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 x2 (x1 x3 x4) = x1 x3 (x1 x2 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.
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.
Apply unknownprop_f0b76402e77112232d36cbd146a3e3efd40fdf823d5b935fe896a6fb8918a817 with
x0,
x1,
x6,
x7,
x3,
x4,
x5,
x8,
λ x9 x10 . x1 x2 x9 = x1 x6 (x1 x2 (x1 x7 (x1 x5 (x1 x3 (x1 x4 x8))))) leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
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 H8.
Apply H1 with
x2,
x6,
x1 x7 (x1 x3 (x1 x4 (x1 x5 x8))),
λ x9 x10 . x10 = x1 x6 (x1 x2 (x1 x7 (x1 x5 (x1 x3 (x1 x4 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x3 (x1 x4 (x1 x5 x8)) leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x3,
x1 x4 (x1 x5 x8) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 x8 leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x8 leaving 2 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H8.
set y9 to be ...
set y10 to be ...
Claim L9: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H9: x11 (x3 x8 (x3 x4 (x3 y9 (x3 x7 (x3 x5 (x3 x6 y10)))))).
set y12 to be ...
set y13 to be ...
set y14 to be ...
Claim L10: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H10: x15 (x5 x6 (x5 x11 (x5 y9 (x5 x7 (x5 x8 y12))))).
set y16 to be ...
set y17 to be ...
set y18 to be ...
Claim L11: ∀ x19 : ι → ο . x19 y18 ⟶ x19 y17
Let x19 of type ι → ο be given.
Assume H11: x19 (x7 y13 (x7 x11 (x7 y9 (x7 y10 y14)))).
set y20 to be ...
set y21 to be λ x21 x22 . y20 (x7 y13 x21) (x7 y13 x22)
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with
x6,
x7,
x11,
y9,
y10,
y14,
λ x22 x23 . y21 x23 x22 leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H5.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
The subproof is completed by applying H11.
set y19 to be λ x19 x20 . y18 (x7 x8 x19) (x7 x8 x20)
Apply L11 with
λ x20 . y19 x20 y18 ⟶ y19 y18 x20 leaving 2 subgoals.
Assume H12: y19 y18 y18.
The subproof is completed by applying H12.
The subproof is completed by applying L11.
set y15 to be λ x15 x16 . y14 (x5 y10 x15) (x5 y10 x16)
Apply L10 with
λ x16 . y15 x16 y14 ⟶ y15 y14 x16 leaving 2 subgoals.
Assume H11: y15 y14 y14.
The subproof is completed by applying H11.
The subproof is completed by applying L10.
Let x11 of type ι → ι → ο be given.
Apply L9 with
λ x12 . x11 x12 y10 ⟶ x11 y10 x12.
Assume H10: x11 y10 y10.
The subproof is completed by applying H10.