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Proof step 10 from proof log-cons-copy-new (previous step)

# kx0 = # ax0 + 1, # kx1 = # ax1, kx0 = key0 + kx, kx1.toSet ⊆ dom ri, kx0.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(kx1), dups(kx0), dups(kx1 + kx0), dups(ax1 + ax0), dups(ax1), dups(ax0), node-cons(fs[ax1, fs[ri[kx1]]][adr ' + ax0, fs[ri[kx0]]][adr].key, fs[ax1, fs[ri[kx1]]][adr ' + ax0, fs[ri[kx0]]][adr]), adr ∈ ax1, adr ∈ ax0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
# kx1 = # ax1, # kx = # ax0, key0 ∈ ri, kx1.toSet ⊆ dom ri, kx.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(ax0), dups(ax1), dups(ax1 + ax0), dups(kx1 + kx), dups(kx1), dups(kx), node-cons(fs[ax1, fs[ri[kx1]]][adr, fs[ri[key0]]][ax0, fs[ri[kx]]][adr].key, fs[ax1, fs[ri[kx1]]][adr, fs[ri[key0]]][ax0, fs[ri[kx]]][adr]), adr ∈ ax0, adr ∈ ax1, key0 ∈ kx1, key0 ∈ kx

Rule argument

Additional information

Used simplifier rules: a: ⊦ ¬ dups(kx + kx0) → ¬ dups(kx) ∧ ¬ dups(kx0); (forward) from specification list-dup[key-list-set]% ⊦ ¬ key ∈ kx → (dups(key ' + kx) ↔ dups(kx)); (s)% ⊦ dups(kx + key ' + kx0) ↔ key ∈ kx ∨ key ∈ kx0 ∨ dups(kx + kx0); (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: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ nd + ndx = nd ' + ndx; (s)% ⊦ fs[adr ' + ax] = fs[adr] + fs[ax]; (s)% ⊦ adr + ax = adr ' + ax; (s)% ⊦ ri[key ' + kx] = ri[key] + ri[kx]; (s)% ⊦ key ∈ dom ri ↔ key ∈ ri; (s)% ⊦ ks ++ key ⊆ ks0 ↔ ks ⊆ ks0 ∧ key ∈ ks0; (ws)%unc: ⊦ ks ∪ ks0 = ks0 ∪ ks; from specification set-union[node-index]%una: ⊦ (ks ∪ ks0) ∪ ks1 = ks ∪ ks0 ∪ ks1; from specification set-union[node-index]% ⊦ ks ∪ ∅ = ks; (s)% ⊦ (ks ++ key) ∪ ks0 = (ks ∪ ks0) ++ key; (s)% ⊦ key '.toSet = ∅ ++ key; (s)% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (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: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(key ' + kx) = (# kx)+1; (s)% ⊦ key + kx = key ' + kx; (s)