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Proof step 10 from proof no-dups-idx-remlast (previous step)

kar[0, kar[m]][m0] = key-index(key, m2), kar[m1] = key-index(key0, m2), m0 + 1 < # kar, m1 < # kar, m1 + 1 < # kar, m < # kar, m0 < # kar, m0 < m, m1 < m, ∀ m0, m1. m0 < # kar ∧ ¬ m < m0 ∧ m1 < # kar ∧ ¬ m < m1 ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ m0 = m1, m1 = 0

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 11:
kar[m1] = key-index(key0, m2), kar[0, kar[m]][m0] = key-index(key, m2), m0 = 0, m1 < m, m0 < m, m0 < # kar, m < # kar, m1 + 1 < # kar, m1 < # kar, m0 + 1 < # kar, ∀ m0, m1. m0 < # kar ∧ ¬ m < m0 ∧ m1 < # kar ∧ ¬ m < m1 ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ m0 = m1, m1 = 0
Proof step 13:
kar[m1] = key-index(key0, m2), kar[0, kar[m]][m0] = key-index(key, m2), m1 < m, m0 < m, m0 < # kar, m < # kar, m1 + 1 < # kar, m1 < # kar, m0 + 1 < # kar, ∀ m0, m1. m0 < # kar ∧ ¬ m < m0 ∧ m1 < # kar ∧ ¬ m < m1 ∧ m0 ≠ m1 → kar[m0].idx ≠ kar[m1].idx ⊦ m0 = m1, m0 = 0, m1 = 0

Rule argument

fma: m0 = 0

Additional information

Used formulas: m0 = 0