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Proof step 30 from proof prepost-decode-orphans (previous step)

# buf1 = ENCODED_NAT_SIZE, ENCODED_NAT_SIZE * m2 = # buf2, # buf0 + ENCODED_NAT_SIZE + ENCODED_NAT_SIZE * m2 = # orphans1 * ENCODED_NAT_SIZE, orphans ∩ orphans0 = ∅, is-encoded(orphans, buf1 + buf2), is-encoded(orphans0, buf0), is-encoded(orphans1, buf0 + buf1 + buf2), is-encoded(n, buf1) ⊦ is-encoded(orphans0 ++ n, buf0 + buf1) ∧ (∃ orphans2. is-encoded(orphans2, buf2) ∧ orphans2 ∩ (orphans0 ++ n) = ∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: encoding-orphans 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] → [buf1 + buf2, orphans] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ns ∩ ∅ = ∅; (s)%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]% ⊦ is-encoded(∅, buf) ↔ buf = mkbuf(0); (s)% ⊦ buf + buf0 = mkbuf(0) ↔ buf = mkbuf(0) ∧ buf0 = mkbuf(0); (ws)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ # mkbuf(n) = n; (s)% ⊦ ENCODED_NAT_SIZE ≠ 0; (s) Used terms: buf1 + buf2, orphans