# mathlibdocumentation

core / init.meta.case_tag

# Case tags #

Case tags are an internal mechanism used by certain tactics to communicate with each other. They are generated by the tactics cases, induction and with_cases ('cases-like tactics'), which generate goals corresponding to the 'cases' of an inductive hypothesis. Each of these goals carries a case tag. They are consumed by the case tactic, which focuses on one of these cases. Their purpose is twofold:

1. Give intuitive names to case goals. For example, when performing induction on a natural number, two cases are generated: one tagged with nat.zero and one tagged with nat.succ. Users can then focus on e.g. the second goal with case succ {...}.
2. Communicate which new hypotheses were introduced by the cases-like tactic that generated the goal. For example, when performing induction on a list α, the cons case introduces two hypotheses corresponding to the two arguments of the cons constructor. case allows users to name these with case cons : x xs {...}. To perform this renaming, case needs to know which hypotheses to rename; this information is contained in the case tag for the cons goal.

## Module contents #

This module defines

1. what a case tag is (see case_tag);
2. how to render a case_tag as a list of names (see render);
3. how to parse a case_tag from a list of names (see parse);
4. how to match a case_tag with a sequence of names given by the user (see match_tag).
inductive tactic.interactive.case_tag  :
Type
• pi :
• hyps :

A case tag carries the following information:

1. A list of names identifying the case ('case names'). This is usually a list of constructor names, one for each case split that was performed. For example, the sequence of tactics cases n; cases xs, where n is a natural number and xs is a list, will generate four cases tagged as follows:

nat.zero, list.nil
nat.zero, list.cons
nat.succ, list.nil
nat.succ, list.cons


Note: In the case tag, the case names are stored in reverse order. Thus, the case names of the first case tag would be list.nil, nat.zero. This is because when printing a goal tag (as part of a goal state), Lean prints all non-internal names in reverse order.

2. Information about the arguments introduced by the cases-like tactic. Different tactics work slightly different in this regard:

1. The with_cases tactic generates goals where the target quantifies over any added hypotheses. For example, with_cases { cases xs }, where xs is a list α, will generate a target of the form α → list α → ... in the cons case, where the two arguments correspond to the two arguments of the cons constructor. Goals of this form are tagged with a pi case tag (since the target is a pi type). In addition to the case names, it contains a natural number, num_arguments, which specifies how many of the arguments that the target quantifies over were introduced by with_cases.

For example, given n : ℕ and xs : list α, the fourth goal generated by with_cases { cases n; induction xs } has this form:

...
⊢ ℕ → α → ∀ (xs' : list α), P xs' → ...


The corresponding case tag is

pi [list.cons, nat.succ] 4


since the first four arguments of the target were introduced by with_cases {...}.

2. The cases and induction tactics do not add arguments to the target, but rather introduce them as hypotheses in the local context. Goals of this form are tagged with a hyps case tag. In addition to the case names, it contains a list of unique names of the hypotheses that were introduced.

For example, given xs : list α, the second goal generated by induction xs has this form:

...
x : α
xs' : list α
ih_xs' : P xs'
⊢ ...


The corresponding goal tag is

hyps [list.cons] [<x>, <xs'>, <ih_xs'>]


where  denotes the unique name of a hypothesis h.

Note: Many tactics do not preserve the unique names of hypotheses (particularly those tactics that use revert). Therefore, a hyps case tag is only guaranteed to be valid directly after it was generated.

@[protected]
@[protected]
Equations
@[protected]
Equations
@[protected, instance]
@[protected, instance]
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@[protected, instance]
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The constructor names associated with a case tag.

Renders a case tag to a goal tag (i.e. a list of names), according to the following schema:

• A pi tag with names N₀ ... Nₙ and number of arguments a is rendered as
[_case.pi.a, N₀, ..., Nₙ]

• A hyps tag with names N₀ ... Nₙ and argument names A₀ ... Aₘ is rendered as
[_case.hyps, A₀._arg, ..., Aₘ._arg, N₀, ..., Nₙ]


Creates a pi case tag from an input tag in_tag. The names of the resulting tag are the non-internal names in in_tag (in the order in which they appear in in_tag). num_arguments is the number of arguments of the resulting tag.

Creates a hyps case tag from an input tag in_tag. The names of the resulting tag are the non-internal names in in_tag (in the order in which they appear in in_tag). arguments is the list of unique hypothesis names of the resulting tag.

Parses a case tag from the list of names produced by render.

Indicates the result of matching a list of names against the names of a case tag. See match_tag.

The 'minimum' of two match results:

• If any of the arguments is no_match, the result is no_match.
• Otherwise, if any of the arguments is fuzzy_match, the result is fuzzy_match.
• Otherwise (iff both arguments are exact_match), the result is exact_match.
Equations

Match the names of a case tag against a user-supplied list of names ns. For this purpose, we consider the names in reverse order, i.e. in the order in which they are displayed to the user. The matching then uses the following rules:

• If ns is exactly the same sequence of names as names, this is an exact match.
• If ns is a suffix of names, this is a fuzzy match. Additionally, each of the names in ns may be a suffix of the corresponding name in names.
• Otherwise, we have no match.

Thus, the tag

nat.zero, list.nil


is matched by any of these tags:

nat.zero, list.nil (exact match)
nat.zero, nil      (fuzzy match)
zero, nil          (fuzzy match)
nil                (fuzzy match)
`