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Proof step 6 from proof ri-closed-gc (previous step)

# kx0 + 1 = # ax, ax = adr + ax1, new(ax, fs), ri-closed(ri, fs, ro), ri-cons(ri, fs), key ∈ ri, kx0.toSet ⊆ dom ri, ∀ ax0, fs0, ri0, ro0. # kx0 = # ax0 ∧ new(ax0, fs0) ∧ ri-closed(ri0, fs0, ro0) ∧ ri-cons(ri0, fs0) ∧ kx0.toSet ⊆ dom ri0 → dups(ax0) ∨ ri-closed(ri0[kx0, ax0], fs0[ax0, fs0[ri0[kx0]]], ro0) ⊦ dups(kx0), dups(ax), ri-closed(ri[key ' + kx0, ax], fs[ax, fs[ri[key]] ' + fs[ri[kx0]]], ro), key ∈ kx0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
# kx0 = # ax1, new(ax1, fs), ri-closed(ri, fs, ro), ri-cons(ri, fs), key ∈ ri, kx0.toSet ⊆ dom ri, ∀ ax0, fs0, ri0, ro0. # ax1 = # ax0 ∧ new(ax0, fs0) ∧ ri-closed(ri0, fs0, ro0) ∧ ri-cons(ri0, fs0) ∧ kx0.toSet ⊆ dom ri0 → dups(ax0) ∨ ri-closed(ri0[kx0, ax0], fs0[ax0, fs0[ri0[kx0]]], ro0) ⊦ dups(kx0), dups(ax1), ri-closed(ri[key, adr][kx0, ax1], fs[adr, fs[ri[key]]][ax1, fs[ri[kx0]]], ro), adr ∈ fs, key ∈ kx0, adr ∈ ax1

Rule argument

Additional information

Used simplifier rules: ⊦ dups(adr ' + ax) ↔ dups(ax) ∨ adr ∈ ax; (ws)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ fs[adr ' + ax, nd ' + ndx] = fs[adr, nd][ax, ndx]; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, ax]; (s)% ⊦ new(adr ' + log, fs) ↔ ¬ adr ∈ fs ∧ new(log, fs); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (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)