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Proofgold Proof

pf
Let x0 of type ιο be given.
Let x1 of type ιιι be given.
Assume H0: ∀ x2 x3 . x0 x2x0 x3x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2x0 x3x0 x4x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Assume H8: x0 x8.
Apply unknownprop_f0b76402e77112232d36cbd146a3e3efd40fdf823d5b935fe896a6fb8918a817 with x0, x1, x6, x7, x3, x4, x5, x8, λ x9 x10 . x1 x2 x9 = x1 x6 (x1 x3 (x1 x2 (x1 x7 (x1 x5 (x1 x4 x8))))) leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
Apply H1 with x2, x6, x1 x7 (x1 x3 (x1 x4 (x1 x5 x8))), λ x9 x10 . x10 = x1 x6 (x1 x3 (x1 x2 (x1 x7 (x1 x5 (x1 x4 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply H0 with x7, x1 x3 (x1 x4 (x1 x5 x8)) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with x3, x1 x4 (x1 x5 x8) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x1 x5 x8 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x5, x8 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
set y9 to be x1 x6 (x1 x2 (x1 x7 (x1 x3 (x1 x4 (x1 x5 x8)))))
set y10 to be x2 x7 (x2 x4 (x2 x3 (x2 x8 (x2 x6 (x2 x5 y9)))))
Claim L9: ∀ x11 : ι → ο . x11 y10x11 y9
Let x11 of type ιο be given.
Assume H9: x11 (x3 x8 (x3 x5 (x3 x4 (x3 y9 (x3 x7 (x3 x6 y10)))))).
set y12 to be λ x12 . x11
Apply unknownprop_4bdb37ff43fa5d0c3ac040d851c0fbfd251035928263b085023cb6769e51270e with x2, x3, x4, y9, x5, x6, x7, y10, λ x13 x14 . y12 (x3 x8 x13) (x3 x8 x14) leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Let x11 of type ιιο be given.
Apply L9 with λ x12 . x11 x12 y10x11 y10 x12.
Assume H10: x11 y10 y10.
The subproof is completed by applying H10.