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Proof step 2 from proof prepost-extract-min (previous step)

ar0 = ar, bh0 = bh, bin-heap-inv(bh, ar) ⊦ ⟪bin_min_heap_extract_min#(; bh, ar, n)⟫ ( bin-heap-inv(bh, ar) ∧ id-idx(ar0, ar) ∧ (∃ key. key-index(key, n) ∈ keyset(bh0) ∧ keyset(bh) = keyset(bh0) -- key-index(key, n) ∧ (∀ ki. ki ∈ keyset(bh0) → ¬ ki.key < key))), keyset(bh) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
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)))

Rule argument

Additional information

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