Goal:
  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  ls : list ((A × B) × C)
  ============================
   (All (λ '(a, b, c), P a b c) ls <~>
    All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) ls)


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

Goals:
  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  ls : list ((A × B) × C)
  ============================
   (All (λ '(a, b, c), P a b c) ls
    → All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) ls)

  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  ls : list ((A × B) × C)
  ============================
   (All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) ls
    → All (λ '(a, b, c), P a b c) ls)


DEBUG:Going to execute:
induction 1

Goals:
    A : Type  B : Type  C : Type  P : A → B → C → Type
  ============================
   (All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) [])

  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  x : (A × B) × C
  l : list ((A × B) × C)
  p : let '(a, b, c) := x in P a b c
  X : All (λ '(a, b, c), P a b c) l
  IHX : All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) l
  ============================
   (All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) (x :: l))

    A : Type  B : Type  C : Type  P : A → B → C → Type
  ============================
   (All (λ '(a, b, c), P a b c) [])

  
  A : Type
  B : Type
  C : Type
  P : A → B → C → Type
  x : (A × B) × C
  l : list ((A × B) × C)
  p : P x.1.1 x.1.2 x.2
  X : All (λ abc : (A × B) × C, P abc.1.1 abc.1.2 abc.2) l
  IHX : All (λ '(a, b, c), P a b c) l
  ============================
   (All (λ '(a, b, c), P a b c) (x :: l))


DEBUG:Going to execute:
constructor => //=