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Proof step 5 from proof keyset-overwrite-add (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 6:
⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n]
Proof step 7:
m0 < # kar, n < # kar, m0 < # kar[n, ki], n ≤ m0, ∀ n, kar, ki. n < m0 ∧ n < # kar ∧ m0 ≤ # kar → keyset(kar[n, ki], m0) ++ kar[n] = keyset(kar, m0) ++ ki ⊦ keyset(kar[n, ki], m0) ++ kar[n, ki][m0] ++ kar[n] = keyset(kar, m0) ++ kar[m0] ++ ki

Rule argument

spec: <toplevel> name: keyset-rec seq: ⊦ n < # kar → keyset(kar, n + 1) = keyset(kar, n) ++ kar[n] subst: [n, kar] → [m0, kar[n, ki]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kar[n, ki] = # kar; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m0, kar[n, ki]