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Proof step 20 from proof is-encoded-split-one (previous step)

# buf1 = ENCODED_NAT_SIZE, # buf2 = ENCODED_NAT_SIZE, # ns = m0 + 1, # buf3 = m0 * ENCODED_NAT_SIZE, # ns1 = 2 + m0, is-encoded(n, buf1), is-encoded(ns1 -- n1, buf3 + buf1), is-encoded(n1, buf2), is-encoded(ns -- n1, buf3), n1 ∈ ns, n1 ∈ ns1, ∀ ns0, buf0, ns, buf1, n. # ns0 < (m0 + 1) + 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

User applied rule apply induction

And obtained the following new goals

Proof step 21:
# ns1 = 2 + m0, # buf3 = m0 * ENCODED_NAT_SIZE, # ns = m0 + 1, # buf2 = ENCODED_NAT_SIZE, # buf1 = ENCODED_NAT_SIZE, is-encoded(ns -- n1, buf3), is-encoded(n1, buf2), is-encoded(ns1 -- n1, buf3 + buf1), is-encoded(n, buf1), n1 ∈ ns1, n1 ∈ ns, (is-encoded(ns1 -- n1, buf3 + buf1) ↔ ns1 -- n1 = (ns -- n1) ++ n) ∨ n ∈ ns -- n1, ∀ ns0, buf0, ns, buf1, n. # ns0 < (m0 + 1) + 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 = ns ++ n, ns1 = ∅, n ∈ ns

Rule argument

subst: [ns0, buf0, ns, buf1, n] → [ns1 -- n1, buf3, ns -- n1, buf1, n]

Additional information

Used simplifier rules: is-encoded-unique: ⊦ is-encoded(ns0, buf) → (is-encoded(ns1, buf) ↔ ns0 = ns1);% ⊦ n ∈ ns → #(ns -- n) = # ns - 1; (s)% ⊦ is-encoded(n, buf) → (is-encoded(n0, buf) ↔ n = n0); (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m * n = m * n0 ↔ m = 0 ∨ n = n0; (ws)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%n < n0 ⊦ (* n) ≤ m0 + (* n0); (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s) Used terms: ns1 -- n1, buf3, ns -- n1, buf1, n