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Proof step 126 from proof refine-gc (previous step)

# ax0 = # ndx3, # lax = # gndx0, ndx3 + ndx2 = nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri)], gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore ↓) = abs-log(nx, gstore2 ↓) + ax0, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, ndx3], remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore2 = ∅, elems nx ∩ jro0 = ∅, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, jmaxino ≠ 0, jmaxino0 ≠ 0, m0 ≠ 0, pmaxino ≠ 0, p_os.keys.inodes?, jro1.keys.inodes?, jro1 < jmaxino0, ri[ndx3.keys, ax0].inos < jmaxino0, jro1 < jmaxino, p_os < pmaxino, ri.inos < jmaxino, fi1.inos < pmaxino, inv(gstore \ jro), inv(groupblock(gndx0, lax, m0)), inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore2, range ri), refsize-cons(gstore1, range fi1), refsize-cons(gstore, range ri[ndx3.keys, ax0]), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(ri[ndx3.keys, ax0]), ri-inj(ri), ri-inj(fi1), n1 ∈ gstore1, n1 ∈ gstore2, pmaxino ≤ jmaxino0, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino, dom gstore2 ⊆ dom gstore, ri.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore1, range ri ⊆ dom nodes(gstore2 ↓, jro0), ri.lebs ⊆ dom gstore, ri[ndx3.keys, ax0].lebs ⊆ dom gstore, ⌊err⌋ , ¬ dups(nx), ¬ dups(nx0), ¬ dups(ax0), ¬ dups(ndx3.keys + ndx2.keys), ¬ dups(ndx3.keys), ¬ dups(ndx2.keys), ¬ n1 ∈ nx, ¬ n1 ∈ jro0, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro), jvalid5 → jvalid4, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0) ⊦ ∃ kx, ax. ¬ dups(kx) ∧ ¬ dups(ax) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ kx.toSet ⊆ dom ri ∧ # kx = # ax ∧ ax.toSet ∩ abs-log(nx, gstore2 ↓).toSet = ∅ ∧ ⟨fs := nodes(gstore2 ↓, jro0)[ax, nodes(gstore2 ↓, jro0)[ri[kx]]], log := abs-log(nx, gstore2 ↓) + ax; ri0 := ri[kx, ax]; aubifs_core_gc_drop#(ri0, log; fs); jvalid := ?⟩ (ri[ndx3.keys, ax0] = ri0 ∧ log = abs-log(nx0, gstore ↓) ∧ fs = nodes(gstore ↓, jro) ∧ (jvalid → jvalid1))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 127:
⊦ fs[ax] = ndx0 + ndx1 ↔ (∃ ax0, ax1. ax = ax0 + ax1 ∧ fs[ax0] = ndx0 ∧ fs[ax1] = ndx1)
Proof step 128:
# lax = # gndx0, # ax0 = # ndx3, gstore2[n1] = groupblock(gndx0, lax, m0), abs-log(nx0, gstore ↓) = abs-log(nx, gstore2 ↓) + ax0, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, ndx3], remove-refsize0(gstore1) = gstore1, elems nx ∩ jro0 = ∅, (nx0.toSet \ nx.toSet) ∩ dom gstore2 = ∅, elems nx0 ∩ jro = ∅, dom gstore1 ∪ elems nx = dom gstore2 ∪ jro0, dom gstore1 ∪ elems nx0 = dom gstore ∪ jro, jro1.keys.inodes?, p_os.keys.inodes?, fi1.inos < pmaxino, ri.inos < jmaxino, p_os < pmaxino, jro1 < jmaxino, ri[ndx3.keys, ax0].inos < jmaxino0, jro1 < jmaxino0, inv(gstore2 \ jro0), inv(gstore1), inv(groupblock(gndx0, lax, m0)), inv(gstore \ jro), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore, range ri[ndx3.keys, ax0]), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(fi1), ri-inj(ri), ri-inj(ri[ndx3.keys, ax0]), n1 ∈ gstore2, n1 ∈ gstore1, pmaxino ≤ jmaxino, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino0, ri[ndx3.keys, ax0].lebs ⊆ dom gstore, ri.lebs ⊆ dom gstore, range ri ⊆ dom nodes(gstore2 ↓, jro0), fi1.lebs ⊆ dom gstore1, ri.lebs ⊆ dom gstore2, dom gstore2 ⊆ dom gstore, ⌊err⌋ , jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0), jvalid5 → jvalid4, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro), ∃ ax0, ax1. referenced-adrs(lax.to-adr(n1), ri) = ax0 + ax1 ∧ nodes(gstore2 ↓, jro0)[ax0] = ndx3 ∧ nodes(gstore2 ↓, jro0)[ax1] = ndx2 ⊦ ∃ kx, ax. ¬ dups(kx) ∧ ¬ dups(ax) ∧ new(ax, nodes(gstore2 ↓, jro0)) ∧ kx.toSet ⊆ dom ri ∧ # kx = # ax ∧ ax.toSet ∩ abs-log(nx, gstore2 ↓).toSet = ∅ ∧ ⟨fs := nodes(gstore2 ↓, jro0)[ax, nodes(gstore2 ↓, jro0)[ri[kx]]], log := abs-log(nx, gstore2 ↓) + ax; ri0 := ri[kx, ax]; aubifs_core_gc_drop#(ri0, log; fs); jvalid := ?⟩ (ri[ndx3.keys, ax0] = ri0 ∧ log = abs-log(nx0, gstore ↓) ∧ fs = nodes(gstore ↓, jro) ∧ (jvalid → jvalid1)), jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, dups(nx), dups(nx0), dups(ax0), dups(ndx3.keys + ndx2.keys), dups(ndx3.keys), dups(ndx2.keys), n1 ∈ nx, n1 ∈ jro0

Rule argument

spec: aubifs-base name: read-split seq: ⊦ fs[ax] = ndx0 + ndx1 ↔ (∃ ax0, ax1. ax = ax0 + ax1 ∧ fs[ax0] = ndx0 ∧ fs[ax1] = ndx1) subst: [ax, fs, ndx1, ndx0] → [referenced-adrs(lax.to-adr(n1), ri), nodes(gstore2 ↓, jro0), ndx2, ndx3] flags: <all paths>

Additional information

Used terms: referenced-adrs(lax.to-adr(n1), ri), nodes(gstore2 ↓, jro0), ndx2, ndx3