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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 H1 with x2, x3, x1 x4 (x1 x5 (x1 x6 (x1 x7 x8))), λ x9 x10 . x10 = x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x2 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with x4, x1 x5 (x1 x6 (x1 x7 x8)) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with x5, x1 x6 (x1 x7 x8) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with x6, x1 x7 x8 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with x7, x8 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
set y9 to be x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 x8)))))
set y10 to be x2 x4 (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x3 y9)))))
Claim L9: ∀ x11 : ι → ο . x11 y10x11 y9
Let x11 of type ιο be given.
Assume H9: x11 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 y9 (x3 x4 y10)))))).
set y12 to be λ x12 . x11
Apply unknownprop_515047c02fed97c50f69ea07f84c55a116d5435d48e441446058ba601add8797 with x2, x3, x4, x6, x7, x8, y9, y10, λ x13 x14 . y12 (x3 x5 x13) (x3 x5 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 H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
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.