mathlib documentation

tactic.reassoc_axiom

Tools to reformulate category-theoretic axioms in a more associativity-friendly way

The reassoc attribute

The reassoc attribute can be applied to a lemma

@[reassoc]
lemma some_lemma : foo  bar = baz := ...

and produce

lemma some_lemma_assoc {Y : C} (f : X  Y) : foo  bar  f = baz  f := ...

The name of the produced lemma can be specified with @[reassoc other_lemma_name]. If simp is added first, the generated lemma will also have the simp attribute.

The reassoc_axiom command

When declaring a class of categories, the axioms can be reformulated to be more amenable to manipulation in right associated expressions:

class some_class (C : Type) [category C] :=
(foo : Π X : C, X  X)
(bar :  {X Y : C} (f : X  Y), foo X  f = f  foo Y)

reassoc_axiom some_class.bar

Here too, the reassoc attribute can be used instead. It works well when combined with simp:

attribute [simp, reassoc] some_class.bar
meta def tactic.get_cat_inst  :

From an expression f ≫ g, extract the expression representing the category instance.

(internals for @[reassoc]) Given a lemma of the form f ≫ g = h, proves a new lemma of the form h : ∀ {W} (k), f ≫ (g ≫ k) = h ≫ k, and returns the type and proof of this lemma.

meta def tactic.reassoc_axiom (n : name) :
(name := n.append_suffix "_assoc")tactic unit

(implementation for @[reassoc]) Given a declaration named n of the form f ≫ g = h, proves a new lemma named n' of the form ∀ {W} (k), f ≫ (g ≫ k) = h ≫ k.

The reassoc attribute can be applied to a lemma

@[reassoc]
lemma some_lemma : foo  bar = baz := ...

to produce

lemma some_lemma_assoc {Y : C} (f : X  Y) : foo  bar  f = baz  f := ...

The name of the produced lemma can be specified with @[reassoc other_lemma_name]. If simp is added first, the generated lemma will also have the simp attribute.

When declaring a class of categories, the axioms can be reformulated to be more amenable to manipulation in right associated expressions:

class some_class (C : Type) [category C] :=
(foo : Π X : C, X  X)
(bar :  {X Y : C} (f : X  Y), foo X  f = f  foo Y)

reassoc_axiom some_class.bar

The above will produce:

lemma some_class.bar_assoc {Z : C} (g : Y  Z) :
  foo X  f  g = f  foo Y  g := ...

Here too, the reassoc attribute can be used instead. It works well when combined with simp:

attribute [simp, reassoc] some_class.bar

reassoc h, for assumption h : x ≫ y = z, creates a new assumption h : ∀ {W} (f : Z ⟶ W), x ≫ y ≫ f = z ≫ f. reassoc! h, does the same but deletes the initial h assumption. (You can also add the attribute @[reassoc] to lemmas to generate new declarations generalized in this way.)

def tactic.calculated_Prop {α : Sort u_1} :
Prop → α → Prop

Equations
theorem category_theory.reassoc_of {α : Sort u_1} (hh : α) {β : Prop} :
(tactic.calculated_Prop β hh . "derive_reassoc_proof") → β

With h : x ≫ y ≫ z = x (with universal quantifiers tolerated), reassoc_of h : ∀ {X'} (f : W ⟶ X'), x ≫ y ≫ z ≫ f = x ≫ f.

The type and proof of reassoc_of h is generated by tactic.derive_reassoc_proof which make reassoc_of meta-programming adjacent. It is not called as a tactic but as an expression. The goal is to avoid creating assumptions that are dismissed after one use:

example (X Y Z W : C) (x : X  Y) (y : Y  Z) (z z' : Z  W) (w : X  Z)
  (h : x  y = w)
  (h' : y  z = y  z') :
  x  y  z = w  z' :=
begin
  rw [h',reassoc_of h],
end