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Proof step 30 from proof is-encoded-split-one (previous step)

# buf1 = ENCODED_NAT_SIZE, # buf0 = # ns * ENCODED_NAT_SIZE, ns1 = ns ++ n, is-encoded(ns, buf0), is-encoded(n, buf1), ∀ ns0, buf0, ns, buf1, n. # ns0 < # ns1 → # buf0 = # ns * ENCODED_NAT_SIZE ∧ # buf1 = ENCODED_NAT_SIZE ∧ is-encoded(n, buf1) ∧ is-encoded(ns, buf0) → (is-encoded(ns0, buf0 + buf1) ↔ ns0 = ns ++ n) ∨ n ∈ ns ⊦ # ns1 = 0, ns1 = ∅, n ∈ ns, ∃ n, buf2, buf3. n ∈ ns1 ∧ buf0 + buf1 = buf2 + buf3 ∧ is-encoded(n, buf2) ∧ is-encoded(ns1 -- n, buf3)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 31:
# buf0 = # ns * ENCODED_NAT_SIZE, # buf1 = ENCODED_NAT_SIZE, is-encoded(n, buf1), is-encoded(ns, buf0), ∀ ns0, buf0, ns2, buf1, n0. # ns0 < #(ns ++ n) → # buf0 = # ns2 * ENCODED_NAT_SIZE ∧ # buf1 = ENCODED_NAT_SIZE ∧ is-encoded(n0, buf1) ∧ is-encoded(ns2, buf0) → (is-encoded(ns0, buf0 + buf1) ↔ ns0 = ns2 ++ n0) ∨ n0 ∈ ns2 ⊦ n ∈ ns, ∃ n0, buf, buf2. (n0 = n ∨ n0 ∈ ns) ∧ buf0 + buf1 = buf + buf2 ∧ is-encoded(n0, buf) ∧ is-encoded((ns ++ n) -- n0, buf2)

Rule argument

Additional information

Used simplifier rules: ⊦ n ∈ ns ++ n0 ↔ n = n0 ∨ n ∈ ns; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ ¬ n ∈ ns → #(ns ++ n) = # ns + 1; (s)% ⊦ ∅ ≠ ns ++ n; (s)