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Require Import Nsatz ZArith Reals List Ring_polynom.
(* Example with a generic domain *)
Variable A: Type.
Variable Ad: Domain A.
Definition Ari : Ring A:= (@domain_ring A Ad).
Existing Instance Ari.
Existing Instance ring_setoid.
Existing Instance ring_plus_comp.
Existing Instance ring_mult_comp.
Existing Instance ring_sub_comp.
Existing Instance ring_opp_comp.
Add Ring Ar: (@ring_ring A (@domain_ring 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}.
Infix "==" := ring_eq (at level 70, no associativity).
Ltac nsatzA := simpl; unfold Ari; nsatz_domain.
Goal forall x y:A, x == y -> x+0 == y*1+0.
nsatzA.
Qed.
Lemma example3 : forall x y z,
x+y+z==0 ->
x*y+x*z+y*z==0->
x*y*z==0 -> x*x*x==0.
Proof.
Time nsatzA.
Admitted.
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*x*x*x==0.
Proof.
Time nsatzA.
Qed.
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*x*x*x*x ==0.
Proof.
Time nsatzA.
Qed.
Goal forall x y:Z, x = y -> (x+0)%Z = (y*1+0)%Z.
nsatz.
Qed.
Goal forall x y:R, x = y -> (x+0)%R = (y*1+0)%R.
nsatz.
Qed.
Goal forall a b c x:R, a = b -> b = c -> (a*a)%R = (c*c)%R.
nsatz.
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.
Qed.
Lemma example2 : forall x, x^2=0 -> x=0.
Proof.
nsatz.
Qed.
(*
Notation X := (PEX Z 3).
Notation Y := (PEX Z 2).
Notation Z_ := (PEX Z 1).
*)
Lemma example3b : 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.
Qed.
(*
Notation X := (PEX Z 4).
Notation Y := (PEX Z 3).
Notation Z_ := (PEX Z 2).
Notation U := (PEX Z 1).
*)
Lemma example4b : 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.
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 example5b : 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.
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.
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.
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 nsatzRpv 2%N 1%Z (@nil R) (@nil R).*)
(*Finished transaction in 3. secs (2.363641u,0.s)*)
(*Time nsatz_domainR. trop long! *)
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 :: (@Datatypes.nil R)) in
nsatz_domainpv ltac:pretacR 2%N 1%Z (@Datatypes.nil R) lv ltac:simplR Rdi;
discrR.
(* Finished transaction in 6. secs (5.579152u,0.001s) *)
Qed.
End Geometry.
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