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Proof step 59 from proof inv-recover (previous step)

# lpt3 = # avol0, # avol = # avol0, # lpt2 = # avol0, # avol2 = # avol0, replay-lpt(replay-log(lpt3, log2), log2, # avol0)[n] = mklp(m, m0, lpf, m1), gc-heap(replay-lpt(replay-log(lpt3, log2), log2, # avol0), log2) = keyset(gch), keyset(gch1) = gc-heap(lpt2, log2), avol-replay-lpt-unchanged(avol, avol2, replay-log(lpt3, log2), log2), awbuf-inv(avol2, nobuffer), awbuf-inv(avol0, bleb1), bin-heap-inv(gch1, lpt2), bin-heap-inv(gch, lpt), buf-ro-rw-mode(isbuf1, bleb1, lpt2, avol0), bufs-log-only(bleb1, log2, lpt2), freelist-cons(lpt2, log5), freelist-cons(replay-lpt(replay-log(lpt3, log2), log2, # avol0), freelist), id-idx(replay-lpt(replay-log(lpt3, log2), log2, # avol0), lpt), log-group-block(log2, lpt2), lpt-free-unref(lpt2), lpt-free-unref(lpt3), lpt-leb-size(lpt2), lpt-leb-size(lpt3), lpt-ram-flash(lpt2, lpt3, log2), synced(avol0), synced(avol), valid-indices(lpt3, log2), avol ↓ avol0, n ∈ log2 ⊦ dups(log2), n < # avol0 ∧ lpf = LP_GROUP_NODES, # lpt = # avol0 ∧ log-group-block(log2, lpt) ∧ lpt-leb-size(replay-lpt(replay-log(lpt3, log2), log2, # avol0)) ∧ lpt-ram-flash(lpt, lpt3, log2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 60:
m < n, n ≤ # lpt ⊦ replay-lpt(lpt, nx, n)[m] = (lpt[m].flags = LP_FREE ∨ m ∈ nx ⊃ lpt[m] ;(lpt[m].flags = LP_INDEX_NODES ∧ lpt[m].ref-size ≠ 0 ⊃ lpt[m].size:= LEB_SIZE ;(lpt[m].ref-size = 0 ⊃ mklp(0, 0, LP_FREE, lpt[m].gcheapidx);lpt[m]) ) )
Proof step 61:
# avol2 = # avol0, # lpt2 = # avol0, # avol = # avol0, # lpt3 = # avol0, replay-lpt(replay-log(lpt3, log2), log2, # avol0)[n] = mklp(m, m0, lpf, m1), gc-heap(replay-lpt(replay-log(lpt3, log2), log2, # avol0), log2) = keyset(gch), keyset(gch1) = gc-heap(lpt2, log2), avol-replay-lpt-unchanged(avol, avol2, replay-log(lpt3, log2), log2), awbuf-inv(avol0, bleb1), awbuf-inv(avol2, nobuffer), bin-heap-inv(gch, lpt), bin-heap-inv(gch1, lpt2), buf-ro-rw-mode(isbuf1, bleb1, lpt2, avol0), bufs-log-only(bleb1, log2, lpt2), freelist-cons(replay-lpt(replay-log(lpt3, log2), log2, # avol0), freelist), freelist-cons(lpt2, log5), id-idx(replay-lpt(replay-log(lpt3, log2), log2, # avol0), lpt), log-group-block(log2, lpt2), lpt-free-unref(lpt3), lpt-free-unref(lpt2), lpt-leb-size(lpt3), lpt-leb-size(lpt2), lpt-ram-flash(lpt2, lpt3, log2), synced(avol), synced(avol0), valid-indices(lpt3, log2), avol ↓ avol0, n ∈ log2 ⊦ dups(log2), # lpt = # avol0 ∧ log-group-block(log2, lpt) ∧ lpt-leb-size(replay-lpt(replay-log(lpt3, log2), log2, # avol0)) ∧ lpt-ram-flash(lpt, lpt3, log2), n < # avol0 ∧ lpf = LP_GROUP_NODES, n < # avol0 ∧ # avol0 ≤ # replay-log(lpt3, log2)

Rule argument

spec: persistence-base name: replay-lpt-at seq: m < n, n ≤ # lpt ⊦ replay-lpt(lpt, nx, n)[m] = (lpt[m].flags = LP_FREE ∨ m ∈ nx ⊃ lpt[m] ;(lpt[m].flags = LP_INDEX_NODES ∧ lpt[m].ref-size ≠ 0 ⊃ lpt[m].size:= LEB_SIZE ;(lpt[m].ref-size = 0 ⊃ mklp(0, 0, LP_FREE, lpt[m].gcheapidx) ;lpt[m] ) ) ) subst: [m, n, nx, lpt] → [n, # avol0, log2, replay-log(lpt3, log2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%n < # lpt ⊦ ¬ n ∈ nx → replay-log(lpt, nx)[n] = lpt[n]; (s)%n < # lpt ⊦ n ∈ nx → replay-log(lpt, nx)[n] = mklp (lpt[n].ref-size, LEB_SIZE, LP_GROUP_NODES, lpt[n].gcheapidx); (s)%⊦ mklp(m, m0, lpf, m1) = mklp(m2, m3, lpf0, m4) ↔ m = m2 ∧ m0 = m3 ∧ lpf = lpf0 ∧ m1 = m4; (ws)% ⊦ # replay-log(lpt, nx) = # lpt; (s) Used terms: n, # avol0, log2, replay-log(lpt3, log2)