The Coq Proof Assistant

# Library Coq.setoid_ring.Cring

Require Export List.
Require Import Setoid.
Require Import BinPos.
Require Import BinList.
Require Import Znumtheory.
Require Export Morphisms Setoid Bool.
Require Import ZArith_base.
Require Export Algebra_syntax.
Require Export Ncring.
Require Export Ncring_initial.
Require Export Ncring_tac.
Require Import InitialRing.
Class Cring {R:Type}`{Rr:Ring R} :=
cring_mul_comm: forall x y:R, x * y == y * x.
Ltac reify_goal lvar lexpr lterm:=
match lexpr with
nil => idtac
| ?e1::?e2::_ =>
match goal with
|- (?op ?u1 ?u2) =>
change (op
(@Ring_polynom.PEeval
_ zero one _+_ _*_ _-_ -_ Z Ncring_initial.gen_phiZ N (fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication) lvar e1)
(@Ring_polynom.PEeval
_ zero one _+_ _*_ _-_ -_ Z Ncring_initial.gen_phiZ N (fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication) lvar e2))
end
end.
Section cring.
Context {R:Type}`{Rr:Cring R}.
Lemma cring_eq_ext: ring_eq_ext _+_ _*_ -_ _==_.
Lemma cring_almost_ring_theory:
almost_ring_theory (R:=R) zero one _+_ _*_ _-_ -_ _==_.
Lemma cring_morph:
ring_morph zero one _+_ _*_ _-_ -_ _==_
0%Z 1%Z Z.add Z.mul Z.sub Z.opp Zeq_bool
Ncring_initial.gen_phiZ.
Lemma cring_power_theory :
@Ring_theory.power_theory R one _*_ _==_ N (fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication).
Lemma cring_div_theory:
div_theory _==_ Z.add Z.mul Ncring_initial.gen_phiZ Z.quotrem.
End cring.
Ltac cring_gen :=
match goal with
|- ?g =>
let lterm := lterm_goal g in
let reif := list_reifyl0 lterm in
match reif with
| (?fv, ?lexpr) =>
reify_goal fv lexpr lterm;
match goal with
|- ?g =>
generalize
(@Ring_polynom.ring_correct _ 0 1 _+_ _*_ _-_ -_ _==_
ring_setoid
cring_eq_ext
cring_almost_ring_theory
Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp Zeq_bool
Ncring_initial.gen_phiZ
cring_morph
N
(fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication)
cring_power_theory
Z.quotrem
cring_div_theory
O fv nil);
let rc := fresh "rc"in
intro rc; apply rc
end
end
end.
Ltac cring_compute:= vm_compute; reflexivity.
Ltac cring:=
intros;
cring_gen;
cring_compute.
#[global]
Instance Zcri: (Cring (Rr:=Zr)).
Ltac cring_simplify_aux lterm fv lexpr hyp :=
match lterm with
| ?t0::?lterm =>
match lexpr with
| ?e::?le =>
let t := constr:(@Ring_polynom.norm_subst
Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp Zeq_bool Z.quotrem O nil e) in
let te :=
constr:(@Ring_polynom.Pphi_dev
_ 0 1 _+_ _*_ _-_ -_
Z 0%Z 1%Z Zeq_bool
Ncring_initial.gen_phiZ
get_signZ fv t) in
let eq1 := fresh "ring" in
let nft := eval vm_compute in t in
let t':= fresh "t" in
pose (t' := nft);
assert (eq1 : t = t');
[vm_cast_no_check (eq_refl t')|
let eq2 := fresh "ring" in
assert (eq2:(@Ring_polynom.PEeval
_ zero _+_ _*_ _-_ -_ Z Ncring_initial.gen_phiZ N (fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication) fv e) == te);
[let eq3 := fresh "ring" in
generalize (@ring_rw_correct _ 0 1 _+_ _*_ _-_ -_ _==_
ring_setoid
cring_eq_ext
cring_almost_ring_theory
Z 0%Z 1%Z Z.add Z.mul Z.sub Z.opp Zeq_bool
Ncring_initial.gen_phiZ
cring_morph
N
(fun n:N => n)
(@Ring_theory.pow_N _ 1 multiplication)
cring_power_theory
Z.quotrem
cring_div_theory
get_signZ get_signZ_th
O nil fv I nil (eq_refl nil) );
intro eq3; apply eq3; reflexivity|
match hyp with
| 1%nat => rewrite eq2
| ?H => try rewrite eq2 in H
end];
let P:= fresh "P" in
match hyp with
| 1%nat =>
rewrite eq1;
pattern (@Ring_polynom.Pphi_dev
_ 0 1 _+_ _*_ _-_ -_
Z 0%Z 1%Z Zeq_bool
Ncring_initial.gen_phiZ
get_signZ fv t');
match goal with
|- (?p ?t) => set (P:=p)
end;
unfold t' in *; clear t' eq1 eq2;
unfold Pphi_dev, Pphi_avoid; simpl;
repeat (unfold mkmult1, mkmultm1, mkmult_c_pos, mkmult_c,
mkadd_mult, mkmult_c_pos, mkmult_pow, mkadd_mult,
mkpow;simpl)
| ?H =>
rewrite eq1 in H;
pattern (@Ring_polynom.Pphi_dev
_ 0 1 _+_ _*_ _-_ -_
Z 0%Z 1%Z Zeq_bool
Ncring_initial.gen_phiZ
get_signZ fv t') in H;
match type of H with
| (?p ?t) => set (P:=p) in H
end;
unfold t' in *; clear t' eq1 eq2;
unfold Pphi_dev, Pphi_avoid in H; simpl in H;
repeat (unfold mkmult1, mkmultm1, mkmult_c_pos, mkmult_c,
mkadd_mult, mkmult_c_pos, mkmult_pow, mkadd_mult,
mkpow in H;simpl in H)
end; unfold P in *; clear P
]; cring_simplify_aux lterm fv le hyp
| nil => idtac
end
| nil => idtac
end.
Ltac set_variables fv :=
match fv with
| nil => idtac
| ?t::?fv =>
let v := fresh "X" in
set (v:=t) in *; set_variables fv
end.
Ltac deset n:=
match n with
| 0%nat => idtac
| S ?n1 =>
match goal with
| h:= ?v : ?t |- ?g => unfold h in *; clear h; deset n1
end
end.
Ltac cring_simplify_gen a hyp :=
let lterm :=
match a with
| _::_ => a
| _ => constr:(a::nil)
end in
let reif := list_reifyl0 lterm in
match reif with
| (?fv, ?lexpr) => idtac lterm; idtac fv; idtac lexpr;
let n := eval compute in (length fv) in
idtac n;
let lt:=fresh "lt" in
set (lt:= lterm);
let lv:=fresh "fv" in
set (lv:= fv);
set_variables fv;
let lterm1 := eval unfold lt in lt in
let lv1 := eval unfold lv in lv in
idtac lterm1; idtac lv1;
cring_simplify_aux lterm1 lv1 lexpr hyp;
clear lt lv;
deset n
end.
Tactic Notation "cring_simplify" constr(lterm):=
cring_simplify_gen lterm 1%nat.
Tactic Notation "cring_simplify" constr(lterm) "in" ident(H):=
cring_simplify_gen lterm H.