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Proof step 123 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[2 * n0 + 1] = key-index(key6, m6), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[4 + 4 * n0] = key-index(key4, m4), kar0[n0] = key-index(key3, m3), kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), kar0[2 * n0 + 2] = key-index(key2, m), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 < # kar, 2 * n0 + 1 < # kar, m3 < # ar, m6 < # ar, n0 + 1 < n2, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar, n2 + n0), no-dups-idx(kar0, n2 + n0), valid-idx(kar, ar1, n2 + n0), valid-idx(kar0, ar, n2 + n0), n2 + n0 ≤ # kar, (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3 ⊦ valid-idx (kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key3, m3)], ar[m6, ar[m6].idx:= n0][m3, ar[m6, ar[m6].idx:= n0][m3].idx:= (2 * n0 + 1)], n2 + n0), n0 + 2 < n2 ∧ key2 < key6

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: valid-idx-xhg seq: ⊦ valid-idx(kar, ar, n) ∧ m0 < n ∧ m1 < n ∧ kar[m0] = key-index(key1, n1) ∧ kar[m1] = key-index(key0, n0) → valid-idx (kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)], ar[n0, ar[n0].idx:= m0] [n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1 ], n) subst: [n, ar0, ar, n1, key1, m1, n0, key0, m0, kar] → [n2 + n0, ar, ar, m3, key3, 2 * n0 + 1, m6, key6, n0, kar0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < m + n ↔ n ≠ 0; (s)% ⊦ m < n → n ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%m2 ≠ 0 ⊦ (* m2) * m + n < m + n0 ↔ (m2 - 1) * m + n < n0; (s)% ⊦ m0 * 1 = m0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult Used terms: n2 + n0, ar, ar, m3, key3, 2 * n0 + 1, m6, key6, n0, kar0