card_union_le
plain-language theorem explainer
The subadditivity bound on the cardinality of a union of two finite sets is recorded to support ball-size estimates inside a minimal causality module. Workers deriving discrete cone growth bounds cite it when splitting a reachable set into an earlier ball and its new neighbors. The proof is a one-line wrapper that invokes the Mathlib primitive Finset.card_union_le.
Claim. $|s| + |t| = |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |t| - |s| + |
background
The ConeBound module supplies lightweight definitions for bounding the size of forward balls under a bounded-degree step relation, avoiding full graph-theory imports. ballFS x n denotes the finite set of points reachable from x in at most n steps via the neighbor relation B. The present lemma supplies the elementary union-cardinality inequality required to control how the ball at step n+1 grows from the ball at step n by adding at most d new neighbors per point.
proof idea
One-line wrapper that applies Finset.card_union_le.
why it matters
This supplies the union-cardinality step inside card_ballFS_succ_le, which yields the linear growth bound (1+d) for successive balls. In the Recognition framework it closes the local estimate required for causal cone volume control before invoking the global forcing chain (T0-T8).
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