Library Coq.ssr.ssrbool

A theory of boolean predicates and operators. A large part of this file is concerned with boolean reflection. Definitions and notations: is_true b == the coercion of b : bool to Prop (:= b = true). This is just input and displayed as `b''. reflect P b == the reflection inductive predicate, asserting that the logical proposition P : Prop holds iff the formula b : bool is equal to true. Lemmas asserting reflect P b are often referred to as "views". iffP, appP, sameP, rwP :: lemmas for direct manipulation of reflection views: iffP is used to prove reflection from logical equivalence, appP to compose views, and sameP and rwP to perform boolean and setoid rewriting. elimT :: coercion reflect >-> Funclass, which allows the direct application of `reflect' views to boolean assertions. decidable P <-> P is effectively decidable (:= {P} + {~ P}). contra, contraL, ... :: contraposition lemmas. altP my_viewP :: natural alternative for reflection; given lemma myviewP: reflect my_Prop my_formula, have [myP | not_myP] := altP my_viewP. generates two subgoals, in which my_formula has been replaced by true and false, resp., with new assumptions myP : my_Prop and not_myP: ~~ my_formula. Caveat: my_formula must be an APPLICATION, not a variable, constant, let-in, etc. (due to the poor behaviour of dependent index matching). boolP my_formula :: boolean disjunction, equivalent to altP (idP my_formula) but circumventing the dependent index capture issue; destructing boolP my_formula generates two subgoals with assumptions my_formula and ~~ my_formula. As with altP, my_formula must be an application. \unless C, P <-> we can assume property P when a something that holds under condition C (such as C itself). := forall G : Prop, (C -> G) -> (P -> G) -> G. This is just C \/ P or rather its impredicative encoding, whose usage better fits the above description: given a lemma UCP whose conclusion is \unless C, P we can assume P by writing: wlog hP: / P by apply/UCP; (prove C -> goal). or even apply: UCP id _ => hP if the goal is C. classically P <-> we can assume P when proving is_true b. := forall b : bool, (P -> b) -> b. This is equivalent to ~ (~ P) when P : Prop. implies P Q == wrapper variant type that coerces to P -> Q and can be used as a P -> Q view unambiguously. Useful to avoid spurious insertion of <-> views when Q is a conjunction of foralls, as in Lemma all_and2 below; conversely, avoids confusion in apply views for impredicative properties, such as \unless C, P. Also supports contrapositives. a && b == the boolean conjunction of a and b. a || b == the boolean disjunction of a and b. a ==> b == the boolean implication of b by a. ~~ a == the boolean negation of a. a (+) b == the boolean exclusive or (or sum) of a and b. [ /\ P1 , P2 & P3 ] == multiway logical conjunction, up to 5 terms. [ \/ P1 , P2 | P3 ] == multiway logical disjunction, up to 4 terms. [&& a, b, c & d] == iterated, right associative boolean conjunction with arbitrary arity. [|| a, b, c | d] == iterated, right associative boolean disjunction with arbitrary arity. [==> a, b, c => d] == iterated, right associative boolean implication with arbitrary arity. and3P, ... == specific reflection lemmas for iterated connectives. andTb, orbAC, ... == systematic names for boolean connective properties (see suffix conventions below). prop_congr == a tactic to move a boolean equality from its coerced form in Prop to the equality in bool. bool_congr == resolution tactic for blindly weeding out like terms from boolean equalities (can fail). This file provides a theory of boolean predicates and relations: pred T == the type of bool predicates (:= T -> bool). simpl_pred T == the type of simplifying bool predicates, based on the simpl_fun type from ssrfun.v. mem_pred T == a specialized form of simpl_pred for "collective" predicates (see below). rel T == the type of bool relations. := T -> pred T or T -> T -> bool. simpl_rel T == type of simplifying relations. := T -> simpl_pred T predType == the generic predicate interface, supported for for lists and sets. pred_sort == the predType >-> Type projection; pred_sort is itself a Coercion target class. Declaring a coercion to pred_sort is an alternative way of equipping a type with a predType structure, which interoperates better with coercion subtyping. This is used, e.g., for finite sets, so that finite groups inherit the membership operation by coercing to sets. {pred T} == a type convertible to pred T, but whose head constant is pred_sort. This type should be used for parameters that can be used as collective predicates (see below), as this will allow passing in directly collections that implement predType by coercion as described above, e.g., finite sets. := pred_sort (predPredType T) If P is a predicate the proposition "x satisfies P" can be written applicatively as (P x), or using an explicit connective as (x \in P); in the latter case we say that P is a "collective" predicate. We use A, B rather than P, Q for collective predicates: x \in A == x satisfies the (collective) predicate A. x \notin A == x doesn't satisfy the (collective) predicate A. The pred T type can be used as a generic predicate type for either kind, but the two kinds of predicates should not be confused. When a "generic" pred T value of one type needs to be passed as the other the following conversions should be used explicitly: SimplPred P == a (simplifying) applicative equivalent of P. mem A == an applicative equivalent of collective predicate A: mem A x simplifies to x \in A, as mem A has in fact type mem_pred T. > In user notation collective predicates only occur as arguments to mem: A only appears as (mem A). This is hidden by notation, e.g., x \in A := in_mem x (mem A) here, enum A := enum_mem (mem A) in fintype. This makes it possible to unify the various ways in which A can be interpreted as a predicate, for both pattern matching and display. Alternatively one can use the syntax for explicit simplifying predicates and relations (in the following x is bound in E): [pred x | E] == simplifying (see ssrfun) predicate x => E. [pred x : T | E] == predicate x => E, with a cast on the argument. [pred : T | P] == constant predicate P on type T. [pred x | E1 & E2] == [pred x | E1 && E2]; an x : T cast is allowed. [pred x in A] == [pred x | x in A]. [pred x in A | E] == [pred x | x in A & E]. [pred x in A | E1 & E2] == [pred x in A | E1 && E2]. [predU A & B] == union of two collective predicates A and B. [predI A & B] == intersection of collective predicates A and B. [predD A & B] == difference of collective predicates A and B. [predC A] == complement of the collective predicate A. [preim f of A] == preimage under f of the collective predicate A. predU P Q, ..., preim f P == union, etc of applicative predicates. pred_oapp A == the predicate A lifted to the option type := [pred x | oapp (mem A) false x]. pred0 == the empty predicate. predT == the total (always true) predicate. if T : predArgType, then T coerces to predT. {: T} == T cast to predArgType (e.g., {: bool * nat}). In the following, x and y are bound in E: [rel x y | E] == simplifying relation x, y => E. [rel x y : T | E] == simplifying relation with arguments cast. [rel x y in A & B | E] == [rel x y | [&& x \in A, y \in B & E] ]. [rel x y in A & B] == [rel x y | (x \in A) && (y \in B) ]. [rel x y in A | E] == [rel x y in A & A | E]. [rel x y in A] == [rel x y in A & A]. relU R S == union of relations R and S. relpre f R == preimage of relation R under f. xpredU, ..., xrelpre == lambda terms implementing predU, ..., etc. Explicit values of type pred T (i.e., lamdba terms) should always be used applicatively, while values of collection types implementing the predType interface, such as sequences or sets should always be used as collective predicates. Defined constants and functions of type pred T or simpl_pred T as well as the explicit simpl_pred T values described below, can generally be used either way. Note however that x \in A will not auto-simplify when A is an explicit simpl_pred T value; the generic simplification rule inE must be used (when A : pred T, the unfold_in rule can be used). Constants of type pred T with an explicit simpl_pred value do not auto-simplify when used applicatively, but can still be expanded with inE. This behavior can be controlled as follows: Let A : collective_pred T := [pred x | ... ]. The collective_pred T type is just an alias for pred T, but this cast stops rewrite inE from expanding the definition of A, thus treating A into an abstract collection (unfold_in or in_collective can be used to expand manually). Let A : applicative_pred T := [pred x | ... ]. This cast causes inE to turn x \in A into the applicative A x form; A will then have to be unfolded explicitly with the /A rule. This will also apply to any definition that reduces to A (e.g., Let B := A). Canonical A_app_pred := ApplicativePred A. This declaration, given after definition of A, similarly causes inE to turn x \in A into A x, but in addition allows the app_predE rule to turn A x back into x \in A; it can be used for any definition of type pred T, which makes it especially useful for ambivalent predicates as the relational transitive closure connect, that are used in both applicative and collective styles. Purely for aesthetics, we provide a subtype of collective predicates: qualifier q T == a pred T pretty-printing wrapper. An A : qualifier q T coerces to pred_sort and thus behaves as a collective predicate, but x \in A and x \notin A are displayed as: x \is A and x \isn't A when q = 0, x \is a A and x \isn't a A when q = 1, x \is an A and x \isn't an A when q = 2, respectively. [qualify x | P] := Qualifier 0 (fun x => P), constructor for the above. [qualify x : T | P], [qualify a x | P], [qualify an X | P], etc. variants of the above with type constraints and different values of q. We provide an internal interface to support attaching properties (such as being multiplicative) to predicates: pred_key p == phantom type that will serve as a support for properties to be attached to p : {pred _}; instances should be created with Fact/Qed so as to be opaque. KeyedPred k_p == an instance of the interface structure that attaches (k_p : pred_key P) to P; the structure projection is a coercion to pred_sort. KeyedQualifier k_q == an instance of the interface structure that attaches (k_q : pred_key q) to (q : qualifier n T). DefaultPredKey p == a default value for pred_key p; the vernacular command Import DefaultKeying attaches this key to all predicates that are not explicitly keyed. Keys can be used to attach properties to predicates, qualifiers and generic nouns in a way that allows them to be used transparently. The key projection of a predicate property structure such as unsignedPred should be a pred_key, not a pred, and corresponding lemmas will have the form Lemma rpredN R S (oppS : @opprPred R S) (kS : keyed_pred oppS) : {mono -%R: x / x \in kS}. Because x \in kS will be displayed as x \in S (or x \is S, etc), the canonical instance of opprPred will not normally be exposed (it will also be erased by /= simplification). In addition each predicate structure should have a DefaultPredKey Canonical instance that simply issues the property as a proof obligation (which can be caught by the Prop-irrelevant feature of the ssreflect plugin). Some properties of predicates and relations: A =i B <-> A and B are extensionally equivalent. {subset A <= B} <-> A is a (collective) subpredicate of B. subpred P Q <-> P is an (applicative) subpredicate or Q. subrel R S <-> R is a subrelation of S. In the following R is in rel T: reflexive R <-> R is reflexive. irreflexive R <-> R is irreflexive. symmetric R <-> R (in rel T) is symmetric (equation). pre_symmetric R <-> R is symmetric (implication). antisymmetric R <-> R is antisymmetric. total R <-> R is total. transitive R <-> R is transitive. left_transitive R <-> R is a congruence on its left hand side. right_transitive R <-> R is a congruence on its right hand side. equivalence_rel R <-> R is an equivalence relation. Localization of (Prop) predicates; if P1 is convertible to forall x, Qx, P2 to forall x y, Qxy and P3 to forall x y z, Qxyz : {for y, P1} <-> Qx{y / x}. {in A, P1} <-> forall x, x \in A -> Qx. {in A1 & A2, P2} <-> forall x y, x \in A1 -> y \in A2 -> Qxy. {in A &, P2} <-> forall x y, x \in A -> y \in A -> Qxy. {in A1 & A2 & A3, Q3} <-> forall x y z, x \in A1 -> y \in A2 -> z \in A3 -> Qxyz. {in A1 & A2 &, Q3} := {in A1 & A2 & A2, Q3}. {in A1 && A3, Q3} := {in A1 & A1 & A3, Q3}. {in A &&, Q3} := {in A & A & A, Q3}. {in A, bijective f} <-> f has a right inverse in A. {on C, P1} <-> forall x, (f x) \in C -> Qx when P1 is also convertible to Pf f, e.g., {on C, involutive f}. {on C &, P2} == forall x y, f x \in C -> f y \in C -> Qxy when P2 is also convertible to Pf f, e.g., {on C &, injective f}. {on C, P1' & g} == forall x, (f x) \in cd -> Qx when P1' is convertible to Pf f and P1' g is convertible to forall x, Qx, e.g., {on C, cancel f & g}. {on C, bijective f} == f has a right inverse on C. This file extends the lemma name suffix conventions of ssrfun as follows: A -- associativity, as in andbA : associative andb. AC -- right commutativity. ACA -- self-interchange (inner commutativity), e.g., orbACA : (a || b) || (c || d) = (a || c) || (b || d). b -- a boolean argument, as in andbb : idempotent andb. C -- commutativity, as in andbC : commutative andb, or predicate complement, as in predC. CA -- left commutativity. D -- predicate difference, as in predD. E -- elimination, as in negbFE : ~~ b = false -> b. F or f -- boolean false, as in andbF : b && false = false. I -- left/right injectivity, as in addbI : right_injective addb, or predicate intersection, as in predI. l -- a left-hand operation, as andb_orl : left_distributive andb orb. N or n -- boolean negation, as in andbN : a && (~~ a) = false. P -- a characteristic property, often a reflection lemma, as in andP : reflect (a /\ b) (a && b). r -- a right-hand operation, as orb_andr : right_distributive orb andb. T or t -- boolean truth, as in andbT: right_id true andb. U -- predicate union, as in predU. W -- weakening, as in in1W : (forall x, P) -> {in D, forall x, P}.

We introduce a number of n-ary "list-style" notations that share a common format, namely [op arg1, arg2, ... last_separator last_arg] This usually denotes a right-associative applications of op, e.g., [&& a, b, c & d] denotes a && (b && (c && d)) The last_separator must be a non-operator token. Here we use &, | or =>; our default is &, but we try to match the intended meaning of op. The separator is a workaround for limitations of the parsing engine; the same limitations mean the separator cannot be omitted even when last_arg can. The Notation declarations are complicated by the separate treatment for some fixed arities (binary for bool operators, and all arities for Prop operators). We also use the square brackets in comprehension-style notations [type var separator expr] where "type" is the type of the comprehension (e.g., pred) and "separator" is | or => . It is important that in other notations a leading square bracket [ is always followed by an operator symbol or a fixed identifier.

Shorter delimiter

An alternative to xorb that behaves somewhat better wrt simplification.

Notation for && and || is declared in Init.Datatypes.

Constant is_true b := b = true is defined in Init.Datatypes.

Lemmas for trivial.

Shorter names.

Negation lemmas. We generally take NEGATION as the standard form of a false condition: negative boolean hypotheses should be of the form ~~ b, rather than ~ b or b = false, as much as possible.

Coercion of sum-style datatypes into bool, which makes it possible to use ssr's boolean if rather than Coq's "generic" if.

Lemmas for ifs with large conditions, which allow reasoning about the condition without repeating it inside the proof (the latter IS preferable when the condition is short). Usage : if the goal contains (if cond then ...) = ... case: ifP => Hcond. generates two subgoal, with the assumption Hcond : cond = true/false Rewrite if_same eliminates redundant ifs Rewrite (fun_if f) moves a function f inside an if Rewrite if_arg moves an argument inside a function-valued if

Turning a boolean "if" form into an application.

Core (internal) reflection lemmas, used for the three kinds of views.

Internal negated reflection lemmas

User-oriented reflection lemmas

Predicate family to reflect excluded middle in bool.

Allow the direct application of a reflection lemma to a boolean assertion.

Impredicative or, which can emulate a classical not-implies.

Classical reasoning becomes directly accessible for any bool subgoal. Note that we cannot use "unless" here for lack of universe polymorphism.

List notations for wider connectives; the Prop connectives have a fixed width so as to avoid iterated destruction (we go up to width 5 for /\, and width 4 for or). The bool connectives have arbitrary widths, but denote expressions that associate to the RIGHT. This is consistent with the right associativity of list expressions and thus more convenient in most proofs.

Shorter, more systematic names for the boolean connectives laws.

Pseudo-cancellation -- i.e, absorption

Imply

Addition (xor)

Resolution tactic for blindly weeding out common terms from boolean equalities. When faced with a goal of the form (andb/orb/addb b1 b2) = b3 they will try to locate b1 in b3 and remove it. This can fail!

Predicates, i.e., packaged functions to bool.

• pred T, the basic type for predicates over a type T, is simply an alias

for T -> bool. We actually distinguish two kinds of predicates, which we call applicative and collective, based on the syntax used to test them at some x in T:

• For an applicative predicate P, one uses prefix syntax: P x Also, most operations on applicative predicates use prefix syntax as well (e.g., predI P Q).
• For a collective predicate A, one uses infix syntax: x \in A and all operations on collective predicates use infix syntax as well (e.g., [predI A & B]).

There are only two kinds of applicative predicates:

• pred T, the alias for T -> bool mentioned above
• simpl_pred T, an alias for simpl_fun T bool with a coercion to pred T that auto-simplifies on application (see ssrfun).

On the other hand, the set of collective predicate types is open-ended via

• predType T, a Structure that can be used to put Canonical collective predicate interpretation on other types, such as lists, tuples, finite sets, etc.

Indeed, we define such interpretations for applicative predicate types, which can therefore also be used with the infix syntax, e.g., x \in predI P Q Moreover these infix forms are convertible to their prefix counterpart (e.g., predI P Q x which in turn simplifies to P x && Q x). The converse is not true, however; collective predicate types cannot, in general, be used applicatively, because of restrictions on implicit coercions. However, we do define an explicit generic coercion

• mem : forall (pT : predType), pT -> mem_pred T where mem_pred T is a variant of simpl_pred T that preserves the infix syntax, i.e., mem A x auto-simplifies to x \in A.

Indeed, the infix "collective" operators are notation for a prefix operator with arguments of type mem_pred T or pred T, applied to coerced collective predicates, e.g., Notation "x \in A" := (in_mem x (mem A)). This prevents the variability in the predicate type from interfering with the application of generic lemmas. Moreover this also makes it much easier to define generic lemmas, because the simplest type -- pred T -- can be used as the type of generic collective predicates, provided one takes care not to use it applicatively; this avoids the burden of having to declare a different predicate type for each predicate parameter of each section or lemma. In detail, we ensure that the head normal form of mem A is always of the eta-long MemPred (fun x => pA x) form, where pA is the pred interpretation of A following its predType pT, i.e., the expansion of topred A. For a pred T evar ?P, (mem ?P) converts MemPred (fun x => ?P x), whose argument is a Miller pattern and therefore always unify: unifying (mem A) with (mem ?P) always yields ?P = pA, because the rigid constant MemPred aligns the unification. Furthermore, we ensure pA is always either A or toP .... A where toP ... is the expansion of @topred T pT, and toP is declared as a Coercion, so pA will display as A in either case, and the instances of @mem T (predPredType T) pA appearing in the premises or right-hand side of a generic lemma parameterized by ?P will be indistinguishable from @mem T pT A. Users should take care not to inadvertently "strip" (mem A) down to the coerced A, since this will expose the internal toP coercion: Coq could then display terms A x that cannot be typed as such. The topredE lemma can be used to restore the x \in A syntax in this case. While -topredE can conversely be used to change x \in P into P x for an applicative P, it is safer to use the inE, unfold_in or and memE lemmas instead, as they do not run the risk of exposing internal coercions. As a consequence it is better to explicitly cast a generic applicative predicate to simpl_pred using the SimplPred constructor when it is used as a collective predicate (see, e.g., Lemma eq_big in bigop). We also sometimes "instantiate" the predType structure by defining a coercion to the sort of the predPredType structure, conveniently denoted {pred T}. This works better for types such as {set T} that have subtypes that coerce to them, since the same coercion will be inserted by the application of mem, or of any lemma that expects a generic collective predicates with type {pred T} := pred_sort (predPredType T) = pred T; thus {pred T} should be the preferred type for generic collective predicate parameters. This device also lets us turn any Type aT : predArgType into the total predicate over that type, i.e., fun _: aT => true. This allows us to write, e.g., #|'I_n| for the cardinal of the (finite) type of integers less than n. Boolean predicates.

The packed class interface for pred-like types.

The type of abstract collective predicates. While {pred T} is convertible to pred T, it presents the pred_sort coercion class, which crucially does not coerce to Funclass. Term whose type P coerces to {pred T} cannot be applied to arguments, but they can be used as if P had a canonical predType instance, as the coercion will be inserted if the unification P =~= pred_sort ?pT fails, changing the problem into the trivial {pred T} =~= pred_sort ?pT (solution ?pT := predPredType P). Additional benefits of this approach are that any type coercing to P will also inherit this behaviour, and that the coercion will be apparent in the elaborated expression. The latter may be important if the coercion is also a canonical structure projector - see mathcomp/fingroup/fingroup.v. The main drawback of implementing predType by coercion in this way is that the type of the value must be known when the unification constraint is imposed: if we only register the constraint and then later discover later that the expression had type P it will be too late to insert a coercion, whereas a canonical instance of predType for P would have solved the deferred constraint. Finally, definitions, lemmas and sections should use type {pred T} for their generic collective type parameters, as this will make it possible to apply such definitions and lemmas directly to values of types that implement predType by coercion to {pred T} (values of types that implement predType without coercing to {pred T} will have to be coerced explicitly using topred).

The type of self-simplifying collective predicates.

Some simpl_pred constructors.

Coercions for simpl_pred. As simpl_pred T values are used both applicatively and collectively we need simpl_pred to coerce to both pred T and {pred T}. However it is undesirable to have two distinct constants for what are essentially identical coercion functions, as this confuses the SSReflect keyed matching algorithm. While the Coq Coercion declarations appear to disallow such Coercion aliasing, it is possible to work around this limitation with a combination of modules and functors, which we do below. In addition we also give a predType instance for simpl_pred, which will be preferred to the {pred T} coercion to solve simpl_pred T =~= pred_sort ?pT constraints; note however that the pred_of_simpl coercion will be used when a simpl_pred T is passed as a {pred T}, since the simplPredType T structure for simpl_pred T is not convertible to predPredType T.

Type to pred coercion. This lets us use types of sort predArgType as a synonym for their universal predicate. We define this predicate as a simpl_pred T rather than a pred T or a {pred T} so that /= and inE reduce (T x) and x \in T to true, respectively. Unfortunately, this can't be used for existing types like bool whose sort is already fixed (at least, not without redefining bool, true, false and all bool operations and lemmas); we provide syntax to recast a given type in predArgType as a workaround.

Boolean relations. Simplifying relations follow the coding pattern of 2-argument simplifying functions: the simplifying type constructor is applied to the last argument. This design choice will let the in_simpl component of inE expand membership in simpl_rel as well. We provide an explicit coercion to rel T to avoid eta-expansion during coercion; this coercion self-simplifies so it should be invisible.

Variant of simpl_pred specialised to the membership operator.

We mainly declare pred_of_mem as a coercion so that it is not displayed. Similarly to pred_of_simpl, it will usually not be inserted by type inference, as all mem_pred mp =~= pred_sort ?pT unification problems will be solve by the memPredType instance below; pred_of_mem will however be used if a mem_pred T is used as a {pred T}, which is desirable as it will avoid a redundant mem in a collective, e.g., passing (mem A) to a lemma exception a generic collective predicate p : {pred T} and premise x \in P will display a subgoal x \in A rather than x \in mem A. Conversely, pred_of_mem will not if it is used id (mem A) is used applicatively or as a pred T; there the simpl_of_mem coercion defined below will be used, resulting in a subgoal that displays as mem A x by simplifies to x \in A.

The simpl_of_mem; pred_of_simpl path provides a new mem_pred >-> pred coercion, but does not override the pred_of_mem : mem_pred >-> pred_sort explicit coercion declaration above.

It is essential to interlock the production of the Mem constructor inside the branch of the predType match, to ensure that unifying mem A with Mem eta ?p sets ?p := toP A (or ?p := P if toP = id and A = eta P), rather than topred pT A, had we put mem A := Mem (topred A).