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Proofgold Proof

pf
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 → ι → ι be given.
Let x5 of type ι → ι → ι be given.
Let x6 of type ι → ι → ι be given.
Assume H0: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x1 x7 x8 = x2 x7 x8.
Assume H1: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x3 x7 x8 = x4 x7 x8.
Assume H2: ∀ x7 . x7 ∈ x0 ⟶ ∀ x8 . x8 ∈ x0 ⟶ x5 x7 x8 = x6 x7 x8.
Claim L3: ...
...
Apply L3 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) (encode_b x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) x7 (If_i (x9 = 2) (encode_b x0 x4) (encode_b x0 x6)))).
Claim L4: encode_b x0 x3 = encode_b ... ...
...
Apply L4 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) (encode_b x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) x7 (encode_b x0 x6)))).
Claim L5: encode_b x0 x5 = encode_b x0 x6
Apply encode_b_ext with x0, x5, x6.
The subproof is completed by applying H2.
Apply L5 with λ x7 x8 . lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) (encode_b x0 x5)))) = lam 4 (λ x9 . If_i (x9 = 0) x0 (If_i (x9 = 1) (encode_b x0 x1) (If_i (x9 = 2) (encode_b x0 x3) x7))).
Let x7 of type ι → ι → ο be given.
Assume H6: x7 (lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_b x0 x3) (encode_b x0 x5))))) (lam 4 (λ x8 . If_i (x8 = 0) x0 (If_i (x8 = 1) (encode_b x0 x1) (If_i (x8 = 2) (encode_b x0 x3) (encode_b x0 x5))))).
The subproof is completed by applying H6.
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