| Theorem |
| @theorem 1: G∈NODES×{{MM↦RR}}∧ |
| CORR∈CORR∧ |
| Card({n∣n∈corr∧G(n)(m)=∅})≠∅∧≠∅∧ |
| G_pre∈NODES→(NODES→ℙ(ℕ×(ℕ×value))) ∧ |
| G_r∈NODES→(NODES→ℙ(ℕ×(ℕ×value))) ∧ |
| G_pre_check∈NODES→ℙ(ℕ×(ℕ×value)) ∧ |
| G_r_check∈NODES→ℙ(ℕ×(ℕ×value)) |
| ⇒ |
| (node2∈NODES∧ |
| m2∈ℕ×(ℕ×value) ∧ |
| message2∈message∧ contents(message2)=prj2(m2) ∧ |
| m2∈G_r_check(node2) ∧ |
| Cache[corr] ≠ ∅∧ |
| message2∈inter(cache[corr]) ∧ |
| message2∉dom(G(node2)) ∧ prj1(prj2(m2))=n)) |
| @theorem2 G∈NODES×{{MM↦RR}}∧ |
| CORR∈CORR∧ |
| Card({n∣n∈corr∧G(n)(m)=∅})≠∅∧ |
| G_pre∈NODES→(NODES⇸ℙ(ℕ×(ℕ×value))) ∧ |
| G_r∈NODES→(NODES→ℙ(ℕ×(ℕ×value))) ∧ |
| G_pre_check∈NODES→ℙ(ℕ×(ℕ×value)) ∧ |
| G_r_check∈NODES→ℙ(ℕ×(ℕ×value)) |
| ⇒ |
| (node∈NODES∧ |
| node1∈corr∧ |
| m∈cache(node) ∧ |
| m∉dom(G(node)) ∧ |
| prj1(contents(m))<n∧ |
| m∈dom(G(node1))) |
| @theorem3 G∈NODES×{{MM↦RR}}∧ |
| CORR∈CORR ∧ |
| Card()=∅ ∧ |
| G_pre∈NODES→(NODES→ℙ(ℕ×(ℕ×value))) ∧ |
| G_r∈NODES→(NODES→ℙ(ℕ×(ℕ×value))) ∧ |
| G_pre_check∈NODES→ℙ(ℕ×(ℕ×value)) ∧ |
| G_r_check∈NODES→ℙ(ℕ×(ℕ×value)) |
| ⇒ |
| 3({i,j·i∈NODES∧j∈dom(G(i))∧(G(i))(j)=True_execute(j∣i})≥2(NODES)+1 |
