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Proof step 36 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(# kar0), 2exp-size(# kar), m < # ar, m0 + 1 < # kar0, m0 < # kar, heap-order(kar0, m0 + 1), heap-order(kar, m0 + 1), no-dups-idx(kar0, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar0, ar, m0 + 1), valid-idx(kar, ar, m0 + 1), key-index(key1, m) ∈ keyset(kar, m0 + 1), m0 + 1 ≤ # kar0, m0 + 1 ≤ # kar, ∀ ki0. ki0 = kar[m0] ∨ ki0 ∈ keyset(kar, m0) → ki0.idx ≠ m ⊦ # keyset(kar, m0 + 1) = 0, keyset(kar, 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))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 37:
kar0[m0 / 2] = key-index(key0, m1), subarray(kar0, 0, m0 + 1) = subarray(kar, 0, m0 + 1), 2exp-size(# kar0), 2exp-size(# kar), m0 < # kar0, m < # ar, m0 + 1 < # kar0, m0 < # kar, heap-order(kar0, m0 + 1), heap-order(kar, m0 + 1), no-dups-idx(kar0, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar0, ar, m0 + 1), valid-idx(kar, ar, m0 + 1), ∀ ki0. ki0 = kar[m0] ∨ ki0 ∈ keyset(kar, m0) → ki0.idx ≠ m, key-index(key1, m) = kar[m0] ∨ key-index(key1, m) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ kar[m0]) = 0, 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))

Rule argument

Additional information

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