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.
Let x13 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.
Assume H13: x0 x13.
Apply unknownprop_20d59f711648743c10e7397b300f029b260270e5ae517b5e58738e37f16d84a2 with
x0,
x1,
x2,
x3,
x4,
x5,
x6,
x7,
x8,
x9,
x1 x10 x11,
x12,
x13,
λ x14 x15 . x15 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 x13)))))))))) leaving 14 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.
The subproof is completed by applying H9.
Apply H0 with
x10,
x11 leaving 2 subgoals.
The subproof is completed by applying H10.
The subproof is completed by applying H11.
The subproof is completed by applying H12.
The subproof is completed by applying H13.
set y14 to be ...
set y15 to be ...
Claim L14: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H14: x16 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 x10 (x3 x11 (x3 x12 (x3 x13 (x3 y14 y15))))))))))).
set y17 to be ...
set y18 to be ...
set y19 to be ...
Claim L15: ∀ x20 : ι → ο . x20 y19 ⟶ x20 y18
Let x20 of type ι → ο be given.
Assume H15: x20 (x5 x7 (x5 x8 (x5 x9 (x5 x10 (x5 x11 (x5 x12 (x5 x13 (x5 y14 (x5 y15 (x5 x16 y17)))))))))).
set y21 to be ...
set y22 to be ...
set y23 to be ...
Claim L16: ∀ x24 : ι → ο . x24 y23 ⟶ x24 y22
Let x24 of type ι → ο be given.
Assume H16: x24 (x7 x10 (x7 x11 (x7 x12 (x7 x13 (x7 y14 (x7 y15 (x7 x16 (x7 y17 (x7 y18 y19))))))))).
set y25 to be ...
set y26 to be ...
set y27 to be ...
set y28 to be ...
Apply L17 with
λ x29 . ... ⟶ y28 y27 x29 leaving 2 subgoals.
set y24 to be λ x24 x25 . y23 (x7 x9 x24) (x7 x9 x25)
Apply L16 with
λ x25 . y24 x25 y23 ⟶ y24 y23 x25 leaving 2 subgoals.
Assume H17: y24 y23 y23.
The subproof is completed by applying H17.
The subproof is completed by applying L16.
set y20 to be λ x20 x21 . y19 (x5 x6 x20) (x5 x6 x21)
Apply L15 with
λ x21 . y20 x21 y19 ⟶ y20 y19 x21 leaving 2 subgoals.
Assume H16: y20 y19 y19.
The subproof is completed by applying H16.
The subproof is completed by applying L15.
Let x16 of type ι → ι → ο be given.
Apply L14 with
λ x17 . x16 x17 y15 ⟶ x16 y15 x17.
Assume H15: x16 y15 y15.
The subproof is completed by applying H15.