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Proof step 8 from proof ri-closed-gc (previous step)

# kx0 = # ax1, new(ax1, fs), ri-closed(ri, fs, ro), ri-cons(ri, fs), key ∈ ri, kx0.toSet ⊆ dom ri, # ax1 = # ax1 ∧ new(ax1, fs[adr, fs[ri[key]]]) ∧ ri-closed(ri[key, adr], fs[adr, fs[ri[key]]], ro) ∧ ri-cons(ri[key, adr], fs[adr, fs[ri[key]]]) ∧ kx0.toSet ⊆ dom ri[key, adr] → dups(ax1) ∨ ri-closed(ri[key, adr][kx0, ax1], fs[adr, fs[ri[key]]][ax1, fs[adr, fs[ri[key]]][ri[key, adr][kx0]]], ro) ⊦ dups(ax1), dups(kx0), ri-closed(ri[key, adr][kx0, ax1], fs[adr, fs[ri[key]]][ax1, fs[ri[kx0]]], ro), adr ∈ ax1, key ∈ kx0, adr ∈ fs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ new(ax, fs) → (new(ax, fs[adr, nd]) ↔ ¬ adr ∈ ax); (ss)% ⊦ ¬ key ∈ kx → ri[key, adr][kx] = ri[kx]; (s)% ⊦ key ∈ ri → dom ri ++ key = dom ri; (s)% ⊦ dom ri[key, adr] = dom ri ++ key; (s)%⊦ key ∈ ri ∧ ¬ adr ∈ fs ∧ ri-cons(ri, fs) → ri-cons(ri[key, adr], fs[adr, fs[ri[key]]]); (s)%⊦ key ∈ ri ∧ ¬ adr ∈ fs ∧ ri-closed(ri, fs, ro) ∧ ri-cons(ri, fs) → ri-closed(ri[key, adr], fs[adr, fs[ri[key]]], ro); (s)