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.
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 unknownprop_2d20c41fcc9a118a0c3b9c5b1baa7c68fc24a7eb196062ddabd928ab672e2748 with
x0,
x1,
x2,
x3,
x1 x4 x5,
x6,
x7,
λ x8 x9 . x9 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 x7)))) leaving 8 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.
Apply H0 with
x4,
x5 leaving 2 subgoals.
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.
set y8 to be x1 x2 (x1 x3 (x1 (x1 x4 x5) (x1 x6 x7)))
set y9 to be x2 x3 (x2 x4 (x2 x5 (x2 x6 (x2 x7 y8))))
Claim L8: ∀ x10 : ι → ο . x10 y9 ⟶ x10 y8
Let x10 of type ι → ο be given.
Assume H8: x10 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 y8 y9))))).
set y11 to be λ x11 . x10
set y12 to be x3 x5 (x3 (x3 x6 x7) (x3 y8 y9))
set y13 to be x4 x6 (x4 x7 (x4 y8 (x4 y9 x10)))
Claim L9: ∀ x14 : ι → ο . x14 y13 ⟶ x14 y12
Let x14 of type ι → ο be given.
Assume H9: x14 (x5 x7 (x5 y8 (x5 y9 (x5 x10 y11)))).
set y15 to be λ x15 . x14
Apply unknownprop_4aef431da355638d092d1af3952763e46a0de88399b3400cacc13c5390d4cf48 with
x4,
x5,
y8,
y9,
x10,
y11,
λ x16 x17 . y15 (x5 x7 x16) (x5 x7 x17) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H9.
set y14 to be λ x14 x15 . y13 (x5 x6 x14) (x5 x6 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.