lemma zero_or_gt: ∀n. n = 0 ∨ 0 < n.
#n elim (lt_or_eq_or_gt 0 n) /2/
#H elim (lt_zero_false … H)
qed. 

lemma pippo: ∀x,y,z. x < z → y < z - x → x + y < z.
/3 width=2/

lemma arith_b1x: ∀e,x1,x2,y. y ≤ x2 → x2 ≤ x1 → 
                 e + (x1 - y) - (x2 - y) = e + (x1 - x2).
#e #x1 #x2 #y #H1 #H2
<(arith_b1 … H1) >le_plus_minus // /2 width=1/
qed-.
		 
lemma arith1: ∀x,y,z,w. z < y → x + (y + w) - z = x + y - z + w.
#x #y #z #w #H <le_plus_minus_comm // /3 width=1 by lt_to_le, le_plus_a/
qed-.

lemma lt_dec: ∀n2,n1. Decidable (n1 < n2).
#n2 elim n2 -n2
[ /4 width=3 by or_intror, absurd, nmk/
| #n2 #IHn2 * /2 width=1 by or_introl/
  #n1 elim (IHn2 n1) -IHn2
  /4 width=1 by le_S_S, monotonic_pred, or_intror, or_introl/
]
qed-.

lemma false_lt_to_le: ∀x,y. (x < y → ⊥) → y ≤ x.
#x #y #H elim (decidable_lt x y) /2 width=1 by not_lt_to_le/
#Hxy elim (H Hxy)
qed-.

lemma arith_m2 (x) (y): x < y → x+(y-↑x) = ↓y.
#x #y #Hxy >minus_minus [|*: // ] <minus_Sn_n //
qed-.