Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι → ο be given.

Let x5 of type ι → ο be given.

Assume H0: 33a0d.. x0 x2 x4 = 33a0d.. x1 x3 x5.

Claim L1: x1 = f482f.. (33a0d.. x0 x2 x4) 4a7ef..

Apply unknownprop_bd647b72bbc4a85252654132c2274ee3e808ed27529cc4ab5374fb42261ed5fd with 33a0d.. 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 unknownprop_9fa32bd638e8c31d6b182a5633791e799b87de2a4ba474c8268d6be18d528c69 with x0, x2, x4.

Apply and3I with x0 = x1, ∀ x6 . prim1 x6 x0 ⟶ ∀ x7 . prim1 x7 x0 ⟶ x2 x6 x7 = x3 x6 x7, ∀ x6 . prim1 x6 x0 ⟶ x4 x6 = x5 x6 leaving 3 subgoals.

The subproof is completed by applying L2.

Let x6 of type ι be given.

Assume H3: prim1 x6 x0.

Let x7 of type ι be given.

Assume H4: prim1 x7 x0.

Apply unknownprop_4e0a023e0994003d0aeb1b6b3ffa0ab2be65253744ded6ca621dc0f96c465796 with x0, x2, x4, x6, x7, λ x8 x9 . x9 = x3 x6 x7 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x6 x1

Apply L2 with λ x8 x9 . prim1 x6 x8.

The subproof is completed by applying H3.

Claim L6: prim1 x7 x1

Apply L2 with λ x8 x9 . prim1 x7 x8.

The subproof is completed by applying H4.

Apply H0 with λ x8 x9 . e3162.. (f482f.. x9 (4ae4a.. 4a7ef..)) x6 x7 = x3 x6 x7.

Let x8 of type ι → ι → ο be given.

Apply unknownprop_4e0a023e0994003d0aeb1b6b3ffa0ab2be65253744ded6ca621dc0f96c465796 with x1, x3, x5, x6, x7, λ x9 x10 . x8 x10 x9 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x6 of type ι be given.

Assume H3: prim1 x6 x0.

Apply unknownprop_fd2efe54fe67f3ab31cabede3eaf251effecfe7cd1887072248950b21a2f3196 with x0, x2, x4, x6, λ x7 x8 : ο . x8 = x5 x6 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x6 x1

Apply L2 with λ x7 x8 . prim1 x6 x7.

The subproof is completed by applying H3.

Apply H0 with λ x7 x8 . decode_p (f482f.. x8 (4ae4a.. (4ae4a.. 4a7ef..))) x6 = x5 x6.

Let x7 of type ο → ο → ο be given.

Apply unknownprop_fd2efe54fe67f3ab31cabede3eaf251effecfe7cd1887072248950b21a2f3196 with x1, x3, x5, x6, λ x8 x9 : ο . x7 x9 x8.

The subproof is completed by applying L4.

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Proofgold Proof