Let x0 of type ι be given.

Let x1 of type ι be given.

Let x2 of type (ι → ο) → ο be given.

Let x3 of type (ι → ο) → ο be given.

Let x4 of type ι → ι be given.

Let x5 of type ι → ι be given.

Let x6 of type ι → ο be given.

Let x7 of type ι → ο be given.

Let x8 of type ι → ο be given.

Let x9 of type ι → ο be given.

Assume H0: f4fb4.. x0 x2 x4 x6 x8 = f4fb4.. x1 x3 x5 x7 x9.

Claim L1: x1 = f482f.. (f4fb4.. x0 x2 x4 x6 x8) 4a7ef..

Apply unknownprop_8f7e1d767a4e108888a6217bd59b2f947ddbe938cdcd115f198c837383b75c99 with f4fb4.. ... ... ... ... ..., ..., ..., ..., ..., ....

...

Claim L2: x0 = x1

Apply L1 with λ x10 x11 . x0 = x11.

The subproof is completed by applying unknownprop_ae678ad878bbb814c7a7959a43e7a6938c60fbfc51ab3f6fdd661988d67e7f0a with x0, x2, x4, x6, x8.

Apply and5I with x0 = x1, ∀ x10 : ι → ο . (∀ x11 . x10 x11 ⟶ prim1 x11 x0) ⟶ x2 x10 = x3 x10, ∀ x10 . prim1 x10 x0 ⟶ x4 x10 = x5 x10, ∀ x10 . prim1 x10 x0 ⟶ x6 x10 = x7 x10, ∀ x10 . prim1 x10 x0 ⟶ x8 x10 = x9 x10 leaving 5 subgoals.

The subproof is completed by applying L2.

Let x10 of type ι → ο be given.

Assume H3: ∀ x11 . x10 x11 ⟶ prim1 x11 x0.

Apply unknownprop_e76823d36ee6db8ba833b34f42a40774cc3044020361dd9910febc7213bfae10 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x3 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: ∀ x11 . x10 x11 ⟶ prim1 x11 x1

Apply L2 with λ x11 x12 . ∀ x13 . x10 x13 ⟶ prim1 x13 x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_c (f482f.. x12 (4ae4a.. 4a7ef..)) x10 = x3 x10.

Let x11 of type ο → ο → ο be given.

Apply unknownprop_e76823d36ee6db8ba833b34f42a40774cc3044020361dd9910febc7213bfae10 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_5e1f761718d285321c4cb3dc55f0919bfa4a4030aa97d5da2bb3f8a34962fa42 with x0, x2, x4, x6, x8, x10, λ x11 x12 . x12 = x5 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . f482f.. (f482f.. x12 (4ae4a.. (4ae4a.. 4a7ef..))) x10 = x5 x10.

Let x11 of type ι → ι → ο be given.

Apply unknownprop_5e1f761718d285321c4cb3dc55f0919bfa4a4030aa97d5da2bb3f8a34962fa42 with x1, x3, x5, x7, x9, x10, λ x12 x13 . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_0aba2d0795a56205879344d672f508bd17da6911a38e8280a31351ebdd8657c2 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x7 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_p (f482f.. x12 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x10 = x7 x10.

Let x11 of type ο → ο → ο be given.

Apply unknownprop_0aba2d0795a56205879344d672f508bd17da6911a38e8280a31351ebdd8657c2 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

Let x10 of type ι be given.

Assume H3: prim1 x10 x0.

Apply unknownprop_1c902ef957d0386ad522c8de47ba741a02d9b45b3972360fdd4aa7b83a097539 with x0, x2, x4, x6, x8, x10, λ x11 x12 : ο . x12 = x9 x10 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x10 x1

Apply L2 with λ x11 x12 . prim1 x10 x11.

The subproof is completed by applying H3.

Apply H0 with λ x11 x12 . decode_p (f482f.. x12 (4ae4a.. (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..))))) x10 = x9 x10.

Let x11 of type ο → ο → ο be given.

Apply unknownprop_1c902ef957d0386ad522c8de47ba741a02d9b45b3972360fdd4aa7b83a097539 with x1, x3, x5, x7, x9, x10, λ x12 x13 : ο . x11 x13 x12.

The subproof is completed by applying L4.

■

Proofgold Proof