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.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Apply unknownprop_3357afebd81483279cf67f1b7f967421c587ef898cabdea64b1348e221c1d50f with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . x8 = x1 x3 (x1 x2 (x1 x4 (x1 x5 (x1 x6 x7)))) leaving 9 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.
Apply unknownprop_3357afebd81483279cf67f1b7f967421c587ef898cabdea64b1348e221c1d50f with
x0,
x1,
x3,
x2,
x4,
x5,
x6,
x7,
λ x8 x9 . x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 x7))) = x8 leaving 9 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.
set y8 to be x1 (x1 x2 x3) (x1 x4 (x1 x5 (x1 x6 x7)))
set y9 to be x2 (x2 x4 x3) (x2 x5 (x2 x6 (x2 x7 y8)))
Claim L9: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H9: x10 (x3 (x3 x5 x4) (x3 x6 (x3 x7 (x3 y8 y9)))).
set y11 to be λ x11 . x10
Apply H2 with
x4,
x5,
λ x12 x13 . y11 (x3 x12 (x3 x6 (x3 x7 (x3 y8 y9)))) (x3 x13 (x3 x6 (x3 x7 (x3 y8 y9)))) leaving 3 subgoals.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H9.
Let x10 of type ι → ι → ο be given.
Apply L9 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H10: x10 y9 y9.
The subproof is completed by applying H10.