Let x0 of type ι be given.

Let x1 of type ι be given.

Assume H0: nat_p x1.

Apply unknownprop_b1346dc02c417c09673835f042d97ea29d427978c21a77e3cf957f238d582b43 with x0, ordsucc (ordsucc x1), λ x2 x3 . x3 = ordsucc (ordsucc (ordsucc (ordsucc (add_nat x0 x1)))) leaving 2 subgoals.

Apply unknownprop_a28d7ee32146a0c35a46897916c6589ba569d17fb166d34d873cce1f81ab1ec9 with x1.

The subproof is completed by applying H0.

set y2 to be ordsucc (ordsucc (add_nat x0 (ordsucc (ordsucc x1))))

set y3 to be ordsucc (ordsucc (ordsucc (ordsucc (add_nat x1 y2))))

Claim L1: ∀ x4 : ι → ο . x4 y3 ⟶ x4 y2

Let x4 of type ι → ο be given.

Assume H1: x4 (ordsucc (ordsucc (ordsucc (ordsucc (add_nat y2 y3))))).

set y5 to be λ x5 . x4

set y6 to be ordsucc (add_nat y2 (ordsucc (ordsucc y3)))

set y7 to be ordsucc (ordsucc (ordsucc (add_nat y3 x4)))

Claim L2: ∀ x8 : ι → ο . x8 y7 ⟶ x8 y6

Let x8 of type ι → ο be given.

Assume H2: x8 (ordsucc (ordsucc (ordsucc (add_nat x4 y5)))).

set y9 to be λ x9 . x8

Apply unknownprop_b1346dc02c417c09673835f042d97ea29d427978c21a77e3cf957f238d582b43 with x4, y5, λ x10 x11 . y9 (ordsucc x10) (ordsucc x11) leaving 2 subgoals.

The subproof is completed by applying H0.

The subproof is completed by applying H2.

set y8 to be λ x8 x9 . y7 (ordsucc x8) (ordsucc x9)

Apply L2 with λ x9 . y8 x9 y7 ⟶ y8 y7 x9 leaving 2 subgoals.

Assume H3: y8 y7 y7.

The subproof is completed by applying H3.

The subproof is completed by applying L2.

Let x4 of type ι → ι → ο be given.

Apply L1 with λ x5 . x4 x5 y3 ⟶ x4 y3 x5.

Assume H2: x4 y3 y3.

The subproof is completed by applying H2.

■

Proofgold Proof