Let x0 of type ι be given.

Assume H0: b610d.. x0.

Apply H0 with λ x1 . x1 = c7d1f.. (f482f.. x1 4a7ef..) (f482f.. (f482f.. x1 (4ae4a.. 4a7ef..))) (2b2e3.. (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 → ι be given.

Assume H1: ∀ x3 . prim1 x3 x1 ⟶ prim1 (x2 x3) x1.

Let x3 of type ι → ι → ο be given.

Let x4 of type ι → ο be given.

Let x5 of type ι be given.

Assume H2: prim1 x5 x1.

Apply unknownprop_1b2e762fd8f25bb145cbb129ef3dfe053d045a0d61a7627c90cf9e56c2249738 with x1, x2, x3, x4, x5, λ x6 x7 . c7d1f.. x1 x2 x3 x4 x5 = c7d1f.. x6 (f482f.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))).

Apply unknownprop_a35ed9f7fda8ace7da0102061fd9347bffcb4f4a1c72e18d0b0708f16009da3f with x1, x2, x3, x4, x5, λ x6 x7 . c7d1f.. x1 x2 x3 x4 x5 = c7d1f.. x1 (f482f.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..))) (2b2e3.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..)))) (decode_p (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x6.

Apply unknownprop_06b85951e7d7d6f9193f0544cc788ac21b3d134cf9b29dcb87d3b3391548c84c with x1, x2, f482f.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. 4a7ef..)), x3, 2b2e3.. (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. 4a7ef..))), x4, decode_p (f482f.. (c7d1f.. x1 x2 x3 x4 x5) (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))), x5 leaving 3 subgoals.

The subproof is completed by applying unknownprop_8482b29062c8a6106f5dab65035d6d3e20a385ea1dc18096a6f4c6cce0ac93d5 with x1, x2, x3, x4, x5.

Let x6 of type ι be given.

Assume H3: prim1 x6 x1.

Let x7 of type ι be given.

Assume H4: prim1 x7 x1.

Apply unknownprop_12c6efd42213654930aac6fa5e2f9a09a7aaaea4c25796a7d4666d9e672b5dea 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 H3.

The subproof is completed by applying H4.

The subproof is completed by applying iff_refl with x3 x6 x7.

Let x6 of type ι be given.

Assume H3: prim1 x6 x1.

Apply unknownprop_a45abd536f0ab7db4999cd12ede789c19c9a8ae2b64f00ebaf91ff94aab6ce95 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