关于Z3中PROOF=true选项下量化变量日志识别的技术问询

Let's break down your questions about Z3's logging behavior when using these flags:

1. Your understanding of unbound identifiers for quantified variables without PROOF=true is correct #

When PROOF=true is not enabled, Z3 uses internal "ghost" identifiers for quantified variables in its TRACE logs. These IDs represent bound variables in the quantifier's scope, but they don't have corresponding [mk-app] entries because they aren't constructed as regular terms—they're just placeholders Z3 uses to track variable bindings internally.

For example, in your log snippet:

[mk-app] #3 f #2
[mk-app] #4 pattern #3
[mk-app] #5 = #3 #2
[mk-quant] #6 quantifier_name #4 #5

The #2 is a bound variable for the quantifier #6, but since no proof generation is enabled, Z3 doesn't log any construction for it. This is expected behavior.

2. Identifying quantified variables when PROOF=true is enabled #

When PROOF=true is turned on, Z3 merges the identifier namespace for proof steps and regular terms, which is why you see overlapping IDs (like #2 being both a proof term and a quantified variable). To distinguish which terms are quantified variables, use these strategies:

- Analyze the [mk-quant] context #

Quantified variables are only used within the scope of the quantifier's pattern (#P) and body (#B) in the [mk-quant] #N quantifier_name #P #B entry. Any ID that appears in #P or #B but is treated as a variable (rather than a fully constructed term) is a bound variable for that quantifier.

- Track ID usage and scope #

• A quantified variable ID will not be referenced outside the quantifier it belongs to (unlike regular terms, which may be reused in other constructions later in the log).
• If an ID was constructed as a proof term (e.g., #2 iff #1 #1 in your example) but is used as a parameter in the quantifier's body/pattern, it's acting as a bound variable for that quantifier—Z3 reuses proof-related IDs for variable bindings in proof mode.

- Cross-reference with your SMT source file #

Match the log entries to the quantifiers in your uploaded SMT file:

• For the quantifier at log line 50, the bound variable #40 (from line 32) corresponds to the quantified variable declared in the SMT source.
• For the quantifier at log line 63, #58 (from line 52) is the bound variable, even though it was constructed as a regular term—this is Z3's way of tying the variable binding to proof steps.

内容的提问来源于stack exchange，提问作者Nils