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Proof step 8 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) ++ kar[n0, kar[n1]][n1, kar[n0]][m0] = keyset(kar, m0) ++ kar[m0]

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 9:
m0 = n1, 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]
Proof step 12:
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

Rule argument

fma: m0 = n1

Additional information

Used formulas: m0 = n1