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Proof step 34 from proof prepost-insert (previous step)

kar0[m0 / 2] = key-index(key0, m1), subarray(kar0, 0, m0 + 1) = subarray(kar, 0, m0 + 1), 2exp-size(# kar), 2exp-size(# kar0), m0 < # kar, m0 + 1 < # kar0, m < # ar, heap-order(kar, m0 + 1), heap-order(kar0, m0 + 1), no-dups-idx(kar, m0 + 1), no-dups-idx(kar0, m0 + 1), valid-idx(kar, ar, m0 + 1), valid-idx(kar0, ar, m0 + 1), key-index(key1, m) ∈ keyset(kar0, m0 + 1), ∀ ki0. ki0 = kar[m0] ∨ ki0 ∈ keyset(kar, m0) → ki0.idx ≠ m ⊦ # keyset(kar0, m0 + 1) = 0, keyset(kar0, m0 + 1) = ∅, no-dups-idx(kar0[m0 + 1, key-index(key, m)], 2 + m0) ∧ valid-idx(kar0[m0 + 1, key-index(key, m)], ar[m, ar[m].idx:= (m0 + 1)], 2 + m0) ∧ (¬ key < key0 → heap-order(kar0[m0 + 1, key-index(key, m)], 2 + m0)) ∧ (key < key0 → heap-order(kar0[m0 + 1, key-index(key0, m1)], 2 + m0))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: keyset-subarray-eq seq: m ≤ # kar0, m ≤ # kar1 ⊦ subarray(kar0, 0, m) = subarray(kar1, 0, m) → keyset(kar0, m) = keyset(kar1, m) subst: [kar1, m, kar0] → [kar, m0 + 1, kar0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ¬ m < n → ¬ m + m0 < n; (s)%# kar = n ⊦ subarray(kar, 0, n) = kar; (s) Used terms: kar, m0 + 1, kar0