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Proof step 54 from proof replayorphans-prepost (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 55:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 56:
ri0[key0] = ri1[key0], ks0.inodes?, ro1.inodes?, key0 ∈ ri1, key0 ∈ ri0, key ∈ ro1, key ∈ ks0, key0 ∈ (ri0 \ datakeys(key, ri0)) -- key, ks0 ⊆ ro1, ∀ key. (key ∈ ri0 ⊖ ks0 ↔ key ∈ ri1 ⊖ ro1) ∧ (ri0 ⊖ ks0)[key] = (ri1 ⊖ ro1)[key] ⊦ # ro1 = 0, # ks0 = 0, ks0 = ∅, ro1 = ∅, key0.data?, key0 ∈ ro1, key0 ∈ ks0, key0 ∈ ks0 -- key, (key0 ∈ ri1 ⊖ ro1 → key0 ∈ ((ri0 \ datakeys(key, ri0)) -- key) ⊖ (ks0 -- key)) ∧ ((ri0 \ datakeys(key, ri0)) -- key)[key0] = ri1[key0]

Rule argument

spec: aubifs-base name: at-dels-ro-nodata seq: ⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key] subst: [key, ks, ri] → [key0, ks0 -- key, (ri0 \ datakeys(key, ri0)) -- key] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ∈ ri -- key0 ↔ key ≠ key0 ∧ key ∈ ri; (s)% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)%⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino; (s)% ⊦ key ∈ ks -- key0 ↔ key ≠ key0 ∧ key ∈ ks; (s) Used terms: key0, ks0 -- key, (ri0 \ datakeys(key, ri0)) -- key