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Proof step 55 from proof prepost-bubble-up (previous step)

# kar0 = # kar, m / 2 = m6 + 1, kar0[m + 1, key-index(key1, m3)][m6 / 2] = key-index(key3, m5), kar0[m + 1] = key-index(key2, m4), kar0[m6 + 1] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), keyset(kar0, m2) = keyset(kar, m2), m6 + 1 < # kar, key3 < key1, m6 + 1 < 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 = 0, m = 1, m2 = 1, key2 < key3, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key2, m4)], m2)

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 56:
# kar0 = # kar, m / 2 = m6 + 1, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m6 + 1] = key-index(key1, m3), kar0[m + 1] = key-index(key2, m4), kar0[m + 1, key-index(key1, m3)][m6 / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m + 1 < # kar, m + 1 < m2, n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, m6 + 1 < m2, key3 < key1, m6 + 1 < # 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, m6 / 2 = m + 1, m = 1, m = 0, m2 = 1, key2 < key3, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key2, m4)], m2)

Rule argument

fma: m6 / 2 = m + 1

Additional information

Used simplifier rules: lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws) Used formulas: m6 / 2 = m + 1