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Proofgold Proof

pf
Let x0 of type ι(ιι) → (ιιο) → (ιο) → ιο be given.
Let x1 of type ι be given.
Let x2 of type ιι be given.
Let x3 of type ιιο be given.
Let x4 of type ιο be given.
Let x5 of type ι be given.
Assume H0: ∀ x6 : ι → ι . (∀ x7 . x7x1x2 x7 = x6 x7)∀ x7 : ι → ι → ο . (∀ x8 . x8x1∀ x9 . x9x1iff (x3 x8 x9) (x7 x8 x9))∀ x8 : ι → ο . (∀ x9 . x9x1iff (x4 x9) (x8 x9))x0 x1 x6 x7 x8 x5 = x0 x1 x2 x3 x4 x5.
Apply pack_u_r_p_e_0_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . x0 x6 (ap (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 1)) (decode_r (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 2)) (decode_p (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 3)) (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 4) = x0 x1 x2 x3 x4 x5.
Apply pack_u_r_p_e_4_eq2 with x1, x2, x3, x4, x5, λ x6 x7 . x0 x1 (ap (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 1)) (decode_r (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 2)) (decode_p (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 3)) x6 = x0 x1 x2 x3 x4 x5.
Apply H0 with ap (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 1), decode_r (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 2), decode_p (ap (pack_u_r_p_e x1 x2 x3 x4 x5) 3) leaving 3 subgoals.
The subproof is completed by applying pack_u_r_p_e_1_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_u_r_p_e_2_eq2 with x1, x2, x3, x4, x5, x6, x7, λ x8 x9 : ο . iff (x3 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 x3 x6 x7.
Let x6 of type ι be given.
Assume H1: x6x1.
Apply pack_u_r_p_e_3_eq2 with x1, x2, x3, x4, x5, x6, λ x7 x8 : ο . iff (x4 x6) x7 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x4 x6.