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Proof step 12 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 ⊦ ⟪bin_min_heap_resize#(m + 1; bh); bh.ar[bh.size] := key-index(key, m0); ar1 := ar[m0, ar[m0].idx:= bh.size]; bh.size := bh.size + 1; bin_min_heap_bubble_up#(bh.size - 1; bh, ar1)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar1, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar, ar1) ∧ keyset(bh.ar, bh.size) = keyset(kar, m) ++ key-index(key, m0))

User applied rule insert program lemma

And obtained the following new goals

Proof step 13:
bin-heap-inv(bh, ar), bh0 = bh ⊦ ⟪bin_min_heap_resize#(minsize; bh)⟫ (bin-heap-inv(bh, ar) ∧ bh.size = bh0.size ∧ # bh.ar ≥ minsize ∧ subarray(bh.ar, 0, bh.size) = subarray(bh0.ar, 0, bh0.size))
Proof step 14:
ar0 = ar, bh = bin-heap(kar, m), bh = bh, bh1 = bh, minsize0 = m + 1, 2exp-size(# kar), m0 < # ar, bin-heap-inv(bh, 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 ⊦ [bin_min_heap_resize#(m + 1; bh)] ( bin-heap-inv(bh, ar0) ∧ bh.size = bh1.size ∧ # bh.ar ≥ minsize0 ∧ subarray(bh.ar, 0, bh.size) = subarray(bh1.ar, 0, bh1.size) → ⟪bh.ar[bh.size] := key-index(key, m0); ar1 := ar[m0, ar[m0].idx:= bh.size]; bh.size := bh.size + 1; bin_min_heap_bubble_up#(bh.size - 1; bh, ar1)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar1, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar, ar1) ∧ keyset(bh.ar, bh.size) = keyset(kar, m) ++ key-index(key, m0)))

Rule argument

spec: <toplevel> name: prepost-resize seq: bin-heap-inv(bh, ar), bh0 = bh ⊦ ⟪bin_min_heap_resize#(minsize; bh)⟫ ( bin-heap-inv(bh, ar) ∧ bh.size = bh0.size ∧ # bh.ar ≥ minsize ∧ subarray(bh.ar, 0, bh.size) = subarray(bh0.ar, 0, bh0.size)) subst: [minsize, bh, ar, bh0] → [m + 1, bh, ar, bh] flags: <all paths>

Additional information

Used simplifier rules: bin-heap-inv-def: ⊦ bin-heap-inv(bh, ar) ↔ bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size);% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)