NO
Problem 1:
Infeasibility Problem:
[(VAR vNonEmpty xs y' ws v ys' w x' xs' zs z y ys vNonEmpty x5 x3 w x4 ys' x6 x1 x2 x7)
(STRATEGY CONTEXTSENSITIVE
(get 1)
(ssp' 1 2)
(sub 1 2)
(0)
(cons 1 2)
(fSNonEmpty)
(nil)
(s 1)
(tp2 1 2)
)
(RULES
get(cons(x',xs')) -> tp2(y,cons(x',zs)) | get(xs') ->* tp2(y,zs)
get(cons(y,ys)) -> tp2(y,ys)
ssp'(cons(y',ws),v) -> cons(y',ys') | sub(v,y') ->* w, ssp'(ws,w) ->* ys'
ssp'(cons(x',xs'),v) -> cons(y',ys') | get(xs') ->* tp2(y',zs), sub(v,y') ->* w, ssp'(cons(x',zs),w) ->* ys'
ssp'(xs,0) -> nil
sub(s(v),s(w)) -> z | sub(v,w) ->* z
sub(z,0) -> z
)
]
Infeasibility Conditions:
sub(x5,x3) ->* w, ssp'(x4,w) ->* ys', get(x4) ->* tp2(x6,x1), sub(x5,x6) ->* x2, ssp'(cons(x3,x1),x2) ->* x7
Problem 1:
Obtaining a proof using Prover9:
-> Prover9 Output:
============================== Prover9 ===============================
Prover9 (64) version 2009-11A, November 2009.
Process 3317521 was started by sandbox on z026.star.cs.uiowa.edu,
Tue Jul 30 09:21:25 2024
The command was "./prover9 -f /tmp/prover93317512-0.in".
============================== end of head ===========================
============================== INPUT =================================
% Reading from file /tmp/prover93317512-0.in
assign(max_seconds,20).
formulas(assumptions).
->_s0(x1,y) -> ->_s0(get(x1),get(y)) # label(congruence).
->_s0(x1,y) -> ->_s0(ssp'(x1,x2),ssp'(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(ssp'(x1,x2),ssp'(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(sub(x1,x2),sub(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(sub(x1,x2),sub(x1,y)) # label(congruence).
->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence).
->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(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(get(x8),tp2(x11,x9)) -> ->_s0(get(cons(x7,x8)),tp2(x11,cons(x7,x9))) # label(replacement).
->_s0(get(cons(x11,x12)),tp2(x11,x12)) # label(replacement).
->*_s0(sub(x4,x2),x6) & ->*_s0(ssp'(x3,x6),x5) -> ->_s0(ssp'(cons(x2,x3),x4),cons(x2,x5)) # label(replacement).
->*_s0(get(x8),tp2(x2,x9)) & ->*_s0(sub(x4,x2),x6) & ->*_s0(ssp'(cons(x7,x9),x6),x5) -> ->_s0(ssp'(cons(x7,x8),x4),cons(x2,x5)) # label(replacement).
->_s0(ssp'(x1,0),nil) # label(replacement).
->*_s0(sub(x4,x6),x10) -> ->_s0(sub(s(x4),s(x6)),x10) # label(replacement).
->_s0(sub(x10,0),x10) # label(replacement).
->*_s0(x,x) # label(reflexivity).
->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity).
end_of_list.
formulas(goals).
(exists x14 exists x15 exists x16 exists x17 exists x18 exists x19 exists x20 exists x21 exists x22 (->*_s0(sub(x14,x15),x16) & ->*_s0(ssp'(x17,x16),x18) & ->*_s0(get(x17),tp2(x19,x20)) & ->*_s0(sub(x14,x19),x21) & ->*_s0(ssp'(cons(x15,x20),x21),x22))) # label(goal).
end_of_list.
============================== end of input ==========================
============================== PROCESS NON-CLAUSAL FORMULAS ==========
% Formulas that are not ordinary clauses:
1 ->_s0(x1,y) -> ->_s0(get(x1),get(y)) # label(congruence) # label(non_clause). [assumption].
2 ->_s0(x1,y) -> ->_s0(ssp'(x1,x2),ssp'(y,x2)) # label(congruence) # label(non_clause). [assumption].
3 ->_s0(x2,y) -> ->_s0(ssp'(x1,x2),ssp'(x1,y)) # label(congruence) # label(non_clause). [assumption].
4 ->_s0(x1,y) -> ->_s0(sub(x1,x2),sub(y,x2)) # label(congruence) # label(non_clause). [assumption].
5 ->_s0(x2,y) -> ->_s0(sub(x1,x2),sub(x1,y)) # label(congruence) # label(non_clause). [assumption].
6 ->_s0(x1,y) -> ->_s0(cons(x1,x2),cons(y,x2)) # label(congruence) # label(non_clause). [assumption].
7 ->_s0(x2,y) -> ->_s0(cons(x1,x2),cons(x1,y)) # label(congruence) # label(non_clause). [assumption].
8 ->_s0(x1,y) -> ->_s0(s(x1),s(y)) # label(congruence) # label(non_clause). [assumption].
9 ->_s0(x1,y) -> ->_s0(tp2(x1,x2),tp2(y,x2)) # label(congruence) # label(non_clause). [assumption].
10 ->_s0(x2,y) -> ->_s0(tp2(x1,x2),tp2(x1,y)) # label(congruence) # label(non_clause). [assumption].
11 ->*_s0(get(x8),tp2(x11,x9)) -> ->_s0(get(cons(x7,x8)),tp2(x11,cons(x7,x9))) # label(replacement) # label(non_clause). [assumption].
12 ->*_s0(sub(x4,x2),x6) & ->*_s0(ssp'(x3,x6),x5) -> ->_s0(ssp'(cons(x2,x3),x4),cons(x2,x5)) # label(replacement) # label(non_clause). [assumption].
13 ->*_s0(get(x8),tp2(x2,x9)) & ->*_s0(sub(x4,x2),x6) & ->*_s0(ssp'(cons(x7,x9),x6),x5) -> ->_s0(ssp'(cons(x7,x8),x4),cons(x2,x5)) # label(replacement) # label(non_clause). [assumption].
14 ->*_s0(sub(x4,x6),x10) -> ->_s0(sub(s(x4),s(x6)),x10) # label(replacement) # label(non_clause). [assumption].
15 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption].
16 (exists x14 exists x15 exists x16 exists x17 exists x18 exists x19 exists x20 exists x21 exists x22 (->*_s0(sub(x14,x15),x16) & ->*_s0(ssp'(x17,x16),x18) & ->*_s0(get(x17),tp2(x19,x20)) & ->*_s0(sub(x14,x19),x21) & ->*_s0(ssp'(cons(x15,x20),x21),x22))) # 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(get(x),get(y)) # label(congruence). [clausify(1)].
-->_s0(x,y) | ->_s0(ssp'(x,z),ssp'(y,z)) # label(congruence). [clausify(2)].
-->_s0(x,y) | ->_s0(ssp'(z,x),ssp'(z,y)) # label(congruence). [clausify(3)].
-->_s0(x,y) | ->_s0(sub(x,z),sub(y,z)) # label(congruence). [clausify(4)].
-->_s0(x,y) | ->_s0(sub(z,x),sub(z,y)) # label(congruence). [clausify(5)].
-->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(6)].
-->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(7)].
-->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(8)].
-->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(9)].
-->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(10)].
-->*_s0(get(x),tp2(y,z)) | ->_s0(get(cons(u,x)),tp2(y,cons(u,z))) # label(replacement). [clausify(11)].
->_s0(get(cons(x,y)),tp2(x,y)) # label(replacement). [assumption].
-->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | ->_s0(ssp'(cons(y,u),x),cons(y,w)) # label(replacement). [clausify(12)].
-->*_s0(get(x),tp2(y,z)) | -->*_s0(sub(u,y),w) | -->*_s0(ssp'(cons(v5,z),w),v6) | ->_s0(ssp'(cons(v5,x),u),cons(y,v6)) # label(replacement). [clausify(13)].
->_s0(ssp'(x,0),nil) # label(replacement). [assumption].
-->*_s0(sub(x,y),z) | ->_s0(sub(s(x),s(y)),z) # label(replacement). [clausify(14)].
->_s0(sub(x,0),x) # label(replacement). [assumption].
->*_s0(x,x) # label(reflexivity). [assumption].
-->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(15)].
-->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | -->*_s0(get(u),tp2(v5,v6)) | -->*_s0(sub(x,v5),v7) | -->*_s0(ssp'(cons(y,v6),v7),v8) # label(goal). [deny(16)].
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, nil, cons, ssp', sub, tp2, get, s ]).
After inverse_order: (no changes).
Unfolding symbols: (none).
Auto_inference settings:
% set(neg_binary_resolution). % (HNE depth_diff=-13)
% clear(ordered_res). % (HNE depth_diff=-13)
% set(ur_resolution). % (HNE depth_diff=-13)
% 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: 17 -->_s0(x,y) | ->_s0(get(x),get(y)) # label(congruence). [clausify(1)].
kept: 18 -->_s0(x,y) | ->_s0(ssp'(x,z),ssp'(y,z)) # label(congruence). [clausify(2)].
kept: 19 -->_s0(x,y) | ->_s0(ssp'(z,x),ssp'(z,y)) # label(congruence). [clausify(3)].
kept: 20 -->_s0(x,y) | ->_s0(sub(x,z),sub(y,z)) # label(congruence). [clausify(4)].
kept: 21 -->_s0(x,y) | ->_s0(sub(z,x),sub(z,y)) # label(congruence). [clausify(5)].
kept: 22 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(6)].
kept: 23 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(7)].
kept: 24 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(8)].
kept: 25 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(9)].
kept: 26 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(10)].
kept: 27 -->*_s0(get(x),tp2(y,z)) | ->_s0(get(cons(u,x)),tp2(y,cons(u,z))) # label(replacement). [clausify(11)].
kept: 28 ->_s0(get(cons(x,y)),tp2(x,y)) # label(replacement). [assumption].
kept: 29 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | ->_s0(ssp'(cons(y,u),x),cons(y,w)) # label(replacement). [clausify(12)].
kept: 30 -->*_s0(get(x),tp2(y,z)) | -->*_s0(sub(u,y),w) | -->*_s0(ssp'(cons(v5,z),w),v6) | ->_s0(ssp'(cons(v5,x),u),cons(y,v6)) # label(replacement). [clausify(13)].
kept: 31 ->_s0(ssp'(x,0),nil) # label(replacement). [assumption].
kept: 32 -->*_s0(sub(x,y),z) | ->_s0(sub(s(x),s(y)),z) # label(replacement). [clausify(14)].
kept: 33 ->_s0(sub(x,0),x) # label(replacement). [assumption].
kept: 34 ->*_s0(x,x) # label(reflexivity). [assumption].
kept: 35 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(15)].
kept: 36 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | -->*_s0(get(u),tp2(v5,v6)) | -->*_s0(sub(x,v5),v7) | -->*_s0(ssp'(cons(y,v6),v7),v8) # label(goal) # answer(goal). [deny(16)].
============================== end of process initial clauses ========
============================== CLAUSES FOR SEARCH ====================
% Clauses after input processing:
formulas(usable).
end_of_list.
formulas(sos).
17 -->_s0(x,y) | ->_s0(get(x),get(y)) # label(congruence). [clausify(1)].
18 -->_s0(x,y) | ->_s0(ssp'(x,z),ssp'(y,z)) # label(congruence). [clausify(2)].
19 -->_s0(x,y) | ->_s0(ssp'(z,x),ssp'(z,y)) # label(congruence). [clausify(3)].
20 -->_s0(x,y) | ->_s0(sub(x,z),sub(y,z)) # label(congruence). [clausify(4)].
21 -->_s0(x,y) | ->_s0(sub(z,x),sub(z,y)) # label(congruence). [clausify(5)].
22 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(6)].
23 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(7)].
24 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(8)].
25 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(9)].
26 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(10)].
27 -->*_s0(get(x),tp2(y,z)) | ->_s0(get(cons(u,x)),tp2(y,cons(u,z))) # label(replacement). [clausify(11)].
28 ->_s0(get(cons(x,y)),tp2(x,y)) # label(replacement). [assumption].
29 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | ->_s0(ssp'(cons(y,u),x),cons(y,w)) # label(replacement). [clausify(12)].
30 -->*_s0(get(x),tp2(y,z)) | -->*_s0(sub(u,y),w) | -->*_s0(ssp'(cons(v5,z),w),v6) | ->_s0(ssp'(cons(v5,x),u),cons(y,v6)) # label(replacement). [clausify(13)].
31 ->_s0(ssp'(x,0),nil) # label(replacement). [assumption].
32 -->*_s0(sub(x,y),z) | ->_s0(sub(s(x),s(y)),z) # label(replacement). [clausify(14)].
33 ->_s0(sub(x,0),x) # label(replacement). [assumption].
34 ->*_s0(x,x) # label(reflexivity). [assumption].
35 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(15)].
36 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | -->*_s0(get(u),tp2(v5,v6)) | -->*_s0(sub(x,v5),v7) | -->*_s0(ssp'(cons(y,v6),v7),v8) # label(goal) # answer(goal). [deny(16)].
end_of_list.
formulas(demodulators).
end_of_list.
============================== end of clauses for search =============
============================== SEARCH ================================
% Starting search at 0.01 seconds.
given #1 (I,wt=8): 17 -->_s0(x,y) | ->_s0(get(x),get(y)) # label(congruence). [clausify(1)].
given #2 (I,wt=10): 18 -->_s0(x,y) | ->_s0(ssp'(x,z),ssp'(y,z)) # label(congruence). [clausify(2)].
given #3 (I,wt=10): 19 -->_s0(x,y) | ->_s0(ssp'(z,x),ssp'(z,y)) # label(congruence). [clausify(3)].
given #4 (I,wt=10): 20 -->_s0(x,y) | ->_s0(sub(x,z),sub(y,z)) # label(congruence). [clausify(4)].
given #5 (I,wt=10): 21 -->_s0(x,y) | ->_s0(sub(z,x),sub(z,y)) # label(congruence). [clausify(5)].
given #6 (I,wt=10): 22 -->_s0(x,y) | ->_s0(cons(x,z),cons(y,z)) # label(congruence). [clausify(6)].
given #7 (I,wt=10): 23 -->_s0(x,y) | ->_s0(cons(z,x),cons(z,y)) # label(congruence). [clausify(7)].
given #8 (I,wt=8): 24 -->_s0(x,y) | ->_s0(s(x),s(y)) # label(congruence). [clausify(8)].
given #9 (I,wt=10): 25 -->_s0(x,y) | ->_s0(tp2(x,z),tp2(y,z)) # label(congruence). [clausify(9)].
given #10 (I,wt=10): 26 -->_s0(x,y) | ->_s0(tp2(z,x),tp2(z,y)) # label(congruence). [clausify(10)].
given #11 (I,wt=16): 27 -->*_s0(get(x),tp2(y,z)) | ->_s0(get(cons(u,x)),tp2(y,cons(u,z))) # label(replacement). [clausify(11)].
given #12 (I,wt=8): 28 ->_s0(get(cons(x,y)),tp2(x,y)) # label(replacement). [assumption].
given #13 (I,wt=19): 29 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | ->_s0(ssp'(cons(y,u),x),cons(y,w)) # label(replacement). [clausify(12)].
given #14 (I,wt=27): 30 -->*_s0(get(x),tp2(y,z)) | -->*_s0(sub(u,y),w) | -->*_s0(ssp'(cons(v5,z),w),v6) | ->_s0(ssp'(cons(v5,x),u),cons(y,v6)) # label(replacement). [clausify(13)].
given #15 (I,wt=5): 31 ->_s0(ssp'(x,0),nil) # label(replacement). [assumption].
given #16 (I,wt=12): 32 -->*_s0(sub(x,y),z) | ->_s0(sub(s(x),s(y)),z) # label(replacement). [clausify(14)].
given #17 (I,wt=5): 33 ->_s0(sub(x,0),x) # label(replacement). [assumption].
given #18 (I,wt=3): 34 ->*_s0(x,x) # label(reflexivity). [assumption].
given #19 (I,wt=9): 35 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(15)].
given #20 (I,wt=28): 36 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | -->*_s0(get(u),tp2(v5,v6)) | -->*_s0(sub(x,v5),v7) | -->*_s0(ssp'(cons(y,v6),v7),v8) # label(goal) # answer(goal). [deny(16)].
============================== PROOF =================================
% Proof 1 at 0.01 (+ 0.00) seconds: goal.
% Length of proof is 9.
% Level of proof is 3.
% Maximum clause weight is 28.000.
% Given clauses 20.
15 ->_s0(x,y) & ->*_s0(y,z) -> ->*_s0(x,z) # label(transitivity) # label(non_clause). [assumption].
16 (exists x14 exists x15 exists x16 exists x17 exists x18 exists x19 exists x20 exists x21 exists x22 (->*_s0(sub(x14,x15),x16) & ->*_s0(ssp'(x17,x16),x18) & ->*_s0(get(x17),tp2(x19,x20)) & ->*_s0(sub(x14,x19),x21) & ->*_s0(ssp'(cons(x15,x20),x21),x22))) # label(goal) # label(non_clause) # label(goal). [goal].
28 ->_s0(get(cons(x,y)),tp2(x,y)) # label(replacement). [assumption].
34 ->*_s0(x,x) # label(reflexivity). [assumption].
35 -->_s0(x,y) | -->*_s0(y,z) | ->*_s0(x,z) # label(transitivity). [clausify(15)].
36 -->*_s0(sub(x,y),z) | -->*_s0(ssp'(u,z),w) | -->*_s0(get(u),tp2(v5,v6)) | -->*_s0(sub(x,v5),v7) | -->*_s0(ssp'(cons(y,v6),v7),v8) # label(goal) # answer(goal). [deny(16)].
71 ->*_s0(get(cons(x,y)),tp2(x,y)). [ur(35,a,28,a,b,34,a)].
81 -->*_s0(get(x),tp2(y,z)) # answer(goal). [ur(36,a,34,a,b,34,a,d,34,a,e,34,a)].
82 $F # answer(goal). [resolve(81,a,71,a)].
============================== end of proof ==========================
============================== STATISTICS ============================
Given=20. Generated=66. Kept=65. proofs=1.
Usable=20. Sos=35. Demods=0. Limbo=9, 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=25. Nonunit_bsub_feature_tests=27.
Megabytes=0.22.
User_CPU=0.01, System_CPU=0.00, Wall_clock=0.
============================== end of statistics =====================
============================== end of search =========================
THEOREM PROVED
Exiting with 1 proof.
Process 3317521 exit (max_proofs) Tue Jul 30 09:21:25 2024
The problem is feasible.