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Proof step 30 from proof inv-gc (previous step)

# kx = # ax, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ ax.toSet = ∅, fo.inodes?, ro.inodes?, file-node-cons(fs0, ri0), 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, ro), replay-cons-strong(fs0, ri0, ro, fi, fo, log0), ri-closed(ri0, fs0, ro), ri-cons(ri0, fs0), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as), ri-inj(fi), ri-inj(ri0), ri-inj(ri0[kx, ax]), fo ⊆ dom fi, ro ⊆ dom ri0, kx.toSet ⊆ dom ri0 ⊦ dups(log0), dups(kx), dups(ax), inodekey(0) ∈ ri0, ri-closed(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as, ro) ∧ replay-safe(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) ∧ replay-log(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) = ri0[kx, ax] ⊖ ro ∧ file-node-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax]) ∧ orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], ro)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ replay-cons-strong(fs, ri, ro, fi, fo, log) ↔ replay-safe(log, fs, fi ⊖ fo) ∧ replay(log, fs, fi, fo) = ri ⊖ ro
Proof step 32:
# kx = # ax, replay(log0, fs0, fi, fo) = ri0 ⊖ ro, as ∩ ax.toSet = ∅, as ∩ log0.toSet = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ range ri0[kx, ax] = ∅, ro.inodes?, fo.inodes?, file-node-cons(fs0, ri0), log-cons(log0, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(ax, fs0[ax, fs0[ri0[kx]]] \ as), log-cons(log0, fs0), new(ax, fs0), orphans-cons(fs0, ri0, ro), replay-safe(log0, fs0, fi ⊖ fo), ri-closed(ri0, fs0, ro), ri-cons(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as), ri-cons(ri0, fs0), ri-inj(ri0[kx, ax]), ri-inj(ri0), ri-inj(fi), kx.toSet ⊆ dom ri0, ro ⊆ dom ri0, fo ⊆ dom fi ⊦ dups(ax), dups(kx), dups(log0), inodekey(0) ∈ ri0, ri-closed(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]] \ as, ro) ∧ replay-safe(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) ∧ replay-log(ax, fs0[ax, fs0[ri0[kx]]], replay-log(log0, fs0, fi ⊖ fo)) = ri0[kx, ax] ⊖ ro ∧ file-node-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax]) ∧ orphans-cons(fs0[ax, fs0[ri0[kx]]] \ as, ri0[kx, ax], ro)

Rule argument

spec: aubifs-replay name: replay-cons-strong seq: ⊦ replay-cons-strong(fs, ri, ro, fi, fo, log) ↔ replay-safe(log, fs, fi ⊖ fo) ∧ replay(log, fs, fi, fo) = ri ⊖ ro subst: [log, fo, fi, ro, ri, fs] → [log0, fo, fi, ro, ri0, fs0] flags: <all paths>

Additional information

Used terms: log0, fo, fi, ro, ri0, fs0