Theorem: LO.FirstOrder.Arith.Model.Seq.mem_seqCons_iff_of_lt (Arithmetization/ISigmaOne/HFS/Seq.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 s x z : M
hi : i < lh s
⊢ ⟪i, x⟫ ∈ s ⁀' z ↔ ⟪i, x⟫ ∈ s
Tactic:
simp [seqCons, hi]
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
i s x z : M
hi : i < lh s
⊢ i = lh s → x = z → ⟪i, x⟫ ∈ s

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 s x z : M
hi : i < lh s
⊢ i = lh s → x = z → ⟪i, x⟫ ∈ s
Tactic:
rintro rfl
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
s x z : M
hi : lh s < lh s
⊢ x = z → ⟪lh s, x⟫ ∈ s

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
s x z : M
hi : lh s < lh s
⊢ x = z → ⟪lh s, x⟫ ∈ s
Tactic:
simp at hi
State after:
no goals