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Proof step 12 from proof is-encoded-split-one (previous step)

# buf = m * ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # buf0 = # ns * ENCODED_NAT_SIZE, # buf1 = ENCODED_NAT_SIZE, # ns1 = m + 1, buf0 + buf1 = buf2 + buf, is-encoded(n0, buf2), is-encoded(ns1 -- n0, buf), is-encoded(n, buf1), is-encoded(ns, buf0), n0 ∈ ns1, ∀ ns0, buf0, ns, buf1, n. # ns0 < m + 1 → # 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 = ns ++ n, ns1 = ∅, n ∈ ns

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ ns ≠ ∅ → (is-encoded(ns, buf) ↔ (∃ n, buf0, buf1. n ∈ ns ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns -- n, buf1)))
Proof step 14:
# ns1 = m + 1, # buf1 = ENCODED_NAT_SIZE, # buf0 = # ns * ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # buf = m * ENCODED_NAT_SIZE, buf0 + buf1 = buf2 + buf, ns = ∅, 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 < m + 1 → # 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 = ∅, ns1 = ns ++ n, n ∈ ns
Proof step 17:
# ns1 = m + 1, # buf1 = ENCODED_NAT_SIZE, # buf0 = # ns * ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # buf = m * ENCODED_NAT_SIZE, buf0 + buf1 = buf2 + buf, is-encoded(n, buf1), is-encoded(ns1 -- n0, buf), is-encoded(n0, buf2), n0 ∈ ns1, ∃ n, buf2, buf1. n ∈ ns ∧ buf0 = buf2 + buf1 ∧ is-encoded(n, buf2) ∧ is-encoded(ns -- n, buf1), ∀ ns0, buf0, ns, buf1, n. # ns0 < m + 1 → # 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 ⊦ ns = ∅, ns1 = ∅, ns1 = ns ++ n, n ∈ ns

Rule argument

spec: <toplevel> name: is-encoded-rec seq: ⊦ ns ≠ ∅ → ( is-encoded(ns, buf) ↔ (∃ n, buf0, buf1. n ∈ ns ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns -- n, buf1))) subst: [buf, ns] → [buf0, ns] flags: <all paths>

Additional information

Used terms: buf0, ns