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_515047c02fed97c50f69ea07f84c55a116d5435d48e441446058ba601add8797 with
x0,
x1,
x7,
x3,
x4,
x5,
x6,
x8,
λ x9 x10 . x1 x2 x9 = x1 x7 (x1 x4 (x1 x3 (x1 x6 (x1 x5 (x1 x2 x8))))) leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H6.
The subproof is completed by applying H8.
Apply H1 with
x2,
x7,
x1 x3 (x1 x4 (x1 x5 (x1 x6 x8))),
λ x9 x10 . x10 = x1 x7 (x1 x4 (x1 x3 (x1 x6 (x1 x5 (x1 x2 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Apply H0 with
x3,
x1 x4 (x1 x5 (x1 x6 x8)) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 x8) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 x8 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x8 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
set y9 to be x1 x7 (x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x8)))))
set y10 to be x2 x8 (x2 x5 (x2 x4 (x2 x7 (x2 x6 (x2 x3 y9)))))
Claim L9: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H9: x11 (x3 y9 (x3 x6 (x3 x5 (x3 x8 (x3 x7 (x3 x4 y10)))))).
set y12 to be λ x12 . x11
set y13 to be λ x13 x14 . y12 (x3 y9 x13) (x3 y9 x14)
Apply unknownprop_a06fd6ee9edfca0fb4d6c6418fcee3224ace46be4fd37196bd92cb85604ae3ff with
x2,
x3,
x6,
x5,
x8,
x7,
x4,
y10,
λ x14 x15 . y13 x15 x14 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H4.
The subproof is completed by applying H3.
The subproof is completed by applying H6.
The subproof is completed by applying H5.
The subproof is completed by applying H2.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
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.