Certification Problem

Input (TPDB TRS_Standard/Rubio_04/elimdupl)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Dependency Graph Processor

The dependency pairs are split into 3 components.

• The 1^st component contains the pair

  purge#(add(N,X))

  →

  purge#(rm(N,X))

  (17)

  1.1.1 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [ifrm#(x1, x2, x3)]	=	0
  [s(x1)]	=	1
  [purge#(x1)]	=	x1 + 0
  [eq(x1, x2)]	=	1
  [false]	=	1
  [true]	=	1
  [purge(x1)]	=	0
  [eq#(x1, x2)]	=	0
  [0]	=	1
  [ifrm(x1, x2, x3)]	=	x2 + x3 + 1
  [nil]	=	1
  [add(x1, x2)]	=	x1 + x2 + 2
  [rm(x1, x2)]	=	x1 + x2 + 1
  [rm#(x1, x2)]	=	0

  together with the usable rules

  eq(s(X),s(Y))	→	eq(X,Y)	(4)
  ifrm(false,N,add(M,X))	→	add(M,rm(N,X))	(8)
  eq(0,0)	→	true	(1)
  eq(s(X),0)	→	false	(3)
  rm(N,nil)	→	nil	(5)
  ifrm(true,N,add(M,X))	→	rm(N,X)	(7)
  rm(N,add(M,X))	→	ifrm(eq(N,M),N,add(M,X))	(6)
  eq(0,s(X))	→	false	(2)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  purge#(add(N,X))

  →

  purge#(rm(N,X))

  (17)

  could be deleted.

  1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 2^nd component contains the pair

  rm#(N,add(M,X))	→	ifrm#(eq(N,M),N,add(M,X))	(16)
  ifrm#(true,N,add(M,X))	→	rm#(N,X)	(15)
  ifrm#(false,N,add(M,X))	→	rm#(N,X)	(12)

  1.1.2 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [ifrm#(x1, x2, x3)]	=	x1 + x2 + x3 + 0
  [s(x1)]	=	1
  [purge#(x1)]	=	x1 + 0
  [eq(x1, x2)]	=	9227
  [false]	=	9227
  [true]	=	9227
  [purge(x1)]	=	0
  [eq#(x1, x2)]	=	0
  [0]	=	1
  [ifrm(x1, x2, x3)]	=	x2 + x3 + 1
  [nil]	=	36459
  [add(x1, x2)]	=	x2 + 2
  [rm(x1, x2)]	=	x1 + x2 + 1
  [rm#(x1, x2)]	=	x1 + x2 + 9228

  together with the usable rules

  eq(s(X),s(Y))	→	eq(X,Y)	(4)
  ifrm(false,N,add(M,X))	→	add(M,rm(N,X))	(8)
  eq(0,0)	→	true	(1)
  eq(s(X),0)	→	false	(3)
  rm(N,nil)	→	nil	(5)
  ifrm(true,N,add(M,X))	→	rm(N,X)	(7)
  rm(N,add(M,X))	→	ifrm(eq(N,M),N,add(M,X))	(6)
  eq(0,s(X))	→	false	(2)

  (w.r.t. the implicit argument filter of the reduction pair), the pairs

  rm#(N,add(M,X))	→	ifrm#(eq(N,M),N,add(M,X))	(16)
  ifrm#(true,N,add(M,X))	→	rm#(N,X)	(15)
  ifrm#(false,N,add(M,X))	→	rm#(N,X)	(12)

  could be deleted.

  1.1.2.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.

• The 3^rd component contains the pair

  eq#(s(X),s(Y))

  →

  eq#(X,Y)

  (13)

  1.1.3 Reduction Pair Processor with Usable Rules

  Using the Max-polynomial interpretation

  [ifrm#(x1, x2, x3)]	=	x1 + 0
  [s(x1)]	=	x1 + 1
  [purge#(x1)]	=	x1 + 0
  [eq(x1, x2)]	=	9227
  [false]	=	1
  [true]	=	9227
  [purge(x1)]	=	0
  [eq#(x1, x2)]	=	x1 + x2 + 0
  [0]	=	1
  [ifrm(x1, x2, x3)]	=	x2 + x3 + 17678
  [nil]	=	1
  [add(x1, x2)]	=	x2 + 1
  [rm(x1, x2)]	=	x1 + x2 + 17678
  [rm#(x1, x2)]	=	9228

  together with the usable rules

  eq(s(X),s(Y))	→	eq(X,Y)	(4)
  ifrm(false,N,add(M,X))	→	add(M,rm(N,X))	(8)
  eq(0,0)	→	true	(1)
  eq(s(X),0)	→	false	(3)
  rm(N,nil)	→	nil	(5)
  ifrm(true,N,add(M,X))	→	rm(N,X)	(7)
  rm(N,add(M,X))	→	ifrm(eq(N,M),N,add(M,X))	(6)
  eq(0,s(X))	→	false	(2)

  (w.r.t. the implicit argument filter of the reduction pair), the pair

  eq#(s(X),s(Y))

  →

  eq#(X,Y)

  (13)

  could be deleted.

  1.1.3.1 Dependency Graph Processor

  The dependency pairs are split into 0 components.