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Proof step 6 from proof ri-inj-index-update (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 7:
⊦ kx = [] ↔ ¬(∃ kx0, key. kx = kx0 + key ')
Proof step 8:
# ax = # kx, ri[kx, ax][key] = ri[kx, ax][key0], ax.toSet ∩ range ri = ∅, ri-inj(ri), key0 ∈ ri[kx, ax], key ∈ ri[kx, ax], ∀ kx0, ax, ri, key, key0. # kx0 < # kx → # ax = # kx0 ∧ ri[kx0, ax][key] = ri[kx0, ax][key0] ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ key ∈ ri[kx0, ax] ∧ key0 ∈ ri[kx0, ax] → key = key0 ∨ dups(ax) ∨ key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0] ⊦ key = key0, dups(ax), ∀ kx0, key. kx ≠ kx0 + key ', key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0]

Rule argument

spec: list, inst: key-list-set name: lastdiv-rew seq: ⊦ kx = [] ↔ ¬(∃ kx0, key. kx = kx0 + key ') subst: [kx] → [kx] flags: <all paths>

Additional information

Used terms: kx