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Proof step 31 from proof prepost-insert (previous step)

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))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 32:
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, ki.idx < # 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), ki ∈ keyset(kar0, m0 + 1), ∀ ki0. ki0 = kar[m0] ∨ ki0 ∈ keyset(kar, m0) → ki0.idx ≠ ki.idx ⊦ # keyset(kar0, m0 + 1) = 0, keyset(kar0, m0 + 1) = ∅, no-dups-idx(kar0[m0 + 1, key-index(key, ki.idx)], 2 + m0) ∧ valid-idx(kar0[m0 + 1, key-index(key, ki.idx)], ar[ki.idx, ar[ki.idx].idx:= (m0 + 1)], 2 + m0) ∧ (¬ key < key0 → heap-order(kar0[m0 + 1, key-index(key, ki.idx)], 2 + m0)) ∧ (key < key0 → heap-order(kar0[m0 + 1, key-index(key0, m1)], 2 + m0))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic[keyindex-set]%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set]%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ ki ∈ s ++ ki0 ↔ ki = ki0 ∨ ki ∈ s; (s)% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m + 1 ≠ 0; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m / 2 ≠ m + 1; (s)