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Proof step 57 from proof prepost-bubble-up (previous step)

# kar0 = # kar, m / 2 = m6 + 1, kar0[m + 1] = key-index(key2, m4), kar0[m6 + 1] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), kar0[m6 / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m6 + 1 < # kar, key3 < key1, m6 + 1 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2) ⊦ # kar = 1, m6 / 2 = m + 1, m = 0, m = 1, m2 = 1, key2 < key3, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key2, m4)], m2)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-overwrite-parent-ok seq: n ≤ # kar, m < n, m ≠ 0 ⊦ heap-order(kar, n) ∧ ¬ kar[m].key < ki.key ∧ ¬ ki.key < kar[(m - 1) / 2].key → heap-order(kar[m, ki], n) subst: [n, ki, m, kar] → [m2, key-index(key2, m4), m6 + 1, kar0[m + 1, key-index(key1, m3)]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kar[n, ki] = # kar; (s)% ⊦ 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 + n = m + n0 ↔ n = n0; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ key-index(key, m).key = key; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)% ⊦ m ≠ n → kar[n, ki][m] = kar[m]; (s)%transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s) Used terms: m2, key-index(key2, m4), m6 + 1, kar0[m + 1, key-index(key1, m3)]