Theorem: LO.FirstOrder.Arith.Model.Nuon.Segment.le_add (Arithmetization/OmegaOne/Nuon.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* 𝐈𝚫₀ + 𝛀₁
L✝ A✝ U L A start intv nₛ nₑ : M
H : Segment U L A start intv nₛ nₑ
⊢ nₑ ≤ nₛ + intv
Tactic:
rcases H with ⟨S, _, hS, rfl, 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* 𝐈𝚫₀ + 𝛀₁
L✝ A✝ U L A start intv S : M
left✝ : S < U
hS : IsSegment L A start intv S
⊢ S{L}[intv] ≤ S{L}[0] + intv

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* 𝐈𝚫₀ + 𝛀₁
L✝ A✝ U L A start intv S : M
left✝ : S < U
hS : IsSegment L A start intv S
⊢ S{L}[intv] ≤ S{L}[0] + intv
Tactic:
exact hS.le_add intv (by rfl)
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* 𝐈𝚫₀ + 𝛀₁
L✝ A✝ U L A start intv S : M
left✝ : S < U
hS : IsSegment L A start intv S
⊢ intv ≤ intv
Tactic:
rfl
State after:
no goals