Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^\* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq\_P}, \mathbb{1}\_P) \in M $ and $ \mathbb{Q} := (Q, {\leq\_Q}, \mathbb{1}\_Q) \in M $ are forcing notions.




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Reminder
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**Definition** ([Kun80, VII.7.1]). A mapping $ i \colon P \to Q $ is a *complete embedding* of $ \mathbb{P} $ into $ \mathbb{Q} $ iff


(i) $ \forall r, s \in P \ (r \leq\_P s \implies i(r) \leq\_Q i(s)) $,


(ii) $ \forall r, s \in P \ (r \perp\_P s \implies i(r) \perp\_Q i(s)) $, and


(iii) $ \forall q \in Q \ \exists p \in P \ \forall r \in P \ (r \leq\_P p \implies i(r) \parallel\_Q q) $.


A condition $ p \in P $ as in (iii) is called a *reduction* of $ q $ to $ \mathbb{P} $.


*Remark.* Note that in clause (ii) equivalence holds because of (i).


**Definition** ([Kun80, VII.7.7]). A mapping $ i \colon P \to Q $ is a *dense embedding* of $ \mathbb{P} $ into $ \mathbb{Q} $ iff


(i) $ \forall r, s \in P \ (r \leq\_P s \implies i(r) \leq\_Q i(s)) $,


(ii) $ \forall r, s \in P \ (r \perp\_P s \implies i(r) \perp\_Q i(s)) $, and


(iii) $ i[P] $ is dense in $ \mathbb{Q} $.


*Remark.* Every dense embedding is a complete embedding.


**Theorem** ([Kun80, II.3.3]). There exists a dense embedding of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{P}) \setminus \{ \mathbb{0} \} $.


**Lemma** ([Kun80, VII.Ex.C2]). If $ \mathbb{P} $ and $ \mathbb{Q} $ are separative and $ i \colon P \to Q $ is a complete embedding of $ \mathbb{P} $ into $ \mathbb{Q} $, then $ i $ is one-to-one, $ i(\mathbb{1}\_P) = \mathbb{1}\_Q $, and $ (r \leq\_P s \iff i(r) \leq\_Q i(s)) $ holds for all $ r, s \in P $.




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Now, consider the following statements:


*(C1)* For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $. (Then $ M[G] \subseteq M[H] $.)


*(C2)* For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $. (Then $ M[G] \subseteq M[H] $.)


*(C3)* There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.


*(C4)* There exists a complete embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.


*(D1)* For each $ \mathbb{Q} $-generic $ H $, there exists a $ G \in M[H] $ such that $ G $ is $ \mathbb{P} $-generic over $ M $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)


*(D2)* For each $ \mathbb{P} $-generic $ G $, there exists a $ \mathbb{Q} $-generic $ H $ such that $ G \in M[H] $ and $ H \in M[G] $. (Then $ M[G] = M[H] $.)


*(D3)* There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \mathbb{Q} $.


*(D4)* There exists a dense embedding $ i \in M $ of $ \mathbb{P} $ into $ \operatorname{RO}(\mathbb{Q}) \setminus \{ \mathbb{0} \} $.


*(Pr)* If $ \mathbb{Q} $ is proper, then $ \mathbb{P} $ is also proper.




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Question
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What implications between the above statements are provable in $ \mathsf{ZFC} $? Which are not?


If it is helpful, you may assume that $ \mathbb{P} $ and $ \mathbb{Q} $ are separative partial orders (in the strict sense).




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Main problem
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Suppose that *(C1)* holds. What additional assumptions do we need to show *(Pr)*?


(Note that *(C2)* implies *(Pr)*. So what additional assumptions does one need to show *(C2)* from *(C1)*?)




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Bibliography
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[Kun80] Kenneth Kunen: *Set Theory: An Introduction to Independence Proofs.* North Holland, 1980



Embeddings of forcing notions - preserve properness?
**The following implications are either trivial or well known:**


*(D$ n $)* implies *(C$ n $)*: Trivial.


*(C3)* implies *(C1)*: Use $ G := i^{-1}[H] $. (See [Kun80, VII.7.5].)


*(C3)* implies *(C4)*: The composition of two complete embeddings is a complete embedding.


*(D3)* implies *(D1)*: Use $ G := i^{-1}[H] $. Then $ H = \{ q \in Q : \exists p \in G \ i(p) \leq q \} $. (See [Kun80, VII.7.11].)


*(D3)* implies *(D2)*: Use $ H := \{ q \in Q : \exists p \in G \ i(p) \leq q \} $. Then $ G = i^{-1}[H] $. (See [Kun80, VII.7.11].)


*(D3)* implies *(D4)*: The composition of two dense embeddings is a dense embedding.




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Regarding ***(C1)* does not imply *(C3)*** and ***(C1)* does not imply *(C4)***:


Let $ \mathbb{Q} $ be the Cohen forcing and let $ \mathbb{P} $ be the lottery sum of $ \mathbb{Q} $ and $ \operatorname{Col}(\omega, \omega\_1) $. Then *(C1)* clearly holds (use $ G := H $), but *(C3)* and *(C4)* do not hold since $ \mathbb{P} $ is not c.c.c. whereas $ \mathbb{Q} $ and hence $ \operatorname{RO}(\mathbb{Q}) $ satisfy the countable chain condition.


(See the comments of **Yair Hayut** and **Andreas Blass**.)


Also see [*A common forcing extension obtained via different forcing notions*](http://jdh.hamkins.org/common-forcing-extension-via-different-forcing-notions/) by J.D. Hamkins or [*Lemma 25.5*](http://books.google.com/books?id=pLxq0myANiEC&pg=PA267) in T. Jech's book *Set Theory* (1978 edition).




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Regarding ***(C3)* implies *(C2)***:


Each $ H \subseteq Q $ which is $ \mathbb{Q} / G $-generic over $ M[G] $ is also $ \mathbb{Q} $-generic over $ M $, and
$$
G \in M[G][H]\_{\mathbb{Q} / G} = M[H]\_{\mathbb{Q}}.
$$


(See [Kun80, VII.Ex.D3] and [Kun80, VII.Ex.D4].)


