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(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
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(*    //   *    This file is distributed under the terms of the         *)
(*         *     GNU Lesser General Public License Version 2.1          *)
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(* This file is (C) Copyright 2006-2015 Microsoft Corporation and Inria. *)
(** #<style> .doc { font-family: monospace; white-space: pre; } </style># **)
(** 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 olift f := Some \o f  - 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.  - The empty type: void == a notation for the Empty_set type of the standard library. of_void T == the canonical injection void -> T.  - 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 (inv y)) op 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.
(** Parsing / printing declarations. *)
Reserved Notation "f ^~ y" (at level 10, y at level 8, no associativity,
format "f ^~  y").
Reserved Notation "@^~ x" (at level 10, x at level 8, no associativity,
format "@^~  x").
Reserved Notation "[ 'eta' f ]" (at level 0, format "[ 'eta'  f ]").
Reserved Notation "'fun' => E" (at level 200, format "'fun' =>  E").
Reserved Notation "[ 'fun' : T => E ]" (at level 0,
format "'[hv' [ 'fun' :  T  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' x => E ]" (at level 0,
x name, format "'[hv' [ 'fun'  x  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' x : T => E ]" (at level 0,
x name, format "'[hv' [ 'fun'  x  :  T  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' x y => E ]" (at level 0,
x name, y name, format "'[hv' [ 'fun'  x  y  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' x y : T => E ]" (at level 0,
x name, y name, format "'[hv' [ 'fun'  x  y  :  T  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' ( x : T ) y => E ]" (at level 0,
x name, y name, format "'[hv' [ 'fun'  ( x  :  T )  y  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' x ( y : T ) => E ]" (at level 0,
x name, y name, format "'[hv' [ 'fun'  x  ( y  :  T )  => '/ '  E ] ']'").
Reserved Notation "[ 'fun' ( x : T ) ( y : U ) => E ]" (at level 0,
x name, y name, format "[ 'fun'  ( x  :  T )  ( y  :  U )  =>  E ]" ).
Reserved Notation "f =1 g" (at level 70, no associativity).
Reserved Notation "f =1 g :> A" (at level 70, g at next level, A at level 90).
Reserved Notation "f =2 g" (at level 70, no associativity).
Reserved Notation "f =2 g :> A" (at level 70, g at next level, A at level 90).
Reserved Notation "f \o g" (at level 50, format "f  \o '/ '  g").
Reserved Notation "f \; g" (at level 60, right associativity,
format "f  \; '/ '  g").
Reserved Notation "{ 'morph' f : x / a >-> r }" (at level 0, f at level 99,
x name, format "{ 'morph'  f  :  x  /  a  >->  r }").
Reserved Notation "{ 'morph' f : x / a }" (at level 0, f at level 99,
x name, format "{ 'morph'  f  :  x  /  a }").
Reserved Notation "{ 'morph' f : x y / a >-> r }" (at level 0, f at level 99,
x name, y name, format "{ 'morph'  f  :  x  y  /  a  >->  r }").
Reserved Notation "{ 'morph' f : x y / a }" (at level 0, f at level 99,
x name, y name, format "{ 'morph'  f  :  x  y  /  a }").
Reserved Notation "{ 'homo' f : x / a >-> r }" (at level 0, f at level 99,
x name, format "{ 'homo'  f  :  x  /  a  >->  r }").
Reserved Notation "{ 'homo' f : x / a }"  (at level 0, f at level 99,
x name, format "{ 'homo'  f  :  x  /  a }").
Reserved Notation "{ 'homo' f : x y / a >-> r }" (at level 0, f at level 99,
x name, y name, format "{ 'homo'  f  :  x  y  /  a  >->  r }").
Reserved Notation "{ 'homo' f : x y / a }" (at level 0, f at level 99,
x name, y name, format "{ 'homo'  f  :  x  y  /  a }").
Reserved Notation "{ 'homo' f : x y /~ a }" (at level 0, f at level 99,
x name, y name, format "{ 'homo'  f  :  x  y  /~  a }").
Reserved Notation "{ 'mono' f : x / a >-> r }" (at level 0, f at level 99,
x name, format "{ 'mono'  f  :  x  /  a  >->  r }").
Reserved Notation "{ 'mono' f : x / a }" (at level 0, f at level 99,
x name, format "{ 'mono'  f  :  x  /  a }").
Reserved Notation "{ 'mono' f : x y / a >-> r }" (at level 0, f at level 99,
x name, y name, format "{ 'mono'  f  :  x  y  /  a  >->  r }").
Reserved Notation "{ 'mono' f : x y / a }" (at level 0, f at level 99,
x name, y name, format "{ 'mono'  f  :  x  y  /  a }").
Reserved Notation "{ 'mono' f : x y /~ a }" (at level 0, f at level 99,
x name, y name, format "{ 'mono'  f  :  x  y  /~  a }").
Reserved Notation "@ 'id' T" (at level 10, T at level 8, format "@ 'id'  T").
#[warning="-closed-notation-not-level-0"]
Reserved Notation "@ 'sval'" (at level 10, format "@ 'sval'").
(** 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' a ]" (at level 0,
format "[ 'rec'  a ]").
Reserved Notation "[ 'rec' a , b ]" (at level 0,
format "[ 'rec'  a ,  b ]").
Reserved Notation "[ 'rec' a , b , c ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ]").
Reserved Notation "[ 'rec' a , b , c , d ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ]").
Reserved Notation "[ 'rec' a , b , c , d , e ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ]").
Reserved Notation "[ 'rec' a , b , c , d , e , f ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ,  f ]").
Reserved Notation "[ 'rec' a , b , c , d , e , f , g ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ,  f ,  g ]").
Reserved Notation "[ 'rec' a , b , c , d , e , f , g , h ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ,  f ,  g ,  h ]").
Reserved Notation "[ 'rec' a , b , c , d , e , f , g , h , i ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ,  f ,  g ,  h ,  i ]").
Reserved Notation "[ 'rec' a , b , c , d , e , f , g , h , i , j ]" (at level 0,
format "[ 'rec'  a ,  b ,  c ,  d ,  e ,  f ,  g ,  h ,  i ,  j ]").
Declare Scope pair_scope.
Delimit Scope pair_scope with PAIR.
Open Scope pair_scope.
(**  Notations for pair/conjunction projections  **)
Notation "p .1" := (fst p) : pair_scope.
Notation "p .2" := (snd p) : 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)).
Definition lift aT rT (f : aT -> rT) := fun x => Some (f x).
End Option.
Notation oapp := Option.apply.
Notation odflt := Option.default.
Notation obind := Option.bind.
Notation omap := Option.map.
Notation olift := Option.lift.
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.
#[universes(template)]
Structure wrapped T := Wrap {unwrap : T}.