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Proof step 7 from proof replay-safe-gc-copy (previous step)

# ax0 = # kx1, ro.inodes?, new(ax0, fs), orphans-cons(fs, ri, ro), ri-closed(ri, fs, ro), ri-cons(ri, fs), key ∈ ri, kx1.toSet ⊆ dom ri, ∀ kx0, ro0, fs0, ri0. # kx1 = # kx0 ∧ ro0.inodes? ∧ new(ax0, fs0) ∧ orphans-cons(fs0, ri0, ro0) ∧ ri-closed(ri0, fs0, ro0) ∧ ri-cons(ri0, fs0) ∧ kx0.toSet ⊆ dom ri0 → dups(kx0) ∨ replay-safe(ax0, fs0[ax0, fs0[ri0[kx0]]], ri0 ⊖ ro0) ⊦ dups(ax0), dups(kx1), adr ∈ fs, adr ∈ ax0, key ∈ kx1, ¬ adr ∈ ri ⊖ ro ∧ node-cons(fs[adr, fs[ri[key]]][ax0, fs[ri[kx1]]][adr].key, fs[adr, fs[ri[key]]][ax0, fs[ri[kx1]]][adr]) ∧ replay-safe(ax0, fs[adr, fs[ri[key]]][ax0, fs[ri[kx1]]], replay-one(adr, fs[adr, fs[ri[key]]][ax0, fs[ri[kx1]]][adr], ri ⊖ ro))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-base name: at-write-other seq: ⊦ # ax = # ndx ∧ ¬ adr ∈ ax → fs[ax, ndx][adr] = fs[adr] subst: [adr, ndx, ax, fs] → [adr, fs[ri[kx1]], ax0, fs[adr, fs[ri[key]]]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # fs[ax] = # ax; (s)% ⊦ # ri[kx] = # kx; (s) Used terms: adr, fs[ri[kx1]], ax0, fs[adr, fs[ri[key]]]