Let x0 of type ι be given.

Let x1 of type ι → ι → ι be given.

Let x2 of type ι → ι → ι be given.

Let x3 of type ι → ι → ι be given.

Let x4 of type ι be given.

Let x5 of type ι → ι → ι be given.

Let x6 of type ι → ι → ι → ι be given.

Let x7 of type ι → ι → ι be given.

Let x8 of type ι → ι → ι → ι be given.

Let x9 of type ι → ι → ι → ι be given.

Let x10 of type ι → ι → ι be given.

Let x11 of type ι → ι → ι be given.

Let x12 of type ι → ι → ι be given.

Let x13 of type ι → ι → ι be given.

Assume H0: Loop_with_defs x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13.

Assume H1: In x4 x0.

Let x14 of type ο be given.

Assume H2: ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ ... ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ ∀ x17 . In x17 x0 ⟶ In (x9 x15 x16 x17) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x10 x15 x16 = x1 x15 (x1 x16 (x2 x15 x4))) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x10 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x12 x15 x16 = x1 (x2 x15 x16) (x2 (x2 x15 x4) x4)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x12 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x11 x15 x16 = x1 (x1 (x3 x4 x15) x16) x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x11 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x13 x15 x16 = x1 (x3 x4 (x3 x4 x15)) (x3 x16 x15)) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ In (x13 x15 x16) x0) ⟶ (∀ x15 . In x15 x0 ⟶ x1 x4 x15 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ x1 x15 x4 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x2 x15 (x1 x15 x16) = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x1 x15 (x2 x15 x16) = x16) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x3 (x1 x15 x16) x16 = x15) ⟶ (∀ x15 . In x15 x0 ⟶ ∀ x16 . In x16 x0 ⟶ x1 (x3 x15 x16) x16 = x15) ⟶ x14.

...

■

Proofgold Proof