I'm trying to import Library Coq.Structures.OrdersFacts as usual with:
Require Import Coq.Structures.OrdersFacts
Then I try to use of the lemmas there with either:
apply CompareFacts.compare_nlt_iff. or apply compare_nlt_iff.
But none work ... what am I missing?
CompareFacts is a Module Type, not a Module. You can see that if you do
Require Import Coq.Structures.OrdersFacts.
Print OrdersFacts.CompareFacts.
Find a Module of this type and apply its Lemmas instead.
EDIT:
I meant that to use the lemmas on i.e. nat, you need a module that shows that nat is a DecStrOrder' (and Nat from PeanoNat is such a module), and also one that specializes CompareFacts for nat .
Perhaps an example is more useful.
Require Import Coq.Structures.OrdersFacts.
Module mymodule (O:DecStrOrder') (T: CompareFacts O).
Import T.
Import O.
Check compare_eq_iff. (* from CompareFacts *)
(* a theorem about terms of type O.t *)
Lemma lem1 a b c: (a ?= b) = Eq -> b == c -> c == a.
intros.
rewrite compare_eq_iff in H. (* here we use the lemma *)
rewrite H.
rewrite H0.
apply eq_equiv.
Qed.
End mymodule.
(* the above module functor can be specialised for i.e. nat *)
Require Import PeanoNat.
Print CompareFacts.
Module M : CompareFacts Nat.
Definition compare_eq_iff := Nat.compare_eq_iff.
Definition compare_eq := Nat.compare_eq.
Definition compare_lt_iff := Nat.compare_lt_iff.
Definition compare_gt_iff := Nat.compare_gt_iff.
Definition compare_nlt_iff := Nat.compare_nlt_iff.
Definition compare_ngt_iff := Nat.compare_ngt_iff.
Definition compare_refl := Nat.compare_refl.
Definition compare_compat: Proper (eq==>eq==>eq) Nat.compare.
intros x y Hxy a b Hab; now subst. Defined.
Definition compare_antisym := Nat.compare_antisym.
End M.
Module natmodule := mymodule Nat M.
Check natmodule.lem1.
Related
The following code states that it defines finite sets in Coq with the axiom of extensionality:
(** A library for finite sets with extensional equality.
Author: Brian Aydemir. *)
Require Import FSets.
Require Import ListFacts.
Require Import Coq.Logic.ProofIrrelevance.
(* *********************************************************************** *)
(** * Interface *)
(** The following interface wraps the standard library's finite set
interface with an additional property: extensional equality. *)
Module Type S.
Declare Module E : UsualOrderedType.
Declare Module F : FSetInterface.S with Module E := E.
Parameter eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
End S.
(* *********************************************************************** *)
(** * Implementation *)
(** For documentation purposes, we hide the implementation of a
functor implementing the above interface. We note only that the
implementation here assumes (as an axiom) that proof irrelevance
holds. *)
Module Make (X : UsualOrderedType) <: S with Module E := X.
(* begin hide *)
Module E := X.
Module F := FSetList.Make E.
Module OFacts := OrderedType.OrderedTypeFacts E.
Lemma eq_if_Equal :
forall s s' : F.t, F.Equal s s' -> s = s'.
Proof.
intros [s1 pf1] [s2 pf2] Eq.
assert (s1 = s2).
unfold F.MSet.Raw.t in *.
eapply Sort_InA_eq_ext; eauto.
intros; eapply E.lt_trans; eauto.
1 : {
apply F.MSet.Raw.isok_iff.
auto.
}
1 : {
apply F.MSet.Raw.isok_iff.
auto.
}
subst s1.
assert (pf1 = pf2).
apply proof_irrelevance.
subst pf2.
reflexivity.
Qed.
(* end hide *)
End Make.
How can I define a function with signature from finite sets to finite sets using this module?
You need to define a Module (call it M) that implements the UsualOrderedType module type for the type you want to make finite sets out of, and then build another Module with Make M which contains an implementation of finite sets for your type.
Module M <: UsualOrderedType.
...
End M.
Module foo := Make M.
Check foo.F.singleton.
Note that you need to declare the module type with <: instead of just :, otherwise you are hiding the fact that the module is defined for (in the example below) nat behind an opaque type t.
Say you want to make finite sets of nats:
(* Print the module type to see all the things you need to define. *)
Print Module Type UsualOrderedType.
Require Import PeanonNat.
Module NatOrdered <: UsualOrderedType . (* note the `<:` *)
Definition t:=nat.
Definition eq:=#eq nat.
Definition lt:=lt.
Definition eq_refl:=#eq_refl nat.
Definition eq_sym:=#eq_sym nat.
Definition eq_trans:=#eq_trans nat.
Definition lt_trans:=Nat.lt_trans.
(* I wrote Admitted where I didn't provide an implementation *)
Definition lt_not_eq : forall x y : t, lt x y -> ~ eq x y. Admitted.
Definition compare : forall x y : t, Compare lt eq x y. Admitted.
Definition eq_dec : forall x y : t, {eq x y} + {~ eq x y}. Admitted.
End NatOrdered.
Now you can create a module that uses this ordered type.
Module foo := Make NatOrdered.
Print foo.F. (* to see everything that is defined for the FSetList *)
Import foo. (* get F in our namespace so we can say F.t instead of foo.F.t, etc. *)
Now we can use our F module. The finite sets have type F.t and the elements have type F.elt which is coercible to nat since we know they come from NatOrdered.
Lets build a function that uses stuff from F.
Definition f: F.elt -> F.t.
intros x. apply (F.singleton x).
Defined.
Print F.
Goal F.cardinal (F.union (f 1) (f 2)) = 2.
compute.
Ok. That gets stuck halfway through the computation because I didn't implement compare above. I was lazy and just wrote Admitted. But you can do it! :-)
It seems that at some point there was a file SfLib.v that was used for the Software Foundations course. However, the case command proposed in this file is not getting through in my case:
Require Omega. (* needed for using the [omega] tactic *)
Require Export Bool.
Require Export List.
Require Export Arith.
Require Export Arith.EqNat. (* Contains [beq_nat], among other things *)
(** * From Basics.v *)
Require String. Open Scope string_scope.
Ltac move_to_top x :=
match reverse goal with
| H : _ |- _ => try move x after H
end.
Tactic Notation "assert_eq" ident(x) constr(v) :=
let H := fresh in
assert (x = v) as H by reflexivity;
clear H.
Tactic Notation "Case_aux" ident(x) constr(name) :=
first [
set (x := name); move_to_top x
| assert_eq x name; move_to_top x
| fail 1 "because we are working on a different case" ].
Tactic Notation "Case" constr(name) := Case_aux Case name.
Tactic Notation "SCase" constr(name) := Case_aux SCase name.
Tactic Notation "SSCase" constr(name) := Case_aux SSCase name.
Tactic Notation "SSSCase" constr(name) := Case_aux SSSCase name.
Tactic Notation "SSSSCase" constr(name) := Case_aux SSSSCase name.
Tactic Notation "SSSSSCase" constr(name) := Case_aux SSSSSCase name.
Tactic Notation "SSSSSSCase" constr(name) := Case_aux SSSSSSCase name.
Tactic Notation "SSSSSSSCase" constr(name) := Case_aux SSSSSSSCase name.
Fixpoint ble_nat (n m : nat) : bool :=
match n with
| O => true
| S n' =>
match m with
| O => false
| S m' => ble_nat n' m'
end
end.
Theorem andb_true_elim1 : forall b c,
andb b c = true -> b = true.
Proof.
intros b c H.
destruct b.
Case "b = true".
reflexivity.
Case "b = false".
rewrite <- H. reflexivity. Qed.
In the first case, I'm getting No interpretation for string "b = true".
This was previously addressed in coq error when trying to use Case. Example from Software Foundations book. However, the solution there does no longer work. Should I get rid of all the case statements?
Maybe something changed since then, but now we need to Import notations.
(* 8.9, 8.10, and newer *)
From Coq Require String.
Export String.StringSyntax. (* [Export] means to [Import] the StringSyntax module, but also make it automatically available to whoever imports this file as well. *)
Open Scope string_scope.
(* 8.8 and older (still works with 8.9, 8.10, but pollutes the namespace) *)
From Coq Require Export String.
Open Scope string_scope.
However, as evidenced by the removal of the module, the benefit of this trick is quite minor. Most people just use bullets, with comments to indicate the cases if they even care to comment.
Above, Open Scope is used, which may be fine for self-contained projects like SF, but for big and open projects, to avoid surprises with notations, I would recommend only using Local Open Scope (which then needs to appear in every file), or keeping Open Scope inside inner modules (that still need to be imported explicitly, but can be reexported together with others).
I was able to create a ComRingMixin, but when I try to declare this type as a canonical ring, Coq complains:
x : phantom (GRing.Zmodule.class_of ?bT) (GRing.Zmodule.class ?bT)
The term "x" has type "phantom (GRing.Zmodule.class_of ?bT) (GRing.Zmodule.class ?bT)"
while it is expected to have type "phantom (GRing.Zmodule.class_of 'I_n) ?b".
This is what I have so far, I was able to define the operations and instantiate the abelian group mixin as well as the canonical declaration, but for the ring, my code fails.
From mathcomp Require Import all_ssreflect all_algebra.
Set Implicit Arguments.
Unset Strict Implicit.
Unset Printing Implicit Defensive.
Import GRing.Theory.
Open Scope ring_scope.
Section Zn.
Variables n :nat.
Axiom one_lt_n : (1 < n)%N.
Axiom z_lt_n : (0 < n)%N.
Lemma mod_lt_n : forall (x : nat), ((x %% n)%N < n)%N.
Proof.
move=> x0; rewrite ltn_mod; by exact: z_lt_n.
Qed.
Definition mulmod (a b : 'I_n) : 'I_n := Ordinal (mod_lt_n (((a*b)%N %% n)%N)).
Definition addmod (a b : 'I_n) : 'I_n := Ordinal (mod_lt_n (((a+b)%N %% n)%N)).
Definition oppmod (x : 'I_n) : 'I_n := Ordinal (mod_lt_n (n - x)%N).
Lemma addmodC : commutative addmod. Admitted.
Lemma addmod0 : left_id (Ordinal z_lt_n) addmod. Admitted.
Lemma oppmodK : involutive oppmod. Admitted.
Lemma addmodA : associative addmod. Admitted.
Lemma addmodN : left_inverse (Ordinal z_lt_n) oppmod addmod. Admitted.
Definition Mixin := ZmodMixin addmodA addmodC addmod0 addmodN.
Canonical ordn_ZmodType := ZmodType 'I_n Mixin.
Lemma mulmodA : associative mulmod. Admitted.
Lemma mulmodC : commutative mulmod. Admitted.
Lemma mulmod1 : left_id (Ordinal one_lt_n) mulmod. Admitted.
Lemma mulmod_addl : left_distributive mulmod addmod. Admitted.
Lemma one_neq_0_ord : (Ordinal one_lt_n) != Ordinal z_lt_n. Proof. by []. Qed.
Definition mcommixin := #ComRingMixin ordn_ZmodType (Ordinal one_lt_n) mulmod mulmodA mulmodC mulmod1 mulmod_addl one_neq_0_ord.
Canonical ordnRing := RingType 'I_n mcommixin.
Canonical ordncomRing := ComRingType int intRing.mulzC.
What am i doing wrong? I'm basing myself on http://www-sop.inria.fr/teams/marelle/advanced-coq-17/lesson5.html.
The problem is that ssralg already declares ordinal as a zmodType instance. There can only be one canonical instance of a structure per head symbol, so your declaration of ordn_ZmodType is effectively ignored.
One solution around it is to introduce a local synonym in this section and use it to define the canonical structures:
(* ... *)
Definition foo := 'I_n.
(* ... *)
Definition ordn_ZmodType := ZmodType foo Mixin.
(* ... *)
Canonical ordnRing := RingType foo mcommixin. (* This now works *)
The other solution is to use the ringType instance defined in MathComp for ordinal. The catch is that it is only defined for types of the form 'I_n.+2.
In principle, one could also have declared these instances assuming the same axioms on n as you did, but this would make the inference of canonical structures more difficult.
Check fun n => [ringType of 'I_n.+2].
(* ... : nat -> ringType *)
What is the exact semantics of Include M1 inside another module, say M ? How is it different from doing Import M1 inside the module M ? More precisely what is the semantics of the following command:
Module Type M := M1 <+ M2 <+ M3.
To summarize the semantics of both vernacular commands:
The command Include M1 (which can be used in the definition of a module or a module type) asks Coq to make a copy of all the fields of M1. Thus, it acts just like a "copy-and-paste" of the contents of M1 inside the ambient module (resp. module type).
The command Import M1 (which can also be used in the definition of a module or a module type, but additionally requires that M1 is a module) allows one to refer to the fields of M1 with their short identifier (i.e., without needing to use qualified identifiers M1.field_name)
Next, the syntax Module Type M := M1 <+ M2 <+ M3 is a shortcut for:
Module Type M.
Include M1.
Include M2.
Include M3.
End M.
See the corresponding sections of Coq's reference manual on the Include and Import commands, and you may also want to take a look at the Export command (a variant of Import).
If at some point you hesitate between Include and Import, you should probably try to use Import in the first place, because it will have a lighter effect (it won't clone the contents of the specified module but just define shorter names).
Finally, below are two comprehensive examples that illustrate the use of modules, module types, and functors in Coq as well as the commands Include and Import:
(********************************************)
(* Example involving a parameterized module *)
(********************************************)
(* A signature *)
Module Type MT.
Parameter t : Type.
End MT.
(* A bigger signature *)
Module Type MT1.
Include MT.
Parameter u : t.
Parameter f : t -> t.
End MT1.
(* A parameterized module *)
Module F1 (M1 : MT1).
Import M1. (* => we can now write f rather than M1.f *)
Definition fu := f u.
End F1.
(* A module implementing MT1 *)
Module M1 <: MT1. (* => check the signature but don't make the module opaque *)
Definition t := nat.
Definition u := O.
Definition f := S.
End M1.
(* Instantiation *)
Module FM1 := F1 M1.
Compute FM1.fu.
and I recall that Compute is a shortcut for Eval vm_compute in
(********************************************)
(* Extra example: a parameterized signature *)
(* *)
(* It can be noted that this feature of Coq *)
(* module types has no equivalent in OCaml… *)
(********************************************)
Module Type MT2 (M : MT).
Parameter u : M.t.
Import M. (* => we can now write t rather than M.t *)
Parameter f : t -> t.
End MT2.
(* Another parameterized module *)
Module F2 (M : MT) (M2 : MT2 M).
Import M2.
Definition fu := f u.
End F2.
(* Modules implementing MT and MT2 *)
Module M <: MT.
Definition t := bool.
End M.
Module M2 <: MT2 M.
Definition u := false.
Definition f := negb.
End M2.
(* Instantiation *)
Module FM2 := F2 M M2.
Compute FM2.fu.
I'm trying to unfold Mem.load and I get the error:
Error: Cannot coerce Mem.load to an evaluable reference.
I wrote the exact same Definition of Mem.load as load1 and is unfoldable.
Require Import compcert.common.AST.
Require Import compcert.common.Memory.
Require Import compcert.common.Values.
Require Import compcert.lib.Coqlib.
Require Import compcert.lib.Maps.
Local Notation "a # b" := (PMap.get b a) (at level 1).
Definition load1 (chunk: memory_chunk) (m: mem) (b: block) (ofs: Z): option val :=
if Mem.valid_access_dec m chunk b ofs Readable
then Some(decode_val chunk (Mem.getN (size_chunk_nat chunk) ofs (m.(Mem.mem_contents)#b)))
else None.
Lemma mem_load_eq: forall (c : memory_chunk) (m : mem) (b : block) (ofs : Z),
(load1 c m b ofs) = (Mem.load c m b ofs).
Proof.
intros.
unfold load1.
unfold Mem.load. (*I get error here when unfolding *)
Admitted.
The compcert.common.Memory module defines several names including Mem.load to be opaque:
Global Opaque Mem.alloc Mem.free Mem.store Mem.load Mem.storebytes Mem.loadbytes.
This means it cannot be unfolded.
Before trying to unfold Mem.load just say that it is Transparent, after that everything will work:
Transparent Mem.load.
unfold Mem.load.