[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 14 from proof refine-add-gnds (previous step)

# lpt0 = # avol, # lpt = # avol, gstore[log1.last] = groupblock(gndx, lax, m), lpt[log1.last] = mklp(m, m0, LP_GROUP_NODES, m1), keyset(gch) = gc-heap(lpt, log1), remove-refsize0(istore0) = istore0, remove-refsize0(gstore0) = gstore0, elems log1 ∩ ginval = ∅, dom gstore0 ∪ elems log1 = dom gstore ∪ ginval, log1.last < # avol, ginval < # avol, abs-gstore(avol, lpt, gstore, ginval), abs-istore(avol, lpt, istore), awbuf-inv(avol, buffered(log1.last)), bin-heap-inv(gch, lpt), buf-ro-rw-mode(false, buffered(log1.last), lpt, avol), bufs-log-only(buffered(log1.last), log1, lpt), flash-lpt-cons(lpt0, istore0, gstore0), freelist-cons(lpt, freelist), inv(groupblock(gndx, lax, m)), inv(istore0), inv(istore), inv(gstore0), inv(gstore \ ginval), log-group-block(log1, lpt), lpt-free-unref(lpt0), lpt-free-unref(lpt), lpt-leb-size(lpt0), lpt-leb-size(lpt), lpt-ram-flash(lpt, lpt0, log1), log1.last ∈ gstore, istore0 ⊑ istore, isbuf → synced(avol) ⊦ dups(log1), ∀ nx0, n. log1 ≠ nx0 + n ', avol[log1.last].mapped? ∧ (gndx0 ≠ [] ∧ m0 + gndx0.flashsize ≤ LEB_SIZE → m0 = # avol[log1.last].data)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
# lpt = # avol, # lpt0 = # avol, (elems nx ∪ dom gstore0) ++ n = dom gstore ∪ ginval, lpt[n] = mklp(m, m0, LP_GROUP_NODES, m1), gstore[n] = groupblock(gndx, lax, m), keyset(gch) = gc-heap(lpt, nx + n '), remove-refsize0(gstore0) = gstore0, remove-refsize0(istore0) = istore0, ginval ∩ elems nx = ∅, ginval < # avol, n < # avol, abs-gstore(avol, lpt, gstore, ginval), abs-istore(avol, lpt, istore), awbuf-inv(avol, buffered(n)), bin-heap-inv(gch, lpt), buf-ro-rw-mode(false, buffered(n), lpt, avol), bufs-log-only(buffered(n), nx + n ', lpt), flash-lpt-cons(lpt0, istore0, gstore0), freelist-cons(lpt, freelist), inv(gstore \ ginval), inv(gstore0), inv(istore), inv(istore0), inv(groupblock(gndx, lax, m)), log-group-block(nx + n ', lpt), lpt-free-unref(lpt), lpt-free-unref(lpt0), lpt-leb-size(lpt), lpt-leb-size(lpt0), lpt-ram-flash(lpt, lpt0, nx + n '), n ∈ gstore, istore0 ⊑ istore, isbuf → synced(avol) ⊦ dups(nx), n ∈ ginval, n ∈ nx, avol[n].mapped? ∧ (gndx0 ≠ [] ∧ m0 + gndx0.flashsize ≤ LEB_SIZE → m0 = # avol[n].data)

Rule argument

Additional information

Used simplifier rules: fwd-in-gstore-lpt: ⊦ n ∈ gstore ∧ abs-gstore(avol, lpt, gstore, ginval) → n < # lpt ∧ lpt[n].ref-size = gstore[n].ref-size ∧ lpt[n].flags = LP_GROUP_NODES;% ⊦ mklp(n2, n3, lpf, m).flags = lpf; (s)% ⊦ groupblock(gndx, lax, m).ref-size = m; (s)% ⊦ mklp(n2, n3, lpf, m).ref-size = n2; (s)%fwd-inv-gblock-rem: ⊦ inv(gstore \ ginval) ∧ n ∈ gstore ∧ ¬ n ∈ ginval ∧ gstore[n] = gblock → inv(gblock); (forward) from specification apersistence-base%inv-buf-in-vol: ⊦ awbuf-inv(avol, buffered(n)) → n < # avol; (forward) from specification awbuf-inv%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ dups(nx + n ') ↔ n ∈ nx ∨ dups(nx); (ws)% ⊦ (nx + n ').last = n; (s)%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]% ⊦ ns ∩ (ns0 ++ n) = ∅ ↔ ns ∩ ns0 = ∅ ∧ ¬ n ∈ ns; (ws)%unc: ⊦ ns ∪ ns0 = ns0 ∪ ns; from specification set-union[nat-set]%una: ⊦ (ns ∪ ns0) ∪ ns1 = ns ∪ ns0 ∪ ns1; from specification set-union[nat-set]% ⊦ (ns ++ n) ∪ ns0 = (ns ∪ ns0) ++ n; (s)%a: ⊦ (nx + nx0) + nx1 = nx + nx0 + nx1; from specification list[nat-list-set]% ⊦ elems(nx0 + n ') = elems nx0 ++ n; (s)