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Proof step 98 from proof prepost-remove (previous step)

# kar0 = # kar, (keyset(kar, m2) -- key-index(key, m)) ++ key-index(key0, m0) = keyset(kar0, m2), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), kar[m2] = key-index(key0, m0), kar[2 * m1 + 1] = key-index(key2, m4), kar[2 * m1 + 2] = key-index(key3, m5), key-index(key, m) = key-index(key0, m0), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, m2 < # kar, m0 < # ar1, m1 < m2, heap-order(kar, m2 + 1), heap-order(kar0, m2), id-idx(ar1, ar0), no-dups-idx(kar, m2 + 1), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2 + 1), valid-idx(kar0, ar0, m2), elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1 + 1) → kar[m3] = key-index(key, m), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, (2 * m1 + 1 < m2 → ¬ key2 < key0) ∧ (2 * m1 + 2 < m2 → ¬ key3 < key0) → heap-order(kar[m1, key-index(key0, m0)], m2), 2 * m1 + 1 < m2 ∧ key2 < key0 ∨ 2 * m1 + 2 < m2 ∧ key3 < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key3 < key2 ⊃ key-index(key3, m5);key-index(key2, m4))], m2), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, (keyset(kar, m2) -- key-index(key, m)) ++ key-index(key0, m0) = (keyset(kar, m2) ++ key-index(key0, m0)) -- key-index(key, m), m2 = m1, m1 ≠ 0 ∧ key0 < key1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 99:
# kar0 = # kar, keyset(kar, m2) ++ key-index(key0, m0) = keyset(kar0, m2), ar1[m0] = elemindex(a, m1), kar[m1] = key-index(key0, m0), kar[m2] = key-index(key0, m0), kar[2 * m1 + 1] = key-index(key2, m4), kar[2 * m1 + 2] = key-index(key3, m5), kar[(m1 - 1) / 2] = key-index(key1, m3), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, m2 < # kar, m0 < # ar1, m1 < m2, heap-order(kar, m2 + 1), heap-order(kar0, m2), heap-order(kar, m2), id-idx(ar1, ar0), no-dups-idx(kar, m2 + 1), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2 + 1), valid-idx(kar0, ar0, m2), 2 * m1 + 1 < m2 ∧ key2 < key0 ∨ 2 * m1 + 2 < m2 ∧ key3 < key0 → heap-order(kar[m1, (2 * m1 + 2 < m2 ∧ key3 < key2 ⊃ key-index(key3, m5);key-index(key2, m4))], m2), elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1 + 1) → kar[m3] = key-index(key0, m0), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1 ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, m1 ≠ 0 ∧ key0 < key1

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% ⊦ s -- ki ≠ s0 ++ ki; (s)% ⊦ (s ++ ki) -- ki = s -- ki; (s)% ⊦ kar[n] = ki → kar[n, ki] = kar; (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)