[Imprint][Data Privacy Statement][E-Mail]

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Proof step 3 from proof valid-idx-xhg (previous step)

kar[m1] = key-index(key0, n0), kar[m0] = key-index(key1, n1), m1 < n, m0 < n, m1 < # kar, m0 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ valid-idx(kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)], ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1], n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m)
Proof step 5:
kar[m0] = key-index(key1, n1), kar[m1] = key-index(key0, n0), m0 < # kar, m1 < # kar, m0 < n, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ n ≤ # kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)] ∧ (∀ m. m < n → kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx < # ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1] ∧ ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1][kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)][m].idx].idx = m)

Rule argument

spec: <toplevel> name: valid-idx seq: ⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m) subst: [n, ar, kar] → [n, ar[n0, ar[n0].idx:= m0] [n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1 ], kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)]] flags: <all paths>

Additional information

Used terms: n, ar[n0, ar[n0].idx:= m0][n1, ar[n0, ar0[n0].idx:= m0][n1].idx:= m1], kar[m0, key-index(key0, n0)][m1, key-index(key1, n1)]