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Proof step 139 from proof inv-gc-copy (previous step)

# ax = # ndx1, # ax1 = # ndx + 1, LEB_SIZE = n1 + n0, abs-log(nx0, gstore0 ↓) = abs-log(nx0, gstore ↓), abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) + ax1, dom gstore0 = dom gstore, nodes(gstore0 ↓, jro3) = nodes(gstore ↓, jro3), nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro)[ax1, nd0 ' + ndx], remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro3 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, fi2[ndx1.keys, ax].inos < jmaxino, fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1].inos < jmaxino, jro2 < jmaxino3, fi2.inos < jmaxino3, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore0 \ jro3), inv(gstore \ jro3), inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), key-cons(ndx3), key-cons(ndx), key-cons(ndx1), key-cons(nd0.key, nd0), new(ax, nodes(gstore2 ↓, jro0)), new(ax1, nodes(gstore3 ↓, jro)), nodes-cons(ax1, nd0 ' + ndx, gstore, jro3), refsize-cons(gstore0, range fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1]), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), ri-inj(fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1]), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), nd0.key ∈ fi2, nx.last ∈ gstore3, pmaxino ≤ jmaxino, jmaxino0 ≤ jmaxino, nd0.flashsize ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ n1, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize ≤ n0, pmaxino ≤ jmaxino0, jmaxino3 ≤ jmaxino, fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1].lebs ⊆ dom gstore, dom gstore3 ⊆ dom gstore, fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, ndx3.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), dom gstore2 ⊆ dom gstore3, fi2.lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, fi1.lebs ⊆ dom gstore1, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore0, jro3), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), ndx3 = [] ∨ n1 < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ #((nx0.toSet \ nx1.toSet) ∩ dom gstore2) = 0, jmaxino = 0, jmaxino0 = 0, jmaxino3 = 0, nx = [], pmaxino = 0, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, dups(nx0), dups(ndx.keys), dups(ndx1.keys), dups(ndx1.keys + ndx.keys + ndx3.keys), dups(ax), dups(nx), dups(nx1), dups(ndx.keys + ndx3.keys), dups(ndx1.keys + ndx.keys), dups(ndx3.keys), dups(ax1), nd0.key ∈ ndx1.keys, nd0.key ∈ ndx.keys, nd0.key ∈ ndx3.keys

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: nat-set name: rem-inter-empty3 seq: ⊦ (ns \ ns0) ∩ ns1 = ∅ ∧ (ns2 \ ns) ∩ ns1 = ∅ → (ns2 \ ns0) ∩ ns1 = ∅ subst: [ns, ns1, ns0, ns2] → [nx.toSet, dom gstore2, nx1.toSet, nx0.toSet] flags: <all paths>

Additional information

Used simplifier rules: int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]% ⊦ ns ⊆ ns0 ∧ ns0 ∩ ns1 = ∅ → ns ∩ ns1 = ∅; (s) Used terms: nx.toSet, dom gstore2, nx1.toSet, nx0.toSet