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Proof step 87 from proof prepost-extract-min (previous step)

m = m0 + 1, 2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), key-index(key, kar[0].idx) ∈ keyset(kar, m), # ar ≤ kar[m0].idx, m ≤ # kar, ∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key ⊦ # kar = 0, # keyset(kar, m) = 0, keyset(kar, m) = ∅, m = 0, m = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 88:
2exp-size(# kar), m0 < # kar, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar, m0 + 1), # ar ≤ kar[m0].idx, ∀ ki. ki = kar[m0] ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, kar[0].idx) = kar[m0] ∨ key-index(key, kar[0].idx) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ kar[m0]) = 0, m0 = 0

Rule argument

Additional information

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