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Proof step 143 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, key2 ∈ ri0, key2.data? → ¬ inodekey(key2.ino) ∈ ro, m3 ≠ 0 ∨ m0 = ROOT_INO ⊦ (ri0[inodekey(m0), adr1] ⊖ ro)[key2] = ((ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2)))[key2], adr1 = adr3, adr1 = adr2, adr2 = adr3, key2 = inodekey(m2), key2 = dentrykey(m0, str), 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, key2 ∈ ro

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 144:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 145:
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), key2 ∈ ri0, ri0[inodekey(m0)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, ri0[dentrykey(m0, str)] ∈ fs0, inodekey(m0) ∈ ri0, inodekey(m2) ∈ ri0, dentrykey(m0, str) ∈ ri0, key2.data? → ¬ inodekey(key2.ino) ∈ ro, m3 ≠ 0 ∨ m0 = ROOT_INO ⊦ (ri0[inodekey(m0), adr1] ⊖ ro)[key2] = ((ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ⊖ (ro ++ inodekey(m2)))[key2], adr1 = adr2, adr1 = adr3, adr2 = adr3, key2 = dentrykey(m0, str), key2 = inodekey(m2), m0 = 0, m2 = ROOT_INO, m2 = 0, key2 ∈ ro, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, inodekey(m0) ∈ ro, inodekey(m2) ∈ ro, adr3 ∈ log0, adr2 ∈ log0, adr1 ∈ log0, key2 ∈ (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3] ∧ ¬ key2 ∈ ro ++ inodekey(m2) ∧ ¬ key2.data?
Proof step 151:
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), key2 ∈ ri0, ri0[inodekey(m0)] ∈ fs0, ri0[inodekey(m2)] ∈ fs0, ri0[dentrykey(m0, str)] ∈ fs0, inodekey(m0) ∈ ri0, inodekey(m2) ∈ ri0, dentrykey(m0, str) ∈ ri0, key2 ∈ (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3], key2.data? → ¬ inodekey(key2.ino) ∈ ro, m3 ≠ 0 ∨ m0 = ROOT_INO ⊦ (ri0[inodekey(m0), adr1] ⊖ ro)[key2] = (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3][key2], adr1 = adr2, adr1 = adr3, adr2 = adr3, key2 = dentrykey(m0, str), key2 = inodekey(m2), m0 = 0, m2 = ROOT_INO, m2 = 0, key2.data?, key2 ∈ ro, adr3 ∈ fs0, adr2 ∈ fs0, adr1 ∈ fs0, inodekey(m0) ∈ ro, inodekey(m2) ∈ ro, adr3 ∈ log0, adr2 ∈ log0, adr1 ∈ log0, key2 ∈ ro ++ inodekey(m2)

Rule argument

spec: aubifs-base name: at-dels-ro-nodata seq: ⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key] subst: [key, ri, ks] → [key2, (ri0[inodekey(m0), adr1] -- dentrykey(m0, str) )[inodekey(m2), adr3 ], ro ++ inodekey(m2)] flags: <all paths>

Additional information

Used terms: key2, (ri0[inodekey(m0), adr1] -- dentrykey(m0, str))[inodekey(m2), adr3], ro ++ inodekey(m2)