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Proof step 34 from proof prepost-extract-min (previous step)

kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), kar[2] = key-index(key2, m2), kar[1] = key-index(key3, m3), 2exp-size(# kar), m0 < # kar, m < # ar1, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), 2 < m0 → ¬ key2 < key0, ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0, m0 = 1, key3 < key0, heap-order(kar[0, key-index(key0, m)], m0)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-overwrite-children-at seq: n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ (m ≠ 0 → ¬ ki.key < kar[(m - 1) / 2].key) ∧ (2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key) ∧ (2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key) → (¬ heap-order(kar[m, ki], n) ↔ false) subst: [n, ki, m, kar] → [m0, key-index(key0, m), 0, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)% ⊦ 0 < m ↔ m ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%n < # kar ⊦ heap-order(kar, n + 1) → heap-order(kar, n); (ss)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ key-index(key, m).key = key; (s) Used terms: m0, key-index(key0, m), 0, kar