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Proof step 18 from proof valid-idx-remlast (previous step)

ar[n, elemindex(a1, 0)][m1] = elemindex(a, m2), ar[n] = elemindex(a1, m0), kar[m0] = key-index(key, n), kar[0, key-index(key, n)][m4] = key-index(key0, m1), ar[m3] = elemindex(a0, m4), kar[m4] = key-index(key1, m3), n < # ar, m0 < # kar, m4 < # kar, m4 < m0, m4 + 1 < # kar, m3 < # ar, ∀ m. ¬ m0 < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m4 = 0, m1 < # ar ∧ m2 = m4

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 19:
ar[m3] = elemindex(a0, m4), kar[m4] = key-index(key1, m3), kar[m0] = key-index(key, n), ar[n] = elemindex(a1, m0), ar[n, elemindex(a1, 0)][m3] = elemindex(a, m2), m3 < # ar, m4 + 1 < # kar, m4 < m0, m4 < # kar, m0 < # kar, n < # ar, ∀ m. ¬ m0 < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 = m4, m4 = 0

Rule argument

Additional information

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