Theorem: LO.FirstOrder.Arith.Model.lh_empty (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
⊢ lh ∅ = 0
Tactic:
have : under (lh ∅ : M) = under 0 := by simpa using Eq.symm <| Seq.domain_eq (M := M) (s := ∅) (by 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
this : under (lh ∅) = under 0
⊢ lh ∅ = 0

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
this : under (lh ∅) = under 0
⊢ lh ∅ = 0
Tactic:
exact under_inj.mp this
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
⊢ under (lh ∅) = under 0
Tactic:
simpa using Eq.symm <| Seq.domain_eq (M := M) (s := ∅) (by simp)
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
⊢ Seq ∅
Tactic:
simp
State after:
no goals