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Proof step 45 from proof prepost-update (previous step)

kar[m4 + 1] = key-index(key4, m6), kar[m4 / 2] = key-index(key5, m7), kar[m1] = key-index(key1, m), ar[m] = elemindex(a, m1), kar[m1, key-index(key, m)][(m1 - 1) / 2] = key-index(key0, m2), kar[m1, key-index(key, m)][m4 + 1] = key-index(key2, m3), 2exp-size(# kar), m4 + 1 < # kar, key < key1, m1 < # kar, m1 < m0, m4 + 1 < m0, key2 < key5, no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, m1 = 0 ∨ ¬ key < key0 ⊦ # keyset(kar, m0) = 0, m4 / 2 = m1, keyset(kar, m0) = ∅, key4 < key5

Heuristic weak cut applied rule cut formula

And closed the goal!

Rule argument

fma: m4 + 1 = m1

Additional information

Used simplifier rules: ⊦ m ≠ n → kar[n, ki][m] = kar[m]; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ m / 2 ≠ m + 1; (s)% ⊦ m + 1 ≠ 0; (s)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key] Used formulas: m4 + 1 = m1