A Unified Framework for DPLL(T) + Certificates

Table 2

The CDPLL(T) procedure for Example  1.

No.	𝑀‖	𝐹	⊕ P	Reason	Explanation
1	∅‖	𝐹0	⊕ 𝑃0	∅	Initial state
2	𝑙^𝖽1	~	⊕~	∅	Decide
3	𝑙^𝖽1,𝑙2	~	⊕~	𝑙2↦𝐶3	UnitPropagate
4	𝑙^𝖽1,𝑙2,¬𝑙3	~	⊕~	𝑙2↦𝐶3¬𝑙3↦𝐶1	UnitPropagate; now 𝐶2 is falsified
5	¬𝑙1	𝐹0𝐶6=¬𝑙1	⊕𝑃0⊲𝖱𝖾𝗌{𝐶1,𝐶2,𝐶3}=𝐶6	¬𝑙1↦𝐶6	𝐶6=¬𝑙1 is learned
					T-Backjump
6	¬𝑙1,¬𝑙2	~	⊕~	¬𝑙1↦𝐶6¬𝑙2↦𝐶4	UnitPropagate
7	⟨Fail⟩‖	~	⊕𝑃0⊲𝖱𝖾𝗌{𝐶1,𝐶2,𝐶3}=𝐶6𝖱𝖾𝗌{𝐶4,𝐶5,𝐶6}=□	∅	Fail