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 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
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 H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply unknownprop_9551c74ff689713f6e29d6039e164b7f427808e036d50b6ddb9edf722f975820 with
x0,
x1,
x3,
x4,
x5,
x6,
x7,
λ x8 x9 . x1 x2 x9 = x1 x5 (x1 x6 (x1 x4 (x1 x2 (x1 x3 x7)))) leaving 8 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.
Apply H1 with
x2,
x5,
x1 x6 (x1 x3 (x1 x4 x7)),
λ x8 x9 . x9 = x1 x5 (x1 x6 (x1 x4 (x1 x2 (x1 x3 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x3 (x1 x4 x7) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x3,
x1 x4 x7 leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x7 leaving 2 subgoals.
The subproof is completed by applying H4.
The subproof is completed by applying H7.
set y8 to be x1 x5 (x1 x2 (x1 x6 (x1 x3 (x1 x4 x7))))
set y9 to be x2 x6 (x2 x7 (x2 x5 (x2 x3 (x2 x4 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H8: x10 (x3 x7 (x3 y8 (x3 x6 (x3 x4 (x3 x5 y9))))).
set y11 to be λ x11 . x10
set y12 to be λ x12 x13 . y11 (x3 x7 x12) (x3 x7 x13)
Apply unknownprop_5d3187157b7b1096324bc8c4956a65dc04e116adcad35fc0fcbac4c7f04ef09e with
x2,
x3,
y8,
x6,
x4,
x5,
y9,
λ x13 x14 . y12 x14 x13 leaving 8 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H6.
The subproof is completed by applying H4.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
Let x10 of type ι → ι → ο be given.
Apply L8 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.