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Proof step 10 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) ⊦ ax = [], range ri[kx0 + key ', ax] = ((range ri \ ri[kx0].toSet) -- ri[key]) ∪ ax.toSet, dups(kx0), dups(ax), key ∈ kx0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ ax = [] ↔ ¬(∃ ax0, adr. ax = ax0 + adr ')
Proof step 12:
# 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 '

Rule argument

spec: list, inst: address-list-set name: lastdiv-rew seq: ⊦ ax = [] ↔ ¬(∃ ax0, adr. ax = ax0 + adr ') subst: [ax] → [ax] flags: <all paths>

Additional information

Used terms: ax