J'ai cru que tu aurais préparé le bilan de l'avancement.

Questions & Answers about J'ai cru que tu aurais préparé le bilan de l'avancement.

Why do we use the conditional past (aurais préparé) here instead of a future tense?
This is an example of the sequence of tenses in reported speech. Because the introductory verb is in a past tense (J'ai cru), the tense of the original thought must shift backwards. If the original thought used the futur antérieur (future perfect) to say "you will have prepared", it must backshift into the conditionnel passé to say "you would have prepared".
What would the direct, present-tense version of this thought look like?
If you were thinking this in the present, you would use the futur antérieur: Je crois que tu auras préparé le bilan (I believe that you will have prepared the assessment). When Je crois changes to the past (J'ai cru), tu auras préparé shifts to tu aurais préparé.
Could I say Je croyais instead of J'ai cru?
Yes, but they carry a slightly different nuance. Je croyais (imperfect) describes an ongoing state of mind in the past ("I was under the impression..."). J'ai cru (passé composé) can imply a belief that was momentary or a belief that has just been proven wrong—which fits perfectly here, as the speaker is realizing the assessment wasn't actually prepared.
How does bilan translate to "assessment"? I thought it meant a balance sheet.
While un bilan literally means a balance sheet in accounting, it is used very broadly in everyday and professional French to mean a review, assessment, or wrap-up. Paired with l'avancement (the progress or advancement of a project), le bilan de l'avancement is the standard way to say "progress report" or "progress assessment".

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