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Proof step 23 from proof key-set-write-nodups (previous step)

kar[m1] = key-index(key0, m3), kar[m] = key-index(key, m2), m2 = m3, m < # kar, m < m0, m1 < # kar, m1 < m0, 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 ⊦ keyset(kar[n, ki], m0) ++ kar[m0] = ((keyset(kar, m0) ++ kar[m0]) -- kar[n]) ++ ki, m = m1, m0 = n

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 24:
kar[m1] = key-index(key0, m3), kar[m] = key-index(key, m3), m < m0, m1 < # kar, m1 < m0, n + 1 < # kar, n < # kar, m0 < # kar, n < m0, m + 1 < # kar, m < # kar, m1 + 1 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx ⊦ keyset(kar[n, ki], m0) ++ kar[m0] = ((keyset(kar, m0) ++ kar[m0]) -- kar[n]) ++ ki, m = m1, m0 = n

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic