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 unknownprop_01b85a5229069244bfc0fa2496398b6a876770b04992bb705aa6f0b7bd0f95bd with
x0,
x1,
x7,
x8,
x3,
x4,
x5,
x6,
x9,
λ x10 x11 . x1 x2 x10 = x1 x7 (x1 x2 (x1 x3 (x1 x5 (x1 x4 (x1 x8 (x1 x6 x9)))))) leaving 10 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
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 H9.
Apply H1 with
x2,
x7,
x1 x8 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x9)))),
λ x10 x11 . x11 = x1 x7 (x1 x2 (x1 x3 (x1 x5 (x1 x4 (x1 x8 (x1 x6 x9)))))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H7.
Apply H0 with
x8,
x1 x3 (x1 x4 (x1 x5 (x1 x6 x9))) leaving 2 subgoals.
The subproof is completed by applying H8.
Apply H0 with
x3,
x1 x4 (x1 x5 (x1 x6 x9)) leaving 2 subgoals.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 x9) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 x9 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x9 leaving 2 subgoals.
The subproof is completed by applying H6.
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 x9 (x3 x4 (x3 x5 (x3 x7 (x3 x6 (x3 y10 (x3 x8 y11))))))).
set y13 to be ...
set y14 to be ...
set y15 to be x4 x5 (x4 x6 ...)
Claim L11: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H11: x16 (x5 x6 (x5 x7 (x5 x9 (x5 x8 (x5 x12 (x5 y10 y13)))))).
set y17 to be λ x17 . x16
set y18 to be λ x18 x19 . y17 (x5 x6 x18) (x5 x6 x19)
Apply unknownprop_90d509bd8706c145adbb622ea79d7b70b4307aa02d76a2427a59b6e040f47cdc with
x4,
x5,
x7,
x9,
x8,
x12,
x5 y10 y13,
λ x19 x20 . y18 x20 x19 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 H5.
The subproof is completed by applying H4.
The subproof is completed by applying H8.
Apply H0 with
y10,
y13 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H9.
The subproof is completed by applying H11.
set y16 to be λ x16 x17 . y15 (x5 y11 x16) (x5 y11 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.