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

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 16 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, ki0.idx < # ar, (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) ⊦ ⟪n1 := (n - 1) / 2; bh := bh.ar:= bh.ar[n, ki0].ar:= (bh.ar:= bh.ar[n, ki0]).ar[n1, ki]; ar := ar[ki0.idx, ar[ki0.idx].idx:= n][ki.idx, ar[ki0.idx, ar[ki0.idx].idx:= n][ki.idx].idx:= n1]⟫ ( ( (n1 ≠ 0 → n1 < # bh.ar ∧ (n1 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n1 ≥ 1) ∧ (n1 ≠ 0 → n1 < # bh.ar ∧ (n1 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n1 ≥ 1) ∧ (n1 ≠ 0 → n1 < # bh.ar ∧ (n1 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n1 ≥ 1) ∧ (n1 ≠ 0 → n1 < # bh.ar ∧ (n1 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n1 ≥ 1) ∧ (n1 ≠ 0 → n1 < # bh.ar ∧ (n1 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n1 ≥ 1) ∧ bubble-up-inv(n1, bh, ar, bin-heap(kar, m2), ar0)) ∧ n1 < n0), 2 = 0, n = 0

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Proof step 17:
kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), ar0 = ar1, bh0 = bin-heap(kar, m2), ki = bh.ar[n], ki0 = bh.ar[(n - 1) / 2], n = n0, n1 < m2, n1 < # kar, bh.ar[n].key < bh.ar[(n - 1) / 2].key, n < # bh.ar, (n - 1) / 2 < # bh.ar, ki0.idx < # ar, 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, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n ≠ 0 → n < # bh.ar ∧ (n - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ n ≥ 1 ⊦ ⟪bh := bh.ar:= bh.ar[n, ki0].ar:= (bh.ar:= bh.ar[n, ki0]).ar[(n - 1) / 2, ki]; ar := ar[ki0.idx, ar[ki0.idx].idx:= n][ki.idx, ar[ki0.idx, ar[ki0.idx].idx:= n][ki.idx].idx:= ((n - 1) / 2)]⟫ ( ( ((n - 1) / 2 ≠ 0 → (n - 1) / 2 < # bh.ar ∧ ((n - 1) / 2 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ (n - 1) / 2 ≥ 1) ∧ ((n - 1) / 2 ≠ 0 → (n - 1) / 2 < # bh.ar ∧ ((n - 1) / 2 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ (n - 1) / 2 ≥ 1) ∧ ((n - 1) / 2 ≠ 0 → (n - 1) / 2 < # bh.ar ∧ ((n - 1) / 2 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ (n - 1) / 2 ≥ 1) ∧ ((n - 1) / 2 ≠ 0 → (n - 1) / 2 < # bh.ar ∧ ((n - 1) / 2 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ (n - 1) / 2 ≥ 1) ∧ ((n - 1) / 2 ≠ 0 → (n - 1) / 2 < # bh.ar ∧ ((n - 1) / 2 - 1) / 2 < # bh.ar ∧ 2 ≠ 0 ∧ (n - 1) / 2 ≥ 1) ∧ bubble-up-inv((n - 1) / 2, bh, ar, bin-heap(kar, m2), ar0)) ∧ (n - 1) / 2 < n0), 2 = 0, 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);% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)