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Proof step 2 from proof inv-commit (previous step)

# lpt + sb.main = # avol, avol0 = avol, m = EB_PAGE_SIZE, m0 = LEB_SIZE, ns = orphans, sb0 = sb, ebm-vol-inv(avol), ∃ lpt, log, orphans. persistence-io-inv(sb, logoff, avol, lpt, log, orphans) ⊦ ⟪persistence_io_commit#(lpt, adr, m1, m, m0; orphans, sb, avol, logoff; err)⟫ ( (∃ lpt, log, orphans. persistence-io-inv(sb, logoff, avol, lpt, log, orphans)) ∧ ebm-vol-inv(avol) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → sb.indexaddr = adr ∧ sb.maxino = m1 ∧ is-encoded([], avol[sb.log].data) ∧ is-encoded(lpt, lebs(avol, sb.lpt, lptlebs(# avol - sb.main, LEB_SIZE))) ∧ is-encoded-pad(ns, sb.orphsize, avol[sb.orph].data)) ∧ ( err ≠ ESUCCESS → sb = sb0 ∧ avol[sb.log] = avol0[sb.log] ∧ avol[sb.orph] = avol0[sb.orph] ∧ lebs(avol, sb.lpt, lptlebs(# avol - sb.main, LEB_SIZE)) = lebs(avol0, sb.lpt, lptlebs(# avol - sb.main, LEB_SIZE))) ∧ sb.main = sb0.main ∧ # avol = # avol0 ∧ subarray(avol, sb.main, # avol - sb.main) = subarray(avol0, sb.main, # avol - sb.main))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
# lpt0 = # lpt, sb.main + # lpt = # avol, avol[0] = mapped(buf), avol[sb.orph] = mapped(buf0), avol[sb.log] = mapped(buf1), avol0 = avol, ns = orphans, sb0 = sb, disk-layout(sb, # lpt), ebm-vol-inv(avol), is-encoded(log, buf1), is-encoded(lpt0, lebs(avol, sb.lpt, lptlebs(# lpt, LEB_SIZE))), is-encoded(sb, buf), is-encoded-pad(orphans0, sb.orphsize, buf0), mapped-full?(avol, sb.lpt, lptlebs(# lpt, LEB_SIZE)), logoff ≠ LEB_SIZE → # buf1 = logoff ⊦ ⟪persistence_io_commit#(lpt, adr, m1, EB_PAGE_SIZE, LEB_SIZE; orphans, sb, avol, logoff; err)⟫ ( (∃ lpt1, log, orphans. disk-layout(sb, # lpt1) ∧ sb.main + # lpt1 = # 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(# lpt1, LEB_SIZE)) ∧ is-encoded(lpt1, lebs(avol, sb.lpt, lptlebs(# lpt1, LEB_SIZE)))) ∧ ebm-vol-inv(avol) ∧ (err = ESUCCESS ∨ ⌊err⌋ ) ∧ ( err = ESUCCESS → sb.indexaddr = adr ∧ sb.maxino = m1 ∧ avol[sb.log].data = mkbuf(0) ∧ is-encoded(lpt, lebs(avol, sb.lpt, lptlebs(# avol - sb.main, LEB_SIZE))) ∧ is-encoded-pad(ns, sb.orphsize, avol[sb.orph].data)) ∧ ( err ≠ ESUCCESS → sb = sb0 ∧ avol[sb0.log] = avol0[sb0.log] ∧ avol[sb0.orph] = avol0[sb0.orph] ∧ lebs(avol, sb0.lpt, lptlebs(# avol - sb0.main, LEB_SIZE)) = lebs(avol0, sb0.lpt, lptlebs(# avol - sb0.main, LEB_SIZE))) ∧ sb.main = sb0.main ∧ # avol = # avol0 ∧ subarray(avol, sb.main, # avol - sb.main) = subarray(avol0, sb.main, # avol - sb.main))

Rule argument

Additional information

Used simplifier rules: ⊦ mapped(buf).data = buf; (s)%eqs: ⊦ m = n + n0 ∧ m = n + n1 → n0 = n1; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ is-encoded([], buf) ↔ buf = mkbuf(0); (s)% ⊦ aleb.mapped? ↔ (∃ buf. aleb = mapped(buf)); (as)%⊦ 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)