6.27/2.52	YES
6.35/2.55	proof of /export/starexec/sandbox/benchmark/theBenchmark.pl
6.35/2.55	# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty
6.35/2.55	
6.35/2.55	
6.35/2.55	Left Termination of the query pattern
6.35/2.55	
6.35/2.55	add(a,a,g)
6.35/2.55	
6.35/2.55	w.r.t. the given Prolog program could successfully be proven:
6.35/2.55	
6.35/2.55	(0) Prolog
6.35/2.55	(1) PrologToPiTRSProof [SOUND, 53 ms]
6.35/2.55	(2) PiTRS
6.35/2.55	(3) DependencyPairsProof [EQUIVALENT, 22 ms]
6.35/2.55	(4) PiDP
6.35/2.55	(5) DependencyGraphProof [EQUIVALENT, 0 ms]
6.35/2.55	(6) AND
6.35/2.55	    (7) PiDP
6.35/2.55	        (8) UsableRulesProof [EQUIVALENT, 0 ms]
6.35/2.55	        (9) PiDP
6.35/2.55	        (10) PiDPToQDPProof [EQUIVALENT, 0 ms]
6.35/2.55	        (11) QDP
6.35/2.55	        (12) QDPSizeChangeProof [EQUIVALENT, 1 ms]
6.35/2.55	        (13) YES
6.35/2.55	    (14) PiDP
6.35/2.55	        (15) UsableRulesProof [EQUIVALENT, 0 ms]
6.35/2.55	        (16) PiDP
6.35/2.55	        (17) PiDPToQDPProof [SOUND, 0 ms]
6.35/2.55	        (18) QDP
6.35/2.55	        (19) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.35/2.55	        (20) YES
6.35/2.55	    (21) PiDP
6.35/2.55	        (22) UsableRulesProof [EQUIVALENT, 0 ms]
6.35/2.55	        (23) PiDP
6.35/2.55	        (24) PiDPToQDPProof [SOUND, 0 ms]
6.35/2.55	        (25) QDP
6.35/2.55	        (26) QDPSizeChangeProof [EQUIVALENT, 0 ms]
6.35/2.55	        (27) YES
6.35/2.55	
6.35/2.55	
6.35/2.55	----------------------------------------
6.35/2.55	
6.35/2.55	(0)
6.35/2.55	Obligation:
6.35/2.55	Clauses:
6.35/2.55	
6.35/2.55	add(b, b, b).
6.35/2.55	add(X, b, X) :- binaryZ(X).
6.35/2.55	add(b, Y, Y) :- binaryZ(Y).
6.35/2.55	add(X, Y, Z) :- addz(X, Y, Z).
6.35/2.55	addx(one(X), b, one(X)) :- binary(X).
6.35/2.55	addx(zero(X), b, zero(X)) :- binaryZ(X).
6.35/2.55	addx(X, Y, Z) :- addz(X, Y, Z).
6.35/2.55	addy(b, one(Y), one(Y)) :- binary(Y).
6.35/2.55	addy(b, zero(Y), zero(Y)) :- binaryZ(Y).
6.35/2.55	addy(X, Y, Z) :- addz(X, Y, Z).
6.35/2.55	addz(zero(X), zero(Y), zero(Z)) :- addz(X, Y, Z).
6.35/2.55	addz(zero(X), one(Y), one(Z)) :- addx(X, Y, Z).
6.35/2.55	addz(one(X), zero(Y), one(Z)) :- addy(X, Y, Z).
6.35/2.55	addz(one(X), one(Y), zero(Z)) :- addc(X, Y, Z).
6.35/2.55	addc(b, b, one(b)).
6.35/2.55	addc(X, b, Z) :- succZ(X, Z).
6.35/2.55	addc(b, Y, Z) :- succZ(Y, Z).
6.35/2.55	addc(X, Y, Z) :- addC(X, Y, Z).
6.35/2.55	addX(zero(X), b, one(X)) :- binaryZ(X).
6.35/2.55	addX(one(X), b, zero(Z)) :- succ(X, Z).
6.35/2.55	addX(X, Y, Z) :- addC(X, Y, Z).
6.35/2.55	addY(b, zero(Y), one(Y)) :- binaryZ(Y).
6.35/2.55	addY(b, one(Y), zero(Z)) :- succ(Y, Z).
6.35/2.55	addY(X, Y, Z) :- addC(X, Y, Z).
6.35/2.55	addC(zero(X), zero(Y), one(Z)) :- addz(X, Y, Z).
6.35/2.55	addC(zero(X), one(Y), zero(Z)) :- addX(X, Y, Z).
6.35/2.55	addC(one(X), zero(Y), zero(Z)) :- addY(X, Y, Z).
6.35/2.55	addC(one(X), one(Y), one(Z)) :- addc(X, Y, Z).
6.35/2.55	binary(b).
6.35/2.55	binary(zero(X)) :- binaryZ(X).
6.35/2.55	binary(one(X)) :- binary(X).
6.35/2.55	binaryZ(zero(X)) :- binaryZ(X).
6.35/2.55	binaryZ(one(X)) :- binary(X).
6.35/2.55	succ(b, one(b)).
6.35/2.55	succ(zero(X), one(X)) :- binaryZ(X).
6.35/2.55	succ(one(X), zero(Z)) :- succ(X, Z).
6.35/2.55	succZ(zero(X), one(X)) :- binaryZ(X).
6.35/2.55	succZ(one(X), zero(Z)) :- succ(X, Z).
6.35/2.55	times(one(b), X, X).
6.35/2.55	times(zero(R), S, zero(RS)) :- times(R, S, RS).
6.35/2.55	times(one(R), S, RSS) :- ','(times(R, S, RS), add(S, zero(RS), RSS)).
6.35/2.55	
6.35/2.55	
6.35/2.55	Query: add(a,a,g)
6.35/2.55	----------------------------------------
6.35/2.55	
6.35/2.55	(1) PrologToPiTRSProof (SOUND)
6.35/2.55	We use the technique of [TOCL09]. With regard to the inferred argument filtering the predicates were used in the following modes:
6.35/2.55	
6.35/2.55	add_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	binaryZ_in_1: (b)
6.35/2.55	
6.35/2.55	binary_in_1: (b)
6.35/2.55	
6.35/2.55	addz_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	addx_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	addy_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	addc_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	succZ_in_2: (f,b)
6.35/2.55	
6.35/2.55	succ_in_2: (f,b)
6.35/2.55	
6.35/2.55	addC_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	addX_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	addY_in_3: (f,f,b)
6.35/2.55	
6.35/2.55	Transforming Prolog into the following Term Rewriting System:
6.35/2.55	
6.35/2.55	Pi-finite rewrite system:
6.35/2.55	The TRS R consists of the following rules:
6.35/2.55	
6.35/2.55	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.55	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.55	   binary_in_g(b) -> binary_out_g(b)
6.35/2.55	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.55	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.55	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.55	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.55	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.55	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.55	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.55	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.55	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.55	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.55	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.55	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.55	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.55	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.55	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.55	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.55	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.55	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.55	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.55	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.55	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.55	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.55	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.55	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.55	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.55	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.55	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.55	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.55	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.55	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.55	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.55	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.55	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.55	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.55	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.55	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.55	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.55	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.55	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.55	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.55	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.55	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.55	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.55	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.55	
6.35/2.55	The argument filtering Pi contains the following mapping:
6.35/2.55	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.55	
6.35/2.55	b  =  b
6.35/2.55	
6.35/2.55	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.55	
6.35/2.55	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.55	
6.35/2.55	zero(x1)  =  zero(x1)
6.35/2.55	
6.35/2.55	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.55	
6.35/2.55	one(x1)  =  one(x1)
6.35/2.55	
6.35/2.55	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.55	
6.35/2.55	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.55	
6.35/2.55	binary_out_g(x1)  =  binary_out_g
6.35/2.55	
6.35/2.55	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.55	
6.35/2.55	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.55	
6.35/2.55	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.55	
6.35/2.55	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.55	
6.35/2.55	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.55	
6.35/2.55	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.55	
6.35/2.55	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.55	
6.35/2.55	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.55	
6.35/2.55	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.55	
6.35/2.55	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.55	
6.35/2.55	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.55	
6.35/2.55	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.55	
6.35/2.55	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.55	
6.35/2.55	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.55	
6.35/2.55	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.55	
6.35/2.55	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.55	
6.35/2.55	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.55	
6.35/2.55	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.55	
6.35/2.55	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.55	
6.35/2.55	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.55	
6.35/2.55	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.55	
6.35/2.55	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.55	
6.35/2.55	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.55	
6.35/2.55	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.55	
6.35/2.55	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.55	
6.35/2.55	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.55	
6.35/2.55	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.55	
6.35/2.55	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.55	
6.35/2.55	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.55	
6.35/2.55	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.55	
6.35/2.55	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.55	
6.35/2.55	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.55	
6.35/2.55	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.55	
6.35/2.55	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.55	
6.35/2.55	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.55	
6.35/2.55	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.55	
6.35/2.55	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.55	
6.35/2.55	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.55	
6.35/2.55	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.55	
6.35/2.55	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.55	
6.35/2.55	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.55	
6.35/2.55	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.55	
6.35/2.55	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.55	
6.35/2.55	
6.35/2.55	
6.35/2.55	
6.35/2.55	
6.35/2.55	Infinitary Constructor Rewriting Termination of PiTRS implies Termination of Prolog
6.35/2.55	
6.35/2.55	
6.35/2.55	
6.35/2.55	----------------------------------------
6.35/2.55	
6.35/2.55	(2)
6.35/2.55	Obligation:
6.35/2.55	Pi-finite rewrite system:
6.35/2.55	The TRS R consists of the following rules:
6.35/2.55	
6.35/2.55	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.55	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.55	   binary_in_g(b) -> binary_out_g(b)
6.35/2.55	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.55	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.55	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.55	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.55	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.55	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.55	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.55	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.55	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.55	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.55	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.55	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.55	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.55	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.55	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.55	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.55	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.55	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.55	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.55	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.55	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.55	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.55	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.55	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.55	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.55	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.55	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.55	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.55	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.55	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.55	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.55	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.55	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.55	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.55	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.55	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.55	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.55	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.55	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.55	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.55	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.55	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.55	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.55	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.55	
6.35/2.55	The argument filtering Pi contains the following mapping:
6.35/2.55	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.55	
6.35/2.55	b  =  b
6.35/2.55	
6.35/2.55	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.55	
6.35/2.55	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.55	
6.35/2.55	zero(x1)  =  zero(x1)
6.35/2.55	
6.35/2.55	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.55	
6.35/2.55	one(x1)  =  one(x1)
6.35/2.55	
6.35/2.55	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.55	
6.35/2.55	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.55	
6.35/2.55	binary_out_g(x1)  =  binary_out_g
6.35/2.55	
6.35/2.55	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.55	
6.35/2.55	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.55	
6.35/2.55	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.55	
6.35/2.55	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.55	
6.35/2.55	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.55	
6.35/2.55	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.55	
6.35/2.55	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.55	
6.35/2.55	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.55	
6.35/2.55	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.55	
6.35/2.55	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.55	
6.35/2.55	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.55	
6.35/2.55	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.55	
6.35/2.55	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.55	
6.35/2.55	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.55	
6.35/2.55	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.55	
6.35/2.55	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.55	
6.35/2.55	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.55	
6.35/2.55	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.55	
6.35/2.55	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.55	
6.35/2.55	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.55	
6.35/2.55	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.55	
6.35/2.55	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.55	
6.35/2.55	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.55	
6.35/2.55	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.55	
6.35/2.55	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.55	
6.35/2.55	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.55	
6.35/2.55	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.55	
6.35/2.55	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.55	
6.35/2.55	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.55	
6.35/2.55	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.55	
6.35/2.55	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.55	
6.35/2.55	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.55	
6.35/2.55	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.55	
6.35/2.55	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.55	
6.35/2.55	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.55	
6.35/2.55	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.55	
6.35/2.55	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.55	
6.35/2.55	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.55	
6.35/2.55	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.55	
6.35/2.55	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.55	
6.35/2.55	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.55	
6.35/2.55	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.55	
6.35/2.55	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.55	
6.35/2.55	
6.35/2.55	
6.35/2.55	----------------------------------------
6.35/2.55	
6.35/2.55	(3) DependencyPairsProof (EQUIVALENT)
6.35/2.55	Using Dependency Pairs [AG00,LOPSTR] we result in the following initial DP problem:
6.35/2.55	Pi DP problem:
6.35/2.55	The TRS P consists of the following rules:
6.35/2.55	
6.35/2.55	   ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X))
6.35/2.55	   BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X))
6.35/2.55	   BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.55	   BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X))
6.35/2.55	   BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X))
6.35/2.55	   BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.55	   ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z)
6.35/2.55	   ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X))
6.35/2.55	   ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X)
6.35/2.55	   ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z)
6.35/2.55	   ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y))
6.35/2.55	   ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z)
6.35/2.55	   SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X))
6.35/2.55	   SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X))
6.35/2.55	   SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z)
6.35/2.55	   ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z)
6.35/2.55	   ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.55	
6.35/2.55	The TRS R consists of the following rules:
6.35/2.55	
6.35/2.55	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.55	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.55	   binary_in_g(b) -> binary_out_g(b)
6.35/2.55	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.55	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.55	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.55	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.55	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.55	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.55	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.55	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.55	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.55	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.55	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.55	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.55	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.55	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.55	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.55	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.55	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.55	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.55	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.55	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.55	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.55	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.55	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.55	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.55	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.55	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.55	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.55	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.55	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.55	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.55	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.55	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.55	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.55	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.55	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.55	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.55	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.55	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.55	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.55	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.55	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.55	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.55	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.55	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.55	
6.35/2.55	The argument filtering Pi contains the following mapping:
6.35/2.55	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.55	
6.35/2.55	b  =  b
6.35/2.55	
6.35/2.55	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.55	
6.35/2.55	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.55	
6.35/2.55	zero(x1)  =  zero(x1)
6.35/2.55	
6.35/2.55	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.55	
6.35/2.55	one(x1)  =  one(x1)
6.35/2.55	
6.35/2.55	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.55	
6.35/2.55	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.55	
6.35/2.55	binary_out_g(x1)  =  binary_out_g
6.35/2.55	
6.35/2.55	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.55	
6.35/2.55	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.55	
6.35/2.55	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.55	
6.35/2.55	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.55	
6.35/2.55	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.55	
6.35/2.55	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.55	
6.35/2.55	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.55	
6.35/2.55	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.55	
6.35/2.55	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.55	
6.35/2.55	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.55	
6.35/2.55	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.55	
6.35/2.55	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.55	
6.35/2.55	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.55	
6.35/2.55	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.55	
6.35/2.55	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.55	
6.35/2.55	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.55	
6.35/2.55	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.55	
6.35/2.55	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.55	
6.35/2.55	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.55	
6.35/2.55	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.55	
6.35/2.55	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.55	
6.35/2.55	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.55	
6.35/2.55	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.55	
6.35/2.55	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.55	
6.35/2.55	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.55	
6.35/2.55	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.55	
6.35/2.55	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.55	
6.35/2.55	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.55	
6.35/2.55	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.55	
6.35/2.55	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.55	
6.35/2.55	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.55	
6.35/2.55	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.55	
6.35/2.55	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.55	
6.35/2.55	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.55	
6.35/2.55	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.55	
6.35/2.55	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.55	
6.35/2.55	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.55	
6.35/2.55	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.55	
6.35/2.55	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.55	
6.35/2.55	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.55	
6.35/2.55	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.55	
6.35/2.55	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.55	
6.35/2.55	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.55	
6.35/2.55	ADD_IN_AAG(x1, x2, x3)  =  ADD_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U1_AAG(x1, x2)  =  U1_AAG(x1, x2)
6.35/2.55	
6.35/2.55	BINARYZ_IN_G(x1)  =  BINARYZ_IN_G(x1)
6.35/2.55	
6.35/2.55	U29_G(x1, x2)  =  U29_G(x2)
6.35/2.55	
6.35/2.55	U30_G(x1, x2)  =  U30_G(x2)
6.35/2.55	
6.35/2.55	BINARY_IN_G(x1)  =  BINARY_IN_G(x1)
6.35/2.55	
6.35/2.55	U27_G(x1, x2)  =  U27_G(x2)
6.35/2.55	
6.35/2.55	U28_G(x1, x2)  =  U28_G(x2)
6.35/2.55	
6.35/2.55	U2_AAG(x1, x2)  =  U2_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U3_AAG(x1, x2, x3, x4)  =  U3_AAG(x4)
6.35/2.55	
6.35/2.55	ADDZ_IN_AAG(x1, x2, x3)  =  ADDZ_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U10_AAG(x1, x2, x3, x4)  =  U10_AAG(x4)
6.35/2.55	
6.35/2.55	U11_AAG(x1, x2, x3, x4)  =  U11_AAG(x4)
6.35/2.55	
6.35/2.55	ADDX_IN_AAG(x1, x2, x3)  =  ADDX_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U4_AAG(x1, x2)  =  U4_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U5_AAG(x1, x2)  =  U5_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U6_AAG(x1, x2, x3, x4)  =  U6_AAG(x4)
6.35/2.55	
6.35/2.55	U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x4)
6.35/2.55	
6.35/2.55	ADDY_IN_AAG(x1, x2, x3)  =  ADDY_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U7_AAG(x1, x2)  =  U7_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U8_AAG(x1, x2)  =  U8_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U9_AAG(x1, x2, x3, x4)  =  U9_AAG(x4)
6.35/2.55	
6.35/2.55	U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x4)
6.35/2.55	
6.35/2.55	ADDC_IN_AAG(x1, x2, x3)  =  ADDC_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U14_AAG(x1, x2, x3)  =  U14_AAG(x3)
6.35/2.55	
6.35/2.55	SUCCZ_IN_AG(x1, x2)  =  SUCCZ_IN_AG(x2)
6.35/2.55	
6.35/2.55	U33_AG(x1, x2)  =  U33_AG(x1, x2)
6.35/2.55	
6.35/2.55	U34_AG(x1, x2, x3)  =  U34_AG(x3)
6.35/2.55	
6.35/2.55	SUCC_IN_AG(x1, x2)  =  SUCC_IN_AG(x2)
6.35/2.55	
6.35/2.55	U31_AG(x1, x2)  =  U31_AG(x1, x2)
6.35/2.55	
6.35/2.55	U32_AG(x1, x2, x3)  =  U32_AG(x3)
6.35/2.55	
6.35/2.55	U15_AAG(x1, x2, x3)  =  U15_AAG(x3)
6.35/2.55	
6.35/2.55	U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x4)
6.35/2.55	
6.35/2.55	ADDC_IN_AAG^1(x1, x2, x3)  =  ADDC_IN_AAG^1(x3)
6.35/2.55	
6.35/2.55	U23_AAG(x1, x2, x3, x4)  =  U23_AAG(x4)
6.35/2.55	
6.35/2.55	U24_AAG(x1, x2, x3, x4)  =  U24_AAG(x4)
6.35/2.55	
6.35/2.55	ADDX_IN_AAG^1(x1, x2, x3)  =  ADDX_IN_AAG^1(x3)
6.35/2.55	
6.35/2.55	U17_AAG(x1, x2)  =  U17_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U18_AAG(x1, x2, x3)  =  U18_AAG(x3)
6.35/2.55	
6.35/2.55	U19_AAG(x1, x2, x3, x4)  =  U19_AAG(x4)
6.35/2.55	
6.35/2.55	U25_AAG(x1, x2, x3, x4)  =  U25_AAG(x4)
6.35/2.55	
6.35/2.55	ADDY_IN_AAG^1(x1, x2, x3)  =  ADDY_IN_AAG^1(x3)
6.35/2.55	
6.35/2.55	U20_AAG(x1, x2)  =  U20_AAG(x1, x2)
6.35/2.55	
6.35/2.55	U21_AAG(x1, x2, x3)  =  U21_AAG(x3)
6.35/2.55	
6.35/2.55	U22_AAG(x1, x2, x3, x4)  =  U22_AAG(x4)
6.35/2.55	
6.35/2.55	U26_AAG(x1, x2, x3, x4)  =  U26_AAG(x4)
6.35/2.55	
6.35/2.55	
6.35/2.55	We have to consider all (P,R,Pi)-chains
6.35/2.55	----------------------------------------
6.35/2.55	
6.35/2.55	(4)
6.35/2.55	Obligation:
6.35/2.55	Pi DP problem:
6.35/2.55	The TRS P consists of the following rules:
6.35/2.55	
6.35/2.55	   ADD_IN_AAG(X, b, X) -> U1_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADD_IN_AAG(X, b, X) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARYZ_IN_G(zero(X)) -> U29_G(X, binaryZ_in_g(X))
6.35/2.55	   BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARYZ_IN_G(one(X)) -> U30_G(X, binary_in_g(X))
6.35/2.55	   BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.55	   BINARY_IN_G(zero(X)) -> U27_G(X, binaryZ_in_g(X))
6.35/2.55	   BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   BINARY_IN_G(one(X)) -> U28_G(X, binary_in_g(X))
6.35/2.55	   BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.55	   ADD_IN_AAG(b, Y, Y) -> U2_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADD_IN_AAG(b, Y, Y) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADD_IN_AAG(X, Y, Z) -> U3_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADD_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> U10_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> U11_AAG(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z)
6.35/2.55	   ADDX_IN_AAG(one(X), b, one(X)) -> U4_AAG(X, binary_in_g(X))
6.35/2.55	   ADDX_IN_AAG(one(X), b, one(X)) -> BINARY_IN_G(X)
6.35/2.55	   ADDX_IN_AAG(zero(X), b, zero(X)) -> U5_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADDX_IN_AAG(zero(X), b, zero(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   ADDX_IN_AAG(X, Y, Z) -> U6_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> U12_AAG(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z)
6.35/2.55	   ADDY_IN_AAG(b, one(Y), one(Y)) -> U7_AAG(Y, binary_in_g(Y))
6.35/2.55	   ADDY_IN_AAG(b, one(Y), one(Y)) -> BINARY_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG(b, zero(Y), zero(Y)) -> U8_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADDY_IN_AAG(b, zero(Y), zero(Y)) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG(X, Y, Z) -> U9_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> U13_AAG(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG(X, b, Z) -> U14_AAG(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   ADDC_IN_AAG(X, b, Z) -> SUCCZ_IN_AG(X, Z)
6.35/2.55	   SUCCZ_IN_AG(zero(X), one(X)) -> U33_AG(X, binaryZ_in_g(X))
6.35/2.55	   SUCCZ_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   SUCCZ_IN_AG(one(X), zero(Z)) -> U34_AG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   SUCCZ_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   SUCC_IN_AG(zero(X), one(X)) -> U31_AG(X, binaryZ_in_g(X))
6.35/2.55	   SUCC_IN_AG(zero(X), one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   SUCC_IN_AG(one(X), zero(Z)) -> U32_AG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   ADDC_IN_AAG(b, Y, Z) -> U15_AAG(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   ADDC_IN_AAG(b, Y, Z) -> SUCCZ_IN_AG(Y, Z)
6.35/2.55	   ADDC_IN_AAG(X, Y, Z) -> U16_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> U23_AAG(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> U24_AAG(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDX_IN_AAG^1(zero(X), b, one(X)) -> U17_AAG(X, binaryZ_in_g(X))
6.35/2.55	   ADDX_IN_AAG^1(zero(X), b, one(X)) -> BINARYZ_IN_G(X)
6.35/2.55	   ADDX_IN_AAG^1(one(X), b, zero(Z)) -> U18_AAG(X, Z, succ_in_ag(X, Z))
6.35/2.55	   ADDX_IN_AAG^1(one(X), b, zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.55	   ADDX_IN_AAG^1(X, Y, Z) -> U19_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> U25_AAG(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> U20_AAG(Y, binaryZ_in_g(Y))
6.35/2.55	   ADDY_IN_AAG^1(b, zero(Y), one(Y)) -> BINARYZ_IN_G(Y)
6.35/2.55	   ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> U21_AAG(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   ADDY_IN_AAG^1(b, one(Y), zero(Z)) -> SUCC_IN_AG(Y, Z)
6.35/2.55	   ADDY_IN_AAG^1(X, Y, Z) -> U22_AAG(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.55	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> U26_AAG(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.55	
6.35/2.55	The TRS R consists of the following rules:
6.35/2.55	
6.35/2.55	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.55	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.55	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.55	   binary_in_g(b) -> binary_out_g(b)
6.35/2.55	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.55	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.55	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.55	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.55	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.55	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.55	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.55	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.55	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.55	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.55	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.55	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.55	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.55	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.55	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.55	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.55	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.55	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.55	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.55	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.55	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.55	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.55	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.55	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.55	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.55	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.55	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.55	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.55	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.55	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.55	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.55	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.55	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.55	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.55	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.55	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.55	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.55	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.55	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.55	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.55	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.55	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.55	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.55	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.55	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.55	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.55	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.55	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.55	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.55	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.55	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.55	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.55	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.55	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.55	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.55	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.55	
6.35/2.55	The argument filtering Pi contains the following mapping:
6.35/2.55	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.55	
6.35/2.55	b  =  b
6.35/2.55	
6.35/2.55	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.55	
6.35/2.55	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.55	
6.35/2.55	zero(x1)  =  zero(x1)
6.35/2.55	
6.35/2.55	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.55	
6.35/2.55	one(x1)  =  one(x1)
6.35/2.55	
6.35/2.55	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.55	
6.35/2.55	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.55	
6.35/2.55	binary_out_g(x1)  =  binary_out_g
6.35/2.55	
6.35/2.55	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.55	
6.35/2.55	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.55	
6.35/2.55	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.55	
6.35/2.55	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.55	
6.35/2.55	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.55	
6.35/2.55	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.55	
6.35/2.55	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.55	
6.35/2.55	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.55	
6.35/2.55	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.55	
6.35/2.55	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.55	
6.35/2.55	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.55	
6.35/2.55	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.55	
6.35/2.55	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.55	
6.35/2.55	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.55	
6.35/2.55	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.55	
6.35/2.55	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.55	
6.35/2.55	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.55	
6.35/2.55	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.55	
6.35/2.55	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.55	
6.35/2.55	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.55	
6.35/2.55	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.55	
6.35/2.55	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.55	
6.35/2.55	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.55	
6.35/2.55	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.55	
6.35/2.55	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.55	
6.35/2.55	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.55	
6.35/2.55	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.55	
6.35/2.55	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.55	
6.35/2.55	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.55	
6.35/2.55	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.55	
6.35/2.55	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.55	
6.35/2.55	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.55	
6.35/2.55	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.55	
6.35/2.55	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.55	
6.35/2.55	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.55	
6.35/2.55	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.55	
6.35/2.55	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.55	
6.35/2.55	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.55	
6.35/2.55	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.55	
6.35/2.55	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.55	
6.35/2.55	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.55	
6.35/2.55	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.55	
6.35/2.55	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.55	
6.35/2.55	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.55	
6.35/2.55	ADD_IN_AAG(x1, x2, x3)  =  ADD_IN_AAG(x3)
6.35/2.55	
6.35/2.55	U1_AAG(x1, x2)  =  U1_AAG(x1, x2)
6.35/2.55	
6.35/2.55	BINARYZ_IN_G(x1)  =  BINARYZ_IN_G(x1)
6.35/2.56	
6.35/2.56	U29_G(x1, x2)  =  U29_G(x2)
6.35/2.56	
6.35/2.56	U30_G(x1, x2)  =  U30_G(x2)
6.35/2.56	
6.35/2.56	BINARY_IN_G(x1)  =  BINARY_IN_G(x1)
6.35/2.56	
6.35/2.56	U27_G(x1, x2)  =  U27_G(x2)
6.35/2.56	
6.35/2.56	U28_G(x1, x2)  =  U28_G(x2)
6.35/2.56	
6.35/2.56	U2_AAG(x1, x2)  =  U2_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U3_AAG(x1, x2, x3, x4)  =  U3_AAG(x4)
6.35/2.56	
6.35/2.56	ADDZ_IN_AAG(x1, x2, x3)  =  ADDZ_IN_AAG(x3)
6.35/2.56	
6.35/2.56	U10_AAG(x1, x2, x3, x4)  =  U10_AAG(x4)
6.35/2.56	
6.35/2.56	U11_AAG(x1, x2, x3, x4)  =  U11_AAG(x4)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG(x1, x2, x3)  =  ADDX_IN_AAG(x3)
6.35/2.56	
6.35/2.56	U4_AAG(x1, x2)  =  U4_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U5_AAG(x1, x2)  =  U5_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U6_AAG(x1, x2, x3, x4)  =  U6_AAG(x4)
6.35/2.56	
6.35/2.56	U12_AAG(x1, x2, x3, x4)  =  U12_AAG(x4)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG(x1, x2, x3)  =  ADDY_IN_AAG(x3)
6.35/2.56	
6.35/2.56	U7_AAG(x1, x2)  =  U7_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U8_AAG(x1, x2)  =  U8_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U9_AAG(x1, x2, x3, x4)  =  U9_AAG(x4)
6.35/2.56	
6.35/2.56	U13_AAG(x1, x2, x3, x4)  =  U13_AAG(x4)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG(x1, x2, x3)  =  ADDC_IN_AAG(x3)
6.35/2.56	
6.35/2.56	U14_AAG(x1, x2, x3)  =  U14_AAG(x3)
6.35/2.56	
6.35/2.56	SUCCZ_IN_AG(x1, x2)  =  SUCCZ_IN_AG(x2)
6.35/2.56	
6.35/2.56	U33_AG(x1, x2)  =  U33_AG(x1, x2)
6.35/2.56	
6.35/2.56	U34_AG(x1, x2, x3)  =  U34_AG(x3)
6.35/2.56	
6.35/2.56	SUCC_IN_AG(x1, x2)  =  SUCC_IN_AG(x2)
6.35/2.56	
6.35/2.56	U31_AG(x1, x2)  =  U31_AG(x1, x2)
6.35/2.56	
6.35/2.56	U32_AG(x1, x2, x3)  =  U32_AG(x3)
6.35/2.56	
6.35/2.56	U15_AAG(x1, x2, x3)  =  U15_AAG(x3)
6.35/2.56	
6.35/2.56	U16_AAG(x1, x2, x3, x4)  =  U16_AAG(x4)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG^1(x1, x2, x3)  =  ADDC_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	U23_AAG(x1, x2, x3, x4)  =  U23_AAG(x4)
6.35/2.56	
6.35/2.56	U24_AAG(x1, x2, x3, x4)  =  U24_AAG(x4)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG^1(x1, x2, x3)  =  ADDX_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	U17_AAG(x1, x2)  =  U17_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U18_AAG(x1, x2, x3)  =  U18_AAG(x3)
6.35/2.56	
6.35/2.56	U19_AAG(x1, x2, x3, x4)  =  U19_AAG(x4)
6.35/2.56	
6.35/2.56	U25_AAG(x1, x2, x3, x4)  =  U25_AAG(x4)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG^1(x1, x2, x3)  =  ADDY_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	U20_AAG(x1, x2)  =  U20_AAG(x1, x2)
6.35/2.56	
6.35/2.56	U21_AAG(x1, x2, x3)  =  U21_AAG(x3)
6.35/2.56	
6.35/2.56	U22_AAG(x1, x2, x3, x4)  =  U22_AAG(x4)
6.35/2.56	
6.35/2.56	U26_AAG(x1, x2, x3, x4)  =  U26_AAG(x4)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(5) DependencyGraphProof (EQUIVALENT)
6.35/2.56	The approximation of the Dependency Graph [LOPSTR] contains 3 SCCs with 50 less nodes.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(6)
6.35/2.56	Complex Obligation (AND)
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(7)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	   BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	
6.35/2.56	The TRS R consists of the following rules:
6.35/2.56	
6.35/2.56	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.56	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.56	   binary_in_g(b) -> binary_out_g(b)
6.35/2.56	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.56	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.56	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.56	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.56	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.56	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.56	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.56	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.56	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.56	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.56	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.56	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.56	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.56	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.56	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.56	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.56	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.56	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.56	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.56	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.56	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.56	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.56	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.56	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.56	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.56	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.56	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.56	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.56	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.56	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.56	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.56	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.56	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.56	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.56	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.56	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.56	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.56	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.56	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.56	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.56	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.56	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.56	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.56	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.56	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.56	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.56	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.56	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.56	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.56	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.56	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.56	
6.35/2.56	The argument filtering Pi contains the following mapping:
6.35/2.56	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.56	
6.35/2.56	b  =  b
6.35/2.56	
6.35/2.56	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.56	
6.35/2.56	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.56	
6.35/2.56	zero(x1)  =  zero(x1)
6.35/2.56	
6.35/2.56	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.56	
6.35/2.56	one(x1)  =  one(x1)
6.35/2.56	
6.35/2.56	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.56	
6.35/2.56	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.56	
6.35/2.56	binary_out_g(x1)  =  binary_out_g
6.35/2.56	
6.35/2.56	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.56	
6.35/2.56	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.56	
6.35/2.56	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.56	
6.35/2.56	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.56	
6.35/2.56	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.56	
6.35/2.56	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.56	
6.35/2.56	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.56	
6.35/2.56	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.56	
6.35/2.56	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.56	
6.35/2.56	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.56	
6.35/2.56	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.56	
6.35/2.56	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.56	
6.35/2.56	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.56	
6.35/2.56	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.56	
6.35/2.56	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.56	
6.35/2.56	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.56	
6.35/2.56	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.56	
6.35/2.56	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.56	
6.35/2.56	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.56	
6.35/2.56	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.56	
6.35/2.56	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.56	
6.35/2.56	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.56	
6.35/2.56	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.56	
6.35/2.56	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.56	
6.35/2.56	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.56	
6.35/2.56	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.56	
6.35/2.56	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.56	
6.35/2.56	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.56	
6.35/2.56	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.56	
6.35/2.56	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.56	
6.35/2.56	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.56	
6.35/2.56	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.56	
6.35/2.56	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.56	
6.35/2.56	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.56	
6.35/2.56	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.56	
6.35/2.56	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.56	
6.35/2.56	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.56	
6.35/2.56	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.56	
6.35/2.56	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.56	
6.35/2.56	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.56	
6.35/2.56	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.56	
6.35/2.56	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.56	
6.35/2.56	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.56	
6.35/2.56	BINARYZ_IN_G(x1)  =  BINARYZ_IN_G(x1)
6.35/2.56	
6.35/2.56	BINARY_IN_G(x1)  =  BINARY_IN_G(x1)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(8) UsableRulesProof (EQUIVALENT)
6.35/2.56	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(9)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	   BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	Pi is empty.
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(10) PiDPToQDPProof (EQUIVALENT)
6.35/2.56	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(11)
6.35/2.56	Obligation:
6.35/2.56	Q DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	   BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	   BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	Q is empty.
6.35/2.56	We have to consider all (P,Q,R)-chains.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(12) QDPSizeChangeProof (EQUIVALENT)
6.35/2.56	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
6.35/2.56	
6.35/2.56	From the DPs we obtained the following set of size-change graphs:
6.35/2.56	*BINARY_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*BINARY_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*BINARYZ_IN_G(zero(X)) -> BINARYZ_IN_G(X)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*BINARYZ_IN_G(one(X)) -> BINARY_IN_G(X)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(13)
6.35/2.56	YES
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(14)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.56	
6.35/2.56	The TRS R consists of the following rules:
6.35/2.56	
6.35/2.56	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.56	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.56	   binary_in_g(b) -> binary_out_g(b)
6.35/2.56	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.56	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.56	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.56	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.56	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.56	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.56	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.56	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.56	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.56	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.56	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.56	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.56	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.56	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.56	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.56	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.56	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.56	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.56	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.56	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.56	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.56	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.56	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.56	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.56	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.56	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.56	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.56	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.56	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.56	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.56	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.56	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.56	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.56	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.56	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.56	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.56	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.56	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.56	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.56	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.56	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.56	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.56	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.56	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.56	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.56	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.56	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.56	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.56	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.56	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.56	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.56	
6.35/2.56	The argument filtering Pi contains the following mapping:
6.35/2.56	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.56	
6.35/2.56	b  =  b
6.35/2.56	
6.35/2.56	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.56	
6.35/2.56	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.56	
6.35/2.56	zero(x1)  =  zero(x1)
6.35/2.56	
6.35/2.56	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.56	
6.35/2.56	one(x1)  =  one(x1)
6.35/2.56	
6.35/2.56	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.56	
6.35/2.56	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.56	
6.35/2.56	binary_out_g(x1)  =  binary_out_g
6.35/2.56	
6.35/2.56	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.56	
6.35/2.56	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.56	
6.35/2.56	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.56	
6.35/2.56	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.56	
6.35/2.56	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.56	
6.35/2.56	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.56	
6.35/2.56	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.56	
6.35/2.56	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.56	
6.35/2.56	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.56	
6.35/2.56	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.56	
6.35/2.56	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.56	
6.35/2.56	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.56	
6.35/2.56	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.56	
6.35/2.56	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.56	
6.35/2.56	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.56	
6.35/2.56	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.56	
6.35/2.56	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.56	
6.35/2.56	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.56	
6.35/2.56	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.56	
6.35/2.56	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.56	
6.35/2.56	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.56	
6.35/2.56	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.56	
6.35/2.56	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.56	
6.35/2.56	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.56	
6.35/2.56	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.56	
6.35/2.56	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.56	
6.35/2.56	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.56	
6.35/2.56	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.56	
6.35/2.56	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.56	
6.35/2.56	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.56	
6.35/2.56	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.56	
6.35/2.56	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.56	
6.35/2.56	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.56	
6.35/2.56	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.56	
6.35/2.56	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.56	
6.35/2.56	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.56	
6.35/2.56	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.56	
6.35/2.56	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.56	
6.35/2.56	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.56	
6.35/2.56	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.56	
6.35/2.56	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.56	
6.35/2.56	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.56	
6.35/2.56	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.56	
6.35/2.56	SUCC_IN_AG(x1, x2)  =  SUCC_IN_AG(x2)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(15) UsableRulesProof (EQUIVALENT)
6.35/2.56	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(16)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   SUCC_IN_AG(one(X), zero(Z)) -> SUCC_IN_AG(X, Z)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	The argument filtering Pi contains the following mapping:
6.35/2.56	zero(x1)  =  zero(x1)
6.35/2.56	
6.35/2.56	one(x1)  =  one(x1)
6.35/2.56	
6.35/2.56	SUCC_IN_AG(x1, x2)  =  SUCC_IN_AG(x2)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(17) PiDPToQDPProof (SOUND)
6.35/2.56	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(18)
6.35/2.56	Obligation:
6.35/2.56	Q DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	Q is empty.
6.35/2.56	We have to consider all (P,Q,R)-chains.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(19) QDPSizeChangeProof (EQUIVALENT)
6.35/2.56	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
6.35/2.56	
6.35/2.56	From the DPs we obtained the following set of size-change graphs:
6.35/2.56	*SUCC_IN_AG(zero(Z)) -> SUCC_IN_AG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(20)
6.35/2.56	YES
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(21)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z)
6.35/2.56	   ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.56	
6.35/2.56	The TRS R consists of the following rules:
6.35/2.56	
6.35/2.56	   add_in_aag(b, b, b) -> add_out_aag(b, b, b)
6.35/2.56	   add_in_aag(X, b, X) -> U1_aag(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(zero(X)) -> U29_g(X, binaryZ_in_g(X))
6.35/2.56	   binaryZ_in_g(one(X)) -> U30_g(X, binary_in_g(X))
6.35/2.56	   binary_in_g(b) -> binary_out_g(b)
6.35/2.56	   binary_in_g(zero(X)) -> U27_g(X, binaryZ_in_g(X))
6.35/2.56	   U27_g(X, binaryZ_out_g(X)) -> binary_out_g(zero(X))
6.35/2.56	   binary_in_g(one(X)) -> U28_g(X, binary_in_g(X))
6.35/2.56	   U28_g(X, binary_out_g(X)) -> binary_out_g(one(X))
6.35/2.56	   U30_g(X, binary_out_g(X)) -> binaryZ_out_g(one(X))
6.35/2.56	   U29_g(X, binaryZ_out_g(X)) -> binaryZ_out_g(zero(X))
6.35/2.56	   U1_aag(X, binaryZ_out_g(X)) -> add_out_aag(X, b, X)
6.35/2.56	   add_in_aag(b, Y, Y) -> U2_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U2_aag(Y, binaryZ_out_g(Y)) -> add_out_aag(b, Y, Y)
6.35/2.56	   add_in_aag(X, Y, Z) -> U3_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), zero(Y), zero(Z)) -> U10_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(zero(X), one(Y), one(Z)) -> U11_aag(X, Y, Z, addx_in_aag(X, Y, Z))
6.35/2.56	   addx_in_aag(one(X), b, one(X)) -> U4_aag(X, binary_in_g(X))
6.35/2.56	   U4_aag(X, binary_out_g(X)) -> addx_out_aag(one(X), b, one(X))
6.35/2.56	   addx_in_aag(zero(X), b, zero(X)) -> U5_aag(X, binaryZ_in_g(X))
6.35/2.56	   U5_aag(X, binaryZ_out_g(X)) -> addx_out_aag(zero(X), b, zero(X))
6.35/2.56	   addx_in_aag(X, Y, Z) -> U6_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), zero(Y), one(Z)) -> U12_aag(X, Y, Z, addy_in_aag(X, Y, Z))
6.35/2.56	   addy_in_aag(b, one(Y), one(Y)) -> U7_aag(Y, binary_in_g(Y))
6.35/2.56	   U7_aag(Y, binary_out_g(Y)) -> addy_out_aag(b, one(Y), one(Y))
6.35/2.56	   addy_in_aag(b, zero(Y), zero(Y)) -> U8_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U8_aag(Y, binaryZ_out_g(Y)) -> addy_out_aag(b, zero(Y), zero(Y))
6.35/2.56	   addy_in_aag(X, Y, Z) -> U9_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   addz_in_aag(one(X), one(Y), zero(Z)) -> U13_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   addc_in_aag(b, b, one(b)) -> addc_out_aag(b, b, one(b))
6.35/2.56	   addc_in_aag(X, b, Z) -> U14_aag(X, Z, succZ_in_ag(X, Z))
6.35/2.56	   succZ_in_ag(zero(X), one(X)) -> U33_ag(X, binaryZ_in_g(X))
6.35/2.56	   U33_ag(X, binaryZ_out_g(X)) -> succZ_out_ag(zero(X), one(X))
6.35/2.56	   succZ_in_ag(one(X), zero(Z)) -> U34_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   succ_in_ag(b, one(b)) -> succ_out_ag(b, one(b))
6.35/2.56	   succ_in_ag(zero(X), one(X)) -> U31_ag(X, binaryZ_in_g(X))
6.35/2.56	   U31_ag(X, binaryZ_out_g(X)) -> succ_out_ag(zero(X), one(X))
6.35/2.56	   succ_in_ag(one(X), zero(Z)) -> U32_ag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U32_ag(X, Z, succ_out_ag(X, Z)) -> succ_out_ag(one(X), zero(Z))
6.35/2.56	   U34_ag(X, Z, succ_out_ag(X, Z)) -> succZ_out_ag(one(X), zero(Z))
6.35/2.56	   U14_aag(X, Z, succZ_out_ag(X, Z)) -> addc_out_aag(X, b, Z)
6.35/2.56	   addc_in_aag(b, Y, Z) -> U15_aag(Y, Z, succZ_in_ag(Y, Z))
6.35/2.56	   U15_aag(Y, Z, succZ_out_ag(Y, Z)) -> addc_out_aag(b, Y, Z)
6.35/2.56	   addc_in_aag(X, Y, Z) -> U16_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(zero(X), zero(Y), one(Z)) -> U23_aag(X, Y, Z, addz_in_aag(X, Y, Z))
6.35/2.56	   U23_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), zero(Y), one(Z))
6.35/2.56	   addC_in_aag(zero(X), one(Y), zero(Z)) -> U24_aag(X, Y, Z, addX_in_aag(X, Y, Z))
6.35/2.56	   addX_in_aag(zero(X), b, one(X)) -> U17_aag(X, binaryZ_in_g(X))
6.35/2.56	   U17_aag(X, binaryZ_out_g(X)) -> addX_out_aag(zero(X), b, one(X))
6.35/2.56	   addX_in_aag(one(X), b, zero(Z)) -> U18_aag(X, Z, succ_in_ag(X, Z))
6.35/2.56	   U18_aag(X, Z, succ_out_ag(X, Z)) -> addX_out_aag(one(X), b, zero(Z))
6.35/2.56	   addX_in_aag(X, Y, Z) -> U19_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), zero(Y), zero(Z)) -> U25_aag(X, Y, Z, addY_in_aag(X, Y, Z))
6.35/2.56	   addY_in_aag(b, zero(Y), one(Y)) -> U20_aag(Y, binaryZ_in_g(Y))
6.35/2.56	   U20_aag(Y, binaryZ_out_g(Y)) -> addY_out_aag(b, zero(Y), one(Y))
6.35/2.56	   addY_in_aag(b, one(Y), zero(Z)) -> U21_aag(Y, Z, succ_in_ag(Y, Z))
6.35/2.56	   U21_aag(Y, Z, succ_out_ag(Y, Z)) -> addY_out_aag(b, one(Y), zero(Z))
6.35/2.56	   addY_in_aag(X, Y, Z) -> U22_aag(X, Y, Z, addC_in_aag(X, Y, Z))
6.35/2.56	   addC_in_aag(one(X), one(Y), one(Z)) -> U26_aag(X, Y, Z, addc_in_aag(X, Y, Z))
6.35/2.56	   U26_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addC_out_aag(one(X), one(Y), one(Z))
6.35/2.56	   U22_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addY_out_aag(X, Y, Z)
6.35/2.56	   U25_aag(X, Y, Z, addY_out_aag(X, Y, Z)) -> addC_out_aag(one(X), zero(Y), zero(Z))
6.35/2.56	   U19_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addX_out_aag(X, Y, Z)
6.35/2.56	   U24_aag(X, Y, Z, addX_out_aag(X, Y, Z)) -> addC_out_aag(zero(X), one(Y), zero(Z))
6.35/2.56	   U16_aag(X, Y, Z, addC_out_aag(X, Y, Z)) -> addc_out_aag(X, Y, Z)
6.35/2.56	   U13_aag(X, Y, Z, addc_out_aag(X, Y, Z)) -> addz_out_aag(one(X), one(Y), zero(Z))
6.35/2.56	   U9_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addy_out_aag(X, Y, Z)
6.35/2.56	   U12_aag(X, Y, Z, addy_out_aag(X, Y, Z)) -> addz_out_aag(one(X), zero(Y), one(Z))
6.35/2.56	   U6_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addx_out_aag(X, Y, Z)
6.35/2.56	   U11_aag(X, Y, Z, addx_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), one(Y), one(Z))
6.35/2.56	   U10_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> addz_out_aag(zero(X), zero(Y), zero(Z))
6.35/2.56	   U3_aag(X, Y, Z, addz_out_aag(X, Y, Z)) -> add_out_aag(X, Y, Z)
6.35/2.56	
6.35/2.56	The argument filtering Pi contains the following mapping:
6.35/2.56	add_in_aag(x1, x2, x3)  =  add_in_aag(x3)
6.35/2.56	
6.35/2.56	b  =  b
6.35/2.56	
6.35/2.56	add_out_aag(x1, x2, x3)  =  add_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U1_aag(x1, x2)  =  U1_aag(x1, x2)
6.35/2.56	
6.35/2.56	binaryZ_in_g(x1)  =  binaryZ_in_g(x1)
6.35/2.56	
6.35/2.56	zero(x1)  =  zero(x1)
6.35/2.56	
6.35/2.56	U29_g(x1, x2)  =  U29_g(x2)
6.35/2.56	
6.35/2.56	one(x1)  =  one(x1)
6.35/2.56	
6.35/2.56	U30_g(x1, x2)  =  U30_g(x2)
6.35/2.56	
6.35/2.56	binary_in_g(x1)  =  binary_in_g(x1)
6.35/2.56	
6.35/2.56	binary_out_g(x1)  =  binary_out_g
6.35/2.56	
6.35/2.56	U27_g(x1, x2)  =  U27_g(x2)
6.35/2.56	
6.35/2.56	binaryZ_out_g(x1)  =  binaryZ_out_g
6.35/2.56	
6.35/2.56	U28_g(x1, x2)  =  U28_g(x2)
6.35/2.56	
6.35/2.56	U2_aag(x1, x2)  =  U2_aag(x1, x2)
6.35/2.56	
6.35/2.56	U3_aag(x1, x2, x3, x4)  =  U3_aag(x4)
6.35/2.56	
6.35/2.56	addz_in_aag(x1, x2, x3)  =  addz_in_aag(x3)
6.35/2.56	
6.35/2.56	U10_aag(x1, x2, x3, x4)  =  U10_aag(x4)
6.35/2.56	
6.35/2.56	U11_aag(x1, x2, x3, x4)  =  U11_aag(x4)
6.35/2.56	
6.35/2.56	addx_in_aag(x1, x2, x3)  =  addx_in_aag(x3)
6.35/2.56	
6.35/2.56	U4_aag(x1, x2)  =  U4_aag(x1, x2)
6.35/2.56	
6.35/2.56	addx_out_aag(x1, x2, x3)  =  addx_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U5_aag(x1, x2)  =  U5_aag(x1, x2)
6.35/2.56	
6.35/2.56	U6_aag(x1, x2, x3, x4)  =  U6_aag(x4)
6.35/2.56	
6.35/2.56	U12_aag(x1, x2, x3, x4)  =  U12_aag(x4)
6.35/2.56	
6.35/2.56	addy_in_aag(x1, x2, x3)  =  addy_in_aag(x3)
6.35/2.56	
6.35/2.56	U7_aag(x1, x2)  =  U7_aag(x1, x2)
6.35/2.56	
6.35/2.56	addy_out_aag(x1, x2, x3)  =  addy_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U8_aag(x1, x2)  =  U8_aag(x1, x2)
6.35/2.56	
6.35/2.56	U9_aag(x1, x2, x3, x4)  =  U9_aag(x4)
6.35/2.56	
6.35/2.56	U13_aag(x1, x2, x3, x4)  =  U13_aag(x4)
6.35/2.56	
6.35/2.56	addc_in_aag(x1, x2, x3)  =  addc_in_aag(x3)
6.35/2.56	
6.35/2.56	addc_out_aag(x1, x2, x3)  =  addc_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U14_aag(x1, x2, x3)  =  U14_aag(x3)
6.35/2.56	
6.35/2.56	succZ_in_ag(x1, x2)  =  succZ_in_ag(x2)
6.35/2.56	
6.35/2.56	U33_ag(x1, x2)  =  U33_ag(x1, x2)
6.35/2.56	
6.35/2.56	succZ_out_ag(x1, x2)  =  succZ_out_ag(x1)
6.35/2.56	
6.35/2.56	U34_ag(x1, x2, x3)  =  U34_ag(x3)
6.35/2.56	
6.35/2.56	succ_in_ag(x1, x2)  =  succ_in_ag(x2)
6.35/2.56	
6.35/2.56	succ_out_ag(x1, x2)  =  succ_out_ag(x1)
6.35/2.56	
6.35/2.56	U31_ag(x1, x2)  =  U31_ag(x1, x2)
6.35/2.56	
6.35/2.56	U32_ag(x1, x2, x3)  =  U32_ag(x3)
6.35/2.56	
6.35/2.56	U15_aag(x1, x2, x3)  =  U15_aag(x3)
6.35/2.56	
6.35/2.56	U16_aag(x1, x2, x3, x4)  =  U16_aag(x4)
6.35/2.56	
6.35/2.56	addC_in_aag(x1, x2, x3)  =  addC_in_aag(x3)
6.35/2.56	
6.35/2.56	U23_aag(x1, x2, x3, x4)  =  U23_aag(x4)
6.35/2.56	
6.35/2.56	addz_out_aag(x1, x2, x3)  =  addz_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	addC_out_aag(x1, x2, x3)  =  addC_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U24_aag(x1, x2, x3, x4)  =  U24_aag(x4)
6.35/2.56	
6.35/2.56	addX_in_aag(x1, x2, x3)  =  addX_in_aag(x3)
6.35/2.56	
6.35/2.56	U17_aag(x1, x2)  =  U17_aag(x1, x2)
6.35/2.56	
6.35/2.56	addX_out_aag(x1, x2, x3)  =  addX_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U18_aag(x1, x2, x3)  =  U18_aag(x3)
6.35/2.56	
6.35/2.56	U19_aag(x1, x2, x3, x4)  =  U19_aag(x4)
6.35/2.56	
6.35/2.56	U25_aag(x1, x2, x3, x4)  =  U25_aag(x4)
6.35/2.56	
6.35/2.56	addY_in_aag(x1, x2, x3)  =  addY_in_aag(x3)
6.35/2.56	
6.35/2.56	U20_aag(x1, x2)  =  U20_aag(x1, x2)
6.35/2.56	
6.35/2.56	addY_out_aag(x1, x2, x3)  =  addY_out_aag(x1, x2)
6.35/2.56	
6.35/2.56	U21_aag(x1, x2, x3)  =  U21_aag(x3)
6.35/2.56	
6.35/2.56	U22_aag(x1, x2, x3, x4)  =  U22_aag(x4)
6.35/2.56	
6.35/2.56	U26_aag(x1, x2, x3, x4)  =  U26_aag(x4)
6.35/2.56	
6.35/2.56	ADDZ_IN_AAG(x1, x2, x3)  =  ADDZ_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG(x1, x2, x3)  =  ADDX_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG(x1, x2, x3)  =  ADDY_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG(x1, x2, x3)  =  ADDC_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG^1(x1, x2, x3)  =  ADDC_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG^1(x1, x2, x3)  =  ADDX_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG^1(x1, x2, x3)  =  ADDY_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(22) UsableRulesProof (EQUIVALENT)
6.35/2.56	For (infinitary) constructor rewriting [LOPSTR] we can delete all non-usable rules from R.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(23)
6.35/2.56	Obligation:
6.35/2.56	Pi DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   ADDX_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(zero(X), zero(Y), zero(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(zero(X), one(Y), one(Z)) -> ADDX_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(one(X), zero(Y), one(Z)) -> ADDY_IN_AAG(X, Y, Z)
6.35/2.56	   ADDY_IN_AAG(X, Y, Z) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDZ_IN_AAG(one(X), one(Y), zero(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(X), zero(Y), one(Z)) -> ADDZ_IN_AAG(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(X), one(Y), zero(Z)) -> ADDX_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDX_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(one(X), zero(Y), zero(Z)) -> ADDY_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDY_IN_AAG^1(X, Y, Z) -> ADDC_IN_AAG^1(X, Y, Z)
6.35/2.56	   ADDC_IN_AAG^1(one(X), one(Y), one(Z)) -> ADDC_IN_AAG(X, Y, Z)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	The argument filtering Pi contains the following mapping:
6.35/2.56	zero(x1)  =  zero(x1)
6.35/2.56	
6.35/2.56	one(x1)  =  one(x1)
6.35/2.56	
6.35/2.56	ADDZ_IN_AAG(x1, x2, x3)  =  ADDZ_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG(x1, x2, x3)  =  ADDX_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG(x1, x2, x3)  =  ADDY_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG(x1, x2, x3)  =  ADDC_IN_AAG(x3)
6.35/2.56	
6.35/2.56	ADDC_IN_AAG^1(x1, x2, x3)  =  ADDC_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	ADDX_IN_AAG^1(x1, x2, x3)  =  ADDX_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	ADDY_IN_AAG^1(x1, x2, x3)  =  ADDY_IN_AAG^1(x3)
6.35/2.56	
6.35/2.56	
6.35/2.56	We have to consider all (P,R,Pi)-chains
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(24) PiDPToQDPProof (SOUND)
6.35/2.56	Transforming (infinitary) constructor rewriting Pi-DP problem [LOPSTR] into ordinary QDP problem [LPAR04] by application of Pi.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(25)
6.35/2.56	Obligation:
6.35/2.56	Q DP problem:
6.35/2.56	The TRS P consists of the following rules:
6.35/2.56	
6.35/2.56	   ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z)
6.35/2.56	   ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z)
6.35/2.56	   ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z)
6.35/2.56	   ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z)
6.35/2.56	   ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z)
6.35/2.56	   ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z)
6.35/2.56	   ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	   ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z)
6.35/2.56	   ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	   ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z)
6.35/2.56	   ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	   ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z)
6.35/2.56	
6.35/2.56	R is empty.
6.35/2.56	Q is empty.
6.35/2.56	We have to consider all (P,Q,R)-chains.
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(26) QDPSizeChangeProof (EQUIVALENT)
6.35/2.56	By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 
6.35/2.56	
6.35/2.56	From the DPs we obtained the following set of size-change graphs:
6.35/2.56	*ADDZ_IN_AAG(one(Z)) -> ADDX_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDZ_IN_AAG(zero(Z)) -> ADDZ_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDX_IN_AAG(Z) -> ADDZ_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 >= 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDY_IN_AAG(Z) -> ADDZ_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 >= 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDC_IN_AAG^1(one(Z)) -> ADDZ_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDZ_IN_AAG(one(Z)) -> ADDY_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDZ_IN_AAG(zero(Z)) -> ADDC_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDC_IN_AAG(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	The graph contains the following edges 1 >= 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDC_IN_AAG^1(one(Z)) -> ADDC_IN_AAG(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDX_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	The graph contains the following edges 1 >= 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDY_IN_AAG^1(Z) -> ADDC_IN_AAG^1(Z)
6.35/2.56	The graph contains the following edges 1 >= 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDC_IN_AAG^1(zero(Z)) -> ADDX_IN_AAG^1(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	*ADDC_IN_AAG^1(zero(Z)) -> ADDY_IN_AAG^1(Z)
6.35/2.56	The graph contains the following edges 1 > 1
6.35/2.56	
6.35/2.56	
6.35/2.56	----------------------------------------
6.35/2.56	
6.35/2.56	(27)
6.35/2.56	YES
6.35/2.59	EOF