Theorem: LO.FirstOrder.Arith.Model.pi₁_le_self (Arithmetization/Basic/IOpen.lean)

State before:
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : M ⊧ₘ* 𝐏𝐀⁻
inst✝ : M ⊧ₘ* 𝐈open
a : M
⊢ π₁ a ≤ a
Tactic:
simp [pi₁, unpair]
State after:
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : M ⊧ₘ* 𝐏𝐀⁻
inst✝ : M ⊧ₘ* 𝐈open
a : M
⊢ (if a - √a * √a < √a then (a - √a * √a, √a) else (√a, a - √a * √a - √a)).1 ≤ a

State before:
M : Type u_1
inst✝⁶ : Zero M
inst✝⁵ : One M
inst✝⁴ : Add M
inst✝³ : Mul M
inst✝² : LT M
inst✝¹ : M ⊧ₘ* 𝐏𝐀⁻
inst✝ : M ⊧ₘ* 𝐈open
a : M
⊢ (if a - √a * √a < √a then (a - √a * √a, √a) else (√a, a - √a * √a - √a)).1 ≤ a
Tactic:
split_ifs <;> simp
State after:
no goals