# Prolog restrictions
[TOC]

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## Materialization of restrictions
A restriction can be set as boolean if a variable is not set:

|? - X #= 10, 
B #= (X#>= 2) + (X#>= 4) + (X #>=8). 

B=3, 
X=10

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## list_to_fdset(+List, -FD_Set).
Transforms a list in a domain (Finite domain set).
### Example

Numbers = [4,8,15,16,23,42], 
list_to_fdset(Numbers, FD_Set), 
Vars in FD_Set. 

The elements inside `Var` are in the specified range.

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## domain(+List_of_variables, + Min, +Max)
Set a numeric range domain for a list of variables.
### Example:

domain([A,B,C], 5, 12). 

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## fd_batch(+Constraints)
When is set a restriction to a variable, the domain of all the other variables is updated. With `fd_batch` we first set all the variables domain, then update all the domains at once.

### Example:

domain([A,B,C], 5, 12), 
fd_batch([A#>8, B+C#<12, A+B+C #= 20]). 

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## sum(+Xs, +RelOp, ?Value).
Corresnpond to `sumlist` from the `list` library.
### Example

|?- domain([X,Y], 1, 10), 
sum([X,Y], #<, 10). 

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## scalar_product
- `Coeffs` is a list of size `n` of integers.
- `Xs` is a list of domain integers with size n.
- `RelOp` is a operational operator.
- `Value` is the answer (domain or variable).
- It's true if sum(Coeffs*Xs) RelOp Value.

### Usage

scalar_product(+Coeffs, +Xs, +RelOp, ?Value, +Options) 
scalar_product(+Coeffs, +Xs, +RelOp, ?Value, ?Reif, +Options)

### Options
`among(Least, Most, Range)` indicates thar in the maximum or min interval `Xs` has values in the range given.
`consistency(Cons)` indicates the level of consistence used by the restriction. Cons can be `domain`, `bounds` or `value`.

### Example

|?- domain([A,B,C], 1,5 ), 
scalar_product([1,2,3], [A,B,C], #=, 10). 

A in 1..5,
B in 1..3,
C in 1..2 

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## minimum(?Value, +Xs) and maximum(?Value, +Xs)
- `Xs` is a list of integers and/or domain variables.
- `Value` is the integer or domain variable.
- It's true if `Value` is the min or max value of `Xs`.
- Corresponds for min_member and max_member of the library(lists).

### Example 1

|? - domain([A,B], 1, 10), C in 5..15,
minimum(C, [A,B]). 

A in 5..10 
B in 5..10 
C in 5..10 

### Example 2

|? - domain([A,B,C], 1, 5),
    sum([A,B,C], #=, 10), 
    maximum(3, [A,B]).  
    
A in 2..3
B in 2..3 
C in 4..5

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## maximum_arg(+Xs, ?Index) and minimum_arg(+Xs, ?Index)
- Xs is a list of integers in a domain.
- Index is a variable of domain or integer.
- It's true if the index of the max/min variable in `Xs` is `Index`.
- If the max/min value is repeated `Index` will be the first occurrence.
- Can't be materialized.

### Example

|? - domain([A,B,C], 1,5),
    sum([A,B], #=, 10), 
    minimum_arg([A,B,C], X).  
    
A in 5 
B in 5 
C in 1..5

X = {1} \/ {3}          % since C can be 5 or 1.  

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## global_cardinality(+Xs, +Vals, + Options)
- Restricts the number of occurrences for each value in a list.
- Xs is a list o integers or domain variables
- Vals is a list of terms `K-V` where K in an unique integer and V is an integer of domain variable
- True if each element of Xs is equals to a K and for each pair K-V, exactly V elements of Xs is equal to K.

### Example

global_cardinality([A,B,C], [1-2, 3-1]). 

## nvalue(?N, +Variables)
- Restricts the list of variables `Variables` in a way that just exists exactly `N` distict values.
- `Variables` is a list of integers an/or variables of domain with finit limis and N is an integer or domain variable.

### Example 1

domain([X,Y], 1, 3), 
domain([Z], 3, 5). 
nvalue(2, [X,Y,Z]), 
X#=Y, X#=1. 

X = 1, 
Y = 3
Z = 3

## all_distinct_except_0(+Variables)
- True whe all the variables have distinct values except if the variables has the number 0.

### Example

|? - L = [A,B, 1, D], domain(L, 0, 2), 
all_distinct_except_0(L). 

A in {0} \/ {2} 
B in {0} \/ {2} 
C in {0} \/ {2} 

## symmetric_all_distinct(+Variables)
- True when all th variables have distinct values and all the variables $X_i = j <=> X_j = i$ .
- Variables is a list of integers or domain variables.

### Example

|? - L [A,B,C,D], 
symmetric_all_distinct(L), A#=3. 

A = 3, 
C = 1, 
B in {2} \/ {4}, 
D in {2} \/ {4}

## assignment(+Xs, +Ys, +Options)
-