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Proof step 47 from proof prepost-insert (previous step)

kar0[m2 / 2] = key-index(key0, m1), subarray(kar0, 0, m2 + 1) = subarray(kar, 0, m2 + 1), 2exp-size(# kar), 2exp-size(# kar0), m2 < # kar, m0 < # ar, m2 + 1 < # kar0, heap-order(kar, m2 + 1), heap-order(kar0, m2 + 1), no-dups-idx(kar0[m2 + 1, key-index(key, m0)], 2 + m2), no-dups-idx(kar, m2 + 1), no-dups-idx(kar0, m2 + 1), valid-idx(kar0, ar, m2 + 1), valid-idx(kar, ar, m2 + 1), ∀ ki. ki = kar[m2] ∨ ki ∈ keyset(kar, m2) → ki.idx ≠ m0 ⊦ key < key0, heap-order(kar0[m2 + 1, key-index(key, m0)], 2 + m2)

User applied rule apply lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: heap-order-add-smaller seq: n + 1 ≤ # kar, n ≠ 0 ⊦ heap-order(kar, n) ∧ ¬ ki.key < kar[(n - 1) / 2].key → heap-order(kar[n, ki], n + 1) subst: [n, kar, ki] → [m2 + 1, kar0, key-index(key, m0)]

Additional information

Used simplifier rules: ⊦ key-index(key, m).key = key; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s)%# kar = n ⊦ subarray(kar, 0, n) = kar; (s)% ⊦ m + 1 ≠ 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)% ⊦ m +1 = m + 1; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n < n; (s)%succ-03: ⊦ m < n ∧ m + 1 ≠ n → m + 1 < n; (forward) from specification nat-basic Used terms: m2 + 1, kar0, key-index(key, m0)