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Proof step 14 from proof is-encoded-split-one (previous step)

# 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
# ns1 = m + 1, # buf = 0, # buf2 = ENCODED_NAT_SIZE, m * ENCODED_NAT_SIZE = 0, is-encoded(n, buf2), is-encoded(ns1 -- n, buf), n ∈ 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 = ∅ ++ n, ns1 = ∅

Rule argument

Additional information

Used simplifier rules: is-encoded-base: ⊦ is-encoded(∅, buf) ↔ buf = mkbuf(0);%is-encoded-size: ⊦ is-encoded(ns, buf) → # buf = # ns * ENCODED_NAT_SIZE;% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n ∈ ns → #(ns -- n) = # ns - 1; (s)% ⊦ is-encoded(n, buf) → (is-encoded(n0, buf) ↔ n = n0); (s)% ⊦ # buf = 0 → buf0 + buf = buf0; (s)% ⊦ n + 0 = n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m = m + n ↔ n = 0; (s)%is-encoded-size: ⊦ is-encoded(n, buf) → # buf = ENCODED_NAT_SIZE; (forward) from specification encoding-fixed-size[encoding-nat]% ⊦ # mkbuf(n) = n; (s)% ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ ¬ n ∈ ∅; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ # buf = 0 → buf + buf0 = buf0; (s)%mult-c: ⊦ m * n = n * m; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult% ⊦ 0 * n = 0; (s)% ⊦ # ∅ = 0; (s)