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Proof step 72 from proof prepost-update (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), ar0 = ar, bh = bh0.ar:= bh0.ar[n, bh0.ar[n].key:= key], bh0 = bin-heap(kar, m0), key0 = bh0.ar[n].key, n = m1, 2exp-size(# kar), n < # bh0.ar, n < bh0.size, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_bubble_down#(n; 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(key1, m)) ++ key-index(key, m)), # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, key < key0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 73:
ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), ar0 = ar, bh = bin-heap(kar[m1, key-index(key, m)], m0), # keyset(kar, m0) ≠ 0, keyset(kar, m0) ≠ ∅, 2exp-size(# kar), m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, ¬ key < key1 ⊦ ⟪bin_min_heap_bubble_down#(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(key1, m)) ++ key-index(key, m))

Rule argument

Additional information

Used simplifier rules: ⊦ key-index(key, m).key = key; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ key-index(key, m).key:= key0 = key-index(key0, m); (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%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% ⊦ bin-heap(kar, m).size = m; (s)% ⊦ bin-heap(kar, m).ar:= kar0 = bin-heap(kar0, m); (s)% ⊦ bin-heap(kar, m).ar = kar; (s)