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Proof step 4 from proof replay-safe-gc-copy (previous step)

#(adr + ax0) = # kx, ro.inodes?, new(adr + ax0, fs), orphans-cons(fs, ri, ro), ri-closed(ri, fs, ro), ri-cons(ri, fs), kx.toSet ⊆ dom ri, ∀ kx, ro, fs, ri. # ax0 = # kx ∧ ro.inodes? ∧ new(ax0, fs) ∧ orphans-cons(fs, ri, ro) ∧ ri-closed(ri, fs, ro) ∧ ri-cons(ri, fs) ∧ kx.toSet ⊆ dom ri → dups(ax0) ∨ dups(kx) ∨ replay-safe(ax0, fs[ax0, fs[ri[kx]]], ri ⊖ ro) ⊦ dups(kx), dups(adr + ax0), replay-safe(adr + ax0, fs[adr + ax0, fs[ri[kx]]], ri ⊖ ro)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
# ax0 + 1 = # kx, ro.inodes?, new(ax0, fs), orphans-cons(fs, ri, ro), ri-closed(ri, fs, ro), ri-cons(ri, fs), kx.toSet ⊆ dom ri, ∀ kx0, ro0, fs0, ri0. # ax0 = # 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(kx), dups(ax0), adr ∈ fs, adr ∈ ax0, ¬ adr ∈ ri ⊖ ro ∧ node-cons(fs[adr ' + ax0, fs[ri[kx]]][adr].key, fs[adr ' + ax0, fs[ri[kx]]][adr]) ∧ replay-safe(ax0, fs[adr ' + ax0, fs[ri[kx]]], replay-one(adr, fs[adr ' + ax0, fs[ri[kx]]][adr], ri ⊖ ro))

Rule argument

Additional information

Used simplifier rules: replay-safe-rec: ⊦ replay-safe(adr ' + log, fs, ri) ↔ adr ∈ fs ∧ ¬ adr ∈ ri ∧ node-cons(fs[adr].key, fs[adr]) ∧ replay-safe(log, fs, replay-one(adr, fs[adr], ri));% ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)% ⊦ adr ∈ adr0 ' + ax ↔ adr = adr0 ∨ adr ∈ ax; (s)%# ax = # ndx ⊦ adr ∈ fs[ax, ndx] ↔ adr ∈ fs ∨ adr ∈ ax; (s)% ⊦ # ri[kx] = # kx; (s)% ⊦ # fs[ax] = # ax; (s)% ⊦ new(adr ' + log, fs) ↔ ¬ adr ∈ fs ∧ new(log, fs); (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ adr + ax = adr ' + ax; (s)