0.12/0.12	% Problem    : theBenchmark.p : TPTP v0.0.0. Released v0.0.0.
0.12/0.13	% Command    : do_cvc5 %s %d
0.13/0.35	% Computer   : n016.cluster.edu
0.13/0.35	% Model      : x86_64 x86_64
0.13/0.35	% CPU        : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
0.13/0.35	% Memory     : 8042.1875MB
0.13/0.35	% OS         : Linux 3.10.0-693.el7.x86_64
0.13/0.35	% CPULimit   : 1200
0.13/0.35	% WCLimit    : 120
0.13/0.35	% DateTime   : Tue Jul 13 10:37:08 EDT 2021
0.13/0.35	% CPUTime    : 
0.20/0.50	%----THF division
0.20/0.51	------- cvc5-thf casc 28 : /export/starexec/sandbox2/benchmark/theBenchmark.p at 120...
0.20/0.51	--- Run --ho-elim --full-saturate-quant at 10...
0.34/0.93	% SZS status Theorem for theBenchmark
0.34/0.94	% SZS output start Proof for theBenchmark
0.34/0.94	(proof
0.34/0.94	(let ((_let_1 (product_snd_a_a x))) (let ((_let_2 (mult_a_ring_ext_a_b r))) (let ((_let_3 (_let_2 _let_1))) (let ((_let_4 (_let_3 r2))) (let ((_let_5 (product_snd_a_a y))) (let ((_let_6 (_let_2 t))) (let ((_let_7 ((_let_2 (_let_6 s3)) _let_5))) (let ((_let_8 (_let_2 _let_7))) (let ((_let_9 (_let_2 (_let_8 t2)))) (let ((_let_10 (_let_9 _let_4))) (let ((_let_11 (_let_2 s3))) (let ((_let_12 (_let_11 r2))) (let ((_let_13 (_let_3 _let_5))) (let ((_let_14 (_let_2 _let_13))) (let ((_let_15 (_let_14 _let_12))) (let ((_let_16 (_let_2 t2))) (let ((_let_17 (_let_16 t))) (let ((_let_18 (_let_2 _let_17))) (let ((_let_19 (_let_18 _let_15))) (let ((_let_20 (_let_2 s2))) (let ((_let_21 (_let_20 r3))) (let ((_let_22 (_let_2 (_let_18 _let_13)))) (let ((_let_23 (add_a_b r))) (let ((_let_24 (_let_14 ((_let_23 _let_12) _let_21)))) (let ((_let_25 (partia1066395285xt_a_b r))) (let ((_let_26 (product_fst_a_a x))) (let ((_let_27 (product_fst_a_a y))) (let ((_let_28 (_let_3 _let_27))) (let ((_let_29 (_let_20 s3))) (let ((_let_30 (_let_2 _let_29))) (let ((_let_31 (_let_30 _let_28))) (let ((_let_32 (_let_14 _let_21))) (let ((_let_33 (a_minus_a_b r))) (let ((_let_34 ((_let_33 _let_32) _let_31))) (let ((_let_35 (_let_18 _let_34))) (let ((_let_36 (_let_2 _let_5))) (let ((_let_37 (_let_36 _let_26))) (let ((_let_38 (_let_30 _let_37))) (let ((_let_39 ((_let_33 _let_15) _let_38))) (let ((_let_40 (_let_18 _let_39))) (let ((_let_41 ((_let_23 _let_40) _let_35))) (let ((_let_42 (_let_18 ((_let_23 _let_39) _let_34)))) (let ((_let_43 (member_a _let_17))) (let ((_let_44 (_let_43 _let_25))) (let ((_let_45 ((member_a _let_34) _let_25))) (let ((_let_46 ((member_a _let_39) _let_25))) (let ((_let_47 (=> _let_46 (=> _let_45 (=> _let_44 (= _let_42 _let_41)))))) (let ((_let_48 (_let_23 _let_15))) (let ((_let_49 (_let_11 _let_27))) (let ((_let_50 (_let_36 r3))) (let ((_let_51 ((_let_33 (_let_6 _let_50)) (_let_6 _let_49)))) (let ((_let_52 ((_let_2 (_let_16 s2)) _let_1))) (let ((_let_53 (_let_2 _let_52))) (let ((_let_54 (_let_6 ((_let_33 _let_50) _let_49)))) (let ((_let_55 (_let_53 _let_54))) (let ((_let_56 (_let_18 _let_38))) (let ((_let_57 ((_let_33 _let_19) _let_56))) (let ((_let_58 (_let_20 _let_26))) (let ((_let_59 (_let_16 ((_let_33 _let_4) _let_58)))) (let ((_let_60 (_let_8 _let_59))) (let ((_let_61 ((_let_23 _let_37) _let_28))) (let ((_let_62 (_let_18 _let_31))) (let ((_let_63 (_let_18 _let_32))) (let ((_let_64 ((_let_33 _let_63) _let_62))) (let ((_let_65 (zero_a_b r))) (let ((_let_66 (a_inv_a_b r))) (let ((_let_67 (member_a t))) (let ((_let_68 (= a_minus_a_b (lambda ((R2 partia1833973666xt_a_b) (X2 a) (Y5 a)) (((add_a_b R2) X2) ((a_inv_a_b R2) Y5)))))) (let ((_let_69 (_let_9 _let_58))) (let ((_let_70 (member_a t2))) (let ((_let_71 (_let_70 _let_25))) (let ((_let_72 (_let_23 _let_57))) (let ((_let_73 ((_let_23 _let_60) _let_55))) (let ((_let_74 (_let_2 (_let_53 t)))) (let ((_let_75 (= ord_less_eq_set_a (lambda ((A4 set_a) (B4 set_a)) (forall ((X2 a)) (let ((_let_1 (member_a X2))) (=> (_let_1 A4) (_let_1 B4)))))))) (let ((_let_76 (semiring_a_b r))) (let ((_let_77 (_let_67 _let_25))) (let ((_let_78 (_let_2 (_let_18 _let_29)))) (let ((_let_79 (= minus_minus_int (lambda ((A3 int) (B2 int)) ((plus_plus_int A3) (uminus_uminus_int B2)))))) (let ((_let_80 ((_let_33 _let_10) _let_69))) (let ((_let_81 (= _let_73 _let_41))) (let ((_let_82 (forall ((R partia1833973666xt_a_b) (X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) ((member_a (((mult_a_ring_ext_a_b R) X) Y)) _let_1)))))))) (let ((_let_83 (not (= _let_73 _let_42)))) (let ((_let_84 (forall ((R partia1833973666xt_a_b) (X a) (Y a)) (let ((_let_1 (ho_28 k_27 R))) (or (not (ho_37 k_36 R)) (not (ho_16 (ho_15 k_14 X) _let_1)) (not (ho_16 (ho_15 k_14 Y) _let_1)) (ho_16 (ho_15 k_14 (ho_32 (ho_31 (ho_30 k_29 R) X) Y)) _let_1)))))) (let ((_let_85 (ho_28 k_27 r))) (let ((_let_86 (ho_30 k_29 r))) (let ((_let_87 (ho_31 _let_86 t2))) (let ((_let_88 (ho_32 _let_87 t))) (let ((_let_89 (ho_16 (ho_15 k_14 _let_88) _let_85))) (let ((_let_90 (ho_16 (ho_15 k_14 t) _let_85))) (let ((_let_91 (not _let_90))) (let ((_let_92 (ho_16 (ho_15 k_14 t2) _let_85))) (let ((_let_93 (not _let_92))) (let ((_let_94 (ho_37 k_36 r))) (let ((_let_95 (not _let_94))) (let ((_let_96 (or _let_95 _let_93 _let_91 _let_89))) (let ((_let_97 (EQ_RESOLVE (ASSUME |:args| (_let_82)) (TRANS (MACRO_SR_EQ_INTRO |:args| (_let_82 7 12)) (PREPROCESS |:args| ((= (forall ((R partia1833973666xt_a_b) (X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b R))) (or (not (semiring_a_b R)) (not ((member_a X) _let_1)) (not ((member_a Y) _let_1)) ((member_a (((mult_a_ring_ext_a_b R) X) Y)) _let_1)))) _let_84))))))) (let ((_let_98 (not _let_96))) (let ((_let_99 (ho_11 k_12 y))) (let ((_let_100 (ho_11 k_10 x))) (let ((_let_101 (ho_31 _let_86 _let_100))) (let ((_let_102 (ho_31 _let_86 s2))) (let ((_let_103 (ho_31 _let_86 (ho_32 _let_102 s3)))) (let ((_let_104 (ho_53 k_52 r))) (let ((_let_105 (ho_11 k_10 y))) (let ((_let_106 (ho_31 _let_86 (ho_32 _let_101 _let_105)))) (let ((_let_107 (ho_30 k_33 r))) (let ((_let_108 (ho_32 (ho_31 _let_107 (ho_32 _let_106 (ho_32 _let_102 r3))) (ho_32 _let_104 (ho_32 _let_103 (ho_32 _let_101 _let_99)))))) (let ((_let_109 (ho_31 _let_86 _let_88))) (let ((_let_110 (ho_11 k_12 x))) (let ((_let_111 (ho_31 _let_86 _let_105))) (let ((_let_112 (ho_31 _let_86 s3))) (let ((_let_113 (ho_32 (ho_31 _let_107 (ho_32 _let_106 (ho_32 _let_112 r2))) (ho_32 _let_104 (ho_32 _let_103 (ho_32 _let_111 _let_110)))))) (let ((_let_114 (ho_32 (ho_31 _let_107 (ho_32 _let_109 _let_113)) (ho_32 _let_109 _let_108)))) (let ((_let_115 (ho_32 _let_109 (ho_32 (ho_31 _let_107 _let_113) _let_108)))) (let ((_let_116 (= _let_115 _let_114))) (let ((_let_117 (=> _let_89 _let_116))) (let ((_let_118 (not _let_89))) (let ((_let_119 (ho_16 (ho_15 k_14 _let_108) _let_85))) (let ((_let_120 (=> _let_119 _let_117))) (let ((_let_121 (ho_16 (ho_15 k_14 _let_113) _let_85))) (let ((_let_122 (_let_48 (_let_66 _let_38)))) (let ((_let_123 ((member_a _let_122) _let_25))) (let ((_let_124 (ASSUME |:args| (_let_79)))) (let ((_let_125 (EQ_RESOLVE (ASSUME |:args| (_let_75)) (MACRO_SR_EQ_INTRO |:args| (_let_75 7 12))))) (let ((_let_126 (ASSUME |:args| (_let_68)))) (let ((_let_127 ((_let_23 _let_32) (_let_66 _let_31)))) (let ((_let_128 ((_let_23 (_let_18 _let_122)) (_let_18 _let_127)))) (let ((_let_129 (_let_18 ((_let_23 _let_122) _let_127)))) (let ((_let_130 ((member_a _let_127) _let_25))) (let ((_let_131 (ho_31 _let_86 t))) (let ((_let_132 (ho_32 (ho_31 _let_107 (ho_32 (ho_31 _let_86 (ho_32 (ho_31 _let_86 (ho_32 _let_131 s3)) _let_105)) (ho_32 _let_87 (ho_32 (ho_31 _let_107 (ho_32 _let_101 r2)) (ho_32 _let_104 (ho_32 _let_102 _let_110)))))) (ho_32 (ho_31 _let_86 (ho_32 (ho_31 _let_86 (ho_32 _let_87 s2)) _let_100)) (ho_32 _let_131 (ho_32 (ho_31 _let_107 (ho_32 _let_111 r3)) (ho_32 _let_104 (ho_32 _let_112 _let_99)))))))) (let ((_let_133 (= _let_132 _let_114))) (let ((_let_134 (= _let_115 _let_132))) (let ((_let_135 (not _let_116))) (let ((_let_136 ((_let_23 (_let_8 (_let_16 ((_let_23 _let_4) (_let_66 _let_58))))) (_let_53 (_let_6 ((_let_23 _let_50) (_let_66 _let_49))))))) (let ((_let_137 (EQ_RESOLVE (ASSUME |:args| (_let_81)) (TRANS (MACRO_SR_EQ_INTRO _let_126 _let_125 _let_124 |:args| (_let_81 7 12)) (PREPROCESS |:args| ((= (= _let_136 _let_128) _let_133))))))) (let ((_let_138 (not _let_134))) (let ((_let_139 (EQ_RESOLVE (ASSUME |:args| (_let_83)) (TRANS (MACRO_SR_EQ_INTRO _let_126 _let_125 _let_124 |:args| (_let_83 7 12)) (PREPROCESS |:args| ((= (not (= _let_136 _let_129)) _let_138))))))) (let ((_let_140 (and _let_138 _let_133))) (SCOPE (MACRO_RESOLUTION_TRUST (IMPLIES_ELIM (SCOPE (INSTANTIATE _let_97 |:args| (r t2 t)) |:args| (_let_84))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_OR_POS |:args| (_let_96)) |:args| ((or _let_89 _let_91 _let_95 _let_93 _let_98))) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_IMPLIES_POS |:args| (_let_117)) |:args| ((or _let_116 _let_118 (not _let_117)))) (MACRO_RESOLUTION_TRUST (EQ_RESOLVE (RESOLUTION (CNF_AND_NEG |:args| (_let_140)) (IMPLIES_ELIM (SCOPE (FALSE_ELIM (TRANS (CONG (REFL |:args| (_let_115)) (SYMM _let_137) |:args| (6)) (FALSE_INTRO _let_139))) |:args| (_let_138 _let_133))) |:args| (true _let_140)) (CONG (MACRO_SR_PRED_INTRO |:args| ((= (not _let_138) _let_134))) (REFL |:args| ((not _let_133))) (REFL |:args| (_let_135)) |:args| (20))) _let_139 _let_137 |:args| (_let_135 true _let_134 false _let_133)) (MACRO_RESOLUTION_TRUST (REORDERING (CNF_IMPLIES_POS |:args| (_let_120)) |:args| ((or _let_117 (not _let_119) (not _let_120)))) (EQ_RESOLVE (ASSUME |:args| (_let_45)) (TRANS (MACRO_SR_EQ_INTRO _let_126 _let_125 _let_124 |:args| (_let_45 7 12)) (PREPROCESS |:args| ((= _let_130 _let_119))))) (MACRO_RESOLUTION_TRUST (REORDERING (IMPLIES_ELIM (EQ_RESOLVE (ASSUME |:args| (_let_47)) (TRANS (MACRO_SR_EQ_INTRO _let_126 _let_125 _let_124 |:args| (_let_47 7 12)) (PREPROCESS |:args| ((= (=> _let_123 (=> _let_130 (=> _let_44 (= _let_129 _let_128)))) (=> _let_121 _let_120))))))) |:args| ((or _let_120 (not _let_121)))) (EQ_RESOLVE (ASSUME |:args| (_let_46)) (TRANS (MACRO_SR_EQ_INTRO _let_126 _let_125 _let_124 |:args| (_let_46 7 12)) (PREPROCESS |:args| ((= _let_123 _let_121))))) |:args| (_let_120 false _let_121)) |:args| (_let_117 false _let_119 false _let_120)) |:args| (_let_118 true _let_116 false _let_117)) (EQ_RESOLVE (ASSUME |:args| (_let_77)) (PREPROCESS |:args| ((= _let_77 _let_90)))) (EQ_RESOLVE (ASSUME |:args| (_let_76)) (PREPROCESS |:args| ((= _let_76 _let_94)))) (EQ_RESOLVE (ASSUME |:args| (_let_71)) (PREPROCESS |:args| ((= _let_71 _let_92)))) |:args| (_let_98 true _let_89 false _let_90 false _let_94 false _let_92)) _let_97 |:args| (false true _let_96 false _let_84)) |:args| (((member_a _let_58) _let_25) (forall ((A int)) (= (uminus_uminus_int (uminus_uminus_int A)) A)) ((member_a _let_29) _let_25) ((member_a _let_52) _let_25) (forall ((X a) (Y a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (= ((_let_1 X) Y) ((_let_1 Y) X))))))) (forall ((A2 set_a) (B3 set_a) (X a)) (let ((_let_1 (member_a X))) (=> ((ord_less_eq_set_a A2) B3) (=> (_let_1 A2) (_let_1 B3))))) (forall ((X a) (Y a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (= ((_let_1 X) Y) ((_let_1 Y) X))))))) ((member_a _let_65) _let_25) (forall ((R partia1833973666xt_a_b) (X a) (Y a)) (let ((_let_1 (add_a_b R))) (let ((_let_2 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (= ((_let_1 X) Y) ((_let_1 Y) X)))))))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (_let_1 Y))) (let ((_let_4 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_4) (=> ((member_a Y) _let_4) (=> ((member_a Z) _let_4) (= (_let_2 (_let_3 Z)) (_let_3 (_let_2 Z))))))))))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (=> ((member_a Z) _let_2) (= (= ((_let_1 Y) X) ((_let_1 Z) X)) (= Y Z)))))))) (forall ((R partia1833973666xt_a_b) (X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) ((member_a (((add_a_b R) X) Y)) _let_1)))))) (forall ((X a) (A a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a A) _let_1) (= (= (((add_a_b r) X) A) X) (= A (zero_a_b r))))))) (= plus_plus_int (lambda ((A3 int) (B2 int)) ((plus_plus_int B2) A3))) (forall ((R partia1833973666xt_a_b) (S2 set_a)) (=> ((locali807230110ng_a_b R) S2) ((locali880295127ng_a_b R) S2))) (forall ((X a) (Y a) (I int)) (let ((_let_1 ((add_pow_a_b_int r) I))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= (_let_1 ((_let_2 X) Y)) ((_let_2 (_let_1 X)) (_let_1 Y))))))))) (forall ((X a) (A a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a A) _let_1) (= (= (((add_a_b r) A) X) X) (= A (zero_a_b r))))))) (forall ((H set_a) (K2 set_a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((ord_less_eq_set_a H) _let_1) (=> ((ord_less_eq_set_a K2) _let_1) ((ord_less_eq_set_a (((set_add_a_b r) H) K2)) _let_1))))) (forall ((A2 set_a)) (= ((minus_minus_set_a A2) bot_bot_set_a) A2)) (forall ((A int) (B int) (D int) (C int)) (=> ((ord_less_eq_int A) B) (=> ((ord_less_eq_int D) C) ((ord_less_eq_int ((minus_minus_int A) C)) ((minus_minus_int B) D))))) ((member_a _let_35) _let_25) (forall ((I int) (K int) (N int) (J int)) (let ((_let_1 ((ord_less_eq_int N) ((plus_plus_int J) K)))) (let ((_let_2 ((ord_less_eq_int ((plus_plus_int I) K)) N))) (=> _let_2 (=> _let_1 (=> _let_2 (=> _let_1 ((ord_less_eq_int ((minus_minus_int N) K)) J)))))))) (forall ((X a) (A a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a A) _let_1) (= (= X (((add_a_b r) X) A)) (= A (zero_a_b r))))))) (forall ((R partia1833973666xt_a_b)) (=> (semiring_a_b R) ((member_a (zero_a_b R)) (partia1066395285xt_a_b R)))) (forall ((P (-> a Bool))) (= (= bot_bot_set_a (collect_a P)) (forall ((X2 a)) (not (P X2))))) (forall ((A int) (B int)) (= ((ord_less_eq_int (uminus_uminus_int A)) B) ((ord_less_eq_int (uminus_uminus_int B)) A))) (forall ((C int) (A int) (B int)) (let ((_let_1 (plus_plus_int C))) (=> ((ord_less_eq_int (_let_1 A)) (_let_1 B)) ((ord_less_eq_int A) B)))) ((member_a _let_5) _let_25) _let_46 (forall ((A a) (B a) (C a)) (let ((_let_1 (a_minus_a_b r))) (let ((_let_2 (_let_1 C))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_3) (=> ((member_a B) _let_3) (=> ((member_a C) _let_3) (= ((_let_1 (_let_2 A)) B) (_let_2 (((add_a_b r) A) B)))))))))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> (= ((_let_1 X) Y) Z) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (=> ((member_a Z) _let_2) (= ((_let_1 ((a_inv_a_b r) X)) Z) Y)))))))) (forall ((A2 set_a)) (= (forall ((X2 a)) (not ((member_a X2) A2))) (= A2 bot_bot_set_a))) _let_83 (not (forall ((T2 a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (=> ((member_a T2) s) (not (= ((_let_1 T2) (((a_minus_a_b r) ((_let_1 (product_snd_a_a y)) r3)) ((_let_1 s3) (product_fst_a_a y)))) (zero_a_b r))))))) (forall ((A2 int) (K int) (A int) (B int)) (let ((_let_1 (plus_plus_int K))) (=> (= A2 (_let_1 A)) (= ((plus_plus_int A2) B) (_let_1 ((plus_plus_int A) B)))))) (forall ((P2 product_prod_a_a) (Q2 product_prod_a_a)) (=> (= (product_fst_a_a P2) (product_fst_a_a Q2)) (=> (= (product_snd_a_a P2) (product_snd_a_a Q2)) (= P2 Q2)))) (forall ((X a) (Y a) (I int)) (let ((_let_1 ((add_pow_a_b_int r) I))) (let ((_let_2 (add_a_b r))) (let ((_let_3 ((_let_2 X) Y))) (let ((_let_4 (partia1066395285xt_a_b r))) (=> (= _let_3 ((_let_2 Y) X)) (=> ((member_a X) _let_4) (=> ((member_a Y) _let_4) (= (_let_1 _let_3) ((_let_2 (_let_1 X)) (_let_1 Y))))))))))) (forall ((B int) (A int) (C int)) (let ((_let_1 (minus_minus_int C))) (=> ((ord_less_eq_int B) A) ((ord_less_eq_int (_let_1 A)) (_let_1 B))))) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((add_a_b r) X) (zero_a_b r)) X))) (forall ((Y a) (X a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> (= (((add_a_b r) Y) X) (zero_a_b r)) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) (= ((a_inv_a_b r) X) Y)))))) (forall ((X a) (Y a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (= ((_let_1 ((a_inv_a_b r) X)) ((_let_1 X) Y)) Y)))))) (forall ((A int) (B int) (C int)) (=> ((ord_less_eq_int A) B) ((ord_less_eq_int ((plus_plus_int A) C)) ((plus_plus_int B) C)))) (_let_70 s) (forall ((H set_a)) (=> ((ord_less_eq_set_a H) (partia1066395285xt_a_b r)) (=> (not (= H bot_bot_set_a)) (=> (forall ((X3 a)) (=> ((member_a X3) H) ((member_a ((a_inv_a_b r) X3)) H))) (=> (forall ((X3 a)) (=> ((member_a X3) H) (forall ((Xa a)) (=> ((member_a Xa) H) ((member_a (((add_a_b r) X3) Xa)) H))))) ((member_a (zero_a_b r)) H)))))) (= _let_54 _let_51) (forall ((I int) (J int) (K int) (L int)) (=> (and (= K L) (= I J)) (= ((plus_plus_int I) K) ((plus_plus_int J) L)))) (forall ((A int) (B int)) (= ((minus_minus_int ((plus_plus_int A) B)) B) A)) (forall ((B int) (A int) (C int)) (=> (= ((plus_plus_int B) A) ((plus_plus_int C) A)) (= B C))) (forall ((A int) (B int)) (= ((plus_plus_int (uminus_uminus_int A)) B) ((minus_minus_int B) A))) _let_82 (forall ((C int) (B int) (A int)) (=> (= ((plus_plus_int C) B) A) (= C ((minus_minus_int A) B)))) (forall ((C int) (A int) (B int)) (let ((_let_1 (plus_plus_int C))) (= ((minus_minus_int (_let_1 A)) (_let_1 B)) ((minus_minus_int A) B)))) (forall ((A int) (B int)) (=> ((ord_less_eq_int A) B) ((ord_less_eq_int (uminus_uminus_int B)) (uminus_uminus_int A)))) (= _let_59 _let_65) (forall ((A a) (B a) (K int)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 ((add_pow_a_b_int r) K))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_3) (=> ((member_a B) _let_3) (= ((_let_1 (_let_2 A)) B) (_let_2 ((_let_1 A) B))))))))) (forall ((A int) (B int)) (= ((plus_plus_int A) ((plus_plus_int (uminus_uminus_int A)) B)) B)) (= _let_57 _let_80) (forall ((X a) (Y a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 (a_inv_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= ((_let_1 (_let_2 X)) Y) (_let_2 ((_let_1 X) Y))))))))) ((locali1648887798ac_a_b r) s) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (= ((plus_plus_int (_let_1 B)) C) (_let_1 ((plus_plus_int B) C))))) ((member_a _let_49) _let_25) ((member_a _let_50) _let_25) (forall ((M partia1833973666xt_a_b) (S2 set_a)) (let ((_let_1 ((locali425460998xt_a_b M) S2))) (=> _let_1 _let_1))) (forall ((A int) (B int) (C int)) (= (= ((minus_minus_int A) B) C) (= A ((plus_plus_int C) B)))) (forall ((I int) (K int) (N int)) (=> ((ord_less_eq_int ((plus_plus_int I) K)) N) ((ord_less_eq_int I) ((minus_minus_int N) K)))) (forall ((M set_a)) (=> ((ord_less_eq_set_a M) (partia1066395285xt_a_b r)) (= (((a_l_coset_a_b r) (zero_a_b r)) M) M))) ((member_a s3) _let_25) (forall ((A int) (B int) (C int) (D int)) (=> ((ord_less_eq_int A) B) (=> ((ord_less_eq_int C) D) ((ord_less_eq_int ((plus_plus_int A) C)) ((plus_plus_int B) D))))) (forall ((X a) (I int)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) ((member_a (((add_pow_a_b_int r) I) X)) _let_1)))) (forall ((X a) (A a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a A) _let_1) (= (= X (((add_a_b r) A) X)) (= A (zero_a_b r))))))) (forall ((B3 int) (K int) (B int) (A int)) (let ((_let_1 (plus_plus_int A))) (let ((_let_2 (plus_plus_int K))) (=> (= B3 (_let_2 B)) (= (_let_1 B3) (_let_2 (_let_1 B))))))) (forall ((X a)) (let ((_let_1 (a_inv_a_b r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (_let_1 (_let_1 X)) X)))) (forall ((I int) (J int) (K int) (L int)) (=> (and (= K L) ((ord_less_eq_int I) J)) ((ord_less_eq_int ((plus_plus_int I) K)) ((plus_plus_int J) L)))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 (_let_2 Y)) Z) (_let_2 ((_let_1 Y) Z)))))))))) (forall ((A2 set_a)) (= ((minus_minus_set_a A2) A2) bot_bot_set_a)) (forall ((A2 set_a) (B3 set_a) (C a)) (let ((_let_1 (member_a C))) (=> ((ord_less_eq_set_a A2) B3) (=> (_let_1 A2) (_let_1 B3))))) (forall ((R partia1833973666xt_a_b) (X a) (Y a) (Z a)) (let ((_let_1 (mult_a_ring_ext_a_b R))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 (_let_2 Y)) Z) (_let_2 ((_let_1 Y) Z))))))))))) (forall ((A2 int) (K int) (A int) (B int)) (let ((_let_1 (plus_plus_int K))) (=> (= A2 (_let_1 A)) (= ((minus_minus_int A2) B) (_let_1 ((minus_minus_int A) B)))))) (forall ((A int) (B int)) (= (= (uminus_uminus_int A) (uminus_uminus_int B)) (= A B))) (forall ((X a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) ((member_a ((a_inv_a_b r) X)) _let_1)))) (forall ((C a)) (not ((member_a C) bot_bot_set_a))) ((member_a _let_37) _let_25) (forall ((A a) (B a) (C a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_2) (=> ((member_a B) _let_2) (=> ((member_a C) _let_2) (= (= A ((_let_1 B) ((a_inv_a_b r) C))) (= B ((_let_1 A) C))))))))) (forall ((A a) (B a) (C a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_2) (=> ((member_a B) _let_2) (=> ((member_a C) _let_2) (= (= A ((_let_1 ((a_inv_a_b r) B)) C)) (= C ((_let_1 B) A))))))))) _let_81 (= _let_60 _let_80) (forall ((X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) ((member_a (((a_minus_a_b r) X) Y)) _let_1))))) (forall ((X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) ((member_a (((add_a_b r) X) Y)) _let_1))))) (forall ((A int) (B int) (C int)) (= ((minus_minus_int A) ((minus_minus_int B) C)) ((minus_minus_int ((plus_plus_int A) C)) B))) _let_79 (forall ((R partia1833973666xt_a_b) (X a) (Y a) (Z a)) (let ((_let_1 (add_a_b R))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 (_let_2 Y)) Z) (_let_2 ((_let_1 Y) Z))))))))))) (forall ((A2 set_a)) ((ord_less_eq_set_a bot_bot_set_a) A2)) (forall ((X a) (Y a)) (let ((_let_1 (a_inv_a_b r))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= (_let_1 ((_let_2 X) Y)) ((_let_2 (_let_1 X)) (_let_1 Y))))))))) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (= (= (_let_1 B) (_let_1 C)) (= B C)))) (forall ((X a) (Y a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_2) (=> ((member_a Y) _let_2) (= ((_let_1 X) ((_let_1 ((a_inv_a_b r) X)) Y)) Y)))))) (forall ((A a) (B a) (C a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_2) (=> ((member_a B) _let_2) (=> ((member_a C) _let_2) (= (= ((_let_1 B) ((a_inv_a_b r) C)) A) (= B ((_let_1 A) C))))))))) (= minus_minus_int (lambda ((A3 int) (B2 int)) ((plus_plus_int A3) (uminus_uminus_int B2)))) (forall ((A int) (C int) (B int)) (= ((ord_less_eq_int ((plus_plus_int A) C)) ((plus_plus_int B) C)) ((ord_less_eq_int A) B))) (forall ((A a) (B a)) (let ((_let_1 (a_inv_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_2) (=> ((member_a B) _let_2) (= (_let_1 (((add_a_b r) A) B)) (((a_minus_a_b r) (_let_1 A)) B))))))) (forall ((A int) (B int)) (= ((plus_plus_int (uminus_uminus_int A)) ((plus_plus_int A) B)) B)) (= (_let_78 _let_61) ((_let_23 (_let_78 _let_37)) (_let_78 _let_28))) (forall ((X a)) (let ((_let_1 (zero_a_b r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((mult_a_ring_ext_a_b r) _let_1) X) _let_1)))) (= _let_41 ((_let_33 (_let_72 _let_63)) _let_62)) (forall ((A a) (B a) (C a)) (let ((_let_1 ((add_a_b r) C))) (=> (= A B) (= (_let_1 A) (_let_1 B))))) _let_77 (forall ((A int) (B int)) (= (= A (uminus_uminus_int B)) (= B (uminus_uminus_int A)))) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((add_a_b r) X) ((a_inv_a_b r) X)) (zero_a_b r)))) (forall ((A a)) (not ((member_a A) bot_bot_set_a))) ((member_a _let_4) _let_25) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (exists ((X3 a)) (and ((member_a X3) (partia1066395285xt_a_b r)) (= (((add_a_b r) X3) X) (zero_a_b r)))))) _let_76 (forall ((A a) (B a) (C a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_2) (=> ((member_a B) _let_2) (=> ((member_a C) _let_2) (= (= ((_let_1 ((a_inv_a_b r) B)) C) A) (= C ((_let_1 B) A))))))))) (forall ((B int) (A int) (C int)) (= (= ((plus_plus_int B) A) ((plus_plus_int C) A)) (= B C))) (forall ((A int) (C int) (B int)) (=> ((ord_less_eq_int ((plus_plus_int A) C)) ((plus_plus_int B) C)) ((ord_less_eq_int A) B))) (forall ((A2 set_a)) (= (collect_a (lambda ((X2 a)) ((member_a X2) A2))) A2)) (forall ((X a) (Y a)) (let ((_let_1 ((mult_a_ring_ext_a_b r) X))) (let ((_let_2 (a_inv_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= (_let_1 (_let_2 Y)) (_let_2 (_let_1 Y))))))))) (forall ((A int) (B int) (C int)) (let ((_let_1 (minus_minus_int A))) (= (_let_1 ((plus_plus_int B) C)) ((minus_minus_int (_let_1 C)) B)))) (forall ((A2 set_a)) (= (exists ((X2 a)) ((member_a X2) A2)) (not (= A2 bot_bot_set_a)))) (forall ((B int) (A int)) (=> ((ord_less_eq_int B) A) (= ((plus_plus_int ((minus_minus_int A) B)) B) A))) _let_75 (forall ((X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) ((member_a (((mult_a_ring_ext_a_b r) X) Y)) _let_1))))) (= _let_59 ((_let_33 (_let_16 _let_4)) (_let_16 _let_58))) (forall ((A int) (C int) (B int)) (= ((minus_minus_int ((plus_plus_int A) C)) ((plus_plus_int B) C)) ((minus_minus_int A) B))) (forall ((I int) (J int) (K int) (L int)) (=> (and ((ord_less_eq_int K) L) (= I J)) ((ord_less_eq_int ((plus_plus_int I) K)) ((plus_plus_int J) L)))) (= _let_54 _let_65) (forall ((A2 set_a) (B3 set_a)) (=> ((ord_less_eq_set_a A2) B3) ((ord_less_eq_set_a (uminus_uminus_set_a B3)) (uminus_uminus_set_a A2)))) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (= (= (_let_1 B) (_let_1 C)) (= B C)))) (forall ((P (-> a a Bool)) (P2 product_prod_a_a)) (=> ((P (product_snd_a_a P2)) (product_fst_a_a P2)) (not (forall ((X3 a) (Y2 a)) (not ((P Y2) X3)))))) ((locali807230110ng_a_b r) s) _let_45 ((member_a _let_57) _let_25) (forall ((U a)) (=> ((member_a U) (partia1066395285xt_a_b r)) (=> (forall ((X3 a)) (=> ((member_a X3) (partia1066395285xt_a_b r)) (= (((add_a_b r) U) X3) X3))) (= U (zero_a_b r))))) (= _let_64 ((_let_33 (_let_74 _let_50)) (_let_74 _let_49))) (forall ((R partia1833973666xt_a_b) (S2 set_a)) (=> ((locali1648887798ac_a_b R) S2) ((locali807230110ng_a_b R) S2))) (forall ((X a)) (let ((_let_1 (zero_a_b r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((mult_a_ring_ext_a_b r) X) _let_1) _let_1)))) (forall ((M set_a) (G a) (H2 a)) (let ((_let_1 (a_l_coset_a_b r))) (let ((_let_2 (partia1066395285xt_a_b r))) (=> ((ord_less_eq_set_a M) _let_2) (=> ((member_a G) _let_2) (=> ((member_a H2) _let_2) (= ((_let_1 G) ((_let_1 H2) M)) ((_let_1 (((add_a_b r) G) H2)) M)))))))) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (= ((plus_plus_int (_let_1 B)) C) (_let_1 ((plus_plus_int B) C))))) (forall ((A int) (B int)) (= (uminus_uminus_int ((plus_plus_int A) B)) ((plus_plus_int (uminus_uminus_int B)) (uminus_uminus_int A)))) (forall ((A a) (B a) (C a) (D a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (a_minus_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_3) (=> ((member_a B) _let_3) (=> ((member_a C) _let_3) (=> ((member_a D) _let_3) (= ((_let_2 ((_let_1 ((_let_2 A) C)) B)) D) ((_let_2 ((_let_2 ((_let_1 A) B)) C)) D)))))))))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 (_let_2 Y)) Z) (_let_2 ((_let_1 Y) Z)))))))))) (forall ((B3 int) (K int) (B int) (A int)) (let ((_let_1 (minus_minus_int A))) (=> (= B3 ((plus_plus_int K) B)) (= (_let_1 B3) ((plus_plus_int (uminus_uminus_int K)) (_let_1 B)))))) ((member_a s2) _let_25) (forall ((X a) (Y a)) (let ((_let_1 (a_inv_a_b r))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= (_let_1 ((_let_2 X) Y)) ((_let_2 (_let_1 Y)) (_let_1 X))))))))) (= _let_73 (_let_72 _let_64)) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (=> (= (_let_1 B) (_let_1 C)) (= B C)))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 ((_let_2 X) Y)) Z) ((_let_2 ((_let_1 X) Z)) ((_let_1 Y) Z)))))))))) (forall ((B int) (A int)) (=> ((ord_less_eq_int B) A) (= ((plus_plus_int B) ((minus_minus_int A) B)) A))) (forall ((X a) (I int) (J int)) (let ((_let_1 (add_pow_a_b_int r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= ((_let_1 ((plus_plus_int I) J)) X) (((add_a_b r) ((_let_1 I) X)) ((_let_1 J) X)))))) _let_71 (forall ((A int) (B int)) (= ((minus_minus_int ((plus_plus_int A) B)) A) B)) (forall ((Y a) (X a) (Y3 a)) (let ((_let_1 (partia1066395285xt_a_b r))) (let ((_let_2 (zero_a_b r))) (let ((_let_3 (add_a_b r))) (=> (= ((_let_3 Y) X) _let_2) (=> (= ((_let_3 X) Y3) _let_2) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) (=> ((member_a Y3) _let_1) (= Y Y3)))))))))) (= _let_57 _let_40) (forall ((X a)) (let ((_let_1 (zero_a_b r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (= ((a_inv_a_b r) X) _let_1) (= X _let_1))))) (forall ((B int) (A int)) (= ((minus_minus_int (uminus_uminus_int B)) A) ((minus_minus_int (uminus_uminus_int A)) B))) (forall ((R partia1833973666xt_a_b) (X a) (Y a) (Z a)) (let ((_let_1 ((mult_a_ring_ext_a_b R) Z))) (let ((_let_2 (add_a_b R))) (let ((_let_3 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= (_let_1 ((_let_2 X) Y)) ((_let_2 (_let_1 X)) (_let_1 Y))))))))))) (forall ((A2 set_a) (B3 set_a)) (= ((ord_less_eq_set_a (uminus_uminus_set_a A2)) (uminus_uminus_set_a B3)) ((ord_less_eq_set_a B3) A2))) (forall ((A a) (B a) (C a)) (let ((_let_1 ((a_minus_a_b r) C))) (=> (= A B) (= (_let_1 A) (_let_1 B))))) (forall ((X a) (Y a)) (let ((_let_1 (zero_a_b r))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> (= ((_let_2 X) Y) _let_1) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (= ((_let_2 Y) X) _let_1)))))))) ((locali880295127ng_a_b r) s) (forall ((A a) (B a) (K int)) (let ((_let_1 ((mult_a_ring_ext_a_b r) A))) (let ((_let_2 ((add_pow_a_b_int r) K))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a A) _let_3) (=> ((member_a B) _let_3) (= (_let_1 (_let_2 B)) (_let_2 (_let_1 B))))))))) (forall ((A int) (B int) (C int)) (= ((ord_less_eq_int ((minus_minus_int A) B)) C) ((ord_less_eq_int A) ((plus_plus_int C) B)))) (= _let_56 _let_69) (forall ((A2 set_a) (B3 set_a)) (=> ((ord_less_eq_set_a A2) B3) (=> ((ord_less_eq_set_a B3) A2) (= A2 B3)))) (forall ((A2 set_a) (B3 set_a)) (= (= ((minus_minus_set_a A2) B3) bot_bot_set_a) ((ord_less_eq_set_a A2) B3))) (forall ((M partia1833973666xt_a_b) (S2 set_a)) (=> ((locali425460998xt_a_b M) S2) ((ord_less_eq_set_a S2) (partia1066395285xt_a_b M)))) (forall ((A2 set_a) (B3 set_a)) (=> (forall ((X3 a)) (let ((_let_1 (member_a X3))) (=> (_let_1 A2) (_let_1 B3)))) ((ord_less_eq_set_a A2) B3))) (forall ((X a) (Y a)) (= (((a_minus_a_b r) X) Y) (((add_a_b r) X) ((a_inv_a_b r) Y)))) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((add_a_b r) ((a_inv_a_b r) X)) X) (zero_a_b r)))) (= _let_55 _let_64) _let_68 (forall ((R partia1833973666xt_a_b) (X a) (Y a) (Z a)) (let ((_let_1 (add_a_b R))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (_let_1 Y))) (let ((_let_4 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_4) (=> ((member_a Y) _let_4) (=> ((member_a Z) _let_4) (= (_let_2 (_let_3 Z)) (_let_3 (_let_2 Z)))))))))))) (_let_67 s) (= (lambda ((Y4 product_prod_a_a) (Z2 product_prod_a_a)) (= Y4 Z2)) (lambda ((S product_prod_a_a) (T3 product_prod_a_a)) (and (= (product_fst_a_a S) (product_fst_a_a T3)) (= (product_snd_a_a S) (product_snd_a_a T3))))) (forall ((A int) (B int)) (= ((plus_plus_int ((minus_minus_int A) B)) B) A)) (forall ((X a) (Y a) (Z a)) (let ((_let_1 (add_a_b r))) (let ((_let_2 (_let_1 X))) (let ((_let_3 (_let_1 Y))) (let ((_let_4 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_4) (=> ((member_a Y) _let_4) (=> ((member_a Z) _let_4) (= (_let_2 (_let_3 Z)) (_let_3 (_let_2 Z))))))))))) (forall ((X a) (N int) (M2 int)) (let ((_let_1 (add_pow_a_b_int r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= ((_let_1 ((minus_minus_int N) M2)) X) (((add_a_b r) ((_let_1 N) X)) ((a_inv_a_b r) ((_let_1 M2) X))))))) (forall ((A int) (B int)) (= (= (uminus_uminus_int A) B) (= (uminus_uminus_int B) A))) (= (_let_66 _let_65) _let_65) (forall ((X a) (I int)) (let ((_let_1 ((add_pow_a_b_int r) I))) (let ((_let_2 (a_inv_a_b r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (_let_1 (_let_2 X)) (_let_2 (_let_1 X))))))) (forall ((Prod product_prod_a_a) (Prod2 product_prod_a_a)) (=> (and (= (product_fst_a_a Prod) (product_fst_a_a Prod2)) (= (product_snd_a_a Prod) (product_snd_a_a Prod2))) (= Prod Prod2))) (forall ((H set_a) (X a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((ord_less_eq_set_a H) _let_1) (=> ((member_a X) _let_1) ((ord_less_eq_set_a (((a_l_coset_a_b r) X) H)) _let_1))))) (= _let_64 _let_35) (forall ((A int) (B int)) (= ((ord_less_eq_int A) (uminus_uminus_int B)) ((ord_less_eq_int B) (uminus_uminus_int A)))) (forall ((R partia1833973666xt_a_b) (X a)) (let ((_let_1 (zero_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) (partia1066395285xt_a_b R)) (= (((mult_a_ring_ext_a_b R) X) _let_1) _let_1))))) (forall ((A int) (B int)) (= ((minus_minus_int ((plus_plus_int A) B)) B) A)) (forall ((A2 int) (K int) (A int)) (=> (= A2 ((plus_plus_int K) A)) (= (uminus_uminus_int A2) ((plus_plus_int (uminus_uminus_int K)) (uminus_uminus_int A))))) (forall ((A2 set_a)) (= ((ord_less_eq_set_a A2) bot_bot_set_a) (= A2 bot_bot_set_a))) (forall ((X a) (Y a)) (=> ((member_a X) s) (=> ((member_a Y) s) ((member_a (((mult_a_ring_ext_a_b r) X) Y)) s)))) (forall ((X a) (Y a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_1) (=> ((member_a Y) _let_1) (=> (= (((add_a_b r) X) Y) (zero_a_b r)) (= X ((a_inv_a_b r) Y))))))) (forall ((Z int)) (let ((_let_1 (zero_a_b r))) (= (((add_pow_a_b_int r) Z) _let_1) _let_1))) (_let_43 s) (forall ((A int) (B int) (C int)) (let ((_let_1 (minus_minus_int A))) (= ((minus_minus_int (_let_1 B)) C) (_let_1 ((plus_plus_int B) C))))) (= (_let_30 _let_61) ((_let_23 _let_38) _let_31)) (forall ((B int) (A int)) (= ((ord_less_eq_int (uminus_uminus_int B)) (uminus_uminus_int A)) ((ord_less_eq_int A) B))) (forall ((I int) (J int) (K int) (L int)) (=> (and ((ord_less_eq_int I) J) ((ord_less_eq_int K) L)) ((ord_less_eq_int ((plus_plus_int I) K)) ((plus_plus_int J) L)))) (forall ((P (-> a Bool))) (= (= (collect_a P) bot_bot_set_a) (forall ((X2 a)) (not (P X2))))) (forall ((P (-> a Bool)) (Q (-> a Bool))) (=> (forall ((X3 a)) (= (P X3) (Q X3))) (= (collect_a P) (collect_a Q)))) ((locali425460998xt_a_b r) s) (forall ((A int) (B int) (C int) (D int)) (=> (= ((minus_minus_int A) B) ((minus_minus_int C) D)) (= (= A B) (= C D)))) (= _let_60 _let_57) (forall ((A int) (B int) (C int) (D int)) (=> (= ((minus_minus_int A) B) ((minus_minus_int C) D)) (= ((ord_less_eq_int A) B) ((ord_less_eq_int C) D)))) (forall ((X a) (I int)) (let ((_let_1 (add_pow_a_b_int r))) (=> ((member_a X) (partia1066395285xt_a_b r)) (= ((_let_1 (uminus_uminus_int I)) X) ((a_inv_a_b r) ((_let_1 I) X)))))) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int A))) (= (_let_1 ((minus_minus_int B) C)) ((minus_minus_int (_let_1 B)) C)))) (= _let_55 (_let_53 _let_51)) (forall ((A int) (B int) (C int)) (= ((plus_plus_int ((minus_minus_int A) B)) C) ((minus_minus_int ((plus_plus_int A) C)) B))) (forall ((A int) (C int) (B int)) (let ((_let_1 (minus_minus_int A))) (= ((minus_minus_int (_let_1 C)) B) ((minus_minus_int (_let_1 B)) C)))) (= _let_24 (_let_48 _let_32)) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (exists ((X3 a)) (and (= (((add_a_b r) X) X3) (zero_a_b r)) ((member_a X3) (partia1066395285xt_a_b r)))))) (forall ((A2 set_a)) (= ((minus_minus_set_a bot_bot_set_a) A2) bot_bot_set_a)) ((member_a _let_40) _let_25) (forall ((R partia1833973666xt_a_b) (X a) (Y a) (Z a)) (let ((_let_1 (mult_a_ring_ext_a_b R))) (let ((_let_2 (add_a_b R))) (let ((_let_3 (partia1066395285xt_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= ((_let_1 ((_let_2 X) Y)) Z) ((_let_2 ((_let_1 X) Z)) ((_let_1 Y) Z))))))))))) (forall ((A int) (B int)) (= ((minus_minus_int A) (uminus_uminus_int B)) ((plus_plus_int A) B))) ((member_a _let_13) _let_25) (forall ((A2 set_a) (B3 set_a)) (=> (= A2 B3) (not (=> ((ord_less_eq_set_a A2) B3) (not ((ord_less_eq_set_a B3) A2)))))) (forall ((X a)) (=> ((member_a X) (partia1066395285xt_a_b r)) (= (((add_a_b r) (zero_a_b r)) X) X))) (forall ((A int) (B int)) (= (uminus_uminus_int ((plus_plus_int A) B)) ((plus_plus_int (uminus_uminus_int A)) (uminus_uminus_int B)))) (forall ((X a) (Y a) (Z a)) (let ((_let_1 ((mult_a_ring_ext_a_b r) Z))) (let ((_let_2 (add_a_b r))) (let ((_let_3 (partia1066395285xt_a_b r))) (=> ((member_a X) _let_3) (=> ((member_a Y) _let_3) (=> ((member_a Z) _let_3) (= (_let_1 ((_let_2 X) Y)) ((_let_2 (_let_1 X)) (_let_1 Y)))))))))) (forall ((R partia1833973666xt_a_b) (X a)) (let ((_let_1 (zero_a_b R))) (=> (semiring_a_b R) (=> ((member_a X) (partia1066395285xt_a_b R)) (= (((mult_a_ring_ext_a_b R) _let_1) X) _let_1))))) ((member_a _let_38) _let_25) (forall ((M partia1833973666xt_a_b) (S2 set_a) (X a) (Y a)) (=> ((locali425460998xt_a_b M) S2) (=> ((member_a X) S2) (=> ((member_a Y) S2) ((member_a (((mult_a_ring_ext_a_b M) X) Y)) S2))))) (forall ((R partia1833973666xt_a_b) (X a)) (=> (semiring_a_b R) (=> ((member_a X) (partia1066395285xt_a_b R)) (= (((add_a_b R) X) (zero_a_b R)) X)))) ((member_a _let_7) _let_25) (not (= _let_25 bot_bot_set_a)) (forall ((A2 set_a)) (=> (forall ((Y2 a)) (not ((member_a Y2) A2))) (= A2 bot_bot_set_a))) (not (forall ((T a)) (let ((_let_1 (mult_a_ring_ext_a_b r))) (=> ((member_a T) s) (not (= ((_let_1 T) (((a_minus_a_b r) ((_let_1 (product_snd_a_a x)) r2)) ((_let_1 s2) (product_fst_a_a x)))) (zero_a_b r))))))) (forall ((B int) (A int) (C int)) (= (= ((plus_plus_int B) A) ((plus_plus_int C) A)) (= B C))) (forall ((B int) (A int) (C int)) (let ((_let_1 (plus_plus_int B))) (let ((_let_2 (plus_plus_int A))) (= (_let_1 (_let_2 C)) (_let_2 (_let_1 C)))))) (forall ((A2 set_a) (A a)) (=> (= A2 bot_bot_set_a) (not ((member_a A) A2)))) (forall ((H set_a) (K2 set_a)) (=> ((additi2104487374up_a_b H) r) (=> ((additi2104487374up_a_b K2) r) ((additi2104487374up_a_b (((set_add_a_b r) H) K2)) r)))) ((ord_less_eq_set_a s) _let_25) _let_47 ((member_a _let_15) _let_25) (forall ((R partia1833973666xt_a_b) (S2 set_a)) (=> ((locali880295127ng_a_b R) S2) ((locali425460998xt_a_b R) S2))) (forall ((R partia1833973666xt_a_b) (X a)) (=> (semiring_a_b R) (=> ((member_a X) (partia1066395285xt_a_b R)) (= (((add_a_b R) (zero_a_b R)) X) X)))) ((member_a _let_26) _let_25) (= (_let_18 _let_24) ((_let_23 (_let_22 _let_12)) (_let_22 _let_21))) (= _let_19 _let_10) (forall ((A a) (P (-> a Bool))) (= ((member_a A) (collect_a P)) (P A))) (forall ((C int) (A int) (B int)) (let ((_let_1 (plus_plus_int C))) (= ((ord_less_eq_int (_let_1 A)) (_let_1 B)) ((ord_less_eq_int A) B)))) (forall ((A int) (C int) (B int)) (= (= A ((minus_minus_int C) B)) (= ((plus_plus_int A) B) C))) (forall ((A2 set_a) (B3 set_a)) (let ((_let_1 (partia1066395285xt_a_b r))) (=> ((ord_less_eq_set_a A2) _let_1) (=> ((ord_less_eq_set_a B3) _let_1) ((ord_less_eq_set_a (((set_add_a_b r) A2) B3)) _let_1))))) (forall ((A int) (B int) (C int)) (let ((_let_1 (plus_plus_int C))) (=> ((ord_less_eq_int A) B) ((ord_less_eq_int (_let_1 A)) (_let_1 B))))) (forall ((A int) (C int) (B int)) (= ((ord_less_eq_int A) ((minus_minus_int C) B)) ((ord_less_eq_int ((plus_plus_int A) B)) C))) (forall ((A int) (B int) (C int)) (=> ((ord_less_eq_int A) B) ((ord_less_eq_int ((minus_minus_int A) C)) ((minus_minus_int B) C)))) (forall ((A int) (B int)) (= (uminus_uminus_int ((minus_minus_int A) B)) ((minus_minus_int B) A))) (not false)))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
0.34/0.94	)
0.34/0.94	% SZS output end Proof for theBenchmark
0.34/0.94	EOF