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Proof step 58 from proof prepost-insert (previous step)

# 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)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 59:
# 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)][m2 / 2] = key-index(key0, m1), m = m2 + 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), heap-order(bh.ar, m + 1), id-idx(ar, ar0), no-dups-idx(kar, m), no-dups-idx(kar0, m), no-dups-idx(kar0[m, key-index(key, m0)], m + 1), no-dups-idx(bh.ar, 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), valid-idx(bh.ar, ar0, 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 ⊦ keyset(kar0, m) ++ key-index(key, m0) = keyset(kar, m) ++ key-index(key, m0), m = 0

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [m, m2]

Additional information

Used simplifier rules: ⊦ subarray(kar, n, 0) = mkkeyarray(0); (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ keyset(kar, 0) = ∅; (s)%# kar = n ⊦ subarray(kar, 0, n) = kar; (s)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m, m2