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Proof step 209 from proof inv-truncate (previous step)

# buf-opt.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m3 % VFS_PAGE_SIZE, fs0[ri0[inodekey(m)]] = inodenode(key, md, false, m0, m1, m2), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, ro.inodes?, fo.inodes?, buf-opt.some?, 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-inj(ri0), ri-inj(fi), ubifs-cons(fs0, ri0, ro, fi, fo, log0), ri0[inodekey(m)] ∈ fs0, inodekey(m) ∈ ri0, key0 ∈ ri0, m3 ≤ n1, key0.data? → ¬ inodekey(key0.ino) ∈ ro, ¬ key0.data? ∨ key0.ino ≠ m ∨ ¬ m3 ≤ key0.part * VFS_PAGE_SIZE ⊦ m3 % VFS_PAGE_SIZE = 0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] = adr3, adr1 = adr3, adr1 = ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0], m = 0, adr1 ∈ log0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] ∈ log0, adr3 ∈ log0, adr1 ∈ fs0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] ∈ fs0, adr3 ∈ fs0, inodekey(m) ∈ ro, key0 ∈ ro, m0 = 0 ∧ m ≠ ROOT_INO

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 210:
⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key]
Proof step 211:
# buf-opt.buf = VFS_PAGE_SIZE, VFS_PAGE_SIZE = n0 + m3 % VFS_PAGE_SIZE, fs0[ri0[inodekey(m)]] = inodenode(key, md, false, m0, m1, m2), replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, fo.inodes?, ro.inodes?, buf-opt.some?, 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-inj(fi), ri-inj(ri0), ubifs-cons(fs0, ri0, ro, fi, fo, log0), key0 ∈ ri0, inodekey(m) ∈ ri0, ri0[inodekey(m)] ∈ fs0, m3 ≤ n1, key0.data? → ¬ inodekey(key0.ino) ∈ ro, ¬ key0.data? ∨ key0.ino ≠ m ∨ ¬ m3 ≤ key0.part * VFS_PAGE_SIZE ⊦ m3 % VFS_PAGE_SIZE = 0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] = adr3, adr1 = ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0], adr1 = adr3, m = 0, key0 ∈ ro, inodekey(m) ∈ ro, adr3 ∈ fs0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] ∈ fs0, adr1 ∈ fs0, adr3 ∈ log0, ((ri0 \ datakeys(inodekey(m), m3, ri0)) ⊖ ro)[key0] ∈ log0, adr1 ∈ log0, m0 = 0 ∧ m ≠ ROOT_INO, key0 ∈ ri0 \ datakeys(inodekey(m), m3, ri0) ∧ ¬ key0 ∈ ro ∧ ¬ key0.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, ri, ks] → [key0, ri0 \ datakeys(inodekey(m), m3, ri0), ro] flags: <all paths>

Additional information

Used simplifier rules: at-diff-ri: ⊦ (ri \ ks)[key] = (key ∈ ks ⊃ (∅)[key];ri[key]);%⊦ key ∈ datakeys(key0, n, ri) ↔ key ∈ datakeys(key0, ri) ∧ n ≤ key.part * VFS_PAGE_SIZE; (s)%⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino; (s)% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)% ⊦ key ∈ ri ∧ ubifs-cons(fs, ri, ro, fi, fo, log) → ri[key] ∈ fs; (ss) Used terms: key0, ri0 \ datakeys(inodekey(m), m3, ri0), ro