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Proof step 195 from proof inv-gc-copy (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 196:
# abs-log(nx0, gstore ↓) = # ndx + 1, abs-log(nx1, gstore2 ↓) = [], abs-log(nx0, gstore0 ↓) = abs-log(nx0, gstore ↓), dom gstore1 = dom gstore3 ∪ jro, dom gstore0 = dom gstore, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0), nodes(gstore0 ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx], remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro3 = ∅, nx0.toSet ∩ dom gstore3 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, fi1.inos < pmaxino, fi2.inos < jmaxino0, p_os < pmaxino, jro2 < jmaxino3, fi2.inos < jmaxino3, fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].inos < jmaxino, fi2.inos < jmaxino, inv(gstore \ jro3), inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(gstore0 \ jro3), key-cons(ndx), key-cons(nd.key, nd), key-cons(ndx1), new(abs-log(nx0, gstore ↓), nodes(gstore2 ↓, jro0)), nodes-cons(abs-log(nx0, gstore ↓), nd ' + ndx, gstore, jro3), refsize-cons(gstore1, range fi1), refsize-cons(gstore, range fi2), refsize-cons(gstore2, range fi2), refsize-cons(gstore3, range fi2), refsize-cons(gstore0, ((range fi2 \ fi2[ndx.keys].toSet) -- fi2[nd.key]) ∪ abs-log(nx0, gstore ↓).toSet), ri-inj(fi2), ri-inj(fi1), ri-inj(fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]), nd.key ∈ fi2, jmaxino3 ≤ jmaxino, pmaxino ≤ jmaxino0, nd.flashsize + ndx.flashsize ≤ LEB_SIZE, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, jmaxino0 ≤ jmaxino, pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore1, fi2.lebs ⊆ dom gstore, dom gstore3 ⊆ dom gstore, dom gstore2 ⊆ dom gstore3, range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, dom gstore2 ⊆ dom gstore, fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)].lebs ⊆ dom gstore, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3), jvalid7 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore0, jro3), ndx1 = [] ∨ LEB_SIZE < (nd.flashsize + ndx.flashsize) + ndx1.head.flashsize ⊦ jmaxino = 0, jmaxino0 = 0, jmaxino3 = 0, pmaxino = 0, dups(abs-log(nx0, gstore ↓)), dups(nx0), dups(nx1), dups(ndx.keys + ndx1.keys), dups(ndx.keys), dups(ndx1.keys), nd.key ∈ ndx1.keys, nd.key ∈ ndx.keys, (range fi2 \ fi2[ndx.keys].toSet) -- fi2[nd.key] ⊆ dom nodes(gstore2 ↓, jro0) ∪ abs-log(nx0, gstore ↓).toSet

Rule argument

Additional information

Used simplifier rules: unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]% ⊦ as ∪ as0 ⊆ as1 ∪ as0 ↔ as ⊆ as1 ∪ as0; (s)%# ax = # ndx ⊦ dom fs[ax, ndx] = dom fs ∪ ax.toSet; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ m +1 = m + 1; (s)% ⊦ #(nd ' + ndx) = (# ndx)+1; (s)