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



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Proof step 36 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[2 * (2 * n0 + 2) + 1] = key-index(key6, m6), kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)][2 * (2 * n0 + 2) + 2] = key-index(key5, m5), 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), 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), key2 < key4 ∨ key3 < key4 ⊦ ( 2 * (2 * n0 + 2) + 1 < n2 + n0 → 2 * (2 * n0 + 2) + 1 < # kar ∧ ( ¬ kar0[n0, key-index(key3, m3)][2 * (2 * n0 + 2) + 1].key < key4 → 2 * (2 * n0 + 2) + 2 < n2 + n0 → 2 * (2 * n0 + 2) + 2 < # kar)) ∧ no-dups-idx(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], n2 + n0) ∧ valid-idx (kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], ar[m3, ar[m3].idx:= n0][m4, ar[m3, ar[m3].idx:= n0][m4].idx:= (2 * n0 + 2)], n2 + n0) ∧ ( (2 * (2 * n0 + 2) + 1 < n2 + n0 → ¬ key6 < key4) ∧ (2 * (2 * n0 + 2) + 2 < n2 + n0 → ¬ key5 < key4) → heap-order(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], n2 + n0)) ∧ ( 2 * (2 * n0 + 2) + 1 < n2 + n0 ∧ key6 < key4 ∨ 2 * (2 * n0 + 2) + 2 < n2 + n0 ∧ key5 < key4 → heap-order (kar0[n0, key-index(key3, m3)][2 * n0 + 2, (2 * (2 * n0 + 2) + 2 < n2 + n0 ∧ key5 < key6 ⊃ key-index(key5, m5);key-index(key6, m6))], n2 + n0)) ∧ keyset(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key4, m4)], n2 + n0) = keyset(kar, n2 + n0)

User applied rule case distinction

And obtained the following new goals

Rule argument

pos: right 1

Additional information

Used simplifier rules: f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n → kar[m, ki][n] = kar[n]; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n < n0 ⊦ m + (* n) < m0 + (* n0) ↔ m < m0 + (n0 - n); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ 2 * m < 2 * n ↔ m < n; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m < n → (n ≤ m ↔ false); (s)%transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]