coq Getting started with coq Example of a proof by induction

Example

Here is a simple proof by induction.

Require Import Coq.Setoids.Setoid.
Require Import Coq.Arith.Lt.

(* A number is less than or equal to itself *)
Theorem aLTEa : forall a,
    a <= a.
    auto with arith. (* This follows by simple arithmetic *)
    Qed.

Theorem simplALTE : forall a b,
    S a <= S b <-> a <= b. (* If a <= b, then a + 1 <= b + 1 *)
Proof.
Admitted.

Theorem ltAlwaysLt: forall a b,
    a <= a + b.
Proof.
  intros. (* Introduce relevant variables *)
  induction a, b. (* Induction on every variable *)
  simpl. apply aLTEa. (* 0 <= 0 + S b *)
  rewrite -> plus_O_n. auto with arith. (* 0 <= S b *)
  rewrite <- plus_n_O. apply aLTEa. (* S a <= S a + 0 *)
  rewrite <- simplALTE in IHa. (* IHa: a <= a + S b. Goal: S a <= S a + S b. *)
  apply IHa. (* We rewrote the induction hypothesis to be in the same form as the goal, so it applies immediately now *)
  Qed.