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Proof step 28 from proof refine-gc (previous step)

# kx = # ax, as ∩ ax.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, fo.inodes?, orphans(fs0, ri0).inodes?, afs-cons(abs-dirs(fs0, ri0), abs-files(fs0, ri0)), file-node-cons(fs0, ri0), file-node-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax]), log-cons(log0, fs0), log-cons(ax, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(log0, fs0[ax, fs0[ri0[kx]]] \ as), new(ax, fs0), orphans-cons(fs0, ri0, orphans(fs0, ri0)), orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], orphans(fs0, ri0)), replay-cons-strong(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), replay-cons-strong(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], orphans(fs0, ri0), fi, fo, log0 + ax), ri-inj(fi), ri-inj(ri0), ri-inj(ri0[kx, ax]), ubifs-cons(fs0, ri0, orphans(fs0, ri0), fi, fo, log0), ubifs-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], orphans(fs0, ri0), fi, fo, log0 + ax), kx.toSet ⊆ dom ri0, asynced0 → boolvar0 ⊦ orphans(abs-dirs(fs0[ax, fs0[ri0[kx]]], ri0[kx, ax]), abs-files(fs0[ax, fs0[ri0[kx]]], ri0[kx, ax])) = abs-keys(orphans(fs0, ri0)), dups(kx), dups(ax)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-abs name: abs-dirs-gc-copy seq: # ax = # kx ⊦ new(ax, fs) ∧ kx.toSet ⊆ dom ri ∧ ¬ dups(kx) ∧ ¬ dups(ax) ∧ ri-cons(ri, fs) → abs-dirs(fs[ax, fs[ri[kx]]], ri[kx, ax]) = abs-dirs(fs, ri) subst: [kx, ri, ax, fs] → [kx, ri0, ax, fs0] flags: <all paths>

Additional information

Used simplifier rules: # ax = # kx ⊦ new(ax, fs) ∧ kx.toSet ⊆ dom ri ∧ ¬ dups(kx) ∧ ¬ dups(ax) ∧ ri-cons(ri, fs) → abs-files(fs[ax, fs[ri[kx]]], ri[kx, ax]) = abs-files(fs, ri); (s)%⊦ afs-cons(abs-dirs(fs, ri), abs-files(fs, ri)) → orphans(abs-dirs(fs, ri), abs-files(fs, ri)) = abs-keys(orphans(fs, ri)); (s)%⊦ abs-keys(orphans(fs0, ri0)) = abs-keys(orphans(fs, ri)) ↔ orphans(fs0, ri0) = orphans(fs, ri); (s)% ⊦ ubifs-cons(fs, ri, ro, fi, fo, log) → ri-cons(ri, fs); (ss) Used terms: kx, ri0, ax, fs0