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

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Proof step 39 from proof prepost-extract-min (previous step)

kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), ar = ar1[m, ar1[m].idx:= 0], ar = ar, ar2 = ar, bh0 = bin-heap(kar[0, key-index(key0, m)], m0), bh0 = bh0, bh1 = bh0, n0 = 0, 2exp-size(# kar), m0 < # kar, m < # ar1, bubble-down-inv(0, bh0, ar, bh0, ar), heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0) ⊦ [bin_min_heap_bubble_down#(0; bh0, ar)] ( bh0.size = bh1.size ∧ # bh0.ar = # bh1.ar ∧ id-idx(ar2, ar) ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ heap-order(bh0.ar, bh0.size) ∧ keyset(bh1) = keyset(bh0) → bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar1, ar) ∧ (∃ key. (key-index(key, m1) = kar[m0] ∨ key-index(key, m1) ∈ keyset(kar, m0)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, m0) ++ kar[m0]) -- key-index(key, m1) ∧ (∀ ki. ki = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key))), #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 40:
kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), ar = ar1[m, ar1[m].idx:= 0], bh0 = bin-heap(kar[0, key-index(key0, m)], m0), #(keyset(kar, m0) ++ key-index(key0, m)) ≠ 0, m0 ≠ 0, 2exp-size(# kar), m0 < # kar, m < # ar1, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), m0 ≠ 1 ∧ kar[1].key < key0 ∨ 2 < m0 ∧ kar[2].key < key0 → heap-order(kar[0, (2 < m0 ∧ kar[2].key < kar[1].key ⊃ kar[2];kar[1])], m0), (m0 ≠ 1 → ¬ kar[1].key < key0) ∧ (2 < m0 → ¬ kar[2].key < key0) → heap-order(kar[0, key-index(key0, m)], m0), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0) ⊦ [bin_min_heap_bubble_down#(0; bh0, ar)] ( bh0.size = m0 ∧ # bh0.ar = # kar ∧ id-idx(ar1, ar) ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar, bh0.size) ∧ heap-order(bh0.ar, bh0.size) ∧ keyset(kar[0, key-index(key0, m)], bh0.size) = keyset(bh0.ar, bh0.size) → bh0.size ≤ # bh0.ar ∧ 2exp-size(# bh0.ar) ∧ (∃ key. (key-index(key, m1) = kar[bh0.size] ∨ key-index(key, m1) ∈ keyset(kar, bh0.size)) ∧ keyset(bh0.ar, bh0.size) = (keyset(kar, bh0.size) ++ kar[bh0.size]) -- key-index(key, m1) ∧ (∀ ki. ki = kar[bh0.size] ∨ ki ∈ keyset(kar, bh0.size) → ¬ ki.key < 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);%keyset: ⊦ keyset(bh) = keyset(bh.ar, bh.size);%m < # kar ⊦ kar[m, ki][m, ki0] = kar[m, ki0]; (s)% ⊦ key-index(key, m).key = key; (s)%m < # kar ⊦ kar[m, ki][m] = ki; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)%m ≠ n ⊦ kar[(* m), ki][(* n)] = kar[n]; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)%mult-c: ⊦ m * n = n * m; from specification nat-mult%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%⊦ m < # kar ∧ valid-idx(kar, ar, m + 1) ∧ kar[m] = key-index(key, m1) → valid-idx(kar[m0, key-index(key, m1)], ar[m1, ar[m1].idx:= m0], m); (s)%⊦ m < # kar ∧ no-dups-idx(kar, m + 1) ∧ kar[m] = ki → no-dups-idx(kar[m0, ki], m); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ id-idx(ar, ar); (s)% ⊦ id-idx(ar1, ar0[n, ar0[n].idx:= m]) ↔ id-idx(ar1, ar0); (s)% ⊦ id-idx(ar1[m, ar1[m].idx:= m0], ar0) ↔ id-idx(ar1, ar0); (s)% ⊦ m < n → m ≤ n; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ bin-heap(kar, m).size = m; (s)