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Proof step 7 from proof is-encoded-split-one (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
# buf0 = # ns * ENCODED_NAT_SIZE, # buf1 = ENCODED_NAT_SIZE, is-encoded(n, buf1), is-encoded(ns, buf0), ∀ 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, (∃ n, buf2, buf3. n ∈ ns1 ∧ buf0 + buf1 = buf2 + buf3 ∧ is-encoded(n, buf2) ∧ is-encoded(ns1 -- n, buf3)) ↔ ns1 = ns ++ n, ns1 = ∅, n ∈ ns

Rule argument

Additional information

Used simplifier rules: size-zero: ⊦ # ns ≠ 0 → ns ≠ ∅; (forward) from specification set-basic[nat-set]%size-zero-01: ⊦ ns ≠ ∅ → # ns ≠ 0; (forward) from specification set-basic[nat-set]