top_intro
plain-language theorem explainer
The always-true predicate on finite traces holds at every trace. Anyone assembling the propositional surface of primitive recognition trace logic cites this as the introduction rule for truth. The proof is a one-line appeal to triviality, since the predicate is constantly True.
Claim. For every finite trace $T$, the top (truth) predicate holds at $T$: its holding relation is the constantly-true predicate on traces.
background
In Primitive Recognition Calculus, a finite trace is built inductively: it is either empty or obtained by extending a prior trace by one distinction act. Trace predicates are stable properties of such traces: if a predicate holds at $T$ and $U$ extends $T$, it continues to hold at $U$.
The top predicate is defined as the stable predicate whose holding relation is $\mathsf{True}$ at every trace. Its stability proof is immediate. This module packages the connectives and quantifiers of a small logic over traces; top is the unit of that logic.
Upstream, the inductive type of traces and the definition of the top predicate fix the meaning of the claim. Name collisions with unrelated top/T symbols elsewhere in the monorepo are irrelevant here.
proof idea
One-line wrapper. Unfolding the top predicate, the goal is True at the given trace, discharged by trivial. No lemmas about extension, stability, or other connectives are required.
why it matters
This is the introduction rule for truth in the trace-logic layer of Primitive Recognition Calculus. Downstream, trace_logic_certificate packages the propositional surface with truth_intro set to this theorem, alongside conjunction introduction and related rules. It is bookkeeping rather than a deep forcing step: it certifies that the constantly-true stable predicate is inhabited at every trace, so the certificate's truth slot is filled without sorry. It does not itself touch the T0–T8 forcing chain, RCL, or the phi ladder; it only supplies the logical unit used when those layers speak in trace predicates.
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