Theorem: LO.FirstOrder.Arith.Model.length_two_mul_add_one (Arithmetization/ISigmaZero/Exponential/Log.lean)

State before:
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
a : M
⊢ ‖2 * a + 1‖ = ‖a‖ + 1
Tactic:
rcases zero_le a with (rfl | pos)
State after:
case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
⊢ ‖2 * 0 + 1‖ = ‖0‖ + 1

case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
a : M
pos : 0 < a
⊢ ‖2 * a + 1‖ = ‖a‖ + 1

State before:
case inl
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
⊢ ‖2 * 0 + 1‖ = ‖0‖ + 1
Tactic:
simp
State after:
no goals

State before:
case inr
M : Type u_1
inst✝⁵ : Zero M
inst✝⁴ : One M
inst✝³ : Add M
inst✝² : Mul M
inst✝¹ : LT M
inst✝ : M ⊧ₘ* 𝐈𝚫₀
a : M
pos : 0 < a
⊢ ‖2 * a + 1‖ = ‖a‖ + 1
Tactic:
simp [pos, length_of_pos, log_two_mul_add_one_of_pos]
State after:
no goals