The Coq Proof Assistant

# Library Coq.setoid_ring.Field_theory

Require Ring.
Import Ring_polynom Ring_tac Ring_theory InitialRing Setoid List Morphisms.
Require Import ZArith_base.
Set Implicit Arguments.
Section MakeFieldPol.
Variable R:Type.
Declare Scope R_scope.
Bind Scope R_scope with R.
Delimit Scope R_scope with ring.
Local Open Scope R_scope.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R->R).
Variable (rdiv : R->R->R) (rinv : R->R).
Variable req : R -> R -> Prop.
Notation "0" := rO : R_scope.
Notation "1" := rI : R_scope.
Infix "+" := radd : R_scope.
Infix "-" := rsub : R_scope.
Infix "*" := rmul : R_scope.
Infix "/" := rdiv : R_scope.
Notation "- x" := (ropp x) : R_scope.
Notation "/ x" := (rinv x) : R_scope.
Infix "==" := req (at level 70, no associativity) : R_scope.
Variable Rsth : Equivalence req.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Variable SRinv_ext : forall p q, p == q -> / p == / q.
Record almost_field_theory : Prop := mk_afield {
AF_AR : almost_ring_theory rO rI radd rmul rsub ropp req;
AF_1_neq_0 : ~ 1 == 0;
AFdiv_def : forall p q, p / q == p * / q;
AFinv_l : forall p, ~ p == 0 -> / p * p == 1
}.
Section AlmostField.
Variable AFth : almost_field_theory.
Let ARth := (AF_AR AFth).
Let rI_neq_rO := (AF_1_neq_0 AFth).
Let rdiv_def := (AFdiv_def AFth).
Let rinv_l := (AFinv_l AFth).
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.
Add Morphism rinv with signature (req ==> req) as rinv_ext.
Let eq_trans := Setoid.Seq_trans _ _ Rsth.
Let eq_sym := Setoid.Seq_sym _ _ Rsth.
Let eq_refl := Setoid.Seq_refl _ _ Rsth.
Let radd_0_l := ARadd_0_l ARth.
Let radd_comm := ARadd_comm ARth.
Let radd_assoc := ARadd_assoc ARth.
Let rmul_1_l := ARmul_1_l ARth.
Let rmul_0_l := ARmul_0_l ARth.
Let rmul_comm := ARmul_comm ARth.
Let rmul_assoc := ARmul_assoc ARth.
Let rdistr_l := ARdistr_l ARth.
Let ropp_mul_l := ARopp_mul_l ARth.
Let ropp_add := ARopp_add ARth.
Let rsub_def := ARsub_def ARth.
Let radd_0_r := ARadd_0_r Rsth ARth.
Let rmul_0_r := ARmul_0_r Rsth ARth.
Let rmul_1_r := ARmul_1_r Rsth ARth.
Let ropp_0 := ARopp_zero Rsth Reqe ARth.
Let rdistr_r := ARdistr_r Rsth Reqe ARth.
Variable C: Type.
Declare Scope C_scope.
Bind Scope C_scope with C.
Delimit Scope C_scope with coef.
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.
Notation "0" := cO : C_scope.
Notation "1" := cI : C_scope.
Infix "+" := cadd : C_scope.
Infix "-" := csub : C_scope.
Infix "*" := cmul : C_scope.
Notation "- x" := (copp x) : C_scope.
Infix "=?" := ceqb : C_scope.
Notation "[ x ]" := (phi x) (at level 0).
Let phi_0 := (morph0 CRmorph).
Let phi_1 := (morph1 CRmorph).
Lemma ceqb_spec c c' : BoolSpec ([c] == [c']) True (c =? c')%coef.
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 get_sign : C -> option C.
Variable get_sign_spec : sign_theory copp ceqb get_sign.
Variable cdiv:C -> C -> C*C.
Variable cdiv_th : div_theory req cadd cmul phi cdiv.
Let rpow_pow := (rpow_pow_N pow_th).
Declare Scope PE_scope.
Bind Scope PE_scope with PExpr.
Delimit Scope PE_scope with poly.
Notation NPEeval := (PEeval rO rI radd rmul rsub ropp phi Cp_phi rpow).
Notation "P @ l" := (NPEeval l P) (at level 10, no associativity).
Arguments PEc _ _%coef.
Notation "0" := (PEc 0) : PE_scope.
Notation "1" := (PEc 1) : PE_scope.
Infix "+" := PEadd : PE_scope.
Infix "-" := PEsub : PE_scope.
Infix "*" := PEmul : PE_scope.
Notation "- e" := (PEopp e) : PE_scope.
Infix "^" := PEpow : PE_scope.
Definition NPEequiv e e' := forall l, e@l == e'@l.
Infix "===" := NPEequiv (at level 70, no associativity) : PE_scope.
Instance NPEequiv_eq : Equivalence NPEequiv.
Instance NPEeval_ext : Proper (eq ==> NPEequiv ==> req) NPEeval.
Notation Nnorm :=
(norm_subst cO cI cadd cmul csub copp ceqb cdiv).
Notation NPphi_dev :=
(Pphi_dev rO rI radd rmul rsub ropp cO cI ceqb phi get_sign).
Notation NPphi_pow :=
(Pphi_pow rO rI radd rmul rsub ropp cO cI ceqb phi Cp_phi rpow get_sign).
Add Ring Rring : (ARth_SRth ARth).
Lemma rsub_0_l r : 0 - r == - r.
Lemma rsub_0_r r : r - 0 == r.
Theorem rdiv_simpl p q : ~ q == 0 -> q * (p / q) == p.
Instance rdiv_ext: Proper (req ==> req ==> req) rdiv.
Lemma rmul_reg_l p q1 q2 :
~ p == 0 -> p * q1 == p * q2 -> q1 == q2.
Theorem field_is_integral_domain r1 r2 :
~ r1 == 0 -> ~ r2 == 0 -> ~ r1 * r2 == 0.
Theorem ropp_neq_0 r :
~ -(1) == 0 -> ~ r == 0 -> ~ -r == 0.
Theorem rdiv_r_r r : ~ r == 0 -> r / r == 1.
Theorem rdiv1 r : r == r / 1.
Theorem rdiv2 a b c d :
~ b == 0 ->
~ d == 0 ->
a / b + c / d == (a * d + c * b) / (b * d).
Theorem rdiv2b a b c d e :
~ (b*e) == 0 ->
~ (d*e) == 0 ->
a / (b*e) + c / (d*e) == (a * d + c * b) / (b * (d * e)).
Theorem rdiv5 a b : - (a / b) == - a / b.
Theorem rdiv3b a b c d e :
~ (b * e) == 0 ->
~ (d * e) == 0 ->
a / (b*e) - c / (d*e) == (a * d - c * b) / (b * (d * e)).
Theorem rdiv6 a b :
~ a == 0 -> ~ b == 0 -> / (a / b) == b / a.
Theorem rdiv4 a b c d :
~ b == 0 ->
~ d == 0 ->
(a / b) * (c / d) == (a * c) / (b * d).
Theorem rdiv4b a b c d e f :
~ b * e == 0 ->
~ d * f == 0 ->
((a * f) / (b * e)) * ((c * e) / (d * f)) == (a * c) / (b * d).
Theorem rdiv7 a b c d :
~ b == 0 ->
~ c == 0 ->
~ d == 0 ->
(a / b) / (c / d) == (a * d) / (b * c).
Theorem rdiv7b a b c d e f :
~ b * f == 0 ->
~ c * e == 0 ->
~ d * f == 0 ->
((a * e) / (b * f)) / ((c * e) / (d * f)) == (a * d) / (b * c).
Theorem rinv_nz a : ~ a == 0 -> ~ /a == 0.
Theorem rdiv8 a b : ~ b == 0 -> a == 0 -> a / b == 0.
Theorem cross_product_eq a b c d :
~ b == 0 -> ~ d == 0 -> a * d == c * b -> a / b == c / d.
Instance pow_ext : Proper (req ==> eq ==> req) (pow_pos rmul).
Instance pow_N_ext : Proper (req ==> eq ==> req) (pow_N rI rmul).
Lemma pow_pos_0 p : pow_pos rmul 0 p == 0.
Lemma pow_pos_1 p : pow_pos rmul 1 p == 1.
Lemma pow_pos_cst c p : pow_pos rmul [c] p == [pow_pos cmul c p].
Lemma pow_pos_mul_l x y p :
pow_pos rmul (x * y) p == pow_pos rmul x p * pow_pos rmul y p.
Lemma pow_pos_add_r x p1 p2 :
pow_pos rmul x (p1+p2) == pow_pos rmul x p1 * pow_pos rmul x p2.
Lemma pow_pos_mul_r x p1 p2 :
pow_pos rmul x (p1*p2) == pow_pos rmul (pow_pos rmul x p1) p2.
Lemma pow_pos_nz x p : ~x==0 -> ~pow_pos rmul x p == 0.
Lemma pow_pos_div a b p : ~ b == 0 ->
pow_pos rmul (a / b) p == pow_pos rmul a p / pow_pos rmul b p.
Instance PEadd_ext : Proper (NPEequiv ==> NPEequiv ==> NPEequiv) (@PEadd C).
Instance PEsub_ext : Proper (NPEequiv ==> NPEequiv ==> NPEequiv) (@PEsub C).
Instance PEmul_ext : Proper (NPEequiv ==> NPEequiv ==> NPEequiv) (@PEmul C).
Instance PEopp_ext : Proper (NPEequiv ==> NPEequiv) (@PEopp C).
Instance PEpow_ext : Proper (NPEequiv ==> eq ==> NPEequiv) (@PEpow C).
Lemma PE_1_l (e : PExpr C) : (1 * e === e)%poly.
Lemma PE_1_r (e : PExpr C) : (e * 1 === e)%poly.
Lemma PEpow_0_r (e : PExpr C) : (e ^ 0 === 1)%poly.
Lemma PEpow_1_r (e : PExpr C) : (e ^ 1 === e)%poly.
Lemma PEpow_1_l n : (1 ^ n === 1)%poly.
Lemma PEpow_add_r (e : PExpr C) n n' :
(e ^ (n+n') === e ^ n * e ^ n')%poly.
Lemma PEpow_mul_l (e e' : PExpr C) n :
((e * e') ^ n === e ^ n * e' ^ n)%poly.
Lemma PEpow_mul_r (e : PExpr C) n n' :
(e ^ (n * n') === (e ^ n) ^ n')%poly.
Lemma PEpow_nz l e n : ~ e @ l == 0 -> ~ (e^n) @ l == 0.
Fixpoint PExpr_eq (e e' : PExpr C) {struct e} : bool :=
match e, e' with
| PEc c, PEc c' => ceqb c c'
| PEX _ p, PEX _ p' => Pos.eqb p p'
| e1 + e2, e1' + e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| e1 - e2, e1' - e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| e1 * e2, e1' * e2' => PExpr_eq e1 e1' &&& PExpr_eq e2 e2'
| - e, - e' => PExpr_eq e e'
| e ^ n, e' ^ n' => N.eqb n n' &&& PExpr_eq e e'
| _, _ => false
end%poly.
Lemma if_true (a b : bool) : a &&& b = true -> a = true /\ b = true.
Theorem PExpr_eq_semi_ok e e' :
PExpr_eq e e' = true -> (e === e')%poly.
Lemma PExpr_eq_spec e e' : BoolSpec (e === e')%poly True (PExpr_eq e e').
Smart constructors for polynomial expression, with reduction of constants
Definition NPEadd e1 e2 :=
match e1, e2 with
| PEc c1, PEc c2 => PEc (c1 + c2)
| PEc c, _ => if (c =? 0)%coef then e2 else e1 + e2
| _, PEc c => if (c =? 0)%coef then e1 else e1 + e2
| _, _ => (e1 + e2)
end%poly.
Infix "++" := NPEadd (at level 60, right associativity).
Theorem NPEadd_ok e1 e2 : (e1 ++ e2 === e1 + e2)%poly.
Definition NPEsub e1 e2 :=
match e1, e2 with
| PEc c1, PEc c2 => PEc (c1 - c2)
| PEc c, _ => if (c =? 0)%coef then - e2 else e1 - e2
| _, PEc c => if (c =? 0)%coef then e1 else e1 - e2
| _, _ => e1 - e2
end%poly.
Infix "--" := NPEsub (at level 50, left associativity).
Theorem NPEsub_ok e1 e2: (e1 -- e2 === e1 - e2)%poly.
Definition NPEopp e1 :=
match e1 with PEc c1 => PEc (- c1) | _ => - e1 end%poly.
Theorem NPEopp_ok e : (NPEopp e === -e)%poly.
Definition NPEpow x n :=
match n with
| N0 => 1
| Npos p =>
if (p =? 1)%positive then x else
match x with
| PEc c =>
if (c =? 1)%coef then 1
else if (c =? 0)%coef then 0
else PEc (pow_pos cmul c p)
| _ => x ^ n
end
end%poly.
Infix "^^" := NPEpow (at level 35, right associativity).
Theorem NPEpow_ok e n : (e ^^ n === e ^ n)%poly.
Fixpoint NPEmul (x y : PExpr C) {struct x} : PExpr C :=
match x, y with
| PEc c1, PEc c2 => PEc (c1 * c2)
| PEc c, _ => if (c =? 1)%coef then y else if (c =? 0)%coef then 0 else x * y
| _, PEc c => if (c =? 1)%coef then x else if (c =? 0)%coef then 0 else x * y
| e1 ^ n1, e2 ^ n2 => if (n1 =? n2)%N then (NPEmul e1 e2)^^n1 else x * y
| _, _ => x * y
end%poly.
Infix "**" := NPEmul (at level 40, left associativity).
Theorem NPEmul_ok e1 e2 : (e1 ** e2 === e1 * e2)%poly.
Fixpoint PEsimp (e : PExpr C) : PExpr C :=
match e with
| e1 + e2 => (PEsimp e1) ++ (PEsimp e2)
| e1 * e2 => (PEsimp e1) ** (PEsimp e2)
| e1 - e2 => (PEsimp e1) -- (PEsimp e2)
| - e1 => NPEopp (PEsimp e1)
| e1 ^ n1 => (PEsimp e1) ^^ n1
| _ => e
end%poly.
Theorem PEsimp_ok e : (PEsimp e === e)%poly.
Inductive FExpr : Type :=
| FEO : FExpr
| FEI : FExpr
| FEc: C -> FExpr
| FEX: positive -> FExpr
| FEadd: FExpr -> FExpr -> FExpr
| FEsub: FExpr -> FExpr -> FExpr
| FEmul: FExpr -> FExpr -> FExpr
| FEopp: FExpr -> FExpr
| FEinv: FExpr -> FExpr
| FEdiv: FExpr -> FExpr -> FExpr
| FEpow: FExpr -> N -> FExpr .
Fixpoint FEeval (l : list R) (pe : FExpr) {struct pe} : R :=
match pe with
| FEO => rO
| FEI => rI
| FEc c => phi c
| FEX x => BinList.nth 0 x l
| FEadd x y => FEeval l x + FEeval l y
| FEsub x y => FEeval l x - FEeval l y
| FEmul x y => FEeval l x * FEeval l y
| FEopp x => - FEeval l x
| FEinv x => / FEeval l x
| FEdiv x y => FEeval l x / FEeval l y
| FEpow x n => rpow (FEeval l x) (Cp_phi n)
end.
Strategy expand [FEeval].
Record linear : Type := mk_linear {
num : PExpr C;
denum : PExpr C;
condition : list (PExpr C) }.
Fixpoint PCond (l : list R) (le : list (PExpr C)) {struct le} : Prop :=
match le with
| nil => True
| e1 :: nil => ~ req (e1 @ l) rO
| e1 :: l1 => ~ req (e1 @ l) rO /\ PCond l l1
end.
Theorem PCond_cons l a l1 :
PCond l (a :: l1) <-> ~ a @ l == 0 /\ PCond l l1.
Theorem PCond_cons_inv_l l a l1 : PCond l (a::l1) -> ~ a @ l == 0.
Theorem PCond_cons_inv_r l a l1 : PCond l (a :: l1) -> PCond l l1.
Theorem PCond_app l l1 l2 :
PCond l (l1 ++ l2) <-> PCond l l1 /\ PCond l l2.
Definition absurd_PCond := cons 0%poly nil.
Lemma absurd_PCond_bottom : forall l, ~ PCond l absurd_PCond.
Definition default_isIn e1 p1 e2 p2 :=
if PExpr_eq e1 e2 then
match Z.pos_sub p1 p2 with
| Zpos p => Some (Npos p, 1%poly)
| Z0 => Some (N0, 1%poly)
| Zneg p => Some (N0, e2 ^^ Npos p)
end
else None.
Fixpoint isIn e1 p1 e2 p2 {struct e2}: option (N * PExpr C) :=
match e2 with
| e3 * e4 =>
match isIn e1 p1 e3 p2 with
| Some (N0, e5) => Some (N0, e5 ** (e4 ^^ Npos p2))
| Some (Npos p, e5) =>
match isIn e1 p e4 p2 with
| Some (n, e6) => Some (n, e5 ** e6)
| None => Some (Npos p, e5 ** (e4 ^^ Npos p2))
end
| None =>
match isIn e1 p1 e4 p2 with
| Some (n, e5) => Some (n, (e3 ^^ Npos p2) ** e5)
| None => None
end
end
| e3 ^ N0 => None
| e3 ^ Npos p3 => isIn e1 p1 e3 (Pos.mul p3 p2)
| _ => default_isIn e1 p1 e2 p2
end%poly.
Definition ZtoN z := match z with Zpos p => Npos p | _ => N0 end.
Definition NtoZ n := match n with Npos p => Zpos p | _ => Z0 end.
Lemma Z_pos_sub_gt p q : (p > q)%positive ->
Z.pos_sub p q = Zpos (p - q).
Ltac simpl_pos_sub := rewrite ?Z_pos_sub_gt in * by assumption.
Lemma default_isIn_ok e1 e2 p1 p2 :
match default_isIn e1 p1 e2 p2 with
| Some(n, e3) =>
let n' := ZtoN (Zpos p1 - NtoZ n) in
(e2 ^ N.pos p2 === e1 ^ n' * e3)%poly
/\ (Zpos p1 > NtoZ n)%Z
| _ => True
end.
Ltac npe_simpl := rewrite ?NPEmul_ok, ?NPEpow_ok, ?PEpow_mul_l.
Ltac npe_ring := intro l; simpl; ring.
Theorem isIn_ok e1 p1 e2 p2 :
match isIn e1 p1 e2 p2 with
| Some(n, e3) =>
let n' := ZtoN (Zpos p1 - NtoZ n) in
(e2 ^ N.pos p2 === e1 ^ n' * e3)%poly
/\ (Zpos p1 > NtoZ n)%Z
| _ => True
end.
Opaque NPEpow.
Record rsplit : Type := mk_rsplit {
rsplit_left : PExpr C;
rsplit_common : PExpr C;
rsplit_right : PExpr C}.
Notation left := rsplit_left.
Notation right := rsplit_right.
Notation common := rsplit_common.
Fixpoint split_aux e1 p e2 {struct e1}: rsplit :=
match e1 with
| e3 * e4 =>
let r1 := split_aux e3 p e2 in
let r2 := split_aux e4 p (right r1) in
mk_rsplit (left r1 ** left r2)
(common r1 ** common r2)
(right r2)
| e3 ^ N0 => mk_rsplit 1 1 e2
| e3 ^ Npos p3 => split_aux e3 (Pos.mul p3 p) e2
| _ =>
match isIn e1 p e2 1 with
| Some (N0,e3) => mk_rsplit 1 (e1 ^^ Npos p) e3
| Some (Npos q, e3) => mk_rsplit (e1 ^^ Npos q) (e1 ^^ Npos (p - q)) e3
| None => mk_rsplit (e1 ^^ Npos p) 1 e2
end
end%poly.
Lemma split_aux_ok1 e1 p e2 :
(let res := match isIn e1 p e2 1 with
| Some (N0,e3) => mk_rsplit 1 (e1 ^^ Npos p) e3
| Some (Npos q, e3) => mk_rsplit (e1 ^^ Npos q) (e1 ^^ Npos (p - q)) e3
| None => mk_rsplit (e1 ^^ Npos p) 1 e2
end
in
e1 ^ Npos p === left res * common res
/\ e2 === right res * common res)%poly.
Opaque NPEpow NPEmul.
Theorem split_aux_ok: forall e1 p e2,
(e1 ^ Npos p === left (split_aux e1 p e2) * common (split_aux e1 p e2)
/\ e2 === right (split_aux e1 p e2) * common (split_aux e1 p e2))%poly.
Definition split e1 e2 := split_aux e1 xH e2.
Theorem split_ok_l e1 e2 :
(e1 === left (split e1 e2) * common (split e1 e2))%poly.
Theorem split_ok_r e1 e2 :
(e2 === right (split e1 e2) * common (split e1 e2))%poly.
Lemma split_nz_l l e1 e2 :
~ e1 @ l == 0 -> ~ left (split e1 e2) @ l == 0.
Lemma split_nz_r l e1 e2 :
~ e2 @ l == 0 -> ~ right (split e1 e2) @ l == 0.
Fixpoint Fnorm (e : FExpr) : linear :=
match e with
| FEO => mk_linear 0 1 nil
| FEI => mk_linear 1 1 nil
| FEc c => mk_linear (PEc c) 1 nil
| FEX x => mk_linear (PEX C x) 1 nil
| FEadd e1 e2 =>
let x := Fnorm e1 in
let y := Fnorm e2 in
let s := split (denum x) (denum y) in
mk_linear
((num x ** right s) ++ (num y ** left s))
(left s ** (right s ** common s))
(condition x ++ condition y)%list
| FEsub e1 e2 =>
let x := Fnorm e1 in
let y := Fnorm e2 in
let s := split (denum x) (denum y) in
mk_linear
((num x ** right s) -- (num y ** left s))
(left s ** (right s ** common s))
(condition x ++ condition y)%list
| FEmul e1 e2 =>
let x := Fnorm e1 in
let y := Fnorm e2 in
let s1 := split (num x) (denum y) in
let s2 := split (num y) (denum x) in
mk_linear (left s1 ** left s2)
(right s2 ** right s1)
(condition x ++ condition y)%list
| FEopp e1 =>
let x := Fnorm e1 in
mk_linear (NPEopp (num x)) (denum x) (condition x)
| FEinv e1 =>
let x := Fnorm e1 in
mk_linear (denum x) (num x) (num x :: condition x)
| FEdiv e1 e2 =>
let x := Fnorm e1 in
let y := Fnorm e2 in
let s1 := split (num x) (num y) in
let s2 := split (denum x) (denum y) in
mk_linear (left s1 ** right s2)
(left s2 ** right s1)
(num y :: condition x ++ condition y)%list
| FEpow e1 n =>
let x := Fnorm e1 in
mk_linear ((num x)^^n) ((denum x)^^n) (condition x)
end.
Theorem Pcond_Fnorm l e :
PCond l (condition (Fnorm e)) -> ~ (denum (Fnorm e))@l == 0.
Ltac uneval :=
repeat match goal with
| |- context [ ?x @ ?l * ?y @ ?l ] => change (x@l * y@l) with ((x*y)@l)
| |- context [ ?x @ ?l + ?y @ ?l ] => change (x@l + y@l) with ((x+y)@l)
end.
Theorem Fnorm_FEeval_PEeval l fe:
PCond l (condition (Fnorm fe)) ->
FEeval l fe == (num (Fnorm fe)) @ l / (denum (Fnorm fe)) @ l.
Theorem Fnorm_crossproduct l fe1 fe2 :
let nfe1 := Fnorm fe1 in
let nfe2 := Fnorm fe2 in
(num nfe1 * denum nfe2) @ l == (num nfe2 * denum nfe1) @ l ->
PCond l (condition nfe1 ++ condition nfe2) ->
FEeval l fe1 == FEeval l fe2.
Notation Ninterp_PElist :=
(interp_PElist rO rI radd rmul rsub ropp req phi Cp_phi rpow).
Notation Nmk_monpol_list :=
(mk_monpol_list cO cI cadd cmul csub copp ceqb cdiv).
Theorem Fnorm_ok:
forall n l lpe fe,
Ninterp_PElist l lpe ->
Peq ceqb (Nnorm n (Nmk_monpol_list lpe) (num (Fnorm fe))) (Pc cO) = true ->
PCond l (condition (Fnorm fe)) -> FEeval l fe == 0.
Notation ring_rw_correct :=
(ring_rw_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec).
Notation ring_rw_pow_correct :=
(ring_rw_pow_correct Rsth Reqe ARth CRmorph pow_th cdiv_th get_sign_spec).
Notation ring_correct :=
(ring_correct Rsth Reqe ARth CRmorph pow_th cdiv_th).
Definition display_linear l num den :=
let lnum := NPphi_dev l num in
match den with
| Pc c => if ceqb c cI then lnum else lnum / NPphi_dev l den
| _ => lnum / NPphi_dev l den
end.
Definition display_pow_linear l num den :=
let lnum := NPphi_pow l num in
match den with
| Pc c => if ceqb c cI then lnum else lnum / NPphi_pow l den
| _ => lnum / NPphi_pow l den
end.
Theorem Field_rw_correct n lpe l :
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall fe nfe, Fnorm fe = nfe ->
PCond l (condition nfe) ->
FEeval l fe ==
display_linear l (Nnorm n lmp (num nfe)) (Nnorm n lmp (denum nfe)).
Theorem Field_rw_pow_correct n lpe l :
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall fe nfe, Fnorm fe = nfe ->
PCond l (condition nfe) ->
FEeval l fe ==
display_pow_linear l (Nnorm n lmp (num nfe)) (Nnorm n lmp (denum nfe)).
Theorem Field_correct n l lpe fe1 fe2 :
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall nfe1, Fnorm fe1 = nfe1 ->
forall nfe2, Fnorm fe2 = nfe2 ->
Peq ceqb (Nnorm n lmp (num nfe1 * denum nfe2))
(Nnorm n lmp (num nfe2 * denum nfe1)) = true ->
PCond l (condition nfe1 ++ condition nfe2) ->
FEeval l fe1 == FEeval l fe2.
This allows rewriting modulo the simplification of PEeval on PMul
Theorem Field_simplify_eq_correct :
forall n l lpe fe1 fe2,
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall nfe1, Fnorm fe1 = nfe1 ->
forall nfe2, Fnorm fe2 = nfe2 ->
forall den, split (denum nfe1) (denum nfe2) = den ->
NPphi_dev l (Nnorm n lmp (num nfe1 * right den)) ==
NPphi_dev l (Nnorm n lmp (num nfe2 * left den)) ->
PCond l (condition nfe1 ++ condition nfe2) ->
FEeval l fe1 == FEeval l fe2.
Theorem Field_simplify_eq_pow_correct :
forall n l lpe fe1 fe2,
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall nfe1, Fnorm fe1 = nfe1 ->
forall nfe2, Fnorm fe2 = nfe2 ->
forall den, split (denum nfe1) (denum nfe2) = den ->
NPphi_pow l (Nnorm n lmp (num nfe1 * right den)) ==
NPphi_pow l (Nnorm n lmp (num nfe2 * left den)) ->
PCond l (condition nfe1 ++ condition nfe2) ->
FEeval l fe1 == FEeval l fe2.
Theorem Field_simplify_aux_ok l fe1 fe2 den :
FEeval l fe1 == FEeval l fe2 ->
split (denum (Fnorm fe1)) (denum (Fnorm fe2)) = den ->
PCond l (condition (Fnorm fe1) ++ condition (Fnorm fe2)) ->
(num (Fnorm fe1) * right den) @ l == (num (Fnorm fe2) * left den) @ l.
Theorem Field_simplify_eq_pow_in_correct :
forall n l lpe fe1 fe2,
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall nfe1, Fnorm fe1 = nfe1 ->
forall nfe2, Fnorm fe2 = nfe2 ->
forall den, split (denum nfe1) (denum nfe2) = den ->
forall np1, Nnorm n lmp (num nfe1 * right den) = np1 ->
forall np2, Nnorm n lmp (num nfe2 * left den) = np2 ->
FEeval l fe1 == FEeval l fe2 ->
PCond l (condition nfe1 ++ condition nfe2) ->
NPphi_pow l np1 ==
NPphi_pow l np2.
Theorem Field_simplify_eq_in_correct :
forall n l lpe fe1 fe2,
Ninterp_PElist l lpe ->
forall lmp, Nmk_monpol_list lpe = lmp ->
forall nfe1, Fnorm fe1 = nfe1 ->
forall nfe2, Fnorm fe2 = nfe2 ->
forall den, split (denum nfe1) (denum nfe2) = den ->
forall np1, Nnorm n lmp (num nfe1 * right den) = np1 ->
forall np2, Nnorm n lmp (num nfe2 * left den) = np2 ->
FEeval l fe1 == FEeval l fe2 ->
PCond l (condition nfe1 ++ condition nfe2) ->
NPphi_dev l np1 == NPphi_dev l np2.
Section Fcons_impl.
Variable Fcons : PExpr C -> list (PExpr C) -> list (PExpr C).
Hypothesis PCond_fcons_inv : forall l a l1,
PCond l (Fcons a l1) -> ~ a @ l == 0 /\ PCond l l1.
Fixpoint Fapp (l m:list (PExpr C)) {struct l} : list (PExpr C) :=
match l with
| nil => m
| cons a l1 => Fcons a (Fapp l1 m)
end.
Lemma fcons_ok : forall l l1,
(forall lock, lock = PCond l -> lock (Fapp l1 nil)) -> PCond l l1.
End Fcons_impl.
Section Fcons_simpl.
Fixpoint Fcons (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) :=
match l with
nil => cons e nil
| cons a l1 => if PExpr_eq e a then l else cons a (Fcons e l1)
end.
Theorem PFcons_fcons_inv:
forall l a l1, PCond l (Fcons a l1) -> ~ a @ l == 0 /\ PCond l l1.
Fixpoint Fcons0 (e:PExpr C) (l:list (PExpr C)) {struct l} : list (PExpr C) :=
match l with
nil => cons e nil
| cons a l1 =>
if Peq ceqb (Nnorm O nil e) (Nnorm O nil a) then l
else cons a (Fcons0 e l1)
end.
Theorem PFcons0_fcons_inv:
forall l a l1, PCond l (Fcons0 a l1) -> ~ a @ l == 0 /\ PCond l l1.
Fixpoint Fcons00 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) :=
match e with
PEmul e1 e2 => Fcons00 e1 (Fcons00 e2 l)
| PEpow e1 _ => Fcons00 e1 l
| _ => Fcons0 e l
end.
Theorem PFcons00_fcons_inv:
forall l a l1, PCond l (Fcons00 a l1) -> ~ a @ l == 0 /\ PCond l l1.
Definition Pcond_simpl_gen :=
fcons_ok _ PFcons00_fcons_inv.
Hypothesis ceqb_complete : forall c1 c2, [c1] == [c2] -> ceqb c1 c2 = true.
Lemma ceqb_spec' c1 c2 : Bool.reflect ([c1] == [c2]) (ceqb c1 c2).
Fixpoint Fcons1 (e:PExpr C) (l:list (PExpr C)) {struct e} : list (PExpr C) :=
match e with
| PEmul e1 e2 => Fcons1 e1 (Fcons1 e2 l)
| PEpow e _ => Fcons1 e l
| PEopp e => if (-(1) =? 0)%coef then absurd_PCond else Fcons1 e l
| PEc c => if (c =? 0)%coef then absurd_PCond else l
| _ => Fcons0 e l
end.
Theorem PFcons1_fcons_inv:
forall l a l1, PCond l (Fcons1 a l1) -> ~ a @ l == 0 /\ PCond l l1.
Definition Fcons2 e l := Fcons1 (PEsimp e) l.
Theorem PFcons2_fcons_inv:
forall l a l1, PCond l (Fcons2 a l1) -> ~ a @ l == 0 /\ PCond l l1.
Definition Pcond_simpl_complete :=
fcons_ok _ PFcons2_fcons_inv.
End Fcons_simpl.
End AlmostField.
Section FieldAndSemiField.
Record field_theory : Prop := mk_field {
F_R : ring_theory rO rI radd rmul rsub ropp req;
F_1_neq_0 : ~ 1 == 0;
Fdiv_def : forall p q, p / q == p * / q;
Finv_l : forall p, ~ p == 0 -> / p * p == 1
}.
Definition F2AF f :=
mk_afield
(Rth_ARth Rsth Reqe (F_R f)) (F_1_neq_0 f) (Fdiv_def f) (Finv_l f).
Record semi_field_theory : Prop := mk_sfield {
SF_SR : semi_ring_theory rO rI radd rmul req;
SF_1_neq_0 : ~ 1 == 0;
SFdiv_def : forall p q, p / q == p * / q;
SFinv_l : forall p, ~ p == 0 -> / p * p == 1
}.
End FieldAndSemiField.
End MakeFieldPol.
Definition SF2AF R (rO rI:R) radd rmul rdiv rinv req Rsth
(sf:semi_field_theory rO rI radd rmul rdiv rinv req) :=
mk_afield _ _
(SRth_ARth Rsth (SF_SR sf))
(SF_1_neq_0 sf)
(SFdiv_def sf)
(SFinv_l sf).
Section Complete.
Variable R : Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R).
Variable (rdiv : R -> R -> R) (rinv : R -> R).
Variable req : R -> R -> Prop.
Notation "0" := rO. Notation "1" := rI.
Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y).
Notation "x - y " := (rsub x y). Notation "- x" := (ropp x).
Notation "x / y " := (rdiv x y). Notation "/ x" := (rinv x).
Notation "x == y" := (req x y) (at level 70, no associativity).
Variable Rsth : Setoid_Theory R req.
Add Parametric Relation : R req
reflexivity proved by (@Equivalence_Reflexive _ _ Rsth)
symmetry proved by (@Equivalence_Symmetric _ _ Rsth)
transitivity proved by (@Equivalence_Transitive _ _ Rsth)
as R_setoid3.
Variable Reqe : ring_eq_ext radd rmul ropp req.
Add Morphism radd with signature (req ==> req ==> req) as radd_ext3.
Add Morphism rmul with signature (req ==> req ==> req) as rmul_ext3.
Add Morphism ropp with signature (req ==> req) as ropp_ext3.
Section AlmostField.
Variable AFth : almost_field_theory rO rI radd rmul rsub ropp rdiv rinv req.
Let ARth := (AF_AR AFth).
Let rI_neq_rO := (AF_1_neq_0 AFth).
Let rdiv_def := (AFdiv_def AFth).
Let rinv_l := (AFinv_l AFth).
Hypothesis S_inj : forall x y, 1+x==1+y -> x==y.
Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0.
Lemma add_inj_r p x y :
gen_phiPOS1 rI radd rmul p + x == gen_phiPOS1 rI radd rmul p + y -> x==y.
Lemma gen_phiPOS_inj x y :
gen_phiPOS rI radd rmul x == gen_phiPOS rI radd rmul y ->
x = y.
Lemma gen_phiN_inj x y :
gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y ->
x = y.
Lemma gen_phiN_complete x y :
gen_phiN rO rI radd rmul x == gen_phiN rO rI radd rmul y ->
N.eqb x y = true.
End AlmostField.
Section Field.
Variable Fth : field_theory rO rI radd rmul rsub ropp rdiv rinv req.
Let Rth := (F_R Fth).
Let rI_neq_rO := (F_1_neq_0 Fth).
Let rdiv_def := (Fdiv_def Fth).
Let rinv_l := (Finv_l Fth).
Let AFth := F2AF Rsth Reqe Fth.
Let ARth := Rth_ARth Rsth Reqe Rth.
Lemma ring_S_inj x y : 1+x==1+y -> x==y.
Hypothesis gen_phiPOS_not_0 : forall p, ~ gen_phiPOS1 rI radd rmul p == 0.
Let gen_phiPOS_inject :=
gen_phiPOS_inj AFth ring_S_inj gen_phiPOS_not_0.
Lemma gen_phiPOS_discr_sgn x y :
~ gen_phiPOS rI radd rmul x == - gen_phiPOS rI radd rmul y.
Lemma gen_phiZ_inj x y :
gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y ->
x = y.
Lemma gen_phiZ_complete x y :
gen_phiZ rO rI radd rmul ropp x == gen_phiZ rO rI radd rmul ropp y ->
Zeq_bool x y = true.
End Field.
End Complete.
Arguments FEO {C}.
Arguments FEI {C}.