Require Import ssreflect.
Require Import Arith.

Goal forall a b:nat,
  a - b < 0 -> a < b.
Proof.
  elim => [|a'].
  - case => [|b']; first by [].
    rewrite //= => H.
    by have HH := PeanoNat.Nat.lt_irrefl 0.
  - move => IH.
    case => [|b']; first by [].
    rewrite //= => H.
    have HH := IH b' H.
    exact: iffLR (PeanoNat.Nat.succ_lt_mono a' b') HH.
Qed.


Goal forall n:nat,
  S n - 1 = n.
Proof.
  move => n.
  by rewrite PeanoNat.Nat.sub_succ_r.
  (* by rewrite (PeanoNat.Nat.sub_succ_r (S n) 0). *)
Qed.
  

Goal forall n:nat,
  S (S n) < 2 -> False.
Proof.
  elim.
  - exact: PeanoNat.Nat.lt_irrefl 2.
  - move => n IH H.
    apply: IH.
    have HH := PeanoNat.Nat.lt_succ_l (S (S n)) 2.
    exact: HH H.
Qed.