diff options
| author |  Benjamin Barenblat <bbaren@debian.org> | 2019-02-05 11:24:23 -0500 |
|---|---|---|
| committer | Benjamin Barenblat <bbaren@debian.org> | 2019-02-05 11:24:23 -0500 |
| commit | 6e23288ca437e8a2761a93da1c09ec52460606aa (patch) | |
| tree | 81d822db80f9724b48f036e725379674412a928f | |
| parent | 956960a408a4fe67c7508bf3101d469b6465686d (diff) | |
Disable ssrmatching and ssreflect
The nonfree file in ssrmatching I removed in
cf916fd97fbac51af6fa68ec2704da2a28ef9ede turns out to be important to
the build process, and I can’t figure out how to replicate its effects
without including the file itself. Disable ssrmatching and ssreflect
again until the license situation gets resolved.
| -rw-r--r-- | debian/libcoq-ocaml.install.in | 4 | ||||
| -rw-r--r-- | debian/patches/remove-ssrmatching.patch | 3462 | ||||
| -rw-r--r-- | debian/patches/series | 1 |
3 files changed, 3463 insertions, 4 deletions
diff --git a/debian/libcoq-ocaml.install.in b/debian/libcoq-ocaml.install.in index 44d87c81..c89a0e53 100644 --- a/debian/libcoq-ocaml.install.in +++ b/debian/libcoq-ocaml.install.in @@ -21,8 +21,6 @@ usr/lib/coq/plugins/syntax/nat_syntax_plugin.cmo usr/lib/coq/plugins/syntax/ascii_syntax_plugin.cmo usr/lib/coq/plugins/syntax/string_syntax_plugin.cmo usr/lib/coq/plugins/syntax/z_syntax_plugin.cmo -usr/lib/coq/plugins/ssr/ssreflect_plugin.cmo -usr/lib/coq/plugins/ssrmatching/ssrmatching_plugin.cmo usr/lib/coq/toploop/proofworkertop.cma usr/lib/coq/toploop/tacworkertop.cma usr/lib/coq/toploop/queryworkertop.cma @@ -51,5 +49,3 @@ DYN: usr/lib/coq/plugins/syntax/r_syntax_plugin.cmxs DYN: usr/lib/coq/plugins/syntax/int31_syntax_plugin.cmxs DYN: usr/lib/coq/plugins/syntax/ascii_syntax_plugin.cmxs DYN: usr/lib/coq/plugins/syntax/string_syntax_plugin.cmxs -DYN: usr/lib/coq/plugins/ssr/ssreflect_plugin.cmxs -DYN: usr/lib/coq/plugins/ssrmatching/ssrmatching_plugin.cmxs diff --git a/debian/patches/remove-ssrmatching.patch b/debian/patches/remove-ssrmatching.patch new file mode 100644 index 00000000..739c643e --- /dev/null +++ b/debian/patches/remove-ssrmatching.patch @@ -0,0 +1,3462 @@ +From: Benjamin Barenblat <bbaren@debian.org> +Subject: Remove ssrmatching +Forwarded: not-needed +Last-Update: 2019-02-02 + +ssrmatching still has a file licensed under CeCILL-B, which I believe is a +nonfree license. I’ve removed it from the Debian source package (see gbp.conf). +This patch disables everything that depends on it. +--- a/Makefile.common ++++ b/Makefile.common +@@ -99,7 +99,7 @@ + setoid_ring extraction \ + cc funind firstorder derive \ + rtauto nsatz syntax btauto \ +- ssrmatching ltac ssr ++ ltac + + SRCDIRS:=\ + $(CORESRCDIRS) \ +@@ -156,7 +156,7 @@ + $(EXTRACTIONCMO) \ + $(CCCMO) $(FOCMO) $(RTAUTOCMO) $(BTAUTOCMO) \ + $(FUNINDCMO) $(NSATZCMO) $(SYNTAXCMO) \ +- $(DERIVECMO) $(SSRMATCHINGCMO) $(SSRCMO) ++ $(DERIVECMO) + + ifeq ($(HASNATDYNLINK)-$(BEST),false-opt) + STATICPLUGINS:=$(PLUGINSCMO) +--- a/plugins/ssr/ssrbool.v ++++ /dev/null +@@ -1,1893 +0,0 @@ +-(************************************************************************) +-(* * The Coq Proof Assistant / The Coq Development Team *) +-(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +-(* <O___,, * (see CREDITS file for the list of authors) *) +-(* \VV/ **************************************************************) +-(* // * This file is distributed under the terms of the *) +-(* * GNU Lesser General Public License Version 2.1 *) +-(* * (see LICENSE file for the text of the license) *) +-(************************************************************************) +- +-(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *) +- +-(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **) +- +-Require Bool. +-Require Import ssreflect ssrfun. +- +-(** +- 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. **) +- +- +-Set Implicit Arguments. +-Unset Strict Implicit. +-Unset Printing Implicit Defensive. +-Set Warnings "-projection-no-head-constant". +- +-Notation reflect := Bool.reflect. +-Notation ReflectT := Bool.ReflectT. +-Notation ReflectF := Bool.ReflectF. +- +-Reserved Notation "~~ b" (at level 35, right associativity). +-Reserved Notation "b ==> c" (at level 55, right associativity). +-Reserved Notation "b1 (+) b2" (at level 50, left associativity). +-Reserved Notation "x \in A" +- (at level 70, format "'[hv' x '/ ' \in A ']'", no associativity). +-Reserved Notation "x \notin A" +- (at level 70, format "'[hv' x '/ ' \notin A ']'", no associativity). +-Reserved Notation "p1 =i p2" +- (at level 70, format "'[hv' p1 '/ ' =i p2 ']'", no associativity). +- +-(** +- 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. **) +- +-Reserved Notation "[ /\ P1 & P2 ]" (at level 0, only parsing). +-Reserved Notation "[ /\ P1 , P2 & P3 ]" (at level 0, format +- "'[hv' [ /\ '[' P1 , '/' P2 ']' '/ ' & P3 ] ']'"). +-Reserved Notation "[ /\ P1 , P2 , P3 & P4 ]" (at level 0, format +- "'[hv' [ /\ '[' P1 , '/' P2 , '/' P3 ']' '/ ' & P4 ] ']'"). +-Reserved Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" (at level 0, format +- "'[hv' [ /\ '[' P1 , '/' P2 , '/' P3 , '/' P4 ']' '/ ' & P5 ] ']'"). +- +-Reserved Notation "[ \/ P1 | P2 ]" (at level 0, only parsing). +-Reserved Notation "[ \/ P1 , P2 | P3 ]" (at level 0, format +- "'[hv' [ \/ '[' P1 , '/' P2 ']' '/ ' | P3 ] ']'"). +-Reserved Notation "[ \/ P1 , P2 , P3 | P4 ]" (at level 0, format +- "'[hv' [ \/ '[' P1 , '/' P2 , '/' P3 ']' '/ ' | P4 ] ']'"). +- +-Reserved Notation "[ && b1 & c ]" (at level 0, only parsing). +-Reserved Notation "[ && b1 , b2 , .. , bn & c ]" (at level 0, format +- "'[hv' [ && '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/ ' & c ] ']'"). +- +-Reserved Notation "[ || b1 | c ]" (at level 0, only parsing). +-Reserved Notation "[ || b1 , b2 , .. , bn | c ]" (at level 0, format +- "'[hv' [ || '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/ ' | c ] ']'"). +- +-Reserved Notation "[ ==> b1 => c ]" (at level 0, only parsing). +-Reserved Notation "[ ==> b1 , b2 , .. , bn => c ]" (at level 0, format +- "'[hv' [ ==> '[' b1 , '/' b2 , '/' .. , '/' bn ']' '/' => c ] ']'"). +- +-Reserved Notation "[ 'pred' : T => E ]" (at level 0, format +- "'[hv' [ 'pred' : T => '/ ' E ] ']'"). +-Reserved Notation "[ 'pred' x => E ]" (at level 0, x at level 8, format +- "'[hv' [ 'pred' x => '/ ' E ] ']'"). +-Reserved Notation "[ 'pred' x : T => E ]" (at level 0, x at level 8, format +- "'[hv' [ 'pred' x : T => '/ ' E ] ']'"). +- +-Reserved Notation "[ 'rel' x y => E ]" (at level 0, x, y at level 8, format +- "'[hv' [ 'rel' x y => '/ ' E ] ']'"). +-Reserved Notation "[ 'rel' x y : T => E ]" (at level 0, x, y at level 8, format +- "'[hv' [ 'rel' x y : T => '/ ' E ] ']'"). +- +-(** Shorter delimiter **) +-Delimit Scope bool_scope with B. +-Open Scope bool_scope. +- +-(** An alternative to xorb that behaves somewhat better wrt simplification. **) +-Definition addb b := if b then negb else id. +- +-(** Notation for && and || is declared in Init.Datatypes. **) +-Notation "~~ b" := (negb b) : bool_scope. +-Notation "b ==> c" := (implb b c) : bool_scope. +-Notation "b1 (+) b2" := (addb b1 b2) : bool_scope. +- +-(** Constant is_true b := b = true is defined in Init.Datatypes. **) +-Coercion is_true : bool >-> Sortclass. (* Prop *) +- +-Lemma prop_congr : forall b b' : bool, b = b' -> b = b' :> Prop. +-Proof. by move=> b b' ->. Qed. +- +-Ltac prop_congr := apply: prop_congr. +- +-(** Lemmas for trivial. **) +-Lemma is_true_true : true. Proof. by []. Qed. +-Lemma not_false_is_true : ~ false. Proof. by []. Qed. +-Lemma is_true_locked_true : locked true. Proof. by unlock. Qed. +-Hint Resolve is_true_true not_false_is_true is_true_locked_true. +- +-(** Shorter names. **) +-Definition isT := is_true_true. +-Definition notF := not_false_is_true. +- +-(** 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. **) +- +-Lemma negbT b : b = false -> ~~ b. Proof. by case: b. Qed. +-Lemma negbTE b : ~~ b -> b = false. Proof. by case: b. Qed. +-Lemma negbF b : (b : bool) -> ~~ b = false. Proof. by case: b. Qed. +-Lemma negbFE b : ~~ b = false -> b. Proof. by case: b. Qed. +-Lemma negbK : involutive negb. Proof. by case. Qed. +-Lemma negbNE b : ~~ ~~ b -> b. Proof. by case: b. Qed. +- +-Lemma negb_inj : injective negb. Proof. exact: can_inj negbK. Qed. +-Lemma negbLR b c : b = ~~ c -> ~~ b = c. Proof. exact: canLR negbK. Qed. +-Lemma negbRL b c : ~~ b = c -> b = ~~ c. Proof. exact: canRL negbK. Qed. +- +-Lemma contra (c b : bool) : (c -> b) -> ~~ b -> ~~ c. +-Proof. by case: b => //; case: c. Qed. +-Definition contraNN := contra. +- +-Lemma contraL (c b : bool) : (c -> ~~ b) -> b -> ~~ c. +-Proof. by case: b => //; case: c. Qed. +-Definition contraTN := contraL. +- +-Lemma contraR (c b : bool) : (~~ c -> b) -> ~~ b -> c. +-Proof. by case: b => //; case: c. Qed. +-Definition contraNT := contraR. +- +-Lemma contraLR (c b : bool) : (~~ c -> ~~ b) -> b -> c. +-Proof. by case: b => //; case: c. Qed. +-Definition contraTT := contraLR. +- +-Lemma contraT b : (~~ b -> false) -> b. Proof. by case: b => // ->. Qed. +- +-Lemma wlog_neg b : (~~ b -> b) -> b. Proof. by case: b => // ->. Qed. +- +-Lemma contraFT (c b : bool) : (~~ c -> b) -> b = false -> c. +-Proof. by move/contraR=> notb_c /negbT. Qed. +- +-Lemma contraFN (c b : bool) : (c -> b) -> b = false -> ~~ c. +-Proof. by move/contra=> notb_notc /negbT. Qed. +- +-Lemma contraTF (c b : bool) : (c -> ~~ b) -> b -> c = false. +-Proof. by move/contraL=> b_notc /b_notc/negbTE. Qed. +- +-Lemma contraNF (c b : bool) : (c -> b) -> ~~ b -> c = false. +-Proof. by move/contra=> notb_notc /notb_notc/negbTE. Qed. +- +-Lemma contraFF (c b : bool) : (c -> b) -> b = false -> c = false. +-Proof. by move/contraFN=> bF_notc /bF_notc/negbTE. Qed. +- +-(** +- Coercion of sum-style datatypes into bool, which makes it possible +- to use ssr's boolean if rather than Coq's "generic" if. **) +- +-Coercion isSome T (u : option T) := if u is Some _ then true else false. +- +-Coercion is_inl A B (u : A + B) := if u is inl _ then true else false. +- +-Coercion is_left A B (u : {A} + {B}) := if u is left _ then true else false. +- +-Coercion is_inleft A B (u : A + {B}) := if u is inleft _ then true else false. +- +-Prenex Implicits isSome is_inl is_left is_inleft. +- +-Definition decidable P := {P} + {~ P}. +- +-(** +- 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 **) +- +-Section BoolIf. +- +-Variables (A B : Type) (x : A) (f : A -> B) (b : bool) (vT vF : A). +- +-Variant if_spec (not_b : Prop) : bool -> A -> Set := +- | IfSpecTrue of b : if_spec not_b true vT +- | IfSpecFalse of not_b : if_spec not_b false vF. +- +-Lemma ifP : if_spec (b = false) b (if b then vT else vF). +-Proof. by case def_b: b; constructor. Qed. +- +-Lemma ifPn : if_spec (~~ b) b (if b then vT else vF). +-Proof. by case def_b: b; constructor; rewrite ?def_b. Qed. +- +-Lemma ifT : b -> (if b then vT else vF) = vT. Proof. by move->. Qed. +-Lemma ifF : b = false -> (if b then vT else vF) = vF. Proof. by move->. Qed. +-Lemma ifN : ~~ b -> (if b then vT else vF) = vF. Proof. by move/negbTE->. Qed. +- +-Lemma if_same : (if b then vT else vT) = vT. +-Proof. by case b. Qed. +- +-Lemma if_neg : (if ~~ b then vT else vF) = if b then vF else vT. +-Proof. by case b. Qed. +- +-Lemma fun_if : f (if b then vT else vF) = if b then f vT else f vF. +-Proof. by case b. Qed. +- +-Lemma if_arg (fT fF : A -> B) : +- (if b then fT else fF) x = if b then fT x else fF x. +-Proof. by case b. Qed. +- +-(** Turning a boolean "if" form into an application. **) +-Definition if_expr := if b then vT else vF. +-Lemma ifE : (if b then vT else vF) = if_expr. Proof. by []. Qed. +- +-End BoolIf. +- +-(** Core (internal) reflection lemmas, used for the three kinds of views. **) +- +-Section ReflectCore. +- +-Variables (P Q : Prop) (b c : bool). +- +-Hypothesis Hb : reflect P b. +- +-Lemma introNTF : (if c then ~ P else P) -> ~~ b = c. +-Proof. by case c; case Hb. Qed. +- +-Lemma introTF : (if c then P else ~ P) -> b = c. +-Proof. by case c; case Hb. Qed. +- +-Lemma elimNTF : ~~ b = c -> if c then ~ P else P. +-Proof. by move <-; case Hb. Qed. +- +-Lemma elimTF : b = c -> if c then P else ~ P. +-Proof. by move <-; case Hb. Qed. +- +-Lemma equivPif : (Q -> P) -> (P -> Q) -> if b then Q else ~ Q. +-Proof. by case Hb; auto. Qed. +- +-Lemma xorPif : Q \/ P -> ~ (Q /\ P) -> if b then ~ Q else Q. +-Proof. by case Hb => [? _ H ? | ? H _]; case: H. Qed. +- +-End ReflectCore. +- +-(** Internal negated reflection lemmas **) +-Section ReflectNegCore. +- +-Variables (P Q : Prop) (b c : bool). +-Hypothesis Hb : reflect P (~~ b). +- +-Lemma introTFn : (if c then ~ P else P) -> b = c. +-Proof. by move/(introNTF Hb) <-; case b. Qed. +- +-Lemma elimTFn : b = c -> if c then ~ P else P. +-Proof. by move <-; apply: (elimNTF Hb); case b. Qed. +- +-Lemma equivPifn : (Q -> P) -> (P -> Q) -> if b then ~ Q else Q. +-Proof. by rewrite -if_neg; apply: equivPif. Qed. +- +-Lemma xorPifn : Q \/ P -> ~ (Q /\ P) -> if b then Q else ~ Q. +-Proof. by rewrite -if_neg; apply: xorPif. Qed. +- +-End ReflectNegCore. +- +-(** User-oriented reflection lemmas **) +-Section Reflect. +- +-Variables (P Q : Prop) (b b' c : bool). +-Hypotheses (Pb : reflect P b) (Pb' : reflect P (~~ b')). +- +-Lemma introT : P -> b. Proof. exact: introTF true _. Qed. +-Lemma introF : ~ P -> b = false. Proof. exact: introTF false _. Qed. +-Lemma introN : ~ P -> ~~ b. Proof. exact: introNTF true _. Qed. +-Lemma introNf : P -> ~~ b = false. Proof. exact: introNTF false _. Qed. +-Lemma introTn : ~ P -> b'. Proof. exact: introTFn true _. Qed. +-Lemma introFn : P -> b' = false. Proof. exact: introTFn false _. Qed. +- +-Lemma elimT : b -> P. Proof. exact: elimTF true _. Qed. +-Lemma elimF : b = false -> ~ P. Proof. exact: elimTF false _. Qed. +-Lemma elimN : ~~ b -> ~P. Proof. exact: elimNTF true _. Qed. +-Lemma elimNf : ~~ b = false -> P. Proof. exact: elimNTF false _. Qed. +-Lemma elimTn : b' -> ~ P. Proof. exact: elimTFn true _. Qed. +-Lemma elimFn : b' = false -> P. Proof. exact: elimTFn false _. Qed. +- +-Lemma introP : (b -> Q) -> (~~ b -> ~ Q) -> reflect Q b. +-Proof. by case b; constructor; auto. Qed. +- +-Lemma iffP : (P -> Q) -> (Q -> P) -> reflect Q b. +-Proof. by case: Pb; constructor; auto. Qed. +- +-Lemma equivP : (P <-> Q) -> reflect Q b. +-Proof. by case; apply: iffP. Qed. +- +-Lemma sumboolP (decQ : decidable Q) : reflect Q decQ. +-Proof. by case: decQ; constructor. Qed. +- +-Lemma appP : reflect Q b -> P -> Q. +-Proof. by move=> Qb; move/introT; case: Qb. Qed. +- +-Lemma sameP : reflect P c -> b = c. +-Proof. by case; [apply: introT | apply: introF]. Qed. +- +-Lemma decPcases : if b then P else ~ P. Proof. by case Pb. Qed. +- +-Definition decP : decidable P. by case: b decPcases; [left | right]. Defined. +- +-Lemma rwP : P <-> b. Proof. by split; [apply: introT | apply: elimT]. Qed. +- +-Lemma rwP2 : reflect Q b -> (P <-> Q). +-Proof. by move=> Qb; split=> ?; [apply: appP | apply: elimT; case: Qb]. Qed. +- +-(** Predicate family to reflect excluded middle in bool. **) +-Variant alt_spec : bool -> Type := +- | AltTrue of P : alt_spec true +- | AltFalse of ~~ b : alt_spec false. +- +-Lemma altP : alt_spec b. +-Proof. by case def_b: b / Pb; constructor; rewrite ?def_b. Qed. +- +-End Reflect. +- +-Hint View for move/ elimTF|3 elimNTF|3 elimTFn|3 introT|2 introTn|2 introN|2. +- +-Hint View for apply/ introTF|3 introNTF|3 introTFn|3 elimT|2 elimTn|2 elimN|2. +- +-Hint View for apply// equivPif|3 xorPif|3 equivPifn|3 xorPifn|3. +- +-(** Allow the direct application of a reflection lemma to a boolean assertion. **) +-Coercion elimT : reflect >-> Funclass. +- +-Variant implies P Q := Implies of P -> Q. +-Lemma impliesP P Q : implies P Q -> P -> Q. Proof. by case. Qed. +-Lemma impliesPn (P Q : Prop) : implies P Q -> ~ Q -> ~ P. +-Proof. by case=> iP ? /iP. Qed. +-Coercion impliesP : implies >-> Funclass. +-Hint View for move/ impliesPn|2 impliesP|2. +-Hint View for apply/ impliesPn|2 impliesP|2. +- +-(** Impredicative or, which can emulate a classical not-implies. **) +-Definition unless condition property : Prop := +- forall goal : Prop, (condition -> goal) -> (property -> goal) -> goal. +- +-Notation "\unless C , P" := (unless C P) +- (at level 200, C at level 100, +- format "'[' \unless C , '/ ' P ']'") : type_scope. +- +-Lemma unlessL C P : implies C (\unless C, P). +-Proof. by split=> hC G /(_ hC). Qed. +- +-Lemma unlessR C P : implies P (\unless C, P). +-Proof. by split=> hP G _ /(_ hP). Qed. +- +-Lemma unless_sym C P : implies (\unless C, P) (\unless P, C). +-Proof. by split; apply; [apply/unlessR | apply/unlessL]. Qed. +- +-Lemma unlessP (C P : Prop) : (\unless C, P) <-> C \/ P. +-Proof. by split=> [|[/unlessL | /unlessR]]; apply; [left | right]. Qed. +- +-Lemma bind_unless C P {Q} : implies (\unless C, P) (\unless (\unless C, Q), P). +-Proof. by split; apply=> [hC|hP]; [apply/unlessL/unlessL | apply/unlessR]. Qed. +- +-Lemma unless_contra b C : implies (~~ b -> C) (\unless C, b). +-Proof. by split; case: b => [_ | hC]; [apply/unlessR | apply/unlessL/hC]. Qed. +- +-(** +- Classical reasoning becomes directly accessible for any bool subgoal. +- Note that we cannot use "unless" here for lack of universe polymorphism. **) +-Definition classically P : Prop := forall b : bool, (P -> b) -> b. +- +-Lemma classicP (P : Prop) : classically P <-> ~ ~ P. +-Proof. +-split=> [cP nP | nnP [] // nP]; last by case nnP; move/nP. +-by have: P -> false; [move/nP | move/cP]. +-Qed. +- +-Lemma classicW P : P -> classically P. Proof. by move=> hP _ ->. Qed. +- +-Lemma classic_bind P Q : (P -> classically Q) -> classically P -> classically Q. +-Proof. by move=> iPQ cP b /iPQ-/cP. Qed. +- +-Lemma classic_EM P : classically (decidable P). +-Proof. +-by case=> // undecP; apply/undecP; right=> notP; apply/notF/undecP; left. +-Qed. +- +-Lemma classic_pick T P : classically ({x : T | P x} + (forall x, ~ P x)). +-Proof. +-case=> // undecP; apply/undecP; right=> x Px. +-by apply/notF/undecP; left; exists x. +-Qed. +- +-Lemma classic_imply P Q : (P -> classically Q) -> classically (P -> Q). +-Proof. +-move=> iPQ []// notPQ; apply/notPQ=> /iPQ-cQ. +-by case: notF; apply: cQ => hQ; apply: notPQ. +-Qed. +- +-(** +- 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. **) +- +-Inductive and3 (P1 P2 P3 : Prop) : Prop := And3 of P1 & P2 & P3. +- +-Inductive and4 (P1 P2 P3 P4 : Prop) : Prop := And4 of P1 & P2 & P3 & P4. +- +-Inductive and5 (P1 P2 P3 P4 P5 : Prop) : Prop := +- And5 of P1 & P2 & P3 & P4 & P5. +- +-Inductive or3 (P1 P2 P3 : Prop) : Prop := Or31 of P1 | Or32 of P2 | Or33 of P3. +- +-Inductive or4 (P1 P2 P3 P4 : Prop) : Prop := +- Or41 of P1 | Or42 of P2 | Or43 of P3 | Or44 of P4. +- +-Notation "[ /\ P1 & P2 ]" := (and P1 P2) (only parsing) : type_scope. +-Notation "[ /\ P1 , P2 & P3 ]" := (and3 P1 P2 P3) : type_scope. +-Notation "[ /\ P1 , P2 , P3 & P4 ]" := (and4 P1 P2 P3 P4) : type_scope. +-Notation "[ /\ P1 , P2 , P3 , P4 & P5 ]" := (and5 P1 P2 P3 P4 P5) : type_scope. +- +-Notation "[ \/ P1 | P2 ]" := (or P1 P2) (only parsing) : type_scope. +-Notation "[ \/ P1 , P2 | P3 ]" := (or3 P1 P2 P3) : type_scope. +-Notation "[ \/ P1 , P2 , P3 | P4 ]" := (or4 P1 P2 P3 P4) : type_scope. +- +-Notation "[ && b1 & c ]" := (b1 && c) (only parsing) : bool_scope. +-Notation "[ && b1 , b2 , .. , bn & c ]" := (b1 && (b2 && .. (bn && c) .. )) +- : bool_scope. +- +-Notation "[ || b1 | c ]" := (b1 || c) (only parsing) : bool_scope. +-Notation "[ || b1 , b2 , .. , bn | c ]" := (b1 || (b2 || .. (bn || c) .. )) +- : bool_scope. +- +-Notation "[ ==> b1 , b2 , .. , bn => c ]" := +- (b1 ==> (b2 ==> .. (bn ==> c) .. )) : bool_scope. +-Notation "[ ==> b1 => c ]" := (b1 ==> c) (only parsing) : bool_scope. +- +-Section AllAnd. +- +-Variables (T : Type) (P1 P2 P3 P4 P5 : T -> Prop). +-Local Notation a P := (forall x, P x). +- +-Lemma all_and2 : implies (forall x, [/\ P1 x & P2 x]) [/\ a P1 & a P2]. +-Proof. by split=> haveP; split=> x; case: (haveP x). Qed. +- +-Lemma all_and3 : implies (forall x, [/\ P1 x, P2 x & P3 x]) +- [/\ a P1, a P2 & a P3]. +-Proof. by split=> haveP; split=> x; case: (haveP x). Qed. +- +-Lemma all_and4 : implies (forall x, [/\ P1 x, P2 x, P3 x & P4 x]) +- [/\ a P1, a P2, a P3 & a P4]. +-Proof. by split=> haveP; split=> x; case: (haveP x). Qed. +- +-Lemma all_and5 : implies (forall x, [/\ P1 x, P2 x, P3 x, P4 x & P5 x]) +- [/\ a P1, a P2, a P3, a P4 & a P5]. +-Proof. by split=> haveP; split=> x; case: (haveP x). Qed. +- +-End AllAnd. +- +-Arguments all_and2 {T P1 P2}. +-Arguments all_and3 {T P1 P2 P3}. +-Arguments all_and4 {T P1 P2 P3 P4}. +-Arguments all_and5 {T P1 P2 P3 P4 P5}. +- +-Lemma pair_andP P Q : P /\ Q <-> P * Q. Proof. by split; case. Qed. +- +-Section ReflectConnectives. +- +-Variable b1 b2 b3 b4 b5 : bool. +- +-Lemma idP : reflect b1 b1. +-Proof. by case b1; constructor. Qed. +- +-Lemma boolP : alt_spec b1 b1 b1. +-Proof. exact: (altP idP). Qed. +- +-Lemma idPn : reflect (~~ b1) (~~ b1). +-Proof. by case b1; constructor. Qed. +- +-Lemma negP : reflect (~ b1) (~~ b1). +-Proof. by case b1; constructor; auto. Qed. +- +-Lemma negPn : reflect b1 (~~ ~~ b1). +-Proof. by case b1; constructor. Qed. +- +-Lemma negPf : reflect (b1 = false) (~~ b1). +-Proof. by case b1; constructor. Qed. +- +-Lemma andP : reflect (b1 /\ b2) (b1 && b2). +-Proof. by case b1; case b2; constructor=> //; case. Qed. +- +-Lemma and3P : reflect [/\ b1, b2 & b3] [&& b1, b2 & b3]. +-Proof. by case b1; case b2; case b3; constructor; try by case. Qed. +- +-Lemma and4P : reflect [/\ b1, b2, b3 & b4] [&& b1, b2, b3 & b4]. +-Proof. by case b1; case b2; case b3; case b4; constructor; try by case. Qed. +- +-Lemma and5P : reflect [/\ b1, b2, b3, b4 & b5] [&& b1, b2, b3, b4 & b5]. +-Proof. +-by case b1; case b2; case b3; case b4; case b5; constructor; try by case. +-Qed. +- +-Lemma orP : reflect (b1 \/ b2) (b1 || b2). +-Proof. by case b1; case b2; constructor; auto; case. Qed. +- +-Lemma or3P : reflect [\/ b1, b2 | b3] [|| b1, b2 | b3]. +-Proof. +-case b1; first by constructor; constructor 1. +-case b2; first by constructor; constructor 2. +-case b3; first by constructor; constructor 3. +-by constructor; case. +-Qed. +- +-Lemma or4P : reflect [\/ b1, b2, b3 | b4] [|| b1, b2, b3 | b4]. +-Proof. +-case b1; first by constructor; constructor 1. +-case b2; first by constructor; constructor 2. +-case b3; first by constructor; constructor 3. +-case b4; first by constructor; constructor 4. +-by constructor; case. +-Qed. +- +-Lemma nandP : reflect (~~ b1 \/ ~~ b2) (~~ (b1 && b2)). +-Proof. by case b1; case b2; constructor; auto; case; auto. Qed. +- +-Lemma norP : reflect (~~ b1 /\ ~~ b2) (~~ (b1 || b2)). +-Proof. by case b1; case b2; constructor; auto; case; auto. Qed. +- +-Lemma implyP : reflect (b1 -> b2) (b1 ==> b2). +-Proof. by case b1; case b2; constructor; auto. Qed. +- +-End ReflectConnectives. +- +-Arguments idP [b1]. +-Arguments idPn [b1]. +-Arguments negP [b1]. +-Arguments negPn [b1]. +-Arguments negPf [b1]. +-Arguments andP [b1 b2]. +-Arguments and3P [b1 b2 b3]. +-Arguments and4P [b1 b2 b3 b4]. +-Arguments and5P [b1 b2 b3 b4 b5]. +-Arguments orP [b1 b2]. +-Arguments or3P [b1 b2 b3]. +-Arguments or4P [b1 b2 b3 b4]. +-Arguments nandP [b1 b2]. +-Arguments norP [b1 b2]. +-Arguments implyP [b1 b2]. +-Prenex Implicits idP idPn negP negPn negPf. +-Prenex Implicits andP and3P and4P and5P orP or3P or4P nandP norP implyP. +- +-(** Shorter, more systematic names for the boolean connectives laws. **) +- +-Lemma andTb : left_id true andb. Proof. by []. Qed. +-Lemma andFb : left_zero false andb. Proof. by []. Qed. +-Lemma andbT : right_id true andb. Proof. by case. Qed. +-Lemma andbF : right_zero false andb. Proof. by case. Qed. +-Lemma andbb : idempotent andb. Proof. by case. Qed. +-Lemma andbC : commutative andb. Proof. by do 2!case. Qed. +-Lemma andbA : associative andb. Proof. by do 3!case. Qed. +-Lemma andbCA : left_commutative andb. Proof. by do 3!case. Qed. +-Lemma andbAC : right_commutative andb. Proof. by do 3!case. Qed. +-Lemma andbACA : interchange andb andb. Proof. by do 4!case. Qed. +- +-Lemma orTb : forall b, true || b. Proof. by []. Qed. +-Lemma orFb : left_id false orb. Proof. by []. Qed. +-Lemma orbT : forall b, b || true. Proof. by case. Qed. +-Lemma orbF : right_id false orb. Proof. by case. Qed. +-Lemma orbb : idempotent orb. Proof. by case. Qed. +-Lemma orbC : commutative orb. Proof. by do 2!case. Qed. +-Lemma orbA : associative orb. Proof. by do 3!case. Qed. +-Lemma orbCA : left_commutative orb. Proof. by do 3!case. Qed. +-Lemma orbAC : right_commutative orb. Proof. by do 3!case. Qed. +-Lemma orbACA : interchange orb orb. Proof. by do 4!case. Qed. +- +-Lemma andbN b : b && ~~ b = false. Proof. by case: b. Qed. +-Lemma andNb b : ~~ b && b = false. Proof. by case: b. Qed. +-Lemma orbN b : b || ~~ b = true. Proof. by case: b. Qed. +-Lemma orNb b : ~~ b || b = true. Proof. by case: b. Qed. +- +-Lemma andb_orl : left_distributive andb orb. Proof. by do 3!case. Qed. +-Lemma andb_orr : right_distributive andb orb. Proof. by do 3!case. Qed. +-Lemma orb_andl : left_distributive orb andb. Proof. by do 3!case. Qed. +-Lemma orb_andr : right_distributive orb andb. Proof. by do 3!case. Qed. +- +-Lemma andb_idl (a b : bool) : (b -> a) -> a && b = b. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma andb_idr (a b : bool) : (a -> b) -> a && b = a. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma andb_id2l (a b c : bool) : (a -> b = c) -> a && b = a && c. +-Proof. by case: a; case: b; case: c => // ->. Qed. +-Lemma andb_id2r (a b c : bool) : (b -> a = c) -> a && b = c && b. +-Proof. by case: a; case: b; case: c => // ->. Qed. +- +-Lemma orb_idl (a b : bool) : (a -> b) -> a || b = b. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma orb_idr (a b : bool) : (b -> a) -> a || b = a. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma orb_id2l (a b c : bool) : (~~ a -> b = c) -> a || b = a || c. +-Proof. by case: a; case: b; case: c => // ->. Qed. +-Lemma orb_id2r (a b c : bool) : (~~ b -> a = c) -> a || b = c || b. +-Proof. by case: a; case: b; case: c => // ->. Qed. +- +-Lemma negb_and (a b : bool) : ~~ (a && b) = ~~ a || ~~ b. +-Proof. by case: a; case: b. Qed. +- +-Lemma negb_or (a b : bool) : ~~ (a || b) = ~~ a && ~~ b. +-Proof. by case: a; case: b. Qed. +- +-(** Pseudo-cancellation -- i.e, absorbtion **) +- +-Lemma andbK a b : a && b || a = a. Proof. by case: a; case: b. Qed. +-Lemma andKb a b : a || b && a = a. Proof. by case: a; case: b. Qed. +-Lemma orbK a b : (a || b) && a = a. Proof. by case: a; case: b. Qed. +-Lemma orKb a b : a && (b || a) = a. Proof. by case: a; case: b. Qed. +- +-(** Imply **) +- +-Lemma implybT b : b ==> true. Proof. by case: b. Qed. +-Lemma implybF b : (b ==> false) = ~~ b. Proof. by case: b. Qed. +-Lemma implyFb b : false ==> b. Proof. by []. Qed. +-Lemma implyTb b : (true ==> b) = b. Proof. by []. Qed. +-Lemma implybb b : b ==> b. Proof. by case: b. Qed. +- +-Lemma negb_imply a b : ~~ (a ==> b) = a && ~~ b. +-Proof. by case: a; case: b. Qed. +- +-Lemma implybE a b : (a ==> b) = ~~ a || b. +-Proof. by case: a; case: b. Qed. +- +-Lemma implyNb a b : (~~ a ==> b) = a || b. +-Proof. by case: a; case: b. Qed. +- +-Lemma implybN a b : (a ==> ~~ b) = (b ==> ~~ a). +-Proof. by case: a; case: b. Qed. +- +-Lemma implybNN a b : (~~ a ==> ~~ b) = b ==> a. +-Proof. by case: a; case: b. Qed. +- +-Lemma implyb_idl (a b : bool) : (~~ a -> b) -> (a ==> b) = b. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma implyb_idr (a b : bool) : (b -> ~~ a) -> (a ==> b) = ~~ a. +-Proof. by case: a; case: b => // ->. Qed. +-Lemma implyb_id2l (a b c : bool) : (a -> b = c) -> (a ==> b) = (a ==> c). +-Proof. by case: a; case: b; case: c => // ->. Qed. +- +-(** Addition (xor) **) +- +-Lemma addFb : left_id false addb. Proof. by []. Qed. +-Lemma addbF : right_id false addb. Proof. by case. Qed. +-Lemma addbb : self_inverse false addb. Proof. by case. Qed. +-Lemma addbC : commutative addb. Proof. by do 2!case. Qed. +-Lemma addbA : associative addb. Proof. by do 3!case. Qed. +-Lemma addbCA : left_commutative addb. Proof. by do 3!case. Qed. +-Lemma addbAC : right_commutative addb. Proof. by do 3!case. Qed. +-Lemma addbACA : interchange addb addb. Proof. by do 4!case. Qed. +-Lemma andb_addl : left_distributive andb addb. Proof. by do 3!case. Qed. +-Lemma andb_addr : right_distributive andb addb. Proof. by do 3!case. Qed. +-Lemma addKb : left_loop id addb. Proof. by do 2!case. Qed. +-Lemma addbK : right_loop id addb. Proof. by do 2!case. Qed. +-Lemma addIb : left_injective addb. Proof. by do 3!case. Qed. +-Lemma addbI : right_injective addb. Proof. by do 3!case. Qed. +- +-Lemma addTb b : true (+) b = ~~ b. Proof. by []. Qed. +-Lemma addbT b : b (+) true = ~~ b. Proof. by case: b. Qed. +- +-Lemma addbN a b : a (+) ~~ b = ~~ (a (+) b). +-Proof. by case: a; case: b. Qed. +-Lemma addNb a b : ~~ a (+) b = ~~ (a (+) b). +-Proof. by case: a; case: b. Qed. +- +-Lemma addbP a b : reflect (~~ a = b) (a (+) b). +-Proof. by case: a; case: b; constructor. Qed. +-Arguments addbP [a b]. +- +-(** +- 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! **) +- +-Ltac bool_congr := +- match goal with +- | |- (?X1 && ?X2 = ?X3) => first +- [ symmetry; rewrite -1?(andbC X1) -?(andbCA X1); congr 1 (andb X1); symmetry +- | case: (X1); [ rewrite ?andTb ?andbT // | by rewrite ?andbF /= ] ] +- | |- (?X1 || ?X2 = ?X3) => first +- [ symmetry; rewrite -1?(orbC X1) -?(orbCA X1); congr 1 (orb X1); symmetry +- | case: (X1); [ by rewrite ?orbT //= | rewrite ?orFb ?orbF ] ] +- | |- (?X1 (+) ?X2 = ?X3) => +- symmetry; rewrite -1?(addbC X1) -?(addbCA X1); congr 1 (addb X1); symmetry +- | |- (~~ ?X1 = ?X2) => congr 1 negb +- end. +- +- +-(** +- 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. **) +- +- +-Definition pred T := T -> bool. +- +-Identity Coercion fun_of_pred : pred >-> Funclass. +- +-Definition rel T := T -> pred T. +- +-Identity Coercion fun_of_rel : rel >-> Funclass. +- +-Notation xpred0 := (fun _ => false). +-Notation xpredT := (fun _ => true). +-Notation xpredI := (fun (p1 p2 : pred _) x => p1 x && p2 x). +-Notation xpredU := (fun (p1 p2 : pred _) x => p1 x || p2 x). +-Notation xpredC := (fun (p : pred _) x => ~~ p x). +-Notation xpredD := (fun (p1 p2 : pred _) x => ~~ p2 x && p1 x). +-Notation xpreim := (fun f (p : pred _) x => p (f x)). +-Notation xrelU := (fun (r1 r2 : rel _) x y => r1 x y || r2 x y). +- +-Section Predicates. +- +-Variables T : Type. +- +-Definition subpred (p1 p2 : pred T) := forall x, p1 x -> p2 x. +- +-Definition subrel (r1 r2 : rel T) := forall x y, r1 x y -> r2 x y. +- +-Definition simpl_pred := simpl_fun T bool. +-Definition applicative_pred := pred T. +-Definition collective_pred := pred T. +- +-Definition SimplPred (p : pred T) : simpl_pred := SimplFun p. +- +-Coercion pred_of_simpl (p : simpl_pred) : pred T := fun_of_simpl p. +-Coercion applicative_pred_of_simpl (p : simpl_pred) : applicative_pred := +- fun_of_simpl p. +-Coercion collective_pred_of_simpl (p : simpl_pred) : collective_pred := +- fun x => (let: SimplFun f := p in fun _ => f x) x. +-(** +- Note: applicative_of_simpl is convertible to pred_of_simpl, while +- collective_of_simpl is not. **) +- +-Definition pred0 := SimplPred xpred0. +-Definition predT := SimplPred xpredT. +-Definition predI p1 p2 := SimplPred (xpredI p1 p2). +-Definition predU p1 p2 := SimplPred (xpredU p1 p2). +-Definition predC p := SimplPred (xpredC p). +-Definition predD p1 p2 := SimplPred (xpredD p1 p2). +-Definition preim rT f (d : pred rT) := SimplPred (xpreim f d). +- +-Definition simpl_rel := simpl_fun T (pred T). +- +-Definition SimplRel (r : rel T) : simpl_rel := [fun x => r x]. +- +-Coercion rel_of_simpl_rel (r : simpl_rel) : rel T := fun x y => r x y. +- +-Definition relU r1 r2 := SimplRel (xrelU r1 r2). +- +-Lemma subrelUl r1 r2 : subrel r1 (relU r1 r2). +-Proof. by move=> *; apply/orP; left. Qed. +- +-Lemma subrelUr r1 r2 : subrel r2 (relU r1 r2). +-Proof. by move=> *; apply/orP; right. Qed. +- +-Variant mem_pred := Mem of pred T. +- +-Definition isMem pT topred mem := mem = (fun p : pT => Mem [eta topred p]). +- +-Structure predType := PredType { +- pred_sort :> Type; +- topred : pred_sort -> pred T; +- _ : {mem | isMem topred mem} +-}. +- +-Definition mkPredType pT toP := PredType (exist (@isMem pT toP) _ (erefl _)). +- +-Canonical predPredType := Eval hnf in @mkPredType (pred T) id. +-Canonical simplPredType := Eval hnf in mkPredType pred_of_simpl. +-Canonical boolfunPredType := Eval hnf in @mkPredType (T -> bool) id. +- +-Coercion pred_of_mem mp : pred_sort predPredType := let: Mem p := mp in [eta p]. +-Canonical memPredType := Eval hnf in mkPredType pred_of_mem. +- +-Definition clone_pred U := +- fun pT & pred_sort pT -> U => +- fun a mP (pT' := @PredType U a mP) & phant_id pT' pT => pT'. +- +-End Predicates. +- +-Arguments pred0 [T]. +-Arguments predT [T]. +-Prenex Implicits pred0 predT predI predU predC predD preim relU. +- +-Notation "[ 'pred' : T | E ]" := (SimplPred (fun _ : T => E%B)) +- (at level 0, format "[ 'pred' : T | E ]") : fun_scope. +-Notation "[ 'pred' x | E ]" := (SimplPred (fun x => E%B)) +- (at level 0, x ident, format "[ 'pred' x | E ]") : fun_scope. +-Notation "[ 'pred' x | E1 & E2 ]" := [pred x | E1 && E2 ] +- (at level 0, x ident, format "[ 'pred' x | E1 & E2 ]") : fun_scope. +-Notation "[ 'pred' x : T | E ]" := (SimplPred (fun x : T => E%B)) +- (at level 0, x ident, only parsing) : fun_scope. +-Notation "[ 'pred' x : T | E1 & E2 ]" := [pred x : T | E1 && E2 ] +- (at level 0, x ident, only parsing) : fun_scope. +-Notation "[ 'rel' x y | E ]" := (SimplRel (fun x y => E%B)) +- (at level 0, x ident, y ident, format "[ 'rel' x y | E ]") : fun_scope. +-Notation "[ 'rel' x y : T | E ]" := (SimplRel (fun x y : T => E%B)) +- (at level 0, x ident, y ident, only parsing) : fun_scope. +- +-Notation "[ 'predType' 'of' T ]" := (@clone_pred _ T _ id _ _ id) +- (at level 0, format "[ 'predType' 'of' T ]") : form_scope. +- +-(** +- 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. **) +- +-Notation pred_class := (pred_sort (predPredType _)). +-Coercion sort_of_simpl_pred T (p : simpl_pred T) : pred_class := p : pred T. +- +-(** +- 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. **) +-Definition predArgType := Type. +-Bind Scope type_scope with predArgType. +-Identity Coercion sort_of_predArgType : predArgType >-> Sortclass. +-Coercion pred_of_argType (T : predArgType) : simpl_pred T := predT. +- +-Notation "{ : T }" := (T%type : predArgType) +- (at level 0, format "{ : T }") : type_scope. +- +-(** +- These must be defined outside a Section because "cooking" kills the +- nosimpl tag. **) +- +-Definition mem T (pT : predType T) : pT -> mem_pred T := +- nosimpl (let: @PredType _ _ _ (exist _ mem _) := pT return pT -> _ in mem). +-Definition in_mem T x mp := nosimpl pred_of_mem T mp x. +- +-Prenex Implicits mem. +- +-Coercion pred_of_mem_pred T mp := [pred x : T | in_mem x mp]. +- +-Definition eq_mem T p1 p2 := forall x : T, in_mem x p1 = in_mem x p2. +-Definition sub_mem T p1 p2 := forall x : T, in_mem x p1 -> in_mem x p2. +- +-Typeclasses Opaque eq_mem. +- +-Lemma sub_refl T (p : mem_pred T) : sub_mem p p. Proof. by []. Qed. +-Arguments sub_refl {T p}. +- +-Notation "x \in A" := (in_mem x (mem A)) : bool_scope. +-Notation "x \in A" := (in_mem x (mem A)) : bool_scope. +-Notation "x \notin A" := (~~ (x \in A)) : bool_scope. +-Notation "A =i B" := (eq_mem (mem A) (mem B)) : type_scope. +-Notation "{ 'subset' A <= B }" := (sub_mem (mem A) (mem B)) +- (at level 0, A, B at level 69, +- format "{ '[hv' 'subset' A '/ ' <= B ']' }") : type_scope. +-Notation "[ 'mem' A ]" := (pred_of_simpl (pred_of_mem_pred (mem A))) +- (at level 0, only parsing) : fun_scope. +-Notation "[ 'rel' 'of' fA ]" := (fun x => [mem (fA x)]) +- (at level 0, format "[ 'rel' 'of' fA ]") : fun_scope. +-Notation "[ 'predI' A & B ]" := (predI [mem A] [mem B]) +- (at level 0, format "[ 'predI' A & B ]") : fun_scope. +-Notation "[ 'predU' A & B ]" := (predU [mem A] [mem B]) +- (at level 0, format "[ 'predU' A & B ]") : fun_scope. +-Notation "[ 'predD' A & B ]" := (predD [mem A] [mem B]) +- (at level 0, format "[ 'predD' A & B ]") : fun_scope. +-Notation "[ 'predC' A ]" := (predC [mem A]) +- (at level 0, format "[ 'predC' A ]") : fun_scope. +-Notation "[ 'preim' f 'of' A ]" := (preim f [mem A]) +- (at level 0, format "[ 'preim' f 'of' A ]") : fun_scope. +- +-Notation "[ 'pred' x 'in' A ]" := [pred x | x \in A] +- (at level 0, x ident, format "[ 'pred' x 'in' A ]") : fun_scope. +-Notation "[ 'pred' x 'in' A | E ]" := [pred x | x \in A & E] +- (at level 0, x ident, format "[ 'pred' x 'in' A | E ]") : fun_scope. +-Notation "[ 'pred' x 'in' A | E1 & E2 ]" := [pred x | x \in A & E1 && E2 ] +- (at level 0, x ident, +- format "[ 'pred' x 'in' A | E1 & E2 ]") : fun_scope. +-Notation "[ 'rel' x y 'in' A & B | E ]" := +- [rel x y | (x \in A) && (y \in B) && E] +- (at level 0, x ident, y ident, +- format "[ 'rel' x y 'in' A & B | E ]") : fun_scope. +-Notation "[ 'rel' x y 'in' A & B ]" := [rel x y | (x \in A) && (y \in B)] +- (at level 0, x ident, y ident, +- format "[ 'rel' x y 'in' A & B ]") : fun_scope. +-Notation "[ 'rel' x y 'in' A | E ]" := [rel x y in A & A | E] +- (at level 0, x ident, y ident, +- format "[ 'rel' x y 'in' A | E ]") : fun_scope. +-Notation "[ 'rel' x y 'in' A ]" := [rel x y in A & A] +- (at level 0, x ident, y ident, +- format "[ 'rel' x y 'in' A ]") : fun_scope. +- +-Section simpl_mem. +- +-Variables (T : Type) (pT : predType T). +-Implicit Types (x : T) (p : pred T) (sp : simpl_pred T) (pp : pT). +- +-(** +- 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). **) +-Structure manifest_applicative_pred p := ManifestApplicativePred { +- manifest_applicative_pred_value :> pred T; +- _ : manifest_applicative_pred_value = p +-}. +-Definition ApplicativePred p := ManifestApplicativePred (erefl p). +-Canonical applicative_pred_applicative sp := +- ApplicativePred (applicative_pred_of_simpl sp). +- +-Structure manifest_simpl_pred p := ManifestSimplPred { +- manifest_simpl_pred_value :> simpl_pred T; +- _ : manifest_simpl_pred_value = SimplPred p +-}. +-Canonical expose_simpl_pred p := ManifestSimplPred (erefl (SimplPred p)). +- +-Structure manifest_mem_pred p := ManifestMemPred { +- manifest_mem_pred_value :> mem_pred T; +- _ : manifest_mem_pred_value= Mem [eta p] +-}. +-Canonical expose_mem_pred p := @ManifestMemPred p _ (erefl _). +- +-Structure applicative_mem_pred p := +- ApplicativeMemPred {applicative_mem_pred_value :> manifest_mem_pred p}. +-Canonical check_applicative_mem_pred p (ap : manifest_applicative_pred p) mp := +- @ApplicativeMemPred ap mp. +- +-Lemma mem_topred (pp : pT) : mem (topred pp) = mem pp. +-Proof. by rewrite /mem; case: pT pp => T1 app1 [mem1 /= ->]. Qed. +- +-Lemma topredE x (pp : pT) : topred pp x = (x \in pp). +-Proof. by rewrite -mem_topred. Qed. +- +-Lemma app_predE x p (ap : manifest_applicative_pred p) : ap x = (x \in p). +-Proof. by case: ap => _ /= ->. Qed. +- +-Lemma in_applicative x p (amp : applicative_mem_pred p) : in_mem x amp = p x. +-Proof. by case: amp => [[_ /= ->]]. Qed. +- +-Lemma in_collective x p (msp : manifest_simpl_pred p) : +- (x \in collective_pred_of_simpl msp) = p x. +-Proof. by case: msp => _ /= ->. Qed. +- +-Lemma in_simpl x p (msp : manifest_simpl_pred p) : +- in_mem x (Mem [eta fun_of_simpl (msp : simpl_pred T)]) = p x. +-Proof. by case: msp => _ /= ->. Qed. +- +-(** +- 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. **) +-Lemma unfold_in x p : (x \in ([eta p] : pred T)) = p x. +-Proof. by []. Qed. +- +-Lemma simpl_predE p : SimplPred p =1 p. +-Proof. by []. Qed. +- +-Definition inE := (in_applicative, in_simpl, simpl_predE). (* to be extended *) +- +-Lemma mem_simpl sp : mem sp = sp :> pred T. +-Proof. by []. Qed. +- +-Definition memE := mem_simpl. (* could be extended *) +- +-Lemma mem_mem (pp : pT) : (mem (mem pp) = mem pp) * (mem [mem pp] = mem pp). +-Proof. by rewrite -mem_topred. Qed. +- +-End simpl_mem. +- +-(** Qualifiers and keyed predicates. **) +- +-Variant qualifier (q : nat) T := Qualifier of predPredType T. +- +-Coercion has_quality n T (q : qualifier n T) : pred_class := +- fun x => let: Qualifier _ p := q in p x. +-Arguments has_quality n [T]. +- +-Lemma qualifE n T p x : (x \in @Qualifier n T p) = p x. Proof. by []. Qed. +- +-Notation "x \is A" := (x \in has_quality 0 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \is A ']'") : bool_scope. +-Notation "x \is 'a' A" := (x \in has_quality 1 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \is 'a' A ']'") : bool_scope. +-Notation "x \is 'an' A" := (x \in has_quality 2 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \is 'an' A ']'") : bool_scope. +-Notation "x \isn't A" := (x \notin has_quality 0 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \isn't A ']'") : bool_scope. +-Notation "x \isn't 'a' A" := (x \notin has_quality 1 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \isn't 'a' A ']'") : bool_scope. +-Notation "x \isn't 'an' A" := (x \notin has_quality 2 A) +- (at level 70, no associativity, +- format "'[hv' x '/ ' \isn't 'an' A ']'") : bool_scope. +-Notation "[ 'qualify' x | P ]" := (Qualifier 0 (fun x => P%B)) +- (at level 0, x at level 99, +- format "'[hv' [ 'qualify' x | '/ ' P ] ']'") : form_scope. +-Notation "[ 'qualify' x : T | P ]" := (Qualifier 0 (fun x : T => P%B)) +- (at level 0, x at level 99, only parsing) : form_scope. +-Notation "[ 'qualify' 'a' x | P ]" := (Qualifier 1 (fun x => P%B)) +- (at level 0, x at level 99, +- format "'[hv' [ 'qualify' 'a' x | '/ ' P ] ']'") : form_scope. +-Notation "[ 'qualify' 'a' x : T | P ]" := (Qualifier 1 (fun x : T => P%B)) +- (at level 0, x at level 99, only parsing) : form_scope. +-Notation "[ 'qualify' 'an' x | P ]" := (Qualifier 2 (fun x => P%B)) +- (at level 0, x at level 99, +- format "'[hv' [ 'qualify' 'an' x | '/ ' P ] ']'") : form_scope. +-Notation "[ 'qualify' 'an' x : T | P ]" := (Qualifier 2 (fun x : T => P%B)) +- (at level 0, x at level 99, only parsing) : form_scope. +- +-(** Keyed predicates: support for property-bearing predicate interfaces. **) +- +-Section KeyPred. +- +-Variable T : Type. +-Variant pred_key (p : predPredType T) := DefaultPredKey. +- +-Variable p : predPredType T. +-Structure keyed_pred (k : pred_key p) := +- PackKeyedPred {unkey_pred :> pred_class; _ : unkey_pred =i p}. +- +-Variable k : pred_key p. +-Definition KeyedPred := @PackKeyedPred k p (frefl _). +- +-Variable k_p : keyed_pred k. +-Lemma keyed_predE : k_p =i p. Proof. by case: k_p. Qed. +- +-(** +- 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 !! **) +-Canonical keyed_mem := +- @PackKeyedPred k (pred_of_mem (mem k_p)) keyed_predE. +-Canonical keyed_mem_simpl := +- @PackKeyedPred k (pred_of_simpl (mem k_p)) keyed_predE. +- +-End KeyPred. +- +-Notation "x \i 'n' S" := (x \in @unkey_pred _ S _ _) +- (at level 70, format "'[hv' x '/ ' \i 'n' S ']'") : bool_scope. +- +-Section KeyedQualifier. +- +-Variables (T : Type) (n : nat) (q : qualifier n T). +- +-Structure keyed_qualifier (k : pred_key q) := +- PackKeyedQualifier {unkey_qualifier; _ : unkey_qualifier = q}. +-Definition KeyedQualifier k := PackKeyedQualifier k (erefl q). +-Variables (k : pred_key q) (k_q : keyed_qualifier k). +-Fact keyed_qualifier_suproof : unkey_qualifier k_q =i q. +-Proof. by case: k_q => /= _ ->. Qed. +-Canonical keyed_qualifier_keyed := PackKeyedPred k keyed_qualifier_suproof. +- +-End KeyedQualifier. +- +-Notation "x \i 's' A" := (x \i n has_quality 0 A) +- (at level 70, format "'[hv' x '/ ' \i 's' A ']'") : bool_scope. +-Notation "x \i 's' 'a' A" := (x \i n has_quality 1 A) +- (at level 70, format "'[hv' x '/ ' \i 's' 'a' A ']'") : bool_scope. +-Notation "x \i 's' 'an' A" := (x \i n has_quality 2 A) +- (at level 70, format "'[hv' x '/ ' \i 's' 'an' A ']'") : bool_scope. +- +-Module DefaultKeying. +- +-Canonical default_keyed_pred T p := KeyedPred (@DefaultPredKey T p). +-Canonical default_keyed_qualifier T n (q : qualifier n T) := +- KeyedQualifier (DefaultPredKey q). +- +-End DefaultKeying. +- +-(** Skolemizing with conditions. **) +- +-Lemma all_tag_cond_dep I T (C : pred I) U : +- (forall x, T x) -> (forall x, C x -> {y : T x & U x y}) -> +- {f : forall x, T x & forall x, C x -> U x (f x)}. +-Proof. +-move=> f0 fP; apply: all_tag (fun x y => C x -> U x y) _ => x. +-by case Cx: (C x); [case/fP: Cx => y; exists y | exists (f0 x)]. +-Qed. +- +-Lemma all_tag_cond I T (C : pred I) U : +- T -> (forall x, C x -> {y : T & U x y}) -> +- {f : I -> T & forall x, C x -> U x (f x)}. +-Proof. by move=> y0; apply: all_tag_cond_dep. Qed. +- +-Lemma all_sig_cond_dep I T (C : pred I) P : +- (forall x, T x) -> (forall x, C x -> {y : T x | P x y}) -> +- {f : forall x, T x | forall x, C x -> P x (f x)}. +-Proof. by move=> f0 /(all_tag_cond_dep f0)[f]; exists f. Qed. +- +-Lemma all_sig_cond I T (C : pred I) P : +- T -> (forall x, C x -> {y : T | P x y}) -> +- {f : I -> T | forall x, C x -> P x (f x)}. +-Proof. by move=> y0; apply: all_sig_cond_dep. Qed. +- +-Section RelationProperties. +- +-(** +- Caveat: reflexive should not be used to state lemmas, as auto and trivial +- will not expand the constant. **) +- +-Variable T : Type. +- +-Variable R : rel T. +- +-Definition total := forall x y, R x y || R y x. +-Definition transitive := forall y x z, R x y -> R y z -> R x z. +- +-Definition symmetric := forall x y, R x y = R y x. +-Definition antisymmetric := forall x y, R x y && R y x -> x = y. +-Definition pre_symmetric := forall x y, R x y -> R y x. +- +-Lemma symmetric_from_pre : pre_symmetric -> symmetric. +-Proof. by move=> symR x y; apply/idP/idP; apply: symR. Qed. +- +-Definition reflexive := forall x, R x x. +-Definition irreflexive := forall x, R x x = false. +- +-Definition left_transitive := forall x y, R x y -> R x =1 R y. +-Definition right_transitive := forall x y, R x y -> R^~ x =1 R^~ y. +- +-Section PER. +- +-Hypotheses (symR : symmetric) (trR : transitive). +- +-Lemma sym_left_transitive : left_transitive. +-Proof. by move=> x y Rxy z; apply/idP/idP; apply: trR; rewrite // symR. Qed. +- +-Lemma sym_right_transitive : right_transitive. +-Proof. by move=> x y /sym_left_transitive Rxy z; rewrite !(symR z) Rxy. Qed. +- +-End PER. +- +-(** +- We define the equivalence property with prenex quantification so that it +- can be localized using the {in ..., ..} form defined below. **) +- +-Definition equivalence_rel := forall x y z, R z z * (R x y -> R x z = R y z). +- +-Lemma equivalence_relP : equivalence_rel <-> reflexive /\ left_transitive. +-Proof. +-split=> [eqiR | [Rxx trR] x y z]; last by split=> [|/trR->]. +-by split=> [x | x y Rxy z]; [rewrite (eqiR x x x) | rewrite (eqiR x y z)]. +-Qed. +- +-End RelationProperties. +- +-Lemma rev_trans T (R : rel T) : transitive R -> transitive (fun x y => R y x). +-Proof. by move=> trR x y z Ryx Rzy; apply: trR Rzy Ryx. Qed. +- +-(** Property localization **) +- +-Local Notation "{ 'all1' P }" := (forall x, P x : Prop) (at level 0). +-Local Notation "{ 'all2' P }" := (forall x y, P x y : Prop) (at level 0). +-Local Notation "{ 'all3' P }" := (forall x y z, P x y z: Prop) (at level 0). +-Local Notation ph := (phantom _). +- +-Section LocalProperties. +- +-Variables T1 T2 T3 : Type. +- +-Variables (d1 : mem_pred T1) (d2 : mem_pred T2) (d3 : mem_pred T3). +-Local Notation ph := (phantom Prop). +- +-Definition prop_for (x : T1) P & ph {all1 P} := P x. +- +-Lemma forE x P phP : @prop_for x P phP = P x. Proof. by []. Qed. +- +-Definition prop_in1 P & ph {all1 P} := +- forall x, in_mem x d1 -> P x. +- +-Definition prop_in11 P & ph {all2 P} := +- forall x y, in_mem x d1 -> in_mem y d2 -> P x y. +- +-Definition prop_in2 P & ph {all2 P} := +- forall x y, in_mem x d1 -> in_mem y d1 -> P x y. +- +-Definition prop_in111 P & ph {all3 P} := +- forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d3 -> P x y z. +- +-Definition prop_in12 P & ph {all3 P} := +- forall x y z, in_mem x d1 -> in_mem y d2 -> in_mem z d2 -> P x y z. +- +-Definition prop_in21 P & ph {all3 P} := +- forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d2 -> P x y z. +- +-Definition prop_in3 P & ph {all3 P} := +- forall x y z, in_mem x d1 -> in_mem y d1 -> in_mem z d1 -> P x y z. +- +-Variable f : T1 -> T2. +- +-Definition prop_on1 Pf P & phantom T3 (Pf f) & ph {all1 P} := +- forall x, in_mem (f x) d2 -> P x. +- +-Definition prop_on2 Pf P & phantom T3 (Pf f) & ph {all2 P} := +- forall x y, in_mem (f x) d2 -> in_mem (f y) d2 -> P x y. +- +-End LocalProperties. +- +-Definition inPhantom := Phantom Prop. +-Definition onPhantom T P (x : T) := Phantom Prop (P x). +- +-Definition bijective_in aT rT (d : mem_pred aT) (f : aT -> rT) := +- exists2 g, prop_in1 d (inPhantom (cancel f g)) +- & prop_on1 d (Phantom _ (cancel g)) (onPhantom (cancel g) f). +- +-Definition bijective_on aT rT (cd : mem_pred rT) (f : aT -> rT) := +- exists2 g, prop_on1 cd (Phantom _ (cancel f)) (onPhantom (cancel f) g) +- & prop_in1 cd (inPhantom (cancel g f)). +- +-Notation "{ 'for' x , P }" := +- (prop_for x (inPhantom P)) +- (at level 0, format "{ 'for' x , P }") : type_scope. +- +-Notation "{ 'in' d , P }" := +- (prop_in1 (mem d) (inPhantom P)) +- (at level 0, format "{ 'in' d , P }") : type_scope. +- +-Notation "{ 'in' d1 & d2 , P }" := +- (prop_in11 (mem d1) (mem d2) (inPhantom P)) +- (at level 0, format "{ 'in' d1 & d2 , P }") : type_scope. +- +-Notation "{ 'in' d & , P }" := +- (prop_in2 (mem d) (inPhantom P)) +- (at level 0, format "{ 'in' d & , P }") : type_scope. +- +-Notation "{ 'in' d1 & d2 & d3 , P }" := +- (prop_in111 (mem d1) (mem d2) (mem d3) (inPhantom P)) +- (at level 0, format "{ 'in' d1 & d2 & d3 , P }") : type_scope. +- +-Notation "{ 'in' d1 & & d3 , P }" := +- (prop_in21 (mem d1) (mem d3) (inPhantom P)) +- (at level 0, format "{ 'in' d1 & & d3 , P }") : type_scope. +- +-Notation "{ 'in' d1 & d2 & , P }" := +- (prop_in12 (mem d1) (mem d2) (inPhantom P)) +- (at level 0, format "{ 'in' d1 & d2 & , P }") : type_scope. +- +-Notation "{ 'in' d & & , P }" := +- (prop_in3 (mem d) (inPhantom P)) +- (at level 0, format "{ 'in' d & & , P }") : type_scope. +- +-Notation "{ 'on' cd , P }" := +- (prop_on1 (mem cd) (inPhantom P) (inPhantom P)) +- (at level 0, format "{ 'on' cd , P }") : type_scope. +- +-Notation "{ 'on' cd & , P }" := +- (prop_on2 (mem cd) (inPhantom P) (inPhantom P)) +- (at level 0, format "{ 'on' cd & , P }") : type_scope. +- +-Local Arguments onPhantom {_%type_scope} _ _. +- +-Notation "{ 'on' cd , P & g }" := +- (prop_on1 (mem cd) (Phantom (_ -> Prop) P) (onPhantom P g)) +- (at level 0, format "{ 'on' cd , P & g }") : type_scope. +- +-Notation "{ 'in' d , 'bijective' f }" := (bijective_in (mem d) f) +- (at level 0, f at level 8, +- format "{ 'in' d , 'bijective' f }") : type_scope. +- +-Notation "{ 'on' cd , 'bijective' f }" := (bijective_on (mem cd) f) +- (at level 0, f at level 8, +- format "{ 'on' cd , 'bijective' f }") : type_scope. +- +-(** +- Weakening and monotonicity lemmas for localized predicates. +- Note that using these lemmas in backward reasoning will force expansion of +- the predicate definition, as Coq needs to expose the quantifier to apply +- these lemmas. We define a few specialized variants to avoid this for some +- of the ssrfun predicates. **) +- +-Section LocalGlobal. +- +-Variables T1 T2 T3 : predArgType. +-Variables (D1 : pred T1) (D2 : pred T2) (D3 : pred T3). +-Variables (d1 d1' : mem_pred T1) (d2 d2' : mem_pred T2) (d3 d3' : mem_pred T3). +-Variables (f f' : T1 -> T2) (g : T2 -> T1) (h : T3). +-Variables (P1 : T1 -> Prop) (P2 : T1 -> T2 -> Prop). +-Variable P3 : T1 -> T2 -> T3 -> Prop. +-Variable Q1 : (T1 -> T2) -> T1 -> Prop. +-Variable Q1l : (T1 -> T2) -> T3 -> T1 -> Prop. +-Variable Q2 : (T1 -> T2) -> T1 -> T1 -> Prop. +- +-Hypothesis sub1 : sub_mem d1 d1'. +-Hypothesis sub2 : sub_mem d2 d2'. +-Hypothesis sub3 : sub_mem d3 d3'. +- +-Lemma in1W : {all1 P1} -> {in D1, {all1 P1}}. +-Proof. by move=> ? ?. Qed. +-Lemma in2W : {all2 P2} -> {in D1 & D2, {all2 P2}}. +-Proof. by move=> ? ?. Qed. +-Lemma in3W : {all3 P3} -> {in D1 & D2 & D3, {all3 P3}}. +-Proof. by move=> ? ?. Qed. +- +-Lemma in1T : {in T1, {all1 P1}} -> {all1 P1}. +-Proof. by move=> ? ?; auto. Qed. +-Lemma in2T : {in T1 & T2, {all2 P2}} -> {all2 P2}. +-Proof. by move=> ? ?; auto. Qed. +-Lemma in3T : {in T1 & T2 & T3, {all3 P3}} -> {all3 P3}. +-Proof. by move=> ? ?; auto. Qed. +- +-Lemma sub_in1 (Ph : ph {all1 P1}) : prop_in1 d1' Ph -> prop_in1 d1 Ph. +-Proof. by move=> allP x /sub1; apply: allP. Qed. +- +-Lemma sub_in11 (Ph : ph {all2 P2}) : prop_in11 d1' d2' Ph -> prop_in11 d1 d2 Ph. +-Proof. by move=> allP x1 x2 /sub1 d1x1 /sub2; apply: allP. Qed. +- +-Lemma sub_in111 (Ph : ph {all3 P3}) : +- prop_in111 d1' d2' d3' Ph -> prop_in111 d1 d2 d3 Ph. +-Proof. by move=> allP x1 x2 x3 /sub1 d1x1 /sub2 d2x2 /sub3; apply: allP. Qed. +- +-Let allQ1 f'' := {all1 Q1 f''}. +-Let allQ1l f'' h' := {all1 Q1l f'' h'}. +-Let allQ2 f'' := {all2 Q2 f''}. +- +-Lemma on1W : allQ1 f -> {on D2, allQ1 f}. Proof. by move=> ? ?. Qed. +- +-Lemma on1lW : allQ1l f h -> {on D2, allQ1l f & h}. Proof. by move=> ? ?. Qed. +- +-Lemma on2W : allQ2 f -> {on D2 &, allQ2 f}. Proof. by move=> ? ?. Qed. +- +-Lemma on1T : {on T2, allQ1 f} -> allQ1 f. Proof. by move=> ? ?; auto. Qed. +- +-Lemma on1lT : {on T2, allQ1l f & h} -> allQ1l f h. +-Proof. by move=> ? ?; auto. Qed. +- +-Lemma on2T : {on T2 &, allQ2 f} -> allQ2 f. +-Proof. by move=> ? ?; auto. Qed. +- +-Lemma subon1 (Phf : ph (allQ1 f)) (Ph : ph (allQ1 f)) : +- prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. +-Proof. by move=> allQ x /sub2; apply: allQ. Qed. +- +-Lemma subon1l (Phf : ph (allQ1l f)) (Ph : ph (allQ1l f h)) : +- prop_on1 d2' Phf Ph -> prop_on1 d2 Phf Ph. +-Proof. by move=> allQ x /sub2; apply: allQ. Qed. +- +-Lemma subon2 (Phf : ph (allQ2 f)) (Ph : ph (allQ2 f)) : +- prop_on2 d2' Phf Ph -> prop_on2 d2 Phf Ph. +-Proof. by move=> allQ x y /sub2=> d2fx /sub2; apply: allQ. Qed. +- +-Lemma can_in_inj : {in D1, cancel f g} -> {in D1 &, injective f}. +-Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. +- +-Lemma canLR_in x y : {in D1, cancel f g} -> y \in D1 -> x = f y -> g x = y. +-Proof. by move=> fK D1y ->; rewrite fK. Qed. +- +-Lemma canRL_in x y : {in D1, cancel f g} -> x \in D1 -> f x = y -> x = g y. +-Proof. by move=> fK D1x <-; rewrite fK. Qed. +- +-Lemma on_can_inj : {on D2, cancel f & g} -> {on D2 &, injective f}. +-Proof. by move=> fK x y /fK{2}<- /fK{2}<- ->. Qed. +- +-Lemma canLR_on x y : {on D2, cancel f & g} -> f y \in D2 -> x = f y -> g x = y. +-Proof. by move=> fK D2fy ->; rewrite fK. Qed. +- +-Lemma canRL_on x y : {on D2, cancel f & g} -> f x \in D2 -> f x = y -> x = g y. +-Proof. by move=> fK D2fx <-; rewrite fK. Qed. +- +-Lemma inW_bij : bijective f -> {in D1, bijective f}. +-Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. +- +-Lemma onW_bij : bijective f -> {on D2, bijective f}. +-Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. +- +-Lemma inT_bij : {in T1, bijective f} -> bijective f. +-Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. +- +-Lemma onT_bij : {on T2, bijective f} -> bijective f. +-Proof. by case=> g' fK g'K; exists g' => * ? *; auto. Qed. +- +-Lemma sub_in_bij (D1' : pred T1) : +- {subset D1 <= D1'} -> {in D1', bijective f} -> {in D1, bijective f}. +-Proof. +-by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K]. +-Qed. +- +-Lemma subon_bij (D2' : pred T2) : +- {subset D2 <= D2'} -> {on D2', bijective f} -> {on D2, bijective f}. +-Proof. +-by move=> subD [g' fK g'K]; exists g' => x; move/subD; [apply: fK | apply: g'K]. +-Qed. +- +-End LocalGlobal. +- +-Lemma sub_in2 T d d' (P : T -> T -> Prop) : +- sub_mem d d' -> forall Ph : ph {all2 P}, prop_in2 d' Ph -> prop_in2 d Ph. +-Proof. by move=> /= sub_dd'; apply: sub_in11. Qed. +- +-Lemma sub_in3 T d d' (P : T -> T -> T -> Prop) : +- sub_mem d d' -> forall Ph : ph {all3 P}, prop_in3 d' Ph -> prop_in3 d Ph. +-Proof. by move=> /= sub_dd'; apply: sub_in111. Qed. +- +-Lemma sub_in12 T1 T d1 d1' d d' (P : T1 -> T -> T -> Prop) : +- sub_mem d1 d1' -> sub_mem d d' -> +- forall Ph : ph {all3 P}, prop_in12 d1' d' Ph -> prop_in12 d1 d Ph. +-Proof. by move=> /= sub1 sub; apply: sub_in111. Qed. +- +-Lemma sub_in21 T T3 d d' d3 d3' (P : T -> T -> T3 -> Prop) : +- sub_mem d d' -> sub_mem d3 d3' -> +- forall Ph : ph {all3 P}, prop_in21 d' d3' Ph -> prop_in21 d d3 Ph. +-Proof. by move=> /= sub sub3; apply: sub_in111. Qed. +- +-Lemma equivalence_relP_in T (R : rel T) (A : pred T) : +- {in A & &, equivalence_rel R} +- <-> {in A, reflexive R} /\ {in A &, forall x y, R x y -> {in A, R x =1 R y}}. +-Proof. +-split=> [eqiR | [Rxx trR] x y z *]; last by split=> [|/trR-> //]; apply: Rxx. +-by split=> [x Ax|x y Ax Ay Rxy z Az]; [rewrite (eqiR x x) | rewrite (eqiR x y)]. +-Qed. +- +-Section MonoHomoMorphismTheory. +- +-Variables (aT rT sT : Type) (f : aT -> rT) (g : rT -> aT). +-Variables (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). +- +-Lemma monoW : {mono f : x / aP x >-> rP x} -> {homo f : x / aP x >-> rP x}. +-Proof. by move=> hf x ax; rewrite hf. Qed. +- +-Lemma mono2W : +- {mono f : x y / aR x y >-> rR x y} -> {homo f : x y / aR x y >-> rR x y}. +-Proof. by move=> hf x y axy; rewrite hf. Qed. +- +-Hypothesis fgK : cancel g f. +- +-Lemma homoRL : +- {homo f : x y / aR x y >-> rR x y} -> forall x y, aR (g x) y -> rR x (f y). +-Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. +- +-Lemma homoLR : +- {homo f : x y / aR x y >-> rR x y} -> forall x y, aR x (g y) -> rR (f x) y. +-Proof. by move=> Hf x y /Hf; rewrite fgK. Qed. +- +-Lemma homo_mono : +- {homo f : x y / aR x y >-> rR x y} -> {homo g : x y / rR x y >-> aR x y} -> +- {mono g : x y / rR x y >-> aR x y}. +-Proof. +-move=> mf mg x y; case: (boolP (rR _ _))=> [/mg //|]. +-by apply: contraNF=> /mf; rewrite !fgK. +-Qed. +- +-Lemma monoLR : +- {mono f : x y / aR x y >-> rR x y} -> forall x y, rR (f x) y = aR x (g y). +-Proof. by move=> mf x y; rewrite -{1}[y]fgK mf. Qed. +- +-Lemma monoRL : +- {mono f : x y / aR x y >-> rR x y} -> forall x y, rR x (f y) = aR (g x) y. +-Proof. by move=> mf x y; rewrite -{1}[x]fgK mf. Qed. +- +-Lemma can_mono : +- {mono f : x y / aR x y >-> rR x y} -> {mono g : x y / rR x y >-> aR x y}. +-Proof. by move=> mf x y /=; rewrite -mf !fgK. Qed. +- +-End MonoHomoMorphismTheory. +- +-Section MonoHomoMorphismTheory_in. +- +-Variables (aT rT sT : predArgType) (f : aT -> rT) (g : rT -> aT). +-Variable (aD : pred aT). +-Variable (aP : pred aT) (rP : pred rT) (aR : rel aT) (rR : rel rT). +- +-Notation rD := [pred x | g x \in aD]. +- +-Lemma monoW_in : +- {in aD &, {mono f : x y / aR x y >-> rR x y}} -> +- {in aD &, {homo f : x y / aR x y >-> rR x y}}. +-Proof. by move=> hf x y hx hy axy; rewrite hf. Qed. +- +-Lemma mono2W_in : +- {in aD, {mono f : x / aP x >-> rP x}} -> +- {in aD, {homo f : x / aP x >-> rP x}}. +-Proof. by move=> hf x hx ax; rewrite hf. Qed. +- +-Hypothesis fgK_on : {on aD, cancel g & f}. +- +-Lemma homoRL_in : +- {in aD &, {homo f : x y / aR x y >-> rR x y}} -> +- {in rD & aD, forall x y, aR (g x) y -> rR x (f y)}. +-Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. +- +-Lemma homoLR_in : +- {in aD &, {homo f : x y / aR x y >-> rR x y}} -> +- {in aD & rD, forall x y, aR x (g y) -> rR (f x) y}. +-Proof. by move=> Hf x y hx hy /Hf; rewrite fgK_on //; apply. Qed. +- +-Lemma homo_mono_in : +- {in aD &, {homo f : x y / aR x y >-> rR x y}} -> +- {in rD &, {homo g : x y / rR x y >-> aR x y}} -> +- {in rD &, {mono g : x y / rR x y >-> aR x y}}. +-Proof. +-move=> mf mg x y hx hy; case: (boolP (rR _ _))=> [/mg //|]; first exact. +-by apply: contraNF=> /mf; rewrite !fgK_on //; apply. +-Qed. +- +-Lemma monoLR_in : +- {in aD &, {mono f : x y / aR x y >-> rR x y}} -> +- {in aD & rD, forall x y, rR (f x) y = aR x (g y)}. +-Proof. by move=> mf x y hx hy; rewrite -{1}[y]fgK_on // mf. Qed. +- +-Lemma monoRL_in : +- {in aD &, {mono f : x y / aR x y >-> rR x y}} -> +- {in rD & aD, forall x y, rR x (f y) = aR (g x) y}. +-Proof. by move=> mf x y hx hy; rewrite -{1}[x]fgK_on // mf. Qed. +- +-Lemma can_mono_in : +- {in aD &, {mono f : x y / aR x y >-> rR x y}} -> +- {in rD &, {mono g : x y / rR x y >-> aR x y}}. +-Proof. by move=> mf x y hx hy /=; rewrite -mf // !fgK_on. Qed. +- +-End MonoHomoMorphismTheory_in. +--- a/plugins/ssr/ssreflect.v ++++ /dev/null +@@ -1,470 +0,0 @@ +-(************************************************************************) +-(* * The Coq Proof Assistant / The Coq Development Team *) +-(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +-(* <O___,, * (see CREDITS file for the list of authors) *) +-(* \VV/ **************************************************************) +-(* // * This file is distributed under the terms of the *) +-(* * GNU Lesser General Public License Version 2.1 *) +-(* * (see LICENSE file for the text of the license) *) +-(************************************************************************) +- +-(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *) +- +-(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **) +- +-Require Import Bool. (* For bool_scope delimiter 'bool'. *) +-Require Import ssrmatching. +-Declare ML Module "ssreflect_plugin". +- +- +-(** +- This file is the Gallina part of the ssreflect plugin implementation. +- Files that use the ssreflect plugin should always Require ssreflect and +- either Import ssreflect or Import ssreflect.SsrSyntax. +- Part of the contents of this file is technical and will only interest +- advanced developers; in addition the following are defined: +- #[#the str of v by f#]# == the Canonical s : str such that f s = v. +- #[#the str of v#]# == the Canonical s : str that coerces to v. +- argumentType c == the T such that c : forall x : T, P x. +- returnType c == the R such that c : T -> R. +- {type of c for s} == P s where c : forall x : T, P x. +- phantom T v == singleton type with inhabitant Phantom T v. +- phant T == singleton type with inhabitant Phant v. +- =^~ r == the converse of rewriting rule r (e.g., in a +- rewrite multirule). +- unkeyed t == t, but treated as an unkeyed matching pattern by +- the ssreflect matching algorithm. +- nosimpl t == t, but on the right-hand side of Definition C := +- nosimpl disables expansion of C by /=. +- locked t == t, but locked t is not convertible to t. +- locked_with k t == t, but not convertible to t or locked_with k' t +- unless k = k' (with k : unit). Coq type-checking +- will be much more efficient if locked_with with a +- bespoke k is used for sealed definitions. +- unlockable v == interface for sealed constant definitions of v. +- Unlockable def == the unlockable that registers def : C = v. +- #[#unlockable of C#]# == a clone for C of the canonical unlockable for the +- definition of C (e.g., if it uses locked_with). +- #[#unlockable fun C#]# == #[#unlockable of C#]# with the expansion forced to be +- an explicit lambda expression. +- -> The usage pattern for ADT operations is: +- Definition foo_def x1 .. xn := big_foo_expression. +- Fact foo_key : unit. Proof. by #[# #]#. Qed. +- Definition foo := locked_with foo_key foo_def. +- Canonical foo_unlockable := #[#unlockable fun foo#]#. +- This minimizes the comparison overhead for foo, while still allowing +- rewrite unlock to expose big_foo_expression. +- More information about these definitions and their use can be found in the +- ssreflect manual, and in specific comments below. **) +- +- +- +-Set Implicit Arguments. +-Unset Strict Implicit. +-Unset Printing Implicit Defensive. +- +-Module SsrSyntax. +- +-(** +- Declare Ssr keywords: 'is' 'of' '//' '/=' and '//='. We also declare the +- parsing level 8, as a workaround for a notation grammar factoring problem. +- Arguments of application-style notations (at level 10) should be declared +- at level 8 rather than 9 or the camlp5 grammar will not factor properly. **) +- +-Reserved Notation "(* x 'is' y 'of' z 'isn't' // /= //= *)" (at level 8). +-Reserved Notation "(* 69 *)" (at level 69). +- +-(** Non ambiguous keyword to check if the SsrSyntax module is imported **) +-Reserved Notation "(* Use to test if 'SsrSyntax_is_Imported' *)" (at level 8). +- +-Reserved Notation "<hidden n >" (at level 200). +-Reserved Notation "T (* n *)" (at level 200, format "T (* n *)"). +- +-End SsrSyntax. +- +-Export SsrMatchingSyntax. +-Export SsrSyntax. +- +-(** +- Make the general "if" into a notation, so that we can override it below. +- The notations are "only parsing" because the Coq decompiler will not +- recognize the expansion of the boolean if; using the default printer +- avoids a spurrious trailing %%GEN_IF. **) +- +-Delimit Scope general_if_scope with GEN_IF. +- +-Notation "'if' c 'then' v1 'else' v2" := +- (if c then v1 else v2) +- (at level 200, c, v1, v2 at level 200, only parsing) : general_if_scope. +- +-Notation "'if' c 'return' t 'then' v1 'else' v2" := +- (if c return t then v1 else v2) +- (at level 200, c, t, v1, v2 at level 200, only parsing) : general_if_scope. +- +-Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := +- (if c as x return t then v1 else v2) +- (at level 200, c, t, v1, v2 at level 200, x ident, only parsing) +- : general_if_scope. +- +-(** Force boolean interpretation of simple if expressions. **) +- +-Delimit Scope boolean_if_scope with BOOL_IF. +- +-Notation "'if' c 'return' t 'then' v1 'else' v2" := +- (if c%bool is true in bool return t then v1 else v2) : boolean_if_scope. +- +-Notation "'if' c 'then' v1 'else' v2" := +- (if c%bool is true in bool return _ then v1 else v2) : boolean_if_scope. +- +-Notation "'if' c 'as' x 'return' t 'then' v1 'else' v2" := +- (if c%bool is true as x in bool return t then v1 else v2) : boolean_if_scope. +- +-Open Scope boolean_if_scope. +- +-(** +- To allow a wider variety of notations without reserving a large number of +- of identifiers, the ssreflect library systematically uses "forms" to +- enclose complex mixfix syntax. A "form" is simply a mixfix expression +- enclosed in square brackets and introduced by a keyword: +- #[#keyword ... #]# +- Because the keyword follows a bracket it does not need to be reserved. +- Non-ssreflect libraries that do not respect the form syntax (e.g., the Coq +- Lists library) should be loaded before ssreflect so that their notations +- do not mask all ssreflect forms. **) +-Delimit Scope form_scope with FORM. +-Open Scope form_scope. +- +-(** +- Allow overloading of the cast (x : T) syntax, put whitespace around the +- ":" symbol to avoid lexical clashes (and for consistency with the parsing +- precedence of the notation, which binds less tightly than application), +- and put printing boxes that print the type of a long definition on a +- separate line rather than force-fit it at the right margin. **) +-Notation "x : T" := (x : T) +- (at level 100, right associativity, +- format "'[hv' x '/ ' : T ']'") : core_scope. +- +-(** +- Allow the casual use of notations like nat * nat for explicit Type +- declarations. Note that (nat * nat : Type) is NOT equivalent to +- (nat * nat)%%type, whose inferred type is legacy type "Set". **) +-Notation "T : 'Type'" := (T%type : Type) +- (at level 100, only parsing) : core_scope. +-(** Allow similarly Prop annotation for, e.g., rewrite multirules. **) +-Notation "P : 'Prop'" := (P%type : Prop) +- (at level 100, only parsing) : core_scope. +- +-(** Constants for abstract: and #[#: name #]# intro pattern **) +-Definition abstract_lock := unit. +-Definition abstract_key := tt. +- +-Definition abstract (statement : Type) (id : nat) (lock : abstract_lock) := +- let: tt := lock in statement. +- +-Notation "<hidden n >" := (abstract _ n _). +-Notation "T (* n *)" := (abstract T n abstract_key). +- +-(** Constants for tactic-views **) +-Inductive external_view : Type := tactic_view of Type. +- +-(** +- Syntax for referring to canonical structures: +- #[#the struct_type of proj_val by proj_fun#]# +- This form denotes the Canonical instance s of the Structure type +- struct_type whose proj_fun projection is proj_val, i.e., such that +- proj_fun s = proj_val. +- Typically proj_fun will be A record field accessors of struct_type, but +- this need not be the case; it can be, for instance, a field of a record +- type to which struct_type coerces; proj_val will likewise be coerced to +- the return type of proj_fun. In all but the simplest cases, proj_fun +- should be eta-expanded to allow for the insertion of implicit arguments. +- In the common case where proj_fun itself is a coercion, the "by" part +- can be omitted entirely; in this case it is inferred by casting s to the +- inferred type of proj_val. Obviously the latter can be fixed by using an +- explicit cast on proj_val, and it is highly recommended to do so when the +- return type intended for proj_fun is "Type", as the type inferred for +- proj_val may vary because of sort polymorphism (it could be Set or Prop). +- Note when using the #[#the _ of _ #]# form to generate a substructure from a +- telescopes-style canonical hierarchy (implementing inheritance with +- coercions), one should always project or coerce the value to the BASE +- structure, because Coq will only find a Canonical derived structure for +- the Canonical base structure -- not for a base structure that is specific +- to proj_value. **) +- +-Module TheCanonical. +- +-Variant put vT sT (v1 v2 : vT) (s : sT) := Put. +- +-Definition get vT sT v s (p : @put vT sT v v s) := let: Put _ _ _ := p in s. +- +-Definition get_by vT sT of sT -> vT := @get vT sT. +- +-End TheCanonical. +- +-Import TheCanonical. (* Note: no export. *) +- +-Local Arguments get_by _%type_scope _%type_scope _ _ _ _. +- +-Notation "[ 'the' sT 'of' v 'by' f ]" := +- (@get_by _ sT f _ _ ((fun v' (s : sT) => Put v' (f s) s) v _)) +- (at level 0, only parsing) : form_scope. +- +-Notation "[ 'the' sT 'of' v ]" := (get ((fun s : sT => Put v (*coerce*)s s) _)) +- (at level 0, only parsing) : form_scope. +- +-(** +- The following are "format only" versions of the above notations. Since Coq +- doesn't provide this facility, we fake it by splitting the "the" keyword. +- We need to do this to prevent the formatter from being be thrown off by +- application collapsing, coercion insertion and beta reduction in the right +- hand side of the notations above. **) +- +-Notation "[ 'th' 'e' sT 'of' v 'by' f ]" := (@get_by _ sT f v _ _) +- (at level 0, format "[ 'th' 'e' sT 'of' v 'by' f ]") : form_scope. +- +-Notation "[ 'th' 'e' sT 'of' v ]" := (@get _ sT v _ _) +- (at level 0, format "[ 'th' 'e' sT 'of' v ]") : form_scope. +- +-(** +- We would like to recognize +-Notation " #[# 'th' 'e' sT 'of' v : 'Type' #]#" := (@get Type sT v _ _) +- (at level 0, format " #[# 'th' 'e' sT 'of' v : 'Type' #]#") : form_scope. +- **) +- +-(** +- Helper notation for canonical structure inheritance support. +- This is a workaround for the poor interaction between delta reduction and +- canonical projections in Coq's unification algorithm, by which transparent +- definitions hide canonical instances, i.e., in +- Canonical a_type_struct := @Struct a_type ... +- Definition my_type := a_type. +- my_type doesn't effectively inherit the struct structure from a_type. Our +- solution is to redeclare the instance as follows +- Canonical my_type_struct := Eval hnf in #[#struct of my_type#]#. +- The special notation #[#str of _ #]# must be defined for each Strucure "str" +- with constructor "Str", typically as follows +- Definition clone_str s := +- let: Str _ x y ... z := s return {type of Str for s} -> str in +- fun k => k _ x y ... z. +- Notation " #[# 'str' 'of' T 'for' s #]#" := (@clone_str s (@Str T)) +- (at level 0, format " #[# 'str' 'of' T 'for' s #]#") : form_scope. +- Notation " #[# 'str' 'of' T #]#" := (repack_str (fun x => @Str T x)) +- (at level 0, format " #[# 'str' 'of' T #]#") : form_scope. +- The notation for the match return predicate is defined below; the eta +- expansion in the second form serves both to distinguish it from the first +- and to avoid the delta reduction problem. +- There are several variations on the notation and the definition of the +- the "clone" function, for telescopes, mixin classes, and join (multiple +- inheritance) classes. We describe a different idiom for clones in ssrfun; +- it uses phantom types (see below) and static unification; see fintype and +- ssralg for examples. **) +- +-Definition argumentType T P & forall x : T, P x := T. +-Definition dependentReturnType T P & forall x : T, P x := P. +-Definition returnType aT rT & aT -> rT := rT. +- +-Notation "{ 'type' 'of' c 'for' s }" := (dependentReturnType c s) +- (at level 0, format "{ 'type' 'of' c 'for' s }") : type_scope. +- +-(** +- A generic "phantom" type (actually, a unit type with a phantom parameter). +- This type can be used for type definitions that require some Structure +- on one of their parameters, to allow Coq to infer said structure so it +- does not have to be supplied explicitly or via the " #[#the _ of _ #]#" notation +- (the latter interacts poorly with other Notation). +- The definition of a (co)inductive type with a parameter p : p_type, that +- needs to use the operations of a structure +- Structure p_str : Type := p_Str {p_repr :> p_type; p_op : p_repr -> ...} +- should be given as +- Inductive indt_type (p : p_str) := Indt ... . +- Definition indt_of (p : p_str) & phantom p_type p := indt_type p. +- Notation "{ 'indt' p }" := (indt_of (Phantom p)). +- Definition indt p x y ... z : {indt p} := @Indt p x y ... z. +- Notation " #[# 'indt' x y ... z #]#" := (indt x y ... z). +- That is, the concrete type and its constructor should be shadowed by +- definitions that use a phantom argument to infer and display the true +- value of p (in practice, the "indt" constructor often performs additional +- functions, like "locking" the representation -- see below). +- We also define a simpler version ("phant" / "Phant") of phantom for the +- common case where p_type is Type. **) +- +-Variant phantom T (p : T) := Phantom. +-Arguments phantom : clear implicits. +-Arguments Phantom : clear implicits. +-Variant phant (p : Type) := Phant. +- +-(** Internal tagging used by the implementation of the ssreflect elim. **) +- +-Definition protect_term (A : Type) (x : A) : A := x. +- +-(** +- The ssreflect idiom for a non-keyed pattern: +- - unkeyed t wiil match any subterm that unifies with t, regardless of +- whether it displays the same head symbol as t. +- - unkeyed t a b will match any application of a term f unifying with t, +- to two arguments unifying with with a and b, repectively, regardless of +- apparent head symbols. +- - unkeyed x where x is a variable will match any subterm with the same +- type as x (when x would raise the 'indeterminate pattern' error). **) +- +-Notation unkeyed x := (let flex := x in flex). +- +-(** Ssreflect converse rewrite rule rule idiom. **) +-Definition ssr_converse R (r : R) := (Logic.I, r). +-Notation "=^~ r" := (ssr_converse r) (at level 100) : form_scope. +- +-(** +- Term tagging (user-level). +- The ssreflect library uses four strengths of term tagging to restrict +- convertibility during type checking: +- nosimpl t simplifies to t EXCEPT in a definition; more precisely, given +- Definition foo := nosimpl bar, foo (or foo t') will NOT be expanded by +- the /= and //= switches unless it is in a forcing context (e.g., in +- match foo t' with ... end, foo t' will be reduced if this allows the +- match to be reduced). Note that nosimpl bar is simply notation for a +- a term that beta-iota reduces to bar; hence rewrite /foo will replace +- foo by bar, and rewrite -/foo will replace bar by foo. +- CAVEAT: nosimpl should not be used inside a Section, because the end of +- section "cooking" removes the iota redex. +- locked t is provably equal to t, but is not convertible to t; 'locked' +- provides support for selective rewriting, via the lock t : t = locked t +- Lemma, and the ssreflect unlock tactic. +- locked_with k t is equal but not convertible to t, much like locked t, +- but supports explicit tagging with a value k : unit. This is used to +- mitigate a flaw in the term comparison heuristic of the Coq kernel, +- which treats all terms of the form locked t as equal and conpares their +- arguments recursively, leading to an exponential blowup of comparison. +- For this reason locked_with should be used rather than locked when +- defining ADT operations. The unlock tactic does not support locked_with +- but the unlock rewrite rule does, via the unlockable interface. +- we also use Module Type ascription to create truly opaque constants, +- because simple expansion of constants to reveal an unreducible term +- doubles the time complexity of a negative comparison. Such opaque +- constants can be expanded generically with the unlock rewrite rule. +- See the definition of card and subset in fintype for examples of this. **) +- +-Notation nosimpl t := (let: tt := tt in t). +- +-Lemma master_key : unit. Proof. exact tt. Qed. +-Definition locked A := let: tt := master_key in fun x : A => x. +- +-Lemma lock A x : x = locked x :> A. Proof. unlock; reflexivity. Qed. +- +-(** Needed for locked predicates, in particular for eqType's. **) +-Lemma not_locked_false_eq_true : locked false <> true. +-Proof. unlock; discriminate. Qed. +- +-(** The basic closing tactic "done". **) +-Ltac done := +- trivial; hnf; intros; solve +- [ do ![solve [trivial | apply: sym_equal; trivial] +- | discriminate | contradiction | split] +- | case not_locked_false_eq_true; assumption +- | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. +- +-(** Quicker done tactic not including split, syntax: /0/ **) +-Ltac ssrdone0 := +- trivial; hnf; intros; solve +- [ do ![solve [trivial | apply: sym_equal; trivial] +- | discriminate | contradiction ] +- | case not_locked_false_eq_true; assumption +- | match goal with H : ~ _ |- _ => solve [case H; trivial] end ]. +- +-(** To unlock opaque constants. **) +-Structure unlockable T v := Unlockable {unlocked : T; _ : unlocked = v}. +-Lemma unlock T x C : @unlocked T x C = x. Proof. by case: C. Qed. +- +-Notation "[ 'unlockable' 'of' C ]" := (@Unlockable _ _ C (unlock _)) +- (at level 0, format "[ 'unlockable' 'of' C ]") : form_scope. +- +-Notation "[ 'unlockable' 'fun' C ]" := (@Unlockable _ (fun _ => _) C (unlock _)) +- (at level 0, format "[ 'unlockable' 'fun' C ]") : form_scope. +- +-(** Generic keyed constant locking. **) +- +-(** The argument order ensures that k is always compared before T. **) +-Definition locked_with k := let: tt := k in fun T x => x : T. +- +-(** +- This can be used as a cheap alternative to cloning the unlockable instance +- below, but with caution as unkeyed matching can be expensive. **) +-Lemma locked_withE T k x : unkeyed (locked_with k x) = x :> T. +-Proof. by case: k. Qed. +- +-(** Intensionaly, this instance will not apply to locked u. **) +-Canonical locked_with_unlockable T k x := +- @Unlockable T x (locked_with k x) (locked_withE k x). +- +-(** More accurate variant of unlock, and safer alternative to locked_withE. **) +-Lemma unlock_with T k x : unlocked (locked_with_unlockable k x) = x :> T. +-Proof. exact: unlock. Qed. +- +-(** The internal lemmas for the have tactics. **) +- +-Definition ssr_have Plemma Pgoal (step : Plemma) rest : Pgoal := rest step. +-Arguments ssr_have Plemma [Pgoal]. +- +-Definition ssr_have_let Pgoal Plemma step +- (rest : let x : Plemma := step in Pgoal) : Pgoal := rest. +-Arguments ssr_have_let [Pgoal]. +- +-Definition ssr_suff Plemma Pgoal step (rest : Plemma) : Pgoal := step rest. +-Arguments ssr_suff Plemma [Pgoal]. +- +-Definition ssr_wlog := ssr_suff. +-Arguments ssr_wlog Plemma [Pgoal]. +- +-(** Internal N-ary congruence lemmas for the congr tactic. **) +- +-Fixpoint nary_congruence_statement (n : nat) +- : (forall B, (B -> B -> Prop) -> Prop) -> Prop := +- match n with +- | O => fun k => forall B, k B (fun x1 x2 : B => x1 = x2) +- | S n' => +- let k' A B e (f1 f2 : A -> B) := +- forall x1 x2, x1 = x2 -> (e (f1 x1) (f2 x2) : Prop) in +- fun k => forall A, nary_congruence_statement n' (fun B e => k _ (k' A B e)) +- end. +- +-Lemma nary_congruence n (k := fun B e => forall y : B, (e y y : Prop)) : +- nary_congruence_statement n k. +-Proof. +-have: k _ _ := _; rewrite {1}/k. +-elim: n k => [|n IHn] k k_P /= A; first exact: k_P. +-by apply: IHn => B e He; apply: k_P => f x1 x2 <-. +-Qed. +- +-Lemma ssr_congr_arrow Plemma Pgoal : Plemma = Pgoal -> Plemma -> Pgoal. +-Proof. by move->. Qed. +-Arguments ssr_congr_arrow : clear implicits. +- +-(** View lemmas that don't use reflection. **) +- +-Section ApplyIff. +- +-Variables P Q : Prop. +-Hypothesis eqPQ : P <-> Q. +- +-Lemma iffLR : P -> Q. Proof. by case: eqPQ. Qed. +-Lemma iffRL : Q -> P. Proof. by case: eqPQ. Qed. +- +-Lemma iffLRn : ~P -> ~Q. Proof. by move=> nP tQ; case: nP; case: eqPQ tQ. Qed. +-Lemma iffRLn : ~Q -> ~P. Proof. by move=> nQ tP; case: nQ; case: eqPQ tP. Qed. +- +-End ApplyIff. +- +-Hint View for move/ iffLRn|2 iffRLn|2 iffLR|2 iffRL|2. +-Hint View for apply/ iffRLn|2 iffLRn|2 iffRL|2 iffLR|2. +- +-(** +- To focus non-ssreflect tactics on a subterm, eg vm_compute. +- Usage: +- elim/abstract_context: (pattern) => G defG. +- vm_compute; rewrite {}defG {G}. +- Note that vm_cast are not stored in the proof term +- for reductions occuring in the context, hence +- set here := pattern; vm_compute in (value of here) +- blows up at Qed time. **) +-Lemma abstract_context T (P : T -> Type) x : +- (forall Q, Q = P -> Q x) -> P x. +-Proof. by move=> /(_ P); apply. Qed. +--- a/plugins/ssr/ssrfun.v ++++ /dev/null +@@ -1,805 +0,0 @@ +-(************************************************************************) +-(* * The Coq Proof Assistant / The Coq Development Team *) +-(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +-(* <O___,, * (see CREDITS file for the list of authors) *) +-(* \VV/ **************************************************************) +-(* // * This file is distributed under the terms of the *) +-(* * GNU Lesser General Public License Version 2.1 *) +-(* * (see LICENSE file for the text of the license) *) +-(************************************************************************) +- +-(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *) +- +-(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **) +- +-Require Import ssreflect. +- +- +-(** +- This file contains the basic definitions and notations for working with +- functions. The definitions provide for: +- +- - Pair projections: +- p.1 == first element of a pair +- p.2 == second element of a pair +- These notations also apply to p : P /\ Q, via an and >-> pair coercion. +- +- - Simplifying functions, beta-reduced by /= and simpl: +- #[#fun : T => E#]# == constant function from type T that returns E +- #[#fun x => E#]# == unary function +- #[#fun x : T => E#]# == unary function with explicit domain type +- #[#fun x y => E#]# == binary function +- #[#fun x y : T => E#]# == binary function with common domain type +- #[#fun (x : T) y => E#]# \ +- #[#fun (x : xT) (y : yT) => E#]# | == binary function with (some) explicit, +- #[#fun x (y : T) => E#]# / independent domain types for each argument +- +- - Partial functions using option type: +- oapp f d ox == if ox is Some x returns f x, d otherwise +- odflt d ox == if ox is Some x returns x, d otherwise +- obind f ox == if ox is Some x returns f x, None otherwise +- omap f ox == if ox is Some x returns Some (f x), None otherwise +- +- - Singleton types: +- all_equal_to x0 == x0 is the only value in its type, so any such value +- can be rewritten to x0. +- +- - A generic wrapper type: +- wrapped T == the inductive type with values Wrap x for x : T. +- unwrap w == the projection of w : wrapped T on T. +- wrap x == the canonical injection of x : T into wrapped T; it is +- equivalent to Wrap x, but is declared as a (default) +- Canonical Structure, which lets the Coq HO unification +- automatically expand x into unwrap (wrap x). The delta +- reduction of wrap x to Wrap can be exploited to +- introduce controlled nondeterminism in Canonical +- Structure inference, as in the implementation of +- the mxdirect predicate in matrix.v. +- +- - Sigma types: +- tag w == the i of w : {i : I & T i}. +- tagged w == the T i component of w : {i : I & T i}. +- Tagged T x == the {i : I & T i} with component x : T i. +- tag2 w == the i of w : {i : I & T i & U i}. +- tagged2 w == the T i component of w : {i : I & T i & U i}. +- tagged2' w == the U i component of w : {i : I & T i & U i}. +- Tagged2 T U x y == the {i : I & T i} with components x : T i and y : U i. +- sval u == the x of u : {x : T | P x}. +- s2val u == the x of u : {x : T | P x & Q x}. +- The properties of sval u, s2val u are given by lemmas svalP, s2valP, and +- s2valP'. We provide coercions sigT2 >-> sigT and sig2 >-> sig >-> sigT. +- A suite of lemmas (all_sig, ...) let us skolemize sig, sig2, sigT, sigT2 +- and pair, e.g., +- have /all_sig#[#f fP#]# (x : T): {y : U | P y} by ... +- yields an f : T -> U such that fP : forall x, P (f x). +- - Identity functions: +- id == NOTATION for the explicit identity function fun x => x. +- @id T == notation for the explicit identity at type T. +- idfun == an expression with a head constant, convertible to id; +- idfun x simplifies to x. +- @idfun T == the expression above, specialized to type T. +- phant_id x y == the function type phantom _ x -> phantom _ y. +- *** In addition to their casual use in functional programming, identity +- functions are often used to trigger static unification as part of the +- construction of dependent Records and Structures. For example, if we need +- a structure sT over a type T, we take as arguments T, sT, and a "dummy" +- function T -> sort sT: +- Definition foo T sT & T -> sort sT := ... +- We can avoid specifying sT directly by calling foo (@id T), or specify +- the call completely while still ensuring the consistency of T and sT, by +- calling @foo T sT idfun. The phant_id type allows us to extend this trick +- to non-Type canonical projections. It also allows us to sidestep +- dependent type constraints when building explicit records, e.g., given +- Record r := R {x; y : T(x)}. +- if we need to build an r from a given y0 while inferring some x0, such +- that y0 : T(x0), we pose +- Definition mk_r .. y .. (x := ...) y' & phant_id y y' := R x y'. +- Calling @mk_r .. y0 .. id will cause Coq to use y' := y0, while checking +- the dependent type constraint y0 : T(x0). +- +- - Extensional equality for functions and relations (i.e. functions of two +- arguments): +- f1 =1 f2 == f1 x is equal to f2 x for all x. +- f1 =1 f2 :> A == ... and f2 is explicitly typed. +- f1 =2 f2 == f1 x y is equal to f2 x y for all x y. +- f1 =2 f2 :> A == ... and f2 is explicitly typed. +- +- - Composition for total and partial functions: +- f^~ y == function f with second argument specialised to y, +- i.e., fun x => f x y +- CAVEAT: conditional (non-maximal) implicit arguments +- of f are NOT inserted in this context +- @^~ x == application at x, i.e., fun f => f x +- #[#eta f#]# == the explicit eta-expansion of f, i.e., fun x => f x +- CAVEAT: conditional (non-maximal) implicit arguments +- of f are NOT inserted in this context. +- fun=> v := the constant function fun _ => v. +- f1 \o f2 == composition of f1 and f2. +- Note: (f1 \o f2) x simplifies to f1 (f2 x). +- f1 \; f2 == categorical composition of f1 and f2. This expands to +- to f2 \o f1 and (f1 \; f2) x simplifies to f2 (f1 x). +- pcomp f1 f2 == composition of partial functions f1 and f2. +- +- +- - Properties of functions: +- injective f <-> f is injective. +- cancel f g <-> g is a left inverse of f / f is a right inverse of g. +- pcancel f g <-> g is a left inverse of f where g is partial. +- ocancel f g <-> g is a left inverse of f where f is partial. +- bijective f <-> f is bijective (has a left and right inverse). +- involutive f <-> f is involutive. +- +- - Properties for operations. +- left_id e op <-> e is a left identity for op (e op x = x). +- right_id e op <-> e is a right identity for op (x op e = x). +- left_inverse e inv op <-> inv is a left inverse for op wrt identity e, +- i.e., (inv x) op x = e. +- right_inverse e inv op <-> inv is a right inverse for op wrt identity e +- i.e., x op (i x) = e. +- self_inverse e op <-> each x is its own op-inverse (x op x = e). +- idempotent op <-> op is idempotent for op (x op x = x). +- associative op <-> op is associative, i.e., +- x op (y op z) = (x op y) op z. +- commutative op <-> op is commutative (x op y = y op x). +- left_commutative op <-> op is left commutative, i.e., +- x op (y op z) = y op (x op z). +- right_commutative op <-> op is right commutative, i.e., +- (x op y) op z = (x op z) op y. +- left_zero z op <-> z is a left zero for op (z op x = z). +- right_zero z op <-> z is a right zero for op (x op z = z). +- left_distributive op1 op2 <-> op1 distributes over op2 to the left: +- (x op2 y) op1 z = (x op1 z) op2 (y op1 z). +- right_distributive op1 op2 <-> op distributes over add to the right: +- x op1 (y op2 z) = (x op1 z) op2 (x op1 z). +- interchange op1 op2 <-> op1 and op2 satisfy an interchange law: +- (x op2 y) op1 (z op2 t) = (x op1 z) op2 (y op1 t). +- Note that interchange op op is a commutativity property. +- left_injective op <-> op is injective in its left argument: +- x op y = z op y -> x = z. +- right_injective op <-> op is injective in its right argument: +- x op y = x op z -> y = z. +- left_loop inv op <-> op, inv obey the inverse loop left axiom: +- (inv x) op (x op y) = y for all x, y, i.e., +- op (inv x) is always a left inverse of op x +- rev_left_loop inv op <-> op, inv obey the inverse loop reverse left +- axiom: x op ((inv x) op y) = y, for all x, y. +- right_loop inv op <-> op, inv obey the inverse loop right axiom: +- (x op y) op (inv y) = x for all x, y. +- rev_right_loop inv op <-> op, inv obey the inverse loop reverse right +- axiom: (x op y) op (inv y) = x for all x, y. +- Note that familiar "cancellation" identities like x + y - y = x or +- x - y + y = x are respectively instances of right_loop and rev_right_loop +- The corresponding lemmas will use the K and NK/VK suffixes, respectively. +- +- - Morphisms for functions and relations: +- {morph f : x / a >-> r} <-> f is a morphism with respect to functions +- (fun x => a) and (fun x => r); if r == R#[#x#]#, +- this states that f a = R#[#f x#]# for all x. +- {morph f : x / a} <-> f is a morphism with respect to the +- function expression (fun x => a). This is +- shorthand for {morph f : x / a >-> a}; note +- that the two instances of a are often +- interpreted at different types. +- {morph f : x y / a >-> r} <-> f is a morphism with respect to functions +- (fun x y => a) and (fun x y => r). +- {morph f : x y / a} <-> f is a morphism with respect to the +- function expression (fun x y => a). +- {homo f : x / a >-> r} <-> f is a homomorphism with respect to the +- predicates (fun x => a) and (fun x => r); +- if r == R#[#x#]#, this states that a -> R#[#f x#]# +- for all x. +- {homo f : x / a} <-> f is a homomorphism with respect to the +- predicate expression (fun x => a). +- {homo f : x y / a >-> r} <-> f is a homomorphism with respect to the +- relations (fun x y => a) and (fun x y => r). +- {homo f : x y / a} <-> f is a homomorphism with respect to the +- relation expression (fun x y => a). +- {mono f : x / a >-> r} <-> f is monotone with respect to projectors +- (fun x => a) and (fun x => r); if r == R#[#x#]#, +- this states that R#[#f x#]# = a for all x. +- {mono f : x / a} <-> f is monotone with respect to the projector +- expression (fun x => a). +- {mono f : x y / a >-> r} <-> f is monotone with respect to relators +- (fun x y => a) and (fun x y => r). +- {mono f : x y / a} <-> f is monotone with respect to the relator +- expression (fun x y => a). +- +- The file also contains some basic lemmas for the above concepts. +- Lemmas relative to cancellation laws use some abbreviated suffixes: +- K - a cancellation rule like esymK : cancel (@esym T x y) (@esym T y x). +- LR - a lemma moving an operation from the left hand side of a relation to +- the right hand side, like canLR: cancel g f -> x = g y -> f x = y. +- RL - a lemma moving an operation from the right to the left, e.g., canRL. +- Beware that the LR and RL orientations refer to an "apply" (back chaining) +- usage; when using the same lemmas with "have" or "move" (forward chaining) +- the directions will be reversed!. **) +- +- +-Set Implicit Arguments. +-Unset Strict Implicit. +-Unset Printing Implicit Defensive. +- +-Delimit Scope fun_scope with FUN. +-Open Scope fun_scope. +- +-(** Notations for argument transpose **) +-Notation "f ^~ y" := (fun x => f x y) +- (at level 10, y at level 8, no associativity, format "f ^~ y") : fun_scope. +-Notation "@^~ x" := (fun f => f x) +- (at level 10, x at level 8, no associativity, format "@^~ x") : fun_scope. +- +-Delimit Scope pair_scope with PAIR. +-Open Scope pair_scope. +- +-(** Notations for pair/conjunction projections **) +-Notation "p .1" := (fst p) +- (at level 2, left associativity, format "p .1") : pair_scope. +-Notation "p .2" := (snd p) +- (at level 2, left associativity, format "p .2") : pair_scope. +- +-Coercion pair_of_and P Q (PandQ : P /\ Q) := (proj1 PandQ, proj2 PandQ). +- +-Definition all_pair I T U (w : forall i : I, T i * U i) := +- (fun i => (w i).1, fun i => (w i).2). +- +-(** +- Complements on the option type constructor, used below to +- encode partial functions. **) +- +-Module Option. +- +-Definition apply aT rT (f : aT -> rT) x u := if u is Some y then f y else x. +- +-Definition default T := apply (fun x : T => x). +- +-Definition bind aT rT (f : aT -> option rT) := apply f None. +- +-Definition map aT rT (f : aT -> rT) := bind (fun x => Some (f x)). +- +-End Option. +- +-Notation oapp := Option.apply. +-Notation odflt := Option.default. +-Notation obind := Option.bind. +-Notation omap := Option.map. +-Notation some := (@Some _) (only parsing). +- +-(** Shorthand for some basic equality lemmas. **) +- +-Notation erefl := refl_equal. +-Notation ecast i T e x := (let: erefl in _ = i := e return T in x). +-Definition esym := sym_eq. +-Definition nesym := sym_not_eq. +-Definition etrans := trans_eq. +-Definition congr1 := f_equal. +-Definition congr2 := f_equal2. +-(** Force at least one implicit when used as a view. **) +-Prenex Implicits esym nesym. +- +-(** A predicate for singleton types. **) +-Definition all_equal_to T (x0 : T) := forall x, unkeyed x = x0. +- +-Lemma unitE : all_equal_to tt. Proof. by case. Qed. +- +-(** A generic wrapper type **) +- +-Structure wrapped T := Wrap {unwrap : T}. +-Canonical wrap T x := @Wrap T x. +- +-Prenex Implicits unwrap wrap Wrap. +- +-(** +- Syntax for defining auxiliary recursive function. +- Usage: +- Section FooDefinition. +- Variables (g1 : T1) (g2 : T2). (globals) +- Fixoint foo_auxiliary (a3 : T3) ... := +- body, using #[#rec e3, ... #]# for recursive calls +- where " #[# 'rec' a3 , a4 , ... #]#" := foo_auxiliary. +- Definition foo x y .. := #[#rec e1, ... #]#. +- + proofs about foo +- End FooDefinition. **) +- +-Reserved Notation "[ 'rec' a0 ]" +- (at level 0, format "[ 'rec' a0 ]"). +-Reserved Notation "[ 'rec' a0 , a1 ]" +- (at level 0, format "[ 'rec' a0 , a1 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 ]" +- (at level 0, format "[ 'rec' a0 , a1 , a2 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 ]" +- (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 ]" +- (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]" +- (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]" +- (at level 0, format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]" +- (at level 0, +- format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]" +- (at level 0, +- format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 ]"). +-Reserved Notation "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]" +- (at level 0, +- format "[ 'rec' a0 , a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 ]"). +- +-(** +- Definitions and notation for explicit functions with simplification, +- i.e., which simpl and /= beta expand (this is complementary to nosimpl). **) +- +-Section SimplFun. +- +-Variables aT rT : Type. +- +-Variant simpl_fun := SimplFun of aT -> rT. +- +-Definition fun_of_simpl f := fun x => let: SimplFun lam := f in lam x. +- +-Coercion fun_of_simpl : simpl_fun >-> Funclass. +- +-End SimplFun. +- +-Notation "[ 'fun' : T => E ]" := (SimplFun (fun _ : T => E)) +- (at level 0, +- format "'[hv' [ 'fun' : T => '/ ' E ] ']'") : fun_scope. +- +-Notation "[ 'fun' x => E ]" := (SimplFun (fun x => E)) +- (at level 0, x ident, +- format "'[hv' [ 'fun' x => '/ ' E ] ']'") : fun_scope. +- +-Notation "[ 'fun' x : T => E ]" := (SimplFun (fun x : T => E)) +- (at level 0, x ident, only parsing) : fun_scope. +- +-Notation "[ 'fun' x y => E ]" := (fun x => [fun y => E]) +- (at level 0, x ident, y ident, +- format "'[hv' [ 'fun' x y => '/ ' E ] ']'") : fun_scope. +- +-Notation "[ 'fun' x y : T => E ]" := (fun x : T => [fun y : T => E]) +- (at level 0, x ident, y ident, only parsing) : fun_scope. +- +-Notation "[ 'fun' ( x : T ) y => E ]" := (fun x : T => [fun y => E]) +- (at level 0, x ident, y ident, only parsing) : fun_scope. +- +-Notation "[ 'fun' x ( y : T ) => E ]" := (fun x => [fun y : T => E]) +- (at level 0, x ident, y ident, only parsing) : fun_scope. +- +-Notation "[ 'fun' ( x : xT ) ( y : yT ) => E ]" := +- (fun x : xT => [fun y : yT => E]) +- (at level 0, x ident, y ident, only parsing) : fun_scope. +- +-(** For delta functions in eqtype.v. **) +-Definition SimplFunDelta aT rT (f : aT -> aT -> rT) := [fun z => f z z]. +- +-(** +- Extensional equality, for unary and binary functions, including syntactic +- sugar. **) +- +-Section ExtensionalEquality. +- +-Variables A B C : Type. +- +-Definition eqfun (f g : B -> A) : Prop := forall x, f x = g x. +- +-Definition eqrel (r s : C -> B -> A) : Prop := forall x y, r x y = s x y. +- +-Lemma frefl f : eqfun f f. Proof. by []. Qed. +-Lemma fsym f g : eqfun f g -> eqfun g f. Proof. by move=> eq_fg x. Qed. +- +-Lemma ftrans f g h : eqfun f g -> eqfun g h -> eqfun f h. +-Proof. by move=> eq_fg eq_gh x; rewrite eq_fg. Qed. +- +-Lemma rrefl r : eqrel r r. Proof. by []. Qed. +- +-End ExtensionalEquality. +- +-Typeclasses Opaque eqfun. +-Typeclasses Opaque eqrel. +- +-Hint Resolve frefl rrefl. +- +-Notation "f1 =1 f2" := (eqfun f1 f2) +- (at level 70, no associativity) : fun_scope. +-Notation "f1 =1 f2 :> A" := (f1 =1 (f2 : A)) +- (at level 70, f2 at next level, A at level 90) : fun_scope. +-Notation "f1 =2 f2" := (eqrel f1 f2) +- (at level 70, no associativity) : fun_scope. +-Notation "f1 =2 f2 :> A" := (f1 =2 (f2 : A)) +- (at level 70, f2 at next level, A at level 90) : fun_scope. +- +-Section Composition. +- +-Variables A B C : Type. +- +-Definition funcomp u (f : B -> A) (g : C -> B) x := let: tt := u in f (g x). +-Definition catcomp u g f := funcomp u f g. +-Local Notation comp := (funcomp tt). +- +-Definition pcomp (f : B -> option A) (g : C -> option B) x := obind f (g x). +- +-Lemma eq_comp f f' g g' : f =1 f' -> g =1 g' -> comp f g =1 comp f' g'. +-Proof. by move=> eq_ff' eq_gg' x; rewrite /= eq_gg' eq_ff'. Qed. +- +-End Composition. +- +-Notation comp := (funcomp tt). +-Notation "@ 'comp'" := (fun A B C => @funcomp A B C tt). +-Notation "f1 \o f2" := (comp f1 f2) +- (at level 50, format "f1 \o '/ ' f2") : fun_scope. +-Notation "f1 \; f2" := (catcomp tt f1 f2) +- (at level 60, right associativity, format "f1 \; '/ ' f2") : fun_scope. +- +-Notation "[ 'eta' f ]" := (fun x => f x) +- (at level 0, format "[ 'eta' f ]") : fun_scope. +- +-Notation "'fun' => E" := (fun _ => E) (at level 200, only parsing) : fun_scope. +- +-Notation id := (fun x => x). +-Notation "@ 'id' T" := (fun x : T => x) +- (at level 10, T at level 8, only parsing) : fun_scope. +- +-Definition id_head T u x : T := let: tt := u in x. +-Definition explicit_id_key := tt. +-Notation idfun := (id_head tt). +-Notation "@ 'idfun' T " := (@id_head T explicit_id_key) +- (at level 10, T at level 8, format "@ 'idfun' T") : fun_scope. +- +-Definition phant_id T1 T2 v1 v2 := phantom T1 v1 -> phantom T2 v2. +- +-(** Strong sigma types. **) +- +-Section Tag. +- +-Variables (I : Type) (i : I) (T_ U_ : I -> Type). +- +-Definition tag := projT1. +-Definition tagged : forall w, T_(tag w) := @projT2 I [eta T_]. +-Definition Tagged x := @existT I [eta T_] i x. +- +-Definition tag2 (w : @sigT2 I T_ U_) := let: existT2 _ _ i _ _ := w in i. +-Definition tagged2 w : T_(tag2 w) := let: existT2 _ _ _ x _ := w in x. +-Definition tagged2' w : U_(tag2 w) := let: existT2 _ _ _ _ y := w in y. +-Definition Tagged2 x y := @existT2 I [eta T_] [eta U_] i x y. +- +-End Tag. +- +-Arguments Tagged [I i]. +-Arguments Tagged2 [I i]. +-Prenex Implicits tag tagged Tagged tag2 tagged2 tagged2' Tagged2. +- +-Coercion tag_of_tag2 I T_ U_ (w : @sigT2 I T_ U_) := +- Tagged (fun i => T_ i * U_ i)%type (tagged2 w, tagged2' w). +- +-Lemma all_tag I T U : +- (forall x : I, {y : T x & U x y}) -> +- {f : forall x, T x & forall x, U x (f x)}. +-Proof. by move=> fP; exists (fun x => tag (fP x)) => x; case: (fP x). Qed. +- +-Lemma all_tag2 I T U V : +- (forall i : I, {y : T i & U i y & V i y}) -> +- {f : forall i, T i & forall i, U i (f i) & forall i, V i (f i)}. +-Proof. by case/all_tag=> f /all_pair[]; exists f. Qed. +- +-(** Refinement types. **) +- +-(** Prenex Implicits and renaming. **) +-Notation sval := (@proj1_sig _ _). +-Notation "@ 'sval'" := (@proj1_sig) (at level 10, format "@ 'sval'"). +- +-Section Sig. +- +-Variables (T : Type) (P Q : T -> Prop). +- +-Lemma svalP (u : sig P) : P (sval u). Proof. by case: u. Qed. +- +-Definition s2val (u : sig2 P Q) := let: exist2 _ _ x _ _ := u in x. +- +-Lemma s2valP u : P (s2val u). Proof. by case: u. Qed. +- +-Lemma s2valP' u : Q (s2val u). Proof. by case: u. Qed. +- +-End Sig. +- +-Prenex Implicits svalP s2val s2valP s2valP'. +- +-Coercion tag_of_sig I P (u : @sig I P) := Tagged P (svalP u). +- +-Coercion sig_of_sig2 I P Q (u : @sig2 I P Q) := +- exist (fun i => P i /\ Q i) (s2val u) (conj (s2valP u) (s2valP' u)). +- +-Lemma all_sig I T P : +- (forall x : I, {y : T x | P x y}) -> +- {f : forall x, T x | forall x, P x (f x)}. +-Proof. by case/all_tag=> f; exists f. Qed. +- +-Lemma all_sig2 I T P Q : +- (forall x : I, {y : T x | P x y & Q x y}) -> +- {f : forall x, T x | forall x, P x (f x) & forall x, Q x (f x)}. +-Proof. by case/all_sig=> f /all_pair[]; exists f. Qed. +- +-Section Morphism. +- +-Variables (aT rT sT : Type) (f : aT -> rT). +- +-(** Morphism property for unary and binary functions **) +-Definition morphism_1 aF rF := forall x, f (aF x) = rF (f x). +-Definition morphism_2 aOp rOp := forall x y, f (aOp x y) = rOp (f x) (f y). +- +-(** Homomorphism property for unary and binary relations **) +-Definition homomorphism_1 (aP rP : _ -> Prop) := forall x, aP x -> rP (f x). +-Definition homomorphism_2 (aR rR : _ -> _ -> Prop) := +- forall x y, aR x y -> rR (f x) (f y). +- +-(** Stability property for unary and binary relations **) +-Definition monomorphism_1 (aP rP : _ -> sT) := forall x, rP (f x) = aP x. +-Definition monomorphism_2 (aR rR : _ -> _ -> sT) := +- forall x y, rR (f x) (f y) = aR x y. +- +-End Morphism. +- +-Notation "{ 'morph' f : x / a >-> r }" := +- (morphism_1 f (fun x => a) (fun x => r)) +- (at level 0, f at level 99, x ident, +- format "{ 'morph' f : x / a >-> r }") : type_scope. +- +-Notation "{ 'morph' f : x / a }" := +- (morphism_1 f (fun x => a) (fun x => a)) +- (at level 0, f at level 99, x ident, +- format "{ 'morph' f : x / a }") : type_scope. +- +-Notation "{ 'morph' f : x y / a >-> r }" := +- (morphism_2 f (fun x y => a) (fun x y => r)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'morph' f : x y / a >-> r }") : type_scope. +- +-Notation "{ 'morph' f : x y / a }" := +- (morphism_2 f (fun x y => a) (fun x y => a)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'morph' f : x y / a }") : type_scope. +- +-Notation "{ 'homo' f : x / a >-> r }" := +- (homomorphism_1 f (fun x => a) (fun x => r)) +- (at level 0, f at level 99, x ident, +- format "{ 'homo' f : x / a >-> r }") : type_scope. +- +-Notation "{ 'homo' f : x / a }" := +- (homomorphism_1 f (fun x => a) (fun x => a)) +- (at level 0, f at level 99, x ident, +- format "{ 'homo' f : x / a }") : type_scope. +- +-Notation "{ 'homo' f : x y / a >-> r }" := +- (homomorphism_2 f (fun x y => a) (fun x y => r)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'homo' f : x y / a >-> r }") : type_scope. +- +-Notation "{ 'homo' f : x y / a }" := +- (homomorphism_2 f (fun x y => a) (fun x y => a)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'homo' f : x y / a }") : type_scope. +- +-Notation "{ 'homo' f : x y /~ a }" := +- (homomorphism_2 f (fun y x => a) (fun x y => a)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'homo' f : x y /~ a }") : type_scope. +- +-Notation "{ 'mono' f : x / a >-> r }" := +- (monomorphism_1 f (fun x => a) (fun x => r)) +- (at level 0, f at level 99, x ident, +- format "{ 'mono' f : x / a >-> r }") : type_scope. +- +-Notation "{ 'mono' f : x / a }" := +- (monomorphism_1 f (fun x => a) (fun x => a)) +- (at level 0, f at level 99, x ident, +- format "{ 'mono' f : x / a }") : type_scope. +- +-Notation "{ 'mono' f : x y / a >-> r }" := +- (monomorphism_2 f (fun x y => a) (fun x y => r)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'mono' f : x y / a >-> r }") : type_scope. +- +-Notation "{ 'mono' f : x y / a }" := +- (monomorphism_2 f (fun x y => a) (fun x y => a)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'mono' f : x y / a }") : type_scope. +- +-Notation "{ 'mono' f : x y /~ a }" := +- (monomorphism_2 f (fun y x => a) (fun x y => a)) +- (at level 0, f at level 99, x ident, y ident, +- format "{ 'mono' f : x y /~ a }") : type_scope. +- +-(** +- In an intuitionistic setting, we have two degrees of injectivity. The +- weaker one gives only simplification, and the strong one provides a left +- inverse (we show in `fintype' that they coincide for finite types). +- We also define an intermediate version where the left inverse is only a +- partial function. **) +- +-Section Injections. +- +-(** +- rT must come first so we can use @ to mitigate the Coq 1st order +- unification bug (e..g., Coq can't infer rT from a "cancel" lemma). **) +-Variables (rT aT : Type) (f : aT -> rT). +- +-Definition injective := forall x1 x2, f x1 = f x2 -> x1 = x2. +- +-Definition cancel g := forall x, g (f x) = x. +- +-Definition pcancel g := forall x, g (f x) = Some x. +- +-Definition ocancel (g : aT -> option rT) h := forall x, oapp h x (g x) = x. +- +-Lemma can_pcan g : cancel g -> pcancel (fun y => Some (g y)). +-Proof. by move=> fK x; congr (Some _). Qed. +- +-Lemma pcan_inj g : pcancel g -> injective. +-Proof. by move=> fK x y /(congr1 g); rewrite !fK => [[]]. Qed. +- +-Lemma can_inj g : cancel g -> injective. +-Proof. by move/can_pcan; apply: pcan_inj. Qed. +- +-Lemma canLR g x y : cancel g -> x = f y -> g x = y. +-Proof. by move=> fK ->. Qed. +- +-Lemma canRL g x y : cancel g -> f x = y -> x = g y. +-Proof. by move=> fK <-. Qed. +- +-End Injections. +- +-Lemma Some_inj {T} : injective (@Some T). Proof. by move=> x y []. Qed. +- +-(** Force implicits to use as a view. **) +-Prenex Implicits Some_inj. +- +-(** cancellation lemmas for dependent type casts. **) +-Lemma esymK T x y : cancel (@esym T x y) (@esym T y x). +-Proof. by case: y /. Qed. +- +-Lemma etrans_id T x y (eqxy : x = y :> T) : etrans (erefl x) eqxy = eqxy. +-Proof. by case: y / eqxy. Qed. +- +-Section InjectionsTheory. +- +-Variables (A B C : Type) (f g : B -> A) (h : C -> B). +- +-Lemma inj_id : injective (@id A). +-Proof. by []. Qed. +- +-Lemma inj_can_sym f' : cancel f f' -> injective f' -> cancel f' f. +-Proof. by move=> fK injf' x; apply: injf'. Qed. +- +-Lemma inj_comp : injective f -> injective h -> injective (f \o h). +-Proof. by move=> injf injh x y /injf; apply: injh. Qed. +- +-Lemma can_comp f' h' : cancel f f' -> cancel h h' -> cancel (f \o h) (h' \o f'). +-Proof. by move=> fK hK x; rewrite /= fK hK. Qed. +- +-Lemma pcan_pcomp f' h' : +- pcancel f f' -> pcancel h h' -> pcancel (f \o h) (pcomp h' f'). +-Proof. by move=> fK hK x; rewrite /pcomp fK /= hK. Qed. +- +-Lemma eq_inj : injective f -> f =1 g -> injective g. +-Proof. by move=> injf eqfg x y; rewrite -2!eqfg; apply: injf. Qed. +- +-Lemma eq_can f' g' : cancel f f' -> f =1 g -> f' =1 g' -> cancel g g'. +-Proof. by move=> fK eqfg eqfg' x; rewrite -eqfg -eqfg'. Qed. +- +-Lemma inj_can_eq f' : cancel f f' -> injective f' -> cancel g f' -> f =1 g. +-Proof. by move=> fK injf' gK x; apply: injf'; rewrite fK. Qed. +- +-End InjectionsTheory. +- +-Section Bijections. +- +-Variables (A B : Type) (f : B -> A). +- +-Variant bijective : Prop := Bijective g of cancel f g & cancel g f. +- +-Hypothesis bijf : bijective. +- +-Lemma bij_inj : injective f. +-Proof. by case: bijf => g fK _; apply: can_inj fK. Qed. +- +-Lemma bij_can_sym f' : cancel f' f <-> cancel f f'. +-Proof. +-split=> fK; first exact: inj_can_sym fK bij_inj. +-by case: bijf => h _ hK x; rewrite -[x]hK fK. +-Qed. +- +-Lemma bij_can_eq f' f'' : cancel f f' -> cancel f f'' -> f' =1 f''. +-Proof. +-by move=> fK fK'; apply: (inj_can_eq _ bij_inj); apply/bij_can_sym. +-Qed. +- +-End Bijections. +- +-Section BijectionsTheory. +- +-Variables (A B C : Type) (f : B -> A) (h : C -> B). +- +-Lemma eq_bij : bijective f -> forall g, f =1 g -> bijective g. +-Proof. by case=> f' fK f'K g eqfg; exists f'; eapply eq_can; eauto. Qed. +- +-Lemma bij_comp : bijective f -> bijective h -> bijective (f \o h). +-Proof. +-by move=> [f' fK f'K] [h' hK h'K]; exists (h' \o f'); apply: can_comp; auto. +-Qed. +- +-Lemma bij_can_bij : bijective f -> forall f', cancel f f' -> bijective f'. +-Proof. by move=> bijf; exists f; first by apply/(bij_can_sym bijf). Qed. +- +-End BijectionsTheory. +- +-Section Involutions. +- +-Variables (A : Type) (f : A -> A). +- +-Definition involutive := cancel f f. +- +-Hypothesis Hf : involutive. +- +-Lemma inv_inj : injective f. Proof. exact: can_inj Hf. Qed. +-Lemma inv_bij : bijective f. Proof. by exists f. Qed. +- +-End Involutions. +- +-Section OperationProperties. +- +-Variables S T R : Type. +- +-Section SopTisR. +-Implicit Type op : S -> T -> R. +-Definition left_inverse e inv op := forall x, op (inv x) x = e. +-Definition right_inverse e inv op := forall x, op x (inv x) = e. +-Definition left_injective op := forall x, injective (op^~ x). +-Definition right_injective op := forall y, injective (op y). +-End SopTisR. +- +- +-Section SopTisS. +-Implicit Type op : S -> T -> S. +-Definition right_id e op := forall x, op x e = x. +-Definition left_zero z op := forall x, op z x = z. +-Definition right_commutative op := forall x y z, op (op x y) z = op (op x z) y. +-Definition left_distributive op add := +- forall x y z, op (add x y) z = add (op x z) (op y z). +-Definition right_loop inv op := forall y, cancel (op^~ y) (op^~ (inv y)). +-Definition rev_right_loop inv op := forall y, cancel (op^~ (inv y)) (op^~ y). +-End SopTisS. +- +-Section SopTisT. +-Implicit Type op : S -> T -> T. +-Definition left_id e op := forall x, op e x = x. +-Definition right_zero z op := forall x, op x z = z. +-Definition left_commutative op := forall x y z, op x (op y z) = op y (op x z). +-Definition right_distributive op add := +- forall x y z, op x (add y z) = add (op x y) (op x z). +-Definition left_loop inv op := forall x, cancel (op x) (op (inv x)). +-Definition rev_left_loop inv op := forall x, cancel (op (inv x)) (op x). +-End SopTisT. +- +-Section SopSisT. +-Implicit Type op : S -> S -> T. +-Definition self_inverse e op := forall x, op x x = e. +-Definition commutative op := forall x y, op x y = op y x. +-End SopSisT. +- +-Section SopSisS. +-Implicit Type op : S -> S -> S. +-Definition idempotent op := forall x, op x x = x. +-Definition associative op := forall x y z, op x (op y z) = op (op x y) z. +-Definition interchange op1 op2 := +- forall x y z t, op1 (op2 x y) (op2 z t) = op2 (op1 x z) (op1 y t). +-End SopSisS. +- +-End OperationProperties. +- +- +- +- +- +- +- +- +- +- +--- a/plugins/ssrmatching/ssrmatching.v ++++ /dev/null +@@ -1,36 +0,0 @@ +-(************************************************************************) +-(* * The Coq Proof Assistant / The Coq Development Team *) +-(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +-(* <O___,, * (see CREDITS file for the list of authors) *) +-(* \VV/ **************************************************************) +-(* // * This file is distributed under the terms of the *) +-(* * GNU Lesser General Public License Version 2.1 *) +-(* * (see LICENSE file for the text of the license) *) +-(************************************************************************) +- +-(* (c) Copyright 2006-2015 Microsoft Corporation and Inria. *) +- +-Declare ML Module "ssrmatching_plugin". +- +-Module SsrMatchingSyntax. +- +-(* Reserve the notation for rewrite patterns so that the user is not allowed *) +-(* to declare it at a different level. *) +-Reserved Notation "( a 'in' b )" (at level 0). +-Reserved Notation "( a 'as' b )" (at level 0). +-Reserved Notation "( a 'in' b 'in' c )" (at level 0). +-Reserved Notation "( a 'as' b 'in' c )" (at level 0). +- +-(* Notation to define shortcuts for the "X in t" part of a pattern. *) +-Notation "( X 'in' t )" := (_ : fun X => t) : ssrpatternscope. +-Delimit Scope ssrpatternscope with pattern. +- +-(* Some shortcuts for recurrent "X in t" parts. *) +-Notation RHS := (X in _ = X)%pattern. +-Notation LHS := (X in X = _)%pattern. +- +-End SsrMatchingSyntax. +- +-Export SsrMatchingSyntax. +- +-Tactic Notation "ssrpattern" ssrpatternarg(p) := ssrpattern p . +--- a/test-suite/Makefile ++++ b/test-suite/Makefile +@@ -99,7 +99,7 @@ + + VSUBSYSTEMS := prerequisite success failure $(BUGS) output \ + output-modulo-time $(INTERACTIVE) micromega $(COMPLEXITY) modules stm \ +- coqdoc ssr ++ coqdoc + + # All subsystems + SUBSYSTEMS := $(VSUBSYSTEMS) misc bugs ide vio coqchk coqwc coq-makefile tools unit-tests +@@ -167,7 +167,6 @@ + $(call summary_dir, "Complexity tests", complexity); \ + $(call summary_dir, "Module tests", modules); \ + $(call summary_dir, "STM tests", stm); \ +- $(call summary_dir, "SSR tests", ssr); \ + $(call summary_dir, "IDE tests", ide); \ + $(call summary_dir, "VI tests", vio); \ + $(call summary_dir, "Coqchk tests", coqchk); \ +--- a/test-suite/bugs/closed/2800.v ++++ /dev/null +@@ -1,19 +0,0 @@ +-Goal False. +- +-intuition +- match goal with +- | |- _ => idtac " foo" +- end. +- +- lazymatch goal with _ => idtac end. +- match goal with _ => idtac end. +- unshelve lazymatch goal with _ => idtac end. +- unshelve match goal with _ => idtac end. +- unshelve (let x := I in idtac). +-Abort. +- +-Require Import ssreflect. +- +-Goal True. +-match goal with _ => idtac end => //. +-Qed. +--- a/test-suite/bugs/closed/5692.v ++++ /dev/null +@@ -1,88 +0,0 @@ +-Set Primitive Projections. +-Require Import ZArith ssreflect. +- +-Module Test1. +- +-Structure semigroup := SemiGroup { +- sg_car :> Type; +- sg_op : sg_car -> sg_car -> sg_car; +-}. +- +-Structure monoid := Monoid { +- monoid_car :> Type; +- monoid_op : monoid_car -> monoid_car -> monoid_car; +- monoid_unit : monoid_car; +-}. +- +-Coercion monoid_sg (X : monoid) : semigroup := +- SemiGroup (monoid_car X) (monoid_op X). +-Canonical Structure monoid_sg. +- +-Parameter X : monoid. +-Parameter x y : X. +- +-Check (sg_op _ x y). +- +-End Test1. +- +-Module Test2. +- +-Structure semigroup := SemiGroup { +- sg_car :> Type; +- sg_op : sg_car -> sg_car -> sg_car; +-}. +- +-Structure monoid := Monoid { +- monoid_car :> Type; +- monoid_op : monoid_car -> monoid_car -> monoid_car; +- monoid_unit : monoid_car; +- monoid_left_id x : monoid_op monoid_unit x = x; +-}. +- +-Coercion monoid_sg (X : monoid) : semigroup := +- SemiGroup (monoid_car X) (monoid_op X). +-Canonical Structure monoid_sg. +- +-Canonical Structure nat_sg := SemiGroup nat plus. +-Canonical Structure nat_monoid := Monoid nat plus 0 plus_O_n. +- +-Lemma foo (x : nat) : 0 + x = x. +-Proof. +-apply monoid_left_id. +-Qed. +- +-End Test2. +- +-Module Test3. +- +-Structure semigroup := SemiGroup { +- sg_car :> Type; +- sg_op : sg_car -> sg_car -> sg_car; +-}. +- +-Structure group := Something { +- group_car :> Type; +- group_op : group_car -> group_car -> group_car; +- group_neg : group_car -> group_car; +- group_neg_op' x y : group_neg (group_op x y) = group_op (group_neg x) (group_neg y) +-}. +- +-Coercion group_sg (X : group) : semigroup := +- SemiGroup (group_car X) (group_op X). +-Canonical Structure group_sg. +- +-Axiom group_neg_op : forall (X : group) (x y : X), +- group_neg X (sg_op (group_sg X) x y) = sg_op (group_sg X) (group_neg X x) (group_neg X y). +- +-Canonical Structure Z_sg := SemiGroup Z Z.add . +-Canonical Structure Z_group := Something Z Z.add Z.opp Z.opp_add_distr. +- +-Lemma foo (x y : Z) : +- sg_op Z_sg (group_neg Z_group x) (group_neg Z_group y) = +- group_neg Z_group (sg_op Z_sg x y). +-Proof. +- rewrite -group_neg_op. +- reflexivity. +-Qed. +- +-End Test3. +--- a/test-suite/bugs/closed/6634.v ++++ /dev/null +@@ -1,6 +0,0 @@ +-From Coq Require Import ssreflect. +- +-Lemma normalizeP (p : tt = tt) : p = p. +-Proof. +-Fail move: {2} tt p. +-Abort. +--- a/test-suite/bugs/closed/6910.v ++++ /dev/null +@@ -1,5 +0,0 @@ +-From Coq Require Import ssreflect ssrfun. +- +-(* We should be able to use Some_inj as a view: *) +-Lemma foo (x y : nat) : Some x = Some y -> x = y. +-Proof. by move/Some_inj. Qed. +--- a/test-suite/bugs/closed/bug_8544.v ++++ /dev/null +@@ -1,6 +0,0 @@ +-Require Import ssreflect. +-Goal True \/ True -> False. +-Proof. +-(* the following should fail: 2 subgoals, but only one intro pattern *) +-Fail case => [a]. +-Abort. +--- a/test-suite/output/ssr_clear.v ++++ /dev/null +@@ -1,6 +0,0 @@ +-Require Import ssreflect. +- +-Example foo : True -> True. +-Proof. +-Fail move=> {NO_SUCH_NAME}. +-Abort. +--- a/test-suite/output/ssr_explain_match.v ++++ /dev/null +@@ -1,23 +0,0 @@ +-(************************************************************************) +-(* * The Coq Proof Assistant / The Coq Development Team *) +-(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *) +-(* <O___,, * (see CREDITS file for the list of authors) *) +-(* \VV/ **************************************************************) +-(* // * This file is distributed under the terms of the *) +-(* * GNU Lesser General Public License Version 2.1 *) +-(* * (see LICENSE file for the text of the license) *) +-(************************************************************************) +- +-Require Import ssrmatching. +-Require Import ssreflect ssrbool TestSuite.ssr_mini_mathcomp. +- +-Definition addnAC := (addnA, addnC). +- +-Lemma test x y z : x + y + z = x + y. +- +-ssrinstancesoftpat (_ + _). +-ssrinstancesofruleL2R addnC. +-ssrinstancesofruleR2L addnA. +-ssrinstancesofruleR2L addnAC. +-Fail ssrinstancesoftpat (_ + _ in RHS). (* Not implemented *) +-Admitted. +--- a/test-suite/success/ssrpattern.v ++++ /dev/null +@@ -1,22 +0,0 @@ +-Require Import ssrmatching. +- +-(*Set Debug SsrMatching.*) +- +-Tactic Notation "at" "[" ssrpatternarg(pat) "]" tactic(t) := +- let name := fresh in +- let def_name := fresh in +- ssrpattern pat; +- intro name; +- pose proof (refl_equal name) as def_name; +- unfold name at 1 in def_name; +- t def_name; +- [ rewrite <- def_name | idtac.. ]; +- clear name def_name. +- +-Lemma test (H : True -> True -> 3 = 7) : 28 = 3 * 4. +-Proof. +-at [ X in X * 4 ] ltac:(fun place => rewrite -> H in place). +-- reflexivity. +-- trivial. +-- trivial. +-Qed. diff --git a/debian/patches/series b/debian/patches/series index 85f4fddd..3a1b8d0f 100644 --- a/debian/patches/series +++ b/debian/patches/series @@ -6,3 +6,4 @@ remove-tests-that-need-coqlib.patch avoid-usr-bin-env.patch python-scripts-libraries.patch skip-dot-pc.patch +remove-ssrmatching.patch |