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.
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.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 x8))),
λ x9 x10 . x10 = x1 x3 (x1 x2 (x1 x4 (x1 x7 (x1 x6 (x1 x5 x8))))) 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 x8)) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 (x1 x7 x8) leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x1 x7 x8 leaving 2 subgoals.
The subproof is completed by applying H6.
Apply H0 with
x7,
x8 leaving 2 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
set y9 to be ...
set y10 to be ...
Claim L9: ∀ x11 : ι → ο . x11 y10 ⟶ x11 y9
Let x11 of type ι → ο be given.
Assume H9: x11 (x3 x5 (x3 x4 (x3 x6 (x3 y9 (x3 x8 (x3 x7 y10)))))).
set y12 to be ...
set y13 to be ...
set y14 to be x4 x5 (x4 ... ...)
Claim L10: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H10: x15 (x5 x6 (x5 x8 (x5 x11 (x5 y10 (x5 y9 y12))))).
set y16 to be λ x16 . x15
set y17 to be x5 x8 (x5 y9 (x5 y10 (x5 x11 y12)))
set y18 to be x6 y9 (x6 y12 (x6 x11 (x6 y10 y13)))
Claim L11: ∀ x19 : ι → ο . x19 y18 ⟶ x19 y17
Let x19 of type ι → ο be given.
Assume H11: x19 (x7 y10 (x7 y13 (x7 y12 (x7 x11 y14)))).
set y20 to be λ x20 . x19
Apply unknownprop_17f2e534568ee7312c417497530472991cbc191bc8362198ef82a32098ba0e8c with
x6,
x7,
x11,
y12,
y13,
y14,
λ x21 x22 . y20 (x7 y10 x21) (x7 y10 x22) leaving 7 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 H11.
set y19 to be λ x19 x20 . y18 (x7 x8 x19) (x7 x8 x20)
Apply L11 with
λ x20 . y19 x20 y18 ⟶ y19 y18 x20 leaving 2 subgoals.
Assume H12: y19 y18 y18.
The subproof is completed by applying H12.
The subproof is completed by applying L11.
set y15 to be λ x15 x16 . y14 (x5 x7 x15) (x5 x7 x16)
Apply L10 with
λ x16 . y15 x16 y14 ⟶ y15 y14 x16 leaving 2 subgoals.
Assume H11: y15 y14 y14.
The subproof is completed by applying H11.
The subproof is completed by applying L10.
Let x11 of type ι → ι → ο be given.
Apply L9 with
λ x12 . x11 x12 y10 ⟶ x11 y10 x12.
Assume H10: x11 y10 y10.
The subproof is completed by applying H10.