Revision 25d7504145c07e4345a472338a8616b023fc8243 authored by Pierre-Yves Strub on 11 July 2015, 19:03:40 UTC, committed by Pierre-Yves Strub on 12 July 2015, 12:14:23 UTC
[fix #17219]
1 parent 9b8598c
Ring.ec
(* --------------------------------------------------------------------
* Copyright (c) - 2012-2015 - IMDEA Software Institute and INRIA
* Distributed under the terms of the CeCILL-B licence.
* -------------------------------------------------------------------- *)
pragma +implicits.
(* -------------------------------------------------------------------- *)
require import Fun Int IntExtra.
(* -------------------------------------------------------------------- *)
abstract theory ZModule.
type t.
op zeror : t.
op ( + ) : t -> t -> t.
op [ - ] : t -> t.
axiom nosmt addrA: associative (+).
axiom nosmt addrC: commutative (+).
axiom nosmt add0r: left_id zeror (+).
axiom nosmt addNr: left_inverse zeror [-] (+).
op ( - ) (x y : t) = x + -y axiomatized by subrE.
lemma nosmt addr0: right_id zeror (+).
proof. by move=> x; rewrite addrC add0r. qed.
lemma nosmt addrN: right_inverse zeror [-] (+).
proof. by move=> x; rewrite addrC addNr. qed.
lemma nosmt addrCA: left_commutative (+).
proof. by move=> x y z; rewrite !addrA @(addrC x y). qed.
lemma nosmt addrAC: right_commutative (+).
proof. by move=> x y z; rewrite -!addrA @(addrC y z). qed.
lemma nosmt subrr (x : t): x - x = zeror.
proof. by rewrite subrE /= addrN. qed.
lemma nosmt addKr: left_loop [-] (+).
proof. by move=> x y; rewrite addrA addNr add0r. qed.
lemma nosmt addNKr: rev_left_loop [-] (+).
proof. by move=> x y; rewrite addrA addrN add0r. qed.
lemma nosmt addrK: right_loop [-] (+).
proof. by move=> x y; rewrite -addrA addrN addr0. qed.
lemma nosmt addrNK: rev_right_loop [-] (+).
proof. by move=> x y; rewrite -addrA addNr addr0. qed.
lemma nosmt addrI: right_injective (+).
proof. by move=> x y z h; rewrite -@(addKr x z) -h addKr. qed.
lemma nosmt addIr: left_injective (+).
proof. by move=> x y z h; rewrite -@(addrK x z) -h addrK. qed.
lemma nosmt opprK: involutive [-].
proof. by move=> x; apply @(addIr (-x)); rewrite addNr addrN. qed.
lemma nosmt oppr0: -zeror = zeror.
proof. by rewrite -@(addr0 (-zeror)) addNr. qed.
lemma nosmt subr0 (x : t): x - zeror = x.
proof. by rewrite subrE /= oppr0 addr0. qed.
lemma nosmt sub0r (x : t): zeror - x = - x.
proof. by rewrite subrE /= add0r. qed.
lemma nosmt opprD (x y : t): -(x + y) = -x + -y.
proof. by apply @(addrI (x + y)); rewrite addrA addrN addrAC addrK addrN. qed.
lemma nosmt opprB (x y : t): -(x - y) = y - x.
proof. by rewrite !subrE opprD opprK addrC. qed.
lemma nosmt subr_eq (x y z : t):
(x - z = y) <=> (x = y + z).
proof.
move: (can2_eq (fun x, x - z) (fun x, x + z) _ _ x y) => //=.
by move=> {x} x /=; rewrite subrE /= addrNK.
by move=> {x} x /=; rewrite subrE /= addrK.
qed.
lemma nosmt subr_eq0 (x y : t): (x - y = zeror) <=> (x = y).
proof. by rewrite subr_eq add0r. qed.
lemma nosmt addr_eq0 (x y : t): (x + y = zeror) <=> (x = -y).
proof. by rewrite -@(subr_eq0 x) subrE /= opprK. qed.
lemma nosmt eqr_opp (x y : t): (- x = - y) <=> (x = y).
proof.
move: (can_eq (fun (z : t), -z) (fun (z : t), -z) _ x y) => //=.
by move=> z /=; rewrite opprK.
qed.
op intmul (x : t) (n : int) =
if n < 0
then -(iterop (-n) ZModule.(+) x zeror)
else (iterop n ZModule.(+) x zeror).
lemma intmul0 (x : t): intmul x 0 = zeror.
proof. by rewrite /intmul /= iterop0. qed.
lemma intmul1 (x : t): intmul x 1 = x.
proof. by rewrite /intmul /= iterop1. qed.
lemma intmulN (x : t) (n : int): intmul x (-n) = -(intmul x n).
proof.
case: (n = 0)=> [-> |nz_n]; 1: by rewrite oppz0 intmul0 oppr0.
rewrite /intmul; have ->: -n < 0 <=> !(n < 0) by smt.
by case: (n < 0)=> //= _; rewrite ?(opprK, oppzK).
qed.
lemma intmulS (x : t) (n : int): 0 <= n =>
intmul x (n+1) = x + intmul x n.
proof.
elim: n=> /= [|i ge0_i ih]; 2: smt.
by rewrite intmul0 intmul1 addr0.
qed.
lemma intmul2 (x : t): intmul x 2 = x + x.
proof. by rewrite /intmul /= @(iteropS 1) // @(iterS 0) // iter0. qed.
end ZModule.
(* -------------------------------------------------------------------- *)
abstract theory ComRing.
type t.
clone include ZModule with type t <- t.
op oner : t.
op ( * ) : t -> t -> t.
op invr : t -> t.
pred unit : t.
op ( / ) (x y : t) = x * (invr y) axiomatized by divrE.
axiom nosmt oner_neq0 : oner <> zeror.
axiom nosmt mulrA : associative ( * ).
axiom nosmt mulrC : commutative ( * ).
axiom nosmt mul1r : left_id oner ( * ).
axiom nosmt mulrDl : left_distributive ( * ) (+).
axiom nosmt mulVr : left_inverse_in unit oner invr ( * ).
axiom nosmt unitP : forall (x y : t), y * x = oner => unit x.
axiom nosmt unitout : forall (x : t), !unit x => invr x = x.
lemma nosmt mulr1: right_id oner ( * ).
proof. by move=> x; rewrite mulrC mul1r. qed.
lemma nosmt mulrDr: right_distributive ( * ) (+).
proof. by move=> x y z; rewrite mulrC mulrDl !@(mulrC _ x). qed.
lemma nosmt mul0r: left_zero zeror ( * ).
proof. by move=> x; apply: @(addIr (oner * x)); rewrite -mulrDl !add0r mul1r. qed.
lemma nosmt mulr0: right_zero zeror ( * ).
proof. by move=> x; apply: @(addIr (x * oner)); rewrite -mulrDr !add0r mulr1. qed.
lemma nosmt mulrN (x y : t): x * (- y) = - (x * y).
proof. by apply: @(addrI (x * y)); rewrite -mulrDr !addrN mulr0. qed.
lemma nosmt mulNr (x y : t): (- x) * y = - (x * y).
proof. by apply: @(addrI (x * y)); rewrite -mulrDl !addrN mul0r. qed.
lemma nosmt mulrNN (x y : t): (- x) * (- y) = x * y.
proof. by rewrite mulrN mulNr opprK. qed.
lemma nosmt mulN1r (x : t): (-oner) * x = -x.
proof. by rewrite mulNr mul1r. qed.
lemma nosmt mulrN1 x: x * -oner = -x.
proof. by rewrite mulrN mulr1. qed.
lemma nosmt mulrV: right_inverse_in unit oner invr ( * ).
proof. by move=> x /mulVr; rewrite mulrC. qed.
lemma nosmt divrr (x : t): unit x => x / x = oner.
proof. by rewrite divrE => /mulrV. qed.
lemma nosmt invr_out (x : t): !unit x => invr x = x.
proof. by apply/unitout. qed.
lemma nosmt unitrP (x : t): unit x <=> (exists y, y * x = oner).
proof. by split=> [/mulVr<- |]; [exists (invr x) | case=> y /unitP]. qed.
lemma nosmt mulKr: left_loop_in unit invr ( * ).
proof. by move=> x un_x y; rewrite mulrA mulVr // mul1r. qed.
lemma nosmt mulrK: right_loop_in unit invr ( * ).
proof. by move=> y un_y x; rewrite -mulrA mulrV // mulr1. qed.
lemma nosmt mulVKr: rev_left_loop_in unit invr ( * ).
proof. by move=> x un_x y; rewrite mulrA mulrV // mul1r. qed.
lemma nosmt mulrVK: rev_right_loop_in unit invr ( * ).
proof. by move=> y nz_y x; rewrite -mulrA mulVr // mulr1. qed.
(* FIXME: have := can_inj _ _ (mulKr _ Ux) *)
lemma nosmt mulrI: right_injective_in unit ( * ).
proof. by move=> x Ux; have /can_inj h := mulKr _ Ux. qed.
lemma nosmt mulIr: left_injective_in unit ( * ).
proof. by move=> x /mulrI h y1 y2; rewrite !@(mulrC _ x) => /h. qed.
lemma nosmt unitrE (x : t): unit x <=> (x / x = oner).
proof.
split=> [Ux|xx1]; 1: by apply/divrr.
by apply/unitrP; exists (invr x); rewrite mulrC -divrE.
qed.
lemma invrK: involutive invr.
proof.
move=> x; case: (unit x)=> Ux; 2: by rewrite !invr_out.
rewrite -(mulrK _ Ux (invr (invr x))) -mulrA.
rewrite @(mulrC x) mulKr //; apply/unitrP.
by exists x; rewrite mulrV.
qed.
lemma nosmt invr_inj: injective invr.
proof. by apply: (can_inj invrK). qed.
lemma nosmt unitrV x: unit (invr x) <=> unit x.
proof. by rewrite !unitrE !divrE invrK mulrC. qed.
lemma nosmt unitr1: unit oner.
proof. by apply/unitrP; exists oner; rewrite mulr1. qed.
lemma invr1: invr oner = oner.
proof. by rewrite -{2}(mulVr _ unitr1) mulr1. qed.
lemma div1r x: oner / x = invr x.
proof. by rewrite divrE mul1r. qed.
lemma divr1 x: x / oner = x.
proof. by rewrite divrE invr1 mulr1. qed.
lemma nosmt unitr0: !unit zeror.
proof. by apply/negP=> /unitrP [y]; rewrite mulr0 eq_sym oner_neq0. qed.
lemma invr0: invr zeror = zeror.
proof. by rewrite invr_out ?unitr0. qed.
lemma nosmt unitrN1: unit (-oner).
proof. by apply/unitrP; exists (-oner); rewrite mulrNN mulr1. qed.
lemma nosmt invrN1: invr (-oner) = -oner.
proof. by rewrite -{2}(divrr unitrN1) divrE mulN1r opprK. qed.
op ofint n = intmul oner n.
lemma ofint0: ofint 0 = zeror.
proof. by apply/intmul0. qed.
lemma ofint1: ofint 1 = oner.
proof. by apply/intmul1. qed.
lemma ofintS (i : int): 0 <= i => ofint (i+1) = oner + ofint i.
proof. by apply/intmulS. qed.
lemma ofintN (i : int): ofint (-i) = - (ofint i).
proof. by apply/intmulN. qed.
lemma mulr0z x: x * ofint 0 = zeror.
proof. by rewrite ofint0 mulr0. qed.
lemma mulr1z x : x * ofint 1 = x.
proof. by rewrite ofint1 mulr1. qed.
lemma mulr2z x : x * ofint 2 = x + x.
proof. by rewrite /ofint intmul2 mulrDr mulr1. qed.
op exp (x : t) (n : int) =
if n < 0
then invr (iterop (-n) ComRing.( * ) x oner)
else iterop n ComRing.( * ) x oner.
lemma expr0 x: exp x 0 = oner.
proof. by rewrite /exp /= iterop0. qed.
lemma expr1 x: exp x 1 = x.
proof. by rewrite /exp /= iterop1. qed.
lemma exprS (x : t) i: 0 <= i => exp x (i+1) = x * (exp x i).
proof.
by elim: i=> /= [|i ge0_i ih]; 2: smt; rewrite expr0 expr1 mulr1.
qed.
lemma exprN (x : t) (i : int): exp x (-i) = invr (exp x i).
proof. case: (i = 0); smt. qed.
end ComRing.
(* -------------------------------------------------------------------- *)
abstract theory BoolRing.
type t.
clone include ComRing with type t <- t.
axiom mulrr : forall (x : t), x * x = x.
lemma nosmt addrr (x : t): x + x = zeror.
proof.
apply @(addrI (x + x)); rewrite addr0 -{1 2 3 4}mulrr.
by rewrite -mulrDr -mulrDl mulrr.
qed.
end BoolRing.
(* -------------------------------------------------------------------- *)
abstract theory IDomain.
type t.
clone include ComRing with type t <- t.
axiom mulf_eq0:
forall (x y : t), x * y = zeror <=> x = zeror \/ y = zeror.
lemma mulf_neq0 (x y : t): x <> zeror => y <> zeror => x * y <> zeror.
proof. by move=> nz_x nz_y; apply/not_def; rewrite mulf_eq0; smt. qed.
lemma mulfI (x : t): x <> zeror => injective (( * ) x).
proof.
move=> ne0_x y y'; rewrite -(opprK (x * y')) -mulrN -addr_eq0.
by rewrite -mulrDr mulf_eq0 ne0_x /= addr_eq0 opprK.
qed.
lemma mulIf x: x <> zeror => injective (fun y => y * x).
proof. by move=> nz_x y z; rewrite -!@(mulrC x); exact: mulfI. qed.
end IDomain.
(* -------------------------------------------------------------------- *)
abstract theory Field.
type t.
clone include IDomain with type t <- t, pred unit (x : t) <- x <> zeror.
lemma mulfV (x : t): x <> zeror => x * (invr x) = oner.
proof. by apply/mulrV. qed.
lemma mulVf (x : t): x <> zeror => (invr x) * x = oner.
proof. by apply/mulVr. qed.
lemma nosmt divff (x : t): x <> zeror => x / x = oner.
proof. by apply/divrr. qed.
end Field.
(* --------------------------------------------------------------------- *)
abstract theory Additive.
type t1, t2.
clone import Self.ZModule as ZM1 with type t <- t1.
clone import Self.ZModule as ZM2 with type t <- t2.
pred additive (f : t1 -> t2) =
forall (x y : t1), f (x - y) = f x - f y.
op f : { t1 -> t2 | additive f } as f_is_additive.
lemma raddf0: f ZM1.zeror = ZM2.zeror.
proof. by rewrite -ZM1.subr0 f_is_additive ZM2.subrr. qed.
lemma raddfB (x y : t1): f (x - y) = f x - f y.
proof. by apply/f_is_additive. qed.
lemma raddfN (x : t1): f (- x) = - (f x).
proof. by rewrite -ZM1.sub0r raddfB raddf0 ZM2.sub0r. qed.
lemma raddfD (x y : t1): f (x + y) = f x + f y.
proof.
rewrite -{1}@(ZM1.opprK y) -ZM1.subrE raddfB raddfN.
by rewrite ZM2.subrE ZM2.opprK.
qed.
end Additive.
(* --------------------------------------------------------------------- *)
abstract theory Multiplicative.
type t1, t2.
clone import Self.ComRing as ZM1 with type t <- t1.
clone import Self.ComRing as ZM2 with type t <- t2.
pred multiplicative (f : t1 -> t2) =
f ZM1.oner = ZM2.oner
/\ forall (x y : t1), f (x * y) = f x * f y.
end Multiplicative.
(* --------------------------------------------------------------------- *)
(* Rewrite database for algebra tactic *)
hint rewrite rw_algebra : .
hint rewrite inj_algebra : .
(* -------------------------------------------------------------------- *)
clone IDomain as IntID with
type t <- int,
pred unit (z : int) <- (z = 1 \/ z = -1),
op zeror <- 0,
op oner <- 1,
op ( + ) <- Int.( + ),
op [ - ] <- Int.([-]),
op ( - ) <- Int.( - ),
op ( * ) <- Int.( * ),
op ( / ) <- Int.( * ),
op invr <- (fun (z : int) => z)
proof * by smt.
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