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Proof step 68 from proof prepost-bubble-up (previous step)

kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), ar0 = ar1, bh0 = bin-heap(kar, m2), ki = bh.ar[n], ki0 = bh.ar[(n - 1) / 2], n = n0, (n - 1) / 2 < # bh.ar, n < # bh.ar, bh.ar[n].key < bh.ar[(n - 1) / 2].key, n1 < # kar, n1 < m2, bubble-up-inv(n, bh, ar, bin-heap(kar, m2), ar0), no-dups-idx(kar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n ≥ 1, n ≠ 0 → n < # bh.ar ∧ (n - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n ≥ 1, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2) ⊦ 2 = 0, n = 0, ki0.idx < # ar ∧ ki0.idx < # ar

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 69:
# bh.ar = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), keyset(bh.ar, bh.size) = keyset(kar, bh.size), n1 < # kar, (n - 1) / 2 < # kar, n < # kar, bh.ar[n].key < bh.ar[(n - 1) / 2].key, n1 < bh.size, n < bh.size, heap-order(bh.ar[n, bh.ar[(n - 1) / 2]], 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), # ar ≤ bh.ar[(n - 1) / 2].idx, bh.size ≤ # kar, 1 ≤ n, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size) ⊦ # kar = 1, bh.size = 1, n = 0

Rule argument

Additional information

Used simplifier rules: bubble-up-inv: ⊦ bubble-up-inv(n, bh, ar, bh0, ar0) ↔ n < bh.size ∧ 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) ∧ ( n = 0 ∨ ¬ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar, bh.size)) ∧ ( n ≠ 0 ∧ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar[n, bh.ar[(n - 1) / 2]], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);% ⊦ m < n → n ≠ 0; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)