Theorem: Nat.le_succ_of_le'' (Fpil/Chap10/Sec04.lean)

State before:
n m : Nat
⊢ n ≤ m → n ≤ succ m
Tactic:
intro h
State after:
n m : Nat
h : n ≤ m
⊢ n ≤ succ m

State before:
n m : Nat
h : n ≤ m
⊢ n ≤ succ m
Tactic:
induction h with
| refl => apply Nat.le.step; exact Nat.le.refl
| step _ ih => apply Nat.le.step; exact ih
State after:
no goals

State before:
case refl
n m : Nat
⊢ n ≤ succ n
Tactic:
apply Nat.le.step
State after:
case refl.a
n m : Nat
⊢ Nat.le n n

State before:
case refl.a
n m : Nat
⊢ Nat.le n n
Tactic:
exact Nat.le.refl
State after:
no goals

State before:
case step
n m m✝ : Nat
a✝ : Nat.le n m✝
ih : n ≤ succ m✝
⊢ n ≤ succ (succ m✝)
Tactic:
apply Nat.le.step
State after:
case step.a
n m m✝ : Nat
a✝ : Nat.le n m✝
ih : n ≤ succ m✝
⊢ Nat.le n (m✝ + 1)

State before:
case step.a
n m m✝ : Nat
a✝ : Nat.le n m✝
ih : n ≤ succ m✝
⊢ Nat.le n (m✝ + 1)
Tactic:
exact ih
State after:
no goals