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Proof step 9 from proof lookup-rem-inj-range-inter-empty (previous step)

ri[key0] = ri[key], ri-inj(ri), key ∈ ri, key ∈ ks, key0 ∈ ri ⊦ #(ri[ks] ∩ range(ri \ ks)) = 0, # ks = 0, ks = ∅, key0 ∈ ks

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ri-inj(ri) ↔ (∀ key0, key1. key0 ∈ ri ∧ key1 ∈ ri ∧ ri[key0] = ri[key1] → key0 = key1)
Proof step 11:
ri[key0] = ri[key], key0 ∈ ri, key ∈ ks, key ∈ ri, ∀ key0, key1. key0 ∈ ri ∧ key1 ∈ ri ∧ ri[key0] = ri[key1] → key0 = key1 ⊦ # ks = 0, #(ri[ks] ∩ range(ri \ ks)) = 0, ks = ∅, key0 ∈ ks

Rule argument

spec: aubifs-inv name: ri-inj seq: ⊦ ri-inj(ri) ↔ (∀ key0, key1. key0 ∈ ri ∧ key1 ∈ ri ∧ ri[key0] = ri[key1] → key0 = key1) subst: [ri] → [ri] flags: <all paths>

Additional information

Used terms: ri