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 with the formula b : bool. 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 generates two subgoals, in which my_formula has been replaced by true and false, resp., with new assumptions myP : my_Prop and not_myP: ~~ my_formula.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 assumtions my_formula and ~~ myformula. 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 unambigously. 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, using the simpl_fun from ssrfun.v. rel T == the type of bool relations. := T -> pred T or T -> T -> bool. simpl_rel T == type of simplifying relations. predType == the generic predicate interface, supported for for lists and sets. pred_class == a coercion class for the predType projection to pred; declaring a coercion to pred_class 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. 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 A: mem A x simplifies to x \in A. 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, ... == union, etc of applicative predicates. 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. 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 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_class 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_class; 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_class. 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. {on C &, P2} == forall x y, f x \in C -> f y \in C -> Qxy when P2 is also convertible to Pf 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. {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 : rightt_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 : {in D, forall x, P} -> 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, absorbtion

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 general, be used applicatively, because of the "uniform inheritance" restriction 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. This trick is made possible by the fact that the constructor of the mem_pred T type aligns the unification process, forcing a generic "collective" predicate A : pred T to unify with the actual collective B, which mem has coerced to pred T via an internal, hidden implicit coercion, supplied by the predType structure for B. Users should take care not to inadvertently "strip" (mem B) down to the coerced B, since this will expose the internal coercion: Coq will display a term B x that cannot be typed as such. The topredE lemma can be used to restore the x \in B syntax in this case. While -topredE can conversely be used to change x \in P into P x, it is safer to use the inE 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 pred T 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. 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. It 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. Collective predicates have a specific extensional equality,

• A =i B,

while applicative predicates use the extensional equality of functions,

• P =1 Q

The two forms are convertible, however. We lift boolean operations to predicates, defining:

• predU (union), predI (intersection), predC (complement), predD (difference), and preim (preimage, i.e., composition)

For each operation we define three forms, typically:

• predU : pred T -> pred T -> simpl_pred T
• [predU A & B], a Notation for predU (mem A) (mem B)
• xpredU, a Notation for the lambda-expression inside predU, which is mostly useful as an argument of =1, since it exposes the head head constant of the expression to the ssreflect matching algorithm.

The syntax for the preimage of a collective predicate A is

• [preim f of A]

Finally, the generic syntax for defining a simpl_pred T is

• [pred x : T | P(x) ], [pred x | P(x) ], [pred x in A | P(x) ], etc.

We also support boolean relations, but only the applicative form, with types

• rel T, an alias for T -> pred T
• simpl_rel T, an auto-simplifying version, and syntax [rel x y | P(x,y) ], [rel x y in A & B | P(x,y) ], etc.

The notation [rel of fA] can be used to coerce a function returning a collective predicate to one returning pred T. Finally, note that there is specific support for ambivalent predicates that can work in either style, as per this file's head descriptor.

Note: applicative_of_simpl is convertible to pred_of_simpl, while collective_of_simpl is not.

This redundant coercion lets us "inherit" the simpl_predType canonical instance by declaring a coercion to simpl_pred. This hack is the only way to put a predType structure on a predArgType. We use simpl_pred rather than pred to ensure that /= removes the identity coercion. Note that the coercion will never be used directly for simpl_pred, since the canonical instance should always be resolved.

This lets us use some types as a synonym for their universal predicate. Unfortunately, this won't work for existing types like bool, unless we redefine bool, true, false and all bool ops.

These must be defined outside a Section because "cooking" kills the nosimpl tag.

Bespoke structures that provide fine-grained control over matching the various forms of the \in predicate; note in particular the different forms of hoisting that are used. We had to work around several bugs in the implementation of unification, notably improper expansion of telescope projections and overwriting of a variable assignment by a later unification (probably due to conversion cache cross-talk).

Because of the explicit eta expansion in the left-hand side, this lemma should only be used in a right-to-left direction. The 8.3 hack allowing partial right-to-left use does not work with the improved expansion heuristics in 8.4.

Qualifiers and keyed predicates.

Keyed predicates: support for property-bearing predicate interfaces.

Instances that strip the mem cast; the first one has "pred_of_mem" as its projection head value, while the second has "pred_of_simpl". The latter has the side benefit of preempting accidental misdeclarations. Note: pred_of_mem is the registered mem >-> pred_class coercion, while simpl_of_mem; pred_of_simpl is the mem >-> pred >=> Funclass coercion. We must write down the coercions explicitly as the Canonical head constant computation does not strip casts !!