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.
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 H2 with
x3,
x4,
x1 x5 (x1 x6 (x1 x7 x8)),
λ x9 x10 . x10 = x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x2 x3 x8)))) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x2 x4 x8)))) leaving 4 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
Apply unknownprop_b48d4480a5526e51a91293fec1b0b9440be4280265441ce358bda14cced12479 with
x0,
x1,
x5,
x6,
x7,
x8 leaving 5 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.
Apply unknownprop_483ab2c4cf352794c6e764ca83196651bc5de6ef598ec843402a4c02baafb47b with
x0,
x1,
x2,
x5,
x6,
x7,
x8,
x3,
λ x9 x10 . x1 x10 (x2 x4 (x1 x5 (x1 x6 (x1 x7 x8)))) = x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x2 x3 x8)))) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x2 x4 x8)))) leaving 8 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 H3.
Apply unknownprop_483ab2c4cf352794c6e764ca83196651bc5de6ef598ec843402a4c02baafb47b with
x0,
x1,
x2,
x5,
x6,
x7,
x8,
x4,
λ x9 x10 . x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x2 x3 x8)))) x10 = x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x2 x3 x8)))) (x1 (x2 x4 x5) (x1 (x2 x4 x6) (x1 (x2 x4 x7) (x2 x4 x8)))) leaving 8 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 H4.
Let x9 of type ι → ι → ο be given.
Assume H9: x9 (x1 (x1 (x2 x3 x5) (x1 (x2 x3 x6) (x1 (x2 x3 x7) (x2 x3 x8)))) (x1 (x2 x4 x5) (x1 (x2 x4 ...) ...))) ....