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Proof step 2 from proof prepost-bubble-down (previous step)

ar0 = ar, bh0 = bh, bubble-down-inv(n, bh, ar, bh0, ar0) ⊦ ⟪bin_min_heap_bubble_down#(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) = keyset(bh))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
ar0 = ar, bh0 = bh, no-dups-idx(bh.ar, bh.size), valid-idx(bh.ar, ar, bh.size), bh.size ≤ # bh.ar, (2 * n + 1 < bh.size → ¬ bh.ar[2 * n + 1].key < bh.ar[n].key) ∧ (2 * n + 2 < bh.size → ¬ bh.ar[2 * n + 2].key < bh.ar[n].key) → heap-order(bh.ar, bh.size), 2 * n + 1 < bh.size ∧ bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order(bh.ar[n, (2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key ⊃ bh.ar[2 * n + 2];bh.ar[2 * n + 1])], bh.size) ⊦ ⟪bin_min_heap_bubble_down#(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))

Rule argument

Additional information

Used simplifier rules: bubble-down-inv: ⊦ bubble-down-inv(n, bh, ar, bh0, ar0) ↔ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ ( ( 2 * n + 1 < bh.size → ¬ bh.ar[2 * n + 1].key < bh.ar[n].key) ∧ ( 2 * n + 2 < bh.size → ¬ bh.ar[2 * n + 2].key < bh.ar[n].key) → heap-order(bh.ar, bh.size)) ∧ ( 2 * n + 1 < bh.size ∧ bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order (bh.ar [n, ( 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key ⊃ bh.ar[2 * n + 2] ;bh.ar[2 * n + 1] ) ], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size);%keyset: ⊦ keyset(bh) = keyset(bh.ar, bh.size);% ⊦ id-idx(ar, ar); (s)