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Proof step 81 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m + 1] = key-index(key1, m3), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, key1 < key2, n1 < m2, m + 1 < m2, heap-order(kar0[m + 1, key-index(key2, m4)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar, ar1, m2), # ar ≤ m4, m2 ≤ # kar, m2 * 2 ≤ m, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2) ⊦ # kar = 1, m2 = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: nat name: le-addex seq: ⊦ m ≤ n ↔ (∃ n0. m + n0 = n) subst: [n, m] → [m, m2 * 2] flags: <all paths>

Additional information

Used simplifier rules: ⊦ (2 * m + n) / 2 = m + n / 2; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m * n + n0 < n ↔ m = 0 ∧ n0 < n; (ws) Used terms: m, m2 * 2