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Proof step 20 from proof prepost-insert (previous step)

ar1 = ar[m0, ar[m0].idx:= (bh1.ar:= bh1.ar[bh1.size, key-index(key, m0)]).size], bh0 = (bh1.ar:= bh1.ar[bh1.size, key-index(key, m0)]).size:= ((bh1.ar:= bh1.ar[bh1.size, key-index(key, m0)]).size + 1), subarray(bh1.ar, 0, bh1.size) = subarray(kar, 0, bh1.size), 2exp-size(# bh1.ar), 2exp-size(# kar), m0 < # ar, bh1.size < # bh1.ar, heap-order(bh1.ar, bh1.size), heap-order(kar, bh1.size), no-dups-idx(bh1.ar, bh1.size), no-dups-idx(kar, bh1.size), valid-idx(bh1.ar, ar, bh1.size), valid-idx(kar, ar, bh1.size), bh1.size ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, bh1.size) → ki0.idx ≠ m0 ⊦ ⟪bin_min_heap_bubble_up#(bh0.size - 1; bh0, ar1)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar1, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar, ar1) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, bh1.size) ++ key-index(key, m0))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 21:
ar1 = ar[m0, ar[m0].idx:= (bh1.ar:= kar0[m, key-index(key, m0)]).size], bh0 = (bh1.ar:= kar0[m, key-index(key, m0)]).size:= ((bh1.ar:= kar0[m, key-index(key, m0)]).size + 1), bh1 = bin-heap(kar0, m), subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar), 2exp-size(# kar0), m < # kar0, m0 < # ar, heap-order(kar, m), heap-order(kar0, m), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ ⟪bin_min_heap_bubble_up#(bh0.size - 1; bh0, ar1)⟫ ( bh0.size ≤ # bh0.ar ∧ no-dups-idx(bh0.ar, bh0.size) ∧ valid-idx(bh0.ar, ar1, bh0.size) ∧ 2exp-size(# bh0.ar) ∧ heap-order(bh0.ar, bh0.size) ∧ id-idx(ar, ar1) ∧ keyset(bh0.ar, bh0.size) = keyset(kar, m) ++ key-index(key, m0))

Rule argument

spec: bin-min-heap name: elim seq: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m) subst: [bh, kar, m] → [bh1, kar0, m]

Additional information

Used simplifier rules: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m); (elim) Used terms: bh1, kar0, m