Theorem: LO.FirstOrder.Arith.Model.log_mem_of_pos (Arithmetization/ISigmaOne/Bit.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
h : 0 < a
⊢ a < exp log a + exp log a
Tactic:
rw [←two_mul]
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* 𝐈𝚺₁
a : M
h : 0 < a
⊢ a < 2 * exp log a

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
h : 0 < a
⊢ a < 2 * exp log a
Tactic:
exact lt_two_mul_exponential_log 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* 𝐈𝚺₁
a : M
h : 0 < a
⊢ a = 0 * exp (log a + 1) + exp log a + (a - exp log a)
Tactic:
simp
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* 𝐈𝚺₁
a : M
h : 0 < a
⊢ a = exp log a + (a - exp log a)

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
h : 0 < a
⊢ a = exp log a + (a - exp log a)
Tactic:
exact Eq.symm <| add_tsub_self_of_le (exp_log_le_self h)
State after:
no goals