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Proof step 4 from proof ri-closed-gc (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
# kx0 + 1 = # ax, new(ax, fs), ri-closed(ri, fs, ro), ri-cons(ri, fs), key ∈ ri, kx0.toSet ⊆ dom ri, ∀ ax0, fs0, ri0, ro0. # kx0 = # ax0 ∧ new(ax0, fs0) ∧ ri-closed(ri0, fs0, ro0) ∧ ri-cons(ri0, fs0) ∧ kx0.toSet ⊆ dom ri0 → dups(ax0) ∨ ri-closed(ri0[kx0, ax0], fs0[ax0, fs0[ri0[kx0]]], ro0) ⊦ dups(ax), dups(kx0), ri-closed(ri[key ' + kx0, ax], fs[ax, fs[ri[key]] ' + fs[ri[kx0]]], ro), key ∈ kx0

Rule argument

Additional information

Used simplifier rules: ⊦ dups(key ' + kx) ↔ dups(kx) ∨ key ∈ kx; (ws)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ nd + ndx = nd ' + ndx; (s)% ⊦ fs[adr ' + ax] = fs[adr] + fs[ax]; (s)% ⊦ adr + ax = adr ' + ax; (s)% ⊦ ri[key ' + kx] = ri[key] + ri[kx]; (s)% ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ ks ++ key ⊆ ks0 ↔ ks ⊆ ks0 ∧ key ∈ ks0; (ws)%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]% ⊦ ks ∪ ∅ = ks; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(key ' + kx) = (# kx)+1; (s)% ⊦ key + kx = key ' + kx; (s)