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Proof step 20 from proof inv-gc-update-index (previous step)

# ax2 = # ndx1, # ax3 = # ndx2, jvalid1 ↔ jvalid2, jvalid3 ↔ jvalid, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), adr0 = @(m, m0, m1), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], fi = fi, fi3 = fi0, fi4 = fi, gstore = gstore, gstore3 = gstore0, gstore4 = gstore, jro2 = fi2[ndx2.keys, ax3].inos, jro3 = jro0, key0 = nd.key, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), nx0 = nx, p_os0 = p_os, pmaxino0 = pmaxino, remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax2.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, jro1 < jmaxino1, fi2.inos < jmaxino1, fi2[ndx2.keys, ax3].inos < jmaxino, p_os < pmaxino, fi.inos < jmaxino, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), inv-journal(nx, gstore, gstore0, jro0), key-cons(ndx1), key-cons(nd.key, nd), key-cons(ndx2), key-cons(nd.key, gstore[@(m, m0, m1)].nd), nodes-cons(ax2, ndx1, gstore, jro0), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi), ri-inj(fi0), nd.key ∈ fi2, @(m, m0, m1) ∈ gstore, jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, fi.lebs ⊆ dom gstore, fi0.lebs ⊆ dom gstore0, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ [gjournal_index_store#(nd.key, @(m, m0, m1), fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore)] ( jro2 < jmaxino ∧ pmaxino0 ≤ jmaxino ∧ fi.inos < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid1 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro3)) ∧ fi.lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi) ∧ inv-journal(nx0, gstore, gstore3, jro3) ∧ ri-inj(fi) ∧ gstore =○refsize gstore4 ∧ fi = fi4[key0, adr0] → ⟪ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi = fi2[ndx.keys, ax]) ∧ dom fi = dom fi2 ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ jmaxino1 ≤ jmaxino ∧ fi.inos < jmaxino ∧ refsize-cons(gstore, range fi) ∧ ri-inj(fi) ∧ fi.lebs ⊆ dom gstore ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ ¬ # ax2 < # ax0)), jmaxino = 0, jmaxino1 = 0, pmaxino = 0, dups(ax2), dups(ax3), dups(ax3 + ax2), dups(nx), @(m, m0, m1) ∈ fi, @(m, m0, m1) ∈ ax3, @(m, m0, m1) ∈ ax2, @(m, m0, m1) ∈ fi2, m ∈ jro0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 21:
# ax2 = # ndx1, # ax3 = # ndx2, gstore[@(m, m0, m1)] = mkgnode(nd, m2, boolvar, boolvar0), abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], gstore4 = gstore, nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax2.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, jmaxino ≠ 0, jmaxino1 ≠ 0, pmaxino ≠ 0, jro1 < jmaxino1, fi2.inos < jmaxino1, p_os < pmaxino, fi2[ndx2.keys, ax3].inos < jmaxino, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), key-cons(ndx1), key-cons(ndx2), key-cons(nd.key, nd), nodes-cons(ax2, ndx1, gstore, jro0), nodes-cons(ax3 + @(m, m0, m1) ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi0), nd.key ∈ fi2, @(m, m0, m1) ∈ gstore, jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2.lebs ⊆ dom gstore2, ndx1.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi0.lebs ⊆ dom gstore0, ¬ dups(ax2), ¬ dups(ax3), ¬ dups(ax3 + ax2), ¬ dups(nx), ¬ @(m, m0, m1) ∈ ax3, ¬ @(m, m0, m1) ∈ ax2, ¬ @(m, m0, m1) ∈ fi2, ¬ m ∈ jro0, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ [gjournal_index_store#(nd.key, @(m, m0, m1), fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore)] ( fi2[ndx2.keys, ax3].inos < jmaxino ∧ pmaxino ≤ jmaxino ∧ fi2[ndx2.keys, ax3].inos ∪ (nd.key.inode? ⊃ ∅ ++ nd.key.ino;∅) < jmaxino ∧ jmaxino ≠ 0 ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)].lebs ⊆ dom gstore ∧ refsize-cons(gstore, range fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ inv(gstore \ jro0) ∧ dom gstore0 ∪ elems nx = dom gstore ∪ jro0 ∧ ri-inj(fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)]) ∧ gstore =○refsize gstore4 ∧ fi = fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] → ⟪ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + @(m, m0, m1) ' + ax2 = ax + ax0 ∧ ndx2 + nd ' + ndx1 = ndx + ndx0 ∧ fi2[ndx2.keys, ax3][nd.key, @(m, m0, m1)] = fi2[ndx.keys, ax]) ∧ (dom fi2 ∪ ndx2.keys.toSet) ++ nd.key = dom fi2 ∧ jmaxino1 ≤ jmaxino ∧ dom gstore = dom gstore2 ∧ nodes-cons(ax0, ndx0, gstore, jro0) ∧ nodes(gstore4 ↓, jro0) = nodes(gstore2 ↓, jro0) ∧ abs-log(nx, gstore4 ↓) = abs-log(nx, gstore2 ↓) ∧ (jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0)) ∧ ¬ # ax2 < # ax0))

Rule argument

Additional information

Used simplifier rules: # kx = # ax ⊦ dom ri[kx, ax] = dom ri ∪ kx.toSet; (s)% ⊦ mkgnode(nd, m, boolvar, boolvar0).nd = nd; (s)%⊦ gstore1 =○refsize gstore0 → abs-log(nx, gstore1 ↓) = abs-log(nx, gstore0 ↓); (s)%⊦ gstore1 =○refsize gstore0 → nodes(gstore1 ↓, ginval) = nodes(gstore0 ↓, ginval); (s)% ⊦ dom ri[key, adr] = dom ri ++ key; (s)% ⊦ ri[key, adr].inos = ri.inos ∪ (key.inode? ⊃ ∅ ++ key.ino;∅); (s)%# kx = # ax ⊦ ¬ adr ∈ ax ∧ ¬ adr ∈ ri → ¬ adr ∈ ri[kx, ax]; (s)%⊦ 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)%# ax = # kx ⊦ ri-inj(ri) ∧ ¬ dups(ax) ∧ ax.toSet ∩ range ri = ∅ → ri-inj(ri[kx, ax]); (s)% ⊦ # ndx.keys = # ndx; (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]