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

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Proof step 19 from proof prepost-bubble-up (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
# bh.ar = # kar, kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), keyset(bh.ar, bh.size) = keyset(kar, bh.size), bh.ar[n0].idx < # ar, bh.ar[n0].key < bh.ar[(n0 - 1) / 2].key, bh.ar[(n0 - 1) / 2].idx < # ar, n1 < bh.size, n1 < # kar, (n0 - 1) / 2 < # kar, n0 < bh.size, n0 < # kar, heap-order(bh.ar[n0, bh.ar[(n0 - 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), bh.size ≤ # kar, 1 ≤ n0, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size) ⊦ # kar = 1, bh.size = 1, n0 = 0, (n0 ≠ 1 ∧ n0 - 1 ≠ 1 → ((n0 - 1) / 2 - 1) / 2 < # kar ∧ 3 ≤ n0) ∧ (n0 - 1) / 2 < bh.size ∧ no-dups-idx(bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]], bh.size) ∧ valid-idx (bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]], ar[bh.ar[(n0 - 1) / 2].idx, ar[bh.ar[(n0 - 1) / 2].idx].idx:= n0] [bh.ar[n0].idx, ar[bh.ar[(n0 - 1) / 2].idx, ar[bh.ar[(n0 - 1) / 2].idx].idx:= n0][bh.ar[n0].idx].idx:= ((n0 - 1) / 2) ], bh.size) ∧ ( n0 = 1 ∨ n0 - 1 = 1 ∨ ¬ bh.ar[n0].key < bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]][((n0 - 1) / 2 - 1) / 2].key → heap-order(bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]], bh.size)) ∧ ( n0 ≠ 1 ∧ n0 - 1 ≠ 1 ∧ bh.ar[n0].key < bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]][((n0 - 1) / 2 - 1) / 2].key → heap-order (bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]][((n0 - 1) / 2 - 1) / 2]], bh.size)) ∧ keyset(bh.ar[n0, bh.ar[(n0 - 1) / 2]][(n0 - 1) / 2, bh.ar[n0]], bh.size) = keyset(kar, bh.size) ∧ (n0 - 1) / 2 < n0

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);%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%m < # kar ⊦ kar[m, ki][m, ki0] = kar[m, ki0]; (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)% ⊦ m * n + n0 < n ↔ m = 0 ∧ n0 < n; (ws)% ⊦ m +1 = m + 1; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ id-idx(ar1, ar0[n, ar0[n].idx:= m]) ↔ id-idx(ar1, ar0); (s)% ⊦ (bh.ar:= kar).size = bh.size; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n; (s)% ⊦ m ≠ 0 → (1 < m ↔ m ≠ 1); (ss)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%mult-c: ⊦ m * n = n * m; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%m0 ≠ 0 ⊦ n ≤ m / m0 ↔ n * m0 ≤ m; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ m / 2 = 0 ↔ m = 0 ∨ m = 1; (s)% ⊦ # ar[n, ei] = # ar; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ (bh.ar:= kar).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ n ≥ m ↔ m ≤ n; (s)