[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 69 from proof inv-gc-update-index (previous step)

# ax2 = # ndx1, # ax3 = # ndx2, gstore3[m3, groupblock(gndx, lax, n1)][m] = groupblock(gndx0, lax0, m6), gstore3[m3] = groupblock(gndx, lax, n1 + m5), fi2[ndx2.keys, ax3][nd.key] = @(m3, m4, m5), gstore3[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore3 ↓) = abs-log(nx, gstore2 ↓), dom gstore3 = dom gstore2, nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore1) = gstore1, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax2.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx = (dom gstore2 ∪ jro0) ++ m, jro1 < jmaxino1, fi2.inos < jmaxino1, p_os < pmaxino, fi2[ndx2.keys, ax3].inos < jmaxino, fi1.inos < pmaxino, fi2[ndx2.keys, ax3].inos ∪ (nd.key.inode? ⊃ ∅ ++ nd.key.ino;∅) < max(nd.key.ino + 1, jmaxino), gstore3[m3, groupblock(gndx, lax, n1)][m, groupblock(gndx0, lax0, m6 + m1)] =○refsize gstore3, inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro0), inv(gstore3[m3, groupblock(gndx, lax, n1)][m, groupblock(gndx0, lax0, m6 + m1)] \ jro0), key-cons(ndx1), key-cons(ndx2), key-cons(nd.key, nd), nodes-cons(ax2, ndx1, gstore3, jro0), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore3[m3, groupblock(gndx, lax, n1)][m, groupblock(gndx0, lax0, m6 + m1)], range fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx2.keys, ax3]), refsize-cons(gstore1, range fi1), ri-inj(fi2), ri-inj(fi1), ri-inj(fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]), m ∈ gstore2, m ∈ gstore3[m3, groupblock(gndx, lax, n1)], m3 ∈ gstore3, m3 ∈ gstore2, nd.key ∈ fi2, @(m, m0, m1) ∈ gstore3, nd.key ∈ fi2[ndx2.keys, ax3], pmaxino ≤ max(nd.key.ino + 1, jmaxino), jmaxino1 ≤ max(nd.key.ino + 1, jmaxino), jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3[m3, groupblock(gndx, lax, n1)][m, groupblock(gndx0, lax0, m6 + m1)], jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro0) ⊦ (dom fi2 ∪ ndx2.keys.toSet) ++ nd.key = dom fi2, jmaxino = 0, jmaxino1 = 0, max(nd.key.ino + 1, jmaxino) = 0, pmaxino = 0, dups(ax2), dups(ax3), dups(ax3 + ax2), dups(nx), @(m, m0, m1) ∈ ax3, @(m, m0, m1) ∈ ax2, @(m, m0, m1) ∈ fi2, m ∈ jro0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: node-index name: union-sub seq: ⊦ ks ⊆ ks0 → ks0 ∪ ks = ks0 subst: [ks, ks0] → [ndx2.keys.toSet, dom fi2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ∈ ri → dom ri ++ key = dom ri; (s) Used terms: ndx2.keys.toSet, dom fi2