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 → ι → ο be given.

Let x5 of type ι → ι → ο be given.

Let x6 of type ι → ο be given.

Let x7 of type ι → ο be given.

Assume H0: 0fd05.. x0 x2 x4 x6 = 0fd05.. x1 x3 x5 x7.

Claim L1: x1 = f482f.. (0fd05.. x0 x2 x4 x6) 4a7ef..

Apply unknownprop_fc5df4b2b872ac739616056ef633a9f68b600ee9303a8eac7aa9a299c0a677d8 with 0fd05.. x0 x2 x4 x6, x1, x3, x5, x7.

The subproof is completed by applying H0.

Claim L2: x0 = x1

Apply L1 with λ x8 x9 . x0 = x9.

The subproof is completed by applying unknownprop_b0f82d1c69f380550644ef11c2dd41f9d5d4f492a768abdd3d6e9adfd74b9c76 with x0, x2, x4, x6.

Apply and4I with x0 = x1, ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x2 x8 x9 = x3 x8 x9, ∀ x8 . prim1 x8 x0 ⟶ ∀ x9 . prim1 x9 x0 ⟶ x4 x8 x9 = x5 x8 x9, ∀ x8 . prim1 x8 x0 ⟶ x6 x8 = x7 x8 leaving 4 subgoals.

The subproof is completed by applying L2.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Let x9 of type ι be given.

Assume H4: prim1 x9 x0.

Apply unknownprop_fed6cf07aa814164a3daa084792cf9b08a4358b581f07c8eb65ac4ff36c71f64 with x0, x2, x4, x6, x8, x9, λ x10 x11 . x11 = x3 x8 x9 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x8 x1

Apply L2 with λ x10 x11 . prim1 x8 x10.

The subproof is completed by applying H3.

Claim L6: prim1 x9 x1

Apply L2 with λ x10 x11 . prim1 x9 x10.

The subproof is completed by applying H4.

Apply H0 with λ x10 x11 . e3162.. (f482f.. x11 (4ae4a.. 4a7ef..)) x8 x9 = x3 x8 x9.

Let x10 of type ι → ι → ο be given.

Apply unknownprop_fed6cf07aa814164a3daa084792cf9b08a4358b581f07c8eb65ac4ff36c71f64 with x1, x3, x5, x7, x8, x9, λ x11 x12 . x10 x12 x11 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Let x9 of type ι be given.

Assume H4: prim1 x9 x0.

Apply unknownprop_5ea683a46d4420d1129d98b68c2d8a411e60a35ab8618f29559598685876fec4 with x0, x2, x4, x6, x8, x9, λ x10 x11 : ο . x11 = x5 x8 x9 leaving 3 subgoals.

The subproof is completed by applying H3.

The subproof is completed by applying H4.

Claim L5: prim1 x8 x1

Apply L2 with λ x10 x11 . prim1 x8 x10.

The subproof is completed by applying H3.

Claim L6: prim1 x9 x1

Apply L2 with λ x10 x11 . prim1 x9 x10.

The subproof is completed by applying H4.

Apply H0 with λ x10 x11 . 2b2e3.. (f482f.. x11 (4ae4a.. (4ae4a.. 4a7ef..))) x8 x9 = x5 x8 x9.

Let x10 of type ο → ο → ο be given.

Apply unknownprop_5ea683a46d4420d1129d98b68c2d8a411e60a35ab8618f29559598685876fec4 with x1, x3, x5, x7, x8, x9, λ x11 x12 : ο . x10 x12 x11 leaving 2 subgoals.

The subproof is completed by applying L5.

The subproof is completed by applying L6.

Let x8 of type ι be given.

Assume H3: prim1 x8 x0.

Apply unknownprop_47bb3b8d7a71a1e8960090c3da1b97fe4880f40eb7b56b3243c5a95be7978365 with x0, x2, x4, x6, x8, λ x9 x10 : ο . x10 = x7 x8 leaving 2 subgoals.

The subproof is completed by applying H3.

Claim L4: prim1 x8 x1

Apply L2 with λ x9 x10 . prim1 x8 x9.

The subproof is completed by applying H3.

Apply H0 with λ x9 x10 . decode_p (f482f.. x10 (4ae4a.. (4ae4a.. (4ae4a.. 4a7ef..)))) x8 = x7 x8.

Let x9 of type ο → ο → ο be given.

Apply unknownprop_47bb3b8d7a71a1e8960090c3da1b97fe4880f40eb7b56b3243c5a95be7978365 with x1, x3, x5, x7, x8, λ x10 x11 : ο . x9 x11 x10.

The subproof is completed by applying L4.

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Proofgold Proof