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Proof step 2 from proof refine-unmap (previous step)

# lpt = # avol, # avol0 = sb.main + # avol, adr = sb.indexaddr, avol = subarray(avol0, sb.main, # avol0 - sb.main), m0 = sb.maxino, m1 = EB_PAGE_SIZE, m2 = LEB_SIZE, m < # avol, ebm-vol-inv(avol0), ebm-vol-inv(avol), is-encoded(lpt, lebs(avol0, sb.lpt, lptlebs(# avol, LEB_SIZE))), is-encoded(plog, avol0[sb.log].data), is-encoded-pad(ns, sb.orphsize, avol0[sb.orph].data), ∃ lpt0, log, orphans. persistence-io-inv(sb, logoff, avol0, lpt0, log, orphans) ⊦ ⟪persistence_io_unmap#(m, sb; avol0)⟫ ⟨apersistence_io_unmap#(m, adr, m0, plog, lpt, ns; avol)⟩ ( m0 = sb.maxino ∧ adr = sb.indexaddr ∧ # avol0 = sb.main + # avol ∧ avol = subarray(avol0, sb.main, # avol0 - sb.main) ∧ is-encoded(plog, avol0[sb.log].data) ∧ is-encoded(lpt, lebs(avol0, sb.lpt, lptlebs(# avol, LEB_SIZE))) ∧ is-encoded-pad(ns, sb.orphsize, avol0[sb.orph].data))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
# lpt = # avol0 - sb.main, sb.main + # lpt0 = # avol0, avol0[0] = mapped(buf), avol0[sb.orph] = mapped(buf0), avol0[sb.log] = mapped(buf1), avol = subarray(avol0, sb.main, # lpt), m < # lpt0, disk-layout(sb, # lpt0), ebm-vol-inv(subarray(avol0, sb.main, # lpt)), ebm-vol-inv(avol0), is-encoded(log, buf1), is-encoded(lpt, lebs(avol0, sb.lpt, lptlebs(# lpt, LEB_SIZE))), is-encoded(sb, buf), is-encoded(lpt0, lebs(avol0, sb.lpt, lptlebs(# lpt0, LEB_SIZE))), is-encoded-pad(orphans, sb.orphsize, buf0), mapped-full?(avol0, sb.lpt, lptlebs(# lpt0, LEB_SIZE)), logoff ≠ LEB_SIZE → # buf1 = logoff ⊦ ⟪persistence_io_unmap#(m, sb; avol0)⟫ ⟨apersistence_io_unmap#(m, sb.indexaddr, sb.maxino, log, lpt, orphans; avol)⟩ ( # avol0 = sb.main + (# avol0 - sb.main) ∧ avol = subarray(avol0, sb.main, # avol0 - sb.main) ∧ is-encoded(log, avol0[sb.log].data) ∧ is-encoded(lpt, lebs(avol0, sb.lpt, lptlebs(# avol0 - sb.main, LEB_SIZE))) ∧ is-encoded-pad(orphans, sb.orphsize, avol0[sb.orph].data))

Rule argument

Additional information

Used simplifier rules: ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s)% ⊦ mapped(buf).data = buf; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)%# avol = sb.main + mainareasize ⊦ disk-layout(sb, mainareasize) → # avol ≠ 0; (s)%n0 ≤ n ⊦ m < n - n0 ↔ m + n0 < n; (s)%n ≤ m ⊦ (m - n) + n0 = n1 ↔ m + n0 = n + n1; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ ¬ m + n < m; (s)% ⊦ m < n → n ≠ 0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ aleb.mapped? ↔ (∃ buf. aleb = mapped(buf)); (as)% ⊦ is-encoded(nx0, buf) → (is-encoded(nx1, buf) ↔ nx0 = nx1); (s)%⊦ is-encoded-pad(ns0, n, buf) → (is-encoded-pad(ns1, n, buf) ↔ ns0 = ns1); (s)%⊦ persistence-io-inv(sb, logoff, avol, lpt, log, orphans) ↔ (∃ mainareasize. disk-layout(sb, mainareasize) ∧ sb.main + mainareasize = # avol ∧ avol[0].mapped? ∧ is-encoded(sb, avol[0].data) ∧ avol[sb.orph].mapped? ∧ is-encoded-pad(orphans, sb.orphsize, avol[sb.orph].data) ∧ avol[sb.log].mapped? ∧ (logoff ≠ LEB_SIZE → # avol[sb.log].data = logoff) ∧ is-encoded(log, avol[sb.log].data) ∧ mapped-full?(avol, sb.lpt, lptlebs(mainareasize, LEB_SIZE)) ∧ # lpt = mainareasize ∧ is-encoded (lpt, lebs(avol, sb.lpt, lptlebs(mainareasize, LEB_SIZE)))); (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ # subarray(avol, m, n) = n; (s)