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Proof step 127 from proof inv-gc-copy (previous step)

# ax1 = # ndx + 1, # ax = # ndx1, LEB_SIZE = n1 + n0, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) + ax1, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, ax2 = ax1, fi = fi2[ndx1.keys, ax], gstore5 = gstore, jmaxino1 = jmaxino0, ndx0 = nd0 ' + ndx, nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro)[ax1, nd0 ' + ndx], 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, fi1.inos < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, p_os < pmaxino, fi2.inos < jmaxino3, jro2 < jmaxino3, 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(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]), ri-inj(fi1), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi2), nx.last ∈ gstore3, nd0.key ∈ fi2, pmaxino ≤ jmaxino0, nd0.flashsize ≤ n0, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, nd0.flashsize + ndx.flashsize ≤ n1, nd0.flashsize ≤ LEB_SIZE, fi1.lebs ⊆ dom gstore1, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2.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, dom gstore3 ⊆ dom gstore, 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 ⊦ jmaxino0 = 0, jmaxino3 = 0, nx = [], pmaxino = 0, dups(ax1), 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, ∀ ax1, ndx0, jmaxino0, fi, gstore. fi2[ndx1.keys, ax].inos < jmaxino0 ∧ pmaxino ≤ jmaxino0 ∧ fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2].inos < jmaxino0 ∧ jmaxino0 ≠ 0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2]) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2]) ∧ fi = fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2] ∧ dom gstore = dom gstore5 ∧ nodes(gstore ↓, jro3) = nodes(gstore5 ↓, jro3) ∧ abs-log(nx0, gstore ↓) = abs-log(nx0, gstore5 ↓) ∧ jmaxino1 ≤ jmaxino0 → jmaxino3 ≤ jmaxino0 ∧ ndx3.keys.toSet ⊆ dom fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2] ∧ range fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2] ⊆ dom nodes(gstore ↓, jro3) ∧ dom gstore2 ⊆ dom gstore ∧ (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax0. new(ax0, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax0) ∧ # ax0 = # ndx1 + # ndx + 1 ∧ fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax2] = fi2[ndx1.keys + nd0.key ' + ndx.keys, ax0] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[ax0, ndx1 + nd0 ' + ndx] ∧ abs-log(nx0, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 128:
# ax1 = # ndx + 1, # ax = # ndx1, LEB_SIZE = n1 + n0, 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 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, fi1.inos < pmaxino, fi2[ndx1.keys, ax].inos < jmaxino0, p_os < pmaxino, fi2.inos < jmaxino3, jro2 < jmaxino3, fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1].inos < jmaxino, fi2[ndx1.keys, ax].inos < jmaxino, inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(gstore3[nx.last]), 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][nd0.key ' + ndx.keys, ax1]), ri-inj(fi1), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1]), nx.last ∈ gstore3, nd0.key ∈ fi2, jmaxino3 ≤ jmaxino, pmaxino ≤ jmaxino0, nd0.flashsize ≤ n0, n0 ≤ LEB_SIZE, pmaxino ≤ jmaxino3, jmaxino3 ≤ jmaxino0, nd0.flashsize + ndx.flashsize ≤ n1, nd0.flashsize ≤ LEB_SIZE, jmaxino0 ≤ jmaxino, pmaxino ≤ jmaxino, fi1.lebs ⊆ dom gstore1, fi2[ndx1.keys, ax].lebs ⊆ dom gstore, dom gstore2 ⊆ dom gstore, fi2.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, dom gstore3 ⊆ dom gstore, fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1].lebs ⊆ dom gstore, 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 = [] ∨ n1 < (nd0.flashsize + ndx.flashsize) + ndx3.head.flashsize ⊦ jmaxino = 0, jmaxino0 = 0, jmaxino3 = 0, nx = [], pmaxino = 0, dups(ax1), 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, range fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1] ⊆ dom nodes(gstore ↓, jro3) ∧ (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅ ∧ (∃ ax0. new(ax0, nodes(gstore2 ↓, jro0)) ∧ ¬ dups(ax0) ∧ # ax0 = # ndx1 + # ax1 ∧ fi2[ndx1.keys, ax][nd0.key ' + ndx.keys, ax1] = fi2[ndx1.keys + nd0.key ' + ndx.keys, ax0] ∧ nodes(gstore ↓, jro3) = nodes(gstore2 ↓, jro0)[ax0, ndx1 + nd0 ' + ndx] ∧ abs-log(nx0, gstore ↓) = abs-log(nx1, gstore2 ↓) + ax0)

Rule argument

Additional information

Used simplifier rules: let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat% ⊦ ks ⊆ ks0 → ks ⊆ (ks0 ∪ ks1) ++ key; (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)%# kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.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)