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

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Proof step 166 from proof prepost-bubble-down (previous step)

# bh.ar = # kar, kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), keyset(bh.ar, bh.size) = keyset(kar, bh.size), id-idx(ar1, ar), no-dups-idx(bh.ar, bh.size), no-dups-idx(kar, bh.size), valid-idx(bh.ar, ar, bh.size), valid-idx(kar, ar1, bh.size), bh.size ≤ # kar, 2 * n + 1 < bh.size → 2 * n + 1 < # kar ∧ n < # kar ∧ (¬ bh.ar[2 * n + 1].key < bh.ar[n].key → 2 * n + 2 < bh.size → 2 * n + 2 < # kar), 2 * n + 1 < bh.size ∧ bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order(bh.ar[n, (2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key ⊃ bh.ar[2 * n + 2];bh.ar[2 * n + 1])], bh.size), (2 * n1 + 1 < bh.size → ¬ key < key1) ∧ (2 * n1 + 2 < bh.size → ¬ key0 < key1) → heap-order(kar, bh.size), 2 * n1 + 1 < bh.size ∧ key < key1 ∨ 2 * n1 + 2 < bh.size ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < bh.size ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], bh.size) ⊦ heap-order(bh.ar, bh.size), (2 * n + 1 < bh.size → ¬ bh.ar[2 * n + 1].key < bh.ar[n].key) ∧ (2 * n + 2 < bh.size → ¬ bh.ar[2 * n + 2].key < bh.ar[n].key), 2 * n + 1 < bh.size ∧ (bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 167:
# kar0 = # kar, kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), bh = bin-heap(kar0, m), keyset(kar0, m) = keyset(kar, m), id-idx(ar1, ar), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar, ar1, m), valid-idx(kar0, ar, m), m ≤ # kar, 2 * n1 + 1 < m ∧ key < key1 ∨ 2 * n1 + 2 < m ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < m ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], m), (2 * n1 + 1 < m → ¬ key < key1) ∧ (2 * n1 + 2 < m → ¬ key0 < key1) → heap-order(kar, m), 2 * n + 1 < m ∧ kar0[2 * n + 1].key < kar0[n].key ∨ 2 * n + 2 < m ∧ kar0[2 * n + 2].key < kar0[n].key → heap-order(kar0[n, (2 * n + 2 < m ∧ kar0[2 * n + 2].key < kar0[2 * n + 1].key ⊃ kar0[2 * n + 2];kar0[2 * n + 1])], m), 2 * n + 1 < m → 2 * n + 1 < # kar ∧ n < # kar ∧ (¬ kar0[2 * n + 1].key < kar0[n].key → 2 * n + 2 < m → 2 * n + 2 < # kar) ⊦ heap-order(kar0, m), 2 * n + 1 < m ∧ (kar0[2 * n + 1].key < kar0[n].key ∨ 2 * n + 2 < m ∧ kar0[2 * n + 2].key < kar0[n].key), (2 * n + 1 < m → ¬ kar0[2 * n + 1].key < kar0[n].key) ∧ (2 * n + 2 < m → ¬ kar0[2 * n + 2].key < kar0[n].key)

Rule argument

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

Additional information

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