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Proof step 25 from proof prepost-extract-min (previous step)

kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), ar = ar1[m, ar1[m].idx:= 0], bh0 = bin-heap(kar[0, key-index(key0, m)], m0), #(keyset(kar, m0) ++ key-index(key0, m)) ≠ 0, m0 ≠ 0, 2exp-size(# kar), m < # ar1, m0 < # kar, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0) ⊦ ⟪bin_min_heap_bubble_down#(0; bh0, ar)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar1, ar) ∧ (∃ key. (key-index(key, m1) = kar[m0] ∨ key-index(key, m1) ∈ keyset(kar, m0)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, m0) ++ kar[m0]) -- key-index(key, m1) ∧ (∀ ki. ki = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key)))

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] → [0, ar, bh0, bh0, ar] flags: <all paths>

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