mathlib documentation

core / init.meta.congr_lemma

inductive congr_arg_kind  :
Type

This is a kind attached to an argument of a congruence lemma that tells the simplifier how to fill it in.

  • fixed: It is a parameter for the congruence lemma, the parameter occurs in the left and right hand sides. For example the α in the congruence generated from f: Π {α : Type} α → α.
  • fixed_no_param: It is not a parameter for the congruence lemma, the lemma was specialized for this parameter. This only happens if the parameter is a subsingleton/proposition, and other parameters depend on it.
  • eq: The lemma contains three parameters for this kind of argument a_i, b_i and (eq_i : a_i = b_i). a_i and b_i represent the left and right hand sides, and eq_i is a proof for their equality. For example the second argument in f: Π {α : Type}, α → α.
  • cast: corresponds to arguments that are subsingletons/propositions. For example the p in the congruence generated from f : Π (x y : ℕ) (p: x < y), ℕ.
  • heq The lemma contains three parameters for this kind of argument a_i, b_i and (eq_i : a_i == b_i). a_i and b_i represent the left and right hand sides, and eq_i is a proof for their heterogeneous equality.
meta structure congr_lemma  :
Type

A congruence lemma is a proof that two terms are equal using a congruence proof generated by mk_congr_lemma_simp and friends. See the docstring for mk_congr_lemma_simp and congr_arg_kind for more information. The conclusion is prepended by a set of arguments. arg_kinds gives a suggestion of how that argument should be filled in using a simplifier.

mk_congr_lemma_simp f nargs md creates a congruence lemma for the simplifier for the given function argument f. If nargs is not none, then it tries to create a lemma for an application of arity nargs. If nargs is none then the number of arguments will be guessed from the type signature of f.

That is, given f : Π {α β γ δ : Type}, α → β → γ → δ and nargs = some 6, we get a congruence lemma:

{ type :=  (α β γ δ : Type),  (a₁ a₂ : α), a₁ = a₂   (b₁ b₂ : β), b₁ = b₂  f a₁ b₁ = f a₂ b₂
, proof := ...
, arg_kinds := [fixed, fixed, fixed, fixed, eq,eq]
}

See the docstrings for the cases of congr_arg_kind for more detail on how arg_kinds are chosen. The system chooses the arg_kinds depending on what the other arguments depend on and whether the arguments have subsingleton types.

Note that the number of arguments that proof takes can be inferred from arg_kinds: arg_kinds.sum (fixed,cast ↦ 1 | eq,heq ↦ 3 | fixed_no_param ↦ 0).

From congr_lemma.cpp:

Create a congruence lemma that is useful for the simplifier. In this kind of lemma, if the i-th argument is a Cast argument, then the lemma contains an input a_i representing the i-th argument in the left-hand-side, and it appears with a cast (e.g., eq.drec ... a_i ...) in the right-hand-side. The idea is that the right-hand-side of this lemma "tells" the simplifier how the resulting term looks like.

Create a specialized theorem using (a prefix of) the arguments of the given application.

An example of usage can be found in tests/lean/simp_subsingleton.lean. For more information on specialization see the comment in the method body for get_specialization_prefix_size in src/library/fun_info.cpp.

Similar to mk_congr_lemma_simp, this will make a congr_lemma object. The difference is that for each congr_arg_kind.cast argument, two proof arguments are generated.

Consider some function f : Π (x : ℕ) (p : x < 4), ℕ.

  • mk_congr_simp will produce a congruence lemma with type ∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4), f x p = f x_1 _.
  • mk_congr will produce a congruence lemma with type ∀ (x x_1 : ℕ) (e_1 : x = x_1) (p : x < 4) (p_1 : x_1 < 4), f x p = f x_1 p_1.

From congr_lemma.cpp:

Create a congruence lemma for the congruence closure module. In this kind of lemma, if the i-th argument is a Cast argument, then the lemma contains two inputs a_i and b_i representing the i-th argument in the left-hand-side and right-hand-side. This lemma is based on the congruence lemma for the simplifier. It uses subsinglenton elimination to show that the congr-simp lemma right-hand-side is equal to the right-hand-side of this lemma.

Create a specialized theorem using (a prefix of) the arguments of the given application.

For more information on specialization see the comment in the method body for get_specialization_prefix_size in src/library/fun_info.cpp.

Make a congruence lemma using hetrogeneous equality heq instead of eq. For example mk_hcongr_lemma (f : Π (α : ℕ → Type) (n:ℕ) (b:α n), ℕ )` will make

{ type :=  α α', α = α'   n n', n = n'   (b : α n) (b' : α' n'), b == b'  f α n b == f α' n' b'
, proof := ...
, arg_kinds := [eq,eq,heq]
}

(Using merely mk_congr_lemma instead will produce [fixed,fixed,eq] instaed.)