########################################################################
#  This program is free software; you can redistribute it and/or
#  modify it under the terms of the GNU General Public License
#  as published by the Free Software Foundation; either version 2
#  of the License, or (at your option) any later version.
#  
#  This program is distributed in the hope that it will be useful,
#  but WITHOUT ANY WARRANTY; without even the implied warranty of
#  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#  GNU General Public License for more details.
#  
#  You should have received a copy of the GNU General Public License
#  along with this program; if not, write to the Free Software
#  Foundation, Inc., 51 Franklin Street, Fifth Floor,
#  Boston, MA  02110-1301, USA.
#  
#  ---
#  Copyright (C) 2011-2015, Simon Hampe <simon.hampe@googlemail.com>
#
#
#  Defines properties and rules for Morphism 
########################################################################

# @Category Combinatorics
object Morphism {

	use overload('*' => 'mult', '""' => 'to_string', bool => 'bool', '+' => 'sum', '-' => 'minus', 'neg'=>'neg', fallback => 1);

	# @category Defining morphisms and functions
	# This property describes the domain of the morphism. I.e. the morphism is
	# defined on this complex and is locally affine integral linear. 
	property DOMAIN : Cycle<Addition>;

	# @category Defining morphisms and functions 
	# The vector at row i describes the function value of vertex [[DOMAIN]]->SEPARATED_VERTICES->row(i). 
	# (In tropical homogenous coordinates, but without leading coordinate).
	# More precisely, if the corresponding vertex is not a far ray, it describes its function value. 
	# If it is a directional ray, it describes the slope on that ray.
	property VERTEX_VALUES : Matrix<Rational>;

	# @category Defining morphisms and functions
	# The vector in row i describes the function value (slope) of [[DOMAIN]]->LINEALITY_SPACE->row(i)
	property LINEALITY_VALUES : Matrix<Rational>; 

	# @category Defining morphisms and functions
	# If the morphism is a global affine linear map x |-> Ax+v, then this contains the matrix A.
	# Note that this must be well-defined on tropical projective coordinates, so the sum of the columns of A
	# must be a multiple of the (1,..,1)-vector.
	# If [[TRANSLATE]] is set, but this property is not set, then it is the identity by default.
	property MATRIX : Matrix<Rational>;

	# @category Defining morphisms and functions
	# If the morphism is a global affine linear map x |-> Ax+v, then this contains the translation vector 
	# v. If [[MATRIX]] is set, but this property is not set, then it is the zero vector by default.
	property TRANSLATE : Vector<Rational>;


	# @category Defining morphisms and functions
	# This is TRUE, if the morphism is defined on the full projective torus by a [[MATRIX]] and a [[TRANSLATE]]
	# The rules do not actually check for completeness of the [[DOMAIN]]. 
	# This property will be set to TRUE, if the morphism is only defined by [[MATRIX]] and [[TRANSLATE]], 
	# otherwise it is false (or you can set it upon creation).
	property IS_GLOBALLY_AFFINE_LINEAR : Bool;

	### RULES ###################################

	rule MATRIX : TRANSLATE {
		if(defined($this->lookup("DOMAIN"))) {
			die "Cannot infer matrix on nontrivial domain";
		}
		$this->MATRIX = unit_matrix($this->TRANSLATE->dim());
	}

	rule TRANSLATE : MATRIX {
		$this->TRANSLATE = zero_vector<Rational>($this->MATRIX->rows());
	}

	rule VERTEX_VALUES : LINEALITY_VALUES {
		$this->VERTEX_VALUES = new Matrix<Rational>(0,$this->LINEALITY_VALUES->cols());
	}

	rule LINEALITY_VALUES : VERTEX_VALUES {
		$this->LINEALITY_VALUES = new Matrix<Rational>(0,$this->VERTEX_VALUES->cols());
	}

	rule DOMAIN, IS_GLOBALLY_AFFINE_LINEAR : MATRIX, TRANSLATE {
		computeDomainFromMatrix($this);
		$this->IS_GLOBALLY_AFFINE_LINEAR = !(defined($this->lookup("VERTEX_VALUES")) || defined($this->lookup("LINEALITY_VALUES")));
	}
	weight 0.10;
	precondition : MATRIX, TRANSLATE {
		$this->MATRIX->rows() == $this->TRANSLATE->dim() or
		die "Matrix and vector dimensions do not match.";
		is_homogeneous_matrix($this->MATRIX) or
		die "Matrix is not homogeneous. Columns must add up to (projective) zero";
	}

	rule IS_GLOBALLY_AFFINE_LINEAR : {
		$this->IS_GLOBALLY_AFFINE_LINEAR = 0;
	}
	precondition : defined(DOMAIN);
	weight 0.1;

	rule VERTEX_VALUES, LINEALITY_VALUES : MATRIX, TRANSLATE, DOMAIN {
		computeValuesFromMatrix($this);
	}
	precondition : MATRIX, TRANSLATE, DOMAIN {
		($this->DOMAIN->AMBIENT_DIM == $this->MATRIX->cols() and $this->MATRIX->rows() == $this->TRANSLATE->dim())
			or die "Dimensions don't match.";
		is_homogeneous_matrix($this->MATRIX) or
		die "Matrix is not homogeneous. Columns must add up to (projective) zero";
	}

	# @category Affine and projective coordinates
	# Computes the representation of a morphism (given by [[MATRIX]] and [[TRANSLATE]])
	# on tropical affine coordinates.
	# @param Int domain_chart Which coordinate index of the homogenized domain is shifted to zero to identify
	# it with the domain of the affine function. 0 by default.
	# @param Int target_chart Which coordinate of the homogenized target space is shifted to zero to identify
	# it with the target of the affine function. 0 by default.
	# @return Pair<Matrix<Rational>, Vector<Rational>> A matrix and a translate in affine coordinates.
	user_method affine_representation(;$=0,$=0) {
		my ($morphism, $domain_chart, $target_chart) = @_;
		if(!defined($morphism->lookup("MATRIX"))) {
			die "Morphism is not defined by matrix and translate.";
		}
		return tdehomog_morphism($morphism->MATRIX,$morphism->TRANSLATE, $domain_chart, $target_chart);
	}

	# @category Morphisms
	# Computes the composition of another morphism g and this morphism.
	# This morphism comes after g.
	# @param Morphism g
	# @return Morphism this after g
	user_method after(Morphism<Addition>) {
		my ($this, $g) = @_;
		return morphism_composition($g,$this);
	}

	# @category Morphisms
	# Computes the composition of this morphism and another morphism g.
	# This morphism comes before g.
	# @param Morphism g
	# @return Morphism g after f
	user_method before(Morphism<Addition>) {
		my ($this,$g) = @_;
		return morphism_composition($this,$g);
	}

	# @category Morphisms
	# Computes the restriction of the morphism to a cycle.
	# The cycle need not be contained in the [[DOMAIN]] of the morphism,
	# the restriction will be computed on the intersection.
	# @param Cycle Some cycle living in the same ambient space as the [[DOMAIN]]
	# @return Morphism The restriction of the morphism to the cycle (or its
	#   intersection with [[DOMAIN]].
	user_method restrict(Cycle<Addition>) {
		my ($f,$X) = @_;

		#Define identity on X
		my $vertexvalues = $X->SEPARATED_VERTICES->minor(All,~scalar2set(0));
		my $linealityvalues = $X->LINEALITY_SPACE->minor(All,~scalar2set(0));

		my $emb = new Morphism<Addition>( DOMAIN => $X,
						  VERTEX_VALUES => $vertexvalues,
						  LINEALITY_VALUES => $linealityvalues,
						  MATRIX => unit_matrix<Rational>($f->DOMAIN->PROJECTIVE_AMBIENT_DIM+1)
						);

		return morphism_composition($emb,$f);
	}

	#----------------------------- OVERLOADED OPERATORS -----------------

	sub bool {
		my $this = shift;
		return !undef($this);
	}


	sub to_string {
		my $this = shift;  
		my $txt = "Morphism on ".($this->DOMAIN->description());
		if(defined($this->lookup("MATRIX"))) {
			$txt = $txt."\nMatrix:\n".$this->MATRIX."\nTranslate:\n".$this->TRANSLATE;	
		}
		return $txt;
	}

	sub sum {
		my ($f,$g) = @_;
		return add_morphisms($f,$g);
	}

	function mult_m(Morphism, RationalFunction) {
		return pullback(@_);
	}

	function mult_m(Morphism, Morphism) {
		return morphism_composition(@_);
	}

	function mult_m(Morphism, $) {
			my ($f,$r) = @_;
			if(defined($f->lookup("MATRIX"))) {
				return new Morphism<Addition>(DOMAIN=>$f->DOMAIN, VERTEX_VALUES=>($r * ($f->VERTEX_VALUES)), LINEALITY_VALUES=>$r * ($f->LINEALITY_VALUES), MATRIX=> $r*$f->MATRIX, TRANSLATE=>$r*$f->TRANSLATE);
			}
			else {
				return new Morphism<Addition>(DOMAIN=>$f->DOMAIN, VERTEX_VALUES=>($r * ($f->VERTEX_VALUES)), LINEALITY_VALUES=>$r * ($f->LINEALITY_VALUES));
			}
	}
	

	sub mult {
		my ($f,$r ) = @_;
		return mult_m($f,$r);
	}


	sub minus {
		my ($f,$g) = @_;
		return add_morphisms($f, (-1) * $g);
	}

	sub neg {
		my $f = shift;
		return mult($f,-1);
	}



}

# @category Affine and projective coordinates
# Takes a representation of a morphism on affine coordinates and converts it to projective ones.
# @param Matrix<Rational> A. The matrix of the morphism x |-> Ax + v in affine coordinates.
# @param Vector<Rational> v. The translate of the morphism x |-> Ax + v in affine coordinates.
# @param Int domain_chart Which coordinate index of the homogenized domain is shifted to zero to identify
# it with the domain of the affine function. 0 by default.
# @param Int target_chart Which coordinate of the homogenized target space is shifted to zero to identify
# it with the target of the affine function. 0 by default.
# @tparam Addition Min or Max
# @return Morphism
user_function morphism_from_affine<Addition>($, $;$=0,$=0) {
	my ($matrix,$translate, $domain_chart, $target_chart) = @_;
	my $homog_rep = thomog_morphism($matrix,$translate, $domain_chart, $target_chart);
	return new Morphism<Addition>(MATRIX=>$homog_rep->first, TRANSLATE=>$homog_rep->second);
}
