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Proof step 104 from proof prepost-remove (previous step)

kar[m1] = key-index(key, m), ar[m] = elemindex(a, m1), ar0 = ar, bh = bh1.ar:= bh1.ar[n, bh1.ar[bh1.size]], bh0 = bin-heap(kar, m0), bh1 = bh0.size:= (bh0.size - 1), n = m1, 2exp-size(# kar), bh1.size < # bh1.ar, n < # bh1.ar, n < bh0.size, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar, bh0.size ≥ 1 ⊦ # keyset(kar, m0) = 0, bh1.size = n, bh1.size = 0, keyset(kar, m0) = ∅, bh.ar[n].idx < # ar ∧ n < # bh.ar ∧ bh.ar[n].idx < # ar ∧ n < # bh.ar

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 105:
kar[m1] = key-index(key, m), ar[m] = elemindex(a, m1), 2exp-size(# kar), m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key, m) ∈ keyset(kar, m0), # ar ≤ kar[m0 - 1].idx, m0 ≤ # kar ⊦ # keyset(kar, m0) = 0, m0 - 1 = m1, keyset(kar, m0) = ∅, m0 = 1

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%n0 ≤ n ⊦ n - n0 < m ↔ n < m + n0; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ m ≤ n → (n - m = 0 ↔ n = m); (s)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ bin-heap(kar, m).size:= m0 = bin-heap(kar, m0); (s)% ⊦ bin-heap(kar, m).size = m; (s)