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Proofgold Proof

pf
Let x0 of type ι be given.
Let x1 of type ι be given.
Let x2 of type (ιο) → ο be given.
Let x3 of type (ιο) → ο be given.
Let x4 of type ιι be given.
Let x5 of type ιι be given.
Assume H0: pack_c_u x0 x2 x4 = pack_c_u x1 x3 x5.
Claim L1: x1 = ap (pack_c_u x0 x2 x4) 0
Apply pack_c_u_0_eq with pack_c_u x0 x2 x4, x1, x3, x5.
The subproof is completed by applying H0.
Claim L2: x0 = x1
Apply L1 with λ x6 x7 . x0 = x7.
The subproof is completed by applying pack_c_u_0_eq2 with x0, x2, x4.
Apply and3I with x0 = x1, ∀ x6 : ι → ο . (∀ x7 . x6 x7x7x0)x2 x6 = x3 x6, ∀ x6 . x6x0x4 x6 = x5 x6 leaving 3 subgoals.
The subproof is completed by applying L2.
Let x6 of type ιο be given.
Assume H3: ∀ x7 . x6 x7x7x0.
Apply pack_c_u_1_eq2 with x0, x2, x4, x6, λ x7 x8 : ο . x8 = x3 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Claim L4: ∀ x7 . x6 x7x7x1
Apply L2 with λ x7 x8 . ∀ x9 . x6 x9x9x7.
The subproof is completed by applying H3.
Apply H0 with λ x7 x8 . decode_c (ap x8 1) x6 = x3 x6.
Let x7 of type οοο be given.
Apply pack_c_u_1_eq2 with x1, x3, x5, x6, λ x8 x9 : ο . x7 x9 x8.
The subproof is completed by applying L4.
Let x6 of type ι be given.
Assume H3: x6x0.
Apply pack_c_u_2_eq2 with x0, x2, x4, x6, λ x7 x8 . x8 = x5 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Claim L4: x6x1
Apply L2 with λ x7 x8 . x6x7.
The subproof is completed by applying H3.
Apply H0 with λ x7 x8 . ap (ap x8 2) x6 = x5 x6.
Let x7 of type ιιο be given.
Apply pack_c_u_2_eq2 with x1, x3, x5, x6, λ x8 x9 . x7 x9 x8.
The subproof is completed by applying L4.