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Proof step 9 from proof is-encoded-split-one (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
# buf1 = ENCODED_NAT_SIZE, # buf0 = # ns * ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # buf = (# ns1 - 1) * ENCODED_NAT_SIZE, buf0 + buf1 = buf2 + buf, is-encoded(ns, buf0), is-encoded(n, buf1), is-encoded(ns1 -- n0, buf), is-encoded(n0, buf2), n0 ∈ ns1, ∀ 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 = ∅, ns1 = ns ++ n, n ∈ ns

Rule argument

Additional information

Used simplifier rules: is-encoded-size: ⊦ is-encoded(ns, buf) → # buf = # ns * ENCODED_NAT_SIZE;% ⊦ n ∈ ns → #(ns -- n) = # ns - 1; (s)%is-encoded-size: ⊦ is-encoded(n, buf) → # buf = ENCODED_NAT_SIZE; (forward) from specification encoding-fixed-size[encoding-nat]