Library ListFacts
Assorted facts about lists.
Implicit arguments are declared by default in this library.
Implicit arguments are declared by default in this library.
Require Import Eqdep_dec.
Require Import List.
Require Import Omega.
Require Import SetoidList.
Require Import Sorting.
Require Import Relations.
Require Import AdditionalTactics.
Open Scope list_scope.
Lemma cons_eq_app : forall (A : Type) (z : A) (xs ys zs : list A),
z :: zs = xs ++ ys ->
(exists qs, xs = z :: qs /\ zs = qs ++ ys) \/
(xs = nil /\ ys = z :: zs).
Lemma app_eq_cons : forall (A : Type) (z : A) (xs ys zs : list A),
xs ++ ys = z :: zs ->
(exists qs, xs = z :: qs /\ zs = qs ++ ys) \/
(xs = nil /\ ys = z :: zs).
Lemma nil_eq_app : forall (A : Type) (xs ys : list A),
nil = xs ++ ys ->
xs = nil /\ ys = nil.
Lemma nil_neq_cons_app : forall (A : Type) (y : A) (xs ys : list A),
nil <> xs ++ y :: ys.
Lemma not_in_cons : forall (A : Type) (ys : list A) x y,
x <> y ->
~ In x ys ->
~ In x (y :: ys).
Lemma not_In_app : forall (A : Type) (xs ys : list A) x,
~ In x xs ->
~ In x ys ->
~ In x (xs ++ ys).
Lemma elim_not_In_cons : forall (A : Type) (y : A) (ys : list A) (x : A),
~ In x (y :: ys) ->
x <> y /\ ~ In x ys.
Lemma elim_not_In_app : forall (A : Type) (xs ys : list A) (x : A),
~ In x (xs ++ ys) ->
~ In x xs /\ ~ In x ys.
Lemma incl_nil : forall (A : Type) (xs : list A),
incl nil xs.
Lemma In_incl : forall (A : Type) (x : A) (ys zs : list A),
In x ys ->
incl ys zs ->
In x zs.
Lemma elim_incl_cons : forall (A : Type) (x : A) (xs zs : list A),
incl (x :: xs) zs ->
In x zs /\ incl xs zs.
Lemma elim_incl_app : forall (A : Type) (xs ys zs : list A),
incl (xs ++ ys) zs ->
incl xs zs /\ incl ys zs.
Lemma InA_iff_In : forall (A : Set) (x : A) (xs : list A),
InA (@eq A) x xs <-> In x xs.
Section EqRectList.
Variable A : Type.
Variable eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Lemma eq_rect_eq_list :
forall (p : list A) (Q : list A -> Type) (x : Q p) (h : p = p),
eq_rect p Q x p h = x.
End EqRectList.
Section DecidableSorting.
Variable A : Set.
Variable leA : relation A.
Hypothesis leA_dec : forall x y, {leA x y} + {~ leA x y}.
Theorem lelistA_dec : forall a xs,
{lelistA leA a xs} + {~ lelistA leA a xs}.
Theorem sort_dec : forall xs,
{sort leA xs} + {~ sort leA xs}.
Section UniqueSortingProofs.
Hypothesis eq_A_dec : forall (x y : A), {x = y} + {x <> y}.
Hypothesis leA_unique : forall (x y : A) (p q : leA x y), p = q.
Scheme lelistA_ind' := Induction for lelistA Sort Prop.
Scheme sort_ind' := Induction for sort Sort Prop.
Theorem lelistA_unique :
forall (x : A) (xs : list A) (p q : lelistA leA x xs), p = q.
Theorem sort_unique :
forall (xs : list A) (p q : sort leA xs), p = q.
End UniqueSortingProofs.
End DecidableSorting.
Section Equality_ext.
Variable A : Set.
Variable ltA : relation A.
Hypothesis ltA_trans : forall x y z, ltA x y -> ltA y z -> ltA x z.
Hypothesis ltA_not_eqA : forall x y, ltA x y -> x <> y.
Hypothesis ltA_eqA : forall x y z, ltA x y -> y = z -> ltA x z.
Hypothesis eqA_ltA : forall x y z, x = y -> ltA y z -> ltA x z.
Hint Resolve ltA_trans.
Hint Immediate ltA_eqA eqA_ltA.
Notation Inf := (lelistA ltA).
Notation Sort := (sort ltA).
Lemma not_InA_if_Sort_Inf : forall xs a,
Sort xs ->
Inf a xs ->
~ InA (@eq A) a xs.
Lemma Sort_eq_head : forall x xs y ys,
Sort (x :: xs) ->
Sort (y :: ys) ->
(forall a, InA (@eq A) a (x :: xs) <-> InA (@eq A) a (y :: ys)) ->
x = y.
Lemma Sort_InA_eq_ext : forall xs ys,
Sort xs ->
Sort ys ->
(forall a, InA (@eq A) a xs <-> InA (@eq A) a ys) ->
xs = ys.
End Equality_ext.
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