# Library mathcomp.ssreflect.ssrnotations

(* (c) Copyright 2006-2016 Microsoft Corporation and Inria.

Distributed under the terms of CeCILL-B. *)

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 "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 ] ']'").

(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").

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 "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 "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 "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 "<[ 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 )").

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 "''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 "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 "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 ']'").

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 ']'").

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 ']'").