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Proof step 3 from proof keyset-xchg (previous step)

n1 < m, n0 < m, n1 < # kar, n0 < # kar, m ≤ # kar ⊦ keyset(kar[n0, kar[n1]][n1, kar[n0]], m) = keyset(kar, m)

User applied rule structural induction

And obtained the following new goals

Proof step 4:
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)

Rule argument

indvar: m

Additional information

Used simplifier rules: key-set-write-other: m ≤ n, n < # kar ⊦ keyset(kar[n, ki], m) = keyset(kar, m);%keyset-base: ⊦ keyset(kar, 0) = ∅;%nat freely generated by 0, +1;;% ⊦ 0 ≤ m; (s)% ⊦ # kar[n, ki] = # kar; (s)