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Proof step 4 from proof prepost-extract-min (previous step)

ar0 = ar, bh = bin-heap(kar, m), bh0 = bh, 2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar ⊦ ⟪bin_min_heap_extract_min#(; bh, ar, n)⟫ ( 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(bh0.ar, bh0.size) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size) -- key-index(key, n) ∧ (∀ ki. ki ∈ keyset(bh0.ar, bh0.size) → ¬ ki.key < key))), # keyset(kar, m) = 0, keyset(kar, m) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
ar0 = ar, bh = bin-heap(kar, m), # keyset(kar, m) ≠ 0, keyset(kar, m) ≠ ∅, 2exp-size(# kar), heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar ⊦ ⟪bin_min_heap_extract_min#(; bh, ar, n)⟫ ( 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)))

Rule argument

Additional information

Used simplifier rules: ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar = kar; (s)%size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic[keyindex-set]%size-zero: ⊦ # s ≠ 0 → s ≠ ∅; (forward) from specification set-basic[keyindex-set]