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Proof step 6 from proof no-dups-idx-mvlast (previous step)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 7:
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
Proof step 9:
kar[m0, kar[m]][m2] = key-index(key0, m3), kar[m0, kar[m]][m1] = key-index(key, m3), 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, m2 = m0

Rule argument

fma: m2 = m0

Additional information

Used formulas: m2 = m0