Let x0 of type ι → ο be given.
Let x1 of type ι → ι → ι be given.
Let x2 of type ι → ι → ι be given.
Assume H0: ∀ x3 x4 . x0 x3 ⟶ x0 x4 ⟶ x0 (x1 x3 x4).
Assume H1: ∀ x3 x4 x5 . x0 x3 ⟶ x0 x4 ⟶ x0 x5 ⟶ x2 (x1 x3 x4) x5 = x1 (x2 x3 x5) (x2 x4 x5).
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.
Let x14 of type ι be given.
Assume H2: x0 x3.
Assume H3: x0 x4.
Assume H4: x0 x5.
Assume H5: x0 x6.
Assume H6: x0 x7.
Assume H7: x0 x8.
Assume H8: x0 x9.
Assume H9: x0 x10.
Assume H10: x0 x11.
Assume H11: x0 x12.
Assume H12: x0 x13.
Assume H13: x0 x14.
Apply H1 with
x3,
x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 (x1 x9 (x1 x10 (x1 x11 (x1 x12 x13)))))))),
x14,
λ x15 x16 . x16 = x1 (x2 x3 x14) (x1 (x2 x4 x14) (x1 (x2 x5 x14) (x1 (x2 x6 x14) (x1 (x2 x7 x14) (x1 (x2 x8 x14) (x1 (x2 x9 x14) (x1 (x2 x10 x14) (x1 (x2 x11 x14) (x1 (x2 x12 x14) (x2 x13 x14)))))))))) leaving 4 subgoals.
The subproof is completed by applying H2.
Apply unknownprop_5b486140b3675dc9982aecd3fd8b54170f576a6488bd9894cf4b35517b5b2768 with
x0,
x1,
x4,
x5,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13 leaving 11 subgoals.
The subproof is completed by applying H0.
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.
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 y15 to be ...
set y16 to be x2 (x3 x4 y15) (x2 (x3 x5 y15) (x2 (x3 x6 y15) ...))
Claim L14: ∀ x17 : ι → ο . x17 y16 ⟶ x17 y15
Let x17 of type ι → ο be given.
Assume H14: x17 (x3 (x4 x5 y16) (x3 (x4 x6 y16) (x3 (x4 x7 y16) (x3 (x4 x8 y16) (x3 (x4 x9 y16) (x3 (x4 x10 y16) (x3 (x4 x11 y16) (x3 (x4 x12 y16) (x3 (x4 x13 y16) (x3 (x4 x14 y16) (x4 y15 y16))))))))))).
set y18 to be λ x18 . x17
Apply unknownprop_e749ba748aa4eeaf5a46cce1bec53a64283c95970d09e2d8f6e92e9995d31b4c with
x2,
x3,
x4,
x6,
x7,
x8,
x9,
x10,
x11,
x12,
x13,
x14,
y15,
y16,
λ x19 x20 . y18 (x3 (x4 x5 y16) x19) (x3 (x4 x5 y16) x20) leaving 14 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 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.
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.
The subproof is completed by applying H14.
Let x17 of type ι → ι → ο be given.
Apply L14 with
λ x18 . x17 x18 y16 ⟶ x17 y16 x18.
Assume H15: x17 y16 y16.
The subproof is completed by applying H15.