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Proof step 88 from proof prepost-remove (previous step)

# bh0.ar = # kar, kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), kar[bh0.size] = key-index(key0, m0), kar[2 * m1 + 1] = key-index(key2, m4), kar[2 * m1 + 2] = key-index(key3, m5), keyset(kar[m1, key-index(key0, m0)], bh0.size) = keyset(bh0.ar, bh0.size), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, bh0.size < # kar, m0 < # ar1, m1 < bh0.size, heap-order(kar, bh0.size + 1), heap-order(bh0.ar, bh0.size), id-idx(ar1, ar0), no-dups-idx(kar, bh0.size + 1), no-dups-idx(bh0.ar, bh0.size), valid-idx(kar, ar1, bh0.size + 1), valid-idx(bh0.ar, ar0, bh0.size), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, (2 * m1 + 1 < bh0.size → ¬ key2 < key0) ∧ (2 * m1 + 2 < bh0.size → ¬ key3 < key0) → heap-order(kar[m1, key-index(key0, m0)], bh0.size), 2 * m1 + 1 < bh0.size ∧ key2 < key0 ∨ 2 * m1 + 2 < bh0.size ∧ key3 < key0 → heap-order(kar[m1, (2 * m1 + 2 < bh0.size ∧ key3 < key2 ⊃ key-index(key3, m5);key-index(key2, m4))], bh0.size), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, bh0.size) ⊦ #(keyset(kar, bh0.size) ++ key-index(key0, m0)) = 0, bh0.size = m1, keyset(bh0.ar, bh0.size) = (keyset(kar, bh0.size) ++ key-index(key0, m0)) -- key-index(key, m), m1 ≠ 0 ∧ key0 < key1

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 89:
# kar0 = # kar, kar[2 * m1 + 2] = key-index(key3, m5), kar[2 * m1 + 1] = key-index(key2, m4), kar[m2] = key-index(key0, m0), kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), bh0 = bin-heap(kar0, m2), keyset(kar[m1, key-index(key0, m0)], m2) = keyset(kar0, m2), 2exp-size(# kar), m1 < m2, m0 < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, heap-order(kar0, m2), heap-order(kar, m2 + 1), id-idx(ar1, ar0), no-dups-idx(kar0, m2), no-dups-idx(kar, m2 + 1), valid-idx(kar0, ar0, m2), valid-idx(kar, ar1, m2 + 1), 2 * m1 + 1 < m2 ∧ key2 < key0 ∨ 2 * m1 + 2 < m2 ∧ key3 < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key3 < key2 ⊃ key-index(key3, m5);key-index(key2, m4))], m2), (2 * m1 + 1 < m2 → ¬ key2 < key0) ∧ (2 * m1 + 2 < m2 → ¬ key3 < key0) → heap-order(kar[m1, key-index(key0, m0)], m2), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, keyset(kar0, m2) = (keyset(kar, m2) ++ key-index(key0, m0)) -- key-index(key, m), m2 = m1, m1 ≠ 0 ∧ key0 < key1

Rule argument

spec: bin-min-heap name: elim seq: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m) subst: [bh, kar, m] → [bh0, kar0, m1]

Additional information

Used simplifier rules: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m); (elim) Used terms: bh0, kar0, m1