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Proof step 24 from proof prepost-insert (previous step)

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(# kar0), 2exp-size(# kar), m0 < # ar, m < # kar0, heap-order(kar0, m), heap-order(kar, m), no-dups-idx(kar0, m), no-dups-idx(kar, m), valid-idx(kar0, ar, m), valid-idx(kar, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ bh0 = bh0 ∧ ar1 = ar1 ∧ bubble-up-inv(bh0.size - 1, bh0, ar1, bh0, ar1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
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), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ no-dups-idx(kar0[m, key-index(key, m0)], m + 1) ∧ valid-idx(kar0[m, key-index(key, m0)], ar[m0, ar[m0].idx:= m], m + 1) ∧ (m = 0 ∨ ¬ key < kar0[m, key-index(key, m0)][(m - 1) / 2].key → heap-order(kar0[m, key-index(key, m0)], m + 1)) ∧ (m ≠ 0 ∧ key < kar0[m, key-index(key, m0)][(m - 1) / 2].key → heap-order(kar0[m, kar0[m, key-index(key, m0)][(m - 1) / 2]], m + 1))

Rule argument

Additional information

Used simplifier rules: bubble-up-inv: ⊦ bubble-up-inv(n, bh, ar, bh0, ar0) ↔ n < bh.size ∧ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ ( n = 0 ∨ ¬ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar, bh.size)) ∧ ( n ≠ 0 ∧ bh.ar[n].key < bh.ar[(n - 1) / 2].key → heap-order(bh.ar[n, bh.ar[(n - 1) / 2]], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);%m < # kar ⊦ kar[m, ki][m, ki0] = kar[m, ki0]; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ key-index(key, m).key = key; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ id-idx(ar, ar); (s)% ⊦ id-idx(ar1, ar0[n, ar0[n].idx:= m]) ↔ id-idx(ar1, ar0); (s)% ⊦ id-idx(ar1[m, ar1[m].idx:= m0], ar0) ↔ id-idx(ar1, ar0); (s)% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ # kar[n, ki] = # kar; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)% ⊦ ¬ n < n; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ n + 0 = n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ bin-heap(kar, m).size = m; (s)