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Proof step 14 from proof no-dups-keyset-size (previous step)

# keyset(kar, m0) = m0, m0 < # kar, kar[m0] ∈ keyset(kar, m0), ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1], ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1] ⊦ keyset(kar, m0) = ∅, m0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])
Proof step 16:
# keyset(kar, m0) = m0, m0 < # kar, m0 ≤ # kar, ∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1], ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1], ∃ m. m < m0 ∧ kar[m0] = kar[m] ⊦ keyset(kar, m0) = ∅, m0 = 0

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] → [m0, kar, kar[m0]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s) Used terms: m0, kar, kar[m0]