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Proof step 3 from proof del-le (previous step)

⊦ (ri -- key0) |≤ key1 = (ri |≤ key1) -- key0, key1 < key0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key]))
Proof step 5:
⊦ key1 < key0, ∀ key. (key ∈ (ri -- key0) |≤ key1 ↔ key ∈ (ri |≤ key1) -- key0) ∧ (key ∈ (ri -- key0) |≤ key1 → ((ri -- key0) |≤ key1)[key] = ((ri |≤ key1) -- key0)[key])

Rule argument

spec: store, inst: node-index name: k-Extension seq: ⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key])) subst: [ri0, ri] → [(ri |≤ key1) -- key0, (ri -- key0) |≤ key1] flags: <all paths>

Additional information

Used terms: (ri |≤ key1) -- key0, (ri -- key0) |≤ key1