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Proof step 3 from proof del-commute-dels (previous step)

⊦ (ri ⊖ key) ⊖ ks = (ri ⊖ ks) ⊖ key, key ∈ ks

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:
⊦ key ∈ ks, ∀ key0. (key0 ∈ (ri ⊖ key) ⊖ ks ↔ key0 ∈ (ri ⊖ ks) ⊖ key) ∧ (key0 ∈ (ri ⊖ key) ⊖ ks → ((ri ⊖ key) ⊖ ks)[key0] = ((ri ⊖ ks) ⊖ key)[key0])

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 ⊖ ks) ⊖ key, (ri ⊖ key) ⊖ ks] flags: <all paths>

Additional information

Used terms: (ri ⊖ ks) ⊖ key, (ri ⊖ key) ⊖ ks