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Proof step 31 from proof valid-idx-xhg (previous step)

ar[n0, elemindex(a2, m0)][m2] = elemindex(a, m3), kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), kar[m0, key-index(key0, n0)][m5] = key-index(key, m2), ar[n0] = elemindex(a2, m1), ar[n1] = elemindex(a1, m0), ar[m4] = elemindex(a0, m5), kar[m5] = key-index(key2, m4), m0 < n, m0 < # kar, n0 < # ar, m1 < n, m5 < # kar, n1 < # ar, m4 < # ar, m5 < n, m1 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 = n1, m5 = m1, m5 = m0, m2 < # ar ∧ m3 = m5

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 32:
ar[m4] = elemindex(a0, m5), ar[n1] = elemindex(a1, m0), ar[n0] = elemindex(a2, m1), kar[m5] = key-index(key2, m4), kar[m1] = key-index(key0, n0), kar[m0] = key-index(key1, n1), ar[n0, elemindex(a2, m0)][m4] = elemindex(a, m3), m1 < # kar, m5 < n, m4 < # ar, n1 < # ar, m5 < # kar, m1 < n, n0 < # ar, m0 < # kar, m0 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m3 = m5, m4 = n1, m5 = m0, m5 = m1

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)