NO

Problem 1: 

Infeasibility Problem:
[(VAR vNonEmpty z y x vNonEmpty x1 x x2 x3)
(STRATEGY CONTEXTSENSITIVE
(gcd 1 2)
(iadd 1)
(u1 1)
(0)
(add 1 2)
(fSNonEmpty)
(s 1)
(tp2 1 2)
)
(RULES
gcd(0,x) -> x
gcd(z,y) -> gcd(x,y) | iadd(z) ->* tp2(x,y)
gcd(y,z) -> gcd(x,y) | iadd(z) ->* tp2(x,y)
gcd(x,0) -> x
iadd(add(x,y)) -> tp2(x,y)
iadd(s(z)) -> u1(iadd(z))
iadd(y) -> tp2(0,y)
u1(tp2(x,y)) -> tp2(s(x),y)
)
]

Infeasibility Conditions:
iadd(x1) ->* tp2(x,x2), iadd(x2) ->* tp2(x3,x1)

Problem 1: 

Obtaining a proof using Prover9:

 -> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3275905 was started by sandbox2 on z034.star.cs.uiowa.edu,
Tue Jul 30 09:22:10 2024
The command was "./prover9 -f /tmp/prover93275898-0.in".
============================== end of head ===========================

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

% Reading from file /tmp/prover93275898-0.in

assign(max_seconds,20).

formulas(assumptions).
->_s0(x1,y) -> ->_s0(gcd(x1,x2),gcd(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(gcd(x1,x2),gcd(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(u1(x1),u1(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(add(x1,x2),add(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(add(x1,x2),add(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence).
->_s0(gcd(0,x3),x3) # label(replacement).
->*_s0(iadd(x1),tp2(x3,x2)) -> ->_s0(gcd(x1,x2),gcd(x3,x2)) # label(replacement).
->*_s0(iadd(x1),tp2(x3,x2)) -> ->_s0(gcd(x2,x1),gcd(x3,x2)) # label(replacement).
->_s0(gcd(x3,0),x3) # label(replacement).
->_s0(iadd(add(x3,x2)),tp2(x3,x2)) # label(replacement).
->_s0(iadd(s(x1)),u1(iadd(x1))) # label(replacement).
->_s0(iadd(x2),tp2(0,x2)) # label(replacement).
->_s0(u1(tp2(x3,x2)),tp2(s(x3),x2)) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.

formulas(goals).
(exists x5 exists x6 exists x7 exists x8 (->*_s0(iadd(x5),tp2(x6,x7)) & ->*_s0(iadd(x7),tp2(x8,x5)))) # label(goal).
end_of_list.

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

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

% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(gcd(x1,x2),gcd(y,x2)) # label(congruence) # label(non_clause).  [assumption].
2 ->_s0(x2,y) -> ->_s0(gcd(x1,x2),gcd(x1,y)) # label(congruence) # label(non_clause).  [assumption].
3 ->_s0(x1,y) -> ->_s0(iadd(x1),iadd(y)) # label(congruence) # label(non_clause).  [assumption].
4 ->_s0(x1,y) -> ->_s0(u1(x1),u1(y)) # label(congruence) # label(non_clause).  [assumption].
5 ->_s0(x1,y) -> ->_s0(add(x1,x2),add(y,x2)) # label(congruence) # label(non_clause).  [assumption].
6 ->_s0(x2,y) -> ->_s0(add(x1,x2),add(x1,y)) # label(congruence) # label(non_clause).  [assumption].
7 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause).  [assumption].
8 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause).  [assumption].
9 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause).  [assumption].
10 ->*_s0(iadd(x1),tp2(x3,x2)) -> ->_s0(gcd(x1,x2),gcd(x3,x2)) # label(replacement) # label(non_clause).  [assumption].
11 ->*_s0(iadd(x1),tp2(x3,x2)) -> ->_s0(gcd(x2,x1),gcd(x3,x2)) # label(replacement) # label(non_clause).  [assumption].
12 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
13 (exists x5 exists x6 exists x7 exists x8 (->*_s0(iadd(x5),tp2(x6,x7)) & ->*_s0(iadd(x7),tp2(x8,x5)))) # label(goal) # label(non_clause) # label(goal).  [goal].

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

============================== PROCESS INITIAL CLAUSES ===============

% Clauses before input processing:

formulas(usable).
end_of_list.

formulas(sos).
-->_s0(x,y) | ->_s0(gcd(x,z),gcd(y,z)) # label(congruence).  [clausify(1)].
-->_s0(x,y) | ->_s0(gcd(z,x),gcd(z,y)) # label(congruence).  [clausify(2)].
-->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence).  [clausify(3)].
-->_s0(x,y) | ->_s0(u1(x),u1(y)) # label(congruence).  [clausify(4)].
-->_s0(x,y) | ->_s0(add(x,z),add(y,z)) # label(congruence).  [clausify(5)].
-->_s0(x,y) | ->_s0(add(z,x),add(z,y)) # label(congruence).  [clausify(6)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(7)].
-->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence).  [clausify(8)].
-->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence).  [clausify(9)].
->_s0(gcd(0,x),x) # label(replacement).  [assumption].
-->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(x,z),gcd(y,z)) # label(replacement).  [clausify(10)].
-->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(z,x),gcd(y,z)) # label(replacement).  [clausify(11)].
->_s0(gcd(x,0),x) # label(replacement).  [assumption].
->_s0(iadd(add(x,y)),tp2(x,y)) # label(replacement).  [assumption].
->_s0(iadd(s(x)),u1(iadd(x))) # label(replacement).  [assumption].
->_s0(iadd(x),tp2(0,x)) # label(replacement).  [assumption].
->_s0(u1(tp2(x,y)),tp2(s(x),y)) # label(replacement).  [assumption].
->*_s0(x,x) # label(reflexivity).  [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(12)].
-->*_s0(iadd(x),tp2(y,z)) | -->*_s0(iadd(z),tp2(u,x)) # label(goal).  [deny(13)].
end_of_list.

formulas(demodulators).
end_of_list.

============================== PREDICATE ELIMINATION =================

No predicates eliminated.

============================== end predicate elimination =============

Auto_denials:
  % copying label goal to answer in negative clause

Term ordering decisions:
Predicate symbol precedence:  predicate_order([ ->_s0, ->*_s0 ]).
Function symbol precedence:  function_order([ 0, gcd, tp2, add, iadd, u1, s ]).
After inverse_order:  (no changes).
Unfolding symbols: (none).

Auto_inference settings:
  % set(neg_binary_resolution).  % (HNE depth_diff=-9)
  % clear(ordered_res).  % (HNE depth_diff=-9)
  % set(ur_resolution).  % (HNE depth_diff=-9)
    % set(ur_resolution) -> set(pos_ur_resolution).
    % set(ur_resolution) -> set(neg_ur_resolution).

Auto_process settings:
  % set(unit_deletion).  % (Horn set with negative nonunits)

kept:      14 -->_s0(x,y) | ->_s0(gcd(x,z),gcd(y,z)) # label(congruence).  [clausify(1)].
kept:      15 -->_s0(x,y) | ->_s0(gcd(z,x),gcd(z,y)) # label(congruence).  [clausify(2)].
kept:      16 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence).  [clausify(3)].
kept:      17 -->_s0(x,y) | ->_s0(u1(x),u1(y)) # label(congruence).  [clausify(4)].
kept:      18 -->_s0(x,y) | ->_s0(add(x,z),add(y,z)) # label(congruence).  [clausify(5)].
kept:      19 -->_s0(x,y) | ->_s0(add(z,x),add(z,y)) # label(congruence).  [clausify(6)].
kept:      20 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(7)].
kept:      21 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence).  [clausify(8)].
kept:      22 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence).  [clausify(9)].
kept:      23 ->_s0(gcd(0,x),x) # label(replacement).  [assumption].
kept:      24 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(x,z),gcd(y,z)) # label(replacement).  [clausify(10)].
kept:      25 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(z,x),gcd(y,z)) # label(replacement).  [clausify(11)].
kept:      26 ->_s0(gcd(x,0),x) # label(replacement).  [assumption].
kept:      27 ->_s0(iadd(add(x,y)),tp2(x,y)) # label(replacement).  [assumption].
kept:      28 ->_s0(iadd(s(x)),u1(iadd(x))) # label(replacement).  [assumption].
kept:      29 ->_s0(iadd(x),tp2(0,x)) # label(replacement).  [assumption].
kept:      30 ->_s0(u1(tp2(x,y)),tp2(s(x),y)) # label(replacement).  [assumption].
kept:      31 ->*_s0(x,x) # label(reflexivity).  [assumption].
kept:      32 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(12)].
kept:      33 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(iadd(z),tp2(u,x)) # label(goal) # answer(goal).  [deny(13)].

============================== end of process initial clauses ========

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

% Clauses after input processing:

formulas(usable).
end_of_list.

formulas(sos).
14 -->_s0(x,y) | ->_s0(gcd(x,z),gcd(y,z)) # label(congruence).  [clausify(1)].
15 -->_s0(x,y) | ->_s0(gcd(z,x),gcd(z,y)) # label(congruence).  [clausify(2)].
16 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence).  [clausify(3)].
17 -->_s0(x,y) | ->_s0(u1(x),u1(y)) # label(congruence).  [clausify(4)].
18 -->_s0(x,y) | ->_s0(add(x,z),add(y,z)) # label(congruence).  [clausify(5)].
19 -->_s0(x,y) | ->_s0(add(z,x),add(z,y)) # label(congruence).  [clausify(6)].
20 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(7)].
21 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence).  [clausify(8)].
22 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence).  [clausify(9)].
23 ->_s0(gcd(0,x),x) # label(replacement).  [assumption].
24 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(x,z),gcd(y,z)) # label(replacement).  [clausify(10)].
25 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(z,x),gcd(y,z)) # label(replacement).  [clausify(11)].
26 ->_s0(gcd(x,0),x) # label(replacement).  [assumption].
27 ->_s0(iadd(add(x,y)),tp2(x,y)) # label(replacement).  [assumption].
28 ->_s0(iadd(s(x)),u1(iadd(x))) # label(replacement).  [assumption].
29 ->_s0(iadd(x),tp2(0,x)) # label(replacement).  [assumption].
30 ->_s0(u1(tp2(x,y)),tp2(s(x),y)) # label(replacement).  [assumption].
31 ->*_s0(x,x) # label(reflexivity).  [assumption].
32 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(12)].
33 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(iadd(z),tp2(u,x)) # label(goal) # answer(goal).  [deny(13)].
end_of_list.

formulas(demodulators).
end_of_list.

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

============================== SEARCH ================================

% Starting search at 0.01 seconds.

given #1 (I,wt=10): 14 -->_s0(x,y) | ->_s0(gcd(x,z),gcd(y,z)) # label(congruence).  [clausify(1)].

given #2 (I,wt=10): 15 -->_s0(x,y) | ->_s0(gcd(z,x),gcd(z,y)) # label(congruence).  [clausify(2)].

given #3 (I,wt=8): 16 -->_s0(x,y) | ->_s0(iadd(x),iadd(y)) # label(congruence).  [clausify(3)].

given #4 (I,wt=8): 17 -->_s0(x,y) | ->_s0(u1(x),u1(y)) # label(congruence).  [clausify(4)].

given #5 (I,wt=10): 18 -->_s0(x,y) | ->_s0(add(x,z),add(y,z)) # label(congruence).  [clausify(5)].

given #6 (I,wt=10): 19 -->_s0(x,y) | ->_s0(add(z,x),add(z,y)) # label(congruence).  [clausify(6)].

given #7 (I,wt=8): 20 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence).  [clausify(7)].

given #8 (I,wt=10): 21 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence).  [clausify(8)].

given #9 (I,wt=10): 22 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence).  [clausify(9)].

given #10 (I,wt=5): 23 ->_s0(gcd(0,x),x) # label(replacement).  [assumption].

given #11 (I,wt=13): 24 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(x,z),gcd(y,z)) # label(replacement).  [clausify(10)].

given #12 (I,wt=13): 25 -->*_s0(iadd(x),tp2(y,z)) | ->_s0(gcd(z,x),gcd(y,z)) # label(replacement).  [clausify(11)].

given #13 (I,wt=5): 26 ->_s0(gcd(x,0),x) # label(replacement).  [assumption].

given #14 (I,wt=8): 27 ->_s0(iadd(add(x,y)),tp2(x,y)) # label(replacement).  [assumption].

given #15 (I,wt=7): 28 ->_s0(iadd(s(x)),u1(iadd(x))) # label(replacement).  [assumption].

given #16 (I,wt=6): 29 ->_s0(iadd(x),tp2(0,x)) # label(replacement).  [assumption].

given #17 (I,wt=9): 30 ->_s0(u1(tp2(x,y)),tp2(s(x),y)) # label(replacement).  [assumption].

given #18 (I,wt=3): 31 ->*_s0(x,x) # label(reflexivity).  [assumption].

given #19 (I,wt=9): 32 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(12)].

given #20 (I,wt=12): 33 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(iadd(z),tp2(u,x)) # label(goal) # answer(goal).  [deny(13)].

given #21 (A,wt=9): 34 ->_s0(tp2(x,gcd(0,y)),tp2(x,y)).  [ur(22,a,23,a)].

given #22 (F,wt=15): 94 -->*_s0(iadd(x),tp2(y,z)) | -->_s0(iadd(z),u) | -->*_s0(u,tp2(w,x)) # answer(goal).  [resolve(33,a,32,c)].

============================== PROOF =================================

% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 10.
% Level of proof is 4.
% Maximum clause weight is 15.000.
% Given clauses 22.

12 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause).  [assumption].
13 (exists x5 exists x6 exists x7 exists x8 (->*_s0(iadd(x5),tp2(x6,x7)) & ->*_s0(iadd(x7),tp2(x8,x5)))) # label(goal) # label(non_clause) # label(goal).  [goal].
29 ->_s0(iadd(x),tp2(0,x)) # label(replacement).  [assumption].
31 ->*_s0(x,x) # label(reflexivity).  [assumption].
32 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity).  [clausify(12)].
33 -->*_s0(iadd(x),tp2(y,z)) | -->*_s0(iadd(z),tp2(u,x)) # label(goal) # answer(goal).  [deny(13)].
89 ->*_s0(iadd(x),tp2(0,x)).  [ur(32,a,29,a,b,31,a)].
94 -->*_s0(iadd(x),tp2(y,z)) | -->_s0(iadd(z),u) | -->*_s0(u,tp2(w,x)) # answer(goal).  [resolve(33,a,32,c)].
112 -->*_s0(iadd(x),tp2(y,x)) # answer(goal).  [ur(94,b,29,a,c,31,a)].
113 $F # answer(goal).  [resolve(112,a,89,a)].

============================== end of proof ==========================

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

Given=22. Generated=100. Kept=99. proofs=1.
Usable=22. Sos=69. Demods=0. Limbo=7, Disabled=20. Hints=0.
Kept_by_rule=0, Deleted_by_rule=0.
Forward_subsumed=1. Back_subsumed=0.
Sos_limit_deleted=0. Sos_displaced=0. Sos_removed=0.
New_demodulators=0 (0 lex), Back_demodulated=0. Back_unit_deleted=0.
Demod_attempts=0. Demod_rewrites=0.
Res_instance_prunes=0. Para_instance_prunes=0. Basic_paramod_prunes=0.
Nonunit_fsub_feature_tests=16. Nonunit_bsub_feature_tests=23.
Megabytes=0.24.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.

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

============================== end of search =========================

THEOREM PROVED

Exiting with 1 proof.

Process 3275905 exit (max_proofs) Tue Jul 30 09:22:10 2024


The problem is feasible.