## TPTP Axioms File: PUZ005-0.ax

```%------------------------------------------------------------------------------
% File     : PUZ005-0 : TPTP v7.5.0. Released v3.2.0.
% Domain   : Puzzles (Sudoku)
% Axioms   : Sudoku axioms
% Version  : [Bau06] axioms : Especial.
% English  :

% Refs     : [Bau06] Baumgartner (2006), Email to G. Sutcliffe
% Source   : [Bau06]
% Names    :

% Status   : Satisfiable
% Syntax   : Number of clauses     :   79 (   4 non-Horn;  54 unit;  79 RR)
%            Number of atoms       :  161 (  39 equality)
%            Maximal clause size   :   11 (   2 average)
%            Number of predicates  :    4 (   0 propositional; 1-3 arity)
%            Number of functors    :    2 (   1 constant; 0-1 arity)
%            Number of variables   :   75 (   0 singleton)
%            Maximal term depth    :   10 (   3 average)
% SPC      :

%------------------------------------------------------------------------------
%----Regarding equality, (un)equality is syntactic (un)equality
%----The domain is the numbers from n1 to n9.
cnf(dom_1,axiom,
( dom(s(n0)) )).

cnf(dom_2,axiom,
( dom(s(s(n0))) )).

cnf(dom_3,axiom,
( dom(s(s(s(n0)))) )).

cnf(dom_4,axiom,
( dom(s(s(s(s(n0))))) )).

cnf(dom_5,axiom,
( dom(s(s(s(s(s(n0)))))) )).

cnf(dom_6,axiom,
( dom(s(s(s(s(s(s(n0))))))) )).

cnf(dom_7,axiom,
( dom(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_8,axiom,
( dom(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_9,axiom,
( dom(s(s(s(s(s(s(s(s(s(n0)))))))))) )).

%----The domain elements are pairwise different;
%----This is the negative part of equality.
cnf(dom_1_not_2,axiom,
( s(n0) != s(s(n0)) )).

cnf(dom_1_not_3,axiom,
( s(n0) != s(s(s(n0))) )).

cnf(dom_1_not_4,axiom,
( s(n0) != s(s(s(s(n0)))) )).

cnf(dom_1_not_5,axiom,
( s(n0) != s(s(s(s(s(n0))))) )).

cnf(dom_1_not_6,axiom,
( s(n0) != s(s(s(s(s(s(n0)))))) )).

cnf(dom_1_not_7,axiom,
( s(n0) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_1_not_8,axiom,
( s(n0) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_1_not_9,axiom,
( s(n0) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_2_not_3,axiom,
( s(s(n0)) != s(s(s(n0))) )).

cnf(dom_2_not_4,axiom,
( s(s(n0)) != s(s(s(s(n0)))) )).

cnf(dom_2_not_5,axiom,
( s(s(n0)) != s(s(s(s(s(n0))))) )).

cnf(dom_2_not_6,axiom,
( s(s(n0)) != s(s(s(s(s(s(n0)))))) )).

cnf(dom_2_not_7,axiom,
( s(s(n0)) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_2_not_8,axiom,
( s(s(n0)) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_2_not_9,axiom,
( s(s(n0)) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_3_not_4,axiom,
( s(s(s(n0))) != s(s(s(s(n0)))) )).

cnf(dom_3_not_5,axiom,
( s(s(s(n0))) != s(s(s(s(s(n0))))) )).

cnf(dom_3_not_6,axiom,
( s(s(s(n0))) != s(s(s(s(s(s(n0)))))) )).

cnf(dom_3_not_7,axiom,
( s(s(s(n0))) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_3_not_8,axiom,
( s(s(s(n0))) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_3_not_9,axiom,
( s(s(s(n0))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_4_not_5,axiom,
( s(s(s(s(n0)))) != s(s(s(s(s(n0))))) )).

cnf(dom_4_not_6,axiom,
( s(s(s(s(n0)))) != s(s(s(s(s(s(n0)))))) )).

cnf(dom_4_not_7,axiom,
( s(s(s(s(n0)))) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_4_not_8,axiom,
( s(s(s(s(n0)))) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_4_not_9,axiom,
( s(s(s(s(n0)))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_5_not_6,axiom,
( s(s(s(s(s(n0))))) != s(s(s(s(s(s(n0)))))) )).

cnf(dom_5_not_7,axiom,
( s(s(s(s(s(n0))))) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_5_not_8,axiom,
( s(s(s(s(s(n0))))) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_5_not_9,axiom,
( s(s(s(s(s(n0))))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_6_not_7,axiom,
( s(s(s(s(s(s(n0)))))) != s(s(s(s(s(s(s(n0))))))) )).

cnf(dom_6_not_8,axiom,
( s(s(s(s(s(s(n0)))))) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_6_not_9,axiom,
( s(s(s(s(s(s(n0)))))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_7_not_8,axiom,
( s(s(s(s(s(s(s(n0))))))) != s(s(s(s(s(s(s(s(n0)))))))) )).

cnf(dom_7_not_9,axiom,
( s(s(s(s(s(s(s(n0))))))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

cnf(dom_8_not_9,axiom,
( s(s(s(s(s(s(s(s(n0)))))))) != s(s(s(s(s(s(s(s(s(n0))))))))) )).

%----Generator:
%----At least one number in each field
%----el(I,J,X) means on row I, column J is the natural number X
cnf(number_in_each_position,axiom,
( ~ dom(I)
| ~ dom(J)
| el(I,J,s(n0))
| el(I,J,s(s(n0)))
| el(I,J,s(s(s(n0))))
| el(I,J,s(s(s(s(n0)))))
| el(I,J,s(s(s(s(s(n0))))))
| el(I,J,s(s(s(s(s(s(n0)))))))
| el(I,J,s(s(s(s(s(s(s(n0))))))))
| el(I,J,s(s(s(s(s(s(s(s(n0)))))))))
| el(I,J,s(s(s(s(s(s(s(s(s(n0)))))))))) )).

%----Restriction:
%----No two same numbers on a field (dual of previous)
%----This is in fact redundant in combination of the previous generator and
%----already one of the following constraints
cnf(only_one_number_in_each_position,axiom,
( ~ el(I,J,X)
| ~ el(I,J,X1)
| X = X1 )).

%----Restriction:
%----No number occurs twice in a row:
%----(J = J1) :- el(I,J,X), el(I,J1,X1), (X = X1).
%----slightly simpler:
cnf(no_duplicates_in_a_row,axiom,
( ~ el(I,J,X)
| ~ el(I,J1,X)
| J = J1 )).

%----Restriction:
%----No number occurs twice in a column:
cnf(no_duplicates_in_a_column,axiom,
( ~ el(I,J,X)
| ~ el(I1,J,X)
| I = I1 )).

%----where different(I,J,I1,J1) means that the field elements at
%----(I,J) and at (I1,J1) are different
%---- different(I,J,I1,J1) ->
%       ( el(I,J,X) & el(I1,J1,X1) -> -(X = X1)).
%----Now, the n3x3 subfields.
cnf(subfield_1_1,hypothesis,
( subfield(s(n0),s(n0)) )).

cnf(subfield_1_4,hypothesis,
( subfield(s(n0),s(s(s(s(n0))))) )).

cnf(subfield_1_7,hypothesis,
( subfield(s(n0),s(s(s(s(s(s(s(n0)))))))) )).

cnf(subfield_4_1,hypothesis,
( subfield(s(s(s(s(n0)))),s(n0)) )).

cnf(subfield_4_4,hypothesis,
( subfield(s(s(s(s(n0)))),s(s(s(s(n0))))) )).

cnf(subfield_4_7,hypothesis,
( subfield(s(s(s(s(n0)))),s(s(s(s(s(s(s(n0)))))))) )).

cnf(subfield_7_1,hypothesis,
( subfield(s(s(s(s(s(s(s(n0))))))),s(n0)) )).

cnf(subfield_7_4,hypothesis,
( subfield(s(s(s(s(s(s(s(n0))))))),s(s(s(s(n0))))) )).

cnf(subfield_7_7,hypothesis,
( subfield(s(s(s(s(s(s(s(n0))))))),s(s(s(s(s(s(s(n0)))))))) )).

%----Each subfield contains no number twice:
%----Note: It is sufficient to specify only along the diagonals,
%----as along the row and columns the general row and column restrictions
%----above subsume the corresponding ones for the subfields.
%----Notice we do a little bit of integer arithmetic (+1 and +2) to talk
%----about the fields in a given subfield.
%----Perhaps more readable formulation of the axioms is like
%----subfield(I,J) ->
%----	( el(I,J,X) & el(s(I),s(J),X1) -> -(X = X1)).
%----which translates to
cnf(subfield_diagonal_1,hypothesis,
( ~ subfield(I,J)
| ~ el(I,J,X)
| ~ el(s(I),s(J),X) )).

cnf(subfield_diagonal_2,hypothesis,
( ~ subfield(I,J)
| ~ el(I,J,X)
| ~ el(s(I),s(s(J)),X) )).

cnf(subfield_diagonal_3,hypothesis,
( ~ subfield(I,J)
| ~ el(I,J,X)
| ~ el(s(s(I)),s(J),X) )).

cnf(subfield_diagonal_4,hypothesis,
( ~ subfield(I,J)
| ~ el(I,J,X)
| ~ el(s(s(I)),s(s(J)),X) )).

cnf(subfield_diagonal_5,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(J),X)
| ~ el(s(I),J,X) )).

cnf(subfield_diagonal_6,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(J),X)
| ~ el(s(I),s(s(J)),X) )).

cnf(subfield_diagonal_7,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(J),X)
| ~ el(s(s(I)),J,X) )).

cnf(subfield_diagonal_8,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(J),X)
| ~ el(s(s(I)),s(s(J)),X) )).

cnf(subfield_diagonal_9,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(s(J)),X)
| ~ el(s(I),J,X) )).

cnf(subfield_diagonal_10,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(s(J)),X)
| ~ el(s(I),s(J),X) )).

cnf(subfield_diagonal_11,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(s(J)),X)
| ~ el(s(s(I)),J,X) )).

cnf(subfield_diagonal_12,hypothesis,
( ~ subfield(I,J)
| ~ el(I,s(s(J)),X)
| ~ el(s(s(I)),s(J),X) )).

cnf(subfield_diagonal_13,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),J,X)
| ~ el(s(s(I)),s(J),X) )).

cnf(subfield_diagonal_14,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),J,X)
| ~ el(s(s(I)),s(s(J)),X) )).

cnf(subfield_diagonal_15,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),s(J),X)
| ~ el(s(s(I)),J,X) )).

cnf(subfield_diagonal_16,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),s(J),X)
| ~ el(s(s(I)),s(s(J)),X) )).

cnf(subfield_diagonal_17,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),s(s(J)),X)
| ~ el(s(s(I)),J,X) )).

cnf(subfield_diagonal_18,hypothesis,
( ~ subfield(I,J)
| ~ el(s(I),s(s(J)),X)
| ~ el(s(s(I)),s(J),X) )).

%----Some additional constraints. They are redundant but help
%----to solve the Sudoku in a deterministic way quite often.
%----I think the underlying heuristics used by people is called
%----'crosshatching'.
%----In every subfield, every value must be put somewhere
cnf(value_somewhere_in_subfield,hypothesis,
( ~ subfield(I,J)
| ~ dom(X)
| el(I,J,X)
| el(I,s(J),X)
| el(I,s(s(J)),X)
| el(s(I),J,X)
| el(s(I),s(J),X)
| el(s(I),s(s(J)),X)
| el(s(s(I)),J,X)
| el(s(s(I)),s(J),X)
| el(s(s(I)),s(s(J)),X) )).

%----In every row, every value must be put somewhere
cnf(value_somewhere_in_row,hypothesis,
( ~ dom(I)
| ~ dom(X)
| el(I,s(n0),X)
| el(I,s(s(n0)),X)
| el(I,s(s(s(n0))),X)
| el(I,s(s(s(s(n0)))),X)
| el(I,s(s(s(s(s(n0))))),X)
| el(I,s(s(s(s(s(s(n0)))))),X)
| el(I,s(s(s(s(s(s(s(n0))))))),X)
| el(I,s(s(s(s(s(s(s(s(n0)))))))),X)
| el(I,s(s(s(s(s(s(s(s(s(n0))))))))),X) )).

%----In every column, every value must be put somewhere
cnf(value_somewhere_in_column,hypothesis,
( ~ dom(J)
| ~ dom(X)
| el(s(n0),J,X)
| el(s(s(n0)),J,X)
| el(s(s(s(n0))),J,X)
| el(s(s(s(s(n0)))),J,X)
| el(s(s(s(s(s(n0))))),J,X)
| el(s(s(s(s(s(s(n0)))))),J,X)
| el(s(s(s(s(s(s(s(n0))))))),J,X)
| el(s(s(s(s(s(s(s(s(n0)))))))),J,X)
| el(s(s(s(s(s(s(s(s(s(n0))))))))),J,X) )).

%------------------------------------------------------------------------------
```