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Proof step 3 from proof ri-cons-copy-new (previous step)

# ax = # kx, new(ax, fs), key ∈ ri[kx, ax], 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 4:
⊦ # kx = # ax → (key ∈ ri[kx, ax] ↔ key ∈ ri ∨ key ∈ kx)
Proof step 5:
# kx = # ax, new(ax, fs), kx.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), key ∈ ri ∨ key ∈ kx ⊦ 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: aubifs-base name: index-update-in seq: ⊦ # kx = # ax → (key ∈ ri[kx, ax] ↔ key ∈ ri ∨ key ∈ kx) subst: [ax, kx, ri, key] → [ax, kx, ri, key] flags: <all paths>

Additional information

Used terms: ax, kx, ri, key