0.00/0.10 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.00/0.10 % Command : do_cvc5 %s %d 0.10/0.30 % Computer : n009.cluster.edu 0.10/0.30 % Model : x86_64 x86_64 0.10/0.30 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.10/0.30 % Memory : 8042.1875MB 0.10/0.30 % OS : Linux 3.10.0-693.el7.x86_64 0.10/0.30 % CPULimit : 960 0.10/0.30 % WCLimit : 120 0.10/0.30 % DateTime : Tue Aug 9 04:31:05 EDT 2022 0.10/0.30 % CPUTime : 0.15/0.53 %----Proving TF0_NAR, FOF, or CNF 2.03/2.22 ------- cvc5-fof casc J11 : /export/starexec/sandbox2/benchmark/theBenchmark.p at /export/starexec/sandbox2/benchmark/theBenchmark.p... 2.03/2.22 --- Run --decision=internal --simplification=none --no-inst-no-entail --no-cbqi --full-saturate-quant at 10... 2.03/2.22 % SZS status Theorem for theBenchmark 2.03/2.22 % SZS output start Proof for theBenchmark 2.03/2.22 (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 _let_1 min))) (let ((_let_4 (ord_less int))) (let ((_let_5 (hAPP int _let_2 _let_4 pls))) (let ((_let_6 (one_one int))) (let ((_let_7 (bit1 pls))) (let ((_let_8 (bit0 _let_7))) (let ((_let_9 (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_8))) m)) _let_6))) (let ((_let_10 (zero_zero int))) (let ((_let_11 (hAPP int _let_2 _let_1 _let_10))) (let ((_let_12 (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 _let_6) t)))) (let ((_let_13 (=> _let_12 (exists ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int)) (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2))))))))) (let ((_let_14 (hAPP int _let_2 _let_1 pls))) (let ((_let_15 (not (exists ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2)) (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int))))))))) (let ((_let_16 (number_number_of int min))) (let ((_let_17 (number_number_of int pls))) (let ((_let_18 (number_number_of nat _let_8))) (let ((_let_19 (hAPP nat int (power_power int s) _let_18))) (let ((_let_20 (hAPP int int (minus_minus int _let_19) _let_16))) (let ((_let_21 (hAPP int _let_2 (dvd_dvd int) _let_9))) (let ((_let_22 (one_one nat))) (let ((_let_23 (hBOOL (hAPP int bool (quadRes _let_9) _let_16)))) (let ((_let_24 (legendre _let_16 _let_9))) (let ((_let_25 (hAPP int _let_2 _let_4 _let_10))) (let ((_let_26 (hAPP int _let_2 _let_4 min))) (let ((_let_27 (hAPP int int (times_times int _let_9) t))) (let ((_let_28 (order int))) (let ((_let_29 (=> (= t _let_6) (exists ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2)) (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int))))))))) (let ((_let_30 (one int))) (let ((_let_31 (hAPP int int (plus_plus int _let_19) _let_6))) (let ((_let_32 (number_number_of nat pls))) (let ((_let_33 (zero_zero nat))) (let ((_let_34 (number_number_of int _let_8))) (let ((_let_35 (forall ((X_a $$unsorted)) (let ((_let_1 (one_one X_a))) (=> (one X_a) (= (ti X_a _let_1) _let_1)))))) (let ((_let_36 (ti int t))) (let ((_let_37 (= _let_36 t))) (let ((_let_38 (hAPP nat int (power_power int s1) _let_18))) (let ((_let_39 (forall ((X_a $$unsorted)) (=> (order X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less_eq X_a) X_1) Y_1)) (=> (not (= (ti X_a Y_1) (ti X_a X_1))) (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less X_a) X_1) Y_1)))))))))) (let ((_let_40 (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 _let_6) t)))) (let ((_let_41 (= t _let_36))) (let ((_let_42 (= _let_6 t))) (let ((_let_43 (ti int _let_6))) (let ((_let_44 (= _let_6 _let_43))) (let ((_let_45 (= _let_36 _let_43))) (let ((_let_46 (SYMM (ASSUME :args (_let_37))))) (let ((_let_47 (forall ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (not (= (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int)) (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2))))))))) (let ((_let_48 (not _let_42))) (let ((_let_49 (MACRO_RESOLUTION_TRUST (EQUIV_ELIM2 (ALPHA_EQUIV :args (_let_47 (= X X) (= Y Y)))) (EQ_RESOLVE (ASSUME :args (_let_15)) (MACRO_SR_EQ_INTRO :args (_let_15 SB_DEFAULT SBA_FIXPOINT))) :args (_let_47 false (forall ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (not (= (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int)) (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2))))))))))) (let ((_let_50 (not _let_30))) (let ((_let_51 (or _let_50 _let_44))) (let ((_let_52 (forall ((X_a $$unsorted)) (let ((_let_1 (one_one X_a))) (or (not (one X_a)) (= _let_1 (ti X_a _let_1))))))) (let ((_let_53 (EQ_RESOLVE (ASSUME :args (_let_35)) (MACRO_SR_EQ_INTRO :args (_let_35 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_54 (not _let_40))) (let ((_let_55 (not _let_28))) (let ((_let_56 (or _let_55 _let_54 _let_45 _let_12))) (let ((_let_57 (forall ((X_a $$unsorted) (BOUND_VARIABLE_28576 $$unsorted) (BOUND_VARIABLE_28574 $$unsorted)) (let ((_let_1 (fun X_a bool))) (or (not (order X_a)) (not (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less_eq X_a) BOUND_VARIABLE_28574) BOUND_VARIABLE_28576))) (= (ti X_a BOUND_VARIABLE_28574) (ti X_a BOUND_VARIABLE_28576)) (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less X_a) BOUND_VARIABLE_28574) BOUND_VARIABLE_28576))))))) (let ((_let_58 (EQ_RESOLVE (ASSUME :args (_let_39)) (MACRO_SR_EQ_INTRO :args (_let_39 SB_DEFAULT SBA_FIXPOINT))))) (let ((_let_59 (forall ((X $$unsorted) (Y $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (not (= (hAPP int int (plus_plus int (hAPP int int (times_times int (number_number_of int (bit0 _let_1))) m)) (one_one int)) (hAPP int int (plus_plus int (hAPP nat int (power_power int X) _let_2)) (hAPP nat int (power_power int Y) _let_2))))))))) (let ((_let_60 (not _let_12))) (let ((_let_61 (not _let_44))) (let ((_let_62 (not _let_45))) (let ((_let_63 (not _let_41))) (let ((_let_64 (ASSUME :args (_let_45)))) (let ((_let_65 (ASSUME :args (_let_44)))) (let ((_let_66 (ASSUME :args (_let_48)))) (SCOPE (MACRO_RESOLUTION_TRUST (REORDERING (EQ_RESOLVE (NOT_AND (MACRO_SR_PRED_TRANSFORM (SCOPE (AND_INTRO _let_46 _let_64 _let_65 _let_66) :args (_let_41 _let_48 _let_44 _let_45)) (SCOPE (MACRO_SR_PRED_ELIM (TRANS (SYMM (FALSE_INTRO _let_66)) (TRUE_INTRO (TRANS (SYMM (SYMM _let_65)) (SYMM _let_64) (SYMM _let_46))))) :args (_let_41 _let_45 _let_44 _let_48)) :args ((not (and _let_41 _let_48 _let_44 _let_45)) SB_LITERAL))) (CONG (REFL :args (_let_63)) (MACRO_SR_PRED_INTRO :args ((= (not _let_48) _let_42))) (REFL :args (_let_61)) (REFL :args (_let_62)) :args (OR))) :args ((or _let_42 _let_63 _let_62 _let_61))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_56)) :args ((or _let_12 _let_54 _let_55 _let_45 (not _let_56)))) (MACRO_RESOLUTION_TRUST (REORDERING (IMPLIES_ELIM (EQ_RESOLVE (ASSUME :args (_let_13)) (MACRO_SR_EQ_INTRO :args (_let_13 SB_DEFAULT SBA_FIXPOINT)))) :args ((or (not _let_59) _let_60))) (MACRO_RESOLUTION_TRUST (EQUIV_ELIM1 (ALPHA_EQUIV :args (_let_47 (= X X) (= Y Y)))) _let_49 :args (_let_59 false _let_47)) :args (_let_60 false _let_59)) (ASSUME :args (_let_40)) (ASSUME :args (_let_28)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (MACRO_SR_PRED_ELIM (SCOPE (INSTANTIATE _let_58 :args (int t _let_6 QUANTIFIERS_INST_E_MATCHING ((hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) BOUND_VARIABLE_28574) BOUND_VARIABLE_28576)))) :args (_let_57)))) _let_58 :args (_let_56 false _let_57)) :args (_let_45 true _let_12 false _let_40 false _let_28 false _let_56)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS :args (_let_51)) :args ((or _let_50 _let_44 (not _let_51)))) (ASSUME :args (_let_30)) (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_53 :args (int QUANTIFIERS_INST_E_MATCHING_SIMPLE ((not (= (one X_a) false))))) :args (_let_52))) _let_53 :args (_let_51 false _let_52)) :args (_let_44 false _let_30 false _let_51)) (MACRO_RESOLUTION_TRUST (REORDERING (IMPLIES_ELIM (EQ_RESOLVE (ASSUME :args (_let_29)) (MACRO_SR_EQ_INTRO :args (_let_29 SB_DEFAULT SBA_FIXPOINT)))) :args ((or (not _let_47) _let_48))) _let_49 :args (_let_48 false _let_47)) _let_46 :args (false false _let_45 false _let_44 true _let_42 false _let_41)) :args (true (= _let_20 _let_31) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((W $$unsorted)) (= (number_number_of X_a (hAPP int int (plus_plus int (bit1 pls)) W)) (hAPP X_a X_a (plus_plus X_a (one_one X_a)) (number_number_of X_a W)))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (zero_zero real))) (let ((_let_2 (number_number_of nat (bit0 (bit1 pls))))) (= (= _let_1 (hAPP real real (plus_plus real (hAPP nat real (power_power real X_2) _let_2)) (hAPP nat real (power_power real Y_2) _let_2))) (and (= _let_1 Y_2) (= X_2 _let_1)))))) (forall ((X_a $$unsorted)) (=> (cancel_semigroup_add X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted) (C_1 $$unsorted)) (let ((_let_1 (plus_plus X_a A_3))) (= (= (hAPP X_a X_a _let_1 C_1) (hAPP X_a X_a _let_1 B_2)) (= (ti X_a C_1) (ti X_a B_2))))))) (forall ((M $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less nat) I_1))) (=> (hBOOL (hAPP nat bool _let_1 J_1)) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat J_1) M)))))) (forall ((X_a $$unsorted) (X_c $$unsorted) (X_b $$unsorted) (P $$unsorted) (Q $$unsorted) (R $$unsorted)) (= (hAPP X_b X_c P (hAPP X_a X_b Q R)) (hAPP X_a X_c (combb X_b X_c X_a P Q) R))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (times_times X_a B_1_1))) (=> (dvd X_a) (= _let_1 (ti (fun X_a X_a) _let_1))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (dvd_dvd int) K_1))) (=> (hBOOL (hAPP int bool _let_1 (hAPP int int (div_mod int M) N_1))) (=> (hBOOL (hAPP int bool _let_1 N_1)) (hBOOL (hAPP int bool _let_1 M)))))) (forall ((X_a $$unsorted) (X_2 $$unsorted) (A_6 $$unsorted)) (= (hBOOL (hAPP (fun X_a bool) bool (member X_a X_2) A_6)) (hBOOL (hAPP X_a bool A_6 X_2)))) (forall ((K $$unsorted) (P_2 $$unsorted) (D $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) _let_1) D)) (=> (forall ((X $$unsorted)) (=> (hBOOL (hAPP int bool P_2 X)) (hBOOL (hAPP int bool P_2 (hAPP int int (minus_minus int X) D))))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) _let_1) K)) (forall ((X $$unsorted)) (=> (hBOOL (hAPP int bool P_2 X)) (hBOOL (hAPP int bool P_2 (hAPP int int (minus_minus int X) (hAPP int int (times_times int K) D)))))))))))) (forall ((N $$unsorted)) (let ((_let_1 (zero_zero nat))) (= (= _let_1 N) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N) _let_1))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (bit0 (hAPP int int (minus_minus int K_1) L_1)) (hAPP int int (minus_minus int (bit1 K_1)) (bit1 L_1)))) (forall ((P_2 $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted)) (= (hBOOL (hAPP nat bool P_2 (hAPP nat nat (minus_minus nat A_3) B_2))) (and (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) A_3) B_2)) (hBOOL (hAPP nat bool P_2 (zero_zero nat)))) (forall ((D_4 $$unsorted)) (=> (= A_3 (hAPP nat nat (plus_plus nat B_2) D_4)) (hBOOL (hAPP nat bool P_2 D_4))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (minus_minus X_a B_1_1))) (=> (group_add X_a) (= _let_1 (ti (fun X_a X_a) _let_1))))) (forall ((S $$unsorted) (T_3 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) S) T_3)) (not (= S T_3)))) (forall ((A_3 $$unsorted) (Ma $$unsorted)) (= (hBOOL (hAPP int bool (zcong A_3 (zero_zero int)) Ma)) (hBOOL (hAPP int bool (hAPP int (fun int bool) (dvd_dvd int) Ma) A_3)))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (let ((_let_1 (multInv B_1_1 B_2_1))) (= (ti int _let_1) _let_1))) (ordere779506340up_add real) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (and (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1))) (not (= X_1 Y_1)))))) (forall ((M $$unsorted)) (= M (hAPP nat nat (plus_plus nat M) (zero_zero nat)))) (forall ((L_1 $$unsorted)) (let ((_let_1 (minus_minus int pls))) (= (hAPP int int _let_1 (bit0 L_1)) (bit0 (hAPP int int _let_1 L_1))))) (ordered_semiring int) (ab_semigroup_add int) (forall ((X_a $$unsorted)) (=> (ring X_a) (forall ((A_3 $$unsorted) (E $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted) (D $$unsorted)) (= (= (ti X_a D) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (minus_minus X_a A_3) B_2)) E)) C_1)) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a A_3) E)) C_1) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a B_2) E)) D)))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (dvd_dvd nat) M) N_1)) (or (= N_1 (zero_zero nat)) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) M) N_1)))))) (forall ((B_1_1 $$unsorted)) (= (zfact (ti int B_1_1)) (zfact B_1_1))) (number_semiring int) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a A_1) C)) (hAPP X_a X_a (plus_plus X_a B) C))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)))))))) (ordere236663937imp_le real) (forall ((I_1 $$unsorted) (J_1 $$unsorted) (K_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat I_1) J_1)) K_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_1) K_1)))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (number_number_of nat (bit0 (bit1 pls))))) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)) (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_1) _let_1)) (hAPP nat X_a (power_power X_a Y_1) _let_1)))))))) (forall ((W_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 W_1) _let_1)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (bit0 W_1)) _let_1))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (plus_plus X_a B_1_1))) (=> (monoid_add X_a) (= (ti (fun X_a X_a) _let_1) _let_1)))) (forall ((I_1 $$unsorted) (J_1 $$unsorted)) (not (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (hAPP nat nat (plus_plus nat I_1) J_1)) I_1)))) (cancel_semigroup_add nat) (forall ((I_2 $$unsorted) (K $$unsorted) (J_2 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 K) J_2)) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_2) (hAPP nat nat (minus_minus nat J_2) K))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus 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(zero_zero nat)) (= N Ma)) (= (hAPP nat nat _let_1 Ma) (hAPP nat nat _let_1 N))))) (semiring_0 int) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (one_one nat))) (= (and (= Ma _let_1) (= _let_1 N)) (= _let_1 (hAPP nat nat (times_times nat Ma) N))))) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) _let_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (plus_plus X_a A_1) B)) _let_1)))))))))) (forall ((K $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) pls))) (= (hBOOL (hAPP int bool _let_1 (bit0 K))) (hBOOL (hAPP int bool _let_1 K))))) (= _let_22 (number_number_of nat _let_7)) (forall ((M $$unsorted) (N_1 $$unsorted) (K_1 $$unsorted)) (= (hAPP nat nat (times_times nat (hAPP nat nat 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(forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (times_times X_a C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a _let_3 A_1)) (hAPP X_a X_a _let_3 B))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)))))))))) (forall ((K_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) M)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP int int (div_mod int M) K_1)) M)))))) (forall ((B_1_1 $$unsorted)) (let ((_let_1 (d22set B_1_1))) (= _let_1 (ti (fun int bool) _let_1)))) (forall ((X_a $$unsorted)) (=> (linordered_ring X_a) (forall ((A_1 $$unsorted)) (not (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (hAPP X_a X_a (times_times X_a A_1) A_1)) (zero_zero X_a))))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) _let_1) B)) (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) _let_1) (hAPP int int (div_mod int A_1) B))))))) (forall ((A_3 $$unsorted) (B_2 $$unsorted) (N $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (not (= N (zero_zero nat))) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (power_power nat A_3) N)) (hAPP nat nat (power_power nat B_2) N))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_3) B_2))))))) (forall ((Ma $$unsorted) (K $$unsorted) (N $$unsorted)) (= (or (= (zero_zero nat) K) (= Ma N)) (= (hAPP nat nat (times_times nat N) K) (hAPP nat nat (times_times nat Ma) K)))) (forall ((A_1 $$unsorted) (N_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) N_1)) (=> (hBOOL (hAPP real bool (hAPP real (fun real bool) (ord_less real) (zero_zero 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(hBOOL (hAPP X_a bool _let_2 (hAPP X_a X_a (plus_plus X_a A_1) C)))))))))) (forall ((X_a $$unsorted)) (let ((_let_1 (ord_less X_a))) (=> (order X_a) (= _let_1 (ti (fun X_a (fun X_a bool)) _let_1))))) (forall ((B_1_1 $$unsorted)) (= (bit1 (ti int B_1_1)) (bit1 B_1_1))) (dvd nat) (forall ((N_1 $$unsorted)) (let ((_let_1 (zero_zero nat))) (= _let_1 (hAPP nat nat (minus_minus nat _let_1) N_1)))) (forall ((K $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) pls) (bit1 K))) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) pls) K))))) (forall ((T $$unsorted) (A $$unsorted)) (let ((_let_1 (ti T A))) (= (ti T _let_1) _let_1))) (order real) (forall ((D_2 $$unsorted) (C $$unsorted) (A_1 $$unsorted) (B $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (zcong A_1 B) M)) (=> (= (ti int B) (ti int C)) (=> (hBOOL (hAPP int bool (zcong C D_2) M)) (hBOOL (hAPP int bool (zcong A_1 D_2) M)))))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 _let_1) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (times_times X_a B) A_1)) _let_1)))))))))) (ring int) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (number_number_of X_a B_1_1))) (=> (number X_a) (= _let_1 (ti X_a _let_1))))) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)))) (= (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat Ma) N))) (or (hBOOL (hAPP nat bool _let_1 Ma)) (hBOOL (hAPP nat bool _let_1 N)))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (or (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M) N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 N_1) M)))))) (forall ((X_a $$unsorted)) (=> (monoid_mult X_a) (forall ((A_1 $$unsorted)) (= (hAPP nat X_a (power_power X_a A_1) (one_one nat)) (ti X_a A_1))))) (ordere216010020id_add int) (linordered_ring real) (forall ((X_2 $$unsorted) (N $$unsorted)) (let ((_let_1 (zero_zero nat))) (let ((_let_2 (hAPP nat (fun nat bool) (ord_less nat) _let_1))) (= (or (hBOOL (hAPP nat bool _let_2 X_2)) (= _let_1 N)) (hBOOL (hAPP nat bool _let_2 (hAPP nat nat (power_power nat X_2) N))))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K)) (bit0 L))))))) (forall ((X_1 $$unsorted)) (let ((_let_1 (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (dvd_dvd nat) X_1) X_1)))) (not (and _let_1 (not _let_1))))) (forall ((Ma $$unsorted) (N $$unsorted)) (= (= (zero_zero nat) (hAPP nat nat (minus_minus nat Ma) N)) (hBOOL (hAPP nat bool (hAPP 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(forall ((A_1 $$unsorted) (J_1 $$unsorted) (K_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (times_times int A_1))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool (zcong J_1 K_1) P_1)) (hBOOL (hAPP int bool (zcong (hAPP int int _let_1 (multInv P_1 J_1)) (hAPP int int _let_1 (multInv P_1 K_1))) P_1)))))) (forall ((K $$unsorted) (Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (times_times nat K))) (= (= (hAPP nat nat _let_1 Ma) (hAPP nat nat _let_1 N)) (or (= (zero_zero nat) K) (= Ma N))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (ord_less_eq real))) (let ((_let_2 (fun real bool))) (= (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 (hAPP real real (minus_minus real X_2) Y_2)) (zero_zero real))) (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 X_2) Y_2)))))) (forall ((X_b $$unsorted) (X_c $$unsorted) (X_a $$unsorted) (B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (combb X_b X_c X_a B_1_1 B_2_1) (combb X_b X_c X_a (ti (fun X_b X_c) B_1_1) B_2_1))) (forall ((U $$unsorted) (Ma $$unsorted) (N $$unsorted) (I_2 $$unsorted) (J_2 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_2) J_2)) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Ma) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat (hAPP nat nat (minus_minus nat J_2) I_2)) U)) N))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat I_2) U)) Ma)) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat J_2) U)) N)))))))) (forall ((W $$unsorted)) (= pls (hAPP int int (times_times int pls) W))) (forall ((K $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K)) pls)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) pls)))))) (real_normed_algebra real) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (times_times X_a A_1) B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (div_mod X_a A_1) C)) B)) C))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (times_times nat K_1))) (= (hAPP nat nat (plus_plus nat (hAPP nat nat _let_1 M)) (hAPP nat nat _let_1 N_1)) (hAPP nat nat _let_1 (hAPP nat nat (plus_plus nat M) N_1))))) (forall ((X_a $$unsorted)) (=> (semiring X_a) (forall ((A_1 $$unsorted) (E_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a A_1) E_1)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a B) E_1)) C)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) E_1)) C))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_1) J_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 K_1) L_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat I_1) K_1)) (hAPP nat nat (plus_plus nat J_1) L_1)))))))) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) _let_1) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a X_2) X_2)) (hAPP X_a X_a (times_times X_a Y_2) Y_2)))) (or (not (= (ti X_a Y_2) _let_1)) (not (= _let_1 (ti X_a X_2))))))))) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (plus_plus X_a C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a _let_3 A_1)) (hAPP X_a X_a _let_3 B))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B))))))))) (hBOOL (hAPP int bool _let_25 _let_6)) (forall ((K1 $$unsorted) (K2 $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) K1) K2)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) (bit0 K1)) (bit1 K2)))))) (linordered_semiring real) (= (number_number_of int _let_7) _let_6) (forall ((N_1 $$unsorted) (P_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (fun int bool))) (let ((_let_2 (hAPP int _let_1 (dvd_dvd int) P_1))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (zero_zero int)) M)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool _let_2 (hAPP int int (times_times int M) N_1))) (or (hBOOL (hAPP int bool _let_2 N_1)) (hBOOL (hAPP int bool _let_2 M))))))))) (forall ((M $$unsorted) (N_1 $$unsorted) (K_1 $$unsorted) (L_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 K_1) L_1)) (=> (= (hAPP nat nat (plus_plus 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(plus_plus X_a (hAPP X_a X_a (times_times X_a A_3) E)) C_1)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a B_2) E)) D))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (minus_minus X_a A_3) B_2)) E)) C_1)) D)))))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (bit1 pls))) (let ((_let_2 (bit1 _let_1))) (let ((_let_3 (number_number_of nat _let_2))) (let ((_let_4 (power_power int B))) (let ((_let_5 (number_number_of nat (bit0 _let_1)))) (let ((_let_6 (times_times int (number_number_of int _let_2)))) (let ((_let_7 (power_power int A_1))) (= (hAPP nat int (power_power int (hAPP int int (minus_minus int A_1) B)) _let_3) (hAPP int int (minus_minus int (hAPP int int (plus_plus int (hAPP int int (minus_minus int (hAPP nat int _let_7 _let_3)) (hAPP int int (times_times int (hAPP int int _let_6 (hAPP nat int _let_7 _let_5))) B))) (hAPP int int (times_times int (hAPP int int _let_6 A_1)) (hAPP nat 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Ma)) Ma)) (hBOOL (hAPP int bool (zcong A_3 B_2) Ma)))) (forall ((X_a $$unsorted)) (=> (ordered_ab_group_add X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted) (C_1 $$unsorted) (D $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (= (hAPP X_a X_a (minus_minus X_a C_1) D) (hAPP X_a X_a (minus_minus X_a A_3) B_2)) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C_1) D))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hAPP nat _let_2 _let_1 (zero_zero nat)))) (=> (hBOOL (hAPP nat bool _let_3 N_1)) (=> (hBOOL (hAPP nat bool _let_3 M)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (minus_minus nat M) N_1)) M)))))))) (ordere236663937imp_le nat) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (div_mod X_a B_1_1))) (=> (semiring_div X_a) (= (ti (fun X_a X_a) _let_1) _let_1)))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((V $$unsorted) (W $$unsorted)) (= (hAPP X_a X_a (minus_minus X_a (number_number_of X_a V)) (number_number_of X_a W)) (number_number_of X_a (hAPP int int (minus_minus int V) W)))))) (forall ((M $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) M)) (not (hBOOL (hAPP int bool (zcong (one_one int) (number_number_of int min)) M))))) (forall ((X_a $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (=> (order X_a) (= (ti (fun X_a (fun X_a bool)) _let_1) _let_1)))) (forall ((A_1 $$unsorted)) (let ((_let_1 (one_one nat))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (not (and (not (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 _let_1) A_1))) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 A_1) _let_1)))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((B $$unsorted) (C $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 _let_1 B))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (zero_zero X_a)) A_1)) (=> (hBOOL (hAPP X_a bool _let_3 C)) (hBOOL (hAPP X_a bool _let_3 (hAPP X_a X_a (plus_plus X_a A_1) C))))))))))) (forall ((Z_1 $$unsorted)) (= (hAPP nat nat (plus_plus nat Z_1) Z_1) (hAPP nat nat (times_times nat Z_1) (number_number_of nat (bit0 (bit1 pls)))))) (hBOOL (hAPP int bool (zcong _let_38 _let_16) _let_9)) (forall ((N $$unsorted) (Ma $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) (zero_zero nat)) Ma)) (= (hBOOL (hAPP nat bool (hAPP nat _let_1 (dvd_dvd nat) (hAPP nat nat (times_times nat N) Ma)) Ma)) (= (one_one nat) N))))) (mult_zero real) (forall ((A_1 $$unsorted) (M $$unsorted) (B $$unsorted)) (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) M) (hAPP int int (times_times int B) M)) M))) (forall ((W $$unsorted)) (hBOOL (hAPP real bool (hAPP real (fun real bool) (ord_less_eq real) W) W))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted)) (=> (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) A_1) B)) (= (zero_zero X_a) (hAPP X_a X_a (div_mod X_a B) A_1)))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) B)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP int int (div_mod int A_1) B)) B)))))) (forall ((X_a $$unsorted)) (=> (linordered_semiring X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B))))))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (div_mod X_a A_1) B) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) (hAPP X_a X_a (times_times X_a B) C))) B))))) (forall ((X_a $$unsorted)) (=> (mult_zero X_a) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hAPP X_a X_a (times_times X_a _let_1) A_1) _let_1))))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_1) J_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat I_1) K_1)) (hAPP nat nat (plus_plus nat J_1) K_1))))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) K) L)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) (bit0 K)) (bit1 L)))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (inv B_1_1 B_2_1) (inv (ti int B_1_1) B_2_1))) (forall ((A_1 $$unsorted) (J_1 $$unsorted) (K_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (times_times int A_1))) (let ((_let_2 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong K_1 _let_2) P_1))) (=> (not (hBOOL (hAPP int bool (zcong J_1 _let_2) P_1))) (=> (hBOOL (hAPP int bool (zcong J_1 (hAPP int int _let_1 (multInv P_1 K_1))) P_1)) (hBOOL (hAPP int bool (zcong K_1 (hAPP int int _let_1 (multInv P_1 J_1))) P_1)))))))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (hAPP int int (minus_minus int P_1) _let_1))) (let ((_let_3 (ord_less int))) (let ((_let_4 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 _let_1) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 A_1) _let_2)) (not (= _let_2 (inv P_1 A_1))))))))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (ord_less real))) (let ((_let_2 (fun real bool))) (=> (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 (zero_zero real)) Z_2)) (= (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 (hAPP real real (times_times real X_2) Z_2)) (hAPP real real (times_times real Y_2) Z_2))) (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 X_2) Y_2))))))) (forall ((M $$unsorted) (N_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (dvd_dvd int) P_1))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool _let_1 (hAPP int int (times_times int M) N_1))) (or (hBOOL (hAPP int bool _let_1 N_1)) (hBOOL (hAPP int bool _let_1 M))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted) (N_1 $$unsorted)) (=> (or (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) N_1)) (= (ti X_a X_1) (one_one X_a))) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) X_1) (hAPP nat X_a (power_power X_a X_1) N_1))))))) (forall ((Z_1 $$unsorted) (W $$unsorted)) (let ((_let_1 (ord_less_eq real))) (let ((_let_2 (fun real bool))) (or (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 Z_1) W)) (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 W) Z_1)))))) (monoid_add real) (= twoSqu658283162sum2sq (ti _let_2 twoSqu658283162sum2sq)) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP nat X_a (power_power X_a (hAPP X_a X_a (minus_minus X_a X_1) Y_1)) _let_2) (hAPP X_a X_a (minus_minus X_a (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_1) _let_2)) (hAPP nat X_a (power_power X_a Y_1) _let_2))) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (times_times X_a (number_number_of X_a _let_1)) X_1)) Y_1)))))))) (forall ((X_a $$unsorted)) (=> (one X_a) (forall ((X_2 $$unsorted)) (let ((_let_1 (one_one X_a))) (let ((_let_2 (ti X_a X_2))) (= (= _let_1 _let_2) (= _let_2 _let_1))))))) (= min (ti int min)) (forall ((N_1 $$unsorted)) (let ((_let_1 (number_number_of int min))) (let ((_let_2 (hAPP nat int (power_power int _let_1) N_1))) (or (= _let_2 _let_1) (= (one_one int) _let_2))))) (forall ((Q_1 $$unsorted) (B $$unsorted) (R_1 $$unsorted) (C $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 (ord_less_eq int) _let_1))) (let ((_let_4 (ord_less int))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 _let_1) C)) (=> (hBOOL (hAPP int bool _let_3 R_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 R_1) B)) (hBOOL (hAPP int bool _let_3 (hAPP int int (plus_plus int (hAPP int int (times_times int B) (hAPP int int (div_mod int Q_1) C))) R_1))))))))))) (comm_monoid_add nat) (forall ((X_a $$unsorted)) (=> (monoid_mult X_a) (forall ((A_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (power_power X_a A_1))) (= (hAPP nat X_a _let_1 (hAPP nat nat (plus_plus nat M) N_1)) (hAPP X_a X_a (times_times X_a (hAPP nat X_a _let_1 M)) (hAPP nat X_a _let_1 N_1))))))) (forall ((I_1 $$unsorted) (K_1 $$unsorted) (J_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) K_1) J_1)) (= (hAPP nat nat (minus_minus nat (hAPP nat nat (plus_plus nat I_1) K_1)) J_1) (hAPP nat nat (minus_minus nat I_1) (hAPP nat nat (minus_minus nat J_1) K_1))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (legendre A_1 P_1))) (let ((_let_2 (hBOOL (hAPP int bool (quadRes P_1) A_1)))) (let ((_let_3 (zero_zero int))) (let ((_let_4 (hBOOL (hAPP int bool (zcong A_1 _let_3) P_1)))) (and (=> _let_4 (= _let_1 _let_3)) (=> (not _let_4) (and (=> (not _let_2) (= _let_1 (number_number_of int min))) (=> _let_2 (= _let_1 (one_one int))))))))))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted) (P_1 $$unsorted)) (=> (not (hBOOL (hAPP int bool (zcong X_1 (zero_zero int)) P_1))) (=> (hBOOL (hAPP int bool (zcong (hAPP nat int (power_power int Y_1) (number_number_of nat (bit0 (bit1 pls)))) X_1) P_1)) (not (hBOOL (hAPP int bool (hAPP int (fun int bool) (dvd_dvd int) P_1) Y_1)))))) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (= A_1 B) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 B) C)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) C))))))) (forall ((P_1 $$unsorted) (Y_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (dvd_dvd int) P_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) N_1)) (=> (hBOOL (hAPP int bool _let_1 Y_1)) (hBOOL (hAPP int bool _let_1 (hAPP nat int (power_power int Y_1) N_1))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((M $$unsorted) (N_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (power_power X_a A_1))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less X_a) (one_one X_a)) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) (hAPP nat X_a _let_1 M)) (hAPP nat X_a _let_1 N_1))) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) M) N_1))))))))) (forall ((K $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less int) min))) (= (hBOOL (hAPP int bool _let_1 K)) (hBOOL (hAPP int bool _let_1 (bit1 K)))))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (Ya $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a X_1) Ya)) (hAPP X_a X_a (times_times X_a Y_1) Ya)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (plus_plus X_a X_1) Y_1)) Ya))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted)) (= (one_one X_a) (hAPP nat X_a (power_power X_a X_1) (zero_zero nat)))))) (forall ((A_1 $$unsorted)) (let ((_let_1 (bit1 pls))) (let ((_let_2 (power_power int A_1))) (= (hAPP nat int _let_2 (number_number_of nat (bit1 _let_1))) (hAPP int int (times_times int A_1) (hAPP nat int _let_2 (number_number_of nat (bit0 _let_1)))))))) (forall ((B_2 $$unsorted) (A_3 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (fun int bool))) (let ((_let_2 (one_one int))) (let ((_let_3 (ord_less int))) (=> (hBOOL (hAPP int bool zprime P_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_1 _let_3 A_3) (hAPP int int (minus_minus int P_3) _let_2))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 _let_3 _let_2) B_2)) (=> (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) B_2) A_3)) (hBOOL (hAPP _let_1 bool (member int B_2) (wset A_3 P_3))))))))))) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((C_1 $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (plus_plus X_a C_1))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_3) B_2)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a _let_1 A_3)) (hAPP X_a X_a _let_1 B_2)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted) (K_1 $$unsorted)) (= (hAPP nat nat (times_times nat (hAPP nat nat (div_mod nat M) N_1)) K_1) (hAPP nat nat (div_mod nat (hAPP nat nat (times_times nat M) K_1)) (hAPP nat nat (times_times nat N_1) K_1)))) (forall ((X_a $$unsorted)) (let ((_let_1 (dvd_dvd X_a))) (=> (dvd X_a) (= (ti (fun X_a (fun X_a bool)) _let_1) _let_1)))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (zcong B_1_1 B_2_1) (zcong B_1_1 (ti int B_2_1)))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (hAPP int int (minus_minus int P_1) _let_1))) (let ((_let_3 (ord_less int))) (let ((_let_4 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 _let_1) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 A_1) _let_2)) (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 (inv P_1 A_1)) _let_2)))))))))) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (zero_zero nat))) (= (and (= Ma _let_1) (not (= N _let_1))) (= _let_1 (hAPP nat nat (power_power nat Ma) N))))) (forall ((X_1 $$unsorted) (P_1 $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong X_1 (zero_zero int)) P_1))) (hBOOL (hAPP int bool (zcong (multInv P_1 (multInv P_1 X_1)) X_1) P_1)))))) (forall ((M $$unsorted) (K_1 $$unsorted) (N_1 $$unsorted)) (= (hAPP nat nat (minus_minus nat M) N_1) (hAPP nat nat (minus_minus nat (hAPP nat nat (plus_plus nat M) K_1)) (hAPP nat nat (plus_plus nat N_1) K_1)))) (not (= _let_6 _let_10)) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (hAPP int int (times_times int (bit0 K_1)) L_1) (bit0 (hAPP int int (times_times int K_1) L_1)))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong X_1 _let_1) P_1))) (=> (not (hBOOL (hAPP int bool (zcong Y_1 _let_1) P_1))) (=> (hBOOL (hAPP int bool (zcong (multInv P_1 X_1) (multInv P_1 Y_1)) P_1)) (hBOOL (hAPP int bool (zcong X_1 Y_1) P_1))))))))) (forall ((B $$unsorted) (A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) _let_1) A_1)) (=> (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) B) _let_1) P_1)) (or (hBOOL (hAPP int bool (zcong B _let_1) P_1)) (hBOOL (hAPP int bool (zcong A_1 _let_1) P_1)))))))) (forall ((Ma $$unsorted) (D $$unsorted)) (= (exists ((Q_2 $$unsorted)) (= Ma (hAPP nat nat (times_times nat D) Q_2))) (= (zero_zero nat) (hAPP nat nat (div_mod nat Ma) D)))) (forall ((B_3 $$unsorted) (Q_4 $$unsorted) (R_3 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 (ord_less_eq int) _let_1))) (let ((_let_4 (ord_less int))) (=> (hBOOL (hAPP int bool _let_3 (hAPP int int (plus_plus int (hAPP int int (times_times int B_3) Q_4)) R_3))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 R_3) B_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 _let_1) B_3)) (hBOOL (hAPP int bool _let_3 Q_4)))))))))) (hBOOL (hAPP int bool (zcong _let_19 _let_38) _let_9)) (forall ((I_1 $$unsorted) (U_1 $$unsorted) (J_1 $$unsorted) (K_1 $$unsorted)) (= (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat (hAPP nat nat (plus_plus nat I_1) J_1)) U_1)) K_1) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat I_1) U_1)) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat J_1) U_1)) K_1)))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 B) A_1)))) (let ((_let_4 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) B)))) (=> (and _let_4 (not _let_3)) (not (and (not _let_4) _let_3)))))))) (forall ((Z1 $$unsorted) (Z2 $$unsorted) (Z3 $$unsorted)) (let ((_let_1 (times_times real Z1))) (= (hAPP real real _let_1 (hAPP real real (times_times real Z2) Z3)) (hAPP real real (times_times real (hAPP real real _let_1 Z2)) Z3)))) (ring_char_0 int) (forall ((X_2 $$unsorted) (P_2 $$unsorted)) (=> (forall ((A_5 $$unsorted)) (let ((_let_1 (one_one int))) (=> (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) _let_1) A_5)) (hBOOL (hAPP int bool P_2 (hAPP int int (minus_minus int A_5) _let_1)))) (hBOOL (hAPP int bool P_2 A_5))))) (hBOOL (hAPP int bool P_2 X_2)))) (forall ((X_a $$unsorted) (B_1_1 $$unsorted)) (= (collect X_a (ti (fun X_a bool) B_1_1)) (collect X_a B_1_1))) (forall ((X_2 $$unsorted)) (let ((_let_1 (zero_zero real))) (= (not (hBOOL (hAPP real bool (hAPP real (fun real bool) (ord_less real) _let_1) (hAPP real real (times_times real X_2) X_2)))) (= X_2 _let_1)))) (forall ((J_1 $$unsorted) (K_1 $$unsorted) (A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (times_times int A_1))) (let ((_let_2 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong A_1 _let_2) P_1))) (=> (not (hBOOL (hAPP int bool (zcong K_1 _let_2) P_1))) (=> (not (hBOOL (hAPP int bool (zcong J_1 _let_2) P_1))) (=> (hBOOL (hAPP int bool (zcong (hAPP int int _let_1 (multInv P_1 J_1)) (hAPP int int _let_1 (multInv P_1 K_1))) P_1)) (hBOOL (hAPP int bool (zcong J_1 K_1) P_1))))))))))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= _let_1 (hAPP X_a X_a (times_times X_a A_1) _let_1)))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (multInv B_1_1 B_2_1) (multInv B_1_1 (ti int B_2_1)))) (forall ((V_2 $$unsorted)) (= (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) V_2) pls)) (= (number_number_of nat V_2) (zero_zero nat)))) (forall ((N_1 $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N_1) M)) (= (hAPP nat nat (plus_plus nat N_1) (hAPP nat nat (minus_minus nat M) N_1)) M))) (forall ((X_a $$unsorted)) (=> (linord219039673up_add X_a) (forall ((A_3 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (= (hAPP X_a X_a (plus_plus X_a A_3) A_3) _let_1) (= (ti X_a A_3) _let_1)))))) (forall ((Z1 $$unsorted) (Z2 $$unsorted) (W $$unsorted)) (= (hAPP int int (times_times int (hAPP int int (minus_minus int Z1) Z2)) W) (hAPP int int (minus_minus int (hAPP int int (times_times int Z1) W)) (hAPP int int (times_times int Z2) W)))) (forall ((X_a $$unsorted)) (=> (semiri456707255roduct X_a) (forall ((C_1 $$unsorted) (D $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (times_times X_a B_2))) (let ((_let_2 (times_times X_a A_3))) (= (and (not (= (ti X_a B_2) (ti X_a A_3))) (not (= (ti X_a C_1) (ti X_a D)))) (not (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_2 D)) (hAPP X_a X_a _let_1 C_1)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_2 C_1)) (hAPP X_a X_a _let_1 D)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) (hAPP nat nat (minus_minus nat M) N_1)) M))) _let_37 (comm_ring int) (cancel_semigroup_add int) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) D_2)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) D_2)))))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (Q_1 $$unsorted)) (= (hAPP X_a X_a (times_times X_a (hAPP nat X_a (power_power X_a X_1) Q_1)) (hAPP nat X_a (power_power X_a Y_1) Q_1)) (hAPP nat X_a (power_power X_a (hAPP 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$$unsorted) (C $$unsorted) (A_4 $$unsorted)) (=> (= (hAPP X_a X_a (div_mod X_a A_1) C) (hAPP X_a X_a (div_mod X_a A_4) C)) (=> (= (hAPP X_a X_a (div_mod X_a B) C) (hAPP X_a X_a (div_mod X_a B_3) C)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_4) B_3)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) C))))))) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less X_a) (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (=> (hBOOL (hAPP X_a bool _let_1 B)) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (plus_plus X_a A_1) B))))))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (number_number_of nat (bit0 (bit1 pls))))) (= (hAPP nat nat (minus_minus nat (hAPP nat nat (power_power nat X_1) _let_1)) (hAPP nat nat (power_power nat Y_1) _let_1)) (hAPP nat nat (times_times nat (hAPP nat nat (plus_plus nat X_1) Y_1)) (hAPP nat nat (minus_minus nat 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(= _let_1 (ti X_a Y_2))) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a X_2) X_2)) (hAPP X_a X_a (times_times X_a Y_2) Y_2))) _let_1))))))) (ordere223160158up_add int) (forall ((Ma $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (times_times nat K))) (let ((_let_2 (ord_less_eq nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 (ord_less nat) (zero_zero nat)) K)) (= (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 Ma) N)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 Ma)) (hAPP nat nat _let_1 N))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (= (hAPP nat nat (times_times nat N_1) M) (hAPP nat nat (times_times nat M) N_1))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (or (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 X_1) Y_1)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 Y_1) X_1)) (= (ti int X_1) (ti int Y_1)))))) (forall ((X_a $$unsorted) (X_c $$unsorted) (B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (hAPP X_a X_c B_1_1 (ti X_a B_2_1)) (hAPP X_a X_c B_1_1 B_2_1))) (forall ((X_a $$unsorted)) (=> (ordered_ring X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (let ((_let_4 (hAPP X_a _let_3 _let_2 _let_1))) (=> (or (and (hBOOL (hAPP X_a bool _let_4 A_1)) (hBOOL (hAPP X_a bool _let_4 B))) (and (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) _let_1)))) (hBOOL (hAPP X_a bool _let_4 (hAPP X_a X_a (times_times X_a A_1) B))))))))))) (ordere1490568538miring real) (forall ((M $$unsorted) (N_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) M) N_1)) (= M (hAPP nat nat (div_mod nat M) N_1)))) (number real) (number_ring real) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((B 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(semiring_div X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (div_mod X_a A_1) B)) C))))) (order nat) (forall ((Z1 $$unsorted) (Z2 $$unsorted) (W $$unsorted)) (= (hAPP real real (plus_plus real (hAPP real real (times_times real Z1) W)) (hAPP real real (times_times real Z2) W)) (hAPP real real (times_times real (hAPP real real (plus_plus real Z1) Z2)) W))) (ab_group_add int) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((N_1 $$unsorted) (M $$unsorted) (X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 X_1) Y_1)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N_1) M)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP nat X_a (power_power X_a X_1) N_1)) (hAPP nat X_a (power_power X_a Y_1) M)))))))))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) _let_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 _let_1) B)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (times_times X_a A_1) B)) _let_1)))))))))) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((A_3 $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a A_3) C_1)) (hAPP X_a X_a (plus_plus X_a B_2) C_1))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2)))))))) (forall ((N_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_1 X_1)) (hBOOL (hAPP int bool _let_1 (hAPP nat int (power_power int X_1) N_1)))))) (forall ((A_3 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (hAPP int int (minus_minus int P_3) _let_1))) (= (hBOOL (hAPP int bool (zcong A_3 _let_2) P_3)) (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_3) _let_2) _let_1) P_3)))))) (forall ((X_2 $$unsorted) (N $$unsorted)) (let ((_let_1 (zero_zero nat))) (let ((_let_2 (hAPP nat (fun nat bool) (ord_less nat) _let_1))) (= (hBOOL (hAPP nat bool _let_2 (hAPP nat nat (power_power nat X_2) N))) (or (= _let_1 N) (hBOOL (hAPP nat bool _let_2 X_2))))))) (monoid_mult real) (forall ((X_a $$unsorted)) (=> (number_semiring X_a) (forall ((V_1 $$unsorted) (V $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) pls))) (=> (hBOOL (hAPP int bool _let_1 V)) (=> (hBOOL (hAPP int bool _let_1 V_1)) (= (hAPP X_a X_a (plus_plus X_a (number_number_of X_a V)) (number_number_of X_a V_1)) (number_number_of X_a (hAPP int int (plus_plus int V) V_1))))))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (= (and (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_2) X_2)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_2) Y_2))) (= X_2 Y_2))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)) (one_one X_a))))) (forall ((X_a $$unsorted)) (=> (power X_a) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (power_power X_a M))) (let ((_let_2 (hAPP nat X_a _let_1 N_1))) (let ((_let_3 (zero_zero nat))) (and (=> (not (= _let_3 N_1)) (= (hAPP X_a X_a (times_times X_a M) (hAPP nat X_a _let_1 (hAPP nat nat (minus_minus nat N_1) (one_one nat)))) _let_2)) (=> (= N_1 _let_3) (= _let_2 (one_one X_a)))))))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (bit0 (hAPP int int (plus_plus int K_1) L_1)) (hAPP int int (plus_plus int (bit0 K_1)) (bit0 L_1)))) (forall ((R_1 $$unsorted) (Q_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) _let_3) A_1)) (=> (= (hAPP int int (plus_plus int R_1) (hAPP int int (times_times int A_1) Q_1)) (ti int A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 _let_3) R_1)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 Q_1) (one_one int)))))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((N_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less X_a) (one_one X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (times_times X_a A_1) (hAPP nat X_a (power_power X_a A_1) N_1))))))))) (forall ((B $$unsorted) (M $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_3 A_1)) (=> (hBOOL (hAPP int bool (hAPP 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bool _let_1 K))))) (forall ((A_1 $$unsorted) (R_1 $$unsorted) (B $$unsorted) (M $$unsorted) (C $$unsorted) (D_2 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (times_times int R_1))) (= (hAPP int int (plus_plus int (hAPP int int (times_times int (hAPP int int (minus_minus int A_1) (hAPP int int _let_1 B))) M)) (hAPP int int (times_times int (hAPP int int (minus_minus int C) (hAPP int int _let_1 D_2))) N_1)) (hAPP int int (minus_minus int (hAPP int int (plus_plus int (hAPP int int (times_times int A_1) M)) (hAPP int int (times_times int C) N_1))) (hAPP int int _let_1 (hAPP int int (plus_plus int (hAPP int int (times_times int B) M)) (hAPP int int (times_times int D_2) N_1))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted) (P_1 $$unsorted) (Q_1 $$unsorted)) (let ((_let_1 (power_power X_a X_1))) (= (hAPP nat X_a _let_1 (hAPP nat nat (plus_plus nat P_1) Q_1)) (hAPP X_a X_a (times_times X_a (hAPP nat X_a _let_1 P_1)) (hAPP nat X_a _let_1 Q_1))))))) (forall ((B $$unsorted) (Q_4 $$unsorted) (R_3 $$unsorted) (Q_1 $$unsorted) (R_1 $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 (ord_less int) B))) (let ((_let_4 (times_times int B))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP int int (plus_plus int (hAPP int int _let_4 Q_4)) R_3)) (hAPP int int (plus_plus int (hAPP int int _let_4 Q_1)) R_1))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 R_1) (zero_zero int))) (=> (hBOOL (hAPP int bool _let_3 R_1)) (=> (hBOOL (hAPP int bool _let_3 R_3)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 Q_1) Q_4))))))))))) (forall ((M $$unsorted) (K_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat M) K_1)) N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M) N_1)))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((A_1 $$unsorted) (N_1 $$unsorted)) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)) (hAPP nat X_a (power_power X_a A_1) (hAPP nat nat (times_times nat (number_number_of nat (bit0 (bit1 pls)))) N_1))))))) (forall ((B_2 $$unsorted) (A_3 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (wset A_3 P_3))) (let ((_let_2 (fun int bool))) (let ((_let_3 (one_one int))) (let ((_let_4 (hAPP int int (minus_minus int P_3) _let_3))) (let ((_let_5 (ord_less int))) (=> (hBOOL (hAPP int bool zprime P_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) (number_number_of int (bit1 (bit0 (bit1 pls))))) P_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_5 A_3) _let_4)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_5 _let_3) B_2)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_5 B_2) _let_4)) (=> (hBOOL (hAPP _let_2 bool (member int (inv P_3 B_2)) _let_1)) (hBOOL (hAPP _let_2 bool (member int B_2) _let_1)))))))))))))) (forall ((X_b $$unsorted)) (=> (and (monoid_mult X_b) (number X_b)) (forall ((W $$unsorted)) (let ((_let_1 (number_number_of X_b W))) (= (hAPP X_b X_b (times_times X_b _let_1) _let_1) (hAPP nat X_b (power_power X_b _let_1) (number_number_of nat (bit0 (bit1 pls))))))))) (forall ((K $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K)) min)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) min)))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((B $$unsorted) (A_1 $$unsorted) (C $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) A_1))) (=> (hBOOL (hAPP X_a bool _let_1 C)) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (times_times X_a B) C)))))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_1) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 A_1) (hAPP int int (minus_minus int P_1) _let_1))) (not (= _let_1 (inv P_1 A_1)))))))))) (forall ((K $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) min)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K)) min)))))) (forall ((N_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_1 (number_number_of int N_1))) (and (hBOOL (hAPP int bool _let_1 (number_number_of int (bit0 N_1)))) (hBOOL (hAPP int bool _let_1 (number_number_of int (bit1 N_1)))))))) (forall ((K $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) pls))) (= (hBOOL (hAPP int bool _let_1 (bit1 K))) (hBOOL (hAPP int bool _let_1 K))))) (not (= (one_one real) (zero_zero real))) (forall ((X_a $$unsorted)) (=> (number_semiring X_a) (forall ((V_1 $$unsorted) (V $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) pls))) (=> (hBOOL (hAPP int bool _let_1 V)) (=> (hBOOL (hAPP int bool _let_1 V_1)) (= (hAPP X_a X_a (times_times X_a (number_number_of X_a V)) (number_number_of X_a V_1)) (number_number_of X_a (hAPP int int (times_times int V) V_1))))))))) (forall ((N_1 $$unsorted) (M $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N_1) (hAPP nat nat (plus_plus nat M) N_1)))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (number_number_of nat (bit0 (bit1 pls))))) (not (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_1) _let_1)) (hAPP nat X_a (power_power X_a Y_1) _let_1))) (zero_zero X_a)))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (cancel_semigroup_add X_a) (= (plus_plus X_a (ti X_a B_1_1)) (plus_plus X_a B_1_1)))) (=> (not _let_23) (not (= _let_24 _let_6))) (forall ((A_1 $$unsorted) (N_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less 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A_1))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) (zero_zero int)) M)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (dvd_dvd int) M) B)) (= (hAPP int int _let_1 M) (hAPP int int (div_mod int (hAPP int int _let_1 B)) M))))))) (zero int) (forall ((X_a $$unsorted)) (=> (and (plus X_a) (dvd X_a) (linorder X_a)) (forall ((D $$unsorted) (Sa $$unsorted)) (exists ((Z $$unsorted)) (forall ((X $$unsorted)) (let ((_let_1 (fun X_a bool))) (let ((_let_2 (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (dvd_dvd X_a) D) (hAPP X_a X_a (plus_plus X_a X) Sa))))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less X_a) X) Z)) (= _let_2 _let_2))))))))) (mult_zero nat) (linorder real) (forall ((N_1 $$unsorted) (K_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (dvd_dvd nat) K_1))) (=> (hBOOL (hAPP nat bool _let_1 M)) (=> (hBOOL (hAPP nat bool _let_1 N_1)) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (minus_minus nat M) N_1))))))) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (let ((_let_4 (hAPP X_a _let_3 _let_2 _let_1))) (= (hBOOL (hAPP X_a bool _let_4 (hAPP X_a X_a (times_times X_a A_3) B_2))) (or (and (hBOOL (hAPP X_a bool _let_4 A_3)) (hBOOL (hAPP X_a bool _let_4 B_2))) (and (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_3) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B_2) _let_1)))))))))))) (forall ((K_1 $$unsorted)) (= (ti int K_1) (hAPP int int (minus_minus int K_1) pls))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (zero_zero X_a))) (let ((_let_4 (ord_less_eq X_a))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_4 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) D_2)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 _let_3) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_4 _let_3) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) D_2)))))))))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP nat X_a (power_power X_a A_1) N_1)) (hAPP nat X_a (power_power X_a B) N_1))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) (zero_zero X_a)) B)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B))))))))) (forall ((X_a $$unsorted)) (let ((_let_1 (one_one X_a))) (=> (number_semiring X_a) (= (hAPP X_a X_a (plus_plus X_a _let_1) _let_1) (number_number_of X_a (bit0 (bit1 pls))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (dvd_dvd X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) B)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) C)))))))) (forall ((Q_1 $$unsorted) (R_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (times_times nat B))) (let ((_let_2 (ord_less nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (zero_zero nat)) C)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 R_1) B)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat (plus_plus nat (hAPP nat nat _let_1 (hAPP nat nat (div_mod nat Q_1) C))) R_1)) (hAPP nat nat _let_1 C))))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((N_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (one_one X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (hBOOL (hAPP X_a bool _let_1 (hAPP nat X_a (power_power X_a A_1) N_1)))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (not (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (one_one X_a)) (zero_zero X_a)))))) (one real) (forall ((K_1 $$unsorted)) (not (= (bit0 K_1) min))) (forall ((Ma $$unsorted) (K $$unsorted) (N $$unsorted)) (= (= (hAPP nat nat (plus_plus nat Ma) K) (hAPP nat nat (plus_plus nat N) K)) (= Ma N))) (forall ((X_2 $$unsorted)) (let ((_let_1 (hAPP int int (div_mod int X_2) (number_number_of int (bit0 (bit1 pls)))))) (= (= (one_one int) _let_1) (not (= (zero_zero int) _let_1))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (bit1 (hAPP int int (plus_plus int K_1) L_1)) (hAPP int int (plus_plus int (bit1 K_1)) (bit0 L_1)))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((Lx $$unsorted) (Ly $$unsorted) (Rx $$unsorted)) (let ((_let_1 (times_times X_a Lx))) (= (hAPP X_a X_a (times_times X_a (hAPP X_a X_a _let_1 Ly)) Rx) (hAPP X_a X_a _let_1 (hAPP X_a X_a (times_times X_a Ly) Rx))))))) (forall ((A_1 $$unsorted)) (let ((_let_1 (one_one nat))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (dvd_dvd nat) A_1) _let_1)) (= _let_1 A_1)))) (forall ((I_1 $$unsorted) (J_1 $$unsorted) (K_1 $$unsorted)) (let ((_let_1 (minus_minus nat I_1))) (= (hAPP nat nat (minus_minus nat (hAPP nat nat _let_1 J_1)) K_1) (hAPP nat nat _let_1 (hAPP nat nat (plus_plus nat J_1) K_1))))) (forall ((Ma $$unsorted)) (let ((_let_1 (one_one nat))) (= (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (dvd_dvd nat) Ma) _let_1)) (= _let_1 Ma)))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (mult_zero X_a) (= (times_times X_a (ti X_a B_1_1)) (times_times X_a B_1_1)))) (ordere223160158up_add nat) (linordered_semiring nat) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less int) B))) (=> (hBOOL (hAPP int bool _let_1 (zero_zero int))) (hBOOL (hAPP int bool _let_1 (hAPP int int (div_mod int A_1) B)))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (div_mod X_a A_1) C)) (hAPP X_a X_a (div_mod X_a B) C))) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) C))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (number_number_of nat (bit0 (bit1 pls))))) (let ((_let_2 (zero_zero X_a))) (= (or (not (= (ti X_a Y_2) _let_2)) (not (= (ti X_a X_2) _let_2))) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) _let_2) (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_2) _let_1)) (hAPP nat X_a (power_power X_a Y_2) _let_1)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M) N_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 N_1) M)) (= M N_1)))))) (= (ti (fun bool (fun bool bool)) fconj) fconj) (idom real) (forall ((A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) A_3) B_2)) (not (hBOOL (hAPP _let_1 bool (member int B_2) (d22set A_3))))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (= (zero_zero X_a) (number_number_of X_a pls)))) (hBOOL (hAPP int bool _let_11 _let_34)) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (times_times X_a C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a _let_3 A_1)) (hAPP X_a X_a _let_3 B))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less X_a) (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted) (K_1 $$unsorted)) (= (hAPP nat nat (times_times nat (hAPP nat nat (plus_plus nat M) N_1)) K_1) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat M) K_1)) (hAPP nat nat (times_times nat N_1) K_1)))) (forall ((P_1 $$unsorted)) (let ((_let_1 (bit1 pls))) (let ((_let_2 (bit0 _let_1))) (let ((_let_3 (ti int P_1))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (= _let_3 (number_number_of int _let_2))) (=> (not (= (number_number_of int (bit1 _let_1)) _let_3)) (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) (number_number_of int (bit1 _let_2))) P_1))))))))) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((C $$unsorted) (B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 B) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) (zero_zero X_a))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C)))))))))) (forall ((N $$unsorted) (K $$unsorted) (Ma $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less_eq nat) K))) (=> (hBOOL (hAPP nat bool _let_1 Ma)) (=> (hBOOL (hAPP nat bool _let_1 N)) (= (= (hAPP nat nat (minus_minus nat Ma) K) (hAPP nat nat (minus_minus nat N) K)) (= Ma N)))))) (forall ((X_1 $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP nat real (power_power real (hAPP real real (times_times real (number_number_of real _let_1)) X_1)) _let_2) (hAPP real real (times_times real (number_number_of real (bit0 _let_1))) (hAPP nat real (power_power real X_1) _let_2)))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) B)))) (=> _let_3 (=> (not (= A_1 B)) (and _let_3 (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 B) A_1)))))))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (zcong B_1_1 B_2_1) (zcong (ti int B_1_1) B_2_1))) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP nat int (power_power int A_1) N_1)) (hAPP nat int (power_power int B) N_1))) (=> (not (= N_1 (zero_zero nat))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 A_1) B))))))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (hAPP real (fun real bool) (ord_less real) (zero_zero real)))) (=> (hBOOL (hAPP real bool _let_1 X_1)) (=> (hBOOL (hAPP real bool _let_1 Y_1)) (hBOOL (hAPP real bool _let_1 (hAPP real real (times_times real X_1) Y_1))))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((A_1 $$unsorted)) (not (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (hAPP nat X_a (power_power X_a A_1) (number_number_of nat (bit0 (bit1 pls))))) (zero_zero X_a))))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (div_mod X_a A_1) C)) B)) C))))) (forall ((X_a $$unsorted)) (=> (monoid_mult X_a) (forall ((X_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (power_power X_a X_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) N_1)) (= (hAPP X_a X_a (times_times X_a (hAPP nat X_a _let_1 (hAPP nat nat (minus_minus nat N_1) (one_one nat)))) X_1) (hAPP nat X_a _let_1 N_1))))))) (semiring_0 nat) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((Lx $$unsorted) (Rx $$unsorted) (Ry $$unsorted)) (let ((_let_1 (times_times X_a Lx))) (= (hAPP X_a X_a _let_1 (hAPP X_a X_a (times_times X_a Rx) Ry)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a _let_1 Rx)) Ry)))))) (forall ((M $$unsorted) (Y_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 _let_1 (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_3 X_1)) (=> (hBOOL (hAPP int bool _let_3 Y_1)) (=> (hBOOL (hAPP int bool _let_3 M)) (=> (hBOOL (hAPP int bool (zcong X_1 Y_1) M)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 X_1) M)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 Y_1) M)) (= (ti int Y_1) (ti int X_1)))))))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (times_times X_a B_1_1))) (=> (no_zero_divisors X_a) (= (ti (fun X_a X_a) _let_1) _let_1)))) (forall ((N_1 $$unsorted)) (= (hAPP nat nat (times_times nat (one_one nat)) N_1) N_1)) (forall ((N_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (dvd_dvd int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_3 M)) (=> (hBOOL (hAPP int bool _let_3 N_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 M) N_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 N_1) M)) (= (ti int M) (ti int N_1)))))))))) (ring_char_0 real) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted)) (= (zero_zero X_a) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (times_times X_a A_1) B)) B))))) (forall ((X_a $$unsorted) (X_b $$unsorted) (X_c $$unsorted) (B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (combc X_a X_b X_c B_1_1 B_2_1) (combc X_a X_b X_c (ti (fun X_a (fun X_b X_c)) B_1_1) B_2_1))) (number int) (forall ((P_5 $$unsorted) (P_2 $$unsorted) (X_2 $$unsorted)) (let ((_let_1 (hBOOL P_5))) (let ((_let_2 (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)) X_2)))) (let ((_let_3 (hBOOL P_2))) (=> (=> _let_2 (= _let_1 _let_3)) (= (=> _let_2 _let_3) (=> _let_2 _let_1))))))) (forall ((X_a $$unsorted)) (=> (ordere453448008miring X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (let ((_let_4 (hAPP X_a _let_3 _let_2 _let_1))) (=> (or (and (hBOOL (hAPP X_a bool _let_4 B)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) _let_1))) (and (hBOOL (hAPP X_a bool _let_4 A_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)))) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (times_times X_a A_1) B)) _let_1)))))))))) (linordered_idom int) (forall ((N_1 $$unsorted)) (let ((_let_1 (zero_zero nat))) (= (hAPP nat nat (times_times nat _let_1) N_1) _let_1))) (linord1278240602ring_1 real) (linord1278240602ring_1 int) (forall ((K_1 $$unsorted)) (= (number_number_of int K_1) (ti int K_1))) (forall ((X_a $$unsorted)) (=> (comm_monoid_mult X_a) (forall ((A_1 $$unsorted)) (= (ti X_a A_1) (hAPP X_a X_a (times_times X_a A_1) (one_one X_a)))))) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) _let_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (plus_plus X_a A_1) B)) _let_1)))))))))) (forall ((Z_1 $$unsorted)) (= (hAPP nat nat (times_times nat (number_number_of nat (bit0 (bit1 pls)))) Z_1) (hAPP nat nat (plus_plus nat Z_1) Z_1))) (forall ((X_a $$unsorted)) (=> (and (power X_a) (no_zero_divisors X_a) (zero_neq_one X_a) (mult_zero X_a)) (forall ((A_3 $$unsorted) (N $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (= _let_1 (hAPP nat X_a (power_power X_a A_3) N)) (and (not (= N (zero_zero nat))) (= _let_1 (ti X_a A_3)))))))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted)) (=> (hBOOL (hAPP int bool twoSqu658283162sum2sq X_1)) (=> (hBOOL (hAPP int bool twoSqu658283162sum2sq Y_1)) (hBOOL (hAPP int bool twoSqu658283162sum2sq (hAPP int int (times_times int X_1) Y_1)))))) (forall ((Ma $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (times_times nat K))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) K)) (= (= N Ma) (= (hAPP nat nat _let_1 N) (hAPP nat nat _let_1 Ma)))))) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 (ord_less_eq X_a) A_1) _let_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (plus_plus X_a A_1) B)) _let_1)))))))))) (forall ((I_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (minus_minus nat N_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) I_1) N_1)) (= I_1 (hAPP nat nat _let_1 (hAPP nat nat _let_1 I_1)))))) (forall ((Q_1 $$unsorted) (B $$unsorted) (R_1 $$unsorted) (C $$unsorted)) (let ((_let_1 (times_times int B))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (let ((_let_4 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_4) C)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less_eq int) _let_4) R_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 R_1) B)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (hAPP int int (plus_plus 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X_a))) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a _let_1 A_1)) (hAPP X_a X_a _let_1 B))))))))))) (forall ((X_a $$unsorted) (X_c $$unsorted) (B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (hAPP X_a X_c B_1_1 B_2_1) (hAPP X_a X_c (ti (fun X_a X_c) B_1_1) B_2_1))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((V $$unsorted) (W $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (number_number_of X_a V)) (number_number_of X_a W)) (number_number_of X_a (hAPP int int (plus_plus int V) W)))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((Lx $$unsorted) (Ly $$unsorted) (Rx $$unsorted) (Ry $$unsorted)) (let ((_let_1 (times_times X_a Rx))) (let ((_let_2 (times_times X_a (hAPP X_a X_a (times_times X_a Lx) Ly)))) (= (hAPP X_a X_a _let_1 (hAPP X_a X_a _let_2 Ry)) (hAPP X_a X_a _let_2 (hAPP X_a X_a _let_1 Ry)))))))) (forall ((P_2 $$unsorted) (X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_2) Y_2)))) (let ((_let_4 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_2) X_2)))) (=> (and (not _let_4) _let_3) (=> (and _let_4 (not _let_3)) (hBOOL P_2)))))))) (forall ((N $$unsorted) (Ma $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Ma) N)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (zero_zero nat)) (hAPP nat nat (minus_minus nat N) Ma))))))) (forall ((X_a $$unsorted)) (=> (idom X_a) (forall ((C_1 $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (dvd_dvd X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (times_times X_a C_1))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a _let_3 A_3)) (hAPP X_a X_a _let_3 B_2))) (or (= (zero_zero X_a) (ti X_a C_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2)))))))))) (forall ((X_a $$unsorted)) (=> (and (power X_a) (semiring_0 X_a)) (forall ((N_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (hAPP nat X_a (power_power X_a _let_1) N_1))) (let ((_let_3 (zero_zero nat))) (and (=> (= N_1 _let_3) (= (one_one X_a) _let_2)) (=> (not (= _let_3 N_1)) (= _let_2 _let_1))))))))) (forall ((X_a $$unsorted)) (=> (ordered_ring X_a) (forall ((A_3 $$unsorted) (E $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted) (D $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C_1) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (minus_minus X_a B_2) A_3)) E)) D))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a A_3) E)) C_1)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a B_2) E)) D))))))))) (forall ((K $$unsorted) (Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (times_times nat K))) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat _let_3 Ma)) (hAPP nat 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(let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (bit1 K)) (bit0 L))) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) K) L))))) (forall ((X_a $$unsorted)) (=> (ordered_ring X_a) (forall ((C $$unsorted) (B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (times_times X_a C))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 C) (zero_zero X_a))) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a _let_1 A_1)) (hAPP X_a X_a _let_1 B))))))))))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_1) J_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (zero_zero nat)) K_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (times_times nat I_1) K_1)) (hAPP nat nat (times_times nat J_1) K_1)))))))) (forall ((P $$unsorted) (Q $$unsorted)) (or (hBOOL Q) (not (hBOOL (hAPP bool bool (hAPP bool (fun bool bool) fconj P) Q))))) (ring_11004092258visors real) (forall ((X_a $$unsorted)) (=> (comm_monoid_add X_a) (forall ((A_1 $$unsorted)) (= (ti X_a A_1) (hAPP X_a X_a (plus_plus X_a (zero_zero X_a)) A_1))))) (forall ((N_1 $$unsorted)) (= N_1 (hAPP nat nat (plus_plus nat (zero_zero nat)) N_1))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((X_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hAPP X_a X_a (times_times X_a X_1) _let_1) _let_1))))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((Y_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= _let_1 (hAPP X_a X_a (times_times X_a _let_1) Y_1)))))) (not (= min pls)) (forall ((B_1_1 $$unsorted)) (= (quadRes (ti int B_1_1)) (quadRes B_1_1))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (zero_zero X_a)) A_1) (ti X_a A_1))))) (forall ((X_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (multInv P_1 X_1))) (let ((_let_2 (multInv P_1 _let_1))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong X_1 (zero_zero int)) P_1))) (hBOOL (hAPP int bool (zcong _let_2 (hAPP int int (times_times int (hAPP int int (times_times int X_1) _let_1)) _let_2)) P_1)))))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted)) (= (ti X_a A_1) (hAPP X_a X_a (div_mod X_a A_1) (zero_zero X_a)))))) (forall ((B $$unsorted) (A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) _let_1) A_1)) (=> (and (not (hBOOL (hAPP int bool (zcong A_1 _let_1) P_1))) (not (hBOOL (hAPP int bool (zcong B _let_1) P_1)))) (not (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) B) _let_1) P_1)))))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (dvd_dvd int) K_1))) (=> (hBOOL (hAPP int bool _let_1 (hAPP int int (minus_minus int M) N_1))) (=> (hBOOL (hAPP int bool _let_1 N_1)) (hBOOL (hAPP int bool _let_1 M)))))) (forall ((V_1 $$unsorted) (V $$unsorted)) (let ((_let_1 (number_number_of nat V_1))) (let ((_let_2 (number_number_of nat V))) (let ((_let_3 (hAPP nat nat (plus_plus nat _let_2) _let_1))) (let ((_let_4 (ord_less int))) (let ((_let_5 (fun int bool))) (let ((_let_6 (hBOOL (hAPP int bool (hAPP int _let_5 _let_4 V) pls)))) (let ((_let_7 (hBOOL (hAPP int bool (hAPP int _let_5 _let_4 V_1) pls)))) (and (=> (not _let_6) (and (=> _let_7 (= _let_3 _let_2)) (=> (not _let_7) (= _let_3 (number_number_of nat (hAPP int int (plus_plus int V) V_1)))))) (=> _let_6 (= _let_3 _let_1))))))))))) (forall ((N_1 $$unsorted)) (let ((_let_1 (zfact N_1))) (let ((_let_2 (one_one int))) (let ((_let_3 (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) N_1) (zero_zero int))))) (and (=> (not _let_3) (= _let_1 (hAPP int int (times_times int N_1) (zfact (hAPP int int (minus_minus int N_1) _let_2))))) (=> _let_3 (= _let_2 _let_1))))))) (plus nat) (forall ((Z_3 $$unsorted) (Z_1 $$unsorted) (W_2 $$unsorted) (W $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 W_2) W)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) Z_3) Z_1)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP int int (plus_plus int W_2) Z_3)) (hAPP int int (plus_plus int W) Z_1)))))))) (forall ((X_a $$unsorted)) (=> (power X_a) (forall ((A_1 $$unsorted)) (= (one_one X_a) (hAPP nat X_a (power_power X_a A_1) (zero_zero nat)))))) (ab_semigroup_mult nat) (forall ((A_3 $$unsorted)) (let ((_let_1 (one_one nat))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (= (not (= A_3 _let_1)) (and (hBOOL (hAPP nat bool (hAPP nat 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$$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_2) Y_2)))) (= (and _let_3 (not (= X_2 Y_2))) (and (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_2) X_2))) _let_3)))))) (linord893533164strict nat) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (ti int A_1))) (let ((_let_2 (one_one int))) (let ((_let_3 (ord_less int))) (let ((_let_4 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 (zero_zero int)) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_4 _let_3 A_1) P_1)) (=> (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) A_1) _let_2) P_1)) (or (= _let_1 _let_2) (= (hAPP int int (minus_minus int P_1) _let_2) _let_1))))))))))) (forall ((J_1 $$unsorted) (I_1 $$unsorted)) (not (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (hAPP nat nat (plus_plus nat J_1) I_1)) I_1)))) (forall ((X_1 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B)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (hAPP nat nat (div_mod nat M) N_1))) (let ((_let_2 (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) M) N_1)))) (and (=> _let_2 (= M _let_1)) (=> (not _let_2) (= _let_1 (hAPP nat nat (div_mod nat (hAPP nat nat (minus_minus nat M) N_1)) N_1))))))) (forall ((W_1 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less int) W_1))) (= (or (= (ti int Z_2) (ti int W_1)) (hBOOL (hAPP int bool _let_1 Z_2))) (hBOOL (hAPP int bool _let_1 (hAPP int int (plus_plus int Z_2) (one_one int))))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (let ((_let_1 (zcong B_1_1 B_2_1))) (= (ti (fun int bool) _let_1) _let_1))) (forall ((X_a $$unsorted)) (=> (ordered_ring X_a) (forall ((A_3 $$unsorted) (E $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted) (D $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a 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A_1) (hAPP X_a X_a (times_times X_a A_1) _let_1)))))) (cancel146912293up_add real) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (zcong A_1 B) M)) (=> (hBOOL (hAPP int bool (zcong B C) M)) (hBOOL (hAPP int bool (zcong A_1 C) M))))) (forall ((A_3 $$unsorted)) (let ((_let_1 (zero_zero nat))) (= (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (dvd_dvd nat) _let_1) A_3)) (= _let_1 A_3)))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (zero_zero X_a)) (one_one X_a))))) (forall ((B_2 $$unsorted) (A_3 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 A_3) (hAPP int int (minus_minus int P_3) _let_1))) (=> (hBOOL (hAPP _let_3 bool (member int B_2) (wset A_3 P_3))) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_1) B_2))))))))) (forall ((K_1 $$unsorted)) (= (hAPP int int (plus_plus int K_1) pls) (ti int K_1))) (order bool) (forall ((X_a $$unsorted)) (=> (group_add X_a) (forall ((A_1 $$unsorted) (B $$unsorted)) (= (ti X_a A_1) (hAPP X_a X_a (minus_minus X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) B))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (let ((_let_2 (hAPP nat _let_1 (dvd_dvd nat) K_1))) (=> (hBOOL (hAPP nat bool _let_2 (hAPP nat nat (minus_minus nat M) N_1))) (=> (hBOOL (hAPP nat bool _let_2 N_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) N_1) M)) (hBOOL (hAPP nat bool _let_2 M)))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted)) (= (ti X_a X_1) (hAPP nat X_a (power_power X_a X_1) (one_one nat)))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (multInv B_1_1 B_2_1) (multInv (ti int B_1_1) B_2_1))) (forall ((Z_1 $$unsorted)) (= (hAPP int int (times_times int (one_one int)) Z_1) (ti int Z_1))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (power_power X_a B_1_1))) (=> (power X_a) (= _let_1 (ti (fun nat X_a) _let_1))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) M) N_1)) (=> (not (= M N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) M) N_1)))))) (forall ((L_1 $$unsorted)) (let ((_let_1 (minus_minus int min))) (= (bit1 (hAPP int int _let_1 L_1)) (hAPP int int _let_1 (bit0 L_1))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (ab_semigroup_mult X_a) (= (times_times X_a B_1_1) (times_times X_a (ti X_a B_1_1))))) (forall ((X_2 $$unsorted) (P_3 $$unsorted)) (=> (hBOOL (hAPP (fun int bool) bool (member int X_2) (sr P_3))) (= (ti int X_2) (standardRes P_3 X_2)))) (semiring real) (forall ((K $$unsorted) (Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (times_times nat K))) (let ((_let_2 (ord_less_eq nat))) (let ((_let_3 (fun nat bool))) (= (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 (ord_less nat) (zero_zero nat)) K)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 Ma) N))) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 Ma)) (hAPP nat nat _let_1 N)))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted) (B $$unsorted)) (= (hAPP X_a X_a (times_times X_a A_1) B) (hAPP X_a X_a (times_times X_a B) A_1))))) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) _let_33) _let_18)) (forall ((L_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (minus_minus nat L_1))) (let ((_let_2 (ord_less nat))) (let ((_let_3 (fun nat bool))) (let ((_let_4 (hAPP nat _let_3 _let_2 M))) (=> (hBOOL (hAPP nat bool _let_4 N_1)) (=> (hBOOL (hAPP nat bool _let_4 L_1)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 N_1)) (hAPP nat nat _let_1 M)))))))))) (forall ((X_1 $$unsorted) (M $$unsorted) (Y_1 $$unsorted)) (= (hAPP int int (div_mod int (hAPP nat int (power_power int (hAPP int int (div_mod int X_1) M)) Y_1)) M) (hAPP int int (div_mod int (hAPP nat int (power_power int X_1) Y_1)) M))) (forall ((X_a $$unsorted)) (=> (semiri456707255roduct X_a) (forall ((W_1 $$unsorted) (Y_2 $$unsorted) (X_2 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (times_times X_a X_2))) (let ((_let_2 (times_times X_a W_1))) (= (or (= (ti X_a Y_2) (ti X_a Z_2)) (= (ti X_a X_2) (ti X_a W_1))) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_2 Y_2)) (hAPP X_a X_a _let_1 Z_2)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_2 Z_2)) (hAPP X_a X_a _let_1 Y_2))))))))) (forall ((L $$unsorted)) (= (= pls (bit0 L)) (= pls (ti int L)))) (forall ((X_a $$unsorted)) (=> (linordered_ring X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a X_1) X_1)) (hAPP X_a X_a (times_times X_a Y_1) Y_1))))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (wset B_1_1 B_2_1) (wset (ti int B_1_1) B_2_1))) (number_ring int) (= _let_27 (twoSqu1929807760sum2sq (product_Pair int int s _let_6))) (forall ((J_1 $$unsorted) (K_1 $$unsorted) (A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (times_times int (multInv P_1 J_1)))) (=> (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int J_1) K_1) A_1) P_1)) (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int (hAPP int int _let_1 J_1)) K_1) (hAPP int int _let_1 A_1)) P_1))))) (ordered_semiring nat) (forall ((V_1 $$unsorted) (K_1 $$unsorted) (V $$unsorted)) (let ((_let_1 (hAPP nat nat (times_times nat (number_number_of nat V)) (hAPP nat nat (times_times nat (number_number_of nat V_1)) K_1)))) (let ((_let_2 (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) V) pls)))) (and (=> _let_2 (= _let_1 (zero_zero nat))) (=> (not _let_2) (= (hAPP nat nat (times_times nat (number_number_of nat (hAPP int int (times_times int V) V_1))) K_1) _let_1)))))) (hBOOL (hAPP int bool zprime _let_34)) (forall ((X_2 $$unsorted) (Y_2 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (times_times real Z_2))) (let ((_let_2 (ord_less_eq real))) (let ((_let_3 (fun real bool))) (=> (hBOOL (hAPP real bool (hAPP real _let_3 (ord_less real) (zero_zero real)) Z_2)) (= (hBOOL (hAPP real bool (hAPP real _let_3 _let_2 X_2) Y_2)) (hBOOL (hAPP real bool (hAPP real _let_3 _let_2 (hAPP real real _let_1 X_2)) (hAPP real real _let_1 Y_2))))))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (not (= X_1 Y_1)) (=> (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1))))))) (forall ((B_2 $$unsorted) (P_3 $$unsorted) (A_3 $$unsorted)) (let ((_let_1 (ti int B_2))) (let ((_let_2 (member int B_2))) (let ((_let_3 (fun int bool))) (let ((_let_4 (one_one int))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less int) _let_4) A_3)) (=> (not (hBOOL (hAPP _let_3 bool _let_2 (wset (hAPP int int (minus_minus int A_3) _let_4) P_3)))) (=> (hBOOL (hAPP _let_3 bool _let_2 (wset A_3 P_3))) (or (= _let_1 (inv P_3 A_3)) (= _let_1 (ti int A_3))))))))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (bit1 pls))) (let ((_let_2 (bit1 _let_1))) (let ((_let_3 (number_number_of nat _let_2))) (let ((_let_4 (power_power int B))) (let ((_let_5 (number_number_of nat (bit0 _let_1)))) (let ((_let_6 (times_times int (number_number_of int _let_2)))) (let ((_let_7 (power_power int A_1))) (= (hAPP int int (plus_plus int (hAPP int int (plus_plus int (hAPP int int (plus_plus int (hAPP nat int _let_7 _let_3)) (hAPP int int (times_times int (hAPP int int _let_6 (hAPP nat int _let_7 _let_5))) B))) (hAPP int int (times_times int (hAPP int int _let_6 A_1)) (hAPP nat int _let_4 _let_5)))) (hAPP nat int _let_4 _let_3)) (hAPP nat int (power_power int (hAPP int int (plus_plus int A_1) B)) _let_3)))))))))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_1 X_1)) (=> (hBOOL (hAPP int bool _let_1 Y_1)) (hBOOL (hAPP int bool _let_1 (hAPP int int (plus_plus int X_1) Y_1))))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (and (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1)) (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1)))) (not (= Y_1 X_1)))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((Y_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 _let_1 (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool _let_3 X_1)) (=> (hBOOL (hAPP X_a bool _let_3 Y_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 Y_1) (one_one X_a))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a Y_1) X_1)) X_1))))))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)))) (let ((_let_2 (number_number_of nat (bit0 (bit1 pls))))) (=> (= (hAPP nat X_a (power_power X_a X_1) _let_2) (hAPP nat X_a (power_power X_a Y_1) _let_2)) (=> (hBOOL (hAPP X_a bool _let_1 X_1)) (=> (hBOOL (hAPP X_a bool _let_1 Y_1)) (= (ti X_a X_1) (ti X_a Y_1)))))))))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 _let_1 I_1))) (=> (hBOOL (hAPP int bool _let_3 J_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 J_1) K_1)) (hBOOL (hAPP int bool _let_3 K_1)))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (=> (= M (hAPP nat nat (plus_plus nat M) N_1)) (= N_1 (zero_zero nat)))) (forall ((X_a $$unsorted)) (=> (ab_semigroup_mult X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (times_times X_a A_1))) (= (hAPP X_a X_a _let_1 (hAPP X_a X_a (times_times X_a B) C)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a _let_1 B)) C)))))) (forall ((W_1 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) W_1) Z_2)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) W_1) (hAPP int int (minus_minus int Z_2) (one_one int))))))) (ordered_ring real) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (wset B_1_1 B_2_1) (wset B_1_1 (ti int B_2_1)))) (= _let_33 _let_32) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) (hAPP X_a X_a (times_times X_a C) B))) B) (hAPP X_a X_a (div_mod X_a A_1) B))))) (one nat) (forall ((P_2 $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (= (zero_zero nat) K))) (= (hBOOL (hAPP nat bool P_2 (hAPP nat nat (div_mod nat N) K))) (and (=> (not _let_1) (forall ((I $$unsorted) (J $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) J) K)) (=> (= N (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat K) I)) J)) (hBOOL (hAPP nat bool P_2 J)))))) (=> _let_1 (hBOOL (hAPP nat bool P_2 N))))))) (forall ((X_a $$unsorted)) (=> (and (linorder X_a) (dvd X_a) (plus X_a)) (forall ((D $$unsorted) (Sa $$unsorted)) (exists ((Z $$unsorted)) (forall ((X $$unsorted)) (let ((_let_1 (fun X_a bool))) (let ((_let_2 (not (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (dvd_dvd X_a) D) (hAPP X_a X_a (plus_plus X_a X) Sa)))))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_1 (ord_less X_a) Z) X)) (= _let_2 _let_2))))))))) (forall ((X_a $$unsorted)) (=> (and (dvd X_a) (comm_ring X_a)) (forall ((Ta $$unsorted) (D $$unsorted) (D_3 $$unsorted)) (=> (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) D) D_3)) (forall ((X $$unsorted) (K_2 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) D))) (= (not (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (minus_minus X_a X) (hAPP X_a X_a (times_times X_a K_2) D_3))) Ta)))) (not (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (plus_plus X_a X) Ta))))))))))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (times_times nat K_1))) (let ((_let_2 (ord_less_eq nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 I_1) J_1)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 I_1)) (hAPP nat nat _let_1 J_1)))))))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) B)))) (=> (not (= B A_1)) (=> _let_3 (and (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 B) A_1))) _let_3))))))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less X_a) (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B))))))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (bit0 (hAPP int int (minus_minus int K_1) L_1)) (hAPP int int (minus_minus int (bit0 K_1)) (bit0 L_1)))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_1) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 A_1) (hAPP int int (minus_minus int P_1) _let_1))) (not (= (inv P_1 A_1) (ti int A_1)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (or (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) M) N_1)) (= N_1 M)) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) M) N_1))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (number_number_of nat (bit0 (bit1 pls))))) (= (and (= (ti X_a X_2) _let_1) (= _let_1 (ti X_a Y_2))) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_2) _let_2)) (hAPP nat X_a (power_power X_a Y_2) _let_2))) _let_1)))))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((V $$unsorted) (W $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (number_number_of X_a V)) (hAPP X_a X_a (minus_minus X_a (number_number_of X_a W)) C)) (hAPP X_a X_a (minus_minus X_a (number_number_of X_a (hAPP int int (plus_plus int V) W))) C))))) (forall ((N $$unsorted) (Ma $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) (zero_zero nat)) Ma)) (= (hBOOL (hAPP nat bool (hAPP nat _let_1 (dvd_dvd nat) (hAPP nat nat (times_times nat Ma) N)) Ma)) (= N (one_one nat)))))) (forall ((K1 $$unsorted) (K2 $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K1) K2)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K1)) (bit0 K2))))))) (forall ((N_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (dvd_dvd int) N_1) M)) (or (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 N_1) M)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 M) (zero_zero int)))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (number X_a) (= (number_number_of X_a B_1_1) (number_number_of X_a (ti int B_1_1))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((Lx $$unsorted) (Ly $$unsorted) (Rx $$unsorted)) (let ((_let_1 (times_times X_a Lx))) (= (hAPP X_a X_a (times_times X_a (hAPP X_a X_a _let_1 Ly)) Rx) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a _let_1 Rx)) Ly)))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (number_number_of int K)) (number_number_of int L))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)))))) (forall ((K $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) pls)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K)) pls)))))) (= _let_31 _let_27) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 I_1) J_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat I_1) K_1)) (hAPP nat nat (plus_plus nat J_1) K_1))))))) (= _let_10 _let_17) (not (hBOOL (hAPP int bool _let_5 pls))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (N_3 $$unsorted)) (let ((_let_1 (power_power X_a A_1))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) N_1) N_3)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (one_one X_a)) A_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP nat X_a _let_1 N_1)) (hAPP nat X_a _let_1 N_3))))))))))) _let_30 (forall ((Ma $$unsorted) (D $$unsorted)) (= (exists ((Q_2 $$unsorted)) (= (ti int Ma) (hAPP int int (times_times int D) Q_2))) (= (hAPP int int (div_mod int Ma) D) (zero_zero int)))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (dvd_dvd nat) A_1) B)) (or (= (zero_zero nat) B) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) A_1) B)))))) (forall ((B_1_1 $$unsorted)) (let ((_let_1 (zfact B_1_1))) (= _let_1 (ti int _let_1)))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (power X_a) (= (power_power X_a (ti X_a B_1_1)) (power_power X_a B_1_1)))) (forall ((K_1 $$unsorted)) (not (= pls (bit1 K_1)))) (forall ((M $$unsorted) (K_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat M) K_1)) N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 K_1) N_1)))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (plus_plus X_a A_1))) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a _let_1 B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a _let_1 (hAPP X_a X_a (div_mod X_a B) C))) C)))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((N_1 $$unsorted) (M $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less X_a) (one_one X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 M)) (=> (hBOOL (hAPP X_a bool _let_1 N_1)) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (times_times X_a M) N_1))))))))) (forall ((K $$unsorted) (N $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (dvd_dvd nat) K))) (= (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat N) K))) (hBOOL (hAPP nat bool _let_1 N))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a A_1) (number_number_of X_a pls)) (ti X_a A_1))))) (forall ((V_2 $$unsorted)) (= (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) pls) V_2)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) (number_number_of nat V_2))))) (forall ((X_a $$unsorted)) (let ((_let_1 (undefined X_a))) (= (ti X_a _let_1) _let_1))) (forall ((X_a $$unsorted)) (=> (number X_a) (forall ((W_1 $$unsorted) (X_2 $$unsorted)) (let ((_let_1 (number_number_of X_a W_1))) (let ((_let_2 (ti X_a X_2))) (= (= _let_1 _let_2) (= _let_2 _let_1))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (= (hAPP nat nat (plus_plus nat M) N_1) (hAPP nat nat (plus_plus nat N_1) M))) (forall ((N_1 $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N_1) M)) (= M (hAPP nat nat (plus_plus nat (hAPP nat nat (minus_minus nat M) N_1)) N_1)))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (not (= (zero_zero nat) N_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (power_power nat X_1) N_1)) Y_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1))))))) (forall ((V $$unsorted) (W $$unsorted)) (= (hAPP int int (plus_plus int (number_number_of int V)) (number_number_of int W)) (number_number_of int (hAPP int int (plus_plus int V) W)))) (forall ((M $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less_eq nat) I_1))) (=> (hBOOL (hAPP nat bool _let_1 J_1)) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat M) J_1)))))) (forall ((K $$unsorted) (L $$unsorted)) (= (= (bit1 L) (bit1 K)) (= (ti int L) (ti int K)))) (forall ((N_1 $$unsorted)) (not (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) N_1) (zero_zero nat))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (plus_plus X_a A_1))) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a _let_1 B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a _let_1 (hAPP X_a X_a (div_mod X_a B) C))) C)))))) (linorder int) (forall ((X_a $$unsorted) (B_1_1 $$unsorted)) (let ((_let_1 (collect X_a B_1_1))) (= _let_1 (ti (fun X_a bool) _let_1)))) (forall ((Y_1 $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_1 X_1)) (=> (hBOOL (hAPP int bool _let_1 Y_1)) (hBOOL (hAPP int bool _let_1 (hAPP int int (div_mod int X_1) Y_1))))))) (forall ((Q_1 $$unsorted) (B $$unsorted) (R_1 $$unsorted) (C $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (ord_less_eq int))) (let ((_let_3 (fun int bool))) (let ((_let_4 (ord_less int))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_4 _let_1) C)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_4 B) R_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 R_1) _let_1)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (hAPP int int (plus_plus int (hAPP int int (times_times int B) (hAPP int int (div_mod int Q_1) C))) R_1)) _let_1)))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M) N_1)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 N_1) M)) (= M N_1)))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (times_times nat K_1))) (= (hAPP nat nat _let_1 (hAPP nat nat (div_mod nat M) N_1)) (hAPP nat nat (div_mod nat (hAPP nat nat _let_1 M)) (hAPP nat nat _let_1 N_1))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((Ma $$unsorted) (K $$unsorted) (N $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) K))) (=> (hBOOL (hAPP X_a bool _let_1 N)) (= (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (div_mod X_a Ma) N))) (hBOOL (hAPP X_a bool _let_1 Ma)))))))) (forall ((X_a $$unsorted)) (=> (zero_neq_one X_a) (not (= (zero_zero X_a) (one_one X_a))))) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP int int (plus_plus int A_1) B))) (let ((_let_2 (zero_zero int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less int) _let_2) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less_eq int) _let_1) _let_2)) (= (hAPP int int (div_mod int A_1) B) _let_1))))))) (power int) (forall ((X_b $$unsorted) (X_c $$unsorted) (X_a $$unsorted) (B_1_1 $$unsorted) (B_2_1 $$unsorted)) (= (combb X_b X_c X_a B_1_1 B_2_1) (combb X_b X_c X_a B_1_1 (ti (fun X_a X_b) B_2_1)))) (forall ((N_1 $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) (zero_zero nat)) N_1))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (Z_1 $$unsorted)) (let ((_let_1 (power_power int X_1))) (= (hAPP nat int (power_power int (hAPP nat int _let_1 Y_1)) Z_1) (hAPP nat int _let_1 (hAPP nat nat (times_times nat Y_1) Z_1))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (group_add X_a) (= (minus_minus X_a (ti X_a B_1_1)) (minus_minus X_a B_1_1)))) (forall ((N_1 $$unsorted)) (not (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) N_1) (zero_zero nat))))) (forall ((X_a $$unsorted)) (=> (and (linordered_idom X_a) (number_ring X_a)) (forall ((Y_2 $$unsorted)) (= (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) pls) Y_2)) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (zero_zero X_a)) (number_number_of X_a Y_2))))))) (forall ((N_1 $$unsorted)) (let ((_let_1 (zero_zero nat))) (=> (not (= N_1 _let_1)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) _let_1) N_1))))) (forall ((Z_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 (dvd_dvd int) Z_1) N_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) (zero_zero int)) N_1)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) Z_1) N_1)))))) (forall ((K_1 $$unsorted) (M $$unsorted)) (hBOOL (hAPP int bool (zcong K_1 K_1) M))) (semiri456707255roduct real) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((N_1 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (zero_zero X_a)) A_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP nat X_a (power_power X_a A_1) N_1)) (hAPP nat X_a (power_power X_a B) N_1)))))))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (zero_zero int)) A_1)) (=> (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) A_1) _let_1) P_1)) (or (hBOOL (hAPP int bool (zcong A_1 (hAPP int int (minus_minus int P_1) _let_1)) P_1)) (hBOOL (hAPP int bool (zcong A_1 _let_1) P_1)))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (= M (hAPP nat nat (minus_minus nat (hAPP nat nat (plus_plus nat M) N_1)) N_1))) (forall ((N_1 $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) N_1) N_1))) (forall ((N $$unsorted) (K $$unsorted) (Ma $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hAPP nat _let_2 (ord_less_eq nat) K))) (=> (hBOOL (hAPP nat bool _let_3 Ma)) (=> (hBOOL (hAPP nat bool _let_3 N)) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (minus_minus nat Ma) K)) (hAPP nat nat (minus_minus nat N) K))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Ma) N))))))))) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (fun nat bool))) (= (and (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) Ma) N)) (not (= Ma N))) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) Ma) N))))) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) B)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (times_times nat A_1) C)) (hAPP nat nat (times_times nat B) C))))))) (forall ((X_a $$unsorted)) (=> (ring X_a) (forall ((A_3 $$unsorted) (E $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted) (D $$unsorted)) (= (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (minus_minus X_a B_2) A_3)) E)) D) (ti X_a C_1)) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a A_3) E)) C_1) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a B_2) E)) D)))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted)) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) A_1) A_1))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (zero_zero X_a)))) (=> (= (hAPP nat X_a (power_power X_a A_1) N_1) (hAPP nat X_a (power_power X_a B) N_1)) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (=> (hBOOL (hAPP X_a bool _let_1 B)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) N_1)) (= (ti X_a B) (ti X_a A_1)))))))))) (forall ((X_a $$unsorted)) (=> (monoid_add X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (zero_zero X_a)) A_1) (ti X_a A_1))))) (forall ((B_1_1 $$unsorted)) (let ((_let_1 (bit1 B_1_1))) (= (ti int _let_1) _let_1))) (forall ((X_2 $$unsorted) (N $$unsorted) (Y_2 $$unsorted)) (= (= (hAPP int int (div_mod int X_2) N) (hAPP int int (div_mod int Y_2) N)) (hBOOL (hAPP int bool (hAPP int (fun int bool) (dvd_dvd int) N) (hAPP int int (minus_minus int X_2) Y_2))))) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (fun nat bool))) (= (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) Ma) N)) (or (= Ma N) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) Ma) N)))))) (forall ((C_1 $$unsorted) (X_2 $$unsorted) (Ta $$unsorted) (A_3 $$unsorted) (D $$unsorted)) (let ((_let_1 (plus_plus int X_2))) (let ((_let_2 (hAPP int (fun int bool) (dvd_dvd int) A_3))) (=> (hBOOL (hAPP int bool _let_2 D)) (= (hBOOL (hAPP int bool _let_2 (hAPP int int (plus_plus int (hAPP int int _let_1 (hAPP int int (times_times int C_1) D))) Ta))) (hBOOL (hAPP int bool _let_2 (hAPP int int _let_1 Ta)))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (=> (= N_1 M) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) M) N_1)))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1)))) (let ((_let_4 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1)))) (=> (and _let_4 (not _let_3)) (not (and (not _let_4) _let_3)))))))) (forall ((A_1 $$unsorted) (B $$unsorted) (P_1 $$unsorted) (Q_1 $$unsorted)) (let ((_let_1 (times_times int B))) (let ((_let_2 (times_times int A_1))) (= (twoSqu1929807760sum2sq (product_Pair int int (hAPP int int (plus_plus int (hAPP int int _let_2 P_1)) (hAPP int int _let_1 Q_1)) (hAPP int int (minus_minus int (hAPP int int _let_2 Q_1)) (hAPP int int _let_1 P_1)))) (hAPP int int (times_times int (twoSqu1929807760sum2sq (product_Pair int int A_1 B))) (twoSqu1929807760sum2sq (product_Pair int int P_1 Q_1))))))) (forall ((M $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less nat) I_1))) (=> (hBOOL (hAPP nat bool _let_1 J_1)) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat M) J_1)))))) (forall ((X_a $$unsorted)) (=> (and (comm_ring X_a) (dvd X_a)) (forall ((Ta $$unsorted) (D $$unsorted) (D_3 $$unsorted)) (=> (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) D) D_3)) (forall ((X $$unsorted) (K_2 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) D))) (= (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (minus_minus X_a X) (hAPP X_a X_a (times_times X_a K_2) D_3))) Ta))) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (plus_plus X_a X) Ta)))))))))) (number_semiring nat) (forall ((R_1 $$unsorted) (Q_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (fun int bool))) (let ((_let_2 (ord_less int))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 _let_2 (zero_zero int)) A_1)) (=> (= (ti int A_1) (hAPP int int (plus_plus int R_1) (hAPP int int (times_times int A_1) Q_1))) (=> (hBOOL (hAPP int bool (hAPP int _let_1 _let_2 R_1) A_1)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (one_one int)) Q_1)))))))) (forall ((P_1 $$unsorted)) (=> (hBOOL (hAPP int bool zprime P_1)) (hBOOL (hAPP int bool (zcong (zfact (hAPP int int (minus_minus int P_1) (one_one int))) (number_number_of int min)) P_1)))) (forall ((W_1 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (hAPP int int (plus_plus int W_1) (one_one int))) Z_2)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) W_1) Z_2))))) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((A_3 $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (zero_zero X_a))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_3) C_1)) (hAPP X_a X_a (times_times X_a B_2) C_1))) (or (and (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 B_2) A_3)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C_1) _let_3))) (and (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 _let_3) C_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2))))))))))) (forall ((X_a $$unsorted)) (=> (and (linordered_idom X_a) (number_ring X_a)) (forall ((X_2 $$unsorted)) (= (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (number_number_of X_a X_2)) (one_one X_a))) (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) X_2) (bit1 pls))))))) (forall ((V_2 $$unsorted) (V_3 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) V_2))) (= (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) (number_number_of nat V_2)) (number_number_of nat V_3))) (=> (not (hBOOL (hAPP int bool _let_1 V_3))) (hBOOL (hAPP int bool _let_1 pls)))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (power X_a) (= (times_times X_a (ti X_a B_1_1)) (times_times X_a B_1_1)))) (semiring int) (= zprime (ti _let_2 zprime)) (comm_monoid_mult int) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) C))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (number_number_of nat (bit0 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(=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (hAPP int int (plus_plus int (hAPP int int _let_4 Q_4)) R_3)) (hAPP int int (plus_plus int (hAPP int int _let_4 Q_1)) R_1))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) R_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_3 R_3) B)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_3 R_1) B)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 Q_4) Q_1))))))))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K)) (bit1 L))))))) (forall ((Z_1 $$unsorted)) (= (ti int Z_1) (hAPP int int (plus_plus int (zero_zero int)) Z_1))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 (ord_less_eq X_a) 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_let_2) (=> (=> (= Ma N) _let_2) (=> (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 _let_3 N) Ma)) _let_2) _let_2))))))) (hBOOL (hAPP int bool (zcong _let_19 _let_16) _let_9)) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero nat))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (not (and (not (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 A_1) _let_1))) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 _let_1) A_1)))))))) (forall ((K $$unsorted)) (= (= (bit0 K) pls) (= pls (ti int K)))) (forall ((M $$unsorted) (K_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less_eq nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat M) K_1)) N_1)) (not (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M) N_1)) (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 K_1) N_1))))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) (hAPP nat nat (div_mod nat M) N_1)) M))) (forall ((Z_1 $$unsorted) (W $$unsorted)) (let ((_let_1 (ord_less_eq real))) (let ((_let_2 (fun real bool))) (=> (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 Z_1) W)) (=> (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 W) Z_1)) (= W Z_1)))))) (forall ((Z_1 $$unsorted)) (= (ti int Z_1) (hAPP int int (times_times int Z_1) (one_one int)))) (ring_n68954251visors int) (comm_monoid_mult nat) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (power_power nat A_1) N_1)) (hAPP nat nat (power_power nat B) N_1))) (=> (not (= N_1 (zero_zero nat))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 A_1) B))))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((W $$unsorted)) (let ((_let_1 (number_number_of X_a W))) (= (number_number_of X_a (bit0 W)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (plus_plus X_a (zero_zero X_a)) _let_1)) _let_1)))))) _let_28 (forall ((Z_1 $$unsorted)) (= (hAPP real real (times_times real (one_one real)) Z_1) Z_1)) (forall ((X_a $$unsorted)) (=> (ordered_semiring X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) C)))))))))) (forall ((X_a $$unsorted)) (=> (and (linorder X_a) (number X_a)) (forall ((V_2 $$unsorted) (W_1 $$unsorted)) (let ((_let_1 (number_number_of X_a W_1))) (let ((_let_2 (number_number_of X_a V_2))) (let ((_let_3 (fun X_a bool))) (= (not (hBOOL (hAPP X_a bool (hAPP X_a _let_3 (ord_less X_a) _let_1) _let_2))) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 (ord_less_eq X_a) _let_2) _let_1))))))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K)) (bit0 L))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP nat real (power_power real (hAPP real real (plus_plus real X_1) Y_1)) _let_2) (hAPP real real (plus_plus real (hAPP real real (plus_plus real (hAPP nat real (power_power real X_1) _let_2)) (hAPP nat real (power_power real Y_1) _let_2))) (hAPP real real (times_times real (hAPP real real (times_times real (number_number_of real _let_1)) X_1)) Y_1)))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (plus_plus X_a A_1) B)) B) (hAPP X_a X_a (div_mod X_a A_1) B))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (number_number_of int Y_2))) (let ((_let_2 (number_number_of int X_2))) (= (= (zero_zero int) (hAPP int int 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(plus_plus X_a (hAPP X_a X_a _let_1 C)) B) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_1 B)) C)))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((A_1 $$unsorted)) (= (ti X_a A_1) (hAPP X_a X_a (plus_plus X_a (number_number_of X_a pls)) A_1))))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (let ((_let_2 (hAPP nat _let_1 (dvd_dvd nat) K_1))) (=> (hBOOL (hAPP nat bool _let_2 (hAPP nat nat (minus_minus nat M) N_1))) (=> (hBOOL (hAPP nat bool _let_2 M)) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) N_1) M)) (hBOOL (hAPP nat bool _let_2 N_1)))))))) (ring_11004092258visors int) (semiring_1 real) (forall ((X_2 $$unsorted) (P_3 $$unsorted)) (= (hBOOL (hAPP int bool (hAPP int (fun int bool) (dvd_dvd int) P_3) X_2)) (hBOOL (hAPP int bool (zcong X_2 (zero_zero int)) P_3)))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((A_3 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less X_a))) (let 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Z_2))) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) Z_2) W_1))) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) Z_2) W_1))))) (forall ((X_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) P_1)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (standardRes P_1 X_1)) P_1)))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (hAPP int int (plus_plus int (bit0 K_1)) (bit1 L_1)) (bit1 (hAPP int int (plus_plus int K_1) L_1)))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (plus_plus X_a B_1_1))) (=> (cancel_semigroup_add X_a) (= _let_1 (ti (fun X_a X_a) _let_1))))) (forall ((B_1_1 $$unsorted)) (= (d22set B_1_1) (d22set (ti int B_1_1)))) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted) (C_1 $$unsorted)) (let ((_let_1 (times_times X_a C_1))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (zero_zero X_a)) C_1)) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_3) B_2)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a _let_1 A_3)) (hAPP X_a X_a _let_1 B_2))))))))))) (forall ((B $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (dvd_dvd nat) D_2))) (=> (hBOOL (hAPP nat bool _let_1 A_1)) (=> (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat A_1) B))) (hBOOL (hAPP nat bool _let_1 B)))))) (forall ((P_2 $$unsorted) (K $$unsorted) (I_2 $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) K) I_2)) (=> (hBOOL (hAPP int bool P_2 K)) (=> (forall ((I $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) K) I)) (=> (hBOOL (hAPP int bool P_2 I)) (hBOOL (hAPP int bool P_2 (hAPP int int (plus_plus int I) (one_one int))))))) (hBOOL (hAPP int bool P_2 I_2)))))) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 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(forall ((X1 $$unsorted) (X2 $$unsorted) (Ma $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (zero_zero int)) Ma)) (= (hBOOL (hAPP int bool (zcong X1 X2) Ma)) (= (standardRes Ma X2) (standardRes Ma X1))))) (semiring_1 int) (forall ((N $$unsorted) (Ma $$unsorted)) (let ((_let_1 (one_one nat))) (= (= (hAPP nat nat (times_times nat N) Ma) _let_1) (and (= N _let_1) (= _let_1 Ma))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K)) (bit1 L))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)))))) (forall ((X_a $$unsorted)) (=> (ordere1490568538miring X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (times_times X_a C))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (zero_zero X_a)) C)) (hBOOL 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N)))))))) (not (= pls min)) (forall ((X_a $$unsorted)) (=> (and (number_ring X_a) (linordered_idom X_a)) (forall ((X_2 $$unsorted)) (= (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (number_number_of X_a X_2)) (zero_zero X_a))) (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) X_2) pls)))))) (forall ((X_a $$unsorted)) (=> (linord626643107strict X_a) (forall ((V $$unsorted) (U_1 $$unsorted) (Y_1 $$unsorted) (X_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 (ord_less_eq X_a) (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 X_1) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 Y_1) A_1)) (=> (hBOOL (hAPP X_a bool _let_3 U_1)) (=> (hBOOL (hAPP X_a bool _let_3 V)) (=> (= (one_one X_a) (hAPP X_a X_a (plus_plus X_a U_1) V)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a U_1) X_1)) (hAPP X_a X_a (times_times X_a V) Y_1))) A_1))))))))))))) (ordere223160158up_add real) (semiring_div nat) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K)) (bit0 L))))))) (hBOOL (hAPP int bool _let_26 pls)) (forall ((K1 $$unsorted) (K2 $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit1 K1)) (bit1 K2))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K1) K2)))))) (power nat) (hBOOL (hAPP int bool _let_11 (number_number_of int (bit1 _let_7)))) (linord893533164strict real) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) M)) (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int (standardRes M X_1)) (standardRes M Y_1)) (hAPP int int 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(hAPP int int (minus_minus int K) L)) (zero_zero int))))))) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((A_3 $$unsorted) (C_1 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a A_3) C_1)) (hAPP X_a X_a (plus_plus X_a B_2) C_1))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2)))))))) (forall ((N $$unsorted)) (let ((_let_1 (zero_zero nat))) (= (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) _let_1) N)) (not (= _let_1 N))))) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (times_times int A_1))) (= (hAPP int int (div_mod int (hAPP int int _let_1 B)) C) (hAPP int int (div_mod int (hAPP int int _let_1 (hAPP int int (div_mod int B) C))) C)))) (forall ((I_1 $$unsorted) (K_1 $$unsorted) (J_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) K_1) J_1)) (= (hAPP nat nat 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(let ((_let_1 (fun int bool))) (and (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (zero_zero int)) X)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) X) M)) (hBOOL (hAPP int bool (zcong A_1 X) M)) (forall ((Y $$unsorted)) (let ((_let_1 (fun int bool))) (=> (and (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) Y) M)) (hBOOL (hAPP int bool (zcong A_1 Y) M)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (zero_zero int)) Y))) (= (ti int X) (ti int Y)))))))))) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (times_times nat C))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 A_1) B)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 A_1)) (hAPP nat nat _let_1 B)))))))) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less int) (zero_zero int)))) (=> (hBOOL (hAPP int bool _let_1 A_1)) (=> (hBOOL (hAPP int bool _let_1 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((_let_2 (hAPP nat nat (times_times nat M) N_1))) (let ((_let_3 (= _let_1 M))) (and (=> (not _let_3) (= _let_2 (hAPP nat nat (plus_plus nat N_1) (hAPP nat nat (times_times nat (hAPP nat nat (minus_minus nat M) (one_one nat))) N_1)))) (=> _let_3 (= _let_2 _let_1))))))) (= _let_6 _let_24) (forall ((P_2 $$unsorted) (I_2 $$unsorted) (K $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) I_2) K)) (=> (hBOOL (hAPP int bool P_2 K)) (=> (forall ((I $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) I) K)) (=> (hBOOL (hAPP int bool P_2 I)) (hBOOL (hAPP int bool P_2 (hAPP int int (minus_minus int I) (one_one int))))))) (hBOOL (hAPP int bool P_2 I_2)))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hAPP X_a X_a (times_times X_a A_1) _let_1) _let_1))))) (forall ((X_a $$unsorted)) (=> (comm_monoid_mult X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (times_times X_a (one_one X_a)) A_1) (ti X_a A_1))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((Z_1 $$unsorted)) (= (hAPP X_a X_a (times_times X_a Z_1) (number_number_of X_a (bit0 (bit1 pls)))) (hAPP X_a X_a (plus_plus X_a Z_1) Z_1))))) (linord626643107strict int) (forall ((A_1 $$unsorted) (N_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (dvd_dvd int) P_1))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool _let_1 (hAPP nat int (power_power int A_1) N_1))) (hBOOL (hAPP int bool _let_1 A_1)))))) (forall ((M $$unsorted) (N_1 $$unsorted) (K_1 $$unsorted)) (let ((_let_1 (plus_plus nat M))) (= (hAPP nat nat _let_1 (hAPP nat nat (plus_plus nat N_1) K_1)) (hAPP nat nat (plus_plus nat (hAPP nat nat _let_1 N_1)) K_1)))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((N_1 $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less X_a) (one_one X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (=> (hBOOL (hAPP nat bool 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_let_1))) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((A_1 $$unsorted) (C $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (plus_plus X_a A_1) C)) (hAPP X_a X_a (plus_plus X_a B) C))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)))))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (not (= (ti X_a X_1) (ti X_a Y_1))) (=> (not (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 X_1) Y_1))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 Y_1) X_1))))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((X_1 $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (power_power X_a X_1))) (let ((_let_2 (hAPP nat X_a _let_1 N_1))) (= (hAPP nat X_a _let_1 (hAPP nat nat (times_times nat (number_number_of nat (bit0 (bit1 pls)))) N_1)) (hAPP X_a X_a (times_times X_a _let_2) _let_2))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a A_1) (zero_zero X_a)) (ti X_a A_1))))) (forall ((B_1_1 $$unsorted)) (= (sr B_1_1) (sr (ti int B_1_1)))) (= (ti int m) m) (zero nat) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP int int (plus_plus int (hAPP int int (plus_plus int (hAPP nat int (power_power int A_1) _let_2)) (hAPP int int (times_times int (hAPP int int (times_times int (number_number_of int _let_1)) A_1)) B))) (hAPP nat int (power_power int B) _let_2)) (hAPP nat int (power_power int (hAPP int int (plus_plus int A_1) B)) _let_2))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted) (Z_1 $$unsorted)) (let ((_let_1 (plus_plus nat Y_1))) (let ((_let_2 (plus_plus nat X_1))) (= (hAPP nat nat _let_1 (hAPP nat nat _let_2 Z_1)) (hAPP nat nat _let_2 (hAPP nat nat _let_1 Z_1)))))) (no_zero_divisors nat) (forall ((Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)))) (= (and (hBOOL (hAPP nat bool _let_1 Ma)) (hBOOL (hAPP nat bool _let_1 N))) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (times_times nat Ma) N)))))) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (number_number_of int K)) (number_number_of int L))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)))))) (forall ((W_1 $$unsorted) (Z_2 $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) W_1) Z_2)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) W_1) (hAPP int int (plus_plus int Z_2) (one_one int))))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (zero_zero nat)) N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_2 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(hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong K_1 (zero_zero int)) P_1))) (=> (hBOOL (hAPP int bool (zcong _let_1 (hAPP int int (times_times int (hAPP int int (times_times int A_1) (multInv P_1 K_1))) K_1)) P_1)) (hBOOL (hAPP int bool (zcong _let_1 A_1) P_1)))))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (let ((_let_1 (legendre B_1_1 B_2_1))) (= (ti int _let_1) _let_1))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (zero_zero X_a))) (let ((_let_4 (ord_less_eq X_a))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_4 C) D_2)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_4 _let_3) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a 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$$unsorted)) (let ((_let_1 (combb X_b X_c X_a B_1_1 B_2_1))) (= (ti (fun X_a X_c) _let_1) _let_1))) (forall ((Ma $$unsorted) (N $$unsorted)) (= (exists ((K_2 $$unsorted)) (= (hAPP nat nat (plus_plus nat Ma) K_2) N)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) Ma) N)))) (forall ((B_2 $$unsorted) (A_3 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (hAPP int int (minus_minus int P_3) (one_one int)))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_3)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 A_3) _let_1)) (=> (hBOOL (hAPP _let_3 bool (member int B_2) (wset A_3 P_3))) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 B_2) _let_1))))))))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (plus_plus int K_1))) (let ((_let_2 (ord_less_eq int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 I_1) J_1)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (hAPP int int _let_1 I_1)) (hAPP int int _let_1 J_1)))))))) (forall ((C_1 $$unsorted) (D $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted) (Ma $$unsorted)) (let ((_let_1 (times_times int D))) (=> (hBOOL (hAPP int bool (zcong A_3 B_2) Ma)) (= (hBOOL (hAPP int bool (zcong C_1 (hAPP int int _let_1 B_2)) Ma)) (hBOOL (hAPP int bool (zcong C_1 (hAPP int int _let_1 A_3)) Ma)))))) (forall ((A_1 $$unsorted)) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (dvd_dvd nat) A_1) (zero_zero nat)))) (forall ((X_a $$unsorted)) (=> (ordered_semiring X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 _let_1 (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) D_2)) (=> (hBOOL (hAPP X_a bool _let_3 B)) (=> (hBOOL (hAPP X_a bool _let_3 C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) D_2))))))))))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (=> (and (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1))) (not (= X_1 Y_1)))))) (forall ((X_a $$unsorted)) (=> (group_add X_a) (forall ((A_1 $$unsorted)) (= (zero_zero X_a) (hAPP X_a X_a (minus_minus X_a A_1) A_1))))) (group_add int) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (one_one int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (let ((_let_4 (hAPP int _let_3 _let_2 _let_1))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool _let_4 A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 A_1) (hAPP int int (minus_minus int P_1) _let_1))) (hBOOL (hAPP int bool _let_4 (inv P_1 A_1))))))))))) _let_23 (forall ((Z1 $$unsorted) (Z2 $$unsorted) (W $$unsorted)) (= (hAPP int int (times_times int (hAPP int int (plus_plus int Z1) Z2)) W) (hAPP int int (plus_plus int (hAPP int int (times_times int Z1) W)) (hAPP int int (times_times int Z2) W)))) (forall ((A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (hAPP int int (div_mod int A_1) B))) (let ((_let_2 (zero_zero int))) (let ((_let_3 (fun int bool))) (let ((_let_4 (ord_less int))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_4 _let_2) B)) (and (hBOOL (hAPP int bool (hAPP int _let_3 _let_4 _let_1) B)) (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less_eq int) _let_2) _let_1))))))))) (forall ((X_a $$unsorted)) (=> (mult_zero X_a) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hAPP X_a X_a (times_times X_a A_1) _let_1) _let_1))))) (forall ((A_3 $$unsorted)) (let ((_let_1 (zero_zero nat))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (= (not (= _let_1 A_3)) (and (not (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 _let_1) A_3))) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 A_3) _let_1)))))))) (forall ((X_a $$unsorted)) (=> (ordered_semiring X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (hAPP X_a _let_2 _let_1 (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) D_2)) (=> (hBOOL (hAPP X_a bool _let_3 A_1)) (=> (hBOOL (hAPP X_a bool _let_3 C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) D_2))))))))))))) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (let ((_let_1 (inv B_1_1 B_2_1))) (= (ti int _let_1) _let_1))) (forall ((X_a $$unsorted)) (=> (number_semiring X_a) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (bit0 (bit1 pls)))) (let ((_let_2 (number_number_of nat _let_1))) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (plus_plus X_a (hAPP nat X_a (power_power X_a X_1) _let_2)) (hAPP nat X_a (power_power X_a Y_1) _let_2))) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (times_times X_a (number_number_of X_a _let_1)) X_1)) Y_1)) (hAPP nat X_a (power_power X_a (hAPP X_a X_a (plus_plus X_a X_1) Y_1)) _let_2))))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) (number_number_of int (bit1 (bit0 (bit1 pls))))) P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 A_1) P_1)) (= (inv P_1 (inv P_1 A_1)) (ti int A_1))))))))) (forall ((M $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (ord_less_eq nat) I_1))) (=> (hBOOL (hAPP nat bool _let_1 J_1)) (hBOOL (hAPP nat bool _let_1 (hAPP nat nat (plus_plus nat J_1) M)))))) (forall ((K_1 $$unsorted) (A_1 $$unsorted) (J_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (multInv P_1 J_1))) (let ((_let_2 (times_times int _let_1))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (zcong J_1 (zero_zero int)) P_1))) (=> (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int (hAPP int int _let_2 J_1)) K_1) (hAPP int int _let_2 A_1)) P_1)) (hBOOL (hAPP int bool (zcong K_1 (hAPP int int (times_times int A_1) _let_1)) P_1))))))))) (forall ((Z_1 $$unsorted) (W $$unsorted)) (= (hAPP int int (plus_plus int Z_1) W) (hAPP int int (plus_plus int W) Z_1))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((Lx $$unsorted) (Rx $$unsorted) (Ry $$unsorted)) (let ((_let_1 (times_times X_a Lx))) (let ((_let_2 (times_times X_a Rx))) (= (hAPP X_a X_a _let_1 (hAPP X_a X_a _let_2 Ry)) (hAPP X_a X_a _let_2 (hAPP X_a X_a _let_1 Ry)))))))) (hBOOL (hAPP int bool _let_11 _let_17)) (linorder nat) (comm_semiring int) (forall ((T_2 $$unsorted) (T_1 $$unsorted)) (=> (order T_1) (order (fun T_2 T_1)))) (forall ((Q_1 $$unsorted) (B $$unsorted) (R_1 $$unsorted) (C $$unsorted)) (let ((_let_1 (times_times int B))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (let ((_let_4 (zero_zero int))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_4) C)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 B) R_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_3 (ord_less_eq int) R_1) _let_4)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (hAPP int int _let_1 C)) (hAPP int int (plus_plus int (hAPP int int _let_1 (hAPP int int (div_mod int Q_1) C))) R_1))))))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (times_times X_a B_1_1))) (=> (ab_semigroup_mult X_a) (= (ti (fun X_a X_a) _let_1) _let_1)))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (B_3 $$unsorted)) (let ((_let_1 (times_times X_a A_1))) (= (hAPP X_a X_a (minus_minus X_a (hAPP X_a X_a _let_1 B)) (hAPP X_a X_a _let_1 B_3)) (hAPP X_a X_a _let_1 (hAPP X_a X_a (minus_minus X_a B) B_3))))))) (= (hAPP nat nat (plus_plus nat _let_22) _let_22) _let_18) (forall ((W_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 W_1) _let_1)) (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 (bit1 W_1)) _let_1))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (let ((_let_1 (times_times X_a B_1_1))) (=> (semiring X_a) (= (ti (fun X_a X_a) _let_1) _let_1)))) (forall ((P_2 $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (= (and (=> (= _let_1 (ti int K)) (hBOOL (hAPP int bool P_2 N))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 K) _let_1)) (forall ((I $$unsorted) (J $$unsorted)) (let ((_let_1 (fun int bool))) (=> (and (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) K) J)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) J) (zero_zero int))) (= (ti int N) (hAPP int int (plus_plus int (hAPP int int (times_times int K) I)) J))) (hBOOL (hAPP int bool P_2 J)))))) (=> (hBOOL (hAPP int bool (hAPP int _let_3 _let_2 _let_1) K)) (forall ((I $$unsorted) (J $$unsorted)) (let ((_let_1 (fun int bool))) (=> (and (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) (zero_zero int)) J)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) J) K)) (= (hAPP int int (plus_plus int (hAPP int int (times_times int K) I)) J) (ti int N))) (hBOOL (hAPP int bool P_2 J))))))) (hBOOL (hAPP int bool P_2 (hAPP int int (div_mod int N) K)))))))) (forall ((Ma $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (times_times nat K))) (let ((_let_2 (ord_less nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (zero_zero nat)) K)) (= (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 Ma) N)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 Ma)) (hAPP nat nat _let_1 N))))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((M $$unsorted) (A_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a M) (hAPP X_a X_a (times_times X_a A_1) M)) (hAPP X_a X_a (times_times X_a (hAPP X_a X_a (plus_plus X_a A_1) (one_one X_a))) M))))) (ring_div int) (forall ((X_a $$unsorted)) (=> (linord581940658strict X_a) (forall ((X_2 $$unsorted) (Y_2 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (= _let_1 (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a (times_times X_a X_2) X_2)) (hAPP X_a X_a (times_times X_a Y_2) Y_2))) (and (= _let_1 (ti X_a X_2)) (= (ti X_a Y_2) _let_1))))))) (forall ((Z_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) _let_1) Z_1)) (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) _let_1) (hAPP int int (plus_plus int (one_one int)) Z_1))))))) (forall ((X_a $$unsorted)) (=> (linordered_semidom X_a) (forall ((Ma $$unsorted) (N $$unsorted) (A_3 $$unsorted)) (let ((_let_1 (power_power X_a A_3))) (=> (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (one_one X_a)) A_3)) (= (= Ma N) (= (hAPP nat X_a _let_1 N) (hAPP nat X_a _let_1 Ma)))))))) (forall ((U $$unsorted) (Ma $$unsorted) (N $$unsorted) (J_2 $$unsorted) (I_2 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 (ord_less_eq nat) J_2) I_2)) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat (hAPP nat nat (minus_minus nat I_2) J_2)) U)) Ma)) N)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat I_2) U)) Ma)) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat J_2) U)) N)))))))) (hBOOL (hAPP int bool _let_21 _let_20)) (forall ((A_1 $$unsorted)) (= (zero_zero int) (hAPP int int (div_mod int A_1) (number_number_of int min)))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (= (zero_zero X_a) (number_number_of X_a pls)))) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (times_times int (number_number_of int (bit0 (bit1 pls)))))) (let ((_let_2 (plus_plus int (one_one int)))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) (zero_zero int)) A_1)) (= (hAPP int int _let_2 (hAPP int int _let_1 (hAPP int int (div_mod int B) A_1))) (hAPP int int (div_mod int (hAPP int int _let_2 (hAPP int int _let_1 B))) (hAPP int int _let_1 A_1))))))) (linordered_semidom int) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (hAPP X_a X_a (times_times X_a _let_1) A_1) _let_1))))) (forall ((L_1 $$unsorted)) (let ((_let_1 (minus_minus int min))) (= (hAPP int int _let_1 (bit1 L_1)) (bit0 (hAPP int int _let_1 L_1))))) (not (= _let_17 _let_16)) (forall ((M $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (ord_less int))) (let ((_let_3 (fun int bool))) (=> (hBOOL (hAPP 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X_2)) (= (= Y_2 X_2) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_2) Y_2))))))) (forall ((P $$unsorted) (Q $$unsorted)) (or (hBOOL P) (not (hBOOL (hAPP bool bool (hAPP bool (fun bool bool) fconj P) Q))))) (linordered_ring int) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (ord_less X_a) (zero_zero X_a)))) (=> (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (times_times X_a B) A_1))) (=> (hBOOL (hAPP X_a bool _let_1 A_1)) (hBOOL (hAPP X_a bool _let_1 B)))))))) (ab_semigroup_add real) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((Y_2 $$unsorted) (X_2 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (hAPP X_a (fun X_a bool) (ord_less_eq X_a) _let_1))) (=> (hBOOL (hAPP X_a bool _let_2 X_2)) (=> (hBOOL (hAPP X_a bool _let_2 Y_2)) (= (= (hAPP X_a X_a (plus_plus X_a X_2) Y_2) _let_1) (and (= (ti X_a Y_2) _let_1) (= _let_1 (ti X_a X_2))))))))))) _let_15 (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_1 $$unsorted)) (= (hAPP X_a X_a (div_mod X_a A_1) A_1) (zero_zero X_a))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((V $$unsorted) (W $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (number_number_of X_a V)) (number_number_of X_a W)) (number_number_of X_a (hAPP int int (plus_plus int V) W)))))) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (zcong A_1 B) M)) (hBOOL (hAPP int bool (zcong (hAPP int int (plus_plus int A_1) C) (hAPP int int (plus_plus int B) C)) M)))) (forall ((Ma $$unsorted) (K $$unsorted) (F_1 $$unsorted)) (=> (forall ((M_2 $$unsorted) (N_2 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 M_2) N_2)) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat F_1 M_2)) (hAPP nat nat F_1 N_2))))))) (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) (hAPP nat nat (plus_plus nat (hAPP nat nat F_1 Ma)) K)) (hAPP nat nat F_1 (hAPP nat nat (plus_plus nat Ma) K)))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (dvd X_a) (= (times_times X_a (ti X_a B_1_1)) (times_times X_a B_1_1)))) (forall ((M $$unsorted)) (= M (hAPP nat nat (minus_minus nat M) (zero_zero nat)))) (forall ((N_1 $$unsorted) (M $$unsorted) (K_1 $$unsorted)) (let ((_let_1 (times_times nat K_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) (zero_zero nat)) K_1)) (=> (= (hAPP nat nat _let_1 N_1) (hAPP nat nat _let_1 M)) (= N_1 M))))) (forall ((X_2 $$unsorted) (P_3 $$unsorted)) (let ((_let_1 (zero_zero int))) (= (not (hBOOL (hAPP int bool (zcong X_2 _let_1) P_3))) (not (= _let_1 (standardRes P_3 X_2)))))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted)) (= (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) A_3) B_2)) (= (hAPP X_a X_a (div_mod X_a B_2) A_3) (zero_zero X_a)))))) (forall ((A_1 $$unsorted)) (= (zero_zero int) (hAPP int int (div_mod int A_1) A_1))) (forall ((K $$unsorted)) (let ((_let_1 (hAPP int (fun int bool) (ord_less_eq int) min))) (= (hBOOL (hAPP int bool _let_1 (bit1 K))) (hBOOL (hAPP int bool _let_1 K))))) (forall ((X_a $$unsorted)) (=> (number_ring X_a) (forall ((V $$unsorted) (W $$unsorted) (Z_1 $$unsorted)) (= (hAPP X_a X_a (plus_plus X_a (number_number_of X_a (hAPP int int (plus_plus int V) W))) Z_1) (hAPP X_a X_a (plus_plus X_a (number_number_of X_a V)) (hAPP X_a X_a (plus_plus X_a (number_number_of X_a W)) Z_1)))))) (forall ((U_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted) (J_1 $$unsorted) (I_1 $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) J_1) I_1)) (= (hAPP nat nat (minus_minus nat (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat (hAPP nat nat (minus_minus nat I_1) J_1)) U_1)) M)) N_1) (hAPP nat nat (minus_minus nat (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat I_1) U_1)) M)) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat J_1) U_1)) N_1))))) (cancel146912293up_add nat) (forall ((B_1_1 $$unsorted) (B_2_1 $$unsorted)) (let ((_let_1 (product_Pair int int B_1_1 B_2_1))) (= _let_1 (ti (product_prod int int) _let_1)))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (plus_plus X_a A_1))) (= (hAPP X_a X_a _let_1 (hAPP X_a X_a (plus_plus X_a B) C)) (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_1 B)) C)))))) (forall ((N_1 $$unsorted) (B $$unsorted) (A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (dvd_dvd int))) (let ((_let_2 (fun int bool))) (let ((_let_3 (hAPP int _let_2 _let_1 (hAPP nat int (power_power int P_1) N_1)))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (not (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 P_1) A_1))) (=> (hBOOL (hAPP int bool _let_3 (hAPP int int (times_times int A_1) B))) (hBOOL (hAPP int bool _let_3 B))))))))) (forall ((B_1_1 $$unsorted) (X_a $$unsorted)) (=> (semiring X_a) (= (times_times X_a B_1_1) (times_times X_a (ti X_a B_1_1))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1)))) (let ((_let_4 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1)))) (=> (and (not _let_4) _let_3) (not (and _let_4 (not _let_3))))))))) (forall ((X_a $$unsorted)) (=> (ring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a A_1) B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a (hAPP X_a X_a (div_mod X_a A_1) C)) (hAPP X_a X_a (div_mod X_a B) C))) C))))) (forall ((X_a $$unsorted)) (=> (ring_div X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a (hAPP X_a X_a (div_mod X_a A_1) C)) B)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a A_1) B)) C))))) (forall ((M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less nat) M) N_1)) (hBOOL (hAPP nat bool (hAPP nat _let_1 (ord_less_eq nat) M) N_1))))) (forall ((Ma $$unsorted) (N $$unsorted) (K $$unsorted)) (let ((_let_1 (times_times nat K))) (let ((_let_2 (dvd_dvd nat))) (let ((_let_3 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_3 (ord_less nat) (zero_zero nat)) K)) (= (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 Ma) N)) (hBOOL (hAPP nat bool (hAPP nat _let_3 _let_2 (hAPP nat nat _let_1 Ma)) (hAPP nat nat _let_1 N))))))))) (forall ((X_a $$unsorted)) (=> (ordered_ab_group_add X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (= (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_3) B_2)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (minus_minus X_a A_3) B_2)) (zero_zero X_a))))))))) (group_add real) (forall ((Z_1 $$unsorted) (X_1 $$unsorted) (Y_1 $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (zcong X_1 Y_1) M)) (hBOOL (hAPP int bool (zcong (hAPP nat int (power_power int X_1) Z_1) (hAPP nat int (power_power int Y_1) Z_1)) M)))) (forall ((K_1 $$unsorted) (I_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (ord_less_eq real))) (let ((_let_2 (fun real bool))) (let ((_let_3 (hAPP real _let_2 _let_1 I_1))) (=> (hBOOL (hAPP real bool _let_3 J_1)) (=> (hBOOL (hAPP real bool (hAPP real _let_2 _let_1 J_1) K_1)) (hBOOL (hAPP real bool _let_3 K_1)))))))) (forall ((X_a $$unsorted)) (let ((_let_1 (one_one X_a))) (=> (linordered_semidom X_a) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less X_a) (zero_zero X_a)) (hAPP X_a X_a (plus_plus X_a _let_1) _let_1)))))) (forall ((Ma $$unsorted) (K $$unsorted) (N $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (= (or (= (zero_zero nat) K) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Ma) N))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (times_times nat Ma) K)) (hAPP nat nat (times_times nat N) K))))))) (forall ((K $$unsorted) (Ma $$unsorted) (N $$unsorted)) (let ((_let_1 (plus_plus nat K))) (= (= (hAPP nat nat _let_1 Ma) (hAPP nat nat _let_1 N)) (= N Ma)))) (forall ((N_1 $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) N_1) M)) (not (= M N_1)))) (forall ((X_a $$unsorted)) (=> (ab_group_add X_a) (forall ((A_3 $$unsorted) (B_2 $$unsorted)) (= (= (hAPP X_a X_a (minus_minus X_a A_3) B_2) (zero_zero X_a)) (= (ti X_a B_2) (ti X_a A_3)))))) (forall ((K_1 $$unsorted)) (= (hAPP int int (plus_plus int K_1) K_1) (bit0 K_1))) (forall ((M $$unsorted) (X_1 $$unsorted)) (let ((_let_1 (zero_zero int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less_eq int) _let_1) X_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) X_1) M)) (=> (hBOOL (hAPP int bool (zcong X_1 _let_1) M)) (= (ti int X_1) _let_1))))))) (forall ((X_a $$unsorted)) (=> (ring_div X_a) (forall ((B $$unsorted) (B_3 $$unsorted) (A_1 $$unsorted) (C $$unsorted) (A_4 $$unsorted)) (=> (= (hAPP X_a X_a (div_mod X_a A_1) C) (hAPP X_a X_a (div_mod X_a A_4) C)) (=> (= (hAPP X_a X_a (div_mod X_a B_3) C) (hAPP X_a X_a (div_mod X_a B) C)) (= (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a A_4) B_3)) C) (hAPP X_a X_a (div_mod X_a (hAPP X_a X_a (minus_minus X_a A_1) B)) C))))))) (forall ((X_1 $$unsorted) (P_1 $$unsorted)) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (number_number_of int (bit0 (bit1 pls)))) P_1)) (=> (hBOOL (hAPP int bool (zcong X_1 (number_number_of int min)) P_1)) (not (hBOOL (hAPP int bool (zcong X_1 (one_one int)) P_1)))))) (forall ((X_a $$unsorted)) (let ((_let_1 (one_one X_a))) (=> (semiring_1 X_a) (= (hAPP nat X_a (power_power X_a _let_1) (number_number_of nat (bit0 (bit1 pls)))) _let_1)))) (forall ((P_2 $$unsorted) (A_3 $$unsorted) (B_2 $$unsorted)) (= (not (or (and (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less nat) A_3) B_2)) (not (hBOOL (hAPP nat bool P_2 (zero_zero nat))))) (exists ((D_4 $$unsorted)) (and (not (hBOOL (hAPP nat bool P_2 D_4))) (= (hAPP nat nat (plus_plus nat B_2) D_4) A_3))))) (hBOOL (hAPP nat bool P_2 (hAPP nat nat (minus_minus nat A_3) B_2))))) (forall ((K_1 $$unsorted) (L_1 $$unsorted)) (= (bit1 (hAPP int int (minus_minus int K_1) L_1)) (hAPP int int (minus_minus int (bit1 K_1)) (bit0 L_1)))) (linord893533164strict int) (forall ((W $$unsorted) (Z1 $$unsorted) (Z2 $$unsorted)) (let ((_let_1 (times_times int W))) (= (hAPP int int (plus_plus int (hAPP int int _let_1 Z1)) (hAPP int int _let_1 Z2)) (hAPP int int _let_1 (hAPP int int (plus_plus int Z1) Z2))))) (forall ((V_1 $$unsorted) (V $$unsorted)) (let ((_let_1 (hAPP nat nat (times_times nat (number_number_of nat V)) (number_number_of nat V_1)))) (let ((_let_2 (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) V) pls)))) (and (=> (not _let_2) (= (number_number_of nat (hAPP int int (times_times int V) V_1)) _let_1)) (=> _let_2 (= _let_1 (zero_zero nat))))))) (forall ((B_1_1 $$unsorted)) (= (hBOOL B_1_1) (hBOOL (ti bool B_1_1)))) (forall ((K_1 $$unsorted) (N_1 $$unsorted) (M $$unsorted)) (= (hAPP nat nat (div_mod nat M) N_1) (hAPP nat nat (div_mod nat (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat K_1) N_1)) M)) N_1))) (forall ((X_a $$unsorted)) (=> (ordere453448008miring X_a) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (let ((_let_2 (ord_less_eq X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 _let_1) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 B) _let_1)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a (times_times X_a A_1) B)) _let_1)))))))))) (= (ti int pls) pls) (forall ((X_a $$unsorted)) (=> (ordere236663937imp_le X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less_eq X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (plus_plus X_a C))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a _let_3 A_1)) (hAPP X_a X_a _let_3 B))) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B))))))))) (forall ((X_a $$unsorted)) (=> (real_normed_algebra X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (B_3 $$unsorted)) (let ((_let_1 (times_times X_a A_1))) (= (hAPP X_a X_a (plus_plus X_a (hAPP X_a X_a _let_1 B)) (hAPP X_a X_a _let_1 B_3)) (hAPP X_a X_a _let_1 (hAPP X_a X_a (plus_plus X_a B) B_3))))))) (forall ((B $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (times_times int (number_number_of int (bit0 (bit1 pls)))))) (let ((_let_2 (one_one int))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less_eq int) A_1) (zero_zero int))) (= (hAPP int int (minus_minus int (hAPP int int _let_1 (hAPP int int (div_mod int (hAPP int int (plus_plus int B) _let_2)) A_1))) _let_2) (hAPP int int (div_mod int (hAPP int int (plus_plus int _let_2) (hAPP int int _let_1 B))) (hAPP int int _let_1 A_1))))))) (forall ((A_1 $$unsorted) (P_1 $$unsorted)) (let ((_let_1 (ord_less int))) (let ((_let_2 (fun int bool))) (=> (hBOOL (hAPP int bool zprime P_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (zero_zero int)) A_1)) (=> (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 A_1) P_1)) (hBOOL (hAPP int bool (zcong (hAPP int int (times_times int A_1) (inv P_1 A_1)) (one_one int)) P_1)))))))) (number nat) (forall ((K $$unsorted)) (let ((_let_1 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less int) (bit0 K)) min)) (hBOOL (hAPP int bool (hAPP int _let_1 (ord_less_eq int) K) min))))) (forall ((X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (dvd_dvd nat))) (let ((_let_2 (fun nat bool))) (let ((_let_3 (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 X_1) Y_1)))) (=> (and (not (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Y_1) X_1))) _let_3) _let_3))))) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted) (M $$unsorted)) (=> (hBOOL (hAPP int bool (zcong A_1 B) M)) (=> (hBOOL (hAPP int bool (zcong C D_2) M)) (hBOOL (hAPP int bool (zcong (hAPP int int (minus_minus int A_1) C) (hAPP int int (minus_minus int B) D_2)) M))))) (hBOOL (hAPP int bool _let_14 pls)) (forall ((Z1 $$unsorted) (Z2 $$unsorted) (Z3 $$unsorted)) (let ((_let_1 (times_times int Z1))) (= (hAPP int int _let_1 (hAPP int int (times_times int Z2) Z3)) (hAPP int int (times_times int (hAPP int int _let_1 Z2)) Z3)))) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (times_times nat K_1))) (= (hAPP nat nat (minus_minus nat (hAPP nat nat _let_1 M)) (hAPP nat nat _let_1 N_1)) (hAPP nat nat _let_1 (hAPP nat nat (minus_minus nat M) N_1))))) _let_13 (linordered_semidom nat) (forall ((I_1 $$unsorted) (K_1 $$unsorted) (J_1 $$unsorted)) (let ((_let_1 (plus_plus nat I_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) K_1) J_1)) (= (hAPP nat nat (minus_minus nat (hAPP nat nat _let_1 J_1)) K_1) (hAPP nat nat _let_1 (hAPP nat nat (minus_minus nat J_1) K_1)))))) (forall ((X_a $$unsorted)) (=> (and (dvd X_a) (semiring_0 X_a)) (forall ((P_2 $$unsorted) (L $$unsorted)) (= (exists ((X $$unsorted)) (and (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) L) (hAPP X_a X_a (plus_plus X_a X) (zero_zero X_a)))) (hBOOL (hAPP X_a bool P_2 X)))) (exists ((X $$unsorted)) (hBOOL (hAPP X_a bool P_2 (hAPP X_a X_a (times_times X_a L) X)))))))) (ordere779506340up_add nat) (forall ((X_a $$unsorted) (X_c $$unsorted) (X_b $$unsorted) (P $$unsorted) (Q $$unsorted) (R $$unsorted)) (= (hAPP X_a X_c (combc X_a X_b X_c P Q) R) (hAPP X_b X_c (hAPP X_a (fun X_b X_c) P R) Q))) (forall ((L $$unsorted)) (= (= (bit1 L) min) (= min (ti int L)))) (forall ((X_a $$unsorted) (P_2 $$unsorted)) (= (ti (fun X_a bool) P_2) (collect X_a P_2))) (and (hBOOL (hAPP int bool _let_11 s)) (hBOOL (hAPP int bool (zcong s1 s) _let_9)) (hBOOL (hAPP int bool (hAPP int _let_2 _let_4 s) _let_9))) (forall ((X_a $$unsorted)) (=> (semiring_div X_a) (forall ((K_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) K_1))) (=> (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (div_mod X_a M) N_1))) (=> (hBOOL (hAPP X_a bool _let_1 N_1)) (hBOOL (hAPP X_a bool _let_1 M)))))))) (forall ((U $$unsorted) (Ma $$unsorted) (N $$unsorted) (I_2 $$unsorted) (J_2 $$unsorted)) (let ((_let_1 (ord_less nat))) (let ((_let_2 (fun nat bool))) (=> (hBOOL (hAPP nat bool (hAPP nat _let_2 (ord_less_eq nat) I_2) J_2)) (= (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 Ma) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat (hAPP nat nat (minus_minus nat J_2) I_2)) U)) N))) (hBOOL (hAPP nat bool (hAPP nat _let_2 _let_1 (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat I_2) U)) Ma)) (hAPP nat nat (plus_plus nat (hAPP nat nat (times_times nat J_2) U)) N)))))))) (forall ((X_a $$unsorted)) (=> (comm_semiring_1 X_a) (forall ((A_1 $$unsorted) (M $$unsorted) (N_1 $$unsorted)) (let ((_let_1 (power_power X_a A_1))) (=> (hBOOL (hAPP nat bool (hAPP nat (fun nat bool) (ord_less_eq nat) M) N_1)) (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (dvd_dvd X_a) (hAPP nat X_a _let_1 M)) (hAPP nat X_a _let_1 N_1)))))))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (D_2 $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (let ((_let_3 (zero_zero X_a))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 C) D_2)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 _let_3) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) _let_3) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (hAPP X_a X_a (times_times X_a A_1) C)) (hAPP X_a X_a (times_times X_a B) D_2))))))))))))) (forall ((X_a $$unsorted)) (=> (comm_ring_1 X_a) (forall ((Z_1 $$unsorted) (X_1 $$unsorted) (Y_1 $$unsorted)) (let ((_let_1 (hAPP X_a (fun X_a bool) (dvd_dvd X_a) X_1))) (=> (hBOOL (hAPP X_a bool _let_1 Y_1)) (=> (hBOOL (hAPP X_a bool _let_1 Z_1)) (hBOOL (hAPP X_a bool _let_1 (hAPP X_a X_a (minus_minus X_a Y_1) Z_1))))))))) (forall ((X_a $$unsorted)) (=> (linordered_idom X_a) (forall ((A_1 $$unsorted) (K_1 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (=> (hBOOL (hAPP X_a bool (hAPP X_a (fun X_a bool) (ord_less_eq X_a) (hAPP nat X_a (power_power X_a A_1) (hAPP nat nat (times_times nat (number_number_of nat (bit0 (bit1 pls)))) K_1))) _let_1)) (= _let_1 (ti X_a A_1))))))) (semiri456707255roduct nat) (not (hBOOL (hAPP int bool _let_5 min))) (forall ((X_a $$unsorted)) (=> (linord20386208strict X_a) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (times_times X_a C))) (let ((_let_2 (ord_less X_a))) (let ((_let_3 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 A_1) B)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (zero_zero X_a)) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_3 _let_2 (hAPP X_a X_a _let_1 A_1)) (hAPP X_a X_a _let_1 B))))))))))) (forall ((B_1_1 $$unsorted)) (= (twoSqu1929807760sum2sq (ti (product_prod int int) B_1_1)) (twoSqu1929807760sum2sq B_1_1))) (hBOOL (hAPP int bool _let_3 min)) (forall ((X_b $$unsorted)) (=> (and (ring X_b) (number X_b)) (forall ((A_1 $$unsorted) (B $$unsorted) (V $$unsorted)) (let ((_let_1 (number_number_of X_b V))) (= (hAPP X_b X_b (minus_minus X_b (hAPP X_b X_b (times_times X_b A_1) _let_1)) (hAPP X_b X_b (times_times X_b B) _let_1)) (hAPP X_b X_b (times_times X_b (hAPP X_b X_b (minus_minus X_b A_1) B)) _let_1)))))) (forall ((N $$unsorted) (Ma $$unsorted)) (let ((_let_1 (one_one int))) (=> (hBOOL (hAPP int bool (hAPP int (fun int bool) (ord_less int) (zero_zero int)) Ma)) (= (= _let_1 (hAPP int int (times_times int Ma) N)) (and (= _let_1 (ti int Ma)) (= _let_1 (ti int N))))))) (forall ((B_2 $$unsorted) (P_3 $$unsorted) (A_3 $$unsorted)) (let ((_let_1 (member int B_2))) (let ((_let_2 (fun int bool))) (let ((_let_3 (one_one int))) (=> (hBOOL (hAPP int bool (hAPP int _let_2 (ord_less int) _let_3) A_3)) (=> (hBOOL (hAPP _let_2 bool _let_1 (wset (hAPP int int (minus_minus int A_3) _let_3) P_3))) (hBOOL (hAPP _let_2 bool _let_1 (wset A_3 P_3))))))))) (forall ((X_a $$unsorted)) (=> (cancel_semigroup_add X_a) (forall ((A_1 $$unsorted) (B $$unsorted) (C $$unsorted)) (let ((_let_1 (plus_plus X_a A_1))) (=> (= (hAPP X_a X_a _let_1 B) (hAPP X_a X_a _let_1 C)) (= (ti X_a C) (ti X_a B))))))) (forall ((K $$unsorted)) (= (= min (bit1 K)) (= min (ti int K)))) (ordere1490568538miring int) (forall ((K $$unsorted) (L $$unsorted)) (let ((_let_1 (ord_less_eq int))) (let ((_let_2 (fun int bool))) (= (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 (bit0 K)) (bit1 L))) (hBOOL (hAPP int bool (hAPP int _let_2 _let_1 K) L)))))) (forall ((X_a $$unsorted)) (=> (ordere216010020id_add X_a) (forall ((B $$unsorted) (C $$unsorted) (A_1 $$unsorted)) (let ((_let_1 (ord_less X_a))) (let ((_let_2 (fun X_a bool))) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 (zero_zero X_a)) A_1)) (=> (hBOOL (hAPP X_a bool (hAPP X_a _let_2 (ord_less_eq X_a) B) C)) (hBOOL (hAPP X_a bool (hAPP X_a _let_2 _let_1 B) (hAPP X_a X_a (plus_plus X_a A_1) C)))))))))) (forall ((X_a $$unsorted)) (=> (ring_11004092258visors X_a) (forall ((A_3 $$unsorted)) (let ((_let_1 (zero_zero X_a))) (= (= _let_1 (hAPP nat X_a (power_power X_a A_3) (number_number_of nat (bit0 (bit1 pls))))) (= _let_1 (ti X_a A_3))))))) (forall ((C $$unsorted) (A_1 $$unsorted) (B $$unsorted)) (let ((_let_1 (hAPP nat (fun nat bool) (dvd_dvd nat) A_1))) (=> (hBOOL (hAPP nat bool _let_1 B)) (=> (= B C) (hBOOL (hAPP nat bool _let_1 C)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) 2.03/2.23 % SZS output end Proof for theBenchmark 2.03/2.23 EOF