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Proof step 181 from proof refine-gc (previous step)

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

Heuristic nonrecursive calls applied rule call right

And obtained the following new goals

Proof step 182:
# ax = # referenced-adrs(lax.to-adr(n1), ri), # lax = # gndx0, gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax, fi = fi1, fo = p_os.keys, fs = nodes(gstore2 ↓, jro0), log = abs-log(nx1, gstore2 ↓), nodes(gstore3 ↓, jro) = nodes(gstore2 ↓, jro0)[ax, nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)]], remove-refsize0(gstore1) = gstore1, ri0 = ri, ro = jro1.keys, 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, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax].inos < jmaxino0, jro1 < jmaxino, p_os < pmaxino, ri.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(ax, nodes(gstore2 ↓, jro0)), refsize-cons(gstore3, range ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax]), refsize-cons(gstore2, range ri), refsize-cons(gstore1, range fi1), refsize-cons(gstore3 -- n1, range ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax]), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax]), ri-inj(ri), ri-inj(fi1), n1 ∈ gstore1, n1 ∈ gstore2, n1 ∈ gstore3, n1 ∈ jro ∪ dom gstore3, pmaxino ≤ jmaxino0, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino, ri.lebs ⊆ dom gstore3, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys, ax].lebs ⊆ dom gstore3, dom gstore2 ⊆ dom gstore3, ri.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, range ri ⊆ 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) ⊦ ⟨if ri-cons(ri, fs) then {aubifs_core_gc_copy#(; ri, fs, log); aubifs_core_gc_drop#(ri, log; fs); jvalid := ?} else abort⟩ ( ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax] = ri ∧ log = abs-log(nx0, gstore3 ↓) ∧ fs = nodes(gstore3 ↓, jro) \ (dom nodes(gstore3 ↓, jro) | n1)), #(jro ∪ dom gstore3) = 0, jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, jro ∪ dom gstore3 = ∅, dups(ax), dups(nx0), dups(nx1), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)].keys), n1 ∈ nx0, n1 ∈ jro0, n1 ∈ nx1, ∃ adr. adr.lnum = n1 ∧ adr ∈ ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(adr.lnum), ri)].keys, ax]

Rule argument

pos: right 1

Additional information

Used simplifier rules: aubifs_core_journal_gc#(FI, RO, FO; RI, FS, LOG; CSYNCED) {if ri-cons(RI, FS) then {aubifs_core_gc_copy#(; RI, FS, LOG); aubifs_core_gc_drop#(RI, LOG; FS); CSYNCED := ?} else abort };