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Proofgold Proof

pf
Let x0 of type ι(ιιι) → (ιιι) → (ιι) → (ιιο) → ι be given.
Let x1 of type ι be given.
Let x2 of type ιιι be given.
Let x3 of type ιιι be given.
Let x4 of type ιι be given.
Let x5 of type ιιο be given.
Assume H0: ∀ x6 : ι → ι → ι . (∀ x7 . x7x1∀ x8 . x8x1x2 x7 x8 = x6 x7 x8)∀ x7 : ι → ι → ι . (∀ x8 . x8x1∀ x9 . x9x1x3 x8 x9 = x7 x8 x9)∀ x8 : ι → ι . (∀ x9 . x9x1x4 x9 = x8 x9)∀ x9 : ι → ι → ο . (∀ x10 . x10x1∀ x11 . x11x1iff (x5 x10 x11) (x9 x10 x11))x0 x1 x6 x7 x8 x9 = x0 x1 x2 x3 x4 x5.
Apply pack_b_b_u_r_0_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . x0 x6 (decode_b (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 1)) (decode_b (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 2)) (ap (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 3)) (decode_r (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 4)) = x0 x1 x2 x3 x4 x5.
Apply H0 with decode_b (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 1), decode_b (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 2), ap (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 3), decode_r (ap (pack_b_b_u_r x1 x2 x3 x4 x5) 4) leaving 4 subgoals.
The subproof is completed by applying pack_b_b_u_r_1_eq2 with x1, x2, x3, x4, x5.
The subproof is completed by applying pack_b_b_u_r_2_eq2 with x1, x2, x3, x4, x5.
The subproof is completed by applying pack_b_b_u_r_3_eq2 with x1, x2, x3, x4, x5.
Let x6 of type ι be given.
Assume H1: x6x1.
Let x7 of type ι be given.
Assume H2: x7x1.
Apply pack_b_b_u_r_4_eq2 with x1, x2, x3, x4, x5, x6, x7, λ x8 x9 : ο . iff (x5 x6 x7) x8 leaving 3 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying iff_refl with x5 x6 x7.