[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 99 from proof prepost-remove (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 100:
⊦ no-dups-idx(kar, n) ↔ (∀ m0, m1. m0 < # kar ∧ m0 < n ∧ m1 < # kar ∧ m1 < n ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx)
Proof step 101:
# kar0 = # kar, keyset(kar, m2) ++ key-index(key0, m0) = keyset(kar0, m2), kar[(m1 - 1) / 2] = key-index(key1, m3), kar[2 * m1 + 2] = key-index(key3, m5), kar[2 * m1 + 1] = key-index(key2, m4), kar[m2] = key-index(key0, m0), kar[m1] = key-index(key0, m0), ar1[m0] = elemindex(a, m1), 2exp-size(# kar), m1 < m2, m0 < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, heap-order(kar, m2), heap-order(kar0, m2), heap-order(kar, m2 + 1), id-idx(ar1, ar0), no-dups-idx(kar0, m2), valid-idx(kar0, ar0, m2), valid-idx(kar, ar1, m2 + 1), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1 + 1) → kar[m3] = key-index(key0, m0), 2 * m1 + 1 < m2 ∧ key2 < key0 ∨ 2 * m1 + 2 < m2 ∧ key3 < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key3 < key2 ⊃ key-index(key3, m5);key-index(key2, m4))], m2), ∀ m0, m1. m0 < # kar ∧ m0 < m2 + 1 ∧ m1 < # kar ∧ m1 < m2 + 1 ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, m1 ≠ 0 ∧ key0 < key1

Rule argument

spec: bin-min-heap-inv name: no-dups-idx seq: ⊦ no-dups-idx(kar, n) ↔ (∀ m0, m1. m0 < # kar ∧ m0 < n ∧ m1 < # kar ∧ m1 < n ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx) subst: [n, kar] → [m2 + 1, kar] flags: <all paths>

Additional information

Used terms: m2 + 1, kar