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Proof step 13 from proof prepost-cleanup-lpt (previous step)

# lpt4 = n1 + n0, # avol1 = n1 + n0, # lpt3 = n1 + n0, lpt3[n0] = mklp(0, m1, LP_GROUP_NODES, m2), keyset(gch0) = gc-heap(lpt4, []), keyset(gch) = gc-heap(cleanup-lpt(lpt4, n0), []), n1 ≠ 0, avol-cleanup-lpt-unchanged(avol1, avol, lpt4), awbuf-inv(avol1, nobuffer), awbuf-inv(avol, nobuffer), bin-heap-inv(gch0, lpt4), bin-heap-inv(gch, lpt3), freelist-cons(cleanup-lpt(lpt4, n0), freelist), freelist-cons(lpt4, freelist0), id-idx(cleanup-lpt(lpt4, n0), lpt3), synced(avol1), synced(avol) ⊦ ⟪gc_heap_remove#(m2; gch, lpt3); lpt := lpt3[n0, lpt3[n0].flags:= LP_FREE]; lpt[n0] := mklp(lpt[n0].ref-size, 0, lpt[n0].flags, lpt[n0].gcheapidx); freelist := freelist + n0 '; awbuf_unmap#(n0, adr, m, log, lpt0, ns, nobuffer, isbuf; avol); n := n0 + 1⟫ ( n ≤ # lpt ∧ # lpt = # lpt4 ∧ id-idx(cleanup-lpt(lpt4, n), lpt) ∧ awbuf-inv(avol, nobuffer) ∧ avol-cleanup-lpt-unchanged(avol1, avol, lpt4) ∧ freelist-cons(cleanup-lpt(lpt4, n), freelist) ∧ keyset(gch) = gc-heap(cleanup-lpt(lpt4, n), []) ∧ bin-heap-inv(gch, lpt) ∧ # lpt < n1 + n)

User applied rule apply lemma

And obtained the following new goals

Proof step 14:
⊦ 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 15:
# lpt3 = n1 + n0, # avol1 = n1 + n0, # lpt4 = n1 + n0, key-index(0, n0) ∈ gc-heap(lpt3, []) ↔ n0 < # lpt3 ∧ ¬ n0 ∈ [] ∧ lpt3[n0].ref-size = 0 ∧ lpt3[n0].flags = LP_GROUP_NODES, lpt3[n0] = mklp(0, m1, LP_GROUP_NODES, m2), keyset(gch) = gc-heap(cleanup-lpt(lpt4, n0), []), keyset(gch0) = gc-heap(lpt4, []), avol-cleanup-lpt-unchanged(avol1, avol, lpt4), awbuf-inv(avol, nobuffer), awbuf-inv(avol1, nobuffer), bin-heap-inv(gch, lpt3), bin-heap-inv(gch0, lpt4), freelist-cons(lpt4, freelist0), freelist-cons(cleanup-lpt(lpt4, n0), freelist), id-idx(cleanup-lpt(lpt4, n0), lpt3), synced(avol), synced(avol1) ⊦ ⟪gc_heap_remove#(m2; gch, lpt3); lpt := lpt3[n0, lpt3[n0].flags:= LP_FREE]; lpt[n0] := mklp(lpt[n0].ref-size, 0, lpt[n0].flags, lpt[n0].gcheapidx); freelist := freelist + n0 '; awbuf_unmap#(n0, adr, m, log, lpt0, ns, nobuffer, isbuf; avol); n := n0 + 1⟫ ( n ≤ # lpt ∧ # lpt = # lpt4 ∧ id-idx(cleanup-lpt(lpt4, n), lpt) ∧ awbuf-inv(avol, nobuffer) ∧ avol-cleanup-lpt-unchanged(avol1, avol, lpt4) ∧ freelist-cons(cleanup-lpt(lpt4, n), freelist) ∧ keyset(gch) = gc-heap(cleanup-lpt(lpt4, n), []) ∧ bin-heap-inv(gch, lpt) ∧ # lpt < n1 + n), n1 = 0

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] → [0, n0, lpt3, []]

Additional information

Used terms: 0, n0, lpt3, []