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Proof step 20 from proof inv-replay (previous step)

replay-log(log0, fs0, fi0 ⊖ fo0) = ri ⊖ ro, fo0.inodes?, ro.inodes?, file-node-cons(fs0, ri), log-cons(log0, fs0), orphans-cons(fs0, ri, ro), replay-safe(log0, fs0, fi0 ⊖ fo0), ri-closed(ri, fs0, ro), ri-cons(ri, fs0), ri-inj(ri), ri-inj(fi0), ro ⊆ dom ri, fo0 ⊆ dom fi0 ⊦ dups(log0), inodekey(0) ∈ ri, inodekey(0) ∈ ri ⊖ ro, ri-closed(ri ⊖ ro, fs0, ∅) ∧ file-node-cons(fs0, ri ⊖ ro) ∧ orphans-cons(fs0, ri ⊖ ro, ∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 21:
⊦ ri-closed(ri, fs, ro) → ( ri-closed(ri0, fs0, ro0) ↔ (∀ key. (key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro)) → key ∈ ri0 → ri0[key] ∈ fs0 ∧ node-closed(fs0[ri0[key]], ri0, fs0, ro0)))
Proof step 22:
replay-log(log0, fs0, fi0 ⊖ fo0) = ri ⊖ ro, ro.inodes?, fo0.inodes?, file-node-cons(fs0, ri), log-cons(log0, fs0), orphans-cons(fs0, ri, ro), replay-safe(log0, fs0, fi0 ⊖ fo0), ri-closed(ri, fs0, ro), ri-cons(ri, fs0), ri-inj(fi0), ri-inj(ri), fo0 ⊆ dom fi0, ro ⊆ dom ri ⊦ dups(log0), inodekey(0) ∈ ri ⊖ ro, inodekey(0) ∈ ri, (∀ key. (key ∈ ri → ri[key] ∈ fs0 ∧ node-closed(fs0[ri[key]], ri, fs0, ro)) → key ∈ ri ⊖ ro → (ri ⊖ ro)[key] ∈ fs0 ∧ node-closed(fs0[(ri ⊖ ro)[key]], ri ⊖ ro, fs0, ∅)) ∧ file-node-cons(fs0, ri ⊖ ro) ∧ orphans-cons(fs0, ri ⊖ ro, ∅)

Rule argument

spec: aubifs-inv name: ri-closed-variant1-01 seq: ⊦ ri-closed(ri, fs, ro) → ( ri-closed(ri0, fs0, ro0) ↔ (∀ key. ( key ∈ ri → ri[key] ∈ fs ∧ node-closed(fs[ri[key]], ri, fs, ro)) → key ∈ ri0 → ri0[key] ∈ fs0 ∧ node-closed(fs0[ri0[key]], ri0, fs0, ro0))) subst: [ri, fs, ro, ro0, fs0, ri0] → [ri, fs0, ro, ∅, fs0, ri ⊖ ro] flags: <all paths>

Additional information

Used terms: ri, fs0, ro, ∅, fs0, ri ⊖ ro