Goal:
    A : Type  P : list A → A → Type  Γ : list A  Δ : list A
  ============================
   (All_fold P (Γ ++ Δ)
    → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ)


DEBUG:Going to execute:
induction Γ in Δ |- *

Goals:
    A : Type  P : list A → A → Type  Δ : list A
  ============================
   (All_fold P ([] ++ Δ)
    → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) [])

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  ============================
   (All_fold P ((a :: Γ) ++ Δ)
    → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) (a :: Γ))


DEBUG:Going to execute:
split
DEBUG LTAC Evaluated term: Γ

Goals:
    A : Type  P : list A → A → Type  Δ : list A
  ============================
   (All_fold P ([] ++ Δ)
    → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) [])

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  ============================
   (All_fold P ((a :: Γ) ++ Δ)
    → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) (a :: Γ))


DEBUG:Going to execute:
fun => <coq-core.plugins.ltac::split@0>

Goals:
    A : Type  P : list A → A → Type  Δ : list A  X : All_fold P ([] ++ Δ)
  ============================
   (All_fold P Δ)

    A : Type  P : list A → A → Type  Δ : list A  X : All_fold P ([] ++ Δ)
  ============================
   (All_fold (λ Γ : list A, P (Γ ++ Δ)) [])

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  X : All_fold P ((a :: Γ) ++ Δ)
  ============================
   (All_fold P Δ)

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  X : All_fold P ((a :: Γ) ++ Δ)
  ============================
   (All_fold (λ Γ : list A, P (Γ ++ Δ)) (a :: Γ))


DEBUG:Going to execute:
auto

Goals:
    A : Type  P : list A → A → Type  Δ : list A  X : All_fold P ([] ++ Δ)
  ============================
   (All_fold P Δ)

    A : Type  P : list A → A → Type  Δ : list A  X : All_fold P ([] ++ Δ)
  ============================
   (All_fold (λ Γ : list A, P (Γ ++ Δ)) [])

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  X : All_fold P ((a :: Γ) ++ Δ)
  ============================
   (All_fold P Δ)

  
  A : Type
  P : list A → A → Type
  a : A
  Γ : list A
  Δ : list A
  IHΓ : ∀ Δ : list A,
          All_fold P (Γ ++ Δ)
          → All_fold P Δ × All_fold (λ Γ : list A, P (Γ ++ Δ)) Γ
  X : All_fold P ((a :: Γ) ++ Δ)
  ============================
   (All_fold (λ Γ : list A, P (Γ ++ Δ)) (a :: Γ))


DEBUG:Going to execute:
<coq-core.plugins.ltac::auto@0> $1 $2 $3