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Proof step 111 from proof prepost-remove (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key, m), m0 = m2 + 1, 2exp-size(# kar), m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key, m) ∈ keyset(kar, m0), 1 ≤ # kar, m0 ≤ # kar, 1 ≤ m0, m0 ≤ # kar + 1 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, m0 = 0, m0 ≠ 1 ∧ m2 ≠ m1, keyset(kar, m2) = keyset(kar, m0) -- key-index(key, m) ∧ heap-order(kar, m2) ∧ valid-idx(kar, ar, m2) ∧ no-dups-idx(kar, m2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 112:
ar[m] = elemindex(a, m1), kar[m1] = key-index(key, m), 2exp-size(# kar), m2 < # kar, m1 < # kar, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar, m2 + 1), m1 ≤ m2, 1 ≤ # kar, key-index(key, m) = kar[m2] ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ kar[m2]) = 0, keyset(kar, m2) = (keyset(kar, m2) ++ kar[m2]) -- key-index(key, m), m2 ≠ 0 ∧ m2 ≠ m1

Rule argument

Additional information

Used simplifier rules: size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]% ⊦ 1 = m + 1 ↔ m = 0; (s)% ⊦ no-dups-idx(kar, n + 1) → no-dups-idx(kar, n); (s)% ⊦ valid-idx(kar, ar, n + 1) → valid-idx(kar, ar, n); (s)%n < # kar ⊦ heap-order(kar, n + 1) → heap-order(kar, n); (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → m ≤ n; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ki ∈ s ++ ki0 ↔ ki = ki0 ∨ ki ∈ s; (s)% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m ≤ m + n; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)