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Proof step 57 from proof prepost-insert (previous step)

kar0[m, key-index(key, m0)][(m - 1) / 2] = key-index(key0, m1), ar1 = ar[m0, ar[m0].idx:= m], bh0 = bin-heap(kar0[m, key-index(key, m0)], m + 1), subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar), 2exp-size(# kar0), m < # kar0, m0 < # ar, heap-order(kar, m), heap-order(kar0, m), no-dups-idx(kar, m), no-dups-idx(kar0, m), no-dups-idx(kar0[m, key-index(key, m0)], m + 1), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), valid-idx(kar0[m, key-index(key, m0)], ar[m0, ar[m0].idx:= m], m + 1), m ≤ # kar, m = 0 ∨ ¬ key < key0 → heap-order(kar0[m, key-index(key, m0)], m + 1), m ≠ 0 ∧ key < key0 → heap-order(kar0[m, key-index(key0, m1)], m + 1), ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ ∀ bh0, ar1. bh0.size = m + 1 ∧ # bh0.ar = # kar0 ∧ id-idx(ar, ar1) ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar1, bh0.size) ∧ heap-order(bh0.ar, bh0.size) ∧ keyset(kar0, m) ++ key-index(key, m0) = keyset(bh0.ar, bh0.size) → bh0.size ≤ # bh0.ar ∧ 2exp-size(# bh0.ar) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, m) ++ key-index(key, m0)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 58:
# bh.ar = # kar0, keyset(kar0, m) ++ key-index(key, m0) = keyset(bh.ar, m) ++ bh.ar[m], bh.size = m + 1, kar0[m, key-index(key, m0)][(m - 1) / 2] = key-index(key0, m1), subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar0), 2exp-size(# kar), m0 < # ar, m < # kar0, heap-order(bh.ar, m + 1), heap-order(kar0, m), heap-order(kar, m), id-idx(ar, ar0), no-dups-idx(bh.ar, m + 1), no-dups-idx(kar0[m, key-index(key, m0)], m + 1), no-dups-idx(kar0, m), no-dups-idx(kar, m), valid-idx(bh.ar, ar0, m + 1), valid-idx(kar0[m, key-index(key, m0)], ar[m0, ar[m0].idx:= m], m + 1), valid-idx(kar0, ar, m), valid-idx(kar, ar, m), m ≤ # kar, m ≠ 0 ∧ key < key0 → heap-order(kar0[m, key-index(key0, m1)], m + 1), m = 0 ∨ ¬ key < key0 → heap-order(kar0[m, key-index(key, m0)], m + 1), ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ keyset(kar0, m) ++ key-index(key, m0) = keyset(kar, m) ++ key-index(key, m0)

Rule argument

Additional information

Used simplifier rules: ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%eqle-one: ⊦ m = n + 1 ∧ n < n0 → m ≤ n0; (forward) from specification nat