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Proof step 54 from proof prepost-bubble-up (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 55:
# 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)

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m / 2 = 0 ↔ m = 0 ∨ m = 1; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m / 2 ≠ m + 1; (s)