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Proof step 32 from proof key-set-write-nodups (previous step)

kar[m] = key-index(key, m2), kar[m1] = key-index(key0, m3), keyset(kar[n, ki], m0) = (keyset(kar, m0) -- kar[n]) ++ ki, ki0 = kar[m0], n < m0, m0 < # kar, n < # kar, n + 1 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx ⊦ ki0 = ki, m0 = n, ki0 ≠ kar[n] ∧ (ki0 = kar[m0] ∨ ki0 ∈ keyset(kar, m0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 33:
kar[m1] = key-index(key0, m3), kar[m] = key-index(key, m2), kar[m0] = kar[n], keyset(kar[n, ki], m0) = (keyset(kar, m0) -- kar[n]) ++ ki, n + 1 < # kar, n < # kar, m0 < # kar, n < m0, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx ⊦ kar[n] = ki, m0 = n

Proof step 32 from proof key-set-write-nodups (previous step)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

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