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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 (x1 x2 x3) x4.
Assume H2: ∀ x2 x3 . x0 x2x0 x3x1 x2 x3 = x1 x3 x2.
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 H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Claim L9: ∀ x8 x9 x10 . x0 x8x0 x9x0 x10x1 x8 (x1 x9 x10) = x1 x9 (x1 x8 x10)
Let x8 of type ι be given.
Let x9 of type ι be given.
Let x10 of type ι be given.
Assume H9: x0 x8.
Assume H10: x0 x9.
Assume H11: x0 x10.
Apply H1 with x9, x8, x10, λ x11 x12 . x1 x8 (x1 x9 x10) = x12 leaving 4 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H9.
The subproof is completed by applying H11.
Apply H2 with x8, x9, λ x11 x12 . x1 x8 (x1 x9 x10) = x1 x11 x10 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
Apply H1 with x8, x9, x10 leaving 3 subgoals.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply unknownprop_92e882ccf713d4f88ea99421a6e9680f28cf16aed50f757a8211b3720aa8bc60 with x0, x1, x2, x3, x4, x5, x6, x7 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying L9.
The subproof is completed by applying H2.
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.
The subproof is completed by applying H8.