Theorem: LO.FirstOrder.Arith.Model.Seq.lh_seqCons (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
x s : M
h : Seq s
⊢ lh (s ⁀' x) = lh s + 1
Tactic:
have : under (lh s + 1) = under (lh (s ⁀' x)) := by
  simpa [seqCons, h.domain_eq] using (h.seqCons x).domain_eq
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
x s : M
h : Seq s
this : under (lh s + 1) = under (lh (s ⁀' x))
⊢ lh (s ⁀' x) = lh s + 1

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
x s : M
h : Seq s
this : under (lh s + 1) = under (lh (s ⁀' x))
⊢ lh (s ⁀' x) = lh s + 1
Tactic:
exact Eq.symm <| 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
x s : M
h : Seq s
⊢ under (lh s + 1) = under (lh (s ⁀' x))
Tactic:
simpa [seqCons, h.domain_eq] using (h.seqCons x).domain_eq
State after:
no goals