Let x0 of type ι → (ι → ι → ι) → (ι → ι → ι) → (ι → ι → ο) → ι → ι → ο be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι → ι → ο be given.

Let x5 of type ι be given.

Let x6 of type ι be given.

Assume H0: ∀ x7 : ι → ι → ι . (∀ x8 . x8 ∈ x1 ⟶ ∀ x9 . x9 ∈ x1 ⟶ x2 x8 x9 = x7 x8 x9) ⟶ ∀ x8 : ι → ι → ι . (∀ x9 . x9 ∈ x1 ⟶ ∀ x10 . x10 ∈ x1 ⟶ x3 x9 x10 = x8 x9 x10) ⟶ ∀ x9 : ι → ι → ο . (∀ x10 . x10 ∈ x1 ⟶ ∀ x11 . x11 ∈ x1 ⟶ iff (x4 x10 x11) (x9 x10 x11)) ⟶ x0 x1 x7 x8 x9 x5 x6 = x0 x1 x2 x3 x4 x5 x6.

Apply pack_b_b_r_e_e_0_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 . x0 x7 (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 1)) (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 2)) (decode_r (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 3)) (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 4) (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 5) = x0 x1 x2 x3 x4 x5 x6.

Apply pack_b_b_r_e_e_4_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 . x0 x1 (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 1)) (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 2)) (decode_r (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 3)) x7 (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 5) = x0 x1 x2 x3 x4 x5 x6.

Apply pack_b_b_r_e_e_5_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 . x0 x1 (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 1)) (decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 2)) (decode_r (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 3)) x5 x7 = x0 x1 x2 x3 x4 x5 x6.

Apply H0 with decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 1), decode_b (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 x6) 2), decode_r (ap (pack_b_b_r_e_e x1 x2 x3 x4 x5 ...) 3) leaving 3 subgoals.

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Proofgold Proof