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Proof step 85 from proof prepost-update (previous step)

kar[2 * m1 + 1] = key-index(key2, m3), kar[2 * m1 + 2] = key-index(key0, m2), ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), 2exp-size(# kar), m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, 2 * m1 + 1 < m0 ∧ key2 < key ∨ 2 * m1 + 2 < m0 ∧ key0 < key ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, key < key1, heap-order(kar[m1, (2 * m1 + 2 < m0 ∧ key0 < key2 ⊃ key-index(key0, m2);key-index(key2, m3))], m0)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-pull-up-child seq: n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ 2 * m + 1 < n ∧ ki = ( 2 * m + 2 < n ∧ kar[2 * m + 2].key < kar[2 * m + 1].key ⊃ kar[2 * m + 2] ;kar[2 * m + 1] ) → (¬ heap-order(kar[m, ki], n) ↔ false) subst: [n, ki, m, kar] → [m0, (2 * m1 + 2 < m0 ∧ key0 < key2 ⊃ key-index(key0, m2) ;key-index(key2, m3) ), m1, kar] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key-index(key, m).key = key; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n < m ↔ false); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ 2 * n < n; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%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) Used terms: m0, (2 * m1 + 2 < m0 ∧ key0 < key2 ⊃ key-index(key0, m2) ;key-index(key2, m3) ), m1, kar