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Proof step 108 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, ⌊err0⌋ , 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 109:
⊦ refsize-cons(gstore, as) → (refsize-cons(gstore[n, gblock], as) ↔ ¬ n ∈ gstore ∨ gstore[n].ref-size = (as | n).size → gblock.ref-size = (as | n).size)
Proof step 110:
# gndx0 = # lax1, # lax = # gndx, 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, ⌊err0⌋ , jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ nx0 = [], pmaxino = 0, dups(lax), dups(nx1), dups(nx0), valid-journal-head(nx0, gstore3[nx0.last, groupblock(gndx0 + gndx, lax1 + lax, m)], jro0) ∧ ( ¬ nx0.last ∈ gstore3 ∨ gstore3[nx0.last].ref-size = (range fi2 | nx0.last).size → groupblock(gndx0 + gndx, lax1 + lax, m).ref-size = (range fi2 | nx0.last).size) ∧ 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))

Rule argument

spec: ablock-store-refsize-cons, inst: group-store name: refsize-cons-write seq: ⊦ refsize-cons(gstore, as) → ( refsize-cons(gstore[n, gblock], as) ↔ ¬ n ∈ gstore ∨ gstore[n].ref-size = (as | n).size → gblock.ref-size = (as | n).size) subst: [as, gblock, n, gstore] → [range fi2, groupblock(gndx0 + gndx, lax1 + lax, m), nx0.last, gstore3] flags: <all paths>

Additional information

Used terms: range fi2, groupblock(gndx0 + gndx, lax1 + lax, m), nx0.last, gstore3