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Proof step 12 from proof index-update-range (previous step)

# kx0 + 1 = # ax, ax.toSet ∩ range ri = ∅, ri-inj(ri), key ∈ ri, kx0.toSet ⊆ dom ri, ∀ kx1, ax, ri. # kx1 < #(kx0 + key ') → # kx1 = # ax ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ kx1.toSet ⊆ dom ri → range ri[kx1, ax] = (range ri \ ri[kx1].toSet) ∪ ax.toSet ∨ dups(kx1) ∨ dups(ax) ⊦ range ri[kx0 + key ', ax] = ((range ri \ ri[kx0].toSet) -- ri[key]) ∪ ax.toSet, dups(ax), dups(kx0), key ∈ kx0, ∀ ax0, adr. ax ≠ ax0 + adr '

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ dups(ax + adr ') ↔ adr ∈ ax ∨ dups(ax); (ws)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%# kx = # ax ⊦ ri[kx + key ', ax + adr '] = ri[kx, ax][key, adr]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ # adr ' = 0 +1; (s)% ⊦ #(ax + ax0) = # ax + # ax0; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ adr ∈ range ri ↔ adr ∈ ri; (s)% ⊦ as ∩ (as0 ++ adr) = ∅ ↔ as ∩ as0 = ∅ ∧ ¬ adr ∈ as; (ws)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]% ⊦ as ∪ ∅ = as; (s)% ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ adr '.toSet = ∅ ++ adr; (s)% ⊦ (ax + ax0).toSet = ax.toSet ∪ ax0.toSet; (s)