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Proof step 24 from proof prepost-get-min (previous step)

kar[0] = key-index(key0, m0), key-index(key, m1) = kar[m2], 2exp-size(# kar), m2 < m, key < key0, m2 < # kar, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar ⊦ # keyset(kar, m) = 0, keyset(kar, m) = ∅

User applied rule apply lemma

And obtained the following new goals

Proof step 25:
⊦ n ≠ 0 ∧ n ≤ # kar ∧ heap-order(kar, n) ∧ m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[0].key

Rule argument

spec: bin-min-heap-inv name: heap-order-min seq: ⊦ n ≠ 0 ∧ n ≤ # kar ∧ heap-order(kar, n) ∧ m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[0].key subst: [n, kar, m] → [m, kar, m2]

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)% ⊦ key-index(key, m).key = key; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ 0 < m ↔ m ≠ 0; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s) Used terms: m, kar, m2