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Proof step 7 from proof no-dups-idx-mvlast (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 8:
kar[m] = key-index(key0, m3), kar[m1] = key-index(key, m3), m0 < # kar, m0 < m, m1 < m, m1 < # kar, m < # kar, m1 + 1 < # kar, m0 + 1 < # kar, ∀ m1, m2. m1 < # kar ∧ ¬ m < m1 ∧ m2 < # kar ∧ ¬ m < m2 ∧ m1 ≠ m2 → kar[m1].idx ≠ kar[m2].idx ⊦ m1 = m0

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% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ m ≠ n → kar[n, ki][m] = kar[m]; (s)