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(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
(* *)
(************************************************************************)
Require Import ZArith Zwf Psatz QArith.
Open Scope Z_scope.
Lemma Zabs_square : forall x, (Z.abs x)^2 = x^2.
Proof.
intros ; case (Zabs_dec x) ; intros ; psatz Z 2.
Qed.
Hint Resolve Z.abs_nonneg Zabs_square.
Lemma integer_statement : ~exists n, exists p, n^2 = 2*p^2 /\ n <> 0.
Proof.
intros [n [p [Heq Hnz]]]; pose (n' := Z.abs n); pose (p':=Z.abs p).
assert (facts : 0 <= Z.abs n /\ 0 <= Z.abs p /\ Z.abs n^2=n^2
/\ Z.abs p^2 = p^2) by auto.
assert (H : (0 < n' /\ 0 <= p' /\ n' ^2 = 2* p' ^2)) by
(destruct facts as [Hf1 [Hf2 [Hf3 Hf4]]]; unfold n', p' ; psatz Z 2).
generalize p' H; elim n' using (well_founded_ind (Zwf_well_founded 0)); clear.
intros n IHn p [Hn [Hp Heq]].
assert (Hzwf : Zwf 0 (2*p-n) n) by (unfold Zwf; psatz Z 2).
assert (Hdecr : 0 < 2*p-n /\ 0 <= n-p /\ (2*p-n)^2=2*(n-p)^2) by psatz Z 2.
apply (IHn (2*p-n) Hzwf (n-p) Hdecr).
Qed.
Open Scope Q_scope.
Lemma QnumZpower : forall x : Q, Qnum (x ^ 2)%Q = ((Qnum x) ^ 2) %Z.
Proof.
intros.
destruct x.
cbv beta iota zeta delta - [Z.mul].
ring.
Qed.
Lemma QdenZpower : forall x : Q, ' Qden (x ^ 2)%Q = ('(Qden x) ^ 2) %Z.
Proof.
intros.
destruct x.
simpl.
unfold Z.pow_pos.
simpl.
rewrite Pos.mul_1_r.
reflexivity.
Qed.
Theorem sqrt2_not_rational : ~exists x:Q, x^2==2#1.
Proof.
unfold Qeq; intros (x,HQeq); simpl (Qden (2#1)) in HQeq; rewrite Z.mul_1_r in HQeq.
assert (Heq : (Qnum x ^ 2 = 2 * ' Qden x ^ 2%Q)%Z) by
(rewrite QnumZpower in HQeq ; rewrite QdenZpower in HQeq ; auto).
assert (Hnx : (Qnum x <> 0)%Z)
by (intros Hx; simpl in HQeq; rewrite Hx in HQeq; discriminate HQeq).
apply integer_statement; exists (Qnum x); exists (' Qden x); auto.
Qed.
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