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 x4 (x1 x6 (x1 x8 (x1 x7 (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 ...
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 x6 (x3 x8 (x3 y10 (x3 x9 (x3 x7 y11))))))).
set y13 to be ...
set y14 to be ...
set y15 to be ...
Claim L11: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H11: x16 (x5 x6 (x5 x8 (x5 y10 (x5 x12 (x5 y11 (x5 x9 y13)))))).
set y17 to be ...
set y18 to be ...
set y19 to be ...
Claim L12: ∀ x20 : ι → ο . x20 y19 ⟶ x20 y18
Let x20 of type ι → ο be given.
Assume H12: x20 (x7 y10 (x7 x12 (x7 y14 (x7 y13 (x7 y11 y15))))).
set y21 to be λ x21 . x20
set y22 to be λ x22 x23 . y21 (x7 y10 x22) (x7 y10 x23)
Apply unknownprop_90d509bd8706c145adbb622ea79d7b70b4307aa02d76a2427a59b6e040f47cdc with
x6,
x7,
x12,
y14,
y13,
y11,
y15,
λ x23 x24 . y22 x24 x23 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 H8.
The subproof is completed by applying H7.
The subproof is completed by applying H5.
The subproof is completed by applying H9.
The subproof is completed by applying H12.
set y20 to be λ x20 x21 . y19 (x7 x8 x20) (x7 x8 x21)
Apply L12 with
λ x21 . y20 x21 y19 ⟶ y20 y19 x21 leaving 2 subgoals.
Assume H13: y20 y19 y19.
The subproof is completed by applying H13.
The subproof is completed by applying L12.
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.