case zero
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
⊢ (Nat.zero + 1) • x = x

case succ
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
IH : (n + 1) • x = x
⊢ (Nat.succ n + 1) • x = x
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
⊢ (n + 1) • x = x
induction' n with n IH
no goals
case zero
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
⊢ (Nat.zero + 1) • x = x
simp
no goals
case succ
R✝ : Type u
inst✝² : LinearOrderedAddCommMonoidWithTop R✝
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : OrderTop R
x : Tropical R
n : ℕ
IH : (n + 1) • x = x
⊢ (Nat.succ n + 1) • x = x
rw [add_nsmul, IH, one_nsmul, add_self]