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.
Let x15 of type ι be given.
Let x16 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.
Assume H15: x0 x15.
Assume H16: x0 x16.
Apply H1 with
x2,
x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 (x1 x14 x15))))))))))),
x16,
λ x17 x18 . x18 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 (x1 x14 (x1 x15 x16))))))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_afb4cfc9f10424b1b5cf811579ab414fa5effe9e0015718aad3c27c7fa587bda with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15 leaving 14 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.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
set y17 to be ...
set y18 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 (x2 x15 ...))))))))))))
Claim L17: ∀ x19 : ι → ο . x19 y18 ⟶ x19 y17
Let x19 of type ι → ο be given.
Assume H17: x19 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 x13 (x3 x14 (x3 x15 (x3 x16 (x3 y17 y18)))))))))))))).
set y20 to be λ x20 . x19
Apply unknownprop_e17fd7f99eac4929eea933d5bda2b26571f309a35e1463906acbcd264ef0fdc5 with
x2,
x3,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
x15,
x16,
y17,
y18,
λ x21 x22 . y20 (x3 x4 x21) (x3 x4 x22) leaving 17 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.
The subproof is completed by applying H16.
The subproof is completed by applying H17.
Let x19 of type ι → ι → ο be given.
Apply L17 with
λ x20 . x19 x20 y18 ⟶ x19 y18 x20.
Assume H18: x19 y18 y18.
The subproof is completed by applying H18.