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Proof step 27 from proof read-lookup-copy (previous step)

# ax0 = # kx0 + # kx + 1, new(ax0, fs), key1 ∈ ri, key ∈ ri, kx0.toSet ⊆ dom ri, kx.toSet ⊆ dom ri, ∀ kx1, fs0, key0, ri0. # ax0 = # kx1 ∧ new(ax0, fs0) ∧ key0 ∈ ri0 ∧ kx1.toSet ⊆ dom ri0 ∧ (∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]])) → fs0[ax0, fs0[ri0[kx1]]][ri0[kx1, ax0][key0]] = fs0[ri0[key0]] ∨ dups(kx1), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∃ ax2, ax1. ax0 = ax2 + ax1 ∧ # ax2 = # kx0 ⊦ fs[ri[key1]] = fs[ri[key]], dups(kx), dups(kx0), dups(kx0 + kx), dups(ax0), ax0[# kx0] ∈ fs, ax0[# kx0] ∈ ax0, key1 ∈ kx, key1 ∈ kx0, key ∈ kx, key ∈ kx0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 28:
# ax1 = # kx0, # ax = # kx + 1, new(ax, fs), new(ax1, fs), key1 ∈ ri, key ∈ ri, kx0.toSet ⊆ dom ri, kx.toSet ⊆ dom ri, ∀ kx1, fs0, key0, ri0. # kx0 + # ax = # kx1 ∧ new(ax1, fs0) ∧ new(ax, fs0) ∧ key0 ∈ ri0 ∧ kx1.toSet ⊆ dom ri0 ∧ (∀ key. key ∈ ri0 → ri0[key] ∈ fs0 ∧ key-cons(key, fs0[ri0[key]])) → fs0[ax1 + ax, fs0[ri0[kx1]]][ri0[kx1, ax1 + ax][key0]] = fs0[ri0[key0]] ∨ dups(kx1), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ fs[ri[key1]] = fs[ri[key]], dups(ax), dups(ax1), dups(kx), dups(kx0), dups(kx0 + kx), dups(ax1 + ax), (ax1 + ax)[# kx0] ∈ ax, (ax1 + ax)[# kx0] ∈ ax1, key1 ∈ kx, key1 ∈ kx0, key ∈ kx, key ∈ kx0, (ax1 + ax)[# kx0] ∈ fs

Rule argument

Additional information

Used simplifier rules: 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)