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

ar[m0] = elemindex(a, m1), m < # kar, key-index(key, m0) ∈ keyset(kar, m), m ≤ m1, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ # keyset(kar, m) = 0, keyset(kar, m) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 5:
ar[m0] = elemindex(a, m1), m < # kar, m ≤ m1, m ≤ # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2, ∃ m1. m1 < m ∧ key-index(key, m0) = kar[m1] ⊦ # keyset(kar, m) = 0, keyset(kar, m) = ∅

Rule argument

spec: <toplevel> name: key-set-extract-index seq: n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m]) subst: [n, kar, ki] → [m, kar, key-index(key, m0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: m, kar, key-index(key, m0)