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Proof step 73 from proof prepost-update (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), ar0 = ar, bh = bin-heap(kar[m1, key-index(key, m)], m0), # keyset(kar, m0) ≠ 0, keyset(kar, m0) ≠ ∅, 2exp-size(# kar), m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, ¬ key < key1 ⊦ ⟪bin_min_heap_bubble_down#(m1; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar0, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m0) -- key-index(key1, m)) ++ key-index(key, m))

User applied rule insert program lemma

And obtained the following new goals

Rule argument

spec: <toplevel> name: prepost-bubble-down seq: bh0 = bh, ar0 = ar, bubble-down-inv(n, bh, ar, bh0, ar0) ⊦ ⟪bin_min_heap_bubble_down#(n; bh, ar)⟫ ( bh.size = bh0.size ∧ # bh.ar = # bh0.ar ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(bh0) = keyset(bh)) subst: [n, ar, bh, bh0, ar0] → [m1, ar, bh, bh, ar] flags: <all paths>

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