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).
Assume H2: ∀ x2 x3 . x0 x2 ⟶ x0 x3 ⟶ x1 x2 x3 = x1 x3 x2.
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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Assume H9: x0 x8.
Assume H10: x0 x9.
Assume H11: x0 x10.
Assume H12: x0 x11.
Apply unknownprop_3d8be1424c68a4fbd2fc7519d72600a7ea80f1b1c348d71d7a49f22dc43384aa with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
λ x12 x13 . x12 = x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11)))))))) leaving 13 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.
Apply unknownprop_3d8be1424c68a4fbd2fc7519d72600a7ea80f1b1c348d71d7a49f22dc43384aa with
x0,
x1,
x3,
x2,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
λ x12 x13 . x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))))))) = x12 leaving 13 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 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.
set y12 to be x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 (x1 ... ...))))
set y13 to be x2 (x2 x4 x3) (x2 x5 (x2 x6 (x2 x7 (x2 x8 (x2 x9 (x2 x10 (x2 x11 y12)))))))
Claim L13: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H13: x14 (x3 (x3 x5 x4) (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 y12 y13)))))))).
set y15 to be λ x15 . x14
Apply H2 with
x4,
x5,
λ x16 x17 . y15 (x3 x16 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 y12 y13)))))))) (x3 x17 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 y12 y13)))))))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H13.
Let x14 of type ι → ι → ο be given.
Apply L13 with
λ x15 . x14 x15 y13 ⟶ x14 y13 x15.
Assume H14: x14 y13 y13.
The subproof is completed by applying H14.