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Proof step 131 from proof inv-rmdir (previous step)

fs0[ri0[inodekey(m0)]] = inodenode(key0, md1, true, m3, m + 1, m1 + 1), fs0[ri0[dentrykey(m0, str)]] = dentrynode(key, m2), fs0[ri0[inodekey(m2)]] = inodenode(key1, md2, true, 1, m5, m6), dentrykeys(inodekey(m2), ri0) = ∅, links(m2, fs0, ri0) = ∅ ++ dentrykey(m0, str), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), orphans-cons(fs0, ri0, ro), replay-cons-strong(fs0, ri0, ro, fi, fo, log0), replay-safe(log0, fs0, fi ⊖ fo), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ubifs-cons(fs0, ri0, ro, fi, fo, log0), dentrykey(m0, str) ∈ ri0, inodekey(m2) ∈ ri0, inodekey(m0) ∈ ri0, ri0[dentrykey(m0, str)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, ri0[inodekey(m0)] ∈ fs0, m3 ≠ 0 ∨ m0 = ROOT_INO ⊦ (ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2) = (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2)), adr1 = adr3, adr1 = adr2, adr2 = adr3, m0 = 0, m2 = 0, m2 = ROOT_INO, adr1 ∈ log0, adr2 ∈ log0, adr3 ∈ log0, inodekey(m2) ∈ ro, inodekey(m0) ∈ ro, adr1 ∈ fs0, adr2 ∈ fs0, adr3 ∈ fs0

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 132:
⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key]))
Proof step 133:
fs0[ri0[inodekey(m2)]] = inodenode(key1, md2, true, 1, m5, m6), fs0[ri0[dentrykey(m0, str)]] = dentrynode(key, m2), fs0[ri0[inodekey(m0)]] = inodenode(key0, md1, true, m3, m + 1, m1 + 1), dentrykeys(inodekey(m2), ri0) = ∅, links(m2, fs0, ri0) = ∅ ++ dentrykey(m0, str), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, fo.inodes?, ro.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0), orphans-cons(fs0, ri0, ro), replay-cons-strong(fs0, ri0, ro, fi, fo, log0), replay-safe(log0, fs0, fi ⊖ fo), ri-cons(ri0, fs0), ri-inj(fi), ri-inj(ri0), ubifs-cons(fs0, ri0, ro, fi, fo, log0), ri0[inodekey(m0)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, ri0[dentrykey(m0, str)] ∈ fs0, inodekey(m0) ∈ ri0, inodekey(m2) ∈ ri0, dentrykey(m0, str) ∈ ri0, m3 ≠ 0 ∨ m0 = ROOT_INO ⊦ adr1 = adr2, adr1 = adr3, adr2 = adr3, m0 = 0, m2 = ROOT_INO, m2 = 0, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, inodekey(m0) ∈ ro, inodekey(m2) ∈ ro, adr3 ∈ log0, adr2 ∈ log0, adr1 ∈ log0, ∀ key. ( key ∈ (ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2) ↔ key ∈ (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2))) ∧ ( key ∈ (ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2) → ((ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2))[key] = ((ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2)))[key])

Rule argument

spec: store, inst: node-index name: k-Extension seq: ⊦ ri = ri0 ↔ (∀ key. (key ∈ ri ↔ key ∈ ri0) ∧ (key ∈ ri → ri[key] = ri0[key])) subst: [ri, ri0] → [ (ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2), (ri0[inodekey(m0), adr1] -- dentrykey(m0, str) )[inodekey(m2), adr3 ] ⊖ (ro ++ inodekey(m2))] flags: <all paths>

Additional information

Used terms: (ri0[inodekey(m0), adr1] ⊖ ro) -- dentrykey(m0, str) -- inodekey(m2), (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2))