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Proof step 5 from proof valid-idx-move-last (previous step)

kar[m] = key-index(key, m1), m < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ m ≤ # kar[m0, key-index(key, m1)] ∧ (∀ m2. m2 < m → kar[m0, key-index(key, m1)][m2].idx < # ar[m1, ar[m1].idx:= m0] ∧ ar[m1, ar[m1].idx:= m0][kar[m0, key-index(key, m1)][m2].idx].idx = m2)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
kar[m] = key-index(key, m1), m2 < # kar, m2 < m, m < # kar, m2 + 1 < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ kar[m0, key-index(key, m1)][m2].idx < # ar ∧ ar[m1, ar[m1].idx:= m0][kar[m0, key-index(key, m1)][m2].idx].idx = m2

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ # ar[n, ei] = # ar; (s)% ⊦ m < n → m ≤ n; (s)% ⊦ # kar[n, ki] = # kar; (s)