Certification Problem

Input (TPDB TRS_Equational/Mixed_AC_and_C/AC47)

The rewrite relation of the following equational TRS is considered.

Associative symbols: or

Commutative symbols: eq, or

Property / Task

Answer / Result

Proof (by NaTT @ termCOMP 2023)

1 AC Dependency Pair Transformation

• The following set of (strict) dependency pairs is constructed for the TRS.

  if_reach_1#(true,x,y,edge(u,v,i),h)	→	eq#(y,v)	(22)
  if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(v,y,union(i,h),empty)	(23)
  union#(edge(x,y,i),h)	→	union#(i,h)	(24)
  reach#(x,y,edge(u,v,i),h)	→	eq#(x,u)	(25)
  if_reach_2#(false,x,y,edge(u,v,i),h)	→	union#(i,h)	(26)
  if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,h)	(27)
  if_reach_1#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,edge(u,v,h))	(28)
  if_reach_1#(true,x,y,edge(u,v,i),h)	→	if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)	(29)
  reach#(x,y,edge(u,v,i),h)	→	if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)	(30)
  eq#(s(x),s(y))	→	eq#(x,y)	(31)
  if_reach_2#(false,x,y,edge(u,v,i),h)	→	or#(reach(x,y,i,h),reach(v,y,union(i,h),empty))	(32)

  Finiteness for these DPs in combination with the equational DPs is proven as follows.

  1.1 Dependency Graph Processor

  The dependency pairs are split into 4 components.

  • The 1^st component contains the pair

    reach#(x,y,edge(u,v,i),h)	→	if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)	(30)
    if_reach_1#(true,x,y,edge(u,v,i),h)	→	if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)	(29)
    if_reach_1#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,edge(u,v,h))	(28)
    if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(v,y,union(i,h),empty)	(23)
    if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,h)	(27)

    1.1.1 AC Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	20818
    [if_reach_1#(x1,...,x5)]	=	x2 + x3 + x4 + x5 + 1
    [edge(x1, x2, x3)]	=	x1 + x2 + x3 + 17067
    [if_reach_2#(x1,...,x5)]	=	x2 + x3 + x4 + x5 + 0
    [eq(x1, x2)]	=	x1 + x2 + 1
    [false]	=	20821
    [reach(x1,...,x4)]	=	0
    [if_reach_2(x1,...,x5)]	=	0
    [true]	=	4
    [reach#(x1,...,x4)]	=	x1 + x2 + x3 + x4 + 1
    [eq#(x1, x2)]	=	0
    [0]	=	1
    [union(x1, x2)]	=	x1 + x2 + 1
    [or(x1, x2)]	=	0
    [empty]	=	17064
    [if_reach_1(x1,...,x5)]	=	0
    [or#(x1, x2)]	=	0
    [union#(x1, x2)]	=	0

    together with the usable rules

    union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
    union(empty,h)	→	h	(7)

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

    if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,h)	(27)
    if_reach_2#(false,x,y,edge(u,v,i),h)	→	reach#(v,y,union(i,h),empty)	(23)
    if_reach_1#(true,x,y,edge(u,v,i),h)	→	if_reach_2#(eq(y,v),x,y,edge(u,v,i),h)	(29)

    could be deleted.

    1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      reach#(x,y,edge(u,v,i),h)	→	if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)	(30)
      if_reach_1#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,edge(u,v,h))	(28)

      1.1.1.1.1 AC Reduction Pair Processor with Usable Rules

      Using the Max-polynomial interpretation

      [s(x1)]	=	20818
      [if_reach_1#(x1,...,x5)]	=	x2 + x4 + 1
      [edge(x1, x2, x3)]	=	x1 + x2 + x3 + 2
      [if_reach_2#(x1,...,x5)]	=	0
      [eq(x1, x2)]	=	x1 + x2 + 1
      [false]	=	20821
      [reach(x1,...,x4)]	=	0
      [if_reach_2(x1,...,x5)]	=	0
      [true]	=	4
      [reach#(x1,...,x4)]	=	x1 + x3 + 2
      [eq#(x1, x2)]	=	0
      [0]	=	1
      [union(x1, x2)]	=	x1 + x2 + 1
      [or(x1, x2)]	=	0
      [empty]	=	17064
      [if_reach_1(x1,...,x5)]	=	0
      [or#(x1, x2)]	=	0
      [union#(x1, x2)]	=	0

      together with the usable rules

      union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
      union(empty,h)	→	h	(7)

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

      if_reach_1#(false,x,y,edge(u,v,i),h)	→	reach#(x,y,i,edge(u,v,h))	(28)
      reach#(x,y,edge(u,v,i),h)	→	if_reach_1#(eq(x,u),x,y,edge(u,v,i),h)	(30)

      could be deleted.

      1.1.1.1.1.1 Dependency Graph Processor

      The dependency pairs are split into 0 components.

  • The 2^nd component contains the pair

    or#(x,or(y,z))	→	or#(x,y)	(19)
    or#(x,or(y,z))	→	or#(or(x,y),z)	(18)
    or#(x,y)	→	or#(y,x)	(21)

    1.1.2 AC Dependency Pair Problem is trivial

    There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
  • The 3^rd component contains the pair

    union#(edge(x,y,i),h)

    →

    union#(i,h)

    (24)

    1.1.3 AC Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	20818
    [if_reach_1#(x1,...,x5)]	=	1
    [edge(x1, x2, x3)]	=	x1 + x2 + x3 + 2
    [if_reach_2#(x1,...,x5)]	=	0
    [eq(x1, x2)]	=	x1 + x2 + 1
    [false]	=	20821
    [reach(x1,...,x4)]	=	0
    [if_reach_2(x1,...,x5)]	=	0
    [true]	=	4
    [reach#(x1,...,x4)]	=	x3 + 2
    [eq#(x1, x2)]	=	0
    [0]	=	1
    [union(x1, x2)]	=	x1 + x2 + 1
    [or(x1, x2)]	=	0
    [empty]	=	17678
    [if_reach_1(x1,...,x5)]	=	0
    [or#(x1, x2)]	=	0
    [union#(x1, x2)]	=	x1 + 0

    together with the usable rules

    union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
    union(empty,h)	→	h	(7)

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

    union#(edge(x,y,i),h)

    →

    union#(i,h)

    (24)

    could be deleted.

    1.1.3.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.

  • The 4^th component contains the pair

    eq#(x,y)	→	eq#(y,x)	(20)
    eq#(s(x),s(y))	→	eq#(x,y)	(31)

    1.1.4 AC Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 20818
    [if_reach_1#(x1,...,x5)]	=	1
    [edge(x1, x2, x3)]	=	x1 + x2 + x3 + 2
    [if_reach_2#(x1,...,x5)]	=	0
    [eq(x1, x2)]	=	x1 + x2 + 1
    [false]	=	20821
    [reach(x1,...,x4)]	=	0
    [if_reach_2(x1,...,x5)]	=	0
    [true]	=	4
    [reach#(x1,...,x4)]	=	x3 + 2
    [eq#(x1, x2)]	=	x1 + x2 + 0
    [0]	=	1
    [union(x1, x2)]	=	x1 + x2 + 1
    [or(x1, x2)]	=	0
    [empty]	=	52868
    [if_reach_1(x1,...,x5)]	=	0
    [or#(x1, x2)]	=	0
    [union#(x1, x2)]	=	0

    together with the usable rules

    union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
    union(empty,h)	→	h	(7)

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

    eq#(s(x),s(y))

    →

    eq#(x,y)

    (31)

    could be deleted.

    1.1.4.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      eq#(x,y)

      →

      eq#(y,x)

      (20)

      1.1.4.1.1 AC Dependency Pair Problem is trivial

      There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.

• The extended rules of the TRS

  or(or(false,y),_1)	→	or(y,_1)	(33)
  or(or(true,y),_1)	→	or(true,_1)	(34)

  give rise to another dependency pair problem. Finiteness for these DPs in combination with the equational DPs is proven as follows.

  1.2 Dependency Graph Processor

  The dependency pairs are split into 2 components.

  • The 1^st component contains the pair

    eq#(x,y)

    →

    eq#(y,x)

    (20)

    1.2.1 AC Dependency Pair Problem is trivial

    There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.
  • The 2^nd component contains the pair

    or#(x,y)	→	or#(y,x)	(21)
    or#(or(false,y),_1)	→	or#(y,_1)	(35)
    or#(x,or(y,z))	→	or#(or(x,y),z)	(18)
    or#(or(true,y),_1)	→	or#(true,_1)	(36)
    or#(x,or(y,z))	→	or#(x,y)	(19)

    1.2.2 AC Reduction Pair Processor with Usable Rules

    Using the Max-polynomial interpretation

    [s(x1)]	=	x1 + 18578
    [if_reach_1#(x1,...,x5)]	=	1
    [edge(x1, x2, x3)]	=	x1 + x2 + x3 + 2
    [if_reach_2#(x1,...,x5)]	=	0
    [eq(x1, x2)]	=	x1 + x2 + 1
    [false]	=	20821
    [reach(x1,...,x4)]	=	0
    [if_reach_2(x1,...,x5)]	=	0
    [true]	=	4484
    [reach#(x1,...,x4)]	=	x3 + 2
    [eq#(x1, x2)]	=	0
    [0]	=	2241
    [union(x1, x2)]	=	x1 + x2 + 1
    [or(x1, x2)]	=	x1 + x2 + 1
    [empty]	=	52868
    [if_reach_1(x1,...,x5)]	=	0
    [or#(x1, x2)]	=	x1 + x2 + 0
    [union#(x1, x2)]	=	0

    together with the usable rules

    union(edge(x,y,i),h)	→	edge(x,y,union(i,h))	(8)
    or(x,or(y,z))	→	or(or(x,y),z)	(16)
    or(x,y)	→	or(y,x)	(15)
    or(true,y)	→	true	(5)
    union(empty,h)	→	h	(7)
    or(false,y)	→	y	(6)

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

    or#(x,or(y,z))	→	or#(x,y)	(19)
    or#(or(true,y),_1)	→	or#(true,_1)	(36)
    or#(or(false,y),_1)	→	or#(y,_1)	(35)

    could be deleted.

    1.2.2.1 Dependency Graph Processor

    The dependency pairs are split into 1 component.

    • The 1^st component contains the pair

      or#(x,or(y,z))	→	or#(or(x,y),z)	(18)
      or#(x,y)	→	or#(y,x)	(21)

      1.2.2.1.1 AC Dependency Pair Problem is trivial

      There are no strict pairs and rules remaining, or there are no DPs remaining. Therefore, finiteness is trivially satisfied.