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(* (c) Copyright 2006-2016 Microsoft Corporation and Inria. *)
(* Distributed under the terms of CeCILL-B. *)
(******************************************************************************)
(* - Reserved notation for various arithmetic and algebraic operations: *)
(* e.[a1, ..., a_n] evaluation (e.g., polynomials). *)
(* e`_i indexing (number list, integer pi-part). *)
(* x^-1 inverse (group, field). *)
(* x *+ n, x *- n integer multiplier (modules and rings). *)
(* x ^+ n, x ^- n integer exponent (groups and rings). *)
(* x *: A, A :* x external product (scaling/module product in rings, *)
(* left/right cosets in groups). *)
(* A :&: B intersection (of sets, groups, subspaces, ...). *)
(* A :|: B, a |: B union, union with a singleton (of sets). *)
(* A :\: B, A :\ b relative complement (of sets, subspaces, ...). *)
(* <<A>>, <[a]> generated group/subspace, generated cycle/line. *)
(* 'C[x], 'C_A[x] point centralisers (in groups and F-algebras). *)
(* 'C(A), 'C_B(A) centralisers (in groups and matrix and F_algebras). *)
(* 'Z(A) centers (in groups and matrix and F-algebras). *)
(* m %/ d, m %% d Euclidean division and remainder (nat, polynomials). *)
(* d %| m Euclidean divisibility (nat, polynomial). *)
(* m = n %[mod d] equality mod d (also defined for <>, ==, and !=). *)
(* e^`(n) nth formal derivative (groups, polynomials). *)
(* e^`() simple formal derivative (polynomials only). *)
(* `|x| norm, absolute value, distance (rings, int, nat). *)
(* x <= y ?= iff C x is less than y, and equal iff C holds (nat, rings). *)
(* x <= y :> T, etc cast comparison (rings, all comparison operators). *)
(* [rec a1, ..., an] standard shorthand for hidden recursor (see prime.v). *)
(* The interpretation of these notations is not defined here, but the *)
(* declarations help maintain consistency across the library. *)
(******************************************************************************)
(* Reserved notation for evaluation *)
Reserved Notation "e .[ x ]"
(at level 2, left associativity, format "e .[ x ]").
Reserved Notation "e .[ x1 , x2 , .. , xn ]" (at level 2, left associativity,
format "e '[ ' .[ x1 , '/' x2 , '/' .. , '/' xn ] ']'").
(* Reserved notation for subscripting and superscripting *)
Reserved Notation "s `_ i" (at level 3, i at level 2, left associativity,
format "s `_ i").
Reserved Notation "x ^-1" (at level 3, left associativity, format "x ^-1").
(* Reserved notation for integer multipliers and exponents *)
Reserved Notation "x *+ n" (at level 40, left associativity).
Reserved Notation "x *- n" (at level 40, left associativity).
Reserved Notation "x ^+ n" (at level 29, left associativity).
Reserved Notation "x ^- n" (at level 29, left associativity).
(* Reserved notation for external multiplication. *)
Reserved Notation "x *: A" (at level 40).
Reserved Notation "A :* x" (at level 40).
(* Reserved notation for set-theoretic operations. *)
Reserved Notation "A :&: B" (at level 48, left associativity).
Reserved Notation "A :|: B" (at level 52, left associativity).
Reserved Notation "a |: A" (at level 52, left associativity).
Reserved Notation "A :\: B" (at level 50, left associativity).
Reserved Notation "A :\ b" (at level 50, left associativity).
(* Reserved notation for generated structures *)
Reserved Notation "<< A >>" (at level 0, format "<< A >>").
Reserved Notation "<[ a ] >" (at level 0, format "<[ a ] >").
(* Reserved notation for the order of an element (group, polynomial, etc) *)
Reserved Notation "#[ x ]" (at level 0, format "#[ x ]").
(* Reserved notation for centralisers and centers. *)
Reserved Notation "''C' [ x ]" (at level 8, format "''C' [ x ]").
Reserved Notation "''C_' A [ x ]"
(at level 8, A at level 2, format "''C_' A [ x ]").
Reserved Notation "''C' ( A )" (at level 8, format "''C' ( A )").
Reserved Notation "''C_' B ( A )"
(at level 8, B at level 2, format "''C_' B ( A )").
Reserved Notation "''Z' ( A )" (at level 8, format "''Z' ( A )").
(* Compatibility with group action centraliser notation. *)
Reserved Notation "''C_' ( A ) [ x ]" (at level 8).
Reserved Notation "''C_' ( B ) ( A )" (at level 8).
(* Reserved notation for Euclidean division and divisibility. *)
Reserved Notation "m %/ d" (at level 40, no associativity).
Reserved Notation "m %% d" (at level 40, no associativity).
Reserved Notation "m %| d" (at level 70, no associativity).
Reserved Notation "m = n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' = n '/' %[mod d ] ']'").
Reserved Notation "m == n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' == n '/' %[mod d ] ']'").
Reserved Notation "m <> n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' <> n '/' %[mod d ] ']'").
Reserved Notation "m != n %[mod d ]" (at level 70, n at next level,
format "'[hv ' m '/' != n '/' %[mod d ] ']'").
(* Reserved notation for derivatives. *)
Reserved Notation "a ^` ()" (at level 8, format "a ^` ()").
Reserved Notation "a ^` ( n )" (at level 8, format "a ^` ( n )").
(* Reserved notation for absolute value. *)
Reserved Notation "`| x |" (at level 0, x at level 99, format "`| x |").
(* Reserved notation for conditional comparison *)
Reserved Notation "x <= y ?= 'iff' c" (at level 70, y, c at next level,
format "x '[hv' <= y '/' ?= 'iff' c ']'").
(* Reserved notation for cast comparison. *)
Reserved Notation "x <= y :> T" (at level 70, y at next level).
Reserved Notation "x >= y :> T" (at level 70, y at next level).
Reserved Notation "x < y :> T" (at level 70, y at next level).
Reserved Notation "x > y :> T" (at level 70, y at next level).
Reserved Notation "x <= y ?= 'iff' c :> T" (at level 70, y, c at next level,
format "x '[hv' <= y '/' ?= 'iff' c :> T ']'").
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