[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 57 from proof inv-addn (previous step)

# lax = # gndx, abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓), nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro1), remove-refsize0(gstore0) = gstore0, elems nx1 ∩ jro1 = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx0 ∩ jro0 = ∅, dom gstore0 ∪ elems nx1 = dom gstore2 ∪ jro1, dom gstore0 ∪ elems nx0 = dom gstore3 ∪ jro0, nx0 ≠ [], pmaxino ≠ 0, p_os < pmaxino, fi2.inos < jmaxino1, fi1.inos < pmaxino, inv(gstore3[nx0.last]), inv((gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax)), inv(gstore0), inv(gstore2 \ jro1), inv(gstore3 \ jro0), inv-flashsize(gndx, lax), refsize-cons(gstore2, range fi2), refsize-cons(gstore0, range fi1), refsize-cons(gstore3, range fi2), ri-inj(fi2), ri-inj(fi1), transaction(ndx, gndx), valid-journal-head(nx0, gstore3, jro0), nx0.last ∈ gstore3, pmaxino ≤ jmaxino1, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore0, dom gstore2 ⊆ dom gstore3, ¬ dups(lax), ¬ dups(gstore3[nx0.last].ladrlist), ¬ dups(nx1), ¬ dups(nx0), ¬ dups(gstore3[nx0.last].ladrlist + lax), jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ ⟪index_sync# (fi2, fi1, jro0, gstore3 [nx0.last, gstore3[nx0.last, gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)][nx0.last] .ladrlist:= (gstore3[nx0.last, gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)][nx0.last].ladrlist + lax) ], gstore0, nx0, pmaxino, p_os ; ; jvalid, err0)⟫ ( valid-journal-head (nx0, gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax)], jro0) ∧ refsize-cons (gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax)], range fi2) ∧ nodes-cons (lax.to-adr(nx0.last), ndx, gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax)], jro0) ∧ abs-log(nx0.butlast, gstore3 ↓) + ((gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax))↓.ladrlist .to-adr (nx0.last ) = abs-log(nx1, gstore2 ↓) + lax.to-adr(nx0.last) ∧ nodes (gstore3 ↓[nx0.last, ((gstore3[nx0.last].gndlist:= (gstore3[nx0.last].gndlist + gndx)).ladrlist:= (gstore3[nx0.last].ladrlist + lax))↓], jro0) = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 58:
# lax = # gndx, gstore3[nx0.last] = groupblock(gndx0, lax0, m), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓), nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro1), remove-refsize0(gstore0) = gstore0, elems nx0 ∩ jro0 = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro1 = ∅, dom gstore0 ∪ elems nx0 = dom gstore3 ∪ jro0, dom gstore0 ∪ elems nx1 = dom gstore2 ∪ jro1, fi1.inos < pmaxino, fi2.inos < jmaxino1, p_os < pmaxino, inv(gstore3 \ jro0), inv(gstore2 \ jro1), inv(gstore0), inv((gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax)), inv(gstore3[nx0.last]), inv-flashsize(gndx, lax), refsize-cons(gstore3, range fi2), refsize-cons(gstore0, range fi1), refsize-cons(gstore2, range fi2), ri-inj(fi1), ri-inj(fi2), transaction(ndx, gndx), valid-journal-head(nx0, gstore3, jro0), nx0.last ∈ gstore3, pmaxino ≤ jmaxino1, dom gstore2 ⊆ dom gstore3, fi1.lebs ⊆ dom gstore0, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ ⟪index_sync# (fi2, fi1, jro0, gstore3 [nx0.last, gstore3[nx0.last, gstore3[nx0.last].gndlist:= (gndx0 + gndx)][nx0.last] .ladrlist:= (gstore3[nx0.last, gstore3[nx0.last].gndlist:= (gndx0 + gndx)][nx0.last].ladrlist + lax) ], gstore0, nx0, pmaxino, p_os ; ; jvalid, err0)⟫ ( valid-journal-head(nx0, gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax)], jro0) ∧ refsize-cons(gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax)], range fi2) ∧ nodes-cons(lax.to-adr(nx0.last), ndx, gstore3[nx0.last, (gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax)], jro0) ∧ abs-log(nx0.butlast, gstore3 ↓) + ((gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax))↓.ladrlist.to-adr(nx0.last) = abs-log(nx1, gstore2 ↓) + lax.to-adr(nx0.last) ∧ nodes(gstore3 ↓[nx0.last, ((gstore3[nx0.last].gndlist:= (gndx0 + gndx)).ladrlist:= (lax0 + lax))↓], jro0) = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1))), nx0 = [], pmaxino = 0, dups(lax), dups(lax0), dups(nx1), dups(nx0), dups(lax0 + lax)

Rule argument

spec: ablock, inst: group-store name: elim seq: ⊦ (gndx = gblock.gndlist ∧ lax = gblock.ladrlist) ∧ m = gblock.ref-size ↔ gblock = groupblock(gndx, lax, m) subst: [gblock, gndx, lax, m] → [gstore3[nx0.last], gndx0, lax0, m]

Additional information

Used simplifier rules: ⊦ (gndx = gblock.gndlist ∧ lax = gblock.ladrlist) ∧ m = gblock.ref-size ↔ gblock = groupblock(gndx, lax, m); (elim) Used terms: gstore3[nx0.last], gndx0, lax0, m