diff options
Diffstat (limited to 'theories7/Lists')
-rwxr-xr-x | theories7/Lists/List.v | 261 | ||||
-rw-r--r-- | theories7/Lists/ListSet.v | 389 | ||||
-rwxr-xr-x | theories7/Lists/MonoList.v | 259 | ||||
-rw-r--r-- | theories7/Lists/PolyList.v | 646 | ||||
-rw-r--r-- | theories7/Lists/PolyListSyntax.v | 10 | ||||
-rwxr-xr-x | theories7/Lists/Streams.v | 170 | ||||
-rwxr-xr-x | theories7/Lists/TheoryList.v | 386 |
7 files changed, 0 insertions, 2121 deletions
diff --git a/theories7/Lists/List.v b/theories7/Lists/List.v deleted file mode 100755 index 574b2688..00000000 --- a/theories7/Lists/List.v +++ /dev/null @@ -1,261 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: List.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -(* This file is a copy of file MonoList.v *) - -(** THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED ***) - -Require Le. - -Parameter List_Dom:Set. -Definition A := List_Dom. - -Inductive list : Set := nil : list | cons : A -> list -> list. - -Fixpoint app [l:list] : list -> list - := [m:list]<list>Cases l of - nil => m - | (cons a l1) => (cons a (app l1 m)) - end. - - -Lemma app_nil_end : (l:list)(l=(app l nil)). -Proof. - Intro l ; Elim l ; Simpl ; Auto. - Induction 1; Auto. -Qed. -Hints Resolve app_nil_end : list v62. - -Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)). -Proof. - Intros l m n ; Elim l ; Simpl ; Auto with list. - Induction 1; Auto with list. -Qed. -Hints Resolve app_ass : list v62. - -Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n). -Proof. - Auto with list. -Qed. -Hints Resolve ass_app : list v62. - -Definition tail := - [l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list. - - -Lemma nil_cons : (a:A)(m:list)~nil=(cons a m). - Intros; Discriminate. -Qed. - -(****************************************) -(* Length of lists *) -(****************************************) - -Fixpoint length [l:list] : nat - := <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end. - -(******************************) -(* Length order of lists *) -(******************************) - -Section length_order. -Definition lel := [l,m:list](le (length l) (length m)). - -Hints Unfold lel : list. - -Variables a,b:A. -Variables l,m,n:list. - -Lemma lel_refl : (lel l l). -Proof. - Unfold lel ; Auto with list. -Qed. - -Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). -Proof. - Unfold lel ; Intros. - Apply le_trans with (length m) ; Auto with list. -Qed. - -Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_cons : (lel l m)->(lel l (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). -Proof. - Intro l' ; Elim l' ; Auto with list arith. - Intros a' y H H0. - (* <list>nil=(cons a' y) - ============================ - H0 : (lel (cons a' y) nil) - H : (lel y nil)->(<list>nil=y) - y : list - a' : A - l' : list *) - Absurd (le (S (length y)) O); Auto with list arith. -Qed. -End length_order. - -Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons : list v62. - -Fixpoint In [a:A;l:list] : Prop := - Cases l of - nil => False - | (cons b m) => (b=a)\/(In a m) - end. - -Lemma in_eq : (a:A)(l:list)(In a (cons a l)). -Proof. - Simpl ; Auto with list. -Qed. -Hints Resolve in_eq : list v62. - -Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). -Proof. - Simpl ; Auto with list. -Qed. -Hints Resolve in_cons : list v62. - -Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Auto with list. - Intros a0 y H H0. - (* ((<A>a0=a)\/(In a y))\/(In a m) - ============================ - H0 : (<A>a0=a)\/(In a (app y m)) - H : (In a (app y m))->((In a y)\/(In a m)) - y : list - a0 : A - a : A - m : list - l : list *) - Elim H0 ; Auto with list. - Intro H1. - (* ((<A>a0=a)\/(In a y))\/(In a m) - ============================ - H1 : (In a (app y m)) *) - Elim (H H1) ; Auto with list. -Qed. -Hints Immediate in_app_or : list v62. - -Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Intro H. - (* 1 (In a m) - ============================ - H : False\/(In a m) - a : A - m : list - l : list *) - Elim H ; Auto with list ; Intro H0. - (* (In a m) - ============================ - H0 : False *) - Elim H0. (* subProof completed *) - Intros y H0 H1. - (* 2 (<A>H=a)\/(In a (app y m)) - ============================ - H1 : ((<A>H=a)\/(In a y))\/(In a m) - H0 : ((In a y)\/(In a m))->(In a (app y m)) - y : list *) - Elim H1 ; Auto 4 with list. - Intro H2. - (* (<A>H=a)\/(In a (app y m)) - ============================ - H2 : (<A>H=a)\/(In a y) *) - Elim H2 ; Auto with list. -Qed. -Hints Resolve in_or_app : list v62. - -Definition incl := [l,m:list](a:A)(In a l)->(In a m). - -Hints Unfold incl : list v62. - -Lemma incl_refl : (l:list)(incl l l). -Proof. - Auto with list. -Qed. -Hints Resolve incl_refl : list v62. - -Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_tl : list v62. - -Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). -Proof. - Auto with list. -Qed. - -Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_appl : list v62. - -Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_appr : list v62. - -Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). -Proof. - Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. - (* (In a0 m) - ============================ - H1 : (<A>a=a0)\/(In a0 l) - a0 : A - H0 : (a:A)(In a l)->(In a m) - H : (In a m) - m : list - l : list - a : A *) - Elim H1. - (* 1 (<A>a=a0)->(In a0 m) *) - Elim H1 ; Auto with list ; Intro H2. - (* (<A>a=a0)->(In a0 m) - ============================ - H2 : <A>a=a0 *) - Elim H2 ; Auto with list. (* solves subgoal *) - (* 2 (In a0 l)->(In a0 m) *) - Auto with list. -Qed. -Hints Resolve incl_cons : list v62. - -Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n). -Proof. - Unfold incl ; Simpl ; Intros l m n H H0 a H1. - (* (In a n) - ============================ - H1 : (In a (app l m)) - a : A - H0 : (a:A)(In a m)->(In a n) - H : (a:A)(In a l)->(In a n) - n : list - m : list - l : list *) - Elim (in_app_or l m a) ; Auto with list. -Qed. -Hints Resolve incl_app : list v62. diff --git a/theories7/Lists/ListSet.v b/theories7/Lists/ListSet.v deleted file mode 100644 index 9bf259da..00000000 --- a/theories7/Lists/ListSet.v +++ /dev/null @@ -1,389 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: ListSet.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -(** A Library for finite sets, implemented as lists - A Library with similar interface will soon be available under - the name TreeSet in the theories/Trees directory *) - -(** PolyList is loaded, but not exported. - This allow to "hide" the definitions, functions and theorems of PolyList - and to see only the ones of ListSet *) - -Require PolyList. - -Set Implicit Arguments. -V7only [Implicits nil [1].]. - -Section first_definitions. - - Variable A : Set. - Hypothesis Aeq_dec : (x,y:A){x=y}+{~x=y}. - - Definition set := (list A). - - Definition empty_set := (!nil ?) : set. - - Fixpoint set_add [a:A; x:set] : set := - Cases x of - | nil => (cons a nil) - | (cons a1 x1) => Cases (Aeq_dec a a1) of - | (left _) => (cons a1 x1) - | (right _) => (cons a1 (set_add a x1)) - end - end. - - - Fixpoint set_mem [a:A; x:set] : bool := - Cases x of - | nil => false - | (cons a1 x1) => Cases (Aeq_dec a a1) of - | (left _) => true - | (right _) => (set_mem a x1) - end - end. - - (** If [a] belongs to [x], removes [a] from [x]. If not, does nothing *) - Fixpoint set_remove [a:A; x:set] : set := - Cases x of - | nil => empty_set - | (cons a1 x1) => Cases (Aeq_dec a a1) of - | (left _) => x1 - | (right _) => (cons a1 (set_remove a x1)) - end - end. - - Fixpoint set_inter [x:set] : set -> set := - Cases x of - | nil => [y]nil - | (cons a1 x1) => [y]if (set_mem a1 y) - then (cons a1 (set_inter x1 y)) - else (set_inter x1 y) - end. - - Fixpoint set_union [x,y:set] : set := - Cases y of - | nil => x - | (cons a1 y1) => (set_add a1 (set_union x y1)) - end. - - (** returns the set of all els of [x] that does not belong to [y] *) - Fixpoint set_diff [x:set] : set -> set := - [y]Cases x of - | nil => nil - | (cons a1 x1) => if (set_mem a1 y) - then (set_diff x1 y) - else (set_add a1 (set_diff x1 y)) - end. - - - Definition set_In : A -> set -> Prop := (In 1!A). - - Lemma set_In_dec : (a:A; x:set){(set_In a x)}+{~(set_In a x)}. - - Proof. - Unfold set_In. - (*** Realizer set_mem. Program_all. ***) - Induction x. - Auto. - Intros a0 x0 Ha0. Case (Aeq_dec a a0); Intro eq. - Rewrite eq; Simpl; Auto with datatypes. - Elim Ha0. - Auto with datatypes. - Right; Simpl; Unfold not; Intros [Hc1 | Hc2 ]; Auto with datatypes. - Qed. - - Lemma set_mem_ind : - (B:Set)(P:B->Prop)(y,z:B)(a:A)(x:set) - ((set_In a x) -> (P y)) - ->(P z) - ->(P (if (set_mem a x) then y else z)). - - Proof. - Induction x; Simpl; Intros. - Assumption. - Elim (Aeq_dec a a0); Auto with datatypes. - Qed. - - Lemma set_mem_ind2 : - (B:Set)(P:B->Prop)(y,z:B)(a:A)(x:set) - ((set_In a x) -> (P y)) - ->(~(set_In a x) -> (P z)) - ->(P (if (set_mem a x) then y else z)). - - Proof. - Induction x; Simpl; Intros. - Apply H0; Red; Trivial. - Case (Aeq_dec a a0); Auto with datatypes. - Intro; Apply H; Intros; Auto. - Apply H1; Red; Intro. - Case H3; Auto. - Qed. - - - Lemma set_mem_correct1 : - (a:A)(x:set)(set_mem a x)=true -> (set_In a x). - Proof. - Induction x; Simpl. - Discriminate. - Intros a0 l; Elim (Aeq_dec a a0); Auto with datatypes. - Qed. - - Lemma set_mem_correct2 : - (a:A)(x:set)(set_In a x) -> (set_mem a x)=true. - Proof. - Induction x; Simpl. - Intro Ha; Elim Ha. - Intros a0 l; Elim (Aeq_dec a a0); Auto with datatypes. - Intros H1 H2 [H3 | H4]. - Absurd a0=a; Auto with datatypes. - Auto with datatypes. - Qed. - - Lemma set_mem_complete1 : - (a:A)(x:set)(set_mem a x)=false -> ~(set_In a x). - Proof. - Induction x; Simpl. - Tauto. - Intros a0 l; Elim (Aeq_dec a a0). - Intros; Discriminate H0. - Unfold not; Intros; Elim H1; Auto with datatypes. - Qed. - - Lemma set_mem_complete2 : - (a:A)(x:set)~(set_In a x) -> (set_mem a x)=false. - Proof. - Induction x; Simpl. - Tauto. - Intros a0 l; Elim (Aeq_dec a a0). - Intros; Elim H0; Auto with datatypes. - Tauto. - Qed. - - Lemma set_add_intro1 : (a,b:A)(x:set) - (set_In a x) -> (set_In a (set_add b x)). - - Proof. - Unfold set_In; Induction x; Simpl. - Auto with datatypes. - Intros a0 l H [ Ha0a | Hal ]. - Elim (Aeq_dec b a0); Left; Assumption. - Elim (Aeq_dec b a0); Right; [ Assumption | Auto with datatypes ]. - Qed. - - Lemma set_add_intro2 : (a,b:A)(x:set) - a=b -> (set_In a (set_add b x)). - - Proof. - Unfold set_In; Induction x; Simpl. - Auto with datatypes. - Intros a0 l H Hab. - Elim (Aeq_dec b a0); - [ Rewrite Hab; Intro Hba0; Rewrite Hba0; Simpl; Auto with datatypes - | Auto with datatypes ]. - Qed. - - Hints Resolve set_add_intro1 set_add_intro2. - - Lemma set_add_intro : (a,b:A)(x:set) - a=b\/(set_In a x) -> (set_In a (set_add b x)). - - Proof. - Intros a b x [H1 | H2] ; Auto with datatypes. - Qed. - - Lemma set_add_elim : (a,b:A)(x:set) - (set_In a (set_add b x)) -> a=b\/(set_In a x). - - Proof. - Unfold set_In. - Induction x. - Simpl; Intros [H1|H2]; Auto with datatypes. - Simpl; Do 3 Intro. - Elim (Aeq_dec b a0). - Simpl; Tauto. - Simpl; Intros; Elim H0. - Trivial with datatypes. - Tauto. - Tauto. - Qed. - - Lemma set_add_elim2 : (a,b:A)(x:set) - (set_In a (set_add b x)) -> ~(a=b) -> (set_In a x). - Intros a b x H; Case (set_add_elim H); Intros; Trivial. - Case H1; Trivial. - Qed. - - Hints Resolve set_add_intro set_add_elim set_add_elim2. - - Lemma set_add_not_empty : (a:A)(x:set)~(set_add a x)=empty_set. - Proof. - Induction x; Simpl. - Discriminate. - Intros; Elim (Aeq_dec a a0); Intros; Discriminate. - Qed. - - - Lemma set_union_intro1 : (a:A)(x,y:set) - (set_In a x) -> (set_In a (set_union x y)). - Proof. - Induction y; Simpl; Auto with datatypes. - Qed. - - Lemma set_union_intro2 : (a:A)(x,y:set) - (set_In a y) -> (set_In a (set_union x y)). - Proof. - Induction y; Simpl. - Tauto. - Intros; Elim H0; Auto with datatypes. - Qed. - - Hints Resolve set_union_intro2 set_union_intro1. - - Lemma set_union_intro : (a:A)(x,y:set) - (set_In a x)\/(set_In a y) -> (set_In a (set_union x y)). - Proof. - Intros; Elim H; Auto with datatypes. - Qed. - - Lemma set_union_elim : (a:A)(x,y:set) - (set_In a (set_union x y)) -> (set_In a x)\/(set_In a y). - Proof. - Induction y; Simpl. - Auto with datatypes. - Intros. - Generalize (set_add_elim H0). - Intros [H1 | H1]. - Auto with datatypes. - Tauto. - Qed. - - Lemma set_union_emptyL : (a:A)(x:set)(set_In a (set_union empty_set x)) -> (set_In a x). - Intros a x H; Case (set_union_elim H); Auto Orelse Contradiction. - Qed. - - - Lemma set_union_emptyR : (a:A)(x:set)(set_In a (set_union x empty_set)) -> (set_In a x). - Intros a x H; Case (set_union_elim H); Auto Orelse Contradiction. - Qed. - - - Lemma set_inter_intro : (a:A)(x,y:set) - (set_In a x) -> (set_In a y) -> (set_In a (set_inter x y)). - Proof. - Induction x. - Auto with datatypes. - Simpl; Intros a0 l Hrec y [Ha0a | Hal] Hy. - Simpl; Rewrite Ha0a. - Generalize (!set_mem_correct1 a y). - Generalize (!set_mem_complete1 a y). - Elim (set_mem a y); Simpl; Intros. - Auto with datatypes. - Absurd (set_In a y); Auto with datatypes. - Elim (set_mem a0 y); [ Right; Auto with datatypes | Auto with datatypes]. - Qed. - - Lemma set_inter_elim1 : (a:A)(x,y:set) - (set_In a (set_inter x y)) -> (set_In a x). - Proof. - Induction x. - Auto with datatypes. - Simpl; Intros a0 l Hrec y. - Generalize (!set_mem_correct1 a0 y). - Elim (set_mem a0 y); Simpl; Intros. - Elim H0; EAuto with datatypes. - EAuto with datatypes. - Qed. - - Lemma set_inter_elim2 : (a:A)(x,y:set) - (set_In a (set_inter x y)) -> (set_In a y). - Proof. - Induction x. - Simpl; Tauto. - Simpl; Intros a0 l Hrec y. - Generalize (!set_mem_correct1 a0 y). - Elim (set_mem a0 y); Simpl; Intros. - Elim H0; [ Intro Hr; Rewrite <- Hr; EAuto with datatypes | EAuto with datatypes ] . - EAuto with datatypes. - Qed. - - Hints Resolve set_inter_elim1 set_inter_elim2. - - Lemma set_inter_elim : (a:A)(x,y:set) - (set_In a (set_inter x y)) -> (set_In a x)/\(set_In a y). - Proof. - EAuto with datatypes. - Qed. - - Lemma set_diff_intro : (a:A)(x,y:set) - (set_In a x) -> ~(set_In a y) -> (set_In a (set_diff x y)). - Proof. - Induction x. - Simpl; Tauto. - Simpl; Intros a0 l Hrec y [Ha0a | Hal] Hay. - Rewrite Ha0a; Generalize (set_mem_complete2 Hay). - Elim (set_mem a y); [ Intro Habs; Discriminate Habs | Auto with datatypes ]. - Elim (set_mem a0 y); Auto with datatypes. - Qed. - - Lemma set_diff_elim1 : (a:A)(x,y:set) - (set_In a (set_diff x y)) -> (set_In a x). - Proof. - Induction x. - Simpl; Tauto. - Simpl; Intros a0 l Hrec y; Elim (set_mem a0 y). - EAuto with datatypes. - Intro; Generalize (set_add_elim H). - Intros [H1 | H2]; EAuto with datatypes. - Qed. - - Lemma set_diff_elim2 : (a:A)(x,y:set) - (set_In a (set_diff x y)) -> ~(set_In a y). - Intros a x y; Elim x; Simpl. - Intros; Contradiction. - Intros a0 l Hrec. - Apply set_mem_ind2; Auto. - Intros H1 H2; Case (set_add_elim H2); Intros; Auto. - Rewrite H; Trivial. - Qed. - - Lemma set_diff_trivial : (a:A)(x:set)~(set_In a (set_diff x x)). - Red; Intros a x H. - Apply (set_diff_elim2 H). - Apply (set_diff_elim1 H). - Qed. - -Hints Resolve set_diff_intro set_diff_trivial. - - -End first_definitions. - -Section other_definitions. - - Variables A,B : Set. - - Definition set_prod : (set A) -> (set B) -> (set A*B) := (list_prod 1!A 2!B). - - (** [B^A], set of applications from [A] to [B] *) - Definition set_power : (set A) -> (set B) -> (set (set A*B)) := - (list_power 1!A 2!B). - - Definition set_map : (A->B) -> (set A) -> (set B) := (map 1!A 2!B). - - Definition set_fold_left : (B -> A -> B) -> (set A) -> B -> B := - (fold_left 1!B 2!A). - - Definition set_fold_right : (A -> B -> B) -> (set A) -> B -> B := - [f][x][b](fold_right f b x). - - -End other_definitions. - -V7only [Implicits nil [].]. -Unset Implicit Arguments. diff --git a/theories7/Lists/MonoList.v b/theories7/Lists/MonoList.v deleted file mode 100755 index 2ab78f7f..00000000 --- a/theories7/Lists/MonoList.v +++ /dev/null @@ -1,259 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: MonoList.v,v 1.1.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -(** THIS IS A OLD CONTRIB. IT IS NO LONGER MAINTAINED ***) - -Require Le. - -Parameter List_Dom:Set. -Definition A := List_Dom. - -Inductive list : Set := nil : list | cons : A -> list -> list. - -Fixpoint app [l:list] : list -> list - := [m:list]<list>Cases l of - nil => m - | (cons a l1) => (cons a (app l1 m)) - end. - - -Lemma app_nil_end : (l:list)(l=(app l nil)). -Proof. - Intro l ; Elim l ; Simpl ; Auto. - Induction 1; Auto. -Qed. -Hints Resolve app_nil_end : list v62. - -Lemma app_ass : (l,m,n : list)(app (app l m) n)=(app l (app m n)). -Proof. - Intros l m n ; Elim l ; Simpl ; Auto with list. - Induction 1; Auto with list. -Qed. -Hints Resolve app_ass : list v62. - -Lemma ass_app : (l,m,n : list)(app l (app m n))=(app (app l m) n). -Proof. - Auto with list. -Qed. -Hints Resolve ass_app : list v62. - -Definition tail := - [l:list] <list>Cases l of (cons _ m) => m | _ => nil end : list->list. - - -Lemma nil_cons : (a:A)(m:list)~nil=(cons a m). - Intros; Discriminate. -Qed. - -(****************************************) -(* Length of lists *) -(****************************************) - -Fixpoint length [l:list] : nat - := <nat>Cases l of (cons _ m) => (S (length m)) | _ => O end. - -(******************************) -(* Length order of lists *) -(******************************) - -Section length_order. -Definition lel := [l,m:list](le (length l) (length m)). - -Hints Unfold lel : list. - -Variables a,b:A. -Variables l,m,n:list. - -Lemma lel_refl : (lel l l). -Proof. - Unfold lel ; Auto with list. -Qed. - -Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). -Proof. - Unfold lel ; Intros. - Apply le_trans with (length m) ; Auto with list. -Qed. - -Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_cons : (lel l m)->(lel l (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). -Proof. - Unfold lel ; Simpl ; Auto with list arith. -Qed. - -Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). -Proof. - Intro l' ; Elim l' ; Auto with list arith. - Intros a' y H H0. - (* <list>nil=(cons a' y) - ============================ - H0 : (lel (cons a' y) nil) - H : (lel y nil)->(<list>nil=y) - y : list - a' : A - l' : list *) - Absurd (le (S (length y)) O); Auto with list arith. -Qed. -End length_order. - -Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons : list v62. - -Fixpoint In [a:A;l:list] : Prop := - Cases l of - nil => False - | (cons b m) => (b=a)\/(In a m) - end. - -Lemma in_eq : (a:A)(l:list)(In a (cons a l)). -Proof. - Simpl ; Auto with list. -Qed. -Hints Resolve in_eq : list v62. - -Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). -Proof. - Simpl ; Auto with list. -Qed. -Hints Resolve in_cons : list v62. - -Lemma in_app_or : (l,m:list)(a:A)(In a (app l m))->((In a l)\/(In a m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Auto with list. - Intros a0 y H H0. - (* ((<A>a0=a)\/(In a y))\/(In a m) - ============================ - H0 : (<A>a0=a)\/(In a (app y m)) - H : (In a (app y m))->((In a y)\/(In a m)) - y : list - a0 : A - a : A - m : list - l : list *) - Elim H0 ; Auto with list. - Intro H1. - (* ((<A>a0=a)\/(In a y))\/(In a m) - ============================ - H1 : (In a (app y m)) *) - Elim (H H1) ; Auto with list. -Qed. -Hints Immediate in_app_or : list v62. - -Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (app l m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Intro H. - (* 1 (In a m) - ============================ - H : False\/(In a m) - a : A - m : list - l : list *) - Elim H ; Auto with list ; Intro H0. - (* (In a m) - ============================ - H0 : False *) - Elim H0. (* subProof completed *) - Intros y H0 H1. - (* 2 (<A>H=a)\/(In a (app y m)) - ============================ - H1 : ((<A>H=a)\/(In a y))\/(In a m) - H0 : ((In a y)\/(In a m))->(In a (app y m)) - y : list *) - Elim H1 ; Auto 4 with list. - Intro H2. - (* (<A>H=a)\/(In a (app y m)) - ============================ - H2 : (<A>H=a)\/(In a y) *) - Elim H2 ; Auto with list. -Qed. -Hints Resolve in_or_app : list v62. - -Definition incl := [l,m:list](a:A)(In a l)->(In a m). - -Hints Unfold incl : list v62. - -Lemma incl_refl : (l:list)(incl l l). -Proof. - Auto with list. -Qed. -Hints Resolve incl_refl : list v62. - -Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_tl : list v62. - -Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). -Proof. - Auto with list. -Qed. - -Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (app n m)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_appl : list v62. - -Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (app m n)). -Proof. - Auto with list. -Qed. -Hints Immediate incl_appr : list v62. - -Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). -Proof. - Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. - (* (In a0 m) - ============================ - H1 : (<A>a=a0)\/(In a0 l) - a0 : A - H0 : (a:A)(In a l)->(In a m) - H : (In a m) - m : list - l : list - a : A *) - Elim H1. - (* 1 (<A>a=a0)->(In a0 m) *) - Elim H1 ; Auto with list ; Intro H2. - (* (<A>a=a0)->(In a0 m) - ============================ - H2 : <A>a=a0 *) - Elim H2 ; Auto with list. (* solves subgoal *) - (* 2 (In a0 l)->(In a0 m) *) - Auto with list. -Qed. -Hints Resolve incl_cons : list v62. - -Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (app l m) n). -Proof. - Unfold incl ; Simpl ; Intros l m n H H0 a H1. - (* (In a n) - ============================ - H1 : (In a (app l m)) - a : A - H0 : (a:A)(In a m)->(In a n) - H : (a:A)(In a l)->(In a n) - n : list - m : list - l : list *) - Elim (in_app_or l m a) ; Auto with list. -Qed. -Hints Resolve incl_app : list v62. diff --git a/theories7/Lists/PolyList.v b/theories7/Lists/PolyList.v deleted file mode 100644 index e69ecd10..00000000 --- a/theories7/Lists/PolyList.v +++ /dev/null @@ -1,646 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: PolyList.v,v 1.2.2.1 2004/07/16 19:31:28 herbelin Exp $ i*) - -Require Le. - - -Section Lists. - -Variable A : Set. - -Set Implicit Arguments. - -Inductive list : Set := nil : list | cons : A -> list -> list. - -Infix "::" cons (at level 7, right associativity) : list_scope - V8only (at level 60, right associativity). - -Open Scope list_scope. - -(*************************) -(** Discrimination *) -(*************************) - -Lemma nil_cons : (a:A)(m:list)~(nil=(cons a m)). -Proof. - Intros; Discriminate. -Qed. - -(*************************) -(** Concatenation *) -(*************************) - -Fixpoint app [l:list] : list -> list - := [m:list]Cases l of - nil => m - | (cons a l1) => (cons a (app l1 m)) - end. - -Infix RIGHTA 7 "^" app : list_scope - V8only RIGHTA 60 "++". - -Lemma app_nil_end : (l:list)l=(l^nil). -Proof. - NewInduction l ; Simpl ; Auto. - Rewrite <- IHl; Auto. -Qed. -Hints Resolve app_nil_end. - -Tactic Definition now_show c := Change c. -V7only [Tactic Definition NowShow := now_show.]. - -Lemma app_ass : (l,m,n : list)((l^m)^ n)=(l^(m^n)). -Proof. - Intros. NewInduction l ; Simpl ; Auto. - NowShow '(cons a (app (app l m) n))=(cons a (app l (app m n))). - Rewrite <- IHl; Auto. -Qed. -Hints Resolve app_ass. - -Lemma ass_app : (l,m,n : list)(l^(m^n))=((l^m)^n). -Proof. - Auto. -Qed. -Hints Resolve ass_app. - -Lemma app_comm_cons : (x,y:list)(a:A) (cons a (x^y))=((cons a x)^y). -Proof. - Auto. -Qed. - -Lemma app_eq_nil: (x,y:list) (x^y)=nil -> x=nil /\ y=nil. -Proof. - NewDestruct x;NewDestruct y;Simpl;Auto. - Intros H;Discriminate H. - Intros;Discriminate H. -Qed. - -Lemma app_cons_not_nil: (x,y:list)(a:A)~nil=(x^(cons a y)). -Proof. -Unfold not . - NewDestruct x;Simpl;Intros. - Discriminate H. - Discriminate H. -Qed. - -Lemma app_eq_unit:(x,y:list)(a:A) - (x^y)=(cons a nil)-> (x=nil)/\ y=(cons a nil) \/ x=(cons a nil)/\ y=nil. - -Proof. - NewDestruct x;NewDestruct y;Simpl. - Intros a H;Discriminate H. - Left;Split;Auto. - Right;Split;Auto. - Generalize H . - Generalize (app_nil_end l) ;Intros E. - Rewrite <- E;Auto. - Intros. - Injection H. - Intro. - Cut nil=(l^(cons a0 l0));Auto. - Intro. - Generalize (app_cons_not_nil H1); Intro. - Elim H2. -Qed. - -Lemma app_inj_tail : (x,y:list)(a,b:A) - (x^(cons a nil))=(y^(cons b nil)) -> x=y /\ a=b. -Proof. - NewInduction x as [|x l IHl];NewDestruct y;Simpl;Auto. - Intros a b H. - Injection H. - Auto. - Intros a0 b H. - Injection H;Intros. - Generalize (app_cons_not_nil H0) ;NewDestruct 1. - Intros a b H. - Injection H;Intros. - Cut nil=(l^(cons a nil));Auto. - Intro. - Generalize (app_cons_not_nil H2) ;NewDestruct 1. - Intros a0 b H. - Injection H;Intros. - NewDestruct (IHl l0 a0 b H0). - Split;Auto. - Rewrite <- H1;Rewrite <- H2;Reflexivity. -Qed. - -(*************************) -(** Head and tail *) -(*************************) - -Definition head := - [l:list]Cases l of - | nil => Error - | (cons x _) => (Value x) - end. - -Definition tail : list -> list := - [l:list]Cases l of - | nil => nil - | (cons a m) => m - end. - -(****************************************) -(** Length of lists *) -(****************************************) - -Fixpoint length [l:list] : nat - := Cases l of nil => O | (cons _ m) => (S (length m)) end. - -(******************************) -(** Length order of lists *) -(******************************) - -Section length_order. -Definition lel := [l,m:list](le (length l) (length m)). - -Variables a,b:A. -Variables l,m,n:list. - -Lemma lel_refl : (lel l l). -Proof. - Unfold lel ; Auto with arith. -Qed. - -Lemma lel_trans : (lel l m)->(lel m n)->(lel l n). -Proof. - Unfold lel ; Intros. - NowShow '(le (length l) (length n)). - Apply le_trans with (length m) ; Auto with arith. -Qed. - -Lemma lel_cons_cons : (lel l m)->(lel (cons a l) (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with arith. -Qed. - -Lemma lel_cons : (lel l m)->(lel l (cons b m)). -Proof. - Unfold lel ; Simpl ; Auto with arith. -Qed. - -Lemma lel_tail : (lel (cons a l) (cons b m)) -> (lel l m). -Proof. - Unfold lel ; Simpl ; Auto with arith. -Qed. - -Lemma lel_nil : (l':list)(lel l' nil)->(nil=l'). -Proof. - Intro l' ; Elim l' ; Auto with arith. - Intros a' y H H0. - NowShow 'nil=(cons a' y). - Absurd (le (S (length y)) O); Auto with arith. -Qed. -End length_order. - -Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons. - -(*********************************) -(** The [In] predicate *) -(*********************************) - -Fixpoint In [a:A;l:list] : Prop := - Cases l of nil => False | (cons b m) => (b=a)\/(In a m) end. - -Lemma in_eq : (a:A)(l:list)(In a (cons a l)). -Proof. - Simpl ; Auto. -Qed. -Hints Resolve in_eq. - -Lemma in_cons : (a,b:A)(l:list)(In b l)->(In b (cons a l)). -Proof. - Simpl ; Auto. -Qed. -Hints Resolve in_cons. - -Lemma in_nil : (a:A)~(In a nil). -Proof. - Unfold not; Intros a H; Inversion_clear H. -Qed. - - -Lemma in_inv : (a,b:A)(l:list) - (In b (cons a l)) -> a=b \/ (In b l). -Proof. - Intros a b l H ; Inversion_clear H ; Auto. -Qed. - -Lemma In_dec : ((x,y:A){x=y}+{~x=y}) -> (a:A)(l:list){(In a l)}+{~(In a l)}. - -Proof. - NewInduction l as [|a0 l IHl]. - Right; Apply in_nil. - NewDestruct (H a0 a); Simpl; Auto. - NewDestruct IHl; Simpl; Auto. - Right; Unfold not; Intros [Hc1 | Hc2]; Auto. -Qed. - -Lemma in_app_or : (l,m:list)(a:A)(In a (l^m))->((In a l)\/(In a m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Auto. - Intros a0 y H H0. - NowShow '(a0=a\/(In a y))\/(In a m). - Elim H0 ; Auto. - Intro H1. - NowShow '(a0=a\/(In a y))\/(In a m). - Elim (H H1) ; Auto. -Qed. -Hints Immediate in_app_or. - -Lemma in_or_app : (l,m:list)(a:A)((In a l)\/(In a m))->(In a (l^m)). -Proof. - Intros l m a. - Elim l ; Simpl ; Intro H. - NowShow '(In a m). - Elim H ; Auto ; Intro H0. - NowShow '(In a m). - Elim H0. (* subProof completed *) - Intros y H0 H1. - NowShow 'H=a\/(In a (app y m)). - Elim H1 ; Auto 4. - Intro H2. - NowShow 'H=a\/(In a (app y m)). - Elim H2 ; Auto. -Qed. -Hints Resolve in_or_app. - -(***************************) -(** Set inclusion on list *) -(***************************) - -Definition incl := [l,m:list](a:A)(In a l)->(In a m). -Hints Unfold incl. - -Lemma incl_refl : (l:list)(incl l l). -Proof. - Auto. -Qed. -Hints Resolve incl_refl. - -Lemma incl_tl : (a:A)(l,m:list)(incl l m)->(incl l (cons a m)). -Proof. - Auto. -Qed. -Hints Immediate incl_tl. - -Lemma incl_tran : (l,m,n:list)(incl l m)->(incl m n)->(incl l n). -Proof. - Auto. -Qed. - -Lemma incl_appl : (l,m,n:list)(incl l n)->(incl l (n^m)). -Proof. - Auto. -Qed. -Hints Immediate incl_appl. - -Lemma incl_appr : (l,m,n:list)(incl l n)->(incl l (m^n)). -Proof. - Auto. -Qed. -Hints Immediate incl_appr. - -Lemma incl_cons : (a:A)(l,m:list)(In a m)->(incl l m)->(incl (cons a l) m). -Proof. - Unfold incl ; Simpl ; Intros a l m H H0 a0 H1. - NowShow '(In a0 m). - Elim H1. - NowShow 'a=a0->(In a0 m). - Elim H1 ; Auto ; Intro H2. - NowShow 'a=a0->(In a0 m). - Elim H2 ; Auto. (* solves subgoal *) - NowShow '(In a0 l)->(In a0 m). - Auto. -Qed. -Hints Resolve incl_cons. - -Lemma incl_app : (l,m,n:list)(incl l n)->(incl m n)->(incl (l^m) n). -Proof. - Unfold incl ; Simpl ; Intros l m n H H0 a H1. - NowShow '(In a n). - Elim (in_app_or H1); Auto. -Qed. -Hints Resolve incl_app. - -(**************************) -(** Nth element of a list *) -(**************************) - -Fixpoint nth [n:nat; l:list] : A->A := - [default]Cases n l of - O (cons x l') => x - | O other => default - | (S m) nil => default - | (S m) (cons x t) => (nth m t default) - end. - -Fixpoint nth_ok [n:nat; l:list] : A->bool := - [default]Cases n l of - O (cons x l') => true - | O other => false - | (S m) nil => false - | (S m) (cons x t) => (nth_ok m t default) - end. - -Lemma nth_in_or_default : - (n:nat)(l:list)(d:A){(In (nth n l d) l)}+{(nth n l d)=d}. -(* Realizer nth_ok. Program_all. *) -Proof. - Intros n l d; Generalize n; NewInduction l; Intro n0. - Right; Case n0; Trivial. - Case n0; Simpl. - Auto. - Intro n1; Elim (IHl n1); Auto. -Qed. - -Lemma nth_S_cons : - (n:nat)(l:list)(d:A)(a:A)(In (nth n l d) l) - ->(In (nth (S n) (cons a l) d) (cons a l)). -Proof. - Simpl; Auto. -Qed. - -Fixpoint nth_error [l:list;n:nat] : (Exc A) := - Cases n l of - | O (cons x _) => (Value x) - | (S n) (cons _ l) => (nth_error l n) - | _ _ => Error - end. - -Definition nth_default : A -> list -> nat -> A := - [default,l,n]Cases (nth_error l n) of - | (Some x) => x - | None => default - end. - -Lemma nth_In : - (n:nat)(l:list)(d:A)(lt n (length l))->(In (nth n l d) l). - -Proof. -Unfold lt; NewInduction n as [|n hn]; Simpl. -NewDestruct l ; Simpl ; [ Inversion 2 | Auto]. -NewDestruct l as [|a l hl] ; Simpl. -Inversion 2. -Intros d ie ; Right ; Apply hn ; Auto with arith. -Qed. - -(********************************) -(** Decidable equality on lists *) -(********************************) - - -Lemma list_eq_dec : ((x,y:A){x=y}+{~x=y})->(x,y:list){x=y}+{~x=y}. -Proof. - NewInduction x as [|a l IHl]; NewDestruct y as [|a0 l0]; Auto. - NewDestruct (H a a0) as [e|e]. - NewDestruct (IHl l0) as [e'|e']. - Left; Rewrite e; Rewrite e'; Trivial. - Right; Red; Intro. - Apply e'; Injection H0; Trivial. - Right; Red; Intro. - Apply e; Injection H0; Trivial. -Qed. - -(*************************) -(** Reverse *) -(*************************) - -Fixpoint rev [l:list] : list := - Cases l of - nil => nil - | (cons x l') => (rev l')^(cons x nil) - end. - -Lemma distr_rev : - (x,y:list) (rev (x^y))=((rev y)^(rev x)). -Proof. - NewInduction x as [|a l IHl]. - NewDestruct y. - Simpl. - Auto. - - Simpl. - Apply app_nil_end;Auto. - - Intro y. - Simpl. - Rewrite (IHl y). - Apply (app_ass (rev y) (rev l) (cons a nil)). -Qed. - -Remark rev_unit : (l:list)(a:A) (rev l^(cons a nil))= (cons a (rev l)). -Proof. - Intros. - Apply (distr_rev l (cons a nil));Simpl;Auto. -Qed. - -Lemma idempot_rev : (l:list)(rev (rev l))=l. -Proof. - NewInduction l as [|a l IHl]. - Simpl;Auto. - - Simpl. - Rewrite (rev_unit (rev l) a). - Rewrite -> IHl;Auto. -Qed. - -(*********************************************) -(** Reverse Induction Principle on Lists *) -(*********************************************) - -Section Reverse_Induction. - -Unset Implicit Arguments. - -Remark rev_list_ind: (P:list->Prop) - (P nil) - ->((a:A)(l:list)(P (rev l))->(P (rev (cons a l)))) - ->(l:list) (P (rev l)). -Proof. - NewInduction l; Auto. -Qed. -Set Implicit Arguments. - -Lemma rev_ind : - (P:list->Prop) - (P nil)-> - ((x:A)(l:list)(P l)->(P l^(cons x nil))) - ->(l:list)(P l). -Proof. - Intros. - Generalize (idempot_rev l) . - Intros E;Rewrite <- E. - Apply (rev_list_ind P). - Auto. - - Simpl. - Intros. - Apply (H0 a (rev l0)). - Auto. -Qed. - -End Reverse_Induction. - -End Lists. - -Implicits nil [1]. - -Hints Resolve nil_cons app_nil_end ass_app app_ass : datatypes v62. -Hints Resolve app_comm_cons app_cons_not_nil : datatypes v62. -Hints Immediate app_eq_nil : datatypes v62. -Hints Resolve app_eq_unit app_inj_tail : datatypes v62. -Hints Resolve lel_refl lel_cons_cons lel_cons lel_nil lel_nil nil_cons - : datatypes v62. -Hints Resolve in_eq in_cons in_inv in_nil in_app_or in_or_app : datatypes v62. -Hints Resolve incl_refl incl_tl incl_tran incl_appl incl_appr incl_cons incl_app - : datatypes v62. - -Section Functions_on_lists. - -(****************************************************************) -(** Some generic functions on lists and basic functions of them *) -(****************************************************************) - -Section Map. -Variables A,B:Set. -Variable f:A->B. -Fixpoint map [l:(list A)] : (list B) := - Cases l of - nil => nil - | (cons a t) => (cons (f a) (map t)) - end. -End Map. - -Lemma in_map : (A,B:Set)(f:A->B)(l:(list A))(x:A) - (In x l) -> (In (f x) (map f l)). -Proof. - NewInduction l as [|a l IHl]; Simpl; - [ Auto - | NewDestruct 1; - [ Left; Apply f_equal with f:=f; Assumption - | Auto] - ]. -Qed. - -Fixpoint flat_map [A,B:Set; f:A->(list B); l:(list A)] : (list B) := - Cases l of - nil => nil - | (cons x t) => (app (f x) (flat_map f t)) - end. - -Fixpoint list_prod [A:Set; B:Set; l:(list A)] : (list B)->(list A*B) := - [l']Cases l of - nil => nil - | (cons x t) => (app (map [y:B](x,y) l') - (list_prod t l')) - end. - -Lemma in_prod_aux : - (A:Set)(B:Set)(x:A)(y:B)(l:(list B)) - (In y l) -> (In (x,y) (map [y0:B](x,y0) l)). -Proof. - NewInduction l; - [ Simpl; Auto - | Simpl; NewDestruct 1 as [H1|]; - [ Left; Rewrite H1; Trivial - | Right; Auto] - ]. -Qed. - -Lemma in_prod : (A:Set)(B:Set)(l:(list A))(l':(list B)) - (x:A)(y:B)(In x l)->(In y l')->(In (x,y) (list_prod l l')). -Proof. - NewInduction l; - [ Simpl; Tauto - | Simpl; Intros; Apply in_or_app; NewDestruct H; - [ Left; Rewrite H; Apply in_prod_aux; Assumption - | Right; Auto] - ]. -Qed. - -(** [(list_power x y)] is [y^x], or the set of sequences of elts of [y] - indexed by elts of [x], sorted in lexicographic order. *) - -Fixpoint list_power [A,B:Set; l:(list A)] : (list B)->(list (list A*B)) := - [l']Cases l of - nil => (cons nil nil) - | (cons x t) => (flat_map [f:(list A*B)](map [y:B](cons (x,y) f) l') - (list_power t l')) - end. - -(************************************) -(** Left-to-right iterator on lists *) -(************************************) - -Section Fold_Left_Recursor. -Variables A,B:Set. -Variable f:A->B->A. -Fixpoint fold_left[l:(list B)] : A -> A := -[a0]Cases l of - nil => a0 - | (cons b t) => (fold_left t (f a0 b)) - end. -End Fold_Left_Recursor. - -(************************************) -(** Right-to-left iterator on lists *) -(************************************) - -Section Fold_Right_Recursor. -Variables A,B:Set. -Variable f:B->A->A. -Variable a0:A. -Fixpoint fold_right [l:(list B)] : A := - Cases l of - nil => a0 - | (cons b t) => (f b (fold_right t)) - end. -End Fold_Right_Recursor. - -Theorem fold_symmetric : - (A:Set)(f:A->A->A) - ((x,y,z:A)(f x (f y z))=(f (f x y) z)) - ->((x,y:A)(f x y)=(f y x)) - ->(a0:A)(l:(list A))(fold_left f l a0)=(fold_right f a0 l). -Proof. -NewDestruct l as [|a l]. -Reflexivity. -Simpl. -Rewrite <- H0. -Generalize a0 a. -NewInduction l as [|a3 l IHl]; Simpl. -Trivial. -Intros. -Rewrite H. -Rewrite (H0 a2). -Rewrite <- (H a1). -Rewrite (H0 a1). -Rewrite IHl. -Reflexivity. -Qed. - -End Functions_on_lists. - -V7only [Implicits nil [].]. - -(** Exporting list notations *) - -V8Infix "::" cons (at level 60, right associativity) : list_scope. - -Infix RIGHTA 7 "^" app : list_scope V8only RIGHTA 60 "++". - -Open Scope list_scope. - -Delimits Scope list_scope with list. - -Bind Scope list_scope with list. diff --git a/theories7/Lists/PolyListSyntax.v b/theories7/Lists/PolyListSyntax.v deleted file mode 100644 index 15c57166..00000000 --- a/theories7/Lists/PolyListSyntax.v +++ /dev/null @@ -1,10 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: PolyListSyntax.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) - diff --git a/theories7/Lists/Streams.v b/theories7/Lists/Streams.v deleted file mode 100755 index ccfc4895..00000000 --- a/theories7/Lists/Streams.v +++ /dev/null @@ -1,170 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: Streams.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) - -Set Implicit Arguments. - -(** Streams *) - -Section Streams. - -Variable A : Set. - -CoInductive Set Stream := Cons : A->Stream->Stream. - - -Definition hd := - [x:Stream] Cases x of (Cons a _) => a end. - -Definition tl := - [x:Stream] Cases x of (Cons _ s) => s end. - - -Fixpoint Str_nth_tl [n:nat] : Stream->Stream := - [s:Stream] Cases n of - O => s - |(S m) => (Str_nth_tl m (tl s)) - end. - -Definition Str_nth : nat->Stream->A := [n:nat][s:Stream](hd (Str_nth_tl n s)). - - -Lemma unfold_Stream :(x:Stream)x=(Cases x of (Cons a s) => (Cons a s) end). -Proof. - Intro x. - Case x. - Trivial. -Qed. - -Lemma tl_nth_tl : (n:nat)(s:Stream)(tl (Str_nth_tl n s))=(Str_nth_tl n (tl s)). -Proof. - Induction n; Simpl; Auto. -Qed. -Hints Resolve tl_nth_tl : datatypes v62. - -Lemma Str_nth_tl_plus -: (n,m:nat)(s:Stream)(Str_nth_tl n (Str_nth_tl m s))=(Str_nth_tl (plus n m) s). -Induction n; Simpl; Intros; Auto with datatypes. -Rewrite <- H. -Rewrite tl_nth_tl; Trivial with datatypes. -Qed. - -Lemma Str_nth_plus - : (n,m:nat)(s:Stream)(Str_nth n (Str_nth_tl m s))=(Str_nth (plus n m) s). -Intros; Unfold Str_nth; Rewrite Str_nth_tl_plus; Trivial with datatypes. -Qed. - -(** Extensional Equality between two streams *) - -CoInductive EqSt : Stream->Stream->Prop := - eqst : (s1,s2:Stream) - ((hd s1)=(hd s2))-> - (EqSt (tl s1) (tl s2)) - ->(EqSt s1 s2). - -(** A coinduction principle *) - -Tactic Definition CoInduction proof := - Cofix proof; Intros; Constructor; - [Clear proof | Try (Apply proof;Clear proof)]. - - -(** Extensional equality is an equivalence relation *) - -Theorem EqSt_reflex : (s:Stream)(EqSt s s). -CoInduction EqSt_reflex. -Reflexivity. -Qed. - -Theorem sym_EqSt : - (s1:Stream)(s2:Stream)(EqSt s1 s2)->(EqSt s2 s1). -(CoInduction Eq_sym). -Case H;Intros;Symmetry;Assumption. -Case H;Intros;Assumption. -Qed. - - -Theorem trans_EqSt : - (s1,s2,s3:Stream)(EqSt s1 s2)->(EqSt s2 s3)->(EqSt s1 s3). -(CoInduction Eq_trans). -Transitivity (hd s2). -Case H; Intros; Assumption. -Case H0; Intros; Assumption. -Apply (Eq_trans (tl s1) (tl s2) (tl s3)). -Case H; Trivial with datatypes. -Case H0; Trivial with datatypes. -Qed. - -(** The definition given is equivalent to require the elements at each - position to be equal *) - -Theorem eqst_ntheq : - (n:nat)(s1,s2:Stream)(EqSt s1 s2)->(Str_nth n s1)=(Str_nth n s2). -Unfold Str_nth; Induction n. -Intros s1 s2 H; Case H; Trivial with datatypes. -Intros m hypind. -Simpl. -Intros s1 s2 H. -Apply hypind. -Case H; Trivial with datatypes. -Qed. - -Theorem ntheq_eqst : - (s1,s2:Stream)((n:nat)(Str_nth n s1)=(Str_nth n s2))->(EqSt s1 s2). -(CoInduction Equiv2). -Apply (H O). -Intros n; Apply (H (S n)). -Qed. - -Section Stream_Properties. - -Variable P : Stream->Prop. - -(*i -Inductive Exists : Stream -> Prop := - | Here : forall x:Stream, P x -> Exists x - | Further : forall x:Stream, ~ P x -> Exists (tl x) -> Exists x. -i*) - -Inductive Exists : Stream -> Prop := - Here : (x:Stream)(P x) ->(Exists x) | - Further : (x:Stream)(Exists (tl x))->(Exists x). - -CoInductive ForAll : Stream -> Prop := - forall : (x:Stream)(P x)->(ForAll (tl x))->(ForAll x). - - -Section Co_Induction_ForAll. -Variable Inv : Stream -> Prop. -Hypothesis InvThenP : (x:Stream)(Inv x)->(P x). -Hypothesis InvIsStable: (x:Stream)(Inv x)->(Inv (tl x)). - -Theorem ForAll_coind : (x:Stream)(Inv x)->(ForAll x). -(CoInduction ForAll_coind);Auto. -Qed. -End Co_Induction_ForAll. - -End Stream_Properties. - -End Streams. - -Section Map. -Variables A,B : Set. -Variable f : A->B. -CoFixpoint map : (Stream A)->(Stream B) := - [s:(Stream A)](Cons (f (hd s)) (map (tl s))). -End Map. - -Section Constant_Stream. -Variable A : Set. -Variable a : A. -CoFixpoint const : (Stream A) := (Cons a const). -End Constant_Stream. - -Unset Implicit Arguments. diff --git a/theories7/Lists/TheoryList.v b/theories7/Lists/TheoryList.v deleted file mode 100755 index f7adda70..00000000 --- a/theories7/Lists/TheoryList.v +++ /dev/null @@ -1,386 +0,0 @@ -(************************************************************************) -(* v * The Coq Proof Assistant / The Coq Development Team *) -(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *) -(* \VV/ **************************************************************) -(* // * This file is distributed under the terms of the *) -(* * GNU Lesser General Public License Version 2.1 *) -(************************************************************************) - -(*i $Id: TheoryList.v,v 1.1.2.1 2004/07/16 19:31:29 herbelin Exp $ i*) - -(** Some programs and results about lists following CAML Manual *) - -Require Export PolyList. -Set Implicit Arguments. -Chapter Lists. - -Variable A : Set. - -(**********************) -(** The null function *) -(**********************) - -Definition Isnil : (list A) -> Prop := [l:(list A)](nil A)=l. - -Lemma Isnil_nil : (Isnil (nil A)). -Red; Auto. -Qed. -Hints Resolve Isnil_nil. - -Lemma not_Isnil_cons : (a:A)(l:(list A))~(Isnil (cons a l)). -Unfold Isnil. -Intros; Discriminate. -Qed. - -Hints Resolve Isnil_nil not_Isnil_cons. - -Lemma Isnil_dec : (l:(list A)){(Isnil l)}+{~(Isnil l)}. -Intro l; Case l;Auto. -(* -Realizer (fun l => match l with - | nil => true - | _ => false - end). -*) -Qed. - -(************************) -(** The Uncons function *) -(************************) - -Lemma Uncons : (l:(list A)){a : A & { m: (list A) | (cons a m)=l}}+{Isnil l}. -Intro l; Case l. -Auto. -Intros a m; Intros; Left; Exists a; Exists m; Reflexivity. -(* -Realizer (fun l => match l with - | nil => error - | (cons a m) => value (a,m) - end). -*) -Qed. - -(********************************) -(** The head function *) -(********************************) - -Lemma Hd : (l:(list A)){a : A | (EX m:(list A) |(cons a m)=l)}+{Isnil l}. -Intro l; Case l. -Auto. -Intros a m; Intros; Left; Exists a; Exists m; Reflexivity. -(* -Realizer (fun l => match l with - | nil => error - | (cons a m) => value a - end). -*) -Qed. - -Lemma Tl : (l:(list A)){m:(list A)| (EX a:A |(cons a m)=l) - \/ ((Isnil l) /\ (Isnil m)) }. -Intro l; Case l. -Exists (nil A); Auto. -Intros a m; Intros; Exists m; Left; Exists a; Reflexivity. -(* -Realizer (fun l => match l with - | nil => nil - | (cons a m) => m - end). -*) -Qed. - -(****************************************) -(** Length of lists *) -(****************************************) - -(* length is defined in List *) -Fixpoint Length_l [l:(list A)] : nat -> nat - := [n:nat] Cases l of - nil => n - | (cons _ m) => (Length_l m (S n)) - end. - -(* A tail recursive version *) -Lemma Length_l_pf : (l:(list A))(n:nat){m:nat|(plus n (length l))=m}. -NewInduction l as [|a m lrec]. -Intro n; Exists n; Simpl; Auto. -Intro n; Elim (lrec (S n)); Simpl; Intros. -Exists x; Transitivity (S (plus n (length m))); Auto. -(* -Realizer Length_l. -*) -Qed. - -Lemma Length : (l:(list A)){m:nat|(length l)=m}. -Intro l. Apply (Length_l_pf l O). -(* -Realizer (fun l -> Length_l_pf l O). -*) -Qed. - -(*******************************) -(** Members of lists *) -(*******************************) -Inductive In_spec [a:A] : (list A) -> Prop := - | in_hd : (l:(list A))(In_spec a (cons a l)) - | in_tl : (l:(list A))(b:A)(In a l)->(In_spec a (cons b l)). -Hints Resolve in_hd in_tl. -Hints Unfold In. -Hints Resolve in_cons. - -Theorem In_In_spec : (a:A)(l:(list A))(In a l) <-> (In_spec a l). -Split. -Elim l; [ Intros; Contradiction - | Intros; Elim H0; - [ Intros; Rewrite H1; Auto - | Auto ]]. -Intros; Elim H; Auto. -Qed. - -Inductive AllS [P:A->Prop] : (list A) -> Prop - := allS_nil : (AllS P (nil A)) - | allS_cons : (a:A)(l:(list A))(P a)->(AllS P l)->(AllS P (cons a l)). -Hints Resolve allS_nil allS_cons. - -Hypothesis eqA_dec : (a,b:A){a=b}+{~a=b}. - -Fixpoint mem [a:A; l:(list A)] : bool := - Cases l of - nil => false - | (cons b m) => if (eqA_dec a b) then [H]true else [H](mem a m) - end. - -Hints Unfold In. -Lemma Mem : (a:A)(l:(list A)){(In a l)}+{(AllS [b:A]~b=a l)}. -Intros a l. -NewInduction l. -Auto. -Elim (eqA_dec a a0). -Auto. -Simpl. Elim IHl; Auto. -(* -Realizer mem. -*) -Qed. - -(*********************************) -(** Index of elements *) -(*********************************) - -Require Le. -Require Lt. - -Inductive nth_spec : (list A)->nat->A->Prop := - nth_spec_O : (a:A)(l:(list A))(nth_spec (cons a l) (S O) a) -| nth_spec_S : (n:nat)(a,b:A)(l:(list A)) - (nth_spec l n a)->(nth_spec (cons b l) (S n) a). -Hints Resolve nth_spec_O nth_spec_S. - -Inductive fst_nth_spec : (list A)->nat->A->Prop := - fst_nth_O : (a:A)(l:(list A))(fst_nth_spec (cons a l) (S O) a) -| fst_nth_S : (n:nat)(a,b:A)(l:(list A))(~a=b)-> - (fst_nth_spec l n a)->(fst_nth_spec (cons b l) (S n) a). -Hints Resolve fst_nth_O fst_nth_S. - -Lemma fst_nth_nth : (l:(list A))(n:nat)(a:A)(fst_nth_spec l n a)->(nth_spec l n a). -NewInduction 1; Auto. -Qed. -Hints Immediate fst_nth_nth. - -Lemma nth_lt_O : (l:(list A))(n:nat)(a:A)(nth_spec l n a)->(lt O n). -NewInduction 1; Auto. -Qed. - -Lemma nth_le_length : (l:(list A))(n:nat)(a:A)(nth_spec l n a)->(le n (length l)). -NewInduction 1; Simpl; Auto with arith. -Qed. - -Fixpoint Nth_func [l:(list A)] : nat -> (Exc A) - := [n:nat] Cases l n of - (cons a _) (S O) => (value A a) - | (cons _ l') (S (S p)) => (Nth_func l' (S p)) - | _ _ => Error - end. - -Lemma Nth : (l:(list A))(n:nat) - {a:A|(nth_spec l n a)}+{(n=O)\/(lt (length l) n)}. -NewInduction l as [|a l IHl]. -Intro n; Case n; Simpl; Auto with arith. -Intro n; NewDestruct n as [|[|n1]]; Simpl; Auto. -Left; Exists a; Auto. -NewDestruct (IHl (S n1)) as [[b]|o]. -Left; Exists b; Auto. -Right; NewDestruct o. -Absurd (S n1)=O; Auto. -Auto with arith. -(* -Realizer Nth_func. -*) -Qed. - -Lemma Item : (l:(list A))(n:nat){a:A|(nth_spec l (S n) a)}+{(le (length l) n)}. -Intros l n; Case (Nth l (S n)); Intro. -Case s; Intro a; Left; Exists a; Auto. -Right; Case o; Intro. -Absurd (S n)=O; Auto. -Auto with arith. -Qed. - -Require Minus. -Require DecBool. - -Fixpoint index_p [a:A;l:(list A)] : nat -> (Exc nat) := - Cases l of nil => [p]Error - | (cons b m) => [p](ifdec (eqA_dec a b) (Value p) (index_p a m (S p))) - end. - -Lemma Index_p : (a:A)(l:(list A))(p:nat) - {n:nat|(fst_nth_spec l (minus (S n) p) a)}+{(AllS [b:A]~a=b l)}. -NewInduction l as [|b m irec]. -Auto. -Intro p. -NewDestruct (eqA_dec a b) as [e|e]. -Left; Exists p. -NewDestruct e; Elim minus_Sn_m; Trivial; Elim minus_n_n; Auto with arith. -NewDestruct (irec (S p)) as [[n H]|]. -Left; Exists n; Auto with arith. -Elim minus_Sn_m; Auto with arith. -Apply lt_le_weak; Apply lt_O_minus_lt; Apply nth_lt_O with m a; Auto with arith. -Auto. -Qed. - -Lemma Index : (a:A)(l:(list A)) - {n:nat|(fst_nth_spec l n a)}+{(AllS [b:A]~a=b l)}. - -Intros a l; Case (Index_p a l (S O)); Auto. -Intros (n,P); Left; Exists n; Auto. -Rewrite (minus_n_O n); Trivial. -(* -Realizer (fun a l -> Index_p a l (S O)). -*) -Qed. - -Section Find_sec. -Variable R,P : A -> Prop. - -Inductive InR : (list A) -> Prop - := inR_hd : (a:A)(l:(list A))(R a)->(InR (cons a l)) - | inR_tl : (a:A)(l:(list A))(InR l)->(InR (cons a l)). -Hints Resolve inR_hd inR_tl. - -Definition InR_inv := - [l:(list A)]Cases l of - nil => False - | (cons b m) => (R b)\/(InR m) - end. - -Lemma InR_INV : (l:(list A))(InR l)->(InR_inv l). -NewInduction 1; Simpl; Auto. -Qed. - -Lemma InR_cons_inv : (a:A)(l:(list A))(InR (cons a l))->((R a)\/(InR l)). -Intros a l H; Exact (InR_INV H). -Qed. - -Lemma InR_or_app : (l,m:(list A))((InR l)\/(InR m))->(InR (app l m)). -Intros l m [|]. -NewInduction 1; Simpl; Auto. -Intro. NewInduction l; Simpl; Auto. -Qed. - -Lemma InR_app_or : (l,m:(list A))(InR (app l m))->((InR l)\/(InR m)). -Intros l m; Elim l; Simpl; Auto. -Intros b l' Hrec IAc; Elim (InR_cons_inv IAc);Auto. -Intros; Elim Hrec; Auto. -Qed. - -Hypothesis RS_dec : (a:A){(R a)}+{(P a)}. - -Fixpoint find [l:(list A)] : (Exc A) := - Cases l of nil => Error - | (cons a m) => (ifdec (RS_dec a) (Value a) (find m)) - end. - -Lemma Find : (l:(list A)){a:A | (In a l) & (R a)}+{(AllS P l)}. -NewInduction l as [|a m [[b H1 H2]|H]]; Auto. -Left; Exists b; Auto. -NewDestruct (RS_dec a). -Left; Exists a; Auto. -Auto. -(* -Realizer find. -*) -Qed. - -Variable B : Set. -Variable T : A -> B -> Prop. - -Variable TS_dec : (a:A){c:B| (T a c)}+{(P a)}. - -Fixpoint try_find [l:(list A)] : (Exc B) := - Cases l of - nil => Error - | (cons a l1) => - Cases (TS_dec a) of - (inleft (exist c _)) => (Value c) - | (inright _) => (try_find l1) - end - end. - -Lemma Try_find : (l:(list A)){c:B|(EX a:A |(In a l) & (T a c))}+{(AllS P l)}. -NewInduction l as [|a m [[b H1]|H]]. -Auto. -Left; Exists b; NewDestruct H1 as [a' H2 H3]; Exists a'; Auto. -NewDestruct (TS_dec a) as [[c H1]|]. -Left; Exists c. -Exists a; Auto. -Auto. -(* -Realizer try_find. -*) -Qed. - -End Find_sec. - -Section Assoc_sec. - -Variable B : Set. -Fixpoint assoc [a:A;l:(list A*B)] : (Exc B) := - Cases l of nil => Error - | (cons (a',b) m) => (ifdec (eqA_dec a a') (Value b) (assoc a m)) - end. - -Inductive AllS_assoc [P:A -> Prop]: (list A*B) -> Prop := - allS_assoc_nil : (AllS_assoc P (nil A*B)) - | allS_assoc_cons : (a:A)(b:B)(l:(list A*B)) - (P a)->(AllS_assoc P l)->(AllS_assoc P (cons (a,b) l)). - -Hints Resolve allS_assoc_nil allS_assoc_cons. - -(* The specification seems too weak: it is enough to return b if the - list has at least an element (a,b); probably the intention is to have - the specification - - (a:A)(l:(list A*B)){b:B|(In_spec (a,b) l)}+{(AllS_assoc [a':A]~(a=a') l)}. -*) - -Lemma Assoc : (a:A)(l:(list A*B))(B+{(AllS_assoc [a':A]~(a=a') l)}). -NewInduction l as [|[a' b] m assrec]. Auto. -NewDestruct (eqA_dec a a'). -Left; Exact b. -NewDestruct assrec as [b'|]. -Left; Exact b'. -Right; Auto. -(* -Realizer assoc. -*) -Qed. - -End Assoc_sec. - -End Lists. - -Hints Resolve Isnil_nil not_Isnil_cons in_hd in_tl in_cons allS_nil allS_cons - : datatypes. -Hints Immediate fst_nth_nth : datatypes. - |