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Proof step 64 from proof prepost-extract-min (previous step)

# 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), key-index(key1, m1) = key-index(key0, m), 2exp-size(# kar), m < # ar1, m0 < # kar, 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), ∀ 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, (keyset(kar, m0) -- key-index(key1, m1)) ++ key-index(key0, m) = (keyset(kar, m0) ++ key-index(key0, m)) -- key-index(key1, m1), m0 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 65:
# kar0 = # kar, kar[0] = key-index(key0, m), kar[m0] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), 2exp-size(# kar), m < # ar1, m0 < # kar, heap-order(kar, m0 + 1), heap-order(kar0, m0), heap-order(kar, 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, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0), ∀ ki. ki = key-index(key0, m) ∨ ki ∈ keyset(kar, m0) → ¬ ki.key < key, key-index(key, m) = key-index(key0, m) ∨ key-index(key, m) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0

Rule argument

Additional information

Used simplifier rules: n < # kar ⊦ heap-order(kar, n + 1) → heap-order(kar, n); (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ kar[n] = ki → kar[n, ki] = kar; (s)% ⊦ s -- ki ≠ s0 ++ ki; (s)% ⊦ (s ++ ki) -- ki = s -- ki; (s)% ⊦ (s -- ki) ++ ki = s ++ ki; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)