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

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Proof step 30 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key1, m3), kar0[m + 1] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), m + 1 < # kar, m + 1 < m2, n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, m / 2 < m2, m / 2 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar, m2), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2), valid-idx(kar0, ar, m2), m2 ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2) ⊦ # kar = 1, m2 = 1, (m ≠ 0 ∧ m ≠ 1 ∧ key2 < key3 → heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key3, m5)], m2)) ∧ (m = 0 ∨ m = 1 ∨ ¬ key2 < key3 → heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)) ∧ no-dups-idx(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2) ∧ (m ≠ 0 ∧ m ≠ 1 → (m / 2 - 1) / 2 < # kar ∧ 2 ≤ m)

User applied rule case distinction

And obtained the following new goals

Proof step 31:
# kar0 = # kar, kar0[m + 1] = key-index(key2, m4), kar0[m / 2] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), key2 < key3, m / 2 < # kar, m / 2 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2) ⊦ # kar = 1, m = 1, m = 0, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key3, m5)], m2)
Proof step 42:
# kar0 = # kar, kar0[m + 1] = key-index(key2, m4), kar0[m / 2] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m / 2 < # kar, m / 2 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), m = 0 ∨ m = 1 ∨ ¬ key2 < key3 ⊦ # kar = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)
Proof step 59:
# kar0 = # kar, kar0[m + 1] = key-index(key2, m4), kar0[m / 2] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m / 2 < # kar, m / 2 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2) ⊦ # kar = 1, m2 = 1, no-dups-idx(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)

Rule argument

pos: right 3

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ m ≠ 0 → (m < 2 ↔ m = 1); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < n → (n ≤ m ↔ false); (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%m0 ≠ 0 ⊦ n ≤ m / m0 ↔ n * m0 ≤ m; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic% ⊦ m / 2 = 0 ↔ m = 0 ∨ m = 1; (s)% ⊦ m +1 = m + 1; (s)% ⊦ m * n + n0 < n ↔ m = 0 ∧ n0 < n; (ws)