phanerozoic/Coq-ExtLib · Datasets at Hugging Face

fact stringlengths 17 6.18k	type stringclasses 17 values	library stringclasses 3 values	imports listlengths 0 12	filename stringclasses 115 values	symbolic_name stringlengths 1 30
value : Type := \| Int : nat -> value \| Bool : bool -> value.	Inductive	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	value
exp : Type := \| ConstI : nat -> exp \| ConstB : bool -> exp \| Plus : exp -> exp -> exp \| If : exp -> exp -> exp -> exp.	Inductive	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	exp
asInt (v : value) : m nat := match v with \| Int n => ret n \| _ => ( if we don't have an integer, signal an error using [raise] from the MoandExc instance **) raise ("expected integer got bool")%string end.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	asInt
asBool (v : value) : m bool := match v with \| Bool b => ret b \| _ => raise ("expected bool got integer")%string end. ( The main evaluator routine returns a [value], but since we are working in the [m] monad, we return [m value] **)	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	asBool
eval' (e : exp) : m value := match e with ( when there is no error, we can just return (i.e. [ret]) the answer ) \| ConstI i => ret (Int i) \| ConstB b => ret (Bool b) \| Plus l r => ( evaluate the sub-terms to numbers ) l <- eval' l ;; l <- asInt l ;; r <- eval' r ;; r <- asInt r ;; ( Combine the result ) ret (Int (l + r)) \| If t tr fa => ( evaluate the test condition to a boolean ) t <- eval' t ;; t <- asBool t ;; ( case split and perform the appropriate recursion **) if (t : bool) then eval' tr else eval' fa end.	Fixpoint	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	eval'
eval : exp -> string + value := eval' (m := sum string). ( Some tests ) Eval compute in eval (Plus (ConstI 1) (ConstI 2)). Eval compute in eval (Plus (ConstI 1) (ConstB false)). ( Other useful monads: * Reader - for handling lexicographic environments ** * State - for handling non-lexical state, like a heap ** * MonadFix - for handling unbounded recursion **)	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/EvalWithExc.v	eval
update1 : istate A B unit := modify_ function1.	Definition	examples	[ "Require Import ExtLib.", "Require Import ExtLib." ]	examples/indexedstate.v	update1
update2 : istate B C unit := modify_ function2.	Definition	examples	[ "Require Import ExtLib.", "Require Import ExtLib." ]	examples/indexedstate.v	update2
compose : istate A C unit := update1 ;; update2.	Definition	examples	[ "Require Import ExtLib.", "Require Import ExtLib." ]	examples/indexedstate.v	compose
proper_eta : forall T U (f : T -> U) (type_T : type T) (type_U : type U), proper f -> proper (fun x => f x). Proof. intros; do 3 red; intros. eapply H. assumption. Qed. Goal forall x : T, proper x -> equal (bind (ret x) (fun x => ret x)) (ret x). Proof. intros. etransitivity. { eapply bind_of_return; eauto. eapply proper_eta. eapply ret_proper; eauto. } { eapply ret_proper; eauto. eapply equiv_prefl; eauto. } Qed. Goal forall x : T, proper x -> equal (bind (ret x) (fun x => ret x)) (ret x). Proof. intros. etransitivity. { eapply bind_of_return; eauto. eapply proper_eta. eapply ret_proper; eauto. } { eapply ret_proper; eauto. eapply equiv_prefl; eauto. } Qed.	Lemma	examples	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/MonadReasoning.v	proper_eta
repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit := match n with \| O => ret tt \| S n => y <- p x;; repeatM n y p end.	Fixpoint	examples	[ "Require Import ExtLib." ]	examples/Notations.v	repeatM
repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit := match n with \| O => ret tt \| S n => let* y := p x in repeatM n y p end.	Fixpoint	examples	[ "Require Import ExtLib." ]	examples/Notations.v	repeatM
PrinterMonad : Type -> Type := writerT (@show_mon _ ShowScheme_string_compose) ident.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/Printing.v	PrinterMonad
print {T : Type} {ST : Show T} (val : T) : PrinterMonad unit := @MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _ (@show _ ST val _ show_inj (@show_mon _ ShowScheme_string_compose)).	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/Printing.v	print
printString (str : string) : PrinterMonad unit := @MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _ (@show_exact str _ show_inj (@show_mon _ ShowScheme_string_compose)).	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/Printing.v	printString
runPrinter {T : Type} (c : PrinterMonad T) : T * string := let '(ppair val str) := unIdent (runWriterT c) in (val, str ""%string). Eval compute in runPrinter (Monad.bind (print 1) (fun _ => print 2)). Eval compute in runPrinter (Monad.bind (print "hello "%string) (fun _ => print 2)). Eval compute in runPrinter (Monad.bind (printString "hello "%string) (fun _ => print 2)).	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/Printing.v	runPrinter
use of State monad Passes a string of dictionary {a,b,c} Game is to produce a number from the string. By default the game is off, a C toggles the game on and off. A 'a' gives +1 and a b gives -1. E.g 'ab' = 0 'ca' = 1 'cabca' = 0 State = game is on or off & current score = (Bool, Int) *) Require Import Coq.ZArith.ZArith_base Coq.Strings.String Coq.Strings.Ascii. Require Import ExtLib.Data.Monads.StateMonad ExtLib.Structures.Monads.	Example	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	use
GameValue : Type := Z.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	GameValue
GameState : Type := (prod bool Z). Variable m : Type -> Type. Context {Monad_m: Monad m}. Context {State_m: MonadState GameState m}.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	GameState
playGame (s: string) {struct s}: m GameValue := match s with \| EmptyString => v <- get ;; let '(on, score) := v in ret score \| String x xs => v <- get ;; let '(on, score) := v in match x, on with \| "a", true => put (on, score + 1) \| "b", true => put (on, score - 1) \| "c", _ => put (negb on, score) \| _, _ => put (on, score) end ;; playGame xs end.	Fixpoint	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	playGame
startState : GameState := (false, 0).	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	startState
main : GameValue := (@evalState GameState GameValue (playGame (state GameState) "abcaaacbbcabbab") startState). (* The following should return '2%Z' *) Compute main.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib." ]	examples/StateGame.v	main
foo : stateT unit option unit := ret tt.	Definition	examples	[]	examples/StateTMonad.v	foo
contains_both (v1 v2 : V) (s : set) : bool := contains v1 s && contains v2 s. ( Iteration requires foldability ) Context {Foldable_set : Foldable set V}.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/UsingSets.v	contains_both
toList (s : set) : list V := fold (@cons _) nil s.	Definition	examples	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	examples/UsingSets.v	toList
RTest : Set := mkRTest { a : bool ; b : nat ; c : bool }. Bind Scope struct_scope with RTest. Global Instance Struct_RTest : Struct RTest := { fields := ((@existT _ _ _ a) :: (@existT _ _ _ b) :: (@existT _ _ _ c):: nil) ; ctor := mkRTest }. Global Instance Acc_RTest_a : Accessor a := { acc := Here }. Global Instance Acc_RTest_b : Accessor b := { acc := Next Here }. Global Instance Acc_RTest_c : Accessor c := { acc := Next (Next Here) }. Eval compute in {$ mkRTest true 1 true with c := false $}%record. Eval compute in forall x : RTest, c {$ x with c := false $}%record = false.	Record	examples	[ "Require Import List.", "Require Import ExtLib." ]	examples/WithDemo.v	RTest
compose (A B C : Type) (f : A -> B) (g : B -> C) : A -> C := fun x => g (f x).	Definition	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	compose
pure (T : Type) : T -> m T := @ret _ _ _.	Definition	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	pure
fapply (T U : Type) (f : m (T -> U)) (x : m T) : m U := bind f (fun f => bind x (fun x => ret (f x))).	Definition	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	fapply
Instance fun_trans.	Existing	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	Instance
Instance fun_refl. Variables A B C : Type. Context (rA : relation A) (rB : relation B) (rC : relation C) (pA : Proper rA) (pB : Proper rB) (pC : Proper rC). Context (Ra : PReflexive rA) (Rb : PReflexive rB) (Rc : PReflexive rC). Context (Ta : PTransitive rA) (Tb : PTransitive rB) (Tc : PTransitive rC).	Existing	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	Instance
fun_app_proper (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) x : proper f -> proper x -> proper (f x). Proof. intros. apply H. auto. Qed.	Instance	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	fun_app_proper
fun_abs (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) : (forall x, proper x -> proper (f x)) -> (forall x y, proper x -> proper y -> rA x y -> rB (f x) (f y)) -> proper (fun x => f x). Proof. intros. split; auto; eapply H. Qed.	Instance	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	fun_abs
prove_proper x k := match x with \| _ => match goal with \| [ H : proper x \|- _ ] => k H end \| bind ?A ?B => prove_proper A ltac:(fun a => prove_proper B ltac:(fun b => let H := fresh in assert (H : proper x); [ eapply bind_proper; eauto with typeclass_instances \| k H ])) \| ret ?A => prove_proper A ltac:(fun a => let H := fresh in assert (H : proper x); [ eapply ret_proper; eauto with typeclass_instances \| k H ]) \| (fun x => _) => let H := fresh in assert (H : proper x); [ eapply fun_abs; intros; [ propers \| repeat red; intros; prove_mleq ] \| k H ] \| _ => let H := fresh in assert (H : proper x); [ eauto with typeclass_instances \| k H ] end with prove_mleq := try match goal with \| \|- proper (fun x => _) => eapply fun_abs; intros; [ propers \| repeat red; intros; prove_mleq ] \| [ R : _ , H' : pfun_ext ?R _ ?F ?G \|- ?R (?F _) (?G _) ] => eapply H'; [ propers \| propers \| prove_mleq ] \| [ H' : proper ?F \|- ?R (?F _) (?F _) ] => eapply H'; [ propers \| propers \| try assumption; prove_mleq ] \| [ \|- mleq _ (bind _ _) (bind _ _) ] => eapply bind_respectful_leq; [ eauto with typeclass_instances \| eauto with typeclass_instances \| prove_mleq \| intros; prove_mleq ] \| [ \|- mleq _ (ret _) (ret _) ] => eapply ret_respectful_leq; [ eauto with typeclass_instances \| eauto with typeclass_instances \| prove_mleq ] \| [ H : proper ?f \|- pfun_ext _ _ ?f ?f ] => apply H \| [ H : proper ?f \|- pfun_ext _ _ (fun x => _) (fun y => _) ] => red; intros; prove_mleq \| _ => eassumption end with propers := match goal with \| \|- proper ?X => prove_proper X ltac:(fun x => eapply x) \| \|- mleq _ ?X ?Y => prove_proper X ltac:(fun x => prove_proper Y ltac:(fun x => idtac)) end.	Ltac	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	prove_proper
PReflexive_stuff : PReflexive (pfun_ext (pfun_ext (pfun_ext rC pA) (Proper_pfun pB pC)) (Proper_pfun pA pB)). Proof. intuition. Qed.	Instance	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	PReflexive_stuff
bind_law : forall (f : A -> B) (g : B -> C), proper f -> proper g -> mleq (pfun_ext rC pA) (fapply (fapply (pure (@compose A B C)) (pure f)) (pure g)) (pure (compose f g)). Proof. unfold fapply, pure, compose; simpl; intros. propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_associativity _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_associativity _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ \| \| \| \| eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) \| ]); eauto with typeclass_instances; propers. eapply preflexive; eauto with typeclass_instances. Qed.	Theorem	scratch	[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	scratch/FunctorFromMonad.v	bind_law
Class RESOLVE. #[global] Hint Extern 0 (RESOLVE _) => unfold RESOLVE : typeclass_instances.	Existing	theories	[]	theories/Core/Any.v	Class
CmpDec (T : Type) (equ : T -> T -> Prop) (ltu : T -> T -> Prop) : Type := { cmp_dec : T -> T -> comparison }.	Class	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	CmpDec
CmpDec_Correct T (equ ltu : T -> T -> Prop) (ED : CmpDec equ ltu) : Prop := { cmp_dec_correct : forall x y : T, match cmp_dec x y with \| Eq => equ x y \| Lt => ltu x y \| Gt => ltu y x end }.	Class	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	CmpDec_Correct
cmp_case (P Q R : Prop) : comparison -> Prop := \| CaseEq : P -> cmp_case P Q R Eq \| CaseLt : Q -> cmp_case P Q R Lt \| CaseGt : R -> cmp_case P Q R Gt.	Inductive	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	cmp_case
eq_pair (a b : T * U) : Prop := eqt (fst a) (fst b) /\ equ (snd a) (snd b).	Definition	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	eq_pair
lt_pair (a b : T * U) : Prop := ltt (fst a) (fst b) \/ (eqt (fst a) (fst b) /\ ltu (snd a) (snd b)). Variable cdt : CmpDec eqt ltt. Variable cdu : CmpDec equ ltu.	Definition	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	lt_pair
CmpDec_pair : CmpDec eq_pair lt_pair := { cmp_dec := fun a b => let '(al,ar) := a in let '(bl,br) := b in match cmp_dec al bl with \| Eq => cmp_dec ar br \| x => x end }. Variable cdtC : CmpDec_Correct cdt. Variable cduC : CmpDec_Correct cdu. Variable Symmetric_eqt : Symmetric eqt.	Instance	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	CmpDec_pair
CmpDec_Correct_pair : CmpDec_Correct CmpDec_pair. Proof. constructor. destruct x; destruct y; unfold eq_pair, lt_pair; simpl in *. generalize (cmp_dec_correct t t0); destruct (cmp_dec t t0); simpl; intros; auto. generalize (cmp_dec_correct u u0); destruct (cmp_dec u u0); simpl; intros; auto. Qed.	Instance	theories	[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]	theories/Core/CmpDec.v	CmpDec_Correct_pair
decideP (P : Prop) {D : Decidable P} : {P} + {~P} := match @Decidable_witness P D as X return (X = true -> P) -> (X = false -> ~P) -> {P} + {~P} with \| true => fun pf _ => left (pf eq_refl) \| false => fun _ pf => right (pf eq_refl) end (@Decidable_sound _ D) (@Decidable_complete_alt _ D).	Definition	theories	[ "From Coq.Classes Require Import DecidableClass." ]	theories/Core/Decision.v	decideP
cases_ifd Hn := match goal with \|- context[if ?d then ?tt else ?ff] => let Hnt := fresh Hn "t" in let Hnf := fresh Hn "f" in destruct d as [Hnt \| Hnf] end.	Ltac	theories	[ "From Coq.Classes Require Import DecidableClass." ]	theories/Core/Decision.v	cases_ifd
decide_decideP {P:Prop }`{Decidable P} {R:Type} (a b : R) : (if (decide P) then a else b) = (if (decideP P) then a else b). Proof. symmetry. unfold decide. destruct (decideP P). - rewrite Decidable_complete; auto. - rewrite Decidable_sound_alt; auto. Qed.	Lemma	theories	[ "From Coq.Classes Require Import DecidableClass." ]	theories/Core/Decision.v	decide_decideP
EquivDec_refl_left {T : Type} {c : EqDec T (@eq T)} : forall (n : T), equiv_dec n n = left (refl_equal _). Proof. intros. destruct (equiv_dec n n); try congruence. Require Eqdep_dec. rewrite (Eqdep_dec.UIP_dec (A := T) (@equiv_dec _ _ _ c) e (refl_equal _)). reflexivity. Qed. Export EquivDec.	Theorem	theories	[ "From Coq.Classes Require Import EquivDec." ]	theories/Core/EquivDec.v	EquivDec_refl_left
RelDec (T : Type) (equ : T -> T -> Prop) : Type := { rel_dec : T -> T -> bool }. Arguments rel_dec {_} {equ} {_} _ _. Arguments rel_dec _ _ _ !x !y.	Class	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	RelDec
RelDec_Correct T (equ : T -> T -> Prop) (ED : RelDec equ) : Prop := { rel_dec_correct : forall x y : T, rel_dec x y = true <-> equ x y }.	Class	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	RelDec_Correct
eq_dec {T : Type} {ED : RelDec (@eq T)} := rel_dec.	Definition	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	eq_dec
neg_rel_dec_correct : forall {x y}, ~R x y <-> rel_dec x y = false. Proof. intros x y. destruct (bool_dec (rel_dec x y) true) ; constructor ; intros ; repeat match goal with \| [ \|- ~ _ ] => unfold not ; intros \| [ H1 : ?P, H2 : ~?P \|- _ ] => specialize (H2 H1) ; contradiction \| [ H1 : ?P = true, H2 : ?P = false \|- _ ] => rewrite H1 in H2 ; discriminate \| [ H1 : ?P <> true \|- ?P = false ] => apply not_true_is_false ; exact H1 \| [ H1 : ?rel_dec ?a ?b = true, H2 : ~?R ?a ?b \|- _ ] => apply rel_dec_correct in H1 \| [ H1 : ?rel_dec ?a ?b = false, H2 : ?R ?a ?b \|- _ ] => apply rel_dec_correct in H2 end. Qed.	Definition	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	neg_rel_dec_correct
rel_dec_p (x:T) (y:T) : {R x y} + {~R x y}. Proof. destruct (bool_dec (rel_dec x y) true) as [H \| H]. apply rel_dec_correct in H ; eauto. apply not_true_is_false in H ; apply neg_rel_dec_correct in H ; eauto. Qed.	Definition	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	rel_dec_p
neg_rel_dec_p (x:T) (y:T) : {~R x y} + {R x y}. Proof. destruct (rel_dec_p x y) ; [ right \| left ] ; auto. Qed.	Definition	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	neg_rel_dec_p
rel_dec_eq_true : forall x y, eqt x y -> rel_dec x y = true. Proof. intros. eapply rel_dec_correct in H. assumption. Qed.	Theorem	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	rel_dec_eq_true
rel_dec_neq_false : forall x y, ~eqt x y -> rel_dec x y = false. Proof. intros. remember (x ?[ eqt ] y). symmetry in Heqb. destruct b; try reflexivity. exfalso. eapply (@rel_dec_correct _ _ _ rc) in Heqb. auto. Qed.	Theorem	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	rel_dec_neq_false
rel_dec_sym : Symmetric eqt -> forall x y, x ?[ eqt ] y = y ?[ eqt ] x. Proof. intros. remember (x ?[ eqt ] y); remember (y ?[ eqt ] x); intuition. destruct b; destruct b0; auto. { symmetry in Heqb; symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb. symmetry in Heqb. eapply (@rel_dec_correct _ _ _ rc) in Heqb. congruence. } { symmetry in Heqb; symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb0. symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb0. congruence. } Qed.	Theorem	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	rel_dec_sym
RelDec_from_dec : RelDec R := {\| rel_dec := fun a b => match f a b with \| left _ => true \| right _ => false end \|}. Global Instance RelDec_Correct_eq_typ : RelDec_Correct RelDec_from_dec. Proof. constructor. intros. unfold rel_dec; simpl. destruct (f x y). - tauto. - split. + inversion 1. + intro. apply n in H. tauto. Qed.	Definition	theories	[ "Require Import Coq.", "Require Import Coq." ]	theories/Core/RelDec.v	RelDec_from_dec
digit2ascii (n:nat) : Ascii.ascii := match n with \| 0 => "0" \| 1 => "1" \| 2 => "2" \| 3 => "3" \| 4 => "4" \| 5 => "5" \| 6 => "6" \| 7 => "7" \| 8 => "8" \| 9 => "9" \| n => ascii_of_nat (n - 10 + nat_of_ascii "A") end%char.	Definition	theories	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Char.v	digit2ascii
chr_newline : ascii := Eval compute in ascii_of_nat 10. Export Ascii.	Definition	theories	[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Char.v	chr_newline
Checked : option T -> Type := \| Success : forall {v}, F v -> Checked (Some v) \| Failure : Checked None.	Inductive	theories	[]	theories/Data/Checked.v	Checked
succeeded (o : option T) (d : Checked o) : bool := match d with \| Success _ _ => true \| Failure => false end.	Definition	theories	[]	theories/Data/Checked.v	succeeded
failed (o : option T) (d : Checked o) : bool := match d with \| Success _ _ => false \| Failure => true end.	Definition	theories	[]	theories/Data/Checked.v	failed
asOption (o : option T) (d : Checked o) : option (match o with \| None => False \| Some x => F x end) := match d in Checked o return option match o with \| None => False \| Some x => F x end with \| Success _ x => Some x \| Failure => None end.	Definition	theories	[]	theories/Data/Checked.v	asOption
eq_sym_eq : forall T (a b : T) (pf : a = b) (F : T -> Type) val, match eq_sym pf in _ = x return F x with \| eq_refl => val end = match pf in _ = x return F x -> F a with \| eq_refl => fun x => x end val. Proof. destruct pf. reflexivity. Defined.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	eq_sym_eq
match_eq_sym_eq : forall T (a b : T) (pf : a = b) F X, match pf in _ = t return F t with \| eq_refl => match eq_sym pf in _ = t return F t with \| eq_refl => X end end = X. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite match_eq_sym_eq : eq_rw.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	match_eq_sym_eq
match_eq_sym_eq' : forall T (a b : T) (pf : a = b) F X, match eq_sym pf in _ = t return F t with \| eq_refl => match pf in _ = t return F t with \| eq_refl => X end end = X. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite match_eq_sym_eq' : eq_rw.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	match_eq_sym_eq'
match_eq_match_eq : forall T F (a b : T) (pf : a = b) X Y, X = Y -> match pf in _ = T return F T with \| eq_refl => X end = match pf in _ = T return F T with \| eq_refl => Y end. Proof. intros. subst. auto. Defined.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	match_eq_match_eq
eq_sym_eq_trans : forall T (a b c : T) (pf : a = b) (pf' : b = c), eq_sym (eq_trans pf pf') = eq_trans (eq_sym pf') (eq_sym pf). Proof. clear. destruct pf. destruct pf'. reflexivity. Defined. ( Particular Instances )	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	eq_sym_eq_trans
eq_Const_eq : forall T (a b : T) (pf : a = b) (R : Type) val, match pf in _ = x return R with \| eq_refl => val end = val. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_Const_eq : eq_rw.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	eq_Const_eq
eq_Arr_eq : forall T (a b : T) (pf : a = b) (F G : T -> Type) val x, match pf in _ = x return F x -> G x with \| eq_refl => val end x = match pf in _ = x return G x with \| eq_refl => val match eq_sym pf in _ = x return F x with \| eq_refl => x end end. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_Arr_eq : eq_rw.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	eq_Arr_eq
eq_sym_eq_sym : forall (T : Type) (a b : T) (pf : a = b), eq_sym (eq_sym pf) = pf. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_sym_eq_sym : eq_rw.	Lemma	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	eq_sym_eq_sym
autorewrite_eq_rw := repeat progress (autorewrite with eq_rw; repeat match goal with \| \|- context [ match ?X in @eq _ _ _ return _ -> _ with \| eq_refl => _ end ] => rewrite (eq_Arr_eq X) end). Require Export ExtLib.Data.Eq.UIP_trans.	Ltac	theories	[ "Require Export ExtLib." ]	theories/Data/Eq.v	autorewrite_eq_rw
fin : nat -> Type := \| F0 : forall {n}, fin (S n) \| FS : forall {n}, fin n -> fin (S n).	Inductive	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin
fin_all (n : nat) : list (fin n) := match n as n return list (fin n) with \| 0 => nil \| S n => @F0 n :: List.map (@FS _) (fin_all n) end%list.	Fixpoint	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin_all
fin_all_In : forall {n} (f : fin n), List.In f (fin_all n). Proof. induction n; intros. inversion f. remember (S n). destruct f. simpl; firstorder. inversion Heqn0. subst. simpl. right. apply List.in_map. auto. Qed.	Theorem	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin_all_In
fin_case : forall n (f : fin (S n)), f = F0 \/ exists f', f = FS f'. Proof. intros. generalize (fin_all_In f). intros. destruct H; auto. eapply List.in_map_iff in H. right. destruct H. exists x. intuition. Qed.	Theorem	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin_case
fin0_elim (f : fin 0) : forall T, T := match f in fin n return match n with \| 0 => forall T, T \| _ => unit end with \| F0 _ => tt \| FS _ _ => tt end.	Definition	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin0_elim
pf_lt (n m : nat) : Prop := match n , m with \| 0 , S _ => True \| S n , S m => pf_lt n m \| _ , _ => False end.	Fixpoint	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	pf_lt
make (m n : nat) {struct m} : pf_lt n m -> fin m := match n as n , m as m return pf_lt n m -> fin m with \| 0 , 0 => @False_rect _ \| 0 , S n => fun _ => F0 \| S n , 0 => @False_rect _ \| S n , S m => fun pf => FS (make m n pf) end.	Fixpoint	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	make
fin_eq_dec {n} (x : fin n) {struct x} : fin n -> bool := match x in fin n' return fin n' -> bool with \| F0 _ => fun y => match y with \| F0 _ => true \| _ => false end \| FS n' x' => fun y : fin (S n') => match y in fin n'' return (match n'' with \| 0 => unit \| S n'' => fin n'' end -> bool) -> bool with \| F0 _ => fun _ => false \| FS _ y' => fun f => f y' end (fun y => fin_eq_dec x' y) end. Global Instance RelDec_fin_eq (n : nat) : RelDec (@eq (fin n)) := { rel_dec := fin_eq_dec }. Global Instance RelDec_Correct_fin_eq (n : nat) : RelDec_Correct (RelDec_fin_eq n). Proof. constructor. induction x. simpl. intro. destruct (fin_case y) ; subst. intuition. destruct H ; subst. intuition; auto with ; try congruence. ( inversion H.*) intro ; destruct (fin_case y) ; subst ; simpl. intuition ; try congruence. inversion H. destruct H ; subst. split ; intro. f_equal ; eauto. eapply IHx. eapply H. inv_all ; subst. apply IHx. reflexivity. Qed.	Fixpoint	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fin.v	fin_eq_dec
Monoid_compose T : Monoid (T -> T) := {\| monoid_plus g f x := g (f x) ; monoid_unit x := x \|}. Export PreFun.	Definition	theories	[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]	theories/Data/Fun.v	Monoid_compose
app_ass_trans @{X} : forall {T : Type@{X} } (a b c : list T), (a ++ b) ++ c = a ++ b ++ c. Proof. induction a; simpl. reflexivity. intros. destruct (IHa b c). reflexivity. Defined.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	app_ass_trans
app_nil_r_trans : forall {T : Type} (a : list T), a ++ nil = a. Proof. induction a; simpl. reflexivity. refine match IHa in _ = X return _ = _ :: X with \| eq_refl => eq_refl end. Defined. Monomorphic Universe hlist_large. ( Core Type and Functions )	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	app_nil_r_trans
hlist : list iT -> Type := \| Hnil : hlist nil \| Hcons : forall l ls, F l -> hlist ls -> hlist (l :: ls).	Inductive	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist
hlist_hd {a b} (hl : hlist (a :: b)) : F a := match hl in hlist x return match x return Type@{Uv} with \| nil => unit \| l :: _ => F l end with \| Hnil => tt \| Hcons _ _ x _ => x end.	Definition	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_hd
hlist_tl {a b} (hl : hlist (a :: b)) : hlist b := match hl in hlist x return match x return Type@{hlist_large} with \| nil => unit \| _ :: ls => hlist ls end with \| Hnil => tt \| Hcons _ _ _ x => x end.	Definition	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_tl
hlist_eta : forall ls (h : hlist ls), h = match ls as ls return hlist ls -> hlist ls with \| nil => fun _ => Hnil \| a :: b => fun h => Hcons (hlist_hd h) (hlist_tl h) end h. Proof. intros. destruct h; auto. Qed.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_eta
hlist_app ll lr (h : hlist ll) : hlist lr -> hlist (ll ++ lr) := match h in hlist ll return hlist lr -> hlist (ll ++ lr) with \| Hnil => fun x => x \| Hcons _ _ hd tl => fun r => Hcons hd (hlist_app tl r) end.	Fixpoint	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_app
hlist_app_nil_r : forall ls (h : hlist ls), hlist_app h Hnil = match eq_sym (app_nil_r_trans ls) in _ = t return hlist t with \| eq_refl => h end. Proof. induction h; simpl; intros; auto. rewrite IHh at 1. unfold eq_trans. unfold f_equal. unfold eq_sym. clear. revert h. generalize dependent (app_nil_r_trans ls). destruct e. reflexivity. Qed.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_app_nil_r
hlist_rev' ls ls' (h : hlist ls) : hlist ls' -> hlist (rev ls ++ ls') := match h in hlist ls return hlist ls' -> hlist (rev ls ++ ls') with \| Hnil => fun h => h \| Hcons l ls0 x h' => fun hacc => match app_ass_trans (rev ls0) (l :: nil) ls' in _ = t return hlist t -> hlist _ with \| eq_refl => fun x => x end (@hlist_rev' _ (l :: ls') h' (Hcons x hacc)) end.	Fixpoint	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_rev'
hlist_rev ls (h : hlist ls) : hlist (rev ls) := match app_nil_r_trans (rev ls) in _ = t return hlist t with \| eq_refl => hlist_rev' h Hnil end.	Definition	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_rev
hlist_rev_nil : hlist_rev Hnil = Hnil. Proof. reflexivity. Qed. ( TODO: I need hlist_rev_cons ) ( Equivalence ) ( TODO: This should change to relations )	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_rev_nil
equiv_hlist : forall ls, hlist ls -> hlist ls -> Prop := \| hlist_eqv_nil : equiv_hlist Hnil Hnil \| hlist_eqv_cons : forall l ls x y h1 h2, eqv x y -> equiv_hlist h1 h2 -> @equiv_hlist (l :: ls) (Hcons x h1) (Hcons y h2). Global Instance Reflexive_equiv_hlist (R : forall t, Reflexive (@eqv t)) ls : Reflexive (@equiv_hlist ls). Proof. red. induction x; constructor; auto. reflexivity. Qed. Global Instance Symmetric_equiv_hlist (R : forall t, Symmetric (@eqv t)) ls : Symmetric (@equiv_hlist ls). Proof. red. induction 1. { constructor. } { constructor. symmetry. assumption. auto. } Qed. Global Instance Transitive_equiv_hlist (R : forall t, Transitive (@eqv t)) ls : Transitive (@equiv_hlist ls). Proof. red. induction 1. { intro; assumption. } { rewrite (hlist_eta z). refine (fun H' => match H' in @equiv_hlist ls X Y return match ls as ls return hlist ls -> hlist ls -> Prop with \| nil => fun _ _ : hlist nil => True \| l :: ls => fun (X Y : hlist (l :: ls)) => forall Z x xs, eqv (hlist_hd Z) (hlist_hd X) -> equiv_hlist xs (hlist_tl X) -> (forall z : hlist ls, equiv_hlist (hlist_tl X) z -> equiv_hlist (hlist_tl Z) z) -> @equiv_hlist (l :: ls) Z Y end X Y with \| hlist_eqv_nil => I \| hlist_eqv_cons l ls x y h1 h2 pf pf' => _ end (Hcons x h1) x _ H H0 (@IHequiv_hlist)). intros. rewrite (hlist_eta Z). constructor. simpl in . etransitivity. eassumption. eassumption. eapply H3. simpl in . eassumption. } Qed.	Inductive	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	equiv_hlist
equiv_hlist_Hcons : forall ls i a b (c : hlist ls) d, equiv_hlist (Hcons a c) (@Hcons i ls b d) -> (@eqv i a b /\ equiv_hlist c d). Proof. clear. intros. refine match H in @equiv_hlist ls' l r return match ls' as ls' return hlist ls' -> hlist ls' -> _ with \| nil => fun _ _ => True \| l :: ls => fun l r => eqv (hlist_hd l) (hlist_hd r) /\ equiv_hlist (hlist_tl l) (hlist_tl r) end l r with \| hlist_eqv_nil => I \| hlist_eqv_cons _ _ _ _ _ _ pf pf' => conj pf pf' end. Defined.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	equiv_hlist_Hcons
equiv_hlist_app : forall a b (c c' : hlist a) (d d' : hlist b), (equiv_hlist c c' /\ equiv_hlist d d') <-> equiv_hlist (hlist_app c d) (hlist_app c' d'). Proof. clear. split. - destruct 1. induction H. + assumption. + simpl. constructor; auto. - induction c. + rewrite (hlist_eta c'). simpl; intros; split; auto. constructor. + rewrite (hlist_eta c'); simpl. specialize (IHc (hlist_tl c')). intro. eapply equiv_hlist_Hcons in H. intuition. constructor; auto. Qed. Global Instance Injection_equiv_hlist_cons ls i a b (c : hlist ls) d : Injective (equiv_hlist (Hcons a c) (@Hcons i ls b d)) := { result := @eqv i a b /\ equiv_hlist c d ; injection := @equiv_hlist_Hcons _ _ _ _ _ _ }. Global Instance Injection_equiv_hlist_app a b (c c' : hlist a) (d d' : hlist b) : Injective (equiv_hlist (hlist_app c d) (hlist_app c' d')) := { result := equiv_hlist c c' /\ equiv_hlist d d' ; injection := fun x => proj2 (@equiv_hlist_app _ _ _ _ _ _) x }.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	equiv_hlist_app
hlist_nil_eta : forall (h : hlist nil), h = Hnil. Proof. intros; rewrite (hlist_eta h); reflexivity. Qed.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_nil_eta
hlist_cons_eta : forall a b (h : hlist (a :: b)), h = Hcons (hlist_hd h) (hlist_tl h). Proof. intros; rewrite (hlist_eta h); reflexivity. Qed.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	hlist_cons_eta
Hcons_inv : forall l ls a b c d, @eq (hlist (l :: ls)) (Hcons a b) (Hcons c d) -> a = c /\ b = d. Proof. intros. refine ( match H as K in _ = Z return match Z in hlist LS return match LS with \| nil => Prop \| l :: ls => F l -> hlist ls -> Prop end with \| Hcons X Y x y => fun a b => a = x /\ b = y \| Hnil => True end a b with \| eq_refl => conj eq_refl eq_refl end). Qed. Global Instance Injection_hlist_cons ls t (a : F t) (b : hlist ls) c d : Injective (Hcons a b = Hcons c d) := { result := a = c /\ b = d ; injection := @Hcons_inv t ls a b c d }.	Lemma	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	Hcons_inv
equiv_eq_eq : forall ls (x y : hlist ls), equiv_hlist (fun x => @eq _) x y <-> x = y. Proof. induction x; simpl; intros. { split. inversion 1. rewrite hlist_nil_eta. reflexivity. intros; subst; constructor. } { split. { intro. rewrite (hlist_eta y). specialize (IHx (hlist_tl y)). refine (match H in @equiv_hlist _ LS X Y return match X in hlist LS return F match LS with \| nil => l \| l :: _ => l end -> hlist match LS with \| nil => ls \| _ :: ls => ls end -> Prop with \| Hnil => fun _ _ => True \| Hcons a b c d => fun x y => (equiv_hlist (fun x0 : iT => eq) d y <-> d = y) -> @Hcons a b c d = Hcons x y end (match LS as LS return hlist LS -> F match LS with \| nil => l \| l :: _ => l end with \| nil => fun _ => f \| l :: ls => hlist_hd end Y) (match LS as LS return hlist LS -> hlist match LS with \| nil => ls \| _ :: ls => ls end with \| nil => fun _ => x \| l :: ls => hlist_tl end Y) with \| hlist_eqv_nil => I \| hlist_eqv_cons l ls x y h1 h2 pf1 pf2 => _ end IHx). simpl. subst. intros. f_equal. apply H0. assumption. } { intros; subst. constructor; auto. reflexivity. } } Qed.	Theorem	theories	[ "From Coq Require Import List PeanoNat.", "Require Import Relations RelationClasses.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import Coq." ]	theories/Data/HList.v	equiv_eq_eq

value : Type := | Int : nat -> value | Bool : bool -> value.

Inductive

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

value

exp : Type := | ConstI : nat -> exp | ConstB : bool -> exp | Plus : exp -> exp -> exp | If : exp -> exp -> exp -> exp.

Inductive

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

exp

asInt (v : value) : m nat := match v with | Int n => ret n | _ => (** if we don't have an integer, signal an error using ** [raise] from the MoandExc instance **) raise ("expected integer got bool")%string end.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

asInt

asBool (v : value) : m bool := match v with | Bool b => ret b | _ => raise ("expected bool got integer")%string end. (** The main evaluator routine returns a [value], but since we are ** working in the [m] monad, we return [m value] **)

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

asBool

eval' (e : exp) : m value := match e with (** when there is no error, we can just return (i.e. [ret]) ** the answer **) | ConstI i => ret (Int i) | ConstB b => ret (Bool b) | Plus l r => (** evaluate the sub-terms to numbers **) l <- eval' l ;; l <- asInt l ;; r <- eval' r ;; r <- asInt r ;; (** Combine the result **) ret (Int (l + r)) | If t tr fa => (** evaluate the test condition to a boolean **) t <- eval' t ;; t <- asBool t ;; (** case split and perform the appropriate recursion **) if (t : bool) then eval' tr else eval' fa end.

Fixpoint

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

eval'

eval : exp -> string + value := eval' (m := sum string). (** Some tests **) Eval compute in eval (Plus (ConstI 1) (ConstI 2)). Eval compute in eval (Plus (ConstI 1) (ConstB false)). (** Other useful monads: ** * Reader - for handling lexicographic environments ** * State - for handling non-lexical state, like a heap ** * MonadFix - for handling unbounded recursion **)

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/EvalWithExc.v

eval

update1 : istate A B unit := modify_ function1.

Definition

examples

[ "Require Import ExtLib.", "Require Import ExtLib." ]

examples/indexedstate.v

update1

update2 : istate B C unit := modify_ function2.

Definition

examples

[ "Require Import ExtLib.", "Require Import ExtLib." ]

examples/indexedstate.v

update2

compose : istate A C unit := update1 ;; update2.

Definition

examples

[ "Require Import ExtLib.", "Require Import ExtLib." ]

examples/indexedstate.v

compose

proper_eta : forall T U (f : T -> U) (type_T : type T) (type_U : type U), proper f -> proper (fun x => f x). Proof. intros; do 3 red; intros. eapply H. assumption. Qed. Goal forall x : T, proper x -> equal (bind (ret x) (fun x => ret x)) (ret x). Proof. intros. etransitivity. { eapply bind_of_return; eauto. eapply proper_eta. eapply ret_proper; eauto. } { eapply ret_proper; eauto. eapply equiv_prefl; eauto. } Qed. Goal forall x : T, proper x -> equal (bind (ret x) (fun x => ret x)) (ret x). Proof. intros. etransitivity. { eapply bind_of_return; eauto. eapply proper_eta. eapply ret_proper; eauto. } { eapply ret_proper; eauto. eapply equiv_prefl; eauto. } Qed.

Lemma

examples

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/MonadReasoning.v

proper_eta

repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit := match n with | O => ret tt | S n => y <- p x;; repeatM n y p end.

Fixpoint

examples

[ "Require Import ExtLib." ]

examples/Notations.v

repeatM

repeatM `{Monad M} (n : nat) `(x : A) (p : A -> M A) : M unit := match n with | O => ret tt | S n => let* y := p x in repeatM n y p end.

Fixpoint

examples

[ "Require Import ExtLib." ]

examples/Notations.v

repeatM

PrinterMonad : Type -> Type := writerT (@show_mon _ ShowScheme_string_compose) ident.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/Printing.v

PrinterMonad

print {T : Type} {ST : Show T} (val : T) : PrinterMonad unit := @MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _ (@show _ ST val _ show_inj (@show_mon _ ShowScheme_string_compose)).

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/Printing.v

print

printString (str : string) : PrinterMonad unit := @MonadWriter.tell _ (@show_mon _ ShowScheme_string_compose) _ _ (@show_exact str _ show_inj (@show_mon _ ShowScheme_string_compose)).

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/Printing.v

printString

runPrinter {T : Type} (c : PrinterMonad T) : T * string := let '(ppair val str) := unIdent (runWriterT c) in (val, str ""%string). Eval compute in runPrinter (Monad.bind (print 1) (fun _ => print 2)). Eval compute in runPrinter (Monad.bind (print "hello "%string) (fun _ => print 2)). Eval compute in runPrinter (Monad.bind (printString "hello "%string) (fun _ => print 2)).

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/Printing.v

runPrinter

use of State monad Passes a string of dictionary {a,b,c} Game is to produce a number from the string. By default the game is off, a C toggles the game on and off. A 'a' gives +1 and a b gives -1. E.g 'ab' = 0 'ca' = 1 'cabca' = 0 State = game is on or off & current score = (Bool, Int) *) Require Import Coq.ZArith.ZArith_base Coq.Strings.String Coq.Strings.Ascii. Require Import ExtLib.Data.Monads.StateMonad ExtLib.Structures.Monads.

Example

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

use

GameValue : Type := Z.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

GameValue

GameState : Type := (prod bool Z). Variable m : Type -> Type. Context {Monad_m: Monad m}. Context {State_m: MonadState GameState m}.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

GameState

playGame (s: string) {struct s}: m GameValue := match s with | EmptyString => v <- get ;; let '(on, score) := v in ret score | String x xs => v <- get ;; let '(on, score) := v in match x, on with | "a", true => put (on, score + 1) | "b", true => put (on, score - 1) | "c", _ => put (negb on, score) | _, _ => put (on, score) end ;; playGame xs end.

Fixpoint

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

playGame

startState : GameState := (false, 0).

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

startState

main : GameValue := (@evalState GameState GameValue (playGame (state GameState) "abcaaacbbcabbab") startState). (* The following should return '2%Z' *) Compute main.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib." ]

examples/StateGame.v

main

foo : stateT unit option unit := ret tt.

Definition

examples

[]

examples/StateTMonad.v

foo

contains_both (v1 v2 : V) (s : set) : bool := contains v1 s && contains v2 s. (** Iteration requires foldability **) Context {Foldable_set : Foldable set V}.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/UsingSets.v

contains_both

toList (s : set) : list V := fold (@cons _) nil s.

Definition

examples

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

examples/UsingSets.v

toList

RTest : Set := mkRTest { a : bool ; b : nat ; c : bool }. Bind Scope struct_scope with RTest. Global Instance Struct_RTest : Struct RTest := { fields := ((@existT _ _ _ a) :: (@existT _ _ _ b) :: (@existT _ _ _ c):: nil) ; ctor := mkRTest }. Global Instance Acc_RTest_a : Accessor a := { acc := Here }. Global Instance Acc_RTest_b : Accessor b := { acc := Next Here }. Global Instance Acc_RTest_c : Accessor c := { acc := Next (Next Here) }. Eval compute in {$ mkRTest true 1 true with c := false $}%record. Eval compute in forall x : RTest, c {$ x with c := false $}%record = false.

Record

examples

[ "Require Import List.", "Require Import ExtLib." ]

examples/WithDemo.v

RTest

compose (A B C : Type) (f : A -> B) (g : B -> C) : A -> C := fun x => g (f x).

Definition

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

compose

pure (T : Type) : T -> m T := @ret _ _ _.

Definition

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

pure

fapply (T U : Type) (f : m (T -> U)) (x : m T) : m U := bind f (fun f => bind x (fun x => ret (f x))).

Definition

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

fapply

Instance fun_trans.

Existing

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

Instance

Instance fun_refl. Variables A B C : Type. Context (rA : relation A) (rB : relation B) (rC : relation C) (pA : Proper rA) (pB : Proper rB) (pC : Proper rC). Context (Ra : PReflexive rA) (Rb : PReflexive rB) (Rc : PReflexive rC). Context (Ta : PTransitive rA) (Tb : PTransitive rB) (Tc : PTransitive rC).

Existing

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

Instance

fun_app_proper (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) x : proper f -> proper x -> proper (f x). Proof. intros. apply H. auto. Qed.

Instance

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

fun_app_proper

fun_abs (A B : Type) (rA : relation A) (rB : relation B) (pA : Proper rA) (pB : Proper rB) (f : A -> B) : (forall x, proper x -> proper (f x)) -> (forall x y, proper x -> proper y -> rA x y -> rB (f x) (f y)) -> proper (fun x => f x). Proof. intros. split; auto; eapply H. Qed.

Instance

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

fun_abs

Ltac

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

prove_proper

PReflexive_stuff : PReflexive (pfun_ext (pfun_ext (pfun_ext rC pA) (Proper_pfun pB pC)) (Proper_pfun pA pB)). Proof. intuition. Qed.

Instance

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

PReflexive_stuff

bind_law : forall (f : A -> B) (g : B -> C), proper f -> proper g -> mleq (pfun_ext rC pA) (fapply (fapply (pure (@compose A B C)) (pure f)) (pure g)) (pure (compose f g)). Proof. unfold fapply, pure, compose; simpl; intros. propers. (eapply ptransitive; [ | | | | eapply (@bind_associativity _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ | | | | eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ | | | | eapply (@bind_associativity _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ | | | | eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ | | | | eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. (eapply ptransitive; [ | | | | eapply (@bind_of_return _ _ _ _ MonadLaws_mleq) | ]); eauto with typeclass_instances; propers. eapply preflexive; eauto with typeclass_instances. Qed.

Theorem

scratch

[ "Require Import Relations.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

scratch/FunctorFromMonad.v

bind_law

Class RESOLVE. #[global] Hint Extern 0 (RESOLVE _) => unfold RESOLVE : typeclass_instances.

Existing

theories

[]

theories/Core/Any.v

Class

CmpDec (T : Type) (equ : T -> T -> Prop) (ltu : T -> T -> Prop) : Type := { cmp_dec : T -> T -> comparison }.

Class

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

CmpDec

CmpDec_Correct T (equ ltu : T -> T -> Prop) (ED : CmpDec equ ltu) : Prop := { cmp_dec_correct : forall x y : T, match cmp_dec x y with | Eq => equ x y | Lt => ltu x y | Gt => ltu y x end }.

Class

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

CmpDec_Correct

cmp_case (P Q R : Prop) : comparison -> Prop := | CaseEq : P -> cmp_case P Q R Eq | CaseLt : Q -> cmp_case P Q R Lt | CaseGt : R -> cmp_case P Q R Gt.

Inductive

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

cmp_case

eq_pair (a b : T * U) : Prop := eqt (fst a) (fst b) /\ equ (snd a) (snd b).

Definition

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

eq_pair

lt_pair (a b : T * U) : Prop := ltt (fst a) (fst b) \/ (eqt (fst a) (fst b) /\ ltu (snd a) (snd b)). Variable cdt : CmpDec eqt ltt. Variable cdu : CmpDec equ ltu.

Definition

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

lt_pair

CmpDec_pair : CmpDec eq_pair lt_pair := { cmp_dec := fun a b => let '(al,ar) := a in let '(bl,br) := b in match cmp_dec al bl with | Eq => cmp_dec ar br | x => x end }. Variable cdtC : CmpDec_Correct cdt. Variable cduC : CmpDec_Correct cdu. Variable Symmetric_eqt : Symmetric eqt.

Instance

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

CmpDec_pair

CmpDec_Correct_pair : CmpDec_Correct CmpDec_pair. Proof. constructor. destruct x; destruct y; unfold eq_pair, lt_pair; simpl in *. generalize (cmp_dec_correct t t0); destruct (cmp_dec t t0); simpl; intros; auto. generalize (cmp_dec_correct u u0); destruct (cmp_dec u u0); simpl; intros; auto. Qed.

Instance

theories

[ "Require Import Coq.", "Require Import Coq.", "Require Import ExtLib." ]

theories/Core/CmpDec.v

CmpDec_Correct_pair

decideP (P : Prop) {D : Decidable P} : {P} + {~P} := match @Decidable_witness P D as X return (X = true -> P) -> (X = false -> ~P) -> {P} + {~P} with | true => fun pf _ => left (pf eq_refl) | false => fun _ pf => right (pf eq_refl) end (@Decidable_sound _ D) (@Decidable_complete_alt _ D).

Definition

theories

[ "From Coq.Classes Require Import DecidableClass." ]

theories/Core/Decision.v

decideP

cases_ifd Hn := match goal with |- context[if ?d then ?tt else ?ff] => let Hnt := fresh Hn "t" in let Hnf := fresh Hn "f" in destruct d as [Hnt | Hnf] end.

Ltac

theories

[ "From Coq.Classes Require Import DecidableClass." ]

theories/Core/Decision.v

cases_ifd

decide_decideP {P:Prop }`{Decidable P} {R:Type} (a b : R) : (if (decide P) then a else b) = (if (decideP P) then a else b). Proof. symmetry. unfold decide. destruct (decideP P). - rewrite Decidable_complete; auto. - rewrite Decidable_sound_alt; auto. Qed.

Lemma

theories

[ "From Coq.Classes Require Import DecidableClass." ]

theories/Core/Decision.v

decide_decideP

EquivDec_refl_left {T : Type} {c : EqDec T (@eq T)} : forall (n : T), equiv_dec n n = left (refl_equal _). Proof. intros. destruct (equiv_dec n n); try congruence. Require Eqdep_dec. rewrite (Eqdep_dec.UIP_dec (A := T) (@equiv_dec _ _ _ c) e (refl_equal _)). reflexivity. Qed. Export EquivDec.

Theorem

theories

[ "From Coq.Classes Require Import EquivDec." ]

theories/Core/EquivDec.v

EquivDec_refl_left

RelDec (T : Type) (equ : T -> T -> Prop) : Type := { rel_dec : T -> T -> bool }. Arguments rel_dec {_} {equ} {_} _ _. Arguments rel_dec _ _ _ !x !y.

Class

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

RelDec

RelDec_Correct T (equ : T -> T -> Prop) (ED : RelDec equ) : Prop := { rel_dec_correct : forall x y : T, rel_dec x y = true <-> equ x y }.

Class

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

RelDec_Correct

eq_dec {T : Type} {ED : RelDec (@eq T)} := rel_dec.

Definition

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

eq_dec

neg_rel_dec_correct : forall {x y}, ~R x y <-> rel_dec x y = false. Proof. intros x y. destruct (bool_dec (rel_dec x y) true) ; constructor ; intros ; repeat match goal with | [ |- ~ _ ] => unfold not ; intros | [ H1 : ?P, H2 : ~?P |- _ ] => specialize (H2 H1) ; contradiction | [ H1 : ?P = true, H2 : ?P = false |- _ ] => rewrite H1 in H2 ; discriminate | [ H1 : ?P <> true |- ?P = false ] => apply not_true_is_false ; exact H1 | [ H1 : ?rel_dec ?a ?b = true, H2 : ~?R ?a ?b |- _ ] => apply rel_dec_correct in H1 | [ H1 : ?rel_dec ?a ?b = false, H2 : ?R ?a ?b |- _ ] => apply rel_dec_correct in H2 end. Qed.

Definition

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

neg_rel_dec_correct

rel_dec_p (x:T) (y:T) : {R x y} + {~R x y}. Proof. destruct (bool_dec (rel_dec x y) true) as [H | H]. apply rel_dec_correct in H ; eauto. apply not_true_is_false in H ; apply neg_rel_dec_correct in H ; eauto. Qed.

Definition

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

rel_dec_p

neg_rel_dec_p (x:T) (y:T) : {~R x y} + {R x y}. Proof. destruct (rel_dec_p x y) ; [ right | left ] ; auto. Qed.

Definition

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

neg_rel_dec_p

rel_dec_eq_true : forall x y, eqt x y -> rel_dec x y = true. Proof. intros. eapply rel_dec_correct in H. assumption. Qed.

Theorem

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

rel_dec_eq_true

rel_dec_neq_false : forall x y, ~eqt x y -> rel_dec x y = false. Proof. intros. remember (x ?[ eqt ] y). symmetry in Heqb. destruct b; try reflexivity. exfalso. eapply (@rel_dec_correct _ _ _ rc) in Heqb. auto. Qed.

Theorem

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

rel_dec_neq_false

rel_dec_sym : Symmetric eqt -> forall x y, x ?[ eqt ] y = y ?[ eqt ] x. Proof. intros. remember (x ?[ eqt ] y); remember (y ?[ eqt ] x); intuition. destruct b; destruct b0; auto. { symmetry in Heqb; symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb. symmetry in Heqb. eapply (@rel_dec_correct _ _ _ rc) in Heqb. congruence. } { symmetry in Heqb; symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb0. symmetry in Heqb0. eapply (@rel_dec_correct _ _ _ rc) in Heqb0. congruence. } Qed.

Theorem

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

rel_dec_sym

RelDec_from_dec : RelDec R := {| rel_dec := fun a b => match f a b with | left _ => true | right _ => false end |}. Global Instance RelDec_Correct_eq_typ : RelDec_Correct RelDec_from_dec. Proof. constructor. intros. unfold rel_dec; simpl. destruct (f x y). - tauto. - split. + inversion 1. + intro. apply n in H. tauto. Qed.

Definition

theories

[ "Require Import Coq.", "Require Import Coq." ]

theories/Core/RelDec.v

RelDec_from_dec

digit2ascii (n:nat) : Ascii.ascii := match n with | 0 => "0" | 1 => "1" | 2 => "2" | 3 => "3" | 4 => "4" | 5 => "5" | 6 => "6" | 7 => "7" | 8 => "8" | 9 => "9" | n => ascii_of_nat (n - 10 + nat_of_ascii "A") end%char.

Definition

theories

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Char.v

digit2ascii

chr_newline : ascii := Eval compute in ascii_of_nat 10. Export Ascii.

Definition

theories

[ "Require Import Coq.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Char.v

chr_newline

Checked : option T -> Type := | Success : forall {v}, F v -> Checked (Some v) | Failure : Checked None.

Inductive

theories

[]

theories/Data/Checked.v

Checked

succeeded (o : option T) (d : Checked o) : bool := match d with | Success _ _ => true | Failure => false end.

Definition

theories

[]

theories/Data/Checked.v

succeeded

failed (o : option T) (d : Checked o) : bool := match d with | Success _ _ => false | Failure => true end.

Definition

theories

[]

theories/Data/Checked.v

failed

Definition

theories

[]

theories/Data/Checked.v

asOption

eq_sym_eq : forall T (a b : T) (pf : a = b) (F : T -> Type) val, match eq_sym pf in _ = x return F x with | eq_refl => val end = match pf in _ = x return F x -> F a with | eq_refl => fun x => x end val. Proof. destruct pf. reflexivity. Defined.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

eq_sym_eq

match_eq_sym_eq : forall T (a b : T) (pf : a = b) F X, match pf in _ = t return F t with | eq_refl => match eq_sym pf in _ = t return F t with | eq_refl => X end end = X. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite match_eq_sym_eq : eq_rw.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

match_eq_sym_eq

match_eq_sym_eq' : forall T (a b : T) (pf : a = b) F X, match eq_sym pf in _ = t return F t with | eq_refl => match pf in _ = t return F t with | eq_refl => X end end = X. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite match_eq_sym_eq' : eq_rw.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

match_eq_sym_eq'

match_eq_match_eq : forall T F (a b : T) (pf : a = b) X Y, X = Y -> match pf in _ = T return F T with | eq_refl => X end = match pf in _ = T return F T with | eq_refl => Y end. Proof. intros. subst. auto. Defined.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

match_eq_match_eq

eq_sym_eq_trans : forall T (a b c : T) (pf : a = b) (pf' : b = c), eq_sym (eq_trans pf pf') = eq_trans (eq_sym pf') (eq_sym pf). Proof. clear. destruct pf. destruct pf'. reflexivity. Defined. (** Particular Instances **)

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

eq_sym_eq_trans

eq_Const_eq : forall T (a b : T) (pf : a = b) (R : Type) val, match pf in _ = x return R with | eq_refl => val end = val. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_Const_eq : eq_rw.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

eq_Const_eq

eq_Arr_eq : forall T (a b : T) (pf : a = b) (F G : T -> Type) val x, match pf in _ = x return F x -> G x with | eq_refl => val end x = match pf in _ = x return G x with | eq_refl => val match eq_sym pf in _ = x return F x with | eq_refl => x end end. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_Arr_eq : eq_rw.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

eq_Arr_eq

eq_sym_eq_sym : forall (T : Type) (a b : T) (pf : a = b), eq_sym (eq_sym pf) = pf. Proof. destruct pf. reflexivity. Defined. #[global] Hint Rewrite eq_sym_eq_sym : eq_rw.

Lemma

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

eq_sym_eq_sym

autorewrite_eq_rw := repeat progress (autorewrite with eq_rw; repeat match goal with | |- context [ match ?X in @eq _ _ _ return _ -> _ with | eq_refl => _ end ] => rewrite (eq_Arr_eq X) end). Require Export ExtLib.Data.Eq.UIP_trans.

Ltac

theories

[ "Require Export ExtLib." ]

theories/Data/Eq.v

autorewrite_eq_rw

fin : nat -> Type := | F0 : forall {n}, fin (S n) | FS : forall {n}, fin n -> fin (S n).

Inductive

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin

fin_all (n : nat) : list (fin n) := match n as n return list (fin n) with | 0 => nil | S n => @F0 n :: List.map (@FS _) (fin_all n) end%list.

Fixpoint

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin_all

fin_all_In : forall {n} (f : fin n), List.In f (fin_all n). Proof. induction n; intros. inversion f. remember (S n). destruct f. simpl; firstorder. inversion Heqn0. subst. simpl. right. apply List.in_map. auto. Qed.

Theorem

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin_all_In

fin_case : forall n (f : fin (S n)), f = F0 \/ exists f', f = FS f'. Proof. intros. generalize (fin_all_In f). intros. destruct H; auto. eapply List.in_map_iff in H. right. destruct H. exists x. intuition. Qed.

Theorem

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin_case

fin0_elim (f : fin 0) : forall T, T := match f in fin n return match n with | 0 => forall T, T | _ => unit end with | F0 _ => tt | FS _ _ => tt end.

Definition

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin0_elim

pf_lt (n m : nat) : Prop := match n , m with | 0 , S _ => True | S n , S m => pf_lt n m | _ , _ => False end.

Fixpoint

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

pf_lt

make (m n : nat) {struct m} : pf_lt n m -> fin m := match n as n , m as m return pf_lt n m -> fin m with | 0 , 0 => @False_rect _ | 0 , S n => fun _ => F0 | S n , 0 => @False_rect _ | S n , S m => fun pf => FS (make m n pf) end.

Fixpoint

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

make

fin_eq_dec {n} (x : fin n) {struct x} : fin n -> bool := match x in fin n' return fin n' -> bool with | F0 _ => fun y => match y with | F0 _ => true | _ => false end | FS n' x' => fun y : fin (S n') => match y in fin n'' return (match n'' with | 0 => unit | S n'' => fin n'' end -> bool) -> bool with | F0 _ => fun _ => false | FS _ y' => fun f => f y' end (fun y => fin_eq_dec x' y) end. Global Instance RelDec_fin_eq (n : nat) : RelDec (@eq (fin n)) := { rel_dec := fin_eq_dec }. Global Instance RelDec_Correct_fin_eq (n : nat) : RelDec_Correct (RelDec_fin_eq n). Proof. constructor. induction x. simpl. intro. destruct (fin_case y) ; subst. intuition. destruct H ; subst. intuition; auto with *; try congruence. (* inversion H.*) intro ; destruct (fin_case y) ; subst ; simpl. intuition ; try congruence. inversion H. destruct H ; subst. split ; intro. f_equal ; eauto. eapply IHx. eapply H. inv_all ; subst. apply IHx. reflexivity. Qed.

Fixpoint

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fin.v

fin_eq_dec

Monoid_compose T : Monoid (T -> T) := {| monoid_plus g f x := g (f x) ; monoid_unit x := x |}. Export PreFun.

Definition

theories

[ "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib.", "Require Import ExtLib." ]

theories/Data/Fun.v

Monoid_compose

app_ass_trans @{X} : forall {T : Type@{X} } (a b c : list T), (a ++ b) ++ c = a ++ b ++ c. Proof. induction a; simpl. reflexivity. intros. destruct (IHa b c). reflexivity. Defined.

Lemma

theories