Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x2 x3 x4).
Assume H2: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 x3 (x1 x4 x5) = x1 x4 (x1 x3 x5).
Assume H3: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x1 (x1 x3 x4) x5 = x1 x3 (x1 x4 x5).
Assume H4: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H5: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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 H6: x0 x3.
Assume H7: x0 x4.
Assume H8: x0 x5.
Assume H9: x0 x6.
Assume H10: x0 x7.
Assume H11: x0 x8.
set y9 to be ...
set y10 to be ...
Claim L12: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H12: x11 (x3 (x4 x5 y9) (x3 (x4 x5 y10) (x3 (x4 x6 y9) (x3 (x4 x6 y10) (x3 (x4 x7 y9) (x3 (x4 x7 y10) (x3 (x4 x8 y9) (x4 x8 y10)))))))).
Apply H4 with
x3 x5 (x3 x6 (x3 x7 x8)),
y9,
y10,
λ x12 . x11 leaving 4 subgoals.
Apply H0 with
x5,
x3 x6 (x3 x7 x8) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x6,
x3 x7 x8 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x7,
x8 leaving 2 subgoals.
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.
set y12 to be ...
set y13 to be ...
set y14 to be ...
Apply L13 with
λ x15 . y14 x15 y13 ⟶ y14 y13 x15 leaving 2 subgoals.
Assume H14: y14 y13 y13.
The subproof is completed by applying H14.
Apply unknownprop_cb47cfc423cf7efda97b4744781e9876675ababf9ff6eee93d13013fa81540c5 with
x5,
x6,
x7 x8 y12,
x7 y9 y12,
x7 y10 y12,
x7 x11 y12,
x7 x8 y13,
x7 y9 y13,
x7 y10 y13,
x7 x11 y13,
λ x15 x16 . x16 = x6 (x7 x8 y12) (x6 (x7 x8 y13) (x6 (x7 y9 y12) (x6 (x7 y9 y13) (x6 (x7 y10 y12) (x6 (x7 y10 y13) (x6 (x7 x11 ...) ...)))))),
... leaving 12 subgoals.
Let x11 of type ι → ι → ο be given.
Apply L12 with
λ x12 . x11 x12 y10 ⟶ x11 y10 x12.
Assume H13: x11 y10 y10.
The subproof is completed by applying H13.