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Proof step 5 from proof valid-idx-xhg (previous step)

kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), m0 < # kar, m1 < # kar, m0 < n, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ n ≤ # kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)] ∧ (∀ m. m < n → kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx < # ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1] ∧ ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx].idx = m)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
kar[m1] = key-index(key0, n0), kar[m0] = key-index(key1, n1), m1 < n, m0 < n, m1 < # kar, m0 < # kar, m < n, m < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx < # ar ∧ ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx].idx = m

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ # ar[n, ei] = # ar; (s)% ⊦ # kar[n, ki] = # kar; (s)