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.
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.
Apply unknownprop_8ac85daae17fdbfc8ff47a3afd09055c02d79bb3aece0a735127f757c600bfc9 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x1 x9 x10,
x11,
x12,
λ x13 x14 . x14 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 x12))))))))) leaving 13 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.
Apply H0 with
x9,
x10 leaving 2 subgoals.
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.
set y13 to be ...
set y14 to be ...
Claim L13: ∀ x15 : ι → ο . x15 y14 ⟶ x15 y13
Let x15 of type ι → ο be given.
Assume H13: x15 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 y13 y14)))))))))).
set y16 to be ...
set y17 to be ...
set y18 to be ...
Claim L14: ∀ x19 : ι → ο . x19 y18 ⟶ x19 y17
Let x19 of type ι → ο be given.
Assume H14: x19 (x5 x7 (x5 x8 (x5 x9 (x5 x10 (x5 x11 (x5 x12 (x5 y13 (x5 y14 (x5 x15 y16))))))))).
set y20 to be ...
set y21 to be ...
set y22 to be ...
Claim L15: ∀ x23 : ι → ο . x23 y22 ⟶ x23 y21
Let x23 of type ι → ο be given.
Assume H15: x23 (x7 x10 (x7 x11 (x7 x12 (x7 y13 (x7 y14 (x7 x15 (x7 y16 (x7 y17 y18)))))))).
set y24 to be ...
set y25 to be ...
set y26 to be ...
set y27 to be λ x27 x28 . y26 (x9 x12 ...) ...
Apply L16 with
λ x28 . y27 x28 y26 ⟶ y27 y26 x28 leaving 2 subgoals.
Assume H17: y27 y26 y26.
The subproof is completed by applying H17.
The subproof is completed by applying L16.
set y23 to be λ x23 x24 . y22 (x7 x9 x23) (x7 x9 x24)
Apply L15 with
λ x24 . y23 x24 y22 ⟶ y23 y22 x24 leaving 2 subgoals.
Assume H16: y23 y22 y22.
The subproof is completed by applying H16.
The subproof is completed by applying L15.
set y19 to be λ x19 x20 . y18 (x5 x6 x19) (x5 x6 x20)
Apply L14 with
λ x20 . y19 x20 y18 ⟶ y19 y18 x20 leaving 2 subgoals.
Assume H15: y19 y18 y18.
The subproof is completed by applying H15.
The subproof is completed by applying L14.
Let x15 of type ι → ι → ο be given.
Apply L13 with
λ x16 . x15 x16 y14 ⟶ x15 y14 x16.
Assume H14: x15 y14 y14.
The subproof is completed by applying H14.