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Proof step 162 from proof inv-addn (previous step)

# gndx0 = # lax0, # gndx = # lax, gstore3[nx0.last] = groupblock(gndx0, lax0, m0), 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(groupblock(gndx0 + gndx, lax0 + lax, m0)), inv(groupblock(gndx0, lax0, m0)), inv-flashsize(gndx, lax), refsize-cons(gstore3, range fi2), refsize-cons(gstore0, range fi1), refsize-cons(gstore2, range fi2), refsize-cons(gstore3[nx0.last, groupblock(gndx0 + gndx, lax0 + lax, m0)], 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) ⊦ gndx = [], nx0 = [], pmaxino = 0, dups(lax), dups(nx0), dups(nx1), new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1)) ∧ nodes(gstore3 ↓[nx0.last, groupblock(gndx0 + gndx, lax0 + lax, m0)], jro0) = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ abs-log(nx0.butlast, gstore3 ↓) + lax0.to-adr(nx0.last) = abs-log(nx1, gstore2 ↓) ∧ valid-journal-head(nx0, gstore3[nx0.last, groupblock(gndx0 + gndx, lax0 + lax, m0)], jro0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 163:
⊦ n ∈ gstore ∧ ¬ n ∈ ginval ∧ transaction(ndx, gndx1) ∧ gstore[n].gndlist = gndx0 ∧ gstore[n].ladrlist = lax0 ∧ # lax0 = # gndx0 ∧ # lax1 = # gndx1 ∧ ¬ dups(lax0 + lax1) → nodes(gstore[n, groupblock(gndx0 + gndx1, lax0 + lax1, m0)], ginval) = nodes(gstore, ginval)[lax1.to-adr(n), ndx]
Proof step 164:
# lax0 = # gndx0, # lax = # gndx, gstore3 ↓[nx0.last].gndlist = gndx0, gstore3 ↓[nx0.last].ladrlist = lax0, gstore3[nx0.last] = groupblock(gndx0, lax0, m0), 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, p_os < pmaxino, fi2.inos < jmaxino1, fi1.inos < pmaxino, inv(groupblock(gndx0, lax0, m0)), inv(groupblock(gndx0 + gndx, lax0 + lax, m0)), inv(gstore0), inv(gstore2 \ jro1), inv(gstore3 \ jro0), inv-flashsize(gndx, lax), refsize-cons(gstore3[nx0.last, groupblock(gndx0 + gndx, lax0 + lax, m0)], range fi2), 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, nx0.last ∈ gstore3 ↓, pmaxino ≤ jmaxino1, fi2.lebs ⊆ dom gstore3, fi2.lebs ⊆ dom gstore2, fi1.lebs ⊆ dom gstore0, dom gstore2 ⊆ dom gstore3, jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ gndx = [], nx0 = [], pmaxino = 0, dups(nx1), dups(nx0), dups(lax), dups(lax0 + lax), nx0.last ∈ jro0, new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1)) ∧ nodes(gstore3 ↓, jro0)[lax.to-adr(nx0.last), ndx] = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ abs-log(nx0.butlast, gstore3 ↓) + lax0.to-adr(nx0.last) = abs-log(nx1, gstore2 ↓) ∧ valid-journal-head(nx0, gstore3[nx0.last, groupblock(gndx0 + gndx, lax0 + lax, m0)], jro0)

Rule argument

spec: gjournal-abs name: nodes-transaction seq: ⊦ n ∈ gstore ∧ ¬ n ∈ ginval ∧ transaction(ndx, gndx1) ∧ gstore[n].gndlist = gndx0 ∧ gstore[n].ladrlist = lax0 ∧ # lax0 = # gndx0 ∧ # lax1 = # gndx1 ∧ ¬ dups(lax0 + lax1) → nodes (gstore[n, groupblock(gndx0 + gndx1, lax0 + lax1, m0)], ginval) = nodes(gstore, ginval)[lax1.to-adr(n), ndx] subst: [ndx, ginval, m0, lax1, lax0, gndx1, gndx0, n, gstore] → [ndx, jro0, m0, lax, lax0, gndx, gndx0, nx0.last, gstore3 ↓] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ∈ gstore ↓ ↔ n ∈ gstore; (s)%int-a: ⊦ (ns ∩ ns0) ∩ ns1 = ns ∩ ns0 ∩ ns1; from specification set-union[nat-set]%int-c: ⊦ ns ∩ ns0 = ns0 ∩ ns; from specification set-union[nat-set]% ⊦ elems nx ∩ ginval = ∅ ∧ nx ≠ [] → ¬ nx.last ∈ ginval; (s)% ⊦ n ∈ gstore → gstore ↓[n] = gstore[n] ↓; (s)% ⊦ inv(groupblock(gndx, lax, m0)) → ¬ dups(lax); (s)%⊦ groupblock(gndx, lax, m0) ↓ = groupblock (prefixn(valid-group-nodes(gndx), gndx), prefixn(valid-group-nodes(gndx), lax), m0); (s)%⊦ valid-journal-head(nx, gstore, ginval) ∧ nx ≠ [] ∧ gstore[nx.last] = groupblock(gndx, lax, lnum) → valid-group-nodes(gndx) = # gndx; (s)% ⊦ prefixn(# gndx, gndx) = gndx; (s)% ⊦ inv(groupblock(gndx, lax, m0)) → prefixn(# gndx, lax) = lax; (s)% ⊦ groupblock(gndx, lax, m).gndlist = gndx; (s)% ⊦ groupblock(gndx, lax, m).ladrlist = lax; (s) Used terms: ndx, jro0, m0, lax, lax0, gndx, gndx0, nx0.last, gstore3 ↓