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Proof step 8 from proof ri-cons-copy-new (previous step)

# kx = # ax, new(ax, fs), key ∈ kx, key ∈ ri, kx.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(kx), dups(ax), (ri[kx, ax][key] ∈ fs ∨ ri[kx, ax][key] ∈ ax) ∧ key-cons(key, fs[ax, fs[ri[kx]]][ri[kx, ax][key]])

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ key ∈ kx ↔ (∃ kx0, kx1. kx = kx0 + (key + kx1))
Proof step 10:
# kx = # ax, new(ax, fs), key ∈ ri, kx.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∃ kx0, kx1. kx = kx0 + (key + kx1) ⊦ dups(ax), dups(kx), (ri[kx, ax][key] ∈ fs ∨ ri[kx, ax][key] ∈ ax) ∧ key-cons(key, fs[ax, fs[ri[kx]]][ri[kx, ax][key]])

Rule argument

spec: list, inst: key-list-set name: in-def seq: ⊦ key ∈ kx ↔ (∃ kx0, kx1. kx = kx0 + (key + kx1)) subst: [kx, key] → [kx, key] flags: <all paths>

Additional information

Used terms: kx, key