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Proof step 33 from proof inv-gc (previous step)

# kx = # ax, replay-log(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) ∧ 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 34:
⊦ range ri ∩ as = ∅ ∧ ri-closed(ri, fs, ro) → ri-closed(ri, fs \ as, ro)
Proof step 35:
# kx = # ax, replay-log(log0, fs0, fi ⊖ fo) = ri0 ⊖ ro, as ∩ range ri0[kx, ax] = ∅, ax.toSet ∩ log0.toSet = ∅, as ∩ log0.toSet = ∅, as ∩ ax.toSet = ∅, range ri0[kx, ax] ∩ as = ∅, 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-safe(log0, fs0, fi ⊖ fo), ri-closed(ri0, fs0, ro), ri-closed(ri0[kx, ax], fs0[ax, fs0[ri0[kx]]], 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, true ∧ 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-inv name: ri-closed-drop seq: ⊦ range ri ∩ as = ∅ ∧ ri-closed(ri, fs, ro) → ri-closed(ri, fs \ as, ro) subst: [ro, as, fs, ri] → [ro, as, fs0[ax, fs0[ri0[kx]]], ri0[kx, ax]] flags: <all paths>

Additional information

Used simplifier rules: int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%# kx = # ax ⊦ kx.toSet ⊆ dom ri ∧ new(ax, fs) ∧ ri-closed(ri, fs, ro) ∧ ri-cons(ri, fs) ∧ ¬ dups(kx) ∧ ¬ dups(ax) → ri-closed(ri[kx, ax], fs[ax, fs[ri[kx]]], ro); (s) Used terms: ro, as, fs0[ax, fs0[ri0[kx]]], ri0[kx, ax]