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Proof step 21 from proof prepost-insert (previous step)

ar1 = ar[m0, ar[m0].idx:= (bh1.ar:= kar0[m, key-index(key, m0)]).size], bh0 = (bh1.ar:= kar0[m, key-index(key, m0)]).size:= ((bh1.ar:= kar0[m, key-index(key, m0)]).size + 1), bh1 = bin-heap(kar0, m), 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 ⊦ ⟪bin_min_heap_bubble_up#(bh0.size - 1; bh0, ar1)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar1, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar, ar1) ∧ 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 22:
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), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ ⟪bin_min_heap_bubble_up#(bh0.size - 1; bh0, ar1)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar1, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar, ar1) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, m) ++ key-index(key, m0))

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ bin-heap(kar, m).size:= m0 = bin-heap(kar, m0); (s)% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)