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Proof step 14 from proof valid-idx-remlast (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
kar[ar[n].idx] = 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), ar[n, ar[n].idx:= 0][m1] = elemindex(a, m2), n < # ar, ar[n].idx < # kar, m4 < # kar, m4 < ar[n].idx, m4 + 1 < # kar, m3 < # ar, ∀ m. ¬ ar[n].idx < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m1 < # ar ∧ m2 = m4

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m +1 = m + 1; (s)% ⊦ key-index(key, m).idx = m; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic% ⊦ ¬ n < n; (s)