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Proof step 13 from proof replay-orphans-dentry (previous step)

⊦ key = dentrykey(ino, str), dentrykey(ino, str) ∈ ro, (key ∈ (ri ⊖ ro)[dentrykey(ino, str), adr] ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro) ∧ (key ∈ (ri ⊖ ro)[dentrykey(ino, str), adr] → (ri ⊖ ro)[dentrykey(ino, str), adr][key] = (ri[dentrykey(ino, str), adr] ⊖ ro)[key])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
⊦ key = dentrykey(ino, str), dentrykey(ino, str) ∈ ro, (key ∈ ri ⊖ ro ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro) ∧ (key ∈ ri ⊖ ro → (ri ⊖ ro)[key] = (ri[dentrykey(ino, str), adr] ⊖ ro)[key])

Rule argument

Additional information

Used simplifier rules: ⊦ key ≠ key0 → ri[key0, adr][key] = ri[key]; (s)% ⊦ key ≠ key0 → (key ∈ ri[key0, adr] ↔ key ∈ ri); (s)