-- Mathlib/.lake/packages/batteries/Batteries/Data/List/Lemmas.lean
theorem dropInfix?_go_eq_some_iff [BEq α] {i l acc p s : List α} :
    dropInfix?.go i l acc = some (p, s) ↔ ∃ p',
      p = acc.reverse ++ p' ∧
      -- `i` is an infix up to `==`
      (∃ i', l = p' ++ i' ++ s ∧ i' == i) ∧
        -- and there is no shorter prefix for which that is the case
        (∀ p'' i'' s'', l = p'' ++ i'' ++ s'' → i'' == i → p''.length ≥ p'.length)

Proof state:
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
  ∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
p'' i'' s'' : List α
h₃ : p' ++ (i' ++ s) = p'' ++ (i'' ++ s'')
h₄ : (i'' == i) = true
⊢ p'.length ≤ p''.length

Tactic:
rw [← append_assoc] at h₃

Result:
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
  ∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
p'' i'' s'' : List α
h₃ : p' ++ i' ++ s = p'' ++ (i'' ++ s'')
h₄ : (i'' == i) = true
⊢ p'.length ≤ p''.length

Proof state:
case h_2.h_1.mp.intro.intro.intro.intro.intro.inl.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
p' i' : List α
h₂✝ : (i' == i) = true
w : ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i'' s'' : List α
h : (i'' ++ s'').dropPrefix? i = none
h₂ : (i'' == i) = true
h₁ : i'' ++ s'' = a :: (p' ++ (i' ++ s))
this : (i'' ++ s'').dropPrefix? i = some s''
⊢ p'.length + 1 ≤ [].length

Tactic:
simp_all

Result:
no goals

Proof state:
case h_2.h_1.mpr.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
h : (a :: t).dropPrefix? i = none
p' : List α
w : ∀ (p'' i'' s'' : List α), a :: t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i' : List α
h₁ : a :: t = p' ++ i' ++ s
h₂ : (i' == i) = true
⊢ ∃ p'_1,
    acc.reverse ++ p' = (a :: acc).reverse ++ p'_1 ∧
      (∃ i', t = p'_1 ++ i' ++ s ∧ (i' == i) = true) ∧
        ∀ (p'' i'' s'' : List α), t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'_1.length

Tactic:
rw [cons_eq_append_iff] at h₁

Result:
case h_2.h_1.mpr.intro.intro.intro.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
h : (a :: t).dropPrefix? i = none
p' : List α
w : ∀ (p'' i'' s'' : List α), a :: t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
i' : List α
h₁ : p' ++ i' = [] ∧ s = a :: t ∨ ∃ a', p' ++ i' = a :: a' ∧ t = a' ++ s
h₂ : (i' == i) = true
⊢ ∃ p'_1,
    acc.reverse ++ p' = (a :: acc).reverse ++ p'_1 ∧
      (∃ i', t = p'_1 ++ i' ++ s ∧ (i' == i) = true) ∧
        ∀ (p'' i'' s'' : List α), t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'_1.length

Proof state:
case h_2.h_2.intro.intro
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
s' p' : List α
h : a :: t = p' ++ s'
w : (p' == i) = true
⊢ acc.reverse = p ∧ s' = s ↔
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ (i' ++ s) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ (i'' ++ s'') → (i'' == i) = true → p'.length ≤ p''.length

Tactic:
constructor

Result:
case h_2.h_2.intro.intro.mp
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
s' p' : List α
h : a :: t = p' ++ s'
w : (p' == i) = true
⊢ acc.reverse = p ∧ s' = s →
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ (i' ++ s) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ (i'' ++ s'') → (i'' == i) = true → p'.length ≤ p''.length

case h_2.h_2.intro.intro.mpr
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
s' p' : List α
h : a :: t = p' ++ s'
w : (p' == i) = true
⊢ (∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ (i' ++ s) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ (i'' ++ s'') → (i'' == i) = true → p'.length ≤ p''.length) →
    acc.reverse = p ∧ s' = s

Proof state:
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
  ∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
⊢ ∃ p'_1,
    acc.reverse ++ a :: p' = (a :: acc).reverse ++ p'_1 ∧
      (∃ i'_1, p' ++ i' ++ s = p'_1 ++ i'_1 ++ s ∧ (i'_1 == i) = true) ∧
        ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'_1.length

Tactic:
refine ⟨p', ?_⟩

Result:
case h_2.h_1.mpr.intro.intro.intro.intro.intro.inr.intro.intro.inr.intro.intro
α : Type u_1
inst✝ : BEq α
i acc s x✝² x✝¹ : List α
a : α
x✝ : Option (List α)
i' : List α
h₂ : (i' == i) = true
p' : List α
h : (a :: (p' ++ i' ++ s)).dropPrefix? i = none
w :
  ∀ (p'' i'' s'' : List α), a :: (p' ++ i' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ (a :: p').length
⊢ acc.reverse ++ a :: p' = (a :: acc).reverse ++ p' ∧
    (∃ i'_1, p' ++ i' ++ s = p' ++ i'_1 ++ s ∧ (i'_1 == i) = true) ∧
      ∀ (p'' i'' s'' : List α), p' ++ i' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

Proof state:
case h_2.h_1
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
h : (a :: t).dropPrefix? i = none
⊢ (∃ p',
      p = (a :: acc).reverse ++ p' ∧
        (∃ i', t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length) ↔
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

Tactic:
constructor

Result:
case h_2.h_1.mp
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
h : (a :: t).dropPrefix? i = none
⊢ (∃ p',
      p = (a :: acc).reverse ++ p' ∧
        (∃ i', t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length) →
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

case h_2.h_1.mpr
α : Type u_1
inst✝ : BEq α
i acc p s x✝² x✝¹ : List α
a : α
t : List α
x✝ : Option (List α)
h : (a :: t).dropPrefix? i = none
⊢ (∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', a :: t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), a :: t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length) →
    ∃ p',
      p = (a :: acc).reverse ++ p' ∧
        (∃ i', t = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), t = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length