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Proof step 12 from proof key-set-write-nodups (previous step)

n < # kar, kar[n] ∈ keyset(kar, n), ∀ m, m1. m < # kar ∧ ¬ n < m ∧ m1 < # kar ∧ ¬ n < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx, ∀ n0, kar, ki. n0 < n ∧ n0 < # kar ∧ n ≤ # kar ∧ (∀ m2, m1. m2 < # kar ∧ m2 < n ∧ m1 < # kar ∧ m1 < n ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx) → keyset(kar[n0, ki], n) = (keyset(kar, n) -- kar[n0]) ++ ki ⊦ keyset(kar, n) ++ ki = (keyset(kar, n) -- kar[n]) ++ ki

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
n < # kar, kar[n] ∈ keyset(kar, n), ∀ m, m1. m < # kar ∧ ¬ n < m ∧ m1 < # kar ∧ ¬ n < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx, ∀ n0, kar, ki. n0 < n ∧ n0 < # kar ∧ n ≤ # kar ∧ (∀ m2, m1. m2 < # kar ∧ m2 < n ∧ m1 < # kar ∧ m1 < n ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx) → keyset(kar[n0, ki], n) = (keyset(kar, n) -- kar[n0]) ++ ki ⊦ # keyset(kar, n) = 0, keyset(kar, n) ++ ki = (keyset(kar, n) -- kar[n]) ++ ki, keyset(kar, n) = ∅

Rule argument

Additional information

Used simplifier rules: size-zero-01: ⊦ s ≠ ∅ → # s ≠ 0; (forward) from specification set-basic[keyindex-set]%ne-01: ⊦ ki ∈ s → s ≠ ∅; (forward) from specification set-basic[keyindex-set]