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Proof step 2 from proof prepost-insert (previous step)

ar0 = ar, bh0 = bh, ki.idx < # ar, bin-heap-inv(bh, ar), ∀ ki0. ki0 ∈ keyset(bh0) → ki0.idx ≠ ki.idx ⊦ ⟪bin_min_heap_insert#(ki; bh, ar)⟫ (bin-heap-inv(bh, ar) ∧ id-idx(ar0, ar) ∧ keyset(bh) = keyset(bh0) ++ ki)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
ar0 = ar, bh0 = bh, 2exp-size(# bh.ar), ki.idx < # ar, heap-order(bh.ar, bh.size), no-dups-idx(bh.ar, bh.size), valid-idx(bh.ar, ar, bh.size), bh.size ≤ # bh.ar, ∀ ki0. ki0 ∈ keyset(bh.ar, bh.size) → ki0.idx ≠ ki.idx ⊦ ⟪bin_min_heap_insert#(ki; 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(bh0.ar, bh0.size) ++ ki)

Rule argument

Additional information

Used simplifier rules: bin-heap-inv-def: ⊦ bin-heap-inv(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);%keyset: ⊦ keyset(bh) = keyset(bh.ar, bh.size);% ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic