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Proof step 4 from proof prepost-update (previous step)

ar0 = ar, bh = bin-heap(kar, m0), bh0 = bh, 2exp-size(# kar), heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key0, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_update#(ar[m].idx, key; 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) ∧ keyset(bh.ar, bh.size) = (keyset(bh0.ar, bh0.size) -- key-index(key0, m)) ++ key-index(key, m)), # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
ar0 = ar, bh = bin-heap(kar, m0), # keyset(kar, m0) ≠ 0, keyset(kar, m0) ≠ ∅, 2exp-size(# kar), heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key0, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_update#(ar[m].idx, key; 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) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m0) -- key-index(key0, m)) ++ key-index(key, m))

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]%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set]