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Proof step 225 from proof refine-gc (previous step)

# ax0 = # referenced-adrs(lax.to-adr(adr.lnum), ri0), # lax = # gndx, gstore2[adr.lnum] = 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(adr.lnum), 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 | adr.lnum)) ∪ ax0.toSet), refsize-cons(gstore2, range ri0), refsize-cons(gstore1, range fi1), refsize-cons(gstore3 -- adr.lnum, (range ri0 \ (range ri0 | adr.lnum)) ∪ ax0.toSet), ri-cons(ri0, nodes(gstore2 ↓, jro0)), ri-inj(ri0), ri-inj(ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(adr.lnum), ri0)].keys, ax0]), ri-inj(fi1), adr.lnum ∈ gstore1, adr ∈ ax0, adr.lnum ∈ gstore2, adr.lnum ∈ gstore3, adr.lnum ∈ jro ∪ dom gstore3, pmaxino ≤ jmaxino0, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino, ri0.lebs ⊆ dom gstore3, ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(adr.lnum), 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 -- adr.lnum, jro ++ adr.lnum) ⊦ #(jro ∪ dom gstore3) = 0, #((ax0.toSet | adr.lnum) ∩ ax0.toSet) = 0, jmaxino = 0, jmaxino0 = 0, m = 0, pmaxino = 0, (ax0.toSet | adr.lnum) ∩ ax0.toSet = ∅, jro ∪ dom gstore3 = ∅, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(adr.lnum), ri0)].keys), dups(ax0), dups(nx0), dups(nx1), adr.lnum ∈ jro0, adr.lnum ∈ nx0, adr.lnum ∈ nx1

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 226:
# lax = # gndx, # ax0 = # referenced-adrs(lax.to-adr(m0), ri0), gstore2[m0] = groupblock(gndx, lax, m), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓) + ax0, adr = @(m0, m1, m2), 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(m0), 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 -- m0, (range ri0 \ (range ri0 | m0)) ∪ ax0.toSet), refsize-cons(gstore1, range fi1), refsize-cons(gstore2, range ri0), refsize-cons(gstore3, (range ri0 \ (range ri0 | m0)) ∪ ax0.toSet), ri-cons(ri0, nodes(gstore2 ↓, jro0)), ri-inj(fi1), ri-inj(ri0[nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(m0), ri0)].keys, ax0]), ri-inj(ri0), m0 ∈ jro ∪ dom gstore3, m0 ∈ gstore3, m0 ∈ gstore2, adr ∈ ax0, m0 ∈ gstore1, 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(m0), ri0)].keys, ax0].lebs ⊆ dom gstore3, ri0.lebs ⊆ dom gstore3, jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore3 -- m0, jro ++ m0), jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore3, jro), jvalid3 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro0), jvalid5 → jvalid4 ⊦ #((ax0.toSet | m0) ∩ ax0.toSet) = 0, #(jro ∪ dom gstore3) = 0, jmaxino = 0, jmaxino0 = 0, m = 0, pmaxino = 0, (ax0.toSet | m0) ∩ ax0.toSet = ∅, jro ∪ dom gstore3 = ∅, dups(nx1), dups(nx0), dups(ax0), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax.to-adr(m0), ri0)].keys), m0 ∈ nx1, m0 ∈ nx0, m0 ∈ jro0

Rule argument

spec: address-data name: elim seq: ⊦ (m = adr.lnum ∧ m0 = adr.pos) ∧ m1 = adr.size ↔ adr = @(m, m0, m1) subst: [adr, m, m0, m1] → [adr, m0, m1, m2]

Additional information

Used simplifier rules: ⊦ (m = adr.lnum ∧ m0 = adr.pos) ∧ m1 = adr.size ↔ adr = @(m, m0, m1); (elim) Used terms: adr, m0, m1, m2