[Imprint][Data Privacy Statement][E-Mail]

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Proof step 81 from proof prepost-remove (previous step)

kar[m2] = key-index(key0, m0), kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar = ar1[m0, ar1[m0].idx:= m1], ar = ar, ar2 = ar, bh = bin-heap(kar[m1, key-index(key0, m0)], m2), bh = bh, bh1 = bh, n0 = m1, 2exp-size(# kar), m1 < m2, m0 < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, bubble-down-inv(m1, bh, ar, bh, ar), 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) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ [bin_min_heap_bubble_down#(m1; bh, ar)] ( bh.size = bh1.size ∧ # bh.ar = # bh1.ar ∧ id-idx(ar2, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(bh1) = keyset(bh) → 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)), #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, m1 ≠ 0 ∧ key0 < key1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 82:
kar[m2] = key-index(key0, m0), kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar = ar1[m0, ar1[m0].idx:= m1], bh = bin-heap(kar[m1, key-index(key0, m0)], m2), #(keyset(kar, m2) ++ key-index(key0, m0)) ≠ 0, m2 ≠ m1, 2exp-size(# kar), m1 < m2, m0 < # 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), 2 * m1 + 1 < m2 ∧ kar[2 * m1 + 1].key < key0 ∨ 2 * m1 + 2 < m2 ∧ kar[2 * m1 + 2].key < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ kar[2 * m1 + 2].key < kar[2 * m1 + 1].key ⊃ kar[2 * m1 + 2];kar[2 * m1 + 1])], m2), (2 * m1 + 1 < m2 → ¬ kar[2 * m1 + 1].key < key0) ∧ (2 * m1 + 2 < m2 → ¬ kar[2 * m1 + 2].key < key0) → heap-order(kar[m1, key-index(key0, m0)], m2), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ [bin_min_heap_bubble_down#(m1; bh, ar)] ( bh.size = m2 ∧ # bh.ar = # kar ∧ id-idx(ar1, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(kar[m1, key-index(key0, m0)], bh.size) = keyset(bh.ar, bh.size) → bh.size ≤ # bh.ar ∧ 2exp-size(# bh.ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, bh.size) ++ kar[bh.size]) -- key-index(key, m)), m1 ≠ 0 ∧ key0 < key1

Rule argument

Additional information

Used simplifier rules: bubble-down-inv: ⊦ bubble-down-inv(n, bh, ar, bh0, ar0) ↔ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ ( ( 2 * n + 1 < bh.size → ¬ bh.ar[2 * n + 1].key < bh.ar[n].key) ∧ ( 2 * n + 2 < bh.size → ¬ bh.ar[2 * n + 2].key < bh.ar[n].key) → heap-order(bh.ar, bh.size)) ∧ ( 2 * n + 1 < bh.size ∧ bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order (bh.ar [n, ( 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key ⊃ bh.ar[2 * n + 2] ;bh.ar[2 * n + 1] ) ], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);%keyset: ⊦ keyset(bh) = keyset(bh.ar, bh.size);%m < # kar ⊦ kar[m, ki][m, ki0] = kar[m, ki0]; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)% ⊦ key-index(key, m).key = key; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m + 1 ≠ 0; (s)%mult-c: ⊦ m * n = n * m; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult% ⊦ m0 * 1 = m0; (s)%m1 ≠ 0 ⊦ m = (* m1) * m + n0 ↔ 0 = (m1 - 1) * m + n0; (s)%⊦ m < # kar ∧ valid-idx(kar, ar, m + 1) ∧ kar[m] = key-index(key, m1) → valid-idx(kar[m0, key-index(key, m1)], ar[m1, ar[m1].idx:= m0], m); (s)%⊦ m < # kar ∧ no-dups-idx(kar, m + 1) ∧ kar[m] = ki → no-dups-idx(kar[m0, ki], m); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ id-idx(ar, ar); (s)% ⊦ id-idx(ar1, ar0[n, ar0[n].idx:= m]) ↔ id-idx(ar1, ar0); (s)% ⊦ id-idx(ar1[m, ar1[m].idx:= m0], ar0) ↔ id-idx(ar1, ar0); (s)% ⊦ m < n → m ≤ n; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)