TSTP Solution File: ITP108^1 by cvc5---1.0.5
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cvc5---1.0.5
% Problem : ITP108^1 : TPTP v8.1.2. Released v7.5.0.
% Transfm : none
% Format : tptp
% Command : do_cvc5 %s %d
% Computer : n019.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 03:18:18 EDT 2023
% Result : Theorem 0.57s 0.88s
% Output : Proof 0.71s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.17 % Problem : ITP108^1 : TPTP v8.1.2. Released v7.5.0.
% 0.17/0.18 % Command : do_cvc5 %s %d
% 0.18/0.39 % Computer : n019.cluster.edu
% 0.18/0.39 % Model : x86_64 x86_64
% 0.18/0.39 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.18/0.39 % Memory : 8042.1875MB
% 0.18/0.39 % OS : Linux 3.10.0-693.el7.x86_64
% 0.18/0.39 % CPULimit : 300
% 0.18/0.39 % WCLimit : 300
% 0.18/0.39 % DateTime : Sun Aug 27 10:17:43 EDT 2023
% 0.18/0.40 % CPUTime :
% 0.25/0.57 %----Proving TH0
% 0.25/0.58 %------------------------------------------------------------------------------
% 0.25/0.58 % File : ITP108^1 : TPTP v8.1.2. Released v7.5.0.
% 0.25/0.58 % Domain : Interactive Theorem Proving
% 0.25/0.58 % Problem : Sledgehammer Localization problem prob_758__8980450_1
% 0.25/0.58 % Version : Especial.
% 0.25/0.58 % English :
% 0.25/0.58
% 0.25/0.58 % Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% 0.25/0.58 % : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% 0.25/0.58 % Source : [Des21]
% 0.25/0.58 % Names : Localization/prob_758__8980450_1 [Des21]
% 0.25/0.58
% 0.25/0.58 % Status : Theorem
% 0.25/0.58 % Rating : 0.23 v8.1.0, 0.27 v7.5.0
% 0.25/0.58 % Syntax : Number of formulae : 287 ( 119 unt; 42 typ; 0 def)
% 0.25/0.58 % Number of atoms : 568 ( 250 equ; 0 cnn)
% 0.25/0.58 % Maximal formula atoms : 8 ( 2 avg)
% 0.25/0.58 % Number of connectives : 3894 ( 18 ~; 0 |; 8 &;3625 @)
% 0.25/0.58 % ( 0 <=>; 243 =>; 0 <=; 0 <~>)
% 0.25/0.58 % Maximal formula depth : 15 ( 6 avg)
% 0.25/0.58 % Number of types : 6 ( 5 usr)
% 0.25/0.58 % Number of type conns : 58 ( 58 >; 0 *; 0 +; 0 <<)
% 0.25/0.58 % Number of symbols : 38 ( 37 usr; 11 con; 0-3 aty)
% 0.25/0.58 % Number of variables : 464 ( 16 ^; 445 !; 3 ?; 464 :)
% 0.25/0.58 % SPC : TH0_THM_EQU_NAR
% 0.25/0.58
% 0.25/0.58 % Comments : This file was generated by Sledgehammer 2021-02-23 15:39:18.132
% 0.25/0.58 %------------------------------------------------------------------------------
% 0.25/0.58 % Could-be-implicit typings (5)
% 0.25/0.58 thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
% 0.25/0.58 partia1833973666xt_a_b: $tType ).
% 0.25/0.58
% 0.25/0.58 thf(ty_n_t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
% 0.25/0.58 product_prod_a_a: $tType ).
% 0.25/0.58
% 0.25/0.58 thf(ty_n_t__Set__Oset_Itf__a_J,type,
% 0.25/0.58 set_a: $tType ).
% 0.25/0.58
% 0.25/0.58 thf(ty_n_t__Int__Oint,type,
% 0.25/0.58 int: $tType ).
% 0.25/0.58
% 0.25/0.58 thf(ty_n_tf__a,type,
% 0.25/0.58 a: $tType ).
% 0.25/0.58
% 0.25/0.58 % Explicit typings (37)
% 0.25/0.58 thf(sy_c_AbelCoset_Oa__l__coset_001tf__a_001tf__b,type,
% 0.25/0.58 a_l_coset_a_b: partia1833973666xt_a_b > a > set_a > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_AbelCoset_Oadditive__subgroup_001tf__a_001tf__b,type,
% 0.25/0.58 additi2104487374up_a_b: set_a > partia1833973666xt_a_b > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
% 0.25/0.58 set_add_a_b: partia1833973666xt_a_b > set_a > set_a > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
% 0.25/0.58 partia1066395285xt_a_b: partia1833973666xt_a_b > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.25/0.58 mult_a_ring_ext_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
% 0.25/0.58 minus_minus_int: int > int > int ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
% 0.25/0.58 minus_minus_set_a: set_a > set_a > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
% 0.25/0.58 plus_plus_int: int > int > int ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
% 0.25/0.58 uminus_uminus_int: int > int ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_Itf__a_J,type,
% 0.25/0.58 uminus_uminus_set_a: set_a > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Oeq__obj__rng__of__frac_001tf__a_001tf__b,type,
% 0.25/0.58 locali1648887798ac_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__crng_001tf__a_001tf__b,type,
% 0.25/0.58 locali807230110ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Omult__submonoid__of__rng_001tf__a_001tf__b,type,
% 0.25/0.58 locali880295127ng_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Localization__Mirabelle__afvchqjmgj_Osubmonoid_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
% 0.25/0.58 locali425460998xt_a_b: partia1833973666xt_a_b > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
% 0.25/0.58 bot_bot_set_a: set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
% 0.25/0.58 ord_less_eq_int: int > int > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
% 0.25/0.58 ord_less_eq_set_a: set_a > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Product__Type_Oprod_Ofst_001tf__a_001tf__a,type,
% 0.25/0.58 product_fst_a_a: product_prod_a_a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Product__Type_Oprod_Osnd_001tf__a_001tf__a,type,
% 0.25/0.58 product_snd_a_a: product_prod_a_a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
% 0.25/0.58 a_inv_a_b: partia1833973666xt_a_b > a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Oa__minus_001tf__a_001tf__b,type,
% 0.25/0.58 a_minus_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Oadd__pow_001tf__a_001tf__b_001t__Int__Oint,type,
% 0.25/0.58 add_pow_a_b_int: partia1833973666xt_a_b > int > a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
% 0.25/0.58 add_a_b: partia1833973666xt_a_b > a > a > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
% 0.25/0.58 zero_a_b: partia1833973666xt_a_b > a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Ring_Osemiring_001tf__a_001tf__b,type,
% 0.25/0.58 semiring_a_b: partia1833973666xt_a_b > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_Set_OCollect_001tf__a,type,
% 0.25/0.58 collect_a: ( a > $o ) > set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_c_member_001tf__a,type,
% 0.25/0.58 member_a: a > set_a > $o ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_R,type,
% 0.25/0.58 r: partia1833973666xt_a_b ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_S,type,
% 0.25/0.58 s: set_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_r,type,
% 0.25/0.58 r2: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_r_H,type,
% 0.25/0.58 r3: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_s,type,
% 0.25/0.58 s2: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_s_H,type,
% 0.25/0.58 s3: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_t_H____,type,
% 0.25/0.58 t: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_t____,type,
% 0.25/0.58 t2: a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_x_H____,type,
% 0.25/0.58 x: product_prod_a_a ).
% 0.25/0.58
% 0.25/0.58 thf(sy_v_y_H____,type,
% 0.25/0.58 y: product_prod_a_a ).
% 0.25/0.58
% 0.25/0.58 % Relevant facts (244)
% 0.25/0.58 thf(fact_0_f18,axiom,
% 0.25/0.58 member_a @ t @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f18
% 0.25/0.58 thf(fact_1_f17,axiom,
% 0.25/0.58 member_a @ t2 @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f17
% 0.25/0.58 thf(fact_2_right__add__eq,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( A = B )
% 0.25/0.58 => ( ( add_a_b @ r @ C @ A )
% 0.25/0.58 = ( add_a_b @ r @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % right_add_eq
% 0.25/0.58 thf(fact_3_local_Oright__minus__eq,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( A = B )
% 0.25/0.58 => ( ( a_minus_a_b @ r @ C @ A )
% 0.25/0.58 = ( a_minus_a_b @ r @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.right_minus_eq
% 0.25/0.58 thf(fact_4_m__assoc,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % m_assoc
% 0.25/0.58 thf(fact_5_m__comm,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % m_comm
% 0.25/0.58 thf(fact_6_m__lcomm,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % m_lcomm
% 0.25/0.58 thf(fact_7_add_Om__assoc,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.25/0.58 = ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.m_assoc
% 0.25/0.58 thf(fact_8_add_Om__comm,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ X @ Y )
% 0.25/0.58 = ( add_a_b @ r @ Y @ X ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.m_comm
% 0.25/0.58 thf(fact_9_add_Om__lcomm,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
% 0.25/0.58 = ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.m_lcomm
% 0.25/0.58 thf(fact_10_f20,axiom,
% 0.25/0.58 member_a @ s3 @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f20
% 0.25/0.58 thf(fact_11_f19,axiom,
% 0.25/0.58 member_a @ s2 @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f19
% 0.25/0.58 thf(fact_12_f41,axiom,
% 0.25/0.58 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f41
% 0.25/0.58 thf(fact_13_f42,axiom,
% 0.25/0.58 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f42
% 0.25/0.58 thf(fact_14_l__distr,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % l_distr
% 0.25/0.58 thf(fact_15_r__distr,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_distr
% 0.25/0.58 thf(fact_16_four__elem__comm,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a,D: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ D @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ A @ C ) @ B ) @ D )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ ( add_a_b @ r @ A @ B ) @ C ) @ D ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % four_elem_comm
% 0.25/0.58 thf(fact_17_right__inv__add,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_minus_a_b @ r @ ( a_minus_a_b @ r @ C @ A ) @ B )
% 0.25/0.58 = ( a_minus_a_b @ r @ C @ ( add_a_b @ r @ A @ B ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % right_inv_add
% 0.25/0.58 thf(fact_18_f21,axiom,
% 0.25/0.58 member_a @ ( product_snd_a_a @ y ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f21
% 0.25/0.58 thf(fact_19_f40,axiom,
% 0.25/0.58 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f40
% 0.25/0.58 thf(fact_20_f22,axiom,
% 0.25/0.58 member_a @ ( product_fst_a_a @ x ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f22
% 0.25/0.58 thf(fact_21__092_060open_062_092_060lbrakk_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060in_062_Acarrier_AR_059_Asnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_A_092_060in_062_Acarrier_AR_059_At_A_092_060otimes_062_At_H_A_092_060in_062_Acarrier_AR_092_060rbrakk_062_A_092_060Longrightarrow_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_A_092_060oplus_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.25/0.58 ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>\<lbrakk>snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<in> carrier R; snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y') \<in> carrier R; t \<otimes> t' \<in> carrier R\<rbrakk> \<Longrightarrow> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x') \<oplus> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.25/0.58 thf(fact_22_f15,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f15
% 0.25/0.58 thf(fact_23__092_060open_062snd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>snd x' \<otimes> snd y' \<in> carrier R\<close>
% 0.25/0.58 thf(fact_24_f28,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f28
% 0.25/0.58 thf(fact_25_f11,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f11
% 0.25/0.58 thf(fact_26_f10,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f10
% 0.25/0.58 thf(fact_27_f27,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f27
% 0.25/0.58 thf(fact_28_f35,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f35
% 0.25/0.58 thf(fact_29__092_060open_062t_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = t \<otimes> (snd x' \<otimes> r) \<ominus> t \<otimes> (s \<otimes> fst x')\<close>
% 0.25/0.58 thf(fact_30__092_060open_062t_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_At_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Afst_Ay_H_J_092_060close_062,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = t' \<otimes> (snd y' \<otimes> r') \<ominus> t' \<otimes> (s' \<otimes> fst y')\<close>
% 0.25/0.58 thf(fact_31_f16,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f16
% 0.25/0.58 thf(fact_32_f24,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f24
% 0.25/0.58 thf(fact_33_f9,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f9
% 0.25/0.58 thf(fact_34__092_060open_062t_A_092_060otimes_062_As_A_092_060otimes_062_Asnd_Ax_H_A_092_060in_062_Acarrier_AR_092_060close_062,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t \<otimes> s \<otimes> snd x' \<in> carrier R\<close>
% 0.25/0.58 thf(fact_35_f23,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f23
% 0.25/0.58 thf(fact_36_f34,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f34
% 0.25/0.58 thf(fact_37_f36,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f36
% 0.25/0.58 thf(fact_38_f13,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f13
% 0.25/0.58 thf(fact_39__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_061_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_At_H_A_092_060otimes_062_As_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_At_A_092_060otimes_062_A_Is_A_092_060otimes_062_Afst_Ax_H_J_092_060close_062,axiom,
% 0.25/0.58 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) = t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (snd x' \<otimes> r) \<ominus> t' \<otimes> s' \<otimes> snd y' \<otimes> t \<otimes> (s \<otimes> fst x')\<close>
% 0.25/0.58 thf(fact_40_f12,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ t2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f12
% 0.25/0.58 thf(fact_41_f25,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f25
% 0.25/0.58 thf(fact_42_f32,axiom,
% 0.25/0.58 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f32
% 0.25/0.58 thf(fact_43_f26,axiom,
% 0.25/0.58 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ t ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f26
% 0.25/0.58 thf(fact_44_f29,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f29
% 0.25/0.58 thf(fact_45_mem__Collect__eq,axiom,
% 0.25/0.58 ! [A: a,P: a > $o] :
% 0.25/0.58 ( ( member_a @ A @ ( collect_a @ P ) )
% 0.25/0.58 = ( P @ A ) ) ).
% 0.25/0.58
% 0.25/0.58 % mem_Collect_eq
% 0.25/0.58 thf(fact_46_Collect__mem__eq,axiom,
% 0.25/0.58 ! [A2: set_a] :
% 0.25/0.58 ( ( collect_a
% 0.25/0.58 @ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
% 0.25/0.58 = A2 ) ).
% 0.25/0.58
% 0.25/0.58 % Collect_mem_eq
% 0.25/0.58 thf(fact_47_Collect__cong,axiom,
% 0.25/0.58 ! [P: a > $o,Q: a > $o] :
% 0.25/0.58 ( ! [X3: a] :
% 0.25/0.58 ( ( P @ X3 )
% 0.25/0.58 = ( Q @ X3 ) )
% 0.25/0.58 => ( ( collect_a @ P )
% 0.25/0.58 = ( collect_a @ Q ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % Collect_cong
% 0.25/0.58 thf(fact_48_f30,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f30
% 0.25/0.58 thf(fact_49_f33,axiom,
% 0.25/0.58 ( ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f33
% 0.25/0.58 thf(fact_50__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_A_092_060oplus_062_As_A_092_060otimes_062_Ar_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_J_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_092_060close_062,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r \<oplus> s \<otimes> r')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s' \<otimes> r) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y') \<otimes> (s \<otimes> r')\<close>
% 0.25/0.58 thf(fact_51_f14,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f14
% 0.25/0.58 thf(fact_52__092_060open_062t_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_A_092_060ominus_062_As_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_A_061_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_H_A_092_060otimes_062_Ar_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Afst_Ax_H_J_J_A_092_060oplus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Asnd_Ay_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_Ar_H_J_J_A_092_060ominus_062_At_A_092_060otimes_062_At_H_A_092_060otimes_062_A_Is_A_092_060otimes_062_As_H_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Afst_Ay_H_J_J_092_060close_062,axiom,
% 0.25/0.58 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r) \<ominus> s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r') \<ominus> s \<otimes> s' \<otimes> (snd x' \<otimes> fst y')) = t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s' \<otimes> r)) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd y' \<otimes> fst x')) \<oplus> t \<otimes> t' \<otimes> (snd x' \<otimes> snd y' \<otimes> (s \<otimes> r')) \<ominus> t \<otimes> t' \<otimes> (s \<otimes> s' \<otimes> (snd x' \<otimes> fst y'))\<close>
% 0.25/0.58 thf(fact_53_f31,axiom,
% 0.25/0.58 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.25/0.58 = ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % f31
% 0.25/0.58 thf(fact_54_f37,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f37
% 0.25/0.58 thf(fact_55_f39,axiom,
% 0.25/0.58 member_a @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f39
% 0.25/0.58 thf(fact_56_f38,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % f38
% 0.25/0.58 thf(fact_57_f5,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ).
% 0.25/0.58
% 0.25/0.58 % f5
% 0.25/0.58 thf(fact_58_f7,axiom,
% 0.25/0.58 ( ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ).
% 0.25/0.58
% 0.25/0.58 % f7
% 0.25/0.58 thf(fact_59_minus__closed,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( member_a @ ( a_minus_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_closed
% 0.25/0.58 thf(fact_60_add_Om__closed,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.m_closed
% 0.25/0.58 thf(fact_61_local_Oadd_Oright__cancel,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ Y @ X )
% 0.25/0.58 = ( add_a_b @ r @ Z @ X ) )
% 0.25/0.58 = ( Y = Z ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.add.right_cancel
% 0.25/0.58 thf(fact_62_semiring__simprules_I3_J,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring_simprules(3)
% 0.25/0.58 thf(fact_63_f4,axiom,
% 0.25/0.58 member_a @ t2 @ s ).
% 0.25/0.58
% 0.25/0.58 % f4
% 0.25/0.58 thf(fact_64_f6,axiom,
% 0.25/0.58 member_a @ t @ s ).
% 0.25/0.58
% 0.25/0.58 % f6
% 0.25/0.58 thf(fact_65_f8,axiom,
% 0.25/0.58 member_a @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ s ).
% 0.25/0.58
% 0.25/0.58 % f8
% 0.25/0.58 thf(fact_66_local_Osemiring__axioms,axiom,
% 0.25/0.58 semiring_a_b @ r ).
% 0.25/0.58
% 0.25/0.58 % local.semiring_axioms
% 0.25/0.58 thf(fact_67_minus__eq,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( a_minus_a_b @ r @ X @ Y )
% 0.25/0.58 = ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_eq
% 0.25/0.58 thf(fact_68_prod__eqI,axiom,
% 0.25/0.58 ! [P2: product_prod_a_a,Q2: product_prod_a_a] :
% 0.25/0.58 ( ( ( product_fst_a_a @ P2 )
% 0.25/0.58 = ( product_fst_a_a @ Q2 ) )
% 0.25/0.58 => ( ( ( product_snd_a_a @ P2 )
% 0.25/0.58 = ( product_snd_a_a @ Q2 ) )
% 0.25/0.58 => ( P2 = Q2 ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % prod_eqI
% 0.25/0.58 thf(fact_69_exE__realizer_H,axiom,
% 0.25/0.58 ! [P: a > a > $o,P2: product_prod_a_a] :
% 0.25/0.58 ( ( P @ ( product_snd_a_a @ P2 ) @ ( product_fst_a_a @ P2 ) )
% 0.25/0.58 => ~ ! [X3: a,Y2: a] :
% 0.25/0.58 ~ ( P @ Y2 @ X3 ) ) ).
% 0.25/0.58
% 0.25/0.58 % exE_realizer'
% 0.25/0.58 thf(fact_70_prod_Oexpand,axiom,
% 0.25/0.58 ! [Prod: product_prod_a_a,Prod2: product_prod_a_a] :
% 0.25/0.58 ( ( ( ( product_fst_a_a @ Prod )
% 0.25/0.58 = ( product_fst_a_a @ Prod2 ) )
% 0.25/0.58 & ( ( product_snd_a_a @ Prod )
% 0.25/0.58 = ( product_snd_a_a @ Prod2 ) ) )
% 0.25/0.58 => ( Prod = Prod2 ) ) ).
% 0.25/0.58
% 0.25/0.58 % prod.expand
% 0.25/0.58 thf(fact_71_local_Ominus__unique,axiom,
% 0.25/0.58 ! [Y: a,X: a,Y3: a] :
% 0.25/0.58 ( ( ( add_a_b @ r @ Y @ X )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ X @ Y3 )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y3 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( Y = Y3 ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.minus_unique
% 0.25/0.58 thf(fact_72_add_Or__inv__ex,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ? [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 & ( ( add_a_b @ r @ X @ X3 )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.r_inv_ex
% 0.25/0.58 thf(fact_73_add_Oone__unique,axiom,
% 0.25/0.58 ! [U: a] :
% 0.25/0.58 ( ( member_a @ U @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ! [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ U @ X3 )
% 0.25/0.58 = X3 ) )
% 0.25/0.58 => ( U
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.one_unique
% 0.25/0.58 thf(fact_74_add_Ol__inv__ex,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ? [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 & ( ( add_a_b @ r @ X3 @ X )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.l_inv_ex
% 0.25/0.58 thf(fact_75_add_Oinv__comm,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ Y @ X )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_comm
% 0.25/0.58 thf(fact_76_r__minus,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
% 0.25/0.58 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_minus
% 0.25/0.58 thf(fact_77_l__minus,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
% 0.25/0.58 = ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % l_minus
% 0.25/0.58 thf(fact_78_r__neg2,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
% 0.25/0.58 = Y ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_neg2
% 0.25/0.58 thf(fact_79_r__neg1,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = Y ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_neg1
% 0.25/0.58 thf(fact_80_local_Ominus__add,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.minus_add
% 0.25/0.58 thf(fact_81_add_Oinv__solve__right_H,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
% 0.25/0.58 = A )
% 0.25/0.58 = ( B
% 0.25/0.58 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_solve_right'
% 0.25/0.58 thf(fact_82_add_Oinv__solve__right,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( A
% 0.25/0.58 = ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
% 0.25/0.58 = ( B
% 0.25/0.58 = ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_solve_right
% 0.25/0.58 thf(fact_83_add_Oinv__solve__left_H,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
% 0.25/0.58 = A )
% 0.25/0.58 = ( C
% 0.25/0.58 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_solve_left'
% 0.25/0.58 thf(fact_84_add_Oinv__solve__left,axiom,
% 0.25/0.58 ! [A: a,B: a,C: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ C @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( A
% 0.25/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
% 0.25/0.58 = ( C
% 0.25/0.58 = ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_solve_left
% 0.25/0.58 thf(fact_85_add_Oinv__mult__group,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_mult_group
% 0.25/0.58 thf(fact_86_a__transpose__inv,axiom,
% 0.25/0.58 ! [X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.25/0.58 = Z )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
% 0.25/0.58 = Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % a_transpose_inv
% 0.25/0.58 thf(fact_87_eq__obj__rng__of__frac__axioms,axiom,
% 0.25/0.58 locali1648887798ac_a_b @ r @ s ).
% 0.25/0.58
% 0.25/0.58 % eq_obj_rng_of_frac_axioms
% 0.25/0.58 thf(fact_88_sum__zero__eq__neg,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ X @ Y )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 => ( X
% 0.25/0.58 = ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % sum_zero_eq_neg
% 0.25/0.58 thf(fact_89_r__neg,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_neg
% 0.25/0.58 thf(fact_90_minus__equality,axiom,
% 0.25/0.58 ! [Y: a,X: a] :
% 0.25/0.58 ( ( ( add_a_b @ r @ Y @ X )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_inv_a_b @ r @ X )
% 0.25/0.58 = Y ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_equality
% 0.25/0.58 thf(fact_91_l__neg,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % l_neg
% 0.25/0.58 thf(fact_92_inv__add,axiom,
% 0.25/0.58 ! [A: a,B: a] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_inv_a_b @ r @ ( add_a_b @ r @ A @ B ) )
% 0.25/0.58 = ( a_minus_a_b @ r @ ( a_inv_a_b @ r @ A ) @ B ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % inv_add
% 0.25/0.58 thf(fact_93__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_O_A_092_060lbrakk_062t_A_092_060in_062_AS_059_At_A_092_060otimes_062_A_Isnd_Ax_H_A_092_060otimes_062_Ar_A_092_060ominus_062_As_A_092_060otimes_062_Afst_Ax_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.25/0.58 ~ ! [T: a] :
% 0.25/0.58 ( ( member_a @ T @ s )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ T @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) )
% 0.25/0.58 != ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>\<And>thesis. (\<And>t. \<lbrakk>t \<in> S; t \<otimes> (snd x' \<otimes> r \<ominus> s \<otimes> fst x') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.25/0.58 thf(fact_94__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062t_H_O_A_092_060lbrakk_062t_H_A_092_060in_062_AS_059_At_H_A_092_060otimes_062_A_Isnd_Ay_H_A_092_060otimes_062_Ar_H_A_092_060ominus_062_As_H_A_092_060otimes_062_Afst_Ay_H_J_A_061_A_092_060zero_062_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
% 0.25/0.58 ~ ! [T2: a] :
% 0.25/0.58 ( ( member_a @ T2 @ s )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ T2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) )
% 0.25/0.58 != ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % \<open>\<And>thesis. (\<And>t'. \<lbrakk>t' \<in> S; t' \<otimes> (snd y' \<otimes> r' \<ominus> s' \<otimes> fst y') = \<zero>\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
% 0.25/0.58 thf(fact_95_zero__closed,axiom,
% 0.25/0.58 member_a @ ( zero_a_b @ r ) @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % zero_closed
% 0.25/0.58 thf(fact_96_local_Ominus__minus,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
% 0.25/0.58 = X ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.minus_minus
% 0.25/0.58 thf(fact_97_add_Oinv__closed,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_closed
% 0.25/0.58 thf(fact_98_local_Ominus__zero,axiom,
% 0.25/0.58 ( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ).
% 0.25/0.58
% 0.25/0.58 % local.minus_zero
% 0.25/0.58 thf(fact_99_m__closed,axiom,
% 0.25/0.58 ! [X: a,Y: a] :
% 0.25/0.58 ( ( member_a @ X @ s )
% 0.25/0.58 => ( ( member_a @ Y @ s )
% 0.25/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ s ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % m_closed
% 0.25/0.58 thf(fact_100_r__null,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_null
% 0.25/0.58 thf(fact_101_l__null,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % l_null
% 0.25/0.58 thf(fact_102_r__zero,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
% 0.25/0.58 = X ) ) ).
% 0.25/0.58
% 0.25/0.58 % r_zero
% 0.25/0.58 thf(fact_103_l__zero,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
% 0.25/0.58 = X ) ) ).
% 0.25/0.58
% 0.25/0.58 % l_zero
% 0.25/0.58 thf(fact_104_add_Or__cancel__one_H,axiom,
% 0.25/0.58 ! [X: a,A: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( X
% 0.25/0.58 = ( add_a_b @ r @ A @ X ) )
% 0.25/0.58 = ( A
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.r_cancel_one'
% 0.25/0.58 thf(fact_105_add_Or__cancel__one,axiom,
% 0.25/0.58 ! [X: a,A: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ A @ X )
% 0.25/0.58 = X )
% 0.25/0.58 = ( A
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.r_cancel_one
% 0.25/0.58 thf(fact_106_add_Ol__cancel__one_H,axiom,
% 0.25/0.58 ! [X: a,A: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( X
% 0.25/0.58 = ( add_a_b @ r @ X @ A ) )
% 0.25/0.58 = ( A
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.l_cancel_one'
% 0.25/0.58 thf(fact_107_add_Ol__cancel__one,axiom,
% 0.25/0.58 ! [X: a,A: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( add_a_b @ r @ X @ A )
% 0.25/0.58 = X )
% 0.25/0.58 = ( A
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.l_cancel_one
% 0.25/0.58 thf(fact_108_add_Oinv__eq__1__iff,axiom,
% 0.25/0.58 ! [X: a] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ( a_inv_a_b @ r @ X )
% 0.25/0.58 = ( zero_a_b @ r ) )
% 0.25/0.58 = ( X
% 0.25/0.58 = ( zero_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inv_eq_1_iff
% 0.25/0.58 thf(fact_109_mult__submonoid__of__rng__axioms,axiom,
% 0.25/0.58 locali880295127ng_a_b @ r @ s ).
% 0.25/0.58
% 0.25/0.58 % mult_submonoid_of_rng_axioms
% 0.25/0.58 thf(fact_110_mult__submonoid__of__crng__axioms,axiom,
% 0.25/0.58 locali807230110ng_a_b @ r @ s ).
% 0.25/0.58
% 0.25/0.58 % mult_submonoid_of_crng_axioms
% 0.25/0.58 thf(fact_111_prod__eq__iff,axiom,
% 0.25/0.58 ( ( ^ [Y4: product_prod_a_a,Z2: product_prod_a_a] : ( Y4 = Z2 ) )
% 0.25/0.58 = ( ^ [S: product_prod_a_a,T3: product_prod_a_a] :
% 0.25/0.58 ( ( ( product_fst_a_a @ S )
% 0.25/0.58 = ( product_fst_a_a @ T3 ) )
% 0.25/0.58 & ( ( product_snd_a_a @ S )
% 0.25/0.58 = ( product_snd_a_a @ T3 ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % prod_eq_iff
% 0.25/0.58 thf(fact_112_submonoid__axioms,axiom,
% 0.25/0.58 locali425460998xt_a_b @ r @ s ).
% 0.25/0.58
% 0.25/0.58 % submonoid_axioms
% 0.25/0.58 thf(fact_113_add_Oint__pow__inv,axiom,
% 0.25/0.58 ! [X: a,I: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( a_inv_a_b @ r @ X ) )
% 0.25/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_inv
% 0.25/0.58 thf(fact_114_semiring_Osemiring__simprules_I11_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( add_a_b @ R @ X @ ( zero_a_b @ R ) )
% 0.25/0.58 = X ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(11)
% 0.25/0.58 thf(fact_115_semiring_Osemiring__simprules_I6_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( add_a_b @ R @ ( zero_a_b @ R ) @ X )
% 0.25/0.58 = X ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(6)
% 0.25/0.58 thf(fact_116_semiring_Ol__null,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ R @ ( zero_a_b @ R ) @ X )
% 0.25/0.58 = ( zero_a_b @ R ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.l_null
% 0.25/0.58 thf(fact_117_semiring_Or__null,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ R @ X @ ( zero_a_b @ R ) )
% 0.25/0.58 = ( zero_a_b @ R ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.r_null
% 0.25/0.58 thf(fact_118_semiring_Or__distr,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ R @ Z @ ( add_a_b @ R @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ Z @ X ) @ ( mult_a_ring_ext_a_b @ R @ Z @ Y ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.r_distr
% 0.25/0.58 thf(fact_119_add__pow__ldistr__int,axiom,
% 0.25/0.58 ! [A: a,B: a,K: int] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ ( add_pow_a_b_int @ r @ K @ A ) @ B )
% 0.25/0.58 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_pow_ldistr_int
% 0.25/0.58 thf(fact_120_add__pow__rdistr__int,axiom,
% 0.25/0.58 ! [A: a,B: a,K: int] :
% 0.25/0.58 ( ( member_a @ A @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ B @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ r @ A @ ( add_pow_a_b_int @ r @ K @ B ) )
% 0.25/0.58 = ( add_pow_a_b_int @ r @ K @ ( mult_a_ring_ext_a_b @ r @ A @ B ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_pow_rdistr_int
% 0.25/0.58 thf(fact_121_add_Oint__pow__distrib,axiom,
% 0.25/0.58 ! [X: a,Y: a,I: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_distrib
% 0.25/0.58 thf(fact_122_add_Oint__pow__mult__distrib,axiom,
% 0.25/0.58 ! [X: a,Y: a,I: int] :
% 0.25/0.58 ( ( ( add_a_b @ r @ X @ Y )
% 0.25/0.58 = ( add_a_b @ r @ Y @ X ) )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ I @ ( add_a_b @ r @ X @ Y ) )
% 0.25/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ I @ Y ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_mult_distrib
% 0.25/0.58 thf(fact_123_add_Oint__pow__closed,axiom,
% 0.25/0.58 ! [X: a,I: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( member_a @ ( add_pow_a_b_int @ r @ I @ X ) @ ( partia1066395285xt_a_b @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_closed
% 0.25/0.58 thf(fact_124_add_Oint__pow__one,axiom,
% 0.25/0.58 ! [Z: int] :
% 0.25/0.58 ( ( add_pow_a_b_int @ r @ Z @ ( zero_a_b @ r ) )
% 0.25/0.58 = ( zero_a_b @ r ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_one
% 0.25/0.58 thf(fact_125_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Ois__submonoid,axiom,
% 0.25/0.58 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 0.25/0.58 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.25/0.58 => ( locali425460998xt_a_b @ M @ S2 ) ) ).
% 0.25/0.58
% 0.25/0.58 % Localization_Mirabelle_afvchqjmgj.submonoid.is_submonoid
% 0.25/0.58 thf(fact_126_mult__submonoid__of__crng_Oaxioms_I2_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.25/0.58 ( ( locali807230110ng_a_b @ R @ S2 )
% 0.25/0.58 => ( locali880295127ng_a_b @ R @ S2 ) ) ).
% 0.25/0.58
% 0.25/0.58 % mult_submonoid_of_crng.axioms(2)
% 0.25/0.58 thf(fact_127_mult__submonoid__of__rng_Oaxioms_I2_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.25/0.58 ( ( locali880295127ng_a_b @ R @ S2 )
% 0.25/0.58 => ( locali425460998xt_a_b @ R @ S2 ) ) ).
% 0.25/0.58
% 0.25/0.58 % mult_submonoid_of_rng.axioms(2)
% 0.25/0.58 thf(fact_128_eq__obj__rng__of__frac_Oaxioms_I2_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,S2: set_a] :
% 0.25/0.58 ( ( locali1648887798ac_a_b @ R @ S2 )
% 0.25/0.58 => ( locali807230110ng_a_b @ R @ S2 ) ) ).
% 0.25/0.58
% 0.25/0.58 % eq_obj_rng_of_frac.axioms(2)
% 0.25/0.58 thf(fact_129_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Om__closed,axiom,
% 0.25/0.58 ! [M: partia1833973666xt_a_b,S2: set_a,X: a,Y: a] :
% 0.25/0.58 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.25/0.58 => ( ( member_a @ X @ S2 )
% 0.25/0.58 => ( ( member_a @ Y @ S2 )
% 0.25/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ M @ X @ Y ) @ S2 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % Localization_Mirabelle_afvchqjmgj.submonoid.m_closed
% 0.25/0.58 thf(fact_130_a__minus__def,axiom,
% 0.25/0.58 ( a_minus_a_b
% 0.25/0.58 = ( ^ [R2: partia1833973666xt_a_b,X2: a,Y5: a] : ( add_a_b @ R2 @ X2 @ ( a_inv_a_b @ R2 @ Y5 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % a_minus_def
% 0.25/0.58 thf(fact_131_semiring_Osemiring__simprules_I8_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ Z )
% 0.25/0.58 = ( mult_a_ring_ext_a_b @ R @ X @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(8)
% 0.25/0.58 thf(fact_132_semiring_Osemiring__simprules_I3_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( member_a @ ( mult_a_ring_ext_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(3)
% 0.25/0.58 thf(fact_133_semiring_Osemiring__simprules_I12_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) )
% 0.25/0.58 = ( add_a_b @ R @ Y @ ( add_a_b @ R @ X @ Z ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(12)
% 0.25/0.58 thf(fact_134_semiring_Osemiring__simprules_I7_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( add_a_b @ R @ X @ Y )
% 0.25/0.58 = ( add_a_b @ R @ Y @ X ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(7)
% 0.25/0.58 thf(fact_135_semiring_Osemiring__simprules_I5_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( add_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 0.25/0.58 = ( add_a_b @ R @ X @ ( add_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(5)
% 0.25/0.58 thf(fact_136_semiring_Osemiring__simprules_I1_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( member_a @ ( add_a_b @ R @ X @ Y ) @ ( partia1066395285xt_a_b @ R ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(1)
% 0.25/0.58 thf(fact_137_semiring_Osemiring__simprules_I2_J,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( member_a @ ( zero_a_b @ R ) @ ( partia1066395285xt_a_b @ R ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.semiring_simprules(2)
% 0.25/0.58 thf(fact_138_semiring_Ol__distr,axiom,
% 0.25/0.58 ! [R: partia1833973666xt_a_b,X: a,Y: a,Z: a] :
% 0.25/0.58 ( ( semiring_a_b @ R )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Y @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( member_a @ Z @ ( partia1066395285xt_a_b @ R ) )
% 0.25/0.58 => ( ( mult_a_ring_ext_a_b @ R @ ( add_a_b @ R @ X @ Y ) @ Z )
% 0.25/0.58 = ( add_a_b @ R @ ( mult_a_ring_ext_a_b @ R @ X @ Z ) @ ( mult_a_ring_ext_a_b @ R @ Y @ Z ) ) ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % semiring.l_distr
% 0.25/0.58 thf(fact_139_add_Oint__pow__diff,axiom,
% 0.25/0.58 ! [X: a,N: int,M2: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ ( minus_minus_int @ N @ M2 ) @ X )
% 0.25/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ N @ X ) @ ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ M2 @ X ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_diff
% 0.25/0.58 thf(fact_140_add_Oint__pow__neg,axiom,
% 0.25/0.58 ! [X: a,I: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ ( uminus_uminus_int @ I ) @ X )
% 0.25/0.58 = ( a_inv_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_neg
% 0.25/0.58 thf(fact_141_add_Oint__pow__mult,axiom,
% 0.25/0.58 ! [X: a,I: int,J: int] :
% 0.25/0.58 ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( add_pow_a_b_int @ r @ ( plus_plus_int @ I @ J ) @ X )
% 0.25/0.58 = ( add_a_b @ r @ ( add_pow_a_b_int @ r @ I @ X ) @ ( add_pow_a_b_int @ r @ J @ X ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.int_pow_mult
% 0.25/0.58 thf(fact_142_subset,axiom,
% 0.25/0.58 ord_less_eq_set_a @ s @ ( partia1066395285xt_a_b @ r ) ).
% 0.25/0.58
% 0.25/0.58 % subset
% 0.25/0.58 thf(fact_143_Localization__Mirabelle__afvchqjmgj_Osubmonoid_Osubset,axiom,
% 0.25/0.58 ! [M: partia1833973666xt_a_b,S2: set_a] :
% 0.25/0.58 ( ( locali425460998xt_a_b @ M @ S2 )
% 0.25/0.58 => ( ord_less_eq_set_a @ S2 @ ( partia1066395285xt_a_b @ M ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % Localization_Mirabelle_afvchqjmgj.submonoid.subset
% 0.25/0.58 thf(fact_144_diff__minus__eq__add,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
% 0.25/0.58 = ( plus_plus_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_minus_eq_add
% 0.25/0.58 thf(fact_145_uminus__add__conv__diff,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
% 0.25/0.58 = ( minus_minus_int @ B @ A ) ) ).
% 0.25/0.58
% 0.25/0.58 % uminus_add_conv_diff
% 0.25/0.58 thf(fact_146_le__add__diff__inverse,axiom,
% 0.25/0.58 ! [B: int,A: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.25/0.58 => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
% 0.25/0.58 = A ) ) ).
% 0.25/0.58
% 0.25/0.58 % le_add_diff_inverse
% 0.25/0.58 thf(fact_147_le__add__diff__inverse2,axiom,
% 0.25/0.58 ! [B: int,A: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.25/0.58 => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.25/0.58 = A ) ) ).
% 0.25/0.58
% 0.25/0.58 % le_add_diff_inverse2
% 0.25/0.58 thf(fact_148_add__right__cancel,axiom,
% 0.25/0.58 ! [B: int,A: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.25/0.58 = ( plus_plus_int @ C @ A ) )
% 0.25/0.58 = ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_right_cancel
% 0.25/0.58 thf(fact_149_add__left__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.25/0.58 = ( plus_plus_int @ A @ C ) )
% 0.25/0.58 = ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_left_cancel
% 0.25/0.58 thf(fact_150_neg__equal__iff__equal,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( ( uminus_uminus_int @ A )
% 0.25/0.58 = ( uminus_uminus_int @ B ) )
% 0.25/0.58 = ( A = B ) ) ).
% 0.25/0.58
% 0.25/0.58 % neg_equal_iff_equal
% 0.25/0.58 thf(fact_151_add_Oinverse__inverse,axiom,
% 0.25/0.58 ! [A: int] :
% 0.25/0.58 ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
% 0.25/0.58 = A ) ).
% 0.25/0.58
% 0.25/0.58 % add.inverse_inverse
% 0.25/0.58 thf(fact_152_add__le__cancel__right,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.25/0.58 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_cancel_right
% 0.25/0.58 thf(fact_153_add__le__cancel__left,axiom,
% 0.25/0.58 ! [C: int,A: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.25/0.58 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_cancel_left
% 0.25/0.58 thf(fact_154_neg__le__iff__le,axiom,
% 0.25/0.58 ! [B: int,A: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
% 0.25/0.58 = ( ord_less_eq_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % neg_le_iff_le
% 0.25/0.58 thf(fact_155_add__diff__cancel__right_H,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.25/0.58 = A ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_cancel_right'
% 0.25/0.58 thf(fact_156_add__diff__cancel__right,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.25/0.58 = ( minus_minus_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_cancel_right
% 0.25/0.58 thf(fact_157_add__diff__cancel__left_H,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
% 0.25/0.58 = B ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_cancel_left'
% 0.25/0.58 thf(fact_158_add__diff__cancel__left,axiom,
% 0.25/0.58 ! [C: int,A: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.25/0.58 = ( minus_minus_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_cancel_left
% 0.25/0.58 thf(fact_159_diff__add__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
% 0.25/0.58 = A ) ).
% 0.25/0.58
% 0.25/0.58 % diff_add_cancel
% 0.25/0.58 thf(fact_160_add__diff__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
% 0.25/0.58 = A ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_cancel
% 0.25/0.58 thf(fact_161_minus__add__distrib,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.25/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_add_distrib
% 0.25/0.58 thf(fact_162_minus__add__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
% 0.25/0.58 = B ) ).
% 0.25/0.58
% 0.25/0.58 % minus_add_cancel
% 0.25/0.58 thf(fact_163_add__minus__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
% 0.25/0.58 = B ) ).
% 0.25/0.58
% 0.25/0.58 % add_minus_cancel
% 0.25/0.58 thf(fact_164_minus__diff__eq,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
% 0.25/0.58 = ( minus_minus_int @ B @ A ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_diff_eq
% 0.25/0.58 thf(fact_165_add__right__imp__eq,axiom,
% 0.25/0.58 ! [B: int,A: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.25/0.58 = ( plus_plus_int @ C @ A ) )
% 0.25/0.58 => ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_right_imp_eq
% 0.25/0.58 thf(fact_166_add__left__imp__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.25/0.58 = ( plus_plus_int @ A @ C ) )
% 0.25/0.58 => ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_left_imp_eq
% 0.25/0.58 thf(fact_167_add_Oleft__commute,axiom,
% 0.25/0.58 ! [B: int,A: int,C: int] :
% 0.25/0.58 ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
% 0.25/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.left_commute
% 0.25/0.58 thf(fact_168_add_Ocommute,axiom,
% 0.25/0.58 ( plus_plus_int
% 0.25/0.58 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.commute
% 0.25/0.58 thf(fact_169_group__add__class_Oadd_Oright__cancel,axiom,
% 0.25/0.58 ! [B: int,A: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ B @ A )
% 0.25/0.58 = ( plus_plus_int @ C @ A ) )
% 0.25/0.58 = ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_add_class.add.right_cancel
% 0.25/0.58 thf(fact_170_add_Oleft__cancel,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ A @ B )
% 0.25/0.58 = ( plus_plus_int @ A @ C ) )
% 0.25/0.58 = ( B = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.left_cancel
% 0.25/0.58 thf(fact_171_add_Oassoc,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.25/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.assoc
% 0.25/0.58 thf(fact_172_group__cancel_Oadd2,axiom,
% 0.25/0.58 ! [B3: int,K: int,B: int,A: int] :
% 0.25/0.58 ( ( B3
% 0.25/0.58 = ( plus_plus_int @ K @ B ) )
% 0.25/0.58 => ( ( plus_plus_int @ A @ B3 )
% 0.25/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_cancel.add2
% 0.25/0.58 thf(fact_173_group__cancel_Oadd1,axiom,
% 0.25/0.58 ! [A2: int,K: int,A: int,B: int] :
% 0.25/0.58 ( ( A2
% 0.25/0.58 = ( plus_plus_int @ K @ A ) )
% 0.25/0.58 => ( ( plus_plus_int @ A2 @ B )
% 0.25/0.58 = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_cancel.add1
% 0.25/0.58 thf(fact_174_add__mono__thms__linordered__semiring_I4_J,axiom,
% 0.25/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.25/0.58 ( ( ( I = J )
% 0.25/0.58 & ( K = L ) )
% 0.25/0.58 => ( ( plus_plus_int @ I @ K )
% 0.25/0.58 = ( plus_plus_int @ J @ L ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_mono_thms_linordered_semiring(4)
% 0.25/0.58 thf(fact_175_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
% 0.25/0.58 = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % ab_semigroup_add_class.add_ac(1)
% 0.25/0.58 thf(fact_176_diff__right__commute,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
% 0.25/0.58 = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_right_commute
% 0.25/0.58 thf(fact_177_diff__eq__diff__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.25/0.58 ( ( ( minus_minus_int @ A @ B )
% 0.25/0.58 = ( minus_minus_int @ C @ D ) )
% 0.25/0.58 => ( ( A = B )
% 0.25/0.58 = ( C = D ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_eq_diff_eq
% 0.25/0.58 thf(fact_178_minus__equation__iff,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( ( uminus_uminus_int @ A )
% 0.25/0.58 = B )
% 0.25/0.58 = ( ( uminus_uminus_int @ B )
% 0.25/0.58 = A ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_equation_iff
% 0.25/0.58 thf(fact_179_equation__minus__iff,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( A
% 0.25/0.58 = ( uminus_uminus_int @ B ) )
% 0.25/0.58 = ( B
% 0.25/0.58 = ( uminus_uminus_int @ A ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % equation_minus_iff
% 0.25/0.58 thf(fact_180_add__le__imp__le__right,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
% 0.25/0.58 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_imp_le_right
% 0.25/0.58 thf(fact_181_add__le__imp__le__left,axiom,
% 0.25/0.58 ! [C: int,A: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
% 0.25/0.58 => ( ord_less_eq_int @ A @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_imp_le_left
% 0.25/0.58 thf(fact_182_add__right__mono,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_right_mono
% 0.25/0.58 thf(fact_183_add__left__mono,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_left_mono
% 0.25/0.58 thf(fact_184_add__mono,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ( ord_less_eq_int @ C @ D )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_mono
% 0.25/0.58 thf(fact_185_add__mono__thms__linordered__semiring_I1_J,axiom,
% 0.25/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.25/0.58 ( ( ( ord_less_eq_int @ I @ J )
% 0.25/0.58 & ( ord_less_eq_int @ K @ L ) )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_mono_thms_linordered_semiring(1)
% 0.25/0.58 thf(fact_186_add__mono__thms__linordered__semiring_I2_J,axiom,
% 0.25/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.25/0.58 ( ( ( I = J )
% 0.25/0.58 & ( ord_less_eq_int @ K @ L ) )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_mono_thms_linordered_semiring(2)
% 0.25/0.58 thf(fact_187_add__mono__thms__linordered__semiring_I3_J,axiom,
% 0.25/0.58 ! [I: int,J: int,K: int,L: int] :
% 0.25/0.58 ( ( ( ord_less_eq_int @ I @ J )
% 0.25/0.58 & ( K = L ) )
% 0.25/0.58 => ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_mono_thms_linordered_semiring(3)
% 0.25/0.58 thf(fact_188_diff__eq__diff__less__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int,D: int] :
% 0.25/0.58 ( ( ( minus_minus_int @ A @ B )
% 0.25/0.58 = ( minus_minus_int @ C @ D ) )
% 0.25/0.58 => ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 = ( ord_less_eq_int @ C @ D ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_eq_diff_less_eq
% 0.25/0.58 thf(fact_189_diff__right__mono,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_right_mono
% 0.25/0.58 thf(fact_190_diff__left__mono,axiom,
% 0.25/0.58 ! [B: int,A: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ B @ A )
% 0.25/0.58 => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_left_mono
% 0.25/0.58 thf(fact_191_diff__mono,axiom,
% 0.25/0.58 ! [A: int,B: int,D: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ( ord_less_eq_int @ D @ C )
% 0.25/0.58 => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_mono
% 0.25/0.58 thf(fact_192_le__imp__neg__le,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ B )
% 0.25/0.58 => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % le_imp_neg_le
% 0.25/0.58 thf(fact_193_minus__le__iff,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
% 0.25/0.58 = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_le_iff
% 0.25/0.58 thf(fact_194_le__minus__iff,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
% 0.25/0.58 = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % le_minus_iff
% 0.25/0.58 thf(fact_195_add__implies__diff,axiom,
% 0.25/0.58 ! [C: int,B: int,A: int] :
% 0.25/0.58 ( ( ( plus_plus_int @ C @ B )
% 0.25/0.58 = A )
% 0.25/0.58 => ( C
% 0.25/0.58 = ( minus_minus_int @ A @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_implies_diff
% 0.25/0.58 thf(fact_196_diff__diff__add,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.25/0.58 = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_diff_add
% 0.25/0.58 thf(fact_197_diff__add__eq__diff__diff__swap,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
% 0.25/0.58 = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_add_eq_diff_diff_swap
% 0.25/0.58 thf(fact_198_diff__add__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.25/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_add_eq
% 0.25/0.58 thf(fact_199_diff__diff__eq2,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.25/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_diff_eq2
% 0.25/0.58 thf(fact_200_add__diff__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
% 0.25/0.58 = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_diff_eq
% 0.25/0.58 thf(fact_201_eq__diff__eq,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( A
% 0.25/0.58 = ( minus_minus_int @ C @ B ) )
% 0.25/0.58 = ( ( plus_plus_int @ A @ B )
% 0.25/0.58 = C ) ) ).
% 0.25/0.58
% 0.25/0.58 % eq_diff_eq
% 0.25/0.58 thf(fact_202_diff__eq__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ( minus_minus_int @ A @ B )
% 0.25/0.58 = C )
% 0.25/0.58 = ( A
% 0.25/0.58 = ( plus_plus_int @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_eq_eq
% 0.25/0.58 thf(fact_203_group__cancel_Osub1,axiom,
% 0.25/0.58 ! [A2: int,K: int,A: int,B: int] :
% 0.25/0.58 ( ( A2
% 0.25/0.58 = ( plus_plus_int @ K @ A ) )
% 0.25/0.58 => ( ( minus_minus_int @ A2 @ B )
% 0.25/0.58 = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_cancel.sub1
% 0.25/0.58 thf(fact_204_add_Oinverse__distrib__swap,axiom,
% 0.25/0.58 ! [A: int,B: int] :
% 0.25/0.58 ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
% 0.25/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.inverse_distrib_swap
% 0.25/0.58 thf(fact_205_group__cancel_Oneg1,axiom,
% 0.25/0.58 ! [A2: int,K: int,A: int] :
% 0.25/0.58 ( ( A2
% 0.25/0.58 = ( plus_plus_int @ K @ A ) )
% 0.25/0.58 => ( ( uminus_uminus_int @ A2 )
% 0.25/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( uminus_uminus_int @ A ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_cancel.neg1
% 0.25/0.58 thf(fact_206_minus__diff__commute,axiom,
% 0.25/0.58 ! [B: int,A: int] :
% 0.25/0.58 ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
% 0.25/0.58 = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).
% 0.25/0.58
% 0.25/0.58 % minus_diff_commute
% 0.25/0.58 thf(fact_207_add__le__add__imp__diff__le,axiom,
% 0.25/0.58 ! [I: int,K: int,N: int,J: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.25/0.58 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.25/0.58 => ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.25/0.58 => ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
% 0.25/0.58 => ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_add_imp_diff_le
% 0.25/0.58 thf(fact_208_add__le__imp__le__diff,axiom,
% 0.25/0.58 ! [I: int,K: int,N: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( plus_plus_int @ I @ K ) @ N )
% 0.25/0.58 => ( ord_less_eq_int @ I @ ( minus_minus_int @ N @ K ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_le_imp_le_diff
% 0.25/0.58 thf(fact_209_le__diff__eq,axiom,
% 0.25/0.58 ! [A: int,C: int,B: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
% 0.25/0.58 = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% 0.25/0.58
% 0.25/0.58 % le_diff_eq
% 0.25/0.58 thf(fact_210_diff__le__eq,axiom,
% 0.25/0.58 ! [A: int,B: int,C: int] :
% 0.25/0.58 ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
% 0.25/0.58 = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_le_eq
% 0.25/0.58 thf(fact_211_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
% 0.25/0.58 ( minus_minus_int
% 0.25/0.58 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % ab_group_add_class.ab_diff_conv_add_uminus
% 0.25/0.58 thf(fact_212_diff__conv__add__uminus,axiom,
% 0.25/0.58 ( minus_minus_int
% 0.25/0.58 = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % diff_conv_add_uminus
% 0.25/0.58 thf(fact_213_group__cancel_Osub2,axiom,
% 0.25/0.58 ! [B3: int,K: int,B: int,A: int] :
% 0.25/0.58 ( ( B3
% 0.25/0.58 = ( plus_plus_int @ K @ B ) )
% 0.25/0.58 => ( ( minus_minus_int @ A @ B3 )
% 0.25/0.58 = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % group_cancel.sub2
% 0.25/0.58 thf(fact_214_add_Oone__in__subset,axiom,
% 0.25/0.58 ! [H: set_a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( H != bot_bot_set_a )
% 0.25/0.58 => ( ! [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ H )
% 0.25/0.58 => ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H ) )
% 0.25/0.58 => ( ! [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ H )
% 0.25/0.58 => ! [Xa: a] :
% 0.25/0.58 ( ( member_a @ Xa @ H )
% 0.25/0.58 => ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H ) ) )
% 0.25/0.58 => ( member_a @ ( zero_a_b @ r ) @ H ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add.one_in_subset
% 0.25/0.58 thf(fact_215_a__lcos__mult__one,axiom,
% 0.25/0.58 ! [M: set_a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_l_coset_a_b @ r @ ( zero_a_b @ r ) @ M )
% 0.25/0.58 = M ) ) ).
% 0.25/0.58
% 0.25/0.58 % a_lcos_mult_one
% 0.25/0.58 thf(fact_216_a__lcos__m__assoc,axiom,
% 0.25/0.58 ! [M: set_a,G: a,H2: a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ M @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ G @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ H2 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( a_l_coset_a_b @ r @ G @ ( a_l_coset_a_b @ r @ H2 @ M ) )
% 0.25/0.58 = ( a_l_coset_a_b @ r @ ( add_a_b @ r @ G @ H2 ) @ M ) ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % a_lcos_m_assoc
% 0.25/0.58 thf(fact_217_carrier__not__empty,axiom,
% 0.25/0.58 ( ( partia1066395285xt_a_b @ r )
% 0.25/0.58 != bot_bot_set_a ) ).
% 0.25/0.58
% 0.25/0.58 % carrier_not_empty
% 0.25/0.58 thf(fact_218_a__l__coset__subset__G,axiom,
% 0.25/0.58 ! [H: set_a,X: a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( member_a @ X @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ord_less_eq_set_a @ ( a_l_coset_a_b @ r @ X @ H ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % a_l_coset_subset_G
% 0.25/0.58 thf(fact_219_setadd__subset__G,axiom,
% 0.25/0.58 ! [H: set_a,K2: set_a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ H @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ord_less_eq_set_a @ K2 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H @ K2 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % setadd_subset_G
% 0.25/0.58 thf(fact_220_set__add__closed,axiom,
% 0.25/0.58 ! [A2: set_a,B3: set_a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ A2 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ( ord_less_eq_set_a @ B3 @ ( partia1066395285xt_a_b @ r ) )
% 0.25/0.58 => ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B3 ) @ ( partia1066395285xt_a_b @ r ) ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % set_add_closed
% 0.25/0.58 thf(fact_221_add__additive__subgroups,axiom,
% 0.25/0.58 ! [H: set_a,K2: set_a] :
% 0.25/0.58 ( ( additi2104487374up_a_b @ H @ r )
% 0.25/0.58 => ( ( additi2104487374up_a_b @ K2 @ r )
% 0.25/0.58 => ( additi2104487374up_a_b @ ( set_add_a_b @ r @ H @ K2 ) @ r ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % add_additive_subgroups
% 0.25/0.58 thf(fact_222_Diff__eq__empty__iff,axiom,
% 0.25/0.58 ! [A2: set_a,B3: set_a] :
% 0.25/0.58 ( ( ( minus_minus_set_a @ A2 @ B3 )
% 0.25/0.58 = bot_bot_set_a )
% 0.25/0.58 = ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.25/0.58
% 0.25/0.58 % Diff_eq_empty_iff
% 0.25/0.58 thf(fact_223_empty__iff,axiom,
% 0.25/0.58 ! [C: a] :
% 0.25/0.58 ~ ( member_a @ C @ bot_bot_set_a ) ).
% 0.25/0.58
% 0.25/0.58 % empty_iff
% 0.25/0.58 thf(fact_224_all__not__in__conv,axiom,
% 0.25/0.58 ! [A2: set_a] :
% 0.25/0.58 ( ( ! [X2: a] :
% 0.25/0.58 ~ ( member_a @ X2 @ A2 ) )
% 0.25/0.58 = ( A2 = bot_bot_set_a ) ) ).
% 0.25/0.58
% 0.25/0.58 % all_not_in_conv
% 0.25/0.58 thf(fact_225_Collect__empty__eq,axiom,
% 0.25/0.58 ! [P: a > $o] :
% 0.25/0.58 ( ( ( collect_a @ P )
% 0.25/0.58 = bot_bot_set_a )
% 0.25/0.58 = ( ! [X2: a] :
% 0.25/0.58 ~ ( P @ X2 ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % Collect_empty_eq
% 0.25/0.58 thf(fact_226_empty__Collect__eq,axiom,
% 0.25/0.58 ! [P: a > $o] :
% 0.25/0.58 ( ( bot_bot_set_a
% 0.25/0.58 = ( collect_a @ P ) )
% 0.25/0.58 = ( ! [X2: a] :
% 0.25/0.58 ~ ( P @ X2 ) ) ) ).
% 0.25/0.58
% 0.25/0.58 % empty_Collect_eq
% 0.25/0.58 thf(fact_227_subsetI,axiom,
% 0.25/0.58 ! [A2: set_a,B3: set_a] :
% 0.25/0.58 ( ! [X3: a] :
% 0.25/0.58 ( ( member_a @ X3 @ A2 )
% 0.25/0.58 => ( member_a @ X3 @ B3 ) )
% 0.25/0.58 => ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% 0.25/0.58
% 0.25/0.58 % subsetI
% 0.25/0.58 thf(fact_228_subset__antisym,axiom,
% 0.25/0.58 ! [A2: set_a,B3: set_a] :
% 0.25/0.58 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.25/0.58 => ( ( ord_less_eq_set_a @ B3 @ A2 )
% 0.25/0.58 => ( A2 = B3 ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % subset_antisym
% 0.25/0.65 thf(fact_229_Compl__subset__Compl__iff,axiom,
% 0.25/0.65 ! [A2: set_a,B3: set_a] :
% 0.25/0.65 ( ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ A2 ) @ ( uminus_uminus_set_a @ B3 ) )
% 0.25/0.65 = ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% 0.25/0.65
% 0.25/0.65 % Compl_subset_Compl_iff
% 0.25/0.65 thf(fact_230_Compl__anti__mono,axiom,
% 0.25/0.65 ! [A2: set_a,B3: set_a] :
% 0.25/0.65 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.25/0.65 => ( ord_less_eq_set_a @ ( uminus_uminus_set_a @ B3 ) @ ( uminus_uminus_set_a @ A2 ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % Compl_anti_mono
% 0.25/0.65 thf(fact_231_subset__empty,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
% 0.25/0.65 = ( A2 = bot_bot_set_a ) ) ).
% 0.25/0.65
% 0.25/0.65 % subset_empty
% 0.25/0.65 thf(fact_232_empty__subsetI,axiom,
% 0.25/0.65 ! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% 0.25/0.65
% 0.25/0.65 % empty_subsetI
% 0.25/0.65 thf(fact_233_Diff__empty,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
% 0.25/0.65 = A2 ) ).
% 0.25/0.65
% 0.25/0.65 % Diff_empty
% 0.25/0.65 thf(fact_234_empty__Diff,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
% 0.25/0.65 = bot_bot_set_a ) ).
% 0.25/0.65
% 0.25/0.65 % empty_Diff
% 0.25/0.65 thf(fact_235_Diff__cancel,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ( minus_minus_set_a @ A2 @ A2 )
% 0.25/0.65 = bot_bot_set_a ) ).
% 0.25/0.65
% 0.25/0.65 % Diff_cancel
% 0.25/0.65 thf(fact_236_emptyE,axiom,
% 0.25/0.65 ! [A: a] :
% 0.25/0.65 ~ ( member_a @ A @ bot_bot_set_a ) ).
% 0.25/0.65
% 0.25/0.65 % emptyE
% 0.25/0.65 thf(fact_237_equals0D,axiom,
% 0.25/0.65 ! [A2: set_a,A: a] :
% 0.25/0.65 ( ( A2 = bot_bot_set_a )
% 0.25/0.65 => ~ ( member_a @ A @ A2 ) ) ).
% 0.25/0.65
% 0.25/0.65 % equals0D
% 0.25/0.65 thf(fact_238_equals0I,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ! [Y2: a] :
% 0.25/0.65 ~ ( member_a @ Y2 @ A2 )
% 0.25/0.65 => ( A2 = bot_bot_set_a ) ) ).
% 0.25/0.65
% 0.25/0.65 % equals0I
% 0.25/0.65 thf(fact_239_ex__in__conv,axiom,
% 0.25/0.65 ! [A2: set_a] :
% 0.25/0.65 ( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
% 0.25/0.65 = ( A2 != bot_bot_set_a ) ) ).
% 0.25/0.65
% 0.25/0.65 % ex_in_conv
% 0.25/0.65 thf(fact_240_in__mono,axiom,
% 0.25/0.65 ! [A2: set_a,B3: set_a,X: a] :
% 0.25/0.65 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.25/0.65 => ( ( member_a @ X @ A2 )
% 0.25/0.65 => ( member_a @ X @ B3 ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % in_mono
% 0.25/0.65 thf(fact_241_subsetD,axiom,
% 0.25/0.65 ! [A2: set_a,B3: set_a,C: a] :
% 0.25/0.65 ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.25/0.65 => ( ( member_a @ C @ A2 )
% 0.25/0.65 => ( member_a @ C @ B3 ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % subsetD
% 0.25/0.65 thf(fact_242_equalityE,axiom,
% 0.25/0.65 ! [A2: set_a,B3: set_a] :
% 0.25/0.65 ( ( A2 = B3 )
% 0.25/0.65 => ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
% 0.25/0.65 => ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % equalityE
% 0.25/0.65 thf(fact_243_subset__eq,axiom,
% 0.25/0.65 ( ord_less_eq_set_a
% 0.25/0.65 = ( ^ [A4: set_a,B4: set_a] :
% 0.25/0.65 ! [X2: a] :
% 0.25/0.65 ( ( member_a @ X2 @ A4 )
% 0.25/0.65 => ( member_a @ X2 @ B4 ) ) ) ) ).
% 0.25/0.65
% 0.25/0.65 % subset_eq
% 0.25/0.65
% 0.25/0.65 % Conjectures (1)
% 0.25/0.65 thf(conj_0,conjecture,
% 0.25/0.65 ( ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t @ s3 ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ t2 @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ r2 ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ ( product_fst_a_a @ x ) ) ) ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ s2 ) @ ( product_snd_a_a @ x ) ) @ ( mult_a_ring_ext_a_b @ r @ t @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ r3 ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ ( product_fst_a_a @ y ) ) ) ) ) )
% 0.25/0.65 = ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ t2 @ t ) @ ( add_a_b @ r @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s3 @ r2 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ y ) @ ( product_fst_a_a @ x ) ) ) ) @ ( a_minus_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_snd_a_a @ y ) ) @ ( mult_a_ring_ext_a_b @ r @ s2 @ r3 ) ) @ ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ s2 @ s3 ) @ ( mult_a_ring_ext_a_b @ r @ ( product_snd_a_a @ x ) @ ( product_fst_a_a @ y ) ) ) ) ) ) ) ).
% 0.25/0.65
% 0.25/0.65 %------------------------------------------------------------------------------
% 0.25/0.65 ------- convert to smt2 : /export/starexec/sandbox/tmp/tmp.59bE3LXaPe/cvc5---1.0.5_15236.p...
% 0.25/0.65 (declare-sort $$unsorted 0)
% 0.25/0.65 (declare-sort tptp.partia1833973666xt_a_b 0)
% 0.25/0.65 (declare-sort tptp.product_prod_a_a 0)
% 0.25/0.65 (declare-sort tptp.set_a 0)
% 0.25/0.65 (declare-sort tptp.int 0)
% 0.25/0.65 (declare-sort tptp.a 0)
% 0.25/0.65 (declare-fun tptp.a_l_coset_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.set_a) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.additi2104487374up_a_b (tptp.set_a tptp.partia1833973666xt_a_b) Bool)
% 0.25/0.65 (declare-fun tptp.set_add_a_b (tptp.partia1833973666xt_a_b tptp.set_a tptp.set_a) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.partia1066395285xt_a_b (tptp.partia1833973666xt_a_b) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.mult_a_ring_ext_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.25/0.65 (declare-fun tptp.minus_minus_int (tptp.int tptp.int) tptp.int)
% 0.25/0.65 (declare-fun tptp.minus_minus_set_a (tptp.set_a tptp.set_a) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.plus_plus_int (tptp.int tptp.int) tptp.int)
% 0.25/0.65 (declare-fun tptp.uminus_uminus_int (tptp.int) tptp.int)
% 0.25/0.65 (declare-fun tptp.uminus_uminus_set_a (tptp.set_a) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.locali1648887798ac_a_b (tptp.partia1833973666xt_a_b tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.locali807230110ng_a_b (tptp.partia1833973666xt_a_b tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.locali880295127ng_a_b (tptp.partia1833973666xt_a_b tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.locali425460998xt_a_b (tptp.partia1833973666xt_a_b tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.bot_bot_set_a () tptp.set_a)
% 0.25/0.65 (declare-fun tptp.ord_less_eq_int (tptp.int tptp.int) Bool)
% 0.25/0.65 (declare-fun tptp.ord_less_eq_set_a (tptp.set_a tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.product_fst_a_a (tptp.product_prod_a_a) tptp.a)
% 0.25/0.65 (declare-fun tptp.product_snd_a_a (tptp.product_prod_a_a) tptp.a)
% 0.25/0.65 (declare-fun tptp.a_inv_a_b (tptp.partia1833973666xt_a_b tptp.a) tptp.a)
% 0.25/0.65 (declare-fun tptp.a_minus_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.25/0.65 (declare-fun tptp.add_pow_a_b_int (tptp.partia1833973666xt_a_b tptp.int tptp.a) tptp.a)
% 0.25/0.65 (declare-fun tptp.add_a_b (tptp.partia1833973666xt_a_b tptp.a tptp.a) tptp.a)
% 0.25/0.65 (declare-fun tptp.zero_a_b (tptp.partia1833973666xt_a_b) tptp.a)
% 0.25/0.65 (declare-fun tptp.semiring_a_b (tptp.partia1833973666xt_a_b) Bool)
% 0.25/0.65 (declare-fun tptp.collect_a ((-> tptp.a Bool)) tptp.set_a)
% 0.25/0.65 (declare-fun tptp.member_a (tptp.a tptp.set_a) Bool)
% 0.25/0.65 (declare-fun tptp.r () tptp.partia1833973666xt_a_b)
% 0.25/0.65 (declare-fun tptp.s () tptp.set_a)
% 0.25/0.65 (declare-fun tptp.r2 () tptp.a)
% 0.25/0.65 (declare-fun tptp.r3 () tptp.a)
% 0.25/0.65 (declare-fun tptp.s2 () tptp.a)
% 0.25/0.65 (declare-fun tptp.s3 () tptp.a)
% 0.25/0.65 (declare-fun tptp.t () tptp.a)
% 0.25/0.65 (declare-fun tptp.t2 () tptp.a)
% 0.25/0.65 (declare-fun tptp.x () tptp.product_prod_a_a)
% 0.25/0.65 (declare-fun tptp.y () tptp.product_prod_a_a)
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.t) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.t2) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ (@ tptp.add_a_b tptp.r) C))) (=> (= A B) (= (@ _let_1 A) (@ _let_1 B))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ (@ tptp.a_minus_a_b tptp.r) C))) (=> (= A B) (= (@ _let_1 A) (@ _let_1 B))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z)))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z))))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z)))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z))))))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.s3) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.s2) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_2 (@ (@ _let_2 (@ tptp.product_snd_a_a tptp.x)) _let_1)) (@ (@ _let_2 tptp.s3) tptp.r2))) (@ (@ _let_2 (@ (@ _let_2 tptp.s2) tptp.s3)) (@ (@ _let_2 _let_1) (@ tptp.product_fst_a_a tptp.x))))) (@ tptp.partia1066395285xt_a_b tptp.r)))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.s2))) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ _let_2 (@ tptp.product_snd_a_a tptp.y))) (@ _let_3 tptp.r3))) (@ (@ _let_1 (@ _let_3 tptp.s3)) (@ _let_2 (@ tptp.product_fst_a_a tptp.y))))) (@ tptp.partia1066395285xt_a_b tptp.r))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ (@ _let_2 X) Y)) Z) (@ (@ _let_2 (@ (@ _let_1 X) Z)) (@ (@ _let_1 Y) Z)))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) Z))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y)))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a) (D tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (=> (@ (@ tptp.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)))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_2 (@ _let_1 C))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (=> (@ (@ tptp.member_a C) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ (@ tptp.add_a_b tptp.r) A) B)))))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ tptp.product_snd_a_a tptp.y)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 _let_2))) (let ((_let_5 (@ _let_3 tptp.s2))) (let ((_let_6 (@ _let_3 (@ _let_5 tptp.s3)))) (let ((_let_7 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_8 (@ _let_3 (@ _let_4 _let_7)))) (let ((_let_9 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_10 (@ _let_3 tptp.t2))) (let ((_let_11 (@ _let_3 (@ _let_10 tptp.t)))) (let ((_let_12 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_13 (@ _let_3 _let_7))) (let ((_let_14 (@ _let_3 tptp.s3))) (let ((_let_15 (@ tptp.add_a_b tptp.r))) (let ((_let_16 (@ _let_3 tptp.t))) (= (@ (@ _let_15 (@ (@ _let_3 (@ (@ _let_3 (@ _let_16 tptp.s3)) _let_7)) (@ _let_10 (@ (@ _let_9 (@ _let_4 tptp.r2)) (@ _let_5 _let_12))))) (@ (@ _let_3 (@ (@ _let_3 (@ _let_10 tptp.s2)) _let_2)) (@ _let_16 (@ (@ _let_9 (@ _let_13 tptp.r3)) (@ _let_14 _let_1))))) (@ (@ _let_15 (@ _let_11 (@ (@ _let_9 (@ _let_8 (@ _let_14 tptp.r2))) (@ _let_6 (@ _let_13 _let_12))))) (@ _let_11 (@ (@ _let_9 (@ _let_8 (@ _let_5 tptp.r3))) (@ _let_6 (@ _let_4 _let_1)))))))))))))))))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ tptp.product_fst_a_a tptp.x)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.s2))) (let ((_let_4 (@ _let_1 (@ _let_3 tptp.s3)))) (let ((_let_5 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_6 (@ _let_1 (@ _let_2 _let_5)))) (let ((_let_7 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_8 (@ (@ _let_7 (@ _let_6 (@ _let_3 tptp.r3))) (@ _let_4 (@ _let_2 (@ tptp.product_fst_a_a tptp.y)))))) (let ((_let_9 (@ (@ _let_1 tptp.t2) tptp.t))) (let ((_let_10 (@ _let_1 _let_9))) (let ((_let_11 (@ (@ _let_7 (@ _let_6 (@ (@ _let_1 tptp.s3) tptp.r2))) (@ _let_4 (@ (@ _let_1 _let_5) (@ tptp.product_fst_a_a tptp.x)))))) (let ((_let_12 (@ tptp.add_a_b tptp.r))) (let ((_let_13 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a _let_11) _let_13) (=> (@ (@ tptp.member_a _let_8) _let_13) (=> (@ (@ tptp.member_a _let_9) _let_13) (= (@ _let_10 (@ (@ _let_12 _let_11) _let_8)) (@ (@ _let_12 (@ _let_10 _let_11)) (@ _let_10 _let_8))))))))))))))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) tptp.s2) tptp.s3)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_snd_a_a tptp.y))) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) tptp.s3) (@ tptp.product_fst_a_a tptp.y))) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) tptp.s2) (@ tptp.product_fst_a_a tptp.x))) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) (@ tptp.product_snd_a_a tptp.x)) tptp.r2)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) (@ tptp.product_snd_a_a tptp.y)) tptp.r3)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_fst_a_a tptp.y)))) (let ((_let_3 (@ _let_1 (@ (@ _let_1 tptp.s2) tptp.s3)))) (let ((_let_4 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) (@ tptp.product_fst_a_a tptp.x)))) (let ((_let_5 (@ tptp.add_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_5 _let_4) _let_2)) (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s2) (@ tptp.product_fst_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.t2))) (let ((_let_4 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) tptp.r2))) (let ((_let_5 (@ tptp.a_minus_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_5 _let_4) _let_2)) (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s3) (@ tptp.product_fst_a_a tptp.y)))) (let ((_let_3 (@ _let_1 tptp.t))) (let ((_let_4 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) tptp.r3))) (let ((_let_5 (@ tptp.a_minus_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_5 _let_4) _let_2)) (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) (@ tptp.product_snd_a_a tptp.y)) (@ tptp.product_fst_a_a tptp.x))) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ tptp.product_snd_a_a tptp.y))) (= (@ (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)) (@ (@ _let_1 (@ _let_2 _let_3)) (@ (@ _let_1 tptp.s3) tptp.r2))) (@ (@ _let_1 (@ (@ _let_1 (@ (@ _let_1 (@ (@ _let_1 tptp.t) tptp.s3)) _let_3)) tptp.t2)) (@ _let_2 tptp.r2)))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ _let_1 (@ (@ _let_1 tptp.t) tptp.s3)) (@ tptp.product_snd_a_a tptp.y))) (@ tptp.partia1066395285xt_a_b tptp.r))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.s2)) (@ tptp.product_snd_a_a tptp.x))) (@ tptp.partia1066395285xt_a_b tptp.r))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ _let_2 tptp.s2))) (let ((_let_4 (@ tptp.product_snd_a_a tptp.y))) (= (@ (@ _let_2 (@ (@ _let_2 tptp.t2) tptp.t)) (@ (@ _let_2 (@ _let_3 tptp.s3)) (@ (@ _let_2 _let_4) _let_1))) (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_2 tptp.t) tptp.s3)) _let_4)) tptp.t2)) (@ _let_3 _let_1))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s2) tptp.r3))) (let ((_let_3 (@ _let_1 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_snd_a_a tptp.y))))) (let ((_let_4 (@ (@ _let_1 tptp.s3) tptp.r2))) (let ((_let_5 (@ tptp.add_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_5 _let_4) _let_2)) (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_fst_a_a tptp.y)))) (let ((_let_3 (@ _let_1 (@ (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)) (@ (@ _let_1 tptp.s2) tptp.s3))))) (let ((_let_4 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) (@ tptp.product_fst_a_a tptp.x)))) (let ((_let_5 (@ tptp.add_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_5 _let_4) _let_2)) (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ _let_1 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_snd_a_a tptp.y))) (@ (@ _let_1 tptp.s3) tptp.r2))) (@ tptp.partia1066395285xt_a_b tptp.r))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ _let_2 tptp.s2))) (let ((_let_4 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_5 (@ _let_2 (@ (@ _let_2 (@ (@ _let_2 (@ (@ _let_2 tptp.t) tptp.s3)) _let_4)) tptp.t2)))) (let ((_let_6 (@ _let_2 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_7 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_8 (@ _let_2 (@ (@ _let_2 tptp.t2) tptp.t)))) (= (@ (@ _let_7 (@ _let_8 (@ (@ _let_2 (@ _let_6 _let_4)) (@ (@ _let_2 tptp.s3) tptp.r2)))) (@ _let_8 (@ (@ _let_2 (@ _let_3 tptp.s3)) (@ (@ _let_2 _let_4) _let_1)))) (@ (@ _let_7 (@ _let_5 (@ _let_6 tptp.r2))) (@ _let_5 (@ _let_3 _let_1)))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s2) (@ tptp.product_fst_a_a tptp.x)))) (let ((_let_3 (@ _let_1 (@ (@ _let_1 (@ (@ _let_1 tptp.t) tptp.s3)) (@ tptp.product_snd_a_a tptp.y))))) (let ((_let_4 (@ _let_1 (@ _let_3 tptp.t2)))) (let ((_let_5 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) tptp.r2))) (let ((_let_6 (@ tptp.a_minus_a_b tptp.r))) (= (@ _let_3 (@ (@ _let_1 tptp.t2) (@ (@ _let_6 _let_5) _let_2))) (@ (@ _let_6 (@ _let_4 _let_5)) (@ _let_4 _let_2))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 tptp.s2))) (let ((_let_5 (@ _let_3 tptp.t2))) (let ((_let_6 (@ _let_3 (@ _let_5 tptp.t)))) (let ((_let_7 (@ _let_3 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_8 (@ tptp.a_minus_a_b tptp.r))) (= (@ (@ _let_3 (@ (@ _let_3 (@ (@ _let_3 tptp.t) tptp.s3)) _let_2)) (@ _let_5 (@ (@ _let_8 (@ _let_7 tptp.r2)) (@ _let_4 _let_1)))) (@ (@ _let_8 (@ _let_6 (@ (@ _let_3 (@ _let_7 _let_2)) (@ (@ _let_3 tptp.s3) tptp.r2)))) (@ _let_6 (@ (@ _let_3 (@ _let_4 tptp.s3)) (@ (@ _let_3 _let_2) _let_1))))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ (@ _let_2 (@ (@ _let_2 tptp.s2) tptp.s3)) (@ (@ _let_2 _let_1) (@ tptp.product_fst_a_a tptp.x))))) (let ((_let_4 (@ (@ _let_2 (@ (@ _let_2 (@ tptp.product_snd_a_a tptp.x)) _let_1)) (@ (@ _let_2 tptp.s3) tptp.r2)))) (let ((_let_5 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_6 (@ _let_2 (@ (@ _let_2 tptp.t2) tptp.t)))) (= (@ (@ _let_5 (@ _let_6 _let_4)) (@ _let_6 _let_3)) (@ _let_6 (@ (@ _let_5 _let_4) _let_3))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_4 (@ _let_2 tptp.t2))) (let ((_let_5 (@ _let_2 (@ (@ _let_2 (@ (@ _let_2 (@ _let_4 tptp.s2)) _let_3)) tptp.t)))) (let ((_let_6 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_7 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_8 (@ _let_2 _let_3))) (let ((_let_9 (@ _let_2 tptp.s2))) (let ((_let_10 (@ _let_2 (@ _let_4 tptp.t)))) (= (@ (@ _let_7 (@ _let_10 (@ (@ _let_2 (@ _let_8 _let_6)) (@ _let_9 tptp.r3)))) (@ _let_10 (@ (@ _let_2 (@ _let_9 tptp.s3)) (@ _let_8 _let_1)))) (@ (@ _let_7 (@ _let_5 (@ (@ _let_2 _let_6) tptp.r3))) (@ _let_5 (@ (@ _let_2 tptp.s3) _let_1)))))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s3) (@ tptp.product_fst_a_a tptp.y)))) (let ((_let_3 (@ _let_1 tptp.t))) (let ((_let_4 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) tptp.r3))) (let ((_let_5 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_6 (@ _let_1 (@ (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.s2)) (@ tptp.product_snd_a_a tptp.x))))) (= (@ _let_6 (@ _let_3 (@ (@ _let_5 _let_4) _let_2))) (@ _let_6 (@ (@ _let_5 (@ _let_3 _let_4)) (@ _let_3 _let_2)))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (P (-> tptp.a Bool))) (= (@ (@ tptp.member_a A) (@ tptp.collect_a P)) (@ P A))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (@ tptp.collect_a (lambda ((X2 tptp.a)) (@ (@ tptp.member_a X2) A2))) A2)))
% 0.25/0.65 (assert (forall ((P (-> tptp.a Bool)) (Q (-> tptp.a Bool))) (=> (forall ((X3 tptp.a)) (= (@ P X3) (@ Q X3))) (= (@ tptp.collect_a P) (@ tptp.collect_a Q)))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 _let_2))) (let ((_let_5 (@ _let_3 tptp.s2))) (let ((_let_6 (@ _let_3 tptp.t2))) (let ((_let_7 (@ _let_3 (@ _let_6 tptp.t)))) (let ((_let_8 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_9 (@ tptp.a_minus_a_b tptp.r))) (= (@ (@ _let_3 (@ (@ _let_3 (@ _let_6 tptp.s2)) _let_2)) (@ (@ _let_3 tptp.t) (@ (@ _let_9 (@ (@ _let_3 _let_8) tptp.r3)) (@ (@ _let_3 tptp.s3) _let_1)))) (@ (@ _let_9 (@ _let_7 (@ (@ _let_3 (@ _let_4 _let_8)) (@ _let_5 tptp.r3)))) (@ _let_7 (@ (@ _let_3 (@ _let_5 tptp.s3)) (@ _let_4 _let_1)))))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.s2))) (let ((_let_4 (@ (@ _let_1 (@ _let_3 tptp.s3)) (@ _let_2 (@ tptp.product_fst_a_a tptp.y))))) (let ((_let_5 (@ (@ _let_1 (@ _let_2 (@ tptp.product_snd_a_a tptp.y))) (@ _let_3 tptp.r3)))) (let ((_let_6 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_7 (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)))) (= (@ (@ _let_6 (@ _let_7 _let_5)) (@ _let_7 _let_4)) (@ _let_7 (@ (@ _let_6 _let_5) _let_4)))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ _let_1 tptp.s2) tptp.r3))) (let ((_let_3 (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) (@ tptp.product_snd_a_a tptp.y)))) (let ((_let_4 (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)))) (let ((_let_5 (@ _let_1 (@ _let_4 _let_3)))) (let ((_let_6 (@ (@ _let_1 tptp.s3) tptp.r2))) (let ((_let_7 (@ tptp.add_a_b tptp.r))) (= (@ _let_4 (@ (@ _let_1 _let_3) (@ (@ _let_7 _let_6) _let_2))) (@ (@ _let_7 (@ _let_5 _let_6)) (@ _let_5 _let_2)))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ _let_1 (@ (@ _let_1 tptp.s2) tptp.s3)) (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) (@ tptp.product_fst_a_a tptp.x)))) (@ tptp.partia1066395285xt_a_b tptp.r))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.s2))) (let ((_let_4 (@ _let_1 (@ _let_3 tptp.s3)))) (let ((_let_5 (@ _let_4 (@ _let_2 (@ tptp.product_fst_a_a tptp.y))))) (let ((_let_6 (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)))) (let ((_let_7 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_8 (@ _let_1 (@ _let_2 _let_7)))) (let ((_let_9 (@ _let_8 (@ _let_3 tptp.r3)))) (let ((_let_10 (@ _let_4 (@ (@ _let_1 _let_7) (@ tptp.product_fst_a_a tptp.x))))) (let ((_let_11 (@ _let_8 (@ (@ _let_1 tptp.s3) tptp.r2)))) (let ((_let_12 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_13 (@ tptp.add_a_b tptp.r))) (= (@ (@ _let_13 (@ _let_6 (@ (@ _let_12 _let_11) _let_10))) (@ _let_6 (@ (@ _let_12 _let_9) _let_5))) (@ (@ _let_12 (@ (@ _let_13 (@ (@ _let_12 (@ _let_6 _let_11)) (@ _let_6 _let_10))) (@ _let_6 _let_9))) (@ _let_6 _let_5)))))))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 _let_2))) (let ((_let_5 (@ _let_3 tptp.s2))) (let ((_let_6 (@ _let_3 (@ _let_5 tptp.s3)))) (let ((_let_7 (@ _let_3 tptp.t2))) (let ((_let_8 (@ _let_3 (@ _let_7 tptp.t)))) (let ((_let_9 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_10 (@ _let_3 (@ _let_4 _let_9)))) (let ((_let_11 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_12 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_13 (@ _let_3 _let_9))) (let ((_let_14 (@ _let_3 tptp.s3))) (let ((_let_15 (@ tptp.add_a_b tptp.r))) (let ((_let_16 (@ _let_3 tptp.t))) (= (@ (@ _let_15 (@ (@ _let_3 (@ (@ _let_3 (@ _let_16 tptp.s3)) _let_9)) (@ _let_7 (@ (@ _let_11 (@ _let_4 tptp.r2)) (@ _let_5 _let_12))))) (@ (@ _let_3 (@ (@ _let_3 (@ _let_7 tptp.s2)) _let_2)) (@ _let_16 (@ (@ _let_11 (@ _let_13 tptp.r3)) (@ _let_14 _let_1))))) (@ (@ _let_15 (@ (@ _let_11 (@ _let_8 (@ _let_10 (@ _let_14 tptp.r2)))) (@ _let_8 (@ _let_6 (@ _let_13 _let_12))))) (@ (@ _let_11 (@ _let_8 (@ _let_10 (@ _let_5 tptp.r3)))) (@ _let_8 (@ _let_6 (@ _let_4 _let_1)))))))))))))))))))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (@ (@ tptp.member_a (@ (@ _let_2 (@ (@ _let_2 tptp.t2) tptp.t)) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_2 (@ (@ _let_2 (@ tptp.product_snd_a_a tptp.x)) _let_1)) (@ (@ _let_2 tptp.s3) tptp.r2))) (@ (@ _let_2 (@ (@ _let_2 tptp.s2) tptp.s3)) (@ (@ _let_2 _let_1) (@ tptp.product_fst_a_a tptp.x)))))) (@ tptp.partia1066395285xt_a_b tptp.r)))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_2 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_3 (@ _let_2 (@ (@ _let_2 tptp.t2) tptp.t)))) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ _let_3 (@ (@ _let_2 (@ (@ _let_2 (@ tptp.product_snd_a_a tptp.x)) _let_1)) (@ (@ _let_2 tptp.s3) tptp.r2)))) (@ _let_3 (@ (@ _let_2 (@ (@ _let_2 tptp.s2) tptp.s3)) (@ (@ _let_2 _let_1) (@ tptp.product_fst_a_a tptp.x)))))) (@ tptp.partia1066395285xt_a_b tptp.r))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ _let_1 (@ tptp.product_snd_a_a tptp.x)))) (let ((_let_3 (@ _let_1 tptp.s2))) (@ (@ tptp.member_a (@ (@ _let_1 (@ (@ _let_1 tptp.t2) tptp.t)) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ _let_2 (@ tptp.product_snd_a_a tptp.y))) (@ _let_3 tptp.r3))) (@ (@ _let_1 (@ _let_3 tptp.s3)) (@ _let_2 (@ tptp.product_fst_a_a tptp.y)))))) (@ tptp.partia1066395285xt_a_b tptp.r))))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (= (@ (@ _let_1 tptp.t2) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) tptp.r2)) (@ (@ _let_1 tptp.s2) (@ tptp.product_fst_a_a tptp.x)))) (@ tptp.zero_a_b tptp.r))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (= (@ (@ _let_1 tptp.t) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) tptp.r3)) (@ (@ _let_1 tptp.s3) (@ tptp.product_fst_a_a tptp.y)))) (@ tptp.zero_a_b tptp.r))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y)) _let_1))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b tptp.r) X) Y)) _let_1))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (= (@ (@ _let_1 Y) X) (@ (@ _let_1 Z) X)) (= Y Z)))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) _let_1))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.t2) tptp.s))
% 0.25/0.65 (assert (@ (@ tptp.member_a tptp.t) tptp.s))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) tptp.t2) tptp.t)) tptp.s))
% 0.25/0.65 (assert (@ tptp.semiring_a_b tptp.r))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (= (@ (@ (@ tptp.a_minus_a_b tptp.r) X) Y) (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) Y)))))
% 0.25/0.65 (assert (forall ((P2 tptp.product_prod_a_a) (Q2 tptp.product_prod_a_a)) (=> (= (@ tptp.product_fst_a_a P2) (@ tptp.product_fst_a_a Q2)) (=> (= (@ tptp.product_snd_a_a P2) (@ tptp.product_snd_a_a Q2)) (= P2 Q2)))))
% 0.25/0.65 (assert (forall ((P (-> tptp.a tptp.a Bool)) (P2 tptp.product_prod_a_a)) (=> (@ (@ P (@ tptp.product_snd_a_a P2)) (@ tptp.product_fst_a_a P2)) (not (forall ((X3 tptp.a) (Y2 tptp.a)) (not (@ (@ P Y2) X3)))))))
% 0.25/0.65 (assert (forall ((Prod tptp.product_prod_a_a) (Prod2 tptp.product_prod_a_a)) (=> (and (= (@ tptp.product_fst_a_a Prod) (@ tptp.product_fst_a_a Prod2)) (= (@ tptp.product_snd_a_a Prod) (@ tptp.product_snd_a_a Prod2))) (= Prod Prod2))))
% 0.25/0.65 (assert (forall ((Y tptp.a) (X tptp.a) (Y3 tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (let ((_let_2 (@ tptp.zero_a_b tptp.r))) (let ((_let_3 (@ tptp.add_a_b tptp.r))) (=> (= (@ (@ _let_3 Y) X) _let_2) (=> (= (@ (@ _let_3 X) Y3) _let_2) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (@ (@ tptp.member_a Y3) _let_1) (= Y Y3)))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) X3) (@ tptp.zero_a_b tptp.r)))))))
% 0.25/0.65 (assert (forall ((U tptp.a)) (=> (@ (@ tptp.member_a U) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) U) X3) X3))) (= U (@ tptp.zero_a_b tptp.r))))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (exists ((X3 tptp.a)) (and (@ (@ tptp.member_a X3) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X3) X) (@ tptp.zero_a_b tptp.r)))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_2 X) Y) _let_1) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_2 Y) X) _let_1)))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ _let_2 Y)) (@ _let_2 (@ _let_1 Y))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ (@ _let_1 (@ _let_2 X)) Y) (@ _let_2 (@ (@ _let_1 X) Y))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) Y)) Y)))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) (@ (@ _let_1 X) Y)) Y)))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= (@ (@ _let_1 B) (@ (@ tptp.a_inv_a_b tptp.r) C)) A) (= B (@ (@ _let_1 A) C))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= A (@ (@ _let_1 B) (@ (@ tptp.a_inv_a_b tptp.r) C))) (= B (@ (@ _let_1 A) C))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C) A) (= C (@ (@ _let_1 B) A))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (C tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (=> (@ (@ tptp.member_a C) _let_2) (= (= A (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) B)) C)) (= C (@ (@ _let_1 B) A))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 Y)) (@ _let_1 X))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ _let_1 X) Y) Z) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (=> (@ (@ tptp.member_a Z) _let_2) (= (@ (@ _let_1 (@ (@ tptp.a_inv_a_b tptp.r) X)) Z) Y)))))))))
% 0.25/0.65 (assert (@ (@ tptp.locali1648887798ac_a_b tptp.r) tptp.s))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) X) Y) (@ tptp.zero_a_b tptp.r)) (= X (@ (@ tptp.a_inv_a_b tptp.r) Y))))))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) (@ (@ tptp.a_inv_a_b tptp.r) X)) (@ tptp.zero_a_b tptp.r)))))
% 0.25/0.65 (assert (forall ((Y tptp.a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= (@ (@ (@ tptp.add_a_b tptp.r) Y) X) (@ tptp.zero_a_b tptp.r)) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (= (@ (@ tptp.a_inv_a_b tptp.r) X) Y)))))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ tptp.a_inv_a_b tptp.r) X)) X) (@ tptp.zero_a_b tptp.r)))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_2) (=> (@ (@ tptp.member_a B) _let_2) (= (@ _let_1 (@ (@ (@ tptp.add_a_b tptp.r) A) B)) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ _let_1 A)) B))))))))
% 0.25/0.65 (assert (not (forall ((T tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (=> (@ (@ tptp.member_a T) tptp.s) (not (= (@ (@ _let_1 T) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.x)) tptp.r2)) (@ (@ _let_1 tptp.s2) (@ tptp.product_fst_a_a tptp.x)))) (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (not (forall ((T2 tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (=> (@ (@ tptp.member_a T2) tptp.s) (not (= (@ (@ _let_1 T2) (@ (@ (@ tptp.a_minus_a_b tptp.r) (@ (@ _let_1 (@ tptp.product_snd_a_a tptp.y)) tptp.r3)) (@ (@ _let_1 tptp.s3) (@ tptp.product_fst_a_a tptp.y)))) (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (@ (@ tptp.member_a (@ tptp.zero_a_b tptp.r)) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_1 X)) X)))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) X)) _let_1)))))
% 0.25/0.65 (assert (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (= (@ (@ tptp.a_inv_a_b tptp.r) _let_1) _let_1)))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a)) (=> (@ (@ tptp.member_a X) tptp.s) (=> (@ (@ tptp.member_a Y) tptp.s) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) tptp.s)))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) _let_1) _let_1)))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) _let_1) X) _let_1)))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) X) (@ tptp.zero_a_b tptp.r)) X))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.add_a_b tptp.r) (@ tptp.zero_a_b tptp.r)) X) X))))
% 0.25/0.65 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.add_a_b tptp.r) A) X)) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.add_a_b tptp.r) A) X) X) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= X (@ (@ (@ tptp.add_a_b tptp.r) X) A)) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (A tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a A) _let_1) (= (= (@ (@ (@ tptp.add_a_b tptp.r) X) A) X) (= A (@ tptp.zero_a_b tptp.r))))))))
% 0.25/0.65 (assert (forall ((X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (= (@ (@ tptp.a_inv_a_b tptp.r) X) _let_1) (= X _let_1))))))
% 0.25/0.65 (assert (@ (@ tptp.locali880295127ng_a_b tptp.r) tptp.s))
% 0.25/0.65 (assert (@ (@ tptp.locali807230110ng_a_b tptp.r) tptp.s))
% 0.25/0.65 (assert (= (lambda ((Y4 tptp.product_prod_a_a) (Z2 tptp.product_prod_a_a)) (= Y4 Z2)) (lambda ((S tptp.product_prod_a_a) (T3 tptp.product_prod_a_a)) (and (= (@ tptp.product_fst_a_a S) (@ tptp.product_fst_a_a T3)) (= (@ tptp.product_snd_a_a S) (@ tptp.product_snd_a_a T3))))))
% 0.25/0.65 (assert (@ (@ tptp.locali425460998xt_a_b tptp.r) tptp.s))
% 0.25/0.65 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.a_inv_a_b tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ _let_1 (@ _let_2 X)) (@ _let_2 (@ _let_1 X))))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R)) (= (@ (@ (@ tptp.add_a_b R) X) (@ tptp.zero_a_b R)) X)))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a)) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R)) (= (@ (@ (@ tptp.add_a_b R) (@ tptp.zero_a_b R)) X) X)))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R) _let_1) X) _let_1))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a)) (let ((_let_1 (@ tptp.zero_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b R)) (= (@ (@ (@ tptp.mult_a_ring_ext_a_b R) X) _let_1) _let_1))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b R) Z))) (let ((_let_2 (@ tptp.add_a_b R))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.int)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_int tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ (@ _let_1 (@ _let_2 A)) B) (@ _let_2 (@ (@ _let_1 A) B))))))))))
% 0.25/0.65 (assert (forall ((A tptp.a) (B tptp.a) (K tptp.int)) (let ((_let_1 (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) A))) (let ((_let_2 (@ (@ tptp.add_pow_a_b_int tptp.r) K))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a A) _let_3) (=> (@ (@ tptp.member_a B) _let_3) (= (@ _let_1 (@ _let_2 B)) (@ _let_2 (@ _let_1 B))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (= (@ _let_1 (@ (@ _let_2 X) Y)) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (Y tptp.a) (I tptp.int)) (let ((_let_1 (@ (@ tptp.add_pow_a_b_int tptp.r) I))) (let ((_let_2 (@ tptp.add_a_b tptp.r))) (let ((_let_3 (@ (@ _let_2 X) Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (= _let_3 (@ (@ _let_2 Y) X)) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (= (@ _let_1 _let_3) (@ (@ _let_2 (@ _let_1 X)) (@ _let_1 Y))))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_pow_a_b_int tptp.r) I) X)) _let_1)))))
% 0.25/0.65 (assert (forall ((Z tptp.int)) (let ((_let_1 (@ tptp.zero_a_b tptp.r))) (= (@ (@ (@ tptp.add_pow_a_b_int tptp.r) Z) _let_1) _let_1))))
% 0.25/0.65 (assert (forall ((M tptp.partia1833973666xt_a_b) (S2 tptp.set_a)) (let ((_let_1 (@ (@ tptp.locali425460998xt_a_b M) S2))) (=> _let_1 _let_1))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (S2 tptp.set_a)) (=> (@ (@ tptp.locali807230110ng_a_b R) S2) (@ (@ tptp.locali880295127ng_a_b R) S2))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (S2 tptp.set_a)) (=> (@ (@ tptp.locali880295127ng_a_b R) S2) (@ (@ tptp.locali425460998xt_a_b R) S2))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (S2 tptp.set_a)) (=> (@ (@ tptp.locali1648887798ac_a_b R) S2) (@ (@ tptp.locali807230110ng_a_b R) S2))))
% 0.25/0.65 (assert (forall ((M tptp.partia1833973666xt_a_b) (S2 tptp.set_a) (X tptp.a) (Y tptp.a)) (=> (@ (@ tptp.locali425460998xt_a_b M) S2) (=> (@ (@ tptp.member_a X) S2) (=> (@ (@ tptp.member_a Y) S2) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b M) X) Y)) S2))))))
% 0.25/0.65 (assert (= tptp.a_minus_a_b (lambda ((R2 tptp.partia1833973666xt_a_b) (X2 tptp.a) (Y5 tptp.a)) (@ (@ (@ tptp.add_a_b R2) X2) (@ (@ tptp.a_inv_a_b R2) Y5)))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b R))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b R) X) Y)) _let_1)))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ _let_1 Y))) (let ((_let_4 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_4) (=> (@ (@ tptp.member_a Y) _let_4) (=> (@ (@ tptp.member_a Z) _let_4) (= (@ _let_2 (@ _let_3 Z)) (@ _let_3 (@ _let_2 Z)))))))))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.add_a_b R))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_2) (=> (@ (@ tptp.member_a Y) _let_2) (= (@ (@ _let_1 X) Y) (@ (@ _let_1 Y) X)))))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.add_a_b R))) (let ((_let_2 (@ _let_1 X))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ _let_2 Y)) Z) (@ _let_2 (@ (@ _let_1 Y) Z))))))))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b R) X) Y)) _let_1)))))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b)) (=> (@ tptp.semiring_a_b R) (@ (@ tptp.member_a (@ tptp.zero_a_b R)) (@ tptp.partia1066395285xt_a_b R)))))
% 0.25/0.65 (assert (forall ((R tptp.partia1833973666xt_a_b) (X tptp.a) (Y tptp.a) (Z tptp.a)) (let ((_let_1 (@ tptp.mult_a_ring_ext_a_b R))) (let ((_let_2 (@ tptp.add_a_b R))) (let ((_let_3 (@ tptp.partia1066395285xt_a_b R))) (=> (@ tptp.semiring_a_b R) (=> (@ (@ tptp.member_a X) _let_3) (=> (@ (@ tptp.member_a Y) _let_3) (=> (@ (@ tptp.member_a Z) _let_3) (= (@ (@ _let_1 (@ (@ _let_2 X) Y)) Z) (@ (@ _let_2 (@ (@ _let_1 X) Z)) (@ (@ _let_1 Y) Z))))))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (N tptp.int) (M2 tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ (@ tptp.minus_minus_int N) M2)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 N) X)) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 M2) X))))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (I tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ tptp.uminus_uminus_int I)) X) (@ (@ tptp.a_inv_a_b tptp.r) (@ (@ _let_1 I) X)))))))
% 0.25/0.65 (assert (forall ((X tptp.a) (I tptp.int) (J tptp.int)) (let ((_let_1 (@ tptp.add_pow_a_b_int tptp.r))) (=> (@ (@ tptp.member_a X) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ _let_1 (@ (@ tptp.plus_plus_int I) J)) X) (@ (@ (@ tptp.add_a_b tptp.r) (@ (@ _let_1 I) X)) (@ (@ _let_1 J) X)))))))
% 0.25/0.65 (assert (@ (@ tptp.ord_less_eq_set_a tptp.s) (@ tptp.partia1066395285xt_a_b tptp.r)))
% 0.25/0.65 (assert (forall ((M tptp.partia1833973666xt_a_b) (S2 tptp.set_a)) (=> (@ (@ tptp.locali425460998xt_a_b M) S2) (@ (@ tptp.ord_less_eq_set_a S2) (@ tptp.partia1066395285xt_a_b M)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ tptp.uminus_uminus_int B)) (@ (@ tptp.plus_plus_int A) B))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B) (@ (@ tptp.minus_minus_int B) A))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int)) (=> (@ (@ tptp.ord_less_eq_int B) A) (= (@ (@ tptp.plus_plus_int B) (@ (@ tptp.minus_minus_int A) B)) A))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int)) (=> (@ (@ tptp.ord_less_eq_int B) A) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) B) A))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (= (@ tptp.uminus_uminus_int A) (@ tptp.uminus_uminus_int B)) (= A B))))
% 0.25/0.65 (assert (forall ((A tptp.int)) (= (@ tptp.uminus_uminus_int (@ tptp.uminus_uminus_int A)) A)))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.ord_less_eq_int A) B))))
% 0.25/0.65 (assert (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.ord_less_eq_int A) B)))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)) (@ (@ tptp.ord_less_eq_int A) B))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int A) B))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) A) B)))
% 0.25/0.65 (assert (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.minus_minus_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_int A) B)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) B) A)))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ tptp.uminus_uminus_int B)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ (@ tptp.plus_plus_int A) B)) B)))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int A) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B)) B)))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.minus_minus_int A) B)) (@ (@ tptp.minus_minus_int B) A))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (=> (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int B))) (let ((_let_2 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))))
% 0.25/0.65 (assert (= tptp.plus_plus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int B2) A3))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.25/0.65 (assert (forall ((B3 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (let ((_let_2 (@ tptp.plus_plus_int K))) (=> (= B3 (@ _let_2 B)) (= (@ _let_1 B3) (@ _let_2 (@ _let_1 B))))))))
% 0.25/0.65 (assert (forall ((A2 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A2 (@ _let_1 A)) (= (@ (@ tptp.plus_plus_int A2) B) (@ _let_1 (@ (@ tptp.plus_plus_int A) B)))))))
% 0.25/0.65 (assert (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (= I J) (= K L)) (= (@ (@ tptp.plus_plus_int I) K) (@ (@ tptp.plus_plus_int J) L)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 C)) B) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (= (@ (@ tptp.minus_minus_int A) B) (@ (@ tptp.minus_minus_int C) D)) (= (= A B) (= C D)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (= (@ tptp.uminus_uminus_int A) B) (= (@ tptp.uminus_uminus_int B) A))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (= A (@ tptp.uminus_uminus_int B)) (= B (@ tptp.uminus_uminus_int A)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (=> (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.ord_less_eq_int A) B))))
% 0.25/0.65 (assert (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (=> (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.ord_less_eq_int A) B)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (=> (@ (@ tptp.ord_less_eq_int C) D) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) D))))))
% 0.25/0.65 (assert (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (@ (@ tptp.ord_less_eq_int I) J) (@ (@ tptp.ord_less_eq_int K) L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))))
% 0.25/0.65 (assert (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (= I J) (@ (@ tptp.ord_less_eq_int K) L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))))
% 0.25/0.65 (assert (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (@ (@ tptp.ord_less_eq_int I) J) (= K L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (= (@ (@ tptp.minus_minus_int A) B) (@ (@ tptp.minus_minus_int C) D)) (= (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int C) D)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) C)) (@ (@ tptp.minus_minus_int B) C)))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int C))) (=> (@ (@ tptp.ord_less_eq_int B) A) (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (D tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (=> (@ (@ tptp.ord_less_eq_int D) C) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) C)) (@ (@ tptp.minus_minus_int B) D))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int A)) B) (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) A))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int A) (@ tptp.uminus_uminus_int B)) (@ (@ tptp.ord_less_eq_int B) (@ tptp.uminus_uminus_int A)))))
% 0.25/0.65 (assert (forall ((C tptp.int) (B tptp.int) (A tptp.int)) (=> (= (@ (@ tptp.plus_plus_int C) B) A) (= C (@ (@ tptp.minus_minus_int A) B)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ _let_1 (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 C)) B)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) C) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (= A (@ (@ tptp.minus_minus_int C) B)) (= (@ (@ tptp.plus_plus_int A) B) C))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (= (@ (@ tptp.minus_minus_int A) B) C) (= A (@ (@ tptp.plus_plus_int C) B)))))
% 0.25/0.65 (assert (forall ((A2 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A2 (@ _let_1 A)) (= (@ (@ tptp.minus_minus_int A2) B) (@ _let_1 (@ (@ tptp.minus_minus_int A) B)))))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))))
% 0.25/0.65 (assert (forall ((A2 tptp.int) (K tptp.int) (A tptp.int)) (=> (= A2 (@ (@ tptp.plus_plus_int K) A)) (= (@ tptp.uminus_uminus_int A2) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ tptp.uminus_uminus_int A))))))
% 0.25/0.65 (assert (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int B)) A) (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int A)) B))))
% 0.25/0.65 (assert (forall ((I tptp.int) (K tptp.int) (N tptp.int) (J tptp.int)) (let ((_let_1 (@ (@ tptp.ord_less_eq_int N) (@ (@ tptp.plus_plus_int J) K)))) (let ((_let_2 (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) N))) (=> _let_2 (=> _let_1 (=> _let_2 (=> _let_1 (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int N) K)) J)))))))))
% 0.25/0.65 (assert (forall ((I tptp.int) (K tptp.int) (N tptp.int)) (=> (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) N) (@ (@ tptp.ord_less_eq_int I) (@ (@ tptp.minus_minus_int N) K)))))
% 0.25/0.65 (assert (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int A) (@ (@ tptp.minus_minus_int C) B)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) B)) C))))
% 0.25/0.65 (assert (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) B)) C) (@ (@ tptp.ord_less_eq_int A) (@ (@ tptp.plus_plus_int C) B)))))
% 0.25/0.65 (assert (= tptp.minus_minus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int A3) (@ tptp.uminus_uminus_int B2)))))
% 0.25/0.65 (assert (= tptp.minus_minus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int A3) (@ tptp.uminus_uminus_int B2)))))
% 0.25/0.65 (assert (forall ((B3 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (=> (= B3 (@ (@ tptp.plus_plus_int K) B)) (= (@ _let_1 B3) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ _let_1 B)))))))
% 0.25/0.65 (assert (forall ((H tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a H) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (not (= H tptp.bot_bot_set_a)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) H) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) X3)) H))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) H) (forall ((Xa tptp.a)) (=> (@ (@ tptp.member_a Xa) H) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b tptp.r) X3) Xa)) H))))) (@ (@ tptp.member_a (@ tptp.zero_a_b tptp.r)) H)))))))
% 0.25/0.65 (assert (forall ((M tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a M) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.a_l_coset_a_b tptp.r) (@ tptp.zero_a_b tptp.r)) M) M))))
% 0.25/0.65 (assert (forall ((M tptp.set_a) (G tptp.a) (H2 tptp.a)) (let ((_let_1 (@ tptp.a_l_coset_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a M) _let_2) (=> (@ (@ tptp.member_a G) _let_2) (=> (@ (@ tptp.member_a H2) _let_2) (= (@ (@ _let_1 G) (@ (@ _let_1 H2) M)) (@ (@ _let_1 (@ (@ (@ tptp.add_a_b tptp.r) G) H2)) M)))))))))
% 0.25/0.65 (assert (not (= (@ tptp.partia1066395285xt_a_b tptp.r) tptp.bot_bot_set_a)))
% 0.25/0.65 (assert (forall ((H tptp.set_a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a H) _let_1) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.a_l_coset_a_b tptp.r) X) H)) _let_1))))))
% 0.25/0.65 (assert (forall ((H tptp.set_a) (K2 tptp.set_a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a H) _let_1) (=> (@ (@ tptp.ord_less_eq_set_a K2) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.set_add_a_b tptp.r) H) K2)) _let_1))))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a A2) _let_1) (=> (@ (@ tptp.ord_less_eq_set_a B3) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.set_add_a_b tptp.r) A2) B3)) _let_1))))))
% 0.25/0.65 (assert (forall ((H tptp.set_a) (K2 tptp.set_a)) (=> (@ (@ tptp.additi2104487374up_a_b H) tptp.r) (=> (@ (@ tptp.additi2104487374up_a_b K2) tptp.r) (@ (@ tptp.additi2104487374up_a_b (@ (@ (@ tptp.set_add_a_b tptp.r) H) K2)) tptp.r)))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (= (= (@ (@ tptp.minus_minus_set_a A2) B3) tptp.bot_bot_set_a) (@ (@ tptp.ord_less_eq_set_a A2) B3))))
% 0.25/0.65 (assert (forall ((C tptp.a)) (not (@ (@ tptp.member_a C) tptp.bot_bot_set_a))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (forall ((X2 tptp.a)) (not (@ (@ tptp.member_a X2) A2))) (= A2 tptp.bot_bot_set_a))))
% 0.25/0.65 (assert (forall ((P (-> tptp.a Bool))) (= (= (@ tptp.collect_a P) tptp.bot_bot_set_a) (forall ((X2 tptp.a)) (not (@ P X2))))))
% 0.25/0.65 (assert (forall ((P (-> tptp.a Bool))) (= (= tptp.bot_bot_set_a (@ tptp.collect_a P)) (forall ((X2 tptp.a)) (not (@ P X2))))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (forall ((X3 tptp.a)) (let ((_let_1 (@ tptp.member_a X3))) (=> (@ _let_1 A2) (@ _let_1 B3)))) (@ (@ tptp.ord_less_eq_set_a A2) B3))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ (@ tptp.ord_less_eq_set_a B3) A2) (= A2 B3)))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (= (@ (@ tptp.ord_less_eq_set_a (@ tptp.uminus_uminus_set_a A2)) (@ tptp.uminus_uminus_set_a B3)) (@ (@ tptp.ord_less_eq_set_a B3) A2))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (@ (@ tptp.ord_less_eq_set_a (@ tptp.uminus_uminus_set_a B3)) (@ tptp.uminus_uminus_set_a A2)))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (@ (@ tptp.ord_less_eq_set_a A2) tptp.bot_bot_set_a) (= A2 tptp.bot_bot_set_a))))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (@ (@ tptp.ord_less_eq_set_a tptp.bot_bot_set_a) A2)))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a A2) tptp.bot_bot_set_a) A2)))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a tptp.bot_bot_set_a) A2) tptp.bot_bot_set_a)))
% 0.25/0.65 (assert (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a A2) A2) tptp.bot_bot_set_a)))
% 0.57/0.88 (assert (forall ((A tptp.a)) (not (@ (@ tptp.member_a A) tptp.bot_bot_set_a))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a) (A tptp.a)) (=> (= A2 tptp.bot_bot_set_a) (not (@ (@ tptp.member_a A) A2)))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a)) (=> (forall ((Y2 tptp.a)) (not (@ (@ tptp.member_a Y2) A2))) (= A2 tptp.bot_bot_set_a))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a)) (= (exists ((X2 tptp.a)) (@ (@ tptp.member_a X2) A2)) (not (= A2 tptp.bot_bot_set_a)))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a) (X tptp.a)) (let ((_let_1 (@ tptp.member_a X))) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ _let_1 A2) (@ _let_1 B3))))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a) (C tptp.a)) (let ((_let_1 (@ tptp.member_a C))) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ _let_1 A2) (@ _let_1 B3))))))
% 0.57/0.88 (assert (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (= A2 B3) (not (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (not (@ (@ tptp.ord_less_eq_set_a B3) A2)))))))
% 0.57/0.88 (assert (= tptp.ord_less_eq_set_a (lambda ((A4 tptp.set_a) (B4 tptp.set_a)) (forall ((X2 tptp.a)) (let ((_let_1 (@ tptp.member_a X2))) (=> (@ _let_1 A4) (@ _let_1 B4)))))))
% 0.57/0.88 (assert (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 _let_2))) (let ((_let_5 (@ _let_3 tptp.s2))) (let ((_let_6 (@ _let_3 (@ _let_5 tptp.s3)))) (let ((_let_7 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_8 (@ _let_3 (@ _let_4 _let_7)))) (let ((_let_9 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_10 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_11 (@ _let_3 _let_7))) (let ((_let_12 (@ _let_3 tptp.s3))) (let ((_let_13 (@ tptp.add_a_b tptp.r))) (let ((_let_14 (@ _let_3 tptp.t2))) (let ((_let_15 (@ _let_3 tptp.t))) (not (= (@ (@ _let_13 (@ (@ _let_3 (@ (@ _let_3 (@ _let_15 tptp.s3)) _let_7)) (@ _let_14 (@ (@ _let_9 (@ _let_4 tptp.r2)) (@ _let_5 _let_10))))) (@ (@ _let_3 (@ (@ _let_3 (@ _let_14 tptp.s2)) _let_2)) (@ _let_15 (@ (@ _let_9 (@ _let_11 tptp.r3)) (@ _let_12 _let_1))))) (@ (@ _let_3 (@ _let_14 tptp.t)) (@ (@ _let_13 (@ (@ _let_9 (@ _let_8 (@ _let_12 tptp.r2))) (@ _let_6 (@ _let_11 _let_10)))) (@ (@ _let_9 (@ _let_8 (@ _let_5 tptp.r3))) (@ _let_6 (@ _let_4 _let_1)))))))))))))))))))))))
% 0.57/0.88 (set-info :filename cvc5---1.0.5_15236)
% 0.57/0.88 (check-sat-assuming ( true ))
% 0.57/0.88 ------- get file name : TPTP file name is ITP108^1
% 0.57/0.88 ------- cvc5-thf : /export/starexec/sandbox/solver/bin/cvc5---1.0.5_15236.smt2...
% 0.57/0.88 --- Run --ho-elim --full-saturate-quant at 10...
% 0.57/0.88 % SZS status Theorem for ITP108^1
% 0.57/0.88 % SZS output start Proof for ITP108^1
% 0.57/0.88 (
% 0.57/0.88 (let ((_let_1 (@ tptp.product_fst_a_a tptp.y))) (let ((_let_2 (@ tptp.product_snd_a_a tptp.x))) (let ((_let_3 (@ tptp.mult_a_ring_ext_a_b tptp.r))) (let ((_let_4 (@ _let_3 _let_2))) (let ((_let_5 (@ _let_4 _let_1))) (let ((_let_6 (@ _let_3 tptp.s2))) (let ((_let_7 (@ _let_6 tptp.s3))) (let ((_let_8 (@ _let_3 _let_7))) (let ((_let_9 (@ _let_8 _let_5))) (let ((_let_10 (@ _let_6 tptp.r3))) (let ((_let_11 (@ tptp.product_snd_a_a tptp.y))) (let ((_let_12 (@ _let_4 _let_11))) (let ((_let_13 (@ _let_3 _let_12))) (let ((_let_14 (@ _let_13 _let_10))) (let ((_let_15 (@ tptp.a_minus_a_b tptp.r))) (let ((_let_16 (@ (@ _let_15 _let_14) _let_9))) (let ((_let_17 (@ tptp.product_fst_a_a tptp.x))) (let ((_let_18 (@ _let_3 _let_11))) (let ((_let_19 (@ _let_18 _let_17))) (let ((_let_20 (@ _let_8 _let_19))) (let ((_let_21 (@ _let_3 tptp.s3))) (let ((_let_22 (@ _let_21 tptp.r2))) (let ((_let_23 (@ _let_13 _let_22))) (let ((_let_24 (@ (@ _let_15 _let_23) _let_20))) (let ((_let_25 (@ tptp.add_a_b tptp.r))) (let ((_let_26 (@ _let_3 tptp.t2))) (let ((_let_27 (@ _let_26 tptp.t))) (let ((_let_28 (@ _let_3 _let_27))) (let ((_let_29 (@ _let_28 (@ (@ _let_25 _let_24) _let_16)))) (let ((_let_30 (@ _let_21 _let_1))) (let ((_let_31 (@ _let_18 tptp.r3))) (let ((_let_32 (@ _let_3 tptp.t))) (let ((_let_33 (@ _let_32 (@ (@ _let_15 _let_31) _let_30)))) (let ((_let_34 (@ (@ _let_3 (@ _let_26 tptp.s2)) _let_2))) (let ((_let_35 (@ _let_3 _let_34))) (let ((_let_36 (@ _let_35 _let_33))) (let ((_let_37 (@ _let_6 _let_17))) (let ((_let_38 (@ _let_4 tptp.r2))) (let ((_let_39 (@ _let_26 (@ (@ _let_15 _let_38) _let_37)))) (let ((_let_40 (@ (@ _let_3 (@ _let_32 tptp.s3)) _let_11))) (let ((_let_41 (@ _let_3 _let_40))) (let ((_let_42 (@ _let_41 _let_39))) (let ((_let_43 (@ (@ _let_25 _let_42) _let_36))) (let ((_let_44 (not (= _let_43 _let_29)))) (let ((_let_45 (= tptp.ord_less_eq_set_a (lambda ((A4 tptp.set_a) (B4 tptp.set_a)) (forall ((X2 tptp.a)) (let ((_let_1 (@ tptp.member_a X2))) (=> (@ _let_1 A4) (@ _let_1 B4)))))))) (let ((_let_46 (@ tptp.partia1066395285xt_a_b tptp.r))) (let ((_let_47 (= tptp.minus_minus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int A3) (@ tptp.uminus_uminus_int B2)))))) (let ((_let_48 (= tptp.a_minus_a_b (lambda ((R2 tptp.partia1833973666xt_a_b) (X2 tptp.a) (Y5 tptp.a)) (@ (@ (@ tptp.add_a_b R2) X2) (@ (@ tptp.a_inv_a_b R2) Y5)))))) (let ((_let_49 (@ tptp.zero_a_b tptp.r))) (let ((_let_50 (@ tptp.a_inv_a_b tptp.r))) (let ((_let_51 (@ tptp.member_a _let_27))) (let ((_let_52 (@ tptp.member_a tptp.t))) (let ((_let_53 (@ tptp.member_a tptp.t2))) (let ((_let_54 (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.member_a X) _let_1) (=> (@ (@ tptp.member_a Y) _let_1) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) _let_1))))))) (let ((_let_55 (@ _let_28 _let_16))) (let ((_let_56 (@ _let_28 _let_20))) (let ((_let_57 (@ _let_28 _let_23))) (let ((_let_58 (@ (@ _let_15 _let_57) _let_56))) (let ((_let_59 (@ _let_28 _let_24))) (let ((_let_60 (@ _let_28 _let_9))) (let ((_let_61 (@ _let_28 _let_14))) (let ((_let_62 (@ (@ _let_15 _let_61) _let_60))) (let ((_let_63 (@ _let_25 _let_58))) (let ((_let_64 (@ (@ _let_25 _let_59) _let_55))) (let ((_let_65 (@ _let_3 (@ _let_28 _let_12)))) (let ((_let_66 (@ _let_13 (@ (@ _let_25 _let_22) _let_10)))) (let ((_let_67 (@ (@ _let_15 (@ _let_32 _let_31)) (@ _let_32 _let_30)))) (let ((_let_68 (@ _let_3 (@ _let_35 tptp.t)))) (let ((_let_69 (@ _let_3 (@ _let_41 tptp.t2)))) (let ((_let_70 (@ _let_69 _let_37))) (let ((_let_71 (@ _let_69 _let_38))) (let ((_let_72 (@ (@ _let_15 _let_71) _let_70))) (let ((_let_73 (@ _let_3 (@ _let_28 _let_7)))) (let ((_let_74 (@ (@ _let_25 _let_19) _let_5))) (let ((_let_75 (@ _let_25 _let_23))) (let ((_let_76 (@ _let_51 _let_46))) (let ((_let_77 (@ (@ tptp.member_a _let_16) _let_46))) (let ((_let_78 (@ (@ tptp.member_a _let_24) _let_46))) (let ((_let_79 (=> _let_78 (=> _let_77 (=> _let_76 (= _let_29 _let_64)))))) (let ((_let_80 (= _let_43 _let_64))) (let ((_let_81 (@ _let_53 _let_46))) (let ((_let_82 (@ _let_52 _let_46))) (let ((_let_83 (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (ho_29 k_28 tptp.r))) (or (not (ho_27 (ho_26 k_25 X) _let_1)) (not (ho_27 (ho_26 k_25 Y) _let_1)) (ho_27 (ho_26 k_25 (ho_33 (ho_32 (ho_31 k_30 tptp.r) X) Y)) _let_1)))))) (let ((_let_84 (ho_29 k_28 tptp.r))) (let ((_let_85 (ho_31 k_30 tptp.r))) (let ((_let_86 (ho_32 _let_85 tptp.t2))) (let ((_let_87 (ho_33 _let_86 tptp.t))) (let ((_let_88 (ho_27 (ho_26 k_25 _let_87) _let_84))) (let ((_let_89 (ho_27 (ho_26 k_25 tptp.t) _let_84))) (let ((_let_90 (not _let_89))) (let ((_let_91 (ho_27 (ho_26 k_25 tptp.t2) _let_84))) (let ((_let_92 (not _let_91))) (let ((_let_93 (or _let_92 _let_90 _let_88))) (let ((_let_94 (EQ_RESOLVE (ASSUME :args (_let_54)) (TRANS (MACRO_SR_EQ_INTRO :args (_let_54 SB_DEFAULT SBA_FIXPOINT)) (PREPROCESS :args ((= (forall ((X tptp.a) (Y tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (or (not (@ (@ tptp.member_a X) _let_1)) (not (@ (@ tptp.member_a Y) _let_1)) (@ (@ tptp.member_a (@ (@ (@ tptp.mult_a_ring_ext_a_b tptp.r) X) Y)) _let_1)))) _let_83))))))) (let ((_let_95 (not _let_93))) (let ((_let_96 (forall ((u |u_(-> tptp.a tptp.a)|) (e tptp.a) (i tptp.a)) (not (forall ((v |u_(-> tptp.a tptp.a)|)) (not (forall ((ii tptp.a)) (= (ho_33 v ii) (ite (= i ii) e (ho_33 u ii)))))))))) (let ((_let_97 (forall ((x |u_(-> tptp.a tptp.a)|) (y |u_(-> tptp.a tptp.a)|)) (or (not (forall ((z tptp.a)) (= (ho_33 x z) (ho_33 y z)))) (= x y))))) (let ((_let_98 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.set_a)|) (e tptp.set_a) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.set_a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_29 v ii) (ite (= i ii) e (ho_29 u ii)))))))))) (let ((_let_99 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.set_a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.set_a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_29 x z) (ho_29 y z)))) (= x y))))) (let ((_let_100 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.int tptp.a tptp.a)|) (e |u_(-> tptp.int tptp.a tptp.a)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.int tptp.a tptp.a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_50 v ii) (ite (= i ii) e (ho_50 u ii)))))))))) (let ((_let_101 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.int tptp.a tptp.a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.int tptp.a tptp.a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_50 x z) (ho_50 y z)))) (= x y))))) (let ((_let_102 (forall ((u |u_(-> tptp.a tptp.set_a Bool)|) (e |u_(-> tptp.set_a Bool)|) (i tptp.a)) (not (forall ((v |u_(-> tptp.a tptp.set_a Bool)|)) (not (forall ((ii tptp.a)) (= (ho_26 v ii) (ite (= i ii) e (ho_26 u ii)))))))))) (let ((_let_103 (forall ((x |u_(-> tptp.a tptp.set_a Bool)|) (y |u_(-> tptp.a tptp.set_a Bool)|)) (or (not (forall ((z tptp.a)) (= (ho_26 x z) (ho_26 y z)))) (= x y))))) (let ((_let_104 (forall ((u |u_(-> tptp.a tptp.a tptp.a)|) (e |u_(-> tptp.a tptp.a)|) (i tptp.a)) (not (forall ((v |u_(-> tptp.a tptp.a tptp.a)|)) (not (forall ((ii tptp.a)) (= (ho_32 v ii) (ite (= i ii) e (ho_32 u ii)))))))))) (let ((_let_105 (forall ((x |u_(-> tptp.a tptp.a tptp.a)|) (y |u_(-> tptp.a tptp.a tptp.a)|)) (or (not (forall ((z tptp.a)) (= (ho_32 x z) (ho_32 y z)))) (= x y))))) (let ((_let_106 (forall ((u |u_(-> tptp.set_a Bool)|) (e Bool) (i tptp.set_a)) (not (forall ((v |u_(-> tptp.set_a Bool)|)) (not (forall ((ii tptp.set_a)) (= (ho_27 v ii) (ite (= i ii) e (ho_27 u ii)))))))))) (let ((_let_107 (forall ((x |u_(-> tptp.set_a Bool)|) (y |u_(-> tptp.set_a Bool)|)) (or (not (forall ((z tptp.set_a)) (= (ho_27 x z) (ho_27 y z)))) (= x y))))) (let ((_let_108 (forall ((u |u_(-> tptp.product_prod_a_a tptp.a)|) (e tptp.a) (i tptp.product_prod_a_a)) (not (forall ((v |u_(-> tptp.product_prod_a_a tptp.a)|)) (not (forall ((ii tptp.product_prod_a_a)) (= (ho_20 v ii) (ite (= i ii) e (ho_20 u ii)))))))))) (let ((_let_109 (forall ((x |u_(-> tptp.product_prod_a_a tptp.a)|) (y |u_(-> tptp.product_prod_a_a tptp.a)|)) (or (not (forall ((z tptp.product_prod_a_a)) (= (ho_20 x z) (ho_20 y z)))) (= x y))))) (let ((_let_110 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.set_a tptp.set_a)|) (e |u_(-> tptp.a tptp.set_a tptp.set_a)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.set_a tptp.set_a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_56 v ii) (ite (= i ii) e (ho_56 u ii)))))))))) (let ((_let_111 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.set_a tptp.set_a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.set_a tptp.set_a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_56 x z) (ho_56 y z)))) (= x y))))) (let ((_let_112 (forall ((u |u_(-> tptp.a Bool)|) (e Bool) (i tptp.a)) (not (forall ((v |u_(-> tptp.a Bool)|)) (not (forall ((ii tptp.a)) (= (ho_24 v ii) (ite (= i ii) e (ho_24 u ii)))))))))) (let ((_let_113 (forall ((x |u_(-> tptp.a Bool)|) (y |u_(-> tptp.a Bool)|)) (or (not (forall ((z tptp.a)) (= (ho_24 x z) (ho_24 y z)))) (= x y))))) (let ((_let_114 (forall ((u |u_(-> tptp.set_a tptp.a Bool)|) (e |u_(-> tptp.a Bool)|) (i tptp.set_a)) (not (forall ((v |u_(-> tptp.set_a tptp.a Bool)|)) (not (forall ((ii tptp.set_a)) (= (ho_23 v ii) (ite (= i ii) e (ho_23 u ii)))))))))) (let ((_let_115 (forall ((x |u_(-> tptp.set_a tptp.a Bool)|) (y |u_(-> tptp.set_a tptp.a Bool)|)) (or (not (forall ((z tptp.set_a)) (= (ho_23 x z) (ho_23 y z)))) (= x y))))) (let ((_let_116 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a tptp.a)|) (e |u_(-> tptp.a tptp.a tptp.a)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a tptp.a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_31 v ii) (ite (= i ii) e (ho_31 u ii)))))))))) (let ((_let_117 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a tptp.a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a tptp.a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_31 x z) (ho_31 y z)))) (= x y))))) (let ((_let_118 (forall ((u |u_(-> tptp.product_prod_a_a tptp.product_prod_a_a Bool)|) (e |u_(-> tptp.product_prod_a_a Bool)|) (i tptp.product_prod_a_a)) (not (forall ((v |u_(-> tptp.product_prod_a_a tptp.product_prod_a_a Bool)|)) (not (forall ((ii tptp.product_prod_a_a)) (= (ho_16 v ii) (ite (= i ii) e (ho_16 u ii)))))))))) (let ((_let_119 (forall ((x |u_(-> tptp.product_prod_a_a tptp.product_prod_a_a Bool)|) (y |u_(-> tptp.product_prod_a_a tptp.product_prod_a_a Bool)|)) (or (not (forall ((z tptp.product_prod_a_a)) (= (ho_16 x z) (ho_16 y z)))) (= x y))))) (let ((_let_120 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.set_a tptp.set_a tptp.set_a)|) (e |u_(-> tptp.set_a tptp.set_a tptp.set_a)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.set_a tptp.set_a tptp.set_a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_60 v ii) (ite (= i ii) e (ho_60 u ii)))))))))) (let ((_let_121 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.set_a tptp.set_a tptp.set_a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.set_a tptp.set_a tptp.set_a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_60 x z) (ho_60 y z)))) (= x y))))) (let ((_let_122 (forall ((u |u_(-> tptp.product_prod_a_a Bool)|) (e Bool) (i tptp.product_prod_a_a)) (not (forall ((v |u_(-> tptp.product_prod_a_a Bool)|)) (not (forall ((ii tptp.product_prod_a_a)) (= (ho_17 v ii) (ite (= i ii) e (ho_17 u ii)))))))))) (let ((_let_123 (forall ((x |u_(-> tptp.product_prod_a_a Bool)|) (y |u_(-> tptp.product_prod_a_a Bool)|)) (or (not (forall ((z tptp.product_prod_a_a)) (= (ho_17 x z) (ho_17 y z)))) (= x y))))) (let ((_let_124 (forall ((u |u_(-> tptp.int tptp.int tptp.int)|) (e |u_(-> tptp.int tptp.int)|) (i tptp.int)) (not (forall ((v |u_(-> tptp.int tptp.int tptp.int)|)) (not (forall ((ii tptp.int)) (= (ho_9 v ii) (ite (= i ii) e (ho_9 u ii)))))))))) (let ((_let_125 (forall ((x |u_(-> tptp.int tptp.int tptp.int)|) (y |u_(-> tptp.int tptp.int tptp.int)|)) (or (not (forall ((z tptp.int)) (= (ho_9 x z) (ho_9 y z)))) (= x y))))) (let ((_let_126 (forall ((u |u_(-> tptp.int tptp.int)|) (e tptp.int) (i tptp.int)) (not (forall ((v |u_(-> tptp.int tptp.int)|)) (not (forall ((ii tptp.int)) (= (ho_10 v ii) (ite (= i ii) e (ho_10 u ii)))))))))) (let ((_let_127 (forall ((x |u_(-> tptp.int tptp.int)|) (y |u_(-> tptp.int tptp.int)|)) (or (not (forall ((z tptp.int)) (= (ho_10 x z) (ho_10 y z)))) (= x y))))) (let ((_let_128 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a)|) (e |u_(-> tptp.a tptp.a)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_36 v ii) (ite (= i ii) e (ho_36 u ii)))))))))) (let ((_let_129 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.a tptp.a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_36 x z) (ho_36 y z)))) (= x y))))) (let ((_let_130 (forall ((u |u_(-> tptp.a tptp.set_a tptp.set_a)|) (e |u_(-> tptp.set_a tptp.set_a)|) (i tptp.a)) (not (forall ((v |u_(-> tptp.a tptp.set_a tptp.set_a)|)) (not (forall ((ii tptp.a)) (= (ho_57 v ii) (ite (= i ii) e (ho_57 u ii)))))))))) (let ((_let_131 (forall ((x |u_(-> tptp.a tptp.set_a tptp.set_a)|) (y |u_(-> tptp.a tptp.set_a tptp.set_a)|)) (or (not (forall ((z tptp.a)) (= (ho_57 x z) (ho_57 y z)))) (= x y))))) (let ((_let_132 (forall ((u |u_(-> _u_(-> tptp.a Bool)_ tptp.set_a)|) (e tptp.set_a) (i |u_(-> tptp.a Bool)|)) (not (forall ((v |u_(-> _u_(-> tptp.a Bool)_ tptp.set_a)|)) (not (forall ((ii |u_(-> tptp.a Bool)|)) (= (ho_38 v ii) (ite (= i ii) e (ho_38 u ii)))))))))) (let ((_let_133 (forall ((x |u_(-> _u_(-> tptp.a Bool)_ tptp.set_a)|) (y |u_(-> _u_(-> tptp.a Bool)_ tptp.set_a)|)) (or (not (forall ((z |u_(-> tptp.a Bool)|)) (= (ho_38 x z) (ho_38 y z)))) (= x y))))) (let ((_let_134 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.a)|) (e tptp.a) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.a)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_40 v ii) (ite (= i ii) e (ho_40 u ii)))))))))) (let ((_let_135 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.a)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.a)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_40 x z) (ho_40 y z)))) (= x y))))) (let ((_let_136 (forall ((u |u_(-> tptp.partia1833973666xt_a_b Bool)|) (e Bool) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b Bool)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_42 v ii) (ite (= i ii) e (ho_42 u ii)))))))))) (let ((_let_137 (forall ((x |u_(-> tptp.partia1833973666xt_a_b Bool)|) (y |u_(-> tptp.partia1833973666xt_a_b Bool)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_42 x z) (ho_42 y z)))) (= x y))))) (let ((_let_138 (forall ((u |u_(-> tptp.a tptp.a Bool)|) (e |u_(-> tptp.a Bool)|) (i tptp.a)) (not (forall ((v |u_(-> tptp.a tptp.a Bool)|)) (not (forall ((ii tptp.a)) (= (ho_43 v ii) (ite (= i ii) e (ho_43 u ii)))))))))) (let ((_let_139 (forall ((x |u_(-> tptp.a tptp.a Bool)|) (y |u_(-> tptp.a tptp.a Bool)|)) (or (not (forall ((z tptp.a)) (= (ho_43 x z) (ho_43 y z)))) (= x y))))) (let ((_let_140 (forall ((u |u_(-> tptp.partia1833973666xt_a_b tptp.set_a Bool)|) (e |u_(-> tptp.set_a Bool)|) (i tptp.partia1833973666xt_a_b)) (not (forall ((v |u_(-> tptp.partia1833973666xt_a_b tptp.set_a Bool)|)) (not (forall ((ii tptp.partia1833973666xt_a_b)) (= (ho_45 v ii) (ite (= i ii) e (ho_45 u ii)))))))))) (let ((_let_141 (forall ((x |u_(-> tptp.partia1833973666xt_a_b tptp.set_a Bool)|) (y |u_(-> tptp.partia1833973666xt_a_b tptp.set_a Bool)|)) (or (not (forall ((z tptp.partia1833973666xt_a_b)) (= (ho_45 x z) (ho_45 y z)))) (= x y))))) (let ((_let_142 (forall ((u |u_(-> tptp.int tptp.a tptp.a)|) (e |u_(-> tptp.a tptp.a)|) (i tptp.int)) (not (forall ((v |u_(-> tptp.int tptp.a tptp.a)|)) (not (forall ((ii tptp.int)) (= (ho_51 v ii) (ite (= i ii) e (ho_51 u ii)))))))))) (let ((_let_143 (forall ((x |u_(-> tptp.int tptp.a tptp.a)|) (y |u_(-> tptp.int tptp.a tptp.a)|)) (or (not (forall ((z tptp.int)) (= (ho_51 x z) (ho_51 y z)))) (= x y))))) (let ((_let_144 (forall ((u |u_(-> tptp.int Bool)|) (e Bool) (i tptp.int)) (not (forall ((v |u_(-> tptp.int Bool)|)) (not (forall ((ii tptp.int)) (= (ho_54 v ii) (ite (= i ii) e (ho_54 u ii)))))))))) (let ((_let_145 (forall ((x |u_(-> tptp.int Bool)|) (y |u_(-> tptp.int Bool)|)) (or (not (forall ((z tptp.int)) (= (ho_54 x z) (ho_54 y z)))) (= x y))))) (let ((_let_146 (forall ((u |u_(-> tptp.set_a tptp.partia1833973666xt_a_b Bool)|) (e |u_(-> tptp.partia1833973666xt_a_b Bool)|) (i tptp.set_a)) (not (forall ((v |u_(-> tptp.set_a tptp.partia1833973666xt_a_b Bool)|)) (not (forall ((ii tptp.set_a)) (= (ho_63 v ii) (ite (= i ii) e (ho_63 u ii)))))))))) (let ((_let_147 (forall ((x |u_(-> tptp.set_a tptp.partia1833973666xt_a_b Bool)|) (y |u_(-> tptp.set_a tptp.partia1833973666xt_a_b Bool)|)) (or (not (forall ((z tptp.set_a)) (= (ho_63 x z) (ho_63 y z)))) (= x y))))) (let ((_let_148 (forall ((u |u_(-> tptp.int tptp.int Bool)|) (e |u_(-> tptp.int Bool)|) (i tptp.int)) (not (forall ((v |u_(-> tptp.int tptp.int Bool)|)) (not (forall ((ii tptp.int)) (= (ho_53 v ii) (ite (= i ii) e (ho_53 u ii)))))))))) (let ((_let_149 (forall ((x |u_(-> tptp.int tptp.int Bool)|) (y |u_(-> tptp.int tptp.int Bool)|)) (or (not (forall ((z tptp.int)) (= (ho_53 x z) (ho_53 y z)))) (= x y))))) (let ((_let_150 (forall ((u |u_(-> tptp.set_a tptp.set_a)|) (e tptp.set_a) (i tptp.set_a)) (not (forall ((v |u_(-> tptp.set_a tptp.set_a)|)) (not (forall ((ii tptp.set_a)) (= (ho_58 v ii) (ite (= i ii) e (ho_58 u ii)))))))))) (let ((_let_151 (forall ((x |u_(-> tptp.set_a tptp.set_a)|) (y |u_(-> tptp.set_a tptp.set_a)|)) (or (not (forall ((z tptp.set_a)) (= (ho_58 x z) (ho_58 y z)))) (= x y))))) (let ((_let_152 (forall ((u |u_(-> tptp.set_a tptp.set_a tptp.set_a)|) (e |u_(-> tptp.set_a tptp.set_a)|) (i tptp.set_a)) (not (forall ((v |u_(-> tptp.set_a tptp.set_a tptp.set_a)|)) (not (forall ((ii tptp.set_a)) (= (ho_61 v ii) (ite (= i ii) e (ho_61 u ii)))))))))) (let ((_let_153 (forall ((x |u_(-> tptp.set_a tptp.set_a tptp.set_a)|) (y |u_(-> tptp.set_a tptp.set_a tptp.set_a)|)) (or (not (forall ((z tptp.set_a)) (= (ho_61 x z) (ho_61 y z)))) (= x y))))) (let ((_let_154 (forall ((BOUND_VARIABLE_6274 tptp.int) (BOUND_VARIABLE_6275 tptp.int)) (= (ho_10 (ho_9 k_8 BOUND_VARIABLE_6274) BOUND_VARIABLE_6275) (ho_10 (ho_9 k_12 BOUND_VARIABLE_6274) (ho_10 k_11 BOUND_VARIABLE_6275)))))) (let ((_let_155 (forall ((BOUND_VARIABLE_6265 tptp.int) (BOUND_VARIABLE_6266 tptp.int)) (= (ho_10 (ho_9 k_13 BOUND_VARIABLE_6265) 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(forall ((BOUND_VARIABLE_6237 tptp.product_prod_a_a) (BOUND_VARIABLE_6238 tptp.product_prod_a_a)) (= (and (= (@ tptp.product_fst_a_a BOUND_VARIABLE_6237) (@ tptp.product_fst_a_a BOUND_VARIABLE_6238)) (= (@ tptp.product_snd_a_a BOUND_VARIABLE_6237) (@ tptp.product_snd_a_a BOUND_VARIABLE_6238))) (ll_3 BOUND_VARIABLE_6237 BOUND_VARIABLE_6238))))) (let ((_let_165 (forall ((BOUND_VARIABLE_6229 tptp.set_a) (BOUND_VARIABLE_6230 tptp.a)) (= (@ (@ tptp.member_a BOUND_VARIABLE_6230) BOUND_VARIABLE_6229) (ll_2 BOUND_VARIABLE_6229 BOUND_VARIABLE_6230))))) (let ((_let_166 (and _let_82 _let_165 _let_164 _let_163 _let_162 _let_161 _let_160))) (let ((_let_167 (ho_20 k_21 tptp.y))) (let ((_let_168 (ho_20 k_19 tptp.x))) (let ((_let_169 (ho_32 _let_85 _let_168))) (let ((_let_170 (ho_32 _let_85 tptp.s2))) (let ((_let_171 (ho_32 _let_85 (ho_33 _let_170 tptp.s3)))) (let ((_let_172 (ho_36 k_35 tptp.r))) (let ((_let_173 (ho_20 k_19 tptp.y))) (let ((_let_174 (ho_32 _let_85 (ho_33 _let_169 _let_173)))) (let ((_let_175 (ho_31 k_34 tptp.r))) (let ((_let_176 (ho_33 (ho_32 _let_175 (ho_33 _let_174 (ho_33 _let_170 tptp.r3))) (ho_33 _let_172 (ho_33 _let_171 (ho_33 _let_169 _let_167)))))) (let ((_let_177 (ho_20 k_21 tptp.x))) (let ((_let_178 (ho_32 _let_85 _let_173))) (let ((_let_179 (ho_32 _let_85 tptp.s3))) (let ((_let_180 (ho_33 (ho_32 _let_175 (ho_33 _let_174 (ho_33 _let_179 tptp.r2))) (ho_33 _let_172 (ho_33 _let_171 (ho_33 _let_178 _let_177)))))) (let ((_let_181 (ho_32 _let_85 _let_87))) (let ((_let_182 (ho_33 _let_181 (ho_33 (ho_32 _let_175 _let_180) _let_176)))) (let ((_let_183 (ho_33 (ho_32 _let_175 (ho_33 _let_181 _let_180)) (ho_33 _let_181 _let_176)))) (let ((_let_184 (= _let_183 _let_182))) (let ((_let_185 (=> _let_88 _let_184))) (let ((_let_186 (not _let_88))) (let ((_let_187 (ho_27 (ho_26 k_25 _let_176) _let_84))) (let ((_let_188 (=> _let_187 _let_185))) (let ((_let_189 (ho_27 (ho_26 k_25 _let_180) _let_84))) (let ((_let_190 (@ _let_75 (@ _let_50 _let_20)))) (let ((_let_191 (@ (@ 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tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.int) (B tptp.int)) (= (= (@ tptp.uminus_uminus_int A) (@ tptp.uminus_uminus_int B)) (= A B))) (forall ((A tptp.int)) (= (@ tptp.uminus_uminus_int (@ tptp.uminus_uminus_int A)) A)) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.ord_less_eq_int A) B))) (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.ord_less_eq_int A) B)))) (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)) (@ (@ tptp.ord_less_eq_int A) B))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int A) B))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) A) B)) (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (= (@ (@ tptp.minus_minus_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.minus_minus_int A) B)))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) B) A)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) B)) B) A)) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ tptp.uminus_uminus_int B)))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) (@ (@ tptp.plus_plus_int A) B)) B)) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.plus_plus_int A) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int A)) B)) B)) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.minus_minus_int A) B)) (@ (@ tptp.minus_minus_int B) A))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (=> (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (=> (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int B))) (let ((_let_2 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ _let_2 C)) (@ _let_2 (@ _let_1 C)))))) (= tptp.plus_plus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int B2) A3))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (= (= (@ (@ tptp.plus_plus_int B) A) (@ (@ tptp.plus_plus_int C) A)) (= B C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (= (@ _let_1 B) (@ _let_1 C)) (= B C)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))) (forall ((B3 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (let ((_let_2 (@ tptp.plus_plus_int K))) (=> (= B3 (@ _let_2 B)) (= (@ _let_1 B3) (@ _let_2 (@ _let_1 B))))))) (forall ((A2 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A2 (@ _let_1 A)) (= (@ (@ tptp.plus_plus_int A2) B) (@ _let_1 (@ (@ tptp.plus_plus_int A) B)))))) (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (= I J) (= K L)) (= (@ (@ tptp.plus_plus_int I) K) (@ (@ tptp.plus_plus_int J) L)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ (@ tptp.plus_plus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 C)) B) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (= (@ (@ tptp.minus_minus_int A) B) (@ (@ tptp.minus_minus_int C) D)) (= (= A B) (= C D)))) (forall ((A tptp.int) (B tptp.int)) (= (= (@ tptp.uminus_uminus_int A) B) (= (@ tptp.uminus_uminus_int B) A))) (forall ((A tptp.int) (B tptp.int)) (= (= A (@ tptp.uminus_uminus_int B)) (= B (@ tptp.uminus_uminus_int A)))) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (=> (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.ord_less_eq_int A) B))) (forall ((C tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (=> (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B)) (@ (@ tptp.ord_less_eq_int A) B)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) C)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int C))) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B))))) (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (=> (@ (@ tptp.ord_less_eq_int C) D) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) C)) (@ (@ tptp.plus_plus_int B) D))))) (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (@ (@ tptp.ord_less_eq_int I) J) (@ (@ tptp.ord_less_eq_int K) L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))) (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (= I J) (@ (@ tptp.ord_less_eq_int K) L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))) (forall ((I tptp.int) (J tptp.int) (K tptp.int) (L tptp.int)) (=> (and (@ (@ tptp.ord_less_eq_int I) J) (= K L)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) (@ (@ tptp.plus_plus_int J) L)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int) (D tptp.int)) (=> (= (@ (@ tptp.minus_minus_int A) B) (@ (@ tptp.minus_minus_int C) D)) (= (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int C) D)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) C)) (@ (@ tptp.minus_minus_int B) C)))) (forall ((B tptp.int) (A tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int C))) (=> (@ (@ tptp.ord_less_eq_int B) A) (@ (@ tptp.ord_less_eq_int (@ _let_1 A)) (@ _let_1 B))))) (forall ((A tptp.int) (B tptp.int) (D tptp.int) (C tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (=> (@ (@ tptp.ord_less_eq_int D) C) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) C)) (@ (@ tptp.minus_minus_int B) D))))) (forall ((A tptp.int) (B tptp.int)) (=> (@ (@ tptp.ord_less_eq_int A) B) (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int A)) B) (@ (@ tptp.ord_less_eq_int (@ tptp.uminus_uminus_int B)) A))) (forall ((A tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int A) (@ tptp.uminus_uminus_int B)) (@ (@ tptp.ord_less_eq_int B) (@ tptp.uminus_uminus_int A)))) (forall ((C tptp.int) (B tptp.int) (A tptp.int)) (=> (= (@ (@ tptp.plus_plus_int C) B) A) (= C (@ (@ tptp.minus_minus_int A) B)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ (@ tptp.minus_minus_int (@ _let_1 B)) C) (@ _let_1 (@ (@ tptp.plus_plus_int B) C))))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (= (@ _let_1 (@ (@ tptp.plus_plus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 C)) B)))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.plus_plus_int (@ (@ tptp.minus_minus_int A) B)) C) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.minus_minus_int A) (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ (@ tptp.plus_plus_int A) C)) B))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int A))) (= (@ _let_1 (@ (@ tptp.minus_minus_int B) C)) (@ (@ tptp.minus_minus_int (@ _let_1 B)) C)))) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (= A (@ (@ tptp.minus_minus_int C) B)) (= (@ (@ tptp.plus_plus_int A) B) C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (= (@ (@ tptp.minus_minus_int A) B) C) (= A (@ (@ tptp.plus_plus_int C) B)))) (forall ((A2 tptp.int) (K tptp.int) (A tptp.int) (B tptp.int)) (let ((_let_1 (@ tptp.plus_plus_int K))) (=> (= A2 (@ _let_1 A)) (= (@ (@ tptp.minus_minus_int A2) B) (@ _let_1 (@ (@ tptp.minus_minus_int A) B)))))) (forall ((A tptp.int) (B tptp.int)) (= (@ tptp.uminus_uminus_int (@ (@ tptp.plus_plus_int A) B)) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int B)) (@ tptp.uminus_uminus_int A)))) (forall ((A2 tptp.int) (K tptp.int) (A tptp.int)) (=> (= A2 (@ (@ tptp.plus_plus_int K) A)) (= (@ tptp.uminus_uminus_int A2) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ tptp.uminus_uminus_int A))))) (forall ((B tptp.int) (A tptp.int)) (= (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int B)) A) (@ (@ tptp.minus_minus_int (@ tptp.uminus_uminus_int A)) B))) (forall ((I tptp.int) (K tptp.int) (N tptp.int) (J tptp.int)) (let ((_let_1 (@ (@ tptp.ord_less_eq_int N) (@ (@ tptp.plus_plus_int J) K)))) (let ((_let_2 (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) N))) (=> _let_2 (=> _let_1 (=> _let_2 (=> _let_1 (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int N) K)) J)))))))) (forall ((I tptp.int) (K tptp.int) (N tptp.int)) (=> (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int I) K)) N) (@ (@ tptp.ord_less_eq_int I) (@ (@ tptp.minus_minus_int N) K)))) (forall ((A tptp.int) (C tptp.int) (B tptp.int)) (= (@ (@ tptp.ord_less_eq_int A) (@ (@ tptp.minus_minus_int C) B)) (@ (@ tptp.ord_less_eq_int (@ (@ tptp.plus_plus_int A) B)) C))) (forall ((A tptp.int) (B tptp.int) (C tptp.int)) (= (@ (@ tptp.ord_less_eq_int (@ (@ tptp.minus_minus_int A) B)) C) (@ (@ tptp.ord_less_eq_int A) (@ (@ tptp.plus_plus_int C) B)))) _let_47 (= tptp.minus_minus_int (lambda ((A3 tptp.int) (B2 tptp.int)) (@ (@ tptp.plus_plus_int A3) (@ tptp.uminus_uminus_int B2)))) (forall ((B3 tptp.int) (K tptp.int) (B tptp.int) (A tptp.int)) (let ((_let_1 (@ tptp.minus_minus_int A))) (=> (= B3 (@ (@ tptp.plus_plus_int K) B)) (= (@ _let_1 B3) (@ (@ tptp.plus_plus_int (@ tptp.uminus_uminus_int K)) (@ _let_1 B)))))) (forall ((H tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a H) (@ tptp.partia1066395285xt_a_b tptp.r)) (=> (not (= H tptp.bot_bot_set_a)) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) H) (@ (@ tptp.member_a (@ (@ tptp.a_inv_a_b tptp.r) X3)) H))) (=> (forall ((X3 tptp.a)) (=> (@ (@ tptp.member_a X3) H) (forall ((Xa tptp.a)) (=> (@ (@ tptp.member_a Xa) H) (@ (@ tptp.member_a (@ (@ (@ tptp.add_a_b tptp.r) X3) Xa)) H))))) (@ (@ tptp.member_a (@ tptp.zero_a_b tptp.r)) H)))))) (forall ((M tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a M) (@ tptp.partia1066395285xt_a_b tptp.r)) (= (@ (@ (@ tptp.a_l_coset_a_b tptp.r) (@ tptp.zero_a_b tptp.r)) M) M))) (forall ((M tptp.set_a) (G tptp.a) (H2 tptp.a)) (let ((_let_1 (@ tptp.a_l_coset_a_b tptp.r))) (let ((_let_2 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a M) _let_2) (=> (@ (@ tptp.member_a G) _let_2) (=> (@ (@ tptp.member_a H2) _let_2) (= (@ (@ _let_1 G) (@ (@ _let_1 H2) M)) (@ (@ _let_1 (@ (@ (@ tptp.add_a_b tptp.r) G) H2)) M)))))))) (not (= _let_46 tptp.bot_bot_set_a)) (forall ((H tptp.set_a) (X tptp.a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a H) _let_1) (=> (@ (@ tptp.member_a X) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.a_l_coset_a_b tptp.r) X) H)) _let_1))))) (forall ((H tptp.set_a) (K2 tptp.set_a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a H) _let_1) (=> (@ (@ tptp.ord_less_eq_set_a K2) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.set_add_a_b tptp.r) H) K2)) _let_1))))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (let ((_let_1 (@ tptp.partia1066395285xt_a_b tptp.r))) (=> (@ (@ tptp.ord_less_eq_set_a A2) _let_1) (=> (@ (@ tptp.ord_less_eq_set_a B3) _let_1) (@ (@ tptp.ord_less_eq_set_a (@ (@ (@ tptp.set_add_a_b tptp.r) A2) B3)) _let_1))))) (forall ((H tptp.set_a) (K2 tptp.set_a)) (=> (@ (@ tptp.additi2104487374up_a_b H) tptp.r) (=> (@ (@ tptp.additi2104487374up_a_b K2) tptp.r) (@ (@ tptp.additi2104487374up_a_b (@ (@ (@ tptp.set_add_a_b tptp.r) H) K2)) tptp.r)))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (= (= (@ (@ tptp.minus_minus_set_a A2) B3) tptp.bot_bot_set_a) (@ (@ tptp.ord_less_eq_set_a A2) B3))) (forall ((C tptp.a)) (not (@ (@ tptp.member_a C) tptp.bot_bot_set_a))) (forall ((A2 tptp.set_a)) (= (forall ((X2 tptp.a)) (not (@ (@ tptp.member_a X2) A2))) (= A2 tptp.bot_bot_set_a))) (forall ((P (-> tptp.a Bool))) (= (= (@ tptp.collect_a P) tptp.bot_bot_set_a) (forall ((X2 tptp.a)) (not (@ P X2))))) (forall ((P (-> tptp.a Bool))) (= (= tptp.bot_bot_set_a (@ tptp.collect_a P)) (forall ((X2 tptp.a)) (not (@ P X2))))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (forall ((X3 tptp.a)) (let ((_let_1 (@ tptp.member_a X3))) (=> (@ _let_1 A2) (@ _let_1 B3)))) (@ (@ tptp.ord_less_eq_set_a A2) B3))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ (@ tptp.ord_less_eq_set_a B3) A2) (= A2 B3)))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (= (@ (@ tptp.ord_less_eq_set_a (@ tptp.uminus_uminus_set_a A2)) (@ tptp.uminus_uminus_set_a B3)) (@ (@ tptp.ord_less_eq_set_a B3) A2))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (@ (@ tptp.ord_less_eq_set_a (@ tptp.uminus_uminus_set_a B3)) (@ tptp.uminus_uminus_set_a A2)))) (forall ((A2 tptp.set_a)) (= (@ (@ tptp.ord_less_eq_set_a A2) tptp.bot_bot_set_a) (= A2 tptp.bot_bot_set_a))) (forall ((A2 tptp.set_a)) (@ (@ tptp.ord_less_eq_set_a tptp.bot_bot_set_a) A2)) (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a A2) tptp.bot_bot_set_a) A2)) (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a tptp.bot_bot_set_a) A2) tptp.bot_bot_set_a)) (forall ((A2 tptp.set_a)) (= (@ (@ tptp.minus_minus_set_a A2) A2) tptp.bot_bot_set_a)) (forall ((A tptp.a)) (not (@ (@ tptp.member_a A) tptp.bot_bot_set_a))) (forall ((A2 tptp.set_a) (A tptp.a)) (=> (= A2 tptp.bot_bot_set_a) (not (@ (@ tptp.member_a A) A2)))) (forall ((A2 tptp.set_a)) (=> (forall ((Y2 tptp.a)) (not (@ (@ tptp.member_a Y2) A2))) (= A2 tptp.bot_bot_set_a))) (forall ((A2 tptp.set_a)) (= (exists ((X2 tptp.a)) (@ (@ tptp.member_a X2) A2)) (not (= A2 tptp.bot_bot_set_a)))) (forall ((A2 tptp.set_a) (B3 tptp.set_a) (X tptp.a)) (let ((_let_1 (@ tptp.member_a X))) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ _let_1 A2) (@ _let_1 B3))))) (forall ((A2 tptp.set_a) (B3 tptp.set_a) (C tptp.a)) (let ((_let_1 (@ tptp.member_a C))) (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (=> (@ _let_1 A2) (@ _let_1 B3))))) (forall ((A2 tptp.set_a) (B3 tptp.set_a)) (=> (= A2 B3) (not (=> (@ (@ tptp.ord_less_eq_set_a A2) B3) (not (@ (@ tptp.ord_less_eq_set_a B3) A2)))))) _let_45 _let_44 true))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))
% 0.71/0.88 )
% 0.71/0.88 % SZS output end Proof for ITP108^1
% 0.71/0.88 % cvc5---1.0.5 exiting
% 0.71/0.88 % cvc5---1.0.5 exiting
%------------------------------------------------------------------------------