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Proof step 26 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, key ∈ ri ∧ ¬ key ∈ ro ∧ ¬ key.data?, ( (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, ∅)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 27:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 28:
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 ∈ 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, ∅), key ∈ ri ∧ ¬ key ∈ ro ∧ ¬ key.data?

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] → [key, ro, ri] flags: <all paths>

Additional information

Used terms: key, ro, ri