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Proof step 68 from proof inv-gc-copy (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 69:
# ax = # ndx1, # ax1 = # ndx + 1, 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 nx ∩ jro = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, fi2[ndx1.keys, ax].inos < jmaxino, fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1].inos < jmaxino, n0 < nd0.flashsize, fi2.inos < jmaxino3, jro2 < jmaxino3, p_os < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, fi1.inos < pmaxino, inv(gstore0 \ jro3), inv(gstore \ jro3), inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(gstore3[nx.last]), 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(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx1.keys, ax]), refsize-cons(gstore1, range fi1), refsize-cons(gstore0, ((range fi2[ndx1.keys, ax] \ fi2[ndx1.keys, ax][ndx.keys].toSet) -- fi2[ndx1.keys, ax][nd0.key]) ∪ ax1.toSet), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1]), ri-inj(fi2), ri-inj(fi1), nd0.key ∈ fi2, nx.last ∈ gstore3, pmaxino ≤ jmaxino, jmaxino0 ≤ jmaxino, nd0.flashsize + ndx.flashsize ≤ LEB_SIZE, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, 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, dom gstore2 ⊆ dom gstore3, range fi2[ndx1.keys, ax] ⊆ dom nodes(gstore3 ↓, jro), range fi2 ⊆ dom nodes(gstore2 ↓, jro0), ndx3.keys.toSet ⊆ dom fi2, ndx.keys.toSet ⊆ dom fi2, ndx1.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, dom gstore2 ⊆ dom gstore, fi2.lebs ⊆ 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 = [] ∨ LEB_SIZE < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ jmaxino = 0, jmaxino0 = 0, jmaxino3 = 0, nx = [], pmaxino = 0, dups(ax1), dups(nx0), dups(ndx.keys), dups(ndx1.keys), dups(ndx1.keys + ndx.keys + ndx3.keys), dups(nx1), dups(nx), dups(ax), dups(ndx.keys + ndx3.keys), dups(ndx1.keys + ndx.keys), dups(ndx3.keys), nd0.key ∈ ndx1.keys, nd0.key ∈ ndx3.keys, nd0.key ∈ ndx.keys, ((range fi2[ndx1.keys, ax] \ fi2[ndx1.keys, ax][ndx.keys].toSet) -- fi2[ndx1.keys, ax][nd0.key]) ∪ ax1.toSet ⊆ dom nodes(gstore ↓, jro3)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ dups(kx) → (dups(key ' + kx) ↔ key ∈ kx); (s)% ⊦ as \ (as0 ++ adr) = (as \ as0) -- adr; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%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 ∪ ∅ = as; (s)% ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ adr '.toSet = ∅ ++ adr; (s)% ⊦ (ax + ax0).toSet = ax.toSet ∪ ax0.toSet; (s)% ⊦ adr + ax = adr ' + ax; (s)% ⊦ ri[key ' + kx] = ri[key] + ri[kx]; (s)% ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ ks ⊆ ks0 → (ks ++ key ⊆ ks0 ∪ ks1 ↔ key ∈ ks0 ∨ key ∈ ks1); (s)%# kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet; (s)%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]% ⊦ ks ∪ ∅ = ks; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ # ndx.keys = # ndx; (s)% ⊦ #(key ' + kx) = (# kx)+1; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ as ⊆ dom fs ∧ new(ax, fs) → ax.toSet ∩ as = ∅; (s)