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Proof step 28 from proof read-lookup-copy (previous step)

# 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

User applied rule constructor cut

And obtained the following new goals

Proof step 29:
# ax = # kx + 1, # ax1 = # kx0, ax = adr + ax0, new(ax1, fs), new(ax, fs), key ∈ ri, key1 ∈ ri, kx.toSet ⊆ dom ri, kx0.toSet ⊆ dom ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ 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) ⊦ fs[ri[key1]] = fs[ri[key]], dups(ax1 + ax), dups(kx0 + kx), dups(kx0), dups(kx), dups(ax1), dups(ax), (ax1 + ax)[# kx0] ∈ fs, key ∈ kx0, key ∈ kx, key1 ∈ kx0, key1 ∈ kx, (ax1 + ax)[# kx0] ∈ ax1, (ax1 + ax)[# kx0] ∈ ax

Rule argument

var: ax

Additional information

Used simplifier rules: address-list freely generated by [], +;;% ⊦ # [] = 0; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)% ⊦ # kx = 0 ↔ kx = []; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic