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Proof step 142 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[2 * n0 + 1] = key-index(key6, m6), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[4 + 4 * n0] = key-index(key8, m4), kar0[n0] = key-index(key3, m3), kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), kar0[2 * n0 + 2] = key-index(key2, m), key-index(key5, m5) = key-index(key7, m7), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), key8 < key5, n0 < # kar, 2 * n0 + 1 < # kar, m3 < # ar, m6 < # ar, n0 + 1 < n2, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar, n2 + n0), no-dups-idx(kar0, n2 + n0), valid-idx(kar, ar1, n2 + n0), valid-idx(kar0, ar, n2 + n0), n2 + n0 ≤ # kar, (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3, 3 * n0 + 3 < n2 ∧ key5 < key3 ∨ 3 * n0 + 4 < n2 ∧ key8 < key3 ⊦ 3 * n0 + 4 < n2, heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key7, m7)], n2 + n0), ¬ key7 < key6 ∧ (3 * n0 + 3 < n2 → ¬ key5 < key7), n0 + 2 < n2 ∧ key2 < key6

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 143:
# kar0 = # kar, kar0[2 * n0 + 2] = key-index(key2, m), kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), kar0[n0] = key-index(key3, m3), kar0[4 + 4 * n0] = key-index(key8, m4), kar0[3 + 4 * n0] = key-index(key7, m7), kar0[2 * n0 + 1] = key-index(key6, m6), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), key8 < key6, key7 < key6, 2 * n0 + 2 < # kar, n0 + 2 < n2, n0 + 1 < n2, m6 < # ar, m3 < # ar, 2 * n0 + 1 < # kar, n0 < # kar, key8 < key7, 3 * n0 + 3 < n2, key8 < key3, key7 < key3, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar0, n2 + n0), no-dups-idx(kar, n2 + n0), valid-idx(kar0, ar, n2 + n0), valid-idx(kar, ar1, n2 + n0), n2 + n0 ≤ # kar, n2 ≤ 3 * n0 + 4, ¬ key6 < key3 ∧ ¬ key2 < key3 → heap-order(kar0, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), key6 < key3 ∨ key2 < key3 ⊦ key2 < key6, heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key7, m7)], n2 + n0)

Rule argument

Additional information

Used simplifier rules: lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ ¬ key < key; (s)% ⊦ m * n + n0 < n ↔ m = 0 ∧ n0 < n; (ws)%n0 < n ⊦ m + (* n) < m0 + (* n0) ↔ m + (n - n0) < m0; (s)%mult-c: ⊦ m * n = n * m; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)%m1 ≠ 0 ⊦ m < (* m1) * m + n0 ↔ (m1 - 1) * m + n0 ≠ 0; (s)%n < n0 ⊦ m + (* n) < m0 + (* n0) ↔ m < m0 + (n0 - n); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n < m + n0 ↔ n < n0; (s)%transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)