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Proof step 5 from proof is-encoded-unique (previous step)

# buf = # ns0 * ENCODED_NAT_SIZE, # buf = # ns1 * ENCODED_NAT_SIZE, is-encoded(ns1, buf), is-encoded(ns0, buf), ∀ ns, buf, ns1. # ns < # ns0 → # buf = # ns1 * ENCODED_NAT_SIZE ∧ # buf = # ns * ENCODED_NAT_SIZE ∧ is-encoded(ns, buf) ∧ is-encoded(ns1, buf) → ns = ns1 ⊦ ns0 = ns1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ ns ≠ ∅ → (is-encoded(ns, buf) ↔ (∃ n, buf0, buf1. n ∈ ns ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns -- n, buf1)))
Proof step 7:
# buf = # ns1 * ENCODED_NAT_SIZE, # buf = # ns0 * ENCODED_NAT_SIZE, is-encoded(ns1, buf), ∃ n, buf0, buf1. n ∈ ns0 ∧ buf = buf0 + buf1 ∧ is-encoded(n, buf0) ∧ is-encoded(ns0 -- n, buf1), ∀ ns, buf, ns1. # ns < # ns0 → # buf = # ns1 * ENCODED_NAT_SIZE ∧ # buf = # ns * ENCODED_NAT_SIZE ∧ is-encoded(ns, buf) ∧ is-encoded(ns1, buf) → ns = ns1 ⊦ ns0 = ns1, ns0 = ∅

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, ns0] 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%size-zero-01: ⊦ ns ≠ ∅ → # ns ≠ 0; (forward) from specification set-basic[nat-set]% ⊦ # mkbuf(n) = n; (s)% ⊦ m ≠ 0 → (m * n = 0 ↔ n = 0); (s)% ⊦ ENCODED_NAT_SIZE ≠ 0; (s) Used terms: buf, ns0