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.
Assume H2: x0 x2.
Assume H3: x0 x3.
Assume H4: x0 x4.
Assume H5: x0 x5.
Assume H6: x0 x6.
Assume H7: x0 x7.
Apply H1 with
x2,
x3,
x1 x4 (x1 x5 (x1 x6 x7)),
λ x8 x9 . x9 = x1 x3 (x1 x2 (x1 x4 (x1 x6 (x1 x5 x7)))) leaving 4 subgoals.
The subproof is completed by applying H2.
The subproof is completed by applying H3.
Apply H0 with
x4,
x1 x5 (x1 x6 x7) leaving 2 subgoals.
The subproof is completed by applying H4.
Apply H0 with
x5,
x1 x6 x7 leaving 2 subgoals.
The subproof is completed by applying H5.
Apply H0 with
x6,
x7 leaving 2 subgoals.
The subproof is completed by applying H6.
The subproof is completed by applying H7.
set y8 to be ...
set y9 to be x2 x4 (x2 x3 (x2 x5 (x2 x7 ...)))
Claim L8: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H8: x10 (x3 x5 (x3 x4 (x3 x6 (x3 y8 (x3 x7 y9))))).
set y11 to be λ x11 . x10
set y12 to be x3 x4 (x3 x6 (x3 x7 (x3 y8 y9)))
set y13 to be x4 x5 (x4 x7 (x4 y9 (x4 y8 x10)))
Claim L9: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H9: x14 (x5 x6 (x5 y8 (x5 x10 (x5 y9 y11)))).
set y15 to be λ x15 . x14
set y16 to be x5 y8 (x5 y9 (x5 x10 y11))
set y17 to be x6 y9 (x6 y11 (x6 x10 y12))
Claim L10: ∀ x18 : ι → ο . x18 y17 ⟶ x18 y16
Let x18 of type ι → ο be given.
Assume H10: x18 (x7 x10 (x7 y12 (x7 y11 y13))).
set y19 to be λ x19 . x18
Apply H1 with
y11,
y12,
y13,
λ x20 x21 . y19 (x7 x10 x20) (x7 x10 x21) leaving 4 subgoals.
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 H10.
set y18 to be λ x18 x19 . y17 (x7 y8 x18) (x7 y8 x19)
Apply L10 with
λ x19 . y18 x19 y17 ⟶ y18 y17 x19 leaving 2 subgoals.
Assume H11: y18 y17 y17.
The subproof is completed by applying H11.
The subproof is completed by applying L10.
set y14 to be λ x14 x15 . y13 (x5 x7 x14) (x5 x7 x15)
Apply L9 with
λ x15 . y14 x15 y13 ⟶ y14 y13 x15 leaving 2 subgoals.
Assume H10: y14 y13 y13.
The subproof is completed by applying H10.
The subproof is completed by applying L9.
Let x10 of type ι → ι → ο be given.
Apply L8 with
λ x11 . x10 x11 y9 ⟶ x10 y9 x11.
Assume H9: x10 y9 y9.
The subproof is completed by applying H9.