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Proof step 19 from proof valid-idx-move-last (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 20:
ar[m1].idx:= m0 = elemindex(a, m4), kar[ar[m1].idx] = key-index(key, m1), kar[m2] = key-index(key0, m1), m2 < # kar, m1 < # ar, ar[m1].idx < # kar, m2 < ar[m1].idx, m2 + 1 < # kar, ∀ m. ¬ ar[m1].idx < m → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 = m0, m4 = m2

Rule argument

Additional information

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