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Proof step 6 from proof no-dups-keyset-size (previous step)

m0 < # kar, m0 ≤ # kar → ((∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1]) ↔ # keyset(kar, m0) = m0) ⊦ (∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1]) ↔ #(keyset(kar, m0) ++ kar[m0]) = m0 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
(∀ m2, m1. m2 < # kar ∧ m2 < m0 ∧ m1 < # kar ∧ m1 < m0 ∧ m2 ≠ m1 → kar[m2] ≠ kar[m1]) ↔ # keyset(kar, m0) = m0, m0 < # kar ⊦ (∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m] ≠ kar[m1]) ↔ #(keyset(kar, m0) ++ kar[m0]) = m0 + 1

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → m ≤ n; (s)