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

# 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, key2 < key3, 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, m = 0, m = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key3, m5)], m2), m2 ≤ # kar0[m + 1, key-index(key1, m3)] ∧ m6 + 1 < m2 ∧ m6 + 1 ≠ 0 ∧ heap-order(kar0[m + 1, key-index(key1, m3)], m2) ∧ ¬ kar0[m + 1, key-index(key1, m3)][m6 + 1].key < key-index(key3, m5).key ∧ ¬ key-index(key3, m5).key < kar0[m + 1, key-index(key1, m3)][(m6 + 1 - 1) / 2].key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 37:
# 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, m6 + 1 < # kar, key1 < key3, key2 < key3, 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, m = 0, m = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key3, m5)], m2)

Rule argument

Additional information

Used simplifier rules: transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s)% ⊦ n + 0 = n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ key-index(key, m).key = key; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m / 2 ≠ m + 1; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)% ⊦ # kar[n, ki] = # kar; (s)