-- 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:
α : Type u_1
inst✝ : BEq α
i l acc p s : List α
⊢ (match l, acc with
      | [], acc => if i.isEmpty = true then some (acc.reverse, []) else none
      | a :: as, acc =>
        match (a :: as).dropPrefix? i with
        | none => dropInfix?.go i as (a :: acc)
        | some s => some (acc.reverse, s)) =
      some (p, s) ↔
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', l = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), l = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

Tactic:
split

Result:
case h_1
α : Type u_1
inst✝ : BEq α
i acc p s x✝¹ x✝ : List α
⊢ (if i.isEmpty = true then some (acc.reverse, []) else none) = some (p, s) ↔
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', [] = p' ++ i' ++ s ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), [] = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length

case h_2
α : Type u_1
inst✝ : BEq α
i acc p s x✝¹ x✝ : List α
a✝ : α
t✝ : List α
⊢ (match (a✝ :: t✝).dropPrefix? i with
      | none => dropInfix?.go i t✝ (a✝ :: acc)
      | some s => some (acc.reverse, s)) =
      some (p, 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

Proof state:
case h_2.h_2
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' : List α
h : (a :: t).dropPrefix? i = some s'
⊢ 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:
rw [dropPrefix?_eq_some_iff] at h

Result:
case h_2.h_2
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' : List α
h : ∃ p', a :: t = p' ++ s' ∧ (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

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

Tactic:
simp only [nil_beq_iff, isEmpty_eq_true] at h₂

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

Proof state:
case h_1
α : Type u_1
inst✝ : BEq α
i acc p s x✝¹ x✝ : List α
⊢ i = [] ∧ acc.reverse = p ∧ s = [] ↔
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', (p' = [] ∧ i' = [] ∧ s = []) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), p'' = [] → i'' = [] → s'' = [] → (i'' == i) = true → p'.length ≤ p''.length

Tactic:
constructor

Result:
case h_1.mp
α : Type u_1
inst✝ : BEq α
i acc p s x✝¹ x✝ : List α
⊢ i = [] ∧ acc.reverse = p ∧ s = [] →
    ∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', (p' = [] ∧ i' = [] ∧ s = []) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), p'' = [] → i'' = [] → s'' = [] → (i'' == i) = true → p'.length ≤ p''.length

case h_1.mpr
α : Type u_1
inst✝ : BEq α
i acc p s x✝¹ x✝ : List α
⊢ (∃ p',
      p = acc.reverse ++ p' ∧
        (∃ i', (p' = [] ∧ i' = [] ∧ s = []) ∧ (i' == i) = true) ∧
          ∀ (p'' i'' s'' : List α), p'' = [] → i'' = [] → s'' = [] → (i'' == i) = true → p'.length ≤ p''.length) →
    i = [] ∧ acc.reverse = p ∧ s = []

Proof state:
α : Type u_1
inst✝ : BEq α
i acc 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
⊢ a :: t = p' ++ s' ∧ (p' == i) = true

Tactic:
simp_all

Result:
no goals

Proof state:
case h_2.h_2.intro.intro.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)
s' p' : List α
h : a :: t = p' ++ s'
p'' i' : List α
w' : p'' = []
h₁ : p' ++ s' = i' ++ s
w : p'.length = i.length
h₂ : i'.length = i.length
⊢ p'.length = i'.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
h₂ : (i' == i) = true
h₁ : (p' = [] ∧ i' = []) ∧ s = a :: t ∨ ∃ a', p' ++ i' = a :: a' ∧ t = a' ++ s
⊢ ∃ 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:
obtain (⟨⟨rfl, rfl⟩, rfl⟩ | ⟨a', h₁, rfl⟩) := h₁

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

case h_2.h_1.mpr.intro.intro.intro.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)
p' i' : List α
h₂ : (i' == i) = true
a' : List α
h₁ : p' ++ i' = a :: a'
h : (a :: (a' ++ s)).dropPrefix? i = none
w : ∀ (p'' i'' s'' : List α), a :: (a' ++ s) = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'.length
⊢ ∃ p'_1,
    acc.reverse ++ p' = (a :: acc).reverse ++ p'_1 ∧
      (∃ i', a' ++ s = p'_1 ++ i' ++ s ∧ (i' == i) = true) ∧
        ∀ (p'' i'' s'' : List α), a' ++ s = p'' ++ i'' ++ s'' → (i'' == i) = true → p''.length ≥ p'_1.length

Proof state:
case h_2.h_2.intro.intro.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)
s' p' : List α
h : a :: t = p' ++ s'
w : (p' == i) = true
p'' i' : List α
h₂ : (i' == i) = true
w' : p'' = []
h₁ : a :: t = i' ++ s
⊢ s' = s

Tactic:
rw [h] at h₁

Result:
case h_2.h_2.intro.intro.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)
s' p' : List α
h : a :: t = p' ++ s'
w : (p' == i) = true
p'' i' : List α
h₂ : (i' == i) = true
w' : p'' = []
h₁ : p' ++ s' = i' ++ s
⊢ s' = s