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Proof step 26 from proof prepost-remove (previous step)

kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), ar = ar1[kar[m0 - 1].idx, ar1[kar[m0 - 1].idx].idx:= m1], bh = bin-heap(kar[m1, kar[m0 - 1]], m0 - 1), m1 = m2 + 1, 2exp-size(# kar), m1 < # kar, m1 < m0, kar[m0 - 1].idx < # ar1, kar[m0 - 1].key < kar[m1, kar[m0 - 1]][m2 / 2].key, m2 / 2 < # kar, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar, 1 ≤ m1 ⊦ ⟪bin_min_heap_bubble_up#(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(ar1, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m)), # keyset(kar, m0) = 0, # kar = 1, m0 - 1 = m1, keyset(kar, m0) = ∅, m0 = 1, m1 = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 27:
ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), ar = ar1[kar[m0 - 1].idx, ar1[kar[m0 - 1].idx].idx:= (m2 + 1)], bh = bin-heap(kar[m2 + 1, kar[m0 - 1]], m0 - 1), # kar ≠ 1, # keyset(kar, m0) ≠ 0, m0 - 1 ≠ m2 + 1, keyset(kar, m0) ≠ ∅, m0 ≠ 1, 2exp-size(# kar), kar[m0 - 1].idx < # ar1, m2 + 1 < m0, m2 / 2 < # kar, kar[m0 - 1].key < kar[m2 / 2].key, m2 + 1 < # kar, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar1, m0), key-index(key, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_bubble_up#(m2 + 1; 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(ar1, ar) ∧ keyset(bh.ar, bh.size) = keyset(kar, m0) -- key-index(key, m))

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%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%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m / 2 ≠ m + 1; (s)% ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)