Theorem: LO.FirstOrder.Arith.Model.LenBit.le (Arithmetization/ISigmaZero/Exponential/Pow2.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* 𝐈open
i a : M
h : LenBit i a
⊢ i ≤ a
Tactic:
by_contra A
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* 𝐈open
i a : M
h : LenBit i a
A : ¬i ≤ a
⊢ False

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* 𝐈open
i a : M
h : LenBit i a
A : ¬i ≤ a
⊢ False
Tactic:
simp [LenBit, show a < i from by simpa using A] at h
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* 𝐈open
i a : M
h : LenBit i a
A : ¬i ≤ a
⊢ a < i
Tactic:
simpa using A
State after:
no goals