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Proof step 8 from proof valid-idx-update-key (previous step)

kar[m, key-index(key, m0)][m1] = key-index(key1, m2), kar[m] = key-index(key0, m0), ar[m2] = elemindex(a, m3), m1 < n, m1 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m1

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 9:
ar[m2] = elemindex(a, m3), kar[m] = key-index(key0, m0), kar[m, key-index(key, m0)][m1] = key-index(key1, m2), m1 = m, m1 < # kar, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m1
Proof step 11:
ar[m2] = elemindex(a, m3), kar[m] = key-index(key0, m0), kar[m, key-index(key, m0)][m1] = key-index(key1, m2), m1 < # kar, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m1 = m, m2 < # ar ∧ m3 = m1

Rule argument

fma: m1 = m

Additional information

Used formulas: m1 = m