Theorem: LO.FirstOrder.Arith.Model.sUnion_defined (Arithmetization/ISigmaOne/HFS/Basic.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
⊢ 𝚺₀-Function₁ fun x => ⋃ʰᶠ x via sUnionDef
Tactic:
intro v
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
v : Fin (Nat.succ 1) → M
⊢ (fun v => v 0 = (fun v => (fun x => ⋃ʰᶠ x) (v 0)) fun x => v x.succ) v ↔
    (Semiformula.Evalbm M v) (HSemiformula.val sUnionDef)

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
v : Fin (Nat.succ 1) → M
⊢ (fun v => v 0 = (fun v => (fun x => ⋃ʰᶠ x) (v 0)) fun x => v x.succ) v ↔
    (Semiformula.Evalbm M v) (HSemiformula.val sUnionDef)
Tactic:
simp [sUnionDef, sUnion_graph]
State after:
no goals