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Proof step 132 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[2 * n0 + 2] = key-index(key2, m), kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), kar0[n0] = key-index(key3, m3), kar0[4 + 4 * n0] = key-index(key4, m4), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[2 * n0 + 1] = key-index(key6, m6), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 + 1 < n2, m6 < # ar, m3 < # ar, 2 * n0 + 1 < # kar, n0 < # kar, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar0, n2 + n0), no-dups-idx(kar, n2 + n0), valid-idx(kar0, ar, n2 + n0), valid-idx(kar, ar1, n2 + n0), n2 + n0 ≤ # kar, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), 3 * n0 + 3 < n2 ∧ key5 < key3 ∨ 3 * n0 + 4 < n2 ∧ key4 < key3, key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3 ⊦ heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5))], n2 + n0), n0 + 2 < n2 ∧ key2 < key6, ¬ (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key < key6 ∧ (3 * n0 + 3 < n2 → ¬ key5 < (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key) ∧ (3 * n0 + 4 < n2 → ¬ key4 < (key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 133:
# kar0 = # kar, (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)) = key-index(key7, m7), kar0[2 * n0 + 1] = key-index(key6, m6), kar0[3 + 4 * n0] = key-index(key5, m5), kar0[4 + 4 * n0] = key-index(key4, m4), kar0[n0] = key-index(key3, m3), kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), kar0[2 * n0 + 2] = key-index(key2, m), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), n0 < # kar, 2 * n0 + 1 < # kar, m3 < # ar, m6 < # ar, n0 + 1 < n2, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar, n2 + n0), no-dups-idx(kar0, n2 + n0), valid-idx(kar, ar1, n2 + n0), valid-idx(kar0, ar, n2 + n0), n2 + n0 ≤ # kar, (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), n0 + 2 < n2 → 2 * n0 + 2 < # kar, ¬ key6 < key3 ∧ (n0 + 2 < n2 → ¬ key2 < key3) → heap-order(kar0, n2 + n0), key6 < key3 ∨ n0 + 2 < n2 ∧ key2 < key3, 3 * n0 + 3 < n2 ∧ key5 < key3 ∨ 3 * n0 + 4 < n2 ∧ key4 < key3 ⊦ heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5))], n2 + n0), ¬ key7 < key6 ∧ (3 * n0 + 3 < n2 → ¬ key5 < key7) ∧ (3 * n0 + 4 < n2 → ¬ key4 < (key4 < key5 ⊃ key-index(key4, m4);key-index(key5, m5)).key), n0 + 2 < n2 ∧ key2 < key6

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [(3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4) ;key-index(key5, m5) ), key7, m7]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: (3 * n0 + 4 < n2 ∧ key4 < key5 ⊃ key-index(key4, m4) ;key-index(key5, m5) ), key7, m7