Theorem: LO.FirstOrder.Arith.Model.mem_seqProduct_bound (Arithmetization/ISigmaOne/HFS/Supplemental.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 a : M
h : x ∈ s ×ˢ a
⊢ x ≤ s + exp ((2 * s + a + 1) ^ 2)
Tactic:
rcases mem_seqProduct_iff.mp h with ⟨v, hv, u, hu, rfl⟩
State after:
case intro.intro.intro.intro
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 a v : M
hv : v ∈ s
u : M
hu : u ∈ a
h : v ⁀' u ∈ s ×ˢ a
⊢ v ⁀' u ≤ s + exp ((2 * s + a + 1) ^ 2)

State before:
case intro.intro.intro.intro
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 a v : M
hv : v ∈ s
u : M
hu : u ∈ a
h : v ⁀' u ∈ s ×ˢ a
⊢ v ⁀' u ≤ s + exp ((2 * s + a + 1) ^ 2)
Tactic:
exact seqCons_le (le_of_lt <| lt_of_mem hu) (le_of_lt <| lt_of_mem hv)
State after:
no goals