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Proof step 5 from proof replay-safe-gc-copy (previous step)

# 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))

User applied rule constructor cut

And obtained the following new goals

Rule argument

var: kx

Additional information

Used simplifier rules: key-list freely generated by [], +;;% ⊦ # [] = 0; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)% ⊦ # ax = 0 ↔ ax = []; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic