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Proof step 19 from proof ri-inj-index-update (previous step)

# ax0 = # kx0, range ri ∩ ax0.toSet = ∅, ri-inj(ri), key0 ∈ ri[kx0, ax0], key0 ∈ kx0, ∀ 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] ⊦ key1 = key0, dups(ax0), ri[kx0, ax0][key0] ∈ ri, ri[kx0, ax0][key0] ∈ ax0, key1 ∈ ri ∧ key0 ∈ ri ∧ ri[key1] = ri[key0]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ key ∈ kx ↔ (∃ kx0, kx1. kx = kx0 + key ' + kx1 ∧ ¬ key ∈ kx1)
Proof step 21:
# ax0 = # kx0, range ri ∩ ax0.toSet = ∅, ri-inj(ri), key0 ∈ ri[kx0, ax0], ∃ kx2, kx1. kx0 = kx2 + key0 ' + kx1 ∧ ¬ key0 ∈ kx1, ∀ 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] ⊦ key1 = key0, dups(ax0), ri[kx0, ax0][key0] ∈ ax0, ri[kx0, ax0][key0] ∈ ri, key1 ∈ ri ∧ key0 ∈ ri ∧ ri[key1] = ri[key0]

Rule argument

spec: list, inst: key-list-set name: lastoc-rew seq: ⊦ key ∈ kx ↔ (∃ kx0, kx1. kx = kx0 + key ' + kx1 ∧ ¬ key ∈ kx1) subst: [kx, key] → [kx0, key0] flags: <all paths>

Additional information

Used terms: kx0, key0