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

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Proof step 21 from proof prepost-bubble-up (previous step)

# bh.ar = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), keyset(bh.ar, bh.size) = keyset(kar, bh.size), n0 = m + 1, n0 < # kar, n0 < bh.size, m / 2 < # kar, n1 < # kar, n1 < bh.size, bh.ar[m / 2].idx < # ar, bh.ar[n0].key < bh.ar[m / 2].key, bh.ar[n0].idx < # ar, heap-order(bh.ar[n0, bh.ar[m / 2]], bh.size), id-idx(ar1, ar), no-dups-idx(kar, bh.size), no-dups-idx(bh.ar, bh.size), valid-idx(kar, ar1, bh.size), valid-idx(bh.ar, ar, bh.size), 1 ≤ n0, bh.size ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size) ⊦ # kar = 1, bh.size = 1, n0 = 0, (n0 ≠ 1 ∧ m ≠ 1 → (m / 2 - 1) / 2 < # kar ∧ 3 ≤ n0) ∧ m / 2 < bh.size ∧ no-dups-idx(bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]], bh.size) ∧ valid-idx (bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]], ar[bh.ar[m / 2].idx, ar[bh.ar[m / 2].idx].idx:= n0] [bh.ar[n0].idx, ar[bh.ar[m / 2].idx, ar[bh.ar[m / 2].idx].idx:= n0][bh.ar[n0].idx].idx:= (m / 2) ], bh.size) ∧ ( n0 = 1 ∨ m = 1 ∨ ¬ bh.ar[n0].key < bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]][(m / 2 - 1) / 2].key → heap-order(bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]], bh.size)) ∧ ( n0 ≠ 1 ∧ m ≠ 1 ∧ bh.ar[n0].key < bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]][(m / 2 - 1) / 2].key → heap-order(bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]][(m / 2 - 1) / 2]], bh.size)) ∧ keyset(bh.ar[n0, bh.ar[m / 2]][m / 2, bh.ar[n0]], bh.size) = keyset(kar, bh.size) ∧ m / 2 < n0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 22:
# bh.ar = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), keyset(bh.ar, bh.size) = keyset(kar, bh.size), m + 1 < # kar, m + 1 < bh.size, n1 < # kar, n1 < bh.size, bh.ar[m / 2].idx < # ar, bh.ar[m + 1].key < bh.ar[m / 2].key, bh.ar[m + 1].idx < # ar, m / 2 < bh.size, m / 2 < # kar, heap-order(bh.ar[m + 1, bh.ar[m / 2]], bh.size), id-idx(ar1, ar), no-dups-idx(kar, bh.size), no-dups-idx(bh.ar, bh.size), valid-idx(kar, ar1, bh.size), valid-idx(bh.ar, ar, bh.size), bh.size ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size) ⊦ # kar = 1, bh.size = 1, keyset(bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]], bh.size) = keyset(kar, bh.size) ∧ ( m ≠ 0 ∧ m ≠ 1 ∧ bh.ar[m + 1].key < bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]][(m / 2 - 1) / 2].key → heap-order(bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]][(m / 2 - 1) / 2]], bh.size)) ∧ ( m = 0 ∨ m = 1 ∨ ¬ bh.ar[m + 1].key < bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]][(m / 2 - 1) / 2].key → heap-order(bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]], bh.size)) ∧ valid-idx (bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]], ar[bh.ar[m / 2].idx, ar[bh.ar[m / 2].idx].idx:= (m + 1)] [bh.ar[m + 1].idx, ar[bh.ar[m / 2].idx, ar[bh.ar[m / 2].idx].idx:= (m + 1)][bh.ar[m + 1].idx].idx:= (m / 2) ], bh.size) ∧ no-dups-idx(bh.ar[m + 1, bh.ar[m / 2]][m / 2, bh.ar[m + 1]], bh.size) ∧ (m ≠ 0 ∧ m ≠ 1 → (m / 2 - 1) / 2 < # kar ∧ 2 ≤ m)

Rule argument

Additional information

Used simplifier rules: ⊦ ¬ m + 1 < m / 2; (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ ¬ m < m / 2; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%n0 < n ⊦ (* n) ≤ m0 + (* n0) ↔ n - n0 ≤ m0; (s)% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)