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Proof step 144 from proof inv-gc-copy (previous step)

# ax = # ndx1, LEB_SIZE = n1 + n0, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓), abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro), nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], remove-refsize0(gstore1) = gstore1, elems nx ∩ jro = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx0 ∩ jro3 = ∅, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, fi2.inos < jmaxino3, jro2 < jmaxino3, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(gstore3[nx.last]), inv(gstore \ jro3), key-cons(nd0.key, nd0), key-cons(ndx1), key-cons(ndx), key-cons(ndx3), new(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore, range fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi1), nx.last ∈ gstore3, nd0.key ∈ fi2, nd0.flashsize ≤ n0, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, nd0.flashsize + ndx.flashsize ≤ n1, nd0.flashsize ≤ LEB_SIZE, pmaxino ≤ jmaxino0, dom gstore2 ⊆ dom gstore, fi2.lebs ⊆ dom gstore, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore3, range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx.keys.toSet ⊆ dom fi2, ndx3.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, fi2[ndx1.keys, ax].lebs ⊆ dom gstore3, fi1.lebs ⊆ dom gstore1, dom gstore3 ⊆ dom gstore, ⌊err⌋ , jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), jvalid1 → valid-journal-head(nx, gstore3, jro), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), ndx3 = [] ∨ n1 < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ jmaxino3 = 0, nx = [], pmaxino = 0, dups(ndx3.keys), dups(ndx1.keys + ndx.keys), dups(ndx.keys + ndx3.keys), dups(nx1), dups(nx), dups(ax), dups(ndx1.keys + ndx.keys + ndx3.keys), dups(ndx1.keys), dups(ndx.keys), dups(nx0), nd0.key ∈ ndx3.keys, nd0.key ∈ ndx.keys, nd0.key ∈ ndx1.keys, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax0, ndx0, ndx2. ndx1 + nd0 ' + ndx + ndx3 = ndx0 + ndx2 ∧ new(ax0, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax0) ∧ # ax0 = # ndx0 ∧ fi2[ndx1.keys, ax] = fi2[ndx0.keys, ax0] ∧ nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, ndx0] ∧ abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 145:
⊦ (ns \ ns0) ∩ ns1 = ∅ ∧ (ns2 \ ns) ∩ ns1 = ∅ → (ns2 \ ns0) ∩ ns1 = ∅
Proof step 146:
# ax = # ndx1, LEB_SIZE = n1 + n0, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓), nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro), remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro3 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx1 ∩ jro0 = ∅, elems nx ∩ jro = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore2 = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, fi1.inos < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, p_os < pmaxino, jro2 < jmaxino3, fi2.inos < jmaxino3, 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)), refsize-cons(gstore, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore2, range fi2), ri-inj(fi1), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi2), nd0.key ∈ fi2, nx.last ∈ gstore3, pmaxino ≤ jmaxino0, nd0.flashsize ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ n1, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize ≤ n0, dom gstore3 ⊆ dom gstore, fi1.lebs ⊆ dom gstore1, 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[ndx1.keys, ax].lebs ⊆ dom gstore, fi2.lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, ⌊err⌋ , 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 ⊦ jmaxino3 = 0, nx = [], pmaxino = 0, 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), nd0.key ∈ ndx1.keys, nd0.key ∈ ndx.keys, nd0.key ∈ ndx3.keys, ∅ = ∅ ∧ (∃ ax0, ndx0, ndx2. ndx1 + nd0 ' + ndx + ndx3 = ndx0 + ndx2 ∧ new(ax0, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax0) ∧ # ax0 = # ndx0 ∧ fi2[ndx1.keys, ax] = fi2[ndx0.keys, ax0] ∧ nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, ndx0] ∧ abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0)

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