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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.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply unknownprop_9551c74ff689713f6e29d6039e164b7f427808e036d50b6ddb9edf722f975820 with x0, x1, x3, x4, x5, x6, x7, λ x8 x9 . x1 x2 x9 = x1 x5 (x1 x4 (x1 x6 (x1 x2 (x1 x3 x7)))) leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H6.
The subproof is completed by applying H7.
Apply H1 with x2, x5, x1 x6 (x1 x3 (x1 x4 x7)), λ x8 x9 . x9 = x1 x5 (x1 x4 (x1 x6 (x1 x2 (x1 x3 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply H0 with x6, x1 x3 (x1 x4 x7) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with x3, x1 x4 x7 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x7 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
set y8 to be x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7))))
set y9 to be x2 x6 (x2 x5 (x2 x7 (x2 x3 (x2 x4 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9x10 y8
Let x10 of type ιο be given.
Assume H8: x10 (x3 x7 (x3 x6 (x3 y8 (x3 x4 (x3 x5 y9))))).
set y11 to be λ x11 . x10
Apply unknownprop_01fcd6886950395663334639e7fa0ba571c79753a62fd50292d889e3bc03b9f3 with x2, x3, x4, y8, x5, x6, y9, λ x12 x13 . y11 (x3 x7 x12) (x3 x7 x13) leaving 8 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 H6.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Let x10 of type ιιο be given.
Apply L8 with λ x11 . x10 x11 y9x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.