[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 139 from proof refine-gc (previous step)

# ax0 = # referenced-adrs(lax1.to-adr(n1), ri), # lax1 + # lax0 = # gndx0, gstore2[n1] = groupblock(gndx0, lax1 + lax0, m0), abs-log(nx0, gstore ↓) = abs-log(nx, gstore2 ↓) + ax0, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)]], 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[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys, ax0].inos < jmaxino0, jro1 < jmaxino0, inv(groupblock(gndx0, lax1 + lax0, m0)), inv(gstore2 \ jro0), inv(gstore1), inv(gstore \ jro), log-cons(abs-log(nx, gstore2 ↓), nodes(gstore2 ↓, jro0)), new(ax0, nodes(gstore2 ↓, jro0)), refsize-cons(gstore, range ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].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[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys, ax0]), n1 ∈ gstore2, n1 ∈ gstore1, pmaxino ≤ jmaxino, jmaxino ≤ jmaxino0, pmaxino ≤ jmaxino0, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].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.toSet ∩ abs-log(nx, gstore2 ↓).toSet) = 0, jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, ax0.toSet ∩ abs-log(nx, gstore2 ↓).toSet = ∅, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax0.to-adr(n1), ri)].keys), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys), dups(nx), dups(nx0), dups(ax0), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys + nodes(gstore2 ↓, jro0)[referenced-adrs(lax0.to-adr(n1), ri)].keys), n1 ∈ nx, n1 ∈ jro0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 140:
⊦ new(ax, nodes(gstore, ginval)) ∧ elems nx ⊆ dom gstore ∧ elems nx ∩ ginval = ∅ → ax.toSet ∩ abs-log(nx, gstore).toSet = ∅
Proof step 141:
# ax0 = # referenced-adrs(lax1.to-adr(n1), ri), # lax1 + # lax0 = # gndx0, gstore2[n1] = groupblock(gndx0, lax1 + lax0, m0), abs-log(nx0, gstore ↓) = abs-log(nx, gstore2 ↓) + ax0, nodes(gstore ↓, jro) = nodes(gstore2 ↓, jro0)[ax0, nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)]], 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, p_os.keys.inodes?, jro1.keys.inodes?, jro1 < jmaxino0, ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys, ax0].inos < jmaxino0, jro1 < jmaxino, p_os < pmaxino, ri.inos < jmaxino, fi1.inos < pmaxino, inv(gstore \ jro), inv(gstore1), inv(gstore2 \ jro0), inv(groupblock(gndx0, lax1 + lax0, m0)), 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[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys, ax0]), ri-cons(ri, nodes(gstore2 ↓, jro0)), ri-inj(ri[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].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[nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys, ax0].lebs ⊆ dom gstore, ⌊err⌋ , jvalid0 ∧ nx0 ≠ [] → valid-journal-head(nx0, gstore, jro), jvalid5 → jvalid4, jvalid3 ∧ nx ≠ [] → valid-journal-head(nx, gstore2, jro0) ⊦ #(ax0.toSet ∩ abs-log(nx, gstore2 ↓).toSet) = 0, jmaxino = 0, jmaxino0 = 0, m0 = 0, pmaxino = 0, ax0.toSet ∩ abs-log(nx, gstore2 ↓).toSet = ∅, dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys + nodes(gstore2 ↓, jro0)[referenced-adrs(lax0.to-adr(n1), ri)].keys), dups(ax0), dups(nx0), dups(nx), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax1.to-adr(n1), ri)].keys), dups(nodes(gstore2 ↓, jro0)[referenced-adrs(lax0.to-adr(n1), ri)].keys), n1 ∈ jro0, n1 ∈ nx, new(ax0, nodes(gstore2 ↓, jro0)) ∧ elems nx ⊆ dom gstore2 ↓ ∧ elems nx ∩ jro0 = ∅

Rule argument

spec: gjournal-abs name: abs-log-new seq: ⊦ new(ax, nodes(gstore, ginval)) ∧ elems nx ⊆ dom gstore ∧ elems nx ∩ ginval = ∅ → ax.toSet ∩ abs-log(nx, gstore).toSet = ∅ subst: [ginval, gstore, nx, ax] → [jro0, gstore2 ↓, nx, ax0] flags: <all paths>

Additional information

Used terms: jro0, gstore2 ↓, nx, ax0