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Proof step 165 from proof prepost-bubble-down (previous step)

kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), ar0 = ar1, bh1 = bin-heap(kar, m3), bubble-down-inv(n, bh, ar, bin-heap(kar, m3), ar0), no-dups-idx(kar, m3), valid-idx(kar, ar1, m3), m3 ≤ # kar, (2 * n1 + 1 < m3 → ¬ key < key1) ∧ (2 * n1 + 2 < m3 → ¬ key0 < key1) → heap-order(kar, m3), 2 * n1 + 1 < m3 ∧ key < key1 ∨ 2 * n1 + 2 < m3 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < m3 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], m3), 2 * n + 1 < bh.size → (2 * n + 1 < # bh.ar ∧ n < # bh.ar) ∧ (¬ bh.ar[2 * n + 1].key < bh.ar[n].key → 2 * n + 2 < bh.size → 2 * n + 2 < # bh.ar ∧ n < # bh.ar) ⊦ bh.size = m3 ∧ # bh.ar = # kar ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(kar, bh.size) = keyset(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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 166:
# 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)

Rule argument

Additional information

Used simplifier rules: bubble-down-inv: ⊦ bubble-down-inv(n, bh, ar, bh0, ar0) ↔ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, 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) → 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 → 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)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)