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Proof step 4 from proof is-encoded-size (previous step)

is-encoded(ns, buf), ∀ ns0, buf. # ns0 < # ns → is-encoded(ns0, buf) → # buf = # ns0 * ENCODED_NAT_SIZE ⊦ # buf = # ns * ENCODED_NAT_SIZE

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ ns ≠ ∅ → (is-encoded(ns, buf) ↔ (∃ n, buf0, buf1. n ∈ ns ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns -- n, buf1)))
Proof step 6:
∃ n, buf0, buf1. n ∈ ns ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns -- n, buf1), ∀ ns0, buf. # ns0 < # ns → is-encoded(ns0, buf) → # buf = # ns0 * ENCODED_NAT_SIZE ⊦ # buf = # ns * ENCODED_NAT_SIZE, 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] → [buf, 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% ⊦ # mkbuf(n) = n; (s) Used terms: buf, ns