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Proof step 4 from proof key-set-write-other (previous step)

n < # kar, m0 +1 < # kar, m0 +1 ≤ n, ∀ kar, n, ki. m0 < # kar ∧ n < # kar ∧ m0 ≤ n → keyset(kar[n, ki], m0) = keyset(kar, m0) ⊦ keyset(kar[n, ki], m0 +1) = keyset(kar, m0 +1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
m0 < # kar, m0 < n, n < # kar, m0 + 1 < # kar, ∀ kar, n, ki. m0 < # kar ∧ n < # kar ∧ m0 ≤ n → keyset(kar[n, ki], m0) = keyset(kar, m0) ⊦ keyset(kar[n, ki], m0 + 1) = keyset(kar, m0) ++ kar[m0]

Rule argument

Additional information

Used simplifier rules: keyset-rec: ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n];% ⊦ m < n → n ≠ 0; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)% ⊦ m +1 = m + 1; (s)