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).
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.
Let x10 of type ι be given.
Let x11 of type ι be given.
Let x12 of type ι be given.
Let x13 of type ι be given.
Let x14 of type ι be given.
Let x15 of type ι be given.
Let x16 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.
Assume H10: x0 x10.
Assume H11: x0 x11.
Assume H12: x0 x12.
Assume H13: x0 x13.
Assume H14: x0 x14.
Assume H15: x0 x15.
Assume H16: x0 x16.
Apply unknownprop_927d1c02e5dece6a17ea62ce01c1bfaa7bf14907c62e5724730dec8a7d01022c with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x1 x13 x14,
x15,
x16,
λ x17 x18 . x18 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 (x1 x13 (x1 x14 (x1 x15 x16))))))))))))) leaving 17 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
Apply H0 with
x13,
x14 leaving 2 subgoals.
The subproof is completed by applying H13.
The subproof is completed by applying H14.
The subproof is completed by applying H15.
The subproof is completed by applying H16.
set y17 to be ...
set y18 to be ...
Claim L17: ∀ x19 : ι → ο . x19 y18 ⟶ x19 y17
Let x19 of type ι → ο be given.
Assume H17: x19 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 x13 (x3 x14 (x3 x15 (x3 x16 (x3 y17 y18)))))))))))))).
set y20 to be ...
set y21 to be ...
set y22 to be ...
Claim L18: ∀ x23 : ι → ο . x23 y22 ⟶ x23 y21
Let x23 of type ι → ο be given.
Assume H18: x23 (x5 x7 (x5 x8 (x5 x9 (x5 x10 (x5 x11 (x5 x12 (x5 x13 (x5 x14 (x5 x15 (x5 x16 (x5 y17 (x5 y18 (x5 x19 y20))))))))))))).
set y24 to be ...
set y25 to be ...
set y26 to be ...
set y27 to be λ x27 x28 . y26 (x7 x9 x27) (x7 x9 x28)
Apply L19 with
λ x28 . y27 x28 y26 ⟶ y27 y26 x28 leaving 2 subgoals.
Assume H20: y27 y26 y26.
The subproof is completed by applying H20.
The subproof is completed by applying L19.
set y23 to be λ x23 x24 . y22 (x5 x6 x23) (x5 x6 x24)
Apply L18 with
λ x24 . y23 x24 y22 ⟶ y23 y22 x24 leaving 2 subgoals.
Assume H19: y23 y22 y22.
The subproof is completed by applying H19.
The subproof is completed by applying L18.
Let x19 of type ι → ι → ο be given.
Apply L17 with
λ x20 . x19 x20 y18 ⟶ x19 y18 x20.
Assume H18: x19 y18 y18.
The subproof is completed by applying H18.