Theorem: LO.FirstOrder.Arith.Model.lt_two_mul_exponential_log (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* 𝐈𝚺₁
a : M
pos : 0 < a
⊢ a < 2 * exp log a
Tactic:
rcases log_pos pos with ⟨_, _, H, _⟩
State after:
case intro.intro.intro
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* 𝐈𝚺₁
a : M
pos : 0 < a
w✝ : M
left✝ : w✝ ≤ a
H : Exponential (log a) w✝
right✝ : a < 2 * w✝
⊢ a < 2 * exp log a

State before:
case intro.intro.intro
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* 𝐈𝚺₁
a : M
pos : 0 < a
w✝ : M
left✝ : w✝ ≤ a
H : Exponential (log a) w✝
right✝ : a < 2 * w✝
⊢ a < 2 * exp log a
Tactic:
rcases H.uniq (exponential_exp (log a))
State after:
case intro.intro.intro.refl
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* 𝐈𝚺₁
a : M
pos : 0 < a
left✝ : exp log a ≤ a
H : Exponential (log a) (exp log a)
right✝ : a < 2 * exp log a
⊢ a < 2 * exp log a

State before:
case intro.intro.intro.refl
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* 𝐈𝚺₁
a : M
pos : 0 < a
left✝ : exp log a ≤ a
H : Exponential (log a) (exp log a)
right✝ : a < 2 * exp log a
⊢ a < 2 * exp log a
Tactic:
assumption
State after:
no goals