fact stringlengths 8 6.34k | type stringclasses 15
values | library stringclasses 5
values | imports listlengths 1 4 | filename stringclasses 33
values | symbolic_name stringlengths 1 59 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
uniq_app_2 :
uniq (E ++ F) -> uniq F. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_app_2 | |
uniq_app_3 :
uniq (E ++ F) -> disjoint E F. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_app_3 | |
uniq_app_4 :
uniq E ->
uniq F ->
disjoint E F ->
uniq (E ++ F). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_app_4 | |
uniq_app_iff :
uniq (E ++ F) <-> uniq E /\ uniq F /\ disjoint E F. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_app_iff | |
uniq_map_1 :
uniq (map f E) ->
uniq E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_map_1 | |
uniq_map_2 :
uniq E ->
uniq (map f E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_map_2 | |
uniq_map_iff :
uniq (map f E) <-> uniq E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_map_iff | |
BindsProperties .
Variable A B : Type.
Variables f : A -> B.
Variables x y : atom.
Variables a b : A.
Variables E F G : list (atom*A). | Section | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | BindsProperties | |
binds_nil_iff :
binds x a nil <-> False. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_nil_iff | |
binds_one_1 :
binds x a (y ~ b) ->
x = y. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_one_1 | |
binds_one_2 :
binds x a (y ~ b) ->
a = b. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_one_2 | |
binds_one_3 :
x = y ->
a = b ->
binds x a (y ~ b). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_one_3 | |
binds_one_iff :
binds x a (y ~ b) <-> x = y /\ a = b. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_one_iff | |
binds_cons_1 :
binds x a ((y, b) :: E) ->
(x = y /\ a = b) \/ binds x a E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_1 | |
binds_cons_2 :
x = y ->
a = b ->
binds x a ((y, b) :: E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_2 | |
binds_cons_3 :
binds x a E ->
binds x a ((y, b) :: E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_3 | |
binds_cons_iff :
binds x a ((y, b) :: E) <-> (x = y /\ a = b) \/ binds x a E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_iff | |
binds_app_1 :
binds x a (E ++ F) ->
binds x a E \/ binds x a F. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_1 | |
binds_app_2 :
binds x a E ->
binds x a (E ++ F). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_2 | |
binds_app_3 :
binds x a F ->
binds x a (E ++ F). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_3 | |
binds_app_iff :
binds x a (E ++ F) <-> binds x a E \/ binds x a F. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_iff | |
binds_map_1 :
(forall a b, f a = f b -> a = b) ->
binds x (f a) (map f E) ->
binds x a E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_map_1 | |
binds_map_2 :
binds x a E ->
binds x (f a) (map f E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_map_2 | |
binds_dom_contradiction : forall (E : list (atom*A)),
binds x a E ->
~ In x (dom E) ->
False. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_dom_contradiction | |
binds_app_uniq_1 :
uniq (E ++ F) ->
binds x a (E ++ F) ->
(binds x a E /\ ~ In x (dom F)) \/ (binds x a F /\ ~ In x (dom E)). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_uniq_1 | |
binds_app_uniq_iff :
uniq (E ++ F) ->
(binds x a (E ++ F) <->
(binds x a E /\ ~ In x (dom F)) \/
(binds x a F /\ ~ In x (dom E))). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_app_uniq_iff | |
BindsProperties2 .
Variable A B : Type.
Variables f : A -> B.
Variables x y : atom.
Variables a b : A.
Variables E F G : list (atom*A). | Section | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | BindsProperties2 | |
binds_cons_uniq_1 :
uniq ((y, b) :: E) ->
binds x a ((y, b) :: E) ->
(x = y /\ a = b /\ ~ In x (dom E)) \/ (binds x a E /\ x <> y). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_uniq_1 | |
binds_cons_uniq_iff :
uniq ((y, b) :: E) ->
(binds x a ((y, b) :: E) <->
(x = y /\ a = b /\ ~ In x (dom E)) \/
(binds x a E /\ x <> y)). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_cons_uniq_iff | |
AssortedListProperties .
Variable X : Type.
Variables x : X.
Variables xs ys zs : list X. | Section | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | AssortedListProperties | |
one_eq_app :
one x ++ xs = ys ++ zs ->
(exists qs, ys = x :: qs /\ xs = qs ++ zs) \/
(ys = nil /\ zs = x :: xs). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | one_eq_app | |
app_eq_one :
ys ++ zs = one x ++ xs ->
(exists qs, ys = x :: qs /\ xs = qs ++ zs) \/
(ys = nil /\ zs = x :: xs). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | app_eq_one | |
nil_neq_one_mid :
nil <> xs ++ one x ++ ys. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | nil_neq_one_mid | |
one_mid_neq_nil :
xs ++ one x ++ ys <> nil. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | one_mid_neq_nil | |
destruct_uniq :=
match goal with
| H : uniq nil |- _ =>
clear H;
destruct_uniq
| H : uniq (?x ~ ?a) |- _ =>
clear H;
destruct_uniq
| H : uniq ((?x, ?a) :: ?E) |- _ =>
let J := fresh "UniqTac" in
pose proof H as J;
apply uniq_cons_1 in H;
apply uniq_cons_2 in... | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | destruct_uniq | |
solve_uniq :=
intros;
destruct_uniq;
repeat first [ apply uniq_push
| apply uniq_cons_3
| apply uniq_app_4
| apply uniq_one_1
| apply uniq_nil ];
auto;
try tauto;
unfold disjoint in *;
try fsetdec;
fail "Not solvable by [solve_uniq]; try [destr... | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | solve_uniq | |
UniqDerived .
Variable A : Type.
Variables x y : atom.
Variables a b : A.
Variables E F G : list (atom*A). | Section | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | UniqDerived | |
uniq_insert_mid :
uniq (G ++ E) ->
~ In x (dom G) ->
~ In x (dom E) ->
uniq (G ++ (x ~ a) ++ E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_insert_mid | |
uniq_remove_mid :
uniq (E ++ F ++ G) ->
uniq (E ++ G). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_remove_mid | |
uniq_reorder_1 :
uniq (E ++ F) ->
uniq (F ++ E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_reorder_1 | |
uniq_reorder_2 :
uniq (E ++ F ++ G) ->
uniq (F ++ E ++ G). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_reorder_2 | |
uniq_map_app_l : forall (f : A -> A),
uniq (F ++ E) ->
uniq (map f F ++ E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | uniq_map_app_l | |
fresh_mid_tail :
uniq (F ++ (x ~ a) ++ E) ->
~ In x (dom E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | fresh_mid_tail | |
fresh_mid_head :
uniq (F ++ (x ~ a) ++ E) ->
~ In x (dom F). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | fresh_mid_head | |
destruct_binds_hyp H :=
match type of H with
| binds ?x ?a nil =>
inversion H
| binds ?x ?a (?y ~ ?b) =>
let J1 := fresh "BindsTacKey" in
let J2 := fresh "BindsTacVal" in
rename H into J1;
pose proof J1 as J2;
apply binds_one_1 in J1;
apply binds_one_2 in J2;
tr... | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | destruct_binds_hyp | |
destruct_binds_hyp_uniq H :=
match type of H with
| binds ?x ?a nil =>
inversion H
| binds ?x ?a (?y ~ ?b) =>
let J1 := fresh "BindsTacKey" in
let J2 := fresh "BindsTacVal" in
rename H into J1;
pose proof J1 as J2;
apply binds_one_1 in J1;
apply binds_one_2 in J2;
... | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | destruct_binds_hyp_uniq | |
analyze_binds_cleanup :=
auto;
try tauto;
try discriminate;
try match goal with
| J : ~ In ?x ?E |- _ =>
match E with
| context [x] => elim J; clear; simpl_env; auto with set
end
end.
(** The [analyze_binds] and [analyze_binds_uniq] tactics decompose a
hypothes... | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | analyze_binds_cleanup | |
analyze_binds H :=
destruct_binds_hyp H;
analyze_binds_cleanup. | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | analyze_binds | |
analyze_binds_uniq H :=
destruct_binds_hyp_uniq H;
analyze_binds_cleanup.
(* *********************************************************************** *)
(** * Facts about [binds] *) | Ltac | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | analyze_binds_uniq | |
BindsDerived .
Variables A B : Type.
Variables f : A -> B.
Variables x y : atom.
Variables a b : A.
Variables E F G : list (atom*A). | Section | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | BindsDerived | |
binds_dec :
(forall a b : A, {a = b} + {a <> b}) ->
{binds x a E} + {~ binds x a E}. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_dec | |
binds_lookup :
{a : A | binds x a E} + (forall a, ~ binds x a E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_lookup | |
binds_lookup_dec :
decidable (exists a, binds x a E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_lookup_dec | |
binds_weaken :
binds x a (E ++ G) ->
binds x a (E ++ F ++ G). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_weaken | |
binds_mid_eq :
binds x a (F ++ (x ~ b) ++ E) ->
uniq (F ++ (x ~ b) ++ E) ->
a = b. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_mid_eq | |
binds_remove_mid :
binds x a (F ++ (y ~ b) ++ G) ->
x <> y ->
binds x a (F ++ G). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_remove_mid | |
binds_In : forall x a (E : list (atom*A)),
binds x a E ->
In x (dom E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_In | |
binds_In_inv : forall x (E : list (atom*A)),
In x (dom E) ->
exists a, binds x a E. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_In_inv | |
binds_unique :
binds x a E ->
binds x b E ->
uniq E ->
a = b. | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | binds_unique | |
fresh_app_l :
uniq (F ++ E) ->
binds x a E ->
~ In x (dom F). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | fresh_app_l | |
fresh_app_r :
uniq (F ++ E) ->
binds x a F ->
~ In x (dom E). | Axiom | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | fresh_app_r | |
Export EnvImpl : ENVIRONMENT := AssocList.Make AtomDT AtomSetImpl. | Module | Attic | [
"Require Import Coq.",
"Require Import Metalib."
] | Attic/MetatheoryEnv.v | Export | |
general_asn (key A B : Type) : Type :=
| VarAsn : key -> A -> general_asn key A B
| AltAsn : B -> general_asn key A B.
Implicit Arguments VarAsn [key A].
Implicit Arguments AltAsn [B].
(* *********************************************************************** *)
(** * Beginning of the functor *) | Inductive | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | general_asn | |
Make (X : UsualDecidableType)
(Import KeySet : FSetInterface.WSfun X). | Module | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | Make | |
Import D := CoqFSetDecide.WDecide_fun X KeySet. | Module | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | Import | |
KeySetProperties := FSetProperties.WProperties_fun X KeySet. | Module | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | KeySetProperties | |
KeySetFacts := FSetFacts.WFacts_fun X KeySet.
(* *********************************************************************** *)
(** * Basic definitions *)
(** Implicit arguments are enabled for the following definitions. *)
Set Implicit Arguments.
(** An assumption maps an [atom] (sometimes called a 'key') to a value
... | Module | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | KeySetFacts | |
asn := (general_asn X.t).
(** [one] constructs a singleton list. We define an infix notation
for it and ensure that the arguments to [app] are interpreted in
the right scope, i.e., [list_scope].
Implementation note: The level associated with the notation gives
it a higher precedence than the "++" not... | Local Notation | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | asn | |
one (C : Type) (item : C) : list C := cons item nil. | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | one | |
dom (A B : Type) (E : list (asn A B))
: KeySet.t :=
match E with
| nil => empty
| VarAsn x _ :: E' => add x (dom E')
| _ :: E' => dom E'
end.
(** [get] looks up a key in the assumption list. *) | Fixpoint | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | dom | |
get (A B : Type) (x : X.t) (E : list (asn A B))
: option A :=
match E with
| nil => None
| VarAsn y c :: F => if X.eq_dec x y then Some c else get x F
| _ :: F => get x F
end.
(** [binds] is a ternary predicate that holds when a key-value pair
appears somewhere in the given assumption list. *) | Fixpoint | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | get | |
binds (A B : Type) (x : X.t) (a : A) (E : list (asn A B))
: Prop :=
List.In (VarAsn _ x a) E.
(** [bindsAlt] is a binary predicate that holds when a key-less
assumption appears somewhere in the given assumption list. *) | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | binds | |
bindsAlt (A B : Type) (b : B) (E : list (asn A B))
: Prop :=
List.In (AltAsn _ _ b) E.
(** [maps] is a ternary predicate that holds when the first binding in
the list for the given key is to the given value. *) | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | bindsAlt | |
maps (A B : Type) (x : X.t) (a : A) (E : list (asn A B))
: Prop :=
get x E = Some a.
(** [disjoint] is a binary predicate that holds when the domains of
two assumption lists are disjoint. *) | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | maps | |
disjoint (A B C D : Type) (E : list (asn A B)) (F : list (asn C D))
: Prop :=
inter (dom E) (dom F) [<=] empty.
(** [map] applies a function to each of the values in an assumption
list. *) | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | disjoint | |
map (A B C D : Type) (f : A -> C) (g : B -> D) (E : list (asn A B))
: list (asn C D) :=
List.map (fun b => match b with
| VarAsn x a => VarAsn _ x (f a)
| AltAsn b => AltAsn _ _ (g b)
end) E.
(** [map_var] is like [map] except that it leave [AltAsn... | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map | |
map_var (A B C : Type) (f : A -> B) (E : list (asn A C))
: list (asn B C) :=
map f (fun x => x) E.
(** [erase_var] deletes all the variable mappings from an assumption
list. *) | Definition | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var | |
erase_var (A B : Type) (E : list (asn A B))
: list B :=
match E with
| nil => nil
| VarAsn x a :: F => erase_var F
| AltAsn b :: F => b :: erase_var F
end.
(** [uniq] is unary predicate that holds if and only if each key is
bound at most once in the given assumption list. Note that
[uniq] is... | Fixpoint | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | erase_var | |
uniq (A B : Type) : list (asn A B) -> Prop :=
| uniq_nil :
uniq nil
| uniq_push : forall x a E,
uniq E ->
~ In x (dom E) ->
uniq (x ~ a ++ E)
| uniq_alt : forall b E,
uniq E ->
uniq (one (AltAsn _ _ b) ++ E).
(** Unless stated otherwise, in the remainder of this file, implicit... | Inductive | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | uniq | |
ListProperties .
Variable X : Type.
Variables x y : X.
Variables l l1 l2 l3 : list X. | Section | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | ListProperties | |
cons_app_one :
cons x l = one x ++ l.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | cons_app_one | |
cons_app_assoc :
(cons x l1) ++ l2 = cons x (l1 ++ l2).
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | cons_app_assoc | |
app_assoc :
(l1 ++ l2) ++ l3 = l1 ++ (l2 ++ l3).
Proof. clear. auto with datatypes. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | app_assoc | |
app_nil_1 :
nil ++ l = l.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | app_nil_1 | |
app_nil_2 :
l ++ nil = l.
Proof. clear. auto with datatypes. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | app_nil_2 | |
in_nil_iff :
List.In x nil <-> False.
Proof. clear. split; inversion 1. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | in_nil_iff | |
in_one_iff :
List.In x (one y) <-> x = y.
Proof. clear. split. inversion 1; intuition. constructor; intuition. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | in_one_iff | |
in_app_iff :
List.In x (l1 ++ l2) <-> List.In x l1 \/ List.In x l2.
Proof. clear. split; auto using List.in_or_app, List.in_app_or. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | in_app_iff | |
Properties .
Variables A B C D : Type.
Variable f : A -> C.
Variable g : B -> D.
Variable x : X.t.
Variable a : A.
Variable b : B.
Variables E F G : list (asn A B). | Section | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | Properties | |
map_nil :
map f g (@nil (asn A B)) = nil.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_nil | |
map_one :
map f g (x ~ a) = (x ~ f a).
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_one | |
map_one_alt :
map f g (one (AltAsn _ _ b)) = one (AltAsn _ _ (g b)).
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_one_alt | |
map_cons :
map f g (VarAsn _ x a :: E) = x ~ f a ++ map f g E.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_cons | |
map_cons_alt :
map f g (AltAsn _ _ b :: E) = one (AltAsn _ _ (g b)) ++ map f g E.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_cons_alt | |
map_app :
map f g (E ++ F) = map f g E ++ map f g F.
Proof. clear. unfold map. rewrite List.map_app. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_app | |
map_var_nil :
map_var f (@nil (asn A B)) = nil.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var_nil | |
map_var_one :
map_var f (x ~ a :> B) = x ~ (f a).
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var_one | |
map_var_one_alt :
map_var f (one (AltAsn _ _ b)) = one (AltAsn _ _ b).
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var_one_alt | |
map_var_cons :
map_var f (VarAsn _ x a :: E) = x ~ f a ++ map_var f E.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var_cons | |
map_var_cons_alt :
map_var f (AltAsn _ _ b :: E) = one (AltAsn _ _ b) ++ map_var f E.
Proof. clear. reflexivity. Qed. | Lemma | Attic | [
"Require Import Coq.",
"Require Import CoqFSetDecide.",
"Require Import CoqListFacts.",
"Require Import LibTactics."
] | Attic/MyAssumeList.v | map_var_cons_alt |
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