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Proof step 41 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 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 (prefixn(valid-group-nodes(gndx0 + prefixn(# lax, gndx)), gndx0 + prefixn(# lax, gndx)), prefixn(valid-group-nodes(gndx0 + prefixn(# lax, gndx)), lax0 + lax), m) ], jro0) = nodes(gstore2 ↓, jro1) ∧ abs-log(nx0.butlast, gstore3 ↓) + prefixn(valid-group-nodes(gndx0 + prefixn(# lax, gndx)), lax0 + lax).to-adr(nx0.last) = abs-log(nx1, gstore2 ↓)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 42:
⊦ transaction-prefix(ndx, gndx1) → valid-group-nodes(gndx0 + gndx1) = valid-group-nodes(gndx0)
Proof step 43:
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 ↓)

Rule argument

spec: gjournal-abs name: valid-group-nodes-app-transaction-prefix seq: ⊦ transaction-prefix(ndx, gndx1) → valid-group-nodes(gndx0 + gndx1) = valid-group-nodes(gndx0) subst: [ndx, gndx1, gndx0] → [ndx, prefixn(# lax, gndx), gndx0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n ≠ n0 → (n < n0 ↔ ¬ n0 < n); (ss)% ⊦ m < n → ¬ n < m; (s)%n < # gndx ⊦ transaction(ndx, gndx) → transaction-prefix(ndx, prefixn(n, gndx)); (s) Used terms: ndx, prefixn(# lax, gndx), gndx0