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Proof step 43 from proof inv-addn (previous step)

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 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, # lax < # gndx, p_os < pmaxino, fi2.inos < jmaxino1, fi1.inos < pmaxino, inv(groupblock(gndx0, lax0, m)), inv(groupblock(gndx0 + prefixn(# lax, gndx), lax0 + lax, m)), inv(gstore0), inv(gstore2 \ jro1), inv(gstore3 \ jro0), inv-flashsize(prefixn(# lax, gndx), lax), refsize-cons(gstore3[nx0.last, groupblock(gndx0 + prefixn(# lax, gndx), lax0 + lax, m)], 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), transaction-prefix(ndx, prefixn(# lax, 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, ⌊err0⌋ , jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ # lax = # gndx, gndx = [], nx0 = [], pmaxino = 0, dups(nx1), dups(nx0), dups(lax), nodes (gstore3 ↓ [nx0.last, groupblock(prefixn(valid-group-nodes(gndx0), gndx0 + prefixn(# lax, gndx)), prefixn(valid-group-nodes(gndx0), lax0 + lax), m) ], jro0) = nodes(gstore2 ↓, jro1) ∧ abs-log(nx0.butlast, gstore3 ↓) + prefixn(valid-group-nodes(gndx0), lax0 + lax).to-adr(nx0.last) = abs-log(nx1, gstore2 ↓)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 44:
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, # lax < # gndx, inv(gstore3 \ jro0), inv(gstore2 \ jro1), inv(gstore0), inv(groupblock(gndx0 + prefixn(# lax, gndx), lax0 + lax, m)), inv(groupblock(gndx0, lax0, m)), inv-flashsize(prefixn(# lax, gndx), lax), refsize-cons(gstore3, range fi2), refsize-cons(gstore0, range fi1), refsize-cons(gstore2, range fi2), refsize-cons(gstore3[nx0.last, groupblock(gndx0 + prefixn(# lax, gndx), lax0 + lax, m)], 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, ⌊err0⌋ , jvalid4 ∧ nx1 ≠ [] → valid-journal-head(nx1, gstore2, jro1) ⊦ # lax = # gndx, gndx = [], nx0 = [], pmaxino = 0, dups(lax), dups(nx0), dups(nx1), nodes(gstore3 ↓[nx0.last, groupblock(gndx0, lax0, m)], jro0) = nodes(gstore2 ↓, jro1) ∧ abs-log(nx0.butlast, gstore3 ↓) + lax0.to-adr(nx0.last) = abs-log(nx1, gstore2 ↓)

Rule argument

Additional information

Used simplifier rules: ls-inv-prefix-app-prefixn: ⊦ inv(groupblock(gndx0 + prefixn(# lax1, gndx1), lax0 + lax1, m)) ∧ # lax1 < # gndx1 → prefixn(# gndx0, lax0 + lax) = lax0;%a: ⊦ (lax + lax0) + lax1 = lax + lax0 + lax1; from specification list[laddress-list]%a: ⊦ (gndx + gndx0) + gndx1 = gndx + gndx0 + gndx1; from specification list[group-store]% ⊦ prefixn(# gndx, gndx) = gndx; (s)%n ≤ # gndx ⊦ prefixn(n, gndx + gndx0) = prefixn(n, gndx); (s)% ⊦ ¬ n < n; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%⊦ valid-journal-head(nx, gstore, ginval) ∧ nx ≠ [] ∧ gstore[nx.last] = groupblock(gndx, lax, lnum) → valid-group-nodes(gndx) = # gndx; (s)%n < # gndx ⊦ transaction(ndx, gndx) → transaction-prefix(ndx, prefixn(n, gndx)); (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ n ≠ n0 → (n < n0 ↔ ¬ n0 < n); (ss)