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Proof step 126 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(key4, 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), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), 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), 3 * n0 + 4 < n2 → ¬ key4 < key3, 3 * n0 + 3 < n2 → ¬ key5 < key3, key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3 ⊦ heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key3, m3)], n2 + n0), n0 + 2 < n2 ∧ key2 < key6

User applied rule insert rewrite lemma

And obtained the following new goals

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(key3, m3), 2 * n0 + 1, kar0[n0, key-index(key6, m6)]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kar[n, ki] = # kar; (s)%m2 ≠ 0 ⊦ (* m2) * m + n < m + n0 ↔ (m2 - 1) * m + n < n0; (s)% ⊦ m0 * 1 = m0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ key-index(key, m).key = key; (s)%m1 ≠ 0 ⊦ m = (* m1) * m + n0 ↔ 0 = (m1 - 1) * m + n0; (s)% ⊦ m + 1 ≠ 0; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m * (n + 1) = m * n + m; (s)%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)%transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s)% ⊦ key < key0 → (key0 < key ↔ false); (ss)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%n < n0 ⊦ m + (* n) < m0 + (* n0) ↔ m < m0 + (n0 - n); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s) Used terms: n2 + n0, key-index(key3, m3), 2 * n0 + 1, kar0[n0, key-index(key6, m6)]