YES
We show the termination of the TRS R:

  app(app(and(),true()),true()) -> true()
  app(app(and(),true()),false()) -> false()
  app(app(and(),false()),true()) -> false()
  app(app(and(),false()),false()) -> false()
  app(app(or(),true()),true()) -> true()
  app(app(or(),true()),false()) -> true()
  app(app(or(),false()),true()) -> true()
  app(app(or(),false()),false()) -> false()
  app(app(forall(),p),nil()) -> true()
  app(app(forall(),p),app(app(cons(),x),xs)) -> app(app(and(),app(p,x)),app(app(forall(),p),xs))
  app(app(forsome(),p),nil()) -> false()
  app(app(forsome(),p),app(app(cons(),x),xs)) -> app(app(or(),app(p,x)),app(app(forsome(),p),xs))

Instead we prove termination of the uncurried system:

  app(and_0(),y) -> and_1(y)
  app(and_1(x1),y) -> and_2(x1,y)
  app(cons_0(),y) -> cons_1(y)
  app(cons_1(x1),y) -> cons_2(x1,y)
  app(forall_0(),y) -> forall_1(y)
  app(forall_1(x1),y) -> forall_2(x1,y)
  app(forsome_0(),y) -> forsome_1(y)
  app(forsome_1(x1),y) -> forsome_2(x1,y)
  app(or_0(),y) -> or_1(y)
  app(or_1(x1),y) -> or_2(x1,y)
  and_2(true_0(),true_0()) -> true_0()
  and_2(true_0(),false_0()) -> false_0()
  and_2(false_0(),true_0()) -> false_0()
  and_2(false_0(),false_0()) -> false_0()
  or_2(true_0(),true_0()) -> true_0()
  or_2(true_0(),false_0()) -> true_0()
  or_2(false_0(),true_0()) -> true_0()
  or_2(false_0(),false_0()) -> false_0()
  forall_2(p,nil_0()) -> true_0()
  forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
  forsome_2(p,nil_0()) -> false_0()
  forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))


-- SCC decomposition.

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

p1: app#(and_1(x1),y) -> and_2#(x1,y)
p2: app#(forall_1(x1),y) -> forall_2#(x1,y)
p3: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p4: app#(or_1(x1),y) -> or_2#(x1,y)
p5: forall_2#(p,cons_2(x,xs)) -> and_2#(app(p,x),forall_2(p,xs))
p6: forall_2#(p,cons_2(x,xs)) -> app#(p,x)
p7: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)
p8: forsome_2#(p,cons_2(x,xs)) -> or_2#(app(p,x),forsome_2(p,xs))
p9: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)
p10: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The estimated dependency graph contains the following SCCs:

  {p2, p3, p6, p7, p9, p10}


-- Reduction pair.

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

p1: app#(forall_1(x1),y) -> forall_2#(x1,y)
p2: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)
p3: forall_2#(p,cons_2(x,xs)) -> app#(p,x)
p4: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p5: forsome_2#(p,cons_2(x,xs)) -> forsome_2#(p,xs)
p6: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The set of usable rules consists of

  r1, r2, r5, r7, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{4, x2 - 8}
        forall_1_A(x1) = max{2, x1 - 18}
        forall_2#_A(x1,x2) = max{4, x2 - 15}
        cons_2_A(x1,x2) = max{14, x1 + 8, x2 + 12}
        forsome_1_A(x1) = max{1, x1 - 18}
        forsome_2#_A(x1,x2) = max{2, x2 - 9}
        app_A(x1,x2) = max{38, x1 - 20, x2 - 17}
        and_0_A = 22
        and_1_A(x1) = max{1, x1 - 18}
        and_2_A(x1,x2) = max{13, x1 - 39, x2 - 39}
        forall_0_A = 1
        forsome_0_A = 19
        or_0_A = 19
        or_1_A(x1) = max{36, x1 - 18}
        or_2_A(x1,x2) = max{15, x1 - 39, x2 - 63}
        true_0_A = 0
        false_0_A = 3
  
    base order 2:
  
      precedence:
      
        or_2 > forsome_1 > app > and_1 = and_2 > cons_2 = and_0 = forsome_0 = or_1 = false_0 > app# = forall_2# = forsome_2# = or_0 = true_0 > forall_1 > forall_0
      
  
    partial status:
  
      pi(app#) = []
      pi(forall_1) = []
      pi(forall_2#) = []
      pi(cons_2) = [2]
      pi(forsome_1) = []
      pi(forsome_2#) = []
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(forall_0) = []
      pi(forsome_0) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p5



-- SCC decomposition.

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

p1: app#(forall_1(x1),y) -> forall_2#(x1,y)
p2: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)
p3: forall_2#(p,cons_2(x,xs)) -> app#(p,x)
p4: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p5: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

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: app#(forall_1(x1),y) -> forall_2#(x1,y)
p2: forall_2#(p,cons_2(x,xs)) -> app#(p,x)
p3: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p4: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)
p5: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The set of usable rules consists of

  r2, r3, r5, r7, r9, r11, r12, r13, r14, r15, r16, r17

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{13, x2 - 11}
        forall_1_A(x1) = max{11, x1 - 28}
        forall_2#_A(x1,x2) = max{12, x2 - 12}
        cons_2_A(x1,x2) = max{25, x1 + 11, x2 + 4}
        forsome_1_A(x1) = max{2, x1 - 28}
        forsome_2#_A(x1,x2) = max{13, x2 - 21}
        app_A(x1,x2) = max{24, x1 - 29, x2 - 27}
        and_1_A(x1) = max{27, x1 - 27}
        and_2_A(x1,x2) = max{24, x1 - 56, x2 - 27}
        cons_0_A = 22
        cons_1_A(x1) = max{23, x1 - 28}
        forall_0_A = 10
        forsome_0_A = 1
        or_0_A = 28
        or_1_A(x1) = 1
        true_0_A = 1
        false_0_A = 4
        or_2_A(x1,x2) = max{2, x1 - 30, x2 - 5}
  
    base order 2:
  
      precedence:
      
        false_0 > app > and_2 = or_2 > cons_2 > forall_2# = forall_0 = or_0 > app# = forall_1 = forsome_1 = forsome_2# = and_1 = cons_1 = or_1 = true_0 > cons_0 = forsome_0
      
  
    partial status:
  
      pi(app#) = []
      pi(forall_1) = []
      pi(forall_2#) = []
      pi(cons_2) = [2]
      pi(forsome_1) = []
      pi(forsome_2#) = []
      pi(app) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(forall_0) = []
      pi(forsome_0) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(true_0) = []
      pi(false_0) = []
      pi(or_2) = []

The following rules are removed:

  p1



-- SCC decomposition.

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

p1: forall_2#(p,cons_2(x,xs)) -> app#(p,x)
p2: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p3: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)
p4: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The estimated dependency graph contains the following SCCs:

  {p4}
  {p2, p3}


-- Reduction pair.

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

p1: forall_2#(p,cons_2(x,xs)) -> forall_2#(p,xs)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The set of usable rules consists of

  r1, r2, r3, r5, r7, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        forall_2#_A(x1,x2) = max{x1 + 5, x2 + 8}
        cons_2_A(x1,x2) = max{3, x1 - 3, x2 + 1}
        app_A(x1,x2) = max{14, x1 - 3, x2 - 11}
        and_0_A = 1
        and_1_A(x1) = max{2, x1 - 12}
        and_2_A(x1,x2) = max{6, x1 - 15, x2 - 16}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 12}
        forall_0_A = 1
        forall_1_A(x1) = max{2, x1 - 12}
        forsome_0_A = 1
        forsome_1_A(x1) = max{2, x1 - 12}
        or_0_A = 1
        or_1_A(x1) = max{2, x1 - 12}
        or_2_A(x1,x2) = max{6, x1 - 16, x2 - 12}
        true_0_A = 0
        false_0_A = 5
  
    base order 2:
  
      precedence:
      
        app > and_2 = or_2 > false_0 > forall_2# = true_0 > cons_2 = and_1 = forall_1 = forsome_1 = or_1 > and_0 = cons_1 = forall_0 = forsome_0 = or_0 > cons_0
      
  
    partial status:
  
      pi(forall_2#) = [2]
      pi(cons_2) = [2]
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(forall_0) = []
      pi(forall_1) = []
      pi(forsome_0) = []
      pi(forsome_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p1

Then no dependency pair remains.

-- Reduction pair.

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

p1: app#(forsome_1(x1),y) -> forsome_2#(x1,y)
p2: forsome_2#(p,cons_2(x,xs)) -> app#(p,x)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The set of usable rules consists of

  r1, r2, r3, r4, r5, r7, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x2 + 7
        forsome_1_A(x1) = 22
        forsome_2#_A(x1,x2) = x2 + 6
        cons_2_A(x1,x2) = max{x1 + 8, x2 - 1}
        app_A(x1,x2) = max{x1 - 11, x2 + 21}
        and_0_A = 10
        and_1_A(x1) = max{20, x1 - 1}
        and_2_A(x1,x2) = max{21, x1 - 25, x2 - 24}
        cons_0_A = 1
        cons_1_A(x1) = x1 + 20
        forall_0_A = 46
        forall_1_A(x1) = max{34, x1 - 1}
        forsome_0_A = 34
        or_0_A = 9
        or_1_A(x1) = max{10, x1 - 1}
        or_2_A(x1,x2) = max{9, x1 - 25, x2 - 25}
        true_0_A = 0
        false_0_A = 8
  
    base order 2:
  
      precedence:
      
        and_0 > cons_2 = app = or_2 > app# = and_2 = cons_1 = forall_1 = or_1 > forsome_2# = and_1 = cons_0 = forall_0 = forsome_0 = or_0 = false_0 > forsome_1 = true_0
      
  
    partial status:
  
      pi(app#) = []
      pi(forsome_1) = []
      pi(forsome_2#) = [2]
      pi(cons_2) = [1]
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(forall_0) = []
      pi(forall_1) = []
      pi(forsome_0) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(true_0) = []
      pi(false_0) = []

The following rules are removed:

  p2



-- SCC decomposition.

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

p1: app#(forsome_1(x1),y) -> forsome_2#(x1,y)

and R consists of:

r1: app(and_0(),y) -> and_1(y)
r2: app(and_1(x1),y) -> and_2(x1,y)
r3: app(cons_0(),y) -> cons_1(y)
r4: app(cons_1(x1),y) -> cons_2(x1,y)
r5: app(forall_0(),y) -> forall_1(y)
r6: app(forall_1(x1),y) -> forall_2(x1,y)
r7: app(forsome_0(),y) -> forsome_1(y)
r8: app(forsome_1(x1),y) -> forsome_2(x1,y)
r9: app(or_0(),y) -> or_1(y)
r10: app(or_1(x1),y) -> or_2(x1,y)
r11: and_2(true_0(),true_0()) -> true_0()
r12: and_2(true_0(),false_0()) -> false_0()
r13: and_2(false_0(),true_0()) -> false_0()
r14: and_2(false_0(),false_0()) -> false_0()
r15: or_2(true_0(),true_0()) -> true_0()
r16: or_2(true_0(),false_0()) -> true_0()
r17: or_2(false_0(),true_0()) -> true_0()
r18: or_2(false_0(),false_0()) -> false_0()
r19: forall_2(p,nil_0()) -> true_0()
r20: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r21: forsome_2(p,nil_0()) -> false_0()
r22: forsome_2(p,cons_2(x,xs)) -> or_2(app(p,x),forsome_2(p,xs))

The estimated dependency graph contains the following SCCs:

  (no SCCs)