YES
We show the termination of the TRS R:

  app(app(and(),true()),true()) -> true()
  app(app(and(),x),false()) -> false()
  app(app(and(),false()),y) -> false()
  app(app(or(),true()),y) -> true()
  app(app(or(),x),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(x,false_0()) -> false_0()
  and_2(false_0(),y) -> false_0()
  or_2(true_0(),y) -> true_0()
  or_2(x,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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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, r9, r10, r11, r12, r13, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = x1 + 2
        forall_1_A(x1) = x1 + 5
        forall_2#_A(x1,x2) = x1 + 4
        cons_2_A(x1,x2) = max{4, x1 - 11, x2 - 11}
        forsome_1_A(x1) = max{4, x1 + 3}
        forsome_2#_A(x1,x2) = max{4, x1 + 3}
        app_A(x1,x2) = max{15, x1 - 6, x2 - 3}
        and_0_A = 4
        and_1_A(x1) = max{5, x1 - 4}
        and_2_A(x1,x2) = max{14, x1 - 16, x2 - 16}
        cons_0_A = 2
        cons_1_A(x1) = max{3, x1 - 4}
        or_0_A = 4
        or_1_A(x1) = max{5, x1 - 4}
        or_2_A(x1,x2) = max{6, x1 - 29, x2 - 19}
        true_0_A = 5
        false_0_A = 5
  
    base order 2:
  
      precedence:
      
        forall_2# = app > forall_1 = and_2 > forsome_1 = and_1 = or_2 > cons_2 = and_0 = or_1 = true_0 = false_0 > forsome_2# = cons_1 = or_0 > app# = cons_0
      
  
    partial status:
  
      pi(app#) = [1]
      pi(forall_1) = [1]
      pi(forall_2#) = [1]
      pi(cons_2) = []
      pi(forsome_1) = [1]
      pi(forsome_2#) = [1]
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(cons_0) = []
      pi(cons_1) = []
      pi(or_0) = []
      pi(or_1) = []
      pi(or_2) = []
      pi(true_0) = []
      pi(false_0) = []

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)) -> forall_2#(p,xs)
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)) -> forsome_2#(p,xs)
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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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}
  {p3, p4, p5}


-- 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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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

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 + 31}
        cons_2_A(x1,x2) = max{x1 - 26, x2 + 5}
        app_A(x1,x2) = max{14, x1 - 3, x2 - 10}
        and_0_A = 1
        and_1_A(x1) = max{2, x1 - 25}
        and_2_A(x1,x2) = max{8, x1 - 28, x2 - 10}
        cons_0_A = 1
        cons_1_A(x1) = max{2, x1 - 10}
        forall_0_A = 1
        forall_1_A(x1) = max{2, x1 - 11}
        forsome_0_A = 1
        forsome_1_A(x1) = max{2, x1 - 11}
        or_0_A = 1
        or_1_A(x1) = max{2, x1 - 11}
        or_2_A(x1,x2) = max{8, x1 - 15, x2 - 11}
        true_0_A = 7
        false_0_A = 7
  
    base order 2:
  
      precedence:
      
        app > forall_2# = and_0 = forall_1 > and_2 = forall_0 = forsome_1 = or_2 > cons_2 = and_1 = forsome_0 = or_1 = true_0 = false_0 > cons_1 = 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)
p3: 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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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, r4, r5, r9, r10, r11, r12, r13, r14, r15, r16

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        app#_A(x1,x2) = max{2, x1 - 3}
        forsome_1_A(x1) = x1 + 7
        forsome_2#_A(x1,x2) = max{3, x1 + 1}
        cons_2_A(x1,x2) = max{1, x1 - 12, x2 - 12}
        app_A(x1,x2) = max{9, x1 - 11, x2 - 1}
        and_0_A = 1
        and_1_A(x1) = max{6, x1 - 2}
        and_2_A(x1,x2) = max{7, x1 - 24, x2 - 21}
        cons_1_A(x1) = max{10, x1 - 1}
        forall_0_A = 8
        forall_1_A(x1) = max{9, x1 - 1}
        or_0_A = 1
        or_1_A(x1) = max{2, x1 - 2}
        or_2_A(x1,x2) = max{7, x1 - 24, x2 - 21}
        true_0_A = 0
        false_0_A = 0
  
    base order 2:
  
      precedence:
      
        forsome_2# > app# = forsome_1 > app > or_2 > and_2 = forall_1 = or_1 > cons_2 = and_1 = forall_0 = or_0 = true_0 = false_0 > and_0 = cons_1
      
  
    partial status:
  
      pi(app#) = []
      pi(forsome_1) = [1]
      pi(forsome_2#) = [1]
      pi(cons_2) = []
      pi(app) = []
      pi(and_0) = []
      pi(and_1) = []
      pi(and_2) = []
      pi(cons_1) = []
      pi(forall_0) = []
      pi(forall_1) = []
      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)
p2: 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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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}


-- Reduction pair.

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

p1: 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(x,false_0()) -> false_0()
r13: and_2(false_0(),y) -> false_0()
r14: or_2(true_0(),y) -> true_0()
r15: or_2(x,true_0()) -> true_0()
r16: or_2(false_0(),false_0()) -> false_0()
r17: forall_2(p,nil_0()) -> true_0()
r18: forall_2(p,cons_2(x,xs)) -> and_2(app(p,x),forall_2(p,xs))
r19: forsome_2(p,nil_0()) -> false_0()
r20: 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

Take the reduction pair:

  generalized weighted path order (>=MG, >MG)
  
    base order 1:
  
      max/plus interpretation on natural numbers:
      
        forsome_2#_A(x1,x2) = max{x1 + 4, x2 + 18}
        cons_2_A(x1,x2) = max{2, x1 - 14, x2 + 1}
        app_A(x1,x2) = max{7, x1 - 9, x2 - 1}
        and_0_A = 3
        and_1_A(x1) = max{4, x1 - 11}
        and_2_A(x1,x2) = max{5, x1 - 20, x2 - 21}
        cons_0_A = 6
        cons_1_A(x1) = max{7, x1 - 1}
        forall_0_A = 8
        forall_1_A(x1) = max{7, x1 - 2}
        forsome_0_A = 1
        forsome_1_A(x1) = max{2, x1 - 12}
        or_0_A = 3
        or_1_A(x1) = max{4, x1 - 2}
        or_2_A(x1,x2) = max{5, x1 - 12, x2 - 5}
        true_0_A = 4
        false_0_A = 4
  
    base order 2:
  
      precedence:
      
        or_2 > forsome_2# > app > forall_1 = forsome_1 = or_1 > and_2 = forall_0 = forsome_0 = or_0 > cons_2 = and_1 = true_0 = false_0 > and_0 = cons_1 > cons_0
      
  
    partial status:
  
      pi(forsome_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.