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Proof step 6 from proof prepost-remove (previous step)

ar[m] = elemindex(a, m1), ar0 = ar, bh = bin-heap(kar, m0), 2exp-size(# kar), heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_remove#(m1; 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(key, m)), # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
ar[m] = elemindex(a, m1), kar[m1] = key-index(key, m), 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(key, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_remove#(m1; 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(key, m))

Rule argument

Additional information

Used simplifier rules: key-index-in-keyset-array-extract: ⊦ ar[m] = elemindex(a, m0) ∧ valid-idx(kar, ar, n) ∧ key-index(key, m) ∈ keyset(kar, n) → kar[m0] = key-index(key, m); (forward) from specification bin-min-heap-inv