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Proof step 648 from proof inv-truncate (previous step)

fs0[ri0[inodekey(m0)]] = inodenode(key, md, false, m, m1, m2), 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-safe(log0, fs0, fi ⊖ fo), ri-cons(ri0, fs0), ri-inj(ri0), ri-inj(fi), ubifs-cons(fs0, ri0, ro, fi, fo, log0), key0 ∈ (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ), inodekey(m0) ∈ ri0, ri0[inodekey(m0)] ∈ fs0, m4 ≤ n0 ⊦ m4 % VFS_PAGE_SIZE = 0, (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ) [key0 ] = ((ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2])) ⊖ ro)[key0], adr1 = adr2, m0 = 0, buf-opt.some?, inodekey(m0) ∈ ro, adr2 ∈ fs0, adr1 ∈ fs0, datakey(m0, m4 / VFS_PAGE_SIZE) ∈ ri0, adr2 ∈ log0, adr1 ∈ log0, m = 0 ∧ m0 ≠ ROOT_INO

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 649:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 650:
fs0[ri0[inodekey(m0)]] = inodenode(key, md, false, m, m1, m2), 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-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, inodekey(m0) ∈ ri0, key0 ∈ (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ), m4 ≤ n0 ⊦ m4 % VFS_PAGE_SIZE = 0, (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ) [key0 ] = ((ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2])) ⊖ ro)[key0], adr1 = adr2, m0 = 0, buf-opt.some?, adr1 ∈ log0, adr2 ∈ log0, datakey(m0, m4 / VFS_PAGE_SIZE) ∈ ri0, adr1 ∈ fs0, adr2 ∈ fs0, inodekey(m0) ∈ ro, m = 0 ∧ m0 ≠ ROOT_INO, key0 ∈ ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2]) ∧ ¬ key0 ∈ ro ∧ ¬ key0.data?
Proof step 660:
fs0[ri0[inodekey(m0)]] = inodenode(key, md, false, m, m1, m2), 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-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, inodekey(m0) ∈ ri0, key0 ∈ (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ), key0 ∈ ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2]), m4 ≤ n0 ⊦ m4 % VFS_PAGE_SIZE = 0, (m ≠ 0 ∨ m0 = ROOT_INO ⊃ (ri0 \ datakeys(inodekey(m0), m4, ri0))[inodekey(m0), adr2] ⊖ ro ;((ri0 \ datakeys(inodekey(m0), m4, ri0)) ⊖ ro) ⊖ inodekey(m0) ) [key0 ] = (ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2]))[key0], adr1 = adr2, m0 = 0, key0.data?, buf-opt.some?, adr1 ∈ log0, adr2 ∈ log0, datakey(m0, m4 / VFS_PAGE_SIZE) ∈ ri0, adr1 ∈ fs0, adr2 ∈ fs0, inodekey(m0) ∈ ro, key0 ∈ ro, m = 0 ∧ m0 ≠ ROOT_INO

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] → [key0, ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2]), ro] flags: <all paths>

Additional information

Used terms: key0, ri0[inodekey(m0), adr2] \ datakeys(inodekey(m0), m4, ri0[inodekey(m0), adr2]), ro