A path locates zero or more subterms of a given term by mimicking the term structure up to:

Warning: the format for a path for a match … with expression is restricted to: match path with [ _ ⇒ path | … | _ ⇒ path ] Its semantics is the following: the n-th "_ ⇒ path" branch is matched against the n-th constructor of the inductive data type. The head λ-abstractions of path are matched against the corresponding constructor arguments.

For instance, the path ∀_,_:?.(? ? % ?)→(? ? ? %) locates at once the subterms x+y and x*y in the term ∀x,y:nat.x+y=1→0=x*y (where the notation A=B hides the term (eq T A B) for some type T).

A simple pattern extends paths to locate subterms in a whole sequent. In particular, the pattern { H: p K: q ⊢ r } locates at once all the subterms located by the pattern r in the conclusion of the sequent and by the patterns p and q in the hypotheses H and K of the sequent.

If no list of hypotheses is provided in a simple pattern, no subterm is selected in the hypothesis. If the ⊢ p part of the pattern is not provided, no subterm will be matched in the conclusion if at least one hypothesis is provided; otherwise the whole conclusion is selected.

Finally, a full pattern is interpreted in three steps. In the first step the match T in part is ignored and a set S of subterms is located as for the case of simple patterns. In the second step the term T is parsed and interpreted in the context of each subterm s ∈ S. In the last term for each s ∈ S the interpreted term T computed in the previous step is looked for. The final set of subterms located by the full pattern is the set of occurrences of the interpreted T in the subterms s.

A full pattern can always be replaced by a simple pattern, often at the cost of increased verbosity or decreased readability.

Example: the pattern { match x+y in ⊢ ∀_,_:?.(? ? % ?) } locates only the first occurrence of x+y in the sequent x,y: nat ⊢ ∀z,w:nat. (x+y) * (z+w) = z * (x+y) + w * (x+y). The corresponding simple pattern is { ⊢ ∀_,_:?.(? ? (? % ?) ?) }.

Every tactic that acts on subterms of the selected sequents have a pattern argument for uniformity. To automatically generate a simple pattern:

Reduction kinds are normalization functions that transform a term to a convertible but simpler one. Each reduction kind can be used both as a tactic argument and as a stand-alone tactic.