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 unknownprop_b7b295f38ec44b7457473010f3621695d26e4e9422bae5a083cc6f30b9abc04b with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x1 x10 x11,
λ x12 x13 . x13 = x1 (x1 (x2 x3 x10) (x2 x3 x11)) (x1 (x1 (x2 x4 x10) (x2 x4 x11)) (x1 (x1 (x2 x5 x10) (x2 x5 x11)) (x1 (x1 (x2 x6 x10) (x2 x6 x11)) (x1 (x1 (x2 x7 x10) (x2 x7 x11)) (x1 (x1 (x2 x8 x10) (x2 x8 x11)) (x1 (x2 x9 x10) (x2 x9 x11))))))) leaving 11 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H2.
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.
Apply H0 with
x10,
x11 leaving 2 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply H1 with
x3,
x10,
x11,
λ x12 x13 . x1 x13 (x1 (x2 x4 (x1 x10 x11)) (x1 (x2 x5 (x1 x10 x11)) (x1 (x2 x6 (x1 x10 x11)) (x1 (x2 x7 (x1 x10 x11)) (x1 (x2 x8 (x1 x10 x11)) (x2 x9 (x1 x10 x11))))))) = x1 (x1 (x2 x3 x10) (x2 x3 x11)) (x1 (x1 (x2 x4 x10) (x2 x4 x11)) (x1 (x1 (x2 x5 x10) (x2 x5 x11)) (x1 (x1 (x2 x6 x10) (x2 x6 x11)) (x1 (x1 (x2 x7 x10) (x2 x7 x11)) (x1 (x1 (x2 x8 x10) (x2 x8 x11)) (x1 (x2 x9 x10) (x2 x9 x11))))))) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
Apply H1 with
x4,
x10,
x11,
λ x12 x13 . x1 ... ... = ... leaving 4 subgoals.