HepLean Documentation

Mathlib.Combinatorics.Quiver.Cast

Rewriting arrows and paths along vertex equalities #

This files defines Hom.cast and Path.cast (and associated lemmas) in order to allow rewriting arrows and paths along equalities of their endpoints.

Rewriting arrows along equalities of vertices #

source
def Quiver.Hom.cast {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
u' ⟶ v'

Change the endpoints of an arrow using equalities.

Equations
Instances For
    source
    theorem Quiver.Hom.cast_eq_cast {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
    Quiver.Hom.cast hu hv e = cast ⋯ e
    source
    @[simp]
    theorem Quiver.Hom.cast_rfl_rfl {U : Type u_1} [Quiver U] {u v : U} (e : u ⟶ v) :
    Quiver.Hom.cast ⋯ ⋯ e = e
    source
    @[simp]
    theorem Quiver.Hom.cast_cast {U : Type u_1} [Quiver U] {u v u' v' u'' v'' : U} (e : u ⟶ v) (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :
    Quiver.Hom.cast hu' hv' (Quiver.Hom.cast hu hv e) = Quiver.Hom.cast ⋯ ⋯ e
    source
    theorem Quiver.Hom.cast_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
    HEq (Quiver.Hom.cast hu hv e) e
    source
    theorem Quiver.Hom.cast_eq_iff_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) (e' : u' ⟶ v') :
    Quiver.Hom.cast hu hv e = e' ↔ HEq e e'
    source
    theorem Quiver.Hom.eq_cast_iff_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) (e' : u' ⟶ v') :
    e' = Quiver.Hom.cast hu hv e ↔ HEq e' e

    Rewriting paths along equalities of vertices #

    source
    def Quiver.Path.cast {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Quiver.Path u v) :
    Quiver.Path u' v'

    Change the endpoints of a path using equalities.

    Equations
    Instances For
      source
      theorem Quiver.Path.cast_eq_cast {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Quiver.Path u v) :
      Quiver.Path.cast hu hv p = cast ⋯ p
      source
      @[simp]
      theorem Quiver.Path.cast_rfl_rfl {U : Type u_1} [Quiver U] {u v : U} (p : Quiver.Path u v) :
      Quiver.Path.cast ⋯ ⋯ p = p
      source
      @[simp]
      theorem Quiver.Path.cast_cast {U : Type u_1} [Quiver U] {u v u' v' u'' v'' : U} (p : Quiver.Path u v) (hu : u = u') (hv : v = v') (hu' : u' = u'') (hv' : v' = v'') :
      Quiver.Path.cast hu' hv' (Quiver.Path.cast hu hv p) = Quiver.Path.cast ⋯ ⋯ p
      source
      @[simp]
      theorem Quiver.Path.cast_nil {U : Type u_1} [Quiver U] {u u' : U} (hu : u = u') :
      Quiver.Path.cast hu hu Quiver.Path.nil = Quiver.Path.nil
      source
      theorem Quiver.Path.cast_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Quiver.Path u v) :
      HEq (Quiver.Path.cast hu hv p) p
      source
      theorem Quiver.Path.cast_eq_iff_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Quiver.Path u v) (p' : Quiver.Path u' v') :
      Quiver.Path.cast hu hv p = p' ↔ HEq p p'
      source
      theorem Quiver.Path.eq_cast_iff_heq {U : Type u_1} [Quiver U] {u v u' v' : U} (hu : u = u') (hv : v = v') (p : Quiver.Path u v) (p' : Quiver.Path u' v') :
      p' = Quiver.Path.cast hu hv p ↔ HEq p' p
      source
      theorem Quiver.Path.cast_cons {U : Type u_1} [Quiver U] {u v w u' w' : U} (p : Quiver.Path u v) (e : v ⟶ w) (hu : u = u') (hw : w = w') :
      Quiver.Path.cast hu hw (p.cons e) = (Quiver.Path.cast hu ⋯ p).cons (Quiver.Hom.cast ⋯ hw e)
      source
      theorem Quiver.cast_eq_of_cons_eq_cons {U : Type u_1} [Quiver U] {u v v' w : U} {p : Quiver.Path u v} {p' : Quiver.Path u v'} {e : v ⟶ w} {e' : v' ⟶ w} (h : p.cons e = p'.cons e') :
      Quiver.Path.cast ⋯ ⋯ p = p'
      source
      theorem Quiver.hom_cast_eq_of_cons_eq_cons {U : Type u_1} [Quiver U] {u v v' w : U} {p : Quiver.Path u v} {p' : Quiver.Path u v'} {e : v ⟶ w} {e' : v' ⟶ w} (h : p.cons e = p'.cons e') :
      Quiver.Hom.cast ⋯ ⋯ e = e'
      source
      theorem Quiver.eq_nil_of_length_zero {U : Type u_1} [Quiver U] {u v : U} (p : Quiver.Path u v) (hzero : p.length = 0) :
      Quiver.Path.cast ⋯ ⋯ p = Quiver.Path.nil