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.
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.
Apply unknownprop_510b31dd0339aec840c19fb69c6737274f8c23d5a45191e11f8aaf7ceaa137c9 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x1 x7 x8,
x9,
x10,
λ x11 x12 . x12 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 x10))))))) leaving 11 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.
Apply H0 with
x7,
x8 leaving 2 subgoals.
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.
set y11 to be ...
set y12 to be ...
Claim L11: ∀ x13 : ι → ο . x13 y12 ⟶ x13 y11
Let x13 of type ι → ο be given.
Assume H11: x13 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 y11 y12)))))))).
set y14 to be ...
set y15 to be ...
set y16 to be ...
Claim L12: ∀ x17 : ι → ο . x17 y16 ⟶ x17 y15
Let x17 of type ι → ο be given.
Assume H12: x17 (x5 x7 (x5 x8 (x5 x9 (x5 x10 (x5 y11 (x5 y12 (x5 x13 y14))))))).
set y18 to be ...
set y19 to be ...
set y20 to be ...
Claim L13: ∀ x21 : ι → ο . x21 y20 ⟶ x21 y19
Let x21 of type ι → ο be given.
Assume H13: x21 (x7 x10 (x7 y11 (x7 y12 (x7 x13 (x7 y14 (x7 y15 y16)))))).
set y22 to be ...
set y23 to be ...
set y24 to be ...
Claim L14: ∀ x25 : ι → ο . x25 y24 ⟶ x25 y23
Let x25 of type ι → ο be given.
Assume H14: x25 (x9 x13 (x9 y14 (x9 y15 (x9 y16 (x9 x17 y18))))).
set y26 to be ...
set y27 to be ...
set y28 to be ...
set y29 to be ...
Apply L15 with
λ x30 . ... leaving 2 subgoals.
set y25 to be λ x25 x26 . y24 (x9 y12 x25) (x9 y12 x26)
Apply L14 with
λ x26 . y25 x26 y24 ⟶ y25 y24 x26 leaving 2 subgoals.
Assume H15: y25 y24 y24.
The subproof is completed by applying H15.
The subproof is completed by applying L14.
set y21 to be λ x21 x22 . y20 (x7 x9 x21) (x7 x9 x22)
Apply L13 with
λ x22 . y21 x22 y20 ⟶ y21 y20 x22 leaving 2 subgoals.
Assume H14: y21 y20 y20.
The subproof is completed by applying H14.
The subproof is completed by applying L13.
set y17 to be λ x17 x18 . y16 (x5 x6 x17) (x5 x6 x18)
Apply L12 with
λ x18 . y17 x18 y16 ⟶ y17 y16 x18 leaving 2 subgoals.
Assume H13: y17 y16 y16.
The subproof is completed by applying H13.
The subproof is completed by applying L12.
Let x13 of type ι → ι → ο be given.
Apply L11 with
λ x14 . x13 x14 y12 ⟶ x13 y12 x14.
Assume H12: x13 y12 y12.
The subproof is completed by applying H12.