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Proof step 16 from proof prepost-insert (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
subarray(bh1.ar, 0, bh1.size) = subarray(kar, 0, bh1.size), 2exp-size(# kar), 2exp-size(# bh1.ar), m0 < # ar, bh1.size < # bh1.ar, heap-order(kar, bh1.size), heap-order(bh1.ar, bh1.size), no-dups-idx(kar, bh1.size), no-dups-idx(bh1.ar, bh1.size), valid-idx(kar, ar, bh1.size), valid-idx(bh1.ar, ar, bh1.size), bh1.size ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, bh1.size) → ki0.idx ≠ m0 ⊦ ⟪bh0 := bh1.ar:= bh1.ar[bh1.size, key-index(key, m0)]; ar1 := ar[m0, ar[m0].idx:= bh0.size]; bh0.size := bh0.size + 1; 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, bh1.size) ++ 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