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Proof step 4 from proof keyset-xchg (previous step)

n0 < # kar, n1 < # kar, n0 < m0 +1, n1 < m0 +1, m0 +1 ≤ # kar, ∀ 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 +1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
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]

Rule argument

Additional information

Used simplifier rules: keyset-rec: ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n];% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m +1 = m + 1; (s)