Theorem: LO.FirstOrder.Arith.Model.mem_bitInsert_iff (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
i j a : M
⊢ i ∈ insert j a ↔ i = j ∨ i ∈ a
Tactic:
by_cases h : j ∈ a <;> simp [h, insert_eq, bitInsert]
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* 𝐈𝚺₁
i j a : M
h : j ∈ a
⊢ i = j → i ∈ a

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* 𝐈𝚺₁
i j a : M
h : j ∉ a
⊢ i ∈ a + exp j ↔ i = j ∨ i ∈ a

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* 𝐈𝚺₁
i j a : M
h : j ∈ a
⊢ i = j → i ∈ a
Tactic:
by_cases e : i = j <;> simp [h, e]
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* 𝐈𝚺₁
i j a : M
h : j ∉ a
⊢ i ∈ a + exp j ↔ i = j ∨ i ∈ a
Tactic:
simpa [exp_inj.eq_iff] using
  lenbit_add_pow2_iff_of_not_lenbit (exp_pow2 i) (exp_pow2 j) h
State after:
no goals