Theorem: LO.FirstOrder.Arith.Model.bexp_exists_unique (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 x : M
⊢ ∃! y, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)
Tactic:
by_cases hx : x < ‖a‖
State after:
case pos
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 x : M
hx : x < ‖a‖
⊢ ∃! y, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)

case neg
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 x : M
hx : ¬x < ‖a‖
⊢ ∃! y, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)

State before:
case pos
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 x : M
hx : x < ‖a‖
⊢ ∃! y, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)
Tactic:
simp [hx, show ¬‖a‖ ≤ x from by simpa using hx, log_exists_unique_pos]
State after:
case pos
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 x : M
hx : x < ‖a‖
⊢ ∃! y, Exponential x y

State before:
case pos
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 x : M
hx : x < ‖a‖
⊢ ∃! y, Exponential x y
Tactic:
exact brange_exists_unique a x hx
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 x : M
hx : x < ‖a‖
⊢ ¬‖a‖ ≤ x
Tactic:
simpa using hx
State after:
no goals

State before:
case neg
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 x : M
hx : ¬x < ‖a‖
⊢ ∃! y, (x < ‖a‖ → Exponential x y) ∧ (‖a‖ ≤ x → y = 0)
Tactic:
simp [hx, show ‖a‖ ≤ x from by simpa using hx]
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 x : M
hx : ¬x < ‖a‖
⊢ ‖a‖ ≤ x
Tactic:
simpa using hx
State after:
no goals