Goal:
    A : Type
  ============================
   (∀ (l : list A) (l0 : ∀ x : A, In x l → bool) (H := CoreTactics.block) 
      (f : A → bool) (l1 : list A) (g : ∀ x : A, In x l1 → bool),
      g = (λ (x : A) (_ : In x l1), f x)
      → {| pr1 := l; pr2 := l0 |} = {| pr1 := l1; pr2 := g |}
        → forallb_InP l l0 = forallb_InP l1 g
          → let H0 := CoreTactics.block in forallb_InP l1 g = forallb f l1)


DEBUG:Going to execute:
<coq-equations.plugin::refine_ho@0> $1
DEBUG LTAC Evaluated term: (fun_elim ?P ?f ?f0)

Goals:
    A : Type
  ============================
   (∀ H : ∀ x : A, In x [] → bool,
      (λ (l : list A) (H0 : ∀ x : A, In x l → bool) (x : bool),
         ∀ (H1 := CoreTactics.block) (f : A → bool) 
           (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
           g = (λ (x0 : A) (_ : In x0 l0), f x0)
           → {| pr1 := l; pr2 := H0 |} = {| pr1 := l0; pr2 := g |}
             → x = forallb_InP l0 g
               → let H2 := CoreTactics.block in
                 forallb_InP l0 g = forallb f l0) [] H true)

    A : Type
  ============================
   (∀ (x : A) (xs : list A) (H : ∀ x0 : A, In x0 (x :: xs) → bool),
      (λ (l : list A) (H0 : ∀ x0 : A, In x0 l → bool) (x0 : bool),
         ∀ (H1 := CoreTactics.block) (f : A → bool) 
           (l0 : list A) (g : ∀ x1 : A, In x1 l0 → bool),
           g = (λ (x1 : A) (_ : In x1 l0), f x1)
           → {| pr1 := l; pr2 := H0 |} = {| pr1 := l0; pr2 := g |}
             → x0 = forallb_InP l0 g
               → let H2 := CoreTactics.block in
                 forallb_InP l0 g = forallb f l0) xs
        (λ (x0 : A) (inx : In x0 xs),
           H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))
        (forallb_InP xs
           (λ (x0 : A) (inx : In x0 xs),
              H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)))
      → (λ (l : list A) (H0 : ∀ x0 : A, In x0 l → bool) (x0 : bool),
           ∀ (H1 := CoreTactics.block) (f : A → bool) 
             (l0 : list A) (g : ∀ x1 : A, In x1 l0 → bool),
             g = (λ (x1 : A) (_ : In x1 l0), f x1)
             → {| pr1 := l; pr2 := H0 |} = {| pr1 := l0; pr2 := g |}
               → x0 = forallb_InP l0 g
                 → let H2 := CoreTactics.block in
                   forallb_InP l0 g = forallb f l0) 
          (x :: xs) H
          (H x (forallb_InP_obligations_obligation_1 x xs) &&
           forallb_InP xs
             (λ (x0 : A) (inx : In x0 xs),
                H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))))


DEBUG:Going to execute:
intros

Goals:
    A : Type  H : ∀ x : A, In x [] → bool
  ============================
   ((λ (l : list A) (H : ∀ x : A, In x l → bool) (x : bool),
       ∀ (H0 := CoreTactics.block) (f : A → bool) 
         (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
         g = (λ (x0 : A) (_ : In x0 l0), f x0)
         → {| pr1 := l; pr2 := H |} = {| pr1 := l0; pr2 := g |}
           → x = forallb_InP l0 g
             → let H1 := CoreTactics.block in forallb_InP l0 g = forallb f l0)
      [] H true)

  
  A : Type
  x : A
  xs : list A
  H : ∀ x0 : A, In x0 (x :: xs) → bool
  H0 : (λ (l : list A) (H : ∀ x : A, In x l → bool) (x : bool),
          ∀ (H0 := CoreTactics.block) (f : A → bool) 
            (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
            g = (λ (x0 : A) (_ : In x0 l0), f x0)
            → {| pr1 := l; pr2 := H |} = {| pr1 := l0; pr2 := g |}
              → x = forallb_InP l0 g
                → let H1 := CoreTactics.block in
                  forallb_InP l0 g = forallb f l0) xs
         (λ (x0 : A) (inx : In x0 xs),
            H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))
         (forallb_InP xs
            (λ (x0 : A) (inx : In x0 xs),
               H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)))
  ============================
   ((λ (l : list A) (H : ∀ x : A, In x l → bool) (x : bool),
       ∀ (H0 := CoreTactics.block) (f : A → bool) 
         (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
         g = (λ (x0 : A) (_ : In x0 l0), f x0)
         → {| pr1 := l; pr2 := H |} = {| pr1 := l0; pr2 := g |}
           → x = forallb_InP l0 g
             → let H1 := CoreTactics.block in forallb_InP l0 g = forallb f l0)
      (x :: xs) H
      (H x (forallb_InP_obligations_obligation_1 x xs) &&
       forallb_InP xs
         (λ (x0 : A) (inx : In x0 xs),
            H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))))


DEBUG:Going to execute:
cbv beta

Goals:
    A : Type  H : ∀ x : A, In x [] → bool
  ============================
   (∀ (H0 := CoreTactics.block) (f : A → bool) (l : list A) 
      (g : ∀ x : A, In x l → bool),
      g = (λ (x : A) (_ : In x l), f x)
      → {| pr1 := []; pr2 := H |} = {| pr1 := l; pr2 := g |}
        → true = forallb_InP l g
          → let H := CoreTactics.block in forallb_InP l g = forallb f l)

  
  A : Type
  x : A
  xs : list A
  H : ∀ x0 : A, In x0 (x :: xs) → bool
  H0 : (λ (l : list A) (H : ∀ x : A, In x l → bool) (x : bool),
          ∀ (H0 := CoreTactics.block) (f : A → bool) 
            (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
            g = (λ (x0 : A) (_ : In x0 l0), f x0)
            → {| pr1 := l; pr2 := H |} = {| pr1 := l0; pr2 := g |}
              → x = forallb_InP l0 g
                → let H1 := CoreTactics.block in
                  forallb_InP l0 g = forallb f l0) xs
         (λ (x0 : A) (inx : In x0 xs),
            H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))
         (forallb_InP xs
            (λ (x0 : A) (inx : In x0 xs),
               H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)))
  ============================
   (∀ (H0 := CoreTactics.block) (f : A → bool) (l : list A) 
      (g : ∀ x : A, In x l → bool),
      g = (λ (x : A) (_ : In x l), f x)
      → {| pr1 := x :: xs; pr2 := H |} = {| pr1 := l; pr2 := g |}
        → H x (forallb_InP_obligations_obligation_1 x xs) &&
          forallb_InP xs
            (λ (x0 : A) (inx : In x0 xs),
               H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)) =
          forallb_InP l g
          → let H := CoreTactics.block in forallb_InP l g = forallb f l)


DEBUG:Going to execute:
DepElim.simplify_dep_elim

Goals:
    A : Type  H : ∀ x : A, In x [] → bool
  ============================
   (∀ (H0 := CoreTactics.block) (f : A → bool) (l : list A) 
      (g : ∀ x : A, In x l → bool),
      g = (λ (x : A) (_ : In x l), f x)
      → {| pr1 := []; pr2 := H |} = {| pr1 := l; pr2 := g |}
        → true = forallb_InP l g
          → let H := CoreTactics.block in forallb_InP l g = forallb f l)

  
  A : Type
  x : A
  xs : list A
  H : ∀ x0 : A, In x0 (x :: xs) → bool
  H0 : (λ (l : list A) (H : ∀ x : A, In x l → bool) (x : bool),
          ∀ (H0 := CoreTactics.block) (f : A → bool) 
            (l0 : list A) (g : ∀ x0 : A, In x0 l0 → bool),
            g = (λ (x0 : A) (_ : In x0 l0), f x0)
            → {| pr1 := l; pr2 := H |} = {| pr1 := l0; pr2 := g |}
              → x = forallb_InP l0 g
                → let H1 := CoreTactics.block in
                  forallb_InP l0 g = forallb f l0) xs
         (λ (x0 : A) (inx : In x0 xs),
            H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx))
         (forallb_InP xs
            (λ (x0 : A) (inx : In x0 xs),
               H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)))
  ============================
   (∀ (H0 := CoreTactics.block) (f : A → bool) (l : list A) 
      (g : ∀ x : A, In x l → bool),
      g = (λ (x : A) (_ : In x l), f x)
      → {| pr1 := x :: xs; pr2 := H |} = {| pr1 := l; pr2 := g |}
        → H x (forallb_InP_obligations_obligation_1 x xs) &&
          forallb_InP xs
            (λ (x0 : A) (inx : In x0 xs),
               H x0 (forallb_InP_obligations_obligation_2 x xs x0 inx)) =
          forallb_InP l g
          → let H := CoreTactics.block in forallb_InP l g = forallb f l)


DEBUG:Going to execute:
repeat DepElim.simplify_one_dep_elim