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Proof step 191 from proof inv-gc-copy (previous step)

# 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(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[abs-log(nx0, gstore ↓), nd ' + ndx], nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0), nodes(gstore0 ↓, jro3) = nodes(gstore ↓, jro3), 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[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)]), ri-inj(fi1), ri-inj(fi2), 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 ∈ ndx.keys, nd.key ∈ ndx1.keys, range fi2[nd.key ' + ndx.keys, abs-log(nx0, gstore ↓)] ⊆ dom nodes(gstore ↓, jro3)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: gjournal-base name: index-update-range seq: # kx = # ax ⊦ ri-inj(ri) ∧ kx.toSet ⊆ dom ri ∧ ax.toSet ∩ range ri = ∅ ∧ ¬ dups(kx) ∧ ¬ dups(ax) → range ri[kx, ax] = (range ri \ ri[kx].toSet) ∪ ax.toSet subst: [ax, kx, ri] → [abs-log(nx0, gstore ↓), nd.key ' + ndx.keys, fi2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(key ' + kx) = (# kx)+1; (s)% ⊦ # ndx.keys = # ndx; (s)% ⊦ m +1 = m + 1; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ ks ∪ ∅ = ks; (s)%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]% ⊦ ks ++ key ⊆ ks0 ↔ ks ⊆ ks0 ∧ key ∈ ks0; (ws)% ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ as ⊆ dom fs ∧ new(ax, fs) → ax.toSet ∩ as = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ dups(key ' + kx) ↔ dups(kx) ∨ key ∈ kx; (ws) Used terms: abs-log(nx0, gstore ↓), nd.key ' + ndx.keys, fi2