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Proof step 29 from proof ri-closed-gc-one (previous step)

ri[key0] = adr, key0 ∈ ri, key ∈ ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ key0 = key, adr ∈ fs, (ri[key0] = adr ∨ ri[key0] ∈ fs) ∧ node-closed(fs[adr, fs[ri[key]]][ri[key0]], ri[key, adr], fs[adr, fs[ri[key]]], ro)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 30:
key ∈ ri, key0 ∈ ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro) ⊦ key0 = key, node-closed(fs[ri[key]], ri[key, ri[key0]], fs[ri[key0], fs[ri[key]]], ro), ri[key0] ∈ fs

Rule argument

Additional information

Used simplifier rules: ⊦ fs[adr, nd][adr] = nd; (s)