Datatypes for bicategory like structures

This file defines the basic datatypes for bicategory like structures. We will use these datatypes to write tactics that can be applied to both monoidal categories and bicategories:

• Obj: objects type
• Atom₁: atomic 1-morphisms type
• Mor₁: 1-morphisms type
• Atom: atomic non-structural 2-morphisms type
• Mor₂: 2-morphisms type
• AtomIso: atomic non-structural 2-isomorphisms type
• Mor₂Iso: 2-isomorphisms type
• NormalizedHom: normalized 1-morphisms type

A term of these datatypes wraps the corresponding Expr term, which can be extracted by e.g. η.e for η : Mor₂.

The operations of these datatypes are defined in a monad m with the corresponding typeclasses:

• MonadMor₁: operations on Mor₁
• MonadMor₂Iso: operations on Mor₂Iso
• MonadMor₂: operations on Mor₂

For example, a monad m with [MonadMor₂ m] provides the operation MonadMor₂.comp₂M : Mor₂Iso → Mor₂Iso → m Mor₂Iso, which constructs the expression for the composition η ≫ θ of 2-morphisms η and θ in the monad m.

structure Mathlib.Tactic.BicategoryLike.Obj : Type

Expressions for objects.

• e? : Option Lean.Expr

  Extracts a lean expression from an Obj term. Return none in the monoidal category context.

instance Mathlib.Tactic.BicategoryLike.instInhabitedObj : Inhabited Obj

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedObj = { default := Mathlib.Tactic.BicategoryLike.instInhabitedObj.default }

def Mathlib.Tactic.BicategoryLike.instInhabitedObj.default : Obj

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedObj.default = { e? := default }

def Mathlib.Tactic.BicategoryLike.Obj.e (a : Obj) : Lean.Expr

Extract a lean expression from an Obj term.

Equations

• a.e = a.e?.get!

structure Mathlib.Tactic.BicategoryLike.Atom₁ : Type

Expressions for atomic 1-morphisms.

• e : Lean.Expr

  Extract a lean expression from an Atom₁ term.

• src : Obj

  The domain of the 1-morphism.

• tgt : Obj

  The codomain of the 1-morphism.

def Mathlib.Tactic.BicategoryLike.instInhabitedAtom₁.default : Atom₁

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtom₁.default = { e := default, src := default, tgt := default }

instance Mathlib.Tactic.BicategoryLike.instInhabitedAtom₁ : Inhabited Atom₁

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtom₁ = { default := Mathlib.Tactic.BicategoryLike.instInhabitedAtom₁.default }

class Mathlib.Tactic.BicategoryLike.MkAtom₁ (m : Type → Type) : Type

A monad equipped with the ability to construct Atom₁ terms.

• ofExpr (e : Lean.Expr) : m Atom₁

  Construct a Atom₁ term from a lean expression.

inductive Mathlib.Tactic.BicategoryLike.Mor₁ : Type

Expressions for 1-morphisms.

• id (e : Lean.Expr) (a : Obj) : Mor₁

  id e a is the expression for 𝟙 a, where e is the underlying lean expression.

• comp (e : Lean.Expr) : Mor₁ → Mor₁ → Mor₁

  comp e f g is the expression for f ≫ g, where e is the underlying lean expression.

• of : Atom₁ → Mor₁

  The expression for an atomic 1-morphism.

instance Mathlib.Tactic.BicategoryLike.instInhabitedMor₁ : Inhabited Mor₁

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₁ = { default := Mathlib.Tactic.BicategoryLike.instInhabitedMor₁.default }

def Mathlib.Tactic.BicategoryLike.instInhabitedMor₁.default : Mor₁

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₁.default = Mathlib.Tactic.BicategoryLike.Mor₁.id default default

class Mathlib.Tactic.BicategoryLike.MkMor₁ (m : Type → Type) : Type

A monad equipped with the ability to construct Mor₁ terms.

• ofExpr (e : Lean.Expr) : m Mor₁

  Construct a Mor₁ term from a lean expression.

def Mathlib.Tactic.BicategoryLike.Mor₁.e : Mor₁ → Lean.Expr

The underlying lean expression of a 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₁.id e a).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₁.comp e a a_1).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₁.of a).e = a.e

def Mathlib.Tactic.BicategoryLike.Mor₁.src : Mor₁ → Obj

The domain of a 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₁.id e a).src = a
• (Mathlib.Tactic.BicategoryLike.Mor₁.comp e a a_1).src = a.src
• (Mathlib.Tactic.BicategoryLike.Mor₁.of a).src = a.src

def Mathlib.Tactic.BicategoryLike.Mor₁.tgt : Mor₁ → Obj

The codomain of a 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₁.id e a).tgt = a
• (Mathlib.Tactic.BicategoryLike.Mor₁.comp e a a_1).tgt = a_1.tgt
• (Mathlib.Tactic.BicategoryLike.Mor₁.of a).tgt = a.tgt

def Mathlib.Tactic.BicategoryLike.Mor₁.toList : Mor₁ → List Atom₁

Converts a 1-morphism into a list of its components.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₁.id e a).toList = []
• (Mathlib.Tactic.BicategoryLike.Mor₁.comp e a a_1).toList = a.toList ++ a_1.toList
• (Mathlib.Tactic.BicategoryLike.Mor₁.of a).toList = [a]

class Mathlib.Tactic.BicategoryLike.MonadMor₁ (m : Type → Type) : Type

A monad equipped with the ability to manipulate 1-morphisms.

• id₁M (a : Obj) : m Mor₁

  The expression for 𝟙 a.

• comp₁M (f g : Mor₁) : m Mor₁

  The expression for f ≫ g.

structure Mathlib.Tactic.BicategoryLike.CoherenceHom : Type

Expressions for coherence isomorphisms (i.e., structural 2-morphisms given by BicategoricalCoherence.iso).

• e : Lean.Expr

  The underlying lean expression of a coherence isomorphism.

• src : Mor₁

  The domain of a coherence isomorphism.

• tgt : Mor₁

  The codomain of a coherence isomorphism.

• inst : Lean.Expr

  The BicategoricalCoherence instance.

• unfold : Lean.Expr

  Extract the structural 2-isomorphism.

instance Mathlib.Tactic.BicategoryLike.instInhabitedCoherenceHom : Inhabited CoherenceHom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedCoherenceHom = { default := Mathlib.Tactic.BicategoryLike.instInhabitedCoherenceHom.default }

def Mathlib.Tactic.BicategoryLike.instInhabitedCoherenceHom.default : CoherenceHom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedCoherenceHom.default = { e := default, src := default, tgt := default, inst := default, unfold := default }

structure Mathlib.Tactic.BicategoryLike.AtomIso : Type

Expressions for atomic non-structural 2-isomorphisms.

• e : Lean.Expr

  The underlying lean expression of an AtomIso term.

• src : Mor₁

  The domain of a 2-isomorphism.

• tgt : Mor₁

  The codomain of a 2-isomorphism.

def Mathlib.Tactic.BicategoryLike.instInhabitedAtomIso.default : AtomIso

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtomIso.default = { e := default, src := default, tgt := default }

instance Mathlib.Tactic.BicategoryLike.instInhabitedAtomIso : Inhabited AtomIso

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtomIso = { default := Mathlib.Tactic.BicategoryLike.instInhabitedAtomIso.default }

inductive Mathlib.Tactic.BicategoryLike.StructuralAtom : Type

Expressions for atomic structural 2-morphisms.

• associator (e : Lean.Expr) (f g h : Mor₁) : StructuralAtom

  The expression for the associator α_ f g h.

• leftUnitor (e : Lean.Expr) (f : Mor₁) : StructuralAtom

  The expression for the left unitor λ_ f.

• rightUnitor (e : Lean.Expr) (f : Mor₁) : StructuralAtom

  The expression for the right unitor ρ_ f.

• id (e : Lean.Expr) (f : Mor₁) : StructuralAtom
• coherenceHom (α : CoherenceHom) : StructuralAtom

def Mathlib.Tactic.BicategoryLike.instInhabitedStructuralAtom.default : StructuralAtom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedStructuralAtom.default = Mathlib.Tactic.BicategoryLike.StructuralAtom.associator default default default default

instance Mathlib.Tactic.BicategoryLike.instInhabitedStructuralAtom : Inhabited StructuralAtom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedStructuralAtom = { default := Mathlib.Tactic.BicategoryLike.instInhabitedStructuralAtom.default }

inductive Mathlib.Tactic.BicategoryLike.Mor₂Iso : Type

Expressions for 2-isomorphisms.

• structuralAtom (α : StructuralAtom) : Mor₂Iso
• comp (e : Lean.Expr) (f g h : Mor₁) (η θ : Mor₂Iso) : Mor₂Iso
• whiskerLeft (e : Lean.Expr) (f g h : Mor₁) (η : Mor₂Iso) : Mor₂Iso
• whiskerRight (e : Lean.Expr) (f g : Mor₁) (η : Mor₂Iso) (h : Mor₁) : Mor₂Iso
• horizontalComp (e : Lean.Expr) (f₁ g₁ f₂ g₂ : Mor₁) (η θ : Mor₂Iso) : Mor₂Iso
• inv (e : Lean.Expr) (f g : Mor₁) (η : Mor₂Iso) : Mor₂Iso
• coherenceComp (e : Lean.Expr) (f g h i : Mor₁) (α : CoherenceHom) (η θ : Mor₂Iso) : Mor₂Iso
• of (η : AtomIso) : Mor₂Iso

instance Mathlib.Tactic.BicategoryLike.instInhabitedMor₂Iso : Inhabited Mor₂Iso

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₂Iso = { default := Mathlib.Tactic.BicategoryLike.instInhabitedMor₂Iso.default }

def Mathlib.Tactic.BicategoryLike.instInhabitedMor₂Iso.default : Mor₂Iso

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₂Iso.default = Mathlib.Tactic.BicategoryLike.Mor₂Iso.structuralAtom default

class Mathlib.Tactic.BicategoryLike.MonadCoherehnceHom (m : Type → Type) : Type

A monad equipped with the ability to unfold BicategoricalCoherence.iso.

• unfoldM (α : CoherenceHom) : m Mor₂Iso

  Unfold a coherence isomorphism.

def Mathlib.Tactic.BicategoryLike.StructuralAtom.e : StructuralAtom → Lean.Expr

The underlying lean expression of a 2-isomorphism.

Equations

• (Mathlib.Tactic.BicategoryLike.StructuralAtom.associator e f g h).e = e
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.leftUnitor e f).e = e
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.rightUnitor e f).e = e
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.id e f).e = e
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.coherenceHom α).e = α.e

def Mathlib.Tactic.BicategoryLike.StructuralAtom.srcM {m : Type → Type} [Monad m] [MonadMor₁ m] : StructuralAtom → m Mor₁

The domain of a 2-isomorphism.

Equations

• One or more equations did not get rendered due to their size.
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.id e f).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.coherenceHom α).srcM = pure α.src

def Mathlib.Tactic.BicategoryLike.StructuralAtom.tgtM {m : Type → Type} [Monad m] [MonadMor₁ m] : StructuralAtom → m Mor₁

The codomain of a 2-isomorphism.

Equations

• One or more equations did not get rendered due to their size.
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.leftUnitor e f).tgtM = pure f
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.rightUnitor e f).tgtM = pure f
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.id e f).tgtM = pure f
• (Mathlib.Tactic.BicategoryLike.StructuralAtom.coherenceHom α).tgtM = pure α.tgt

def Mathlib.Tactic.BicategoryLike.Mor₂Iso.e : Mor₂Iso → Lean.Expr

The underlying lean expression of a 2-isomorphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.structuralAtom α).e = α.e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.comp e f g h η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerLeft e f g h η).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerRight e f g η h).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.horizontalComp e f₁ g₁ f₂ g₂ η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.inv e f g η).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.coherenceComp e f g h i α η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.of η).e = η.e

def Mathlib.Tactic.BicategoryLike.Mor₂Iso.srcM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂Iso → m Mor₁

The domain of a 2-isomorphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.structuralAtom α).srcM = α.srcM
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.comp e f g h η θ).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerLeft e f g h η).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f g
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerRight e f g η h).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f h
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.horizontalComp e f₁ g₁ f₂ g₂ η θ).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f₁ f₂
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.inv e f g η).srcM = pure g
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.coherenceComp e f g h i α η θ).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.of η).srcM = pure η.src

def Mathlib.Tactic.BicategoryLike.Mor₂Iso.tgtM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂Iso → m Mor₁

The codomain of a 2-isomorphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.structuralAtom α).tgtM = α.tgtM
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.comp e f g h η θ).tgtM = pure h
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerLeft e f g h η).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f h
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.whiskerRight e f g η h).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M g h
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.horizontalComp e f₁ g₁ f₂ g₂ η θ).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M g₁ g₂
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.inv e f g η).tgtM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.coherenceComp e f g h i α η θ).tgtM = pure i
• (Mathlib.Tactic.BicategoryLike.Mor₂Iso.of η).tgtM = pure η.tgt

class Mathlib.Tactic.BicategoryLike.MonadMor₂Iso (m : Type → Type) : Type

A monad equipped with the ability to construct Mor₂Iso terms.

• associatorM (f g h : Mor₁) : m StructuralAtom

  The expression for the associator α_ f g h.

• leftUnitorM (f : Mor₁) : m StructuralAtom

  The expression for the left unitor λ_ f.

• rightUnitorM (f : Mor₁) : m StructuralAtom

  The expression for the right unitor ρ_ f.

• id₂M (f : Mor₁) : m StructuralAtom

  The expression for the identity Iso.refl f.

• coherenceHomM (f g : Mor₁) (inst : Lean.Expr) : m CoherenceHom

  The expression for the coherence isomorphism ⊗𝟙 : f ⟶ g.

• comp₂M (η θ : Mor₂Iso) : m Mor₂Iso

  The expression for the composition η ≪≫ θ.

• whiskerLeftM (f : Mor₁) (η : Mor₂Iso) : m Mor₂Iso

  The expression for the left whiskering whiskerLeftIso f η.

• whiskerRightM (η : Mor₂Iso) (h : Mor₁) : m Mor₂Iso

  The expression for the right whiskering whiskerRightIso η h.

• horizontalCompM (η θ : Mor₂Iso) : m Mor₂Iso

  The expression for the horizontal composition η ◫ θ.

• symmM (η : Mor₂Iso) : m Mor₂Iso

  The expression for the inverse Iso.symm η.

• coherenceCompM (α : CoherenceHom) (η θ : Mor₂Iso) : m Mor₂Iso

  The expression for the coherence composition η ≪⊗≫ θ := η ≪≫ α ≪≫ θ.

def Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.associatorM' {m : Type → Type} [Monad m] [MonadMor₂Iso m] (f g h : Mor₁) : m Mor₂Iso

The expression for the associator α_ f g h.

Equations

• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.leftUnitorM' {m : Type → Type} [Monad m] [MonadMor₂Iso m] (f : Mor₁) : m Mor₂Iso

The expression for the left unitor λ_ f.

Equations

• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.rightUnitorM' {m : Type → Type} [Monad m] [MonadMor₂Iso m] (f : Mor₁) : m Mor₂Iso

The expression for the right unitor ρ_ f.

Equations

• One or more equations did not get rendered due to their size.

def Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.id₂M' {m : Type → Type} [Monad m] [MonadMor₂Iso m] (f : Mor₁) : m Mor₂Iso

The expression for the identity Iso.refl f.

Equations

• Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.id₂M' f = do let __do_lift ← Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.id₂M f pure (Mathlib.Tactic.BicategoryLike.Mor₂Iso.structuralAtom __do_lift)

def Mathlib.Tactic.BicategoryLike.MonadMor₂Iso.coherenceHomM' {m : Type → Type} [Monad m] [MonadMor₂Iso m] (f g : Mor₁) (inst : Lean.Expr) : m Mor₂Iso

The expression for the coherence isomorphism ⊗𝟙 : f ⟶ g.

Equations

• One or more equations did not get rendered due to their size.

structure Mathlib.Tactic.BicategoryLike.Atom : Type

Expressions for atomic non-structural 2-morphisms.

• e : Lean.Expr

  Extract a lean expression from an Atom expression.

• src : Mor₁

  The domain of a 2-morphism.

• tgt : Mor₁

  The codomain of a 2-morphism.

def Mathlib.Tactic.BicategoryLike.instInhabitedAtom.default : Atom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtom.default = { e := default, src := default, tgt := default }

instance Mathlib.Tactic.BicategoryLike.instInhabitedAtom : Inhabited Atom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedAtom = { default := Mathlib.Tactic.BicategoryLike.instInhabitedAtom.default }

structure Mathlib.Tactic.BicategoryLike.IsoLift : Type

Mor₂ expressions defined below will have the isoLift? : Option IsoLift field. For η : Mor₂ such that η.isoLift? = some isoLift, we have the following data:

• isoLift.e: an expression for a 2-isomorphism η', given as a Mor₂Iso term,
• isoLift.eq: a lean expression for the proof that η'.hom = η.

• e : Mor₂Iso

  The expression for the 2-isomorphism.

• eq : Lean.Expr

  The expression for the proof that the forward direction of the 2-isomorphism is equal to the original 2-morphism.

inductive Mathlib.Tactic.BicategoryLike.Mor₂ : Type

Expressions for 2-morphisms.

• isoHom (e : Lean.Expr) (isoLift : IsoLift) (iso : Mor₂Iso) : Mor₂

  The expression for Iso.hom.

• isoInv (e : Lean.Expr) (isoLift : IsoLift) (iso : Mor₂Iso) : Mor₂

  The expression for Iso.inv.

• id (e : Lean.Expr) (isoLift : IsoLift) (f : Mor₁) : Mor₂

  The expression for the identity 𝟙 f.

• comp (e : Lean.Expr) (isoLift? : Option IsoLift) (f g h : Mor₁) (η θ : Mor₂) : Mor₂

  The expression for the composition η ≫ θ.

• whiskerLeft (e : Lean.Expr) (isoLift? : Option IsoLift) (f g h : Mor₁) (η : Mor₂) : Mor₂

  The expression for the left whiskering f ◁ η with η : g ⟶ h.

• whiskerRight (e : Lean.Expr) (isoLift? : Option IsoLift) (f g : Mor₁) (η : Mor₂) (h : Mor₁) : Mor₂

  The expression for the right whiskering η ▷ h with η : f ⟶ g.

• horizontalComp (e : Lean.Expr) (isoLift? : Option IsoLift) (f₁ g₁ f₂ g₂ : Mor₁) (η θ : Mor₂) : Mor₂

  The expression for the horizontal composition η ◫ θ with η : f₁ ⟶ g₁ and θ : f₂ ⟶ g₂.

• coherenceComp (e : Lean.Expr) (isoLift? : Option IsoLift) (f g h i : Mor₁) (α : CoherenceHom) (η θ : Mor₂) : Mor₂

  The expression for the coherence composition η ⊗≫ θ := η ≫ α ≫ θ with η : f ⟶ g and θ : h ⟶ i.

• of (η : Atom) : Mor₂

  The expression for an atomic non-structural 2-morphism.

def Mathlib.Tactic.BicategoryLike.instInhabitedMor₂.default : Mor₂

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₂.default = Mathlib.Tactic.BicategoryLike.Mor₂.of default

instance Mathlib.Tactic.BicategoryLike.instInhabitedMor₂ : Inhabited Mor₂

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedMor₂ = { default := Mathlib.Tactic.BicategoryLike.instInhabitedMor₂.default }

class Mathlib.Tactic.BicategoryLike.MkMor₂ (m : Type → Type) : Type

A monad equipped with the ability to construct Mor₂ terms.

• ofExpr (e : Lean.Expr) : m Mor₂

  Construct a Mor₂ term from a lean expression.

def Mathlib.Tactic.BicategoryLike.Mor₂.e : Mor₂ → Lean.Expr

The underlying lean expression of a 2-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂.isoHom e isoLift iso).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.isoInv e isoLift iso).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.id e isoLift f).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.comp e isoLift? f g h η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerLeft e isoLift? f g h η).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerRight e isoLift? f g η h).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.horizontalComp e isoLift? f₁ g₁ f₂ g₂ η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.coherenceComp e isoLift? f g h i α η θ).e = e
• (Mathlib.Tactic.BicategoryLike.Mor₂.of η).e = η.e

def Mathlib.Tactic.BicategoryLike.Mor₂.isoLift? : Mor₂ → Option IsoLift

η.isoLift? is a pair of a 2-isomorphism η' and a proof that η'.hom = η. If no such η' is found, returns none. This function does not seek IsIso instance.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂.isoHom e isoLift iso).isoLift? = some isoLift
• (Mathlib.Tactic.BicategoryLike.Mor₂.isoInv e isoLift iso).isoLift? = some isoLift
• (Mathlib.Tactic.BicategoryLike.Mor₂.id e isoLift f).isoLift? = some isoLift
• (Mathlib.Tactic.BicategoryLike.Mor₂.comp e isoLift? f g h η θ).isoLift? = isoLift?
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerLeft e isoLift? f g h η).isoLift? = isoLift?
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerRight e isoLift? f g η h).isoLift? = isoLift?
• (Mathlib.Tactic.BicategoryLike.Mor₂.horizontalComp e isoLift? f₁ g₁ f₂ g₂ η θ).isoLift? = isoLift?
• (Mathlib.Tactic.BicategoryLike.Mor₂.coherenceComp e isoLift? f g h i α η θ).isoLift? = isoLift?
• (Mathlib.Tactic.BicategoryLike.Mor₂.of η).isoLift? = none

def Mathlib.Tactic.BicategoryLike.Mor₂.srcM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂ → m Mor₁

The domain of a 2-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂.isoHom e isoLift iso).srcM = iso.srcM
• (Mathlib.Tactic.BicategoryLike.Mor₂.isoInv e isoLift iso).srcM = iso.tgtM
• (Mathlib.Tactic.BicategoryLike.Mor₂.id e isoLift f).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂.comp e isoLift? f g h η θ).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerLeft e isoLift? f g h η).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f g
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerRight e isoLift? f g η h).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f h
• (Mathlib.Tactic.BicategoryLike.Mor₂.horizontalComp e isoLift? f₁ g₁ f₂ g₂ η θ).srcM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f₁ f₂
• (Mathlib.Tactic.BicategoryLike.Mor₂.coherenceComp e isoLift? f g h i α η θ).srcM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂.of η).srcM = pure η.src

def Mathlib.Tactic.BicategoryLike.Mor₂.tgtM {m : Type → Type} [Monad m] [MonadMor₁ m] : Mor₂ → m Mor₁

The codomain of a 2-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.Mor₂.isoHom e isoLift iso).tgtM = iso.tgtM
• (Mathlib.Tactic.BicategoryLike.Mor₂.isoInv e isoLift iso).tgtM = iso.srcM
• (Mathlib.Tactic.BicategoryLike.Mor₂.id e isoLift f).tgtM = pure f
• (Mathlib.Tactic.BicategoryLike.Mor₂.comp e isoLift? f g h η θ).tgtM = pure h
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerLeft e isoLift? f g h η).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M f h
• (Mathlib.Tactic.BicategoryLike.Mor₂.whiskerRight e isoLift? f g η h).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M g h
• (Mathlib.Tactic.BicategoryLike.Mor₂.horizontalComp e isoLift? f₁ g₁ f₂ g₂ η θ).tgtM = Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M g₁ g₂
• (Mathlib.Tactic.BicategoryLike.Mor₂.coherenceComp e isoLift? f g h i α η θ).tgtM = pure i
• (Mathlib.Tactic.BicategoryLike.Mor₂.of η).tgtM = pure η.tgt

class Mathlib.Tactic.BicategoryLike.MonadMor₂ (m : Type → Type) : Type

A monad equipped with the ability to manipulate 2-morphisms.

• homM (η : Mor₂Iso) : m Mor₂

  The expression for Iso.hom η.

• atomHomM (η : AtomIso) : m Atom

  The expression for Iso.hom η.

• invM (η : Mor₂Iso) : m Mor₂

  The expression for Iso.inv η.

• atomInvM (η : AtomIso) : m Atom

  The expression for Iso.inv η.

• id₂M (f : Mor₁) : m Mor₂

  The expression for the identity 𝟙 f.

• comp₂M (η θ : Mor₂) : m Mor₂

  The expression for the composition η ≫ θ.

• whiskerLeftM (f : Mor₁) (η : Mor₂) : m Mor₂

  The expression for the left whiskering f ◁ η.

• whiskerRightM (η : Mor₂) (h : Mor₁) : m Mor₂

  The expression for the right whiskering η ▷ h.

• horizontalCompM (η θ : Mor₂) : m Mor₂

  The expression for the horizontal composition η ◫ θ.

• coherenceCompM (α : CoherenceHom) (η θ : Mor₂) : m Mor₂

  The expression for the coherence composition η ⊗≫ θ := η ≫ α ≫ θ.

inductive Mathlib.Tactic.BicategoryLike.NormalizedHom : Type

Type of normalized 1-morphisms ((... ≫ h) ≫ g) ≫ f.

• nil (e : Mor₁) (a : Obj) : NormalizedHom

  The identity 1-morphism 𝟙 a.

• cons (e : Mor₁) : NormalizedHom → Atom₁ → NormalizedHom

  The cons composes an atomic 1-morphism at the end of a normalized 1-morphism.

instance Mathlib.Tactic.BicategoryLike.instInhabitedNormalizedHom : Inhabited NormalizedHom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedNormalizedHom = { default := Mathlib.Tactic.BicategoryLike.instInhabitedNormalizedHom.default }

def Mathlib.Tactic.BicategoryLike.instInhabitedNormalizedHom.default : NormalizedHom

Equations

• Mathlib.Tactic.BicategoryLike.instInhabitedNormalizedHom.default = Mathlib.Tactic.BicategoryLike.NormalizedHom.nil default default

def Mathlib.Tactic.BicategoryLike.NormalizedHom.e : NormalizedHom → Mor₁

The underlying expression of a normalized 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.NormalizedHom.nil e a).e = e
• (Mathlib.Tactic.BicategoryLike.NormalizedHom.cons e a a_1).e = e

def Mathlib.Tactic.BicategoryLike.NormalizedHom.src : NormalizedHom → Obj

The domain of a normalized 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.NormalizedHom.nil e a).src = a
• (Mathlib.Tactic.BicategoryLike.NormalizedHom.cons e a a_1).src = a.src

def Mathlib.Tactic.BicategoryLike.NormalizedHom.tgt : NormalizedHom → Obj

The codomain of a normalized 1-morphism.

Equations

• (Mathlib.Tactic.BicategoryLike.NormalizedHom.nil e a).tgt = a
• (Mathlib.Tactic.BicategoryLike.NormalizedHom.cons e a a_1).tgt = a_1.tgt

def Mathlib.Tactic.BicategoryLike.normalizedHom.nilM {m : Type → Type} [Monad m] [MonadMor₁ m] (a : Obj) : m NormalizedHom

Construct the NormalizedHom.nil term in m.

Equations

• Mathlib.Tactic.BicategoryLike.normalizedHom.nilM a = do let __do_lift ← Mathlib.Tactic.BicategoryLike.MonadMor₁.id₁M a pure (Mathlib.Tactic.BicategoryLike.NormalizedHom.nil __do_lift a)

def Mathlib.Tactic.BicategoryLike.NormalizedHom.consM {m : Type → Type} [Monad m] [MonadMor₁ m] (p : NormalizedHom) (f : Atom₁) : m NormalizedHom

Construct a NormalizedHom.cons term in m.

Equations

• p.consM f = do let __do_lift ← Mathlib.Tactic.BicategoryLike.MonadMor₁.comp₁M p.e (Mathlib.Tactic.BicategoryLike.Mor₁.of f) pure (Mathlib.Tactic.BicategoryLike.NormalizedHom.cons __do_lift p f)

class Mathlib.Tactic.BicategoryLike.Context (ρ : Type) : Type

Context ρ provides the context for manipulating 2-morphisms in a monoidal category or bicategory. In particular, we will store MonoidalCategory or Bicategory instance in a context, and use this through a reader monad when we construct the lean expressions for 2-morphisms.

• mkContext? : Lean.Expr → Lean.MetaM (Option ρ)

  Construct a context from a lean expression for a 2-morphism.

def Mathlib.Tactic.BicategoryLike.mkContext {ρ : Type} [Context ρ] (e : Lean.Expr) : Lean.MetaM ρ

Construct a context from a lean expression for a 2-morphism.

Equations

• One or more equations did not get rendered due to their size.

structure Mathlib.Tactic.BicategoryLike.State : Type

The state for the CoherenceM ρ monad.

• cache : Lean.PersistentExprMap Mor₁

  The cache for evaluating lean expressions of 1-morphisms into Mor₁ terms.

@[reducible, inline]

abbrev Mathlib.Tactic.BicategoryLike.CoherenceM (ρ α : Type) : Type

The monad for manipulating 2-morphisms in a monoidal category or bicategory.

Equations

• Mathlib.Tactic.BicategoryLike.CoherenceM ρ = ReaderT ρ (StateT Mathlib.Tactic.BicategoryLike.State Lean.MetaM)

def Mathlib.Tactic.BicategoryLike.CoherenceM.run {α ρ : Type} (x : CoherenceM ρ α) (ctx : ρ) (s : State := { }) : Lean.MetaM α

Run the CoherenceM ρ monad.

Equations

• x.run ctx s = Prod.fst <$> ReaderT.run x ctx s