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 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 x3 (x1 x4 x5) = x1 (x2 x3 x4) (x2 x3 x5).
Assume H2: ∀ 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.
Let x9 of type ι be given.
Let x10 of type ι be given.
Let x11 of type ι be given.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Assume H8: x0 x8.
Assume H9: x0 x9.
Assume H10: x0 x10.
Assume H11: x0 x11.
Apply H2 with
x3,
x4,
x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))))),
λ x12 x13 . x13 = x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x1 (x2 x3 x8) (x1 (x2 x3 x9) (x1 (x2 x3 x10) (x2 x3 x11))))))) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x1 (x2 x4 x8) (x1 (x2 x4 x9) (x1 (x2 x4 x10) (x2 x4 x11))))))) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply unknownprop_025d233877239fdf8667e3ba4d630729f1334dc236b8bf7cefec04c2fd303300 with
x0,
x1,
x5,
x6,
x7,
x8,
x9,
x10,
x11 leaving 8 subgoals.
The subproof is completed by applying H0.
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.
Apply unknownprop_657d0f30d73a17b9383982246e5348296a7187df8b5edb44a156fba1faa41105 with
x0,
x1,
x2,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x3,
λ x12 x13 . x1 x13 (x2 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 x11))))))) = x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x1 (x2 x3 x8) (x1 (x2 x3 x9) (x1 (x2 x3 x10) (x2 x3 x11))))))) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x1 (x2 x4 x8) (x1 (x2 x4 x9) (x1 (x2 x4 x10) (x2 x4 x11))))))) leaving 11 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H3.
Apply unknownprop_657d0f30d73a17b9383982246e5348296a7187df8b5edb44a156fba1faa41105 with
x0,
x1,
x2,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x4,
λ x12 x13 . ... = ... leaving 11 subgoals.