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Proof step 7 from proof replay-log-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 → replay-log(ax0, fs0[ax0, fs0[ri0[kx0]]], ri0 ⊖ ro0) = ri0[kx0, ax0] ⊖ ro0 ∨ dups(kx0) ⊦ replay-log(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)) = ri[key, adr][kx1, ax0] ⊖ ro, dups(ax0), dups(kx1), adr ∈ fs, adr ∈ ax0, key ∈ kx1

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

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

Additional information

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