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Proof step 125 from proof inv-gc-copy (previous step)

# ax = # ndx1, # ax1 = # ndx + 1, # ax1 = # ndx0, jvalid2 ↔ jvalid0, jvalid3 ↔ jvalid, LEB_SIZE = n1 + n0, abs-log(nx, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, abs-log(nx0, gstore ↓) = abs-log(nx, gstore3 ↓) + ax1, ax1 = ax1, ax2 = ax1, fi = fi2[ndx1.keys, ax], fi = fi, fi3 = fi1, fi4 = fi, gstore = gstore, gstore4 = gstore1, gstore5 = gstore, jmaxino0 = jmaxino0, jmaxino1 = jmaxino0, jro3 = jro3, jro4 = fi2[ndx1.keys, ax].inos, jro5 = jro3, jro6 = jro3, ndx0 = nd0 ' + ndx, ndx0 = ndx0, ndx2 = ndx0, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, ndx1], nodes(gstore ↓, jro3) = nodes(gstore3 ↓, jro)[ax1, nd0 ' + ndx], nx0 = nx0, nx2 = nx0, nx3 = nx0, p_os0 = p_os, pmaxino0 = pmaxino, remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro3 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore3 = ∅, elems nx1 ∩ jro0 = ∅, (nx.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro = ∅, ax1.toSet ∩ range fi = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx = dom gstore3 ∪ jro, jro2 < jmaxino3, fi2.inos < jmaxino3, fi2[ndx1.keys, ax].inos < jmaxino0, p_os < pmaxino, fi.inos < jmaxino0, fi1.inos < pmaxino, inv(gstore \ jro3), inv(gstore3[nx.last]), inv(gstore2 \ jro0), inv(gstore1), inv(gstore3 \ jro), inv-journal(nx0, gstore, gstore1, jro3), key-cons(ndx3), key-cons(ndx), key-cons(ndx1), key-cons(nd0.key, nd0), key-cons(ndx0), new(ax, nodes(gstore2 ↓, jro0)), new(ax1, nodes(gstore3 ↓, jro)), nodes-cons(ax1, nd0 ' + ndx, gstore, jro3), nodes-cons(ax1, ndx0, gstore, jro3), refsize-cons(gstore, range fi2[ndx1.keys, ax]), refsize-cons(gstore3, range fi2[ndx1.keys, ax]), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi), refsize-cons(gstore1, range fi1), ri-inj(fi2[ndx1.keys, ax]), ri-inj(fi2), ri-inj(fi), ri-inj(fi1), nd0.key ∈ fi2, nx.last ∈ gstore3, nd0.flashsize ≤ LEB_SIZE, nd0.flashsize + ndx.flashsize ≤ n1, jmaxino3 ≤ jmaxino0, pmaxino ≤ jmaxino3, n0 ≤ LEB_SIZE, nd0.flashsize ≤ n0, pmaxino ≤ jmaxino0, dom gstore3 ⊆ dom gstore, 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, fi.lebs ⊆ dom gstore, fi1.lebs ⊆ dom gstore1, ndx0.keys.toSet ⊆ dom fi, 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 ⊦ [gjournal_gc_update_index#(jvalid0, fi1, jro3, gstore1, nx0, pmaxino, p_os, jvalid; ax1, ndx0, jmaxino0, fi, gstore)] ( jro4 < jmaxino0 ∧ pmaxino0 ≤ jmaxino0 ∧ fi.inos < jmaxino0 ∧ jmaxino0 ≠ 0 ∧ (jvalid2 ∧ nx2 ≠ [] → valid-journal-head(nx2, gstore, jro5)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx2, gstore, gstore4, jro5) ∧ ri-inj(fi) ∧ fi = fi4[ndx2.keys, ax2] ∧ dom gstore = dom gstore5 ∧ nodes(gstore ↓, jro5) = nodes(gstore5 ↓, jro6) ∧ abs-log(nx2, gstore ↓) = abs-log(nx3, gstore5 ↓) ∧ jmaxino1 ≤ jmaxino0 → jmaxino3 ≤ jmaxino0 ∧ (jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi.inos < jmaxino0 ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv(gstore \ jro3) ∧ dom gstore1 ∪ elems nx0 = dom gstore ∪ jro3 ∧ ri-inj(fi) ∧ ndx3.keys.toSet ⊆ dom fi ∧ range fi ⊆ 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 ∧ fi = 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)), 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ nd ∈ nd0 ' + ndx ↔ nd = nd0 ∨ nd ∈ ndx; (s)%in-14: ⊦ nd ' + ndx = ndx0 → nd ∈ ndx0; (forward) from specification list[node-list]%⊦ inv-journal(nx, gstore, gstore0, ginval) ↔ inv(gstore \ ginval) ∧ inv(gstore0) ∧ elems nx ∩ ginval = ∅ ∧ ¬ dups(nx) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ ginval ∧ remove-refsize0(gstore0) = gstore0; (s)% ⊦ key-cons(nd ' + ndx) ↔ ¬ ¬(key-cons(nd.key, nd) ∧ key-cons(ndx)); (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)% ⊦ # ndx.keys = # ndx; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%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)% ⊦ key + kx = key ' + kx; (s)% ⊦ (nd ' + ndx).keys = nd.key + ndx.keys; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ m +1 = m + 1; (s)% ⊦ #(nd ' + ndx) = (# ndx)+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)