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Proof step 20 from proof index-update-range (previous step)

# kx0 = # ax0, range ri ∩ ax0.toSet = ∅, 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 \ ri[kx0].toSet) -- ri[key]) ∪ (ax0.toSet -- ri[key])) ++ adr = (ax0.toSet ∪ ((range ri \ ri[kx0].toSet) -- ri[key])) ++ adr, dups(kx0), dups(ax0), adr ∈ ri, key ∈ kx0, adr ∈ ax0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: set-union, inst: node-store name: del-in seq: ⊦ ¬ adr ∈ as → as -- adr = as subst: [adr, as] → [ri[key], ax0.toSet] flags: <all paths>

Additional information

Used simplifier rules: unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]% ⊦ adr ∈ as → as ∪ (as0 -- adr) = as ∪ as0; (s)%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ key ∈ ri ∧ as ∩ range ri = ∅ → ¬ ri[key] ∈ as; (s) Used terms: ri[key], ax0.toSet