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,
x3,
x4,
x5,
x6,
x7,
x8,
λ x9 x10 . x1 x2 x10 = x1 x4 (x1 x2 (x1 x6 (x1 x5 (x1 x7 (x1 x3 x8))))) leaving 9 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.
Apply H1 with
x2,
x4,
x1 x5 (x1 x6 (x1 x7 (x1 x3 x8))),
λ x9 x10 . x10 = x1 x4 (x1 x2 (x1 x6 (x1 x5 (x1 x7 (x1 x3 x8))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 (x1 x3 x8)) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 (x1 x3 x8) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x3 x8 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x3,
x8 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H8.
set y9 to be ...
set y10 to be x2 x5 (x2 x3 ...)
Claim L9: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H9: x11 (x3 x6 (x3 x4 (x3 x8 (x3 x7 (x3 y9 (x3 x5 y10)))))).
set y12 to be λ x12 . x11
set y13 to be x3 x4 (x3 x7 (x3 x8 (x3 y9 (x3 x5 y10))))
set y14 to be x4 x5 (x4 y9 (x4 x8 (x4 y10 (x4 x6 x11))))
Claim L10: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H10: x15 (x5 x6 (x5 y10 (x5 y9 (x5 x11 (x5 x7 y12))))).
set y16 to be λ x16 . x15
Apply H1 with
y9,
y10,
x5 x11 (x5 x7 y12),
λ x17 x18 . y16 (x5 x6 x17) (x5 x6 x18) leaving 4 subgoals.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
Apply H0 with
x11,
x5 x7 y12 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x7,
y12 leaving 2 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H8.
The subproof is completed by applying H10.
set y15 to be λ x15 x16 . y14 (x5 x8 x15) (x5 x8 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.