(* 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.