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Proof step 80 from proof prepost-extract-min (previous step)

# kar0 = # kar, kar[0] = key-index(key1, m1), kar[m0] = key-index(key0, m), kar[1] = key-index(key2, m2), kar[2] = key-index(key3, m3), keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), key4 < key1, m < # ar1, m0 < # kar, key < key1, heap-order(kar, m0 + 1), heap-order(kar0, m0), id-idx(ar1, ar0), no-dups-idx(kar, m0 + 1), no-dups-idx(kar0, m0), valid-idx(kar, ar1, m0 + 1), valid-idx(kar0, ar0, m0), key-index(key, m1) ∈ keyset(kar, m0), key-index(key4, m4) ∈ keyset(kar, m0), (m0 ≠ 1 → ¬ key2 < key0) ∧ (2 < m0 → ¬ key3 < key0) → heap-order(kar[0, key-index(key0, m)], m0), 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) ⊦ # keyset(kar, m0) = 0, #(keyset(kar, m0) ++ key-index(key0, m)) = 0, keyset(kar, m0) = ∅, m0 = 0, key4 < key

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 81:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 82:
# 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, m1), keyset(kar[0, key-index(key0, m)], m0) = keyset(kar0, m0), 2exp-size(# kar), key < 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), key-index(key, m1) ∈ keyset(kar, m0), m0 ≤ # kar, 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), ∃ m. m < m0 ∧ key-index(key4, m4) = kar[m] ⊦ #(keyset(kar, m0) ++ key-index(key0, m)) = 0, # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, m0 = 0, key4 < key

Rule argument

spec: bin-min-heap-inv name: key-set-extract-index seq: n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m]) subst: [n, kar, ki] → [m0, kar, key-index(key4, m4)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: m0, kar, key-index(key4, m4)