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Proof step 42 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] = ∅, fs0[ri0[inodekey(ino)]].directory, 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), inodekey(ino) ∈ ri0, inodekey(ino) ∈ ri0[kx, ax], kx.toSet ⊆ dom ri0, ro ⊆ dom ri0, fo ⊆ dom fi ⊦ fs0[ri0[inodekey(ino)]].nsubdirs = 0, (fs0[ax, fs0[ri0[kx]]] \ as)[ri0[kx, ax][inodekey(ino)]].directory, dups(ax), dups(kx), dups(log0), inodekey(0) ∈ ri0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: store+set, inst: node-store name: Diff-get seq: ⊦ ¬ adr ∈ as → (fs \ as)[adr] = fs[adr] subst: [adr, as, fs] → [ri0[kx, ax][inodekey(ino)], as, fs0[ax, fs0[ri0[kx]]]] flags: <all paths>

Additional information

Used simplifier rules: int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]% ⊦ key ∈ ri ∧ as ∩ range ri = ∅ → ¬ ri[key] ∈ as; (s)%# ax = # kx ⊦ new(ax, fs) ∧ ¬ dups(ax) ∧ ¬ dups(kx) ∧ ri-cons(ri, fs) ∧ kx.toSet ⊆ dom ri ∧ key ∈ ri → fs[ax, fs[ri[kx]]][ri[kx, ax][key]] = fs[ri[key]]; (s) Used terms: ri0[kx, ax][inodekey(ino)], as, fs0[ax, fs0[ri0[kx]]]