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Proof step 1 from proof key-set-extract-index (conclusion)

n ≤ # kar ⊦ ki ∈ keyset(kar, n) ↔ (∃ m. m < n ∧ ki = kar[m])

User applied rule structural induction

And obtained the following new goals

Proof step 2:
n0 +1 ≤ # kar, ∀ kar, ki. n0 ≤ # kar → (ki ∈ keyset(kar, n0) ↔ (∃ m. m < n0 ∧ ki = kar[m])) ⊦ ki ∈ keyset(kar, n0 +1) ↔ (∃ m. m < n0 +1 ∧ ki = kar[m])

Rule argument

indvar: n

Additional information

Used simplifier rules: keyset-base: ⊦ keyset(kar, 0) = ∅;%nat freely generated by 0, +1;;% ⊦ ¬ ki ∈ ∅; (s)% ⊦ ¬ m < 0; (s)