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Proof step 23 from proof inv-deallocate-gnd (previous step)

# lpt2 = # avol, # lpt0 = # avol, lpt2[n] = mklp(m3, m1, LP_GROUP_NODES, m2), keyset(gch) = gc-heap(lpt2, log0), lpt1 = lpt2[n, mklp(0, 0, LP_FREE, m2)], buffered(n) ≠ bleb, n < # avol, awbuf-inv(avol, bleb), bin-heap-inv(gch, lpt2), buf-ro-rw-mode(false, bleb, lpt2, avol), bufs-log-only(bleb, log0, lpt2), freelist-cons(lpt2, freelist), log-group-block(log0, lpt2), lpt-free-unref(lpt2), lpt-free-unref(lpt0), lpt-leb-size(lpt2), lpt-leb-size(lpt0), lpt-ram-flash(lpt2, lpt0, log0), synced(avol), ¬ dups(log0), ¬ n ∈ log0 ⊦ ⟪gc_heap_remove#(lpt1[n].gcheapidx; gch, lpt1); err := ESUCCESS⟫ ( lpt-free-unref(lpt1) ∧ freelist-cons(lpt1, freelist + n ') ∧ bin-heap-inv(gch, lpt1) ∧ keyset(gch) = gc-heap(lpt1, log0) ∧ bufs-log-only(bleb, log0, lpt1) ∧ # lpt1 = # avol ∧ log-group-block(log0, lpt1) ∧ buf-ro-rw-mode(false, bleb, lpt1, avol) ∧ lpt-leb-size(lpt1) ∧ lpt-ram-flash(lpt1, lpt0, log0))

User applied rule apply lemma

And obtained the following new goals

Proof step 24:
⊦ 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 25:
# lpt0 = # avol, # lpt2 = # avol, key-index(m3, n) ∈ gc-heap(lpt2, log0) ↔ n < # lpt2 ∧ ¬ n ∈ log0 ∧ lpt2[n].ref-size = m3 ∧ lpt2[n].flags = LP_GROUP_NODES, lpt2[n] = mklp(m3, m1, LP_GROUP_NODES, m2), keyset(gch) = gc-heap(lpt2, log0), lpt1 = lpt2[n, mklp(0, 0, LP_FREE, m2)], n < # avol, awbuf-inv(avol, bleb), bin-heap-inv(gch, lpt2), buf-ro-rw-mode(false, bleb, lpt2, avol), bufs-log-only(bleb, log0, lpt2), freelist-cons(lpt2, freelist), log-group-block(log0, lpt2), lpt-free-unref(lpt0), lpt-free-unref(lpt2), lpt-leb-size(lpt0), lpt-leb-size(lpt2), lpt-ram-flash(lpt2, lpt0, log0), synced(avol) ⊦ ⟪gc_heap_remove#(lpt1[n].gcheapidx; gch, lpt1); err := ESUCCESS⟫ ( lpt-free-unref(lpt1) ∧ freelist-cons(lpt1, freelist + n ') ∧ bin-heap-inv(gch, lpt1) ∧ keyset(gch) = gc-heap(lpt1, log0) ∧ bufs-log-only(bleb, log0, lpt1) ∧ # lpt1 = # avol ∧ log-group-block(log0, lpt1) ∧ buf-ro-rw-mode(false, bleb, lpt1, avol) ∧ lpt-leb-size(lpt1) ∧ lpt-ram-flash(lpt1, lpt0, log0)), buffered(n) = bleb, dups(log0), n ∈ log0

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] → [m3, n, lpt2, log0]

Additional information

Used terms: m3, n, lpt2, log0