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Proof step 9 from proof key-set-write-nodups (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
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

Rule argument

Additional information

Used simplifier rules: key-set-write-other: m ≤ n, n < # kar ⊦ keyset(kar[n, ki], m) = keyset(kar, m);% ⊦ m ≤ m; (s)% ⊦ (s ++ ki) -- ki = s -- ki; (s)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)