Lean.Meta.Tactic.Grind.Parser

The grind_pattern command can be used to manually select a pattern for theorem instantiation. Enabling the option trace.grind.ematch.instance causes grind to print a trace message for each theorem instance it generates, which can be helpful when determining patterns.

When multiple patterns are specified together, all of them must match in the current context before grind attempts to instantiate the theorem. This is referred to as a multi-pattern. This is useful for theorems such as transitivity rules, where multiple premises must be simultaneously present for the rule to apply.

In the following example, R is a transitive binary relation over Int.

opaque R : Int → Int → Prop
axiom Rtrans {x y z : Int} : R x y → R y z → R x z

To use the fact that R is transitive, grind must already be able to satisfy both premises. This is represented using a multi-pattern:

grind_pattern Rtrans => R x y, R y z

example {a b c d} : R a b → R b c → R c d → R a d := by
  grind

The multi-pattern R x y, R y z instructs grind to instantiate Rtrans only when both R x y and R y z are available in the context. In the example, grind applies Rtrans to derive R a c from R a b and R b c, and can then repeat the same reasoning to deduce R a d from R a c and R c d.

You can add constraints to restrict theorem instantiation. For example:

grind_pattern extract_extract => (as.extract i j).extract k l where
  as =/= #[]

The constraint instructs grind to instantiate the theorem only if as is not definitionally equal to #[].

Constraints #

x =/= term: The term bound to x (one of the theorem parameters) is not definitionally equal to term. The term may contain holes (i.e., _).
x =?= term: The term bound to x is definitionally equal to term. The term may contain holes (i.e., _).
size x < n: The term bound to x has size less than n. Implicit arguments and binder types are ignored when computing the size.
depth x < n: The term bound to x has depth less than n.
is_ground x: The term bound to x does not contain local variables or meta-variables.
is_value x: The term bound to x is a value. That is, it is a constructor fully applied to value arguments, a literal (Nat, Int, String, etc.), or a lambda fun x => t.
is_strict_value x: Similar to is_value, but without lambdas.
not_value x: The term bound to x is a not value (see is_value).
not_strict_value x: Similar to not_value, but without lambdas.
gen < n: The theorem instance has generation less than n. Recall that each term is assigned a generation, and terms produced by theorem instantiation have a generation that is one greater than the maximal generation of all the terms used to instantiate the theorem. This constraint complements the gen option available in grind.
max_insts < n: A new instance is generated only if less than n instances have been generated so far.
guard e: The instantiation is delayed until grind learns that e is true in this state.
check e: Similar to guard e, but grind checks whether e is implied by its current state by assuming ¬ e and trying to deduce an inconsistency.

Example #

Consider the following example where f is a monotonic function

opaque f : Nat → Nat
axiom fMono : x ≤ y → f x ≤ f y

and you want to instruct grind to instantiate fMono for every pair of terms f x and f y when x ≤ y and x is not definitionally equal to y. You can use

grind_pattern fMono => f x, f y where
  guard x ≤ y
  x =/= y

Then, in the following example, only three instances are generated.

/--
trace: [grind.ematch.instance] fMono: a ≤ f a → f a ≤ f (f a)
[grind.ematch.instance] fMono: f a ≤ f (f a) → f (f a) ≤ f (f (f a))
[grind.ematch.instance] fMono: a ≤ f (f a) → f a ≤ f (f (f a))
-/
#guard_msgs in
example : f b = f c → a ≤ f a → f (f a) ≤ f (f (f a)) := by
  set_option trace.grind.ematch.instance true in
  grind

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source

def Lean.Parser.Command.initGrindNorm :

Parser

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