[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 12 from proof ri-inj-index-update (previous step)

# ax = # kx0 + 1, ri[kx0 + key1 ', ax][key] = ri[kx0 + key1 ', ax][key0], ax.toSet ∩ range ri = ∅, ri-inj(ri), key ∈ ri[kx0 + key1 ', ax], key0 ∈ ri[kx0 + key1 ', ax], ∀ kx1, ax, ri, key, key0. # kx1 < #(kx0 + key1 ') → # ax = # kx1 ∧ ri[kx1, ax][key] = ri[kx1, ax][key0] ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ key ∈ ri[kx1, ax] ∧ key0 ∈ ri[kx1, ax] → key = key0 ∨ dups(ax) ∨ key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0] ⊦ key = key0, dups(ax), ∀ ax0, adr. ax ≠ ax0 + adr ', key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 13:
# ax0 = # kx0, ri[kx0, ax0][key1, adr][key] = ri[kx0, ax0][key1, adr][key0], range ri ∩ ax0.toSet = ∅, ri-inj(ri), key0 ∈ ri[kx0, ax0][key1, adr], key ∈ ri[kx0, ax0][key1, adr], ∀ kx1, ax, ri, key, key0. # kx1 < #(kx0 + key1 ') → # ax = # kx1 ∧ ri[kx1, ax][key] = ri[kx1, ax][key0] ∧ ax.toSet ∩ range ri = ∅ ∧ ri-inj(ri) ∧ key ∈ ri[kx1, ax] ∧ key0 ∈ ri[kx1, ax] → key = key0 ∨ dups(ax) ∨ key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0] ⊦ key = key0, dups(ax0), adr ∈ ri, adr ∈ ax0, key ∈ ri ∧ key0 ∈ ri ∧ ri[key] = ri[key0]

Rule argument

Additional information

Used simplifier rules: ⊦ dups(ax + adr ') ↔ adr ∈ ax ∨ dups(ax); (ws)%a: ⊦ (kx + kx0) + kx1 = kx + kx0 + kx1; from specification list[key-list-set]%# kx = # ax ⊦ ri[kx + key ', ax + adr '] = ri[kx, ax][key, adr]; (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)% ⊦ # adr ' = 0 +1; (s)% ⊦ #(ax + ax0) = # ax + # ax0; (s)%int-c: ⊦ as ∩ as0 = as0 ∩ as; from specification set-union[node-store]%int-a: ⊦ (as ∩ as0) ∩ as1 = as ∩ as0 ∩ as1; from specification set-union[node-store]% ⊦ adr ∈ range ri ↔ adr ∈ ri; (s)% ⊦ as ∩ (as0 ++ adr) = ∅ ↔ as ∩ as0 = ∅ ∧ ¬ adr ∈ as; (ws)%a: ⊦ (ax + ax0) + ax1 = ax + ax0 + ax1; from specification list[address-list-set]%unc: ⊦ as ∪ as0 = as0 ∪ as; from specification set-union[node-store]%una: ⊦ (as ∪ as0) ∪ as1 = as ∪ as0 ∪ as1; from specification set-union[node-store]% ⊦ as ∪ ∅ = as; (s)% ⊦ (as ++ adr) ∪ as0 = (as ∪ as0) ++ adr; (s)% ⊦ adr '.toSet = ∅ ++ adr; (s)% ⊦ (ax + ax0).toSet = ax.toSet ∪ ax0.toSet; (s)