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.
Let x8 of type ι be given.
Let x9 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.
Assume H8: x0 x8.
Assume H9: x0 x9.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x9)))),
λ x10 x11 . x11 = x1 x3 (x1 x2 (x1 x7 (x1 x8 (x1 x4 (x1 x6 (x1 x5 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 (x1 x7 (x1 x8 x9))) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 (x1 x8 x9)) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 (x1 x8 x9) leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x1 x8 x9 leaving 2 subgoals.
The subproof is completed by applying H7.
Apply H0 with
x8,
x9 leaving 2 subgoals.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
set y10 to be ...
set y11 to be x2 x4 (x2 x3 (x2 x8 (x2 x9 (x2 ... ...))))
Claim L10: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H10: x12 (x3 x5 (x3 x4 (x3 x9 (x3 y10 (x3 x6 (x3 x8 (x3 x7 y11))))))).
set y13 to be λ x13 . x12
set y14 to be x3 x4 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 y10 y11)))))
set y15 to be x4 x5 (x4 y10 (x4 y11 (x4 x7 (x4 x9 (x4 x8 x12)))))
Claim L11: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H11: x16 (x5 x6 (x5 y11 (x5 x12 (x5 x8 (x5 y10 (x5 x9 y13)))))).
set y17 to be λ x17 . x16
Apply unknownprop_fa08f5d0f2b98fa34fa2b6dffc9b2231cf292f2333a188ec6f1bed1ac520e55f with
x4,
x5,
x8,
x9,
y10,
y11,
x12,
y13,
λ x18 x19 . y17 (x5 x6 x18) (x5 x6 x19) 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 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 H11.
set y16 to be λ x16 x17 . y15 (x5 x7 x16) (x5 x7 x17)
Apply L11 with
λ x17 . y16 x17 y15 ⟶ y16 y15 x17 leaving 2 subgoals.
Assume H12: y16 y15 y15.
The subproof is completed by applying H12.
The subproof is completed by applying L11.
Let x12 of type ι → ι → ο be given.
Apply L10 with
λ x13 . x12 x13 y11 ⟶ x12 y11 x13.
Assume H11: x12 y11 y11.
The subproof is completed by applying H11.