# Library Coq.Init.Notations

These are the notations whose level and associativity are imposed by Coq
Notations for propositional connectives

Reserved Notation "x -> y" (at level 99, right associativity, y at level 200).
Reserved Notation "x <-> y" (at level 95, no associativity).
Reserved Notation "x /\ y" (at level 80, right associativity).
Reserved Notation "x \/ y" (at level 85, right associativity).
Reserved Notation "~ x" (at level 75, right associativity).

Notations for equality and inequalities

Reserved Notation "x = y :> T"
(at level 70, y at next level, no associativity).
Reserved Notation "x = y" (at level 70, no associativity).
Reserved Notation "x = y = z"
(at level 70, no associativity, y at next level).

Reserved Notation "x <> y :> T"
(at level 70, y at next level, no associativity).
Reserved Notation "x <> y" (at level 70, no associativity).

Reserved Notation "x <= y" (at level 70, no associativity).
Reserved Notation "x < y" (at level 70, no associativity).
Reserved Notation "x >= y" (at level 70, no associativity).
Reserved Notation "x > y" (at level 70, no associativity).

Reserved Notation "x <= y <= z" (at level 70, y at next level).
Reserved Notation "x <= y < z" (at level 70, y at next level).
Reserved Notation "x < y < z" (at level 70, y at next level).
Reserved Notation "x < y <= z" (at level 70, y at next level).

Arithmetical notations (also used for type constructors)

Reserved Notation "x + y" (at level 50, left associativity).
Reserved Notation "x - y" (at level 50, left associativity).
Reserved Notation "x * y" (at level 40, left associativity).
Reserved Notation "x / y" (at level 40, left associativity).
Reserved Notation "- x" (at level 35, right associativity).
Reserved Notation "/ x" (at level 35, right associativity).
Reserved Notation "x ^ y" (at level 30, right associativity).

Notations for booleans

Reserved Notation "x || y" (at level 50, left associativity).
Reserved Notation "x && y" (at level 40, left associativity).

Notations for pairs

Reserved Notation "( x , y , .. , z )"
(at level 0, format "( '[' x , '/' y , '/' .. , '/' z ']' )").

Notation "{ x }" is reserved and has a special status as component of other notations such as "{ A } + { B }" and "A + { B }" (which are at the same level as "x + y"); "{ x }" is at level 0 to factor with "{ x : A | P }"

Reserved Notation "{ x }" (at level 0, x at level 99).

Notations for sigma-types or subsets

Reserved Notation "{ A } + { B }" (at level 50, left associativity).
Reserved Notation "A + { B }" (at level 50, left associativity).

Reserved Notation "{ x | P }" (at level 0, x at level 99).
Reserved Notation "{ x | P & Q }" (at level 0, x at level 99).

Reserved Notation "{ x : A | P }" (at level 0, x at level 99).
Reserved Notation "{ x : A | P & Q }" (at level 0, x at level 99).

Reserved Notation "{ x & P }" (at level 0, x at level 99).
Reserved Notation "{ x & P & Q }" (at level 0, x at level 99).

Reserved Notation "{ x : A & P }" (at level 0, x at level 99).
Reserved Notation "{ x : A & P & Q }" (at level 0, x at level 99).

Reserved Notation "{ ' pat | P }"
(at level 0, pat strict pattern, format "{ ' pat | P }").
Reserved Notation "{ ' pat | P & Q }"
(at level 0, pat strict pattern, format "{ ' pat | P & Q }").

Reserved Notation "{ ' pat : A | P }"
(at level 0, pat strict pattern, format "{ ' pat : A | P }").
Reserved Notation "{ ' pat : A | P & Q }"
(at level 0, pat strict pattern, format "{ ' pat : A | P & Q }").

Reserved Notation "{ ' pat & P }"
(at level 0, pat strict pattern, format "{ ' pat & P }").
Reserved Notation "{ ' pat & P & Q }"
(at level 0, pat strict pattern, format "{ ' pat & P & Q }").

Reserved Notation "{ ' pat : A & P }"
(at level 0, pat strict pattern, format "{ ' pat : A & P }").
Reserved Notation "{ ' pat : A & P & Q }"
(at level 0, pat strict pattern, format "{ ' pat : A & P & Q }").

Support for Gonthier-Ssreflect's "if c is pat then u else v"

Module IfNotations.

Notation "'if' c 'is' p 'then' u 'else' v" :=
(match c with p => u | _ => v end)
(at level 200, p pattern at level 100).

End IfNotations.

Scopes

Declare Scope core_scope.
Delimit Scope core_scope with core.

Declare Scope function_scope.
Delimit Scope function_scope with function.
Bind Scope function_scope with Funclass.

Declare Scope type_scope.
Delimit Scope type_scope with type.
Bind Scope type_scope with Sortclass.

Open Scope core_scope.
Open Scope function_scope.
Open Scope type_scope.