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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.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with x0, x1, x5, x3, x4, x6, λ x7 x8 . x1 x2 x7 = x1 x5 (x1 x2 (x1 x4 (x1 x3 x6))) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
Apply H1 with x2, x5, x1 x3 (x1 x4 x6), λ x7 x8 . x8 = x1 x5 (x1 x2 (x1 x4 (x1 x3 x6))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply H0 with x3, x1 x4 x6 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with x4, x6 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
set y7 to be x1 x5 (x1 x2 (x1 x3 (x1 x4 x6)))
set y8 to be x2 x6 (x2 x3 (x2 x5 (x2 x4 y7)))
Claim L7: ∀ x9 : ι → ο . x9 y8x9 y7
Let x9 of type ιο be given.
Assume H7: x9 (x3 y7 (x3 x4 (x3 x6 (x3 x5 y8)))).
set y10 to be λ x10 . x9
set y11 to be x3 x4 (x3 x5 (x3 x6 y8))
set y12 to be x4 x5 (x4 y7 (x4 x6 x9))
Claim L8: ∀ x13 : ι → ο . x13 y12x13 y11
Let x13 of type ιο be given.
Assume H8: x13 (x5 x6 (x5 y8 (x5 y7 y10))).
set y14 to be λ x14 . x13
Apply H1 with y7, y8, y10, λ x15 x16 . y14 (x5 x6 x15) (x5 x6 x16) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
set y13 to be λ x13 x14 . y12 (x5 x9 x13) (x5 x9 x14)
Apply L8 with λ x14 . y13 x14 y12y13 y12 x14 leaving 2 subgoals.
Assume H9: y13 y12 y12.
The subproof is completed by applying H9.
The subproof is completed by applying L8.
Let x9 of type ιιο be given.
Apply L7 with λ x10 . x9 x10 y8x9 y8 x10.
Assume H8: x9 y8 y8.
The subproof is completed by applying H8.