[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 21 from proof prepost-remove (previous step)

kar[m1] = key-index(key, m), ar[m] = elemindex(a, m1), ar0 = ar, bh = bh0.size:= (bh0.size - 1), bh0 = bin-heap(kar, m0), n = m1, 2exp-size(# kar), n < bh0.size, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar, bh0.size ≥ 1 ⊦ ⟪bh.ar[n] := bh.ar[bh.size]; ar[bh.ar[n].idx].idx := n; if n ≠ 0 ∧ bh.ar[n].key < bh.ar[(n - 1) / 2].key then bin_min_heap_bubble_up#(n; bh, ar) else bin_min_heap_bubble_down#(n; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar0, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m)), # keyset(kar, m0) = 0, bh.size = n, bh.size = 0, keyset(kar, m0) = ∅

Heuristic symbolic execution applied rule assign right

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).size:= m0 = bin-heap(kar, m0); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%let: ⊦ n ≤ n0 ∧ n0 ≤ n1 → n ≤ n1; (forward) from specification nat