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.
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.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11)))))),
x12,
λ x13 x14 . x14 = x1 (x2 x3 x12) (x1 (x2 x4 x12) (x1 (x2 x5 x12) (x1 (x2 x6 x12) (x1 (x2 x7 x12) (x1 (x2 x8 x12) (x1 (x2 x9 x12) (x1 (x2 x10 x12) (x2 x11 x12)))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_edbdc31c8b550a683544e2ad315a13cf7bd7f44068be39efa27faf89c5105937 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11 leaving 9 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.
set y13 to be x1 (x2 x3 x12) (x2 (x1 ... ...) ...)
set y14 to be x2 (x3 x4 y13) (x2 (x3 x5 y13) (x2 (x3 x6 y13) (x2 (x3 x7 y13) (x2 (x3 x8 y13) (x2 (x3 x9 y13) (x2 (x3 x10 y13) (x2 (x3 x11 y13) (x3 x12 y13))))))))
Claim L12: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H12: x15 (x3 (x4 x5 y14) (x3 (x4 x6 y14) (x3 (x4 x7 y14) (x3 (x4 x8 y14) (x3 (x4 x9 y14) (x3 (x4 x10 y14) (x3 (x4 x11 y14) (x3 (x4 x12 y14) (x4 y13 y14))))))))).
set y16 to be λ x16 . x15
Apply unknownprop_96890bb6437669c5e09c9ab608ee6937f060bce73200a8371786914ccb14f8e2 with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
y13,
y14,
λ x17 x18 . y16 (x3 (x4 x5 y14) x17) (x3 (x4 x5 y14) x18) 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.
The subproof is completed by applying H12.
Let x15 of type ι → ι → ο be given.
Apply L12 with
λ x16 . x15 x16 y14 ⟶ x15 y14 x16.
Assume H13: x15 y14 y14.
The subproof is completed by applying H13.