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Proof step 28 from proof key-set-write-nodups (previous step)

kar[m1] = key-index(key0, m3), kar[m] = key-index(key, m2), keyset(kar[n, ki], m0) = (keyset(kar, m0) -- kar[n]) ++ ki, n + 1 < # kar, n < # kar, m0 < # kar, n < m0, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx ⊦ (keyset(kar, m0) -- kar[n]) ++ kar[m0] ++ ki = ((keyset(kar, m0) ++ kar[m0]) -- kar[n]) ++ ki, m0 = n

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ s1 = s0 ++ ki ↔ (∀ ki0. ki0 ∈ s1 ↔ ki0 = ki ∨ ki0 ∈ s0)
Proof step 30:
kar[m] = key-index(key, m2), kar[m1] = key-index(key0, m3), keyset(kar[n, ki], m0) = (keyset(kar, m0) -- kar[n]) ++ ki, n < m0, m0 < # kar, n < # kar, n + 1 < # kar, ∀ m, m1. m < # kar ∧ ¬ m0 < m ∧ m1 < # kar ∧ ¬ m0 < m1 ∧ m ≠ m1 → kar[m].idx ≠ kar[m1].idx ⊦ m0 = n, ∀ ki0. ki0 ∈ (keyset(kar, m0) -- kar[n]) ++ kar[m0] ++ ki ↔ ki0 = ki ∨ ki0 ∈ (keyset(kar, m0) ++ kar[m0]) -- kar[n]

Rule argument

spec: set-basic, inst: keyindex-set name: ext-rec seq: ⊦ s1 = s0 ++ ki ↔ (∀ ki0. ki0 ∈ s1 ↔ ki0 = ki ∨ ki0 ∈ s0) subst: [ki, s0, s1] → [ki, (keyset(kar, m0) ++ kar[m0]) -- kar[n], (keyset(kar, m0) -- kar[n]) ++ kar[m0] ++ ki] flags: <all paths>

Additional information

Used terms: ki, (keyset(kar, m0) ++ kar[m0]) -- kar[n], (keyset(kar, m0) -- kar[n]) ++ kar[m0] ++ ki