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Proof step 26 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)], # gc-heap(lpt2, log0) ≠ 0, buffered(n) ≠ bleb, gc-heap(lpt2, log0) ≠ ∅, 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), key-index(m3, n) ∈ gc-heap(lpt2, log0), ¬ 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 insert program lemma

And obtained the following new goals

Proof step 27:
bin-heap-inv(bh, lpt), lpt0 = lpt, bh0 = bh, key-index(natkey, m) ∈ keyset(bh0), lpt[m].gcheapidx = n ⊦ ⟪gc_heap_remove#(n; bh, lpt)⟫ (bin-heap-inv(bh, lpt) ∧ id-idx(lpt0, lpt) ∧ keyset(bh) = keyset(bh0) -- key-index(natkey, m))
Proof step 28:
# lpt0 = # avol, # lpt2 = # avol, lpt1[n].gcheapidx = lpt1[n].gcheapidx, lpt2[n] = mklp(m3, m1, LP_GROUP_NODES, m2), bh1 = gch, gch = gch, keyset(gch) = gc-heap(lpt2, log0), lpt1 = lpt2[n, mklp(0, 0, LP_FREE, m2)], lpt1 = lpt1, lpt3 = lpt1, m0 = n, n0 = lpt1[n].gcheapidx, natkey0 = m3, n < # avol, awbuf-inv(avol, bleb), bin-heap-inv(gch, lpt2), bin-heap-inv(gch, lpt1), 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), key-index(m3, n) ∈ gc-heap(lpt2, log0), key-index(m3, n) ∈ keyset(gch) ⊦ [gc_heap_remove#(lpt1[n].gcheapidx; gch, lpt1)] ( bin-heap-inv(gch, lpt1) ∧ id-idx(lpt3, lpt1) ∧ keyset(gch) = keyset(bh1) -- key-index(natkey0, m0) → ⟪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))), # gc-heap(lpt2, log0) = 0, buffered(n) = bleb, gc-heap(lpt2, log0) = ∅, dups(log0), n ∈ log0

Rule argument

spec: bin-min-heap-prepost, inst: gc-heap-rename-gc-heap-base name: prepost-remove seq: bin-heap-inv(bh, lpt), lpt0 = lpt, bh0 = bh, key-index(natkey, m) ∈ keyset(bh0), lpt[m].gcheapidx = n ⊦ ⟪gc_heap_remove#(n; bh, lpt)⟫ ( bin-heap-inv(bh, lpt) ∧ id-idx(lpt0, lpt) ∧ keyset(bh) = keyset(bh0) -- key-index(natkey, m)) subst: [n, lpt, bh, natkey, m, lpt0, bh0] → [lpt1[n].gcheapidx, lpt1, gch, m3, n, lpt1, gch] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%n < # lpt ⊦ lpt[n] = mklp(m0, m1, lpf0, m2) → ( bin-heap-inv(gch, lpt[n, mklp(n0, n1, lpf1, m2)]) ↔ bin-heap-inv(gch, lpt)); (s)