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Proof step 33 from proof prepost-extract-min (previous step)

kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), kar[2] = key-index(key2, m2), kar[1] = key-index(key3, m3), 2exp-size(# kar), m0 < # kar, m < # ar1, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), m0 ≠ 1 → ¬ key3 < key0, 2 < m0 → ¬ key2 < key0, ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, 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, heap-order(kar[0, key-index(key0, m)], m0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 34:
kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m1), kar[2] = key-index(key2, m2), kar[1] = key-index(key3, m3), 2exp-size(# kar), m0 < # kar, m < # ar1, heap-order(kar, m0 + 1), no-dups-idx(kar, m0 + 1), valid-idx(kar, ar1, m0 + 1), 2 < m0 → ¬ key2 < key0, ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, 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, m0 = 1, key3 < key0, heap-order(kar[0, key-index(key0, m)], m0)

Rule argument

Additional information

Used simplifier rules: ⊦ heap-order(kar, 1); (s)%# kar ≠ 0 ⊦ keyset(kar, 1) = ∅ ++ kar[0]; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic