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Proof step 58 from proof prepost-remove (previous step)

kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), ar = ar1[kar[m2].idx, ar1[kar[m2].idx].idx:= m1], bh = bin-heap(kar[m1, kar[m2]], m2), m0 = m2 + 1, 2exp-size(# kar), m1 < # kar, m1 < m0, kar[m2].idx < # ar1, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar, m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1 ⊦ ⟪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(ar1, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m)), # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, m0 = 0, m0 = 1, m2 = m1, m1 ≠ 0 ∧ kar[m2].key < kar[m1, kar[m2]][(m1 - 1) / 2].key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 59:
kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), ar = ar1[kar[m2].idx, ar1[kar[m2].idx].idx:= m1], bh = bin-heap(kar[m1, kar[m2]], m2), #(keyset(kar, m2) ++ kar[m2]) ≠ 0, m2 ≠ m1, 2exp-size(# kar), m1 < m2, kar[m2].idx < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = kar[m2] ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ ⟪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(ar1, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m2) ++ kar[m2]) -- key-index(key, m)), m1 ≠ 0 ∧ kar[m2].key < kar[m1, kar[m2]][(m1 - 1) / 2].key

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ m ≠ n → (m < n + 1 ↔ m < n); (s)% ⊦ ki ∈ s ++ ki0 ↔ ki = ki0 ∨ ki ∈ s; (s)% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m + 1 ≠ 0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = 1 ↔ m = 0 ∧ n = 1 ∨ n = 0 ∧ m = 1; (ws)