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+Require Import Nsatz_domain ZArith Reals List Ring_polynom.
+
+Variable A: Type.
+Variable Ad: Domain A.
+
+Add Ring Ar1: (@ring_ring A (@domain_ring _ Ad)).
+
+Instance Ari : Ring A := {
+  ring0 := @ring0 A (@domain_ring _ Ad);
+  ring1 := @ring1 A (@domain_ring _ Ad);
+  ring_plus := @ring_plus A (@domain_ring _ Ad);
+  ring_mult := @ring_mult A (@domain_ring _ Ad);
+  ring_sub := @ring_sub A (@domain_ring _ Ad);
+  ring_opp := @ring_opp A (@domain_ring _ Ad);
+  ring_ring := @ring_ring A (@domain_ring _ Ad)}.
+
+Instance Adi : Domain A := {
+  domain_ring := Ari;
+  domain_axiom_product := @domain_axiom_product A Ad;
+  domain_axiom_one_zero := @domain_axiom_one_zero A Ad}.
+
+Instance zero_ring2 : Zero A := {zero := ring0}.
+Instance one_ring2 : One A := {one := ring1}.
+Instance addition_ring2 : Addition A := {addition x y := ring_plus x y}.
+Instance multiplication_ring2 : Multiplication A := {multiplication x y := ring_mult x y}.
+Instance subtraction_ring2 : Subtraction A := {subtraction x y := ring_sub x y}.
+Instance opposite_ring2 : Opposite A := {opposite x := ring_opp x}.
+
+Goal forall x y:A,  x = y -> x+0 = y*1+0.
+nsatz_domain.
+Qed.
+
+Goal forall a b c:A, a = b -> b = c -> c = a.
+nsatz_domain.
+Qed.
+
+Goal forall a b c:A, a = b -> b = c -> a = c.
+nsatz_domain.
+Qed.
+
+Goal forall a b c x:A, a = b -> b = c -> a*a = c*c.
+nsatz_domain.
+Qed.
+
+Goal forall x y:Z,  x = y -> (x+0)%Z = (y*1+0)%Z.
+nsatz_domainZ.
+Qed.
+
+Goal forall x y:R,  x = y -> (x+0)%R = (y*1+0)%R.
+nsatz_domainR.
+Qed.
+
+Goal forall a b c x:R, a = b -> b = c -> (a*a)%R = (c*c)%R.
+nsatz_domainR.
+Qed.
+
+Section Examples.
+
+Delimit Scope PE_scope with PE.
+Infix "+" := PEadd : PE_scope.
+Infix "*" := PEmul : PE_scope.
+Infix "-" := PEsub : PE_scope.
+Infix "^" := PEpow : PE_scope.
+Notation "[ n ]" := (@PEc Z n) (at level 0).
+
+Open Scope R_scope.
+
+Lemma example1 : forall x y,
+  x+y=0 ->
+  x*y=0 ->
+  x^2=0.
+Proof.
+ nsatz_domainR.
+Qed.
+
+Lemma example2 : forall x, x^2=0 -> x=0.
+Proof.
+ nsatz_domainR.
+Qed.
+
+(*
+Notation X  := (PEX Z 3).
+Notation Y  := (PEX Z 2).
+Notation Z_ := (PEX Z 1).
+*)
+Lemma example3 : forall x y z,
+  x+y+z=0 ->
+  x*y+x*z+y*z=0->
+  x*y*z=0 -> x^3=0.
+Proof.
+Time nsatz_domainR.
+simpl.
+discrR.
+Qed.
+
+(*
+Notation X  := (PEX Z 4).
+Notation Y  := (PEX Z 3).
+Notation Z_ := (PEX Z 2).
+Notation U  := (PEX Z 1).
+*)
+Lemma example4 : forall x y z u,
+  x+y+z+u=0 ->
+  x*y+x*z+x*u+y*z+y*u+z*u=0->
+  x*y*z+x*y*u+x*z*u+y*z*u=0->
+  x*y*z*u=0 -> x^4=0.
+Proof.
+Time nsatz_domainR.
+Qed.
+
+(*
+Notation x_ := (PEX Z 5).
+Notation y_ := (PEX Z 4).
+Notation z_ := (PEX Z 3).
+Notation u_ := (PEX Z 2).
+Notation v_ := (PEX Z 1).
+Notation "x :: y" := (List.cons x y)
+(at level 60, right associativity, format "'[hv' x  ::  '/' y ']'").
+Notation "x :: y" := (List.app x y)
+(at level 60, right associativity, format "x :: y").
+*)
+
+Lemma example5 : forall x y z u v,
+  x+y+z+u+v=0 ->
+  x*y+x*z+x*u+x*v+y*z+y*u+y*v+z*u+z*v+u*v=0->
+  x*y*z+x*y*u+x*y*v+x*z*u+x*z*v+x*u*v+y*z*u+y*z*v+y*u*v+z*u*v=0->
+  x*y*z*u+y*z*u*v+z*u*v*x+u*v*x*y+v*x*y*z=0 ->
+  x*y*z*u*v=0 -> x^5=0.
+Proof.
+Time nsatz_domainR.
+Qed.
+
+End Examples.
+
+Section Geometry.
+
+Open Scope R_scope.
+
+Record point:Type:={
+ X:R;
+ Y:R}.
+
+Definition collinear(A B C:point):=
+  (X A - X B)*(Y C - Y B)-(Y A - Y B)*(X C - X B)=0.
+
+Definition parallel (A B C D:point):=
+  ((X A)-(X B))*((Y C)-(Y D))=((Y A)-(Y B))*((X C)-(X D)).
+
+Definition notparallel (A B C D:point)(x:R):=
+  x*(((X A)-(X B))*((Y C)-(Y D))-((Y A)-(Y B))*((X C)-(X D)))=1.
+
+Definition orthogonal (A B C D:point):=
+  ((X A)-(X B))*((X C)-(X D))+((Y A)-(Y B))*((Y C)-(Y D))=0.
+
+Definition equal2(A B:point):=
+  (X A)=(X B) /\ (Y A)=(Y B).
+
+Definition equal3(A B:point):=
+  ((X A)-(X B))^2+((Y A)-(Y B))^2 = 0.
+
+Definition nequal2(A B:point):=
+  (X A)<>(X B) \/ (Y A)<>(Y B).
+
+Definition nequal3(A B:point):=
+  not (((X A)-(X B))^2+((Y A)-(Y B))^2 = 0).
+
+Definition middle(A B I:point):=
+  2*(X I)=(X A)+(X B) /\ 2*(Y I)=(Y A)+(Y B).
+
+Definition distance2(A B:point):=
+  (X B - X A)^2 + (Y B - Y A)^2.
+
+(* AB = CD *)
+Definition samedistance2(A B C D:point):=
+  (X B - X A)^2 + (Y B - Y A)^2 = (X D - X C)^2 + (Y D - Y C)^2.
+Definition determinant(A O B:point):=
+  (X A - X O)*(Y B - Y O) - (Y A - Y O)*(X B - X O).
+Definition scalarproduct(A O B:point):=
+  (X A - X O)*(X B - X O) + (Y A - Y O)*(Y B - Y O).
+Definition norm2(A O B:point):=
+  ((X A - X O)^2+(Y A - Y O)^2)*((X B - X O)^2+(Y B - Y O)^2).
+
+
+Lemma a1:forall A B C:Prop, ((A\/B)/\(A\/C)) -> (A\/(B/\C)).
+intuition.
+Qed.
+
+Lemma a2:forall A B C:Prop, ((A\/C)/\(B\/C)) -> ((A/\B)\/C).
+intuition.
+Qed.
+
+Lemma a3:forall a b c d:R, (a-b)*(c-d)=0 -> (a=b \/ c=d).
+intros.
+assert ( (a-b = 0) \/ (c-d = 0)).
+apply Rmult_integral.
+trivial.
+destruct H0.
+left; nsatz_domainR.
+right; nsatz_domainR.
+Qed.
+
+Ltac geo_unfold :=
+  unfold collinear; unfold parallel; unfold notparallel; unfold orthogonal;
+  unfold equal2; unfold equal3; unfold nequal2; unfold nequal3;
+  unfold middle; unfold samedistance2;
+  unfold determinant; unfold scalarproduct; unfold norm2; unfold distance2.
+
+Ltac geo_end :=
+  repeat (
+  repeat (match goal with h:_/\_ |- _ => decompose [and] h; clear h end);
+  repeat (apply a1 || apply a2 || apply a3);
+  repeat split).
+
+Ltac geo_rewrite_hyps:=
+  repeat (match goal with
+           | h:X _ = _ |- _ => rewrite h in *; clear h
+           | h:Y _ = _ |- _ => rewrite h in *; clear h
+          end).
+
+Ltac geo_begin:=
+  geo_unfold;
+  intros;
+  geo_rewrite_hyps;
+  geo_end.
+
+(* Examples *)
+
+Lemma Thales: forall O A B C D:point,
+  collinear O A C -> collinear O B D ->
+  parallel A B C D ->
+  (distance2 O B * distance2 O C = distance2 O D * distance2 O A
+  /\ distance2 O B * distance2 C D = distance2 O D * distance2 A B)
+ \/ collinear O A B.
+repeat geo_begin.
+
+Time nsatz_domainR.
+simpl;discrR.
+Time nsatz_domainR.
+simpl;discrR.
+Qed.
+
+Require Import NsatzR.
+
+Lemma hauteurs:forall A B C A1 B1 C1 H:point,
+  collinear B C A1 ->  orthogonal A A1 B C ->
+  collinear A C B1 -> orthogonal B B1 A C ->
+  collinear A B C1 -> orthogonal C C1 A B ->
+  collinear A A1 H -> collinear B B1 H ->
+
+  collinear C C1 H
+  \/ collinear A B C.
+
+geo_begin.
+(* Time nsatzRpv 2%N 1%Z (@nil R) (@nil R).*)
+(*Finished transaction in 3. secs (2.363641u,0.s)*)
+(*Time nsatz_domainR. trop long! *)
+(* en fait nsatz_domain ne tient pas encore compte de la liste des variables! ;-) *)
+Time
+ let lv := constr:(Y A1
+ :: X A1
+    :: Y B1
+       :: X B1
+          :: Y A0
+             :: Y B
+                :: X B
+                   :: X A0
+                      :: X H
+                         :: Y C
+                            :: Y C1 :: Y H :: X C1 :: X C ::nil) in
+ nsatz_domainpv 2%N 1%Z (@List.nil R) lv ltac:simplR Rdi.
+(* Finished transaction in 6. secs (5.579152u,0.001s) *)
+Qed.
+
+End Geometry.