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Proof step 22 from proof prepost-extract-min (previous step)

kar[m0] = key-index(key0, m), ar = ar1[m, ar1[m].idx:= 0], bh0 = bin-heap(kar[0, kar[m0]], m0), 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 = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, kar[0].idx) = kar[m0] ∨ key-index(key, kar[0].idx) ∈ 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, kar[0].idx) = kar[m0] ∨ key-index(key, kar[0].idx) ∈ keyset(kar, m0)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, m0) ++ kar[m0]) -- key-index(key, kar[0].idx) ∧ (∀ ki. ki = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key))), #(keyset(kar, m0) ++ kar[m0]) = 0, m0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 23:
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, kar[0].idx) = key-index(key0, m) ∨ key-index(key, kar[0].idx) ∈ 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, kar[0].idx) = kar[m0] ∨ key-index(key, kar[0].idx) ∈ keyset(kar, m0)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, m0) ++ kar[m0]) -- key-index(key, kar[0].idx) ∧ (∀ ki. ki = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key)))

Rule argument

Additional information

Used simplifier rules: ⊦ ∅ ≠ s ++ ki; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]