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Proof step 217 from proof refine-gc (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 218:
⊦ (as ∪ as0) ∩ as1 = (as ∩ as1) ∪ (as0 ∩ as1)
Proof step 219:
# ax0 = # referenced-adrs(lax.to-adr(n1), ri0), # lax = # gndx, gstore2[n1] = groupblock(gndx, lax, m), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0, 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(groupblock(gndx, lax, m)), inv(gstore3 \ jro), inv(gstore1), inv(gstore2 \ jro0), log-cons(abs-log(nx1, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore3, (range ri0 \ (range ri0 | n1)) ∪ ax0.toSet), refsize-cons(gstore2, range ri0), refsize-cons(gstore1, range fi1), refsize-cons(gstore3 -- n1, (range ri0 \ (range ri0 | n1)) ∪ ax0.toSet), ri-cons(ri0, nodes(gstore2 ↓, jro0)), ri-inj(ri0), ri-inj(ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys, ax0]), ri-inj(fi1), n1 ∈ gstore2, n1 ∈ gstore1, 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(gstore2 ↓, jro0) | n1) ∩ ax0.toSet) ∪ ((ax0.toSet | n1) ∩ ax0.toSet)) = 0, #(jro ∪ dom gstore3) = 0, jmaxino = 0, jmaxino0 = 0, m = 0, pmaxino = 0, jro ∪ dom gstore3 = ∅, ((dom nodes(gstore2 ↓, jro0) | n1) ∩ ax0.toSet) ∪ ((ax0.toSet | n1) ∩ ax0.toSet) = ∅, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(n1), ri0)].keys), dups(ax0), dups(nx0), dups(nx1), n1 ∈ jro0, n1 ∈ nx0, n1 ∈ nx1

Rule argument

spec: set-union, inst: node-store name: union-inter-inter-union seq: ⊦ (as ∪ as0) ∩ as1 = (as ∩ as1) ∪ (as0 ∩ as1) subst: [as1, as0, as] → [ax0.toSet, ax0.toSet | n1, dom nodes(gstore2 ↓, jro0) | n1] flags: <all paths>

Additional information

Used terms: ax0.toSet, ax0.toSet | n1, dom nodes(gstore2 ↓, jro0) | n1