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Proofgold Proof

pf
Let x0 of type ι((ιο) → ο) → (ιιο) → ο be given.
Let x1 of type ι be given.
Let x2 of type (ιο) → ο be given.
Let x3 of type ιιο be given.
Assume H0: ∀ x4 : (ι → ο) → ο . (∀ x5 : ι → ο . (∀ x6 . x5 x6x6x1)iff (x2 x5) (x4 x5))∀ x5 : ι → ι → ο . (∀ x6 . x6x1∀ x7 . x7x1iff (x3 x6 x7) (x5 x6 x7))x0 x1 x4 x5 = x0 x1 x2 x3.
Apply pack_c_r_0_eq2 with x1, x2, x3, λ x4 x5 . x0 x4 (decode_c (ap (pack_c_r x1 x2 x3) 1)) (decode_r (ap (pack_c_r x1 x2 x3) 2)) = x0 x1 x2 x3.
Apply H0 with decode_c (ap (pack_c_r x1 x2 x3) 1), decode_r (ap (pack_c_r x1 x2 x3) 2) leaving 2 subgoals.
Let x4 of type ιο be given.
Assume H1: ∀ x5 . x4 x5x5x1.
Apply pack_c_r_1_eq2 with x1, x2, x3, x4, λ x5 x6 : ο . iff (x2 x4) x5 leaving 2 subgoals.
The subproof is completed by applying H1.
The subproof is completed by applying iff_refl with x2 x4.
Let x4 of type ι be given.
Assume H1: x4x1.
Let x5 of type ι be given.
Assume H2: x5x1.
Apply pack_c_r_2_eq2 with x1, x2, x3, x4, x5, λ x6 x7 : ο . iff (x3 x4 x5) x6 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 x4 x5.