diff options
Diffstat (limited to 'plugins/ssr/ssrfun.v')
| -rw-r--r-- | plugins/ssr/ssrfun.v | 796 |
1 files changed, 796 insertions, 0 deletions
diff --git a/plugins/ssr/ssrfun.v b/plugins/ssr/ssrfun.v new file mode 100644 index 00000000..ac2c7824 --- /dev/null +++ b/plugins/ssr/ssrfun.v @@ -0,0 +1,796 @@ +(************************************************************************) +(* * 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. *) + +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. + +CoInductive 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). + +CoInductive 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. + + + + + + + + + + |