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Proof step 21 from proof inv-gc-update-index (previous step)

# ax2 = # ndx1, # ax3 = # ndx2, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], gstore4 = gstore, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax2.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, jmaxino ≠ 0, jmaxino1 ≠ 0, pmaxino ≠ 0, jro1 < jmaxino1, fi2.inos < jmaxino1, p_os < pmaxino, fi2[ndx2.keys, ax3].inos < jmaxino, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), key-cons(ndx1), key-cons(ndx2), key-cons(nd.key, nd), nodes-cons(ax2, ndx1, gstore, jro0), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi0), nd.key ∈ fi2, @(m, m0, m1) ∈ gstore, 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, fi0.lebs ⊆ dom gstore0, ¬ dups(ax2), ¬ dups(ax3), ¬ dups(ax3 + ax2), ¬ dups(nx), ¬ @(m, m0, m1) ∈ ax3, ¬ @(m, m0, m1) ∈ ax2, ¬ @(m, m0, m1) ∈ fi2, ¬ m ∈ jro0, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ [gjournal_index_store#(nd.key, @(m, m0, m1), fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore)] ( fi2[ndx2.keys, ax3].inos < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi2[ndx2.keys, ax3].inos ∪ (nd.key.inode? ⊃ ∅ ++ nd.key.ino;∅) < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ ri-inj(fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ gstore =○refsize gstore4 ∧ fi = fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] → ⟪ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] = fi2[ndx.keys, ax]) ∧ (dom fi2 ∪ ndx2.keys.toSet) ++ nd.key = dom fi2 ∧ jmaxino1 ≤ jmaxino ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore4 ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore4 ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ ¬ # ax2 < # ax0))

Heuristic nonrecursive calls applied rule call right

And obtained the following new goals

Proof step 22:
# ax3 = # ndx2, # ax2 = # ndx1, jvalid0 ↔ jvalid, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), adr0 = @(m, m0, m1), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], gstore4 = gstore, jro = jro0, key = nd.key, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, elems nx ∩ jro0 = ∅, range fi2 ∩ ax2.toSet = ∅, range fi2 ∩ ax3.toSet = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, fi0.inos < pmaxino, fi2[ndx2.keys, ax3].inos < jmaxino, p_os < pmaxino, fi2.inos < jmaxino1, jro1 < jmaxino1, inv(gstore \ jro0), inv(gstore0), inv(gstore2 \ jro0), key-cons(nd.key, nd), key-cons(ndx2), key-cons(ndx1), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), nodes-cons(ax2, ndx1, gstore, jro0), refsize-cons(gstore0, range fi0), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore2, range fi2), ri-inj(fi0), ri-inj(fi2), @(m, m0, m1) ∈ gstore, nd.key ∈ fi2, pmaxino ≤ jmaxino, pmaxino ≤ jmaxino1, jmaxino1 ≤ jmaxino, fi0.lebs ⊆ dom gstore0, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, ndx2.keys.toSet ⊆ dom fi2, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0) ⊦ [if ¬ adr0 ∈ fi ∧ adr0 ∈ gstore ∧ key-cons(key, gstore[adr0].nd) then let jvalid = ?, adr = ? in {index_index_store#(key, adr0, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; adr, jvalid, fi); let n = ? in {if jvalid then {index_get_gblock_refsize#(adr.lnum, fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; n); index_set_gblock_refsize#(adr.lnum, n - adr.size, fi, fi0, jro, gstore0, nx, pmaxino, p_os, jvalid0; gstore)}; index_get_gblock_refsize#(adr0.lnum, fi, fi0, jro, gstore, gstore0, nx, pmaxino, p_os, jvalid0; ; n); index_set_gblock_refsize#(adr0.lnum, n + adr0.size, fi, fi0, jro, gstore0, nx, pmaxino, p_os, jvalid0; gstore)}; jmaxino := max(key.ino + 1, jmaxino)} else abort] ( fi2[ndx2.keys, ax3].inos < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi2[ndx2.keys, ax3].inos ∪ (nd.key.inode? ⊃ ∅ ++ nd.key.ino;∅) < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ ri-inj(fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ gstore =○refsize gstore4 ∧ fi = fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] → ⟪ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] = fi2[ndx.keys, ax]) ∧ (dom fi2 ∪ ndx2.keys.toSet) ++ nd.key = dom fi2 ∧ jmaxino1 ≤ jmaxino ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore4 ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore4 ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ ¬ # ax2 < # ax0)), jmaxino = 0, jmaxino1 = 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

Rule argument

pos: right 1

Additional information

Used simplifier rules: store: ⊦ ;