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Proof step 70 from proof prepost-extract-min (previous step)

# kar0 = # kar, kar[2] = key-index(key3, m3), kar[1] = key-index(key2, m2), kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), m0 < # kar, m < # ar1, key4 < key1, heap-order(kar0, m0), heap-order(kar, m0 + 1), id-idx(ar1, ar0), no-dups-idx(kar0, m0), no-dups-idx(kar, m0 + 1), valid-idx(kar0, ar0, m0), valid-idx(kar, ar1, m0 + 1), m0 ≠ 1 ∧ key2 < key0 ∨ 2 < m0 ∧ key3 < key0 → heap-order(kar[0, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), key-index(key4, m4) = key-index(key0, m) ∨ key-index(key4, m4) ∈ keyset(kar, m0) → ¬ key-index(key4, m4).key < key, key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0), key-index(key4, m4) = key-index(key0, m) ∨ key-index(key4, m4) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 71:
# kar0 = # kar, kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), key4 < key1, m < # ar1, m0 < # kar, key < key1, heap-order(kar, m0 + 1), heap-order(kar0, m0), id-idx(ar1, ar0), no-dups-idx(kar, m0 + 1), no-dups-idx(kar0, m0), valid-idx(kar, ar1, m0 + 1), valid-idx(kar0, ar0, m0), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), m0 ≠ 1 ∧ key2 < key0 ∨ 2 < m0 ∧ key3 < key0 → heap-order(kar[0, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0), key-index(key4, m4) = key-index(key0, m) ∨ key-index(key4, m4) ∈ keyset(kar, m0), key-index(key, m1) = key-index(key0, m) ∨ key-index(key, m1) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0, key4 < key

Rule argument

Additional information

Used simplifier rules: nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]% ⊦ key-index(key, m).key = key; (s)