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Proof step 10 from proof inv-gc-update-index (previous step)

# ax1 = # ndx1 + 1, # ax3 = # ndx2, abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), ax1 = adr + ax2, dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, range fi2 ∩ ax3.toSet = ∅, range fi2 ∩ ax1.toSet = ∅, elems nx ∩ jro0 = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, fi2[ndx2.keys, ax3].inos < jmaxino, jro1 < jmaxino1, p_os < pmaxino, fi2.inos < jmaxino1, fi0.inos < pmaxino, inv(gstore2 \ jro0), inv(gstore0), inv(gstore \ jro0), key-cons(ndx1), key-cons(nd.key, nd), key-cons(ndx2), nodes-cons(ax1, nd ' + ndx1, gstore, jro0), nodes-cons(ax3 + ax1, ndx2 + nd ' + ndx1, gstore2, jro0), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), refsize-cons(gstore2, range fi2), refsize-cons(gstore0, range fi0), ri-inj(fi2), ri-inj(fi0), nd.key ∈ fi2, jmaxino1 ≤ jmaxino, pmaxino ≤ jmaxino1, pmaxino ≤ jmaxino, ndx2.keys.toSet ⊆ dom fi2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore2, fi0.lebs ⊆ dom gstore0, ndx1.keys.toSet ⊆ dom fi2, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0) ⊦ ⟪gjournal_index_store#(nd.key, adr, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore); ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + ax1 = 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)) ∧ # ax0 < # ax1), ax1 = [], jmaxino1 = 0, pmaxino = 0, dups(nx), dups(ax3 + ax1), dups(ax3), dups(ax1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
# ax3 = # ndx2, # ax2 = # ndx1, abs-log(nx, gstore ↓) = abs-log(nx, gstore2 ↓), dom gstore = dom gstore2, fi = fi2[ndx2.keys, ax3], nodes(gstore ↓, jro0) = nodes(gstore2 ↓, jro0), remove-refsize0(gstore0) = gstore0, elems nx ∩ jro0 = ∅, range fi2 ∩ ax2.toSet = ∅, range fi2 ∩ ax3.toSet = ∅, dom gstore0 ∪ elems nx = dom gstore2 ∪ jro0, jmaxino1 ≠ 0, pmaxino ≠ 0, fi0.inos < pmaxino, fi2.inos < jmaxino1, p_os < pmaxino, jro1 < jmaxino1, fi2[ndx2.keys, ax3].inos < jmaxino, inv(gstore \ jro0), inv(gstore0), inv(gstore2 \ jro0), key-cons(ndx2), key-cons(nd.key, nd), key-cons(ndx1), nodes-cons(ax3 + adr ' + ax2, ndx2 + nd ' + ndx1, gstore2, jro0), nodes-cons(adr ' + ax2, nd ' + ndx1, gstore, jro0), refsize-cons(gstore0, range fi0), refsize-cons(gstore2, range fi2), refsize-cons(gstore, range fi2[ndx2.keys, ax3]), ri-inj(fi0), ri-inj(fi2), nd.key ∈ fi2, pmaxino ≤ jmaxino, pmaxino ≤ jmaxino1, jmaxino1 ≤ jmaxino, ndx1.keys.toSet ⊆ dom fi2, fi0.lebs ⊆ dom gstore0, fi2.lebs ⊆ dom gstore2, fi2[ndx2.keys, ax3].lebs ⊆ dom gstore2, ndx2.keys.toSet ⊆ dom fi2, ¬ dups(nx), ¬ dups(ax3 + ax2), ¬ dups(ax3), ¬ dups(ax2), ¬ adr ∈ fi2, ¬ adr ∈ ax2, ¬ adr ∈ ax3, jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid2 ∧ nx ≠ [] → valid-journal-head(nx, gstore, jro0) ⊦ ⟪gjournal_index_store#(nd.key, adr, fi0, jro0, gstore0, nx, pmaxino, p_os, jvalid; jmaxino, fi, gstore); ax0 := ax2; ndx0 := ndx1⟫ ( (∃ ax, ndx. # ax = # ndx ∧ ax3 + adr ' + 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)

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(ax + ax0) → ¬ dups(ax) ∧ ¬ dups(ax0); (forward) from specification list-dup[address-list-set]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ dups(ax + adr ' + ax0) ↔ adr ∈ ax ∨ adr ∈ ax0 ∨ dups(ax + ax0); (ws)% ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)% ⊦ adr ' + ax ≠ []; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+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]% ⊦ adr ∈ range ri ↔ adr ∈ ri; (s)% ⊦ as ∩ (as0 ++ adr) = ∅ ↔ as ∩ as0 = ∅ ∧ ¬ adr ∈ as; (ws)%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)