Library Coq.Logic.EqdepFacts



This file defines dependent equality and shows its equivalence with equality on dependent pairs (inhabiting sigma-types). It derives the consequence of axiomatizing the invariance by substitution of reflexive equality proofs and shows the equivalence between the 4 following statements
  • Invariance by Substitution of Reflexive Equality Proofs.
  • Injectivity of Dependent Equality
  • Uniqueness of Identity Proofs
  • Uniqueness of Reflexive Identity Proofs
  • Streicher's Axiom K
These statements are independent of the calculus of constructions 2.
References:
1 T. Streicher, Semantical Investigations into Intensional Type Theory, Habilitationsschrift, LMU M√ľnchen, 1993. 2 M. Hofmann, T. Streicher, The groupoid interpretation of type theory, Proceedings of the meeting Twenty-five years of constructive type theory, Venice, Oxford University Press, 1998
Table of contents:
1. Definition of dependent equality and equivalence with equality of dependent pairs and with dependent pair of equalities
2. Eq_rect_eq <-> Eq_dep_eq <-> UIP <-> UIP_refl <-> K
3. Definition of the functor that builds properties of dependent equalities assuming axiom eq_rect_eq

Definition of dependent equality and equivalence with equality of dependent pairs


Import EqNotations.


Section Dependent_Equality.

  Variable U : Type.
  Variable P : U -> Type.

Dependent equality

  Inductive eq_dep (p:U) (x:P p) : forall q:U, P q -> Prop :=
    eq_dep_intro : eq_dep p x p x.
  Hint Constructors eq_dep: core.

  Lemma eq_dep_refl : forall (p:U) (x:P p), eq_dep p x p x.

  Lemma eq_dep_sym :
    forall (p q:U) (x:P p) (y:P q), eq_dep p x q y -> eq_dep q y p x.
  Hint Immediate eq_dep_sym: core.

  Lemma eq_dep_trans :
    forall (p q r:U) (x:P p) (y:P q) (z:P r),
      eq_dep p x q y -> eq_dep q y r z -> eq_dep p x r z.

  Scheme eq_indd := Induction for eq Sort Prop.

Equivalent definition of dependent equality as a dependent pair of equalities

  Inductive eq_dep1 (p:U) (x:P p) (q:U) (y:P q) : Prop :=
    eq_dep1_intro : forall h:q = p, x = rew h in y -> eq_dep1 p x q y.

  Lemma eq_dep1_dep :
    forall (p:U) (x:P p) (q:U) (y:P q), eq_dep1 p x q y -> eq_dep p x q y.

  Lemma eq_dep_dep1 :
    forall (p q:U) (x:P p) (y:P q), eq_dep p x q y -> eq_dep1 p x q y.

End Dependent_Equality.


Dependent equality is equivalent to equality on dependent pairs

Lemma eq_sigT_eq_dep :
  forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
    existT P p x = existT P q y -> eq_dep p x q y.

Notation eq_sigS_eq_dep := eq_sigT_eq_dep (compat "8.2").
Lemma eq_dep_eq_sigT :
  forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
    eq_dep p x q y -> existT P p x = existT P q y.

Lemma eq_sigT_iff_eq_dep :
  forall (U:Type) (P:U -> Type) (p q:U) (x:P p) (y:P q),
    existT P p x = existT P q y <-> eq_dep p x q y.

Notation equiv_eqex_eqdep := eq_sigT_iff_eq_dep (only parsing).
Lemma eq_sig_eq_dep :
  forall (U:Type) (P:U -> Prop) (p q:U) (x:P p) (y:P q),
    exist P p x = exist P q y -> eq_dep p x q y.

Lemma eq_dep_eq_sig :
  forall (U:Type) (P:U -> Prop) (p q:U) (x:P p) (y:P q),
    eq_dep p x q y -> exist P p x = exist P q y.

Lemma eq_sig_iff_eq_dep :
  forall (U:Type) (P:U -> Prop) (p q:U) (x:P p) (y:P q),
    exist P p x = exist P q y <-> eq_dep p x q y.

Dependent equality is equivalent to a dependent pair of equalities

Set Implicit Arguments.

Lemma eq_sigT_sig_eq : forall X P (x1 x2:X) H1 H2, existT P x1 H1 = existT P x2 H2 <->
                                                   {H:x1=x2 | rew H in H1 = H2}.

Lemma eq_sigT_fst :
  forall X P (x1 x2:X) H1 H2 (H:existT P x1 H1 = existT P x2 H2), x1 = x2.

Lemma eq_sigT_snd :
  forall X P (x1 x2:X) H1 H2 (H:existT P x1 H1 = existT P x2 H2), rew (eq_sigT_fst H) in H1 = H2.

Lemma eq_sig_fst :
  forall X P (x1 x2:X) H1 H2 (H:exist P x1 H1 = exist P x2 H2), x1 = x2.

Lemma eq_sig_snd :
  forall X P (x1 x2:X) H1 H2 (H:exist P x1 H1 = exist P x2 H2), rew (eq_sig_fst H) in H1 = H2.

Unset Implicit Arguments.

Exported hints

Hint Resolve eq_dep_intro: core.
Hint Immediate eq_dep_sym: core.

Eq_rect_eq <-> Eq_dep_eq <-> UIP <-> UIP_refl <-> K


Section Equivalences.

  Variable U:Type.

Invariance by Substitution of Reflexive Equality Proofs

  Definition Eq_rect_eq_on (p : U) (Q : U -> Type) (x : Q p) :=
    forall (h : p = p), x = eq_rect p Q x p h.
  Definition Eq_rect_eq := forall p Q x, Eq_rect_eq_on p Q x.

Injectivity of Dependent Equality

  Definition Eq_dep_eq_on (P : U -> Type) (p : U) (x : P p) :=
    forall (y : P p), eq_dep p x p y -> x = y.
  Definition Eq_dep_eq := forall P p x, Eq_dep_eq_on P p x.

Uniqueness of Identity Proofs (UIP)

  Definition UIP_on_ (x y : U) (p1 : x = y) :=
    forall (p2 : x = y), p1 = p2.
  Definition UIP_ := forall x y p1, UIP_on_ x y p1.

Uniqueness of Reflexive Identity Proofs

  Definition UIP_refl_on_ (x : U) :=
    forall (p : x = x), p = eq_refl x.
  Definition UIP_refl_ := forall x, UIP_refl_on_ x.

Streicher's axiom K

  Definition Streicher_K_on_ (x : U) (P : x = x -> Prop) :=
    P (eq_refl x) -> forall p : x = x, P p.
  Definition Streicher_K_ := forall x P, Streicher_K_on_ x P.

Injectivity of Dependent Equality is a consequence of Invariance by Substitution of Reflexive Equality Proof

  Lemma eq_rect_eq_on__eq_dep1_eq_on (p : U) (P : U -> Type) (y : P p) :
    Eq_rect_eq_on p P y -> forall (x : P p), eq_dep1 p x p y -> x = y.
  Lemma eq_rect_eq__eq_dep1_eq :
    Eq_rect_eq -> forall (P:U->Type) (p:U) (x y:P p), eq_dep1 p x p y -> x = y.

  Lemma eq_rect_eq_on__eq_dep_eq_on (p : U) (P : U -> Type) (x : P p) :
    Eq_rect_eq_on p P x -> Eq_dep_eq_on P p x.
  Lemma eq_rect_eq__eq_dep_eq : Eq_rect_eq -> Eq_dep_eq.

Uniqueness of Identity Proofs (UIP) is a consequence of Injectivity of Dependent Equality

  Lemma eq_dep_eq_on__UIP_on (x y : U) (p1 : x = y) :
    Eq_dep_eq_on (fun y => x = y) x eq_refl -> UIP_on_ x y p1.
  Lemma eq_dep_eq__UIP : Eq_dep_eq -> UIP_.

Uniqueness of Reflexive Identity Proofs is a direct instance of UIP

  Lemma UIP_on__UIP_refl_on (x : U) :
    UIP_on_ x x eq_refl -> UIP_refl_on_ x.
  Lemma UIP__UIP_refl : UIP_ -> UIP_refl_.

Streicher's axiom K is a direct consequence of Uniqueness of Reflexive Identity Proofs
We finally recover from K the Invariance by Substitution of Reflexive Equality Proofs

  Lemma Streicher_K_on__eq_rect_eq_on (p : U) (P : U -> Type) (x : P p) :
    Streicher_K_on_ p (fun h => x = rew -> [P] h in x)
    -> Eq_rect_eq_on p P x.
  Lemma Streicher_K__eq_rect_eq : Streicher_K_ -> Eq_rect_eq.

Remark: It is reasonable to think that eq_rect_eq is strictly stronger than eq_rec_eq (which is eq_rect_eq restricted on Set):
Definition Eq_rec_eq := forall (P:U -> Set) (p:U) (x:P p) (h:p = p), x = eq_rec p P x p h.
Typically, eq_rect_eq allows proving UIP and Streicher's K what does not seem possible with eq_rec_eq. In particular, the proof of UIP requires to use eq_rect_eq on fun y -> x=y which is in Type but not in Set.
UIP_refl is downward closed (a short proof of the key lemma of Voevodsky's proof of inclusion of h-level n into h-level n+1; see hlevelntosn in https://github.com/vladimirias/Foundations.git).

Theorem UIP_shift_on (X : Type) (x : X) :
  UIP_refl_on_ X x -> forall y : x = x, UIP_refl_on_ (x = x) y.
Theorem UIP_shift : forall U, UIP_refl_ U -> forall x:U, UIP_refl_ (x = x).

Section Corollaries.

  Variable U:Type.

UIP implies the injectivity of equality on dependent pairs in Type

 Definition Inj_dep_pair_on (P : U -> Type) (p : U) (x : P p) :=
   forall (y : P p), existT P p x = existT P p y -> x = y.
 Definition Inj_dep_pair := forall P p x, Inj_dep_pair_on P p x.

 Lemma eq_dep_eq_on__inj_pair2_on (P : U -> Type) (p : U) (x : P p) :
   Eq_dep_eq_on U P p x -> Inj_dep_pair_on P p x.
 Lemma eq_dep_eq__inj_pair2 : Eq_dep_eq U -> Inj_dep_pair.

End Corollaries.

Notation Inj_dep_pairS := Inj_dep_pair.
Notation Inj_dep_pairT := Inj_dep_pair.
Notation eq_dep_eq__inj_pairT2 := eq_dep_eq__inj_pair2.

Definition of the functor that builds properties of dependent equalities assuming axiom eq_rect_eq


Module Type EqdepElimination.

  Axiom eq_rect_eq :
    forall (U:Type) (p:U) (Q:U -> Type) (x:Q p) (h:p = p),
      x = eq_rect p Q x p h.

End EqdepElimination.

Module EqdepTheory (M:EqdepElimination).

  Section Axioms.

    Variable U:Type.

Invariance by Substitution of Reflexive Equality Proofs

Lemma eq_rect_eq :
  forall (p:U) (Q:U -> Type) (x:Q p) (h:p = p), x = eq_rect p Q x p h.

Lemma eq_rec_eq :
  forall (p:U) (Q:U -> Set) (x:Q p) (h:p = p), x = eq_rec p Q x p h.

Injectivity of Dependent Equality

Lemma eq_dep_eq : forall (P:U->Type) (p:U) (x y:P p), eq_dep p x p y -> x = y.

Uniqueness of Identity Proofs (UIP) is a consequence of Injectivity of Dependent Equality

Lemma UIP : forall (x y:U) (p1 p2:x = y), p1 = p2.

Uniqueness of Reflexive Identity Proofs is a direct instance of UIP

Lemma UIP_refl : forall (x:U) (p:x = x), p = eq_refl x.

Streicher's axiom K is a direct consequence of Uniqueness of Reflexive Identity Proofs

Lemma Streicher_K :
  forall (x:U) (P:x = x -> Prop), P (eq_refl x) -> forall p:x = x, P p.

End Axioms.

UIP implies the injectivity of equality on dependent pairs in Type

Lemma inj_pair2 :
 forall (U:Type) (P:U -> Type) (p:U) (x y:P p),
   existT P p x = existT P p y -> x = y.

Notation inj_pairT2 := inj_pair2.

End EqdepTheory.

Basic facts about eq_dep

Lemma f_eq_dep :
  forall U (P:U->Type) R p q x y (f:forall p, P p -> R p),
    eq_dep p x q y -> eq_dep p (f p x) q (f q y).

Lemma eq_dep_non_dep :
  forall U P p q x y, @eq_dep U (fun _ => P) p x q y -> x = y.

Lemma f_eq_dep_non_dep :
  forall U (P:U->Type) R p q x y (f:forall p, P p -> R),
    eq_dep p x q y -> f p x = f q y.