Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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.
Assume H2: x0 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Assume H6: x0 x7.
Assume H7: x0 x8.
Assume H8: x0 x9.
Assume H9: x0 x10.
Assume H10: x0 x11.
Assume H11: x0 x12.
Assume H12: x0 x13.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))))))),
x13,
λ x14 x15 . x15 = x1 (x2 x3 x13) (x1 (x2 x4 x13) (x1 (x2 x5 x13) (x1 (x2 x6 x13) (x1 (x2 x7 x13) (x1 (x2 x8 x13) (x1 (x2 x9 x13) (x1 (x2 x10 x13) (x1 (x2 x11 x13) (x2 x12 x13))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_e3761daa583c62dd19e21ac5ecc36db62b544996e686d7070ae47678e3d87642 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12 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 y14 to be ...
set y15 to be x2 (x3 x4 y14) (x2 (x3 x5 y14) (x2 (x3 x6 y14) (x2 (x3 x7 y14) (x2 (x3 x8 y14) (x2 (x3 x9 y14) (x2 (x3 x10 ...) ...))))))
Claim L13: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H13: x16 (x3 (x4 x5 y15) (x3 (x4 x6 y15) (x3 (x4 x7 y15) (x3 (x4 x8 y15) (x3 (x4 x9 y15) (x3 (x4 x10 y15) (x3 (x4 x11 y15) (x3 (x4 x12 y15) (x3 (x4 x13 y15) (x4 y14 y15)))))))))).
set y17 to be λ x17 . x16
Apply unknownprop_4b8d128ecdccf0be1b2d65d4c7bbdd0e9890ef8c5ba924340bc3c5556f35f320 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
y14,
y15,
λ x18 x19 . y17 (x3 (x4 x5 y15) x18) (x3 (x4 x5 y15) x19) 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 x16 of type ι → ι → ο be given.
Apply L13 with
λ x17 . x16 x17 y15 ⟶ x16 y15 x17.
Assume H14: x16 y15 y15.
The subproof is completed by applying H14.