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Proof step 7 from proof valid-idx-rec (previous step)

kar[n, key-index(key, m)][m0] = key-index(key0, m2), m0 < # kar, m < # ar, n < # kar, m0 ≤ n, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m, ∀ m0, m1. m0 < # kar ∧ ¬ n < m0 ∧ m1 < # kar ∧ ¬ n < m1 ∧ m0 ≠ m1 → kar[n, key-index(key, m)][m0].idx ≠ kar[n, key-index(key, m)][m1].idx ⊦ m2 < # ar ∧ ar[m, ar[m].idx:= n][m2].idx = m0

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 8:
kar[n, key-index(key, m)][m0] = key-index(key0, m2), n < # kar, m < # ar, m0 < # kar, m0 ≤ n, ∀ m0, m1. m0 < # kar ∧ ¬ n < m0 ∧ m1 < # kar ∧ ¬ n < m1 ∧ m0 ≠ m1 → kar[n, key-index(key, m)][m0].idx ≠ kar[n, key-index(key, m)][m1].idx, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m0 = n, m2 < # ar ∧ ar[m, ar[m].idx:= n][m2].idx = m0

Rule argument

fma: m0 = n

Additional information

Used simplifier rules: ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ m ≤ m; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < # ar → ar[m, ei][m] = ei; (s)% ⊦ (ei.idx:= m).idx = m; (s) Used formulas: m0 = n