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Proof step 95 from proof inv-gc-update-index (previous step)

# ax2 = # ndx1, # ax3 = # ndx2, gstore3[m] = groupblock(gndx, lax, m3), 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, fi2[ndx2.keys, ax3].inos ∪ (nd.key.inode? ⊃ ∅ ++ nd.key.ino;∅) < max(nd.key.ino + 1, jmaxino), jro1 < jmaxino1, fi2.inos < jmaxino1, p_os < pmaxino, fi2[ndx2.keys, ax3].inos < jmaxino, fi1.inos < pmaxino, gstore3[m, groupblock(gndx, lax, m3 + m1)] =○refsize gstore3, inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro0), inv(groupblock(gndx, lax, m3)), 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[m, groupblock(gndx, lax, m3 + m1)], range fi2[ndx2.keys, ax3] ++ @(m, m0, m1)), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx2.keys, ax3]), refsize-cons(gstore1, range fi1), ri-inj(fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]), ri-inj(fi2), ri-inj(fi1), nd.key ∈ fi2, @(m, m0, m1) ∈ gstore3, m ∈ gstore3, jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, jmaxino1 ≤ max(nd.key.ino + 1, jmaxino), pmaxino ≤ max(nd.key.ino + 1, 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, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore3[m, groupblock(gndx, lax, m3 + m1)], jro0) ⊦ (dom fi2 ∪ ndx2.keys.toSet) ++ nd.key = dom fi2, jmaxino = 0, jmaxino1 = 0, max(nd.key.ino + 1, jmaxino) = 0, pmaxino = 0, dups(nx), dups(ax3 + ax2), dups(ax3), dups(ax2), m ∈ jro0, @(m, m0, m1) ∈ fi2, @(m, m0, m1) ∈ ax2, @(m, m0, m1) ∈ ax3, nd.key ∈ fi2[ndx2.keys, ax3]

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