YES

Problem 1: 

(VAR v_NonEmpty:S x:S y:S z:S)
(RULES
.(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
.(i(x:S),x:S) -> 1
.(i(y:S),.(y:S,z:S)) -> z:S
.(1,x:S) -> x:S
.(x:S,i(x:S)) -> 1
.(x:S,1) -> x:S
.(y:S,.(i(y:S),z:S)) -> z:S
i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
i(i(x:S)) -> x:S
i(1) -> 1
)

Problem 1: 

Dependency Pairs Processor:
-> Pairs:
 .#(.(x:S,y:S),z:S) -> .#(x:S,.(y:S,z:S))
 .#(.(x:S,y:S),z:S) -> .#(y:S,z:S)
 I(.(x:S,y:S)) -> .#(i(y:S),i(x:S))
 I(.(x:S,y:S)) -> I(x:S)
 I(.(x:S,y:S)) -> I(y:S)
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1

Problem 1: 

SCC Processor:
-> Pairs:
 .#(.(x:S,y:S),z:S) -> .#(x:S,.(y:S,z:S))
 .#(.(x:S,y:S),z:S) -> .#(y:S,z:S)
 I(.(x:S,y:S)) -> .#(i(y:S),i(x:S))
 I(.(x:S,y:S)) -> I(x:S)
 I(.(x:S,y:S)) -> I(y:S)
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->Strongly Connected Components:
->->Cycle:
->->-> Pairs:
 .#(.(x:S,y:S),z:S) -> .#(x:S,.(y:S,z:S))
 .#(.(x:S,y:S),z:S) -> .#(y:S,z:S)
->->-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->->Cycle:
->->-> Pairs:
 I(.(x:S,y:S)) -> I(x:S)
 I(.(x:S,y:S)) -> I(y:S)
->->-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1


The problem is decomposed in 2 subproblems.

Problem 1.1: 

Subterm Processor:
-> Pairs:
 .#(.(x:S,y:S),z:S) -> .#(x:S,.(y:S,z:S))
 .#(.(x:S,y:S),z:S) -> .#(y:S,z:S)
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->Projection:
 pi(.#) = 1

Problem 1.1: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.

Problem 1.2: 

Subterm Processor:
-> Pairs:
 I(.(x:S,y:S)) -> I(x:S)
 I(.(x:S,y:S)) -> I(y:S)
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->Projection:
 pi(I) = 1

Problem 1.2: 

SCC Processor:
-> Pairs:
 Empty
-> Rules:
 .(.(x:S,y:S),z:S) -> .(x:S,.(y:S,z:S))
 .(i(x:S),x:S) -> 1
 .(i(y:S),.(y:S,z:S)) -> z:S
 .(1,x:S) -> x:S
 .(x:S,i(x:S)) -> 1
 .(x:S,1) -> x:S
 .(y:S,.(i(y:S),z:S)) -> z:S
 i(.(x:S,y:S)) -> .(i(y:S),i(x:S))
 i(i(x:S)) -> x:S
 i(1) -> 1
->Strongly Connected Components:
 There is no strongly connected component

The problem is finite.