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 → ο be given.

Let x4 of type ι → ι → ο be given.

Let x5 of type ι be given.

Assume H0: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ x1 x6 x7 = x2 x6 x7.

Assume H1: ∀ x6 . x6 ∈ x0 ⟶ ∀ x7 . x7 ∈ x0 ⟶ iff (x3 x6 x7) (x4 x6 x7).

Claim L2: encode_b x0 x1 = encode_b x0 x2

Apply encode_b_ext with x0, x1, x2.

The subproof is completed by applying H0.

Apply L2 with λ x6 x7 . lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_r x0 x3) x5))) = lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) x6 (If_i (x8 = 2) (encode_r x0 x4) x5))).

Claim L3: encode_r x0 x3 = encode_r x0 x4

Apply encode_r_ext with x0, x3, x4.

The subproof is completed by applying H1.

Apply L3 with λ x6 x7 . lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_r x0 x3) x5))) = lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) x6 x5))).

Let x6 of type ι → ι → ο be given.

Assume H4: x6 (lam 4 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) (encode_b x0 x1) (If_i (x7 = 2) (encode_r x0 x3) x5)))) (lam 4 (λ x7 . If_i (x7 = 0) x0 (If_i (x7 = 1) (encode_b x0 x1) (If_i (x7 = 2) (encode_r x0 x3) x5)))).

The subproof is completed by applying H4.

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Proofgold Proof