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Proof step 14 from proof replay-orphans-dentry (previous step)

⊦ 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])

User applied rule cut formula

And obtained the following new goals

Proof step 15:
key ∈ ri ⊖ ro ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro ⊦ 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])
Proof step 30:
⊦ key ∈ ri ⊖ ro ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro, 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

fma: key ∈ ri ⊖ ro ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro

Additional information

Used formulas: key ∈ ri ⊖ ro ↔ key ∈ ri[dentrykey(ino, str), adr] ⊖ ro