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Proof step 17 from proof valid-idx-xhg (previous step)

ar[n1] = elemindex(a1, m0), ar[m4] = elemindex(a0, m5), kar[m5] = key-index(key2, m4), ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][m2] = elemindex(a, m3), kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m5] = key-index(key, m2), m1 < n, m5 < # kar, n1 < # ar, m4 < # ar, m5 < n, m1 < # kar, m0 < n, m0 < # kar, n ≤ # kar, m1 < n → kar[m1].idx < # ar ∧ ar[kar[m1].idx].idx = m1, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m5

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 18:
ar[n1] = elemindex(a1, m0), ar[m4] = elemindex(a0, m5), kar[m5] = key-index(key2, m4), ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= ar[n0].idx][m2] = elemindex(a, m3), kar[m0] = key-index(key1, n1), kar[ar[n0].idx] = key-index(key0, n0), kar[m0, key-index(key0, n0)][ar[n0].idx, key-index(key1, n1)][m5] = key-index(key, m2), m0 < n, m0 < # kar, n0 < # ar, ar[n0].idx < n, m5 < # kar, n1 < # ar, m4 < # ar, m5 < n, ar[n0].idx < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m5

Rule argument

Additional information

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