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Proof step 207 from proof refine-gc (previous step)

# ax0 = # referenced-adrs(lax.to-adr(n1), ri0), # lax = # gndx0, gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0, nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)]], remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro0 = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx0 = jro ∪ dom gstore3, p_os.keys.inodes?, jro1.keys.inodes?, jro1 < jmaxino0, ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0].inos < jmaxino0, jro1 < jmaxino, p_os < pmaxino, ri0.inos < jmaxino, fi1.inos < pmaxino, inv(gstore3 \ jro), inv(groupblock(gndx0, lax, m0)), inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx1, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore3, range ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0]), refsize-cons(gstore2, range ri0), refsize-cons(gstore1, range fi1), refsize-cons(gstore3 -- n1, range ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0]), ri-cons(ri0, nodes(gstore2 ↓, jro0)), ri-inj(ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0]), ri-inj(ri0), ri-inj(fi1), n1 ∈ gstore1, n1 ∈ gstore2, n1 ∈ gstore3, n1 ∈ jro ∪ dom gstore3, pmaxino ≤ jmaxino0, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino, ri0.lebs ⊆ dom gstore3, ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0].lebs ⊆ dom gstore3, dom gstore2 ⊆ dom gstore3, ri0.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, range ri0 ⊆ dom nodes(gstore2 ↓, jro0), jvalid5 → jvalid4, jvalid3 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore3, jro), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore3 -- n1, jro ++ n1) ⊦ #((dom nodes(gstore3 ↓, jro) | n1) ∩ range ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0]) = 0, #(jro ∪ dom gstore3) = 0, jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, (dom nodes(gstore3 ↓, jro) | n1) ∩ range ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0] = ∅, jro ∪ dom gstore3 = ∅, dups(ax0), dups(nx0), dups(nx1), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys), n1 ∈ nx0, n1 ∈ jro0, n1 ∈ nx1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: gjournal-base name: index-update-range seq: # kx = # ax ⊦ ri-inj(ri) ∧ kx.toSet ⊆ dom ri ∧ ax.toSet ∩ range ri = ∅ ∧ ¬ dups(kx) ∧ ¬ dups(ax) → range ri[kx, ax] = (range ri \ ri[kx].toSet) ∪ ax.toSet subst: [ax, kx, ri] → [ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)] .keys, ri0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # ndx.keys = # ndx; (s)% ⊦ # fs[ax] = # ax; (s)% ⊦ ri-cons(ri, fs) → ri-inj(ri); (ss)% ⊦ ri-cons(ri, fs) → fs[referenced-adrs(ax, ri)].keys.toSet ⊆ dom ri; (s)% ⊦ as ⊆ dom fs ∧ new(ax, fs) → ax.toSet ∩ as = ∅; (s)%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store] Used terms: ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ri0