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Proof step 3 from proof key-set-write-nodups (previous step)

n < m, n < # kar, m ≤ # kar, ∀ m0, m1. m0 < # kar ∧ m0 < m ∧ m1 < # kar ∧ m1 < m ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ keyset(kar[n, ki], m) = (keyset(kar, m) -- kar[n]) ++ ki

User applied rule structural induction

And obtained the following new goals

Proof step 4:
n < # kar, n < m0 +1, m0 +1 ≤ # kar, ∀ m2, m1. m2 < # kar ∧ m2 < m0 +1 ∧ m1 < # kar ∧ m1 < m0 +1 ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx, ∀ n, kar, ki. n < m0 ∧ n < # kar ∧ m0 ≤ # kar ∧ (∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx) → keyset(kar[n, ki], m0) = (keyset(kar, m0) -- kar[n]) ++ ki ⊦ keyset(kar[n, ki], m0 +1) = (keyset(kar, m0 +1) -- kar[n]) ++ ki

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)% ⊦ ∅ -- ki = ∅; (s)% ⊦ ∅ ≠ s ++ ki; (s)% ⊦ ¬ m < 0; (s)