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Proof step 73 from proof prepost-extract-min (previous step)

# kar0 = # kar, kar[2] = key-index(key3, m3), kar[1] = key-index(key2, m2), kar[m0] = key-index(key0, m), kar[0] = key-index(key1, m), keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), key0 < key1, m0 < # kar, m < # ar1, key4 < key1, heap-order(kar0, m0), heap-order(kar, m0 + 1), id-idx(ar1, ar0), no-dups-idx(kar0, m0), no-dups-idx(kar, m0 + 1), valid-idx(kar0, ar0, m0), valid-idx(kar, ar1, m0 + 1), m0 ≠ 1 ∧ key2 < key0 ∨ 2 < m0 ∧ key3 < key0 → heap-order(kar[0, (2 < m0 ∧ key3 < key2 ⊃ key-index(key3, m3);key-index(key2, m2))], m0), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), key-index(key4, m4) = key-index(key0, m) ∨ key-index(key4, m4) ∈ keyset(kar, m0) ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, m0 = 0, key4 < key0

User applied rule apply lemma

And obtained the following new goals

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] → [m0 + 1, kar, m0]

Additional information

Used simplifier rules: ⊦ m + 1 ≠ 0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m < n + 1 ↔ ¬ n < m; (s)% ⊦ ¬ n < n; (s)% ⊦ key-index(key, m).key = key; (s) Used terms: m0 + 1, kar, m0