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Proof step 3 from proof prepost-extract-min (previous step)

ar0 = ar, bh0 = bh, # keyset(bh.ar, bh.size) ≠ 0, keyset(bh.ar, bh.size) ≠ ∅, 2exp-size(# bh.ar), heap-order(bh.ar, bh.size), no-dups-idx(bh.ar, bh.size), valid-idx(bh.ar, ar, bh.size), bh.size ≤ # bh.ar ⊦ ⟪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)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 4:
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) = ∅

Rule argument

spec: bin-min-heap name: elim seq: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m) subst: [bh, kar, m] → [bh, kar, m]

Additional information

Used simplifier rules: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m); (elim) Used terms: bh, kar, m