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Proof step 12 from proof keyset-xchg (previous step)

m0 < # kar, n0 < # kar, n1 < # kar, 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], m0 = n1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
n1 < m0, m0 < # kar, n0 < # kar, n1 < # kar, n1 + 1 < # kar, 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) ++ kar[n0, kar[n1]][m0] = keyset(kar, m0) ++ kar[m0], m0 = n1

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≠ n → kar[n, ki][m] = kar[m]; (s)