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diff --git a/test-suite/success/Nsatz.v b/test-suite/success/Nsatz.v new file mode 100644 index 00000000..fde9f470 --- /dev/null +++ b/test-suite/success/Nsatz.v @@ -0,0 +1,216 @@ +Require Import NsatzR ZArith Reals List Ring_polynom. + +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. + nsatzR. +Qed. + +Lemma example2 : forall x, x^2=0 -> x=0. +Proof. + nsatzR. +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 nsatzR. +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 nsatzR. +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 nsatzR. +Qed. + +End Examples. + +Section Geometry. +Require Export Reals NsatzR. +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. +right; nsatz. +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. +*) +Time nsatz without sugar. +(* +Time nsatz with lexico sugar. +Time nsatz with lexico. +*) +(* +Time nsatzRpv 1%N 1%Z (@nil R) (@nil R). (* revlex, sugar, no div *) +(*Finished transaction in 1. secs (0.479927u,0.s)*) +Time nsatzRpv 1%N 0%Z (@nil R) (@nil R). (* revlex, no sugar, no div *) +(*Finished transaction in 0. secs (0.543917u,0.s)*) +Time nsatzRpv 1%N 2%Z (@nil R) (@nil R). (* lex, no sugar, no div *) +(*Finished transaction in 0. secs (0.586911u,0.s)*) +Time nsatzRpv 1%N 3%Z (@nil R) (@nil R). (* lex, sugar, no div *) +(*Finished transaction in 0. secs (0.481927u,0.s)*) +Time nsatzRpv 1%N 5%Z (@nil R) (@nil R). (* revlex, sugar, div *) +(*Finished transaction in 1. secs (0.601909u,0.s)*) +*) +Time nsatz. +Qed. + +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 nsatz. +(*Finished transaction in 3. secs (2.43263u,0.010998s)*) +Qed. + +End Geometry. |