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Proof step 29 from proof prepost-insert (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
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))

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic[keyindex-set]%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ key-index(key, m).idx = m; (s)%⊦ n < # kar ∧ no-dups-idx(kar, n) → ( no-dups-idx(kar[n, ki], n + 1) ↔ ¬(∃ ki0. ki0 ∈ keyset(kar, n) ∧ ki0.idx = ki.idx)); (ss)