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Proof step 45 from proof inv-writepage (previous step)

# buf = VFS_PAGE_SIZE, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri), log-cons(log0, fs0), orphans-cons(fs0, ri, ro), replay-cons-strong(fs0, ri, ro, fi, fo, log0), ri-inj(ri), ri-inj(fi), ubifs-cons(fs0, ri, ro, fi, fo, log0), inodekey(m) ∈ ri ⊦ fs0[ri[inodekey(m)]].directory, replay-cons-strong(fs0[adr1, datanode(datakey(m, pageno), buf)], ri[datakey(m, pageno), adr1], ro, fi, fo, log0 + adr1 '), adr1 ∈ fs0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-replay name: replay-cons-strong-append seq: nd.key = key ⊦ ¬ adr ∈ fs ∧ log-cons(log, fs) ∧ node-cons(key, nd) ∧ ri-cons(ri, fs) ∧ replay-cons-strong(fs, ri, ro, fi, fo, log) → ( replay-cons-strong (fs[adr, nd], ri[key, adr], ro, fi, fo, log + adr ') ↔ replay-one(adr, nd, ri ⊖ ro) = ri[nd.key, adr] ⊖ ro) subst: [log, fo, fi, ro, key, ri, nd, adr, fs] → [log0, fo, fi, ro, datakey(m, pageno), ri, datanode(datakey(m, pageno), buf), adr1, fs0] flags: <all paths>

Additional information

Used simplifier rules: at-fs: ⊦ fs[adr0, nd][adr1] = (adr0 = adr1 ⊃ nd;fs[adr1]);%⊦ replay-cons-strong(fs, ri, ro, fi, fo, log) → ( replay-cons-strong(fs0, ri0, ro0, fi0, fo0, log0) ↔ replay-safe(log, fs, fi ⊖ fo) ∧ replay(log, fs, fi, fo) = ri ⊖ ro → replay-safe(log0, fs0, fi0 ⊖ fo0) ∧ replay(log0, fs0, fi0, fo0) = ri0 ⊖ ro0); (ss)% ⊦ replay(log, fs, fi, fo) = replay-log(log, fs, fi ⊖ fo); (s)%⊦ replay-safe(log0 + log1, fs, ri) ↔ replay-safe(log0, fs, ri) ∧ replay-safe(log1, fs, replay-log(log0, fs, ri)); (s)%⊦ replay-safe(adr ', fs, ri) ↔ adr ∈ fs ∧ ¬ adr ∈ ri ∧ node-cons(fs[adr].key, fs[adr]); (s)% ⊦ adr ∈ fs[adr1, nd] ↔ adr = adr1 ∨ adr ∈ fs; (s)% ⊦ datanode(key, buf).key = key; (s)% ⊦ node-cons(key0, datanode(key1, buf)) ↔ key0.data?; (s)% ⊦ datakey(m, m0).data?; (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%⊦ replay(ax0 + ax1, fs, fi, fo) = replay-log(ax1, fs, replay(ax0, fs, fi, fo)); (s)% ⊦ replay-log(adr ', fs, ri) = replay-one(adr, fs[adr], ri); (s)%⊦ replay-one(adr, datanode(key, buf), ri) = (inodekey(key.ino) ∈ ri ⊃ ri[key, adr];ri); (s)% ⊦ datakey(m, m0).ino = m; (s)% ⊦ ubifs-cons(fs, ri, ro, fi, fo, log) → log-cons(log, fs); (ss)% ⊦ ubifs-cons(fs, ri, ro, fi, fo, log) → ri-cons(ri, fs); (ss) Used terms: log0, fo, fi, ro, datakey(m, pageno), ri, datanode(datakey(m, pageno), buf), adr1, fs0