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Proof step 30 from proof prepost-insert (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 31:
kar0[m, key-index(key, ki.idx)][m0 / 2] = key-index(key0, m1), m = m0 + 1, subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar), 2exp-size(# kar0), m < # kar0, ki.idx < # ar, heap-order(kar, m), heap-order(kar0, m), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), ki ∈ keyset(kar0, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ ki.idx ⊦ # keyset(kar0, m) = 0, keyset(kar0, m) = ∅, m = 0, no-dups-idx(kar0[m, key-index(key, ki.idx)], m + 1) ∧ valid-idx(kar0[m, key-index(key, ki.idx)], ar[ki.idx, ar[ki.idx].idx:= m], m + 1) ∧ (m = 0 ∨ ¬ key < key0 → heap-order(kar0[m, key-index(key, ki.idx)], m + 1)) ∧ (m ≠ 0 ∧ key < key0 → heap-order(kar0[m, key-index(key0, m1)], m + 1))

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [m, m0]

Additional information

Used simplifier rules: ⊦ subarray(kar, n, 0) = mkkeyarray(0); (s)% ⊦ keyset(kar, 0) = ∅; (s)%# kar = n ⊦ subarray(kar, 0, n) = kar; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m, m0