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

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Proof step 47 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[n0] = key-index(key4, m4), kar0[2 * n0 + 2] = key-index(key3, m3), kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), kar0[2 * n0 + 1] = key-index(key2, m), kar0[5 + 4 * n0] = key-index(key6, m6), kar0[6 + 4 * n0] = key-index(key5, m5), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), m4 < # ar, m3 < # ar, 2 * n0 + 2 < # kar, 2 * n0 + 1 < # kar, n0 < # kar, n0 + 1 < n2, key3 < key2, n0 + 2 < n2, heap-order(kar0[n0, key-index(key3, m3)], 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[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), ¬ key2 < key4 ∧ ¬ key3 < key4 → heap-order(kar0, n2 + n0), 3 * n0 + 6 < n2 → ¬ key5 < key4, 3 * n0 + 5 < n2 → ¬ key6 < key4, key2 < key4 ∨ key3 < key4 ⊦ heap-order(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], n2 + n0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 48:
n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ ¬ ki.key < kar[m].key ∧ (2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key) ∧ (2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key) → (¬ heap-order(kar[m, ki], n) ↔ false)
Proof step 49:
# kar0 = # kar, kar0[6 + 4 * n0] = key-index(key5, m5), kar0[5 + 4 * n0] = key-index(key6, m6), kar0[2 * n0 + 1] = key-index(key2, m), 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(key3, m3), kar0[n0] = key-index(key4, m4), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 + 2 < n2, key3 < key2, n0 + 1 < n2, n0 < # kar, 2 * n0 + 1 < # kar, 2 * n0 + 2 < # kar, m3 < # ar, m4 < # ar, heap-order(kar0[n0, key-index(key3, m3)], 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, 3 * n0 + 5 < n2 → ¬ key6 < key4, 3 * n0 + 6 < n2 → ¬ key5 < key4, ¬ key2 < key4 ∧ ¬ key3 < key4 → heap-order(kar0, n2 + n0), (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), key2 < key4 ∨ key3 < key4 ⊦ heap-order(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], n2 + n0), n2 + n0 ≤ # kar0[n0, key-index(key3, m3)] ∧ 2 * n0 + 2 < n2 + n0 ∧ heap-order(kar0[n0, key-index(key3, m3)], n2 + n0) ∧ ¬ key-index(key4, m4).key < kar0[n0, key-index(key3, m3)][2 * n0 + 2].key ∧ (2 * (2 * n0 + 2) + 1 < n2 + n0 → ¬ kar0[n0, key-index(key3, m3)][2 * (2 * n0 + 2) + 1].key < key-index(key4, m4).key) ∧ (2 * (2 * n0 + 2) + 2 < n2 + n0 → ¬ kar0[n0, key-index(key3, m3)][2 * (2 * n0 + 2) + 2].key < key-index(key4, m4).key)

Rule argument

spec: bin-min-heap-inv name: heap-order-overwrite-children-ok seq: n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ ¬ ki.key < kar[m].key ∧ (2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key) ∧ (2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key) → (¬ heap-order(kar[m, ki], n) ↔ false) subst: [n, ki, m, kar] → [n2 + n0, key-index(key4, m4), 2 * n0 + 2, kar0[n0, key-index(key3, m3)]] flags: <all paths>

Additional information

Used terms: n2 + n0, key-index(key4, m4), 2 * n0 + 2, kar0[n0, key-index(key3, m3)]