[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 83 from proof prepost-extract-min (previous step)

# kar0 = # kar, kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), key-index(key4, m4) = kar[m5], keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), m5 + 1 < # kar, m5 < m0, key4 < key1, m < # ar1, m0 < # kar, key < key1, m5 < # kar, heap-order(kar, m0 + 1), heap-order(kar0, m0), id-idx(ar1, ar0), no-dups-idx(kar, m0 + 1), no-dups-idx(kar0, m0), valid-idx(kar, ar1, m0 + 1), valid-idx(kar0, ar0, m0), key-index(key, m1) ∈ keyset(kar, m0), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), m0 ≠ 1 ∧ key2 < key0 ∨ 2 < m0 ∧ key3 < key0 → heap-order(kar[0, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0) ⊦ # keyset(kar, m0) = 0, #(keyset(kar, m0) ++ key-index(key0, m)) = 0, keyset(kar, m0) = ∅, key4 < key

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-min seq: ⊦ n ≠ 0 ∧ n ≤ # kar ∧ heap-order(kar, n) ∧ m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[0].key subst: [n, kar, m] → [m0 + 1, kar, m5]

Additional information

Used simplifier rules: ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ m < n → ¬ n < m; (s)% ⊦ key-index(key, m).key = key; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n < m ↔ false); (s) Used terms: m0 + 1, kar, m5