YES

We show the termination of the TRS R:

  and(false(),false()) -> false()
  and(true(),false()) -> false()
  and(false(),true()) -> false()
  and(true(),true()) -> true()
  eq(nil(),nil()) -> true()
  eq(cons(T,L),nil()) -> false()
  eq(nil(),cons(T,L)) -> false()
  eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
  eq(var(L),var(Lp)) -> eq(L,Lp)
  eq(var(L),apply(T,S)) -> false()
  eq(var(L),lambda(X,T)) -> false()
  eq(apply(T,S),var(L)) -> false()
  eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
  eq(apply(T,S),lambda(X,Tp)) -> false()
  eq(lambda(X,T),var(L)) -> false()
  eq(lambda(X,T),apply(Tp,Sp)) -> false()
  eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
  if(true(),var(K),var(L)) -> var(K)
  if(false(),var(K),var(L)) -> var(L)
  ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
  ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
  ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> and#(eq(T,Tp),eq(L,Lp))
p2: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(T,Tp)
p3: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p4: eq#(var(L),var(Lp)) -> eq#(L,Lp)
p5: eq#(apply(T,S),apply(Tp,Sp)) -> and#(eq(T,Tp),eq(S,Sp))
p6: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p7: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p8: eq#(lambda(X,T),lambda(Xp,Tp)) -> and#(eq(T,Tp),eq(X,Xp))
p9: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p10: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(X,Xp)
p11: ren#(var(L),var(K),var(Lp)) -> if#(eq(L,Lp),var(K),var(Lp))
p12: ren#(var(L),var(K),var(Lp)) -> eq#(L,Lp)
p13: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)
p14: ren#(X,Y,apply(T,S)) -> ren#(X,Y,S)
p15: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p16: ren#(X,Y,lambda(Z,T)) -> ren#(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p13, p14, p15, p16}
  {p2, p3, p4, p6, p7, p9, p10}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)
p2: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p3: ren#(X,Y,apply(T,S)) -> ren#(X,Y,S)
p4: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        ren#_A(x1,x2,x3) = max{102, x2 + 23, x3 + 74}
        lambda_A(x1,x2) = x2 + 97
        var_A(x1) = 3
        cons_A(x1,x2) = max{12, x2 + 3}
        nil_A = 1
        ren_A(x1,x2,x3) = x3
        apply_A(x1,x2) = max{x1 + 31, x2 + 31}
        and_A(x1,x2) = 142
        false_A = 30
        true_A = 46
        eq_A(x1,x2) = 142
        if_A(x1,x2,x3) = max{3, x2, x3 - 5}
  
    precedence:
    
      nil > ren# = lambda = var = cons = ren = apply = and = false = true = eq = if
    
    partial status:
  
      pi(ren#) = [2]
      pi(lambda) = []
      pi(var) = []
      pi(cons) = [2]
      pi(nil) = []
      pi(ren) = []
      pi(apply) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(eq) = []
      pi(if) = []

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p2: ren#(X,Y,apply(T,S)) -> ren#(X,Y,S)
p3: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p2: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)
p3: ren#(X,Y,apply(T,S)) -> ren#(X,Y,S)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        ren#_A(x1,x2,x3) = max{12, x3 + 11}
        lambda_A(x1,x2) = max{x1 + 39, x2 + 114}
        ren_A(x1,x2,x3) = x3
        var_A(x1) = 10
        cons_A(x1,x2) = max{x1 + 13, x2 + 5}
        nil_A = 23
        apply_A(x1,x2) = max{2, x1 + 1, x2 + 1}
        and_A(x1,x2) = max{1, x1, x2}
        false_A = 0
        true_A = 2
        eq_A(x1,x2) = 10
        if_A(x1,x2,x3) = max{10, x2, x3 - 37}
  
    precedence:
    
      ren# = lambda = ren > cons > var = nil = apply = and = false = true = eq = if
    
    partial status:
  
      pi(ren#) = []
      pi(lambda) = []
      pi(ren) = []
      pi(var) = []
      pi(cons) = [2]
      pi(nil) = []
      pi(apply) = [1]
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(eq) = []
      pi(if) = [2]

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p2: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))
p2: ren#(X,Y,apply(T,S)) -> ren#(X,Y,T)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        ren#_A(x1,x2,x3) = max{3, x3 + 2}
        lambda_A(x1,x2) = max{x1 + 65, x2 + 90}
        ren_A(x1,x2,x3) = x3
        var_A(x1) = 0
        cons_A(x1,x2) = max{29, x1, x2 + 13}
        nil_A = 53
        apply_A(x1,x2) = max{x1 + 1, x2}
        and_A(x1,x2) = max{1, x1}
        false_A = 0
        true_A = 2
        eq_A(x1,x2) = 2
        if_A(x1,x2,x3) = max{x1 - 2, x2, x3 - 1}
  
    precedence:
    
      lambda = ren > ren# > apply > var = cons = nil = and = false = true = eq = if
    
    partial status:
  
      pi(ren#) = [3]
      pi(lambda) = []
      pi(ren) = []
      pi(var) = []
      pi(cons) = [2]
      pi(nil) = []
      pi(apply) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(eq) = []
      pi(if) = []

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: ren#(X,Y,lambda(Z,T)) -> ren#(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T))

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        ren#_A(x1,x2,x3) = x3 + 33
        lambda_A(x1,x2) = max{x1 + 120, x2 + 85}
        ren_A(x1,x2,x3) = max{35, x3}
        var_A(x1) = 0
        cons_A(x1,x2) = max{x1, x2 + 9}
        nil_A = 57
        and_A(x1,x2) = max{83, x1 - 2, x2}
        false_A = 84
        true_A = 87
        eq_A(x1,x2) = 248
        apply_A(x1,x2) = max{72, x1 - 1, x2 + 37}
        if_A(x1,x2,x3) = max{x2, x3 + 35}
  
    precedence:
    
      lambda = ren = eq > ren# > var = cons = nil = and = false = true = apply = if
    
    partial status:
  
      pi(ren#) = [3]
      pi(lambda) = []
      pi(ren) = []
      pi(var) = []
      pi(cons) = [2]
      pi(nil) = []
      pi(and) = []
      pi(false) = []
      pi(true) = []
      pi(eq) = []
      pi(apply) = [2]
      pi(if) = [3]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(T,Tp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(X,Xp)
p3: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p5: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p6: eq#(var(L),var(Lp)) -> eq#(L,Lp)
p7: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{2, x1 + 1, x2 - 2}
        cons_A(x1,x2) = max{x1 + 1, x2}
        lambda_A(x1,x2) = max{x1 + 1, x2}
        apply_A(x1,x2) = max{x1 + 1, x2}
        var_A(x1) = max{1, x1}
  
    precedence:
    
      eq# = cons = lambda = apply = var
    
    partial status:
  
      pi(eq#) = [1]
      pi(cons) = [2]
      pi(lambda) = [2]
      pi(apply) = [2]
      pi(var) = [1]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(X,Xp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p5: eq#(var(L),var(Lp)) -> eq#(L,Lp)
p6: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(X,Xp)
p2: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p3: eq#(var(L),var(Lp)) -> eq#(L,Lp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p5: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p6: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{x1 - 1, x2 + 1}
        lambda_A(x1,x2) = max{x1, x2}
        cons_A(x1,x2) = x2
        var_A(x1) = x1
        apply_A(x1,x2) = max{x1, x2}
  
    precedence:
    
      eq# = lambda = cons = var = apply
    
    partial status:
  
      pi(eq#) = [2]
      pi(lambda) = [1, 2]
      pi(cons) = [2]
      pi(var) = [1]
      pi(apply) = [1, 2]

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(var(L),var(Lp)) -> eq#(L,Lp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p5: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p5: eq#(var(L),var(Lp)) -> eq#(L,Lp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{0, x2 - 3}
        cons_A(x1,x2) = max{x1 + 4, x2 + 4}
        lambda_A(x1,x2) = max{3, x1 - 1, x2}
        apply_A(x1,x2) = max{x1 + 4, x2 + 4}
        var_A(x1) = max{4, x1 + 2}
  
    precedence:
    
      eq# = cons = lambda = apply = var
    
    partial status:
  
      pi(eq#) = []
      pi(cons) = [2]
      pi(lambda) = [2]
      pi(apply) = [2]
      pi(var) = [1]

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p4: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(S,Sp)
p4: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{0, x1 - 3, x2 - 1}
        cons_A(x1,x2) = max{x1, x2}
        apply_A(x1,x2) = max{4, x1, x2 + 2}
        lambda_A(x1,x2) = max{1, x2}
  
    precedence:
    
      eq# = cons = apply = lambda
    
    partial status:
  
      pi(eq#) = []
      pi(cons) = [1, 2]
      pi(apply) = [2]
      pi(lambda) = [2]

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)
p3: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)
p3: eq#(apply(T,S),apply(Tp,Sp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{0, x2 - 3}
        cons_A(x1,x2) = max{3, x2}
        lambda_A(x1,x2) = max{3, x2}
        apply_A(x1,x2) = max{x1 + 1, x2 + 4}
  
    precedence:
    
      eq# = cons = lambda = apply
    
    partial status:
  
      pi(eq#) = []
      pi(cons) = [2]
      pi(lambda) = [2]
      pi(apply) = [1]

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)
p2: eq#(lambda(X,T),lambda(Xp,Tp)) -> eq#(T,Tp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{0, x1 - 3}
        cons_A(x1,x2) = max{x1 + 3, x2}
        lambda_A(x1,x2) = max{x1 + 4, x2 + 1}
  
    precedence:
    
      eq# = cons = lambda
    
    partial status:
  
      pi(eq#) = []
      pi(cons) = [1]
      pi(lambda) = [2]

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: eq#(cons(T,L),cons(Tp,Lp)) -> eq#(L,Lp)

and R consists of:

r1: and(false(),false()) -> false()
r2: and(true(),false()) -> false()
r3: and(false(),true()) -> false()
r4: and(true(),true()) -> true()
r5: eq(nil(),nil()) -> true()
r6: eq(cons(T,L),nil()) -> false()
r7: eq(nil(),cons(T,L)) -> false()
r8: eq(cons(T,L),cons(Tp,Lp)) -> and(eq(T,Tp),eq(L,Lp))
r9: eq(var(L),var(Lp)) -> eq(L,Lp)
r10: eq(var(L),apply(T,S)) -> false()
r11: eq(var(L),lambda(X,T)) -> false()
r12: eq(apply(T,S),var(L)) -> false()
r13: eq(apply(T,S),apply(Tp,Sp)) -> and(eq(T,Tp),eq(S,Sp))
r14: eq(apply(T,S),lambda(X,Tp)) -> false()
r15: eq(lambda(X,T),var(L)) -> false()
r16: eq(lambda(X,T),apply(Tp,Sp)) -> false()
r17: eq(lambda(X,T),lambda(Xp,Tp)) -> and(eq(T,Tp),eq(X,Xp))
r18: if(true(),var(K),var(L)) -> var(K)
r19: if(false(),var(K),var(L)) -> var(L)
r20: ren(var(L),var(K),var(Lp)) -> if(eq(L,Lp),var(K),var(Lp))
r21: ren(X,Y,apply(T,S)) -> apply(ren(X,Y,T),ren(X,Y,S))
r22: ren(X,Y,lambda(Z,T)) -> lambda(var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),ren(X,Y,ren(Z,var(cons(X,cons(Y,cons(lambda(Z,T),nil())))),T)))

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  weighted path order
  
    base order:
  
      max/plus interpretations on natural numbers:
      
        eq#_A(x1,x2) = max{0, x1 - 2, x2 - 2}
        cons_A(x1,x2) = max{x1 + 3, x2 + 1}
  
    precedence:
    
      eq# = cons
    
    partial status:
  
      pi(eq#) = []
      pi(cons) = [2]

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.