[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 76 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m / 2] = key-index(key2, m4), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m + 1] = key-index(key1, m3), keyset(kar0, m2) = keyset(kar, m2), m + 1 < m2, n1 < m2, key1 < key2, m + 1 < # kar, m / 2 < # kar, n1 < # kar, heap-order(kar0[m + 1, key-index(key2, m4)], m2), id-idx(ar1, ar), no-dups-idx(kar, m2), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2), valid-idx(kar0, ar, m2), m2 ≤ # kar, # ar ≤ m4, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2) ⊦ # kar = 1, m2 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 77:
⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m)
Proof step 78:
# kar0 = # kar, kar0[m + 1] = key-index(key1, m3), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, key1 < key2, n1 < m2, m + 1 < m2, heap-order(kar0[m + 1, key-index(key2, m4)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar, ar1, m2), # ar ≤ m4, m2 ≤ # kar, m2 ≤ # kar0, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), ∀ m. m < m2 → kar0[m].idx < # ar ∧ ar[kar0[m].idx].idx = m ⊦ # kar = 1, m2 = 1

Rule argument

spec: bin-min-heap-inv name: valid-idx seq: ⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m) subst: [n, ar, kar] → [m2, ar, kar0] flags: <all paths>

Additional information

Used terms: m2, ar, kar0