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(* -*- coding: utf-8 -*- *)
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(*i $Id$ i*)
(** 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 München, 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
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 *)
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.
Proof eq_dep_intro.
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.
Proof.
destruct 1; auto.
Qed.
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.
Proof.
destruct 1; auto.
Qed.
Scheme eq_indd := Induction for eq Sort Prop.
(** Equivalent definition of dependent equality expressed as a non
dependent inductive type *)
Inductive eq_dep1 (p:U) (x:P p) (q:U) (y:P q) : Prop :=
eq_dep1_intro : forall h:q = p, x = eq_rect q P y p h -> 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.
Proof.
destruct 1 as (eq_qp, H).
destruct eq_qp using eq_indd.
rewrite H.
apply eq_dep_intro.
Qed.
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.
Proof.
destruct 1.
apply eq_dep1_intro with (refl_equal p).
simpl in |- *; trivial.
Qed.
End Dependent_Equality.
Implicit Arguments eq_dep [U P].
Implicit Arguments eq_dep1 [U P].
(** 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.
Proof.
intros.
dependent rewrite H.
apply eq_dep_intro.
Qed.
Notation eq_sigS_eq_dep := eq_sigT_eq_dep (only parsing). (* Compatibility *)
Lemma equiv_eqex_eqdep :
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.
Proof.
split.
(* -> *)
apply eq_sigT_eq_dep.
(* <- *)
destruct 1; reflexivity.
Qed.
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.
Proof.
destruct 1; reflexivity.
Qed.
(** 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 :=
forall (p:U) (Q:U -> Type) (x:Q p) (h:p = p), x = eq_rect p Q x p h.
(** Injectivity of Dependent Equality *)
Definition 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) *)
Definition UIP_ :=
forall (x y:U) (p1 p2:x = y), p1 = p2.
(** Uniqueness of Reflexive Identity Proofs *)
Definition UIP_refl_ :=
forall (x:U) (p:x = x), p = refl_equal x.
(** Streicher's axiom K *)
Definition Streicher_K_ :=
forall (x:U) (P:x = x -> Prop), P (refl_equal x) -> forall p:x = x, P p.
(** Injectivity of Dependent Equality is a consequence of *)
(** Invariance by Substitution of Reflexive Equality Proof *)
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.
Proof.
intro eq_rect_eq.
simple destruct 1; intro.
rewrite <- eq_rect_eq; auto.
Qed.
Lemma eq_rect_eq__eq_dep_eq : Eq_rect_eq -> Eq_dep_eq.
Proof.
intros eq_rect_eq; red; intros.
apply (eq_rect_eq__eq_dep1_eq eq_rect_eq); apply eq_dep_dep1; trivial.
Qed.
(** Uniqueness of Identity Proofs (UIP) is a consequence of *)
(** Injectivity of Dependent Equality *)
Lemma eq_dep_eq__UIP : Eq_dep_eq -> UIP_.
Proof.
intro eq_dep_eq; red.
intros; apply eq_dep_eq with (P := fun y => x = y).
elim p2 using eq_indd.
elim p1 using eq_indd.
apply eq_dep_intro.
Qed.
(** Uniqueness of Reflexive Identity Proofs is a direct instance of UIP *)
Lemma UIP__UIP_refl : UIP_ -> UIP_refl_.
Proof.
intro UIP; red; intros; apply UIP.
Qed.
(** Streicher's axiom K is a direct consequence of Uniqueness of
Reflexive Identity Proofs *)
Lemma UIP_refl__Streicher_K : UIP_refl_ -> Streicher_K_.
Proof.
intro UIP_refl; red; intros; rewrite UIP_refl; assumption.
Qed.
(** We finally recover from K the Invariance by Substitution of
Reflexive Equality Proofs *)
Lemma Streicher_K__eq_rect_eq : Streicher_K_ -> Eq_rect_eq.
Proof.
intro Streicher_K; red; intros.
apply Streicher_K with (p := h).
reflexivity.
Qed.
(** 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 to prove 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].
*)
End Equivalences.
Section Corollaries.
Variable U:Type.
(** UIP implies the injectivity of equality on dependent pairs in Type *)
Definition Inj_dep_pair :=
forall (P:U -> Type) (p:U) (x y:P p), existT P p x = existT P p y -> x = y.
Lemma eq_dep_eq__inj_pair2 : Eq_dep_eq U -> Inj_dep_pair.
Proof.
intro eq_dep_eq; red; intros.
apply eq_dep_eq.
apply eq_sigT_eq_dep.
assumption.
Qed.
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.
Proof M.eq_rect_eq U.
Lemma eq_rec_eq :
forall (p:U) (Q:U -> Set) (x:Q p) (h:p = p), x = eq_rec p Q x p h.
Proof (fun p Q => M.eq_rect_eq U p Q).
(** 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.
Proof (eq_rect_eq__eq_dep_eq U eq_rect_eq).
(** 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.
Proof (eq_dep_eq__UIP U eq_dep_eq).
(** Uniqueness of Reflexive Identity Proofs is a direct instance of UIP *)
Lemma UIP_refl : forall (x:U) (p:x = x), p = refl_equal x.
Proof (UIP__UIP_refl U UIP).
(** 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 (refl_equal x) -> forall p:x = x, P p.
Proof (UIP_refl__Streicher_K U UIP_refl).
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.
Proof (fun U => eq_dep_eq__inj_pair2 U (eq_dep_eq U)).
Notation inj_pairT2 := inj_pair2.
End EqdepTheory.
Implicit Arguments eq_dep [].
Implicit Arguments eq_dep1 [].
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