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Proof step 2 from proof ri-closed-variant1-01 (previous step)

ri-closed(ri, fs, ro) ⊦ ri-closed(ri0, fs0, ro0) ↔ (∀ key. (key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro)) → key ∈ ri0 → ri0[key] ∈ fs0 ∧ node-closed(fs0[ri0[key]], ri0, fs0, ro0))

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro) ⊦ (∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ node-closed(fs0[ri0[key]], ri0, fs0, ro0)) ↔ (∀ key. (key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro)) → key ∈ ri0 → ri0[key] ∈ fs0 ∧ node-closed(fs0[ri0[key]], ri0, fs0, ro0))

Rule argument

Additional information

Used simplifier rules: ri-closed: ⊦ ri-closed(ri, fs, ro) ↔ (∀ key. key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro));