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Proof step 20 from proof valid-idx-xhg (previous step)

kar[m5] = key-index(key2, m4), ar[m4] = elemindex(a0, m5), ar[n1] = elemindex(a1, m0), ar[n0] = elemindex(a2, m1), kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m5] = key-index(key, m2), kar[m1] = key-index(key0, n0), kar[m0] = key-index(key1, n1), ar[n0, elemindex(a2, m0)][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][m2] = elemindex(a, m3), m1 < # kar, m5 < n, m4 < # ar, n1 < # ar, m5 < # kar, m1 < n, n0 < # ar, m0 < # kar, m0 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m5

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 21:
ar[n0, elemindex(a2, m0)][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][m2] = elemindex(a, m3), kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m5] = key-index(key, m2), ar[n0] = elemindex(a2, m1), ar[n1] = elemindex(a1, m0), ar[m4] = elemindex(a0, m5), kar[m5] = key-index(key2, m4), m5 = m1, m0 < n, m0 < # kar, n0 < # ar, m1 < n, m5 < # kar, n1 < # ar, m4 < # ar, m5 < n, m1 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m5
Proof step 25:
ar[n0, elemindex(a2, m0)][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][m2] = elemindex(a, m3), kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m5] = key-index(key, m2), ar[n0] = elemindex(a2, m1), ar[n1] = elemindex(a1, m0), ar[m4] = elemindex(a0, m5), kar[m5] = key-index(key2, m4), m0 < n, m0 < # kar, n0 < # ar, m1 < n, m5 < # kar, n1 < # ar, m4 < # ar, m5 < n, m1 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m5 = m1, m2 < # ar ∧ m3 = m5

Rule argument

fma: m5 = m1

Additional information

Used formulas: m5 = m1