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Proof step 6 from proof prepost-extract-min (previous step)

ar0 = ar, bh = bin-heap(kar, m), 2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar ⊦ ⟪bin_min_heap_get_min#(bh; n); bh.size := bh.size - 1; if bh.size ≠ 0 then {bh.ar[0] := bh.ar[bh.size]; ar[bh.ar[0].idx].idx := 0; bin_min_heap_bubble_down#(0; 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) ∧ id-idx(ar0, ar) ∧ (∃ key. key-index(key, n) ∈ keyset(kar, m) ∧ keyset(bh.ar, bh.size) = keyset(kar, m) -- key-index(key, n) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key))), # keyset(kar, m) = 0, keyset(kar, m) = ∅

User applied rule insert program lemma

And obtained the following new goals

Proof step 7:
bin-heap-inv(bh, ar), keyset(bh) ≠ ∅ ⊦ ⟪bin_min_heap_get_min#(bh; n)⟫ ∃ key. key-index(key, n) ∈ keyset(bh) ∧ (∀ ki. ki ∈ keyset(bh) → ¬ ki.key < key)
Proof step 8:
ar0 = ar, ar1 = ar, bh = bin-heap(kar, m), bh0 = bh, 2exp-size(# kar), bin-heap-inv(bh, ar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar ⊦ [bin_min_heap_get_min#(bh; n)] ( (∃ key. key-index(key, n) ∈ keyset(bh0) ∧ (∀ ki. ki ∈ keyset(bh0) → ¬ ki.key < key)) → ⟪bh.size := bh.size - 1; if bh.size ≠ 0 then {bh.ar[0] := bh.ar[bh.size]; ar[bh.ar[0].idx].idx := 0; bin_min_heap_bubble_down#(0; 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) ∧ id-idx(ar0, ar) ∧ (∃ key. key-index(key, n) ∈ keyset(kar, m) ∧ keyset(bh.ar, bh.size) = keyset(kar, m) -- key-index(key, n) ∧ (∀ ki. ki ∈ keyset(kar, m) → ¬ ki.key < key)))), # keyset(kar, m) = 0, keyset(kar, m) = ∅, keyset(bh) = ∅

Rule argument

spec: <toplevel> name: prepost-get-min seq: bin-heap-inv(bh, ar), keyset(bh) ≠ ∅ ⊦ ⟪bin_min_heap_get_min#(bh; n)⟫ ∃ key. key-index(key, n) ∈ keyset(bh) ∧ (∀ ki. ki ∈ keyset(bh) → ¬ ki.key < key) subst: [bh, n, ar] → [bh, n, ar] 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);%keyset: ⊦ keyset(bh) = keyset(bh.ar, bh.size);% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)