Left Termination proof of /tmp/tmpnYQanm/binary4.pl

Left Termination of the query pattern times(f,f,b) w.r.t. the given Prolog program could not be shown:

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

add₃(b₀, b₀, b₀).
add₃(X, b₀, X) :- binaryZ₁(X).
add₃(b₀, Y, Y) :- binaryZ₁(Y).
add₃(X, Y, Z) :- addz₃(X, Y, Z).
addx₃(one₁(X), b₀, one₁(X)) :- binary₁(X).
addx₃(zero₁(X), b₀, zero₁(X)) :- binaryZ₁(X).
addx₃(X, Y, Z) :- addz₃(X, Y, Z).
addy₃(b₀, one₁(Y), one₁(Y)) :- binary₁(Y).
addy₃(b₀, zero₁(Y), zero₁(Y)) :- binaryZ₁(Y).
addy₃(X, Y, Z) :- addz₃(X, Y, Z).
addz₃(zero₁(X), zero₁(Y), zero₁(Z)) :- addz₃(X, Y, Z).
addz₃(zero₁(X), one₁(Y), one₁(Z)) :- addx₃(X, Y, Z).
addz₃(one₁(X), zero₁(Y), one₁(Z)) :- addy₃(X, Y, Z).
addz₃(one₁(X), one₁(Y), zero₁(Z)) :- addc₃(X, Y, Z).
addc₃(b₀, b₀, one₁(b₀)).
addc₃(X, b₀, Z) :- succZ₂(X, Z).
addc₃(b₀, Y, Z) :- succZ₂(Y, Z).
addc₃(X, Y, Z) :- addC₃(X, Y, Z).
addX₃(zero₁(X), b₀, one₁(X)) :- binaryZ₁(X).
addX₃(one₁(X), b₀, zero₁(Z)) :- succ₂(X, Z).
addX₃(X, Y, Z) :- addC₃(X, Y, Z).
addY₃(b₀, zero₁(Y), one₁(Y)) :- binaryZ₁(Y).
addY₃(b₀, one₁(Y), zero₁(Z)) :- succ₂(Y, Z).
addY₃(X, Y, Z) :- addC₃(X, Y, Z).
addC₃(zero₁(X), zero₁(Y), one₁(Z)) :- addz₃(X, Y, Z).
addC₃(zero₁(X), one₁(Y), zero₁(Z)) :- addX₃(X, Y, Z).
addC₃(one₁(X), zero₁(Y), zero₁(Z)) :- addY₃(X, Y, Z).
addC₃(one₁(X), one₁(Y), one₁(Z)) :- addc₃(X, Y, Z).
binary₁(b₀).
binary₁(zero₁(X)) :- binaryZ₁(X).
binary₁(one₁(X)) :- binary₁(X).
binaryZ₁(zero₁(X)) :- binaryZ₁(X).
binaryZ₁(one₁(X)) :- binary₁(X).
succ₂(b₀, one₁(b₀)).
succ₂(zero₁(X), one₁(X)) :- binaryZ₁(X).
succ₂(one₁(X), zero₁(Z)) :- succ₂(X, Z).
succZ₂(zero₁(X), one₁(X)) :- binaryZ₁(X).
succZ₂(one₁(X), zero₁(Z)) :- succ₂(X, Z).
times₃(one₁(b₀), X, X).
times₃(zero₁(R), S, zero₁(RS)) :- times₃(R, S, RS).
times₃(one₁(R), S, RSS) :- times₃(R, S, RS), add₃(S, zero₁(RS), RSS).

With regard to the inferred argument filtering the predicates were used in the following modes:
times₃: (f,f,b) (f,f,f)
add₃: (f,f,f) (f,f,b)
binaryZ₁: (f) (b)
binary₁: (f) (b)
addz₃: (f,f,f) (f,f,b)
addx₃: (f,f,f) (f,f,b)
addy₃: (f,f,f) (f,f,b)
addc₃: (f,f,f) (f,f,b)
succZ₂: (f,f) (f,b)
succ₂: (f,f) (f,b)
addC₃: (f,f,f) (f,f,b)
addX₃: (f,f,f) (f,f,b)
addY₃: (f,f,f) (f,f,b)
Transforming PROLOG into the following Term Rewriting System:
Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

  ↳ PrologToPiTRSProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAG₄(R, S, RS, times_3_in_aag₃(R, S, RS))
TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAA₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAA₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAA₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAA₃(X, b_0, X) -> IF_ADD_3_IN_1_AAA₂(X, binaryZ_1_in_a₁(X))
ADD_3_IN_AAA₃(X, b_0, X) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> IF_BINARY_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADD_3_IN_AAA₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADD_3_IN_AAA₃(b_0, Y, Y) -> BINARYZ_1_IN_A₁(Y)
ADD_3_IN_AAA₃(X, Y, Z) -> IF_ADD_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADD_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAA₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA₂(X, binary_1_in_a₁(X))
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAA₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAA₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA₂(Y, binary_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAA₃(X, Z, succZ_2_in_aa₂(X, Z))
ADDC_3_IN_AAA₃(X, b_0, Z) -> SUCCZ_2_IN_AA₂(X, Z)
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDC_3_IN_AAA₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAA₃(Y, Z, succZ_2_in_aa₂(Y, Z))
ADDC_3_IN_AAA₃(b_0, Y, Z) -> SUCCZ_2_IN_AA₂(Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAA₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAA₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA¹₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAA₃(X, Z, succ_2_in_aa₂(X, Z))
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAA¹₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA¹₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAA₃(Y, Z, succ_2_in_aa₂(Y, Z))
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AA₂(Y, Z)
ADDY_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAG₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAG₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAG₃(X, b_0, X) -> IF_ADD_3_IN_1_AAG₂(X, binaryZ_1_in_g₁(X))
ADD_3_IN_AAG₃(X, b_0, X) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> IF_BINARY_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADD_3_IN_AAG₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADD_3_IN_AAG₃(b_0, Y, Y) -> BINARYZ_1_IN_G₁(Y)
ADD_3_IN_AAG₃(X, Y, Z) -> IF_ADD_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADD_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAG₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG₂(X, binary_1_in_g₁(X))
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAG₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAG₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG₂(Y, binary_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAG₃(X, Z, succZ_2_in_ag₂(X, Z))
ADDC_3_IN_AAG₃(X, b_0, Z) -> SUCCZ_2_IN_AG₂(X, Z)
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDC_3_IN_AAG₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAG₃(Y, Z, succZ_2_in_ag₂(Y, Z))
ADDC_3_IN_AAG₃(b_0, Y, Z) -> SUCCZ_2_IN_AG₂(Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAG₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAG₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG¹₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAG₃(X, Z, succ_2_in_ag₂(X, Z))
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAG¹₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG¹₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAG₃(Y, Z, succ_2_in_ag₂(Y, Z))
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AG₂(Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

The argument filtering Pi contains the following mapping:
times_3_in_aag₃(x1, x2, x3) = times_3_in_aag₁(x3)
b_0 = b_0
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
times_3_out_aag₃(x1, x2, x3) = times_3_out_aag₂(x1, x2)
if_times_3_in_1_aag₄(x1, x2, x3, x4) = if_times_3_in_1_aag₁(x4)
if_times_3_in_2_aag₄(x1, x2, x3, x4) = if_times_3_in_2_aag₂(x3, x4)
times_3_in_aaa₃(x1, x2, x3) = times_3_in_aaa
times_3_out_aaa₃(x1, x2, x3) = times_3_out_aaa₁(x1)
if_times_3_in_1_aaa₄(x1, x2, x3, x4) = if_times_3_in_1_aaa₁(x4)
if_times_3_in_2_aaa₄(x1, x2, x3, x4) = if_times_3_in_2_aaa₁(x4)
if_times_3_in_3_aaa₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aaa₂(x1, x5)
add_3_in_aaa₃(x1, x2, x3) = add_3_in_aaa
add_3_out_aaa₃(x1, x2, x3) = add_3_out_aaa₃(x1, x2, x3)
if_add_3_in_1_aaa₂(x1, x2) = if_add_3_in_1_aaa₁(x2)
binaryZ_1_in_a₁(x1) = binaryZ_1_in_a
if_binaryZ_1_in_1_a₂(x1, x2) = if_binaryZ_1_in_1_a₁(x2)
if_binaryZ_1_in_2_a₂(x1, x2) = if_binaryZ_1_in_2_a₁(x2)
binary_1_in_a₁(x1) = binary_1_in_a
binary_1_out_a₁(x1) = binary_1_out_a₁(x1)
if_binary_1_in_1_a₂(x1, x2) = if_binary_1_in_1_a₁(x2)
binaryZ_1_out_a₁(x1) = binaryZ_1_out_a₁(x1)
if_binary_1_in_2_a₂(x1, x2) = if_binary_1_in_2_a₁(x2)
if_add_3_in_2_aaa₂(x1, x2) = if_add_3_in_2_aaa₁(x2)
if_add_3_in_3_aaa₄(x1, x2, x3, x4) = if_add_3_in_3_aaa₁(x4)
addz_3_in_aaa₃(x1, x2, x3) = addz_3_in_aaa
if_addz_3_in_1_aaa₄(x1, x2, x3, x4) = if_addz_3_in_1_aaa₁(x4)
if_addz_3_in_2_aaa₄(x1, x2, x3, x4) = if_addz_3_in_2_aaa₁(x4)
addx_3_in_aaa₃(x1, x2, x3) = addx_3_in_aaa
if_addx_3_in_1_aaa₂(x1, x2) = if_addx_3_in_1_aaa₁(x2)
addx_3_out_aaa₃(x1, x2, x3) = addx_3_out_aaa₃(x1, x2, x3)
if_addx_3_in_2_aaa₂(x1, x2) = if_addx_3_in_2_aaa₁(x2)
if_addx_3_in_3_aaa₄(x1, x2, x3, x4) = if_addx_3_in_3_aaa₁(x4)
if_addz_3_in_3_aaa₄(x1, x2, x3, x4) = if_addz_3_in_3_aaa₁(x4)
addy_3_in_aaa₃(x1, x2, x3) = addy_3_in_aaa
if_addy_3_in_1_aaa₂(x1, x2) = if_addy_3_in_1_aaa₁(x2)
addy_3_out_aaa₃(x1, x2, x3) = addy_3_out_aaa₃(x1, x2, x3)
if_addy_3_in_2_aaa₂(x1, x2) = if_addy_3_in_2_aaa₁(x2)
if_addy_3_in_3_aaa₄(x1, x2, x3, x4) = if_addy_3_in_3_aaa₁(x4)
if_addz_3_in_4_aaa₄(x1, x2, x3, x4) = if_addz_3_in_4_aaa₁(x4)
addc_3_in_aaa₃(x1, x2, x3) = addc_3_in_aaa
addc_3_out_aaa₃(x1, x2, x3) = addc_3_out_aaa₃(x1, x2, x3)
if_addc_3_in_1_aaa₃(x1, x2, x3) = if_addc_3_in_1_aaa₁(x3)
succZ_2_in_aa₂(x1, x2) = succZ_2_in_aa
if_succZ_2_in_1_aa₂(x1, x2) = if_succZ_2_in_1_aa₁(x2)
succZ_2_out_aa₂(x1, x2) = succZ_2_out_aa₂(x1, x2)
if_succZ_2_in_2_aa₃(x1, x2, x3) = if_succZ_2_in_2_aa₁(x3)
succ_2_in_aa₂(x1, x2) = succ_2_in_aa
succ_2_out_aa₂(x1, x2) = succ_2_out_aa₂(x1, x2)
if_succ_2_in_1_aa₂(x1, x2) = if_succ_2_in_1_aa₁(x2)
if_succ_2_in_2_aa₃(x1, x2, x3) = if_succ_2_in_2_aa₁(x3)
if_addc_3_in_2_aaa₃(x1, x2, x3) = if_addc_3_in_2_aaa₁(x3)
if_addc_3_in_3_aaa₄(x1, x2, x3, x4) = if_addc_3_in_3_aaa₁(x4)
addC_3_in_aaa₃(x1, x2, x3) = addC_3_in_aaa
if_addC_3_in_1_aaa₄(x1, x2, x3, x4) = if_addC_3_in_1_aaa₁(x4)
addz_3_out_aaa₃(x1, x2, x3) = addz_3_out_aaa₃(x1, x2, x3)
addC_3_out_aaa₃(x1, x2, x3) = addC_3_out_aaa₃(x1, x2, x3)
if_addC_3_in_2_aaa₄(x1, x2, x3, x4) = if_addC_3_in_2_aaa₁(x4)
addX_3_in_aaa₃(x1, x2, x3) = addX_3_in_aaa
if_addX_3_in_1_aaa₂(x1, x2) = if_addX_3_in_1_aaa₁(x2)
addX_3_out_aaa₃(x1, x2, x3) = addX_3_out_aaa₃(x1, x2, x3)
if_addX_3_in_2_aaa₃(x1, x2, x3) = if_addX_3_in_2_aaa₁(x3)
if_addX_3_in_3_aaa₄(x1, x2, x3, x4) = if_addX_3_in_3_aaa₁(x4)
if_addC_3_in_3_aaa₄(x1, x2, x3, x4) = if_addC_3_in_3_aaa₁(x4)
addY_3_in_aaa₃(x1, x2, x3) = addY_3_in_aaa
if_addY_3_in_1_aaa₂(x1, x2) = if_addY_3_in_1_aaa₁(x2)
addY_3_out_aaa₃(x1, x2, x3) = addY_3_out_aaa₃(x1, x2, x3)
if_addY_3_in_2_aaa₃(x1, x2, x3) = if_addY_3_in_2_aaa₁(x3)
if_addY_3_in_3_aaa₄(x1, x2, x3, x4) = if_addY_3_in_3_aaa₁(x4)
if_addC_3_in_4_aaa₄(x1, x2, x3, x4) = if_addC_3_in_4_aaa₁(x4)
if_times_3_in_3_aag₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aag₂(x1, x5)
add_3_in_aag₃(x1, x2, x3) = add_3_in_aag₁(x3)
add_3_out_aag₃(x1, x2, x3) = add_3_out_aag₂(x1, x2)
if_add_3_in_1_aag₂(x1, x2) = if_add_3_in_1_aag₂(x1, x2)
binaryZ_1_in_g₁(x1) = binaryZ_1_in_g₁(x1)
if_binaryZ_1_in_1_g₂(x1, x2) = if_binaryZ_1_in_1_g₁(x2)
if_binaryZ_1_in_2_g₂(x1, x2) = if_binaryZ_1_in_2_g₁(x2)
binary_1_in_g₁(x1) = binary_1_in_g₁(x1)
binary_1_out_g₁(x1) = binary_1_out_g
if_binary_1_in_1_g₂(x1, x2) = if_binary_1_in_1_g₁(x2)
binaryZ_1_out_g₁(x1) = binaryZ_1_out_g
if_binary_1_in_2_g₂(x1, x2) = if_binary_1_in_2_g₁(x2)
if_add_3_in_2_aag₂(x1, x2) = if_add_3_in_2_aag₂(x1, x2)
if_add_3_in_3_aag₄(x1, x2, x3, x4) = if_add_3_in_3_aag₁(x4)
addz_3_in_aag₃(x1, x2, x3) = addz_3_in_aag₁(x3)
if_addz_3_in_1_aag₄(x1, x2, x3, x4) = if_addz_3_in_1_aag₁(x4)
if_addz_3_in_2_aag₄(x1, x2, x3, x4) = if_addz_3_in_2_aag₁(x4)
addx_3_in_aag₃(x1, x2, x3) = addx_3_in_aag₁(x3)
if_addx_3_in_1_aag₂(x1, x2) = if_addx_3_in_1_aag₂(x1, x2)
addx_3_out_aag₃(x1, x2, x3) = addx_3_out_aag₂(x1, x2)
if_addx_3_in_2_aag₂(x1, x2) = if_addx_3_in_2_aag₂(x1, x2)
if_addx_3_in_3_aag₄(x1, x2, x3, x4) = if_addx_3_in_3_aag₁(x4)
if_addz_3_in_3_aag₄(x1, x2, x3, x4) = if_addz_3_in_3_aag₁(x4)
addy_3_in_aag₃(x1, x2, x3) = addy_3_in_aag₁(x3)
if_addy_3_in_1_aag₂(x1, x2) = if_addy_3_in_1_aag₂(x1, x2)
addy_3_out_aag₃(x1, x2, x3) = addy_3_out_aag₂(x1, x2)
if_addy_3_in_2_aag₂(x1, x2) = if_addy_3_in_2_aag₂(x1, x2)
if_addy_3_in_3_aag₄(x1, x2, x3, x4) = if_addy_3_in_3_aag₁(x4)
if_addz_3_in_4_aag₄(x1, x2, x3, x4) = if_addz_3_in_4_aag₁(x4)
addc_3_in_aag₃(x1, x2, x3) = addc_3_in_aag₁(x3)
addc_3_out_aag₃(x1, x2, x3) = addc_3_out_aag₂(x1, x2)
if_addc_3_in_1_aag₃(x1, x2, x3) = if_addc_3_in_1_aag₁(x3)
succZ_2_in_ag₂(x1, x2) = succZ_2_in_ag₁(x2)
if_succZ_2_in_1_ag₂(x1, x2) = if_succZ_2_in_1_ag₂(x1, x2)
succZ_2_out_ag₂(x1, x2) = succZ_2_out_ag₁(x1)
if_succZ_2_in_2_ag₃(x1, x2, x3) = if_succZ_2_in_2_ag₁(x3)
succ_2_in_ag₂(x1, x2) = succ_2_in_ag₁(x2)
succ_2_out_ag₂(x1, x2) = succ_2_out_ag₁(x1)
if_succ_2_in_1_ag₂(x1, x2) = if_succ_2_in_1_ag₂(x1, x2)
if_succ_2_in_2_ag₃(x1, x2, x3) = if_succ_2_in_2_ag₁(x3)
if_addc_3_in_2_aag₃(x1, x2, x3) = if_addc_3_in_2_aag₁(x3)
if_addc_3_in_3_aag₄(x1, x2, x3, x4) = if_addc_3_in_3_aag₁(x4)
addC_3_in_aag₃(x1, x2, x3) = addC_3_in_aag₁(x3)
if_addC_3_in_1_aag₄(x1, x2, x3, x4) = if_addC_3_in_1_aag₁(x4)
addz_3_out_aag₃(x1, x2, x3) = addz_3_out_aag₂(x1, x2)
addC_3_out_aag₃(x1, x2, x3) = addC_3_out_aag₂(x1, x2)
if_addC_3_in_2_aag₄(x1, x2, x3, x4) = if_addC_3_in_2_aag₁(x4)
addX_3_in_aag₃(x1, x2, x3) = addX_3_in_aag₁(x3)
if_addX_3_in_1_aag₂(x1, x2) = if_addX_3_in_1_aag₂(x1, x2)
addX_3_out_aag₃(x1, x2, x3) = addX_3_out_aag₂(x1, x2)
if_addX_3_in_2_aag₃(x1, x2, x3) = if_addX_3_in_2_aag₁(x3)
if_addX_3_in_3_aag₄(x1, x2, x3, x4) = if_addX_3_in_3_aag₁(x4)
if_addC_3_in_3_aag₄(x1, x2, x3, x4) = if_addC_3_in_3_aag₁(x4)
addY_3_in_aag₃(x1, x2, x3) = addY_3_in_aag₁(x3)
if_addY_3_in_1_aag₂(x1, x2) = if_addY_3_in_1_aag₂(x1, x2)
addY_3_out_aag₃(x1, x2, x3) = addY_3_out_aag₂(x1, x2)
if_addY_3_in_2_aag₃(x1, x2, x3) = if_addY_3_in_2_aag₁(x3)
if_addY_3_in_3_aag₄(x1, x2, x3, x4) = if_addY_3_in_3_aag₁(x4)
if_addC_3_in_4_aag₄(x1, x2, x3, x4) = if_addC_3_in_4_aag₁(x4)
IF_ADDY_3_IN_1_AAG₂(x1, x2) = IF_ADDY_3_IN_1_AAG₂(x1, x2)
IF_TIMES_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAA₁(x4)
IF_ADDZ_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAA₁(x4)
IF_ADDC_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG₁(x4)
IF_ADDZ_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAA₁(x4)
IF_BINARYZ_1_IN_1_G₂(x1, x2) = IF_BINARYZ_1_IN_1_G₁(x2)
ADDZ_3_IN_AAG₃(x1, x2, x3) = ADDZ_3_IN_AAG₁(x3)
IF_ADDX_3_IN_1_AAG₂(x1, x2) = IF_ADDX_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_2_AAA₂(x1, x2) = IF_ADD_3_IN_2_AAA₁(x2)
IF_ADDX_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG₁(x4)
IF_SUCC_2_IN_2_AA₃(x1, x2, x3) = IF_SUCC_2_IN_2_AA₁(x3)
IF_ADDX_3_IN_1_AAA¹₂(x1, x2) = IF_ADDX_3_IN_1_AAA¹₁(x2)
IF_ADDC_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAG₁(x3)
SUCCZ_2_IN_AA₂(x1, x2) = SUCCZ_2_IN_AA
IF_ADDX_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG¹₁(x4)
IF_ADDX_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAG₁(x3)
IF_BINARYZ_1_IN_2_G₂(x1, x2) = IF_BINARYZ_1_IN_2_G₁(x2)
IF_BINARY_1_IN_2_A₂(x1, x2) = IF_BINARY_1_IN_2_A₁(x2)
IF_TIMES_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAG₂(x3, x4)
IF_ADDY_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG¹₁(x4)
IF_TIMES_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAA₁(x4)
ADDC_3_IN_AAG₃(x1, x2, x3) = ADDC_3_IN_AAG₁(x3)
ADDX_3_IN_AAA₃(x1, x2, x3) = ADDX_3_IN_AAA
IF_ADDX_3_IN_2_AAA₂(x1, x2) = IF_ADDX_3_IN_2_AAA₁(x2)
IF_ADDC_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA¹₁(x4)
SUCC_2_IN_AA₂(x1, x2) = SUCC_2_IN_AA
BINARYZ_1_IN_G₁(x1) = BINARYZ_1_IN_G₁(x1)
IF_ADDC_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAA₁(x3)
IF_BINARY_1_IN_1_A₂(x1, x2) = IF_BINARY_1_IN_1_A₁(x2)
IF_ADDY_3_IN_1_AAG¹₂(x1, x2) = IF_ADDY_3_IN_1_AAG¹₂(x1, x2)
IF_SUCCZ_2_IN_2_AG₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AG₁(x3)
IF_ADD_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAA₁(x4)
IF_BINARY_1_IN_2_G₂(x1, x2) = IF_BINARY_1_IN_2_G₁(x2)
IF_ADDZ_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAG₁(x4)
TIMES_3_IN_AAA₃(x1, x2, x3) = TIMES_3_IN_AAA
IF_ADDY_3_IN_2_AAG₂(x1, x2) = IF_ADDY_3_IN_2_AAG₂(x1, x2)
ADDY_3_IN_AAA₃(x1, x2, x3) = ADDY_3_IN_AAA
ADDC_3_IN_AAA¹₃(x1, x2, x3) = ADDC_3_IN_AAA¹
IF_BINARYZ_1_IN_1_A₂(x1, x2) = IF_BINARYZ_1_IN_1_A₁(x2)
IF_TIMES_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAG₁(x4)
IF_ADDC_3_IN_1_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAG₁(x3)
IF_ADDX_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAA₁(x3)
IF_ADDY_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG₁(x4)
BINARYZ_1_IN_A₁(x1) = BINARYZ_1_IN_A
ADDY_3_IN_AAA¹₃(x1, x2, x3) = ADDY_3_IN_AAA¹
IF_ADD_3_IN_2_AAG₂(x1, x2) = IF_ADD_3_IN_2_AAG₂(x1, x2)
ADDX_3_IN_AAG₃(x1, x2, x3) = ADDX_3_IN_AAG₁(x3)
SUCCZ_2_IN_AG₂(x1, x2) = SUCCZ_2_IN_AG₁(x2)
IF_ADDY_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAG₁(x3)
ADDC_3_IN_AAG¹₃(x1, x2, x3) = ADDC_3_IN_AAG¹₁(x3)
IF_SUCCZ_2_IN_2_AA₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AA₁(x3)
IF_SUCCZ_2_IN_1_AA₂(x1, x2) = IF_SUCCZ_2_IN_1_AA₁(x2)
IF_ADDZ_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAG₁(x4)
ADDX_3_IN_AAG¹₃(x1, x2, x3) = ADDX_3_IN_AAG¹₁(x3)
IF_ADDC_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAA₁(x4)
IF_ADDY_3_IN_1_AAA₂(x1, x2) = IF_ADDY_3_IN_1_AAA₁(x2)
IF_SUCCZ_2_IN_1_AG₂(x1, x2) = IF_SUCCZ_2_IN_1_AG₂(x1, x2)
IF_BINARY_1_IN_1_G₂(x1, x2) = IF_BINARY_1_IN_1_G₁(x2)
ADD_3_IN_AAG₃(x1, x2, x3) = ADD_3_IN_AAG₁(x3)
ADDY_3_IN_AAG¹₃(x1, x2, x3) = ADDY_3_IN_AAG¹₁(x3)
IF_TIMES_3_IN_3_AAA₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAA₂(x1, x5)
ADDX_3_IN_AAA¹₃(x1, x2, x3) = ADDX_3_IN_AAA¹
IF_ADDY_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA¹₁(x4)
IF_ADDX_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA¹₁(x4)
IF_TIMES_3_IN_3_AAG₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAG₂(x1, x5)
IF_ADDX_3_IN_1_AAA₂(x1, x2) = IF_ADDX_3_IN_1_AAA₁(x2)
IF_ADDC_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAG₁(x4)
IF_ADDC_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAG₁(x4)
IF_ADD_3_IN_1_AAG₂(x1, x2) = IF_ADD_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_1_AAA₂(x1, x2) = IF_ADD_3_IN_1_AAA₁(x2)
IF_ADDY_3_IN_1_AAA¹₂(x1, x2) = IF_ADDY_3_IN_1_AAA¹₁(x2)
ADDZ_3_IN_AAA₃(x1, x2, x3) = ADDZ_3_IN_AAA
IF_ADD_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAG₁(x4)
IF_ADDZ_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAG₁(x4)
ADDC_3_IN_AAA₃(x1, x2, x3) = ADDC_3_IN_AAA
IF_ADDC_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAG₁(x4)
IF_ADDY_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAA₁(x3)
SUCC_2_IN_AG₂(x1, x2) = SUCC_2_IN_AG₁(x2)
IF_SUCC_2_IN_1_AA₂(x1, x2) = IF_SUCC_2_IN_1_AA₁(x2)
IF_ADDC_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAA₁(x4)
IF_BINARYZ_1_IN_2_A₂(x1, x2) = IF_BINARYZ_1_IN_2_A₁(x2)
IF_ADDY_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAG₂(x1, x2) = IF_ADDX_3_IN_2_AAG₂(x1, x2)
IF_ADDC_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG¹₁(x4)
TIMES_3_IN_AAG₃(x1, x2, x3) = TIMES_3_IN_AAG₁(x3)
IF_ADDC_3_IN_1_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAA₁(x3)
IF_ADDY_3_IN_2_AAA₂(x1, x2) = IF_ADDY_3_IN_2_AAA₁(x2)
IF_ADDZ_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAG₁(x4)
IF_ADDX_3_IN_1_AAG¹₂(x1, x2) = IF_ADDX_3_IN_1_AAG¹₂(x1, x2)
BINARY_1_IN_A₁(x1) = BINARY_1_IN_A
BINARY_1_IN_G₁(x1) = BINARY_1_IN_G₁(x1)
ADDY_3_IN_AAG₃(x1, x2, x3) = ADDY_3_IN_AAG₁(x3)
IF_SUCC_2_IN_2_AG₃(x1, x2, x3) = IF_SUCC_2_IN_2_AG₁(x3)
ADD_3_IN_AAA₃(x1, x2, x3) = ADD_3_IN_AAA
IF_ADDC_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAA₁(x4)
IF_SUCC_2_IN_1_AG₂(x1, x2) = IF_SUCC_2_IN_1_AG₂(x1, x2)
IF_ADDC_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA₁(x4)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAG₄(R, S, RS, times_3_in_aag₃(R, S, RS))
TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAA₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAA₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAA₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAA₃(X, b_0, X) -> IF_ADD_3_IN_1_AAA₂(X, binaryZ_1_in_a₁(X))
ADD_3_IN_AAA₃(X, b_0, X) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> IF_BINARY_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADD_3_IN_AAA₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADD_3_IN_AAA₃(b_0, Y, Y) -> BINARYZ_1_IN_A₁(Y)
ADD_3_IN_AAA₃(X, Y, Z) -> IF_ADD_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADD_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAA₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA₂(X, binary_1_in_a₁(X))
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAA₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAA₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA₂(Y, binary_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAA₃(X, Z, succZ_2_in_aa₂(X, Z))
ADDC_3_IN_AAA₃(X, b_0, Z) -> SUCCZ_2_IN_AA₂(X, Z)
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDC_3_IN_AAA₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAA₃(Y, Z, succZ_2_in_aa₂(Y, Z))
ADDC_3_IN_AAA₃(b_0, Y, Z) -> SUCCZ_2_IN_AA₂(Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAA₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAA₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA¹₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAA₃(X, Z, succ_2_in_aa₂(X, Z))
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAA¹₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA¹₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAA₃(Y, Z, succ_2_in_aa₂(Y, Z))
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AA₂(Y, Z)
ADDY_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAG₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAG₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAG₃(X, b_0, X) -> IF_ADD_3_IN_1_AAG₂(X, binaryZ_1_in_g₁(X))
ADD_3_IN_AAG₃(X, b_0, X) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> IF_BINARY_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADD_3_IN_AAG₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADD_3_IN_AAG₃(b_0, Y, Y) -> BINARYZ_1_IN_G₁(Y)
ADD_3_IN_AAG₃(X, Y, Z) -> IF_ADD_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADD_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAG₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG₂(X, binary_1_in_g₁(X))
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAG₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAG₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG₂(Y, binary_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAG₃(X, Z, succZ_2_in_ag₂(X, Z))
ADDC_3_IN_AAG₃(X, b_0, Z) -> SUCCZ_2_IN_AG₂(X, Z)
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDC_3_IN_AAG₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAG₃(Y, Z, succZ_2_in_ag₂(Y, Z))
ADDC_3_IN_AAG₃(b_0, Y, Z) -> SUCCZ_2_IN_AG₂(Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAG₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAG₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG¹₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAG₃(X, Z, succ_2_in_ag₂(X, Z))
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAG¹₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG¹₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAG₃(Y, Z, succ_2_in_ag₂(Y, Z))
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AG₂(Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)

R is empty.
Pi is empty.
We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {BINARY_1_IN_G₁, BINARYZ_1_IN_G₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
The graph contains the following edges 1 > 1
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
The graph contains the following edges 1 > 1
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
The graph contains the following edges 1 > 1
BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
SUCC_2_IN_AG₂(x1, x2) = SUCC_2_IN_AG₁(x2)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AG₁(zero_1₁(Z)) -> SUCC_2_IN_AG₁(Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SUCC_2_IN_AG₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

SUCC_2_IN_AG₁(zero_1₁(Z)) -> SUCC_2_IN_AG₁(Z)
The graph contains the following edges 1 > 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
ADDZ_3_IN_AAG₃(x1, x2, x3) = ADDZ_3_IN_AAG₁(x3)
ADDC_3_IN_AAG₃(x1, x2, x3) = ADDC_3_IN_AAG₁(x3)
ADDX_3_IN_AAG₃(x1, x2, x3) = ADDX_3_IN_AAG₁(x3)
ADDC_3_IN_AAG¹₃(x1, x2, x3) = ADDC_3_IN_AAG¹₁(x3)
ADDX_3_IN_AAG¹₃(x1, x2, x3) = ADDX_3_IN_AAG¹₁(x3)
ADDY_3_IN_AAG¹₃(x1, x2, x3) = ADDY_3_IN_AAG¹₁(x3)
ADDY_3_IN_AAG₃(x1, x2, x3) = ADDY_3_IN_AAG₁(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

ADDZ_3_IN_AAG₁(one_1₁(Z)) -> ADDX_3_IN_AAG₁(Z)
ADDZ_3_IN_AAG₁(zero_1₁(Z)) -> ADDZ_3_IN_AAG₁(Z)
ADDY_3_IN_AAG₁(Z) -> ADDZ_3_IN_AAG₁(Z)
ADDC_3_IN_AAG¹₁(one_1₁(Z)) -> ADDZ_3_IN_AAG₁(Z)
ADDC_3_IN_AAG¹₁(one_1₁(Z)) -> ADDC_3_IN_AAG₁(Z)
ADDZ_3_IN_AAG₁(one_1₁(Z)) -> ADDY_3_IN_AAG₁(Z)
ADDY_3_IN_AAG¹₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
ADDC_3_IN_AAG₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
ADDX_3_IN_AAG¹₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
ADDC_3_IN_AAG¹₁(zero_1₁(Z)) -> ADDY_3_IN_AAG¹₁(Z)
ADDZ_3_IN_AAG₁(zero_1₁(Z)) -> ADDC_3_IN_AAG₁(Z)
ADDC_3_IN_AAG¹₁(zero_1₁(Z)) -> ADDX_3_IN_AAG¹₁(Z)
ADDX_3_IN_AAG₁(Z) -> ADDZ_3_IN_AAG₁(Z)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ADDX_3_IN_AAG₁, ADDZ_3_IN_AAG₁, ADDY_3_IN_AAG₁, ADDC_3_IN_AAG¹₁, ADDC_3_IN_AAG₁, ADDY_3_IN_AAG¹₁, ADDX_3_IN_AAG¹₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

ADDX_3_IN_AAG₁(Z) -> ADDZ_3_IN_AAG₁(Z)
The graph contains the following edges 1 >= 1
ADDZ_3_IN_AAG₁(zero_1₁(Z)) -> ADDZ_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDZ_3_IN_AAG₁(one_1₁(Z)) -> ADDY_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDC_3_IN_AAG₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
The graph contains the following edges 1 >= 1
ADDY_3_IN_AAG₁(Z) -> ADDZ_3_IN_AAG₁(Z)
The graph contains the following edges 1 >= 1
ADDC_3_IN_AAG¹₁(one_1₁(Z)) -> ADDZ_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDC_3_IN_AAG¹₁(zero_1₁(Z)) -> ADDY_3_IN_AAG¹₁(Z)
The graph contains the following edges 1 > 1
ADDZ_3_IN_AAG₁(zero_1₁(Z)) -> ADDC_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDZ_3_IN_AAG₁(one_1₁(Z)) -> ADDX_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDC_3_IN_AAG¹₁(one_1₁(Z)) -> ADDC_3_IN_AAG₁(Z)
The graph contains the following edges 1 > 1
ADDC_3_IN_AAG¹₁(zero_1₁(Z)) -> ADDX_3_IN_AAG¹₁(Z)
The graph contains the following edges 1 > 1
ADDY_3_IN_AAG¹₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
The graph contains the following edges 1 >= 1
ADDX_3_IN_AAG¹₁(Z) -> ADDC_3_IN_AAG¹₁(Z)
The graph contains the following edges 1 >= 1

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
BINARYZ_1_IN_A₁(x1) = BINARYZ_1_IN_A
BINARY_1_IN_A₁(x1) = BINARY_1_IN_A

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

BINARYZ_1_IN_A -> BINARYZ_1_IN_A
BINARYZ_1_IN_A -> BINARY_1_IN_A
BINARY_1_IN_A -> BINARY_1_IN_A
BINARY_1_IN_A -> BINARYZ_1_IN_A

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {BINARYZ_1_IN_A, BINARY_1_IN_A}.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
SUCC_2_IN_AA₂(x1, x2) = SUCC_2_IN_AA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

SUCC_2_IN_AA -> SUCC_2_IN_AA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {SUCC_2_IN_AA}.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
ADDX_3_IN_AAA₃(x1, x2, x3) = ADDX_3_IN_AAA
ADDY_3_IN_AAA₃(x1, x2, x3) = ADDY_3_IN_AAA
ADDC_3_IN_AAA¹₃(x1, x2, x3) = ADDC_3_IN_AAA¹
ADDY_3_IN_AAA¹₃(x1, x2, x3) = ADDY_3_IN_AAA¹
ADDX_3_IN_AAA¹₃(x1, x2, x3) = ADDX_3_IN_AAA¹
ADDZ_3_IN_AAA₃(x1, x2, x3) = ADDZ_3_IN_AAA
ADDC_3_IN_AAA₃(x1, x2, x3) = ADDC_3_IN_AAA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

ADDY_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDY_3_IN_AAA
ADDX_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDX_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDY_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDX_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDC_3_IN_AAA
ADDY_3_IN_AAA -> ADDZ_3_IN_AAA
ADDZ_3_IN_AAA -> ADDX_3_IN_AAA
ADDZ_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDC_3_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {ADDC_3_IN_AAA¹, ADDY_3_IN_AAA¹, ADDY_3_IN_AAA, ADDZ_3_IN_AAA, ADDX_3_IN_AAA¹, ADDX_3_IN_AAA, ADDC_3_IN_AAA}.
By using a polynomial ordering, at least one Dependency Pair or term rewrite system rule of this QDP problem can be strictly oriented.

Used ordering: POLO with Polynomial interpretation:

POL(ADDY_3_IN_AAA) = 0
POL(ADDY_3_IN_AAA¹) = 0
POL(ADDZ_3_IN_AAA) = 0
POL(ADDC_3_IN_AAA¹) = 0
POL(ADDX_3_IN_AAA) = 0
POL(ADDC_3_IN_AAA) = 0
POL(ADDX_3_IN_AAA¹) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

ADDY_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDY_3_IN_AAA
ADDX_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDX_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDY_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDX_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDC_3_IN_AAA
ADDY_3_IN_AAA -> ADDZ_3_IN_AAA
ADDZ_3_IN_AAA -> ADDX_3_IN_AAA
ADDZ_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDC_3_IN_AAA

POL(ADDY_3_IN_AAA) = 0
POL(ADDY_3_IN_AAA¹) = 0
POL(ADDZ_3_IN_AAA) = 0
POL(ADDC_3_IN_AAA¹) = 0
POL(ADDX_3_IN_AAA) = 0
POL(ADDC_3_IN_AAA) = 0
POL(ADDX_3_IN_AAA¹) = 0

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ RuleRemovalProof

                          ↳ QDP

                            ↳ RuleRemovalProof

                              ↳ QDP

              ↳ PiDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

ADDY_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDY_3_IN_AAA
ADDX_3_IN_AAA¹ -> ADDC_3_IN_AAA¹
ADDX_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA¹ -> ADDY_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDX_3_IN_AAA¹
ADDC_3_IN_AAA¹ -> ADDC_3_IN_AAA
ADDY_3_IN_AAA -> ADDZ_3_IN_AAA
ADDZ_3_IN_AAA -> ADDX_3_IN_AAA
ADDZ_3_IN_AAA -> ADDZ_3_IN_AAA
ADDC_3_IN_AAA -> ADDC_3_IN_AAA¹
ADDZ_3_IN_AAA -> ADDC_3_IN_AAA

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

              ↳ PiDP

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)

R is empty.
The argument filtering Pi contains the following mapping:
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
TIMES_3_IN_AAA₃(x1, x2, x3) = TIMES_3_IN_AAA

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

              ↳ PiDP

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAA -> TIMES_3_IN_AAA

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {TIMES_3_IN_AAA}.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

  ↳ PrologToPiTRSProof

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)

R is empty.
The argument filtering Pi contains the following mapping:
zero_1₁(x1) = zero_1₁(x1)
TIMES_3_IN_AAG₃(x1, x2, x3) = TIMES_3_IN_AAG₁(x3)

We have to consider all (P,R,Pi)-chains
Transforming (infinitary) constructor rewriting Pi-DP problem into ordinary QDP problem by application of Pi.

↳ PROLOG

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

          ↳ DependencyGraphProof

            ↳ AND

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

              ↳ PiDP

                ↳ UsableRulesProof

                  ↳ PiDP

                    ↳ PiDPToQDPProof

                      ↳ QDP

                        ↳ QDPSizeChangeProof

  ↳ PrologToPiTRSProof

Q DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAG₁(zero_1₁(RS)) -> TIMES_3_IN_AAG₁(RS)

R is empty.
Q is empty.
We have to consider all (P,Q,R)-chains.
The head symbols of this DP problem are {TIMES_3_IN_AAG₁}.
By using the subterm criterion together with the size-change analysis we have proven that there are no infinite chains for this DP problem.

From the DPs we obtained the following set of size-change graphs:

TIMES_3_IN_AAG₁(zero_1₁(RS)) -> TIMES_3_IN_AAG₁(RS)
The graph contains the following edges 1 > 1

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

Infinitary Constructor Rewriting Termination of PiTRS implies Termination of PROLOG

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

Pi-finite rewrite system:
The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAG₄(R, S, RS, times_3_in_aag₃(R, S, RS))
TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAA₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAA₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAA₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAA₃(X, b_0, X) -> IF_ADD_3_IN_1_AAA₂(X, binaryZ_1_in_a₁(X))
ADD_3_IN_AAA₃(X, b_0, X) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> IF_BINARY_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADD_3_IN_AAA₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADD_3_IN_AAA₃(b_0, Y, Y) -> BINARYZ_1_IN_A₁(Y)
ADD_3_IN_AAA₃(X, Y, Z) -> IF_ADD_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADD_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAA₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA₂(X, binary_1_in_a₁(X))
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAA₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAA₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA₂(Y, binary_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAA₃(X, Z, succZ_2_in_aa₂(X, Z))
ADDC_3_IN_AAA₃(X, b_0, Z) -> SUCCZ_2_IN_AA₂(X, Z)
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDC_3_IN_AAA₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAA₃(Y, Z, succZ_2_in_aa₂(Y, Z))
ADDC_3_IN_AAA₃(b_0, Y, Z) -> SUCCZ_2_IN_AA₂(Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAA₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAA₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA¹₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAA₃(X, Z, succ_2_in_aa₂(X, Z))
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAA¹₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA¹₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAA₃(Y, Z, succ_2_in_aa₂(Y, Z))
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AA₂(Y, Z)
ADDY_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAG₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAG₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAG₃(X, b_0, X) -> IF_ADD_3_IN_1_AAG₂(X, binaryZ_1_in_g₁(X))
ADD_3_IN_AAG₃(X, b_0, X) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> IF_BINARY_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADD_3_IN_AAG₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADD_3_IN_AAG₃(b_0, Y, Y) -> BINARYZ_1_IN_G₁(Y)
ADD_3_IN_AAG₃(X, Y, Z) -> IF_ADD_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADD_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAG₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG₂(X, binary_1_in_g₁(X))
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAG₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAG₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG₂(Y, binary_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAG₃(X, Z, succZ_2_in_ag₂(X, Z))
ADDC_3_IN_AAG₃(X, b_0, Z) -> SUCCZ_2_IN_AG₂(X, Z)
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDC_3_IN_AAG₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAG₃(Y, Z, succZ_2_in_ag₂(Y, Z))
ADDC_3_IN_AAG₃(b_0, Y, Z) -> SUCCZ_2_IN_AG₂(Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAG₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAG₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG¹₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAG₃(X, Z, succ_2_in_ag₂(X, Z))
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAG¹₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG¹₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAG₃(Y, Z, succ_2_in_ag₂(Y, Z))
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AG₂(Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

The argument filtering Pi contains the following mapping:
times_3_in_aag₃(x1, x2, x3) = times_3_in_aag₁(x3)
b_0 = b_0
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
times_3_out_aag₃(x1, x2, x3) = times_3_out_aag₃(x1, x2, x3)
if_times_3_in_1_aag₄(x1, x2, x3, x4) = if_times_3_in_1_aag₂(x3, x4)
if_times_3_in_2_aag₄(x1, x2, x3, x4) = if_times_3_in_2_aag₂(x3, x4)
times_3_in_aaa₃(x1, x2, x3) = times_3_in_aaa
times_3_out_aaa₃(x1, x2, x3) = times_3_out_aaa₁(x1)
if_times_3_in_1_aaa₄(x1, x2, x3, x4) = if_times_3_in_1_aaa₁(x4)
if_times_3_in_2_aaa₄(x1, x2, x3, x4) = if_times_3_in_2_aaa₁(x4)
if_times_3_in_3_aaa₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aaa₂(x1, x5)
add_3_in_aaa₃(x1, x2, x3) = add_3_in_aaa
add_3_out_aaa₃(x1, x2, x3) = add_3_out_aaa₃(x1, x2, x3)
if_add_3_in_1_aaa₂(x1, x2) = if_add_3_in_1_aaa₁(x2)
binaryZ_1_in_a₁(x1) = binaryZ_1_in_a
if_binaryZ_1_in_1_a₂(x1, x2) = if_binaryZ_1_in_1_a₁(x2)
if_binaryZ_1_in_2_a₂(x1, x2) = if_binaryZ_1_in_2_a₁(x2)
binary_1_in_a₁(x1) = binary_1_in_a
binary_1_out_a₁(x1) = binary_1_out_a₁(x1)
if_binary_1_in_1_a₂(x1, x2) = if_binary_1_in_1_a₁(x2)
binaryZ_1_out_a₁(x1) = binaryZ_1_out_a₁(x1)
if_binary_1_in_2_a₂(x1, x2) = if_binary_1_in_2_a₁(x2)
if_add_3_in_2_aaa₂(x1, x2) = if_add_3_in_2_aaa₁(x2)
if_add_3_in_3_aaa₄(x1, x2, x3, x4) = if_add_3_in_3_aaa₁(x4)
addz_3_in_aaa₃(x1, x2, x3) = addz_3_in_aaa
if_addz_3_in_1_aaa₄(x1, x2, x3, x4) = if_addz_3_in_1_aaa₁(x4)
if_addz_3_in_2_aaa₄(x1, x2, x3, x4) = if_addz_3_in_2_aaa₁(x4)
addx_3_in_aaa₃(x1, x2, x3) = addx_3_in_aaa
if_addx_3_in_1_aaa₂(x1, x2) = if_addx_3_in_1_aaa₁(x2)
addx_3_out_aaa₃(x1, x2, x3) = addx_3_out_aaa₃(x1, x2, x3)
if_addx_3_in_2_aaa₂(x1, x2) = if_addx_3_in_2_aaa₁(x2)
if_addx_3_in_3_aaa₄(x1, x2, x3, x4) = if_addx_3_in_3_aaa₁(x4)
if_addz_3_in_3_aaa₄(x1, x2, x3, x4) = if_addz_3_in_3_aaa₁(x4)
addy_3_in_aaa₃(x1, x2, x3) = addy_3_in_aaa
if_addy_3_in_1_aaa₂(x1, x2) = if_addy_3_in_1_aaa₁(x2)
addy_3_out_aaa₃(x1, x2, x3) = addy_3_out_aaa₃(x1, x2, x3)
if_addy_3_in_2_aaa₂(x1, x2) = if_addy_3_in_2_aaa₁(x2)
if_addy_3_in_3_aaa₄(x1, x2, x3, x4) = if_addy_3_in_3_aaa₁(x4)
if_addz_3_in_4_aaa₄(x1, x2, x3, x4) = if_addz_3_in_4_aaa₁(x4)
addc_3_in_aaa₃(x1, x2, x3) = addc_3_in_aaa
addc_3_out_aaa₃(x1, x2, x3) = addc_3_out_aaa₃(x1, x2, x3)
if_addc_3_in_1_aaa₃(x1, x2, x3) = if_addc_3_in_1_aaa₁(x3)
succZ_2_in_aa₂(x1, x2) = succZ_2_in_aa
if_succZ_2_in_1_aa₂(x1, x2) = if_succZ_2_in_1_aa₁(x2)
succZ_2_out_aa₂(x1, x2) = succZ_2_out_aa₂(x1, x2)
if_succZ_2_in_2_aa₃(x1, x2, x3) = if_succZ_2_in_2_aa₁(x3)
succ_2_in_aa₂(x1, x2) = succ_2_in_aa
succ_2_out_aa₂(x1, x2) = succ_2_out_aa₂(x1, x2)
if_succ_2_in_1_aa₂(x1, x2) = if_succ_2_in_1_aa₁(x2)
if_succ_2_in_2_aa₃(x1, x2, x3) = if_succ_2_in_2_aa₁(x3)
if_addc_3_in_2_aaa₃(x1, x2, x3) = if_addc_3_in_2_aaa₁(x3)
if_addc_3_in_3_aaa₄(x1, x2, x3, x4) = if_addc_3_in_3_aaa₁(x4)
addC_3_in_aaa₃(x1, x2, x3) = addC_3_in_aaa
if_addC_3_in_1_aaa₄(x1, x2, x3, x4) = if_addC_3_in_1_aaa₁(x4)
addz_3_out_aaa₃(x1, x2, x3) = addz_3_out_aaa₃(x1, x2, x3)
addC_3_out_aaa₃(x1, x2, x3) = addC_3_out_aaa₃(x1, x2, x3)
if_addC_3_in_2_aaa₄(x1, x2, x3, x4) = if_addC_3_in_2_aaa₁(x4)
addX_3_in_aaa₃(x1, x2, x3) = addX_3_in_aaa
if_addX_3_in_1_aaa₂(x1, x2) = if_addX_3_in_1_aaa₁(x2)
addX_3_out_aaa₃(x1, x2, x3) = addX_3_out_aaa₃(x1, x2, x3)
if_addX_3_in_2_aaa₃(x1, x2, x3) = if_addX_3_in_2_aaa₁(x3)
if_addX_3_in_3_aaa₄(x1, x2, x3, x4) = if_addX_3_in_3_aaa₁(x4)
if_addC_3_in_3_aaa₄(x1, x2, x3, x4) = if_addC_3_in_3_aaa₁(x4)
addY_3_in_aaa₃(x1, x2, x3) = addY_3_in_aaa
if_addY_3_in_1_aaa₂(x1, x2) = if_addY_3_in_1_aaa₁(x2)
addY_3_out_aaa₃(x1, x2, x3) = addY_3_out_aaa₃(x1, x2, x3)
if_addY_3_in_2_aaa₃(x1, x2, x3) = if_addY_3_in_2_aaa₁(x3)
if_addY_3_in_3_aaa₄(x1, x2, x3, x4) = if_addY_3_in_3_aaa₁(x4)
if_addC_3_in_4_aaa₄(x1, x2, x3, x4) = if_addC_3_in_4_aaa₁(x4)
if_times_3_in_3_aag₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aag₃(x1, x3, x5)
add_3_in_aag₃(x1, x2, x3) = add_3_in_aag₁(x3)
add_3_out_aag₃(x1, x2, x3) = add_3_out_aag₃(x1, x2, x3)
if_add_3_in_1_aag₂(x1, x2) = if_add_3_in_1_aag₂(x1, x2)
binaryZ_1_in_g₁(x1) = binaryZ_1_in_g₁(x1)
if_binaryZ_1_in_1_g₂(x1, x2) = if_binaryZ_1_in_1_g₂(x1, x2)
if_binaryZ_1_in_2_g₂(x1, x2) = if_binaryZ_1_in_2_g₂(x1, x2)
binary_1_in_g₁(x1) = binary_1_in_g₁(x1)
binary_1_out_g₁(x1) = binary_1_out_g₁(x1)
if_binary_1_in_1_g₂(x1, x2) = if_binary_1_in_1_g₂(x1, x2)
binaryZ_1_out_g₁(x1) = binaryZ_1_out_g₁(x1)
if_binary_1_in_2_g₂(x1, x2) = if_binary_1_in_2_g₂(x1, x2)
if_add_3_in_2_aag₂(x1, x2) = if_add_3_in_2_aag₂(x1, x2)
if_add_3_in_3_aag₄(x1, x2, x3, x4) = if_add_3_in_3_aag₂(x3, x4)
addz_3_in_aag₃(x1, x2, x3) = addz_3_in_aag₁(x3)
if_addz_3_in_1_aag₄(x1, x2, x3, x4) = if_addz_3_in_1_aag₂(x3, x4)
if_addz_3_in_2_aag₄(x1, x2, x3, x4) = if_addz_3_in_2_aag₂(x3, x4)
addx_3_in_aag₃(x1, x2, x3) = addx_3_in_aag₁(x3)
if_addx_3_in_1_aag₂(x1, x2) = if_addx_3_in_1_aag₂(x1, x2)
addx_3_out_aag₃(x1, x2, x3) = addx_3_out_aag₃(x1, x2, x3)
if_addx_3_in_2_aag₂(x1, x2) = if_addx_3_in_2_aag₂(x1, x2)
if_addx_3_in_3_aag₄(x1, x2, x3, x4) = if_addx_3_in_3_aag₂(x3, x4)
if_addz_3_in_3_aag₄(x1, x2, x3, x4) = if_addz_3_in_3_aag₂(x3, x4)
addy_3_in_aag₃(x1, x2, x3) = addy_3_in_aag₁(x3)
if_addy_3_in_1_aag₂(x1, x2) = if_addy_3_in_1_aag₂(x1, x2)
addy_3_out_aag₃(x1, x2, x3) = addy_3_out_aag₃(x1, x2, x3)
if_addy_3_in_2_aag₂(x1, x2) = if_addy_3_in_2_aag₂(x1, x2)
if_addy_3_in_3_aag₄(x1, x2, x3, x4) = if_addy_3_in_3_aag₂(x3, x4)
if_addz_3_in_4_aag₄(x1, x2, x3, x4) = if_addz_3_in_4_aag₂(x3, x4)
addc_3_in_aag₃(x1, x2, x3) = addc_3_in_aag₁(x3)
addc_3_out_aag₃(x1, x2, x3) = addc_3_out_aag₃(x1, x2, x3)
if_addc_3_in_1_aag₃(x1, x2, x3) = if_addc_3_in_1_aag₂(x2, x3)
succZ_2_in_ag₂(x1, x2) = succZ_2_in_ag₁(x2)
if_succZ_2_in_1_ag₂(x1, x2) = if_succZ_2_in_1_ag₂(x1, x2)
succZ_2_out_ag₂(x1, x2) = succZ_2_out_ag₂(x1, x2)
if_succZ_2_in_2_ag₃(x1, x2, x3) = if_succZ_2_in_2_ag₂(x2, x3)
succ_2_in_ag₂(x1, x2) = succ_2_in_ag₁(x2)
succ_2_out_ag₂(x1, x2) = succ_2_out_ag₂(x1, x2)
if_succ_2_in_1_ag₂(x1, x2) = if_succ_2_in_1_ag₂(x1, x2)
if_succ_2_in_2_ag₃(x1, x2, x3) = if_succ_2_in_2_ag₂(x2, x3)
if_addc_3_in_2_aag₃(x1, x2, x3) = if_addc_3_in_2_aag₂(x2, x3)
if_addc_3_in_3_aag₄(x1, x2, x3, x4) = if_addc_3_in_3_aag₂(x3, x4)
addC_3_in_aag₃(x1, x2, x3) = addC_3_in_aag₁(x3)
if_addC_3_in_1_aag₄(x1, x2, x3, x4) = if_addC_3_in_1_aag₂(x3, x4)
addz_3_out_aag₃(x1, x2, x3) = addz_3_out_aag₃(x1, x2, x3)
addC_3_out_aag₃(x1, x2, x3) = addC_3_out_aag₃(x1, x2, x3)
if_addC_3_in_2_aag₄(x1, x2, x3, x4) = if_addC_3_in_2_aag₂(x3, x4)
addX_3_in_aag₃(x1, x2, x3) = addX_3_in_aag₁(x3)
if_addX_3_in_1_aag₂(x1, x2) = if_addX_3_in_1_aag₂(x1, x2)
addX_3_out_aag₃(x1, x2, x3) = addX_3_out_aag₃(x1, x2, x3)
if_addX_3_in_2_aag₃(x1, x2, x3) = if_addX_3_in_2_aag₂(x2, x3)
if_addX_3_in_3_aag₄(x1, x2, x3, x4) = if_addX_3_in_3_aag₂(x3, x4)
if_addC_3_in_3_aag₄(x1, x2, x3, x4) = if_addC_3_in_3_aag₂(x3, x4)
addY_3_in_aag₃(x1, x2, x3) = addY_3_in_aag₁(x3)
if_addY_3_in_1_aag₂(x1, x2) = if_addY_3_in_1_aag₂(x1, x2)
addY_3_out_aag₃(x1, x2, x3) = addY_3_out_aag₃(x1, x2, x3)
if_addY_3_in_2_aag₃(x1, x2, x3) = if_addY_3_in_2_aag₂(x2, x3)
if_addY_3_in_3_aag₄(x1, x2, x3, x4) = if_addY_3_in_3_aag₂(x3, x4)
if_addC_3_in_4_aag₄(x1, x2, x3, x4) = if_addC_3_in_4_aag₂(x3, x4)
IF_ADDY_3_IN_1_AAG₂(x1, x2) = IF_ADDY_3_IN_1_AAG₂(x1, x2)
IF_TIMES_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAA₁(x4)
IF_ADDZ_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAA₁(x4)
IF_ADDC_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG₂(x3, x4)
IF_ADDZ_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAA₁(x4)
IF_BINARYZ_1_IN_1_G₂(x1, x2) = IF_BINARYZ_1_IN_1_G₂(x1, x2)
ADDZ_3_IN_AAG₃(x1, x2, x3) = ADDZ_3_IN_AAG₁(x3)
IF_ADDX_3_IN_1_AAG₂(x1, x2) = IF_ADDX_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_2_AAA₂(x1, x2) = IF_ADD_3_IN_2_AAA₁(x2)
IF_ADDX_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG₂(x3, x4)
IF_SUCC_2_IN_2_AA₃(x1, x2, x3) = IF_SUCC_2_IN_2_AA₁(x3)
IF_ADDX_3_IN_1_AAA¹₂(x1, x2) = IF_ADDX_3_IN_1_AAA¹₁(x2)
IF_ADDC_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAG₂(x2, x3)
SUCCZ_2_IN_AA₂(x1, x2) = SUCCZ_2_IN_AA
IF_ADDX_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG¹₂(x3, x4)
IF_ADDX_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAG₂(x2, x3)
IF_BINARYZ_1_IN_2_G₂(x1, x2) = IF_BINARYZ_1_IN_2_G₂(x1, x2)
IF_BINARY_1_IN_2_A₂(x1, x2) = IF_BINARY_1_IN_2_A₁(x2)
IF_TIMES_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAG₂(x3, x4)
IF_ADDY_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG¹₂(x3, x4)
IF_TIMES_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAA₁(x4)
ADDC_3_IN_AAG₃(x1, x2, x3) = ADDC_3_IN_AAG₁(x3)
ADDX_3_IN_AAA₃(x1, x2, x3) = ADDX_3_IN_AAA
IF_ADDX_3_IN_2_AAA₂(x1, x2) = IF_ADDX_3_IN_2_AAA₁(x2)
IF_ADDC_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA¹₁(x4)
SUCC_2_IN_AA₂(x1, x2) = SUCC_2_IN_AA
BINARYZ_1_IN_G₁(x1) = BINARYZ_1_IN_G₁(x1)
IF_ADDC_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAA₁(x3)
IF_BINARY_1_IN_1_A₂(x1, x2) = IF_BINARY_1_IN_1_A₁(x2)
IF_ADDY_3_IN_1_AAG¹₂(x1, x2) = IF_ADDY_3_IN_1_AAG¹₂(x1, x2)
IF_SUCCZ_2_IN_2_AG₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AG₂(x2, x3)
IF_ADD_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAA₁(x4)
IF_BINARY_1_IN_2_G₂(x1, x2) = IF_BINARY_1_IN_2_G₂(x1, x2)
IF_ADDZ_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAG₂(x3, x4)
TIMES_3_IN_AAA₃(x1, x2, x3) = TIMES_3_IN_AAA
IF_ADDY_3_IN_2_AAG₂(x1, x2) = IF_ADDY_3_IN_2_AAG₂(x1, x2)
ADDY_3_IN_AAA₃(x1, x2, x3) = ADDY_3_IN_AAA
ADDC_3_IN_AAA¹₃(x1, x2, x3) = ADDC_3_IN_AAA¹
IF_BINARYZ_1_IN_1_A₂(x1, x2) = IF_BINARYZ_1_IN_1_A₁(x2)
IF_TIMES_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAG₂(x3, x4)
IF_ADDC_3_IN_1_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAG₂(x2, x3)
IF_ADDX_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAA₁(x3)
IF_ADDY_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG₂(x3, x4)
BINARYZ_1_IN_A₁(x1) = BINARYZ_1_IN_A
ADDY_3_IN_AAA¹₃(x1, x2, x3) = ADDY_3_IN_AAA¹
IF_ADD_3_IN_2_AAG₂(x1, x2) = IF_ADD_3_IN_2_AAG₂(x1, x2)
ADDX_3_IN_AAG₃(x1, x2, x3) = ADDX_3_IN_AAG₁(x3)
SUCCZ_2_IN_AG₂(x1, x2) = SUCCZ_2_IN_AG₁(x2)
IF_ADDY_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAG₂(x2, x3)
ADDC_3_IN_AAG¹₃(x1, x2, x3) = ADDC_3_IN_AAG¹₁(x3)
IF_SUCCZ_2_IN_2_AA₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AA₁(x3)
IF_SUCCZ_2_IN_1_AA₂(x1, x2) = IF_SUCCZ_2_IN_1_AA₁(x2)
IF_ADDZ_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAG₂(x3, x4)
ADDX_3_IN_AAG¹₃(x1, x2, x3) = ADDX_3_IN_AAG¹₁(x3)
IF_ADDC_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAA₁(x4)
IF_ADDY_3_IN_1_AAA₂(x1, x2) = IF_ADDY_3_IN_1_AAA₁(x2)
IF_SUCCZ_2_IN_1_AG₂(x1, x2) = IF_SUCCZ_2_IN_1_AG₂(x1, x2)
IF_BINARY_1_IN_1_G₂(x1, x2) = IF_BINARY_1_IN_1_G₂(x1, x2)
ADD_3_IN_AAG₃(x1, x2, x3) = ADD_3_IN_AAG₁(x3)
ADDY_3_IN_AAG¹₃(x1, x2, x3) = ADDY_3_IN_AAG¹₁(x3)
IF_TIMES_3_IN_3_AAA₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAA₂(x1, x5)
ADDX_3_IN_AAA¹₃(x1, x2, x3) = ADDX_3_IN_AAA¹
IF_ADDY_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA¹₁(x4)
IF_ADDX_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA¹₁(x4)
IF_TIMES_3_IN_3_AAG₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAG₃(x1, x3, x5)
IF_ADDX_3_IN_1_AAA₂(x1, x2) = IF_ADDX_3_IN_1_AAA₁(x2)
IF_ADDC_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAG₂(x3, x4)
IF_ADDC_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAG₂(x3, x4)
IF_ADD_3_IN_1_AAG₂(x1, x2) = IF_ADD_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_1_AAA₂(x1, x2) = IF_ADD_3_IN_1_AAA₁(x2)
IF_ADDY_3_IN_1_AAA¹₂(x1, x2) = IF_ADDY_3_IN_1_AAA¹₁(x2)
ADDZ_3_IN_AAA₃(x1, x2, x3) = ADDZ_3_IN_AAA
IF_ADD_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAG₂(x3, x4)
IF_ADDZ_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAG₂(x3, x4)
ADDC_3_IN_AAA₃(x1, x2, x3) = ADDC_3_IN_AAA
IF_ADDC_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAG₂(x3, x4)
IF_ADDY_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAA₁(x3)
SUCC_2_IN_AG₂(x1, x2) = SUCC_2_IN_AG₁(x2)
IF_SUCC_2_IN_1_AA₂(x1, x2) = IF_SUCC_2_IN_1_AA₁(x2)
IF_ADDC_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAA₁(x4)
IF_BINARYZ_1_IN_2_A₂(x1, x2) = IF_BINARYZ_1_IN_2_A₁(x2)
IF_ADDY_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAG₂(x1, x2) = IF_ADDX_3_IN_2_AAG₂(x1, x2)
IF_ADDC_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG¹₂(x3, x4)
TIMES_3_IN_AAG₃(x1, x2, x3) = TIMES_3_IN_AAG₁(x3)
IF_ADDC_3_IN_1_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAA₁(x3)
IF_ADDY_3_IN_2_AAA₂(x1, x2) = IF_ADDY_3_IN_2_AAA₁(x2)
IF_ADDZ_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAG₂(x3, x4)
IF_ADDX_3_IN_1_AAG¹₂(x1, x2) = IF_ADDX_3_IN_1_AAG¹₂(x1, x2)
BINARY_1_IN_A₁(x1) = BINARY_1_IN_A
BINARY_1_IN_G₁(x1) = BINARY_1_IN_G₁(x1)
ADDY_3_IN_AAG₃(x1, x2, x3) = ADDY_3_IN_AAG₁(x3)
IF_SUCC_2_IN_2_AG₃(x1, x2, x3) = IF_SUCC_2_IN_2_AG₂(x2, x3)
ADD_3_IN_AAA₃(x1, x2, x3) = ADD_3_IN_AAA
IF_ADDC_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAA₁(x4)
IF_SUCC_2_IN_1_AG₂(x1, x2) = IF_SUCC_2_IN_1_AG₂(x1, x2)
IF_ADDC_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA₁(x4)

We have to consider all (P,R,Pi)-chains

↳ PROLOG

  ↳ PrologToPiTRSProof

  ↳ PrologToPiTRSProof

    ↳ PiTRS

      ↳ DependencyPairsProof

        ↳ PiDP

Pi DP problem:
The TRS P consists of the following rules:

TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAG₄(R, S, RS, times_3_in_aag₃(R, S, RS))
TIMES_3_IN_AAG₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAG₃(R, S, RS)
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAG₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> IF_TIMES_3_IN_1_AAA₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(zero_1₁(R), S, zero_1₁(RS)) -> TIMES_3_IN_AAA₃(R, S, RS)
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
TIMES_3_IN_AAA₃(one_1₁(R), S, RSS) -> TIMES_3_IN_AAA₃(R, S, RS)
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAA₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAA₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAA₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAA₃(X, b_0, X) -> IF_ADD_3_IN_1_AAA₂(X, binaryZ_1_in_a₁(X))
ADD_3_IN_AAA₃(X, b_0, X) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARYZ_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARYZ_1_IN_A₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARYZ_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
BINARY_1_IN_A₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_A₂(X, binaryZ_1_in_a₁(X))
BINARY_1_IN_A₁(zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
BINARY_1_IN_A₁(one_1₁(X)) -> IF_BINARY_1_IN_2_A₂(X, binary_1_in_a₁(X))
BINARY_1_IN_A₁(one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADD_3_IN_AAA₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADD_3_IN_AAA₃(b_0, Y, Y) -> BINARYZ_1_IN_A₁(Y)
ADD_3_IN_AAA₃(X, Y, Z) -> IF_ADD_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADD_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAA₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAA₃(X, Y, Z)
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA₂(X, binary_1_in_a₁(X))
ADDX_3_IN_AAA₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_A₁(X)
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAA₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAA₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAA₃(X, Y, Z)
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA₂(Y, binary_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAA₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA₃(X, Y, Z) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDZ_3_IN_AAA₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAA₃(X, Z, succZ_2_in_aa₂(X, Z))
ADDC_3_IN_AAA₃(X, b_0, Z) -> SUCCZ_2_IN_AA₂(X, Z)
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCCZ_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCCZ_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AA₂(X, binaryZ_1_in_a₁(X))
SUCC_2_IN_AA₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AA₃(X, Z, succ_2_in_aa₂(X, Z))
SUCC_2_IN_AA₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDC_3_IN_AAA₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAA₃(Y, Z, succZ_2_in_aa₂(Y, Z))
ADDC_3_IN_AAA₃(b_0, Y, Z) -> SUCCZ_2_IN_AA₂(Y, Z)
ADDC_3_IN_AAA₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAA₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAA₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAA₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAA₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAA¹₃(X, Y, Z)
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAA¹₂(X, binaryZ_1_in_a₁(X))
ADDX_3_IN_AAA¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_A₁(X)
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAA₃(X, Z, succ_2_in_aa₂(X, Z))
ADDX_3_IN_AAA¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AA₂(X, Z)
ADDX_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDX_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAA¹₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAA¹₃(X, Y, Z)
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAA¹₂(Y, binaryZ_1_in_a₁(Y))
ADDY_3_IN_AAA¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_A₁(Y)
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAA₃(Y, Z, succ_2_in_aa₂(Y, Z))
ADDY_3_IN_AAA¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AA₂(Y, Z)
ADDY_3_IN_AAA¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAA¹₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
ADDY_3_IN_AAA¹₃(X, Y, Z) -> ADDC_3_IN_AAA¹₃(X, Y, Z)
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAA₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
ADDC_3_IN_AAA¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAA₃(X, Y, Z)
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> IF_TIMES_3_IN_3_AAG₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
IF_TIMES_3_IN_2_AAG₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> ADD_3_IN_AAG₃(S, zero_1₁(RS), RSS)
ADD_3_IN_AAG₃(X, b_0, X) -> IF_ADD_3_IN_1_AAG₂(X, binaryZ_1_in_g₁(X))
ADD_3_IN_AAG₃(X, b_0, X) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(zero_1₁(X)) -> IF_BINARYZ_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARYZ_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARYZ_1_IN_G₁(one_1₁(X)) -> IF_BINARYZ_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARYZ_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
BINARY_1_IN_G₁(zero_1₁(X)) -> IF_BINARY_1_IN_1_G₂(X, binaryZ_1_in_g₁(X))
BINARY_1_IN_G₁(zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
BINARY_1_IN_G₁(one_1₁(X)) -> IF_BINARY_1_IN_2_G₂(X, binary_1_in_g₁(X))
BINARY_1_IN_G₁(one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADD_3_IN_AAG₃(b_0, Y, Y) -> IF_ADD_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADD_3_IN_AAG₃(b_0, Y, Y) -> BINARYZ_1_IN_G₁(Y)
ADD_3_IN_AAG₃(X, Y, Z) -> IF_ADD_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADD_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_2_AAG₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDX_3_IN_AAG₃(X, Y, Z)
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG₂(X, binary_1_in_g₁(X))
ADDX_3_IN_AAG₃(one_1₁(X), b_0, one_1₁(X)) -> BINARY_1_IN_G₁(X)
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> IF_ADDX_3_IN_2_AAG₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG₃(zero_1₁(X), b_0, zero_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDZ_3_IN_3_AAG₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDY_3_IN_AAG₃(X, Y, Z)
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG₂(Y, binary_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, one_1₁(Y), one_1₁(Y)) -> BINARY_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> IF_ADDY_3_IN_2_AAG₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG₃(X, Y, Z) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDZ_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDZ_3_IN_AAG₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG₃(X, b_0, Z) -> IF_ADDC_3_IN_1_AAG₃(X, Z, succZ_2_in_ag₂(X, Z))
ADDC_3_IN_AAG₃(X, b_0, Z) -> SUCCZ_2_IN_AG₂(X, Z)
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCCZ_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCCZ_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCCZ_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCCZ_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> IF_SUCC_2_IN_1_AG₂(X, binaryZ_1_in_g₁(X))
SUCC_2_IN_AG₂(zero_1₁(X), one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> IF_SUCC_2_IN_2_AG₃(X, Z, succ_2_in_ag₂(X, Z))
SUCC_2_IN_AG₂(one_1₁(X), zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDC_3_IN_AAG₃(b_0, Y, Z) -> IF_ADDC_3_IN_2_AAG₃(Y, Z, succZ_2_in_ag₂(Y, Z))
ADDC_3_IN_AAG₃(b_0, Y, Z) -> SUCCZ_2_IN_AG₂(Y, Z)
ADDC_3_IN_AAG₃(X, Y, Z) -> IF_ADDC_3_IN_3_AAG₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_1_AAG₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> ADDZ_3_IN_AAG₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_2_AAG₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> ADDX_3_IN_AAG¹₃(X, Y, Z)
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> IF_ADDX_3_IN_1_AAG¹₂(X, binaryZ_1_in_g₁(X))
ADDX_3_IN_AAG¹₃(zero_1₁(X), b_0, one_1₁(X)) -> BINARYZ_1_IN_G₁(X)
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> IF_ADDX_3_IN_2_AAG₃(X, Z, succ_2_in_ag₂(X, Z))
ADDX_3_IN_AAG¹₃(one_1₁(X), b_0, zero_1₁(Z)) -> SUCC_2_IN_AG₂(X, Z)
ADDX_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDX_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDX_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> IF_ADDC_3_IN_3_AAG¹₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> ADDY_3_IN_AAG¹₃(X, Y, Z)
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> IF_ADDY_3_IN_1_AAG¹₂(Y, binaryZ_1_in_g₁(Y))
ADDY_3_IN_AAG¹₃(b_0, zero_1₁(Y), one_1₁(Y)) -> BINARYZ_1_IN_G₁(Y)
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> IF_ADDY_3_IN_2_AAG₃(Y, Z, succ_2_in_ag₂(Y, Z))
ADDY_3_IN_AAG¹₃(b_0, one_1₁(Y), zero_1₁(Z)) -> SUCC_2_IN_AG₂(Y, Z)
ADDY_3_IN_AAG¹₃(X, Y, Z) -> IF_ADDY_3_IN_3_AAG¹₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
ADDY_3_IN_AAG¹₃(X, Y, Z) -> ADDC_3_IN_AAG¹₃(X, Y, Z)
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> IF_ADDC_3_IN_4_AAG₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
ADDC_3_IN_AAG¹₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> ADDC_3_IN_AAG₃(X, Y, Z)

The TRS R consists of the following rules:

times_3_in_aag₃(one_1₁(b_0), X, X) -> times_3_out_aag₃(one_1₁(b_0), X, X)
times_3_in_aag₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aag₄(R, S, RS, times_3_in_aag₃(R, S, RS))
times_3_in_aag₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aag₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(b_0), X, X) -> times_3_out_aaa₃(one_1₁(b_0), X, X)
times_3_in_aaa₃(zero_1₁(R), S, zero_1₁(RS)) -> if_times_3_in_1_aaa₄(R, S, RS, times_3_in_aaa₃(R, S, RS))
times_3_in_aaa₃(one_1₁(R), S, RSS) -> if_times_3_in_2_aaa₄(R, S, RSS, times_3_in_aaa₃(R, S, RS))
if_times_3_in_2_aaa₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_in_aaa₃(S, zero_1₁(RS), RSS))
add_3_in_aaa₃(b_0, b_0, b_0) -> add_3_out_aaa₃(b_0, b_0, b_0)
add_3_in_aaa₃(X, b_0, X) -> if_add_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(zero_1₁(X)) -> if_binaryZ_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
binaryZ_1_in_a₁(one_1₁(X)) -> if_binaryZ_1_in_2_a₂(X, binary_1_in_a₁(X))
binary_1_in_a₁(b_0) -> binary_1_out_a₁(b_0)
binary_1_in_a₁(zero_1₁(X)) -> if_binary_1_in_1_a₂(X, binaryZ_1_in_a₁(X))
if_binary_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binary_1_out_a₁(zero_1₁(X))
binary_1_in_a₁(one_1₁(X)) -> if_binary_1_in_2_a₂(X, binary_1_in_a₁(X))
if_binary_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binary_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_2_a₂(X, binary_1_out_a₁(X)) -> binaryZ_1_out_a₁(one_1₁(X))
if_binaryZ_1_in_1_a₂(X, binaryZ_1_out_a₁(X)) -> binaryZ_1_out_a₁(zero_1₁(X))
if_add_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> add_3_out_aaa₃(X, b_0, X)
add_3_in_aaa₃(b_0, Y, Y) -> if_add_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_add_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> add_3_out_aaa₃(b_0, Y, Y)
add_3_in_aaa₃(X, Y, Z) -> if_add_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_in_aaa₃(X, Y, Z))
addx_3_in_aaa₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aaa₂(X, binary_1_in_a₁(X))
if_addx_3_in_1_aaa₂(X, binary_1_out_a₁(X)) -> addx_3_out_aaa₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aaa₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aaa₂(X, binaryZ_1_in_a₁(X))
if_addx_3_in_2_aaa₂(X, binaryZ_1_out_a₁(X)) -> addx_3_out_aaa₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aaa₃(X, Y, Z) -> if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_in_aaa₃(X, Y, Z))
addy_3_in_aaa₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aaa₂(Y, binary_1_in_a₁(Y))
if_addy_3_in_1_aaa₂(Y, binary_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addy_3_in_2_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addy_3_out_aaa₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aaa₃(X, Y, Z) -> if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
addz_3_in_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
addc_3_in_aaa₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aaa₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aaa₃(X, b_0, Z) -> if_addc_3_in_1_aaa₃(X, Z, succZ_2_in_aa₂(X, Z))
succZ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succZ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succZ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succZ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
succ_2_in_aa₂(b_0, one_1₁(b_0)) -> succ_2_out_aa₂(b_0, one_1₁(b_0))
succ_2_in_aa₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_aa₂(X, binaryZ_1_in_a₁(X))
if_succ_2_in_1_aa₂(X, binaryZ_1_out_a₁(X)) -> succ_2_out_aa₂(zero_1₁(X), one_1₁(X))
succ_2_in_aa₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_aa₃(X, Z, succ_2_in_aa₂(X, Z))
if_succ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_aa₃(X, Z, succ_2_out_aa₂(X, Z)) -> succZ_2_out_aa₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aaa₃(X, Z, succZ_2_out_aa₂(X, Z)) -> addc_3_out_aaa₃(X, b_0, Z)
addc_3_in_aaa₃(b_0, Y, Z) -> if_addc_3_in_2_aaa₃(Y, Z, succZ_2_in_aa₂(Y, Z))
if_addc_3_in_2_aaa₃(Y, Z, succZ_2_out_aa₂(Y, Z)) -> addc_3_out_aaa₃(b_0, Y, Z)
addc_3_in_aaa₃(X, Y, Z) -> if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_in_aaa₃(X, Y, Z))
if_addC_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_in_aaa₃(X, Y, Z))
addX_3_in_aaa₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aaa₂(X, binaryZ_1_in_a₁(X))
if_addX_3_in_1_aaa₂(X, binaryZ_1_out_a₁(X)) -> addX_3_out_aaa₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aaa₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aaa₃(X, Z, succ_2_in_aa₂(X, Z))
if_addX_3_in_2_aaa₃(X, Z, succ_2_out_aa₂(X, Z)) -> addX_3_out_aaa₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aaa₃(X, Y, Z) -> if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_in_aaa₃(X, Y, Z))
addY_3_in_aaa₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aaa₂(Y, binaryZ_1_in_a₁(Y))
if_addY_3_in_1_aaa₂(Y, binaryZ_1_out_a₁(Y)) -> addY_3_out_aaa₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aaa₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aaa₃(Y, Z, succ_2_in_aa₂(Y, Z))
if_addY_3_in_2_aaa₃(Y, Z, succ_2_out_aa₂(Y, Z)) -> addY_3_out_aaa₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aaa₃(X, Y, Z) -> if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_in_aaa₃(X, Y, Z))
addC_3_in_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_in_aaa₃(X, Y, Z))
if_addC_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addY_3_out_aaa₃(X, Y, Z)
if_addC_3_in_3_aaa₄(X, Y, Z, addY_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addX_3_out_aaa₃(X, Y, Z)
if_addC_3_in_2_aaa₄(X, Y, Z, addX_3_out_aaa₃(X, Y, Z)) -> addC_3_out_aaa₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aaa₄(X, Y, Z, addC_3_out_aaa₃(X, Y, Z)) -> addc_3_out_aaa₃(X, Y, Z)
if_addz_3_in_4_aaa₄(X, Y, Z, addc_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addy_3_out_aaa₃(X, Y, Z)
if_addz_3_in_3_aaa₄(X, Y, Z, addy_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addx_3_out_aaa₃(X, Y, Z)
if_addz_3_in_2_aaa₄(X, Y, Z, addx_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> addz_3_out_aaa₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aaa₄(X, Y, Z, addz_3_out_aaa₃(X, Y, Z)) -> add_3_out_aaa₃(X, Y, Z)
if_times_3_in_3_aaa₅(R, S, RSS, RS, add_3_out_aaa₃(S, zero_1₁(RS), RSS)) -> times_3_out_aaa₃(one_1₁(R), S, RSS)
if_times_3_in_1_aaa₄(R, S, RS, times_3_out_aaa₃(R, S, RS)) -> times_3_out_aaa₃(zero_1₁(R), S, zero_1₁(RS))
if_times_3_in_2_aag₄(R, S, RSS, times_3_out_aaa₃(R, S, RS)) -> if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_in_aag₃(S, zero_1₁(RS), RSS))
add_3_in_aag₃(b_0, b_0, b_0) -> add_3_out_aag₃(b_0, b_0, b_0)
add_3_in_aag₃(X, b_0, X) -> if_add_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(zero_1₁(X)) -> if_binaryZ_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
binaryZ_1_in_g₁(one_1₁(X)) -> if_binaryZ_1_in_2_g₂(X, binary_1_in_g₁(X))
binary_1_in_g₁(b_0) -> binary_1_out_g₁(b_0)
binary_1_in_g₁(zero_1₁(X)) -> if_binary_1_in_1_g₂(X, binaryZ_1_in_g₁(X))
if_binary_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binary_1_out_g₁(zero_1₁(X))
binary_1_in_g₁(one_1₁(X)) -> if_binary_1_in_2_g₂(X, binary_1_in_g₁(X))
if_binary_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binary_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_2_g₂(X, binary_1_out_g₁(X)) -> binaryZ_1_out_g₁(one_1₁(X))
if_binaryZ_1_in_1_g₂(X, binaryZ_1_out_g₁(X)) -> binaryZ_1_out_g₁(zero_1₁(X))
if_add_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> add_3_out_aag₃(X, b_0, X)
add_3_in_aag₃(b_0, Y, Y) -> if_add_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_add_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> add_3_out_aag₃(b_0, Y, Y)
add_3_in_aag₃(X, Y, Z) -> if_add_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addz_3_in_2_aag₄(X, Y, Z, addx_3_in_aag₃(X, Y, Z))
addx_3_in_aag₃(one_1₁(X), b_0, one_1₁(X)) -> if_addx_3_in_1_aag₂(X, binary_1_in_g₁(X))
if_addx_3_in_1_aag₂(X, binary_1_out_g₁(X)) -> addx_3_out_aag₃(one_1₁(X), b_0, one_1₁(X))
addx_3_in_aag₃(zero_1₁(X), b_0, zero_1₁(X)) -> if_addx_3_in_2_aag₂(X, binaryZ_1_in_g₁(X))
if_addx_3_in_2_aag₂(X, binaryZ_1_out_g₁(X)) -> addx_3_out_aag₃(zero_1₁(X), b_0, zero_1₁(X))
addx_3_in_aag₃(X, Y, Z) -> if_addx_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addz_3_in_3_aag₄(X, Y, Z, addy_3_in_aag₃(X, Y, Z))
addy_3_in_aag₃(b_0, one_1₁(Y), one_1₁(Y)) -> if_addy_3_in_1_aag₂(Y, binary_1_in_g₁(Y))
if_addy_3_in_1_aag₂(Y, binary_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, one_1₁(Y), one_1₁(Y))
addy_3_in_aag₃(b_0, zero_1₁(Y), zero_1₁(Y)) -> if_addy_3_in_2_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addy_3_in_2_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addy_3_out_aag₃(b_0, zero_1₁(Y), zero_1₁(Y))
addy_3_in_aag₃(X, Y, Z) -> if_addy_3_in_3_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
addz_3_in_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addz_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
addc_3_in_aag₃(b_0, b_0, one_1₁(b_0)) -> addc_3_out_aag₃(b_0, b_0, one_1₁(b_0))
addc_3_in_aag₃(X, b_0, Z) -> if_addc_3_in_1_aag₃(X, Z, succZ_2_in_ag₂(X, Z))
succZ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succZ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succZ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succZ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succZ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succZ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
succ_2_in_ag₂(b_0, one_1₁(b_0)) -> succ_2_out_ag₂(b_0, one_1₁(b_0))
succ_2_in_ag₂(zero_1₁(X), one_1₁(X)) -> if_succ_2_in_1_ag₂(X, binaryZ_1_in_g₁(X))
if_succ_2_in_1_ag₂(X, binaryZ_1_out_g₁(X)) -> succ_2_out_ag₂(zero_1₁(X), one_1₁(X))
succ_2_in_ag₂(one_1₁(X), zero_1₁(Z)) -> if_succ_2_in_2_ag₃(X, Z, succ_2_in_ag₂(X, Z))
if_succ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_succZ_2_in_2_ag₃(X, Z, succ_2_out_ag₂(X, Z)) -> succZ_2_out_ag₂(one_1₁(X), zero_1₁(Z))
if_addc_3_in_1_aag₃(X, Z, succZ_2_out_ag₂(X, Z)) -> addc_3_out_aag₃(X, b_0, Z)
addc_3_in_aag₃(b_0, Y, Z) -> if_addc_3_in_2_aag₃(Y, Z, succZ_2_in_ag₂(Y, Z))
if_addc_3_in_2_aag₃(Y, Z, succZ_2_out_ag₂(Y, Z)) -> addc_3_out_aag₃(b_0, Y, Z)
addc_3_in_aag₃(X, Y, Z) -> if_addc_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z)) -> if_addC_3_in_1_aag₄(X, Y, Z, addz_3_in_aag₃(X, Y, Z))
if_addC_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), zero_1₁(Y), one_1₁(Z))
addC_3_in_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_2_aag₄(X, Y, Z, addX_3_in_aag₃(X, Y, Z))
addX_3_in_aag₃(zero_1₁(X), b_0, one_1₁(X)) -> if_addX_3_in_1_aag₂(X, binaryZ_1_in_g₁(X))
if_addX_3_in_1_aag₂(X, binaryZ_1_out_g₁(X)) -> addX_3_out_aag₃(zero_1₁(X), b_0, one_1₁(X))
addX_3_in_aag₃(one_1₁(X), b_0, zero_1₁(Z)) -> if_addX_3_in_2_aag₃(X, Z, succ_2_in_ag₂(X, Z))
if_addX_3_in_2_aag₃(X, Z, succ_2_out_ag₂(X, Z)) -> addX_3_out_aag₃(one_1₁(X), b_0, zero_1₁(Z))
addX_3_in_aag₃(X, Y, Z) -> if_addX_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z)) -> if_addC_3_in_3_aag₄(X, Y, Z, addY_3_in_aag₃(X, Y, Z))
addY_3_in_aag₃(b_0, zero_1₁(Y), one_1₁(Y)) -> if_addY_3_in_1_aag₂(Y, binaryZ_1_in_g₁(Y))
if_addY_3_in_1_aag₂(Y, binaryZ_1_out_g₁(Y)) -> addY_3_out_aag₃(b_0, zero_1₁(Y), one_1₁(Y))
addY_3_in_aag₃(b_0, one_1₁(Y), zero_1₁(Z)) -> if_addY_3_in_2_aag₃(Y, Z, succ_2_in_ag₂(Y, Z))
if_addY_3_in_2_aag₃(Y, Z, succ_2_out_ag₂(Y, Z)) -> addY_3_out_aag₃(b_0, one_1₁(Y), zero_1₁(Z))
addY_3_in_aag₃(X, Y, Z) -> if_addY_3_in_3_aag₄(X, Y, Z, addC_3_in_aag₃(X, Y, Z))
addC_3_in_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z)) -> if_addC_3_in_4_aag₄(X, Y, Z, addc_3_in_aag₃(X, Y, Z))
if_addC_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), one_1₁(Y), one_1₁(Z))
if_addY_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addY_3_out_aag₃(X, Y, Z)
if_addC_3_in_3_aag₄(X, Y, Z, addY_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(one_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_addX_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addX_3_out_aag₃(X, Y, Z)
if_addC_3_in_2_aag₄(X, Y, Z, addX_3_out_aag₃(X, Y, Z)) -> addC_3_out_aag₃(zero_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addc_3_in_3_aag₄(X, Y, Z, addC_3_out_aag₃(X, Y, Z)) -> addc_3_out_aag₃(X, Y, Z)
if_addz_3_in_4_aag₄(X, Y, Z, addc_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), one_1₁(Y), zero_1₁(Z))
if_addy_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addy_3_out_aag₃(X, Y, Z)
if_addz_3_in_3_aag₄(X, Y, Z, addy_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(one_1₁(X), zero_1₁(Y), one_1₁(Z))
if_addx_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addx_3_out_aag₃(X, Y, Z)
if_addz_3_in_2_aag₄(X, Y, Z, addx_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), one_1₁(Y), one_1₁(Z))
if_addz_3_in_1_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> addz_3_out_aag₃(zero_1₁(X), zero_1₁(Y), zero_1₁(Z))
if_add_3_in_3_aag₄(X, Y, Z, addz_3_out_aag₃(X, Y, Z)) -> add_3_out_aag₃(X, Y, Z)
if_times_3_in_3_aag₅(R, S, RSS, RS, add_3_out_aag₃(S, zero_1₁(RS), RSS)) -> times_3_out_aag₃(one_1₁(R), S, RSS)
if_times_3_in_1_aag₄(R, S, RS, times_3_out_aag₃(R, S, RS)) -> times_3_out_aag₃(zero_1₁(R), S, zero_1₁(RS))

The argument filtering Pi contains the following mapping:
times_3_in_aag₃(x1, x2, x3) = times_3_in_aag₁(x3)
b_0 = b_0
one_1₁(x1) = one_1₁(x1)
zero_1₁(x1) = zero_1₁(x1)
times_3_out_aag₃(x1, x2, x3) = times_3_out_aag₃(x1, x2, x3)
if_times_3_in_1_aag₄(x1, x2, x3, x4) = if_times_3_in_1_aag₂(x3, x4)
if_times_3_in_2_aag₄(x1, x2, x3, x4) = if_times_3_in_2_aag₂(x3, x4)
times_3_in_aaa₃(x1, x2, x3) = times_3_in_aaa
times_3_out_aaa₃(x1, x2, x3) = times_3_out_aaa₁(x1)
if_times_3_in_1_aaa₄(x1, x2, x3, x4) = if_times_3_in_1_aaa₁(x4)
if_times_3_in_2_aaa₄(x1, x2, x3, x4) = if_times_3_in_2_aaa₁(x4)
if_times_3_in_3_aaa₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aaa₂(x1, x5)
add_3_in_aaa₃(x1, x2, x3) = add_3_in_aaa
add_3_out_aaa₃(x1, x2, x3) = add_3_out_aaa₃(x1, x2, x3)
if_add_3_in_1_aaa₂(x1, x2) = if_add_3_in_1_aaa₁(x2)
binaryZ_1_in_a₁(x1) = binaryZ_1_in_a
if_binaryZ_1_in_1_a₂(x1, x2) = if_binaryZ_1_in_1_a₁(x2)
if_binaryZ_1_in_2_a₂(x1, x2) = if_binaryZ_1_in_2_a₁(x2)
binary_1_in_a₁(x1) = binary_1_in_a
binary_1_out_a₁(x1) = binary_1_out_a₁(x1)
if_binary_1_in_1_a₂(x1, x2) = if_binary_1_in_1_a₁(x2)
binaryZ_1_out_a₁(x1) = binaryZ_1_out_a₁(x1)
if_binary_1_in_2_a₂(x1, x2) = if_binary_1_in_2_a₁(x2)
if_add_3_in_2_aaa₂(x1, x2) = if_add_3_in_2_aaa₁(x2)
if_add_3_in_3_aaa₄(x1, x2, x3, x4) = if_add_3_in_3_aaa₁(x4)
addz_3_in_aaa₃(x1, x2, x3) = addz_3_in_aaa
if_addz_3_in_1_aaa₄(x1, x2, x3, x4) = if_addz_3_in_1_aaa₁(x4)
if_addz_3_in_2_aaa₄(x1, x2, x3, x4) = if_addz_3_in_2_aaa₁(x4)
addx_3_in_aaa₃(x1, x2, x3) = addx_3_in_aaa
if_addx_3_in_1_aaa₂(x1, x2) = if_addx_3_in_1_aaa₁(x2)
addx_3_out_aaa₃(x1, x2, x3) = addx_3_out_aaa₃(x1, x2, x3)
if_addx_3_in_2_aaa₂(x1, x2) = if_addx_3_in_2_aaa₁(x2)
if_addx_3_in_3_aaa₄(x1, x2, x3, x4) = if_addx_3_in_3_aaa₁(x4)
if_addz_3_in_3_aaa₄(x1, x2, x3, x4) = if_addz_3_in_3_aaa₁(x4)
addy_3_in_aaa₃(x1, x2, x3) = addy_3_in_aaa
if_addy_3_in_1_aaa₂(x1, x2) = if_addy_3_in_1_aaa₁(x2)
addy_3_out_aaa₃(x1, x2, x3) = addy_3_out_aaa₃(x1, x2, x3)
if_addy_3_in_2_aaa₂(x1, x2) = if_addy_3_in_2_aaa₁(x2)
if_addy_3_in_3_aaa₄(x1, x2, x3, x4) = if_addy_3_in_3_aaa₁(x4)
if_addz_3_in_4_aaa₄(x1, x2, x3, x4) = if_addz_3_in_4_aaa₁(x4)
addc_3_in_aaa₃(x1, x2, x3) = addc_3_in_aaa
addc_3_out_aaa₃(x1, x2, x3) = addc_3_out_aaa₃(x1, x2, x3)
if_addc_3_in_1_aaa₃(x1, x2, x3) = if_addc_3_in_1_aaa₁(x3)
succZ_2_in_aa₂(x1, x2) = succZ_2_in_aa
if_succZ_2_in_1_aa₂(x1, x2) = if_succZ_2_in_1_aa₁(x2)
succZ_2_out_aa₂(x1, x2) = succZ_2_out_aa₂(x1, x2)
if_succZ_2_in_2_aa₃(x1, x2, x3) = if_succZ_2_in_2_aa₁(x3)
succ_2_in_aa₂(x1, x2) = succ_2_in_aa
succ_2_out_aa₂(x1, x2) = succ_2_out_aa₂(x1, x2)
if_succ_2_in_1_aa₂(x1, x2) = if_succ_2_in_1_aa₁(x2)
if_succ_2_in_2_aa₃(x1, x2, x3) = if_succ_2_in_2_aa₁(x3)
if_addc_3_in_2_aaa₃(x1, x2, x3) = if_addc_3_in_2_aaa₁(x3)
if_addc_3_in_3_aaa₄(x1, x2, x3, x4) = if_addc_3_in_3_aaa₁(x4)
addC_3_in_aaa₃(x1, x2, x3) = addC_3_in_aaa
if_addC_3_in_1_aaa₄(x1, x2, x3, x4) = if_addC_3_in_1_aaa₁(x4)
addz_3_out_aaa₃(x1, x2, x3) = addz_3_out_aaa₃(x1, x2, x3)
addC_3_out_aaa₃(x1, x2, x3) = addC_3_out_aaa₃(x1, x2, x3)
if_addC_3_in_2_aaa₄(x1, x2, x3, x4) = if_addC_3_in_2_aaa₁(x4)
addX_3_in_aaa₃(x1, x2, x3) = addX_3_in_aaa
if_addX_3_in_1_aaa₂(x1, x2) = if_addX_3_in_1_aaa₁(x2)
addX_3_out_aaa₃(x1, x2, x3) = addX_3_out_aaa₃(x1, x2, x3)
if_addX_3_in_2_aaa₃(x1, x2, x3) = if_addX_3_in_2_aaa₁(x3)
if_addX_3_in_3_aaa₄(x1, x2, x3, x4) = if_addX_3_in_3_aaa₁(x4)
if_addC_3_in_3_aaa₄(x1, x2, x3, x4) = if_addC_3_in_3_aaa₁(x4)
addY_3_in_aaa₃(x1, x2, x3) = addY_3_in_aaa
if_addY_3_in_1_aaa₂(x1, x2) = if_addY_3_in_1_aaa₁(x2)
addY_3_out_aaa₃(x1, x2, x3) = addY_3_out_aaa₃(x1, x2, x3)
if_addY_3_in_2_aaa₃(x1, x2, x3) = if_addY_3_in_2_aaa₁(x3)
if_addY_3_in_3_aaa₄(x1, x2, x3, x4) = if_addY_3_in_3_aaa₁(x4)
if_addC_3_in_4_aaa₄(x1, x2, x3, x4) = if_addC_3_in_4_aaa₁(x4)
if_times_3_in_3_aag₅(x1, x2, x3, x4, x5) = if_times_3_in_3_aag₃(x1, x3, x5)
add_3_in_aag₃(x1, x2, x3) = add_3_in_aag₁(x3)
add_3_out_aag₃(x1, x2, x3) = add_3_out_aag₃(x1, x2, x3)
if_add_3_in_1_aag₂(x1, x2) = if_add_3_in_1_aag₂(x1, x2)
binaryZ_1_in_g₁(x1) = binaryZ_1_in_g₁(x1)
if_binaryZ_1_in_1_g₂(x1, x2) = if_binaryZ_1_in_1_g₂(x1, x2)
if_binaryZ_1_in_2_g₂(x1, x2) = if_binaryZ_1_in_2_g₂(x1, x2)
binary_1_in_g₁(x1) = binary_1_in_g₁(x1)
binary_1_out_g₁(x1) = binary_1_out_g₁(x1)
if_binary_1_in_1_g₂(x1, x2) = if_binary_1_in_1_g₂(x1, x2)
binaryZ_1_out_g₁(x1) = binaryZ_1_out_g₁(x1)
if_binary_1_in_2_g₂(x1, x2) = if_binary_1_in_2_g₂(x1, x2)
if_add_3_in_2_aag₂(x1, x2) = if_add_3_in_2_aag₂(x1, x2)
if_add_3_in_3_aag₄(x1, x2, x3, x4) = if_add_3_in_3_aag₂(x3, x4)
addz_3_in_aag₃(x1, x2, x3) = addz_3_in_aag₁(x3)
if_addz_3_in_1_aag₄(x1, x2, x3, x4) = if_addz_3_in_1_aag₂(x3, x4)
if_addz_3_in_2_aag₄(x1, x2, x3, x4) = if_addz_3_in_2_aag₂(x3, x4)
addx_3_in_aag₃(x1, x2, x3) = addx_3_in_aag₁(x3)
if_addx_3_in_1_aag₂(x1, x2) = if_addx_3_in_1_aag₂(x1, x2)
addx_3_out_aag₃(x1, x2, x3) = addx_3_out_aag₃(x1, x2, x3)
if_addx_3_in_2_aag₂(x1, x2) = if_addx_3_in_2_aag₂(x1, x2)
if_addx_3_in_3_aag₄(x1, x2, x3, x4) = if_addx_3_in_3_aag₂(x3, x4)
if_addz_3_in_3_aag₄(x1, x2, x3, x4) = if_addz_3_in_3_aag₂(x3, x4)
addy_3_in_aag₃(x1, x2, x3) = addy_3_in_aag₁(x3)
if_addy_3_in_1_aag₂(x1, x2) = if_addy_3_in_1_aag₂(x1, x2)
addy_3_out_aag₃(x1, x2, x3) = addy_3_out_aag₃(x1, x2, x3)
if_addy_3_in_2_aag₂(x1, x2) = if_addy_3_in_2_aag₂(x1, x2)
if_addy_3_in_3_aag₄(x1, x2, x3, x4) = if_addy_3_in_3_aag₂(x3, x4)
if_addz_3_in_4_aag₄(x1, x2, x3, x4) = if_addz_3_in_4_aag₂(x3, x4)
addc_3_in_aag₃(x1, x2, x3) = addc_3_in_aag₁(x3)
addc_3_out_aag₃(x1, x2, x3) = addc_3_out_aag₃(x1, x2, x3)
if_addc_3_in_1_aag₃(x1, x2, x3) = if_addc_3_in_1_aag₂(x2, x3)
succZ_2_in_ag₂(x1, x2) = succZ_2_in_ag₁(x2)
if_succZ_2_in_1_ag₂(x1, x2) = if_succZ_2_in_1_ag₂(x1, x2)
succZ_2_out_ag₂(x1, x2) = succZ_2_out_ag₂(x1, x2)
if_succZ_2_in_2_ag₃(x1, x2, x3) = if_succZ_2_in_2_ag₂(x2, x3)
succ_2_in_ag₂(x1, x2) = succ_2_in_ag₁(x2)
succ_2_out_ag₂(x1, x2) = succ_2_out_ag₂(x1, x2)
if_succ_2_in_1_ag₂(x1, x2) = if_succ_2_in_1_ag₂(x1, x2)
if_succ_2_in_2_ag₃(x1, x2, x3) = if_succ_2_in_2_ag₂(x2, x3)
if_addc_3_in_2_aag₃(x1, x2, x3) = if_addc_3_in_2_aag₂(x2, x3)
if_addc_3_in_3_aag₄(x1, x2, x3, x4) = if_addc_3_in_3_aag₂(x3, x4)
addC_3_in_aag₃(x1, x2, x3) = addC_3_in_aag₁(x3)
if_addC_3_in_1_aag₄(x1, x2, x3, x4) = if_addC_3_in_1_aag₂(x3, x4)
addz_3_out_aag₃(x1, x2, x3) = addz_3_out_aag₃(x1, x2, x3)
addC_3_out_aag₃(x1, x2, x3) = addC_3_out_aag₃(x1, x2, x3)
if_addC_3_in_2_aag₄(x1, x2, x3, x4) = if_addC_3_in_2_aag₂(x3, x4)
addX_3_in_aag₃(x1, x2, x3) = addX_3_in_aag₁(x3)
if_addX_3_in_1_aag₂(x1, x2) = if_addX_3_in_1_aag₂(x1, x2)
addX_3_out_aag₃(x1, x2, x3) = addX_3_out_aag₃(x1, x2, x3)
if_addX_3_in_2_aag₃(x1, x2, x3) = if_addX_3_in_2_aag₂(x2, x3)
if_addX_3_in_3_aag₄(x1, x2, x3, x4) = if_addX_3_in_3_aag₂(x3, x4)
if_addC_3_in_3_aag₄(x1, x2, x3, x4) = if_addC_3_in_3_aag₂(x3, x4)
addY_3_in_aag₃(x1, x2, x3) = addY_3_in_aag₁(x3)
if_addY_3_in_1_aag₂(x1, x2) = if_addY_3_in_1_aag₂(x1, x2)
addY_3_out_aag₃(x1, x2, x3) = addY_3_out_aag₃(x1, x2, x3)
if_addY_3_in_2_aag₃(x1, x2, x3) = if_addY_3_in_2_aag₂(x2, x3)
if_addY_3_in_3_aag₄(x1, x2, x3, x4) = if_addY_3_in_3_aag₂(x3, x4)
if_addC_3_in_4_aag₄(x1, x2, x3, x4) = if_addC_3_in_4_aag₂(x3, x4)
IF_ADDY_3_IN_1_AAG₂(x1, x2) = IF_ADDY_3_IN_1_AAG₂(x1, x2)
IF_TIMES_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAA₁(x4)
IF_ADDZ_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAA₁(x4)
IF_ADDC_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG₂(x3, x4)
IF_ADDZ_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAA₁(x4)
IF_BINARYZ_1_IN_1_G₂(x1, x2) = IF_BINARYZ_1_IN_1_G₂(x1, x2)
ADDZ_3_IN_AAG₃(x1, x2, x3) = ADDZ_3_IN_AAG₁(x3)
IF_ADDX_3_IN_1_AAG₂(x1, x2) = IF_ADDX_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_2_AAA₂(x1, x2) = IF_ADD_3_IN_2_AAA₁(x2)
IF_ADDX_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG₂(x3, x4)
IF_SUCC_2_IN_2_AA₃(x1, x2, x3) = IF_SUCC_2_IN_2_AA₁(x3)
IF_ADDX_3_IN_1_AAA¹₂(x1, x2) = IF_ADDX_3_IN_1_AAA¹₁(x2)
IF_ADDC_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAG₂(x2, x3)
SUCCZ_2_IN_AA₂(x1, x2) = SUCCZ_2_IN_AA
IF_ADDX_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAG¹₂(x3, x4)
IF_ADDX_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAG₂(x2, x3)
IF_BINARYZ_1_IN_2_G₂(x1, x2) = IF_BINARYZ_1_IN_2_G₂(x1, x2)
IF_BINARY_1_IN_2_A₂(x1, x2) = IF_BINARY_1_IN_2_A₁(x2)
IF_TIMES_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAG₂(x3, x4)
IF_ADDY_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG¹₂(x3, x4)
IF_TIMES_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_TIMES_3_IN_2_AAA₁(x4)
ADDC_3_IN_AAG₃(x1, x2, x3) = ADDC_3_IN_AAG₁(x3)
ADDX_3_IN_AAA₃(x1, x2, x3) = ADDX_3_IN_AAA
IF_ADDX_3_IN_2_AAA₂(x1, x2) = IF_ADDX_3_IN_2_AAA₁(x2)
IF_ADDC_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA¹₁(x4)
SUCC_2_IN_AA₂(x1, x2) = SUCC_2_IN_AA
BINARYZ_1_IN_G₁(x1) = BINARYZ_1_IN_G₁(x1)
IF_ADDC_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_2_AAA₁(x3)
IF_BINARY_1_IN_1_A₂(x1, x2) = IF_BINARY_1_IN_1_A₁(x2)
IF_ADDY_3_IN_1_AAG¹₂(x1, x2) = IF_ADDY_3_IN_1_AAG¹₂(x1, x2)
IF_SUCCZ_2_IN_2_AG₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AG₂(x2, x3)
IF_ADD_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAA₁(x4)
IF_BINARY_1_IN_2_G₂(x1, x2) = IF_BINARY_1_IN_2_G₂(x1, x2)
IF_ADDZ_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_1_AAG₂(x3, x4)
TIMES_3_IN_AAA₃(x1, x2, x3) = TIMES_3_IN_AAA
IF_ADDY_3_IN_2_AAG₂(x1, x2) = IF_ADDY_3_IN_2_AAG₂(x1, x2)
ADDY_3_IN_AAA₃(x1, x2, x3) = ADDY_3_IN_AAA
ADDC_3_IN_AAA¹₃(x1, x2, x3) = ADDC_3_IN_AAA¹
IF_BINARYZ_1_IN_1_A₂(x1, x2) = IF_BINARYZ_1_IN_1_A₁(x2)
IF_TIMES_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_TIMES_3_IN_1_AAG₂(x3, x4)
IF_ADDC_3_IN_1_AAG₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAG₂(x2, x3)
IF_ADDX_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDX_3_IN_2_AAA₁(x3)
IF_ADDY_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAG₂(x3, x4)
BINARYZ_1_IN_A₁(x1) = BINARYZ_1_IN_A
ADDY_3_IN_AAA¹₃(x1, x2, x3) = ADDY_3_IN_AAA¹
IF_ADD_3_IN_2_AAG₂(x1, x2) = IF_ADD_3_IN_2_AAG₂(x1, x2)
ADDX_3_IN_AAG₃(x1, x2, x3) = ADDX_3_IN_AAG₁(x3)
SUCCZ_2_IN_AG₂(x1, x2) = SUCCZ_2_IN_AG₁(x2)
IF_ADDY_3_IN_2_AAG₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAG₂(x2, x3)
ADDC_3_IN_AAG¹₃(x1, x2, x3) = ADDC_3_IN_AAG¹₁(x3)
IF_SUCCZ_2_IN_2_AA₃(x1, x2, x3) = IF_SUCCZ_2_IN_2_AA₁(x3)
IF_SUCCZ_2_IN_1_AA₂(x1, x2) = IF_SUCCZ_2_IN_1_AA₁(x2)
IF_ADDZ_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAA₁(x4)
IF_ADDZ_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_3_AAG₂(x3, x4)
ADDX_3_IN_AAG¹₃(x1, x2, x3) = ADDX_3_IN_AAG¹₁(x3)
IF_ADDC_3_IN_4_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAA₁(x4)
IF_ADDY_3_IN_1_AAA₂(x1, x2) = IF_ADDY_3_IN_1_AAA₁(x2)
IF_SUCCZ_2_IN_1_AG₂(x1, x2) = IF_SUCCZ_2_IN_1_AG₂(x1, x2)
IF_BINARY_1_IN_1_G₂(x1, x2) = IF_BINARY_1_IN_1_G₂(x1, x2)
ADD_3_IN_AAG₃(x1, x2, x3) = ADD_3_IN_AAG₁(x3)
ADDY_3_IN_AAG¹₃(x1, x2, x3) = ADDY_3_IN_AAG¹₁(x3)
IF_TIMES_3_IN_3_AAA₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAA₂(x1, x5)
ADDX_3_IN_AAA¹₃(x1, x2, x3) = ADDX_3_IN_AAA¹
IF_ADDY_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA¹₁(x4)
IF_ADDX_3_IN_3_AAA¹₄(x1, x2, x3, x4) = IF_ADDX_3_IN_3_AAA¹₁(x4)
IF_TIMES_3_IN_3_AAG₅(x1, x2, x3, x4, x5) = IF_TIMES_3_IN_3_AAG₃(x1, x3, x5)
IF_ADDX_3_IN_1_AAA₂(x1, x2) = IF_ADDX_3_IN_1_AAA₁(x2)
IF_ADDC_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAG₂(x3, x4)
IF_ADDC_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_4_AAG₂(x3, x4)
IF_ADD_3_IN_1_AAG₂(x1, x2) = IF_ADD_3_IN_1_AAG₂(x1, x2)
IF_ADD_3_IN_1_AAA₂(x1, x2) = IF_ADD_3_IN_1_AAA₁(x2)
IF_ADDY_3_IN_1_AAA¹₂(x1, x2) = IF_ADDY_3_IN_1_AAA¹₁(x2)
ADDZ_3_IN_AAA₃(x1, x2, x3) = ADDZ_3_IN_AAA
IF_ADD_3_IN_3_AAG₄(x1, x2, x3, x4) = IF_ADD_3_IN_3_AAG₂(x3, x4)
IF_ADDZ_3_IN_4_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_4_AAG₂(x3, x4)
ADDC_3_IN_AAA₃(x1, x2, x3) = ADDC_3_IN_AAA
IF_ADDC_3_IN_1_AAG₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAG₂(x3, x4)
IF_ADDY_3_IN_2_AAA₃(x1, x2, x3) = IF_ADDY_3_IN_2_AAA₁(x3)
SUCC_2_IN_AG₂(x1, x2) = SUCC_2_IN_AG₁(x2)
IF_SUCC_2_IN_1_AA₂(x1, x2) = IF_SUCC_2_IN_1_AA₁(x2)
IF_ADDC_3_IN_1_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_1_AAA₁(x4)
IF_BINARYZ_1_IN_2_A₂(x1, x2) = IF_BINARYZ_1_IN_2_A₁(x2)
IF_ADDY_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDY_3_IN_3_AAA₁(x4)
IF_ADDX_3_IN_2_AAG₂(x1, x2) = IF_ADDX_3_IN_2_AAG₂(x1, x2)
IF_ADDC_3_IN_3_AAG¹₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAG¹₂(x3, x4)
TIMES_3_IN_AAG₃(x1, x2, x3) = TIMES_3_IN_AAG₁(x3)
IF_ADDC_3_IN_1_AAA₃(x1, x2, x3) = IF_ADDC_3_IN_1_AAA₁(x3)
IF_ADDY_3_IN_2_AAA₂(x1, x2) = IF_ADDY_3_IN_2_AAA₁(x2)
IF_ADDZ_3_IN_2_AAG₄(x1, x2, x3, x4) = IF_ADDZ_3_IN_2_AAG₂(x3, x4)
IF_ADDX_3_IN_1_AAG¹₂(x1, x2) = IF_ADDX_3_IN_1_AAG¹₂(x1, x2)
BINARY_1_IN_A₁(x1) = BINARY_1_IN_A
BINARY_1_IN_G₁(x1) = BINARY_1_IN_G₁(x1)
ADDY_3_IN_AAG₃(x1, x2, x3) = ADDY_3_IN_AAG₁(x3)
IF_SUCC_2_IN_2_AG₃(x1, x2, x3) = IF_SUCC_2_IN_2_AG₂(x2, x3)
ADD_3_IN_AAA₃(x1, x2, x3) = ADD_3_IN_AAA
IF_ADDC_3_IN_2_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_2_AAA₁(x4)
IF_SUCC_2_IN_1_AG₂(x1, x2) = IF_SUCC_2_IN_1_AG₂(x1, x2)
IF_ADDC_3_IN_3_AAA₄(x1, x2, x3, x4) = IF_ADDC_3_IN_3_AAA₁(x4)

We have to consider all (P,R,Pi)-chains