1 files changed, 0 insertions, 168 deletions
diff --git a/phox/root2.phx b/phox/root2.phx
deleted file mode 100644
index 19259993..00000000
--- a/phox/root2.phx
+++ /dev/null
@@ -1,168 +0,0 @@
-(* Example proof by Paul Roziere.  See http://www.cs.kun.nl/~freek/comparison/ *)
-
-Import nat.
-
-flag auto_lvl 1.
-
-theorem minimal.element /\X (\/n:N X n -> \/n:N (X n & /\p:N (X p -> n <= p))).
-intro 2.
-by_absurd.
-rewrite_hyp H0 demorganl.
-use /\n:N ~ X n.
-trivial.
-intros.
-elim well_founded.N with H1.
-intros.
-intro.
-apply H0 with H4.
-lefts G $& $\/.
-elim H3 with H5.
-elim not.lesseq.imply.less.N.
-save.
-
-theorem not_odd_and_even.N  /\x,y,z:N (~ (x = N2 * y & x = N1 + N2 * z)).
-intro 2.
-elim H.
-trivial.
-intros.
-intro.
-left H4.
-elim H2 with [case].
-trivial =H0 H4 H6.
-elim H1.
-axiom H3.
-axiom H6.
-intro.
-rmh H4.
-left H5.
-intros.
-rmh H5.
-left H4.
-axiom H4.
-intros.
-save.
-
-theorem sum_square.N /\x,y:N (x + y)^N2 = x^N2 + N2*x*y + y^N2.
-intros.
-intro.
-save.
-
-fact less.exp.N /\n,x,y:N( x <= y -> x^n <= y^n).
-intros.
-elim H.
-trivial.
-rewrite calcul.N.
-trivial.
-save.
-
-fact less_r.exp.N  /\n,x,y:N( x^n < y^n -> x < y).
-intros.
-elim lesseq.case1.N with y and x.
-apply less.exp.N with n and H3.
-elim lesseq.imply.not.greater.N with y^n and x^n ;; Try intros.
-save.
-
-fact less.ladd.N /\x,y:N (N0 < y -> x < x + y).
-intros.
-elim H.
-rewrite calcul.N.
-trivial.
-save.
-
-theorem n.square.pair /\n:N (\/p:N n^N2=N2*p -> \/q:N n=N2*q).
-intros.
-lefts H0 $\/ $&.
-apply odd_or_even.N with H.
-lefts G $\/ $& $or.
-trivial.
-prove  n^N2 = N1 + N2*(N2*y^N2+N2*y).
-rewrite H3 sum_square.N.
-from N1 + N2 * N2 * y + (N2 * y) ^ N2 = N1 + N4 * y + N2 * N2 * y ^ N2.
-intro.
-elim not_odd_and_even.N with N (n^N2).
-intros.
-select 3.
-intro.
-axiom H1.
-axiom G.
-axiom H0.
-intros.
-save.
-
-lem decrease /\m,n : N (m^ N2 = N2 * n^ N2 -> N0 < n -> n < m).
-intros.
-elim less_r.exp.N with N2 ;; Try intros.
-prove m^N2 = n^N2 + n^N2. axiom H1.
-elim less.ladd.N ;; Try intros.
-trivial  =H0 H2.
-save.
-
-lem sup_zero /\m,n : N (m^ N2 = N2 * n^ N2 -> N0 < m -> N0 < n).
-intros.
-elim neq.less_or_sup.N with N0 and n ;; Try intros.
-rewrite_hyp H1 H3 calcul.N.
-trivial.
-trivial.
-save.
-
-def  Q m = m > N0 & \/n:N (m^ N2 = N2 * n^ N2).
-
-lem dec /\m:N (Q m -> \/m':N (Q m' & m' < m)).
-intros.
-lefts H0 $Q $\/ $&.
-apply sup_zero with H2 and H0.
-intro.
-instance ?1 n.
-intros.
-intros.
-trivial.
-prove \/p:N (m ^ N2 = N2 * p).
-intro.
-instance ?2  n^N2.
-trivial.
-apply n.square.pair with G0.
-lefts G1 $& $\/.
-prove n ^N2 = N2 * q ^N2.
-rewrite_hyp H2 H4.
-prove N2 * N2 * q ^N2 = N2 * n ^ N2.
-from H2.
-left G1.
-trivial =.
-intro.
-intros.
-intros.
-intros.
-trivial =H3 G1.
-elim decrease.
-save.
-
-lem sq2_irrat /\m:N ~Q m.
-intros.
-intro.
-elim minimal.element with Q.
-intros $\/ $&.
-axiom H.
-axiom H0.
-lefts H1.
-elim dec with H2.
-lefts H4.
-apply H3 with H5.
-elim lesseq.imply.not.greater.N with n and m'.
-save.
-
-theorem square2_irrat /\m,n : N (m^ N2 = N2 * n^ N2 -> m = N0 & n = N0).
-intros.
-apply sq2_irrat  with H.
-elim H with [case].
-intro.
-intro.
-elim H0 with [case].
-intro.
-rewrite_hyp H1 H2 H4 calcul.N.
-left H1;; intros.
-prove Q m.
-intros $Q $\/ $& ;; Try axiom H1.
-trivial.
-elim G.
-save.
-