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Proof step 10 from proof ri-inj-index-update (previous step)

# ax = # kx0 + 1, ri[kx0 + key1 ', ax][key] = ri[kx0 + key1 ', ax][key0], ax.toSet ∩ range ri = ∅, ri-inj(ri), key0 ∈ ri[kx0 + key1 ', ax], key ∈ ri[kx0 + key1 ', ax], ∀ kx1, ax, ri, key, key0. # kx1 < #(kx0 + key1 ') → # ax = # kx1 ∧ ri[kx1, ax][key] = ri[kx1, ax][key0] ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ key ∈ ri[kx1, ax] ∧ key0 ∈ ri[kx1, ax] → key = key0 ∨ dups(ax) ∨ key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0] ⊦ ax = [], key = key0, dups(ax), key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 11:
⊦ ax = [] ↔ ¬(∃ ax0, adr. ax = ax0 + adr ')
Proof step 12:
# ax = # kx0 + 1, ri[kx0 + key1 ', ax][key] = ri[kx0 + key1 ', ax][key0], ax.toSet ∩ range ri = ∅, ri-inj(ri), key ∈ ri[kx0 + key1 ', ax], key0 ∈ ri[kx0 + key1 ', ax], ∀ kx1, ax, ri, key, key0. # kx1 < #(kx0 + key1 ') → # ax = # kx1 ∧ ri[kx1, ax][key] = ri[kx1, ax][key0] ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ key ∈ ri[kx1, ax] ∧ key0 ∈ ri[kx1, ax] → key = key0 ∨ dups(ax) ∨ key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0] ⊦ key = key0, dups(ax), ∀ ax0, adr. ax ≠ ax0 + adr ', key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0]

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