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Proof step 13 from proof key-set-write-nodups (previous step)

n < # kar, kar[n] ∈ keyset(kar, n), ∀ m, m1. m < # kar ∧ ¬ n < m ∧ m1 < # kar ∧ ¬ n < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx, ∀ n0, kar, ki. n0 < n ∧ n0 < # kar ∧ n ≤ # kar ∧ (∀ m2, m1. m2 < # kar ∧ m2 < n ∧ m1 < # kar ∧ m1 < n ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx) → keyset(kar[n0, ki], n) = (keyset(kar, n) -- kar[n0]) ++ ki ⊦ # keyset(kar, n) = 0, keyset(kar, n) ++ ki = (keyset(kar, n) -- kar[n]) ++ ki, keyset(kar, n) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 15:
n < # kar, n ≤ # kar, ∀ n0, kar, ki. n0 < n ∧ n0 < # kar ∧ n ≤ # kar ∧ (∀ m2, m1. m2 < # kar ∧ m2 < n ∧ m1 < # kar ∧ m1 < n ∧ m2 ≠ m1 → kar[m2].idx ≠ kar[m1].idx) → keyset(kar[n0, ki], n) = (keyset(kar, n) -- kar[n0]) ++ ki, ∀ m, m1. m < # kar ∧ ¬ n < m ∧ m1 < # kar ∧ ¬ n < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx, ∃ m. m < n ∧ kar[n] = kar[m] ⊦ # keyset(kar, n) = 0, keyset(kar, n) ++ ki = (keyset(kar, n) -- kar[n]) ++ ki, keyset(kar, n) = ∅

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] → [n, kar, kar[n]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: n, kar, kar[n]