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

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Proof step 131 from proof prepost-bubble-down (previous step)

# 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(key4, m4), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[2 * n0 + 1] = key-index(key6, m6), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 + 1 < n2, m6 < # ar, m3 < # ar, 2 * n0 + 1 < # kar, n0 < # kar, 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, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, 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), 3 * n0 + 3 < n2 ∧ key5 < key3 ∨ 3 * n0 + 4 < n2 ∧ key4 < key3, key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3 ⊦ heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5))], n2 + n0), n0 + 2 < n2 ∧ key2 < key6, n2 + n0 ≤ # kar0[n0, key-index(key6, m6)] ∧ 2 * n0 + 1 < n2 + n0 ∧ heap-order(kar0[n0, key-index(key6, m6)], n2 + n0) ∧ ¬ (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key < kar0[n0, key-index(key6, m6)][2 * n0 + 1].key ∧ ( 2 * (2 * n0 + 1) + 1 < n2 + n0 → ¬ kar0[n0, key-index(key6, m6)][2 * (2 * n0 + 1) + 1].key < (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key) ∧ ( 2 * (2 * n0 + 1) + 2 < n2 + n0 → ¬ kar0[n0, key-index(key6, m6)][2 * (2 * n0 + 1) + 2].key < (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 132:
# 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(key4, m4), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[2 * n0 + 1] = key-index(key6, m6), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 + 1 < n2, m6 < # ar, m3 < # ar, 2 * n0 + 1 < # kar, n0 < # kar, 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, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, 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), 3 * n0 + 3 < n2 ∧ key5 < key3 ∨ 3 * n0 + 4 < n2 ∧ key4 < key3, key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3 ⊦ heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5))], n2 + n0), n0 + 2 < n2 ∧ key2 < key6, ¬ (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key < key6 ∧ (3 * n0 + 3 < n2 → ¬ key5 < (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key) ∧ (3 * n0 + 4 < n2 → ¬ key4 < (key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key)

Rule argument

Additional information

Used simplifier rules: ⊦ key-index(key, m).key = key; (s)%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)% ⊦ m * (n + 1) = m * n + m; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m + 1 ≠ 0; (s)%m1 ≠ 0 ⊦ m = (* m1) * m + n0 ↔ 0 = (m1 - 1) * m + n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%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)%m2 ≠ 0 ⊦ (* m2) * m + n < m + n0 ↔ (m2 - 1) * m + n < n0; (s)% ⊦ # kar[n, ki] = # kar; (s)