpoint Cond0 (A:Type) (Interp:A->R) (l:list A) : Prop :=
match l with
| List.nil => True
| a::l => Interp a == 0 /\ Cond0 A Interp l
end.
Lemma mult_l_correct : forall l la lp,
Cond0 PolZ (PhiR l) lp ->
PhiR l (mult_l la lp) == 0.
Lemma compute_list_correct : forall l lla lp,
Cond0 PolZ (PhiR l) lp ->
Cond0 PolZ (PhiR l) (compute_list lla lp).
Lemma check_correct :
forall l lpe qe certif,
check lpe qe certif = true ->
Cond0 PEZ (PEevalR l) lpe ->
PEevalR l qe == 0.
Lemma pow_not_zero: forall p n, pow p n == 0 -> p == 0.
Lemma Rdomain_pow: forall c p r, ~c == ring0 -> ring_mult c (pow p r) == ring0 -> p == ring0.
Definition R2:= ring_plus ring1 ring1.
Fixpoint IPR p {struct p}: R :=
match p with
xH => ring1
| xO xH => ring_plus ring1 ring1
| xO p1 => ring_mult R2 (IPR p1)
| xI xH => ring_plus ring1 (ring_plus ring1 ring1)
| xI p1 => ring_plus ring1 (ring_mult R2 (IPR p1))
end.
Definition IZR1 z :=
match z with Z0 => ring0
| Zpos p => IPR p
| Zneg p => ring_opp(IPR p)
end.
Fixpoint interpret3 t fv {struct t}: R :=
match t with
| (PEadd t1 t2) =>
let v1 := interpret3 t1 fv in
let v2 := interpret3 t2 fv in (ring_plus v1 v2)
| (PEmul t1 t2) =>
let v1 := interpret3 t1 fv in
let v2 := interpret3 t2 fv in (ring_mult v1 v2)
| (PEsub t1 t2) =>
let v1 := interpret3 t1 fv in
let v2 := interpret3 t2 fv in (ring_sub v1 v2)
| (PEopp t1) =>
let v1 := interpret3 t1 fv in (ring_opp v1)
| (PEpow t1 t2) =>
let v1 := interpret3 t1 fv in pow v1 (Nnat.nat_of_N t2)
| (PEc t1) => (IZR1 t1)
| (PEX n) => List.nth (pred (nat_of_P n)) fv 0
end.
End domain.
Ltac equalities_to_goal :=
lazymatch goal with
| H: (@ring_eq _ _ ?x ?y) |- _ =>
try generalize (@psos_r1 _ _ _ _ H); clear H
end.
Ltac nsatz_domain_begin tacsimpl :=
intros;
try apply (@psos_r1b _ _);
repeat equalities_to_goal;
tacsimpl.
Ltac generalise_eq_hyps:=
repeat
(match goal with
|h : (@ring_eq _ _ ?p ?q)|- _ => revert h
end).
Ltac lpol_goal t :=
match t with
| ?a = ring0 -> ?b =>
let r:= lpol_goal b in
constr:(a::r)
| ?a = ring0 => constr:(a::nil)
end.
Ltac parametres_en_tete fv lp :=
match fv with
| (@nil _) => lp
| (@cons _ ?x ?fv1) =>
let res := AddFvTail x lp in
parametres_en_tete fv1 res
end.
Ltac append1 a l :=
match l with
| (@nil _) => constr:(cons a l)
| (cons ?x ?l) => let l' := append1 a l in constr:(cons x l')
end.
Ltac rev l :=
match l with
|(@nil _) => l
| (cons ?x ?l) => let l' := rev l in append1 x l'
end.
Ltac nsatz_call_n info nparam p rr lp kont :=
let ll := constr:(PEc info :: PEc nparam :: PEpow p rr :: lp) in
nsatz_compute ll;
match goal with
| |- (?c::PEpow _ ?r::?lq0)::?lci0 = _ -> _ =>
intros _;
set (lci:=lci0);
set (lq:=lq0);
kont c rr lq lci
end.
Ltac nsatz_call radicalmax info nparam p lp kont :=
let rec try_n n :=
lazymatch n with
| 0%N => fail
| _ =>
(let r := eval compute in (Nminus radicalmax (Npred n)) in
nsatz_call_n info nparam p r lp kont) ||
let n' := eval compute in (Npred n) in try_n n'
end in
try_n radicalmax.
Class Cclosed_seq T (l:list T) := {}.
Instance Iclosed_nil T : Cclosed_seq (T:=T) nil.
Instance Iclosed_cons T t l `{Cclosed_seq (T:=T) l} : Cclosed_seq (T:=T) (t::l).
Class Cfind_at (R:Type) (b:R) (l:list R) (i:nat) := {}.
Instance Ifind0 (R:Type) (b:R) l: Cfind_at b (b::l) 0.
Instance IfindS (R:Type) (b2 b1:R) l i `{Cfind_at R b1 l i} : Cfind_at b1 (b2::l) (S i) | 1.
Definition Ifind0' := Ifind0.
Definition IfindS' := IfindS.
Definition li_find_at (R:Type) (b:R) l i `{Cfind_at R b l i} {H:Cclosed_seq (T:=R) l} := (l,i).
Class Creify (R:Type) (e:PExpr Z) (l:list R) (b:R) := {}.
Instance Ireify_zero (R:Type) (Rd:Domain R) l : Creify (PEc 0%Z) l ring0.
Instance Ireify_one (R:Type) (Rd:Domain R) l : Creify (PEc 1%Z) l ring1.
Instance Ireify_plus (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2}
: Creify (PEadd e1 e2) l (ring_plus b1 b2).
Instance Ireify_mult (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2}
: Creify (PEmul e1 e2) l (ring_mult b1 b2).
Instance Ireify_sub (R:Type) (Rd:Domain R) e1 l b1 e2 b2 `{Creify R e1 l b1} `{Creify R e2 l b2}
: Creify (PEsub e1 e2) l (ring_sub b1 b2).
Instance Ireify_opp (R:Type) (Rd:Domain R) e1 l b1 `{Creify R e1 l b1}
: Creify (PEopp e1) l (ring_opp b1).
Instance Ireify_var (R:Type) b l i `{Cfind_at R b l i}
: Creify (PEX _ (P_of_succ_nat i)) l b | 100.
Class Creifylist (R:Type) (le:list (PExpr Z)) (l:list R) (lb:list R) := {}.
Instance Creify_nil (R:Type) l : Creifylist nil l (@nil R).
Instance Creify_cons (R:Type) e1 l b1 le2 lb2 `{Creify R e1 l b1} `{Creifylist R le2 l lb2}
: Creifylist (e1::le2) l (b1::lb2).
Definition li_reifyl (R:Type) le l lb `{Creifylist R le l lb}
{H:Cclosed_seq (T:=R) l} := (l,le).
Ltac lterm_goal g :=
match g with
ring_eq ?b1 ?b2 => constr:(b1::b2::nil)
| ring_eq ?b1 ?b2 -> ?g => let l := lterm_goal g in constr:(b1::b2::l)
end.
Ltac reify_goal l le lb Rd:=
match le with
nil => idtac
| ?e::?le1 =>
match lb with
?b::?lb1 =>
let x := fresh "B" in
set (x:= b) at 1;
change x with (@interpret3 _ Rd e l);
clear x;
reify_goal l le1 lb1 Rd
end
end.
Ltac get_lpol g :=
match g with
ring_eq (interpret3 _ _ ?p _) _ => constr:(p::nil)
| ring_eq (interpret3 _ _ ?p _) _ -> ?g =>
let l := get_lpol g in constr:(p::l)
end.
Ltac nsatz_domain_generic radicalmax info lparam lvar tacsimpl Rd :=
match goal with
|- ?g => let lb := lterm_goal g in
match eval red in (li_reifyl (lb:=lb)) with
| (?fv, ?le) =>
let fv := match lvar with
(@nil _) => fv
| _ => lvar
end in
let nparam := eval compute in (Z_of_nat (List.length lparam)) in
let fv := parametres_en_tete fv lparam in
match eval red in (li_reifyl (l:=fv) (lb:=lb)) with
| (?fv, ?le) =>
reify_goal fv le lb Rd;
match goal with
|- ?g =>
let lp := get_lpol g in
let lpol := eval compute in (List.rev lp) in
tacsimpl; intros;
let SplitPolyList kont :=
match lpol with
| ?p2::?lp2 => kont p2 lp2
| _ => idtac "polynomial not in the ideal"
end in
tacsimpl;
SplitPolyList ltac:(fun p lp =>
set (p21:=p) ;
set (lp21:=lp);
nsatz_call radicalmax info nparam p lp ltac:(fun c r lq lci =>
set (q := PEmul c (PEpow p21 r));
let Hg := fresh "Hg" in
assert (Hg:check lp21 q (lci,lq) = true);
[ (vm_compute;reflexivity) || idtac "invalid nsatz certificate"
| let Hg2 := fresh "Hg" in
assert (Hg2: ring_eq (interpret3 _ Rd q fv) ring0);
[ tacsimpl;
apply (@check_correct _ Rd fv lp21 q (lci,lq) Hg);
tacsimpl;
repeat (split;[assumption|idtac]); exact I
| simpl in Hg2; tacsimpl;
apply Rdomain_pow with (interpret3 _ Rd c fv) (Nnat.nat_of_N r); auto with domain;
tacsimpl; apply domain_axiom_one_zero
|| (simpl) || idtac "could not prove discrimination result"
]
]
)
)
end end end end .
Ltac nsatz_domainpv pretac radicalmax info lparam lvar tacsimpl rd :=
pretac;
nsatz_domain_begin tacsimpl; auto with domain;
nsatz_domain_generic radicalmax info lparam lvar tacsimpl rd.
Ltac nsatz_domain:=
intros;
match goal with
|- (@ring_eq _ (@domain_ring ?r ?rd) _ _ ) =>
nsatz_domainpv ltac:idtac 6%N 1%Z (@nil r) (@nil r) ltac:(simpl) rd
end.
Require Import Reals.
Require Import RealField.
Instance Rri : Ring R := {
ring0 := 0%R;
ring1 := 1%R;
ring_plus := Rplus;
ring_mult := Rmult;
ring_sub := Rminus;
ring_opp := Ropp;
ring_eq := @eq R;
ring_ring := RTheory}.
Lemma Raxiom_one_zero: 1%R <> 0%R.
Instance Rdi : Domain R := {
domain_ring := Rri;
domain_axiom_product := Rmult_integral;
domain_axiom_one_zero := Raxiom_one_zero}.
Hint Resolve ring_setoid ring_plus_comp ring_mult_comp ring_sub_comp ring_opp_comp: domain.
Ltac replaceR:=
replace 0%R with (@ring0 _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace 1%R with (@ring1 _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace Rplus with (@ring_plus _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace Rmult with (@ring_mult _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace Rminus with (@ring_sub _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace Ropp with (@ring_opp _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity];
replace (@eq R) with (@ring_eq _ (@domain_ring _ Rdi)) in *;[idtac|reflexivity].
Ltac simplR:=
simpl; replaceR.
Ltac pretacR:=
replaceR;
replace Rri with (@domain_ring _ Rdi) in *; [idtac | reflexivity].
Ltac nsatz_domainR:=
nsatz_domainpv ltac:pretacR 6%N 1%Z (@Datatypes.nil R) (@Datatypes.nil R)
ltac:simplR Rdi;
discrR.
Goal forall x y:R, x = y -> (x*x-x+1)%R = ((y*y-y)+1+0)%R.
Instance Zri : Ring Z := {
ring0 := 0%Z;
ring1 := 1%Z;
ring_plus := Zplus;
ring_mult := Zmult;
ring_sub := Zminus;
ring_opp := Zopp;
ring_eq := (@eq Z);
ring_ring := Zth}.
Lemma Zaxiom_one_zero: 1%Z <> 0%Z.
Instance Zdi : Domain Z := {
domain_ring := Zri;
domain_axiom_product := Zmult_integral;
domain_axiom_one_zero := Zaxiom_one_zero}.
Ltac replaceZ :=
replace 0%Z with (@ring0 _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace 1%Z with (@ring1 _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace Zplus with (@ring_plus _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace Zmult with (@ring_mult _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace Zminus with (@ring_sub _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace Zopp with (@ring_opp _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity];
replace (@eq Z) with (@ring_eq _ (@domain_ring _ Zdi)) in *;[idtac|reflexivity].
Ltac simplZ:=
simpl; replaceZ.
Ltac pretacZ :=
replaceZ;
replace Zri with (@domain_ring _ Zdi) in *; [idtac | reflexivity].
Ltac nsatz_domainZ:=
nsatz_domainpv ltac:pretacZ 6%N 1%Z (@Datatypes.nil Z) (@Datatypes.nil Z) ltac:simplZ Zdi.
Require Import QArith.
Instance Qri : Ring Q := {
ring0 := 0%Q;
ring1 := 1%Q;
ring_plus := Qplus;
ring_mult := Qmult;
ring_sub := Qminus;
ring_opp := Qopp;
ring_eq := Qeq;
ring_ring := Qsrt}.
Lemma Qaxiom_one_zero: not (Qeq 1%Q 0%Q).
Instance Qdi : Domain Q := {
domain_ring := Qri;
domain_axiom_product := Qmult_integral;
domain_axiom_one_zero := Qaxiom_one_zero}.
Ltac replaceQ :=
replace 0%Q with (@ring0 _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace 1%Q with (@ring1 _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace Qplus with (@ring_plus _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace Qmult with (@ring_mult _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace Qminus with (@ring_sub _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace Qopp with (@ring_opp _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity];
replace Qeq with (@ring_eq _ (@domain_ring _ Qdi)) in *;[idtac|reflexivity].
Ltac simplQ:=
simpl; replaceQ.
Ltac pretacQ :=
replaceQ;
replace Qri with (@domain_ring _ Qdi) in *; [idtac | reflexivity].
Ltac nsatz_domainQ:=
nsatz_domainpv ltac:pretacQ 6%N 1%Z (@Datatypes.nil Q) (@Datatypes.nil Q) ltac:simplQ Qdi.
Ltac nsatz :=
intros;
match goal with
| |- (@eq R _ _) => nsatz_domainR
| |- (@eq Z _ _) => nsatz_domainZ
| |- (@Qeq _ _) => nsatz_domainQ
| |- _ => nsatz_domain
end.