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Proof step 20 from proof inv-get-gc-block (previous step)

# lpt = # avol, lpt[m3] = mklp(m0, m1, lpf, m2), kar[0] = key-index(natkey, m3), keyset(bin-heap(kar, m)) = gc-heap(lpt, log), bin-heap-inv(bin-heap(kar, m), lpt), buf-ro-rw-mode(isbuf, bleb, lpt, avol), bufs-log-only(bleb, log, lpt), freelist-cons(lpt, freelist), log-group-block(log, lpt), lpt-free-unref(lpt), lpt-leb-size(lpt), lpt-ram-flash(lpt, lpt0, log) ⊦ # kar = 0, m = 0, dups(log), m3 < # avol ∧ lpf = LP_GROUP_NODES ∧ ¬ m3 ∈ log

User applied rule apply lemma

And obtained the following new goals

Proof step 21:
⊦ key-index(n0, n1) ∈ gc-heap(lpt, nx) ↔ n1 < # lpt ∧ ¬ n1 ∈ nx ∧ lpt[n1].ref-size = n0 ∧ lpt[n1].flags = LP_GROUP_NODES
Proof step 22:
# lpt = # avol, key-index(natkey, m3) ∈ gc-heap(lpt, log) ↔ m3 < # lpt ∧ ¬ m3 ∈ log ∧ lpt[m3].ref-size = natkey ∧ lpt[m3].flags = LP_GROUP_NODES, kar[0] = key-index(natkey, m3), lpt[m3] = mklp(m0, m1, lpf, m2), keyset(bin-heap(kar, m)) = gc-heap(lpt, log), bin-heap-inv(bin-heap(kar, m), lpt), buf-ro-rw-mode(isbuf, bleb, lpt, avol), bufs-log-only(bleb, log, lpt), freelist-cons(lpt, freelist), log-group-block(log, lpt), lpt-free-unref(lpt), lpt-leb-size(lpt), lpt-ram-flash(lpt, lpt0, log) ⊦ # kar = 0, m = 0, dups(log), m3 < # avol ∧ lpf = LP_GROUP_NODES ∧ ¬ m3 ∈ log

Rule argument

spec: persistence-base name: gc-heap-in seq: ⊦ key-index(n0, n1) ∈ gc-heap(lpt, nx) ↔ n1 < # lpt ∧ ¬ n1 ∈ nx ∧ lpt[n1].ref-size = n0 ∧ lpt[n1].flags = LP_GROUP_NODES subst: [n0, n1, lpt, nx] → [natkey, m3, lpt, log]

Additional information

Used terms: natkey, m3, lpt, log