1 (*********************************************************************************************************************************)
2 (* General: general data structures *)
3 (*********************************************************************************************************************************)
5 Require Import Coq.Unicode.Utf8.
6 Require Import Coq.Classes.RelationClasses.
7 Require Import Coq.Classes.Morphisms.
8 Require Import Coq.Setoids.Setoid.
9 Require Import Coq.Strings.String.
11 Require Import Coq.Lists.List.
12 Require Import Preamble.
13 Generalizable All Variables.
16 Definition EqDecider T := forall (n1 n2:T), sumbool (n1=n2) (not (n1=n2)).
17 Class EqDecidable (T:Type) :=
19 ; eqd_dec : forall v1 v2:T, sumbool (v1=v2) (not (v1=v2))
21 Coercion eqd_type : EqDecidable >-> Sortclass.
24 Class ToString (T:Type) := { toString : T -> string }.
25 Instance StringToString : ToString string := { toString := fun x => x }.
27 Notation "a +++ b" := (append (toString a) (toString b)) (at level 100).
29 (*******************************************************************************)
32 Inductive Tree (a:Type) : Type :=
33 | T_Leaf : a -> Tree a
34 | T_Branch : Tree a -> Tree a -> Tree a.
35 Implicit Arguments T_Leaf [ a ].
36 Implicit Arguments T_Branch [ a ].
38 Notation "a ,, b" := (@T_Branch _ a b) : tree_scope.
40 (* tree-of-option-of-X comes up a lot, so we give it special notations *)
41 Notation "[ q ]" := (@T_Leaf (option _) (Some q)) : tree_scope.
42 Notation "[ ]" := (@T_Leaf (option _) None) : tree_scope.
43 Notation "[]" := (@T_Leaf (option _) None) : tree_scope.
45 Fixpoint InT {A} (a:A) (t:Tree ??A) {struct t} : Prop :=
47 | T_Leaf None => False
48 | T_Leaf (Some x) => x = a
49 | T_Branch b1 b2 => InT a b1 \/ InT a b2
52 Open Scope tree_scope.
54 Fixpoint mapTree {a b:Type}(f:a->b)(t:@Tree a) : @Tree b :=
56 | T_Leaf x => T_Leaf (f x)
57 | T_Branch l r => T_Branch (mapTree f l) (mapTree f r)
59 Fixpoint mapOptionTree {a b:Type}(f:a->b)(t:@Tree ??a) : @Tree ??b :=
61 | T_Leaf None => T_Leaf None
62 | T_Leaf (Some x) => T_Leaf (Some (f x))
63 | T_Branch l r => T_Branch (mapOptionTree f l) (mapOptionTree f r)
65 Fixpoint mapTreeAndFlatten {a b:Type}(f:a->@Tree b)(t:@Tree a) : @Tree b :=
68 | T_Branch l r => T_Branch (mapTreeAndFlatten f l) (mapTreeAndFlatten f r)
70 Fixpoint mapOptionTreeAndFlatten {a b:Type}(f:a->@Tree ??b)(t:@Tree ??a) : @Tree ??b :=
73 | T_Leaf (Some x) => f x
74 | T_Branch l r => T_Branch (mapOptionTreeAndFlatten f l) (mapOptionTreeAndFlatten f r)
76 Fixpoint treeReduce {T:Type}{R:Type}(mapLeaf:T->R)(mergeBranches:R->R->R) (t:Tree T) :=
78 | T_Leaf x => mapLeaf x
79 | T_Branch y z => mergeBranches (treeReduce mapLeaf mergeBranches y) (treeReduce mapLeaf mergeBranches z)
81 Definition treeDecomposition {D T:Type} (mapLeaf:T->D) (mergeBranches:D->D->D) :=
82 forall d:D, { tt:Tree T & d = treeReduce mapLeaf mergeBranches tt }.
85 forall {Q}(t1 t2:Tree ??Q),
86 (forall q1 q2:Q, sumbool (q1=q2) (not (q1=q2))) ->
87 sumbool (t1=t2) (not (t1=t2)).
92 destruct a; destruct t2.
97 right; unfold not; intro; apply H. inversion H0; subst. auto.
98 right. unfold not; intro; inversion H.
99 right. unfold not; intro; inversion H.
101 right. unfold not; intro; inversion H.
103 right. unfold not; intro; inversion H.
106 right. unfold not; intro; inversion H.
107 set (IHt1_1 t2_1 X) as X1.
108 set (IHt1_2 t2_2 X) as X2.
109 destruct X1; destruct X2; subst.
128 Lemma mapOptionTree_compose : forall A B C (f:A->B)(g:B->C)(l:Tree ??A),
129 (mapOptionTree (g ○ f) l) = (mapOptionTree g (mapOptionTree f l)).
140 Lemma mapOptionTree_extensional {A}{B}(f g:A->B) : (forall a, f a = g a) -> (forall t, mapOptionTree f t = mapOptionTree g t).
144 simpl; rewrite H; auto.
145 simpl; rewrite IHt1; rewrite IHt2; auto.
148 Open Scope string_scope.
149 Fixpoint treeToString {T}{TT:ToString T}(t:Tree ??T) : string :=
151 | T_Leaf None => "[]"
152 | T_Leaf (Some s) => "["+++s+++"]"
153 | T_Branch b1 b2 => treeToString b1 +++ ",," +++ treeToString b2
155 Instance TreeToString {T}{TT:ToString T} : ToString (Tree ??T) := { toString := treeToString }.
157 (*******************************************************************************)
160 Notation "a :: b" := (cons a b) : list_scope.
161 Open Scope list_scope.
162 Fixpoint leaves {a:Type}(t:Tree (option a)) : list a :=
164 | (T_Leaf l) => match l with
168 | (T_Branch l r) => app (leaves l) (leaves r)
170 (* weak inverse of "leaves" *)
171 Fixpoint unleaves {A:Type}(l:list A) : Tree (option A) :=
174 | (a::b) => [a],,(unleaves b)
177 (* a map over a list and the conjunction of the results *)
178 Fixpoint mapProp {A:Type} (f:A->Prop) (l:list A) : Prop :=
181 | (a::al) => f a /\ mapProp f al
184 Lemma map_id : forall A (l:list A), (map (fun x:A => x) l) = l.
191 Lemma map_app : forall `(f:A->B) l l', map f (app l l') = app (map f l) (map f l').
198 Lemma map_compose : forall A B C (f:A->B)(g:B->C)(l:list A),
199 (map (g ○ f) l) = (map g (map f l)).
207 Lemma list_cannot_be_longer_than_itself : forall `(a:A)(b:list A), b = (a::b) -> False.
211 inversion H. apply IHb in H2.
214 Lemma list_cannot_be_longer_than_itself' : forall A (b:list A) (a c:A), b = (a::c::b) -> False.
223 Lemma mapOptionTree_on_nil : forall `(f:A->B) h, [] = mapOptionTree f h -> h=[].
226 destruct o. inversion H.
231 Lemma mapOptionTree_comp a b c (f:a->b) (g:b->c) q : (mapOptionTree g (mapOptionTree f q)) = mapOptionTree (g ○ f) q.
242 Lemma leaves_unleaves {T}(t:list T) : leaves (unleaves t) = t.
248 Lemma mapleaves' {T:Type}(t:list T){Q}{f:T->Q} : unleaves (map f t) = mapOptionTree f (unleaves t).
249 induction t; simpl in *; auto.
253 (* handy facts: map preserves the length of a list *)
254 Lemma map_on_nil : forall A B (s1:list A) (f:A->B), nil = map f s1 -> s1=nil.
263 Lemma map_on_singleton : forall A B (f:A->B) x (s1:list A), (cons x nil) = map f s1 -> { y : A & s1=(cons y nil) & (f y)=x }.
266 simpl in H; assert False. inversion H. inversion H0.
272 inversion H. apply map_on_nil in H2. auto.
276 inversion H. apply map_on_nil in H2. auto.
282 Lemma map_on_doubleton : forall A B (f:A->B) x y (s1:list A), ((x::y::nil) = map f s1) ->
283 { z : A*A & s1=((fst z)::(snd z)::nil) & (f (fst z))=x /\ (f (snd z))=y }.
297 Lemma mapTree_compose : forall A B C (f:A->B)(g:B->C)(l:Tree A),
298 (mapTree (g ○ f) l) = (mapTree g (mapTree f l)).
307 Lemma lmap_compose : forall A B C (f:A->B)(g:B->C)(l:list A),
308 (map (g ○ f) l) = (map g (map f l)).
317 (* sends a::b::c::nil to [[[[],,c],,b],,a] *)
318 Fixpoint unleaves' {A:Type}(l:list A) : Tree (option A) :=
321 | (a::b) => (unleaves' b),,[a]
324 (* sends a::b::c::nil to [[[[],,c],,b],,a] *)
325 Fixpoint unleaves'' {A:Type}(l:list ??A) : Tree ??A :=
328 | (a::b) => (unleaves'' b),,(T_Leaf a)
331 Lemma mapleaves {T:Type}(t:Tree ??T){Q}{f:T->Q} : leaves (mapOptionTree f t) = map f (leaves t).
341 Fixpoint filter {T:Type}(l:list ??T) : list T :=
344 | (None::b) => filter b
345 | ((Some x)::b) => x::(filter b)
348 Inductive distinct {T:Type} : list T -> Prop :=
349 | distinct_nil : distinct nil
350 | distinct_cons : forall a ax, (@In _ a ax -> False) -> distinct ax -> distinct (a::ax).
352 Inductive distinctT {T:Type} : Tree ??T -> Prop :=
353 | distinctT_nil : distinctT []
354 | distinctT_leaf : forall t, distinctT [t]
355 | distinctT_cons : forall t1 t2, (forall v, InT v t1 -> InT v t2 -> False) -> distinctT (t1,,t2).
357 Lemma in_decidable {VV:Type}{eqdVV:EqDecidable VV} : forall (v:VV)(lv:list VV), sumbool (In v lv) (not (In v lv)).
368 set (eqd_dec v a) as dec.
381 Lemma distinct_decidable {VV:Type}{eqdVV:EqDecidable VV} : forall (lv:list VV), sumbool (distinct lv) (not (distinct lv)).
384 left; apply distinct_nil.
386 set (in_decidable a lv) as dec.
388 right; unfold not; intros.
393 apply distinct_cons; auto.
401 Lemma map_preserves_length {A}{B}(f:A->B)(l:list A) : (length l) = (length (map f l)).
407 (* decidable quality on a list of elements which have decidable equality *)
408 Definition list_eq_dec : forall {T:Type}(l1 l2:list T)(dec:forall t1 t2:T, sumbool (eq t1 t2) (not (eq t1 t2))),
409 sumbool (eq l1 l2) (not (eq l1 l2)).
412 induction l1; intros.
415 right; intro H; inversion H.
416 destruct l2 as [| b l2].
417 right; intro H; inversion H.
418 set (IHl1 l2 dec) as eqx.
421 set (dec a b) as eqy.
425 right. intro. inversion H. subst. apply n. auto.
433 Instance EqDecidableList {T:Type}(eqd:EqDecidable T) : EqDecidable (list T).
434 apply Build_EqDecidable.
440 (*******************************************************************************)
441 (* Length-Indexed Lists *)
443 Inductive vec (A:Type) : nat -> Type :=
445 | vec_cons : forall n, A -> vec A n -> vec A (S n).
447 Fixpoint vec2list {n:nat}{t:Type}(v:vec t n) : list t :=
450 | vec_cons n a va => a::(vec2list va)
453 Require Import Omega.
454 Definition vec_get : forall {T:Type}{l:nat}(v:vec T l)(n:nat)(pf:lt n l), T.
469 Definition vec_zip {n:nat}{A B:Type}(va:vec A n)(vb:vec B n) : vec (A*B) n.
474 apply vec_cons; auto.
477 Definition vec_map {n:nat}{A B:Type}(f:A->B)(v:vec A n) : vec B n.
481 apply vec_cons; auto.
484 Fixpoint vec_In {A:Type} {n:nat} (a:A) (l:vec A n) : Prop :=
487 | vec_cons _ n m => (n = a) \/ vec_In a m
489 Implicit Arguments vec_nil [ A ].
490 Implicit Arguments vec_cons [ A n ].
492 Definition append_vec {n:nat}{T:Type}(v:vec T n)(t:T) : vec T (S n).
494 apply (vec_cons t vec_nil).
495 apply vec_cons; auto.
498 Definition list2vec {T:Type}(l:list T) : vec T (length l).
501 apply vec_cons; auto.
504 Definition vec_head {n:nat}{T}(v:vec T (S n)) : T.
507 Definition vec_tail {n:nat}{T}(v:vec T (S n)) : vec T n.
511 Lemma vec_chop {T}{l1 l2:list T}{Q}(v:vec Q (length (app l1 l2))) : vec Q (length l1).
516 inversion v; subst; auto.
518 inversion v; subst; auto.
521 Lemma vec_chop' {T}{l1 l2:list T}{Q}(v:vec Q (length (app l1 l2))) : vec Q (length l2).
525 apply IHl1; clear IHl1.
526 inversion v; subst; auto.
529 Lemma vec2list_len {T}{n}(v:vec T n) : length (vec2list v) = n.
535 Lemma vec2list_map_list2vec {A B}{n}(f:A->B)(v:vec A n) : map f (vec2list v) = vec2list (vec_map f v).
541 Lemma vec2list_list2vec {A}(v:list A) : vec2list (list2vec v) = v.
543 set (vec2list (list2vec (a :: v))) as q.
551 Notation "a ::: b" := (@vec_cons _ _ a b) (at level 20).
555 (*******************************************************************************)
558 (* a ShapedTree is a tree indexed by the shape (but not the leaf values) of another tree; isomorphic to (ITree (fun _ => Q)) *)
559 Inductive ShapedTree {T:Type}(Q:Type) : Tree ??T -> Type :=
560 | st_nil : @ShapedTree T Q []
561 | st_leaf : forall {t}, Q -> @ShapedTree T Q [t]
562 | st_branch : forall {t1}{t2}, @ShapedTree T Q t1 -> @ShapedTree T Q t2 -> @ShapedTree T Q (t1,,t2).
564 Fixpoint unshape {T:Type}{Q:Type}{idx:Tree ??T}(st:@ShapedTree T Q idx) : Tree ??Q :=
568 | st_branch _ _ b1 b2 => (unshape b1),,(unshape b2)
571 Definition mapShapedTree {T}{idx:Tree ??T}{V}{Q}(f:V->Q)(st:ShapedTree V idx) : ShapedTree Q idx.
574 apply st_leaf. apply f. apply q.
575 apply st_branch; auto.
578 Definition zip_shapedTrees {T:Type}{Q1 Q2:Type}{idx:Tree ??T}
579 (st1:ShapedTree Q1 idx)(st2:ShapedTree Q2 idx) : ShapedTree (Q1*Q2) idx.
587 apply st_branch; auto.
588 inversion st1; subst; apply IHidx1; auto.
589 inversion st2; subst; auto.
590 inversion st2; subst; apply IHidx2; auto.
591 inversion st1; subst; auto.
594 Definition build_shapedTree {T:Type}(idx:Tree ??T){Q:Type}(f:T->Q) : ShapedTree Q idx.
599 apply st_branch; auto.
602 Lemma unshape_map : forall {Q}{b}(f:Q->b){T}{idx:Tree ??T}(t:ShapedTree Q idx),
603 mapOptionTree f (unshape t) = unshape (mapShapedTree f t).
615 (*******************************************************************************)
616 (* Type-Indexed Lists *)
618 (* an indexed list *)
619 Inductive IList (I:Type)(F:I->Type) : list I -> Type :=
620 | INil : IList I F nil
621 | ICons : forall i is, F i -> IList I F is -> IList I F (i::is).
622 Implicit Arguments INil [ I F ].
623 Implicit Arguments ICons [ I F ].
625 Notation "a :::: b" := (@ICons _ _ _ _ a b) (at level 20).
627 Definition ilist_head {T}{F}{x}{y} : IList T F (x::y) -> F x.
634 Definition ilist_tail {T}{F}{x}{y} : IList T F (x::y) -> IList T F y.
641 Definition ilmap {I}{F}{G}{il:list I}(f:forall i:I, F i -> G i) : IList I F il -> IList I G il.
642 induction il; intros; auto.
648 Lemma ilist_chop {T}{F}{l1 l2:list T}(v:IList T F (app l1 l2)) : IList T F l1.
657 Lemma ilist_chop' {T}{F}{l1 l2:list T}(v:IList T F (app l1 l2)) : IList T F l2.
660 inversion v; subst; auto.
663 Fixpoint ilist_to_list {T}{Z}{l:list T}(il:IList T (fun _ => Z) l) : list Z :=
666 | a::::b => a::(ilist_to_list b)
669 (* a tree indexed by a (Tree (option X)) *)
670 Inductive ITree (I:Type)(F:I->Type) : Tree ??I -> Type :=
671 | INone : ITree I F []
672 | ILeaf : forall i: I, F i -> ITree I F [i]
673 | IBranch : forall it1 it2:Tree ??I, ITree I F it1 -> ITree I F it2 -> ITree I F (it1,,it2).
674 Implicit Arguments INil [ I F ].
675 Implicit Arguments ILeaf [ I F ].
676 Implicit Arguments IBranch [ I F ].
678 Definition itmap {I}{F}{G}{il:Tree ??I}(f:forall i:I, F i -> G i) : ITree I F il -> ITree I G il.
679 induction il; intros; auto.
685 apply IBranch; inversion X; auto.
688 Fixpoint itree_to_tree {T}{Z}{l:Tree ??T}(il:ITree T (fun _ => Z) l) : Tree ??Z :=
692 | IBranch _ _ b1 b2 => (itree_to_tree b1),,(itree_to_tree b2)
696 (*******************************************************************************)
697 (* Extensional equality on functions *)
699 Definition extensionality := fun (t1 t2:Type) => (fun (f:t1->t2) g => forall x:t1, (f x)=(g x)).
700 Hint Transparent extensionality.
701 Instance extensionality_Equivalence : forall t1 t2, Equivalence (extensionality t1 t2).
702 intros; apply Build_Equivalence;
703 intros; compute; intros; auto.
704 rewrite H; rewrite H0; auto.
706 Add Parametric Morphism (A B C:Type) : (fun f g => g ○ f)
707 with signature (extensionality A B ==> extensionality B C ==> extensionality A C) as parametric_morphism_extensionality.
708 unfold extensionality; intros; rewrite (H x1); rewrite (H0 (y x1)); auto.
710 Lemma extensionality_composes : forall t1 t2 t3 (f f':t1->t2) (g g':t2->t3),
711 (extensionality _ _ f f') ->
712 (extensionality _ _ g g') ->
713 (extensionality _ _ (g ○ f) (g' ○ f')).
715 unfold extensionality.
716 unfold extensionality in H.
717 unfold extensionality in H0.
728 CoInductive Fresh A T :=
729 { next : forall a:A, (T a * Fresh A T)
730 ; split : Fresh A T * Fresh A T
737 Definition map2 {A}{B}(f:A->B)(t:A*A) : (B*B) := ((f (fst t)), (f (snd t))).
741 Variable eol : string.
742 Extract Constant eol => "'\n':[]".
744 Class Monad {T:Type->Type} :=
745 { returnM : forall {a}, a -> T a
746 ; bindM : forall {a}{b}, T a -> (a -> T b) -> T b
748 Implicit Arguments Monad [ ].
749 Notation "a >>>= b" := (@bindM _ _ _ _ a b) (at level 50, left associativity).
751 (* the Error monad *)
752 Inductive OrError (T:Type) :=
753 | Error : forall error_message:string, OrError T
754 | OK : T -> OrError T.
755 Notation "??? T" := (OrError T) (at level 10).
756 Implicit Arguments Error [T].
757 Implicit Arguments OK [T].
759 Definition orErrorBind {T:Type} (oe:OrError T) {Q:Type} (f:T -> OrError Q) :=
764 Notation "a >>= b" := (@orErrorBind _ a _ b) (at level 20).
766 Open Scope string_scope.
767 Definition orErrorBindWithMessage {T:Type} (oe:OrError T) {Q:Type} (f:T -> OrError Q) err_msg :=
769 | Error s => Error (err_msg +++ eol +++ " " +++ s)
773 Notation "a >>=[ S ] b" := (@orErrorBindWithMessage _ a _ b S) (at level 20).
775 Definition addErrorMessage s {T} (x:OrError T) :=
776 x >>=[ s ] (fun y => OK y).
778 Inductive Indexed {T:Type}(f:T -> Type) : ???T -> Type :=
779 | Indexed_Error : forall error_message:string, Indexed f (Error error_message)
780 | Indexed_OK : forall t, f t -> Indexed f (OK t)
784 Require Import Coq.Arith.EqNat.
785 Instance EqDecidableNat : EqDecidable nat.
786 apply Build_EqDecidable.
791 (* for a type with decidable equality, we can maintain lists of distinct elements *)
792 Section DistinctList.
793 Context `{V:EqDecidable}.
795 Fixpoint addToDistinctList (cv:V)(cvl:list V) :=
798 | cv'::cvl' => if eqd_dec cv cv' then cvl' else cv'::(addToDistinctList cv cvl')
801 Fixpoint removeFromDistinctList (cv:V)(cvl:list V) :=
804 | cv'::cvl' => if eqd_dec cv cv' then removeFromDistinctList cv cvl' else cv'::(removeFromDistinctList cv cvl')
807 Fixpoint removeFromDistinctList' (cvrem:list V)(cvl:list V) :=
810 | rem::cvrem' => removeFromDistinctList rem (removeFromDistinctList' cvrem' cvl)
813 Fixpoint mergeDistinctLists (cvl1:list V)(cvl2:list V) :=
816 | cv'::cvl' => mergeDistinctLists cvl' (addToDistinctList cv' cvl2)
820 Lemma list2vecOrFail {T}(l:list T)(n:nat)(error_message:nat->nat->string) : ???(vec T n).
821 set (list2vec l) as v.
822 destruct (eqd_dec (length l) n).
823 rewrite e in v; apply OK; apply v.
824 apply (Error (error_message (length l) n)).
828 Variable UniqSupply : Type. Extract Inlined Constant UniqSupply => "UniqSupply.UniqSupply".
829 Variable Unique : Type. Extract Inlined Constant Unique => "Unique.Unique".
830 Variable uniqFromSupply : UniqSupply -> Unique. Extract Inlined Constant uniqFromSupply => "UniqSupply.uniqFromSupply".
831 Variable splitUniqSupply : UniqSupply -> UniqSupply * UniqSupply.
832 Extract Inlined Constant splitUniqSupply => "UniqSupply.splitUniqSupply".
833 Variable unique_eq : forall u1 u2:Unique, sumbool (u1=u2) (u1≠u2).
834 Extract Inlined Constant unique_eq => "(==)".
835 Variable unique_toString : Unique -> string.
836 Extract Inlined Constant unique_toString => "show".
837 Instance EqDecidableUnique : EqDecidable Unique :=
838 { eqd_dec := unique_eq }.
839 Instance EqDecidableToString : ToString Unique :=
840 { toString := unique_toString }.
842 Inductive UniqM {T:Type} : Type :=
843 | uniqM : (UniqSupply -> ???(UniqSupply * T)) -> UniqM.
844 Implicit Arguments UniqM [ ].
846 Instance UniqMonad : Monad UniqM :=
847 { returnM := fun T (x:T) => uniqM (fun u => OK (u,x))
848 ; bindM := fun a b (x:UniqM a) (f:a->UniqM b) =>
854 | OK (u',va) => match f va with
861 Definition getU : UniqM Unique :=
862 uniqM (fun us => let (us1,us2) := splitUniqSupply us in OK (us1,(uniqFromSupply us2))).
864 Notation "'bind' x = e ; f" := (@bindM _ _ _ _ e (fun x => f)) (x ident, at level 60, right associativity).
865 Notation "'return' x" := (returnM x) (at level 100).
866 Notation "'failM' x" := (uniqM (fun _ => Error x)) (at level 100).
873 Record FreshMonad {T:Type} :=
875 ; FMT_Monad :> Monad FMT
876 ; FMT_fresh : forall tl:list T, FMT { t:T & @In _ t tl -> False }
878 Implicit Arguments FreshMonad [ ].
879 Coercion FMT : FreshMonad >-> Funclass.
883 Variable Prelude_error : forall {A}, string -> A. Extract Inlined Constant Prelude_error => "Prelude.error".
888 Ltac eqd_dec_refl X :=
889 destruct (eqd_dec X X) as [eqd_dec1 | eqd_dec2];
890 [ clear eqd_dec1 | set (eqd_dec2 (refl_equal _)) as eqd_dec2'; inversion eqd_dec2' ].
892 Lemma unleaves_injective : forall T (t1 t2:list T), unleaves t1 = unleaves t2 -> t1 = t2.
894 induction t1; intros.
902 set (IHt1 _ H2) as q.
907 Lemma fst_zip : forall T Q n (v1:vec T n)(v2:vec Q n), vec_map (@fst _ _) (vec_zip v1 v2) = v1.
911 Lemma snd_zip : forall T Q n (v1:vec T n)(v2:vec Q n), vec_map (@snd _ _) (vec_zip v1 v2) = v2.
915 (* escapifies any characters which might cause trouble for LaTeX *)
916 Variable sanitizeForLatex : string -> string.
917 Extract Inlined Constant sanitizeForLatex => "sanitizeForLatex".
918 Inductive Latex : Type := latex : string -> Latex.
919 Instance LatexToString : ToString Latex := { toString := fun x => match x with latex s => s end }.
920 Class ToLatex (T:Type) := { toLatex : T -> Latex }.
921 Instance StringToLatex : ToLatex string := { toLatex := fun x => latex (sanitizeForLatex x) }.
922 Instance LatexToLatex : ToLatex Latex := { toLatex := fun x => x }.
923 Definition concatLatex {T1}{T2}(l1:T1)(l2:T2){L1:ToLatex T1}{L2:ToLatex T2} : Latex :=
924 match toLatex l1 with
926 match toLatex l2 with
931 Notation "a +=+ b" := (concatLatex a b) (at level 60, right associativity).