/export/starexec/sandbox/solver/bin/starexec_run_default /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S)
(RULES
a -> h(b)
a -> h(c)
f(x:S) -> y:S | a ->* h(y:S)
g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
)

Problem 1: 
Valid CTRS Processor:
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> The system is a deterministic 3-CTRS.

Problem 1: 

Dependency Pairs Processor:

Conditional Termination Problem 1:
-> Pairs:
 G(x:S,b) -> F(c) | f(b) ->* x:S, x:S ->* c
 G(x:S,b) -> G(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c

Conditional Termination Problem 2:
-> Pairs:
 F(x:S) -> A
 G(x:S,b) -> F(b)
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c


The problem is decomposed in 2 subproblems.

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(x:S,b) -> F(c) | f(b) ->* x:S, x:S ->* c
 G(x:S,b) -> G(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(x:S,b) -> G(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> QPairs:
 Empty
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c

Problem 1.1: 

Simplification and Narrowing on Condition Processor:
-> Pairs:
 G(x:S,b) -> G(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Narrowed Pairs:
->->Original Pair:
 G(x:S,b) -> G(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->-> Narrowed pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(b), b ->* x3:S, x3:S ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c

Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(b), b ->* x3:S, x3:S ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(b), b ->* x3:S, x3:S ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> QPairs:
 Empty
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(b), b ->* x3:S, x3:S ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> Needed rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
->Mace4 Output:
 ============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 28710 was started by sandbox on n099.star.cs.uiowa.edu,
Wed Jul  1 10:36:44 2020
The command was "./mace4 -c -f /tmp/mace45726603361159126505.in".
============================== end of head ===========================

============================== INPUT =================================

% Reading from file /tmp/mace45726603361159126505.in

assign(max_seconds,20).

formulas(assumptions).
arrowStar_s0(x,x) # label(reflexivity).
arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(compatibility).
gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility).
arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f8(x1),f8(y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f14(x1,x2),f14(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f14(x1,x2),f14(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f3(x1),f3(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f4(x1,x2),f4(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f4(x1,x2),f4(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f8(x1),f8(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f12(x1),f12(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f13(x1,x2),f13(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f13(x1,x2),f13(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f14(x1,x2),f14(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f14(x1,x2),f14(x1,y)) # label(congruence).
arrow_s0(f2,f8(f5)) # label(replacement).
arrow_s0(f2,f8(f6)) # label(replacement).
arrowStar_s0(f2,f8(x2)) -> arrow_s0(f3(x1),x2) # label(replacement).
arrow_s0(f14(x4,x5),x4) # label(replacement).
arrow_s0(f14(x4,x5),x5) # label(replacement).
arrowN_s0(f2,f8(f5)) # label(replacement).
arrowN_s0(f2,f8(f6)) # label(replacement).
arrowStar_s0(f2,f8(x2)) -> arrowN_s0(f3(x1),x2) # label(replacement).
arrowN_s0(f14(x4,x5),x4) # label(replacement).
arrowN_s0(f14(x4,x5),x5) # label(replacement).
arrowN_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion).
arrowStar_s0(f2,f8(f5)) & arrowStar_s0(f5,x3) & arrowStar_s0(x3,f6) -> sqsupset_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement).
arrowStar_s0(f3(f5),f6) -> succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement).
arrowStar_s0(f2,f8(f6)) & arrowStar_s0(f6,x3) & arrowStar_s0(x3,f6) -> succeq_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement).
sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion).
sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility).
end_of_list.

formulas(goals).
(exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness).
end_of_list.

============================== end of input ==========================

============================== PROCESS NON-CLAUSAL FORMULAS ==========

% Formulas that are not ordinary clauses:
1 arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
2 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
3 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
4 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
5 arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause).  [assumption].
6 arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause).  [assumption].
7 arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause).  [assumption].
8 arrow_s0(x1,y) -> arrow_s0(f8(x1),f8(y)) # label(congruence) # label(non_clause).  [assumption].
9 arrow_s0(x1,y) -> arrow_s0(f14(x1,x2),f14(y,x2)) # label(congruence) # label(non_clause).  [assumption].
10 arrow_s0(x2,y) -> arrow_s0(f14(x1,x2),f14(x1,y)) # label(congruence) # label(non_clause).  [assumption].
11 arrowN_s0(x1,y) -> arrowN_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause).  [assumption].
12 arrowN_s0(x1,y) -> arrowN_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause).  [assumption].
13 arrowN_s0(x2,y) -> arrowN_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause).  [assumption].
14 arrowN_s0(x1,y) -> arrowN_s0(f8(x1),f8(y)) # label(congruence) # label(non_clause).  [assumption].
15 arrowN_s0(x1,y) -> arrowN_s0(f12(x1),f12(y)) # label(congruence) # label(non_clause).  [assumption].
16 arrowN_s0(x1,y) -> arrowN_s0(f13(x1,x2),f13(y,x2)) # label(congruence) # label(non_clause).  [assumption].
17 arrowN_s0(x2,y) -> arrowN_s0(f13(x1,x2),f13(x1,y)) # label(congruence) # label(non_clause).  [assumption].
18 arrowN_s0(x1,y) -> arrowN_s0(f14(x1,x2),f14(y,x2)) # label(congruence) # label(non_clause).  [assumption].
19 arrowN_s0(x2,y) -> arrowN_s0(f14(x1,x2),f14(x1,y)) # label(congruence) # label(non_clause).  [assumption].
20 arrowStar_s0(f2,f8(x2)) -> arrow_s0(f3(x1),x2) # label(replacement) # label(non_clause).  [assumption].
21 arrowStar_s0(f2,f8(x2)) -> arrowN_s0(f3(x1),x2) # label(replacement) # label(non_clause).  [assumption].
22 arrowN_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
23 arrowStar_s0(f2,f8(f5)) & arrowStar_s0(f5,x3) & arrowStar_s0(x3,f6) -> sqsupset_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement) # label(non_clause).  [assumption].
24 arrowStar_s0(f3(f5),f6) -> succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement) # label(non_clause).  [assumption].
25 arrowStar_s0(f2,f8(f6)) & arrowStar_s0(f6,x3) & arrowStar_s0(x3,f6) -> succeq_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement) # label(non_clause).  [assumption].
26 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
27 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
28 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

============================== CLAUSES FOR SEARCH ====================

formulas(mace4_clauses).
arrowStar_s0(x,x) # label(reflexivity).
-arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(compatibility).
-gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility).
-arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(x,z),f4(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(z,x),f4(z,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f8(x),f8(y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f14(x,z),f14(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f14(z,x),f14(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f3(x),f3(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f4(x,z),f4(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f4(z,x),f4(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f8(x),f8(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f12(x),f12(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f13(x,z),f13(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f13(z,x),f13(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f14(x,z),f14(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f14(z,x),f14(z,y)) # label(congruence).
arrow_s0(f2,f8(f5)) # label(replacement).
arrow_s0(f2,f8(f6)) # label(replacement).
-arrowStar_s0(f2,f8(x)) | arrow_s0(f3(y),x) # label(replacement).
arrow_s0(f14(x,y),x) # label(replacement).
arrow_s0(f14(x,y),y) # label(replacement).
arrowN_s0(f2,f8(f5)) # label(replacement).
arrowN_s0(f2,f8(f6)) # label(replacement).
-arrowStar_s0(f2,f8(x)) | arrowN_s0(f3(y),x) # label(replacement).
arrowN_s0(f14(x,y),x) # label(replacement).
arrowN_s0(f14(x,y),y) # label(replacement).
-arrowN_s0(x,y) | gtrsim_s0(x,y) # label(inclusion).
-arrowStar_s0(f2,f8(f5)) | -arrowStar_s0(f5,x) | -arrowStar_s0(x,f6) | sqsupset_s0(f13(x,f5),f13(f3(f6),x)) # label(replacement).
-arrowStar_s0(f3(f5),f6) | succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement).
-arrowStar_s0(f2,f8(f6)) | -arrowStar_s0(f6,x) | -arrowStar_s0(x,f6) | succeq_s0(f13(x,f5),f13(f3(f6),x)) # label(replacement).
-sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion).
-sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility).
-sqsupsetStar_s0(x,x) # label(wellfoundedness).
end_of_list.

============================== end of clauses for search =============

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

============================== MODEL =================================

interpretation( 2, [number=1, seconds=0], [

        function(f2, [ 0 ]),

        function(f5, [ 0 ]),

        function(f6, [ 1 ]),

        function(f12(_), [ 0, 0 ]),

        function(f3(_), [ 1, 1 ]),

        function(f8(_), [ 0, 0 ]),

        function(f13(_,_), [
			   0, 0,
			   0, 1 ]),

        function(f14(_,_), [
			   0, 1,
			   1, 1 ]),

        function(f4(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(arrowN_s0(_,_), [
			   1, 0,
			   1, 1 ]),

        relation(arrowStar_s0(_,_), [
			   1, 0,
			   1, 1 ]),

        relation(arrow_s0(_,_), [
			   1, 0,
			   1, 1 ]),

        relation(gtrsim_s0(_,_), [
			   1, 1,
			   1, 1 ]),

        relation(sqsupsetStar_s0(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(sqsupset_s0(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(succeq_s0(_,_), [
			   0, 1,
			   0, 0 ])
]).

============================== end of model ==========================

============================== STATISTICS ============================

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=185, kept=181.
Selections=197, assignments=374, propagations=981, current_models=1.
Rewrite_terms=4840, rewrite_bools=7708, indexes=1226.
Rules_from_neg_clauses=228, cross_offs=228.

============================== end of statistics =====================

User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 28710 exit (max_models) Wed Jul  1 10:36:44 2020
The process finished Wed Jul  1 10:36:44 2020


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

f2 = 0.

f5 = 0.

f6 = 1.

f12(0) = 0.
f12(1) = 0.

f3(0) = 1.
f3(1) = 1.

f8(0) = 0.
f8(1) = 0.

f13(0,0) = 0.
f13(0,1) = 0.
f13(1,0) = 0.
f13(1,1) = 1.

f14(0,0) = 0.
f14(0,1) = 1.
f14(1,0) = 1.
f14(1,1) = 1.

f4(0,0) = 0.
f4(0,1) = 0.
f4(1,0) = 0.
f4(1,1) = 0.

  arrowN_s0(0,0).
- arrowN_s0(0,1).
  arrowN_s0(1,0).
  arrowN_s0(1,1).

  arrowStar_s0(0,0).
- arrowStar_s0(0,1).
  arrowStar_s0(1,0).
  arrowStar_s0(1,1).

  arrow_s0(0,0).
- arrow_s0(0,1).
  arrow_s0(1,0).
  arrow_s0(1,1).

  gtrsim_s0(0,0).
  gtrsim_s0(0,1).
  gtrsim_s0(1,0).
  gtrsim_s0(1,1).

- sqsupsetStar_s0(0,0).
- sqsupsetStar_s0(0,1).
- sqsupsetStar_s0(1,0).
- sqsupsetStar_s0(1,1).

- sqsupset_s0(0,0).
- sqsupset_s0(0,1).
- sqsupset_s0(1,0).
- sqsupset_s0(1,1).

- succeq_s0(0,0).
  succeq_s0(0,1).
- succeq_s0(1,0).
- succeq_s0(1,1).


Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> QPairs:
 Empty
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c

Problem 1.1: 

Reduction Pair Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
 G(x3:S,b) -> G(f(c),x3:S) | a ->* h(c), c ->* x3:S, x3:S ->* c
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> Needed rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
->Mace4 Output:
 ============================== Mace4 =================================
Mace4 (64) version 2009-11A, November 2009.
Process 28736 was started by sandbox on n099.star.cs.uiowa.edu,
Wed Jul  1 10:37:04 2020
The command was "./mace4 -c -f /tmp/mace41780695788709393584.in".
============================== end of head ===========================

============================== INPUT =================================

% Reading from file /tmp/mace41780695788709393584.in

assign(max_seconds,20).

formulas(assumptions).
arrowStar_s0(x,x) # label(reflexivity).
arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(compatibility).
gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility).
gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility).
arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f8(x1),f8(y)) # label(congruence).
arrow_s0(x1,y) -> arrow_s0(f14(x1,x2),f14(y,x2)) # label(congruence).
arrow_s0(x2,y) -> arrow_s0(f14(x1,x2),f14(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f3(x1),f3(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f4(x1,x2),f4(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f4(x1,x2),f4(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f8(x1),f8(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f12(x1),f12(y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f13(x1,x2),f13(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f13(x1,x2),f13(x1,y)) # label(congruence).
arrowN_s0(x1,y) -> arrowN_s0(f14(x1,x2),f14(y,x2)) # label(congruence).
arrowN_s0(x2,y) -> arrowN_s0(f14(x1,x2),f14(x1,y)) # label(congruence).
arrow_s0(f2,f8(f5)) # label(replacement).
arrow_s0(f2,f8(f6)) # label(replacement).
arrowStar_s0(f2,f8(x2)) -> arrow_s0(f3(x1),x2) # label(replacement).
arrow_s0(f14(x4,x5),x4) # label(replacement).
arrow_s0(f14(x4,x5),x5) # label(replacement).
arrowN_s0(f2,f8(f5)) # label(replacement).
arrowN_s0(f2,f8(f6)) # label(replacement).
arrowStar_s0(f2,f8(x2)) -> arrowN_s0(f3(x1),x2) # label(replacement).
arrowN_s0(f14(x4,x5),x4) # label(replacement).
arrowN_s0(f14(x4,x5),x5) # label(replacement).
arrowN_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion).
arrowStar_s0(f2,f8(f6)) & arrowStar_s0(f6,x3) & arrowStar_s0(x3,f6) -> sqsupset_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement).
arrowStar_s0(f3(f5),f6) -> succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement).
sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion).
sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility).
end_of_list.

formulas(goals).
(exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness).
end_of_list.

============================== end of input ==========================

============================== PROCESS NON-CLAUSAL FORMULAS ==========

% Formulas that are not ordinary clauses:
1 arrow_s0(x,y) & arrowStar_s0(y,z) -> arrowStar_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
2 gtrsim_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
3 succeq_s0(x,y) & sqsupset_s0(y,z) -> sqsupset_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
4 gtrsim_s0(x,y) & succeq_s0(y,z) -> gtrsim_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
5 arrow_s0(x1,y) -> arrow_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause).  [assumption].
6 arrow_s0(x1,y) -> arrow_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause).  [assumption].
7 arrow_s0(x2,y) -> arrow_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause).  [assumption].
8 arrow_s0(x1,y) -> arrow_s0(f8(x1),f8(y)) # label(congruence) # label(non_clause).  [assumption].
9 arrow_s0(x1,y) -> arrow_s0(f14(x1,x2),f14(y,x2)) # label(congruence) # label(non_clause).  [assumption].
10 arrow_s0(x2,y) -> arrow_s0(f14(x1,x2),f14(x1,y)) # label(congruence) # label(non_clause).  [assumption].
11 arrowN_s0(x1,y) -> arrowN_s0(f3(x1),f3(y)) # label(congruence) # label(non_clause).  [assumption].
12 arrowN_s0(x1,y) -> arrowN_s0(f4(x1,x2),f4(y,x2)) # label(congruence) # label(non_clause).  [assumption].
13 arrowN_s0(x2,y) -> arrowN_s0(f4(x1,x2),f4(x1,y)) # label(congruence) # label(non_clause).  [assumption].
14 arrowN_s0(x1,y) -> arrowN_s0(f8(x1),f8(y)) # label(congruence) # label(non_clause).  [assumption].
15 arrowN_s0(x1,y) -> arrowN_s0(f12(x1),f12(y)) # label(congruence) # label(non_clause).  [assumption].
16 arrowN_s0(x1,y) -> arrowN_s0(f13(x1,x2),f13(y,x2)) # label(congruence) # label(non_clause).  [assumption].
17 arrowN_s0(x2,y) -> arrowN_s0(f13(x1,x2),f13(x1,y)) # label(congruence) # label(non_clause).  [assumption].
18 arrowN_s0(x1,y) -> arrowN_s0(f14(x1,x2),f14(y,x2)) # label(congruence) # label(non_clause).  [assumption].
19 arrowN_s0(x2,y) -> arrowN_s0(f14(x1,x2),f14(x1,y)) # label(congruence) # label(non_clause).  [assumption].
20 arrowStar_s0(f2,f8(x2)) -> arrow_s0(f3(x1),x2) # label(replacement) # label(non_clause).  [assumption].
21 arrowStar_s0(f2,f8(x2)) -> arrowN_s0(f3(x1),x2) # label(replacement) # label(non_clause).  [assumption].
22 arrowN_s0(x,y) -> gtrsim_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
23 arrowStar_s0(f2,f8(f6)) & arrowStar_s0(f6,x3) & arrowStar_s0(x3,f6) -> sqsupset_s0(f13(x3,f5),f13(f3(f6),x3)) # label(replacement) # label(non_clause).  [assumption].
24 arrowStar_s0(f3(f5),f6) -> succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement) # label(non_clause).  [assumption].
25 sqsupset_s0(x,y) -> sqsupsetStar_s0(x,y) # label(inclusion) # label(non_clause).  [assumption].
26 sqsupset_s0(x,y) & sqsupsetStar_s0(y,z) -> sqsupsetStar_s0(x,z) # label(compatibility) # label(non_clause).  [assumption].
27 (exists x sqsupsetStar_s0(x,x)) # label(wellfoundedness) # label(non_clause) # label(goal).  [goal].

============================== end of process non-clausal formulas ===

============================== CLAUSES FOR SEARCH ====================

formulas(mace4_clauses).
arrowStar_s0(x,x) # label(reflexivity).
-arrow_s0(x,y) | -arrowStar_s0(y,z) | arrowStar_s0(x,z) # label(compatibility).
-gtrsim_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-succeq_s0(x,y) | -sqsupset_s0(y,z) | sqsupset_s0(x,z) # label(compatibility).
-gtrsim_s0(x,y) | -succeq_s0(y,z) | gtrsim_s0(x,z) # label(compatibility).
-arrow_s0(x,y) | arrow_s0(f3(x),f3(y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(x,z),f4(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f4(z,x),f4(z,y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f8(x),f8(y)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f14(x,z),f14(y,z)) # label(congruence).
-arrow_s0(x,y) | arrow_s0(f14(z,x),f14(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f3(x),f3(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f4(x,z),f4(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f4(z,x),f4(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f8(x),f8(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f12(x),f12(y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f13(x,z),f13(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f13(z,x),f13(z,y)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f14(x,z),f14(y,z)) # label(congruence).
-arrowN_s0(x,y) | arrowN_s0(f14(z,x),f14(z,y)) # label(congruence).
arrow_s0(f2,f8(f5)) # label(replacement).
arrow_s0(f2,f8(f6)) # label(replacement).
-arrowStar_s0(f2,f8(x)) | arrow_s0(f3(y),x) # label(replacement).
arrow_s0(f14(x,y),x) # label(replacement).
arrow_s0(f14(x,y),y) # label(replacement).
arrowN_s0(f2,f8(f5)) # label(replacement).
arrowN_s0(f2,f8(f6)) # label(replacement).
-arrowStar_s0(f2,f8(x)) | arrowN_s0(f3(y),x) # label(replacement).
arrowN_s0(f14(x,y),x) # label(replacement).
arrowN_s0(f14(x,y),y) # label(replacement).
-arrowN_s0(x,y) | gtrsim_s0(x,y) # label(inclusion).
-arrowStar_s0(f2,f8(f6)) | -arrowStar_s0(f6,x) | -arrowStar_s0(x,f6) | sqsupset_s0(f13(x,f5),f13(f3(f6),x)) # label(replacement).
-arrowStar_s0(f3(f5),f6) | succeq_s0(f13(f3(f5),f5),f13(f3(f6),f3(f5))) # label(replacement).
-sqsupset_s0(x,y) | sqsupsetStar_s0(x,y) # label(inclusion).
-sqsupset_s0(x,y) | -sqsupsetStar_s0(y,z) | sqsupsetStar_s0(x,z) # label(compatibility).
-sqsupsetStar_s0(x,x) # label(wellfoundedness).
end_of_list.

============================== end of clauses for search =============

% There are no natural numbers in the input.

============================== DOMAIN SIZE 2 =========================

============================== MODEL =================================

interpretation( 2, [number=1, seconds=0], [

        function(f2, [ 0 ]),

        function(f5, [ 0 ]),

        function(f6, [ 1 ]),

        function(f12(_), [ 0, 0 ]),

        function(f3(_), [ 0, 0 ]),

        function(f8(_), [ 0, 0 ]),

        function(f13(_,_), [
			   0, 1,
			   0, 1 ]),

        function(f14(_,_), [
			   0, 0,
			   0, 0 ]),

        function(f4(_,_), [
			   0, 0,
			   0, 0 ]),

        relation(arrowN_s0(_,_), [
			   1, 1,
			   0, 1 ]),

        relation(arrowStar_s0(_,_), [
			   1, 1,
			   0, 1 ]),

        relation(arrow_s0(_,_), [
			   1, 1,
			   0, 0 ]),

        relation(gtrsim_s0(_,_), [
			   1, 1,
			   0, 1 ]),

        relation(sqsupsetStar_s0(_,_), [
			   0, 1,
			   0, 0 ]),

        relation(sqsupset_s0(_,_), [
			   0, 1,
			   0, 0 ]),

        relation(succeq_s0(_,_), [
			   1, 0,
			   0, 0 ])
]).

============================== end of model ==========================

============================== STATISTICS ============================

For domain size 2.

Current CPU time: 0.00 seconds (total CPU time: 0.00 seconds).
Ground clauses: seen=183, kept=179.
Selections=41, assignments=64, propagations=175, current_models=1.
Rewrite_terms=805, rewrite_bools=1456, indexes=206.
Rules_from_neg_clauses=25, cross_offs=25.

============================== end of statistics =====================

User_CPU=0.00, System_CPU=0.00, Wall_clock=0.

Exiting with 1 model.

Process 28736 exit (max_models) Wed Jul  1 10:37:04 2020
The process finished Wed Jul  1 10:37:04 2020


Mace4 cooked interpretation:

% number = 1
% seconds = 0

% Interpretation of size 2

f2 = 0.

f5 = 0.

f6 = 1.

f12(0) = 0.
f12(1) = 0.

f3(0) = 0.
f3(1) = 0.

f8(0) = 0.
f8(1) = 0.

f13(0,0) = 0.
f13(0,1) = 1.
f13(1,0) = 0.
f13(1,1) = 1.

f14(0,0) = 0.
f14(0,1) = 0.
f14(1,0) = 0.
f14(1,1) = 0.

f4(0,0) = 0.
f4(0,1) = 0.
f4(1,0) = 0.
f4(1,1) = 0.

  arrowN_s0(0,0).
  arrowN_s0(0,1).
- arrowN_s0(1,0).
  arrowN_s0(1,1).

  arrowStar_s0(0,0).
  arrowStar_s0(0,1).
- arrowStar_s0(1,0).
  arrowStar_s0(1,1).

  arrow_s0(0,0).
  arrow_s0(0,1).
- arrow_s0(1,0).
- arrow_s0(1,1).

  gtrsim_s0(0,0).
  gtrsim_s0(0,1).
- gtrsim_s0(1,0).
  gtrsim_s0(1,1).

- sqsupsetStar_s0(0,0).
  sqsupsetStar_s0(0,1).
- sqsupsetStar_s0(1,0).
- sqsupsetStar_s0(1,1).

- sqsupset_s0(0,0).
  sqsupset_s0(0,1).
- sqsupset_s0(1,0).
- sqsupset_s0(1,1).

  succeq_s0(0,0).
- succeq_s0(0,1).
- succeq_s0(1,0).
- succeq_s0(1,1).


Problem 1.1: 

SCC Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
-> QPairs:
 Empty
->->-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c

Problem 1.1: 

Reduction Triple Processor:
-> Pairs:
 G(f(b),b) -> G(f(c),f(b)) | f(b) ->* c
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
-> Usable rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[a] = 2
[f](X) = 2.X + 2
[g](X1,X2) = 0
[b] = 1
[c] = 0
[fSNonEmpty] = 0
[h](X) = X + 1
[A] = 0
[F](X) = 0
[G](X1,X2) = 2.X1 + X2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

SCC Processor:
-> Pairs:
 F(x:S) -> A
 G(x:S,b) -> F(b)
-> QPairs:
 Empty
-> Rules:
 a -> h(b)
 a -> h(c)
 f(x:S) -> y:S | a ->* h(y:S)
 g(x:S,b) -> g(f(c),x:S) | f(b) ->* x:S, x:S ->* c
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.