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Proof step 31 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, ns2, buf1, n0. # ns0 < #(ns ++ n) → # buf0 = # ns2 * ENCODED_NAT_SIZE ∧ # buf1 = ENCODED_NAT_SIZE ∧ is-encoded(n0, buf1) ∧ is-encoded(ns2, buf0) → (is-encoded(ns0, buf0 + buf1) ↔ ns0 = ns2 ++ n0) ∨ n0 ∈ ns2 ⊦ n ∈ ns, ∃ n0, buf, buf2. (n0 = n ∨ n0 ∈ ns) ∧ buf0 + buf1 = buf + buf2 ∧ is-encoded(n0, buf) ∧ is-encoded((ns ++ n) -- n0, buf2)

User applied rule insert rewrite lemma

And obtained the following new goals

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