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Proof step 4 from proof prepost-bubble-up (previous step)

ar0 = ar, bh = bin-heap(kar, m), bh0 = bh, n < # kar, n < m, no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, n ≠ 0 ∧ kar[n].key < kar[(n - 1) / 2].key → heap-order(kar[n, kar[(n - 1) / 2]], m), n = 0 ∨ ¬ kar[n].key < kar[(n - 1) / 2].key → heap-order(kar, m) ⊦ ⟪bin_min_heap_bubble_up#(n; bh, ar)⟫ ( bh.size = bh0.size ∧ # bh.ar = # bh0.ar ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(bh0.ar, bh0.size) = keyset(bh.ar, bh.size))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
ar0 = ar, bh = bin-heap(kar, m), n < # kar, n < m, no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, n ≠ 0 ∧ kar[n].key < kar[(n - 1) / 2].key → heap-order(kar[n, kar[(n - 1) / 2]], m), n = 0 ∨ ¬ kar[n].key < kar[(n - 1) / 2].key → heap-order(kar, m) ⊦ ⟪bin_min_heap_bubble_up#(n; bh, ar)⟫ ( bh.size = m ∧ # bh.ar = # kar ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ heap-order(bh.ar, bh.size) ∧ keyset(kar, bh.size) = keyset(bh.ar, bh.size))

Rule argument

Additional information

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