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Proof step 28 from proof replayorphans-prepost (previous step)

key0 ∈ ri0 ⊖ ks0 ↔ key0 ∈ ri1 ⊖ ro1, (ri0 ⊖ ks0)[key0] = (ri1 ⊖ ro1)[key0], ro1.inodes?, ks0.inodes?, key ∈ ks0, key ∈ ro1, ks0 ⊆ ro1, ∀ key. (key ∈ ri0 ⊖ ks0 ↔ key ∈ ri1 ⊖ ro1) ∧ (ri0 ⊖ ks0)[key] = (ri1 ⊖ ro1)[key] ⊦ # ks0 = 0, # ro1 = 0, ks0 = ∅, ro1 = ∅, (key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key) ↔ key0 ∈ ri1 ⊖ ro1) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = (ri1 ⊖ ro1)[key0], key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ ri ⊖ ks = (ri \ ks) \ datakeys(ks, ri)
Proof step 30:
key0 ∈ (ri0 \ ks0) \ datakeys(ks0, ri0) ↔ key0 ∈ ri1 ⊖ ro1, ((ri0 \ ks0) \ datakeys(ks0, ri0))[key0] = (ri1 ⊖ ro1)[key0], ks0.inodes?, ro1.inodes?, key ∈ ro1, key ∈ ks0, ks0 ⊆ ro1, ∀ key. (key ∈ (ri0 \ ks0) \ datakeys(ks0, ri0) ↔ key ∈ ri1 ⊖ ro1) ∧ ((ri0 \ ks0) \ datakeys(ks0, ri0))[key] = (ri1 ⊖ ro1)[key] ⊦ # ro1 = 0, # ks0 = 0, ks0 = ∅, ro1 = ∅, key0 ∈ ri0 ∧ ¬ key0 ∈ ks0 ∧ ¬ key0.data?, (key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key) ↔ key0 ∈ ri1 ⊖ ro1) ∧ (((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key))[key0] = (ri1 ⊖ ro1)[key0]

Rule argument

spec: aubifs-base name: dels-with-data seq: ⊦ ri ⊖ ks = (ri \ ks) \ datakeys(ks, ri) subst: [ks, ri] → [ks0, ri0] flags: <all paths>

Additional information

Used terms: ks0, ri0