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.
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.
Apply unknownprop_3ca134a19cf015f53883d5110009b2043bd1804f66e2ef223744b96b4812e6cd with
x0,
x1,
x2,
x3,
x4,
x5,
x1 x6 x7,
x8,
x9,
λ x10 x11 . x11 = x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x6 (x1 x7 (x1 x8 x9)))))) leaving 10 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.
Apply H0 with
x6,
x7 leaving 2 subgoals.
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.
set y10 to be ...
set y11 to be ...
Claim L10: ∀ x12 : ι → ο . x12 y11 ⟶ x12 y10
Let x12 of type ι → ο be given.
Assume H10: x12 (x3 x4 (x3 x5 (x3 x6 (x3 x7 (x3 x8 (x3 x9 (x3 y10 y11))))))).
set y13 to be ...
set y14 to be ...
set y15 to be ...
Claim L11: ∀ x16 : ι → ο . x16 y15 ⟶ x16 y14
Let x16 of type ι → ο be given.
Assume H11: x16 (x5 x7 (x5 x8 (x5 x9 (x5 y10 (x5 y11 (x5 x12 y13)))))).
set y17 to be ...
set y18 to be ...
set y19 to be ...
Claim L12: ∀ x20 : ι → ο . x20 y19 ⟶ x20 y18
Let x20 of type ι → ο be given.
Assume H12: x20 (x7 y10 (x7 y11 (x7 x12 (x7 y13 (x7 y14 y15))))).
set y21 to be ...
set y22 to be ...
set y23 to be ...
Claim L13: ∀ x24 : ι → ο . x24 y23 ⟶ x24 y22
Let x24 of type ι → ο be given.
Assume H13: x24 (x9 y13 (x9 y14 (x9 y15 (x9 x16 y17)))).
set y25 to be ...
Apply unknownprop_4aef431da355638d092d1af3952763e46a0de88399b3400cacc13c5390d4cf48 with
x8,
x9,
y14,
y15,
x16,
y17,
λ x26 x27 . y25 (x9 y13 x26) (x9 y13 x27) leaving 7 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
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 H13.
set y24 to be λ x24 x25 . y23 (x9 x12 x24) (x9 x12 x25)
Apply L13 with
λ x25 . y24 x25 y23 ⟶ y24 y23 x25 leaving 2 subgoals.
Assume H14: y24 y23 y23.
The subproof is completed by applying H14.
The subproof is completed by applying L13.
set y20 to be λ x20 x21 . y19 (x7 x9 x20) (x7 x9 x21)
Apply L12 with
λ x21 . y20 x21 y19 ⟶ y20 y19 x21 leaving 2 subgoals.
Assume H13: y20 y19 y19.
The subproof is completed by applying H13.
The subproof is completed by applying L12.
set y16 to be λ x16 x17 . y15 (x5 x6 x16) (x5 x6 x17)
Apply L11 with
λ x17 . y16 x17 y15 ⟶ y16 y15 x17 leaving 2 subgoals.
Assume H12: y16 y15 y15.
The subproof is completed by applying H12.
The subproof is completed by applying L11.
Let x12 of type ι → ι → ο be given.
Apply L10 with
λ x13 . x12 x13 y11 ⟶ x12 y11 x13.
Assume H11: x12 y11 y11.
The subproof is completed by applying H11.