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Proof step 5 from proof keyset-xchg (previous step)

n1 < # kar, n0 < # kar, m0 < # kar, n1 ≤ m0, n0 ≤ m0, ∀ n1, n0, kar. n1 < m0 ∧ n0 < m0 ∧ n1 < # kar ∧ n0 < # kar ∧ m0 ≤ # kar → keyset(kar[n0, kar[n1]][n1, kar[n0]], m0) = keyset(kar, m0) ⊦ keyset(kar[n0, kar[n1]][n1, kar[n0]], m0 + 1) = keyset(kar, m0) ++ kar[m0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]
Proof step 7:
m0 < # kar, n0 < # kar, n1 < # kar, m0 < # kar[n0, kar[n1]][n1, kar[n0]], n0 ≤ m0, n1 ≤ m0, ∀ n1, n0, kar. n1 < m0 ∧ n0 < m0 ∧ n1 < # kar ∧ n0 < # kar ∧ m0 ≤ # kar → keyset(kar[n0, kar[n1]][n1, kar[n0]], m0) = keyset(kar, m0) ⊦ keyset(kar[n0, kar[n1]][n1, kar[n0]], m0) ++ kar[n0, kar[n1]][n1, kar[n0]][m0] = keyset(kar, m0) ++ kar[m0]

Rule argument

spec: <toplevel> name: keyset-rec seq: ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n] subst: [n, kar] → [m0, kar[n0, kar[n1]][n1, kar[n0]]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kar[n, ki] = # kar; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m0, kar[n0, kar[n1]][n1, kar[n0]]