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Proof step 10 from proof key-set-write-nodups (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ ¬ ki ∈ s → s -- ki = s
Proof step 12:
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

Rule argument

spec: set-union, inst: keyindex-set name: del-in seq: ⊦ ¬ ki ∈ s → s -- ki = s subst: [ki, s] → [kar[n], keyset(kar, n)] flags: <all paths>

Additional information

Used terms: kar[n], keyset(kar, n)