Let x0 of type ι be given.

Assume H0: 0670d.. x0.

Apply H0 with λ x1 . x1 = 2e40c.. (f482f.. x1 4a7ef..) (decode_c (f482f.. x1 (4ae4a.. 4a7ef..))) (f482f.. (f482f.. x1 (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. x1 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. x1 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Let x1 of type ι be given.

Let x2 of type (ι → ο) → ο be given.

Let x3 of type ι → ι be given.

Assume H1: ∀ x4 . prim1 x4 x1 ⟶ prim1 (x3 x4) x1.

Let x4 of type ι → ο be given.

Let x5 of type ι be given.

Assume H2: prim1 x5 x1.

Apply unknownprop_5091b23ace185de1439c9993ecf5d34c3889f1adbf8fc4314bc1ac1fed4972cf with x1, x2, x3, x4, x5, λ x6 x7 . 2e40c.. x1 x2 x3 x4 x5 = 2e40c.. x6 (decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Apply unknownprop_3ab7806cf0ff3d8bee638a9c7ebd5be410ba51ca429236345648ab7856c58a31 with x1, x2, x3, x4, x5, λ x6 x7 . 2e40c.. x1 x2 x3 x4 x5 = 2e40c.. x1 (decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6.

Apply unknownprop_9c2e2050d75651e529e614d48de9f4a1525a704bd77c968b514b12a36a37d271 with x1, x2, decode_c (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)), x3, f482f.. (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))), x4, decode_p (f482f.. (2e40c.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))), x5 leaving 3 subgoals.

Let x6 of type ι → ο be given.

Assume H3: ∀ x7 . x6 x7 ⟶ prim1 x7 x1.

Apply unknownprop_a13b376ead7440cfb8068d51b0fc26b8bee9b2fd7f0c4983ba28169597604378 with x1, x2, x3, x4, x5, x6, λ x7 x8 : ο . iff (x2 x6) x7 leaving 2 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying iff_refl with x2 x6.

The subproof is completed by applying unknownprop_08751a2182e392d896e3e3f4a36e5bf5e3e32f0eec6892705c231ffefc7cc6f0 with x1, x2, x3, x4, x5.

Let x6 of type ι be given.

Assume H3: prim1 x6 x1.

Apply unknownprop_04e910d1de9e4552e9264adbcc682f73b876009ed78e0a9d5f152658b15ea3ba with x1, x2, x3, x4, x5, x6, λ x7 x8 : ο . iff (x4 x6) x7 leaving 2 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying iff_refl with x4 x6.

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Proofgold Proof