/export/starexec/sandbox2/solver/bin/starexec_run_default /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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YES

Problem 1: 

(VAR A B F F1 F2 Fs G1 G2 Gs Left NF Right V X X0 X1 Xs Ys)
(RULES
intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
u'1'1(intersect'ii'out) -> intersect'ii'out
u'10'1(reduce'ii'out) -> reduce'ii'out
u'11'1(reduce'ii'out) -> reduce'ii'out
u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
u'12'2(reduce'ii'out) -> reduce'ii'out
u'13'1(reduce'ii'out) -> reduce'ii'out
u'14'1(reduce'ii'out) -> reduce'ii'out
u'15'1(intersect'ii'out) -> reduce'ii'out
u'16'1(reduce'ii'out) -> tautology'i'out
u'2'1(intersect'ii'out) -> intersect'ii'out
u'3'1(reduce'ii'out) -> reduce'ii'out
u'4'1(reduce'ii'out) -> reduce'ii'out
u'5'1(reduce'ii'out) -> reduce'ii'out
u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
u'6'2(reduce'ii'out) -> reduce'ii'out
u'7'1(reduce'ii'out) -> reduce'ii'out
u'8'1(reduce'ii'out) -> reduce'ii'out
u'9'1(reduce'ii'out) -> reduce'ii'out
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> U'2'1(intersect'ii'in(Xs,Ys))
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> U'1'1(intersect'ii'in(Xs,Ys))
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> U'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> U'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> U'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> U'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> U'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> U'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 REDUCE'II'IN(sequent(nil,nil),sequent(F1,F2)) -> INTERSECT'II'IN(F1,F2)
 REDUCE'II'IN(sequent(nil,nil),sequent(F1,F2)) -> U'15'1(intersect'ii'in(F1,F2))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> U'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> U'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> U'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> U'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 TAUTOLOGY'I'IN(F) -> REDUCE'II'IN(sequent(nil,cons(F,nil)),sequent(nil,nil))
 TAUTOLOGY'I'IN(F) -> U'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> U'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> U'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1: 

SCC Processor:
-> Pairs:
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> U'2'1(intersect'ii'in(Xs,Ys))
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> U'1'1(intersect'ii'in(Xs,Ys))
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> U'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> U'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> U'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> U'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> U'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> U'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 REDUCE'II'IN(sequent(nil,nil),sequent(F1,F2)) -> INTERSECT'II'IN(F1,F2)
 REDUCE'II'IN(sequent(nil,nil),sequent(F1,F2)) -> U'15'1(intersect'ii'in(F1,F2))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> U'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> U'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> U'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> U'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 TAUTOLOGY'I'IN(F) -> REDUCE'II'IN(sequent(nil,cons(F,nil)),sequent(nil,nil))
 TAUTOLOGY'I'IN(F) -> U'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> U'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> U'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 INTERSECT'II'IN(cons(X0,Xs),Ys) -> INTERSECT'II'IN(Xs,Ys)
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Projection:
 pi(INTERSECT'II'IN) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.1: 

Subterm Processor:
-> Pairs:
 INTERSECT'II'IN(Xs,cons(X0,Ys)) -> INTERSECT'II'IN(Xs,Ys)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Projection:
 pi(INTERSECT'II'IN) = 2

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(if(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 2.X1 + 2
[reduce'ii'in](X1,X2) = 0
[u'1'1](X) = 2
[u'10'1](X) = 2.X
[u'11'1](X) = 2.X
[u'12'1](X1,X2,X3,X4,X5) = 2.X1
[u'12'2](X) = X
[u'13'1](X) = 2.X
[u'14'1](X) = 2.X
[u'15'1](X) = 0
[u'2'1](X) = X
[u'3'1](X) = 2.X
[u'4'1](X) = 2.X
[u'5'1](X) = 2.X
[u'6'1](X1,X2,X3,X4,X5) = 2.X1
[u'6'2](X) = 2.X
[u'7'1](X) = 2.X
[u'8'1](X) = 2.X
[u'9'1](X) = 2.X
[cons](X1,X2) = X1 + X2
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 2
[nil] = 0
[p](X) = 2.X
[reduce'ii'out] = 0
[sequent](X1,X2) = 2.X1 + 2.X2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = X1 + X2
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + X2 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + X5 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X1 + 2.X2 + 2.X3 + 2.X4 + X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(iff(A,B),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 0
[u'1'1](X) = 2.X
[u'10'1](X) = 0
[u'11'1](X) = 2.X
[u'12'1](X1,X2,X3,X4,X5) = 2.X1
[u'12'2](X) = 0
[u'13'1](X) = 0
[u'14'1](X) = 2.X
[u'15'1](X) = 0
[u'2'1](X) = 0
[u'3'1](X) = 2.X
[u'4'1](X) = 0
[u'5'1](X) = 2.X
[u'6'1](X1,X2,X3,X4,X5) = 2.X1
[u'6'2](X) = 0
[u'7'1](X) = 2.X
[u'8'1](X) = 2.X
[u'9'1](X) = 2.X
[cons](X1,X2) = 2.X1 + X2
[if](X1,X2) = X1 + X2
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 2
[p](X) = 2.X + 2
[reduce'ii'out] = 0
[sequent](X1,X2) = X1 + X2
[x'2a](X1,X2) = X1 + X2 + 1
[x'2b](X1,X2) = X1 + X2
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + 2
[U'12'1](X1,X2,X3,X4,X5) = X2 + 2.X3 + X4 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X1 + 2.X2 + X3 + X4 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> REDUCE'II'IN(sequent(Fs,Gs),sequent(cons(p(V),Left),Right))
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 2
[reduce'ii'in](X1,X2) = 2
[u'1'1](X) = X
[u'10'1](X) = 2
[u'11'1](X) = X
[u'12'1](X1,X2,X3,X4,X5) = X1
[u'12'2](X) = 2
[u'13'1](X) = 2
[u'14'1](X) = 2
[u'15'1](X) = X
[u'2'1](X) = X
[u'3'1](X) = X
[u'4'1](X) = 2
[u'5'1](X) = X
[u'6'1](X1,X2,X3,X4,X5) = X1
[u'6'2](X) = X
[u'7'1](X) = X
[u'8'1](X) = 2
[u'9'1](X) = 2
[cons](X1,X2) = X1 + X2
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 2
[nil] = 0
[p](X) = 2
[reduce'ii'out] = 2
[sequent](X1,X2) = 2.X1 + 2.X2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = X1 + X2 + 1
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X1 + 2.X2 + 2.X3 + 2.X4

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,cons(F2,Fs)),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 2.X1 + 1
[u'1'1](X) = 0
[u'10'1](X) = X + 2
[u'11'1](X) = 2
[u'12'1](X1,X2,X3,X4,X5) = 2.X3 + 2.X4 + 2
[u'12'2](X) = 2
[u'13'1](X) = X
[u'14'1](X) = X + 2
[u'15'1](X) = 2.X + 2
[u'2'1](X) = 2.X
[u'3'1](X) = 2
[u'4'1](X) = 2
[u'5'1](X) = 2
[u'6'1](X1,X2,X3,X4,X5) = X1
[u'6'2](X) = 2
[u'7'1](X) = 2
[u'8'1](X) = 2
[u'9'1](X) = 2
[cons](X1,X2) = 2.X1 + X2
[if](X1,X2) = X1 + X2
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 0
[p](X) = X + 1
[reduce'ii'out] = 2
[sequent](X1,X2) = X1 + X2 + 1
[x'2a](X1,X2) = X1 + X2 + 2
[x'2b](X1,X2) = X1 + X2
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + 1
[U'12'1](X1,X2,X3,X4,X5) = X2 + 2.X3 + X4 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X2 + X3 + X4 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> REDUCE'II'IN(sequent(cons(F1,Fs),Gs),NF)
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 2.X1 + 2
[u'1'1](X) = 0
[u'10'1](X) = X + 2
[u'11'1](X) = 2
[u'12'1](X1,X2,X3,X4,X5) = X1 + X3
[u'12'2](X) = 2
[u'13'1](X) = X
[u'14'1](X) = X + 1
[u'15'1](X) = 2
[u'2'1](X) = 2.X
[u'3'1](X) = 2
[u'4'1](X) = 2
[u'5'1](X) = 2
[u'6'1](X1,X2,X3,X4,X5) = X1 + 2.X2 + 1
[u'6'2](X) = 2
[u'7'1](X) = X
[u'8'1](X) = 2
[u'9'1](X) = X
[cons](X1,X2) = X1 + X2 + 1
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 2
[p](X) = X + 2
[reduce'ii'out] = 2
[sequent](X1,X2) = 2.X1 + 2.X2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = X1 + X2 + 1
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + X2
[U'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + X5 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> U'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 2.X1 + 2.X2 + 2
[u'1'1](X) = 0
[u'10'1](X) = 2
[u'11'1](X) = X + 1
[u'12'1](X1,X2,X3,X4,X5) = X1 + X3
[u'12'2](X) = 2
[u'13'1](X) = X
[u'14'1](X) = X
[u'15'1](X) = 2.X + 2
[u'2'1](X) = 2.X
[u'3'1](X) = X
[u'4'1](X) = 2
[u'5'1](X) = 2
[u'6'1](X1,X2,X3,X4,X5) = X3 + 2.X4 + 2.X5 + 2
[u'6'2](X) = 2
[u'7'1](X) = 2
[u'8'1](X) = X
[u'9'1](X) = 2
[cons](X1,X2) = X1 + X2
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 2
[p](X) = 0
[reduce'ii'out] = 2
[sequent](X1,X2) = 2.X1 + 2.X2 + 2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = X1 + X2 + 1
[x'2d](X) = X
[REDUCE'II'IN](X1,X2) = X1 + 2.X2
[U'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2.X5 + 2
[U'6'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2.X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
 U'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> REDUCE'II'IN(sequent(cons(F2,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(cons(x'2d(F1),Fs),Gs),NF) -> REDUCE'II'IN(sequent(Fs,cons(F1,Gs)),NF)
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 2
[reduce'ii'in](X1,X2) = 2.X1 + 2
[u'1'1](X) = X
[u'10'1](X) = 1
[u'11'1](X) = X
[u'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2
[u'12'2](X) = 2
[u'13'1](X) = 2
[u'14'1](X) = 2
[u'15'1](X) = 1
[u'2'1](X) = 2
[u'3'1](X) = 2
[u'4'1](X) = 2
[u'5'1](X) = X
[u'6'1](X1,X2,X3,X4,X5) = X2 + 2.X4 + 2
[u'6'2](X) = 2
[u'7'1](X) = X + 2
[u'8'1](X) = 1
[u'9'1](X) = 2
[cons](X1,X2) = 2.X1 + X2
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 2
[nil] = 2
[p](X) = 2
[reduce'ii'out] = 1
[sequent](X1,X2) = X1 + X2 + 2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = X1 + X2
[x'2d](X) = X + 1
[REDUCE'II'IN](X1,X2) = X1 + X2
[U'12'1](X1,X2,X3,X4,X5) = X2 + 2.X3 + X4 + X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> REDUCE'II'IN(sequent(nil,Gs),sequent(Left,cons(p(V),Right)))
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 2.X2 + 2
[reduce'ii'in](X1,X2) = 0
[u'1'1](X) = 2
[u'10'1](X) = 0
[u'11'1](X) = 0
[u'12'1](X1,X2,X3,X4,X5) = 2.X1
[u'12'2](X) = 2.X
[u'13'1](X) = 2.X
[u'14'1](X) = 0
[u'15'1](X) = 0
[u'2'1](X) = 2
[u'3'1](X) = 2.X
[u'4'1](X) = 2.X
[u'5'1](X) = 2.X
[u'6'1](X1,X2,X3,X4,X5) = 2.X1
[u'6'2](X) = 2.X
[u'7'1](X) = 0
[u'8'1](X) = 0
[u'9'1](X) = 2.X
[cons](X1,X2) = X1 + X2
[if](X1,X2) = 2.X1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 2
[nil] = 0
[p](X) = X + 2
[reduce'ii'out] = 0
[sequent](X1,X2) = 2.X2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = 2.X1 + X2
[x'2d](X) = 0
[REDUCE'II'IN](X1,X2) = X1 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X3 + 2.X4 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(if(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 2
[reduce'ii'in](X1,X2) = 2
[u'1'1](X) = X
[u'10'1](X) = 2
[u'11'1](X) = 2
[u'12'1](X1,X2,X3,X4,X5) = X1
[u'12'2](X) = 1
[u'13'1](X) = X
[u'14'1](X) = 2
[u'15'1](X) = X
[u'2'1](X) = X
[u'3'1](X) = 2
[u'4'1](X) = 2
[u'5'1](X) = X
[u'6'1](X1,X2,X3,X4,X5) = X1
[u'6'2](X) = 1
[u'7'1](X) = 1
[u'8'1](X) = 2
[u'9'1](X) = 1
[cons](X1,X2) = X1 + X2
[if](X1,X2) = X1 + X2 + 1
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 2
[nil] = 2
[p](X) = 0
[reduce'ii'out] = 1
[sequent](X1,X2) = 2.X2
[x'2a](X1,X2) = X1 + X2
[x'2b](X1,X2) = 2.X1 + X2
[x'2d](X) = 0
[REDUCE'II'IN](X1,X2) = X1 + 2.X2 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X3 + 2.X4 + 2.X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(iff(A,B),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 0
[u'1'1](X) = 2.X
[u'10'1](X) = X
[u'11'1](X) = 2.X
[u'12'1](X1,X2,X3,X4,X5) = X1
[u'12'2](X) = 2.X
[u'13'1](X) = 2.X
[u'14'1](X) = 2.X
[u'15'1](X) = 0
[u'2'1](X) = 2.X
[u'3'1](X) = 2.X
[u'4'1](X) = 2.X
[u'5'1](X) = X
[u'6'1](X1,X2,X3,X4,X5) = 0
[u'6'2](X) = X
[u'7'1](X) = 2.X
[u'8'1](X) = 2.X
[u'9'1](X) = 2.X
[cons](X1,X2) = X1 + X2
[if](X1,X2) = 0
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 2
[p](X) = 2
[reduce'ii'out] = 0
[sequent](X1,X2) = 2.X1 + 2.X2
[x'2a](X1,X2) = 2.X1 + 2.X2 + 1
[x'2b](X1,X2) = 2.X1 + X2 + 1
[x'2d](X) = 2.X + 2
[REDUCE'II'IN](X1,X2) = X1 + 2.X2 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X2 + 2.X3 + 2.X4 + 2.X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,Gs)),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = 0
[reduce'ii'in](X1,X2) = 0
[u'1'1](X) = 2.X
[u'10'1](X) = 2.X
[u'11'1](X) = 2.X
[u'12'1](X1,X2,X3,X4,X5) = 2.X1
[u'12'2](X) = 0
[u'13'1](X) = X
[u'14'1](X) = 2.X
[u'15'1](X) = 0
[u'2'1](X) = 0
[u'3'1](X) = 2.X
[u'4'1](X) = 0
[u'5'1](X) = 2.X
[u'6'1](X1,X2,X3,X4,X5) = 2.X1
[u'6'2](X) = 2.X
[u'7'1](X) = 2.X
[u'8'1](X) = 2.X
[u'9'1](X) = 2.X
[cons](X1,X2) = X1 + X2
[if](X1,X2) = X1 + 2.X2 + 2
[iff](X1,X2) = X1 + 2.X2 + 2
[intersect'ii'out] = 0
[nil] = 1
[p](X) = 2.X
[reduce'ii'out] = 0
[sequent](X1,X2) = X1 + X2
[x'2a](X1,X2) = 2.X1 + 2.X2 + 2
[x'2b](X1,X2) = X1 + 2.X2 + 2
[x'2d](X) = 2.X + 2
[REDUCE'II'IN](X1,X2) = 2.X1 + 2.X2 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X1 + 2.X2 + 2.X3 + 2.X4 + 2.X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> U'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[intersect'ii'in](X1,X2) = X2 + 1
[reduce'ii'in](X1,X2) = 2.X2 + 2
[u'1'1](X) = X
[u'10'1](X) = 2
[u'11'1](X) = 2
[u'12'1](X1,X2,X3,X4,X5) = 2.X5 + 2
[u'12'2](X) = 2
[u'13'1](X) = X
[u'14'1](X) = X
[u'15'1](X) = X + 1
[u'2'1](X) = 1
[u'3'1](X) = X
[u'4'1](X) = 2
[u'5'1](X) = 2
[u'6'1](X1,X2,X3,X4,X5) = 2.X5 + 2
[u'6'2](X) = 2
[u'7'1](X) = X
[u'8'1](X) = 2
[u'9'1](X) = 2
[cons](X1,X2) = 2.X1 + X2
[if](X1,X2) = 2
[iff](X1,X2) = 2.X1 + 2.X2 + 2
[intersect'ii'out] = 1
[nil] = 0
[p](X) = 0
[reduce'ii'out] = 2
[sequent](X1,X2) = X2
[x'2a](X1,X2) = 2.X2 + 2
[x'2b](X1,X2) = 2.X1 + 2.X2
[x'2d](X) = 2
[REDUCE'II'IN](X1,X2) = X1 + 2.X2 + 2
[U'12'1](X1,X2,X3,X4,X5) = 2.X3 + X4 + 2.X5 + 2

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
 U'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> REDUCE'II'IN(sequent(Fs,cons(G2,Gs)),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> REDUCE'II'IN(sequent(Fs,cons(G1,cons(G2,Gs))),NF)
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons](X1,X2) = X1 + X2
[sequent](X1,X2) = X2
[x'2b](X1,X2) = X1 + 2.X2 + 2
[x'2d](X) = 1
[REDUCE'II'IN](X1,X2) = X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
->->-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out

Problem 1.2: 

Reduction Pair Processor:
-> Pairs:
 REDUCE'II'IN(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> REDUCE'II'IN(sequent(cons(G1,Fs),Gs),NF)
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
-> Usable rules:
 Empty
->Interpretation type:
 Linear
->Coefficients:
 Natural Numbers
->Dimension:
 1
->Bound:
 2
->Interpretation:
 
[cons](X1,X2) = 2.X2 + 2
[sequent](X1,X2) = 2.X2
[x'2d](X) = 2
[REDUCE'II'IN](X1,X2) = 2.X1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 intersect'ii'in(cons(X,X0),cons(X,X1)) -> intersect'ii'out
 intersect'ii'in(cons(X0,Xs),Ys) -> u'2'1(intersect'ii'in(Xs,Ys))
 intersect'ii'in(Xs,cons(X0,Ys)) -> u'1'1(intersect'ii'in(Xs,Ys))
 reduce'ii'in(sequent(cons(if(A,B),Fs),Gs),NF) -> u'3'1(reduce'ii'in(sequent(cons(x'2b(x'2d(B),A),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(iff(A,B),Fs),Gs),NF) -> u'4'1(reduce'ii'in(sequent(cons(x'2a(if(A,B),if(B,A)),Fs),Gs),NF))
 reduce'ii'in(sequent(cons(p(V),Fs),Gs),sequent(Left,Right)) -> u'10'1(reduce'ii'in(sequent(Fs,Gs),sequent(cons(p(V),Left),Right)))
 reduce'ii'in(sequent(cons(x'2a(F1,F2),Fs),Gs),NF) -> u'5'1(reduce'ii'in(sequent(cons(F1,cons(F2,Fs)),Gs),NF))
 reduce'ii'in(sequent(cons(x'2b(F1,F2),Fs),Gs),NF) -> u'6'1(reduce'ii'in(sequent(cons(F1,Fs),Gs),NF),F2,Fs,Gs,NF)
 reduce'ii'in(sequent(cons(x'2d(F1),Fs),Gs),NF) -> u'7'1(reduce'ii'in(sequent(Fs,cons(F1,Gs)),NF))
 reduce'ii'in(sequent(nil,cons(p(V),Gs)),sequent(Left,Right)) -> u'14'1(reduce'ii'in(sequent(nil,Gs),sequent(Left,cons(p(V),Right))))
 reduce'ii'in(sequent(nil,nil),sequent(F1,F2)) -> u'15'1(intersect'ii'in(F1,F2))
 reduce'ii'in(sequent(Fs,cons(if(A,B),Gs)),NF) -> u'8'1(reduce'ii'in(sequent(Fs,cons(x'2b(x'2d(B),A),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(iff(A,B),Gs)),NF) -> u'9'1(reduce'ii'in(sequent(Fs,cons(x'2a(if(A,B),if(B,A)),Gs)),NF))
 reduce'ii'in(sequent(Fs,cons(x'2a(G1,G2),Gs)),NF) -> u'12'1(reduce'ii'in(sequent(Fs,cons(G1,Gs)),NF),Fs,G2,Gs,NF)
 reduce'ii'in(sequent(Fs,cons(x'2b(G1,G2),Gs)),NF) -> u'11'1(reduce'ii'in(sequent(Fs,cons(G1,cons(G2,Gs))),NF))
 reduce'ii'in(sequent(Fs,cons(x'2d(G1),Gs)),NF) -> u'13'1(reduce'ii'in(sequent(cons(G1,Fs),Gs),NF))
 tautology'i'in(F) -> u'16'1(reduce'ii'in(sequent(nil,cons(F,nil)),sequent(nil,nil)))
 u'1'1(intersect'ii'out) -> intersect'ii'out
 u'10'1(reduce'ii'out) -> reduce'ii'out
 u'11'1(reduce'ii'out) -> reduce'ii'out
 u'12'1(reduce'ii'out,Fs,G2,Gs,NF) -> u'12'2(reduce'ii'in(sequent(Fs,cons(G2,Gs)),NF))
 u'12'2(reduce'ii'out) -> reduce'ii'out
 u'13'1(reduce'ii'out) -> reduce'ii'out
 u'14'1(reduce'ii'out) -> reduce'ii'out
 u'15'1(intersect'ii'out) -> reduce'ii'out
 u'16'1(reduce'ii'out) -> tautology'i'out
 u'2'1(intersect'ii'out) -> intersect'ii'out
 u'3'1(reduce'ii'out) -> reduce'ii'out
 u'4'1(reduce'ii'out) -> reduce'ii'out
 u'5'1(reduce'ii'out) -> reduce'ii'out
 u'6'1(reduce'ii'out,F2,Fs,Gs,NF) -> u'6'2(reduce'ii'in(sequent(cons(F2,Fs),Gs),NF))
 u'6'2(reduce'ii'out) -> reduce'ii'out
 u'7'1(reduce'ii'out) -> reduce'ii'out
 u'8'1(reduce'ii'out) -> reduce'ii'out
 u'9'1(reduce'ii'out) -> reduce'ii'out
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.