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).
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.
Let x11 of type ι be given.
Let x12 of type ι be given.
Let x13 of type ι be given.
Let x14 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.
Assume H9: x0 x9.
Assume H10: x0 x10.
Assume H11: x0 x11.
Assume H12: x0 x12.
Assume H13: x0 x13.
Assume H14: x0 x14.
Apply H1 with
x2,
x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 x13))))))))),
x14,
λ x15 x16 . x16 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 x14))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_3110297454f8d445696fa1ff6d16fb16e6639939f8e803a0cc7ed7a2132c96cf with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13 leaving 12 subgoals.
The subproof is completed by applying H0.
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.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
set y15 to be x1 x2 (x1 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 ... ...)))))))))) ...)
set y16 to be x2 x3 (x2 x4 (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 (x2 x10 (x2 x11 (x2 x12 (x2 x13 (x2 x14 y15)))))))))))
Claim L15: ∀ x17 : ι → ο . x17 y16 ⟶ x17 y15
Let x17 of type ι → ο be given.
Assume H15: x17 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 x13 (x3 x14 (x3 y15 y16)))))))))))).
set y18 to be λ x18 . x17
Apply unknownprop_4d8682f84af9c39bce68aeddde79980ed18f2bc7d072be75d3440e1ad037c2b8 with
x2,
x3,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
y15,
y16,
λ x19 x20 . y18 (x3 x4 x19) (x3 x4 x20) leaving 15 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.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
Let x17 of type ι → ι → ο be given.
Apply L15 with
λ x18 . x17 x18 y16 ⟶ x17 y16 x18.
Assume H16: x17 y16 y16.
The subproof is completed by applying H16.