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Proof step 74 from proof inv-addn (previous step)

# lax = # gndx, # gndx0 = # lax1, gstore3[nx0.last] = groupblock(gndx0, lax1, m), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓), nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro1), remove-refsize0(gstore1) = gstore1, elems nx0 ∩ jro0 = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx1 ∩ jro1 = ∅, dom gstore1 ∪ elems nx0 = dom gstore3 ∪ jro0, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro1, fi1.inos < pmaxino, fi2.inos < jmaxino1, p_os < pmaxino, inv(groupblock(gndx0, lax1, m)), inv(gstore3 \ jro0), inv(gstore2 \ jro1), inv(gstore1), inv(groupblock(gndx0 + gndx, lax1 + lax, m)), inv-flashsize(gndx, lax), refsize-cons(gstore3, range fi2), refsize-cons(gstore1, 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 gstore1, fi2.lebs ⊆ dom gstore2, fi2.lebs ⊆ dom gstore3, jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ nx0 = [], pmaxino = 0, dups(nx0), dups(nx1), dups(lax), valid-journal-head(nx0, gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], jro0) ∧ refsize-cons(gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], range fi2) ∧ nodes-cons(lax.to-adr(nx0.last), ndx, gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], jro0) ∧ abs-log(nx0.butlast, gstore3 ↓) + prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), lax1 + lax).to-adr(nx0.last) = abs-log(nx1, gstore2 ↓) + lax.to-adr(nx0.last) ∧ nodes (gstore3 ↓ [nx0.last, groupblock (prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), gndx0 + gndx), prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), lax1 + lax), m) ], jro0) = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
⊦ n ∈ gstore ∧ ¬ n ∈ ginval ∧ gstore[n].gndlist = gndx0 ∧ gstore[n].ladrlist = lax0 ∧ # gndx0 = # lax0 ∧ # gndx1 = # lax1 ∧ ¬ dups(lax0 + lax1) ∧ transaction(ndx, gndx1) → nodes-cons(lax1.to-adr(n), ndx, gstore[n, groupblock(gndx0 + gndx1, lax0 + lax1, m)], ginval)
Proof step 76:
# gndx0 = # lax1, # gndx = # lax, gstore3[nx0.last].gndlist = gndx0, gstore3[nx0.last].ladrlist = lax1, gstore3[nx0.last] = groupblock(gndx0, lax1, m), abs-log(nx0, gstore3 ↓) = abs-log(nx1, gstore2 ↓), nodes(gstore3 ↓, jro0) = nodes(gstore2 ↓, jro1), remove-refsize0(gstore1) = gstore1, elems nx1 ∩ jro1 = ∅, (nx0.toSet \ nx1.toSet) ∩ dom gstore2 = ∅, elems nx0 ∩ jro0 = ∅, dom gstore1 ∪ elems nx1 = dom gstore2 ∪ jro1, dom gstore1 ∪ elems nx0 = dom gstore3 ∪ jro0, p_os < pmaxino, fi2.inos < jmaxino1, fi1.inos < pmaxino, inv(groupblock(gndx0 + gndx, lax1 + lax, m)), inv(gstore1), inv(gstore2 \ jro1), inv(gstore3 \ jro0), inv(groupblock(gndx0, lax1, m)), inv-flashsize(gndx, lax), refsize-cons(gstore2, range fi2), refsize-cons(gstore1, 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 gstore1, dom gstore2 ⊆ dom gstore3, jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ nx0 = [], pmaxino = 0, dups(lax), dups(nx1), dups(nx0), dups(lax1 + lax), nx0.last ∈ jro0, valid-journal-head(nx0, gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], jro0) ∧ refsize-cons(gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], range fi2) ∧ true ∧ abs-log(nx0.butlast, gstore3 ↓) + prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), lax1 + lax).to-adr(nx0.last) = abs-log(nx1, gstore2 ↓) + lax.to-adr(nx0.last) ∧ nodes (gstore3 ↓ [nx0.last, groupblock (prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), gndx0 + gndx), prefixn((gndx = [] ⊃ # lax1;# lax1 + # gndx), lax1 + lax), m) ], jro0) = nodes(gstore2 ↓, jro1)[lax.to-adr(nx0.last), ndx] ∧ new(lax.to-adr(nx0.last), nodes(gstore2 ↓, jro1))

Rule argument

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

Additional information

Used simplifier rules: 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)% ⊦ inv(groupblock(gndx, lax, m0)) → ¬ dups(lax); (s)% ⊦ groupblock(gndx, lax, m).gndlist = gndx; (s)% ⊦ groupblock(gndx, lax, m).ladrlist = lax; (s) Used terms: jro0, m, lax1, gndx, gndx0, gstore3, ndx, nx0.last, lax