diff --git a/Makefile b/Makefile
index c56e771..ef5b76f 100644
--- a/Makefile
+++ b/Makefile
@@ -6,7 +6,7 @@ theories: $(COQMAKEFILE) FORCE
validate: $(COQMAKEFILE) FORCE
$(MAKE) -f $(COQMAKEFILE) validate
-$(COQMAKEFILE) : theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/unscoped.v
+$(COQMAKEFILE) : theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v
$(COQBIN)coq_makefile -f _CoqProject -o $(COQMAKEFILE)
install: $(COQMAKEFILE)
@@ -15,13 +15,13 @@ install: $(COQMAKEFILE)
uninstall: $(COQMAKEFILE)
$(MAKE) -f $(COQMAKEFILE) uninstall
-theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/unscoped.v : syntax.sig
- autosubst -f -v ge813 -s ucoq -o theories/Autosubst2/syntax.v syntax.sig
+theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v : syntax.sig
+ autosubst -f -v ge813 -s coq -o theories/Autosubst2/syntax.v syntax.sig
.PHONY: clean FORCE
clean:
test ! -f $(COQMAKEFILE) || ( $(MAKE) -f $(COQMAKEFILE) clean && rm $(COQMAKEFILE) )
- rm -f theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v theories/Autosubst2/unscoped.v
+ rm -f theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v
FORCE:
diff --git a/syntax.sig b/syntax.sig
index e63c3b4..4549f7f 100644
--- a/syntax.sig
+++ b/syntax.sig
@@ -1,29 +1,16 @@
nat : Type
-PTm(VarPTm) : Type
Tm(VarTm) : Type
PTag : Type
TTag : Type
-bool : Type
-PL : PTag
-PR : PTag
TPi : TTag
TSig : TTag
-
-PAbs : (bind PTm in PTm) -> PTm
-PApp : PTm -> PTm -> PTm
-PPair : PTm -> PTm -> PTm
-PProj : PTag -> PTm -> PTm
-PConst : TTag -> PTm
-PUniv : nat -> PTm
-PBot : PTm
-
+PL : PTag
+PR : PTag
Abs : (bind Tm in Tm) -> Tm
App : Tm -> Tm -> Tm
Pair : Tm -> Tm -> Tm
Proj : PTag -> Tm -> Tm
TBind : TTag -> Tm -> (bind Tm in Tm) -> Tm
+Bot : Tm
Univ : nat -> Tm
-BVal : bool -> Tm
-Bool : Tm
-If : Tm -> Tm -> Tm -> Tm
diff --git a/theories/Autosubst2/fintype.v b/theories/Autosubst2/fintype.v
new file mode 100644
index 0000000..99508b6
--- /dev/null
+++ b/theories/Autosubst2/fintype.v
@@ -0,0 +1,419 @@
+(** * Autosubst Header for Scoped Syntax
+ Our development utilises well-scoped de Bruijn syntax. This means that the de Bruijn indices are taken from finite types. As a consequence, any kind of substitution or environment used in conjunction with well-scoped syntax takes the form of a mapping from some finite type _I^n_. In particular, _renamings_ are mappings _I^n -> I^m_. Here we develop the theory of how these parts interact.
+
+Version: December 11, 2019.
+*)
+Require Import core.
+Require Import Setoid Morphisms Relation_Definitions.
+
+Set Implicit Arguments.
+Unset Strict Implicit.
+
+Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
+ match p with eq_refl => eq_refl end.
+
+Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
+ match q with eq_refl => match p with eq_refl => eq_refl end end.
+
+(** ** Primitives of the Sigma Calculus
+ We implement the finite type with _n_ elements, _I^n_, as the _n_-fold iteration of the Option Type. _I^0_ is implemented as the empty type.
+*)
+
+Fixpoint fin (n : nat) : Type :=
+ match n with
+ | 0 => False
+ | S m => option (fin m)
+ end.
+
+(** Renamings and Injective Renamings
+ _Renamings_ are mappings between finite types.
+*)
+Definition ren (m n : nat) : Type := fin m -> fin n.
+
+Definition id {X} := @Datatypes.id X.
+
+Definition idren {k: nat} : ren k k := @Datatypes.id (fin k).
+
+(** We give a special name, to the newest element in a non-empty finite type, as it usually corresponds to a freshly bound variable. *)
+Definition var_zero {n : nat} : fin (S n) := None.
+
+Definition null {T} (i : fin 0) : T := match i with end.
+
+Definition shift {n : nat} : ren n (S n) :=
+ Some.
+
+(** Extension of Finite Mappings
+ Assume we are given a mapping _f_ from _I^n_ to some type _X_, then we can _extend_ this mapping with a new value from _x : X_ to a mapping from _I^n+1_ to _X_. We denote this operation by _x . f_ and define it as follows:
+*)
+Definition scons {X : Type} {n : nat} (x : X) (f : fin n -> X) (m : fin (S n)) : X :=
+ match m with
+ | None => x
+ | Some i => f i
+ end.
+
+#[ export ]
+Hint Opaque scons : rewrite.
+
+(** ** Type Class Instances for Notation *)
+
+(** *** Type classes for renamings. *)
+
+Class Ren1 (X1 : Type) (Y Z : Type) :=
+ ren1 : X1 -> Y -> Z.
+
+Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
+ ren2 : X1 -> X2 -> Y -> Z.
+
+Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
+ ren3 : X1 -> X2 -> X3 -> Y -> Z.
+
+Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
+ ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
+
+Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
+ ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
+
+Module RenNotations.
+ Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s ⟨ xi1 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4 xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
+
+ Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5 xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
+
+ Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
+
+ Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
+End RenNotations.
+
+(** *** Type Classes for Substiution *)
+
+Class Subst1 (X1 : Type) (Y Z: Type) :=
+ subst1 : X1 -> Y -> Z.
+
+Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
+ subst2 : X1 -> X2 -> Y -> Z.
+
+Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
+ subst3 : X1 -> X2 -> X3 -> Y -> Z.
+
+Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
+ subst4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
+
+Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
+ subst5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
+
+Module SubstNotations.
+ Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
+
+ Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/' tau ]") : subst_scope.
+End SubstNotations.
+
+(** ** Type Class for Variables *)
+Class Var X Y :=
+ ids : X -> Y.
+
+
+(** ** Proofs for substitution primitives *)
+
+(** Forward Function Composition
+ Substitutions represented as functions are ubiquitious in this development and we often have to compose them, without talking about their pointwise behaviour.
+ That is, we are interested in the forward compostion of functions, _f o g_, for which we introduce a convenient notation, "f >> g". The direction of the arrow serves as a reminder of the _forward_ nature of this composition, that is first apply _f_, then _g_. *)
+
+Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
+
+Module CombineNotations.
+ Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
+
+ Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
+
+ #[ global ]
+ Open Scope fscope.
+ #[ global ]
+ Open Scope subst_scope.
+End CombineNotations.
+
+Import CombineNotations.
+
+
+(** Generic lifting operation for renamings *)
+Definition up_ren {m n} (xi : ren m n) : ren (S m) (S n) :=
+ var_zero .: xi >> shift.
+
+(** Generic proof that lifting of renamings composes. *)
+Lemma up_ren_ren k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
+ forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
+Proof.
+ intros [x|].
+ - cbn. unfold funcomp. now rewrite <- E.
+ - reflexivity.
+Qed.
+
+Arguments up_ren_ren {k l m} xi zeta rho E.
+
+Lemma fin_eta {X} (f g : fin 0 -> X) :
+ pointwise_relation _ eq f g.
+Proof. intros []. Qed.
+
+(** Eta laws *)
+Lemma scons_eta' {T} {n : nat} (f : fin (S n) -> T) :
+ pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_eta_id' {n : nat} :
+ pointwise_relation (fin (S n)) eq (var_zero .: shift) id.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_comp' {T:Type} {U} {m} (s: T) (sigma: fin m -> T) (tau: T -> U) :
+ pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
+Proof. intros x. destruct x; reflexivity. Qed.
+
+(* Lemma scons_tail'_pointwise {X} {n} (s : X) (f : fin n -> X) : *)
+(* pointwise_relation _ eq (funcomp (scons s f) shift) f. *)
+(* Proof. *)
+(* reflexivity. *)
+(* Qed. *)
+
+(* Lemma scons_tail' {X} {n} (s : X) (f : fin n -> X) x : *)
+(* (scons s f (shift x)) = f x. *)
+(* Proof. *)
+(* reflexivity. *)
+(* Qed. *)
+
+(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
+#[export] Instance scons_morphism {X: Type} {n:nat} :
+ Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X n).
+Proof.
+ intros t t' -> sigma tau H.
+ intros [x|].
+ cbn. apply H.
+ reflexivity.
+Qed.
+
+#[export] Instance scons_morphism2 {X: Type} {n: nat} :
+ Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X n).
+Proof.
+ intros ? t -> sigma tau H ? x ->.
+ destruct x as [x|].
+ cbn. apply H.
+ reflexivity.
+Qed.
+
+(** ** Variadic Substitution Primitives *)
+
+Fixpoint shift_p (p : nat) {n} : ren n (p + n) :=
+ fun n => match p with
+ | 0 => n
+ | S p => Some (shift_p p n)
+ end.
+
+Fixpoint scons_p {X: Type} {m : nat} : forall {n} (f : fin m -> X) (g : fin n -> X), fin (m + n) -> X.
+Proof.
+ destruct m.
+ - intros n f g. exact g.
+ - intros n f g. cbn. apply scons.
+ + exact (f var_zero).
+ + apply scons_p.
+ * intros z. exact (f (Some z)).
+ * exact g.
+Defined.
+
+#[export] Hint Opaque scons_p : rewrite.
+
+#[export] Instance scons_p_morphism {X: Type} {m n:nat} :
+ Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons_p X m n).
+Proof.
+ intros sigma sigma' Hsigma tau tau' Htau.
+ intros x.
+ induction m.
+ - cbn. apply Htau.
+ - cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
+ destruct x as [x|].
+ + cbn. apply IHm.
+ intros ?. apply Hsigma.
+ + cbn. apply Hsigma.
+Qed.
+
+#[export] Instance scons_p_morphism2 {X: Type} {m n:nat} :
+ Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons_p X m n).
+Proof.
+ intros sigma sigma' Hsigma tau tau' Htau ? x ->.
+ induction m.
+ - cbn. apply Htau.
+ - cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
+ destruct x as [x|].
+ + cbn. apply IHm.
+ intros ?. apply Hsigma.
+ + cbn. apply Hsigma.
+Qed.
+
+Definition zero_p {m : nat} {n} : fin m -> fin (m + n).
+Proof.
+ induction m.
+ - intros [].
+ - intros [x|].
+ + exact (shift_p 1 (IHm x)).
+ + exact var_zero.
+Defined.
+
+Lemma scons_p_head' {X} {m n} (f : fin m -> X) (g : fin n -> X) z:
+ (scons_p f g) (zero_p z) = f z.
+Proof.
+ induction m.
+ - inversion z.
+ - destruct z.
+ + simpl. simpl. now rewrite IHm.
+ + reflexivity.
+Qed.
+
+Lemma scons_p_head_pointwise {X} {m n} (f : fin m -> X) (g : fin n -> X) :
+ pointwise_relation _ eq (funcomp (scons_p f g) zero_p) f.
+Proof.
+ intros z.
+ unfold funcomp.
+ induction m.
+ - inversion z.
+ - destruct z.
+ + simpl. now rewrite IHm.
+ + reflexivity.
+Qed.
+
+Lemma scons_p_tail' X m n (f : fin m -> X) (g : fin n -> X) z :
+ scons_p f g (shift_p m z) = g z.
+Proof. induction m; cbn; eauto. Qed.
+
+Lemma scons_p_tail_pointwise X m n (f : fin m -> X) (g : fin n -> X) :
+ pointwise_relation _ eq (funcomp (scons_p f g) (shift_p m)) g.
+Proof. intros z. induction m; cbn; eauto. Qed.
+
+Lemma destruct_fin {m n} (x : fin (m + n)):
+ (exists x', x = zero_p x') \/ exists x', x = shift_p m x'.
+Proof.
+ induction m; simpl in *.
+ - right. eauto.
+ - destruct x as [x|].
+ + destruct (IHm x) as [[x' ->] |[x' ->]].
+ * left. now exists (Some x').
+ * right. eauto.
+ + left. exists None. eauto.
+Qed.
+
+Lemma scons_p_comp' X Y m n (f : fin m -> X) (g : fin n -> X) (h : X -> Y) :
+ pointwise_relation _ eq (funcomp h (scons_p f g)) (scons_p (f >> h) (g >> h)).
+Proof.
+ intros x.
+ destruct (destruct_fin x) as [[x' ->]|[x' ->]].
+ (* TODO better way to solve this? *)
+ - revert x'.
+ apply pointwise_forall.
+ change (fun x => (scons_p f g >> h) (zero_p x)) with (zero_p >> scons_p f g >> h).
+ now setoid_rewrite scons_p_head_pointwise.
+ - revert x'.
+ apply pointwise_forall.
+ change (fun x => (scons_p f g >> h) (shift_p m x)) with (shift_p m >> scons_p f g >> h).
+ change (fun x => scons_p (f >> h) (g >> h) (shift_p m x)) with (shift_p m >> scons_p (f >> h) (g >> h)).
+ now rewrite !scons_p_tail_pointwise.
+Qed.
+
+
+Lemma scons_p_congr {X} {m n} (f f' : fin m -> X) (g g': fin n -> X) z:
+ (forall x, f x = f' x) -> (forall x, g x = g' x) -> scons_p f g z = scons_p f' g' z.
+Proof. intros H1 H2. induction m; eauto. cbn. destruct z; eauto. Qed.
+
+(** Generic n-ary lifting operation. *)
+Definition upRen_p p { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : fin (p + m) -> fin (p + n) :=
+ scons_p (zero_p ) (xi >> shift_p _).
+
+Arguments upRen_p p {m n} xi.
+
+(** Generic proof for composition of n-ary lifting. *)
+Lemma up_ren_ren_p p k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
+ forall x, (upRen_p p xi >> upRen_p p zeta) x = upRen_p p rho x.
+Proof.
+ intros x. destruct (destruct_fin x) as [[? ->]|[? ->]].
+ - unfold upRen_p. unfold funcomp. now repeat rewrite scons_p_head'.
+ - unfold upRen_p. unfold funcomp. repeat rewrite scons_p_tail'.
+ now rewrite <- E.
+Qed.
+
+
+Arguments zero_p m {n}.
+Arguments scons_p {X} m {n} f g.
+
+Lemma scons_p_eta {X} {m n} {f : fin m -> X}
+ {g : fin n -> X} (h: fin (m + n) -> X) {z: fin (m + n)}:
+ (forall x, g x = h (shift_p m x)) -> (forall x, f x = h (zero_p m x)) -> scons_p m f g z = h z.
+Proof.
+ intros H1 H2. destruct (destruct_fin z) as [[? ->] |[? ->]].
+ - rewrite scons_p_head'. eauto.
+ - rewrite scons_p_tail'. eauto.
+Qed.
+
+Arguments scons_p_eta {X} {m n} {f g} h {z}.
+Arguments scons_p_congr {X} {m n} {f f'} {g g'} {z}.
+
+(** ** Notations for Scoped Syntax *)
+
+Module ScopedNotations.
+ Include RenNotations.
+ Include SubstNotations.
+ Include CombineNotations.
+
+(* Notation "s , sigma" := (scons s sigma) (at level 60, format "s , sigma", right associativity) : subst_scope. *)
+
+ Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
+
+ Notation "↑" := (shift) : subst_scope.
+
+ #[global]
+ Open Scope fscope.
+ #[global]
+ Open Scope subst_scope.
+End ScopedNotations.
+
+
+(** ** Tactics for Scoped Syntax *)
+
+Tactic Notation "auto_case" tactic(t) := (match goal with
+ | [|- forall (i : fin 0), _] => intros []; t
+ | [|- forall (i : fin (S (S (S (S _))))), _] => intros [[[[|]|]|]|]; t
+ | [|- forall (i : fin (S (S (S _)))), _] => intros [[[|]|]|]; t
+ | [|- forall (i : fin (S (S _))), _] => intros [[?|]|]; t
+ | [|- forall (i : fin (S _)), _] => intros [?|]; t
+ end).
+
+#[export] Hint Rewrite @scons_p_head' @scons_p_tail' : FunctorInstances.
+
+(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
+Ltac fsimpl :=
+ repeat match goal with
+ | [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
+ | [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
+ | [|- context [id ?s]] => change (id s) with s
+ | [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h)) (* AsimplComp *)
+ (* | [|- zero_p >> scons_p ?f ?g] => rewrite scons_p_head *)
+ | [|- context[(?s .: ?sigma) var_zero]] => change ((s.:sigma) var_zero) with s
+ | [|- context[(?s .: ?sigma) (shift ?m)]] => change ((s.:sigma) (shift m)) with (sigma m)
+ (* first [progress setoid_rewrite scons_tail' | progress setoid_rewrite scons_tail'_pointwise ] *)
+ | [|- context[idren >> ?f]] => change (idren >> f) with f
+ | [|- context[?f >> idren]] => change (f >> idren) with f
+ | [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
+ | [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f); eta_reduce
+ | [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite scons_eta'; eta_reduce
+ | [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
+ | [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
+ | [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
+ | [|- context[funcomp _ (scons_p _ _ _)]] => setoid_rewrite scons_p_comp'; eta_reduce
+ | [|- context[scons (@var_zero _) shift]] => setoid_rewrite scons_eta_id'; eta_reduce
+ (* | _ => progress autorewrite with FunctorInstances *)
+ | [|- context[funcomp (scons_p _ _ _) (zero_p _)]] =>
+ first [progress setoid_rewrite scons_p_head_pointwise | progress setoid_rewrite scons_p_head' ]
+ | [|- context[scons_p _ _ _ (zero_p _ _)]] => setoid_rewrite scons_p_head'
+ | [|- context[funcomp (scons_p _ _ _) (shift_p _)]] =>
+ first [progress setoid_rewrite scons_p_tail_pointwise | progress setoid_rewrite scons_p_tail' ]
+ | [|- context[scons_p _ _ _ (shift_p _ _)]] => setoid_rewrite scons_p_tail'
+ | _ => first [progress minimize | progress cbn [shift scons scons_p] ]
+ end.
diff --git a/theories/Autosubst2/syntax.v b/theories/Autosubst2/syntax.v
index 659d8b0..b972476 100644
--- a/theories/Autosubst2/syntax.v
+++ b/theories/Autosubst2/syntax.v
@@ -1,4 +1,4 @@
-Require Import core unscoped.
+Require Import core fintype.
Require Import Setoid Morphisms Relation_Definitions.
@@ -33,806 +33,353 @@ Proof.
exact (eq_refl).
Qed.
-Inductive PTm : Type :=
- | VarPTm : nat -> PTm
- | PAbs : PTm -> PTm
- | PApp : PTm -> PTm -> PTm
- | PPair : PTm -> PTm -> PTm
- | PProj : PTag -> PTm -> PTm
- | PConst : TTag -> PTm
- | PUniv : nat -> PTm
- | PBot : PTm.
+Inductive Tm (n_Tm : nat) : Type :=
+ | VarTm : fin n_Tm -> Tm n_Tm
+ | Abs : Tm (S n_Tm) -> Tm n_Tm
+ | App : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
+ | Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
+ | Proj : PTag -> Tm n_Tm -> Tm n_Tm
+ | TBind : TTag -> Tm n_Tm -> Tm (S n_Tm) -> Tm n_Tm
+ | Bot : Tm n_Tm
+ | Univ : nat -> Tm n_Tm.
-Lemma congr_PAbs {s0 : PTm} {t0 : PTm} (H0 : s0 = t0) : PAbs s0 = PAbs t0.
+Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
+ (H0 : s0 = t0) : Abs m_Tm s0 = Abs m_Tm t0.
Proof.
-exact (eq_trans eq_refl (ap (fun x => PAbs x) H0)).
+exact (eq_trans eq_refl (ap (fun x => Abs m_Tm x) H0)).
Qed.
-Lemma congr_PApp {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
- (H1 : s1 = t1) : PApp s0 s1 = PApp t0 t1.
+Lemma congr_App {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {t0 : Tm m_Tm}
+ {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ App m_Tm s0 s1 = App m_Tm t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PApp x s1) H0))
- (ap (fun x => PApp t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => App m_Tm x s1) H0))
+ (ap (fun x => App m_Tm t0 x) H1)).
Qed.
-Lemma congr_PPair {s0 : PTm} {s1 : PTm} {t0 : PTm} {t1 : PTm} (H0 : s0 = t0)
- (H1 : s1 = t1) : PPair s0 s1 = PPair t0 t1.
+Lemma congr_Pair {m_Tm : nat} {s0 : Tm m_Tm} {s1 : Tm m_Tm} {t0 : Tm m_Tm}
+ {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ Pair m_Tm s0 s1 = Pair m_Tm t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PPair x s1) H0))
- (ap (fun x => PPair t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair m_Tm x s1) H0))
+ (ap (fun x => Pair m_Tm t0 x) H1)).
Qed.
-Lemma congr_PProj {s0 : PTag} {s1 : PTm} {t0 : PTag} {t1 : PTm}
- (H0 : s0 = t0) (H1 : s1 = t1) : PProj s0 s1 = PProj t0 t1.
+Lemma congr_Proj {m_Tm : nat} {s0 : PTag} {s1 : Tm m_Tm} {t0 : PTag}
+ {t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
+ Proj m_Tm s0 s1 = Proj m_Tm t0 t1.
Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => PProj x s1) H0))
- (ap (fun x => PProj t0 x) H1)).
+exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj m_Tm x s1) H0))
+ (ap (fun x => Proj m_Tm t0 x) H1)).
Qed.
-Lemma congr_PConst {s0 : TTag} {t0 : TTag} (H0 : s0 = t0) :
- PConst s0 = PConst t0.
+Lemma congr_TBind {m_Tm : nat} {s0 : TTag} {s1 : Tm m_Tm} {s2 : Tm (S m_Tm)}
+ {t0 : TTag} {t1 : Tm m_Tm} {t2 : Tm (S m_Tm)} (H0 : s0 = t0) (H1 : s1 = t1)
+ (H2 : s2 = t2) : TBind m_Tm s0 s1 s2 = TBind m_Tm t0 t1 t2.
Proof.
-exact (eq_trans eq_refl (ap (fun x => PConst x) H0)).
+exact (eq_trans
+ (eq_trans (eq_trans eq_refl (ap (fun x => TBind m_Tm x s1 s2) H0))
+ (ap (fun x => TBind m_Tm t0 x s2) H1))
+ (ap (fun x => TBind m_Tm t0 t1 x) H2)).
Qed.
-Lemma congr_PUniv {s0 : nat} {t0 : nat} (H0 : s0 = t0) : PUniv s0 = PUniv t0.
-Proof.
-exact (eq_trans eq_refl (ap (fun x => PUniv x) H0)).
-Qed.
-
-Lemma congr_PBot : PBot = PBot.
+Lemma congr_Bot {m_Tm : nat} : Bot m_Tm = Bot m_Tm.
Proof.
exact (eq_refl).
Qed.
-Lemma upRen_PTm_PTm (xi : nat -> nat) : nat -> nat.
+Lemma congr_Univ {m_Tm : nat} {s0 : nat} {t0 : nat} (H0 : s0 = t0) :
+ Univ m_Tm s0 = Univ m_Tm t0.
+Proof.
+exact (eq_trans eq_refl (ap (fun x => Univ m_Tm x) H0)).
+Qed.
+
+Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
+ fin (S m) -> fin (S n).
Proof.
exact (up_ren xi).
Defined.
-Fixpoint ren_PTm (xi_PTm : nat -> nat) (s : PTm) {struct s} : PTm :=
- match s with
- | VarPTm s0 => VarPTm (xi_PTm s0)
- | PAbs s0 => PAbs (ren_PTm (upRen_PTm_PTm xi_PTm) s0)
- | PApp s0 s1 => PApp (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
- | PPair s0 s1 => PPair (ren_PTm xi_PTm s0) (ren_PTm xi_PTm s1)
- | PProj s0 s1 => PProj s0 (ren_PTm xi_PTm s1)
- | PConst s0 => PConst s0
- | PUniv s0 => PUniv s0
- | PBot => PBot
- end.
-
-Lemma up_PTm_PTm (sigma : nat -> PTm) : nat -> PTm.
+Lemma upRen_list_Tm_Tm (p : nat) {m : nat} {n : nat} (xi : fin m -> fin n) :
+ fin (plus p m) -> fin (plus p n).
Proof.
-exact (scons (VarPTm var_zero) (funcomp (ren_PTm shift) sigma)).
+exact (upRen_p p xi).
Defined.
-Fixpoint subst_PTm (sigma_PTm : nat -> PTm) (s : PTm) {struct s} : PTm :=
+Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
+(s : Tm m_Tm) {struct s} : Tm n_Tm :=
match s with
- | VarPTm s0 => sigma_PTm s0
- | PAbs s0 => PAbs (subst_PTm (up_PTm_PTm sigma_PTm) s0)
- | PApp s0 s1 => PApp (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
- | PPair s0 s1 => PPair (subst_PTm sigma_PTm s0) (subst_PTm sigma_PTm s1)
- | PProj s0 s1 => PProj s0 (subst_PTm sigma_PTm s1)
- | PConst s0 => PConst s0
- | PUniv s0 => PUniv s0
- | PBot => PBot
+ | VarTm _ s0 => VarTm n_Tm (xi_Tm s0)
+ | Abs _ s0 => Abs n_Tm (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
+ | App _ s0 s1 => App n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
+ | Pair _ s0 s1 => Pair n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
+ | Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
+ | TBind _ s0 s1 s2 =>
+ TBind n_Tm s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
+ | Bot _ => Bot n_Tm
+ | Univ _ s0 => Univ n_Tm s0
end.
-Lemma upId_PTm_PTm (sigma : nat -> PTm) (Eq : forall x, sigma x = VarPTm x) :
- forall x, up_PTm_PTm sigma x = VarPTm x.
+Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
+ fin (S m) -> Tm (S n_Tm).
Proof.
-exact (fun n =>
- match n with
- | S n' => ap (ren_PTm shift) (Eq n')
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint idSubst_PTm (sigma_PTm : nat -> PTm)
-(Eq_PTm : forall x, sigma_PTm x = VarPTm x) (s : PTm) {struct s} :
-subst_PTm sigma_PTm s = s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (idSubst_PTm (up_PTm_PTm sigma_PTm) (upId_PTm_PTm _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (idSubst_PTm sigma_PTm Eq_PTm s0)
- (idSubst_PTm sigma_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (idSubst_PTm sigma_PTm Eq_PTm s0)
- (idSubst_PTm sigma_PTm Eq_PTm s1)
- | PProj s0 s1 => congr_PProj (eq_refl s0) (idSubst_PTm sigma_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma upExtRen_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
- (Eq : forall x, xi x = zeta x) :
- forall x, upRen_PTm_PTm xi x = upRen_PTm_PTm zeta x.
-Proof.
-exact (fun n => match n with
- | S n' => ap shift (Eq n')
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint extRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
-(Eq_PTm : forall x, xi_PTm x = zeta_PTm x) (s : PTm) {struct s} :
-ren_PTm xi_PTm s = ren_PTm zeta_PTm s :=
- match s with
- | VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
- | PAbs s0 =>
- congr_PAbs
- (extRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
- (upExtRen_PTm_PTm _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
- (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (extRen_PTm xi_PTm zeta_PTm Eq_PTm s0)
- (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0) (extRen_PTm xi_PTm zeta_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma upExt_PTm_PTm (sigma : nat -> PTm) (tau : nat -> PTm)
- (Eq : forall x, sigma x = tau x) :
- forall x, up_PTm_PTm sigma x = up_PTm_PTm tau x.
-Proof.
-exact (fun n =>
- match n with
- | S n' => ap (ren_PTm shift) (Eq n')
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint ext_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
-(Eq_PTm : forall x, sigma_PTm x = tau_PTm x) (s : PTm) {struct s} :
-subst_PTm sigma_PTm s = subst_PTm tau_PTm s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (ext_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
- (upExt_PTm_PTm _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
- (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (ext_PTm sigma_PTm tau_PTm Eq_PTm s0)
- (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0) (ext_PTm sigma_PTm tau_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma up_ren_ren_PTm_PTm (xi : nat -> nat) (zeta : nat -> nat)
- (rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
- forall x,
- funcomp (upRen_PTm_PTm zeta) (upRen_PTm_PTm xi) x = upRen_PTm_PTm rho x.
-Proof.
-exact (up_ren_ren xi zeta rho Eq).
-Qed.
-
-Fixpoint compRenRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat)
-(rho_PTm : nat -> nat)
-(Eq_PTm : forall x, funcomp zeta_PTm xi_PTm x = rho_PTm x) (s : PTm) {struct
- s} : ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm rho_PTm s :=
- match s with
- | VarPTm s0 => ap (VarPTm) (Eq_PTm s0)
- | PAbs s0 =>
- congr_PAbs
- (compRenRen_PTm (upRen_PTm_PTm xi_PTm) (upRen_PTm_PTm zeta_PTm)
- (upRen_PTm_PTm rho_PTm) (up_ren_ren _ _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
- (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s0)
- (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0)
- (compRenRen_PTm xi_PTm zeta_PTm rho_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma up_ren_subst_PTm_PTm (xi : nat -> nat) (tau : nat -> PTm)
- (theta : nat -> PTm) (Eq : forall x, funcomp tau xi x = theta x) :
- forall x,
- funcomp (up_PTm_PTm tau) (upRen_PTm_PTm xi) x = up_PTm_PTm theta x.
-Proof.
-exact (fun n =>
- match n with
- | S n' => ap (ren_PTm shift) (Eq n')
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint compRenSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm)
-(theta_PTm : nat -> PTm)
-(Eq_PTm : forall x, funcomp tau_PTm xi_PTm x = theta_PTm x) (s : PTm) {struct
- s} : subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm theta_PTm s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (compRenSubst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm tau_PTm)
- (up_PTm_PTm theta_PTm) (up_ren_subst_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
- (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s0)
- (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0)
- (compRenSubst_PTm xi_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma up_subst_ren_PTm_PTm (sigma : nat -> PTm) (zeta_PTm : nat -> nat)
- (theta : nat -> PTm)
- (Eq : forall x, funcomp (ren_PTm zeta_PTm) sigma x = theta x) :
- forall x,
- funcomp (ren_PTm (upRen_PTm_PTm zeta_PTm)) (up_PTm_PTm sigma) x =
- up_PTm_PTm theta x.
-Proof.
-exact (fun n =>
- match n with
- | S n' =>
- eq_trans
- (compRenRen_PTm shift (upRen_PTm_PTm zeta_PTm)
- (funcomp shift zeta_PTm) (fun x => eq_refl) (sigma n'))
- (eq_trans
- (eq_sym
- (compRenRen_PTm zeta_PTm shift (funcomp shift zeta_PTm)
- (fun x => eq_refl) (sigma n')))
- (ap (ren_PTm shift) (Eq n')))
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint compSubstRen_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
-(theta_PTm : nat -> PTm)
-(Eq_PTm : forall x, funcomp (ren_PTm zeta_PTm) sigma_PTm x = theta_PTm x)
-(s : PTm) {struct s} :
-ren_PTm zeta_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (compSubstRen_PTm (up_PTm_PTm sigma_PTm) (upRen_PTm_PTm zeta_PTm)
- (up_PTm_PTm theta_PTm) (up_subst_ren_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
- (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s0)
- (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0)
- (compSubstRen_PTm sigma_PTm zeta_PTm theta_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma up_subst_subst_PTm_PTm (sigma : nat -> PTm) (tau_PTm : nat -> PTm)
- (theta : nat -> PTm)
- (Eq : forall x, funcomp (subst_PTm tau_PTm) sigma x = theta x) :
- forall x,
- funcomp (subst_PTm (up_PTm_PTm tau_PTm)) (up_PTm_PTm sigma) x =
- up_PTm_PTm theta x.
-Proof.
-exact (fun n =>
- match n with
- | S n' =>
- eq_trans
- (compRenSubst_PTm shift (up_PTm_PTm tau_PTm)
- (funcomp (up_PTm_PTm tau_PTm) shift) (fun x => eq_refl)
- (sigma n'))
- (eq_trans
- (eq_sym
- (compSubstRen_PTm tau_PTm shift
- (funcomp (ren_PTm shift) tau_PTm) (fun x => eq_refl)
- (sigma n'))) (ap (ren_PTm shift) (Eq n')))
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint compSubstSubst_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
-(theta_PTm : nat -> PTm)
-(Eq_PTm : forall x, funcomp (subst_PTm tau_PTm) sigma_PTm x = theta_PTm x)
-(s : PTm) {struct s} :
-subst_PTm tau_PTm (subst_PTm sigma_PTm s) = subst_PTm theta_PTm s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (compSubstSubst_PTm (up_PTm_PTm sigma_PTm) (up_PTm_PTm tau_PTm)
- (up_PTm_PTm theta_PTm) (up_subst_subst_PTm_PTm _ _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
- (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s0)
- (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0)
- (compSubstSubst_PTm sigma_PTm tau_PTm theta_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma renRen_PTm (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) (s : PTm) :
- ren_PTm zeta_PTm (ren_PTm xi_PTm s) = ren_PTm (funcomp zeta_PTm xi_PTm) s.
-Proof.
-exact (compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma renRen'_PTm_pointwise (xi_PTm : nat -> nat) (zeta_PTm : nat -> nat) :
- pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (ren_PTm xi_PTm))
- (ren_PTm (funcomp zeta_PTm xi_PTm)).
-Proof.
-exact (fun s => compRenRen_PTm xi_PTm zeta_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma renSubst_PTm (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) (s : PTm) :
- subst_PTm tau_PTm (ren_PTm xi_PTm s) = subst_PTm (funcomp tau_PTm xi_PTm) s.
-Proof.
-exact (compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma renSubst_PTm_pointwise (xi_PTm : nat -> nat) (tau_PTm : nat -> PTm) :
- pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (ren_PTm xi_PTm))
- (subst_PTm (funcomp tau_PTm xi_PTm)).
-Proof.
-exact (fun s => compRenSubst_PTm xi_PTm tau_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma substRen_PTm (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat) (s : PTm)
- :
- ren_PTm zeta_PTm (subst_PTm sigma_PTm s) =
- subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm) s.
-Proof.
-exact (compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma substRen_PTm_pointwise (sigma_PTm : nat -> PTm) (zeta_PTm : nat -> nat)
- :
- pointwise_relation _ eq (funcomp (ren_PTm zeta_PTm) (subst_PTm sigma_PTm))
- (subst_PTm (funcomp (ren_PTm zeta_PTm) sigma_PTm)).
-Proof.
-exact (fun s => compSubstRen_PTm sigma_PTm zeta_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma substSubst_PTm (sigma_PTm : nat -> PTm) (tau_PTm : nat -> PTm)
- (s : PTm) :
- subst_PTm tau_PTm (subst_PTm sigma_PTm s) =
- subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm) s.
-Proof.
-exact (compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma substSubst_PTm_pointwise (sigma_PTm : nat -> PTm)
- (tau_PTm : nat -> PTm) :
- pointwise_relation _ eq (funcomp (subst_PTm tau_PTm) (subst_PTm sigma_PTm))
- (subst_PTm (funcomp (subst_PTm tau_PTm) sigma_PTm)).
-Proof.
-exact (fun s => compSubstSubst_PTm sigma_PTm tau_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma rinstInst_up_PTm_PTm (xi : nat -> nat) (sigma : nat -> PTm)
- (Eq : forall x, funcomp (VarPTm) xi x = sigma x) :
- forall x, funcomp (VarPTm) (upRen_PTm_PTm xi) x = up_PTm_PTm sigma x.
-Proof.
-exact (fun n =>
- match n with
- | S n' => ap (ren_PTm shift) (Eq n')
- | O => eq_refl
- end).
-Qed.
-
-Fixpoint rinst_inst_PTm (xi_PTm : nat -> nat) (sigma_PTm : nat -> PTm)
-(Eq_PTm : forall x, funcomp (VarPTm) xi_PTm x = sigma_PTm x) (s : PTm)
-{struct s} : ren_PTm xi_PTm s = subst_PTm sigma_PTm s :=
- match s with
- | VarPTm s0 => Eq_PTm s0
- | PAbs s0 =>
- congr_PAbs
- (rinst_inst_PTm (upRen_PTm_PTm xi_PTm) (up_PTm_PTm sigma_PTm)
- (rinstInst_up_PTm_PTm _ _ Eq_PTm) s0)
- | PApp s0 s1 =>
- congr_PApp (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
- (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PPair s0 s1 =>
- congr_PPair (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s0)
- (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PProj s0 s1 =>
- congr_PProj (eq_refl s0) (rinst_inst_PTm xi_PTm sigma_PTm Eq_PTm s1)
- | PConst s0 => congr_PConst (eq_refl s0)
- | PUniv s0 => congr_PUniv (eq_refl s0)
- | PBot => congr_PBot
- end.
-
-Lemma rinstInst'_PTm (xi_PTm : nat -> nat) (s : PTm) :
- ren_PTm xi_PTm s = subst_PTm (funcomp (VarPTm) xi_PTm) s.
-Proof.
-exact (rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma rinstInst'_PTm_pointwise (xi_PTm : nat -> nat) :
- pointwise_relation _ eq (ren_PTm xi_PTm)
- (subst_PTm (funcomp (VarPTm) xi_PTm)).
-Proof.
-exact (fun s => rinst_inst_PTm xi_PTm _ (fun n => eq_refl) s).
-Qed.
-
-Lemma instId'_PTm (s : PTm) : subst_PTm (VarPTm) s = s.
-Proof.
-exact (idSubst_PTm (VarPTm) (fun n => eq_refl) s).
-Qed.
-
-Lemma instId'_PTm_pointwise : pointwise_relation _ eq (subst_PTm (VarPTm)) id.
-Proof.
-exact (fun s => idSubst_PTm (VarPTm) (fun n => eq_refl) s).
-Qed.
-
-Lemma rinstId'_PTm (s : PTm) : ren_PTm id s = s.
-Proof.
-exact (eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
-Qed.
-
-Lemma rinstId'_PTm_pointwise : pointwise_relation _ eq (@ren_PTm id) id.
-Proof.
-exact (fun s =>
- eq_ind_r (fun t => t = s) (instId'_PTm s) (rinstInst'_PTm id s)).
-Qed.
-
-Lemma varL'_PTm (sigma_PTm : nat -> PTm) (x : nat) :
- subst_PTm sigma_PTm (VarPTm x) = sigma_PTm x.
-Proof.
-exact (eq_refl).
-Qed.
-
-Lemma varL'_PTm_pointwise (sigma_PTm : nat -> PTm) :
- pointwise_relation _ eq (funcomp (subst_PTm sigma_PTm) (VarPTm)) sigma_PTm.
-Proof.
-exact (fun x => eq_refl).
-Qed.
-
-Lemma varLRen'_PTm (xi_PTm : nat -> nat) (x : nat) :
- ren_PTm xi_PTm (VarPTm x) = VarPTm (xi_PTm x).
-Proof.
-exact (eq_refl).
-Qed.
-
-Lemma varLRen'_PTm_pointwise (xi_PTm : nat -> nat) :
- pointwise_relation _ eq (funcomp (ren_PTm xi_PTm) (VarPTm))
- (funcomp (VarPTm) xi_PTm).
-Proof.
-exact (fun x => eq_refl).
-Qed.
-
-Inductive Tm : Type :=
- | VarTm : nat -> Tm
- | Abs : Tm -> Tm
- | App : Tm -> Tm -> Tm
- | Pair : Tm -> Tm -> Tm
- | Proj : PTag -> Tm -> Tm
- | TBind : TTag -> Tm -> Tm -> Tm
- | Univ : nat -> Tm
- | BVal : bool -> Tm
- | Bool : Tm
- | If : Tm -> Tm -> Tm -> Tm.
-
-Lemma congr_Abs {s0 : Tm} {t0 : Tm} (H0 : s0 = t0) : Abs s0 = Abs t0.
-Proof.
-exact (eq_trans eq_refl (ap (fun x => Abs x) H0)).
-Qed.
-
-Lemma congr_App {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
- (H1 : s1 = t1) : App s0 s1 = App t0 t1.
-Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => App x s1) H0))
- (ap (fun x => App t0 x) H1)).
-Qed.
-
-Lemma congr_Pair {s0 : Tm} {s1 : Tm} {t0 : Tm} {t1 : Tm} (H0 : s0 = t0)
- (H1 : s1 = t1) : Pair s0 s1 = Pair t0 t1.
-Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair x s1) H0))
- (ap (fun x => Pair t0 x) H1)).
-Qed.
-
-Lemma congr_Proj {s0 : PTag} {s1 : Tm} {t0 : PTag} {t1 : Tm} (H0 : s0 = t0)
- (H1 : s1 = t1) : Proj s0 s1 = Proj t0 t1.
-Proof.
-exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj x s1) H0))
- (ap (fun x => Proj t0 x) H1)).
-Qed.
-
-Lemma congr_TBind {s0 : TTag} {s1 : Tm} {s2 : Tm} {t0 : TTag} {t1 : Tm}
- {t2 : Tm} (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) :
- TBind s0 s1 s2 = TBind t0 t1 t2.
-Proof.
-exact (eq_trans
- (eq_trans (eq_trans eq_refl (ap (fun x => TBind x s1 s2) H0))
- (ap (fun x => TBind t0 x s2) H1))
- (ap (fun x => TBind t0 t1 x) H2)).
-Qed.
-
-Lemma congr_Univ {s0 : nat} {t0 : nat} (H0 : s0 = t0) : Univ s0 = Univ t0.
-Proof.
-exact (eq_trans eq_refl (ap (fun x => Univ x) H0)).
-Qed.
-
-Lemma congr_BVal {s0 : bool} {t0 : bool} (H0 : s0 = t0) : BVal s0 = BVal t0.
-Proof.
-exact (eq_trans eq_refl (ap (fun x => BVal x) H0)).
-Qed.
-
-Lemma congr_Bool : Bool = Bool.
-Proof.
-exact (eq_refl).
-Qed.
-
-Lemma congr_If {s0 : Tm} {s1 : Tm} {s2 : Tm} {t0 : Tm} {t1 : Tm} {t2 : Tm}
- (H0 : s0 = t0) (H1 : s1 = t1) (H2 : s2 = t2) : If s0 s1 s2 = If t0 t1 t2.
-Proof.
-exact (eq_trans
- (eq_trans (eq_trans eq_refl (ap (fun x => If x s1 s2) H0))
- (ap (fun x => If t0 x s2) H1)) (ap (fun x => If t0 t1 x) H2)).
-Qed.
-
-Lemma upRen_Tm_Tm (xi : nat -> nat) : nat -> nat.
-Proof.
-exact (up_ren xi).
+exact (scons (VarTm (S n_Tm) var_zero) (funcomp (ren_Tm shift) sigma)).
Defined.
-Fixpoint ren_Tm (xi_Tm : nat -> nat) (s : Tm) {struct s} : Tm :=
- match s with
- | VarTm s0 => VarTm (xi_Tm s0)
- | Abs s0 => Abs (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
- | App s0 s1 => App (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
- | Pair s0 s1 => Pair (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
- | Proj s0 s1 => Proj s0 (ren_Tm xi_Tm s1)
- | TBind s0 s1 s2 =>
- TBind s0 (ren_Tm xi_Tm s1) (ren_Tm (upRen_Tm_Tm xi_Tm) s2)
- | Univ s0 => Univ s0
- | BVal s0 => BVal s0
- | Bool => Bool
- | If s0 s1 s2 => If (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1) (ren_Tm xi_Tm s2)
- end.
-
-Lemma up_Tm_Tm (sigma : nat -> Tm) : nat -> Tm.
+Lemma up_list_Tm_Tm (p : nat) {m : nat} {n_Tm : nat}
+ (sigma : fin m -> Tm n_Tm) : fin (plus p m) -> Tm (plus p n_Tm).
Proof.
-exact (scons (VarTm var_zero) (funcomp (ren_Tm shift) sigma)).
+exact (scons_p p (funcomp (VarTm (plus p n_Tm)) (zero_p p))
+ (funcomp (ren_Tm (shift_p p)) sigma)).
Defined.
-Fixpoint subst_Tm (sigma_Tm : nat -> Tm) (s : Tm) {struct s} : Tm :=
+Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
+(s : Tm m_Tm) {struct s} : Tm n_Tm :=
match s with
- | VarTm s0 => sigma_Tm s0
- | Abs s0 => Abs (subst_Tm (up_Tm_Tm sigma_Tm) s0)
- | App s0 s1 => App (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
- | Pair s0 s1 => Pair (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
- | Proj s0 s1 => Proj s0 (subst_Tm sigma_Tm s1)
- | TBind s0 s1 s2 =>
- TBind s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
- | Univ s0 => Univ s0
- | BVal s0 => BVal s0
- | Bool => Bool
- | If s0 s1 s2 =>
- If (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1) (subst_Tm sigma_Tm s2)
+ | VarTm _ s0 => sigma_Tm s0
+ | Abs _ s0 => Abs n_Tm (subst_Tm (up_Tm_Tm sigma_Tm) s0)
+ | App _ s0 s1 => App n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
+ | Pair _ s0 s1 => Pair n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
+ | Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
+ | TBind _ s0 s1 s2 =>
+ TBind n_Tm s0 (subst_Tm sigma_Tm s1) (subst_Tm (up_Tm_Tm sigma_Tm) s2)
+ | Bot _ => Bot n_Tm
+ | Univ _ s0 => Univ n_Tm s0
end.
-Lemma upId_Tm_Tm (sigma : nat -> Tm) (Eq : forall x, sigma x = VarTm x) :
- forall x, up_Tm_Tm sigma x = VarTm x.
+Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
+ (Eq : forall x, sigma x = VarTm m_Tm x) :
+ forall x, up_Tm_Tm sigma x = VarTm (S m_Tm) x.
Proof.
exact (fun n =>
match n with
- | S n' => ap (ren_Tm shift) (Eq n')
- | O => eq_refl
+ | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
+ | None => eq_refl
end).
Qed.
-Fixpoint idSubst_Tm (sigma_Tm : nat -> Tm)
-(Eq_Tm : forall x, sigma_Tm x = VarTm x) (s : Tm) {struct s} :
+Lemma upId_list_Tm_Tm {p : nat} {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
+ (Eq : forall x, sigma x = VarTm m_Tm x) :
+ forall x, up_list_Tm_Tm p sigma x = VarTm (plus p m_Tm) x.
+Proof.
+exact (fun n =>
+ scons_p_eta (VarTm (plus p m_Tm))
+ (fun n => ap (ren_Tm (shift_p p)) (Eq n)) (fun n => eq_refl)).
+Qed.
+
+Fixpoint idSubst_Tm {m_Tm : nat} (sigma_Tm : fin m_Tm -> Tm m_Tm)
+(Eq_Tm : forall x, sigma_Tm x = VarTm m_Tm x) (s : Tm m_Tm) {struct s} :
subst_Tm sigma_Tm s = s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs (idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm sigma_Tm Eq_Tm s1)
- | Proj s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | Proj _ s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
(idSubst_Tm (up_Tm_Tm sigma_Tm) (upId_Tm_Tm _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (idSubst_Tm sigma_Tm Eq_Tm s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
- (idSubst_Tm sigma_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma upExtRen_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
- (Eq : forall x, xi x = zeta x) :
+Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
+ (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
forall x, upRen_Tm_Tm xi x = upRen_Tm_Tm zeta x.
Proof.
-exact (fun n => match n with
- | S n' => ap shift (Eq n')
- | O => eq_refl
- end).
+exact (fun n =>
+ match n with
+ | Some fin_n => ap shift (Eq fin_n)
+ | None => eq_refl
+ end).
Qed.
-Fixpoint extRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
-(Eq_Tm : forall x, xi_Tm x = zeta_Tm x) (s : Tm) {struct s} :
-ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
+Lemma upExtRen_list_Tm_Tm {p : nat} {m : nat} {n : nat} (xi : fin m -> fin n)
+ (zeta : fin m -> fin n) (Eq : forall x, xi x = zeta x) :
+ forall x, upRen_list_Tm_Tm p xi x = upRen_list_Tm_Tm p zeta x.
+Proof.
+exact (fun n =>
+ scons_p_congr (fun n => eq_refl) (fun n => ap (shift_p p) (Eq n))).
+Qed.
+
+Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
+(zeta_Tm : fin m_Tm -> fin n_Tm) (Eq_Tm : forall x, xi_Tm x = zeta_Tm x)
+(s : Tm m_Tm) {struct s} : ren_Tm xi_Tm s = ren_Tm zeta_Tm s :=
match s with
- | VarTm s0 => ap (VarTm) (Eq_Tm s0)
- | Abs s0 =>
+ | VarTm _ s0 => ap (VarTm n_Tm) (Eq_Tm s0)
+ | Abs _ s0 =>
congr_Abs
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | Proj s0 s1 => congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | Proj _ s0 s1 =>
+ congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
(extRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upExtRen_Tm_Tm _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
- (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma upExt_Tm_Tm (sigma : nat -> Tm) (tau : nat -> Tm)
- (Eq : forall x, sigma x = tau x) :
+Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
+ (tau : fin m -> Tm n_Tm) (Eq : forall x, sigma x = tau x) :
forall x, up_Tm_Tm sigma x = up_Tm_Tm tau x.
Proof.
exact (fun n =>
match n with
- | S n' => ap (ren_Tm shift) (Eq n')
- | O => eq_refl
+ | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
+ | None => eq_refl
end).
Qed.
-Fixpoint ext_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
-(Eq_Tm : forall x, sigma_Tm x = tau_Tm x) (s : Tm) {struct s} :
-subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
+Lemma upExt_list_Tm_Tm {p : nat} {m : nat} {n_Tm : nat}
+ (sigma : fin m -> Tm n_Tm) (tau : fin m -> Tm n_Tm)
+ (Eq : forall x, sigma x = tau x) :
+ forall x, up_list_Tm_Tm p sigma x = up_list_Tm_Tm p tau x.
+Proof.
+exact (fun n =>
+ scons_p_congr (fun n => eq_refl)
+ (fun n => ap (ren_Tm (shift_p p)) (Eq n))).
+Qed.
+
+Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
+(tau_Tm : fin m_Tm -> Tm n_Tm) (Eq_Tm : forall x, sigma_Tm x = tau_Tm x)
+(s : Tm m_Tm) {struct s} : subst_Tm sigma_Tm s = subst_Tm tau_Tm s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | Proj s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | Proj _ s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
(ext_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm) (upExt_Tm_Tm _ _ Eq_Tm)
s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
- (ext_Tm sigma_Tm tau_Tm Eq_Tm s1) (ext_Tm sigma_Tm tau_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma up_ren_ren_Tm_Tm (xi : nat -> nat) (zeta : nat -> nat)
- (rho : nat -> nat) (Eq : forall x, funcomp zeta xi x = rho x) :
+Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
+ (zeta : fin l -> fin m) (rho : fin k -> fin m)
+ (Eq : forall x, funcomp zeta xi x = rho x) :
forall x, funcomp (upRen_Tm_Tm zeta) (upRen_Tm_Tm xi) x = upRen_Tm_Tm rho x.
Proof.
exact (up_ren_ren xi zeta rho Eq).
Qed.
-Fixpoint compRenRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat)
-(rho_Tm : nat -> nat) (Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x)
-(s : Tm) {struct s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
+Lemma up_ren_ren_list_Tm_Tm {p : nat} {k : nat} {l : nat} {m : nat}
+ (xi : fin k -> fin l) (zeta : fin l -> fin m) (rho : fin k -> fin m)
+ (Eq : forall x, funcomp zeta xi x = rho x) :
+ forall x,
+ funcomp (upRen_list_Tm_Tm p zeta) (upRen_list_Tm_Tm p xi) x =
+ upRen_list_Tm_Tm p rho x.
+Proof.
+exact (up_ren_ren_p Eq).
+Qed.
+
+Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+(xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
+(rho_Tm : fin m_Tm -> fin l_Tm)
+(Eq_Tm : forall x, funcomp zeta_Tm xi_Tm x = rho_Tm x) (s : Tm m_Tm) {struct
+ s} : ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm rho_Tm s :=
match s with
- | VarTm s0 => ap (VarTm) (Eq_Tm s0)
- | Abs s0 =>
+ | VarTm _ s0 => ap (VarTm l_Tm) (Eq_Tm s0)
+ | Abs _ s0 =>
congr_Abs
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | Proj s0 s1 =>
+ | Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
(compRenRen_Tm (upRen_Tm_Tm xi_Tm) (upRen_Tm_Tm zeta_Tm)
(upRen_Tm_Tm rho_Tm) (up_ren_ren _ _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
- (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
- (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma up_ren_subst_Tm_Tm (xi : nat -> nat) (tau : nat -> Tm)
- (theta : nat -> Tm) (Eq : forall x, funcomp tau xi x = theta x) :
+Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
+ (xi : fin k -> fin l) (tau : fin l -> Tm m_Tm) (theta : fin k -> Tm m_Tm)
+ (Eq : forall x, funcomp tau xi x = theta x) :
forall x, funcomp (up_Tm_Tm tau) (upRen_Tm_Tm xi) x = up_Tm_Tm theta x.
Proof.
exact (fun n =>
match n with
- | S n' => ap (ren_Tm shift) (Eq n')
- | O => eq_refl
+ | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
+ | None => eq_refl
end).
Qed.
-Fixpoint compRenSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm)
-(theta_Tm : nat -> Tm)
-(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm) {struct s} :
-subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
+Lemma up_ren_subst_list_Tm_Tm {p : nat} {k : nat} {l : nat} {m_Tm : nat}
+ (xi : fin k -> fin l) (tau : fin l -> Tm m_Tm) (theta : fin k -> Tm m_Tm)
+ (Eq : forall x, funcomp tau xi x = theta x) :
+ forall x,
+ funcomp (up_list_Tm_Tm p tau) (upRen_list_Tm_Tm p xi) x =
+ up_list_Tm_Tm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ _ n)
+ (scons_p_congr (fun z => scons_p_head' _ _ z)
+ (fun z =>
+ eq_trans (scons_p_tail' _ _ (xi z))
+ (ap (ren_Tm (shift_p p)) (Eq z))))).
+Qed.
+
+Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+(xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
+(theta_Tm : fin m_Tm -> Tm l_Tm)
+(Eq_Tm : forall x, funcomp tau_Tm xi_Tm x = theta_Tm x) (s : Tm m_Tm) {struct
+ s} : subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Proj s0 s1 =>
+ | Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
(compRenSubst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_ren_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
- (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
- (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma up_subst_ren_Tm_Tm (sigma : nat -> Tm) (zeta_Tm : nat -> nat)
- (theta : nat -> Tm)
+Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma : fin k -> Tm l_Tm) (zeta_Tm : fin l_Tm -> fin m_Tm)
+ (theta : fin k -> Tm m_Tm)
(Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
forall x,
funcomp (ren_Tm (upRen_Tm_Tm zeta_Tm)) (up_Tm_Tm sigma) x =
@@ -840,257 +387,326 @@ Lemma up_subst_ren_Tm_Tm (sigma : nat -> Tm) (zeta_Tm : nat -> nat)
Proof.
exact (fun n =>
match n with
- | S n' =>
+ | Some fin_n =>
eq_trans
(compRenRen_Tm shift (upRen_Tm_Tm zeta_Tm)
- (funcomp shift zeta_Tm) (fun x => eq_refl) (sigma n'))
+ (funcomp shift zeta_Tm) (fun x => eq_refl) (sigma fin_n))
(eq_trans
(eq_sym
(compRenRen_Tm zeta_Tm shift (funcomp shift zeta_Tm)
- (fun x => eq_refl) (sigma n')))
- (ap (ren_Tm shift) (Eq n')))
- | O => eq_refl
+ (fun x => eq_refl) (sigma fin_n)))
+ (ap (ren_Tm shift) (Eq fin_n)))
+ | None => eq_refl
end).
Qed.
-Fixpoint compSubstRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat)
-(theta_Tm : nat -> Tm)
-(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
-(s : Tm) {struct s} :
+Lemma up_subst_ren_list_Tm_Tm {p : nat} {k : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma : fin k -> Tm l_Tm) (zeta_Tm : fin l_Tm -> fin m_Tm)
+ (theta : fin k -> Tm m_Tm)
+ (Eq : forall x, funcomp (ren_Tm zeta_Tm) sigma x = theta x) :
+ forall x,
+ funcomp (ren_Tm (upRen_list_Tm_Tm p zeta_Tm)) (up_list_Tm_Tm p sigma) x =
+ up_list_Tm_Tm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ _ n)
+ (scons_p_congr
+ (fun x => ap (VarTm (plus p m_Tm)) (scons_p_head' _ _ x))
+ (fun n =>
+ eq_trans
+ (compRenRen_Tm (shift_p p) (upRen_list_Tm_Tm p zeta_Tm)
+ (funcomp (shift_p p) zeta_Tm)
+ (fun x => scons_p_tail' _ _ x) (sigma n))
+ (eq_trans
+ (eq_sym
+ (compRenRen_Tm zeta_Tm (shift_p p)
+ (funcomp (shift_p p) zeta_Tm) (fun x => eq_refl)
+ (sigma n))) (ap (ren_Tm (shift_p p)) (Eq n)))))).
+Qed.
+
+Fixpoint compSubstRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+(sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
+(theta_Tm : fin m_Tm -> Tm l_Tm)
+(Eq_Tm : forall x, funcomp (ren_Tm zeta_Tm) sigma_Tm x = theta_Tm x)
+(s : Tm m_Tm) {struct s} :
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | Proj s0 s1 =>
+ | Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
(compSubstRen_Tm (up_Tm_Tm sigma_Tm) (upRen_Tm_Tm zeta_Tm)
(up_Tm_Tm theta_Tm) (up_subst_ren_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
- (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
- (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma up_subst_subst_Tm_Tm (sigma : nat -> Tm) (tau_Tm : nat -> Tm)
- (theta : nat -> Tm)
+Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma : fin k -> Tm l_Tm) (tau_Tm : fin l_Tm -> Tm m_Tm)
+ (theta : fin k -> Tm m_Tm)
(Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
forall x,
funcomp (subst_Tm (up_Tm_Tm tau_Tm)) (up_Tm_Tm sigma) x = up_Tm_Tm theta x.
Proof.
exact (fun n =>
match n with
- | S n' =>
+ | Some fin_n =>
eq_trans
(compRenSubst_Tm shift (up_Tm_Tm tau_Tm)
(funcomp (up_Tm_Tm tau_Tm) shift) (fun x => eq_refl)
- (sigma n'))
+ (sigma fin_n))
(eq_trans
(eq_sym
(compSubstRen_Tm tau_Tm shift
(funcomp (ren_Tm shift) tau_Tm) (fun x => eq_refl)
- (sigma n'))) (ap (ren_Tm shift) (Eq n')))
- | O => eq_refl
+ (sigma fin_n))) (ap (ren_Tm shift) (Eq fin_n)))
+ | None => eq_refl
end).
Qed.
-Fixpoint compSubstSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm)
-(theta_Tm : nat -> Tm)
+Lemma up_subst_subst_list_Tm_Tm {p : nat} {k : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma : fin k -> Tm l_Tm) (tau_Tm : fin l_Tm -> Tm m_Tm)
+ (theta : fin k -> Tm m_Tm)
+ (Eq : forall x, funcomp (subst_Tm tau_Tm) sigma x = theta x) :
+ forall x,
+ funcomp (subst_Tm (up_list_Tm_Tm p tau_Tm)) (up_list_Tm_Tm p sigma) x =
+ up_list_Tm_Tm p theta x.
+Proof.
+exact (fun n =>
+ eq_trans
+ (scons_p_comp' (funcomp (VarTm (plus p l_Tm)) (zero_p p)) _ _ n)
+ (scons_p_congr
+ (fun x => scons_p_head' _ (fun z => ren_Tm (shift_p p) _) x)
+ (fun n =>
+ eq_trans
+ (compRenSubst_Tm (shift_p p) (up_list_Tm_Tm p tau_Tm)
+ (funcomp (up_list_Tm_Tm p tau_Tm) (shift_p p))
+ (fun x => eq_refl) (sigma n))
+ (eq_trans
+ (eq_sym
+ (compSubstRen_Tm tau_Tm (shift_p p) _
+ (fun x => eq_sym (scons_p_tail' _ _ x)) (sigma n)))
+ (ap (ren_Tm (shift_p p)) (Eq n)))))).
+Qed.
+
+Fixpoint compSubstSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+(sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
+(theta_Tm : fin m_Tm -> Tm l_Tm)
(Eq_Tm : forall x, funcomp (subst_Tm tau_Tm) sigma_Tm x = theta_Tm x)
-(s : Tm) {struct s} :
+(s : Tm m_Tm) {struct s} :
subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | Proj s0 s1 =>
+ | Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
(compSubstSubst_Tm (up_Tm_Tm sigma_Tm) (up_Tm_Tm tau_Tm)
(up_Tm_Tm theta_Tm) (up_subst_subst_Tm_Tm _ _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
- (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
- (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma renRen_Tm (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) (s : Tm) :
+Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
+ (s : Tm m_Tm) :
ren_Tm zeta_Tm (ren_Tm xi_Tm s) = ren_Tm (funcomp zeta_Tm xi_Tm) s.
Proof.
exact (compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renRen'_Tm_pointwise (xi_Tm : nat -> nat) (zeta_Tm : nat -> nat) :
+Lemma renRen'_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm) :
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (ren_Tm xi_Tm))
(ren_Tm (funcomp zeta_Tm xi_Tm)).
Proof.
exact (fun s => compRenRen_Tm xi_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_Tm (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) (s : Tm) :
- subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
+Lemma renSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) (s : Tm m_Tm)
+ : subst_Tm tau_Tm (ren_Tm xi_Tm s) = subst_Tm (funcomp tau_Tm xi_Tm) s.
Proof.
exact (compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma renSubst_Tm_pointwise (xi_Tm : nat -> nat) (tau_Tm : nat -> Tm) :
+Lemma renSubst_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) :
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (ren_Tm xi_Tm))
(subst_Tm (funcomp tau_Tm xi_Tm)).
Proof.
exact (fun s => compRenSubst_Tm xi_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_Tm (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) (s : Tm) :
+Lemma substRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm)
+ (s : Tm m_Tm) :
ren_Tm zeta_Tm (subst_Tm sigma_Tm s) =
subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm) s.
Proof.
exact (compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substRen_Tm_pointwise (sigma_Tm : nat -> Tm) (zeta_Tm : nat -> nat) :
+Lemma substRen_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma_Tm : fin m_Tm -> Tm k_Tm) (zeta_Tm : fin k_Tm -> fin l_Tm) :
pointwise_relation _ eq (funcomp (ren_Tm zeta_Tm) (subst_Tm sigma_Tm))
(subst_Tm (funcomp (ren_Tm zeta_Tm) sigma_Tm)).
Proof.
exact (fun s => compSubstRen_Tm sigma_Tm zeta_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_Tm (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) (s : Tm) :
+Lemma substSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm)
+ (s : Tm m_Tm) :
subst_Tm tau_Tm (subst_Tm sigma_Tm s) =
subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm) s.
Proof.
exact (compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma substSubst_Tm_pointwise (sigma_Tm : nat -> Tm) (tau_Tm : nat -> Tm) :
+Lemma substSubst_Tm_pointwise {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
+ (sigma_Tm : fin m_Tm -> Tm k_Tm) (tau_Tm : fin k_Tm -> Tm l_Tm) :
pointwise_relation _ eq (funcomp (subst_Tm tau_Tm) (subst_Tm sigma_Tm))
(subst_Tm (funcomp (subst_Tm tau_Tm) sigma_Tm)).
Proof.
exact (fun s => compSubstSubst_Tm sigma_Tm tau_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst_up_Tm_Tm (xi : nat -> nat) (sigma : nat -> Tm)
- (Eq : forall x, funcomp (VarTm) xi x = sigma x) :
- forall x, funcomp (VarTm) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
+Lemma rinstInst_up_Tm_Tm {m : nat} {n_Tm : nat} (xi : fin m -> fin n_Tm)
+ (sigma : fin m -> Tm n_Tm)
+ (Eq : forall x, funcomp (VarTm n_Tm) xi x = sigma x) :
+ forall x, funcomp (VarTm (S n_Tm)) (upRen_Tm_Tm xi) x = up_Tm_Tm sigma x.
Proof.
exact (fun n =>
match n with
- | S n' => ap (ren_Tm shift) (Eq n')
- | O => eq_refl
+ | Some fin_n => ap (ren_Tm shift) (Eq fin_n)
+ | None => eq_refl
end).
Qed.
-Fixpoint rinst_inst_Tm (xi_Tm : nat -> nat) (sigma_Tm : nat -> Tm)
-(Eq_Tm : forall x, funcomp (VarTm) xi_Tm x = sigma_Tm x) (s : Tm) {struct s}
- : ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
+Lemma rinstInst_up_list_Tm_Tm {p : nat} {m : nat} {n_Tm : nat}
+ (xi : fin m -> fin n_Tm) (sigma : fin m -> Tm n_Tm)
+ (Eq : forall x, funcomp (VarTm n_Tm) xi x = sigma x) :
+ forall x,
+ funcomp (VarTm (plus p n_Tm)) (upRen_list_Tm_Tm p xi) x =
+ up_list_Tm_Tm p sigma x.
+Proof.
+exact (fun n =>
+ eq_trans (scons_p_comp' _ _ (VarTm (plus p n_Tm)) n)
+ (scons_p_congr (fun z => eq_refl)
+ (fun n => ap (ren_Tm (shift_p p)) (Eq n)))).
+Qed.
+
+Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
+(xi_Tm : fin m_Tm -> fin n_Tm) (sigma_Tm : fin m_Tm -> Tm n_Tm)
+(Eq_Tm : forall x, funcomp (VarTm n_Tm) xi_Tm x = sigma_Tm x) (s : Tm m_Tm)
+{struct s} : ren_Tm xi_Tm s = subst_Tm sigma_Tm s :=
match s with
- | VarTm s0 => Eq_Tm s0
- | Abs s0 =>
+ | VarTm _ s0 => Eq_Tm s0
+ | Abs _ s0 =>
congr_Abs
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s0)
- | App s0 s1 =>
+ | App _ s0 s1 =>
congr_App (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | Pair s0 s1 =>
+ | Pair _ s0 s1 =>
congr_Pair (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | Proj s0 s1 =>
+ | Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- | TBind s0 s1 s2 =>
+ | TBind _ s0 s1 s2 =>
congr_TBind (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
(rinst_inst_Tm (upRen_Tm_Tm xi_Tm) (up_Tm_Tm sigma_Tm)
(rinstInst_up_Tm_Tm _ _ Eq_Tm) s2)
- | Univ s0 => congr_Univ (eq_refl s0)
- | BVal s0 => congr_BVal (eq_refl s0)
- | Bool => congr_Bool
- | If s0 s1 s2 =>
- congr_If (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
- (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
- (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s2)
+ | Bot _ => congr_Bot
+ | Univ _ s0 => congr_Univ (eq_refl s0)
end.
-Lemma rinstInst'_Tm (xi_Tm : nat -> nat) (s : Tm) :
- ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm) xi_Tm) s.
+Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
+ (s : Tm m_Tm) : ren_Tm xi_Tm s = subst_Tm (funcomp (VarTm n_Tm) xi_Tm) s.
Proof.
exact (rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma rinstInst'_Tm_pointwise (xi_Tm : nat -> nat) :
- pointwise_relation _ eq (ren_Tm xi_Tm) (subst_Tm (funcomp (VarTm) xi_Tm)).
+Lemma rinstInst'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin n_Tm) :
+ pointwise_relation _ eq (ren_Tm xi_Tm)
+ (subst_Tm (funcomp (VarTm n_Tm) xi_Tm)).
Proof.
exact (fun s => rinst_inst_Tm xi_Tm _ (fun n => eq_refl) s).
Qed.
-Lemma instId'_Tm (s : Tm) : subst_Tm (VarTm) s = s.
+Lemma instId'_Tm {m_Tm : nat} (s : Tm m_Tm) : subst_Tm (VarTm m_Tm) s = s.
Proof.
-exact (idSubst_Tm (VarTm) (fun n => eq_refl) s).
+exact (idSubst_Tm (VarTm m_Tm) (fun n => eq_refl) s).
Qed.
-Lemma instId'_Tm_pointwise : pointwise_relation _ eq (subst_Tm (VarTm)) id.
+Lemma instId'_Tm_pointwise {m_Tm : nat} :
+ pointwise_relation _ eq (subst_Tm (VarTm m_Tm)) id.
Proof.
-exact (fun s => idSubst_Tm (VarTm) (fun n => eq_refl) s).
+exact (fun s => idSubst_Tm (VarTm m_Tm) (fun n => eq_refl) s).
Qed.
-Lemma rinstId'_Tm (s : Tm) : ren_Tm id s = s.
+Lemma rinstId'_Tm {m_Tm : nat} (s : Tm m_Tm) : ren_Tm id s = s.
Proof.
exact (eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
Qed.
-Lemma rinstId'_Tm_pointwise : pointwise_relation _ eq (@ren_Tm id) id.
+Lemma rinstId'_Tm_pointwise {m_Tm : nat} :
+ pointwise_relation _ eq (@ren_Tm m_Tm m_Tm id) id.
Proof.
exact (fun s => eq_ind_r (fun t => t = s) (instId'_Tm s) (rinstInst'_Tm id s)).
Qed.
-Lemma varL'_Tm (sigma_Tm : nat -> Tm) (x : nat) :
- subst_Tm sigma_Tm (VarTm x) = sigma_Tm x.
+Lemma varL'_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
+ (x : fin m_Tm) : subst_Tm sigma_Tm (VarTm m_Tm x) = sigma_Tm x.
Proof.
exact (eq_refl).
Qed.
-Lemma varL'_Tm_pointwise (sigma_Tm : nat -> Tm) :
- pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm)) sigma_Tm.
+Lemma varL'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
+ (sigma_Tm : fin m_Tm -> Tm n_Tm) :
+ pointwise_relation _ eq (funcomp (subst_Tm sigma_Tm) (VarTm m_Tm)) sigma_Tm.
Proof.
exact (fun x => eq_refl).
Qed.
-Lemma varLRen'_Tm (xi_Tm : nat -> nat) (x : nat) :
- ren_Tm xi_Tm (VarTm x) = VarTm (xi_Tm x).
+Lemma varLRen'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
+ (x : fin m_Tm) : ren_Tm xi_Tm (VarTm m_Tm x) = VarTm n_Tm (xi_Tm x).
Proof.
exact (eq_refl).
Qed.
-Lemma varLRen'_Tm_pointwise (xi_Tm : nat -> nat) :
- pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm))
- (funcomp (VarTm) xi_Tm).
+Lemma varLRen'_Tm_pointwise {m_Tm : nat} {n_Tm : nat}
+ (xi_Tm : fin m_Tm -> fin n_Tm) :
+ pointwise_relation _ eq (funcomp (ren_Tm xi_Tm) (VarTm m_Tm))
+ (funcomp (VarTm n_Tm) xi_Tm).
Proof.
exact (fun x => eq_refl).
Qed.
@@ -1098,25 +714,17 @@ Qed.
Class Up_Tm X Y :=
up_Tm : X -> Y.
-Class Up_PTm X Y :=
- up_PTm : X -> Y.
-
-#[global] Instance Subst_Tm : (Subst1 _ _ _) := @subst_Tm.
-
-#[global] Instance Up_Tm_Tm : (Up_Tm _ _) := @up_Tm_Tm.
-
-#[global] Instance Ren_Tm : (Ren1 _ _ _) := @ren_Tm.
-
-#[global] Instance VarInstance_Tm : (Var _ _) := @VarTm.
-
-#[global] Instance Subst_PTm : (Subst1 _ _ _) := @subst_PTm.
-
-#[global] Instance Up_PTm_PTm : (Up_PTm _ _) := @up_PTm_PTm.
-
-#[global] Instance Ren_PTm : (Ren1 _ _ _) := @ren_PTm.
+#[global]
+Instance Subst_Tm {m_Tm n_Tm : nat}: (Subst1 _ _ _) := (@subst_Tm m_Tm n_Tm).
#[global]
-Instance VarInstance_PTm : (Var _ _) := @VarPTm.
+Instance Up_Tm_Tm {m n_Tm : nat}: (Up_Tm _ _) := (@up_Tm_Tm m n_Tm).
+
+#[global]
+Instance Ren_Tm {m_Tm n_Tm : nat}: (Ren1 _ _ _) := (@ren_Tm m_Tm n_Tm).
+
+#[global]
+Instance VarInstance_Tm {n_Tm : nat}: (Var _ _) := (@VarTm n_Tm).
Notation "[ sigma_Tm ]" := (subst_Tm sigma_Tm)
( at level 1, left associativity, only printing) : fscope.
@@ -1142,34 +750,10 @@ Notation "x '__Tm'" := (@ids _ _ VarInstance_Tm x)
Notation "x '__Tm'" := (VarTm x) ( at level 5, format "x __Tm") :
subst_scope.
-Notation "[ sigma_PTm ]" := (subst_PTm sigma_PTm)
-( at level 1, left associativity, only printing) : fscope.
-
-Notation "s [ sigma_PTm ]" := (subst_PTm sigma_PTm s)
-( at level 7, left associativity, only printing) : subst_scope.
-
-Notation "↑__PTm" := up_PTm (only printing) : subst_scope.
-
-Notation "↑__PTm" := up_PTm_PTm (only printing) : subst_scope.
-
-Notation "⟨ xi_PTm ⟩" := (ren_PTm xi_PTm)
-( at level 1, left associativity, only printing) : fscope.
-
-Notation "s ⟨ xi_PTm ⟩" := (ren_PTm xi_PTm s)
-( at level 7, left associativity, only printing) : subst_scope.
-
-Notation "'var'" := VarPTm ( at level 1, only printing) : subst_scope.
-
-Notation "x '__PTm'" := (@ids _ _ VarInstance_PTm x)
-( at level 5, format "x __PTm", only printing) : subst_scope.
-
-Notation "x '__PTm'" := (VarPTm x) ( at level 5, format "x __PTm") :
-subst_scope.
-
#[global]
-Instance subst_Tm_morphism :
+Instance subst_Tm_morphism {m_Tm : nat} {n_Tm : nat}:
(Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@subst_Tm)).
+ (@subst_Tm m_Tm n_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
eq_ind s (fun t' => subst_Tm f_Tm s = subst_Tm g_Tm t')
@@ -1177,16 +761,17 @@ exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
Qed.
#[global]
-Instance subst_Tm_morphism2 :
+Instance subst_Tm_morphism2 {m_Tm : nat} {n_Tm : nat}:
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@subst_Tm)).
+ (@subst_Tm m_Tm n_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s => ext_Tm f_Tm g_Tm Eq_Tm s).
Qed.
#[global]
-Instance ren_Tm_morphism :
- (Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_Tm)).
+Instance ren_Tm_morphism {m_Tm : nat} {n_Tm : nat}:
+ (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
+ (@ren_Tm m_Tm n_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
eq_ind s (fun t' => ren_Tm f_Tm s = ren_Tm g_Tm t')
@@ -1194,62 +779,22 @@ exact (fun f_Tm g_Tm Eq_Tm s t Eq_st =>
Qed.
#[global]
-Instance ren_Tm_morphism2 :
+Instance ren_Tm_morphism2 {m_Tm : nat} {n_Tm : nat}:
(Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@ren_Tm)).
+ (@ren_Tm m_Tm n_Tm)).
Proof.
exact (fun f_Tm g_Tm Eq_Tm s => extRen_Tm f_Tm g_Tm Eq_Tm s).
Qed.
-#[global]
-Instance subst_PTm_morphism :
- (Proper (respectful (pointwise_relation _ eq) (respectful eq eq))
- (@subst_PTm)).
-Proof.
-exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
- eq_ind s (fun t' => subst_PTm f_PTm s = subst_PTm g_PTm t')
- (ext_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
-Qed.
-
-#[global]
-Instance subst_PTm_morphism2 :
- (Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@subst_PTm)).
-Proof.
-exact (fun f_PTm g_PTm Eq_PTm s => ext_PTm f_PTm g_PTm Eq_PTm s).
-Qed.
-
-#[global]
-Instance ren_PTm_morphism :
- (Proper (respectful (pointwise_relation _ eq) (respectful eq eq)) (@ren_PTm)).
-Proof.
-exact (fun f_PTm g_PTm Eq_PTm s t Eq_st =>
- eq_ind s (fun t' => ren_PTm f_PTm s = ren_PTm g_PTm t')
- (extRen_PTm f_PTm g_PTm Eq_PTm s) t Eq_st).
-Qed.
-
-#[global]
-Instance ren_PTm_morphism2 :
- (Proper (respectful (pointwise_relation _ eq) (pointwise_relation _ eq))
- (@ren_PTm)).
-Proof.
-exact (fun f_PTm g_PTm Eq_PTm s => extRen_PTm f_PTm g_PTm Eq_PTm s).
-Qed.
-
Ltac auto_unfold := repeat
- unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
- Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
- VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
- Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
+ unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
+ Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1.
Tactic Notation "auto_unfold" "in" "*" := repeat
- unfold VarInstance_PTm, Var, ids,
- Ren_PTm, Ren1, ren1, Up_PTm_PTm,
- Up_PTm, up_PTm, Subst_PTm,
- Subst1, subst1, VarInstance_Tm,
- Var, ids, Ren_Tm, Ren1, ren1,
- Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm,
- Subst1, subst1 in *.
+ unfold VarInstance_Tm, Var, ids,
+ Ren_Tm, Ren1, ren1, Up_Tm_Tm,
+ Up_Tm, up_Tm, Subst_Tm, Subst1,
+ subst1 in *.
Ltac asimpl' := repeat (first
[ progress setoid_rewrite substSubst_Tm_pointwise
@@ -1260,14 +805,6 @@ Ltac asimpl' := repeat (first
| progress setoid_rewrite renSubst_Tm
| progress setoid_rewrite renRen'_Tm_pointwise
| progress setoid_rewrite renRen_Tm
- | progress setoid_rewrite substSubst_PTm_pointwise
- | progress setoid_rewrite substSubst_PTm
- | progress setoid_rewrite substRen_PTm_pointwise
- | progress setoid_rewrite substRen_PTm
- | progress setoid_rewrite renSubst_PTm_pointwise
- | progress setoid_rewrite renSubst_PTm
- | progress setoid_rewrite renRen'_PTm_pointwise
- | progress setoid_rewrite renRen_PTm
| progress setoid_rewrite varLRen'_Tm_pointwise
| progress setoid_rewrite varLRen'_Tm
| progress setoid_rewrite varL'_Tm_pointwise
@@ -1276,55 +813,54 @@ Ltac asimpl' := repeat (first
| progress setoid_rewrite rinstId'_Tm
| progress setoid_rewrite instId'_Tm_pointwise
| progress setoid_rewrite instId'_Tm
- | progress setoid_rewrite varLRen'_PTm_pointwise
- | progress setoid_rewrite varLRen'_PTm
- | progress setoid_rewrite varL'_PTm_pointwise
- | progress setoid_rewrite varL'_PTm
- | progress setoid_rewrite rinstId'_PTm_pointwise
- | progress setoid_rewrite rinstId'_PTm
- | progress setoid_rewrite instId'_PTm_pointwise
- | progress setoid_rewrite instId'_PTm
| progress
- unfold up_Tm_Tm, upRen_Tm_Tm, up_PTm_PTm, upRen_PTm_PTm,
- up_ren
- | progress cbn[subst_Tm ren_Tm subst_PTm ren_PTm]
+ unfold up_list_Tm_Tm, up_Tm_Tm, upRen_list_Tm_Tm,
+ upRen_Tm_Tm, up_ren
+ | progress cbn[subst_Tm ren_Tm]
| progress fsimpl ]).
Ltac asimpl := check_no_evars;
repeat
- unfold VarInstance_PTm, Var, ids, Ren_PTm, Ren1, ren1,
- Up_PTm_PTm, Up_PTm, up_PTm, Subst_PTm, Subst1, subst1,
- VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1, Up_Tm_Tm,
- Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *; asimpl';
- minimize.
+ unfold VarInstance_Tm, Var, ids, Ren_Tm, Ren1, ren1,
+ Up_Tm_Tm, Up_Tm, up_Tm, Subst_Tm, Subst1, subst1 in *;
+ asimpl'; minimize.
Tactic Notation "asimpl" "in" hyp(J) := revert J; asimpl; intros J.
Tactic Notation "auto_case" := auto_case ltac:(asimpl; cbn; eauto).
Ltac substify := auto_unfold; try setoid_rewrite rinstInst'_Tm_pointwise;
- try setoid_rewrite rinstInst'_Tm;
- try setoid_rewrite rinstInst'_PTm_pointwise;
- try setoid_rewrite rinstInst'_PTm.
+ try setoid_rewrite rinstInst'_Tm.
Ltac renamify := auto_unfold; try setoid_rewrite_left rinstInst'_Tm_pointwise;
- try setoid_rewrite_left rinstInst'_Tm;
- try setoid_rewrite_left rinstInst'_PTm_pointwise;
- try setoid_rewrite_left rinstInst'_PTm.
+ try setoid_rewrite_left rinstInst'_Tm.
End Core.
Module Extra.
-Import Core.
+Import
+Core.
-#[global] Hint Opaque subst_Tm: rewrite.
+Arguments VarTm {n_Tm}.
-#[global] Hint Opaque ren_Tm: rewrite.
+Arguments Univ {n_Tm}.
-#[global] Hint Opaque subst_PTm: rewrite.
+Arguments Bot {n_Tm}.
-#[global] Hint Opaque ren_PTm: rewrite.
+Arguments TBind {n_Tm}.
+
+Arguments Proj {n_Tm}.
+
+Arguments Pair {n_Tm}.
+
+Arguments App {n_Tm}.
+
+Arguments Abs {n_Tm}.
+
+#[global]Hint Opaque subst_Tm: rewrite.
+
+#[global]Hint Opaque ren_Tm: rewrite.
End Extra.
diff --git a/theories/Autosubst2/unscoped.v b/theories/Autosubst2/unscoped.v
deleted file mode 100644
index 55f8172..0000000
--- a/theories/Autosubst2/unscoped.v
+++ /dev/null
@@ -1,213 +0,0 @@
-(** * Autosubst Header for Unnamed Syntax
-
-Version: December 11, 2019.
- *)
-
-(* Adrian:
- I changed this library a bit to work better with my generated code.
- 1. I use nat directly instead of defining fin to be nat and using Some/None as S/O
- 2. I removed the "s, sigma" notation for scons because it interacts with dependent function types "forall x, A"*)
-Require Import core.
-Require Import Setoid Morphisms Relation_Definitions.
-
-Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
- match p with eq_refl => eq_refl end.
-
-Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
- match q with eq_refl => match p with eq_refl => eq_refl end end.
-
-(** ** Primitives of the Sigma Calculus. *)
-
-Definition shift := S.
-
-Definition var_zero := 0.
-
-Definition id {X} := @Datatypes.id X.
-
-Definition scons {X: Type} (x : X) (xi : nat -> X) :=
- fun n => match n with
- | 0 => x
- | S n => xi n
- end.
-
-#[ export ]
-Hint Opaque scons : rewrite.
-
-(** ** Type Class Instances for Notation
-Required to make notation work. *)
-
-(** *** Type classes for renamings. *)
-
-Class Ren1 (X1 : Type) (Y Z : Type) :=
- ren1 : X1 -> Y -> Z.
-
-Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
- ren2 : X1 -> X2 -> Y -> Z.
-
-Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
- ren3 : X1 -> X2 -> X3 -> Y -> Z.
-
-Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
- ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
-
-Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
- ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
-
-Module RenNotations.
- Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s ⟨ xi1 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4 xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
-
- Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5 xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
-
- Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
-
- Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
-End RenNotations.
-
-(** *** Type Classes for Substiution *)
-
-Class Subst1 (X1 : Type) (Y Z: Type) :=
- subst1 : X1 -> Y -> Z.
-
-Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
- subst2 : X1 -> X2 -> Y -> Z.
-
-Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
- subst3 : X1 -> X2 -> X3 -> Y -> Z.
-
-Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
- subst4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
-
-Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
- subst5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
-
-Module SubstNotations.
- Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
-
- Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/' tau ]") : subst_scope.
-End SubstNotations.
-
-(** *** Type Class for Variables *)
-
-Class Var X Y :=
- ids : X -> Y.
-
-#[export] Instance idsRen : Var nat nat := id.
-
-(** ** Proofs for the substitution primitives. *)
-
-Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
-
-Module CombineNotations.
- Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
-
- Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
-
- #[ global ]
- Open Scope fscope.
- #[ global ]
- Open Scope subst_scope.
-End CombineNotations.
-
-Import CombineNotations.
-
-
-(** A generic lifting of a renaming. *)
-Definition up_ren (xi : nat -> nat) :=
- 0 .: (xi >> S).
-
-(** A generic proof that lifting of renamings composes. *)
-Lemma up_ren_ren (xi: nat -> nat) (zeta : nat -> nat) (rho: nat -> nat) (E: forall x, (xi >> zeta) x = rho x) :
- forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
-Proof.
- intros [|x].
- - reflexivity.
- - unfold up_ren. cbn. unfold funcomp. f_equal. apply E.
-Qed.
-
-(** Eta laws. *)
-Lemma scons_eta' {T} (f : nat -> T) :
- pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
-Proof. intros x. destruct x; reflexivity. Qed.
-
-Lemma scons_eta_id' :
- pointwise_relation _ eq (var_zero .: shift) id.
-Proof. intros x. destruct x; reflexivity. Qed.
-
-Lemma scons_comp' (T: Type) {U} (s: T) (sigma: nat -> T) (tau: T -> U) :
- pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
-Proof. intros x. destruct x; reflexivity. Qed.
-
-(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
-#[export] Instance scons_morphism {X: Type} :
- Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X).
-Proof.
- intros ? t -> sigma tau H.
- intros [|x].
- cbn. reflexivity.
- apply H.
-Qed.
-
-#[export] Instance scons_morphism2 {X: Type} :
- Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X).
-Proof.
- intros ? t -> sigma tau H ? x ->.
- destruct x as [|x].
- cbn. reflexivity.
- apply H.
-Qed.
-
-(** ** Generic lifting of an allfv predicate *)
-Definition up_allfv (p: nat -> Prop) : nat -> Prop := scons True p.
-
-(** ** Notations for unscoped syntax *)
-Module UnscopedNotations.
- Include RenNotations.
- Include SubstNotations.
- Include CombineNotations.
-
- (* Notation "s , sigma" := (scons s sigma) (at level 60, format "s , sigma", right associativity) : subst_scope. *)
-
- Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
-
- Notation "↑" := (shift) : subst_scope.
-
- #[global]
- Open Scope fscope.
- #[global]
- Open Scope subst_scope.
-End UnscopedNotations.
-
-(** ** Tactics for unscoped syntax *)
-
-(** Automatically does a case analysis on a natural number, useful for proofs with context renamings/context morphisms. *)
-Tactic Notation "auto_case" tactic(t) := (match goal with
- | [|- forall (i : nat), _] => intros []; t
- end).
-
-
-(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
-Ltac fsimpl :=
- repeat match goal with
- | [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
- | [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
- | [|- context [id ?s]] => change (id s) with s
- | [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h))
- | [|- context[(?v .: ?g) var_zero]] => change ((v .: g) var_zero) with v
- | [|- context[(?v .: ?g) 0]] => change ((v .: g) 0) with v
- | [|- context[(?v .: ?g) (S ?n)]] => change ((v .: g) (S n)) with (g n)
- | [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
- | [|- context[var_zero]] => change var_zero with 0
- | [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f)
- | [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); rewrite scons_eta'
- | [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
- | [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
- (* DONE had to put an underscore as the last argument to scons. This might be an argument against unfolding funcomp *)
- | [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
- | [|- context[scons var_zero shift]] => setoid_rewrite scons_eta_id'; eta_reduce
- end.
\ No newline at end of file
diff --git a/theories/compile.v b/theories/compile.v
deleted file mode 100644
index e481d71..0000000
--- a/theories/compile.v
+++ /dev/null
@@ -1,186 +0,0 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red.
-Require Import ssreflect ssrbool.
-From Hammer Require Import Tactics.
-From stdpp Require Import relations (rtc(..)).
-
-Module Compile.
- Fixpoint F {n} (a : Tm n) : Tm n :=
- match a with
- | TBind p A B => Pair (Pair (Const p) (F A)) (Abs (F B))
- | Const k => Const k
- | Univ i => Univ i
- | Abs a => Abs (F a)
- | App a b => App (F a) (F b)
- | VarTm i => VarTm i
- | Pair a b => Pair (F a) (F b)
- | Proj t a => Proj t (F a)
- | Bot => Bot
- | If a b c => App (App (F a) (F b)) (F c)
- | BVal b => if b then (Abs (Abs (VarTm (shift var_zero)))) else (Abs (Abs (VarTm var_zero)))
- | Bool => Bool
- end.
-
- Lemma renaming n m (a : Tm n) (ξ : fin n -> fin m) :
- F (ren_Tm ξ a)= ren_Tm ξ (F a).
- Proof. move : m ξ. elim : n / a => //=; hauto lq:on. Qed.
-
- #[local]Hint Rewrite Compile.renaming : compile.
-
- Lemma morphing n m (a : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
- (forall i, ρ0 i = F (ρ1 i)) ->
- subst_Tm ρ0 (F a) = F (subst_Tm ρ1 a).
- Proof.
- move : m ρ0 ρ1. elim : n / a => n//=.
- - hauto lq:on inv:option rew:db:compile unfold:funcomp.
- - hauto lq:on rew:off.
- - hauto lq:on rew:off.
- - hauto lq:on.
- - hauto lq:on inv:option rew:db:compile unfold:funcomp.
- - hauto lq:on rew:off.
- - hauto lq:on rew:off.
- Qed.
-
- Lemma substing n b (a : Tm (S n)) :
- subst_Tm (scons (F b) VarTm) (F a) = F (subst_Tm (scons b VarTm) a).
- Proof.
- apply morphing.
- case => //=.
- Qed.
-
-End Compile.
-
-#[export] Hint Rewrite Compile.renaming Compile.morphing : compile.
-
-
-Module Join.
- Definition R {n} (a b : Tm n) := join (Compile.F a) (Compile.F b).
-
- Lemma BindInj n p0 p1 (A0 A1 : Tm n) B0 B1 :
- R (TBind p0 A0 B0) (TBind p1 A1 B1) -> p0 = p1 /\ R A0 A1 /\ R B0 B1.
- Proof.
- rewrite /R /= !join_pair_inj.
- move => [[/join_const_inj h0 h1] h2].
- apply abs_eq in h2.
- evar (t : Tm (S n)).
- have : join (App (ren_Tm shift (Abs (Compile.F B1))) (VarTm var_zero)) t by
- apply Join.FromPar; apply Par.AppAbs; auto using Par.refl.
- subst t. rewrite -/ren_Tm.
- move : h2. move /join_transitive => /[apply]. asimpl => h2.
- tauto.
- Qed.
-
- Lemma UnivInj n i j : R (Univ i : Tm n) (Univ j) -> i = j.
- Proof. hauto l:on use:join_univ_inj. Qed.
-
- Lemma transitive n (a b c : Tm n) :
- R a b -> R b c -> R a c.
- Proof. hauto l:on use:join_transitive unfold:R. Qed.
-
- Lemma symmetric n (a b : Tm n) :
- R a b -> R b a.
- Proof. hauto l:on use:join_symmetric. Qed.
-
- Lemma reflexive n (a : Tm n) :
- R a a.
- Proof. hauto l:on use:join_refl. Qed.
-
- Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
- R a b -> R (subst_Tm ρ a) (subst_Tm ρ b).
- Proof.
- rewrite /R.
- rewrite /join.
- move => [C [h0 h1]].
- repeat (rewrite <- Compile.morphing with (ρ0 := funcomp Compile.F ρ); last by reflexivity).
- hauto lq:on use:join_substing.
- Qed.
-
-End Join.
-
-Module Equiv.
- Inductive R {n} : Tm n -> Tm n -> Prop :=
- (***************** Beta ***********************)
- | AppAbs a b :
- R (App (Abs a) b) (subst_Tm (scons b VarTm) a)
- | ProjPair p a b :
- R (Proj p (Pair a b)) (if p is PL then a else b)
-
- (****************** Eta ***********************)
- | AppEta a :
- R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
- | PairEta a :
- R a (Pair (Proj PL a) (Proj PR a))
-
- (*************** Congruence ********************)
- | Var i : R (VarTm i) (VarTm i)
- | AbsCong a b :
- R a b ->
- R (Abs a) (Abs b)
- | AppCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (App a0 b0) (App a1 b1)
- | PairCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (Pair a0 b0) (Pair a1 b1)
- | ProjCong p a0 a1 :
- R a0 a1 ->
- R (Proj p a0) (Proj p a1)
- | BindCong p A0 A1 B0 B1:
- R A0 A1 ->
- R B0 B1 ->
- R (TBind p A0 B0) (TBind p A1 B1)
- | UnivCong i :
- R (Univ i) (Univ i).
-End Equiv.
-
-Module EquivJoin.
- Lemma FromEquiv n (a b : Tm n) : Equiv.R a b -> Join.R a b.
- Proof.
- move => h. elim : n a b /h => n.
- - move => a b.
- rewrite /Join.R /join /=.
- eexists. split. apply relations.rtc_once.
- apply Par.AppAbs; auto using Par.refl.
- rewrite Compile.substing.
- apply relations.rtc_refl.
- - move => p a b.
- apply Join.FromPar.
- simpl. apply : Par.ProjPair'; auto using Par.refl.
- case : p => //=.
- - move => a. apply Join.FromPar => /=.
- apply : Par.AppEta'; auto using Par.refl.
- by autorewrite with compile.
- - move => a. apply Join.FromPar => /=.
- apply : Par.PairEta; auto using Par.refl.
- - hauto l:on use:Join.FromPar, Par.Var.
- - hauto lq:on use:Join.AbsCong.
- - qauto l:on use:Join.AppCong.
- - qauto l:on use:Join.PairCong.
- - qauto use:Join.ProjCong.
- - rewrite /Join.R => p A0 A1 B0 B1 _ hA _ hB /=.
- sfirstorder use:Join.PairCong,Join.AbsCong,Join.FromPar,Par.ConstCong.
- - hauto l:on.
- Qed.
-End EquivJoin.
-
-Lemma compile_rpar n (a b : Tm n) : RPar'.R a b -> RPar'.R (Compile.F a) (Compile.F b).
-Proof.
- move => h. elim : n a b /h.
- - move => n a0 a1 b0 b1 ha iha hb ihb /=.
- rewrite -Compile.substing.
- apply RPar'.AppAbs => //.
- - hauto q:on use:RPar'.ProjPair'.
- - qauto ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
- - hauto lq:on ctrs:RPar'.R.
-Qed.
-
-Lemma compile_rpars n (a b : Tm n) : rtc RPar'.R a b -> rtc RPar'.R (Compile.F a) (Compile.F b).
-Proof. induction 1; hauto lq:on ctrs:rtc use:compile_rpar. Qed.
diff --git a/theories/diagram.txt b/theories/diagram.txt
index ab47f3d..2cf16e8 100644
--- a/theories/diagram.txt
+++ b/theories/diagram.txt
@@ -18,25 +18,3 @@ a0 >> a1
| |
v v
b0 >> b1
-
-
-prov x (x, x)
-
-prov x b
-
-
-a => b
-
-prov x a
-
-prov y b
-
-prov x c
-prov y c
-
-extract c = x
-extract c = y
-
-prov x b
-
-pr
diff --git a/theories/fp_red.v b/theories/fp_red.v
index f9abc08..0c207be 100644
--- a/theories/fp_red.v
+++ b/theories/fp_red.v
@@ -7,7 +7,7 @@ Require Import Arith.Wf_nat.
Require Import Psatz.
From stdpp Require Import relations (rtc (..), rtc_once, rtc_r).
From Hammer Require Import Tactics.
-Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax.
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
Ltac2 spec_refl () :=
List.iter
@@ -22,85 +22,88 @@ Ltac spec_refl := ltac2:(spec_refl ()).
(* Trying my best to not write C style module_funcname *)
Module Par.
- Inductive R : PTm -> PTm -> Prop :=
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
+ R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
| AppPair a0 a1 b0 b1 c0 c1:
R a0 a1 ->
R b0 b1 ->
R c0 c1 ->
- R (PApp (PPair a0 b0) c0) (PPair (PApp a1 c1) (PApp b1 c1))
+ R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
| ProjAbs p a0 a1 :
R a0 a1 ->
- R (PProj p (PAbs a0)) (PAbs (PProj p a1))
+ R (Proj p (Abs a0)) (Abs (Proj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
+ R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
(****************** Eta ***********************)
| AppEta a0 a1 :
R a0 a1 ->
- R a0 (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero)))
+ R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
- R a0 (PPair (PProj PL a1) (PProj PR a1))
+ R a0 (Pair (Proj PL a1) (Proj PR a1))
(*************** Congruence ********************)
- | Var i : R (VarPTm i) (VarPTm i)
+ | Var i : R (VarTm i) (VarTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (PAbs a0) (PAbs a1)
+ R (Abs a0) (Abs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PApp a0 b0) (PApp a1 b1)
+ R (App a0 b0) (App a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PPair a0 b0) (PPair a1 b1)
+ R (Pair a0 b0) (Pair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (PProj p a0) (PProj p a1)
- | ConstCong k :
- R (PConst k) (PConst k)
- | Univ i :
- R (PUniv i) (PUniv i)
- | Bot :
- R PBot PBot.
+ R (Proj p a0) (Proj p a1)
+ | BindCong p A0 A1 B0 B1:
+ R A0 A1 ->
+ R B0 B1 ->
+ R (TBind p A0 B0) (TBind p A1 B1)
+ (* Bot is useful for making the prov function computable *)
+ | BotCong :
+ R Bot Bot
+ | UnivCong i :
+ R (Univ i) (Univ i).
- Lemma refl (a : PTm) : R a a.
- elim : a; hauto ctrs:R.
+ Lemma refl n (a : Tm n) : R a a.
+ elim : n /a; hauto ctrs:R.
Qed.
- Lemma AppAbs' a0 a1 (b0 b1 t : PTm) :
- t = subst_PTm (scons b1 VarPTm) a1 ->
+ Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
+ t = subst_Tm (scons b1 VarTm) a1 ->
R a0 a1 ->
R b0 b1 ->
- R (PApp (PAbs a0) b0) t.
+ R (App (Abs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' p (a0 a1 b0 b1 : PTm) t :
+ Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
- R (PProj p (PPair a0 b0)) t.
+ R (Proj p (Pair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma AppEta' (a0 a1 b : PTm) :
- b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) ->
+ Lemma AppEta' n (a0 a1 b : Tm n) :
+ b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
R a0 a1 ->
R a0 b.
Proof. move => ->; apply AppEta. Qed.
- Lemma renaming (a b : PTm) (ξ : nat -> nat) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
- move => h. move : ξ.
- elim : a b /h.
+ move => h. move : m ξ.
+ elim : n a b /h.
move => *; apply : AppAbs'; eauto; by asimpl.
all : match goal with
| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
@@ -109,41 +112,41 @@ Module Par.
Qed.
- Lemma morphing (a b : PTm) (ρ0 ρ1 : nat -> PTm) :
+ Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
+ R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
Proof.
- move => + h. move : ρ0 ρ1. elim : a b/h.
- - move => a0 a1 b0 b1 ha iha hb ihb ρ0 ρ1 hρ /=.
- eapply AppAbs' with (a1 := subst_PTm (up_PTm_PTm ρ1) a1); eauto.
+ move => + h. move : m ρ0 ρ1. elim : n a b/h.
+ - move => n a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
+ eapply AppAbs' with (a1 := subst_Tm (up_Tm_Tm ρ1) a1); eauto.
by asimpl.
- hauto l:on use:renaming inv:nat.
+ hauto l:on use:renaming inv:option.
- hauto lq:on rew:off ctrs:R.
- - hauto l:on inv:nat use:renaming ctrs:R.
+ - hauto l:on inv:option use:renaming ctrs:R.
- hauto lq:on use:ProjPair'.
- - move => a0 a1 ha iha ρ0 ρ1 hρ /=.
+ - move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
- sfirstorder.
- - hauto l:on inv:nat ctrs:R use:renaming.
+ - hauto l:on inv:option ctrs:R use:renaming.
- hauto q:on ctrs:R.
- qauto l:on ctrs:R.
- qauto l:on ctrs:R.
- hauto l:on inv:option ctrs:R use:renaming.
- - qauto l:on ctrs:R.
- - qauto l:on ctrs:R.
+ - sfirstorder.
+ - sfirstorder.
Qed.
- Lemma substing (a b : PTm) (ρ : nat -> PTm) :
- R a b -> R (subst_PTm ρ a) (subst_PTm ρ b).
+ Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
+ R a b -> R (subst_Tm ρ a) (subst_Tm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma antirenaming (a : PTm) (b : PTm) (ξ : nat -> nat) :
- R (ren_PTm ξ a) b -> exists b0, R a b0 /\ ren_PTm ξ b0 = b.
+ Lemma antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
+ R (ren_Tm ξ a) b -> exists b0, R a b0 /\ ren_Tm ξ b0 = b.
Proof.
- move E : (ren_PTm ξ a) => u h.
- move : ξ a E. elim : u b/h.
- - move => a0 a1 b0 b1 ha iha hb ihb ξ []//=.
+ move E : (ren_Tm ξ a) => u h.
+ move : n ξ a E. elim : m u b/h.
+ - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//=.
move => c c0 [+ ?]. subst.
case : c => //=.
move => c [?]. subst.
@@ -153,7 +156,7 @@ Module Par.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- - move => a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc ξ []//=.
+ - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//=.
move => []//= t t0 t1 [*]. subst.
spec_refl.
move : iha => [? [*]].
@@ -162,71 +165,76 @@ Module Par.
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
- - move => p a0 a1 ha iha ξ []//= p0 []//= t [*]. subst.
+ - move => n p a0 a1 ha iha m ξ []//= p0 []//= t [*]. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
- - move => p a0 a1 b0 b1 ha iha hb ihb ξ []//= p0 []//= t t0[*].
+ - move => n p a0 a1 b0 b1 ha iha hb ihb m ξ []//= p0 []//= t t0[*].
subst. spec_refl.
move : iha => [b0 [? ?]].
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
- - move => a0 a1 ha iha ξ a ?. subst.
+ - move => n a0 a1 ha iha m ξ a ?. subst.
spec_refl. move : iha => [a0 [? ?]]. subst.
eexists. split. apply AppEta; eauto.
by asimpl.
- - move => a0 a1 ha iha ξ a ?. subst.
+ - move => n a0 a1 ha iha m ξ a ?. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply PairEta; eauto.
by asimpl.
- - move => i ξ []//=.
+ - move => n i m ξ []//=.
hauto l:on.
- - move => a0 a1 ha iha ξ []//= t [*]. subst.
+ - move => n a0 a1 ha iha m ξ []//= t [*]. subst.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
- done.
- - move => a0 a1 b0 b1 ha iha hb ihb ξ []//= t t0 [*]. subst.
+ by asimpl.
+ - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply AppCong; eauto.
done.
- - move => a0 a1 b0 b1 ha iha hb ihb ξ []//= t t0[*]. subst.
+ - move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0[*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
- eexists. split=>/=. by apply PairCong; eauto.
- done.
- - move => p a0 a1 ha iha ξ []//= p0 t [*]. subst.
+ eexists. split. by apply PairCong; eauto.
+ by asimpl.
+ - move => n p a0 a1 ha iha m ξ []//= p0 t [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. by apply ProjCong; eauto.
- done.
- - hauto q:on inv:PTm ctrs:R.
- - hauto q:on inv:PTm ctrs:R.
- - hauto q:on inv:PTm ctrs:R.
+ by asimpl.
+ - move => n p A0 A1 B0 B1 ha iha hB ihB m ξ []//= ? t t0 [*]. subst.
+ spec_refl.
+ move : iha => [b0 [? ?]].
+ move : ihB => [c0 [? ?]]. subst.
+ eexists. split. by apply BindCong; eauto.
+ by asimpl.
+ - move => n n0 ξ []//=. hauto l:on.
+ - move => n i n0 ξ []//=. hauto l:on.
Qed.
End Par.
Module Pars.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- rtc Par.R a b -> rtc Par.R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ rtc Par.R a b -> rtc Par.R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
induction 1; hauto lq:on ctrs:rtc use:Par.renaming.
Qed.
- Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+ Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
rtc Par.R a b ->
- rtc Par.R (subst_PTm ρ a) (subst_PTm ρ b).
+ rtc Par.R (subst_Tm ρ a) (subst_Tm ρ b).
induction 1; hauto l:on ctrs:rtc use:Par.substing.
Qed.
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ξ : fin n -> fin m) :
- rtc Par.R (ren_PTm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_PTm ξ b0 = b.
+ Lemma antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
+ rtc Par.R (ren_Tm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_Tm ξ b0 = b.
Proof.
- move E :(ren_PTm ξ a) => u h.
+ move E :(ren_Tm ξ a) => u h.
move : a E.
elim : u b /h.
- sfirstorder.
@@ -235,114 +243,78 @@ Module Pars.
move => [b0 [h2 ?]]. subst.
hauto lq:on rew:off ctrs:rtc.
Qed.
-
- #[local]Ltac solve_s_rec :=
- move => *; eapply rtc_l; eauto;
- hauto lq:on ctrs:Par.R use:Par.refl.
-
- #[local]Ltac solve_s :=
- repeat (induction 1; last by solve_s_rec); apply rtc_refl.
-
- Lemma ProjCong n p (a0 a1 : PTm n) :
- rtc Par.R a0 a1 ->
- rtc Par.R (PProj p a0) (PProj p a1).
- Proof. solve_s. Qed.
-
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
- rtc Par.R a0 a1 ->
- rtc Par.R b0 b1 ->
- rtc Par.R (PPair a0 b0) (PPair a1 b1).
- Proof. solve_s. Qed.
-
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
- rtc Par.R a0 a1 ->
- rtc Par.R b0 b1 ->
- rtc Par.R (PApp a0 b0) (PApp a1 b1).
- Proof. solve_s. Qed.
-
- Lemma AbsCong n (a b : PTm (S n)) :
- rtc Par.R a b ->
- rtc Par.R (PAbs a) (PAbs b).
- Proof. solve_s. Qed.
-
End Pars.
-Definition var_or_const {n} (a : PTm n) :=
- match a with
- | VarPTm _ => true
- | PBot => true
- | _ => false
- end.
-
-
(***************** Beta rules only ***********************)
Module RPar.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
(***************** Beta ***********************)
| AppAbs a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
+ R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
| AppPair a0 a1 b0 b1 c0 c1:
R a0 a1 ->
R b0 b1 ->
R c0 c1 ->
- R (PApp (PPair a0 b0) c0) (PPair (PApp a1 c1) (PApp b1 c1))
+ R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
| ProjAbs p a0 a1 :
R a0 a1 ->
- R (PProj p (PAbs a0)) (PAbs (PProj p a1))
+ R (Proj p (Abs a0)) (Abs (Proj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
+ R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
(*************** Congruence ********************)
- | Var i : R (VarPTm i) (VarPTm i)
+ | Var i : R (VarTm i) (VarTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (PAbs a0) (PAbs a1)
+ R (Abs a0) (Abs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PApp a0 b0) (PApp a1 b1)
+ R (App a0 b0) (App a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PPair a0 b0) (PPair a1 b1)
+ R (Pair a0 b0) (Pair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (PProj p a0) (PProj p a1)
- | ConstCong k :
- R (PConst k) (PConst k)
- | Univ i :
- R (PUniv i) (PUniv i)
- | Bot :
- R PBot PBot.
+ R (Proj p a0) (Proj p a1)
+ | BindCong p A0 A1 B0 B1:
+ R A0 A1 ->
+ R B0 B1 ->
+ R (TBind p A0 B0) (TBind p A1 B1)
+ | BotCong :
+ R Bot Bot
+ | UnivCong i :
+ R (Univ i) (Univ i).
- Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
- Lemma refl n (a : PTm n) : R a a.
+ Lemma refl n (a : Tm n) : R a a.
Proof.
induction a; hauto lq:on ctrs:R.
Qed.
- Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
- t = subst_PTm (scons b1 VarPTm) a1 ->
+ Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
+ t = subst_Tm (scons b1 VarTm) a1 ->
R a0 a1 ->
R b0 b1 ->
- R (PApp (PAbs a0) b0) t.
+ R (App (Abs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
- Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
+ Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
- R (PProj p (PPair a0 b0)) t.
+ R (Proj p (Pair a0 b0)) t.
Proof. move => > ->. apply ProjPair. Qed.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -350,25 +322,25 @@ Module RPar.
all : qauto ctrs:R use:ProjPair'.
Qed.
- Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
+ Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
+ (forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)).
Proof. eauto using renaming. Qed.
- Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b :
+ Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b :
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
- Proof. hauto q:on inv:nat. Qed.
+ Proof. hauto q:on inv:option. Qed.
- Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
+ Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
- Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
+ (forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)).
+ Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed.
- Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
+ Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
(forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
+ R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
Proof.
move => + h. move : m ρ0 ρ1.
elim : n a b /h.
@@ -388,388 +360,62 @@ Module RPar.
- hauto lq:on ctrs:R.
Qed.
- Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+ Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
R a b ->
- R (subst_PTm ρ a) (subst_PTm ρ b).
+ R (subst_Tm ρ a) (subst_Tm ρ b).
Proof. hauto l:on use:morphing, refl. Qed.
- Lemma cong n (a b : PTm (S n)) c d :
+ Lemma cong n (a b : Tm (S n)) c d :
R a b ->
R c d ->
- R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
+ R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b).
Proof.
move => h0 h1. apply morphing => //=.
- qauto l:on ctrs:R inv:nat.
- Qed.
-
- Lemma var_or_const_imp {n} (a b : PTm n) :
- var_or_const a ->
- a = b -> ~~ var_or_const b -> False.
- Proof.
- hauto lq:on inv:PTm.
- Qed.
-
- Lemma var_or_const_up n m (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- (forall i, var_or_const (up_PTm_PTm ρ i)).
- Proof.
- move => h /= [i|].
- - asimpl.
- move /(_ i) in h.
- rewrite /funcomp.
- move : (ρ i) h.
- case => //=.
- - sfirstorder.
- Qed.
-
- Local Ltac antiimp := qauto l:on use:var_or_const_imp.
-
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b.
- Proof.
- move E : (subst_PTm ρ a) => u hρ h.
- move : n ρ hρ a E. elim : m u b/h.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => c c0 [+ ?]. subst.
- case : c => //=; first by antiimp.
- move => c [?]. subst.
- spec_refl.
- have /var_or_const_up hρ' := hρ.
- move : iha hρ' => /[apply] iha.
- move : ihb hρ => /[apply] ihb.
- spec_refl.
- move : iha => [c1][ih0]?. subst.
- move : ihb => [c2][ih1]?. subst.
- eexists. split.
- apply AppAbs; eauto.
- by asimpl.
- - move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ.
- move => []//=;
- first by antiimp.
- move => []//=; first by antiimp.
- move => t t0 t1 [*]. subst.
- have {}/iha := hρ => iha.
- have {}/ihb := hρ => ihb.
- have {}/ihc := hρ => ihc.
- spec_refl.
- move : iha => [? [*]].
- move : ihb => [? [*]].
- move : ihc => [? [*]].
- eexists. split.
- apply AppPair; hauto. subst.
- by asimpl.
- - move => n p a0 a1 ha iha m ρ hρ []//=;
- first by antiimp.
- move => p0 []//= t [*]; first by antiimp. subst.
- have /var_or_const_up {}/iha := hρ => iha.
- spec_refl. move : iha => [b0 [? ?]]. subst.
- eexists. split. apply ProjAbs; eauto. by asimpl.
- - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => p0 []//=; first by antiimp. move => t t0[*].
- subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by eauto using ProjPair.
- hauto q:on.
- - move => n i m ρ hρ []//=.
- hauto l:on.
- - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
- move => t [*]. subst.
- have /var_or_const_up {}/iha := hρ => iha.
- spec_refl.
- move :iha => [b0 [? ?]]. subst.
- eexists. split. by apply AbsCong; eauto.
- by asimpl.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply AppCong; eauto.
- done.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0[*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply PairCong; eauto.
- by asimpl.
- - move => n p a0 a1 ha iha m ρ hρ []//=;
- first by antiimp.
- move => p0 t [*]. subst.
- have {}/iha := (hρ) => iha.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- eexists. split. apply ProjCong; eauto. reflexivity.
- - hauto q:on ctrs:R inv:PTm.
- - hauto q:on ctrs:R inv:PTm.
- - hauto q:on ctrs:R inv:PTm.
+ qauto l:on ctrs:R inv:option.
Qed.
End RPar.
-(***************** Beta rules only ***********************)
-Module RPar'.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
- (***************** Beta ***********************)
- | AppAbs a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (PApp (PAbs a0) b0) (subst_PTm (scons b1 VarPTm) a1)
- | ProjPair p a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (PProj p (PPair a0 b0)) (if p is PL then a1 else b1)
-
-
- (*************** Congruence ********************)
- | Var i : R (VarPTm i) (VarPTm i)
- | AbsCong a0 a1 :
- R a0 a1 ->
- R (PAbs a0) (PAbs a1)
- | AppCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (PApp a0 b0) (PApp a1 b1)
- | PairCong a0 a1 b0 b1 :
- R a0 a1 ->
- R b0 b1 ->
- R (PPair a0 b0) (PPair a1 b1)
- | ProjCong p a0 a1 :
- R a0 a1 ->
- R (PProj p a0) (PProj p a1)
- | ConstCong k :
- R (PConst k) (PConst k)
- | UnivCong i :
- R (PUniv i) (PUniv i)
- | BotCong :
- R PBot PBot.
-
- Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
-
- Lemma refl n (a : PTm n) : R a a.
- Proof.
- induction a; hauto lq:on ctrs:R.
- Qed.
-
- Lemma AppAbs' n a0 a1 (b0 b1 t : PTm n) :
- t = subst_PTm (scons b1 VarPTm) a1 ->
- R a0 a1 ->
- R b0 b1 ->
- R (PApp (PAbs a0) b0) t.
- Proof. move => ->. apply AppAbs. Qed.
-
- Lemma ProjPair' n p (a0 a1 b0 b1 : PTm n) t :
- t = (if p is PL then a1 else b1) ->
- R a0 a1 ->
- R b0 b1 ->
- R (PProj p (PPair a0 b0)) t.
- Proof. move => > ->. apply ProjPair. Qed.
-
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
- Proof.
- move => h. move : m ξ.
- elim : n a b /h.
- move => *; apply : AppAbs'; eauto; by asimpl.
- all : qauto ctrs:R use:ProjPair'.
- Qed.
-
- Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> PTm m) (ξ : fin m -> fin p) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((funcomp (ren_PTm ξ) ρ0) i) ((funcomp (ren_PTm ξ) ρ1) i)).
- Proof. eauto using renaming. Qed.
-
- Lemma morphing_ext n m (ρ0 ρ1 : fin n -> PTm m) a b :
- R a b ->
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
- Proof. hauto q:on inv:nat. Qed.
-
- Lemma morphing_up n m (ρ0 ρ1 : fin n -> PTm m) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- (forall i, R (up_PTm_PTm ρ0 i) (up_PTm_PTm ρ1 i)).
- Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_PTm_PTm. Qed.
-
- Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
- (forall i, R (ρ0 i) (ρ1 i)) ->
- R a b -> R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
- Proof.
- move => + h. move : m ρ0 ρ1.
- elim : n a b /h.
- - move => *.
- apply : AppAbs'; eauto using morphing_up.
- by asimpl.
- - hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
- - hauto l:on ctrs:R use:morphing_up.
- - hauto lq:on ctrs:R.
- - hauto lq:on ctrs:R.
- Qed.
-
- Lemma substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
- R a b ->
- R (subst_PTm ρ a) (subst_PTm ρ b).
- Proof. hauto l:on use:morphing, refl. Qed.
-
- Lemma cong n (a b : PTm (S n)) c d :
- R a b ->
- R c d ->
- R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons d VarPTm) b).
- Proof.
- move => h0 h1. apply morphing => //=.
- qauto l:on ctrs:R inv:nat.
- Qed.
-
- Lemma var_or_const_imp {n} (a b : PTm n) :
- var_or_const a ->
- a = b -> ~~ var_or_const b -> False.
- Proof.
- hauto lq:on inv:PTm.
- Qed.
-
- Lemma var_or_const_up n m (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- (forall i, var_or_const (up_PTm_PTm ρ i)).
- Proof.
- move => h /= [i|].
- - asimpl.
- move /(_ i) in h.
- rewrite /funcomp.
- move : (ρ i) h.
- case => //=.
- - sfirstorder.
- Qed.
-
- Local Ltac antiimp := qauto l:on use:var_or_const_imp.
-
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- R (subst_PTm ρ a) b -> exists b0, R a b0 /\ subst_PTm ρ b0 = b.
- Proof.
- move E : (subst_PTm ρ a) => u hρ h.
- move : n ρ hρ a E. elim : m u b/h.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => c c0 [+ ?]. subst.
- case : c => //=; first by antiimp.
- move => c [?]. subst.
- spec_refl.
- have /var_or_const_up hρ' := hρ.
- move : iha hρ' => /[apply] iha.
- move : ihb hρ => /[apply] ihb.
- spec_refl.
- move : iha => [c1][ih0]?. subst.
- move : ihb => [c2][ih1]?. subst.
- eexists. split.
- apply AppAbs; eauto.
- by asimpl.
- - move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => p0 []//=; first by antiimp. move => t t0[*].
- subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]].
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by eauto using ProjPair.
- hauto q:on.
- - move => n i m ρ hρ []//=.
- hauto l:on.
- - move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
- move => t [*]. subst.
- have /var_or_const_up {}/iha := hρ => iha.
- spec_refl.
- move :iha => [b0 [? ?]]. subst.
- eexists. split. by apply AbsCong; eauto.
- by asimpl.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0 [*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply AppCong; eauto.
- done.
- - move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
- first by antiimp.
- move => t t0[*]. subst.
- have {}/iha := (hρ) => iha.
- have {}/ihb := (hρ) => ihb.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- move : ihb => [c0 [? ?]]. subst.
- eexists. split. by apply PairCong; eauto.
- by asimpl.
- - move => n p a0 a1 ha iha m ρ hρ []//=;
- first by antiimp.
- move => p0 t [*]. subst.
- have {}/iha := (hρ) => iha.
- spec_refl.
- move : iha => [b0 [? ?]]. subst.
- eexists. split. apply ProjCong; eauto. reflexivity.
- - hauto q:on ctrs:R inv:PTm.
- - move => n i n0 ρ hρ []//=; first by antiimp.
- hauto l:on.
- - hauto q:on inv:PTm ctrs:R.
- Qed.
-End RPar'.
-
Module ERed.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
(****************** Eta ***********************)
| AppEta a :
- R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero)))
+ R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
| PairEta a :
- R a (PPair (PProj PL a) (PProj PR a))
+ R a (Pair (Proj PL a) (Proj PR a))
(*************** Congruence ********************)
| AbsCong a0 a1 :
R a0 a1 ->
- R (PAbs a0) (PAbs a1)
+ R (Abs a0) (Abs a1)
| AppCong0 a0 a1 b :
R a0 a1 ->
- R (PApp a0 b) (PApp a1 b)
+ R (App a0 b) (App a1 b)
| AppCong1 a b0 b1 :
R b0 b1 ->
- R (PApp a b0) (PApp a b1)
+ R (App a b0) (App a b1)
| PairCong0 a0 a1 b :
R a0 a1 ->
- R (PPair a0 b) (PPair a1 b)
+ R (Pair a0 b) (Pair a1 b)
| PairCong1 a b0 b1 :
R b0 b1 ->
- R (PPair a b0) (PPair a b1)
+ R (Pair a b0) (Pair a b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (PProj p a0) (PProj p a1).
+ R (Proj p a0) (Proj p a1)
+ | BindCong0 p A0 A1 B:
+ R A0 A1 ->
+ R (TBind p A0 B) (TBind p A1 B)
+ | BindCong1 p A B0 B1:
+ R B0 B1 ->
+ R (TBind p A B0) (TBind p A B1).
- Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
-
- Lemma AppEta' n a (u : PTm n) :
- u = (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))) ->
+ Lemma AppEta' n a (u : Tm n) :
+ u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) ->
R a u.
Proof. move => ->. apply AppEta. Qed.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -779,14 +425,14 @@ Module ERed.
all : qauto ctrs:R.
Qed.
- Lemma substing n m (a : PTm n) b (ρ : fin n -> PTm m) :
+ Lemma substing n m (a : Tm n) b (ρ : fin n -> Tm m) :
R a b ->
- R (subst_PTm ρ a) (subst_PTm ρ b).
+ R (subst_Tm ρ a) (subst_Tm ρ b).
Proof.
move => h. move : m ρ. elim : n a b / h => n.
move => a m ρ /=.
apply : AppEta'; eauto. by asimpl.
- all : hauto ctrs:R inv:nat use:renaming.
+ all : hauto ctrs:R inv:option use:renaming.
Qed.
End ERed.
@@ -800,69 +446,77 @@ Module EReds.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AbsCong n (a b : PTm (S n)) :
+ Lemma AbsCong n (a b : Tm (S n)) :
rtc ERed.R a b ->
- rtc ERed.R (PAbs a) (PAbs b).
+ rtc ERed.R (Abs a) (Abs b).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
- rtc ERed.R (PApp a0 b0) (PApp a1 b1).
+ rtc ERed.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
- rtc ERed.R (PPair a0 b0) (PPair a1 b1).
+ rtc ERed.R (TBind p a0 b0) (TBind p a1 b1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : PTm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc ERed.R a0 a1 ->
- rtc ERed.R (PProj p a0) (PProj p a1).
+ rtc ERed.R b0 b1 ->
+ rtc ERed.R (Pair a0 b0) (Pair a1 b1).
+ Proof. solve_s. Qed.
+
+ Lemma ProjCong n p (a0 a1 : Tm n) :
+ rtc ERed.R a0 a1 ->
+ rtc ERed.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
End EReds.
Module EPar.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
(****************** Eta ***********************)
| AppEta a0 a1 :
R a0 a1 ->
- R a0 (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero)))
+ R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
- R a0 (PPair (PProj PL a1) (PProj PR a1))
+ R a0 (Pair (Proj PL a1) (Proj PR a1))
(*************** Congruence ********************)
- | Var i : R (VarPTm i) (VarPTm i)
+ | Var i : R (VarTm i) (VarTm i)
| AbsCong a0 a1 :
R a0 a1 ->
- R (PAbs a0) (PAbs a1)
+ R (Abs a0) (Abs a1)
| AppCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PApp a0 b0) (PApp a1 b1)
+ R (App a0 b0) (App a1 b1)
| PairCong a0 a1 b0 b1 :
R a0 a1 ->
R b0 b1 ->
- R (PPair a0 b0) (PPair a1 b1)
+ R (Pair a0 b0) (Pair a1 b1)
| ProjCong p a0 a1 :
R a0 a1 ->
- R (PProj p a0) (PProj p a1)
- | ConstCong k :
- R (PConst k) (PConst k)
- | UnivCong i :
- R (PUniv i) (PUniv i)
+ R (Proj p a0) (Proj p a1)
+ | BindCong p A0 A1 B0 B1:
+ R A0 A1 ->
+ R B0 B1 ->
+ R (TBind p A0 B0) (TBind p A1 B1)
| BotCong :
- R PBot PBot.
+ R Bot Bot
+ | UnivCong i :
+ R (Univ i) (Univ i).
- Lemma refl n (a : PTm n) : EPar.R a a.
+ Lemma refl n (a : Tm n) : EPar.R a a.
Proof.
induction a; hauto lq:on ctrs:EPar.R.
Qed.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
move => h. move : m ξ.
elim : n a b /h.
@@ -874,54 +528,54 @@ Module EPar.
all : qauto ctrs:R.
Qed.
- Derive Dependent Inversion inv with (forall n (a b : PTm n), R a b) Sort Prop.
+ Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
- Lemma AppEta' n (a0 a1 b : PTm n) :
- b = (PAbs (PApp (ren_PTm shift a1) (VarPTm var_zero))) ->
+ Lemma AppEta' n (a0 a1 b : Tm n) :
+ b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
R a0 a1 ->
R a0 b.
Proof. move => ->; apply AppEta. Qed.
- Lemma morphing n m (a b : PTm n) (ρ0 ρ1 : fin n -> PTm m) :
+ Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
R a b ->
(forall i, R (ρ0 i) (ρ1 i)) ->
- R (subst_PTm ρ0 a) (subst_PTm ρ1 b).
+ R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
Proof.
move => h. move : m ρ0 ρ1. elim : n a b / h => n.
- move => a0 a1 ha iha m ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- - hauto l:on ctrs:R use:renaming inv:nat.
+ - hauto l:on ctrs:R use:renaming inv:option.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- - hauto l:on ctrs:R use:renaming inv:nat.
+ - hauto l:on ctrs:R use:renaming inv:option.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
Qed.
- Lemma substing n a0 a1 (b0 b1 : PTm n) :
+ Lemma substing n a0 a1 (b0 b1 : Tm n) :
R a0 a1 ->
R b0 b1 ->
- R (subst_PTm (scons b0 VarPTm) a0) (subst_PTm (scons b1 VarPTm) a1).
+ R (subst_Tm (scons b0 VarTm) a0) (subst_Tm (scons b1 VarTm) a1).
Proof.
move => h0 h1. apply morphing => //.
- hauto lq:on ctrs:R inv:nat.
+ hauto lq:on ctrs:R inv:option.
Qed.
End EPar.
Module OExp.
- Inductive R {n} : PTm n -> PTm n -> Prop :=
+ Inductive R {n} : Tm n -> Tm n -> Prop :=
(****************** Eta ***********************)
| AppEta a :
- R a (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero)))
+ R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
| PairEta a :
- R a (PPair (PProj PL a) (PProj PR a)).
+ R a (Pair (Proj PL a) (Proj PR a)).
- Lemma merge n (t a b : PTm n) :
+ Lemma merge n (t a b : Tm n) :
rtc R a b ->
EPar.R t a ->
EPar.R t b.
@@ -931,7 +585,7 @@ Module OExp.
- hauto q:on ctrs:EPar.R inv:R.
Qed.
- Lemma commutativity n (a b c : PTm n) :
+ Lemma commutativity n (a b c : Tm n) :
EPar.R a b -> R a c -> exists d, R b d /\ EPar.R c d.
Proof.
move => h.
@@ -940,7 +594,7 @@ Module OExp.
- hauto lq:on ctrs:EPar.R, R.
Qed.
- Lemma commutativity0 n (a b c : PTm n) :
+ Lemma commutativity0 n (a b c : Tm n) :
EPar.R a b -> rtc R a c -> exists d, rtc R b d /\ EPar.R c d.
Proof.
move => + h. move : b.
@@ -965,168 +619,78 @@ Module RPars.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AbsCong n (a b : PTm (S n)) :
+ Lemma AbsCong n (a b : Tm (S n)) :
rtc RPar.R a b ->
- rtc RPar.R (PAbs a) (PAbs b).
+ rtc RPar.R (Abs a) (Abs b).
Proof. solve_s. Qed.
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
- rtc RPar.R (PApp a0 b0) (PApp a1 b1).
+ rtc RPar.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
- rtc RPar.R (PPair a0 b0) (PPair a1 b1).
+ rtc RPar.R (TBind p a0 b0) (TBind p a1 b1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : PTm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc RPar.R a0 a1 ->
- rtc RPar.R (PProj p a0) (PProj p a1).
+ rtc RPar.R b0 b1 ->
+ rtc RPar.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
- Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
+ Lemma ProjCong n p (a0 a1 : Tm n) :
rtc RPar.R a0 a1 ->
- rtc RPar.R (ren_PTm ξ a0) (ren_PTm ξ a1).
+ rtc RPar.R (Proj p a0) (Proj p a1).
+ Proof. solve_s. Qed.
+
+ Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
+ rtc RPar.R a0 a1 ->
+ rtc RPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
- eauto using RPar.renaming, rtc_l.
Qed.
- Lemma weakening n (a0 a1 : PTm n) :
+ Lemma weakening n (a0 a1 : Tm n) :
rtc RPar.R a0 a1 ->
- rtc RPar.R (ren_PTm shift a0) (ren_PTm shift a1).
+ rtc RPar.R (ren_Tm shift a0) (ren_Tm shift a1).
Proof. apply renaming. Qed.
- Lemma Abs_inv n (a : PTm (S n)) b :
- rtc RPar.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar.R a a'.
+ Lemma Abs_inv n (a : Tm (S n)) b :
+ rtc RPar.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar.R a a'.
Proof.
- move E : (PAbs a) => b0 h. move : a E.
+ move E : (Abs a) => b0 h. move : a E.
elim : b0 b / h.
- hauto lq:on ctrs:rtc.
- hauto lq:on ctrs:rtc inv:RPar.R, rtc.
Qed.
- Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+ Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) :
rtc RPar.R a b ->
- rtc RPar.R (subst_PTm ρ a) (subst_PTm ρ b).
+ rtc RPar.R (subst_Tm ρ a) (subst_Tm ρ b).
Proof. induction 1; qauto l:on ctrs:rtc use:RPar.substing. Qed.
- Lemma substing n (a b : PTm (S n)) c :
+ Lemma substing n (a b : Tm (S n)) c :
rtc RPar.R a b ->
- rtc RPar.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
- Proof. hauto lq:on use:morphing inv:nat. Qed.
-
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- rtc RPar.R (subst_PTm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_PTm ρ b0 = b.
- Proof.
- move E :(subst_PTm ρ a) => u hρ h.
- move : a E.
- elim : u b /h.
- - sfirstorder.
- - move => a b c h0 h1 ih1 a0 ?. subst.
- move /RPar.antirenaming : h0.
- move /(_ hρ).
- move => [b0 [h2 ?]]. subst.
- hauto lq:on rew:off ctrs:rtc.
- Qed.
+ rtc RPar.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
+ Proof. hauto lq:on use:morphing inv:option. Qed.
End RPars.
-Module RPars'.
-
- #[local]Ltac solve_s_rec :=
- move => *; eapply rtc_l; eauto;
- hauto lq:on ctrs:RPar'.R use:RPar'.refl.
-
- #[local]Ltac solve_s :=
- repeat (induction 1; last by solve_s_rec); apply rtc_refl.
-
- Lemma AbsCong n (a b : PTm (S n)) :
- rtc RPar'.R a b ->
- rtc RPar'.R (PAbs a) (PAbs b).
- Proof. solve_s. Qed.
-
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (PApp a0 b0) (PApp a1 b1).
- Proof. solve_s. Qed.
-
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R b0 b1 ->
- rtc RPar'.R (PPair a0 b0) (PPair a1 b1).
- Proof. solve_s. Qed.
-
- Lemma ProjCong n p (a0 a1 : PTm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (PProj p a0) (PProj p a1).
- Proof. solve_s. Qed.
-
- Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_PTm ξ a0) (ren_PTm ξ a1).
- Proof.
- induction 1.
- - apply rtc_refl.
- - eauto using RPar'.renaming, rtc_l.
- Qed.
-
- Lemma weakening n (a0 a1 : PTm n) :
- rtc RPar'.R a0 a1 ->
- rtc RPar'.R (ren_PTm shift a0) (ren_PTm shift a1).
- Proof. apply renaming. Qed.
-
- Lemma Abs_inv n (a : PTm (S n)) b :
- rtc RPar'.R (PAbs a) b -> exists a', b = PAbs a' /\ rtc RPar'.R a a'.
- Proof.
- move E : (PAbs a) => b0 h. move : a E.
- elim : b0 b / h.
- - hauto lq:on ctrs:rtc.
- - hauto lq:on ctrs:rtc inv:RPar'.R, rtc.
- Qed.
-
- Lemma morphing n m (a b : PTm n) (ρ : fin n -> PTm m) :
- rtc RPar'.R a b ->
- rtc RPar'.R (subst_PTm ρ a) (subst_PTm ρ b).
- Proof. induction 1; qauto l:on ctrs:rtc use:RPar'.substing. Qed.
-
- Lemma substing n (a b : PTm (S n)) c :
- rtc RPar'.R a b ->
- rtc RPar'.R (subst_PTm (scons c VarPTm) a) (subst_PTm (scons c VarPTm) b).
- Proof. hauto lq:on use:morphing inv:nat. Qed.
-
- Lemma antirenaming n m (a : PTm n) (b : PTm m) (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- rtc RPar'.R (subst_PTm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_PTm ρ b0 = b.
- Proof.
- move E :(subst_PTm ρ a) => u hρ h.
- move : a E.
- elim : u b /h.
- - sfirstorder.
- - move => a b c h0 h1 ih1 a0 ?. subst.
- move /RPar'.antirenaming : h0.
- move /(_ hρ).
- move => [b0 [h2 ?]]. subst.
- hauto lq:on rew:off ctrs:rtc.
- Qed.
-
-End RPars'.
-
-Lemma Abs_EPar n a (b : PTm n) :
- EPar.R (PAbs a) b ->
+Lemma Abs_EPar n a (b : Tm n) :
+ EPar.R (Abs a) b ->
(exists d, EPar.R a d /\
- rtc RPar.R (PApp (ren_PTm shift b) (VarPTm var_zero)) d) /\
+ rtc RPar.R (App (ren_Tm shift b) (VarTm var_zero)) d) /\
(exists d,
EPar.R a d /\ forall p,
- rtc RPar.R (PProj p b) (PAbs (PProj p d))).
+ rtc RPar.R (Proj p b) (Abs (Proj p d))).
Proof.
- move E : (PAbs a) => u h.
+ move E : (Abs a) => u h.
move : a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha b ?. subst.
@@ -1147,14 +711,14 @@ Proof.
- move => n ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl).
move : iha => [_ [d [ih0 ih1]]].
split.
- + exists (PPair (PProj PL d) (PProj PR d)).
+ + exists (Pair (Proj PL d) (Proj PR d)).
split; first by apply EPar.PairEta.
apply : rtc_l.
apply RPar.AppPair; eauto using RPar.refl.
- suff h : forall p, rtc RPar.R (PApp (PProj p (ren_PTm shift a1)) (VarPTm var_zero)) (PProj p d) by
+ suff h : forall p, rtc RPar.R (App (Proj p (ren_Tm shift a1)) (VarTm var_zero)) (Proj p d) by
sfirstorder use:RPars.PairCong.
move => p. move /(_ p) /RPars.weakening in ih1.
- apply relations.rtc_transitive with (y := PApp (ren_PTm shift (PAbs (PProj p d))) (VarPTm var_zero)).
+ apply relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj p d))) (VarTm var_zero)).
by eauto using RPars.AppCong, rtc_refl.
apply relations.rtc_once => /=.
apply : RPar.AppAbs'; eauto using RPar.refl.
@@ -1170,21 +734,21 @@ Proof.
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
Qed.
-Lemma Pair_EPar n (a b c : PTm n) :
- EPar.R (PPair a b) c ->
- (forall p, exists d, rtc RPar.R (PProj p c) d /\ EPar.R (if p is PL then a else b) d) /\
- (exists d0 d1, rtc RPar.R (PApp (ren_PTm shift c) (VarPTm var_zero))
- (PPair (PApp (ren_PTm shift d0) (VarPTm var_zero))(PApp (ren_PTm shift d1) (VarPTm var_zero))) /\
+Lemma Pair_EPar n (a b c : Tm n) :
+ EPar.R (Pair a b) c ->
+ (forall p, exists d, rtc RPar.R (Proj p c) d /\ EPar.R (if p is PL then a else b) d) /\
+ (exists d0 d1, rtc RPar.R (App (ren_Tm shift c) (VarTm var_zero))
+ (Pair (App (ren_Tm shift d0) (VarTm var_zero))(App (ren_Tm shift d1) (VarTm var_zero))) /\
EPar.R a d0 /\ EPar.R b d1).
Proof.
- move E : (PPair a b) => u h. move : a b E.
+ move E : (Pair a b) => u h. move : a b E.
elim : n u c /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
move : iha => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
split.
+ move => p.
- exists (PAbs (PApp (ren_PTm shift (if p is PL then d0 else d1)) (VarPTm var_zero))).
+ exists (Abs (App (ren_Tm shift (if p is PL then d0 else d1)) (VarTm var_zero))).
split.
* apply : relations.rtc_transitive.
** apply RPars.ProjCong. apply RPars.AbsCong. eassumption.
@@ -1202,7 +766,7 @@ Proof.
exists d. split=>//.
apply : rtc_l. apply RPar.ProjPair; eauto using RPar.refl.
set q := (X in rtc RPar.R X d).
- by have -> : q = PProj p a1 by hauto lq:on.
+ by have -> : q = Proj p a1 by hauto lq:on.
+ move :iha => [iha _].
move : (iha PL) => [d0 [ih0 ih0']].
move : (iha PR) => [d1 [ih1 ih1']] {iha}.
@@ -1223,7 +787,7 @@ Proof.
split => //.
Qed.
-Lemma commutativity0 n (a b0 b1 : PTm n) :
+Lemma commutativity0 n (a b0 b1 : Tm n) :
EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
Proof.
move => h. move : b1.
@@ -1231,13 +795,13 @@ Proof.
- move => n a b0 ha iha b1 hb.
move : iha (hb) => /[apply].
move => [c [ih0 ih1]].
- exists (PAbs (PApp (ren_PTm shift c) (VarPTm var_zero))).
+ exists (Abs (App (ren_Tm shift c) (VarTm var_zero))).
split.
+ hauto lq:on ctrs:rtc use:RPars.AbsCong, RPars.AppCong, RPars.renaming.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
- move => n a b0 hb0 ihb0 b1 /[dup] hb1 {}/ihb0.
move => [c [ih0 ih1]].
- exists (PPair (PProj PL c) (PProj PR c)). split.
+ exists (Pair (Proj PL c) (Proj PR c)). split.
+ apply RPars.PairCong;
by apply RPars.ProjCong.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
@@ -1250,9 +814,9 @@ Proof.
elim /RPar.inv => //= _.
+ move => a2 a3 b3 b4 h0 h1 [*]. subst.
move /(_ _ ltac:(by eauto)) : ihb => [b [ihb0 ihb1]].
- have {}/iha : RPar.R (PAbs a2) (PAbs a3) by hauto lq:on ctrs:RPar.R.
+ have {}/iha : RPar.R (Abs a2) (Abs a3) by hauto lq:on ctrs:RPar.R.
move => [c [ih0 /Abs_EPar [[d [ih1 ih2]] _]]].
- exists (subst_PTm (scons b VarPTm) d).
+ exists (subst_Tm (scons b VarTm) d).
split.
(* By substitution *)
* move /RPars.substing : ih2.
@@ -1268,7 +832,7 @@ Proof.
move /Pair_EPar : ihc1 => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
move /RPars.substing : ih0. move /(_ d).
asimpl => h.
- exists (PPair (PApp d0 d) (PApp d1 d)).
+ exists (Pair (App d0 d) (App d1 d)).
split.
hauto lq:on use:relations.rtc_transitive, RPars.AppCong.
apply EPar.PairCong; by apply EPar.AppCong.
@@ -1279,7 +843,7 @@ Proof.
+ move => ? a0 a1 h [*]. subst.
move /(_ _ ltac:(by eauto using RPar.AbsCong)) : ih0 => [c [ih0 ih1]].
move /Abs_EPar : ih1 => [_ [d [ih1 ih2]]].
- exists (PAbs (PProj p d)).
+ exists (Abs (Proj p d)).
qauto l:on ctrs:EPar.R use:RPars.ProjCong, @relations.rtc_transitive.
+ move => p0 a0 a1 b2 b3 h1 h2 [*]. subst.
move /(_ _ ltac:(by eauto using RPar.PairCong)) : ih0 => [c [ih0 ih1]].
@@ -1287,12 +851,12 @@ Proof.
exists d. split => //.
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
- - hauto l:on ctrs:EPar.R inv:RPar.R.
+ - hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
Qed.
-Lemma commutativity1 n (a b0 b1 : PTm n) :
+Lemma commutativity1 n (a b0 b1 : Tm n) :
EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
Proof.
move => + h. move : b0.
@@ -1301,7 +865,7 @@ Proof.
- qauto l:on use:relations.rtc_transitive, commutativity0.
Qed.
-Lemma commutativity n (a b0 b1 : PTm n) :
+Lemma commutativity n (a b0 b1 : Tm n) :
rtc EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ rtc EPar.R b1 c.
move => h. move : b1. elim : a b0 /h.
- sfirstorder.
@@ -1310,12 +874,12 @@ Lemma commutativity n (a b0 b1 : PTm n) :
hauto q:on ctrs:rtc.
Qed.
-Lemma Abs_EPar' n a (b : PTm n) :
- EPar.R (PAbs a) b ->
+Lemma Abs_EPar' n a (b : Tm n) :
+ EPar.R (Abs a) b ->
(exists d, EPar.R a d /\
- rtc OExp.R (PAbs d) b).
+ rtc OExp.R (Abs d) b).
Proof.
- move E : (PAbs a) => u h.
+ move E : (Abs a) => u h.
move : a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha a ?. subst.
@@ -1327,12 +891,12 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Proj_EPar' n p a (b : PTm n) :
- EPar.R (PProj p a) b ->
+Lemma Proj_EPar' n p a (b : Tm n) :
+ EPar.R (Proj p a) b ->
(exists d, EPar.R a d /\
- rtc OExp.R (PProj p d) b).
+ rtc OExp.R (Proj p d) b).
Proof.
- move E : (PProj p a) => u h.
+ move E : (Proj p a) => u h.
move : p a E.
elim : n u b /h => //=.
- move => n a0 a1 ha iha a p ?. subst.
@@ -1344,11 +908,11 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma App_EPar' n (a b u : PTm n) :
- EPar.R (PApp a b) u ->
- (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PApp a0 b0) u).
+Lemma App_EPar' n (a b u : Tm n) :
+ EPar.R (App a b) u ->
+ (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (App a0 b0) u).
Proof.
- move E : (PApp a b) => t h.
+ move E : (App a b) => t h.
move : a b E. elim : n t u /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
@@ -1359,11 +923,11 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Pair_EPar' n (a b u : PTm n) :
- EPar.R (PPair a b) u ->
- exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (PPair a0 b0) u.
+Lemma Bind_EPar' n p (a : Tm n) b u :
+ EPar.R (TBind p a b) u ->
+ (exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (TBind p a0 b0) u).
Proof.
- move E : (PPair a b) => t h.
+ move E : (TBind p a b) => t h.
move : a b E. elim : n t u /h => //=.
- move => n a0 a1 ha iha a b ?. subst.
specialize iha with (1 := eq_refl).
@@ -1374,24 +938,25 @@ Proof.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Const_EPar' n k (u : PTm n) :
- EPar.R (PConst k) u ->
- rtc OExp.R (PConst k) u.
- move E : (PConst k) => t h.
- move : k E. elim : n t u /h => //=.
- - move => n a0 a1 h ih k ?. subst.
- specialize ih with (1 := eq_refl).
+Lemma Pair_EPar' n (a b u : Tm n) :
+ EPar.R (Pair a b) u ->
+ exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (Pair a0 b0) u.
+Proof.
+ move E : (Pair a b) => t h.
+ move : a b E. elim : n t u /h => //=.
+ - move => n a0 a1 ha iha a b ?. subst.
+ specialize iha with (1 := eq_refl).
hauto lq:on ctrs:OExp.R use:rtc_r.
- - move => n a0 a1 h ih k ?. subst.
- specialize ih with (1 := eq_refl).
+ - move => n a0 a1 ha iha a b ?. subst.
+ specialize iha with (1 := eq_refl).
hauto lq:on ctrs:OExp.R use:rtc_r.
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Bot_EPar' n (u : PTm n) :
- EPar.R (PBot) u ->
- rtc OExp.R (PBot) u.
- move E : (PBot) => t h.
+Lemma Bot_EPar' n (u : Tm n) :
+ EPar.R Bot u ->
+ rtc OExp.R Bot u.
+ move E : Bot => t h.
move : E. elim : n t u /h => //=.
- move => n a0 a1 h ih ?. subst.
specialize ih with (1 := eq_refl).
@@ -1402,10 +967,10 @@ Lemma Bot_EPar' n (u : PTm n) :
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma Univ_EPar' n i (u : PTm n) :
- EPar.R (PUniv i) u ->
- rtc OExp.R (PUniv i) u.
- move E : (PUniv i) => t h.
+Lemma Univ_EPar' n i (u : Tm n) :
+ EPar.R (Univ i) u ->
+ rtc OExp.R (Univ i) u.
+ move E : (Univ i) => t h.
move : E. elim : n t u /h => //=.
- move => n a0 a1 h ih ?. subst.
specialize ih with (1 := eq_refl).
@@ -1416,14 +981,14 @@ Lemma Univ_EPar' n i (u : PTm n) :
- hauto l:on ctrs:OExp.R.
Qed.
-Lemma EPar_diamond n (c a1 b1 : PTm n) :
+Lemma EPar_diamond n (c a1 b1 : Tm n) :
EPar.R c a1 ->
EPar.R c b1 ->
exists d2, EPar.R a1 d2 /\ EPar.R b1 d2.
Proof.
move => h. move : b1. elim : n c a1 / h.
- move => n c a1 ha iha b1 /iha [d2 [hd0 hd1]].
- exists(PAbs (PApp (ren_PTm shift d2) (VarPTm var_zero))).
+ exists(Abs (App (ren_Tm shift d2) (VarTm var_zero))).
hauto lq:on ctrs:EPar.R use:EPar.renaming.
- hauto lq:on rew:off ctrs:EPar.R.
- hauto lq:on use:EPar.refl.
@@ -1431,54 +996,60 @@ Proof.
move /Abs_EPar' => [d [hd0 hd1]].
move : iha hd0; repeat move/[apply].
move => [d2 [h0 h1]].
- have : EPar.R (PAbs d) (PAbs d2) by eauto using EPar.AbsCong.
+ have : EPar.R (Abs d) (Abs d2) by eauto using EPar.AbsCong.
move : OExp.commutativity0 hd1; repeat move/[apply].
move => [d1 [hd1 hd2]].
exists d1. hauto lq:on ctrs:EPar.R use:OExp.merge.
- move => n a0 a1 b0 b1 ha iha hb ihb c.
move /App_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
- have : EPar.R (PApp a2 b2)(PApp a3 b3)
+ have : EPar.R (App a2 b2)(App a3 b3)
by hauto l:on use:EPar.AppCong.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- move => n a0 a1 b0 b1 ha iha hb ihb c.
move /Pair_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
- have : EPar.R (PPair a2 b2)(PPair a3 b3)
+ have : EPar.R (Pair a2 b2)(Pair a3 b3)
by hauto l:on use:EPar.PairCong.
move : OExp.commutativity0 h2; repeat move/[apply].
move => [d h].
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- move => n p a0 a1 ha iha b.
move /Proj_EPar' => [d [/iha [d2 h] h1]] {iha}.
- have : EPar.R (PProj p d) (PProj p d2)
+ have : EPar.R (Proj p d) (Proj p d2)
by hauto l:on use:EPar.ProjCong.
move : OExp.commutativity0 h1; repeat move/[apply].
move => [d1 h1].
exists d1. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- - qauto use:Const_EPar', EPar.refl.
- - qauto use:Univ_EPar', EPar.refl.
+ - move => n p a0 a1 b0 b1 ha iha hb ihb c.
+ move /Bind_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
+ have : EPar.R (TBind p a2 b2)(TBind p a3 b3)
+ by hauto l:on use:EPar.BindCong.
+ move : OExp.commutativity0 h2; repeat move/[apply].
+ move => [d h].
+ exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
- qauto use:Bot_EPar', EPar.refl.
+ - qauto use:Univ_EPar', EPar.refl.
Qed.
-Function tstar {n} (a : PTm n) :=
+Function tstar {n} (a : Tm n) :=
match a with
- | VarPTm i => a
- | PAbs a => PAbs (tstar a)
- | PApp (PAbs a) b => subst_PTm (scons (tstar b) VarPTm) (tstar a)
- | PApp (PPair a b) c =>
- PPair (PApp (tstar a) (tstar c)) (PApp (tstar b) (tstar c))
- | PApp a b => PApp (tstar a) (tstar b)
- | PPair a b => PPair (tstar a) (tstar b)
- | PProj p (PPair a b) => if p is PL then (tstar a) else (tstar b)
- | PProj p (PAbs a) => (PAbs (PProj p (tstar a)))
- | PProj p a => PProj p (tstar a)
- | PConst k => PConst k
- | PUniv i => PUniv i
- | PBot => PBot
+ | VarTm i => a
+ | Abs a => Abs (tstar a)
+ | App (Abs a) b => subst_Tm (scons (tstar b) VarTm) (tstar a)
+ | App (Pair a b) c =>
+ Pair (App (tstar a) (tstar c)) (App (tstar b) (tstar c))
+ | App a b => App (tstar a) (tstar b)
+ | Pair a b => Pair (tstar a) (tstar b)
+ | Proj p (Pair a b) => if p is PL then (tstar a) else (tstar b)
+ | Proj p (Abs a) => (Abs (Proj p (tstar a)))
+ | Proj p a => Proj p (tstar a)
+ | TBind p a b => TBind p (tstar a) (tstar b)
+ | Bot => Bot
+ | Univ i => Univ i
end.
-Lemma RPar_triangle n (a : PTm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
+Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
Proof.
apply tstar_ind => {n a}.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
@@ -1496,49 +1067,13 @@ Proof.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
Qed.
-Function tstar' {n} (a : PTm n) :=
- match a with
- | VarPTm i => a
- | PAbs a => PAbs (tstar' a)
- | PApp (PAbs a) b => subst_PTm (scons (tstar' b) VarPTm) (tstar' a)
- | PApp a b => PApp (tstar' a) (tstar' b)
- | PPair a b => PPair (tstar' a) (tstar' b)
- | PProj p (PPair a b) => if p is PL then (tstar' a) else (tstar' b)
- | PProj p a => PProj p (tstar' a)
- | PConst k => PConst k
- | PUniv i => PUniv i
- | PBot => PBot
- end.
-
-Lemma RPar'_triangle n (a : PTm n) : forall b, RPar'.R a b -> RPar'.R b (tstar' a).
-Proof.
- apply tstar'_ind => {n a}.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on use:RPar'.cong, RPar'.refl ctrs:RPar'.R inv:RPar'.R.
- - hauto lq:on rew:off ctrs:RPar'.R inv:RPar'.R.
- - hauto lq:on rew:off inv:RPar'.R ctrs:RPar'.R.
- - hauto drew:off inv:RPar'.R use:RPar'.refl, RPar'.ProjPair'.
- - hauto drew:off inv:RPar'.R use:RPar'.refl, RPar'.ProjPair'.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
- - hauto lq:on inv:RPar'.R ctrs:RPar'.R.
-Qed.
-
-Lemma RPar_diamond n (c a1 b1 : PTm n) :
+Lemma RPar_diamond n (c a1 b1 : Tm n) :
RPar.R c a1 ->
RPar.R c b1 ->
exists d2, RPar.R a1 d2 /\ RPar.R b1 d2.
Proof. hauto l:on use:RPar_triangle. Qed.
-Lemma RPar'_diamond n (c a1 b1 : PTm n) :
- RPar'.R c a1 ->
- RPar'.R c b1 ->
- exists d2, RPar'.R a1 d2 /\ RPar'.R b1 d2.
-Proof. hauto l:on use:RPar'_triangle. Qed.
-
-Lemma RPar_confluent n (c a1 b1 : PTm n) :
+Lemma RPar_confluent n (c a1 b1 : Tm n) :
rtc RPar.R c a1 ->
rtc RPar.R c b1 ->
exists d2, rtc RPar.R a1 d2 /\ rtc RPar.R b1 d2.
@@ -1546,7 +1081,7 @@ Proof.
sfirstorder use:relations.diamond_confluent, RPar_diamond.
Qed.
-Lemma EPar_confluent n (c a1 b1 : PTm n) :
+Lemma EPar_confluent n (c a1 b1 : Tm n) :
rtc EPar.R c a1 ->
rtc EPar.R c b1 ->
exists d2, rtc EPar.R a1 d2 /\ rtc EPar.R b1 d2.
@@ -1554,35 +1089,107 @@ Proof.
sfirstorder use:relations.diamond_confluent, EPar_diamond.
Qed.
-Inductive prov {n} : PTm n -> PTm n -> Prop :=
+Fixpoint depth_tm {n} (a : Tm n) :=
+ match a with
+ | VarTm _ => 1
+ | TBind _ A B => 1 + max (depth_tm A) (depth_tm B)
+ | Abs a => 1 + depth_tm a
+ | App a b => 1 + max (depth_tm a) (depth_tm b)
+ | Proj p a => 1 + depth_tm a
+ | Pair a b => 1 + max (depth_tm a) (depth_tm b)
+ | Bot => 1
+ | Univ i => 1
+ end.
+
+Lemma depth_ren n m (ξ: fin n -> fin m) a :
+ depth_tm a = depth_tm (ren_Tm ξ a).
+Proof.
+ move : m ξ. elim : n / a; scongruence.
+Qed.
+
+Lemma depth_subst n m (ρ : fin n -> Tm m) a :
+ (forall i, depth_tm (ρ i) = 1) ->
+ depth_tm a = depth_tm (subst_Tm ρ a).
+Proof.
+ move : m ρ. elim : n / a.
+ - sfirstorder.
+ - move => n a iha m ρ hρ.
+ simpl.
+ f_equal. apply iha.
+ destruct i as [i|].
+ + simpl.
+ by rewrite -depth_ren.
+ + by simpl.
+ - hauto lq:on rew:off.
+ - hauto lq:on rew:off.
+ - hauto lq:on rew:off.
+ - move => n p a iha b ihb m ρ hρ.
+ simpl. f_equal.
+ f_equal.
+ by apply iha.
+ apply ihb.
+ destruct i as [i|].
+ + simpl.
+ by rewrite -depth_ren.
+ + by simpl.
+ - sfirstorder.
+ - sfirstorder.
+Qed.
+
+Lemma depth_subst_bool n (a : Tm (S n)) :
+ depth_tm a = depth_tm (subst_Tm (scons Bot VarTm) a).
+Proof.
+ apply depth_subst.
+ destruct i as [i|] => //=.
+Qed.
+
+Local Ltac prov_tac := sfirstorder use:depth_ren.
+Local Ltac extract_tac := rewrite -?depth_subst_bool;hauto use:depth_subst_bool.
+
+Definition prov_bind {n} p0 A0 B0 (a : Tm n) :=
+ match a with
+ | TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
+ | _ => False
+ end.
+
+Definition prov_univ {n} i0 (a : Tm n) :=
+ match a with
+ | Univ i => i = i0
+ | _ => False
+ end.
+
+
+Inductive prov {n} : Tm n -> Tm n -> Prop :=
+| P_Bind p A A0 B B0 :
+ rtc Par.R A A0 ->
+ rtc Par.R B B0 ->
+ prov (TBind p A B) (TBind p A0 B0)
| P_Abs h a :
- (forall b, prov h (subst_PTm (scons b VarPTm) a)) ->
- prov h (PAbs a)
+ (forall b, prov h (subst_Tm (scons b VarTm) a)) ->
+ prov h (Abs a)
| P_App h a b :
prov h a ->
- prov h (PApp a b)
+ prov h (App a b)
| P_Pair h a b :
prov h a ->
prov h b ->
- prov h (PPair a b)
+ prov h (Pair a b)
| P_Proj h p a :
prov h a ->
- prov h (PProj p a)
-| P_Const k :
- prov (PConst k) (PConst k)
+ prov h (Proj p a)
+| P_Bot :
+ prov Bot Bot
| P_Var i :
- prov (VarPTm i) (VarPTm i)
+ prov (VarTm i) (VarTm i)
| P_Univ i :
- prov (PUniv i) (PUniv i)
-| P_Bot :
- prov PBot PBot.
+ prov (Univ i) (Univ i).
-Lemma ERed_EPar n (a b : PTm n) : ERed.R a b -> EPar.R a b.
+Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
Proof.
induction 1; hauto lq:on ctrs:EPar.R use:EPar.refl.
Qed.
-Lemma EPar_ERed n (a b : PTm n) : EPar.R a b -> rtc ERed.R a b.
+Lemma EPar_ERed n (a b : Tm n) : EPar.R a b -> rtc ERed.R a b.
Proof.
move => h. elim : n a b /h.
- eauto using rtc_r, ERed.AppEta.
@@ -1592,84 +1199,95 @@ Proof.
- eauto using EReds.AppCong.
- eauto using EReds.PairCong.
- eauto using EReds.ProjCong.
- - auto using rtc_refl.
+ - eauto using EReds.BindCong.
- auto using rtc_refl.
- auto using rtc_refl.
Qed.
-Lemma EPar_Par n (a b : PTm n) : EPar.R a b -> Par.R a b.
+Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
Proof.
move => h. elim : n a b /h; qauto ctrs:Par.R.
Qed.
-Lemma RPar_Par n (a b : PTm n) : RPar.R a b -> Par.R a b.
+Lemma RPar_Par n (a b : Tm n) : RPar.R a b -> Par.R a b.
Proof.
move => h. elim : n a b /h; hauto lq:on ctrs:Par.R.
Qed.
-Lemma rtc_idem n (R : PTm n -> PTm n -> Prop) (a b : PTm n) : rtc (rtc R) a b -> rtc R a b.
+Lemma rtc_idem n (R : Tm n -> Tm n -> Prop) (a b : Tm n) : rtc (rtc R) a b -> rtc R a b.
Proof.
induction 1; hauto l:on use:@relations.rtc_transitive, @rtc_r.
Qed.
-Lemma EPars_EReds {n} (a b : PTm n) : rtc EPar.R a b <-> rtc ERed.R a b.
+Lemma EPars_EReds {n} (a b : Tm n) : rtc EPar.R a b <-> rtc ERed.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel, EPar_ERed, rtc_idem, ERed_EPar.
Qed.
-Lemma prov_rpar n (u : PTm n) a b : prov u a -> RPar.R a b -> prov u b.
+Lemma prov_rpar n (u : Tm n) a b : prov u a -> RPar.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
- (* - qauto l:on ctrs:prov inv:RPar.R use:@rtc_r, RPar_Par. *)
+ - qauto l:on ctrs:prov inv:RPar.R use:@rtc_r, RPar_Par.
- hauto lq:on ctrs:prov inv:RPar.R use:RPar.substing.
- move => h a b ha iha b0.
elim /RPar.inv => //= _.
+ move => a0 a1 b1 b2 h0 h1 [*]. subst.
- have {}iha : prov h (PAbs a1) by hauto lq:on ctrs:RPar.R.
+ have {}iha : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov use:RPar.substing.
+ move => a0 a1 b1 b2 c0 c1.
move => h0 h1 h2 [*]. subst.
- have {}iha : prov h (PPair a1 b2) by hauto lq:on ctrs:RPar.R.
+ have {}iha : prov h (Pair a1 b2) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov ctrs:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
- move => h p a ha iha b.
elim /RPar.inv => //= _.
+ move => p0 a0 a1 h0 [*]. subst.
- have {iha} : prov h (PAbs a1) by hauto lq:on ctrs:RPar.R.
+ have {iha} : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on ctrs:prov inv:prov use:RPar.substing.
+ move => p0 a0 a1 b0 b1 h0 h1 [*]. subst.
- have {iha} : prov h (PPair a1 b1) by hauto lq:on ctrs:RPar.R.
+ have {iha} : prov h (Pair a1 b1) by hauto lq:on ctrs:RPar.R.
qauto l:on inv:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
- - hauto l:on ctrs:RPar.R inv:RPar.R.
Qed.
-Lemma prov_lam n (u : PTm n) a : prov u a <-> prov u (PAbs (PApp (ren_PTm shift a) (VarPTm var_zero))).
+Lemma prov_lam n (u : Tm n) a : prov u a <-> prov u (Abs (App (ren_Tm shift a) (VarTm var_zero))).
Proof.
split.
move => h. constructor. move => b. asimpl. by constructor.
inversion 1; subst.
- specialize H2 with (b := PBot).
+ specialize H2 with (b := Bot).
move : H2. asimpl. inversion 1; subst. done.
Qed.
-Lemma prov_pair n (u : PTm n) a : prov u a <-> prov u (PPair (PProj PL a) (PProj PR a)).
+Lemma prov_pair n (u : Tm n) a : prov u a <-> prov u (Pair (Proj PL a) (Proj PR a)).
Proof. hauto lq:on inv:prov ctrs:prov. Qed.
-Lemma prov_ered n (u : PTm n) a b : prov u a -> ERed.R a b -> prov u b.
+Derive Dependent Inversion inv with (forall n (a b : Tm n), ERed.R a b) Sort Prop.
+
+Lemma prov_ered n (u : Tm n) a b : prov u a -> ERed.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
+ - move => p A A0 B B0 hA hB b.
+ elim /inv => // _.
+ + move => a0 *. subst.
+ rewrite -prov_lam.
+ by constructor.
+ + move => a0 *. subst.
+ rewrite -prov_pair.
+ by constructor.
+ + qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
+ + qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
- move => h a ha iha b.
- elim /ERed.inv => // _.
+ elim /inv => // _.
+ move => a0 *. subst.
rewrite -prov_lam.
by constructor.
@@ -1679,7 +1297,7 @@ Proof.
+ hauto lq:on ctrs:prov use:ERed.substing.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- move => h a b ha iha hb ihb b0.
- elim /ERed.inv => //_.
+ elim /inv => //_.
+ move => a0 *. subst.
rewrite -prov_lam.
by constructor.
@@ -1692,28 +1310,27 @@ Proof.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- - hauto lq:on inv:ERed.R, prov ctrs:prov.
Qed.
-Lemma prov_ereds n (u : PTm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
+Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
Proof.
induction 2; sfirstorder use:prov_ered.
Qed.
-Fixpoint extract {n} (a : PTm n) : PTm n :=
+Fixpoint extract {n} (a : Tm n) : Tm n :=
match a with
- | PAbs a => subst_PTm (scons PBot VarPTm) (extract a)
- | PApp a b => extract a
- | PPair a b => extract a
- | PProj p a => extract a
- | PConst k => PConst k
- | VarPTm i => VarPTm i
- | PUniv i => PUniv i
- | PBot => PBot
+ | TBind p A B => TBind p A B
+ | Abs a => subst_Tm (scons Bot VarTm) (extract a)
+ | App a b => extract a
+ | Pair a b => extract a
+ | Proj p a => extract a
+ | Bot => Bot
+ | VarTm i => VarTm i
+ | Univ i => Univ i
end.
-Lemma ren_extract n m (a : PTm n) (ξ : fin n -> fin m) :
- extract (ren_PTm ξ a) = ren_PTm ξ (extract a).
+Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
+ extract (ren_Tm ξ a) = ren_Tm ξ (extract a).
Proof.
move : m ξ. elim : n/a.
- sfirstorder.
@@ -1728,9 +1345,9 @@ Proof.
- sfirstorder.
Qed.
-Lemma ren_morphing n m (a : PTm n) (ρ : fin n -> PTm m) :
+Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
(forall i, ρ i = extract (ρ i)) ->
- extract (subst_PTm ρ a) = subst_PTm ρ (extract a).
+ extract (subst_Tm ρ a) = subst_Tm ρ (extract a).
Proof.
move : m ρ.
elim : n /a => n //=.
@@ -1743,43 +1360,42 @@ Proof.
- by asimpl.
Qed.
-Lemma ren_subst_bot n (a : PTm (S n)) :
- extract (subst_PTm (scons PBot VarPTm) a) = subst_PTm (scons PBot VarPTm) (extract a).
+Lemma ren_subst_bot n (a : Tm (S n)) :
+ extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
Proof.
apply ren_morphing. destruct i as [i|] => //=.
Qed.
-Definition prov_extract_spec {n} u (a : PTm n) :=
+Definition prov_extract_spec {n} u (a : Tm n) :=
match u with
- | PUniv i => extract a = PUniv i
- | VarPTm i => extract a = VarPTm i
- | (PConst i) => extract a = (PConst i)
- | PBot => extract a = PBot
+ | TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
+ | Univ i => extract a = Univ i
+ | VarTm i => extract a = VarTm i
| _ => True
end.
-Lemma prov_extract n u (a : PTm n) :
+Lemma prov_extract n u (a : Tm n) :
prov u a -> prov_extract_spec u a.
Proof.
move => h.
elim : u a /h.
+ - sfirstorder.
- move => h a ha ih.
case : h ha ih => //=.
+ move => i ha ih.
- move /(_ PBot) in ih.
+ move /(_ Bot) in ih.
rewrite -ih.
by rewrite ren_subst_bot.
- + move => p _ /(_ PBot).
- by rewrite ren_subst_bot.
- + move => i h /(_ PBot).
+ + move => p A B h ih.
+ move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
+ rewrite ren_subst_bot in h0.
+ rewrite h0.
+ eauto.
+ + move => i h /(_ Bot).
by rewrite ren_subst_bot => ->.
- + move /(_ PBot).
- move => h /(_ PBot).
- by rewrite ren_subst_bot.
- hauto lq:on.
- hauto lq:on.
- hauto lq:on.
- - case => //=.
- sfirstorder.
- sfirstorder.
- sfirstorder.
@@ -1789,21 +1405,21 @@ Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b :=
R0 a b \/ R1 a b.
Module ERPar.
- Definition R {n} (a b : PTm n) := union RPar.R EPar.R a b.
- Lemma RPar {n} (a b : PTm n) : RPar.R a b -> R a b.
+ Definition R {n} (a b : Tm n) := union RPar.R EPar.R a b.
+ Lemma RPar {n} (a b : Tm n) : RPar.R a b -> R a b.
Proof. sfirstorder. Qed.
- Lemma EPar {n} (a b : PTm n) : EPar.R a b -> R a b.
+ Lemma EPar {n} (a b : Tm n) : EPar.R a b -> R a b.
Proof. sfirstorder. Qed.
- Lemma refl {n} ( a : PTm n) : ERPar.R a a.
+ Lemma refl {n} ( a : Tm n) : ERPar.R a a.
Proof.
sfirstorder use:RPar.refl, EPar.refl.
Qed.
- Lemma ProjCong n p (a0 a1 : PTm n) :
+ Lemma ProjCong n p (a0 a1 : Tm n) :
R a0 a1 ->
- rtc R (PProj p a0) (PProj p a1).
+ rtc R (Proj p a0) (Proj p a1).
Proof.
move => [].
- move => h.
@@ -1816,9 +1432,9 @@ Module ERPar.
by apply EPar.ProjCong.
Qed.
- Lemma AbsCong n (a0 a1 : PTm (S n)) :
+ Lemma AbsCong n (a0 a1 : Tm (S n)) :
R a0 a1 ->
- rtc R (PAbs a0) (PAbs a1).
+ rtc R (Abs a0) (Abs a1).
Proof.
move => [].
- move => h.
@@ -1831,10 +1447,10 @@ Module ERPar.
by apply EPar.AbsCong.
Qed.
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
R a0 a1 ->
R b0 b1 ->
- rtc R (PApp a0 b0) (PApp a1 b1).
+ rtc R (App a0 b0) (App a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.AppCong, @rtc_once.
@@ -1851,10 +1467,30 @@ Module ERPar.
- sfirstorder use:EPar.AppCong, @rtc_once.
Qed.
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
R a0 a1 ->
R b0 b1 ->
- rtc R (PPair a0 b0) (PPair a1 b1).
+ rtc R (TBind p a0 b0) (TBind p a1 b1).
+ Proof.
+ move => [] + [].
+ - sfirstorder use:RPar.BindCong, @rtc_once.
+ - move => h0 h1.
+ apply : rtc_l.
+ left. apply RPar.BindCong; eauto; apply RPar.refl.
+ apply rtc_once.
+ hauto l:on use:EPar.BindCong, EPar.refl.
+ - move => h0 h1.
+ apply : rtc_l.
+ left. apply RPar.BindCong; eauto; apply RPar.refl.
+ apply rtc_once.
+ hauto l:on use:EPar.BindCong, EPar.refl.
+ - sfirstorder use:EPar.BindCong, @rtc_once.
+ Qed.
+
+ Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
+ R a0 a1 ->
+ R b0 b1 ->
+ rtc R (Pair a0 b0) (Pair a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.PairCong, @rtc_once.
@@ -1871,15 +1507,15 @@ Module ERPar.
- sfirstorder use:EPar.PairCong, @rtc_once.
Qed.
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- R a b -> R (ren_PTm ξ a) (ren_PTm ξ b).
+ Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
+ R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
sfirstorder use:EPar.renaming, RPar.renaming.
Qed.
End ERPar.
-Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong : erpar.
+Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong ERPar.BindCong : erpar.
Module ERPars.
#[local]Ltac solve_s_rec :=
@@ -1888,31 +1524,37 @@ Module ERPars.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
- rtc ERPar.R (PApp a0 b0) (PApp a1 b1).
+ rtc ERPar.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
- Lemma AbsCong n (a0 a1 : PTm (S n)) :
+ Lemma AbsCong n (a0 a1 : Tm (S n)) :
rtc ERPar.R a0 a1 ->
- rtc ERPar.R (PAbs a0) (PAbs a1).
+ rtc ERPar.R (Abs a0) (Abs a1).
Proof. solve_s. Qed.
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
+ Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
- rtc ERPar.R (PPair a0 b0) (PPair a1 b1).
+ rtc ERPar.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
- Lemma ProjCong n p (a0 a1 : PTm n) :
+ Lemma ProjCong n p (a0 a1 : Tm n) :
rtc ERPar.R a0 a1 ->
- rtc ERPar.R (PProj p a0) (PProj p a1).
+ rtc ERPar.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
- Lemma renaming n (a0 a1 : PTm n) m (ξ : fin n -> fin m) :
+ Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
rtc ERPar.R a0 a1 ->
- rtc ERPar.R (ren_PTm ξ a0) (ren_PTm ξ a1).
+ rtc ERPar.R b0 b1 ->
+ rtc ERPar.R (TBind p a0 b0) (TBind p a1 b1).
+ Proof. solve_s. Qed.
+
+ Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
+ rtc ERPar.R a0 a1 ->
+ rtc ERPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
@@ -1921,16 +1563,16 @@ Module ERPars.
End ERPars.
-Lemma ERPar_Par n (a b : PTm n) : ERPar.R a b -> Par.R a b.
+Lemma ERPar_Par n (a b : Tm n) : ERPar.R a b -> Par.R a b.
Proof.
sfirstorder use:EPar_Par, RPar_Par.
Qed.
-Lemma Par_ERPar n (a b : PTm n) : Par.R a b -> rtc ERPar.R a b.
+Lemma Par_ERPar n (a b : Tm n) : Par.R a b -> rtc ERPar.R a b.
Proof.
move => h. elim : n a b /h.
- move => n a0 a1 b0 b1 ha iha hb ihb.
- suff ? : rtc ERPar.R (PApp (PAbs a0) b0) (PApp (PAbs a1) b1).
+ suff ? : rtc ERPar.R (App (Abs a0) b0) (App (Abs a1) b1).
apply : relations.rtc_transitive; eauto.
apply rtc_once. apply ERPar.RPar.
by apply RPar.AppAbs; eauto using RPar.refl.
@@ -1957,29 +1599,29 @@ Proof.
- sfirstorder use:ERPars.AppCong.
- sfirstorder use:ERPars.PairCong.
- sfirstorder use:ERPars.ProjCong.
- - sfirstorder.
+ - sfirstorder use:ERPars.BindCong.
- sfirstorder.
- sfirstorder.
Qed.
-Lemma Pars_ERPar n (a b : PTm n) : rtc Par.R a b -> rtc ERPar.R a b.
+Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
Proof.
induction 1; hauto l:on use:Par_ERPar, @relations.rtc_transitive.
Qed.
-Lemma Par_ERPar_iff n (a b : PTm n) : rtc Par.R a b <-> rtc ERPar.R a b.
+Lemma Par_ERPar_iff n (a b : Tm n) : rtc Par.R a b <-> rtc ERPar.R a b.
Proof.
split.
sfirstorder use:Pars_ERPar, @relations.rtc_subrel.
sfirstorder use:ERPar_Par, @relations.rtc_subrel.
Qed.
-Lemma RPar_ERPar n (a b : PTm n) : rtc RPar.R a b -> rtc ERPar.R a b.
+Lemma RPar_ERPar n (a b : Tm n) : rtc RPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
-Lemma EPar_ERPar n (a b : PTm n) : rtc EPar.R a b -> rtc ERPar.R a b.
+Lemma EPar_ERPar n (a b : Tm n) : rtc EPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
@@ -2045,7 +1687,7 @@ Module HindleyRosenFacts (M : HindleyRosen).
End HindleyRosenFacts.
Module HindleyRosenER <: HindleyRosen.
- Definition A := PTm.
+ Definition A := Tm.
Definition R0 n := rtc (@RPar.R n).
Definition R1 n := rtc (@EPar.R n).
Lemma diamond_R0 : forall n, relations.diamond (R0 n).
@@ -2063,7 +1705,7 @@ End HindleyRosenER.
Module ERFacts := HindleyRosenFacts HindleyRosenER.
-Lemma rtc_union n (a b : PTm n) :
+Lemma rtc_union n (a b : Tm n) :
rtc (union RPar.R EPar.R) a b <->
rtc (union (rtc RPar.R) (rtc EPar.R)) a b.
Proof.
@@ -2085,7 +1727,7 @@ Proof.
sfirstorder.
Qed.
-Lemma prov_erpar n (u : PTm n) a b : prov u a -> ERPar.R a b -> prov u b.
+Lemma prov_erpar n (u : Tm n) a b : prov u a -> ERPar.R a b -> prov u b.
Proof.
move => h [].
- sfirstorder use:prov_rpar.
@@ -2093,7 +1735,7 @@ Proof.
sfirstorder use:prov_ereds.
Qed.
-Lemma prov_pars n (u : PTm n) a b : prov u a -> rtc Par.R a b -> prov u b.
+Lemma prov_pars n (u : Tm n) a b : prov u a -> rtc Par.R a b -> prov u b.
Proof.
move => h /Pars_ERPar.
move => h0.
@@ -2103,15 +1745,15 @@ Proof.
- hauto lq:on use:prov_erpar.
Qed.
-Lemma Par_confluent n (a b c : PTm n) :
+Lemma Par_confluent n (a b c : Tm n) :
rtc Par.R a b ->
rtc Par.R a c ->
exists d, rtc Par.R b d /\ rtc Par.R c d.
Proof.
move : n a b c.
- suff : forall (n : nat) (a b c : PTm n),
+ suff : forall (n : nat) (a b c : Tm n),
rtc ERPar.R a b ->
- rtc ERPar.R a c -> exists d : PTm n, rtc ERPar.R b d /\ rtc ERPar.R c d.
+ rtc ERPar.R a c -> exists d : Tm n, rtc ERPar.R b d /\ rtc ERPar.R c d.
move => h n a b c h0 h1.
apply Par_ERPar_iff in h0, h1.
move : h h0 h1; repeat move/[apply].
@@ -2122,60 +1764,64 @@ Proof.
specialize h with (n := n).
rewrite /HindleyRosenER.A in h.
rewrite /ERPar.R.
- have eq : (fun a0 b0 : PTm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity.
+ have eq : (fun a0 b0 : Tm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity.
rewrite !{}eq.
move /rtc_union => + /rtc_union.
move : h; repeat move/[apply].
hauto lq:on use:rtc_union.
Qed.
-Lemma pars_univ_inv n i (c : PTm n) :
- rtc Par.R (PUniv i) c ->
- extract c = PUniv i.
+Lemma pars_univ_inv n i (c : Tm n) :
+ rtc Par.R (Univ i) c ->
+ extract c = Univ i.
Proof.
- have : prov (PUniv i) (PUniv i : PTm n) by sfirstorder.
+ have : prov (Univ i) (Univ i : Tm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
-Lemma pars_const_inv n i (c : PTm n) :
- rtc Par.R (PConst i) c ->
- extract c = PConst i.
+Lemma pars_pi_inv n p (A : Tm n) B C :
+ rtc Par.R (TBind p A B) C ->
+ exists A0 B0, extract C = TBind p A0 B0 /\
+ rtc Par.R A A0 /\ rtc Par.R B B0.
Proof.
- have : prov (PConst i) (PConst i : PTm n) by sfirstorder.
+ have : prov (TBind p A B) (TBind p A B) by hauto lq:on ctrs:prov, rtc.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_var_inv n (i : fin n) C :
- rtc Par.R (VarPTm i) C ->
- extract C = VarPTm i.
+ rtc Par.R (VarTm i) C ->
+ extract C = VarTm i.
Proof.
- have : prov (VarPTm i) (VarPTm i) by hauto lq:on ctrs:prov, rtc.
+ have : prov (VarTm i) (VarTm i) by hauto lq:on ctrs:prov, rtc.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
-Lemma pars_univ_inj n i j (C : PTm n) :
- rtc Par.R (PUniv i) C ->
- rtc Par.R (PUniv j) C ->
+Lemma pars_univ_inj n i j (C : Tm n) :
+ rtc Par.R (Univ i) C ->
+ rtc Par.R (Univ j) C ->
i = j.
Proof.
sauto l:on use:pars_univ_inv.
Qed.
-Lemma pars_const_inj n i j (C : PTm n) :
- rtc Par.R (PConst i) C ->
- rtc Par.R (PConst j) C ->
- i = j.
+Lemma pars_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 C :
+ rtc Par.R (TBind p0 A0 B0) C ->
+ rtc Par.R (TBind p1 A1 B1) C ->
+ exists A2 B2, p1 = p0 /\ rtc Par.R A0 A2 /\ rtc Par.R A1 A2 /\
+ rtc Par.R B0 B2 /\ rtc Par.R B1 B2.
Proof.
- sauto l:on use:pars_const_inv.
+ move /pars_pi_inv => [A2 [B2 [? [h0 h1]]]].
+ move /pars_pi_inv => [A3 [B3 [? [h2 h3]]]].
+ exists A2, B2. hauto l:on.
Qed.
-Definition join {n} (a b : PTm n) :=
+Definition join {n} (a b : Tm n) :=
exists c, rtc Par.R a c /\ rtc Par.R b c.
-Lemma join_transitive n (a b c : PTm n) :
+Lemma join_transitive n (a b c : Tm n) :
join a b -> join b c -> join a c.
Proof.
rewrite /join.
@@ -2185,234 +1831,40 @@ Proof.
eauto using relations.rtc_transitive.
Qed.
-Lemma join_symmetric n (a b : PTm n) :
+Lemma join_symmetric n (a b : Tm n) :
join a b -> join b a.
Proof. sfirstorder unfold:join. Qed.
-Lemma join_refl n (a : PTm n) : join a a.
+Lemma join_refl n (a : Tm n) : join a a.
Proof. hauto lq:on ctrs:rtc unfold:join. Qed.
Lemma join_univ_inj n i j :
- join (PUniv i : PTm n) (PUniv j) -> i = j.
+ join (Univ i : Tm n) (Univ j) -> i = j.
Proof.
sfirstorder use:pars_univ_inj.
Qed.
-Lemma join_const_inj n i j :
- join (PConst i : PTm n) (PConst j) -> i = j.
+Lemma join_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 :
+ join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
+ p0 = p1 /\ join A0 A1 /\ join B0 B1.
Proof.
- sfirstorder use:pars_const_inj.
+ move => [c []].
+ move : pars_pi_inj; repeat move/[apply].
+ sfirstorder unfold:join.
Qed.
-Lemma join_substing n m (a b : PTm n) (ρ : fin n -> PTm m) :
+Lemma join_univ_pi_contra n p (A : Tm n) B i :
+ join (TBind p A B) (Univ i) -> False.
+Proof.
+ rewrite /join.
+ move => [c [h0 h1]].
+ move /pars_univ_inv : h1.
+ move /pars_pi_inv : h0.
+ hauto l:on.
+Qed.
+
+
+Lemma join_substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
join a b ->
- join (subst_PTm ρ a) (subst_PTm ρ b).
+ join (subst_Tm ρ a) (subst_Tm ρ b).
Proof. hauto lq:on unfold:join use:Pars.substing. Qed.
-
-Fixpoint ne {n} (a : PTm n) :=
- match a with
- | VarPTm i => true
- | PApp a b => ne a && nf b
- | PAbs a => false
- | PUniv _ => false
- | PProj _ a => ne a
- | PPair _ _ => false
- | PConst _ => false
- | PBot => true
- end
-with nf {n} (a : PTm n) :=
- match a with
- | VarPTm i => true
- | PApp a b => ne a && nf b
- | PAbs a => nf a
- | PUniv _ => true
- | PProj _ a => ne a
- | PPair a b => nf a && nf b
- | PConst _ => true
- | PBot => true
-end.
-
-Lemma ne_nf n a : @ne n a -> nf a.
-Proof. elim : a => //=. Qed.
-
-Definition wn {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ nf b.
-Definition wne {n} (a : PTm n) := exists b, rtc RPar'.R a b /\ ne b.
-
-(* Weakly neutral implies weakly normal *)
-Lemma wne_wn n a : @wne n a -> wn a.
-Proof. sfirstorder use:ne_nf. Qed.
-
-(* Normal implies weakly normal *)
-Lemma nf_wn n v : @nf n v -> wn v.
-Proof. sfirstorder ctrs:rtc. Qed.
-
-Lemma nf_refl n (a b : PTm n) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a).
-Proof.
- elim : a b /h => //=; solve [hauto b:on].
-Qed.
-
-Lemma ne_nf_ren n m (a : PTm n) (ξ : fin n -> fin m) :
- (ne a <-> ne (ren_PTm ξ a)) /\ (nf a <-> nf (ren_PTm ξ a)).
-Proof.
- move : m ξ. elim : n / a => //=; solve [hauto b:on].
-Qed.
-
-Lemma wne_app n (a b : PTm n) :
- wne a -> wn b -> wne (PApp a b).
-Proof.
- move => [a0 [? ?]] [b0 [? ?]].
- exists (PApp a0 b0). hauto b:on drew:off use:RPars'.AppCong.
-Qed.
-
-Lemma wn_abs n a (h : wn a) : @wn n (PAbs a).
-Proof.
- move : h => [v [? ?]].
- exists (PAbs v).
- eauto using RPars'.AbsCong.
-Qed.
-
-Lemma wn_pair n (a b : PTm n) : wn a -> wn b -> wn (PPair a b).
-Proof.
- move => [a0 [? ?]] [b0 [? ?]].
- exists (PPair a0 b0).
- hauto lqb:on use:RPars'.PairCong.
-Qed.
-
-Lemma wne_proj n p (a : PTm n) : wne a -> wne (PProj p a).
-Proof.
- move => [a0 [? ?]].
- exists (PProj p a0). hauto lqb:on use:RPars'.ProjCong.
-Qed.
-
-Create HintDb nfne.
-#[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
-
-Lemma ne_nf_antiren n m (a : PTm n) (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- (ne (subst_PTm ρ a) -> ne a) /\ (nf (subst_PTm ρ a) -> nf a).
-Proof.
- move : m ρ. elim : n / a => //;
- hauto b:on drew:off use:RPar.var_or_const_up.
-Qed.
-
-Lemma wn_antirenaming n m a (ρ : fin n -> PTm m) :
- (forall i, var_or_const (ρ i)) ->
- wn (subst_PTm ρ a) -> wn a.
-Proof.
- rewrite /wn => hρ.
- move => [v [rv nfv]].
- move /RPars'.antirenaming : rv.
- move /(_ hρ) => [b [hb ?]]. subst.
- exists b. split => //=.
- move : nfv.
- by eapply ne_nf_antiren.
-Qed.
-
-Lemma ext_wn n (a : PTm n) :
- wn (PApp a PBot) ->
- wn a.
-Proof.
- move E : (PApp a (PBot)) => a0 [v [hr hv]].
- move : a E.
- move : hv.
- elim : a0 v / hr.
- - hauto q:on inv:PTm ctrs:rtc b:on db: nfne.
- - move => a0 a1 a2 hr0 hr1 ih hnfa2.
- move /(_ hnfa2) in ih.
- move => a.
- case : a0 hr0=>// => b0 b1.
- elim /RPar'.inv=>// _.
- + move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
- have ? : b3 = (PBot) by hauto lq:on inv:RPar'.R. subst.
- suff : wn (PAbs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
- have : wn (subst_PTm (scons (PBot) VarPTm) a3) by sfirstorder.
- move => h. apply wn_abs.
- move : h. apply wn_antirenaming.
- hauto lq:on rew:off inv:nat.
- + hauto q:on inv:RPar'.R ctrs:rtc b:on.
-Qed.
-
-Module Join.
- Lemma ProjCong p n (a0 a1 : PTm n) :
- join a0 a1 ->
- join (PProj p a0) (PProj p a1).
- Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed.
-
- Lemma PairCong n (a0 a1 b0 b1 : PTm n) :
- join a0 a1 ->
- join b0 b1 ->
- join (PPair a0 b0) (PPair a1 b1).
- Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed.
-
- Lemma AppCong n (a0 a1 b0 b1 : PTm n) :
- join a0 a1 ->
- join b0 b1 ->
- join (PApp a0 b0) (PApp a1 b1).
- Proof. hauto lq:on use:Pars.AppCong. Qed.
-
- Lemma AbsCong n (a b : PTm (S n)) :
- join a b ->
- join (PAbs a) (PAbs b).
- Proof. hauto lq:on use:Pars.AbsCong. Qed.
-
- Lemma renaming n m (a b : PTm n) (ξ : fin n -> fin m) :
- join a b -> join (ren_PTm ξ a) (ren_PTm ξ b).
- Proof.
- induction 1; hauto lq:on use:Pars.renaming.
- Qed.
-
- Lemma weakening n (a b : PTm n) :
- join a b -> join (ren_PTm shift a) (ren_PTm shift b).
- Proof.
- apply renaming.
- Qed.
-
- Lemma FromPar n (a b : PTm n) :
- Par.R a b ->
- join a b.
- Proof.
- hauto lq:on ctrs:rtc use:rtc_once.
- Qed.
-End Join.
-
-Lemma abs_eq n a (b : PTm n) :
- join (PAbs a) b <-> join a (PApp (ren_PTm shift b) (VarPTm var_zero)).
-Proof.
- split.
- - move => /Join.weakening h.
- have {h} : join (PApp (ren_PTm shift (PAbs a)) (VarPTm var_zero)) (PApp (ren_PTm shift b) (VarPTm var_zero))
- by hauto l:on use:Join.AppCong, join_refl.
- simpl.
- move => ?. apply : join_transitive; eauto.
- apply join_symmetric. apply Join.FromPar.
- apply : Par.AppAbs'; eauto using Par.refl. by asimpl.
- - move /Join.AbsCong.
- move /join_transitive. apply.
- apply join_symmetric. apply Join.FromPar. apply Par.AppEta. apply Par.refl.
-Qed.
-
-Lemma pair_eq n (a0 a1 b : PTm n) :
- join (PPair a0 a1) b <-> join a0 (PProj PL b) /\ join a1 (PProj PR b).
-Proof.
- split.
- - move => h.
- have /Join.ProjCong {}h := h.
- have h0 : forall p, join (if p is PL then a0 else a1) (PProj p (PPair a0 a1))
- by hauto lq:on use:join_symmetric, Join.FromPar, Par.ProjPair', Par.refl.
- hauto lq:on rew:off use:join_transitive, join_symmetric.
- - move => [h0 h1].
- move : h0 h1.
- move : Join.PairCong; repeat move/[apply].
- move /join_transitive. apply. apply join_symmetric.
- apply Join.FromPar. hauto lq:on ctrs:Par.R use:Par.refl.
-Qed.
-
-Lemma join_pair_inj n (a0 a1 b0 b1 : PTm n) :
- join (PPair a0 a1) (PPair b0 b1) <-> join a0 b0 /\ join a1 b1.
-Proof.
- split; last by hauto lq:on use:Join.PairCong.
- move /pair_eq => [h0 h1].
- have : join (PProj PL (PPair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
- have : join (PProj PR (PPair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
- eauto using join_transitive.
-Qed.
diff --git a/theories/logrel.v b/theories/logrel.v
index b9c854d..9a8f1af 100644
--- a/theories/logrel.v
+++ b/theories/logrel.v
@@ -1,26 +1,23 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax fp_red compile.
+Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
+Require Import fp_red.
From Hammer Require Import Tactics.
From Equations Require Import Equations.
Require Import ssreflect ssrbool.
Require Import Logic.PropExtensionality (propositional_extensionality).
From stdpp Require Import relations (rtc(..), rtc_subrel).
Import Psatz.
-
-Definition ProdSpace {n} (PA : Tm n -> Prop)
- (PF : Tm n -> (Tm n -> Prop) -> Prop) b : Prop :=
+Definition ProdSpace (PA : Tm 0 -> Prop)
+ (PF : Tm 0 -> (Tm 0 -> Prop) -> Prop) b : Prop :=
forall a PB, PA a -> PF a PB -> PB (App b a).
-Definition SumSpace {n} (PA : Tm n -> Prop)
- (PF : Tm n -> (Tm n -> Prop) -> Prop) t : Prop :=
- wne t \/ exists a b, rtc RPar'.R t (Pair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).
+Definition SumSpace (PA : Tm 0 -> Prop)
+ (PF : Tm 0 -> (Tm 0 -> Prop) -> Prop) t : Prop :=
+ exists a b, rtc RPar.R t (Pair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).
-Definition BindSpace {n} p := if p is TPi then @ProdSpace n else SumSpace.
+Definition BindSpace p := if p is TPi then ProdSpace else SumSpace.
Reserved Notation "⟦ A ⟧ i ;; I ↘ S" (at level 70).
-Inductive InterpExt {n} i (I : nat -> Tm n -> Prop) : Tm n -> (Tm n -> Prop) -> Prop :=
-| InterpExt_Ne A :
- ne A ->
- ⟦ A ⟧ i ;; I ↘ wne
+Inductive InterpExt i (I : nat -> Tm 0 -> Prop) : Tm 0 -> (Tm 0 -> Prop) -> Prop :=
| InterpExt_Bind p A B PA PF :
⟦ A ⟧ i ;; I ↘ PA ->
(forall a, PA a -> exists PB, PF a PB) ->
@@ -32,12 +29,12 @@ Inductive InterpExt {n} i (I : nat -> Tm n -> Prop) : Tm n -> (Tm n -> Prop) ->
⟦ Univ j ⟧ i ;; I ↘ (I j)
| InterpExt_Step A A0 PA :
- RPar'.R A A0 ->
+ RPar.R A A0 ->
⟦ A0 ⟧ i ;; I ↘ PA ->
⟦ A ⟧ i ;; I ↘ PA
where "⟦ A ⟧ i ;; I ↘ S" := (InterpExt i I A S).
-Lemma InterpExt_Univ' n i I j (PF : Tm n -> Prop) :
+Lemma InterpExt_Univ' i I j (PF : Tm 0 -> Prop) :
PF = I j ->
j < i ->
⟦ Univ j ⟧ i ;; I ↘ PF.
@@ -45,29 +42,28 @@ Proof. hauto lq:on ctrs:InterpExt. Qed.
Infix "<?" := Compare_dec.lt_dec (at level 60).
-Equations InterpUnivN n (i : nat) : Tm n -> (Tm n -> Prop) -> Prop by wf i lt :=
- InterpUnivN n i := @InterpExt n i
+Equations InterpUnivN (i : nat) : Tm 0 -> (Tm 0 -> Prop) -> Prop by wf i lt :=
+ InterpUnivN i := @InterpExt i
(fun j A =>
match j <? i with
- | left _ => exists PA, InterpUnivN n j A PA
+ | left _ => exists PA, InterpUnivN j A PA
| right _ => False
end).
-Arguments InterpUnivN {n}.
+Arguments InterpUnivN .
-Lemma InterpExt_lt_impl n i I I' A (PA : Tm n -> Prop) :
+Lemma InterpExt_lt_impl i I I' A (PA : Tm 0 -> Prop) :
(forall j, j < i -> I j = I' j) ->
⟦ A ⟧ i ;; I ↘ PA ->
⟦ A ⟧ i ;; I' ↘ PA.
Proof.
move => hI h.
elim : A PA /h.
- - hauto q:on ctrs:InterpExt.
- hauto lq:on rew:off ctrs:InterpExt.
- hauto q:on ctrs:InterpExt.
- hauto lq:on ctrs:InterpExt.
Qed.
-Lemma InterpExt_lt_eq n i I I' A (PA : Tm n -> Prop) :
+Lemma InterpExt_lt_eq i I I' A (PA : Tm 0 -> Prop) :
(forall j, j < i -> I j = I' j) ->
⟦ A ⟧ i ;; I ↘ PA =
⟦ A ⟧ i ;; I' ↘ PA.
@@ -79,8 +75,8 @@ Qed.
Notation "⟦ A ⟧ i ↘ S" := (InterpUnivN i A S) (at level 70).
-Lemma InterpUnivN_nolt n i :
- @InterpUnivN n i = @InterpExt n i (fun j (A : Tm n) => exists PA, ⟦ A ⟧ j ↘ PA).
+Lemma InterpUnivN_nolt i :
+ InterpUnivN i = InterpExt i (fun j (A : Tm 0) => exists PA, ⟦ A ⟧ j ↘ PA).
Proof.
simp InterpUnivN.
extensionality A. extensionality PA.
@@ -93,12 +89,12 @@ Qed.
#[export]Hint Rewrite @InterpUnivN_nolt : InterpUniv.
Lemma RPar_substone n (a b : Tm (S n)) (c : Tm n):
- RPar'.R a b -> RPar'.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
- Proof. hauto l:on inv:option use:RPar'.substing, RPar'.refl. Qed.
+ RPar.R a b -> RPar.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
+ Proof. hauto l:on inv:option use:RPar.substing, RPar.refl. Qed.
-Lemma InterpExt_Bind_inv n p i I (A : Tm n) B P
+Lemma InterpExt_Bind_inv p i I (A : Tm 0) B P
(h : ⟦ TBind p A B ⟧ i ;; I ↘ P) :
- exists (PA : Tm n -> Prop) (PF : Tm n -> (Tm n -> Prop) -> Prop),
+ exists (PA : Tm 0 -> Prop) (PF : Tm 0 -> (Tm 0 -> Prop) -> Prop),
⟦ A ⟧ i ;; I ↘ PA /\
(forall a, PA a -> exists PB, PF a PB) /\
(forall a PB, PF a PB -> ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) /\
@@ -107,35 +103,24 @@ Proof.
move E : (TBind p A B) h => T h.
move : A B E.
elim : T P / h => //.
- - move => //= *. scongruence.
- hauto l:on.
- move => A A0 PA hA hA0 hPi A1 B ?. subst.
- elim /RPar'.inv : hA => //= _ p0 A2 A3 B0 B1 hA1 hB0 [*]. subst.
+ elim /RPar.inv : hA => //= _ p0 A2 A3 B0 B1 hA1 hB0 [*]. subst.
hauto lq:on ctrs:InterpExt use:RPar_substone.
Qed.
-Lemma InterpExt_Ne_inv n i A I P
- (h : ⟦ A : Tm n ⟧ i ;; I ↘ P) :
- ne A ->
- P = wne.
-Proof.
- elim : A P / h => //=.
- qauto l:on ctrs:prov inv:prov use:nf_refl.
-Qed.
-
-Lemma InterpExt_Univ_inv n i I j P
- (h : ⟦ Univ j : Tm n ⟧ i ;; I ↘ P) :
+Lemma InterpExt_Univ_inv i I j P
+ (h : ⟦ Univ j ⟧ i ;; I ↘ P) :
P = I j /\ j < i.
Proof.
move : h.
move E : (Univ j) => T h. move : j E.
elim : T P /h => //.
- - move => //= *. scongruence.
- hauto l:on.
- - hauto lq:on rew:off inv:RPar'.R.
+ - hauto lq:on rew:off inv:RPar.R.
Qed.
-Lemma InterpExt_Bind_nopf n p i I (A : Tm n) B PA :
+Lemma InterpExt_Bind_nopf p i I (A : Tm 0) B PA :
⟦ A ⟧ i ;; I ↘ PA ->
(forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) ->
⟦ TBind p A B ⟧ i ;; I ↘ (BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB)).
@@ -143,7 +128,7 @@ Proof.
move => h0 h1. apply InterpExt_Bind =>//.
Qed.
-Lemma InterpUnivN_Fun_nopf n p i (A : Tm n) B PA :
+Lemma InterpUnivN_Fun_nopf p i (A : Tm 0) B PA :
⟦ A ⟧ i ↘ PA ->
(forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) ->
⟦ TBind p A B ⟧ i ↘ (BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB)).
@@ -151,7 +136,7 @@ Proof.
hauto l:on use:InterpExt_Bind_nopf rew:db:InterpUniv.
Qed.
-Lemma InterpExt_cumulative n i j I (A : Tm n) PA :
+Lemma InterpExt_cumulative i j I (A : Tm 0) PA :
i <= j ->
⟦ A ⟧ i ;; I ↘ PA ->
⟦ A ⟧ j ;; I ↘ PA.
@@ -161,87 +146,61 @@ Proof.
hauto l:on ctrs:InterpExt solve+:(by lia).
Qed.
-Lemma InterpUnivN_cumulative n i (A : Tm n) PA :
+Lemma InterpUnivN_cumulative i (A : Tm 0) PA :
⟦ A ⟧ i ↘ PA -> forall j, i <= j ->
⟦ A ⟧ j ↘ PA.
Proof.
hauto l:on rew:db:InterpUniv use:InterpExt_cumulative.
Qed.
-Lemma InterpExt_preservation n i I (A : Tm n) B P (h : InterpExt i I A P) :
- RPar'.R A B ->
+Lemma InterpExt_preservation i I (A : Tm 0) B P (h : InterpExt i I A P) :
+ RPar.R A B ->
⟦ B ⟧ i ;; I ↘ P.
Proof.
move : B.
elim : A P / h; auto.
- - hauto lq:on use:nf_refl ctrs:InterpExt.
- move => p A B PA PF hPA ihPA hPB hPB' ihPB T hT.
- elim /RPar'.inv : hT => //.
+ elim /RPar.inv : hT => //.
move => hPar p0 A0 A1 B0 B1 h0 h1 [? ?] ? ?; subst.
apply InterpExt_Bind; auto => a PB hPB0.
apply : ihPB; eauto.
- sfirstorder use:RPar'.cong, RPar'.refl.
- - hauto lq:on inv:RPar'.R ctrs:InterpExt.
+ sfirstorder use:RPar.cong, RPar.refl.
+ - hauto lq:on inv:RPar.R ctrs:InterpExt.
- move => A B P h0 h1 ih1 C hC.
- have [D [h2 h3]] := RPar'_diamond _ _ _ _ h0 hC.
+ have [D [h2 h3]] := RPar_diamond _ _ _ _ h0 hC.
hauto lq:on ctrs:InterpExt.
Qed.
-Lemma InterpUnivN_preservation n i (A : Tm n) B P (h : ⟦ A ⟧ i ↘ P) :
- RPar'.R A B ->
+Lemma InterpUnivN_preservation i (A : Tm 0) B P (h : ⟦ A ⟧ i ↘ P) :
+ RPar.R A B ->
⟦ B ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use: InterpExt_preservation. Qed.
-Lemma InterpExt_back_preservation_star n i I (A : Tm n) B P (h : ⟦ B ⟧ i ;; I ↘ P) :
- rtc RPar'.R A B ->
+Lemma InterpExt_back_preservation_star i I (A : Tm 0) B P (h : ⟦ B ⟧ i ;; I ↘ P) :
+ rtc RPar.R A B ->
⟦ A ⟧ i ;; I ↘ P.
Proof. induction 1; hauto l:on ctrs:InterpExt. Qed.
-Lemma InterpExt_preservation_star n i I (A : Tm n) B P (h : ⟦ A ⟧ i ;; I ↘ P) :
- rtc RPar'.R A B ->
+Lemma InterpExt_preservation_star i I (A : Tm 0) B P (h : ⟦ A ⟧ i ;; I ↘ P) :
+ rtc RPar.R A B ->
⟦ B ⟧ i ;; I ↘ P.
Proof. induction 1; hauto l:on use:InterpExt_preservation. Qed.
-Lemma InterpUnivN_preservation_star n i (A : Tm n) B P (h : ⟦ A ⟧ i ↘ P) :
- rtc RPar'.R A B ->
+Lemma InterpUnivN_preservation_star i (A : Tm 0) B P (h : ⟦ A ⟧ i ↘ P) :
+ rtc RPar.R A B ->
⟦ B ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use:InterpExt_preservation_star. Qed.
-Lemma InterpUnivN_back_preservation_star n i (A : Tm n) B P (h : ⟦ B ⟧ i ↘ P) :
- rtc RPar'.R A B ->
+Lemma InterpUnivN_back_preservation_star i (A : Tm 0) B P (h : ⟦ B ⟧ i ↘ P) :
+ rtc RPar.R A B ->
⟦ A ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use:InterpExt_back_preservation_star. Qed.
-Function hfb {n} (A : Tm n) :=
- match A with
- | TBind _ _ _ => true
- | Univ _ => true
- | _ => ne A
- end.
-
-Inductive hfb_case {n} : Tm n -> Prop :=
-| hfb_bind p A B :
- hfb_case (TBind p A B)
-| hfb_univ i :
- hfb_case (Univ i)
-| hfb_ne A :
- ne A ->
- hfb_case A.
-
-Derive Dependent Inversion hfb_inv with (forall n (a : Tm n), hfb_case a) Sort Prop.
-
-Lemma ne_hfb {n} (A : Tm n) : ne A -> hfb A.
-Proof. case : A => //=. Qed.
-
-Lemma hfb_caseP {n} (A : Tm n) : hfb A -> hfb_case A.
-Proof. hauto lq:on ctrs:hfb_case inv:Tm use:ne_hfb. Qed.
-
-Lemma InterpExtInv n i I (A : Tm n) PA :
+Lemma InterpExtInv i I (A : Tm 0) PA :
⟦ A ⟧ i ;; I ↘ PA ->
- exists B, hfb B /\ rtc RPar'.R A B /\ ⟦ B ⟧ i ;; I ↘ PA.
+ exists B, hfb B /\ rtc RPar.R A B /\ ⟦ B ⟧ i ;; I ↘ PA.
Proof.
move => h. elim : A PA /h.
- - hauto q:on ctrs:InterpExt, rtc use:ne_hfb.
- move => p A B PA PF hPA _ hPF hPF0 _.
exists (TBind p A B). repeat split => //=.
apply rtc_refl.
@@ -251,26 +210,17 @@ Proof.
- hauto lq:on ctrs:rtc.
Qed.
-Lemma RPar'_Par n (A B : Tm n) :
- RPar'.R A B ->
- Par.R A B.
-Proof. induction 1; hauto lq:on ctrs:Par.R. Qed.
-
-Lemma RPar's_Pars n (A B : Tm n) :
- rtc RPar'.R A B ->
+Lemma RPars_Pars (A B : Tm 0) :
+ rtc RPar.R A B ->
rtc Par.R A B.
-Proof. hauto lq:on use:RPar'_Par, rtc_subrel. Qed.
+Proof. hauto lq:on use:RPar_Par, rtc_subrel. Qed.
-Lemma RPar's_join n (A B : Tm n) :
- rtc RPar'.R A B -> Join.R A B.
-Proof.
- rewrite /Join.R => h.
- have {}h : rtc RPar'.R (Compile.F A) (Compile.F B) by eauto using compile_rpars.
- rewrite /join. eauto using RPar's_Pars, rtc_refl.
-Qed.
+Lemma RPars_join (A B : Tm 0) :
+ rtc RPar.R A B -> join A B.
+Proof. hauto lq:on ctrs:rtc use:RPars_Pars. Qed.
-Lemma bindspace_iff n p (PA : Tm n -> Prop) PF PF0 b :
- (forall (a : Tm n) (PB PB0 : Tm n -> Prop), PF a PB -> PF0 a PB0 -> PB = PB0) ->
+Lemma bindspace_iff p (PA : Tm 0 -> Prop) PF PF0 b :
+ (forall (a : Tm 0) (PB PB0 : Tm 0 -> Prop), PF a PB -> PF0 a PB0 -> PB = PB0) ->
(forall a, PA a -> exists PB, PF a PB) ->
(forall a, PA a -> exists PB0, PF0 a PB0) ->
(BindSpace p PA PF b <-> BindSpace p PA PF0 b).
@@ -291,105 +241,23 @@ Proof.
hauto lq:on rew:off.
Qed.
-Lemma ne_prov_inv n (a : Tm n) :
- ne a -> (exists i, prov (VarTm i) a) \/ prov Bot a.
-Proof.
- elim : n /a => //=.
- - hauto lq:on ctrs:prov.
- - hauto lq:on rew:off ctrs:prov b:on.
- - hauto lq:on ctrs:prov.
- - move => n k.
- have : @prov n Bot Bot by auto using P_Bot.
- eauto.
-Qed.
-
-Lemma ne_pars_inv n (a b : Tm n) :
- ne a -> rtc Par.R a b -> (exists i, prov (VarTm i) b) \/ prov Bot b.
-Proof.
- move /ne_prov_inv.
- sfirstorder use:prov_pars.
-Qed.
-
-Lemma ne_pars_extract n (a b : Tm n) :
- ne a -> rtc Par.R a b -> (exists i, extract b = (VarTm i)) \/ extract b = Bot.
-Proof. hauto lq:on rew:off use:ne_pars_inv, prov_extract. Qed.
-
-Lemma const_pars_extract n k b :
- rtc Par.R (Const k : Tm n) b -> extract b = Const k.
-Proof. hauto l:on use:pars_const_inv, prov_extract. Qed.
-
-Lemma compile_ne n (a : Tm n) :
- ne a = ne (Compile.F a) /\ nf a = nf (Compile.F a).
-Proof.
- elim : n / a => //=; sfirstorder b:on.
-Qed.
-
-Lemma join_univ_pi_contra n p (A : Tm n) B i :
- Join.R (TBind p A B) (Univ i) -> False.
-Proof.
- rewrite /Join.R /=.
- rewrite !pair_eq.
- move => [[h _ ]_ ].
- move : h => [C [h0 h1]].
- have ? : extract C = Const p by hauto l:on use:pars_const_inv.
- have h : prov (Univ i : Tm n) (Proj PL (Proj PL (Univ i))) by hauto lq:on ctrs:prov.
- have {h} : prov (Univ i) C by eauto using prov_pars.
- move /prov_extract=>/=. congruence.
-Qed.
-
-Lemma join_bind_ne_contra n p (A : Tm n) B C :
- ne C ->
- Join.R (TBind p A B) C -> False.
-Proof.
- rewrite /Join.R. move => hC /=.
- rewrite !pair_eq.
- move => [[[D [h0 h1]] _] _].
- have {}hC : ne (Compile.F C) by hauto lq:on use:compile_ne.
- have {}hC : ne (Proj PL (Proj PL (Compile.F C))) by scongruence.
- have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
- have : extract D = Const p by eauto using const_pars_extract.
- sfirstorder.
-Qed.
-
-Lemma join_univ_ne_contra n i C :
- ne C ->
- Join.R (Univ i : Tm n) C -> False.
-Proof.
- move => hC [D [h0 h1]].
- move /pars_univ_inv : h0 => ?.
- have : (exists i, extract D = (VarTm i)) \/ exists k, extract D =(Const k) by hauto q:on use:ne_pars_extract, compile_ne.
- sfirstorder.
-Qed.
-
-#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra Join.symmetric Join.transitive : join.
-
-Lemma InterpExt_Join n i I (A B : Tm n) PA PB :
+Lemma InterpExt_Join i I (A B : Tm 0) PA PB :
⟦ A ⟧ i ;; I ↘ PA ->
⟦ B ⟧ i ;; I ↘ PB ->
- Join.R A B ->
+ join A B ->
PA = PB.
Proof.
move => h. move : B PB. elim : A PA /h.
- - move => A hA B PB /InterpExtInv.
- move => [B0 []].
- move /hfb_caseP. elim/hfb_inv => _.
- + move => p A0 B1 ? [/RPar's_join h0 h1] h2. subst. exfalso.
- eauto with join.
- + move => ? ? [/RPar's_join *]. subst. exfalso.
- eauto with join.
- + hauto lq:on use:InterpExt_Ne_inv.
- move => p A B PA PF hPA ihPA hTot hRes ihPF U PU /InterpExtInv.
move => [B0 []].
- move /hfb_caseP.
- elim /hfb_inv => _.
- rename B0 into B00.
- + move => p0 A0 B0 ? [hr hPi]. subst.
+ case : B0 => //=.
+ + move => p0 A0 B0 _ [hr hPi].
move /InterpExt_Bind_inv : hPi.
move => [PA0][PF0][hPA0][hTot0][hRes0]?. subst.
move => hjoin.
- have{}hr : Join.R U (TBind p0 A0 B0) by auto using RPar's_join.
- have hj : Join.R (TBind p A B) (TBind p0 A0 B0) by eauto using Join.transitive.
- have {hj} : p0 = p /\ Join.R A A0 /\ Join.R B B0 by hauto l:on use:Join.BindInj.
+ have{}hr : join U (TBind p0 A0 B0) by auto using RPars_join.
+ have hj : join (TBind p A B) (TBind p0 A0 B0) by eauto using join_transitive.
+ have {hj} : p0 = p /\ join A A0 /\ join B B0 by hauto l:on use:join_pi_inj.
move => [? [h0 h1]]. subst.
have ? : PA0 = PA by hauto l:on. subst.
rewrite /ProdSpace.
@@ -398,70 +266,66 @@ Proof.
apply bindspace_iff; eauto.
move => a PB PB0 hPB hPB0.
apply : ihPF; eauto.
- rewrite /Join.R.
- rewrite -!Compile.substing.
- hauto l:on use:join_substing.
- + move => j ?. subst.
+ by apply join_substing.
+ + move => j _.
move => [h0 h1] h.
- have ? : Join.R U (Univ j) by eauto using RPar's_join.
- have : Join.R (TBind p A B) (Univ j) by eauto using Join.transitive.
+ have ? : join U (Univ j) by eauto using RPars_join.
+ have : join (TBind p A B) (Univ j) by eauto using join_transitive.
move => ?. exfalso.
- hauto l: on use: join_univ_pi_contra.
- + move => A0 ? ? [/RPar's_join ?]. subst.
- move => _ ?. exfalso. eauto with join.
+ eauto using join_univ_pi_contra.
- move => j ? B PB /InterpExtInv.
- move => [? []]. move/hfb_caseP.
- elim /hfb_inv => //= _.
+ move => [+ []]. case => //=.
+ move => p A0 B0 _ [].
- move /RPar's_join => *.
- exfalso. eauto with join.
- + move => m _ [/RPar's_join h0 + h1].
- have /join_univ_inj {h0 h1} ? : Join.R (Univ j : Tm n) (Univ m) by eauto using Join.transitive.
+ move /RPars_join => *.
+ have ? : join (TBind p A0 B0) (Univ j) by eauto using join_symmetric, join_transitive.
+ exfalso.
+ eauto using join_univ_pi_contra.
+ + move => m _ [/RPars_join h0 + h1].
+ have /join_univ_inj {h0 h1} ? : join (Univ j : Tm 0) (Univ m) by eauto using join_transitive.
subst.
move /InterpExt_Univ_inv. firstorder.
- + move => A ? ? [/RPar's_join] *. subst. exfalso. eauto with join.
- move => A A0 PA h.
- have /Join.symmetric {}h : Join.R A A0 by hauto lq:on ctrs:rtc use:RPar's_join, relations.rtc_once.
- eauto with join.
+ have /join_symmetric {}h : join A A0 by hauto lq:on ctrs:rtc use:RPar_Par, relations.rtc_once.
+ eauto using join_transitive.
Qed.
-Lemma InterpUniv_Join n i (A B : Tm n) PA PB :
+Lemma InterpUniv_Join i (A B : Tm 0) PA PB :
⟦ A ⟧ i ↘ PA ->
⟦ B ⟧ i ↘ PB ->
- Join.R A B ->
+ join A B ->
PA = PB.
Proof. hauto l:on use:InterpExt_Join rew:db:InterpUniv. Qed.
-Lemma InterpUniv_Bind_inv n p i (A : Tm n) B P
+Lemma InterpUniv_Bind_inv p i (A : Tm 0) B P
(h : ⟦ TBind p A B ⟧ i ↘ P) :
- exists (PA : Tm n -> Prop) (PF : Tm n -> (Tm n -> Prop) -> Prop),
+ exists (PA : Tm 0 -> Prop) (PF : Tm 0 -> (Tm 0 -> Prop) -> Prop),
⟦ A ⟧ i ↘ PA /\
(forall a, PA a -> exists PB, PF a PB) /\
(forall a PB, PF a PB -> ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) /\
P = BindSpace p PA PF.
Proof. hauto l:on use:InterpExt_Bind_inv rew:db:InterpUniv. Qed.
-Lemma InterpUniv_Univ_inv n i j P
+Lemma InterpUniv_Univ_inv i j P
(h : ⟦ Univ j ⟧ i ↘ P) :
- P = (fun (A : Tm n) => exists PA, ⟦ A ⟧ j ↘ PA) /\ j < i.
+ P = (fun (A : Tm 0) => exists PA, ⟦ A ⟧ j ↘ PA) /\ j < i.
Proof. hauto l:on use:InterpExt_Univ_inv rew:db:InterpUniv. Qed.
-Lemma InterpExt_Functional n i I (A B : Tm n) PA PB :
+Lemma InterpExt_Functional i I (A B : Tm 0) PA PB :
⟦ A ⟧ i ;; I ↘ PA ->
⟦ A ⟧ i ;; I ↘ PB ->
PA = PB.
Proof. hauto use:InterpExt_Join, join_refl. Qed.
-Lemma InterpUniv_Functional n i (A : Tm n) PA PB :
+Lemma InterpUniv_Functional i (A : Tm 0) PA PB :
⟦ A ⟧ i ↘ PA ->
⟦ A ⟧ i ↘ PB ->
PA = PB.
Proof. hauto use:InterpExt_Functional rew:db:InterpUniv. Qed.
-Lemma InterpUniv_Join' n i j (A B : Tm n) PA PB :
+Lemma InterpUniv_Join' i j (A B : Tm 0) PA PB :
⟦ A ⟧ i ↘ PA ->
⟦ B ⟧ j ↘ PB ->
- Join.R A B ->
+ join A B ->
PA = PB.
Proof.
have [? ?] : i <= max i j /\ j <= max i j by lia.
@@ -471,16 +335,16 @@ Proof.
eauto using InterpUniv_Join.
Qed.
-Lemma InterpUniv_Functional' n i j A PA PB :
- ⟦ A : Tm n ⟧ i ↘ PA ->
+Lemma InterpUniv_Functional' i j A PA PB :
+ ⟦ A ⟧ i ↘ PA ->
⟦ A ⟧ j ↘ PB ->
PA = PB.
Proof.
hauto l:on use:InterpUniv_Join', join_refl.
Qed.
-Lemma InterpExt_Bind_inv_nopf i n I p A B P (h : ⟦TBind p A B ⟧ i ;; I ↘ P) :
- exists (PA : Tm n -> Prop),
+Lemma InterpExt_Bind_inv_nopf i I p A B P (h : ⟦TBind p A B ⟧ i ;; I ↘ P) :
+ exists (PA : Tm 0 -> Prop),
⟦ A ⟧ i ;; I ↘ PA /\
(forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) /\
P = BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB).
@@ -501,42 +365,34 @@ Proof.
split; hauto q:on use:InterpExt_Functional.
Qed.
-Lemma InterpUniv_Bind_inv_nopf n i p A B P (h : ⟦TBind p A B ⟧ i ↘ P) :
- exists (PA : Tm n -> Prop),
+Lemma InterpUniv_Bind_inv_nopf i p A B P (h : ⟦TBind p A B ⟧ i ↘ P) :
+ exists (PA : Tm 0 -> Prop),
⟦ A ⟧ i ↘ PA /\
(forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) /\
P = BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB).
Proof. hauto l:on use:InterpExt_Bind_inv_nopf rew:db:InterpUniv. Qed.
-Lemma InterpExt_back_clos n i I (A : Tm n) PA :
- (forall j, j < i -> forall a b, (RPar'.R a b) -> I j b -> I j a) ->
+Lemma InterpExt_back_clos i I (A : Tm 0) PA :
+ (forall j, forall a b, (RPar.R a b) -> I j b -> I j a) ->
⟦ A ⟧ i ;; I ↘ PA ->
- forall a b, (RPar'.R a b) ->
+ forall a b, (RPar.R a b) ->
PA b -> PA a.
Proof.
move => hI h.
elim : A PA /h.
- - hauto q:on ctrs:rtc unfold:wne.
- move => p A B PA PF hPA ihPA hTot hRes ihPF a b hr.
case : p => //=.
- + have : forall b0 b1 a, RPar'.R b0 b1 -> RPar'.R (App b0 a) (App b1 a)
- by hauto lq:on ctrs:RPar'.R use:RPar'.refl.
+ + have : forall b0 b1 a, RPar.R b0 b1 -> RPar.R (App b0 a) (App b1 a)
+ by hauto lq:on ctrs:RPar.R use:RPar.refl.
hauto lq:on rew:off unfold:ProdSpace.
+ hauto lq:on ctrs:rtc unfold:SumSpace.
- eauto.
- eauto.
Qed.
-Lemma InterpExt_back_clos_star n i I (A : Tm n) PA :
- (forall j, j < i -> forall a b, (RPar'.R a b) -> I j b -> I j a) ->
- ⟦ A ⟧ i ;; I ↘ PA ->
- forall a b, (rtc RPar'.R a b) ->
- PA b -> PA a.
-Proof. induction 3; hauto l:on use:InterpExt_back_clos. Qed.
-
-Lemma InterpUniv_back_clos n i (A : Tm n) PA :
+Lemma InterpUniv_back_clos i (A : Tm 0) PA :
⟦ A ⟧ i ↘ PA ->
- forall a b, (RPar'.R a b) ->
+ forall a b, (RPar.R a b) ->
PA b -> PA a.
Proof.
simp InterpUniv.
@@ -544,9 +400,9 @@ Proof.
hauto lq:on ctrs:rtc use:InterpUnivN_back_preservation_star.
Qed.
-Lemma InterpUniv_back_clos_star n i (A : Tm n) PA :
+Lemma InterpUniv_back_clos_star i (A : Tm 0) PA :
⟦ A ⟧ i ↘ PA ->
- forall a b, rtc RPar'.R a b ->
+ forall a b, rtc RPar.R a b ->
PA b -> PA a.
Proof.
move => h a b.
@@ -554,101 +410,30 @@ Proof.
hauto lq:on use:InterpUniv_back_clos.
Qed.
-Lemma pars'_wn {n} a b :
- rtc RPar'.R a b ->
- @wn n b ->
- wn a.
-Proof. sfirstorder unfold:wn use:@relations.rtc_transitive. Qed.
+Definition ρ_ok {n} Γ (ρ : fin n -> Tm 0) := forall i m PA,
+ ⟦ subst_Tm ρ (Γ i) ⟧ m ↘ PA -> PA (ρ i).
-(* P identifies a set of "reducibility candidates" *)
-Definition CR {n} (P : Tm n -> Prop) :=
- (forall a, P a -> wn a) /\
- (forall a, ne a -> P a).
-
-Lemma adequacy_ext i n I A PA
- (hI0 : forall j, j < i -> forall a b, (RPar'.R a b) -> I j b -> I j a)
- (hI : forall j, j < i -> CR (I j))
- (h : ⟦ A : Tm n ⟧ i ;; I ↘ PA) :
- CR PA /\ wn A.
-Proof.
- elim : A PA / h.
- - hauto unfold:wne use:wne_wn.
- - move => p A B PA PF hA hPA hTot hRes ihPF.
- rewrite /CR.
- have hb : PA Bot by firstorder.
- repeat split.
- + case : p => /=.
- * qauto l:on use:ext_wn unfold:ProdSpace, CR.
- * rewrite /SumSpace => a []; first by eauto with nfne.
- move => [q0][q1]*.
- have : wn q0 /\ wn q1 by hauto q:on.
- qauto l:on use:wn_pair, pars'_wn.
- + case : p => /=.
- * rewrite /ProdSpace.
- move => a ha c PB hc hPB.
- have hc' : wn c by sfirstorder.
- have : wne (App a c) by hauto lq:on use:wne_app ctrs:rtc.
- have h : (forall a, ne a -> PB a) by sfirstorder.
- suff : (forall a, wne a -> PB a) by hauto l:on.
- move => a0 [a1 [h0 h1]].
- eapply InterpExt_back_clos_star with (b := a1); eauto.
- * rewrite /SumSpace.
- move => a ha. left.
- sfirstorder ctrs:rtc.
- + have wnA : wn A by firstorder.
- apply wn_bind => //.
- apply wn_antirenaming with (ρ := scons Bot VarTm);first by hauto q:on inv:option.
- hauto lq:on.
- - hauto l:on.
- - hauto lq:on rew:off ctrs:rtc.
-Qed.
-
-Lemma adequacy i n A PA
- (h : ⟦ A : Tm n ⟧ i ↘ PA) :
- CR PA /\ wn A.
-Proof.
- move : i A PA h.
- elim /Wf_nat.lt_wf_ind => i ih A PA.
- simp InterpUniv.
- apply adequacy_ext.
- hauto lq:on ctrs:rtc use:InterpUnivN_back_preservation_star.
- hauto l:on use:InterpExt_Ne rew:db:InterpUniv.
-Qed.
-
-Lemma adequacy_wne i n A PA a : ⟦ A : Tm n ⟧ i ↘ PA -> wne a -> PA a.
-Proof. qauto l:on use:InterpUniv_back_clos_star, adequacy unfold:CR. Qed.
-
-Lemma adequacy_wn i n A PA (h : ⟦ A : Tm n ⟧ i ↘ PA) a : PA a -> wn a.
-Proof. hauto q:on use:adequacy. Qed.
-
-Definition ρ_ok {n} (Γ : fin n -> Tm n) (ρ : fin n -> Tm 0) := forall i k PA,
- ⟦ subst_Tm ρ (Γ i) ⟧ k ↘ PA -> PA (ρ i).
-
-Definition SemWt {n} Γ (a A : Tm n) := forall ρ, ρ_ok Γ ρ -> exists k PA, ⟦ subst_Tm ρ A ⟧ k ↘ PA /\ PA (subst_Tm ρ a).
+Definition SemWt {n} Γ (a A : Tm n) := forall ρ, ρ_ok Γ ρ -> exists m PA, ⟦ subst_Tm ρ A ⟧ m ↘ PA /\ PA (subst_Tm ρ a).
Notation "Γ ⊨ a ∈ A" := (SemWt Γ a A) (at level 70).
(* Semantic context wellformedness *)
Definition SemWff {n} Γ := forall (i : fin n), exists j, Γ ⊨ Γ i ∈ Univ j.
Notation "⊨ Γ" := (SemWff Γ) (at level 70).
-Lemma ρ_ok_bot n (Γ : fin n -> Tm n) :
- ρ_ok Γ (fun _ => Bot).
-Proof.
- rewrite /ρ_ok.
- hauto q:on use:adequacy.
-Qed.
+Lemma ρ_ok_nil ρ :
+ ρ_ok null ρ.
+Proof. rewrite /ρ_ok. inversion i; subst. Qed.
Lemma ρ_ok_cons n i (Γ : fin n -> Tm n) ρ a PA A :
⟦ subst_Tm ρ A ⟧ i ↘ PA -> PA a ->
ρ_ok Γ ρ ->
- ρ_ok (funcomp (ren_Tm shift) (scons A Γ)) (scons a ρ).
+ ρ_ok (funcomp (ren_Tm shift) (scons A Γ)) ((scons a ρ)).
Proof.
move => h0 h1 h2.
rewrite /ρ_ok.
move => j.
destruct j as [j|].
- move => m PA0. asimpl => ?.
- asimpl.
firstorder.
- move => m PA0. asimpl => h3.
have ? : PA0 = PA by eauto using InterpUniv_Functional'.
@@ -670,7 +455,7 @@ Proof.
rewrite /ρ_ok in hρ.
move => h.
rewrite /funcomp.
- apply hρ with (k := m').
+ apply hρ with (m := m').
move : h. rewrite -hξ.
by asimpl.
Qed.
@@ -695,17 +480,6 @@ Proof.
hauto lq:on inv:option unfold:renaming_ok.
Qed.
-Lemma SemWt_Wn n Γ (a : Tm n) A :
- Γ ⊨ a ∈ A ->
- wn a /\ wn A.
-Proof.
- move => h.
- have {}/h := ρ_ok_bot _ Γ => h.
- have h0 : wn (subst_Tm (fun _ : fin n => (Bot : Tm 0)) A) by hauto l:on use:adequacy.
- have h1 : wn (subst_Tm (fun _ : fin n => (Bot : Tm 0)) a)by hauto l:on use:adequacy_wn.
- move {h}. hauto lq:on use:wn_antirenaming.
-Qed.
-
Lemma SemWt_Univ n Γ (A : Tm n) i :
Γ ⊨ A ∈ Univ i <->
forall ρ, ρ_ok Γ ρ -> exists S, ⟦ subst_Tm ρ A ⟧ i ↘ S.
@@ -768,12 +542,12 @@ Qed.
Lemma ST_Conv n Γ (a : Tm n) A B i :
Γ ⊨ a ∈ A ->
Γ ⊨ B ∈ Univ i ->
- Join.R A B ->
+ join A B ->
Γ ⊨ a ∈ B.
Proof.
move => ha /SemWt_Univ h h0.
move => ρ hρ.
- have {}h0 : Join.R (subst_Tm ρ A) (subst_Tm ρ B) by eauto using Join.substing.
+ have {}h0 : join (subst_Tm ρ A) (subst_Tm ρ B) by eauto using join_substing.
move /ha : (hρ){ha} => [m [PA [h1 h2]]].
move /h : (hρ){h} => [S hS].
have ? : PA = S by eauto using InterpUniv_Join'. subst.
@@ -798,7 +572,7 @@ Proof.
intros (m & PB0 & hPB0 & hPB0').
replace PB0 with PB in * by hauto l:on use:InterpUniv_Functional'.
apply : InterpUniv_back_clos; eauto.
- apply : RPar'.AppAbs'; eauto using RPar'.refl.
+ apply : RPar.AppAbs'; eauto using RPar.refl.
by asimpl.
Qed.
@@ -830,7 +604,7 @@ Proof.
simpl in hPPi.
move /InterpUniv_Bind_inv_nopf : hPPi.
move => [PA [hPA [hTot ?]]]. subst=>/=.
- rewrite /SumSpace. right.
+ rewrite /SumSpace.
exists (subst_Tm ρ a), (subst_Tm ρ b).
split.
- hauto l:on use:Pars.substing.
@@ -852,25 +626,24 @@ Proof.
move : h0 => [S][h2][h3]?. subst.
move : h1 => /=.
rewrite /SumSpace.
- case; first by hauto lq:on use:adequacy_wne, wne_proj.
move => [a0 [b0 [h4 [h5 h6]]]].
exists m, S. split => //=.
- have {}h4 : rtc RPar'.R (Proj PL (subst_Tm ρ a)) (Proj PL (Pair a0 b0)) by eauto using RPars'.ProjCong.
- have ? : RPar'.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar'.refl, RPar'.ProjPair'.
- have : rtc RPar'.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
+ have {}h4 : rtc RPar.R (Proj PL (subst_Tm ρ a)) (Proj PL (Pair a0 b0)) by eauto using RPars.ProjCong.
+ have ? : RPar.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar.refl, RPar.ProjPair'.
+ have : rtc RPar.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
move => h.
apply : InterpUniv_back_clos_star; eauto.
Qed.
-Lemma substing_RPar' n m (A : Tm (S n)) ρ (B : Tm m) C :
- RPar'.R B C ->
- RPar'.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
-Proof. hauto lq:on inv:option use:RPar'.morphing, RPar'.refl. Qed.
+Lemma substing_RPar n m (A : Tm (S n)) ρ (B : Tm m) C :
+ RPar.R B C ->
+ RPar.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
+Proof. hauto lq:on inv:option use:RPar.morphing, RPar.refl. Qed.
-Lemma substing_RPar's n m (A : Tm (S n)) ρ (B : Tm m) C :
- rtc RPar'.R B C ->
- rtc RPar'.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
-Proof. induction 1; hauto lq:on ctrs:rtc use:substing_RPar'. Qed.
+Lemma substing_RPars n m (A : Tm (S n)) ρ (B : Tm m) C :
+ rtc RPar.R B C ->
+ rtc RPar.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
+Proof. induction 1; hauto lq:on ctrs:rtc use:substing_RPar. Qed.
Lemma ST_Proj2 n Γ (a : Tm n) A B :
Γ ⊨ a ∈ TBind TSig A B ->
@@ -881,156 +654,17 @@ Proof.
move : h0 => [S][h2][h3]?. subst.
move : h1 => /=.
rewrite /SumSpace.
- case.
- - move => h.
- have hp : forall p, wne (Proj p (subst_Tm ρ a)) by auto using wne_proj.
- have hp0 := hp PL. have hp1 := hp PR => {hp}.
- have : S (Proj PL (subst_Tm ρ a)) by hauto q:on use:adequacy_wne.
- move /h3 => [PB]. asimpl. hauto lq:on use:adequacy_wne.
- - move => [a0 [b0 [h4 [h5 h6]]]].
- specialize h3 with (1 := h5).
- move : h3 => [PB hPB].
- have hr : forall p, rtc RPar'.R (Proj p (subst_Tm ρ a)) (Proj p (Pair a0 b0)) by eauto using RPars'.ProjCong.
- have hrl : RPar'.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar'.ProjPair', RPar'.refl.
- have hrr : RPar'.R (Proj PR (Pair a0 b0)) b0 by hauto l:on use:RPar'.ProjPair', RPar'.refl.
- exists m, PB.
- asimpl. split.
- + have h : rtc RPar'.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
- have {}h : rtc RPar'.R (subst_Tm (scons (Proj PL (subst_Tm ρ a)) ρ) B) (subst_Tm (scons a0 ρ) B) by eauto using substing_RPar's.
- move : hPB. asimpl.
- eauto using InterpUnivN_back_preservation_star.
- + hauto lq:on use:@relations.rtc_r, InterpUniv_back_clos_star.
+ move => [a0 [b0 [h4 [h5 h6]]]].
+ specialize h3 with (1 := h5).
+ move : h3 => [PB hPB].
+ have hr : forall p, rtc RPar.R (Proj p (subst_Tm ρ a)) (Proj p (Pair a0 b0)) by eauto using RPars.ProjCong.
+ have hrl : RPar.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar.ProjPair', RPar.refl.
+ have hrr : RPar.R (Proj PR (Pair a0 b0)) b0 by hauto l:on use:RPar.ProjPair', RPar.refl.
+ exists m, PB.
+ asimpl. split.
+ - have h : rtc RPar.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
+ have {}h : rtc RPar.R (subst_Tm (scons (Proj PL (subst_Tm ρ a)) ρ) B) (subst_Tm (scons a0 ρ) B) by eauto using substing_RPars.
+ move : hPB. asimpl.
+ eauto using InterpUnivN_back_preservation_star.
+ - hauto lq:on use:@relations.rtc_r, InterpUniv_back_clos_star.
Qed.
-
-Lemma ne_nf_preservation n (a b : Tm n) : ERed.R b a -> (ne a -> ne b) /\ (nf a -> nf b).
-Proof.
- move => h. elim : n b a /h => //=.
- - move => n a.
- split => //=.
- hauto lqb:on use:ne_nf_ren db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
- - hauto lqb:on db:nfne.
-Qed.
-
-Fixpoint size_tm {n} (a : Tm n) :=
- match a with
- | VarTm _ => 1
- | TBind _ A B => 1 + Nat.add (size_tm A) (size_tm B)
- | Abs a => 1 + size_tm a
- | App a b => 1 + Nat.add (size_tm a) (size_tm b)
- | Proj p a => 1 + size_tm a
- | Pair a b => 1 + Nat.add (size_tm a) (size_tm b)
- | Bot => 1
- | Const _ => 1
- | Univ _ => 1
- end.
-
-Lemma size_tm_ren n m (ξ : fin n -> fin m) a : size_tm (ren_Tm ξ a) = size_tm a.
-Proof.
- move : m ξ. elim : n / a => //=; scongruence.
-Qed.
-
-#[export]Hint Rewrite size_tm_ren : size_tm.
-
-Lemma size_η_lt n (a b : Tm n) :
- ERed.R b a ->
- size_tm b < size_tm a.
-Proof.
- move => h. elim : b a / h => //=; hauto l:on rew:db:size_tm.
-Qed.
-
-Lemma ered_local_confluence n (a b c : Tm n) :
- ERed.R b a ->
- ERed.R c a ->
- exists d, rtc ERed.R d b /\ rtc ERed.R d c.
-Proof.
- move => h. move : c.
- elim : n b a / h => n.
- - move => a c.
- elim /ERed.inv => //= _.
- + move => ? ? [*]. subst.
- have : subst_Tm (scons Bot VarTm) (ren_Tm shift c) = (subst_Tm (scons Bot VarTm) (ren_Tm shift a))
- by congruence.
- asimpl => ?. subst.
- eauto using rtc_refl.
- + move => a0 a1 ha ? [*]. subst.
- elim /ERed.inv : ha => //= _.
- * move => a1 a2 b0 ha ? [*]. subst.
- have [a2 [h0 h1]] : exists a2, ERed.R a2 a /\ a1 = ren_Tm shift a2 by admit. subst.
- eexists. split; cycle 1.
- apply : relations.rtc_r; cycle 1.
- apply ERed.AppEta.
- apply rtc_refl.
- eauto using relations.rtc_once.
- * hauto q:on ctrs:rtc, ERed.R inv:ERed.R.
- - move => a c ha.
- elim /ERed.inv : ha => //= _.
- + hauto l:on.
- + move => a0 a1 b0 ha ? [*]. subst.
- elim /ERed.inv : ha => //= _.
- move => p a1 a2 ha ? [*]. subst.
- exists a1. split. by apply relations.rtc_once.
- apply : rtc_l. apply ERed.PairEta.
- apply : rtc_l. apply ERed.PairCong1. eauto using ERed.ProjCong.
- apply rtc_refl.
- + move => a0 b0 b1 ha ? [*]. subst.
- elim /ERed.inv : ha => //= _ p a0 a1 h ? [*]. subst.
- exists a0. split; first by apply relations.rtc_once.
- apply : rtc_l; first by apply ERed.PairEta.
- apply relations.rtc_once.
- hauto lq:on ctrs:ERed.R.
- - move => a0 a1 ha iha c.
- elim /ERed.inv => //= _.
- + move => a2 ? [*]. subst.
- elim /ERed.inv : ha => //=_.
- * move => a1 a2 b0 ha ? [*] {iha}. subst.
- have [a0 [h0 h1]] : exists a0, ERed.R a0 c /\ a1 = ren_Tm shift a0 by admit. subst.
- exists a0. split; last by apply relations.rtc_once.
- apply relations.rtc_once. apply ERed.AppEta.
- * hauto q:on inv:ERed.R.
- + hauto l:on use:EReds.AbsCong.
- - move => a0 a1 b ha iha c.
- elim /ERed.inv => //= _.
- + hauto lq:on ctrs:rtc use:EReds.AppCong.
- + hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
- - move => a b0 b1 hb ihb c.
- elim /ERed.inv => //=_.
- + move => a0 a1 a2 ha ? [*]. subst.
- move {ihb}. exists (App a0 b0).
- hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
- + hauto lq:on ctrs:rtc use:EReds.AppCong.
- - move => a0 a1 b ha iha c.
- elim /ERed.inv => //= _.
- + move => ? ?[*]. subst.
- elim /ERed.inv : ha => //= _ p a1 a2 h ? [*]. subst.
- exists a1. split; last by apply relations.rtc_once.
- apply : rtc_l. apply ERed.PairEta.
- apply relations.rtc_once. hauto lq:on ctrs:ERed.R.
- + hauto lq:on ctrs:rtc use:EReds.PairCong.
- + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
- - move => a b0 b1 hb hc c. elim /ERed.inv => //= _.
- + move => ? ? [*]. subst.
- elim /ERed.inv : hb => //= _ p a0 a1 ha ? [*]. subst.
- move {hc}.
- exists a0. split; last by apply relations.rtc_once.
- apply : rtc_l; first by apply ERed.PairEta.
- hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
- + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
- + hauto lq:on ctrs:rtc use:EReds.PairCong.
- - qauto l:on inv:ERed.R use:EReds.ProjCong.
- - move => p A0 A1 B hA ihA.
- move => c. elim/ERed.inv => //=.
- + hauto lq:on ctrs:rtc use:EReds.BindCong.
- + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
- - move => p A B0 B1 hB ihB c.
- elim /ERed.inv => //=.
- + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
- + hauto lq:on ctrs:rtc use:EReds.BindCong.
-Admitted.
diff --git a/theories/typing.v b/theories/typing.v
deleted file mode 100644
index d41facd..0000000
--- a/theories/typing.v
+++ /dev/null
@@ -1,251 +0,0 @@
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
-
-Reserved Notation "Γ ⊢ a ∈ A" (at level 70).
-Reserved Notation "Γ ⊢ a ≡ b ∈ A" (at level 70).
-Reserved Notation "Γ ⊢ A ≲ B" (at level 70).
-Reserved Notation "⊢ Γ" (at level 70).
-Inductive Wt : list PTm -> PTm -> PTm -> Prop :=
-| T_Var i Γ A :
- ⊢ Γ ->
- lookup i Γ A ->
- Γ ⊢ VarPTm i ∈ A
-
-| T_Bind Γ i p (A : PTm) (B : PTm) :
- Γ ⊢ A ∈ PUniv i ->
- cons A Γ ⊢ B ∈ PUniv i ->
- Γ ⊢ PBind p A B ∈ PUniv i
-
-| T_Abs Γ (a : PTm) A B i :
- Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
- (cons A Γ) ⊢ a ∈ B ->
- Γ ⊢ PAbs a ∈ PBind PPi A B
-
-| T_App Γ (b a : PTm) A B :
- Γ ⊢ b ∈ PBind PPi A B ->
- Γ ⊢ a ∈ A ->
- Γ ⊢ PApp b a ∈ subst_PTm (scons a VarPTm) B
-
-| T_Pair Γ (a b : PTm) A B i :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a ∈ A ->
- Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
- Γ ⊢ PPair a b ∈ PBind PSig A B
-
-| T_Proj1 Γ (a : PTm) A B :
- Γ ⊢ a ∈ PBind PSig A B ->
- Γ ⊢ PProj PL a ∈ A
-
-| T_Proj2 Γ (a : PTm) A B :
- Γ ⊢ a ∈ PBind PSig A B ->
- Γ ⊢ PProj PR a ∈ subst_PTm (scons (PProj PL a) VarPTm) B
-
-| T_Univ Γ i :
- ⊢ Γ ->
- Γ ⊢ PUniv i ∈ PUniv (S i)
-
-| T_Nat Γ i :
- ⊢ Γ ->
- Γ ⊢ PNat ∈ PUniv i
-
-| T_Zero Γ :
- ⊢ Γ ->
- Γ ⊢ PZero ∈ PNat
-
-| T_Suc Γ (a : PTm) :
- Γ ⊢ a ∈ PNat ->
- Γ ⊢ PSuc a ∈ PNat
-
-| T_Ind Γ P (a : PTm) b c i :
- cons PNat Γ ⊢ P ∈ PUniv i ->
- Γ ⊢ a ∈ PNat ->
- Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
- (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
- Γ ⊢ PInd P a b c ∈ subst_PTm (scons a VarPTm) P
-
-| T_Conv Γ (a : PTm) A B :
- Γ ⊢ a ∈ A ->
- Γ ⊢ A ≲ B ->
- Γ ⊢ a ∈ B
-
-with Eq : list PTm -> PTm -> PTm -> PTm -> Prop :=
-(* Structural *)
-| E_Refl Γ (a : PTm ) A :
- Γ ⊢ a ∈ A ->
- Γ ⊢ a ≡ a ∈ A
-
-| E_Symmetric Γ (a b : PTm) A :
- Γ ⊢ a ≡ b ∈ A ->
- Γ ⊢ b ≡ a ∈ A
-
-| E_Transitive Γ (a b c : PTm) A :
- Γ ⊢ a ≡ b ∈ A ->
- Γ ⊢ b ≡ c ∈ A ->
- Γ ⊢ a ≡ c ∈ A
-
-(* Congruence *)
-| E_Bind Γ i p (A0 A1 : PTm) B0 B1 :
- Γ ⊢ A0 ∈ PUniv i ->
- Γ ⊢ A0 ≡ A1 ∈ PUniv i ->
- (cons A0 Γ) ⊢ B0 ≡ B1 ∈ PUniv i ->
- Γ ⊢ PBind p A0 B0 ≡ PBind p A1 B1 ∈ PUniv i
-
-| E_Abs Γ (a b : PTm) A B i :
- Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
- (cons A Γ) ⊢ a ≡ b ∈ B ->
- Γ ⊢ PAbs a ≡ PAbs b ∈ PBind PPi A B
-
-| E_App Γ i (b0 b1 a0 a1 : PTm) A B :
- Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
- Γ ⊢ b0 ≡ b1 ∈ PBind PPi A B ->
- Γ ⊢ a0 ≡ a1 ∈ A ->
- Γ ⊢ PApp b0 a0 ≡ PApp b1 a1 ∈ subst_PTm (scons a0 VarPTm) B
-
-| E_Pair Γ (a0 a1 b0 b1 : PTm) A B i :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a0 ≡ a1 ∈ A ->
- Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons a0 VarPTm) B ->
- Γ ⊢ PPair a0 b0 ≡ PPair a1 b1 ∈ PBind PSig A B
-
-| E_Proj1 Γ (a b : PTm) A B :
- Γ ⊢ a ≡ b ∈ PBind PSig A B ->
- Γ ⊢ PProj PL a ≡ PProj PL b ∈ A
-
-| E_Proj2 Γ i (a b : PTm) A B :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a ≡ b ∈ PBind PSig A B ->
- Γ ⊢ PProj PR a ≡ PProj PR b ∈ subst_PTm (scons (PProj PL a) VarPTm) B
-
-| E_IndCong Γ P0 P1 (a0 a1 : PTm) b0 b1 c0 c1 i :
- (cons PNat Γ) ⊢ P0 ∈ PUniv i ->
- (cons PNat Γ) ⊢ P0 ≡ P1 ∈ PUniv i ->
- Γ ⊢ a0 ≡ a1 ∈ PNat ->
- Γ ⊢ b0 ≡ b1 ∈ subst_PTm (scons PZero VarPTm) P0 ->
- (cons P0 ((cons PNat Γ))) ⊢ c0 ≡ c1 ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P0) ->
- Γ ⊢ PInd P0 a0 b0 c0 ≡ PInd P1 a1 b1 c1 ∈ subst_PTm (scons a0 VarPTm) P0
-
-| E_SucCong Γ (a b : PTm) :
- Γ ⊢ a ≡ b ∈ PNat ->
- Γ ⊢ PSuc a ≡ PSuc b ∈ PNat
-
-| E_Conv Γ (a b : PTm) A B :
- Γ ⊢ a ≡ b ∈ A ->
- Γ ⊢ A ≲ B ->
- Γ ⊢ a ≡ b ∈ B
-
-(* Beta *)
-| E_AppAbs Γ (a : PTm) b A B i:
- Γ ⊢ PBind PPi A B ∈ PUniv i ->
- Γ ⊢ b ∈ A ->
- (cons A Γ) ⊢ a ∈ B ->
- Γ ⊢ PApp (PAbs a) b ≡ subst_PTm (scons b VarPTm) a ∈ subst_PTm (scons b VarPTm ) B
-
-| E_ProjPair1 Γ (a b : PTm) A B i :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a ∈ A ->
- Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
- Γ ⊢ PProj PL (PPair a b) ≡ a ∈ A
-
-| E_ProjPair2 Γ (a b : PTm) A B i :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a ∈ A ->
- Γ ⊢ b ∈ subst_PTm (scons a VarPTm) B ->
- Γ ⊢ PProj PR (PPair a b) ≡ b ∈ subst_PTm (scons a VarPTm) B
-
-| E_IndZero Γ P i (b : PTm) c :
- (cons PNat Γ) ⊢ P ∈ PUniv i ->
- Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
- (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
- Γ ⊢ PInd P PZero b c ≡ b ∈ subst_PTm (scons PZero VarPTm) P
-
-| E_IndSuc Γ P (a : PTm) b c i :
- (cons PNat Γ) ⊢ P ∈ PUniv i ->
- Γ ⊢ a ∈ PNat ->
- Γ ⊢ b ∈ subst_PTm (scons PZero VarPTm) P ->
- (cons P (cons PNat Γ)) ⊢ c ∈ ren_PTm shift (subst_PTm (scons (PSuc (VarPTm var_zero)) (funcomp VarPTm shift) ) P) ->
- Γ ⊢ PInd P (PSuc a) b c ≡ (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c) ∈ subst_PTm (scons (PSuc a) VarPTm) P
-
-(* Eta *)
-| E_AppEta Γ (b : PTm) A B i :
- Γ ⊢ PBind PPi A B ∈ (PUniv i) ->
- Γ ⊢ b ∈ PBind PPi A B ->
- Γ ⊢ PAbs (PApp (ren_PTm shift b) (VarPTm var_zero)) ≡ b ∈ PBind PPi A B
-
-| E_PairEta Γ (a : PTm ) A B i :
- Γ ⊢ PBind PSig A B ∈ (PUniv i) ->
- Γ ⊢ a ∈ PBind PSig A B ->
- Γ ⊢ a ≡ PPair (PProj PL a) (PProj PR a) ∈ PBind PSig A B
-
-with LEq : list PTm -> PTm -> PTm -> Prop :=
-(* Structural *)
-| Su_Transitive Γ (A B C : PTm) :
- Γ ⊢ A ≲ B ->
- Γ ⊢ B ≲ C ->
- Γ ⊢ A ≲ C
-
-(* Congruence *)
-| Su_Univ Γ i j :
- ⊢ Γ ->
- i <= j ->
- Γ ⊢ PUniv i ≲ PUniv j
-
-| Su_Pi Γ (A0 A1 : PTm) B0 B1 i :
- Γ ⊢ A0 ∈ PUniv i ->
- Γ ⊢ A1 ≲ A0 ->
- (cons A0 Γ) ⊢ B0 ≲ B1 ->
- Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1
-
-| Su_Sig Γ (A0 A1 : PTm) B0 B1 i :
- Γ ⊢ A1 ∈ PUniv i ->
- Γ ⊢ A0 ≲ A1 ->
- (cons A1 Γ) ⊢ B0 ≲ B1 ->
- Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1
-
-(* Injecting from equalities *)
-| Su_Eq Γ (A : PTm) B i :
- Γ ⊢ A ≡ B ∈ PUniv i ->
- Γ ⊢ A ≲ B
-
-(* Projection axioms *)
-| Su_Pi_Proj1 Γ (A0 A1 : PTm) B0 B1 :
- Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 ->
- Γ ⊢ A1 ≲ A0
-
-| Su_Sig_Proj1 Γ (A0 A1 : PTm) B0 B1 :
- Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 ->
- Γ ⊢ A0 ≲ A1
-
-| Su_Pi_Proj2 Γ (a0 a1 A0 A1 : PTm ) B0 B1 :
- Γ ⊢ PBind PPi A0 B0 ≲ PBind PPi A1 B1 ->
- Γ ⊢ a0 ≡ a1 ∈ A1 ->
- Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1
-
-| Su_Sig_Proj2 Γ (a0 a1 A0 A1 : PTm) B0 B1 :
- Γ ⊢ PBind PSig A0 B0 ≲ PBind PSig A1 B1 ->
- Γ ⊢ a0 ≡ a1 ∈ A0 ->
- Γ ⊢ subst_PTm (scons a0 VarPTm) B0 ≲ subst_PTm (scons a1 VarPTm) B1
-
-with Wff : list PTm -> Prop :=
-| Wff_Nil :
- ⊢ nil
-| Wff_Cons Γ (A : PTm) i :
- ⊢ Γ ->
- Γ ⊢ A ∈ PUniv i ->
- (* -------------------------------- *)
- ⊢ (cons A Γ)
-
-where
-"Γ ⊢ a ∈ A" := (Wt Γ a A) and "⊢ Γ" := (Wff Γ) and "Γ ⊢ a ≡ b ∈ A" := (Eq Γ a b A) and "Γ ⊢ A ≲ B" := (LEq Γ A B).
-
-Scheme wf_ind := Induction for Wff Sort Prop
- with wt_ind := Induction for Wt Sort Prop
- with eq_ind := Induction for Eq Sort Prop
- with le_ind := Induction for LEq Sort Prop.
-
-Combined Scheme wt_mutual from wf_ind, wt_ind, eq_ind, le_ind.
-
-(* Lemma lem : *)
-(* (forall n (Γ : fin n -> PTm n), ⊢ Γ -> ...) /\ *)
-(* (forall n Γ (a A : PTm n), Γ ⊢ a ∈ A -> ...) /\ *)
-(* (forall n Γ (a b A : PTm n), Γ ⊢ a ≡ b ∈ A -> ...) /\ *)
-(* (forall n Γ (A B : PTm n), Γ ⊢ A ≲ B -> ...). *)
-(* Proof. apply wt_mutual. ... *)