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.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 x7)),
λ x8 x9 . x9 = x1 x3 (x1 x2 (x1 x6 (x1 x4 (x1 x5 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 x7) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 x7 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x7 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
set y8 to be x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 x7))))
set y9 to be x2 x4 (x2 x3 (x2 x7 (x2 x5 (x2 x6 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H8: x10 (x3 x5 (x3 x4 (x3 y8 (x3 x6 (x3 x7 y9))))).
set y11 to be λ x11 . x10
set y12 to be x3 x4 (x3 x6 (x3 x7 (x3 y8 y9)))
set y13 to be x4 x5 (x4 y9 (x4 x7 (x4 y8 x10)))
Claim L9: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H9: x14 (x5 x6 (x5 x10 (x5 y8 (x5 y9 y11)))).
set y15 to be λ x15 . x14
set y16 to be λ x16 x17 . y15 (x5 x6 x16) (x5 x6 x17)
Apply unknownprop_6df806693864a23a378ddbca02cda4bb4bc233ff1daa8914d51c06eb72ff2550 with
x4,
x5,
x10,
y8,
y9,
y11,
λ x17 x18 . y16 x18 x17 leaving 7 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 H4.
The subproof is completed by applying H5.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
set y14 to be λ x14 x15 . y13 (x5 x7 x14) (x5 x7 x15)
Apply L9 with
λ x15 . y14 x15 y13 ⟶ y14 y13 x15 leaving 2 subgoals.
Assume H10: y14 y13 y13.
The subproof is completed by applying H10.
The subproof is completed by applying L9.
Let x10 of type ι → ι → ο be given.
Apply L8 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.