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Proof step 13 from proof ri-cons-copy-new (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 14:
# ax1 = # kx1, # kx0 + 1 = # ax0, new(ax0, fs), new(ax1, fs), key ∈ ri, kx1.toSet ⊆ dom ri, kx0.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ dups(ax0), dups(ax1), dups(kx0), dups(kx1), dups(kx1 + kx0), dups(ax1 + ax0), key ∈ kx0, key ∈ kx1, ((ax1 + ax0)[# kx1] ∈ fs ∨ (ax1 + ax0)[# kx1] ∈ ax1 ∨ (ax1 + ax0)[# kx1] ∈ ax0) ∧ key-cons(key, fs[ax1, fs[ri[kx1]]][ax0, fs[ri[key]] ' + fs[ri[kx0]]][(ax1 + ax0)[# kx1]])

Rule argument

Additional information

Used simplifier rules: # ax0 = # ndx0, # ax1 = # ndx1 ⊦ fs[ax0 + ax1, ndx0 + ndx1] = fs[ax0, ndx0][ax1, ndx1]; (s)%a: ⊦ (ndx + ndx0) + ndx1 = ndx + ndx0 + ndx1; from specification list[node-list]% ⊦ m +1 = m + 1; (s)% ⊦ #(nd ' + ndx) = (# ndx)+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(ax + ax0) → ¬ dups(ax) ∧ ¬ dups(ax0); (forward) from specification list-dup[address-list-set]% ⊦ adr ∈ ax + ax0 ↔ adr ∈ ax ∨ adr ∈ ax0; (s)% ⊦ new(ax0 + ax1, fs) ↔ new(ax0, fs) ∧ new(ax1, fs); (s)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ #(ax + ax0) = # ax + # ax0; (s)