Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Assume H0: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2 ⟶ x0 x3 ⟶ x0 x4 ⟶ x1 (x1 x2 x3) x4 = x1 x2 (x1 x3 x4).
Assume H2: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 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.
Let x8 of type ι be given.
Let x9 of type ι be given.
Let x10 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.
Assume H9: x0 x8.
Assume H10: x0 x9.
Assume H11: x0 x10.
Apply unknownprop_e4b074928db308f7714399b2450e738f50ff691adae6c3a8e91b656e119c69d9 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
λ x11 x12 . x11 = x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10))))))) leaving 12 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.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply unknownprop_e4b074928db308f7714399b2450e738f50ff691adae6c3a8e91b656e119c69d9 with
x0,
x1,
x3,
x2,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
λ x11 x12 . x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10)))))) = x11 leaving 12 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
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.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
set y11 to be x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10))))))
set y12 to be x2 (x2 x4 x3) (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 (x2 x10 y11))))))
Claim L12: ∀ x13 : ι → ο . x13 y12 ⟶ x13 y11
Let x13 of type ι → ο be given.
Assume H12: x13 (x3 (x3 x5 x4) (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 y11 y12))))))).
set y14 to be λ x14 . x13
Apply H2 with
x4,
x5,
λ x15 x16 . y14 (x3 x15 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 y11 y12))))))) (x3 x16 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 y11 y12))))))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H12.
Let x13 of type ι → ι → ο be given.
Apply L12 with
λ x14 . x13 x14 y12 ⟶ x13 y12 x14.
Assume H13: x13 y12 y12.
The subproof is completed by applying H13.