Theorem: LO.FirstOrder.Arith.Model.le_log_of_lt_length (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 b : M
h : a < ‖b‖
⊢ a ≤ log b
Tactic:
have : 0 < b := pos_of_lt_length h
State after:
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 b : M
h : a < ‖b‖
this : 0 < b
⊢ a ≤ log b

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 b : M
h : a < ‖b‖
this : 0 < b
⊢ a ≤ log b
Tactic:
exact le_iff_lt_succ.mpr (by simpa [length_of_pos this] using h)
State after:
no goals

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 b : M
h : a < ‖b‖
this : 0 < b
⊢ a < log b + 1
Tactic:
simpa [length_of_pos this] using h
State after:
no goals