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Proof step 13 from proof inv-set-gblock-refsize (previous step)

# lpt1 = # avol, lpt1[m] = mklp(m0, m1, LP_GROUP_NODES, m2), keyset(gch) = gc-heap(lpt1, log), lpt = lpt1[m, mklp(n, m1, LP_GROUP_NODES, m2)], # gc-heap(lpt1, log) ≠ 0, gc-heap(lpt1, log) ≠ ∅, m < # avol, bin-heap-inv(gch, lpt1), buf-ro-rw-mode(isbuf, bleb, lpt1, avol), bufs-log-only(bleb, log, lpt1), freelist-cons(lpt1, freelist), log-group-block(log, lpt1), lpt-free-unref(lpt1), lpt-leb-size(lpt1), lpt-ram-flash(lpt1, lpt0, log), key-index(m0, m) ∈ gc-heap(lpt1, log), ¬ dups(log), ¬ m ∈ log ⊦ ⟪gc_heap_update#(lpt[m].gcheapidx, n; gch, lpt)⟫ ( lpt-free-unref(lpt) ∧ freelist-cons(lpt, freelist) ∧ bin-heap-inv(gch, lpt) ∧ keyset(gch) = gc-heap(lpt, log) ∧ bufs-log-only(bleb, log, lpt) ∧ # lpt = # avol ∧ log-group-block(log, lpt) ∧ buf-ro-rw-mode(isbuf, bleb, lpt, avol) ∧ lpt-leb-size(lpt) ∧ lpt-ram-flash(lpt, lpt0, log))

User applied rule insert program lemma

And obtained the following new goals

Proof step 14:
bin-heap-inv(bh, lpt), lpt0 = lpt, bh0 = bh, key-index(natkey, m) ∈ keyset(bh0), lpt[m].gcheapidx = n ⊦ ⟪gc_heap_update#(n, natkey0; bh, lpt)⟫ (bin-heap-inv(bh, lpt) ∧ id-idx(lpt0, lpt) ∧ keyset(bh) = (keyset(bh0) -- key-index(natkey, m)) ++ key-index(natkey0, m))
Proof step 15:
# lpt1 = # avol, lpt[m].gcheapidx = lpt[m].gcheapidx, lpt1[m] = mklp(m0, m1, LP_GROUP_NODES, m2), bh1 = gch, gch = gch, keyset(gch) = gc-heap(lpt1, log), lpt = lpt1[m, mklp(n, m1, LP_GROUP_NODES, m2)], lpt = lpt, lpt2 = lpt, m3 = m, n0 = lpt[m].gcheapidx, natkey1 = m0, natkey2 = n, m < # avol, bin-heap-inv(gch, lpt1), bin-heap-inv(gch, lpt), buf-ro-rw-mode(isbuf, bleb, lpt1, avol), bufs-log-only(bleb, log, lpt1), freelist-cons(lpt1, freelist), log-group-block(log, lpt1), lpt-free-unref(lpt1), lpt-leb-size(lpt1), lpt-ram-flash(lpt1, lpt0, log), key-index(m0, m) ∈ gc-heap(lpt1, log), key-index(m0, m) ∈ keyset(gch) ⊦ [gc_heap_update#(lpt[m].gcheapidx, n; gch, lpt)] ( bin-heap-inv(gch, lpt) ∧ id-idx(lpt2, lpt) ∧ keyset(gch) = (keyset(bh1) -- key-index(natkey1, m3)) ++ key-index(natkey2, m3) → lpt-free-unref(lpt) ∧ freelist-cons(lpt, freelist) ∧ bin-heap-inv(gch, lpt) ∧ keyset(gch) = gc-heap(lpt, log) ∧ bufs-log-only(bleb, log, lpt) ∧ # lpt = # avol ∧ log-group-block(log, lpt) ∧ buf-ro-rw-mode(isbuf, bleb, lpt, avol) ∧ lpt-leb-size(lpt) ∧ lpt-ram-flash(lpt, lpt0, log)), # gc-heap(lpt1, log) = 0, gc-heap(lpt1, log) = ∅, dups(log), m ∈ log

Rule argument

spec: bin-min-heap-prepost, inst: gc-heap-rename-gc-heap-base name: prepost-update seq: bin-heap-inv(bh, lpt), lpt0 = lpt, bh0 = bh, key-index(natkey, m) ∈ keyset(bh0), lpt[m].gcheapidx = n ⊦ ⟪gc_heap_update#(n, natkey0; bh, lpt)⟫ ( bin-heap-inv(bh, lpt) ∧ id-idx(lpt0, lpt) ∧ keyset(bh) = (keyset(bh0) -- key-index(natkey, m)) ++ key-index(natkey0, m)) subst: [natkey0, n, lpt, bh, natkey, m, lpt0, bh0] → [n, lpt[m].gcheapidx, lpt, gch, m0, m, lpt, 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)