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.
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.
Apply H1 with
x2,
x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))))))),
x12,
λ x13 x14 . x14 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_e3761daa583c62dd19e21ac5ecc36db62b544996e686d7070ae47678e3d87642 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11 leaving 10 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.
set y13 to be x1 x2 (x1 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11)))))))) x12)
set y14 to be x2 x3 (x2 x4 (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 (x2 x10 (x2 x11 (x2 x12 y13)))))))))
Claim L13: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H13: x15 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 y13 y14)))))))))).
set y16 to be λ x16 . x15
Apply unknownprop_f9821c5bf17fe69a7bbbaf6eeae5ac13f7453dcebf12a7c80cb773df0edf4ab9 with
x2,
x3,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
y13,
y14,
λ x17 x18 . y16 (x3 x4 x17) (x3 x4 x18) leaving 13 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.
Let x15 of type ι → ι → ο be given.
Apply L13 with
λ x16 . x15 x16 y14 ⟶ x15 y14 x16.
Assume H14: x15 y14 y14.
The subproof is completed by applying H14.