diff options
Diffstat (limited to 'test-suite/success/Nsatz_domain.v')
| -rw-r--r-- | test-suite/success/Nsatz_domain.v | 274 |
1 files changed, 0 insertions, 274 deletions
diff --git a/test-suite/success/Nsatz_domain.v b/test-suite/success/Nsatz_domain.v deleted file mode 100644 index 8a30b47f..00000000 --- a/test-suite/success/Nsatz_domain.v +++ /dev/null @@ -1,274 +0,0 @@ -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. |