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Proof step 67 from proof prepost-bubble-down (previous step)

# kar0 = # kar, (3 * n0 + 6 < n2 ∧ key5 < key6 ⊃ key-index(key5, m5);key-index(key6, m6)) = key-index(key9, m9), kar0[n0, key-index(key3, m3)][(2 * 2 * n0 + 2 * 2) + 1] = key-index(key8, m8), 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), kar0[n0, key-index(key3, m3)][(2 * 2 * n0 + 2 * 2) + 2] = key-index(key7, m7), 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, 3 * n0 + 5 < n2 ∧ key6 < key4 ∨ 3 * n0 + 6 < n2 ∧ key5 < key4 ⊦ heap-order(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key9, m9)], n2 + n0), ¬ key9 < key3 ∧ ((2 * 2 * n0 + 2 * 2) + 1 < n2 + n0 → ¬ key8 < key9) ∧ ((2 * 2 * n0 + 2 * 2) + 2 < n2 + n0 → ¬ key7 < key9)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 68:
# kar0 = # kar, (3 * n0 + 6 < n2 ∧ key7 < key6 ⊃ key-index(key7, m7);key-index(key6, m6)) = key-index(key9, m9), kar0[6 + 4 * n0] = key-index(key7, m7), 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), kar0[5 + 4 * n0] = key-index(key6, m6), 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, ¬ 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), 3 * n0 + 5 < n2 ∧ key6 < key4 ∨ 3 * n0 + 6 < n2 ∧ key7 < key4, key2 < key4 ∨ key3 < key4 ⊦ heap-order(kar0[n0, key-index(key3, m3)][2 * n0 + 2, key-index(key9, m9)], n2 + n0), ¬ key9 < key3 ∧ (3 * n0 + 5 < n2 → ¬ key6 < key9) ∧ (3 * n0 + 6 < n2 → ¬ key7 < key9)

Rule argument

Additional information

Used simplifier rules: ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%m2 ≠ 0 ⊦ (* m2) * m + n < m + n0 ↔ (m2 - 1) * m + n < n0; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%n < n0 ⊦ (* n) ≠ m0 + (* n0); (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