Datasets:

phanerozoic

/

Coq-MetaCoq

like 0

Set := | nAnon | nNamed (_ : ident).

Inductive

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

name

Set := Relevant | Irrelevant.

Inductive

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

relevance

(A : Type) := mkBindAnn { binder_name : A; binder_relevance : relevance }.

Record

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

binder_annot

(A : Type) (e : Classes.EqDec A) : Classes.EqDec (binder_annot A). Proof. ltac:(Equations.Prop.Tactics.eqdec_proof). Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

eqdec_binder_annot

{A B} (f : A -> B) (b : binder_annot A) : binder_annot B := {| binder_name := f b.(binder_name); binder_relevance := b.(binder_relevance) |}.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_binder_annot

{A B} (b : binder_annot A) (b' : binder_annot B) : Prop := b.(binder_relevance) = b'.(binder_relevance).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

eq_binder_annot

binder_annot name.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

aname

Classes.EqDec aname := _.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

anqme_eqdec

{A} (b b' : binder_annot A) : bool := match Classes.eq_dec b.(binder_relevance) b'.(binder_relevance) with | left _ => true | right _ => false end.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

eqb_binder_annot

(na : name) := match na with | nAnon => "_" | nNamed n => n end.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

string_of_name

(r : relevance) := match r with | Relevant => "Relevant" | Irrelevant => "Irrelevant" end.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

string_of_relevance

Set := | VmCast | NativeCast | Cast.

Inductive

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

cast_kind

mk_case_info { ci_ind : inductive; ci_npar : nat; ci_cstr_ndecls : list nat; *) ci_relevance : relevance }.

Record

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

case_info

ci := "(" ^ string_of_inductive ci.(ci_ind) ^ "," ^ string_of_nat ci.(ci_npar) ^ "," ^ string_of_relevance ci.(ci_relevance) ^ ")".

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

string_of_case_info

| Finite | CoFinite | BiFinite .

Inductive

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

recursivity_kind

| Conv | Cumul.

Inductive

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

conv_pb

(pb1 pb2 : conv_pb) : bool := match pb1, pb2 with | Cumul, Conv => false | _, _ => true end.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

conv_pb_leqb

nat. exact 0. Qed.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fresh_evar_id

term := mkdef { dname : aname; dtype : term; dbody : term; rarg : nat }.

Record

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

def

{A} : Classes.EqDec A -> Classes.EqDec (def A). Proof. ltac:(Equations.Prop.Tactics.eqdec_proof). Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

def_eq_dec

{A} (f : A -> string) (def : def A) := "(" ^ string_of_name (binder_name (dname def)) ^ "," ^ string_of_relevance (binder_relevance (dname def)) ^ "," ^ f (dtype def) ^ "," ^ f (dbody def) ^ "," ^ string_of_nat (rarg def) ^ ")".

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

string_of_def

{A} (f : A -> string) (g : A -> string) (def : def A) := string_of_name (binder_name (dname def)) ^ " { struct " ^ string_of_nat (rarg def) ^ " }" ^ " : " ^ f (dtype def) ^ " := " ^ nl ^ g (dbody def).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

print_def

{A B} (tyf bodyf : A -> B) (d : def A) := {| dname := d.(dname); dtype := tyf d.(dtype); dbody := bodyf d.(dbody); rarg := d.(rarg) |}.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def

{A B} (f : A -> B) (g : A -> B) (d : def A) : f (dtype d) = dtype (map_def f g d). Proof. destruct d; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_dtype

{A B} (f : A -> B) (g : A -> B) (d : def A) : g (dbody d) = dbody (map_def f g d). Proof. destruct d; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_dbody

term := list (def term).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mfixpoint

{A} (tyf bodyf : A -> bool) (d : def A) := tyf d.(dtype) && bodyf d.(dbody).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_def

{A} (P P' : A -> Type) (m : mfixpoint A) := All (fun x : def A => P x.(dtype) * P' x.(dbody))%type m.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

tFixProp

{A B C} (f f' : B -> C) (g g' : A -> B) (d : def A) : map_def f f' (map_def g g' d) = map_def (f ∘ g) (f' ∘ g') d. Proof. destruct d; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def_map_def

{A B C} (f f' : B -> C) (g g' : A -> B) : (map_def f f') ∘ (map_def g g') = map_def (f ∘ g) (f' ∘ g'). Proof. reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

compose_map_def

{t} x : map_def (@id t) (@id t) x = id x. Proof. now destruct x. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def_id

{A B} (P P' : A -> Type) (f f' g g' : A -> B) (x : def A) : P' x.(dbody) -> P x.(dtype) -> (forall x, P x -> f x = g x) -> (forall x, P' x -> f' x = g' x) -> map_def f f' x = map_def g g' x. Proof. intros. destruct x. unfold map_def. simpl. now rewrite !H // !H0. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def_spec

{A B} (f : B -> B) (x : A * B) : f (snd x) = snd x <-> on_snd f x = x. Proof. destruct x; simpl; unfold on_snd; simpl. split; congruence. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

on_snd_eq_id_spec

{A B} (f f' g g' : A -> B) (x : def A) : f (dtype x) = g (dtype x) -> f' (dbody x) = g' (dbody x) -> map_def f f' x = map_def g g' x. Proof. intros. unfold map_def; f_equal; auto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def_eq_spec

{A} (f f' : A -> A) (x : def A) : f (dtype x) = (dtype x) -> f' (dbody x) = (dbody x) -> map_def f f' x = x. Proof. intros. rewrite (map_def_eq_spec _ _ id id); auto. destruct x; auto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_def_id_spec

{A B} {P P' : A -> Type} {l} {f f' g g' : A -> B} : tFixProp P P' l -> (forall x, P x -> f x = g x) -> (forall x, P' x -> f' x = g' x) -> map (map_def f f') l = map (map_def g g') l. Proof. intros. eapply All_map_eq. red in X. eapply All_impl; eauto. simpl. intros. destruct X0; eapply map_def_spec; eauto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

tfix_map_spec

{universe Term} := Judge { j_term : option Term; j_typ : Term; j_univ : option universe; }.

Record

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

judgment_

{univ T A} (f: T -> A) (j : judgment_ univ T) := Judge (option_map f (j_term j)) (f (j_typ j)) (j_univ j) .

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

judgment_map

mkdecl { decl_name : aname ; decl_body : option term ; decl_type : term }.

Record

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

context_decl

{term term'} (f : term -> term') (d : context_decl term) : context_decl term' := {| decl_name := d.(decl_name); decl_body := option_map f d.(decl_body); decl_type := f d.(decl_type) |}.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl

{term term' term''} (g : term -> term') (f : term' -> term'') x : map_decl f (map_decl g x) = map_decl (f ∘ g) x. Proof. destruct x as [? [?|] ?]; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

compose_map_decl

{term term'} (f g : term -> term') x : (forall x, f x = g x) -> map_decl f x = map_decl g x. Proof. intros H; destruct x as [? [?|] ?]; rewrite /map_decl /=; f_equal; auto. now rewrite (H t). Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_ext

{term term'} : Proper (`=1` ==> Logic.eq ==> Logic.eq) (@map_decl term term'). Proof. intros f g Hfg x y ->. now apply map_decl_ext. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_proper

{term term'} : Proper (`=1` ==> `=1`) (@map_decl term term'). Proof. intros f g Hfg x. rewrite /map_decl. destruct x => /=. f_equal. - now rewrite Hfg. - apply Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_pointwise

{A B} : subrelation (`=1`) (@Logic.eq A ==> @Logic.eq B)%signature. Proof. intros f g Hfg x y ->. now rewrite Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

pointwise_subrelation

{A B} : subrelation (@Logic.eq A ==> @Logic.eq B)%signature (`=1`). Proof. intros f g Hfg x. now specialize (Hfg x x eq_refl). Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

pointwise_subrelation_inv

{term term'} (f : term -> term') (c : list (context_decl term)) := List.map (map_decl f) c.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_context

{term term'} : Proper (`=1` ==> Logic.eq ==> Logic.eq) (@map_context term term'). Proof. intros f g Hfg x y ->. now rewrite /map_context Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_context_proper

{term term'} (f : term -> term') l : #|map_context f l| = #|l|. Proof. now unfold map_context; rewrite length_map. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_context_length

{term} (f : term -> bool) (d : context_decl term) : bool := option_default f d.(decl_body) true && f d.(decl_type).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_decl

{term} : Proper (`=1` ==> Logic.eq ==> Logic.eq) (@test_decl term). Proof. intros f g Hfg [na [b|] ty] ? <- => /=; rewrite /test_decl /=; now rewrite Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_decl_proper

{A} (Γ : list A) (d : A) := d :: Γ.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

snoc

{A} (Γ Γ': list A) := Γ' ++ Γ.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context

{T} Γ : [] ,,, Γ = Γ :> list T. Proof. unfold app_context. rewrite app_nil_r. reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context_nil_l

{T} Γ Γ' Γ'' : Γ ,,, (Γ' ,,, Γ'') = Γ ,,, Γ' ,,, Γ'' :> list T. Proof. unfold app_context; now rewrite app_assoc. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context_assoc

{T} Γ Γ' A : Γ ,,, (Γ' ,, A) = (Γ ,,, Γ') ,, A :> list T. Proof. exact (app_context_assoc _ _ [A]). Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context_cons

{T} Γ Δ Δ' d : (Γ ,,, Δ ,,, Δ') ,, d = (Γ ,,, Δ ,,, (Δ' ,, d)) :> list T. Proof using Type. reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context_push

{T Γ Δ d} : (Γ ,,, (d :: Δ)) = (Γ ,,, Δ) ,,, [d] :> list T. Proof using Type. reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

snoc_app_context

{T} (Γ Γ' : list T) : #|Γ ,,, Γ'| = #|Γ'| + #|Γ|. Proof. unfold app_context. now rewrite length_app. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

app_context_length

{T} v Γ Γ' : #|Γ'| <= v -> nth_error (Γ ,,, Γ') v = nth_error Γ (v - #|Γ'|) :> option T. Proof. apply nth_error_app_ge. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

nth_error_app_context_ge

{T} v Γ Γ' : v < #|Γ'| -> nth_error (Γ ,,, Γ') v = nth_error Γ' v :> option T. Proof. apply nth_error_app_lt. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

nth_error_app_context_lt

{A} (P : A -> Type) (d : context_decl A) := option_default P d.(decl_body) unit × P d.(decl_type).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

ondecl

(c : list (context_decl term)) : list (context_decl term') := match c with | d :: Γ => map_decl (f #|Γ|) d :: mapi_context Γ | [] => [] end.

Fixpoint

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context

{term term'} : Proper (`=2` ==> Logic.eq ==> Logic.eq) (@mapi_context term term'). Proof. intros f g Hfg Γ ? <-. induction Γ as [|[na [b|] ty] Γ]; simpl; auto; f_equal; auto; now rewrite Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context_proper

{term} (f : nat -> term -> term) l : #|mapi_context f l| = #|l|. Proof. induction l; simpl; auto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context_length

(c : list (context_decl term)) : bool := match c with | d :: Γ => test_context Γ && test_decl f d | [] => true end.

Fixpoint

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_context

{term} : Proper (`=1` ==> Logic.eq ==> Logic.eq) (@test_context term). Proof. intros f g Hfg Γ ? <-. induction Γ as [|[na [b|] ty] Γ]; simpl; auto; f_equal; auto; now rewrite Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_context_proper

(c : list (context_decl term)) : bool := match c with | d :: Γ => test_context_k Γ && test_decl (f (#|Γ| + k)) d | [] => true end.

Fixpoint

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_context_k

{term} : Proper (`=1` ==> Logic.eq ==> Logic.eq ==> Logic.eq) (@test_context_k term). Proof. intros f g Hfg k ? <- Γ ? <-. induction Γ as [|[na [b|] ty] Γ]; simpl; auto; f_equal; auto; now rewrite Hfg. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_context_k_proper

(f g : term -> bool) x : (forall x, f x -> g x) -> test_decl f x -> test_decl g x. Proof using Type. intros Hf; rewrite /test_decl. move/andb_and=> [Hd Hb]. apply/andb_and; split; eauto. destruct (decl_body x); simpl in *; eauto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_decl_impl

(P : nat -> term -> Type) k (ctx : context term) := Alli (fun i d => ondecl (P (Nat.pred #|ctx| - i + k)) d) 0 ctx.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

onctx_k

{P : term -> Type} {p : term -> bool} {d : context_decl term} : (forall x, reflectT (P x) (p x)) -> reflectT (ondecl P d) (test_decl p d). Proof using Type. intros hr. rewrite /ondecl /test_decl; destruct d as [decl_name decl_body decl_type]; cbn. destruct (hr decl_type) => //; destruct (reflect_option_default hr decl_body) => /= //; now constructor. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

ondeclP

{p : term -> bool} {ctx : context term} : reflectT (onctx p ctx) (test_context p ctx). Proof using Type. eapply equiv_reflectT. - induction 1; simpl; auto. rewrite IHX /= //. now move/(ondeclP reflectT_pred): p0. - induction ctx. * constructor. * move => /= /andb_and [Hctx Hd]; constructor; eauto. now move/(ondeclP reflectT_pred): Hd. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

onctxP

(f : term -> term') decl : f (decl_type decl) = decl_type (map_decl f decl). Proof using Type. destruct decl; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_type

(f : term -> term') decl : option_map f (decl_body decl) = decl_body (map_decl f decl). Proof using Type. destruct decl; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_body

@map_decl term term id =1 id. Proof using Type. intros d; now destruct d as [? [] ?]. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_decl_id

(f : term -> term') x : option_map decl_body (option_map (map_decl f) x) = option_map (option_map f) (option_map decl_body x). Proof using Type. destruct x; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

option_map_decl_body_map_decl

(f : term -> term') x : option_map decl_type (option_map (map_decl f) x) = option_map f (option_map decl_type x). Proof using Type. destruct x; reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

option_map_decl_type_map_decl

(f : nat -> term -> term') Γ := List.rev (mapi (fun k' decl => map_decl (f k') decl) (List.rev Γ)).

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k

f Γ : fold_context_k f Γ = mapi (fun k' d => map_decl (f (Nat.pred (length Γ) - k')) d) Γ. Proof using Type. unfold fold_context_k. rewrite rev_mapi. rewrite List.rev_involutive. apply mapi_ext. intros. f_equal. now rewrite List.length_rev. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_alt

f Γ : mapi_context f Γ = fold_context_k f Γ. Proof using Type. setoid_replace f with (fun k => f (k - 0)) using relation (pointwise_relation nat (pointwise_relation term (@Logic.eq term')))%signature at 1. rewrite fold_context_k_alt. unfold mapi. generalize 0. induction Γ as [|d Γ]; intros n; simpl; auto. f_equal. rewrite IHΓ. rewrite mapi_rec_Sk. apply mapi_rec_ext => k x. intros. apply map_decl_ext => t. lia_f_equal. intros k. now rewrite Nat.sub_0_r. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context_fold

f d : fold_context_k f [d] = [map_decl (f 0) d]. Proof using Type. reflexivity. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_tip

f Γ : length (fold_context_k f Γ) = length Γ. Proof using Type. unfold fold_context_k. now rewrite !List.length_rev mapi_length List.length_rev. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_length

f Γ d : fold_context_k f (d :: Γ) = fold_context_k f Γ ,, map_decl (f (length Γ)) d. Proof using Type. unfold fold_context_k. rewrite !rev_mapi !rev_involutive. unfold mapi; rewrite mapi_rec_eqn. unfold snoc. f_equal. now rewrite Nat.sub_0_r List.length_rev. rewrite mapi_rec_Sk. simpl. apply mapi_rec_ext. intros. rewrite length_app !List.length_rev. simpl. f_equal. f_equal. lia. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_snoc0

f Γ Δ : fold_context_k f (Δ ++ Γ) = fold_context_k (fun k => f (length Γ + k)) Δ ++ fold_context_k f Γ. Proof using Type. unfold fold_context_k. rewrite List.rev_app_distr. rewrite mapi_app. rewrite <- List.rev_app_distr. f_equal. f_equal. apply mapi_ext. intros. f_equal. rewrite List.length_rev. f_equal. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_app

(f : nat -> term -> term) (ctx : context term) : mapi_context_In ctx (fun n (x : context_decl term) (_ : In x ctx) => map_decl (f n) x) = mapi_context f ctx. Proof using Type. remember (fun n (x : context_decl term) (_ : In x ctx) => map_decl (f n) x) as g. funelim (mapi_context_In ctx g) => //=; rewrite (H f0) ; trivial. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context_In_spec

f Γ : #|fold_context f Γ| = #|Γ|. Proof using Type. now apply_funelim (fold_context f Γ); intros; simpl; auto; f_equal. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_length

(f : context term -> context_decl term -> context_decl term) (ctx : context term) : fold_context_In ctx (fun n (x : context_decl term) (_ : In x ctx) => f n x) = fold_context f ctx. Proof using Type. remember (fun n (x : context_decl term) (_ : In x ctx) => f n x) as g. funelim (fold_context_In ctx g) => //=; rewrite (H f0); trivial. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_In_spec

Proper (`=2` ==> `=1`) fold_context. Proof using Type. intros f f' Hff' x. funelim (fold_context f x); simpl; auto. simp fold_context. now rewrite (H f' Hff'). Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_Proper

(c : list (BasicAst.context_decl term)) : list aname := map decl_name c.

Definition

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

forget_types

(x : context term) : fold_context_k (fun i x => x) x = x. Proof using Type. rewrite fold_context_k_alt. rewrite /mapi. generalize 0. induction x; simpl; auto. intros n. f_equal; auto. now rewrite map_decl_id. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_id

(f : nat -> term' -> term) (g : nat -> term'' -> term') Γ : fold_context_k f (fold_context_k g Γ) = fold_context_k (fun i => f i ∘ g i) Γ. Proof using Type. rewrite !fold_context_k_alt mapi_mapi. apply mapi_ext => i d. rewrite compose_map_decl. apply map_decl_ext => t. now len. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_compose

(f g : nat -> term' -> term) Γ : f =2 g -> fold_context_k f Γ = fold_context_k g Γ. Proof using Type. intros hfg. induction Γ; simpl; auto; rewrite !fold_context_k_snoc0. simpl. rewrite IHΓ. f_equal. apply map_decl_ext. intros. now apply hfg. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_ext

Proper (pointwise_relation nat (pointwise_relation _ Logic.eq) ==> Logic.eq ==> Logic.eq) (@fold_context_k term' term). Proof using Type. intros f g Hfg x y <-. now apply fold_context_k_ext. Qed.

Instance

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

fold_context_k_proper

(f : nat -> context_decl term -> bool) (g : nat -> term' -> term) ctx : alli f 0 (fold_context_k g ctx) = alli (fun i x => f i (map_decl (g (Nat.pred #|ctx| - i)) x)) 0 ctx. Proof using Type. now rewrite fold_context_k_alt /mapi alli_mapi. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

alli_fold_context_k_prop

f g x : (@test_decl term) f (map_decl g x) = @test_decl term (f ∘ g) x. Proof using Type. rewrite /test_decl /map_decl /=. f_equal. rewrite /option_default. destruct (decl_body x) => //. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

test_decl_map_decl

(f : term' -> term) (g : nat -> term'' -> term') ctx : map (map_decl f) (fold_context_k g ctx) = fold_context_k (fun i => f ∘ g i) ctx. Proof using Type. rewrite !fold_context_k_alt map_mapi. apply mapi_ext => i d. now rewrite compose_map_decl. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_fold_context_k

(f : term' -> term) (g : nat -> term'' -> term') (ctx : list (BasicAst.context_decl term'')) : map_context f (mapi_context g ctx) = mapi_context (fun i => f ∘ g i) ctx. Proof using Type. rewrite !mapi_context_fold. now unfold map_context; rewrite map_fold_context_k. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_context_mapi_context

(f : nat -> term' -> term) (g : context_decl term'' -> context_decl term') ctx : mapi_context f (map g ctx) = mapi (fun i => map_decl (f (Nat.pred #|ctx| - i)) ∘ g) ctx. Proof using Type. rewrite mapi_context_fold fold_context_k_alt mapi_map. now len. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

mapi_context_map

(f : term' -> term) (g : context_decl term'' -> context_decl term') ctx : map_context f (map g ctx) = map (map_decl f ∘ g) ctx. Proof using Type. induction ctx; simpl; f_equal; auto. Qed.

Lemma

common

From Coq Require Import ssreflect Morphisms Orders Setoid. From MetaCoq.Utils Require Import utils. From MetaCoq.Common Require Export Kernames. From Coq Require Floats.SpecFloat. From Equations Require Import Equations.

common\theories\BasicAst.v

map_context_map