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Proof step 8 from proof log-cons-copy-new (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 9:
# kx1 = # ax1, # kx0 = # ax0 + 1, kx0.toSet ⊆ dom ri, kx1.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(ax0), dups(ax1), dups(ax1 + ax0), dups(kx1 + kx0), dups(kx0), dups(kx1), 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 ∈ ax0, adr ∈ ax1

Rule argument

Additional information

Used simplifier rules: a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]%# ax0 = # ndx0, # ax1 = # ndx1 ⊦ fs[ax0 + ax1, ndx0 + ndx1] = fs[ax0, ndx0][ax1, ndx1]; (s)% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ # ri[kx] = # kx; (s)% ⊦ # fs[ax] = # ax; (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: ⊦ ¬ dups(kx + kx0) → ¬ dups(kx) ∧ ¬ dups(kx0); (forward) from specification list-dup[key-list-set]%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ fs[ax0 + ax1] = fs[ax0] + fs[ax1]; (s)% ⊦ ri[kx0 + kx1] = ri[kx0] + ri[kx1]; (s)%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 ∪ ks0 ⊆ ks1 ↔ ks ⊆ ks1 ∧ ks0 ⊆ ks1; (ws)% ⊦ (kx + kx0).toSet = kx.toSet ∪ kx0.toSet; (s)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ #(kx + kx0) = # kx + # kx0; (s)