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Proof step 6 from proof in-ri-update-in-ri (previous step)

# ax1 + 1 = # ax, ax = adr0 + ax0, adr ∈ ri, ax1.toSet ⊆ range ri, ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]), ∀ ax, ri, fs, key. # ax1 = # ax ∧ ax1.toSet ⊆ range ri ∧ (∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]])) → (key ∈ ri[keys(fs, ax1), ax] ↔ key ∈ ri) ⊦ key ∈ ri[fs[adr].key ' + keys(fs, ax1), ax] ↔ key ∈ ri

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 7:
# ax1 = # ax0, adr ∈ ri, ax1.toSet ⊆ range ri, ∀ ax, ri, fs, key. # ax0 = # ax ∧ ax1.toSet ⊆ range ri ∧ (∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]])) → (key ∈ ri[keys(fs, ax1), ax] ↔ key ∈ ri), ∀ key. key ∈ ri → ri[key] ∈ fs ∧ key-cons(key, fs[ri[key]]) ⊦ key ∈ ri[fs[adr].key, adr0][keys(fs, ax1), ax0] ↔ key ∈ ri

Rule argument

Additional information

Used simplifier rules: a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]% ⊦ ri[key ' + kx, adr ' + ax] = ri[key, adr][kx, 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: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]% ⊦ m +1 = m + 1; (s)% ⊦ #(adr ' + ax) = (# ax)+1; (s)% ⊦ adr + ax = adr ' + ax; (s)