TPTP Problem File: SLH0499^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : CRYSTALS-Kyber/0017_Abs_Qr/prob_00058_002024__25634166_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1699 ( 653 unt; 416 typ; 0 def)
% Number of atoms : 3338 (1424 equ; 0 cnn)
% Maximal formula atoms : 9 ( 2 avg)
% Number of connectives : 8955 ( 413 ~; 90 |; 257 &;7049 @)
% ( 0 <=>;1146 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Number of types : 76 ( 75 usr)
% Number of type conns : 1327 (1327 >; 0 *; 0 +; 0 <<)
% Number of symbols : 344 ( 341 usr; 82 con; 0-6 aty)
% Number of variables : 3083 ( 332 ^;2590 !; 161 ?;3083 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 12:35:49.985
%------------------------------------------------------------------------------
% Could-be-implicit typings (75)
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image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Int__Oint_J,type,
image_3115417691219061956_a_int: ( nat > sum_sum_a_int ) > set_nat > set_Sum_sum_a_int ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
image_95481332237249532um_a_a: ( nat > sum_sum_a_a ) > set_nat > set_Sum_sum_a_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__k_J,type,
image_95481375269537542um_a_k: ( nat > sum_sum_a_k ) > set_nat > set_Sum_sum_a_k ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__a,type,
image_nat_a: ( nat > a ) > set_nat > set_a ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__k,type,
image_nat_k: ( nat > k ) > set_nat > set_k ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Complex__Ocomplex,type,
image_3082061952195111286omplex: ( product_unit > complex ) > set_Product_unit > set_complex ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Int__Oint,type,
image_873079544045703924it_int: ( product_unit > int ) > set_Product_unit > set_int ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001tf__a,type,
image_Product_unit_a: ( product_unit > a ) > set_Product_unit > set_a ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001tf__k,type,
image_Product_unit_k: ( product_unit > k ) > set_Product_unit > set_k ).
thf(sy_c_Set_Oimage_001tf__a_001t__Int__Oint,type,
image_a_int: ( a > int ) > set_a > set_int ).
thf(sy_c_Set_Oimage_001tf__a_001t__Nat__Onat,type,
image_a_nat: ( a > nat ) > set_a > set_nat ).
thf(sy_c_Set_Oimage_001tf__k_001t__Int__Oint,type,
image_k_int: ( k > int ) > set_k > set_int ).
thf(sy_c_Set_Oimage_001tf__k_001t__Nat__Onat,type,
image_k_nat: ( k > nat ) > set_k > set_nat ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Int__Oint,type,
set_or4662586982721622107an_int: int > int > set_int ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat,type,
set_or4665077453230672383an_nat: nat > nat > set_nat ).
thf(sy_c_Transcendental_Oln__class_Oln_001t__Real__Oreal,type,
ln_ln_real: real > real ).
thf(sy_c_member_001t__Complex__Ocomplex,type,
member_complex: complex > set_complex > $o ).
thf(sy_c_member_001t__Finite____Field__Omod____ring_Itf__a_J,type,
member3034048621153491438ring_a: finite_mod_ring_a > set_Fi2982333969990053029ring_a > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Complex__Ocomplex_J,type,
member8603132577197391238omplex: sum_sum_a_complex > set_Su8486473086189545021omplex > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Int__Oint_J,type,
member_Sum_sum_a_int: sum_sum_a_int > set_Sum_sum_a_int > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__a_J,type,
member_Sum_sum_a_a: sum_sum_a_a > set_Sum_sum_a_a > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__k_J,type,
member_Sum_sum_a_k: sum_sum_a_k > set_Sum_sum_a_k > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__k,type,
member_k: k > set_k > $o ).
thf(sy_v_k,type,
k2: nat ).
thf(sy_v_n,type,
n: int ).
thf(sy_v_n_H,type,
n2: nat ).
thf(sy_v_q,type,
q: int ).
thf(sy_v_s,type,
s: int ).
thf(sy_v_x,type,
x: kyber_qr_a ).
% Relevant facts (1269)
thf(fact_0_finite__Max,axiom,
! [X: kyber_qr_a] :
( finite_finite_int
@ ( image_nat_int
@ ^ [Xa: nat] : ( abs_ky7385543178848499077ty_q_a @ q @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ X ) @ Xa ) )
@ top_top_set_nat ) ) ).
% finite_Max
thf(fact_1__092_060open_062_092_060And_062xa_O_Aof__int__mod__ring_As_A_K_Apoly_Ocoeff_A_Iof__qr_Ax_J_Axa_A_061_Apoly_Ocoeff_A_Iof__qr_A_Ito__module_As_A_K_Ax_J_J_Axa_092_060close_062,axiom,
! [Xa2: nat] :
( ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ s ) @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ x ) @ Xa2 ) )
= ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ ( times_2095635435063429214r_qr_a @ ( kyber_to_module_a @ s ) @ x ) ) @ Xa2 ) ) ).
% \<open>\<And>xa. of_int_mod_ring s * poly.coeff (of_qr x) xa = poly.coeff (of_qr (to_module s * x)) xa\<close>
thf(fact_2_kyber__spec_Oabs__infty__q_Ocong,axiom,
abs_ky7385543178848499077ty_q_a = abs_ky7385543178848499077ty_q_a ).
% kyber_spec.abs_infty_q.cong
thf(fact_3_finite__Collect__not,axiom,
! [P: complex > $o] :
( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
~ ( P @ X2 ) ) )
= ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).
% finite_Collect_not
thf(fact_4_finite__Collect__not,axiom,
! [P: product_unit > $o] :
( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X2: product_unit] :
~ ( P @ X2 ) ) )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Collect_not
thf(fact_5_finite__Collect__not,axiom,
! [P: nat > $o] :
( ( finite_finite_nat @ ( collect_nat @ P ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
~ ( P @ X2 ) ) )
= ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Collect_not
thf(fact_6_finite__Collect__not,axiom,
! [P: int > $o] :
( ( finite_finite_int @ ( collect_int @ P ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
~ ( P @ X2 ) ) )
= ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Collect_not
thf(fact_7_finite__Collect__not,axiom,
! [P: k > $o] :
( ( finite_finite_k @ ( collect_k @ P ) )
=> ( ( finite_finite_k
@ ( collect_k
@ ^ [X2: k] :
~ ( P @ X2 ) ) )
= ( finite_finite_k @ top_top_set_k ) ) ) ).
% finite_Collect_not
thf(fact_8_finite__Collect__not,axiom,
! [P: a > $o] :
( ( finite_finite_a @ ( collect_a @ P ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
~ ( P @ X2 ) ) )
= ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Collect_not
thf(fact_9_finite__Collect__not,axiom,
! [P: sum_sum_a_complex > $o] :
( ( finite6208533171692660870omplex @ ( collec1363137248864034504omplex @ P ) )
=> ( ( finite6208533171692660870omplex
@ ( collec1363137248864034504omplex
@ ^ [X2: sum_sum_a_complex] :
~ ( P @ X2 ) ) )
= ( finite6208533171692660870omplex @ top_to8676441370508295053omplex ) ) ) ).
% finite_Collect_not
thf(fact_10_finite__Collect__not,axiom,
! [P: sum_sum_a_nat > $o] :
( ( finite502105017643426984_a_nat @ ( collec7073057861543223018_a_nat @ P ) )
=> ( ( finite502105017643426984_a_nat
@ ( collec7073057861543223018_a_nat
@ ^ [X2: sum_sum_a_nat] :
~ ( P @ X2 ) ) )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ) ).
% finite_Collect_not
thf(fact_11_finite__Collect__not,axiom,
! [P: sum_sum_a_int > $o] :
( ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ P ) )
=> ( ( finite5547626034989006084_a_int
@ ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] :
~ ( P @ X2 ) ) )
= ( finite5547626034989006084_a_int @ top_to7528907356895570187_a_int ) ) ) ).
% finite_Collect_not
thf(fact_12_finite__Collect__not,axiom,
! [P: sum_sum_a_k > $o] :
( ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ P ) )
=> ( ( finite51705190296372934um_a_k
@ ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] :
~ ( P @ X2 ) ) )
= ( finite51705190296372934um_a_k @ top_to335874364214223893um_a_k ) ) ) ).
% finite_Collect_not
thf(fact_13_finite__imageI,axiom,
! [F: set_int,H: int > int] :
( ( finite_finite_int @ F )
=> ( finite_finite_int @ ( image_int_int @ H @ F ) ) ) ).
% finite_imageI
thf(fact_14_finite__imageI,axiom,
! [F: set_int,H: int > nat] :
( ( finite_finite_int @ F )
=> ( finite_finite_nat @ ( image_int_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_15_finite__imageI,axiom,
! [F: set_int,H: int > complex] :
( ( finite_finite_int @ F )
=> ( finite3207457112153483333omplex @ ( image_int_complex @ H @ F ) ) ) ).
% finite_imageI
thf(fact_16_finite__imageI,axiom,
! [F: set_nat,H: nat > int] :
( ( finite_finite_nat @ F )
=> ( finite_finite_int @ ( image_nat_int @ H @ F ) ) ) ).
% finite_imageI
thf(fact_17_finite__imageI,axiom,
! [F: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_18_finite__imageI,axiom,
! [F: set_nat,H: nat > complex] :
( ( finite_finite_nat @ F )
=> ( finite3207457112153483333omplex @ ( image_nat_complex @ H @ F ) ) ) ).
% finite_imageI
thf(fact_19_finite__imageI,axiom,
! [F: set_complex,H: complex > int] :
( ( finite3207457112153483333omplex @ F )
=> ( finite_finite_int @ ( image_complex_int @ H @ F ) ) ) ).
% finite_imageI
thf(fact_20_finite__imageI,axiom,
! [F: set_complex,H: complex > nat] :
( ( finite3207457112153483333omplex @ F )
=> ( finite_finite_nat @ ( image_complex_nat @ H @ F ) ) ) ).
% finite_imageI
thf(fact_21_finite__imageI,axiom,
! [F: set_complex,H: complex > complex] :
( ( finite3207457112153483333omplex @ F )
=> ( finite3207457112153483333omplex @ ( image_1468599708987790691omplex @ H @ F ) ) ) ).
% finite_imageI
thf(fact_22_finite__imageI,axiom,
! [F: set_Fi2982333969990053029ring_a,H: finite_mod_ring_a > int] :
( ( finite8151975738636043502ring_a @ F )
=> ( finite_finite_int @ ( image_4238506139956901036_a_int @ H @ F ) ) ) ).
% finite_imageI
thf(fact_23_finite__option__UNIV,axiom,
( ( finite2569390945932476949omplex @ top_to6180147692022559204omplex )
= ( finite3207457112153483333omplex @ top_top_set_complex ) ) ).
% finite_option_UNIV
thf(fact_24_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_25_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_26_finite__option__UNIV,axiom,
( ( finite1345302120164226195on_int @ top_to6430115241214627170on_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% finite_option_UNIV
thf(fact_27_finite__option__UNIV,axiom,
( ( finite1674126261360186615tion_k @ top_top_set_option_k )
= ( finite_finite_k @ top_top_set_k ) ) ).
% finite_option_UNIV
thf(fact_28_finite__option__UNIV,axiom,
( ( finite1674126218327898605tion_a @ top_top_set_option_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% finite_option_UNIV
thf(fact_29_finite__option__UNIV,axiom,
( ( finite6148646586853160780omplex @ top_to3173469198872825171omplex )
= ( finite6208533171692660870omplex @ top_to8676441370508295053omplex ) ) ).
% finite_option_UNIV
thf(fact_30_finite__option__UNIV,axiom,
( ( finite1805530552687753326_a_nat @ top_to5660197770879767925_a_nat )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ).
% finite_option_UNIV
thf(fact_31_finite__option__UNIV,axiom,
( ( finite8538819444610802378_a_int @ top_to846649352091849169_a_int )
= ( finite5547626034989006084_a_int @ top_to7528907356895570187_a_int ) ) ).
% finite_option_UNIV
thf(fact_32_finite__option__UNIV,axiom,
( ( finite2356166643258104854um_a_k @ top_to939405602112015205um_a_k )
= ( finite51705190296372934um_a_k @ top_to335874364214223893um_a_k ) ) ).
% finite_option_UNIV
thf(fact_33_finite__Plus__UNIV__iff,axiom,
( ( finite8634735631094037202omplex @ top_to3399062453794030625omplex )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_34_finite__Plus__UNIV__iff,axiom,
( ( finite9064798490750863615t_unit @ top_to6452945619387671430t_unit )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_35_finite__Plus__UNIV__iff,axiom,
( ( finite975246697100726004ex_nat @ top_to6881960113029525315ex_nat )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_36_finite__Plus__UNIV__iff,axiom,
( ( finite6020767714446305104ex_int @ top_to4391876968097798559ex_int )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_37_finite__Plus__UNIV__iff,axiom,
( ( finite4578673722322966394plex_k @ top_to3852329554344915073plex_k )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_k @ top_top_set_k ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_38_finite__Plus__UNIV__iff,axiom,
( ( finite4578673679290678384plex_a @ top_to3141989153881455223plex_a )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_39_finite__Plus__UNIV__iff,axiom,
( ( finite1078897678113127317omplex @ top_to4440894058499500828omplex )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_40_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_41_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_42_finite__Plus__UNIV__iff,axiom,
( ( finite224101892120063507it_int @ top_to404534460850747034it_int )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_43_finite__Collect__conjI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
| ( finite4290736615968046902t_unit @ ( collect_Product_unit @ Q ) ) )
=> ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X2: product_unit] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_44_finite__Collect__conjI,axiom,
! [P: sum_sum_a_int > $o,Q: sum_sum_a_int > $o] :
( ( ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ P ) )
| ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ Q ) ) )
=> ( finite5547626034989006084_a_int
@ ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_45_finite__Collect__conjI,axiom,
! [P: sum_sum_a_k > $o,Q: sum_sum_a_k > $o] :
( ( ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ P ) )
| ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ Q ) ) )
=> ( finite51705190296372934um_a_k
@ ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_46_finite__Collect__conjI,axiom,
! [P: sum_sum_a_a > $o,Q: sum_sum_a_a > $o] :
( ( ( finite51705147264084924um_a_a @ ( collect_Sum_sum_a_a @ P ) )
| ( finite51705147264084924um_a_a @ ( collect_Sum_sum_a_a @ Q ) ) )
=> ( finite51705147264084924um_a_a
@ ( collect_Sum_sum_a_a
@ ^ [X2: sum_sum_a_a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_47_finite__Collect__conjI,axiom,
! [P: k > $o,Q: k > $o] :
( ( ( finite_finite_k @ ( collect_k @ P ) )
| ( finite_finite_k @ ( collect_k @ Q ) ) )
=> ( finite_finite_k
@ ( collect_k
@ ^ [X2: k] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_48_finite__Collect__conjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( ( finite_finite_a @ ( collect_a @ P ) )
| ( finite_finite_a @ ( collect_a @ Q ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_49_finite__Collect__conjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( ( finite_finite_int @ ( collect_int @ P ) )
| ( finite_finite_int @ ( collect_int @ Q ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_50_finite__Collect__conjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( ( finite_finite_nat @ ( collect_nat @ P ) )
| ( finite_finite_nat @ ( collect_nat @ Q ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_51_finite__Collect__conjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
| ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( P @ X2 )
& ( Q @ X2 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_52_finite__Collect__disjI,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X2: product_unit] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
& ( finite4290736615968046902t_unit @ ( collect_Product_unit @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_53_finite__Collect__disjI,axiom,
! [P: sum_sum_a_int > $o,Q: sum_sum_a_int > $o] :
( ( finite5547626034989006084_a_int
@ ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ P ) )
& ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_54_finite__Collect__disjI,axiom,
! [P: sum_sum_a_k > $o,Q: sum_sum_a_k > $o] :
( ( finite51705190296372934um_a_k
@ ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ P ) )
& ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_55_finite__Collect__disjI,axiom,
! [P: sum_sum_a_a > $o,Q: sum_sum_a_a > $o] :
( ( finite51705147264084924um_a_a
@ ( collect_Sum_sum_a_a
@ ^ [X2: sum_sum_a_a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite51705147264084924um_a_a @ ( collect_Sum_sum_a_a @ P ) )
& ( finite51705147264084924um_a_a @ ( collect_Sum_sum_a_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_56_finite__Collect__disjI,axiom,
! [P: k > $o,Q: k > $o] :
( ( finite_finite_k
@ ( collect_k
@ ^ [X2: k] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_k @ ( collect_k @ P ) )
& ( finite_finite_k @ ( collect_k @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_57_finite__Collect__disjI,axiom,
! [P: a > $o,Q: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_a @ ( collect_a @ P ) )
& ( finite_finite_a @ ( collect_a @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_58_finite__Collect__disjI,axiom,
! [P: int > $o,Q: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_int @ ( collect_int @ P ) )
& ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_59_finite__Collect__disjI,axiom,
! [P: nat > $o,Q: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite_finite_nat @ ( collect_nat @ P ) )
& ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_60_finite__Collect__disjI,axiom,
! [P: complex > $o,Q: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] :
( ( P @ X2 )
| ( Q @ X2 ) ) ) )
= ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
& ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_61_image__ident,axiom,
! [Y: set_nat] :
( ( image_nat_nat
@ ^ [X2: nat] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_62_image__ident,axiom,
! [Y: set_int] :
( ( image_int_int
@ ^ [X2: int] : X2
@ Y )
= Y ) ).
% image_ident
thf(fact_63_finite__range__imageI,axiom,
! [G: product_unit > int,F2: int > int] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_int
@ ( image_873079544045703924it_int
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_64_finite__range__imageI,axiom,
! [G: product_unit > int,F2: int > nat] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_nat
@ ( image_875570014554754200it_nat
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_65_finite__range__imageI,axiom,
! [G: product_unit > int,F2: int > complex] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) )
=> ( finite3207457112153483333omplex
@ ( image_3082061952195111286omplex
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_66_finite__range__imageI,axiom,
! [G: product_unit > int,F2: int > k] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_k
@ ( image_Product_unit_k
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_67_finite__range__imageI,axiom,
! [G: product_unit > int,F2: int > a] :
( ( finite_finite_int @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_a
@ ( image_Product_unit_a
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_68_finite__range__imageI,axiom,
! [G: product_unit > nat,F2: nat > int] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_int
@ ( image_873079544045703924it_int
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_69_finite__range__imageI,axiom,
! [G: product_unit > nat,F2: nat > nat] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_nat
@ ( image_875570014554754200it_nat
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_70_finite__range__imageI,axiom,
! [G: product_unit > nat,F2: nat > complex] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) )
=> ( finite3207457112153483333omplex
@ ( image_3082061952195111286omplex
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_71_finite__range__imageI,axiom,
! [G: product_unit > nat,F2: nat > k] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_k
@ ( image_Product_unit_k
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_72_finite__range__imageI,axiom,
! [G: product_unit > nat,F2: nat > a] :
( ( finite_finite_nat @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_a
@ ( image_Product_unit_a
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit ) ) ) ).
% finite_range_imageI
thf(fact_73_finite__code,axiom,
( finite51705190296372934um_a_k
= ( ^ [A: set_Sum_sum_a_k] : $true ) ) ).
% finite_code
thf(fact_74_finite__code,axiom,
( finite51705147264084924um_a_a
= ( ^ [A: set_Sum_sum_a_a] : $true ) ) ).
% finite_code
thf(fact_75_finite__code,axiom,
( finite_finite_k
= ( ^ [A: set_k] : $true ) ) ).
% finite_code
thf(fact_76_finite__code,axiom,
( finite_finite_a
= ( ^ [A: set_a] : $true ) ) ).
% finite_code
thf(fact_77_image__eqI,axiom,
! [B: int,F2: finite_mod_ring_a > int,X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a] :
( ( B
= ( F2 @ X ) )
=> ( ( member3034048621153491438ring_a @ X @ A2 )
=> ( member_int @ B @ ( image_4238506139956901036_a_int @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_78_image__eqI,axiom,
! [B: complex,F2: int > complex,X: int,A2: set_int] :
( ( B
= ( F2 @ X ) )
=> ( ( member_int @ X @ A2 )
=> ( member_complex @ B @ ( image_int_complex @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_79_image__eqI,axiom,
! [B: int,F2: int > int,X: int,A2: set_int] :
( ( B
= ( F2 @ X ) )
=> ( ( member_int @ X @ A2 )
=> ( member_int @ B @ ( image_int_int @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_80_image__eqI,axiom,
! [B: nat,F2: int > nat,X: int,A2: set_int] :
( ( B
= ( F2 @ X ) )
=> ( ( member_int @ X @ A2 )
=> ( member_nat @ B @ ( image_int_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_81_image__eqI,axiom,
! [B: int,F2: nat > int,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_int @ B @ ( image_nat_int @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_82_image__eqI,axiom,
! [B: complex,F2: nat > complex,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_complex @ B @ ( image_nat_complex @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_83_image__eqI,axiom,
! [B: nat,F2: nat > nat,X: nat,A2: set_nat] :
( ( B
= ( F2 @ X ) )
=> ( ( member_nat @ X @ A2 )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% image_eqI
thf(fact_84_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_85_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_86_UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% UNIV_I
thf(fact_87_UNIV__I,axiom,
! [X: k] : ( member_k @ X @ top_top_set_k ) ).
% UNIV_I
thf(fact_88_UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% UNIV_I
thf(fact_89_UNIV__I,axiom,
! [X: sum_sum_a_complex] : ( member8603132577197391238omplex @ X @ top_to8676441370508295053omplex ) ).
% UNIV_I
thf(fact_90_UNIV__I,axiom,
! [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ top_to795618464972521135_a_nat ) ).
% UNIV_I
thf(fact_91_UNIV__I,axiom,
! [X: sum_sum_a_int] : ( member_Sum_sum_a_int @ X @ top_to7528907356895570187_a_int ) ).
% UNIV_I
thf(fact_92_UNIV__I,axiom,
! [X: sum_sum_a_k] : ( member_Sum_sum_a_k @ X @ top_to335874364214223893um_a_k ) ).
% UNIV_I
thf(fact_93_UNIV__I,axiom,
! [X: sum_sum_a_a] : ( member_Sum_sum_a_a @ X @ top_to8848906000605539851um_a_a ) ).
% UNIV_I
thf(fact_94_iso__tuple__UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_95_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_96_iso__tuple__UNIV__I,axiom,
! [X: int] : ( member_int @ X @ top_top_set_int ) ).
% iso_tuple_UNIV_I
thf(fact_97_iso__tuple__UNIV__I,axiom,
! [X: k] : ( member_k @ X @ top_top_set_k ) ).
% iso_tuple_UNIV_I
thf(fact_98_iso__tuple__UNIV__I,axiom,
! [X: a] : ( member_a @ X @ top_top_set_a ) ).
% iso_tuple_UNIV_I
thf(fact_99_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_a_complex] : ( member8603132577197391238omplex @ X @ top_to8676441370508295053omplex ) ).
% iso_tuple_UNIV_I
thf(fact_100_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X @ top_to795618464972521135_a_nat ) ).
% iso_tuple_UNIV_I
thf(fact_101_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_a_int] : ( member_Sum_sum_a_int @ X @ top_to7528907356895570187_a_int ) ).
% iso_tuple_UNIV_I
thf(fact_102_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_a_k] : ( member_Sum_sum_a_k @ X @ top_to335874364214223893um_a_k ) ).
% iso_tuple_UNIV_I
thf(fact_103_iso__tuple__UNIV__I,axiom,
! [X: sum_sum_a_a] : ( member_Sum_sum_a_a @ X @ top_to8848906000605539851um_a_a ) ).
% iso_tuple_UNIV_I
thf(fact_104_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_105_UNIV__witness,axiom,
? [X3: product_unit] : ( member_Product_unit @ X3 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_106_UNIV__witness,axiom,
? [X3: nat] : ( member_nat @ X3 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_107_UNIV__witness,axiom,
? [X3: int] : ( member_int @ X3 @ top_top_set_int ) ).
% UNIV_witness
thf(fact_108_UNIV__witness,axiom,
? [X3: k] : ( member_k @ X3 @ top_top_set_k ) ).
% UNIV_witness
thf(fact_109_UNIV__witness,axiom,
? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).
% UNIV_witness
thf(fact_110_UNIV__witness,axiom,
? [X3: sum_sum_a_complex] : ( member8603132577197391238omplex @ X3 @ top_to8676441370508295053omplex ) ).
% UNIV_witness
thf(fact_111_UNIV__witness,axiom,
? [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ top_to795618464972521135_a_nat ) ).
% UNIV_witness
thf(fact_112_UNIV__witness,axiom,
? [X3: sum_sum_a_int] : ( member_Sum_sum_a_int @ X3 @ top_to7528907356895570187_a_int ) ).
% UNIV_witness
thf(fact_113_UNIV__witness,axiom,
? [X3: sum_sum_a_k] : ( member_Sum_sum_a_k @ X3 @ top_to335874364214223893um_a_k ) ).
% UNIV_witness
thf(fact_114_UNIV__witness,axiom,
? [X3: sum_sum_a_a] : ( member_Sum_sum_a_a @ X3 @ top_to8848906000605539851um_a_a ) ).
% UNIV_witness
thf(fact_115_UNIV__eq__I,axiom,
! [A2: set_Product_unit] :
( ! [X3: product_unit] : ( member_Product_unit @ X3 @ A2 )
=> ( top_to1996260823553986621t_unit = A2 ) ) ).
% UNIV_eq_I
thf(fact_116_UNIV__eq__I,axiom,
! [A2: set_nat] :
( ! [X3: nat] : ( member_nat @ X3 @ A2 )
=> ( top_top_set_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_117_UNIV__eq__I,axiom,
! [A2: set_int] :
( ! [X3: int] : ( member_int @ X3 @ A2 )
=> ( top_top_set_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_118_UNIV__eq__I,axiom,
! [A2: set_k] :
( ! [X3: k] : ( member_k @ X3 @ A2 )
=> ( top_top_set_k = A2 ) ) ).
% UNIV_eq_I
thf(fact_119_UNIV__eq__I,axiom,
! [A2: set_a] :
( ! [X3: a] : ( member_a @ X3 @ A2 )
=> ( top_top_set_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_120_UNIV__eq__I,axiom,
! [A2: set_Su8486473086189545021omplex] :
( ! [X3: sum_sum_a_complex] : ( member8603132577197391238omplex @ X3 @ A2 )
=> ( top_to8676441370508295053omplex = A2 ) ) ).
% UNIV_eq_I
thf(fact_121_UNIV__eq__I,axiom,
! [A2: set_Sum_sum_a_nat] :
( ! [X3: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X3 @ A2 )
=> ( top_to795618464972521135_a_nat = A2 ) ) ).
% UNIV_eq_I
thf(fact_122_UNIV__eq__I,axiom,
! [A2: set_Sum_sum_a_int] :
( ! [X3: sum_sum_a_int] : ( member_Sum_sum_a_int @ X3 @ A2 )
=> ( top_to7528907356895570187_a_int = A2 ) ) ).
% UNIV_eq_I
thf(fact_123_UNIV__eq__I,axiom,
! [A2: set_Sum_sum_a_k] :
( ! [X3: sum_sum_a_k] : ( member_Sum_sum_a_k @ X3 @ A2 )
=> ( top_to335874364214223893um_a_k = A2 ) ) ).
% UNIV_eq_I
thf(fact_124_UNIV__eq__I,axiom,
! [A2: set_Sum_sum_a_a] :
( ! [X3: sum_sum_a_a] : ( member_Sum_sum_a_a @ X3 @ A2 )
=> ( top_to8848906000605539851um_a_a = A2 ) ) ).
% UNIV_eq_I
thf(fact_125_rev__image__eqI,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,B: int,F2: finite_mod_ring_a > int] :
( ( member3034048621153491438ring_a @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_4238506139956901036_a_int @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_126_rev__image__eqI,axiom,
! [X: int,A2: set_int,B: complex,F2: int > complex] :
( ( member_int @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_complex @ B @ ( image_int_complex @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_127_rev__image__eqI,axiom,
! [X: int,A2: set_int,B: int,F2: int > int] :
( ( member_int @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_int_int @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_128_rev__image__eqI,axiom,
! [X: int,A2: set_int,B: nat,F2: int > nat] :
( ( member_int @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_129_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: int,F2: nat > int] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_130_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: complex,F2: nat > complex] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_131_rev__image__eqI,axiom,
! [X: nat,A2: set_nat,B: nat,F2: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) ) ) ) ).
% rev_image_eqI
thf(fact_132_ball__imageD,axiom,
! [F2: nat > int,A2: set_nat,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_nat_int @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_133_ball__imageD,axiom,
! [F2: finite_mod_ring_a > int,A2: set_Fi2982333969990053029ring_a,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_4238506139956901036_a_int @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_134_ball__imageD,axiom,
! [F2: nat > complex,A2: set_nat,P: complex > $o] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( image_nat_complex @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_135_ball__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_136_ball__imageD,axiom,
! [F2: int > complex,A2: set_int,P: complex > $o] :
( ! [X3: complex] :
( ( member_complex @ X3 @ ( image_int_complex @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_137_ball__imageD,axiom,
! [F2: int > nat,A2: set_int,P: nat > $o] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( image_int_nat @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_138_ball__imageD,axiom,
! [F2: int > int,A2: set_int,P: int > $o] :
( ! [X3: int] :
( ( member_int @ X3 @ ( image_int_int @ F2 @ A2 ) )
=> ( P @ X3 ) )
=> ! [X4: int] :
( ( member_int @ X4 @ A2 )
=> ( P @ ( F2 @ X4 ) ) ) ) ).
% ball_imageD
thf(fact_139_image__cong,axiom,
! [M: set_Fi2982333969990053029ring_a,N: set_Fi2982333969990053029ring_a,F2: finite_mod_ring_a > int,G: finite_mod_ring_a > int] :
( ( M = N )
=> ( ! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_4238506139956901036_a_int @ F2 @ M )
= ( image_4238506139956901036_a_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_140_image__cong,axiom,
! [M: set_int,N: set_int,F2: int > complex,G: int > complex] :
( ( M = N )
=> ( ! [X3: int] :
( ( member_int @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_int_complex @ F2 @ M )
= ( image_int_complex @ G @ N ) ) ) ) ).
% image_cong
thf(fact_141_image__cong,axiom,
! [M: set_int,N: set_int,F2: int > nat,G: int > nat] :
( ( M = N )
=> ( ! [X3: int] :
( ( member_int @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_int_nat @ F2 @ M )
= ( image_int_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_142_image__cong,axiom,
! [M: set_int,N: set_int,F2: int > int,G: int > int] :
( ( M = N )
=> ( ! [X3: int] :
( ( member_int @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_int_int @ F2 @ M )
= ( image_int_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_143_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > int,G: nat > int] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_int @ F2 @ M )
= ( image_nat_int @ G @ N ) ) ) ) ).
% image_cong
thf(fact_144_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > complex,G: nat > complex] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_complex @ F2 @ M )
= ( image_nat_complex @ G @ N ) ) ) ) ).
% image_cong
thf(fact_145_image__cong,axiom,
! [M: set_nat,N: set_nat,F2: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) )
=> ( ( image_nat_nat @ F2 @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_146_bex__imageD,axiom,
! [F2: nat > int,A2: set_nat,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_nat_int @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_147_bex__imageD,axiom,
! [F2: finite_mod_ring_a > int,A2: set_Fi2982333969990053029ring_a,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_4238506139956901036_a_int @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_148_bex__imageD,axiom,
! [F2: nat > complex,A2: set_nat,P: complex > $o] :
( ? [X4: complex] :
( ( member_complex @ X4 @ ( image_nat_complex @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_149_bex__imageD,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_150_bex__imageD,axiom,
! [F2: int > complex,A2: set_int,P: complex > $o] :
( ? [X4: complex] :
( ( member_complex @ X4 @ ( image_int_complex @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_151_bex__imageD,axiom,
! [F2: int > nat,A2: set_int,P: nat > $o] :
( ? [X4: nat] :
( ( member_nat @ X4 @ ( image_int_nat @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_152_bex__imageD,axiom,
! [F2: int > int,A2: set_int,P: int > $o] :
( ? [X4: int] :
( ( member_int @ X4 @ ( image_int_int @ F2 @ A2 ) )
& ( P @ X4 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ( P @ ( F2 @ X3 ) ) ) ) ).
% bex_imageD
thf(fact_153_image__iff,axiom,
! [Z: int,F2: nat > int,A2: set_nat] :
( ( member_int @ Z @ ( image_nat_int @ F2 @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_154_image__iff,axiom,
! [Z: int,F2: finite_mod_ring_a > int,A2: set_Fi2982333969990053029ring_a] :
( ( member_int @ Z @ ( image_4238506139956901036_a_int @ F2 @ A2 ) )
= ( ? [X2: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_155_image__iff,axiom,
! [Z: complex,F2: nat > complex,A2: set_nat] :
( ( member_complex @ Z @ ( image_nat_complex @ F2 @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_156_image__iff,axiom,
! [Z: complex,F2: int > complex,A2: set_int] :
( ( member_complex @ Z @ ( image_int_complex @ F2 @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_157_image__iff,axiom,
! [Z: int,F2: int > int,A2: set_int] :
( ( member_int @ Z @ ( image_int_int @ F2 @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_158_image__iff,axiom,
! [Z: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F2 @ A2 ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_159_image__iff,axiom,
! [Z: nat,F2: int > nat,A2: set_int] :
( ( member_nat @ Z @ ( image_int_nat @ F2 @ A2 ) )
= ( ? [X2: int] :
( ( member_int @ X2 @ A2 )
& ( Z
= ( F2 @ X2 ) ) ) ) ) ).
% image_iff
thf(fact_160_imageI,axiom,
! [X: finite_mod_ring_a,A2: set_Fi2982333969990053029ring_a,F2: finite_mod_ring_a > int] :
( ( member3034048621153491438ring_a @ X @ A2 )
=> ( member_int @ ( F2 @ X ) @ ( image_4238506139956901036_a_int @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_161_imageI,axiom,
! [X: int,A2: set_int,F2: int > complex] :
( ( member_int @ X @ A2 )
=> ( member_complex @ ( F2 @ X ) @ ( image_int_complex @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_162_imageI,axiom,
! [X: int,A2: set_int,F2: int > int] :
( ( member_int @ X @ A2 )
=> ( member_int @ ( F2 @ X ) @ ( image_int_int @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_163_imageI,axiom,
! [X: int,A2: set_int,F2: int > nat] :
( ( member_int @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_int_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_164_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > int] :
( ( member_nat @ X @ A2 )
=> ( member_int @ ( F2 @ X ) @ ( image_nat_int @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_165_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > complex] :
( ( member_nat @ X @ A2 )
=> ( member_complex @ ( F2 @ X ) @ ( image_nat_complex @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_166_imageI,axiom,
! [X: nat,A2: set_nat,F2: nat > nat] :
( ( member_nat @ X @ A2 )
=> ( member_nat @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ A2 ) ) ) ).
% imageI
thf(fact_167_finite,axiom,
! [A2: set_Sum_sum_a_k] : ( finite51705190296372934um_a_k @ A2 ) ).
% finite
thf(fact_168_finite,axiom,
! [A2: set_Sum_sum_a_a] : ( finite51705147264084924um_a_a @ A2 ) ).
% finite
thf(fact_169_finite,axiom,
! [A2: set_k] : ( finite_finite_k @ A2 ) ).
% finite
thf(fact_170_finite,axiom,
! [A2: set_a] : ( finite_finite_a @ A2 ) ).
% finite
thf(fact_171_UNIV__def,axiom,
( top_top_set_complex
= ( collect_complex
@ ^ [X2: complex] : $true ) ) ).
% UNIV_def
thf(fact_172_UNIV__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit
@ ^ [X2: product_unit] : $true ) ) ).
% UNIV_def
thf(fact_173_UNIV__def,axiom,
( top_top_set_nat
= ( collect_nat
@ ^ [X2: nat] : $true ) ) ).
% UNIV_def
thf(fact_174_UNIV__def,axiom,
( top_top_set_int
= ( collect_int
@ ^ [X2: int] : $true ) ) ).
% UNIV_def
thf(fact_175_UNIV__def,axiom,
( top_top_set_k
= ( collect_k
@ ^ [X2: k] : $true ) ) ).
% UNIV_def
thf(fact_176_UNIV__def,axiom,
( top_top_set_a
= ( collect_a
@ ^ [X2: a] : $true ) ) ).
% UNIV_def
thf(fact_177_UNIV__def,axiom,
( top_to8676441370508295053omplex
= ( collec1363137248864034504omplex
@ ^ [X2: sum_sum_a_complex] : $true ) ) ).
% UNIV_def
thf(fact_178_UNIV__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat
@ ^ [X2: sum_sum_a_nat] : $true ) ) ).
% UNIV_def
thf(fact_179_UNIV__def,axiom,
( top_to7528907356895570187_a_int
= ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] : $true ) ) ).
% UNIV_def
thf(fact_180_UNIV__def,axiom,
( top_to335874364214223893um_a_k
= ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] : $true ) ) ).
% UNIV_def
thf(fact_181_Compr__image__eq,axiom,
! [F2: int > int,A2: set_int,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_int_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_int @ F2
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_182_Compr__image__eq,axiom,
! [F2: nat > int,A2: set_nat,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_nat_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_int @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_183_Compr__image__eq,axiom,
! [F2: complex > int,A2: set_complex,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_complex_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_complex_int @ F2
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_184_Compr__image__eq,axiom,
! [F2: a > int,A2: set_a,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_a_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_int @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_185_Compr__image__eq,axiom,
! [F2: k > int,A2: set_k,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_k_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_k_int @ F2
@ ( collect_k
@ ^ [X2: k] :
( ( member_k @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_186_Compr__image__eq,axiom,
! [F2: product_unit > int,A2: set_Product_unit,P: int > $o] :
( ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ ( image_873079544045703924it_int @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_873079544045703924it_int @ F2
@ ( collect_Product_unit
@ ^ [X2: product_unit] :
( ( member_Product_unit @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_187_Compr__image__eq,axiom,
! [F2: int > nat,A2: set_int,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_int_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_int_nat @ F2
@ ( collect_int
@ ^ [X2: int] :
( ( member_int @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_188_Compr__image__eq,axiom,
! [F2: nat > nat,A2: set_nat,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_189_Compr__image__eq,axiom,
! [F2: complex > nat,A2: set_complex,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_complex_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_complex_nat @ F2
@ ( collect_complex
@ ^ [X2: complex] :
( ( member_complex @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_190_Compr__image__eq,axiom,
! [F2: a > nat,A2: set_a,P: nat > $o] :
( ( collect_nat
@ ^ [X2: nat] :
( ( member_nat @ X2 @ ( image_a_nat @ F2 @ A2 ) )
& ( P @ X2 ) ) )
= ( image_a_nat @ F2
@ ( collect_a
@ ^ [X2: a] :
( ( member_a @ X2 @ A2 )
& ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_191_image__image,axiom,
! [F2: complex > int,G: nat > complex,A2: set_nat] :
( ( image_complex_int @ F2 @ ( image_nat_complex @ G @ A2 ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_192_image__image,axiom,
! [F2: complex > complex,G: nat > complex,A2: set_nat] :
( ( image_1468599708987790691omplex @ F2 @ ( image_nat_complex @ G @ A2 ) )
= ( image_nat_complex
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_193_image__image,axiom,
! [F2: complex > nat,G: nat > complex,A2: set_nat] :
( ( image_complex_nat @ F2 @ ( image_nat_complex @ G @ A2 ) )
= ( image_nat_nat
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_194_image__image,axiom,
! [F2: complex > complex,G: int > complex,A2: set_int] :
( ( image_1468599708987790691omplex @ F2 @ ( image_int_complex @ G @ A2 ) )
= ( image_int_complex
@ ^ [X2: int] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_195_image__image,axiom,
! [F2: complex > nat,G: int > complex,A2: set_int] :
( ( image_complex_nat @ F2 @ ( image_int_complex @ G @ A2 ) )
= ( image_int_nat
@ ^ [X2: int] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_196_image__image,axiom,
! [F2: complex > int,G: int > complex,A2: set_int] :
( ( image_complex_int @ F2 @ ( image_int_complex @ G @ A2 ) )
= ( image_int_int
@ ^ [X2: int] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_197_image__image,axiom,
! [F2: nat > int,G: nat > nat,A2: set_nat] :
( ( image_nat_int @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_int
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_198_image__image,axiom,
! [F2: nat > int,G: int > nat,A2: set_int] :
( ( image_nat_int @ F2 @ ( image_int_nat @ G @ A2 ) )
= ( image_int_int
@ ^ [X2: int] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_199_image__image,axiom,
! [F2: nat > complex,G: nat > nat,A2: set_nat] :
( ( image_nat_complex @ F2 @ ( image_nat_nat @ G @ A2 ) )
= ( image_nat_complex
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_200_image__image,axiom,
! [F2: nat > complex,G: int > nat,A2: set_int] :
( ( image_nat_complex @ F2 @ ( image_int_nat @ G @ A2 ) )
= ( image_int_complex
@ ^ [X2: int] : ( F2 @ ( G @ X2 ) )
@ A2 ) ) ).
% image_image
thf(fact_201_imageE,axiom,
! [B: int,F2: finite_mod_ring_a > int,A2: set_Fi2982333969990053029ring_a] :
( ( member_int @ B @ ( image_4238506139956901036_a_int @ F2 @ A2 ) )
=> ~ ! [X3: finite_mod_ring_a] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member3034048621153491438ring_a @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_202_imageE,axiom,
! [B: complex,F2: int > complex,A2: set_int] :
( ( member_complex @ B @ ( image_int_complex @ F2 @ A2 ) )
=> ~ ! [X3: int] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_int @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_203_imageE,axiom,
! [B: int,F2: int > int,A2: set_int] :
( ( member_int @ B @ ( image_int_int @ F2 @ A2 ) )
=> ~ ! [X3: int] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_int @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_204_imageE,axiom,
! [B: int,F2: nat > int,A2: set_nat] :
( ( member_int @ B @ ( image_nat_int @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_205_imageE,axiom,
! [B: complex,F2: nat > complex,A2: set_nat] :
( ( member_complex @ B @ ( image_nat_complex @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_206_imageE,axiom,
! [B: nat,F2: int > nat,A2: set_int] :
( ( member_nat @ B @ ( image_int_nat @ F2 @ A2 ) )
=> ~ ! [X3: int] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_int @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_207_imageE,axiom,
! [B: nat,F2: nat > nat,A2: set_nat] :
( ( member_nat @ B @ ( image_nat_nat @ F2 @ A2 ) )
=> ~ ! [X3: nat] :
( ( B
= ( F2 @ X3 ) )
=> ~ ( member_nat @ X3 @ A2 ) ) ) ).
% imageE
thf(fact_208_pigeonhole__infinite__rel,axiom,
! [A2: set_Product_unit,B2: set_int,R: product_unit > int > $o] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_209_pigeonhole__infinite__rel,axiom,
! [A2: set_Product_unit,B2: set_nat,R: product_unit > nat > $o] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_210_pigeonhole__infinite__rel,axiom,
! [A2: set_Product_unit,B2: set_complex,R: product_unit > complex > $o] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite3207457112153483333omplex @ B2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ? [Xa3: complex] :
( ( member_complex @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_211_pigeonhole__infinite__rel,axiom,
! [A2: set_Product_unit,B2: set_k,R: product_unit > k > $o] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_k @ B2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ? [Xa3: k] :
( ( member_k @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: k] :
( ( member_k @ X3 @ B2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_212_pigeonhole__infinite__rel,axiom,
! [A2: set_Product_unit,B2: set_a,R: product_unit > a > $o] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_213_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_int,R: int > int > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa3: int] :
( ( member_int @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: int] :
( ( member_int @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_214_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_nat,R: int > nat > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa3: nat] :
( ( member_nat @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: nat] :
( ( member_nat @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_215_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_complex,R: int > complex > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite3207457112153483333omplex @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa3: complex] :
( ( member_complex @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: complex] :
( ( member_complex @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_216_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_k,R: int > k > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_k @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa3: k] :
( ( member_k @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: k] :
( ( member_k @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_217_pigeonhole__infinite__rel,axiom,
! [A2: set_int,B2: set_a,R: int > a > $o] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_a @ B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ A2 )
=> ? [Xa3: a] :
( ( member_a @ Xa3 @ B2 )
& ( R @ X3 @ Xa3 ) ) )
=> ? [X3: a] :
( ( member_a @ X3 @ B2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( R @ A3 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_218_not__finite__existsD,axiom,
! [P: product_unit > $o] :
( ~ ( finite4290736615968046902t_unit @ ( collect_Product_unit @ P ) )
=> ? [X_1: product_unit] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_219_not__finite__existsD,axiom,
! [P: int > $o] :
( ~ ( finite_finite_int @ ( collect_int @ P ) )
=> ? [X_1: int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_220_not__finite__existsD,axiom,
! [P: nat > $o] :
( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
=> ? [X_1: nat] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_221_not__finite__existsD,axiom,
! [P: complex > $o] :
( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
=> ? [X_1: complex] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_222_not__finite__existsD,axiom,
! [P: sum_sum_a_int > $o] :
( ~ ( finite5547626034989006084_a_int @ ( collec2895206842034026310_a_int @ P ) )
=> ? [X_1: sum_sum_a_int] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_223_not__finite__existsD,axiom,
! [P: sum_sum_a_k > $o] :
( ~ ( finite51705190296372934um_a_k @ ( collect_Sum_sum_a_k @ P ) )
=> ? [X_1: sum_sum_a_k] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_224_not__finite__existsD,axiom,
! [P: sum_sum_a_a > $o] :
( ~ ( finite51705147264084924um_a_a @ ( collect_Sum_sum_a_a @ P ) )
=> ? [X_1: sum_sum_a_a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_225_not__finite__existsD,axiom,
! [P: k > $o] :
( ~ ( finite_finite_k @ ( collect_k @ P ) )
=> ? [X_1: k] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_226_not__finite__existsD,axiom,
! [P: a > $o] :
( ~ ( finite_finite_a @ ( collect_a @ P ) )
=> ? [X_1: a] : ( P @ X_1 ) ) ).
% not_finite_existsD
thf(fact_227_range__eqI,axiom,
! [B: nat,F2: product_unit > nat,X: product_unit] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_228_range__eqI,axiom,
! [B: int,F2: nat > int,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_nat_int @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_229_range__eqI,axiom,
! [B: complex,F2: nat > complex,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member_complex @ B @ ( image_nat_complex @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_230_range__eqI,axiom,
! [B: nat,F2: nat > nat,X: nat] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_231_range__eqI,axiom,
! [B: complex,F2: int > complex,X: int] :
( ( B
= ( F2 @ X ) )
=> ( member_complex @ B @ ( image_int_complex @ F2 @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_232_range__eqI,axiom,
! [B: int,F2: int > int,X: int] :
( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_int_int @ F2 @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_233_range__eqI,axiom,
! [B: nat,F2: int > nat,X: int] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_int_nat @ F2 @ top_top_set_int ) ) ) ).
% range_eqI
thf(fact_234_range__eqI,axiom,
! [B: nat,F2: k > nat,X: k] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_k_nat @ F2 @ top_top_set_k ) ) ) ).
% range_eqI
thf(fact_235_range__eqI,axiom,
! [B: nat,F2: a > nat,X: a] :
( ( B
= ( F2 @ X ) )
=> ( member_nat @ B @ ( image_a_nat @ F2 @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_236_range__eqI,axiom,
! [B: int,F2: finite_mod_ring_a > int,X: finite_mod_ring_a] :
( ( B
= ( F2 @ X ) )
=> ( member_int @ B @ ( image_4238506139956901036_a_int @ F2 @ top_to2069866484006881781ring_a ) ) ) ).
% range_eqI
thf(fact_237_rangeI,axiom,
! [F2: product_unit > nat,X: product_unit] : ( member_nat @ ( F2 @ X ) @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_238_rangeI,axiom,
! [F2: nat > int,X: nat] : ( member_int @ ( F2 @ X ) @ ( image_nat_int @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_239_rangeI,axiom,
! [F2: nat > complex,X: nat] : ( member_complex @ ( F2 @ X ) @ ( image_nat_complex @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_240_rangeI,axiom,
! [F2: nat > nat,X: nat] : ( member_nat @ ( F2 @ X ) @ ( image_nat_nat @ F2 @ top_top_set_nat ) ) ).
% rangeI
thf(fact_241_rangeI,axiom,
! [F2: int > complex,X: int] : ( member_complex @ ( F2 @ X ) @ ( image_int_complex @ F2 @ top_top_set_int ) ) ).
% rangeI
thf(fact_242_rangeI,axiom,
! [F2: int > int,X: int] : ( member_int @ ( F2 @ X ) @ ( image_int_int @ F2 @ top_top_set_int ) ) ).
% rangeI
thf(fact_243_rangeI,axiom,
! [F2: int > nat,X: int] : ( member_nat @ ( F2 @ X ) @ ( image_int_nat @ F2 @ top_top_set_int ) ) ).
% rangeI
thf(fact_244_rangeI,axiom,
! [F2: k > nat,X: k] : ( member_nat @ ( F2 @ X ) @ ( image_k_nat @ F2 @ top_top_set_k ) ) ).
% rangeI
thf(fact_245_rangeI,axiom,
! [F2: a > nat,X: a] : ( member_nat @ ( F2 @ X ) @ ( image_a_nat @ F2 @ top_top_set_a ) ) ).
% rangeI
thf(fact_246_rangeI,axiom,
! [F2: finite_mod_ring_a > int,X: finite_mod_ring_a] : ( member_int @ ( F2 @ X ) @ ( image_4238506139956901036_a_int @ F2 @ top_to2069866484006881781ring_a ) ) ).
% rangeI
thf(fact_247_infinite__UNIV__char__0,axiom,
~ ( finite3207457112153483333omplex @ top_top_set_complex ) ).
% infinite_UNIV_char_0
thf(fact_248_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_249_infinite__UNIV__char__0,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_char_0
thf(fact_250_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( finite2973822078027888486omplex @ top_to4859276455607160429omplex ) ) ) ).
% finite_Prod_UNIV
thf(fact_251_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite1507344369527211371t_unit @ top_to8925207929171064890t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_252_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite7637020569396982664ex_nat @ top_to2670425159793856911ex_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_253_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite3459169549887785956ex_int @ top_to180342014862130155ex_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_254_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite_finite_k @ top_top_set_k )
=> ( finite7821853734178680806plex_k @ top_to6389705578148759861plex_k ) ) ) ).
% finite_Prod_UNIV
thf(fact_255_finite__Prod__UNIV,axiom,
( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite_finite_a @ top_top_set_a )
=> ( finite7821853691146392796plex_a @ top_to5679365177685300011plex_a ) ) ) ).
% finite_Prod_UNIV
thf(fact_256_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( finite2744815593744250881omplex @ top_to6913156368282894288omplex ) ) ) ).
% finite_Prod_UNIV
thf(fact_257_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_258_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_259_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_int @ top_top_set_int )
=> ( finite1009671796988969599it_int @ top_to3484027333181043534it_int ) ) ) ).
% finite_Prod_UNIV
thf(fact_260_ex__new__if__finite,axiom,
! [A2: set_complex] :
( ~ ( finite3207457112153483333omplex @ top_top_set_complex )
=> ( ( finite3207457112153483333omplex @ A2 )
=> ? [A4: complex] :
~ ( member_complex @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_261_ex__new__if__finite,axiom,
! [A2: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A2 )
=> ? [A4: product_unit] :
~ ( member_Product_unit @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_262_ex__new__if__finite,axiom,
! [A2: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A2 )
=> ? [A4: nat] :
~ ( member_nat @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_263_ex__new__if__finite,axiom,
! [A2: set_int] :
( ~ ( finite_finite_int @ top_top_set_int )
=> ( ( finite_finite_int @ A2 )
=> ? [A4: int] :
~ ( member_int @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_264_ex__new__if__finite,axiom,
! [A2: set_k] :
( ~ ( finite_finite_k @ top_top_set_k )
=> ( ( finite_finite_k @ A2 )
=> ? [A4: k] :
~ ( member_k @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_265_ex__new__if__finite,axiom,
! [A2: set_a] :
( ~ ( finite_finite_a @ top_top_set_a )
=> ( ( finite_finite_a @ A2 )
=> ? [A4: a] :
~ ( member_a @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_266_ex__new__if__finite,axiom,
! [A2: set_Su8486473086189545021omplex] :
( ~ ( finite6208533171692660870omplex @ top_to8676441370508295053omplex )
=> ( ( finite6208533171692660870omplex @ A2 )
=> ? [A4: sum_sum_a_complex] :
~ ( member8603132577197391238omplex @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_267_ex__new__if__finite,axiom,
! [A2: set_Sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat )
=> ( ( finite502105017643426984_a_nat @ A2 )
=> ? [A4: sum_sum_a_nat] :
~ ( member_Sum_sum_a_nat @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_268_ex__new__if__finite,axiom,
! [A2: set_Sum_sum_a_int] :
( ~ ( finite5547626034989006084_a_int @ top_to7528907356895570187_a_int )
=> ( ( finite5547626034989006084_a_int @ A2 )
=> ? [A4: sum_sum_a_int] :
~ ( member_Sum_sum_a_int @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_269_ex__new__if__finite,axiom,
! [A2: set_Sum_sum_a_k] :
( ~ ( finite51705190296372934um_a_k @ top_to335874364214223893um_a_k )
=> ( ( finite51705190296372934um_a_k @ A2 )
=> ? [A4: sum_sum_a_k] :
~ ( member_Sum_sum_a_k @ A4 @ A2 ) ) ) ).
% ex_new_if_finite
thf(fact_270_finite__prod,axiom,
( ( finite2973822078027888486omplex @ top_to4859276455607160429omplex )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).
% finite_prod
thf(fact_271_finite__prod,axiom,
( ( finite1507344369527211371t_unit @ top_to8925207929171064890t_unit )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_272_finite__prod,axiom,
( ( finite7637020569396982664ex_nat @ top_to2670425159793856911ex_nat )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_273_finite__prod,axiom,
( ( finite3459169549887785956ex_int @ top_to180342014862130155ex_int )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_274_finite__prod,axiom,
( ( finite7821853734178680806plex_k @ top_to6389705578148759861plex_k )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_k @ top_top_set_k ) ) ) ).
% finite_prod
thf(fact_275_finite__prod,axiom,
( ( finite7821853691146392796plex_a @ top_to5679365177685300011plex_a )
= ( ( finite3207457112153483333omplex @ top_top_set_complex )
& ( finite_finite_a @ top_top_set_a ) ) ) ).
% finite_prod
thf(fact_276_finite__prod,axiom,
( ( finite2744815593744250881omplex @ top_to6913156368282894288omplex )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite3207457112153483333omplex @ top_top_set_complex ) ) ) ).
% finite_prod
thf(fact_277_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_278_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_279_finite__prod,axiom,
( ( finite1009671796988969599it_int @ top_to3484027333181043534it_int )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_int @ top_top_set_int ) ) ) ).
% finite_prod
thf(fact_280_Finite__Set_Ofinite__set,axiom,
( ( finite6551019134538273531omplex @ top_to4650676778325599690omplex )
= ( finite3207457112153483333omplex @ top_top_set_complex ) ) ).
% Finite_Set.finite_set
thf(fact_281_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_282_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_283_Finite__Set_Ofinite__set,axiom,
( ( finite6197958912794628473et_int @ top_top_set_set_int )
= ( finite_finite_int @ top_top_set_int ) ) ).
% Finite_Set.finite_set
thf(fact_284_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_k @ top_top_set_set_k )
= ( finite_finite_k @ top_top_set_k ) ) ).
% Finite_Set.finite_set
thf(fact_285_Finite__Set_Ofinite__set,axiom,
( ( finite_finite_set_a @ top_top_set_set_a )
= ( finite_finite_a @ top_top_set_a ) ) ).
% Finite_Set.finite_set
thf(fact_286_Finite__Set_Ofinite__set,axiom,
( ( finite4865751890316344550omplex @ top_to7557742135517130733omplex )
= ( finite6208533171692660870omplex @ top_to8676441370508295053omplex ) ) ).
% Finite_Set.finite_set
thf(fact_287_Finite__Set_Ofinite__set,axiom,
( ( finite4842507993062306312_a_nat @ top_to9085961846241471503_a_nat )
= ( finite502105017643426984_a_nat @ top_to795618464972521135_a_nat ) ) ).
% Finite_Set.finite_set
thf(fact_288_Finite__Set_Ofinite__set,axiom,
( ( finite2352424848130579556_a_int @ top_to4272413427453552747_a_int )
= ( finite5547626034989006084_a_int @ top_to7528907356895570187_a_int ) ) ).
% Finite_Set.finite_set
thf(fact_289_Finite__Set_Ofinite__set,axiom,
( ( finite4622999583732487420um_a_k @ top_to133642053589428043um_a_k )
= ( finite51705190296372934um_a_k @ top_to335874364214223893um_a_k ) ) ).
% Finite_Set.finite_set
thf(fact_290_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_291_finite__class_Ofinite__UNIV,axiom,
finite_finite_k @ top_top_set_k ).
% finite_class.finite_UNIV
thf(fact_292_finite__class_Ofinite__UNIV,axiom,
finite_finite_a @ top_top_set_a ).
% finite_class.finite_UNIV
thf(fact_293_finite__class_Ofinite__UNIV,axiom,
finite51705190296372934um_a_k @ top_to335874364214223893um_a_k ).
% finite_class.finite_UNIV
thf(fact_294_finite__class_Ofinite__UNIV,axiom,
finite51705147264084924um_a_a @ top_to8848906000605539851um_a_a ).
% finite_class.finite_UNIV
thf(fact_295_range__composition,axiom,
! [F2: nat > int,G: product_unit > nat] :
( ( image_873079544045703924it_int
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_nat_int @ F2 @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_296_range__composition,axiom,
! [F2: nat > complex,G: product_unit > nat] :
( ( image_3082061952195111286omplex
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_nat_complex @ F2 @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_297_range__composition,axiom,
! [F2: nat > nat,G: product_unit > nat] :
( ( image_875570014554754200it_nat
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_nat_nat @ F2 @ ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_298_range__composition,axiom,
! [F2: int > complex,G: product_unit > int] :
( ( image_3082061952195111286omplex
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_int_complex @ F2 @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_299_range__composition,axiom,
! [F2: int > nat,G: product_unit > int] :
( ( image_875570014554754200it_nat
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_int_nat @ F2 @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_300_range__composition,axiom,
! [F2: int > int,G: product_unit > int] :
( ( image_873079544045703924it_int
@ ^ [X2: product_unit] : ( F2 @ ( G @ X2 ) )
@ top_to1996260823553986621t_unit )
= ( image_int_int @ F2 @ ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit ) ) ) ).
% range_composition
thf(fact_301_range__composition,axiom,
! [F2: complex > int,G: nat > complex] :
( ( image_nat_int
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_complex_int @ F2 @ ( image_nat_complex @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_302_range__composition,axiom,
! [F2: nat > int,G: nat > nat] :
( ( image_nat_int
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_nat_int @ F2 @ ( image_nat_nat @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_303_range__composition,axiom,
! [F2: int > int,G: nat > int] :
( ( image_nat_int
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_int_int @ F2 @ ( image_nat_int @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_304_range__composition,axiom,
! [F2: complex > complex,G: nat > complex] :
( ( image_nat_complex
@ ^ [X2: nat] : ( F2 @ ( G @ X2 ) )
@ top_top_set_nat )
= ( image_1468599708987790691omplex @ F2 @ ( image_nat_complex @ G @ top_top_set_nat ) ) ) ).
% range_composition
thf(fact_305_rangeE,axiom,
! [B: nat,F2: product_unit > nat] :
( ( member_nat @ B @ ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit ) )
=> ~ ! [X3: product_unit] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_306_rangeE,axiom,
! [B: int,F2: nat > int] :
( ( member_int @ B @ ( image_nat_int @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_307_rangeE,axiom,
! [B: complex,F2: nat > complex] :
( ( member_complex @ B @ ( image_nat_complex @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_308_rangeE,axiom,
! [B: nat,F2: nat > nat] :
( ( member_nat @ B @ ( image_nat_nat @ F2 @ top_top_set_nat ) )
=> ~ ! [X3: nat] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_309_rangeE,axiom,
! [B: complex,F2: int > complex] :
( ( member_complex @ B @ ( image_int_complex @ F2 @ top_top_set_int ) )
=> ~ ! [X3: int] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_310_rangeE,axiom,
! [B: int,F2: int > int] :
( ( member_int @ B @ ( image_int_int @ F2 @ top_top_set_int ) )
=> ~ ! [X3: int] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_311_rangeE,axiom,
! [B: nat,F2: int > nat] :
( ( member_nat @ B @ ( image_int_nat @ F2 @ top_top_set_int ) )
=> ~ ! [X3: int] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_312_rangeE,axiom,
! [B: nat,F2: k > nat] :
( ( member_nat @ B @ ( image_k_nat @ F2 @ top_top_set_k ) )
=> ~ ! [X3: k] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_313_rangeE,axiom,
! [B: nat,F2: a > nat] :
( ( member_nat @ B @ ( image_a_nat @ F2 @ top_top_set_a ) )
=> ~ ! [X3: a] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_314_rangeE,axiom,
! [B: int,F2: finite_mod_ring_a > int] :
( ( member_int @ B @ ( image_4238506139956901036_a_int @ F2 @ top_to2069866484006881781ring_a ) )
=> ~ ! [X3: finite_mod_ring_a] :
( B
!= ( F2 @ X3 ) ) ) ).
% rangeE
thf(fact_315_pigeonhole__infinite,axiom,
! [A2: set_Product_unit,F2: product_unit > int] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_int @ ( image_873079544045703924it_int @ F2 @ A2 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_316_pigeonhole__infinite,axiom,
! [A2: set_Product_unit,F2: product_unit > nat] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_nat @ ( image_875570014554754200it_nat @ F2 @ A2 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_317_pigeonhole__infinite,axiom,
! [A2: set_Product_unit,F2: product_unit > complex] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite3207457112153483333omplex @ ( image_3082061952195111286omplex @ F2 @ A2 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_318_pigeonhole__infinite,axiom,
! [A2: set_Product_unit,F2: product_unit > k] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_k @ ( image_Product_unit_k @ F2 @ A2 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_319_pigeonhole__infinite,axiom,
! [A2: set_Product_unit,F2: product_unit > a] :
( ~ ( finite4290736615968046902t_unit @ A2 )
=> ( ( finite_finite_a @ ( image_Product_unit_a @ F2 @ A2 ) )
=> ? [X3: product_unit] :
( ( member_Product_unit @ X3 @ A2 )
& ~ ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [A3: product_unit] :
( ( member_Product_unit @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_320_pigeonhole__infinite,axiom,
! [A2: set_int,F2: int > int] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_int @ ( image_int_int @ F2 @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_321_pigeonhole__infinite,axiom,
! [A2: set_int,F2: int > nat] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_nat @ ( image_int_nat @ F2 @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_322_pigeonhole__infinite,axiom,
! [A2: set_int,F2: int > complex] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite3207457112153483333omplex @ ( image_int_complex @ F2 @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_323_pigeonhole__infinite,axiom,
! [A2: set_int,F2: int > k] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_k @ ( image_int_k @ F2 @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_324_pigeonhole__infinite,axiom,
! [A2: set_int,F2: int > a] :
( ~ ( finite_finite_int @ A2 )
=> ( ( finite_finite_a @ ( image_int_a @ F2 @ A2 ) )
=> ? [X3: int] :
( ( member_int @ X3 @ A2 )
& ~ ( finite_finite_int
@ ( collect_int
@ ^ [A3: int] :
( ( member_int @ A3 @ A2 )
& ( ( F2 @ A3 )
= ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_325_mem__Collect__eq,axiom,
! [A5: int,P: int > $o] :
( ( member_int @ A5 @ ( collect_int @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_326_mem__Collect__eq,axiom,
! [A5: nat,P: nat > $o] :
( ( member_nat @ A5 @ ( collect_nat @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_327_mem__Collect__eq,axiom,
! [A5: complex,P: complex > $o] :
( ( member_complex @ A5 @ ( collect_complex @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_328_mem__Collect__eq,axiom,
! [A5: a,P: a > $o] :
( ( member_a @ A5 @ ( collect_a @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_329_mem__Collect__eq,axiom,
! [A5: k,P: k > $o] :
( ( member_k @ A5 @ ( collect_k @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_330_mem__Collect__eq,axiom,
! [A5: product_unit,P: product_unit > $o] :
( ( member_Product_unit @ A5 @ ( collect_Product_unit @ P ) )
= ( P @ A5 ) ) ).
% mem_Collect_eq
thf(fact_331_Collect__mem__eq,axiom,
! [A2: set_int] :
( ( collect_int
@ ^ [X2: int] : ( member_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_332_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_333_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X2: complex] : ( member_complex @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_334_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_335_Collect__mem__eq,axiom,
! [A2: set_k] :
( ( collect_k
@ ^ [X2: k] : ( member_k @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_336_Collect__mem__eq,axiom,
! [A2: set_Product_unit] :
( ( collect_Product_unit
@ ^ [X2: product_unit] : ( member_Product_unit @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_337_Collect__cong,axiom,
! [P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_int @ P )
= ( collect_int @ Q ) ) ) ).
% Collect_cong
thf(fact_338_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_339_Collect__cong,axiom,
! [P: complex > $o,Q: complex > $o] :
( ! [X3: complex] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_complex @ P )
= ( collect_complex @ Q ) ) ) ).
% Collect_cong
thf(fact_340_Collect__cong,axiom,
! [P: a > $o,Q: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_a @ P )
= ( collect_a @ Q ) ) ) ).
% Collect_cong
thf(fact_341_Collect__cong,axiom,
! [P: k > $o,Q: k > $o] :
( ! [X3: k] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_k @ P )
= ( collect_k @ Q ) ) ) ).
% Collect_cong
thf(fact_342_Collect__cong,axiom,
! [P: product_unit > $o,Q: product_unit > $o] :
( ! [X3: product_unit] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_Product_unit @ P )
= ( collect_Product_unit @ Q ) ) ) ).
% Collect_cong
thf(fact_343_q__nonzero,axiom,
q != zero_zero_int ).
% q_nonzero
thf(fact_344_surj__def,axiom,
! [F2: product_unit > product_unit] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
= ( ! [Y2: product_unit] :
? [X2: product_unit] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_345_surj__def,axiom,
! [F2: product_unit > nat] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: product_unit] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_346_surj__def,axiom,
! [F2: product_unit > int] :
( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_int )
= ( ! [Y2: int] :
? [X2: product_unit] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_347_surj__def,axiom,
! [F2: product_unit > k] :
( ( ( image_Product_unit_k @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_k )
= ( ! [Y2: k] :
? [X2: product_unit] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_348_surj__def,axiom,
! [F2: product_unit > a] :
( ( ( image_Product_unit_a @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_a )
= ( ! [Y2: a] :
? [X2: product_unit] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_349_surj__def,axiom,
! [F2: nat > complex] :
( ( ( image_nat_complex @ F2 @ top_top_set_nat )
= top_top_set_complex )
= ( ! [Y2: complex] :
? [X2: nat] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_350_surj__def,axiom,
! [F2: nat > product_unit] :
( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
= top_to1996260823553986621t_unit )
= ( ! [Y2: product_unit] :
? [X2: nat] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_351_surj__def,axiom,
! [F2: nat > nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y2: nat] :
? [X2: nat] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_352_surj__def,axiom,
! [F2: nat > int] :
( ( ( image_nat_int @ F2 @ top_top_set_nat )
= top_top_set_int )
= ( ! [Y2: int] :
? [X2: nat] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_353_surj__def,axiom,
! [F2: nat > k] :
( ( ( image_nat_k @ F2 @ top_top_set_nat )
= top_top_set_k )
= ( ! [Y2: k] :
? [X2: nat] :
( Y2
= ( F2 @ X2 ) ) ) ) ).
% surj_def
thf(fact_354_surjI,axiom,
! [G: product_unit > product_unit,F2: product_unit > product_unit] :
( ! [X3: product_unit] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_355_surjI,axiom,
! [G: product_unit > nat,F2: nat > product_unit] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ).
% surjI
thf(fact_356_surjI,axiom,
! [G: product_unit > int,F2: int > product_unit] :
( ! [X3: int] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
= top_top_set_int ) ) ).
% surjI
thf(fact_357_surjI,axiom,
! [G: product_unit > k,F2: k > product_unit] :
( ! [X3: k] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_Product_unit_k @ G @ top_to1996260823553986621t_unit )
= top_top_set_k ) ) ).
% surjI
thf(fact_358_surjI,axiom,
! [G: product_unit > a,F2: a > product_unit] :
( ! [X3: a] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_Product_unit_a @ G @ top_to1996260823553986621t_unit )
= top_top_set_a ) ) ).
% surjI
thf(fact_359_surjI,axiom,
! [G: nat > complex,F2: complex > nat] :
( ! [X3: complex] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_complex @ G @ top_top_set_nat )
= top_top_set_complex ) ) ).
% surjI
thf(fact_360_surjI,axiom,
! [G: nat > product_unit,F2: product_unit > nat] :
( ! [X3: product_unit] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_361_surjI,axiom,
! [G: nat > nat,F2: nat > nat] :
( ! [X3: nat] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_362_surjI,axiom,
! [G: nat > int,F2: int > nat] :
( ! [X3: int] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int ) ) ).
% surjI
thf(fact_363_surjI,axiom,
! [G: nat > k,F2: k > nat] :
( ! [X3: k] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( image_nat_k @ G @ top_top_set_nat )
= top_top_set_k ) ) ).
% surjI
thf(fact_364_surjE,axiom,
! [F2: product_unit > product_unit,Y3: product_unit] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ~ ! [X3: product_unit] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_365_surjE,axiom,
! [F2: product_unit > nat,Y3: nat] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ~ ! [X3: product_unit] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_366_surjE,axiom,
! [F2: product_unit > int,Y3: int] :
( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_int )
=> ~ ! [X3: product_unit] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_367_surjE,axiom,
! [F2: product_unit > k,Y3: k] :
( ( ( image_Product_unit_k @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_k )
=> ~ ! [X3: product_unit] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_368_surjE,axiom,
! [F2: product_unit > a,Y3: a] :
( ( ( image_Product_unit_a @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ~ ! [X3: product_unit] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_369_surjE,axiom,
! [F2: nat > complex,Y3: complex] :
( ( ( image_nat_complex @ F2 @ top_top_set_nat )
= top_top_set_complex )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_370_surjE,axiom,
! [F2: nat > product_unit,Y3: product_unit] :
( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_371_surjE,axiom,
! [F2: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_372_surjE,axiom,
! [F2: nat > int,Y3: int] :
( ( ( image_nat_int @ F2 @ top_top_set_nat )
= top_top_set_int )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_373_surjE,axiom,
! [F2: nat > k,Y3: k] :
( ( ( image_nat_k @ F2 @ top_top_set_nat )
= top_top_set_k )
=> ~ ! [X3: nat] :
( Y3
!= ( F2 @ X3 ) ) ) ).
% surjE
thf(fact_374_surjD,axiom,
! [F2: product_unit > product_unit,Y3: product_unit] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ? [X3: product_unit] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_375_surjD,axiom,
! [F2: product_unit > nat,Y3: nat] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ? [X3: product_unit] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_376_surjD,axiom,
! [F2: product_unit > int,Y3: int] :
( ( ( image_873079544045703924it_int @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_int )
=> ? [X3: product_unit] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_377_surjD,axiom,
! [F2: product_unit > k,Y3: k] :
( ( ( image_Product_unit_k @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_k )
=> ? [X3: product_unit] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_378_surjD,axiom,
! [F2: product_unit > a,Y3: a] :
( ( ( image_Product_unit_a @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ? [X3: product_unit] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_379_surjD,axiom,
! [F2: nat > complex,Y3: complex] :
( ( ( image_nat_complex @ F2 @ top_top_set_nat )
= top_top_set_complex )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_380_surjD,axiom,
! [F2: nat > product_unit,Y3: product_unit] :
( ( ( image_8730104196221521654t_unit @ F2 @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_381_surjD,axiom,
! [F2: nat > nat,Y3: nat] :
( ( ( image_nat_nat @ F2 @ top_top_set_nat )
= top_top_set_nat )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_382_surjD,axiom,
! [F2: nat > int,Y3: int] :
( ( ( image_nat_int @ F2 @ top_top_set_nat )
= top_top_set_int )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_383_surjD,axiom,
! [F2: nat > k,Y3: k] :
( ( ( image_nat_k @ F2 @ top_top_set_nat )
= top_top_set_k )
=> ? [X3: nat] :
( Y3
= ( F2 @ X3 ) ) ) ).
% surjD
thf(fact_384_abs__infty__poly__def,axiom,
! [P2: kyber_qr_a] :
( ( abs_ky5074908690697402296poly_a @ q @ P2 )
= ( lattic8263393255366662781ax_int @ ( image_nat_int @ ( comp_F5721690436324261920nt_nat @ ( abs_ky7385543178848499077ty_q_a @ q ) @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ P2 ) ) ) @ top_top_set_nat ) ) ) ).
% abs_infty_poly_def
thf(fact_385_q__prime,axiom,
factor1798656936486255268me_int @ q ).
% q_prime
thf(fact_386_Sup_OSUP__identity__eq,axiom,
! [Sup: set_nat > nat,A2: set_nat] :
( ( Sup
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_387_Sup_OSUP__identity__eq,axiom,
! [Sup: set_int > int,A2: set_int] :
( ( Sup
@ ( image_int_int
@ ^ [X2: int] : X2
@ A2 ) )
= ( Sup @ A2 ) ) ).
% Sup.SUP_identity_eq
thf(fact_388_Inf_OINF__identity__eq,axiom,
! [Inf: set_nat > nat,A2: set_nat] :
( ( Inf
@ ( image_nat_nat
@ ^ [X2: nat] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_389_Inf_OINF__identity__eq,axiom,
! [Inf: set_int > int,A2: set_int] :
( ( Inf
@ ( image_int_int
@ ^ [X2: int] : X2
@ A2 ) )
= ( Inf @ A2 ) ) ).
% Inf.INF_identity_eq
thf(fact_390_kyber__spec_Ofinite__Max,axiom,
! [N2: int,Q2: int,K: nat,N3: nat,X: kyber_qr_a] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( finite_finite_int
@ ( image_nat_int
@ ^ [Xa: nat] : ( abs_ky7385543178848499077ty_q_a @ Q2 @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ X ) @ Xa ) )
@ top_top_set_nat ) ) ) ).
% kyber_spec.finite_Max
thf(fact_391_kyber__spec_Ofinite__Max,axiom,
! [N2: int,Q2: int,K: nat,N3: nat,X: kyber_qr_a] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( finite_finite_int
@ ( image_nat_int
@ ^ [Xa: nat] : ( abs_ky7385543178848499077ty_q_a @ Q2 @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ X ) @ Xa ) )
@ top_top_set_nat ) ) ) ).
% kyber_spec.finite_Max
thf(fact_392_mult__commute__abs,axiom,
! [C: finite_mod_ring_a] :
( ( ^ [X2: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ X2 @ C ) )
= ( times_5121417576591743744ring_a @ C ) ) ).
% mult_commute_abs
thf(fact_393_mult__commute__abs,axiom,
! [C: kyber_qr_a] :
( ( ^ [X2: kyber_qr_a] : ( times_2095635435063429214r_qr_a @ X2 @ C ) )
= ( times_2095635435063429214r_qr_a @ C ) ) ).
% mult_commute_abs
thf(fact_394_mult__commute__abs,axiom,
! [C: int] :
( ( ^ [X2: int] : ( times_times_int @ X2 @ C ) )
= ( times_times_int @ C ) ) ).
% mult_commute_abs
thf(fact_395_mult__commute__abs,axiom,
! [C: nat] :
( ( ^ [X2: nat] : ( times_times_nat @ X2 @ C ) )
= ( times_times_nat @ C ) ) ).
% mult_commute_abs
thf(fact_396_mult__commute__abs,axiom,
! [C: real] :
( ( ^ [X2: real] : ( times_times_real @ X2 @ C ) )
= ( times_times_real @ C ) ) ).
% mult_commute_abs
thf(fact_397_comp__apply,axiom,
( comp_F5721690436324261920nt_nat
= ( ^ [F3: finite_mod_ring_a > int,G2: nat > finite_mod_ring_a,X2: nat] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_apply
thf(fact_398_comp__eq__dest__lhs,axiom,
! [A5: finite_mod_ring_a > int,B: nat > finite_mod_ring_a,C: nat > int,V: nat] :
( ( ( comp_F5721690436324261920nt_nat @ A5 @ B )
= C )
=> ( ( A5 @ ( B @ V ) )
= ( C @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_399_comp__eq__elim,axiom,
! [A5: finite_mod_ring_a > int,B: nat > finite_mod_ring_a,C: finite_mod_ring_a > int,D: nat > finite_mod_ring_a] :
( ( ( comp_F5721690436324261920nt_nat @ A5 @ B )
= ( comp_F5721690436324261920nt_nat @ C @ D ) )
=> ! [V2: nat] :
( ( A5 @ ( B @ V2 ) )
= ( C @ ( D @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_400_comp__eq__dest,axiom,
! [A5: finite_mod_ring_a > int,B: nat > finite_mod_ring_a,C: finite_mod_ring_a > int,D: nat > finite_mod_ring_a,V: nat] :
( ( ( comp_F5721690436324261920nt_nat @ A5 @ B )
= ( comp_F5721690436324261920nt_nat @ C @ D ) )
=> ( ( A5 @ ( B @ V ) )
= ( C @ ( D @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_401_comp__assoc,axiom,
! [F2: finite_mod_ring_a > int,G: nat > finite_mod_ring_a,H: nat > nat] :
( ( comp_nat_int_nat @ ( comp_F5721690436324261920nt_nat @ F2 @ G ) @ H )
= ( comp_F5721690436324261920nt_nat @ F2 @ ( comp_n739488747675965474_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_402_comp__assoc,axiom,
! [F2: int > int,G: finite_mod_ring_a > int,H: nat > finite_mod_ring_a] :
( ( comp_F5721690436324261920nt_nat @ ( comp_i1216107289310836680ring_a @ F2 @ G ) @ H )
= ( comp_int_int_nat @ F2 @ ( comp_F5721690436324261920nt_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_403_comp__assoc,axiom,
! [F2: finite_mod_ring_a > int,G: finite_mod_ring_a > finite_mod_ring_a,H: nat > finite_mod_ring_a] :
( ( comp_F5721690436324261920nt_nat @ ( comp_F2690252154722880373ring_a @ F2 @ G ) @ H )
= ( comp_F5721690436324261920nt_nat @ F2 @ ( comp_F1116550632444010611_a_nat @ G @ H ) ) ) ).
% comp_assoc
thf(fact_404_comp__def,axiom,
( comp_F5721690436324261920nt_nat
= ( ^ [F3: finite_mod_ring_a > int,G2: nat > finite_mod_ring_a,X2: nat] : ( F3 @ ( G2 @ X2 ) ) ) ) ).
% comp_def
thf(fact_405_K__record__comp,axiom,
! [C: int,F2: nat > finite_mod_ring_a] :
( ( comp_F5721690436324261920nt_nat
@ ^ [X2: finite_mod_ring_a] : C
@ F2 )
= ( ^ [X2: nat] : C ) ) ).
% K_record_comp
thf(fact_406_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: complex > int,F2: nat > complex,A2: set_nat] :
( ( Inf @ ( image_complex_int @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_int @ ( comp_complex_int_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_407_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G: complex > complex,F2: nat > complex,A2: set_nat] :
( ( Inf @ ( image_1468599708987790691omplex @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_408_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: complex > nat,F2: nat > complex,A2: set_nat] :
( ( Inf @ ( image_complex_nat @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_nat @ ( comp_complex_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_409_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G: complex > complex,F2: int > complex,A2: set_int] :
( ( Inf @ ( image_1468599708987790691omplex @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_int_complex @ ( comp_c7265521452430408557ex_int @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_410_Inf_OINF__image,axiom,
! [Inf: set_nat > nat,G: complex > nat,F2: int > complex,A2: set_int] :
( ( Inf @ ( image_complex_nat @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_int_nat @ ( comp_complex_nat_int @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_411_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: complex > int,F2: int > complex,A2: set_int] :
( ( Inf @ ( image_complex_int @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Inf @ ( image_int_int @ ( comp_complex_int_int @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_412_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: nat > int,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_int @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_413_Inf_OINF__image,axiom,
! [Inf: set_int > int,G: nat > int,F2: int > nat,A2: set_int] :
( ( Inf @ ( image_nat_int @ G @ ( image_int_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_int_int @ ( comp_nat_int_int @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_414_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G: nat > complex,F2: nat > nat,A2: set_nat] :
( ( Inf @ ( image_nat_complex @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_nat_complex @ ( comp_nat_complex_nat @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_415_Inf_OINF__image,axiom,
! [Inf: set_complex > complex,G: nat > complex,F2: int > nat,A2: set_int] :
( ( Inf @ ( image_nat_complex @ G @ ( image_int_nat @ F2 @ A2 ) ) )
= ( Inf @ ( image_int_complex @ ( comp_nat_complex_int @ G @ F2 ) @ A2 ) ) ) ).
% Inf.INF_image
thf(fact_416_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: complex > int,F2: nat > complex,A2: set_nat] :
( ( Sup @ ( image_complex_int @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_int @ ( comp_complex_int_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_417_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G: complex > complex,F2: nat > complex,A2: set_nat] :
( ( Sup @ ( image_1468599708987790691omplex @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_418_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: complex > nat,F2: nat > complex,A2: set_nat] :
( ( Sup @ ( image_complex_nat @ G @ ( image_nat_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_nat @ ( comp_complex_nat_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_419_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G: complex > complex,F2: int > complex,A2: set_int] :
( ( Sup @ ( image_1468599708987790691omplex @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_int_complex @ ( comp_c7265521452430408557ex_int @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_420_Sup_OSUP__image,axiom,
! [Sup: set_nat > nat,G: complex > nat,F2: int > complex,A2: set_int] :
( ( Sup @ ( image_complex_nat @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_int_nat @ ( comp_complex_nat_int @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_421_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: complex > int,F2: int > complex,A2: set_int] :
( ( Sup @ ( image_complex_int @ G @ ( image_int_complex @ F2 @ A2 ) ) )
= ( Sup @ ( image_int_int @ ( comp_complex_int_int @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_422_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: nat > int,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_int @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_int @ ( comp_nat_int_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_423_Sup_OSUP__image,axiom,
! [Sup: set_int > int,G: nat > int,F2: int > nat,A2: set_int] :
( ( Sup @ ( image_nat_int @ G @ ( image_int_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_int_int @ ( comp_nat_int_int @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_424_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G: nat > complex,F2: nat > nat,A2: set_nat] :
( ( Sup @ ( image_nat_complex @ G @ ( image_nat_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_nat_complex @ ( comp_nat_complex_nat @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_425_Sup_OSUP__image,axiom,
! [Sup: set_complex > complex,G: nat > complex,F2: int > nat,A2: set_int] :
( ( Sup @ ( image_nat_complex @ G @ ( image_int_nat @ F2 @ A2 ) ) )
= ( Sup @ ( image_int_complex @ ( comp_nat_complex_int @ G @ F2 ) @ A2 ) ) ) ).
% Sup.SUP_image
thf(fact_426_image__comp,axiom,
! [F2: complex > int,G: nat > complex,R2: set_nat] :
( ( image_complex_int @ F2 @ ( image_nat_complex @ G @ R2 ) )
= ( image_nat_int @ ( comp_complex_int_nat @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_427_image__comp,axiom,
! [F2: complex > complex,G: nat > complex,R2: set_nat] :
( ( image_1468599708987790691omplex @ F2 @ ( image_nat_complex @ G @ R2 ) )
= ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_428_image__comp,axiom,
! [F2: complex > nat,G: nat > complex,R2: set_nat] :
( ( image_complex_nat @ F2 @ ( image_nat_complex @ G @ R2 ) )
= ( image_nat_nat @ ( comp_complex_nat_nat @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_429_image__comp,axiom,
! [F2: complex > complex,G: int > complex,R2: set_int] :
( ( image_1468599708987790691omplex @ F2 @ ( image_int_complex @ G @ R2 ) )
= ( image_int_complex @ ( comp_c7265521452430408557ex_int @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_430_image__comp,axiom,
! [F2: complex > nat,G: int > complex,R2: set_int] :
( ( image_complex_nat @ F2 @ ( image_int_complex @ G @ R2 ) )
= ( image_int_nat @ ( comp_complex_nat_int @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_431_image__comp,axiom,
! [F2: complex > int,G: int > complex,R2: set_int] :
( ( image_complex_int @ F2 @ ( image_int_complex @ G @ R2 ) )
= ( image_int_int @ ( comp_complex_int_int @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_432_image__comp,axiom,
! [F2: nat > int,G: nat > nat,R2: set_nat] :
( ( image_nat_int @ F2 @ ( image_nat_nat @ G @ R2 ) )
= ( image_nat_int @ ( comp_nat_int_nat @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_433_image__comp,axiom,
! [F2: nat > int,G: int > nat,R2: set_int] :
( ( image_nat_int @ F2 @ ( image_int_nat @ G @ R2 ) )
= ( image_int_int @ ( comp_nat_int_int @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_434_image__comp,axiom,
! [F2: nat > complex,G: nat > nat,R2: set_nat] :
( ( image_nat_complex @ F2 @ ( image_nat_nat @ G @ R2 ) )
= ( image_nat_complex @ ( comp_nat_complex_nat @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_435_image__comp,axiom,
! [F2: nat > complex,G: int > nat,R2: set_int] :
( ( image_nat_complex @ F2 @ ( image_int_nat @ G @ R2 ) )
= ( image_int_complex @ ( comp_nat_complex_int @ F2 @ G ) @ R2 ) ) ).
% image_comp
thf(fact_436_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > int] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_nat_int @ G @ B2 ) )
=> ( ( image_nat_int @ ( comp_int_int_nat @ H @ F2 ) @ A2 )
= ( image_nat_int @ ( comp_int_int_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_437_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > complex] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_nat_int @ G @ B2 ) )
=> ( ( image_nat_complex @ ( comp_int_complex_nat @ H @ F2 ) @ A2 )
= ( image_nat_complex @ ( comp_int_complex_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_438_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: nat > int,B2: set_nat,H: int > nat] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_nat_int @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_int_nat_nat @ H @ F2 ) @ A2 )
= ( image_nat_nat @ ( comp_int_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_439_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: int > int,B2: set_int,H: int > int] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_int_int @ G @ B2 ) )
=> ( ( image_nat_int @ ( comp_int_int_nat @ H @ F2 ) @ A2 )
= ( image_int_int @ ( comp_int_int_int @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_440_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: int > int,B2: set_int,H: int > complex] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_int_int @ G @ B2 ) )
=> ( ( image_nat_complex @ ( comp_int_complex_nat @ H @ F2 ) @ A2 )
= ( image_int_complex @ ( comp_int_complex_int @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_441_image__eq__imp__comp,axiom,
! [F2: nat > int,A2: set_nat,G: int > int,B2: set_int,H: int > nat] :
( ( ( image_nat_int @ F2 @ A2 )
= ( image_int_int @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_int_nat_nat @ H @ F2 ) @ A2 )
= ( image_int_nat @ ( comp_int_nat_int @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_442_image__eq__imp__comp,axiom,
! [F2: nat > complex,A2: set_nat,G: nat > complex,B2: set_nat,H: complex > int] :
( ( ( image_nat_complex @ F2 @ A2 )
= ( image_nat_complex @ G @ B2 ) )
=> ( ( image_nat_int @ ( comp_complex_int_nat @ H @ F2 ) @ A2 )
= ( image_nat_int @ ( comp_complex_int_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_443_image__eq__imp__comp,axiom,
! [F2: nat > complex,A2: set_nat,G: nat > complex,B2: set_nat,H: complex > complex] :
( ( ( image_nat_complex @ F2 @ A2 )
= ( image_nat_complex @ G @ B2 ) )
=> ( ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ H @ F2 ) @ A2 )
= ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_444_image__eq__imp__comp,axiom,
! [F2: nat > complex,A2: set_nat,G: nat > complex,B2: set_nat,H: complex > nat] :
( ( ( image_nat_complex @ F2 @ A2 )
= ( image_nat_complex @ G @ B2 ) )
=> ( ( image_nat_nat @ ( comp_complex_nat_nat @ H @ F2 ) @ A2 )
= ( image_nat_nat @ ( comp_complex_nat_nat @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_445_image__eq__imp__comp,axiom,
! [F2: nat > complex,A2: set_nat,G: int > complex,B2: set_int,H: complex > int] :
( ( ( image_nat_complex @ F2 @ A2 )
= ( image_int_complex @ G @ B2 ) )
=> ( ( image_nat_int @ ( comp_complex_int_nat @ H @ F2 ) @ A2 )
= ( image_int_int @ ( comp_complex_int_int @ H @ G ) @ B2 ) ) ) ).
% image_eq_imp_comp
thf(fact_446_kyber__spec_Oabs__infty__poly_Ocong,axiom,
abs_ky5074908690697402296poly_a = abs_ky5074908690697402296poly_a ).
% kyber_spec.abs_infty_poly.cong
thf(fact_447_top__set__def,axiom,
( top_top_set_complex
= ( collect_complex @ top_top_complex_o ) ) ).
% top_set_def
thf(fact_448_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_449_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_450_top__set__def,axiom,
( top_top_set_int
= ( collect_int @ top_top_int_o ) ) ).
% top_set_def
thf(fact_451_top__set__def,axiom,
( top_top_set_k
= ( collect_k @ top_top_k_o ) ) ).
% top_set_def
thf(fact_452_top__set__def,axiom,
( top_top_set_a
= ( collect_a @ top_top_a_o ) ) ).
% top_set_def
thf(fact_453_top__set__def,axiom,
( top_to8676441370508295053omplex
= ( collec1363137248864034504omplex @ top_to8738565331869774224plex_o ) ) ).
% top_set_def
thf(fact_454_top__set__def,axiom,
( top_to795618464972521135_a_nat
= ( collec7073057861543223018_a_nat @ top_to1565196397637005550_nat_o ) ) ).
% top_set_def
thf(fact_455_top__set__def,axiom,
( top_to7528907356895570187_a_int
= ( collec2895206842034026310_a_int @ top_to8543423466821335442_int_o ) ) ).
% top_set_def
thf(fact_456_top__set__def,axiom,
( top_to335874364214223893um_a_k
= ( collect_Sum_sum_a_k @ top_to1258784492157183664_a_k_o ) ) ).
% top_set_def
thf(fact_457_comp__surj,axiom,
! [F2: product_unit > product_unit,G: product_unit > product_unit] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ ( comp_P7645380973975430442t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ) ).
% comp_surj
thf(fact_458_comp__surj,axiom,
! [F2: product_unit > product_unit,G: product_unit > nat] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( image_875570014554754200it_nat @ ( comp_P8756859368360471761t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_459_comp__surj,axiom,
! [F2: product_unit > product_unit,G: product_unit > int] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( image_873079544045703924it_int @ G @ top_to1996260823553986621t_unit )
= top_top_set_int )
=> ( ( image_873079544045703924it_int @ ( comp_P5873281315562652021t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_int ) ) ) ).
% comp_surj
thf(fact_460_comp__surj,axiom,
! [F2: product_unit > product_unit,G: product_unit > k] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( image_Product_unit_k @ G @ top_to1996260823553986621t_unit )
= top_top_set_k )
=> ( ( image_Product_unit_k @ ( comp_P8501286339450937071t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_k ) ) ) ).
% comp_surj
thf(fact_461_comp__surj,axiom,
! [F2: product_unit > product_unit,G: product_unit > a] :
( ( ( image_405062704495631173t_unit @ F2 @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ( image_Product_unit_a @ G @ top_to1996260823553986621t_unit )
= top_top_set_a )
=> ( ( image_Product_unit_a @ ( comp_P5885858130684155513t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_a ) ) ) ).
% comp_surj
thf(fact_462_comp__surj,axiom,
! [F2: product_unit > nat,G: nat > complex] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( image_nat_complex @ G @ top_top_set_nat )
= top_top_set_complex )
=> ( ( image_3082061952195111286omplex @ ( comp_n3164391556643521898t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_complex ) ) ) ).
% comp_surj
thf(fact_463_comp__surj,axiom,
! [F2: product_unit > nat,G: nat > product_unit] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ ( comp_n6361575188714886131t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ) ).
% comp_surj
thf(fact_464_comp__surj,axiom,
! [F2: product_unit > nat,G: nat > nat] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_875570014554754200it_nat @ ( comp_n7656859423886344392t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ) ).
% comp_surj
thf(fact_465_comp__surj,axiom,
! [F2: product_unit > nat,G: nat > int] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( image_nat_int @ G @ top_top_set_nat )
= top_top_set_int )
=> ( ( image_873079544045703924it_int @ ( comp_n4773281371088524652t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_int ) ) ) ).
% comp_surj
thf(fact_466_comp__surj,axiom,
! [F2: product_unit > nat,G: nat > k] :
( ( ( image_875570014554754200it_nat @ F2 @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ( image_nat_k @ G @ top_top_set_nat )
= top_top_set_k )
=> ( ( image_Product_unit_k @ ( comp_n7999534625695914680t_unit @ G @ F2 ) @ top_to1996260823553986621t_unit )
= top_top_set_k ) ) ) ).
% comp_surj
thf(fact_467_kyber__spec_Oabs__infty__poly__def,axiom,
! [N2: int,Q2: int,K: nat,N3: nat,P2: kyber_qr_a] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( ( abs_ky5074908690697402296poly_a @ Q2 @ P2 )
= ( lattic8263393255366662781ax_int @ ( image_nat_int @ ( comp_F5721690436324261920nt_nat @ ( abs_ky7385543178848499077ty_q_a @ Q2 ) @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ P2 ) ) ) @ top_top_set_nat ) ) ) ) ).
% kyber_spec.abs_infty_poly_def
thf(fact_468_kyber__spec_Oabs__infty__poly__def,axiom,
! [N2: int,Q2: int,K: nat,N3: nat,P2: kyber_qr_a] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( ( abs_ky5074908690697402296poly_a @ Q2 @ P2 )
= ( lattic8263393255366662781ax_int @ ( image_nat_int @ ( comp_F5721690436324261920nt_nat @ ( abs_ky7385543178848499077ty_q_a @ Q2 ) @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ P2 ) ) ) @ top_top_set_nat ) ) ) ) ).
% kyber_spec.abs_infty_poly_def
thf(fact_469_Inf_OINF__cong,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,C2: finite_mod_ring_a > int,D2: finite_mod_ring_a > int,Inf: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_4238506139956901036_a_int @ C2 @ A2 ) )
= ( Inf @ ( image_4238506139956901036_a_int @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_470_Inf_OINF__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > complex,D2: int > complex,Inf: set_complex > complex] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_int_complex @ C2 @ A2 ) )
= ( Inf @ ( image_int_complex @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_471_Inf_OINF__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > nat,D2: int > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_int_nat @ C2 @ A2 ) )
= ( Inf @ ( image_int_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_472_Inf_OINF__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > int,D2: int > int,Inf: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_int_int @ C2 @ A2 ) )
= ( Inf @ ( image_int_int @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_473_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > int,D2: nat > int,Inf: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_int @ C2 @ A2 ) )
= ( Inf @ ( image_nat_int @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_474_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > complex,D2: nat > complex,Inf: set_complex > complex] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_complex @ C2 @ A2 ) )
= ( Inf @ ( image_nat_complex @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_475_Inf_OINF__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A2 ) )
= ( Inf @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_476_Sup_OSUP__cong,axiom,
! [A2: set_Fi2982333969990053029ring_a,B2: set_Fi2982333969990053029ring_a,C2: finite_mod_ring_a > int,D2: finite_mod_ring_a > int,Sup: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_4238506139956901036_a_int @ C2 @ A2 ) )
= ( Sup @ ( image_4238506139956901036_a_int @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_477_Sup_OSUP__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > complex,D2: int > complex,Sup: set_complex > complex] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_int_complex @ C2 @ A2 ) )
= ( Sup @ ( image_int_complex @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_478_Sup_OSUP__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > nat,D2: int > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_int_nat @ C2 @ A2 ) )
= ( Sup @ ( image_int_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_479_Sup_OSUP__cong,axiom,
! [A2: set_int,B2: set_int,C2: int > int,D2: int > int,Sup: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: int] :
( ( member_int @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_int_int @ C2 @ A2 ) )
= ( Sup @ ( image_int_int @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_480_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > int,D2: nat > int,Sup: set_int > int] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_int @ C2 @ A2 ) )
= ( Sup @ ( image_nat_int @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_481_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > complex,D2: nat > complex,Sup: set_complex > complex] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_complex @ C2 @ A2 ) )
= ( Sup @ ( image_nat_complex @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_482_Sup_OSUP__cong,axiom,
! [A2: set_nat,B2: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A2 = B2 )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ B2 )
=> ( ( C2 @ X3 )
= ( D2 @ X3 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A2 ) )
= ( Sup @ ( image_nat_nat @ D2 @ B2 ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_483_coeff__0,axiom,
! [N2: nat] :
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a @ N2 )
= zero_z7902377541816115708ring_a ) ).
% coeff_0
thf(fact_484_coeff__0,axiom,
! [N2: nat] :
( ( coeff_int @ zero_zero_poly_int @ N2 )
= zero_zero_int ) ).
% coeff_0
thf(fact_485_coeff__0,axiom,
! [N2: nat] :
( ( coeff_nat @ zero_zero_poly_nat @ N2 )
= zero_zero_nat ) ).
% coeff_0
thf(fact_486_coeff__0,axiom,
! [N2: nat] :
( ( coeff_complex @ zero_z2709840015065127615omplex @ N2 )
= zero_zero_complex ) ).
% coeff_0
thf(fact_487_coeff__0,axiom,
! [N2: nat] :
( ( coeff_real @ zero_zero_poly_real @ N2 )
= zero_zero_real ) ).
% coeff_0
thf(fact_488_mult__hom_Ohom__zero,axiom,
! [C: complex] :
( ( times_times_complex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_489_mult__hom_Ohom__zero,axiom,
! [C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ C @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_hom.hom_zero
thf(fact_490_mult__hom_Ohom__zero,axiom,
! [C: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ C @ zero_zero_Kyber_qr_a )
= zero_zero_Kyber_qr_a ) ).
% mult_hom.hom_zero
thf(fact_491_mult__hom_Ohom__zero,axiom,
! [C: int] :
( ( times_times_int @ C @ zero_zero_int )
= zero_zero_int ) ).
% mult_hom.hom_zero
thf(fact_492_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_493_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_494_mult__zero__left,axiom,
! [A5: complex] :
( ( times_times_complex @ zero_zero_complex @ A5 )
= zero_zero_complex ) ).
% mult_zero_left
thf(fact_495_mult__zero__left,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a @ A5 )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_left
thf(fact_496_mult__zero__left,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ zero_zero_Kyber_qr_a @ A5 )
= zero_zero_Kyber_qr_a ) ).
% mult_zero_left
thf(fact_497_mult__zero__left,axiom,
! [A5: int] :
( ( times_times_int @ zero_zero_int @ A5 )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_498_mult__zero__left,axiom,
! [A5: nat] :
( ( times_times_nat @ zero_zero_nat @ A5 )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_499_mult__zero__left,axiom,
! [A5: real] :
( ( times_times_real @ zero_zero_real @ A5 )
= zero_zero_real ) ).
% mult_zero_left
thf(fact_500_mult__zero__right,axiom,
! [A5: complex] :
( ( times_times_complex @ A5 @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_zero_right
thf(fact_501_mult__zero__right,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A5 @ zero_z7902377541816115708ring_a )
= zero_z7902377541816115708ring_a ) ).
% mult_zero_right
thf(fact_502_mult__zero__right,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ A5 @ zero_zero_Kyber_qr_a )
= zero_zero_Kyber_qr_a ) ).
% mult_zero_right
thf(fact_503_mult__zero__right,axiom,
! [A5: int] :
( ( times_times_int @ A5 @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_504_mult__zero__right,axiom,
! [A5: nat] :
( ( times_times_nat @ A5 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_505_mult__zero__right,axiom,
! [A5: real] :
( ( times_times_real @ A5 @ zero_zero_real )
= zero_zero_real ) ).
% mult_zero_right
thf(fact_506_mult__eq__0__iff,axiom,
! [A5: complex,B: complex] :
( ( ( times_times_complex @ A5 @ B )
= zero_zero_complex )
= ( ( A5 = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_507_mult__eq__0__iff,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A5 @ B )
= zero_z7902377541816115708ring_a )
= ( ( A5 = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% mult_eq_0_iff
thf(fact_508_mult__eq__0__iff,axiom,
! [A5: int,B: int] :
( ( ( times_times_int @ A5 @ B )
= zero_zero_int )
= ( ( A5 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_509_mult__eq__0__iff,axiom,
! [A5: nat,B: nat] :
( ( ( times_times_nat @ A5 @ B )
= zero_zero_nat )
= ( ( A5 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_510_mult__eq__0__iff,axiom,
! [A5: real,B: real] :
( ( ( times_times_real @ A5 @ B )
= zero_zero_real )
= ( ( A5 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_511_mult__cancel__left,axiom,
! [C: complex,A5: complex,B: complex] :
( ( ( times_times_complex @ C @ A5 )
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( A5 = B ) ) ) ).
% mult_cancel_left
thf(fact_512_mult__cancel__left,axiom,
! [C: finite_mod_ring_a,A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A5 )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A5 = B ) ) ) ).
% mult_cancel_left
thf(fact_513_mult__cancel__left,axiom,
! [C: int,A5: int,B: int] :
( ( ( times_times_int @ C @ A5 )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A5 = B ) ) ) ).
% mult_cancel_left
thf(fact_514_mult__cancel__left,axiom,
! [C: nat,A5: nat,B: nat] :
( ( ( times_times_nat @ C @ A5 )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A5 = B ) ) ) ).
% mult_cancel_left
thf(fact_515_mult__cancel__left,axiom,
! [C: real,A5: real,B: real] :
( ( ( times_times_real @ C @ A5 )
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( A5 = B ) ) ) ).
% mult_cancel_left
thf(fact_516_mult__cancel__right,axiom,
! [A5: complex,C: complex,B: complex] :
( ( ( times_times_complex @ A5 @ C )
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( A5 = B ) ) ) ).
% mult_cancel_right
thf(fact_517_mult__cancel__right,axiom,
! [A5: finite_mod_ring_a,C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A5 @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A5 = B ) ) ) ).
% mult_cancel_right
thf(fact_518_mult__cancel__right,axiom,
! [A5: int,C: int,B: int] :
( ( ( times_times_int @ A5 @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A5 = B ) ) ) ).
% mult_cancel_right
thf(fact_519_mult__cancel__right,axiom,
! [A5: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A5 @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A5 = B ) ) ) ).
% mult_cancel_right
thf(fact_520_mult__cancel__right,axiom,
! [A5: real,C: real,B: real] :
( ( ( times_times_real @ A5 @ C )
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( A5 = B ) ) ) ).
% mult_cancel_right
thf(fact_521_kyber__spec_Oq__prime,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( factor1798656936486255268me_int @ Q2 ) ) ).
% kyber_spec.q_prime
thf(fact_522_kyber__spec_Oq__prime,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( factor1798656936486255268me_int @ Q2 ) ) ).
% kyber_spec.q_prime
thf(fact_523_kyber__spec_Oq__nonzero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( Q2 != zero_zero_int ) ) ).
% kyber_spec.q_nonzero
thf(fact_524_kyber__spec_Oq__nonzero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( Q2 != zero_zero_int ) ) ).
% kyber_spec.q_nonzero
thf(fact_525_kyber__spec_On__nonzero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( N2 != zero_zero_int ) ) ).
% kyber_spec.n_nonzero
thf(fact_526_kyber__spec_On__nonzero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( N2 != zero_zero_int ) ) ).
% kyber_spec.n_nonzero
thf(fact_527_fun_Oset__map,axiom,
! [F2: nat > int,V: product_unit > nat] :
( ( image_873079544045703924it_int @ ( comp_n4773281371088524652t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_nat_int @ F2 @ ( image_875570014554754200it_nat @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_528_fun_Oset__map,axiom,
! [F2: nat > complex,V: product_unit > nat] :
( ( image_3082061952195111286omplex @ ( comp_n3164391556643521898t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_nat_complex @ F2 @ ( image_875570014554754200it_nat @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_529_fun_Oset__map,axiom,
! [F2: nat > nat,V: product_unit > nat] :
( ( image_875570014554754200it_nat @ ( comp_n7656859423886344392t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_nat_nat @ F2 @ ( image_875570014554754200it_nat @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_530_fun_Oset__map,axiom,
! [F2: int > complex,V: product_unit > int] :
( ( image_3082061952195111286omplex @ ( comp_i1572510429124248078t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_int_complex @ F2 @ ( image_873079544045703924it_int @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_531_fun_Oset__map,axiom,
! [F2: int > nat,V: product_unit > int] :
( ( image_875570014554754200it_nat @ ( comp_i2267726487915595116t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_int_nat @ F2 @ ( image_873079544045703924it_int @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_532_fun_Oset__map,axiom,
! [F2: int > int,V: product_unit > int] :
( ( image_873079544045703924it_int @ ( comp_i8607520471972551184t_unit @ F2 @ V ) @ top_to1996260823553986621t_unit )
= ( image_int_int @ F2 @ ( image_873079544045703924it_int @ V @ top_to1996260823553986621t_unit ) ) ) ).
% fun.set_map
thf(fact_533_fun_Oset__map,axiom,
! [F2: complex > int,V: nat > complex] :
( ( image_nat_int @ ( comp_complex_int_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_complex_int @ F2 @ ( image_nat_complex @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_534_fun_Oset__map,axiom,
! [F2: nat > int,V: nat > nat] :
( ( image_nat_int @ ( comp_nat_int_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_nat_int @ F2 @ ( image_nat_nat @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_535_fun_Oset__map,axiom,
! [F2: int > int,V: nat > int] :
( ( image_nat_int @ ( comp_int_int_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_int_int @ F2 @ ( image_nat_int @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_536_fun_Oset__map,axiom,
! [F2: complex > complex,V: nat > complex] :
( ( image_nat_complex @ ( comp_c7268011922939458833ex_nat @ F2 @ V ) @ top_top_set_nat )
= ( image_1468599708987790691omplex @ F2 @ ( image_nat_complex @ V @ top_top_set_nat ) ) ) ).
% fun.set_map
thf(fact_537_of__qr__0,axiom,
( ( kyber_of_qr_a @ zero_zero_Kyber_qr_a )
= zero_z1830546546923837194ring_a ) ).
% of_qr_0
thf(fact_538_of__qr__eq__0__iff,axiom,
! [P2: kyber_qr_a] :
( ( ( kyber_of_qr_a @ P2 )
= zero_z1830546546923837194ring_a )
= ( P2 = zero_zero_Kyber_qr_a ) ) ).
% of_qr_eq_0_iff
thf(fact_539_fun_Omap__comp,axiom,
! [G: int > int,F2: finite_mod_ring_a > int,V: nat > finite_mod_ring_a] :
( ( comp_int_int_nat @ G @ ( comp_F5721690436324261920nt_nat @ F2 @ V ) )
= ( comp_F5721690436324261920nt_nat @ ( comp_i1216107289310836680ring_a @ G @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_540_fun_Omap__comp,axiom,
! [G: finite_mod_ring_a > int,F2: nat > finite_mod_ring_a,V: nat > nat] :
( ( comp_F5721690436324261920nt_nat @ G @ ( comp_n739488747675965474_a_nat @ F2 @ V ) )
= ( comp_nat_int_nat @ ( comp_F5721690436324261920nt_nat @ G @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_541_fun_Omap__comp,axiom,
! [G: finite_mod_ring_a > int,F2: finite_mod_ring_a > finite_mod_ring_a,V: nat > finite_mod_ring_a] :
( ( comp_F5721690436324261920nt_nat @ G @ ( comp_F1116550632444010611_a_nat @ F2 @ V ) )
= ( comp_F5721690436324261920nt_nat @ ( comp_F2690252154722880373ring_a @ G @ F2 ) @ V ) ) ).
% fun.map_comp
thf(fact_542_poly__eq__iff,axiom,
( ( ^ [Y4: poly_F3299452240248304339ring_a,Z2: poly_F3299452240248304339ring_a] : ( Y4 = Z2 ) )
= ( ^ [P3: poly_F3299452240248304339ring_a,Q3: poly_F3299452240248304339ring_a] :
! [N4: nat] :
( ( coeff_1607515655354303335ring_a @ P3 @ N4 )
= ( coeff_1607515655354303335ring_a @ Q3 @ N4 ) ) ) ) ).
% poly_eq_iff
thf(fact_543_poly__eqI,axiom,
! [P2: poly_F3299452240248304339ring_a,Q2: poly_F3299452240248304339ring_a] :
( ! [N5: nat] :
( ( coeff_1607515655354303335ring_a @ P2 @ N5 )
= ( coeff_1607515655354303335ring_a @ Q2 @ N5 ) )
=> ( P2 = Q2 ) ) ).
% poly_eqI
thf(fact_544_coeff__inject,axiom,
! [X: poly_F3299452240248304339ring_a,Y3: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ X )
= ( coeff_1607515655354303335ring_a @ Y3 ) )
= ( X = Y3 ) ) ).
% coeff_inject
thf(fact_545_mult__right__cancel,axiom,
! [C: complex,A5: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A5 @ C )
= ( times_times_complex @ B @ C ) )
= ( A5 = B ) ) ) ).
% mult_right_cancel
thf(fact_546_mult__right__cancel,axiom,
! [C: finite_mod_ring_a,A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ A5 @ C )
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( A5 = B ) ) ) ).
% mult_right_cancel
thf(fact_547_mult__right__cancel,axiom,
! [C: int,A5: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A5 @ C )
= ( times_times_int @ B @ C ) )
= ( A5 = B ) ) ) ).
% mult_right_cancel
thf(fact_548_mult__right__cancel,axiom,
! [C: nat,A5: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A5 @ C )
= ( times_times_nat @ B @ C ) )
= ( A5 = B ) ) ) ).
% mult_right_cancel
thf(fact_549_mult__right__cancel,axiom,
! [C: real,A5: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A5 @ C )
= ( times_times_real @ B @ C ) )
= ( A5 = B ) ) ) ).
% mult_right_cancel
thf(fact_550_mult__left__cancel,axiom,
! [C: complex,A5: complex,B: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A5 )
= ( times_times_complex @ C @ B ) )
= ( A5 = B ) ) ) ).
% mult_left_cancel
thf(fact_551_mult__left__cancel,axiom,
! [C: finite_mod_ring_a,A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C != zero_z7902377541816115708ring_a )
=> ( ( ( times_5121417576591743744ring_a @ C @ A5 )
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( A5 = B ) ) ) ).
% mult_left_cancel
thf(fact_552_mult__left__cancel,axiom,
! [C: int,A5: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A5 )
= ( times_times_int @ C @ B ) )
= ( A5 = B ) ) ) ).
% mult_left_cancel
thf(fact_553_mult__left__cancel,axiom,
! [C: nat,A5: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A5 )
= ( times_times_nat @ C @ B ) )
= ( A5 = B ) ) ) ).
% mult_left_cancel
thf(fact_554_mult__left__cancel,axiom,
! [C: real,A5: real,B: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A5 )
= ( times_times_real @ C @ B ) )
= ( A5 = B ) ) ) ).
% mult_left_cancel
thf(fact_555_no__zero__divisors,axiom,
! [A5: complex,B: complex] :
( ( A5 != zero_zero_complex )
=> ( ( B != zero_zero_complex )
=> ( ( times_times_complex @ A5 @ B )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_556_no__zero__divisors,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( A5 != zero_z7902377541816115708ring_a )
=> ( ( B != zero_z7902377541816115708ring_a )
=> ( ( times_5121417576591743744ring_a @ A5 @ B )
!= zero_z7902377541816115708ring_a ) ) ) ).
% no_zero_divisors
thf(fact_557_no__zero__divisors,axiom,
! [A5: int,B: int] :
( ( A5 != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A5 @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_558_no__zero__divisors,axiom,
! [A5: nat,B: nat] :
( ( A5 != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A5 @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_559_no__zero__divisors,axiom,
! [A5: real,B: real] :
( ( A5 != zero_zero_real )
=> ( ( B != zero_zero_real )
=> ( ( times_times_real @ A5 @ B )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_560_divisors__zero,axiom,
! [A5: complex,B: complex] :
( ( ( times_times_complex @ A5 @ B )
= zero_zero_complex )
=> ( ( A5 = zero_zero_complex )
| ( B = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_561_divisors__zero,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A5 @ B )
= zero_z7902377541816115708ring_a )
=> ( ( A5 = zero_z7902377541816115708ring_a )
| ( B = zero_z7902377541816115708ring_a ) ) ) ).
% divisors_zero
thf(fact_562_divisors__zero,axiom,
! [A5: int,B: int] :
( ( ( times_times_int @ A5 @ B )
= zero_zero_int )
=> ( ( A5 = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_563_divisors__zero,axiom,
! [A5: nat,B: nat] :
( ( ( times_times_nat @ A5 @ B )
= zero_zero_nat )
=> ( ( A5 = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_564_divisors__zero,axiom,
! [A5: real,B: real] :
( ( ( times_times_real @ A5 @ B )
= zero_zero_real )
=> ( ( A5 = zero_zero_real )
| ( B = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_565_mult__not__zero,axiom,
! [A5: complex,B: complex] :
( ( ( times_times_complex @ A5 @ B )
!= zero_zero_complex )
=> ( ( A5 != zero_zero_complex )
& ( B != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_566_mult__not__zero,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A5 @ B )
!= zero_z7902377541816115708ring_a )
=> ( ( A5 != zero_z7902377541816115708ring_a )
& ( B != zero_z7902377541816115708ring_a ) ) ) ).
% mult_not_zero
thf(fact_567_mult__not__zero,axiom,
! [A5: kyber_qr_a,B: kyber_qr_a] :
( ( ( times_2095635435063429214r_qr_a @ A5 @ B )
!= zero_zero_Kyber_qr_a )
=> ( ( A5 != zero_zero_Kyber_qr_a )
& ( B != zero_zero_Kyber_qr_a ) ) ) ).
% mult_not_zero
thf(fact_568_mult__not__zero,axiom,
! [A5: int,B: int] :
( ( ( times_times_int @ A5 @ B )
!= zero_zero_int )
=> ( ( A5 != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_569_mult__not__zero,axiom,
! [A5: nat,B: nat] :
( ( ( times_times_nat @ A5 @ B )
!= zero_zero_nat )
=> ( ( A5 != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_570_mult__not__zero,axiom,
! [A5: real,B: real] :
( ( ( times_times_real @ A5 @ B )
!= zero_zero_real )
=> ( ( A5 != zero_zero_real )
& ( B != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_571_zero__poly_Orep__eq,axiom,
( ( coeff_1607515655354303335ring_a @ zero_z1830546546923837194ring_a )
= ( ^ [Uu: nat] : zero_z7902377541816115708ring_a ) ) ).
% zero_poly.rep_eq
thf(fact_572_zero__poly_Orep__eq,axiom,
( ( coeff_int @ zero_zero_poly_int )
= ( ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly.rep_eq
thf(fact_573_zero__poly_Orep__eq,axiom,
( ( coeff_nat @ zero_zero_poly_nat )
= ( ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly.rep_eq
thf(fact_574_zero__poly_Orep__eq,axiom,
( ( coeff_complex @ zero_z2709840015065127615omplex )
= ( ^ [Uu: nat] : zero_zero_complex ) ) ).
% zero_poly.rep_eq
thf(fact_575_zero__poly_Orep__eq,axiom,
( ( coeff_real @ zero_zero_poly_real )
= ( ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly.rep_eq
thf(fact_576_lambda__zero,axiom,
( ( ^ [H2: complex] : zero_zero_complex )
= ( times_times_complex @ zero_zero_complex ) ) ).
% lambda_zero
thf(fact_577_lambda__zero,axiom,
( ( ^ [H2: finite_mod_ring_a] : zero_z7902377541816115708ring_a )
= ( times_5121417576591743744ring_a @ zero_z7902377541816115708ring_a ) ) ).
% lambda_zero
thf(fact_578_lambda__zero,axiom,
( ( ^ [H2: kyber_qr_a] : zero_zero_Kyber_qr_a )
= ( times_2095635435063429214r_qr_a @ zero_zero_Kyber_qr_a ) ) ).
% lambda_zero
thf(fact_579_lambda__zero,axiom,
( ( ^ [H2: int] : zero_zero_int )
= ( times_times_int @ zero_zero_int ) ) ).
% lambda_zero
thf(fact_580_lambda__zero,axiom,
( ( ^ [H2: nat] : zero_zero_nat )
= ( times_times_nat @ zero_zero_nat ) ) ).
% lambda_zero
thf(fact_581_lambda__zero,axiom,
( ( ^ [H2: real] : zero_zero_real )
= ( times_times_real @ zero_zero_real ) ) ).
% lambda_zero
thf(fact_582_fun_Omap__ident__strong,axiom,
! [T: product_unit > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_875570014554754200it_nat @ T @ top_to1996260823553986621t_unit ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_n7656859423886344392t_unit @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_583_fun_Omap__ident__strong,axiom,
! [T: nat > int,F2: int > int] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_nat_int @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_int_int_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_584_fun_Omap__ident__strong,axiom,
! [T: nat > complex,F2: complex > complex] :
( ! [Z3: complex] :
( ( member_complex @ Z3 @ ( image_nat_complex @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_c7268011922939458833ex_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_585_fun_Omap__ident__strong,axiom,
! [T: nat > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_nat_nat @ T @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_nat @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_586_fun_Omap__ident__strong,axiom,
! [T: int > complex,F2: complex > complex] :
( ! [Z3: complex] :
( ( member_complex @ Z3 @ ( image_int_complex @ T @ top_top_set_int ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_c7265521452430408557ex_int @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_587_fun_Omap__ident__strong,axiom,
! [T: int > int,F2: int > int] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_int_int @ T @ top_top_set_int ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_int_int_int @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_588_fun_Omap__ident__strong,axiom,
! [T: int > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_int_nat @ T @ top_top_set_int ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_int @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_589_fun_Omap__ident__strong,axiom,
! [T: k > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_k_nat @ T @ top_top_set_k ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_k @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_590_fun_Omap__ident__strong,axiom,
! [T: a > nat,F2: nat > nat] :
( ! [Z3: nat] :
( ( member_nat @ Z3 @ ( image_a_nat @ T @ top_top_set_a ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_nat_nat_a @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_591_fun_Omap__ident__strong,axiom,
! [T: finite_mod_ring_a > int,F2: int > int] :
( ! [Z3: int] :
( ( member_int @ Z3 @ ( image_4238506139956901036_a_int @ T @ top_to2069866484006881781ring_a ) )
=> ( ( F2 @ Z3 )
= Z3 ) )
=> ( ( comp_i1216107289310836680ring_a @ F2 @ T )
= T ) ) ).
% fun.map_ident_strong
thf(fact_592_fun_Oinj__map__strong,axiom,
! [X: nat > finite_mod_ring_a,Xa2: nat > finite_mod_ring_a,F2: finite_mod_ring_a > int,Fa: finite_mod_ring_a > int] :
( ! [Z3: finite_mod_ring_a,Za: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ Z3 @ ( image_1980459031860794542ring_a @ X @ top_top_set_nat ) )
=> ( ( member3034048621153491438ring_a @ Za @ ( image_1980459031860794542ring_a @ Xa2 @ top_top_set_nat ) )
=> ( ( ( F2 @ Z3 )
= ( Fa @ Za ) )
=> ( Z3 = Za ) ) ) )
=> ( ( ( comp_F5721690436324261920nt_nat @ F2 @ X )
= ( comp_F5721690436324261920nt_nat @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% fun.inj_map_strong
thf(fact_593_fun_Omap__cong0,axiom,
! [X: nat > finite_mod_ring_a,F2: finite_mod_ring_a > int,G: finite_mod_ring_a > int] :
( ! [Z3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ Z3 @ ( image_1980459031860794542ring_a @ X @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_F5721690436324261920nt_nat @ F2 @ X )
= ( comp_F5721690436324261920nt_nat @ G @ X ) ) ) ).
% fun.map_cong0
thf(fact_594_fun_Omap__cong,axiom,
! [X: nat > finite_mod_ring_a,Ya: nat > finite_mod_ring_a,F2: finite_mod_ring_a > int,G: finite_mod_ring_a > int] :
( ( X = Ya )
=> ( ! [Z3: finite_mod_ring_a] :
( ( member3034048621153491438ring_a @ Z3 @ ( image_1980459031860794542ring_a @ Ya @ top_top_set_nat ) )
=> ( ( F2 @ Z3 )
= ( G @ Z3 ) ) )
=> ( ( comp_F5721690436324261920nt_nat @ F2 @ X )
= ( comp_F5721690436324261920nt_nat @ G @ Ya ) ) ) ) ).
% fun.map_cong
thf(fact_595_of__int__mod__ring__hom_Ohom__zero,axiom,
( ( finite8272632373135393572ring_a @ zero_zero_int )
= zero_z7902377541816115708ring_a ) ).
% of_int_mod_ring_hom.hom_zero
thf(fact_596_surj__fun__eq,axiom,
! [F2: nat > finite_mod_ring_a,X5: set_nat,G1: finite_mod_ring_a > int,G22: finite_mod_ring_a > int] :
( ( ( image_1980459031860794542ring_a @ F2 @ X5 )
= top_to2069866484006881781ring_a )
=> ( ! [X3: nat] :
( ( member_nat @ X3 @ X5 )
=> ( ( comp_F5721690436324261920nt_nat @ G1 @ F2 @ X3 )
= ( comp_F5721690436324261920nt_nat @ G22 @ F2 @ X3 ) ) )
=> ( G1 = G22 ) ) ) ).
% surj_fun_eq
thf(fact_597_q__gt__zero,axiom,
ord_less_int @ zero_zero_int @ q ).
% q_gt_zero
thf(fact_598_top__empty__eq,axiom,
( top_to2465898995584390880unit_o
= ( ^ [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ) ) ).
% top_empty_eq
thf(fact_599_top__empty__eq,axiom,
( top_top_nat_o
= ( ^ [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ) ) ).
% top_empty_eq
thf(fact_600_top__empty__eq,axiom,
( top_top_int_o
= ( ^ [X2: int] : ( member_int @ X2 @ top_top_set_int ) ) ) ).
% top_empty_eq
thf(fact_601_top__empty__eq,axiom,
( top_top_k_o
= ( ^ [X2: k] : ( member_k @ X2 @ top_top_set_k ) ) ) ).
% top_empty_eq
thf(fact_602_top__empty__eq,axiom,
( top_top_a_o
= ( ^ [X2: a] : ( member_a @ X2 @ top_top_set_a ) ) ) ).
% top_empty_eq
thf(fact_603_top__empty__eq,axiom,
( top_to8738565331869774224plex_o
= ( ^ [X2: sum_sum_a_complex] : ( member8603132577197391238omplex @ X2 @ top_to8676441370508295053omplex ) ) ) ).
% top_empty_eq
thf(fact_604_top__empty__eq,axiom,
( top_to1565196397637005550_nat_o
= ( ^ [X2: sum_sum_a_nat] : ( member_Sum_sum_a_nat @ X2 @ top_to795618464972521135_a_nat ) ) ) ).
% top_empty_eq
thf(fact_605_top__empty__eq,axiom,
( top_to8543423466821335442_int_o
= ( ^ [X2: sum_sum_a_int] : ( member_Sum_sum_a_int @ X2 @ top_to7528907356895570187_a_int ) ) ) ).
% top_empty_eq
thf(fact_606_top__empty__eq,axiom,
( top_to1258784492157183664_a_k_o
= ( ^ [X2: sum_sum_a_k] : ( member_Sum_sum_a_k @ X2 @ top_to335874364214223893um_a_k ) ) ) ).
% top_empty_eq
thf(fact_607_top__empty__eq,axiom,
( top_to1463224201319616570_a_a_o
= ( ^ [X2: sum_sum_a_a] : ( member_Sum_sum_a_a @ X2 @ top_to8848906000605539851um_a_a ) ) ) ).
% top_empty_eq
thf(fact_608_nat__not__finite,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% nat_not_finite
thf(fact_609_n__nonzero,axiom,
n != zero_zero_int ).
% n_nonzero
thf(fact_610_not__prime__0,axiom,
~ ( factor5938532291743052070omplex @ zero_zero_complex ) ).
% not_prime_0
thf(fact_611_not__prime__0,axiom,
~ ( factor1868233065478773540e_real @ zero_zero_real ) ).
% not_prime_0
thf(fact_612_not__prime__0,axiom,
~ ( factor1798656936486255268me_int @ zero_zero_int ) ).
% not_prime_0
thf(fact_613_not__prime__0,axiom,
~ ( factor1801147406995305544me_nat @ zero_zero_nat ) ).
% not_prime_0
thf(fact_614_mult__delta__right,axiom,
! [B: $o,X: complex,Y3: complex] :
( ( B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y3 @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_complex @ X @ ( if_complex @ B @ Y3 @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_615_mult__delta__right,axiom,
! [B: $o,X: finite_mod_ring_a,Y3: finite_mod_ring_a] :
( ( B
=> ( ( times_5121417576591743744ring_a @ X @ ( if_Finite_mod_ring_a @ B @ Y3 @ zero_z7902377541816115708ring_a ) )
= ( times_5121417576591743744ring_a @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_5121417576591743744ring_a @ X @ ( if_Finite_mod_ring_a @ B @ Y3 @ zero_z7902377541816115708ring_a ) )
= zero_z7902377541816115708ring_a ) ) ) ).
% mult_delta_right
thf(fact_616_mult__delta__right,axiom,
! [B: $o,X: kyber_qr_a,Y3: kyber_qr_a] :
( ( B
=> ( ( times_2095635435063429214r_qr_a @ X @ ( if_Kyber_qr_a @ B @ Y3 @ zero_zero_Kyber_qr_a ) )
= ( times_2095635435063429214r_qr_a @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_2095635435063429214r_qr_a @ X @ ( if_Kyber_qr_a @ B @ Y3 @ zero_zero_Kyber_qr_a ) )
= zero_zero_Kyber_qr_a ) ) ) ).
% mult_delta_right
thf(fact_617_mult__delta__right,axiom,
! [B: $o,X: int,Y3: int] :
( ( B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y3 @ zero_zero_int ) )
= ( times_times_int @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_int @ X @ ( if_int @ B @ Y3 @ zero_zero_int ) )
= zero_zero_int ) ) ) ).
% mult_delta_right
thf(fact_618_mult__delta__right,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ X @ ( if_nat @ B @ Y3 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_619_mult__delta__right,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ X @ ( if_real @ B @ Y3 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_620_mult__delta__left,axiom,
! [B: $o,X: complex,Y3: complex] :
( ( B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y3 )
= ( times_times_complex @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_complex @ ( if_complex @ B @ X @ zero_zero_complex ) @ Y3 )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_621_mult__delta__left,axiom,
! [B: $o,X: finite_mod_ring_a,Y3: finite_mod_ring_a] :
( ( B
=> ( ( times_5121417576591743744ring_a @ ( if_Finite_mod_ring_a @ B @ X @ zero_z7902377541816115708ring_a ) @ Y3 )
= ( times_5121417576591743744ring_a @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_5121417576591743744ring_a @ ( if_Finite_mod_ring_a @ B @ X @ zero_z7902377541816115708ring_a ) @ Y3 )
= zero_z7902377541816115708ring_a ) ) ) ).
% mult_delta_left
thf(fact_622_mult__delta__left,axiom,
! [B: $o,X: kyber_qr_a,Y3: kyber_qr_a] :
( ( B
=> ( ( times_2095635435063429214r_qr_a @ ( if_Kyber_qr_a @ B @ X @ zero_zero_Kyber_qr_a ) @ Y3 )
= ( times_2095635435063429214r_qr_a @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_2095635435063429214r_qr_a @ ( if_Kyber_qr_a @ B @ X @ zero_zero_Kyber_qr_a ) @ Y3 )
= zero_zero_Kyber_qr_a ) ) ) ).
% mult_delta_left
thf(fact_623_mult__delta__left,axiom,
! [B: $o,X: int,Y3: int] :
( ( B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y3 )
= ( times_times_int @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_int @ ( if_int @ B @ X @ zero_zero_int ) @ Y3 )
= zero_zero_int ) ) ) ).
% mult_delta_left
thf(fact_624_mult__delta__left,axiom,
! [B: $o,X: nat,Y3: nat] :
( ( B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= ( times_times_nat @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_nat @ ( if_nat @ B @ X @ zero_zero_nat ) @ Y3 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_625_mult__delta__left,axiom,
! [B: $o,X: real,Y3: real] :
( ( B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= ( times_times_real @ X @ Y3 ) ) )
& ( ~ B
=> ( ( times_times_real @ ( if_real @ B @ X @ zero_zero_real ) @ Y3 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_626_n__gt__zero,axiom,
ord_less_int @ zero_zero_int @ n ).
% n_gt_zero
thf(fact_627_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_628_kyber__spec__axioms,axiom,
kyber_kyber_spec_a_k @ type_a @ type_k @ n @ q @ k2 @ n2 ).
% kyber_spec_axioms
thf(fact_629_linorder__neqE__linordered__idom,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_630_linorder__neqE__linordered__idom,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_631_lt__ex,axiom,
! [X: int] :
? [Y5: int] : ( ord_less_int @ Y5 @ X ) ).
% lt_ex
thf(fact_632_lt__ex,axiom,
! [X: real] :
? [Y5: real] : ( ord_less_real @ Y5 @ X ) ).
% lt_ex
thf(fact_633_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_634_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_635_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_636_dense,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ? [Z3: real] :
( ( ord_less_real @ X @ Z3 )
& ( ord_less_real @ Z3 @ Y3 ) ) ) ).
% dense
thf(fact_637_less__imp__neq,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_638_less__imp__neq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_639_less__imp__neq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% less_imp_neq
thf(fact_640_order_Oasym,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ A5 @ B )
=> ~ ( ord_less_int @ B @ A5 ) ) ).
% order.asym
thf(fact_641_order_Oasym,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ A5 @ B )
=> ~ ( ord_less_nat @ B @ A5 ) ) ).
% order.asym
thf(fact_642_order_Oasym,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ A5 @ B )
=> ~ ( ord_less_real @ B @ A5 ) ) ).
% order.asym
thf(fact_643_ord__eq__less__trans,axiom,
! [A5: int,B: int,C: int] :
( ( A5 = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A5 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_644_ord__eq__less__trans,axiom,
! [A5: nat,B: nat,C: nat] :
( ( A5 = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A5 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_645_ord__eq__less__trans,axiom,
! [A5: real,B: real,C: real] :
( ( A5 = B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A5 @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_646_ord__less__eq__trans,axiom,
! [A5: int,B: int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( B = C )
=> ( ord_less_int @ A5 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_647_ord__less__eq__trans,axiom,
! [A5: nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A5 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_648_ord__less__eq__trans,axiom,
! [A5: real,B: real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( B = C )
=> ( ord_less_real @ A5 @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_649_less__induct,axiom,
! [P: nat > $o,A5: nat] :
( ! [X3: nat] :
( ! [Y6: nat] :
( ( ord_less_nat @ Y6 @ X3 )
=> ( P @ Y6 ) )
=> ( P @ X3 ) )
=> ( P @ A5 ) ) ).
% less_induct
thf(fact_650_antisym__conv3,axiom,
! [Y3: int,X: int] :
( ~ ( ord_less_int @ Y3 @ X )
=> ( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_651_antisym__conv3,axiom,
! [Y3: nat,X: nat] :
( ~ ( ord_less_nat @ Y3 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_652_antisym__conv3,axiom,
! [Y3: real,X: real] :
( ~ ( ord_less_real @ Y3 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( X = Y3 ) ) ) ).
% antisym_conv3
thf(fact_653_linorder__cases,axiom,
! [X: int,Y3: int] :
( ~ ( ord_less_int @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_654_linorder__cases,axiom,
! [X: nat,Y3: nat] :
( ~ ( ord_less_nat @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_655_linorder__cases,axiom,
! [X: real,Y3: real] :
( ~ ( ord_less_real @ X @ Y3 )
=> ( ( X != Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_cases
thf(fact_656_dual__order_Oasym,axiom,
! [B: int,A5: int] :
( ( ord_less_int @ B @ A5 )
=> ~ ( ord_less_int @ A5 @ B ) ) ).
% dual_order.asym
thf(fact_657_dual__order_Oasym,axiom,
! [B: nat,A5: nat] :
( ( ord_less_nat @ B @ A5 )
=> ~ ( ord_less_nat @ A5 @ B ) ) ).
% dual_order.asym
thf(fact_658_dual__order_Oasym,axiom,
! [B: real,A5: real] :
( ( ord_less_real @ B @ A5 )
=> ~ ( ord_less_real @ A5 @ B ) ) ).
% dual_order.asym
thf(fact_659_dual__order_Oirrefl,axiom,
! [A5: int] :
~ ( ord_less_int @ A5 @ A5 ) ).
% dual_order.irrefl
thf(fact_660_dual__order_Oirrefl,axiom,
! [A5: nat] :
~ ( ord_less_nat @ A5 @ A5 ) ).
% dual_order.irrefl
thf(fact_661_dual__order_Oirrefl,axiom,
! [A5: real] :
~ ( ord_less_real @ A5 @ A5 ) ).
% dual_order.irrefl
thf(fact_662_exists__least__iff,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [N4: nat] :
( ( P5 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P5 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_663_linorder__less__wlog,axiom,
! [P: int > int > $o,A5: int,B: int] :
( ! [A4: int,B3: int] :
( ( ord_less_int @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B3: int] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A5 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_664_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A5: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A5 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_665_linorder__less__wlog,axiom,
! [P: real > real > $o,A5: real,B: real] :
( ! [A4: real,B3: real] :
( ( ord_less_real @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: real] : ( P @ A4 @ A4 )
=> ( ! [A4: real,B3: real] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A5 @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_666_order_Ostrict__trans,axiom,
! [A5: int,B: int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A5 @ C ) ) ) ).
% order.strict_trans
thf(fact_667_order_Ostrict__trans,axiom,
! [A5: nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A5 @ C ) ) ) ).
% order.strict_trans
thf(fact_668_order_Ostrict__trans,axiom,
! [A5: real,B: real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_real @ B @ C )
=> ( ord_less_real @ A5 @ C ) ) ) ).
% order.strict_trans
thf(fact_669_not__less__iff__gr__or__eq,axiom,
! [X: int,Y3: int] :
( ( ~ ( ord_less_int @ X @ Y3 ) )
= ( ( ord_less_int @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_670_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y3: nat] :
( ( ~ ( ord_less_nat @ X @ Y3 ) )
= ( ( ord_less_nat @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_671_not__less__iff__gr__or__eq,axiom,
! [X: real,Y3: real] :
( ( ~ ( ord_less_real @ X @ Y3 ) )
= ( ( ord_less_real @ Y3 @ X )
| ( X = Y3 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_672_dual__order_Ostrict__trans,axiom,
! [B: int,A5: int,C: int] :
( ( ord_less_int @ B @ A5 )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A5 ) ) ) ).
% dual_order.strict_trans
thf(fact_673_dual__order_Ostrict__trans,axiom,
! [B: nat,A5: nat,C: nat] :
( ( ord_less_nat @ B @ A5 )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A5 ) ) ) ).
% dual_order.strict_trans
thf(fact_674_dual__order_Ostrict__trans,axiom,
! [B: real,A5: real,C: real] :
( ( ord_less_real @ B @ A5 )
=> ( ( ord_less_real @ C @ B )
=> ( ord_less_real @ C @ A5 ) ) ) ).
% dual_order.strict_trans
thf(fact_675_order_Ostrict__implies__not__eq,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ A5 @ B )
=> ( A5 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_676_order_Ostrict__implies__not__eq,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( A5 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_677_order_Ostrict__implies__not__eq,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ A5 @ B )
=> ( A5 != B ) ) ).
% order.strict_implies_not_eq
thf(fact_678_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A5: int] :
( ( ord_less_int @ B @ A5 )
=> ( A5 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_679_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A5: nat] :
( ( ord_less_nat @ B @ A5 )
=> ( A5 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_680_dual__order_Ostrict__implies__not__eq,axiom,
! [B: real,A5: real] :
( ( ord_less_real @ B @ A5 )
=> ( A5 != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_681_linorder__neqE,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_682_linorder__neqE,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_683_linorder__neqE,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
=> ( ~ ( ord_less_real @ X @ Y3 )
=> ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neqE
thf(fact_684_order__less__asym,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_685_order__less__asym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_686_order__less__asym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_asym
thf(fact_687_linorder__neq__iff,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
= ( ( ord_less_int @ X @ Y3 )
| ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_688_linorder__neq__iff,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
= ( ( ord_less_nat @ X @ Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_689_linorder__neq__iff,axiom,
! [X: real,Y3: real] :
( ( X != Y3 )
= ( ( ord_less_real @ X @ Y3 )
| ( ord_less_real @ Y3 @ X ) ) ) ).
% linorder_neq_iff
thf(fact_690_order__less__asym_H,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ A5 @ B )
=> ~ ( ord_less_int @ B @ A5 ) ) ).
% order_less_asym'
thf(fact_691_order__less__asym_H,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ A5 @ B )
=> ~ ( ord_less_nat @ B @ A5 ) ) ).
% order_less_asym'
thf(fact_692_order__less__asym_H,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ A5 @ B )
=> ~ ( ord_less_real @ B @ A5 ) ) ).
% order_less_asym'
thf(fact_693_order__less__trans,axiom,
! [X: int,Y3: int,Z: int] :
( ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_int @ Y3 @ Z )
=> ( ord_less_int @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_694_order__less__trans,axiom,
! [X: nat,Y3: nat,Z: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_695_order__less__trans,axiom,
! [X: real,Y3: real,Z: real] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% order_less_trans
thf(fact_696_ord__eq__less__subst,axiom,
! [A5: int,F2: int > int,B: int,C: int] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_697_ord__eq__less__subst,axiom,
! [A5: nat,F2: int > nat,B: int,C: int] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_698_ord__eq__less__subst,axiom,
! [A5: real,F2: int > real,B: int,C: int] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_699_ord__eq__less__subst,axiom,
! [A5: int,F2: nat > int,B: nat,C: nat] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_700_ord__eq__less__subst,axiom,
! [A5: nat,F2: nat > nat,B: nat,C: nat] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_701_ord__eq__less__subst,axiom,
! [A5: real,F2: nat > real,B: nat,C: nat] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_702_ord__eq__less__subst,axiom,
! [A5: int,F2: real > int,B: real,C: real] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_703_ord__eq__less__subst,axiom,
! [A5: nat,F2: real > nat,B: real,C: real] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_704_ord__eq__less__subst,axiom,
! [A5: real,F2: real > real,B: real,C: real] :
( ( A5
= ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_705_ord__less__eq__subst,axiom,
! [A5: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_706_ord__less__eq__subst,axiom,
! [A5: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_707_ord__less__eq__subst,axiom,
! [A5: int,B: int,F2: int > real,C: real] :
( ( ord_less_int @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_708_ord__less__eq__subst,axiom,
! [A5: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_709_ord__less__eq__subst,axiom,
! [A5: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_710_ord__less__eq__subst,axiom,
! [A5: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_711_ord__less__eq__subst,axiom,
! [A5: real,B: real,F2: real > int,C: int] :
( ( ord_less_real @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_712_ord__less__eq__subst,axiom,
! [A5: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_713_ord__less__eq__subst,axiom,
! [A5: real,B: real,F2: real > real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ( F2 @ B )
= C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_714_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_715_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_716_order__less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% order_less_irrefl
thf(fact_717_order__less__subst1,axiom,
! [A5: int,F2: int > int,B: int,C: int] :
( ( ord_less_int @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_718_order__less__subst1,axiom,
! [A5: int,F2: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_719_order__less__subst1,axiom,
! [A5: int,F2: real > int,B: real,C: real] :
( ( ord_less_int @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_720_order__less__subst1,axiom,
! [A5: nat,F2: int > nat,B: int,C: int] :
( ( ord_less_nat @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_721_order__less__subst1,axiom,
! [A5: nat,F2: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_722_order__less__subst1,axiom,
! [A5: nat,F2: real > nat,B: real,C: real] :
( ( ord_less_nat @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_723_order__less__subst1,axiom,
! [A5: real,F2: int > real,B: int,C: int] :
( ( ord_less_real @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_724_order__less__subst1,axiom,
! [A5: real,F2: nat > real,B: nat,C: nat] :
( ( ord_less_real @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_725_order__less__subst1,axiom,
! [A5: real,F2: real > real,B: real,C: real] :
( ( ord_less_real @ A5 @ ( F2 @ B ) )
=> ( ( ord_less_real @ B @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ A5 @ ( F2 @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_726_order__less__subst2,axiom,
! [A5: int,B: int,F2: int > int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_727_order__less__subst2,axiom,
! [A5: int,B: int,F2: int > nat,C: nat] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_728_order__less__subst2,axiom,
! [A5: int,B: int,F2: int > real,C: real] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X3: int,Y5: int] :
( ( ord_less_int @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_729_order__less__subst2,axiom,
! [A5: nat,B: nat,F2: nat > int,C: int] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_730_order__less__subst2,axiom,
! [A5: nat,B: nat,F2: nat > nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_731_order__less__subst2,axiom,
! [A5: nat,B: nat,F2: nat > real,C: real] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_732_order__less__subst2,axiom,
! [A5: real,B: real,F2: real > int,C: int] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_int @ ( F2 @ B ) @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_int @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_int @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_733_order__less__subst2,axiom,
! [A5: real,B: real,F2: real > nat,C: nat] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_nat @ ( F2 @ B ) @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_nat @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_nat @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_734_order__less__subst2,axiom,
! [A5: real,B: real,F2: real > real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_real @ ( F2 @ B ) @ C )
=> ( ! [X3: real,Y5: real] :
( ( ord_less_real @ X3 @ Y5 )
=> ( ord_less_real @ ( F2 @ X3 ) @ ( F2 @ Y5 ) ) )
=> ( ord_less_real @ ( F2 @ A5 ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_735_order__less__not__sym,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_736_order__less__not__sym,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_737_order__less__not__sym,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_not_sym
thf(fact_738_order__less__imp__triv,axiom,
! [X: int,Y3: int,P: $o] :
( ( ord_less_int @ X @ Y3 )
=> ( ( ord_less_int @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_739_order__less__imp__triv,axiom,
! [X: nat,Y3: nat,P: $o] :
( ( ord_less_nat @ X @ Y3 )
=> ( ( ord_less_nat @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_740_order__less__imp__triv,axiom,
! [X: real,Y3: real,P: $o] :
( ( ord_less_real @ X @ Y3 )
=> ( ( ord_less_real @ Y3 @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_741_linorder__less__linear,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_int @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_742_linorder__less__linear,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_nat @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_743_linorder__less__linear,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
| ( X = Y3 )
| ( ord_less_real @ Y3 @ X ) ) ).
% linorder_less_linear
thf(fact_744_order__less__imp__not__eq,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_745_order__less__imp__not__eq,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_746_order__less__imp__not__eq,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( X != Y3 ) ) ).
% order_less_imp_not_eq
thf(fact_747_order__less__imp__not__eq2,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_748_order__less__imp__not__eq2,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_749_order__less__imp__not__eq2,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ( Y3 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_750_order__less__imp__not__less,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ X @ Y3 )
=> ~ ( ord_less_int @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_751_order__less__imp__not__less,axiom,
! [X: nat,Y3: nat] :
( ( ord_less_nat @ X @ Y3 )
=> ~ ( ord_less_nat @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_752_order__less__imp__not__less,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ X @ Y3 )
=> ~ ( ord_less_real @ Y3 @ X ) ) ).
% order_less_imp_not_less
thf(fact_753_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_754_gr__implies__not__zero,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_755_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_756_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_757_top_Oextremum__strict,axiom,
! [A5: set_Product_unit] :
~ ( ord_le8056459307392131481t_unit @ top_to1996260823553986621t_unit @ A5 ) ).
% top.extremum_strict
thf(fact_758_top_Oextremum__strict,axiom,
! [A5: set_nat] :
~ ( ord_less_set_nat @ top_top_set_nat @ A5 ) ).
% top.extremum_strict
thf(fact_759_top_Oextremum__strict,axiom,
! [A5: set_int] :
~ ( ord_less_set_int @ top_top_set_int @ A5 ) ).
% top.extremum_strict
thf(fact_760_top_Oextremum__strict,axiom,
! [A5: set_k] :
~ ( ord_less_set_k @ top_top_set_k @ A5 ) ).
% top.extremum_strict
thf(fact_761_top_Oextremum__strict,axiom,
! [A5: set_a] :
~ ( ord_less_set_a @ top_top_set_a @ A5 ) ).
% top.extremum_strict
thf(fact_762_top_Oextremum__strict,axiom,
! [A5: set_Su8486473086189545021omplex] :
~ ( ord_le8093745868852885225omplex @ top_to8676441370508295053omplex @ A5 ) ).
% top.extremum_strict
thf(fact_763_top_Oextremum__strict,axiom,
! [A5: set_Sum_sum_a_nat] :
~ ( ord_le4638968407124865547_a_nat @ top_to795618464972521135_a_nat @ A5 ) ).
% top.extremum_strict
thf(fact_764_top_Oextremum__strict,axiom,
! [A5: set_Sum_sum_a_int] :
~ ( ord_le2148885262193138791_a_int @ top_to7528907356895570187_a_int @ A5 ) ).
% top.extremum_strict
thf(fact_765_top_Oextremum__strict,axiom,
! [A5: set_Sum_sum_a_k] :
~ ( ord_le1054565356735369145um_a_k @ top_to335874364214223893um_a_k @ A5 ) ).
% top.extremum_strict
thf(fact_766_top_Oextremum__strict,axiom,
! [A5: set_Sum_sum_a_a] :
~ ( ord_le344224956271909295um_a_a @ top_to8848906000605539851um_a_a @ A5 ) ).
% top.extremum_strict
thf(fact_767_top_Onot__eq__extremum,axiom,
! [A5: set_Product_unit] :
( ( A5 != top_to1996260823553986621t_unit )
= ( ord_le8056459307392131481t_unit @ A5 @ top_to1996260823553986621t_unit ) ) ).
% top.not_eq_extremum
thf(fact_768_top_Onot__eq__extremum,axiom,
! [A5: set_nat] :
( ( A5 != top_top_set_nat )
= ( ord_less_set_nat @ A5 @ top_top_set_nat ) ) ).
% top.not_eq_extremum
thf(fact_769_top_Onot__eq__extremum,axiom,
! [A5: set_int] :
( ( A5 != top_top_set_int )
= ( ord_less_set_int @ A5 @ top_top_set_int ) ) ).
% top.not_eq_extremum
thf(fact_770_top_Onot__eq__extremum,axiom,
! [A5: set_k] :
( ( A5 != top_top_set_k )
= ( ord_less_set_k @ A5 @ top_top_set_k ) ) ).
% top.not_eq_extremum
thf(fact_771_top_Onot__eq__extremum,axiom,
! [A5: set_a] :
( ( A5 != top_top_set_a )
= ( ord_less_set_a @ A5 @ top_top_set_a ) ) ).
% top.not_eq_extremum
thf(fact_772_top_Onot__eq__extremum,axiom,
! [A5: set_Su8486473086189545021omplex] :
( ( A5 != top_to8676441370508295053omplex )
= ( ord_le8093745868852885225omplex @ A5 @ top_to8676441370508295053omplex ) ) ).
% top.not_eq_extremum
thf(fact_773_top_Onot__eq__extremum,axiom,
! [A5: set_Sum_sum_a_nat] :
( ( A5 != top_to795618464972521135_a_nat )
= ( ord_le4638968407124865547_a_nat @ A5 @ top_to795618464972521135_a_nat ) ) ).
% top.not_eq_extremum
thf(fact_774_top_Onot__eq__extremum,axiom,
! [A5: set_Sum_sum_a_int] :
( ( A5 != top_to7528907356895570187_a_int )
= ( ord_le2148885262193138791_a_int @ A5 @ top_to7528907356895570187_a_int ) ) ).
% top.not_eq_extremum
thf(fact_775_top_Onot__eq__extremum,axiom,
! [A5: set_Sum_sum_a_k] :
( ( A5 != top_to335874364214223893um_a_k )
= ( ord_le1054565356735369145um_a_k @ A5 @ top_to335874364214223893um_a_k ) ) ).
% top.not_eq_extremum
thf(fact_776_top_Onot__eq__extremum,axiom,
! [A5: set_Sum_sum_a_a] :
( ( A5 != top_to8848906000605539851um_a_a )
= ( ord_le344224956271909295um_a_a @ A5 @ top_to8848906000605539851um_a_a ) ) ).
% top.not_eq_extremum
thf(fact_777_mult__neg__neg,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ A5 @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A5 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_778_mult__neg__neg,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ A5 @ zero_zero_real )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A5 @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_779_not__square__less__zero,axiom,
! [A5: int] :
~ ( ord_less_int @ ( times_times_int @ A5 @ A5 ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_780_not__square__less__zero,axiom,
! [A5: real] :
~ ( ord_less_real @ ( times_times_real @ A5 @ A5 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_781_mult__less__0__iff,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A5 @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A5 )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A5 @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_782_mult__less__0__iff,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A5 @ B ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A5 )
& ( ord_less_real @ B @ zero_zero_real ) )
| ( ( ord_less_real @ A5 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_783_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ A5 @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A5 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_784_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ A5 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A5 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_785_linordered__semiring__strict__class_Omult__neg__pos,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ A5 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ ( times_times_real @ A5 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_neg_pos
thf(fact_786_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A5 @ B ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_787_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A5 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A5 @ B ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_788_linordered__semiring__strict__class_Omult__pos__neg,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A5 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A5 @ B ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg
thf(fact_789_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A5 )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A5 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_790_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A5 )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A5 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_791_linordered__semiring__strict__class_Omult__pos__pos,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A5 )
=> ( ( ord_less_real @ zero_zero_real @ B )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A5 @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_pos_pos
thf(fact_792_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A5 )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A5 ) @ zero_zero_int ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_793_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A5 )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A5 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_794_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ A5 )
=> ( ( ord_less_real @ B @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B @ A5 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_795_zero__less__mult__iff,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A5 @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A5 )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A5 @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_796_zero__less__mult__iff,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A5 @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ A5 )
& ( ord_less_real @ zero_zero_real @ B ) )
| ( ( ord_less_real @ A5 @ zero_zero_real )
& ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_797_zero__less__mult__pos,axiom,
! [A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A5 @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A5 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_798_zero__less__mult__pos,axiom,
! [A5: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A5 @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A5 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_799_zero__less__mult__pos,axiom,
! [A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A5 @ B ) )
=> ( ( ord_less_real @ zero_zero_real @ A5 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_800_zero__less__mult__pos2,axiom,
! [B: int,A5: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A5 ) )
=> ( ( ord_less_int @ zero_zero_int @ A5 )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_801_zero__less__mult__pos2,axiom,
! [B: nat,A5: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A5 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A5 )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_802_zero__less__mult__pos2,axiom,
! [B: real,A5: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A5 ) )
=> ( ( ord_less_real @ zero_zero_real @ A5 )
=> ( ord_less_real @ zero_zero_real @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_803_mult__less__cancel__left__neg,axiom,
! [C: int,A5: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A5 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_804_mult__less__cancel__left__neg,axiom,
! [C: real,A5: real,B: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ B @ A5 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_805_mult__less__cancel__left__pos,axiom,
! [C: int,A5: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A5 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_806_mult__less__cancel__left__pos,axiom,
! [C: real,A5: real,B: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) )
= ( ord_less_real @ A5 @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_807_mult__strict__left__mono__neg,axiom,
! [B: int,A5: int,C: int] :
( ( ord_less_int @ B @ A5 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_808_mult__strict__left__mono__neg,axiom,
! [B: real,A5: real,C: real] :
( ( ord_less_real @ B @ A5 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_809_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A5: int,B: int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_810_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A5: nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A5 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_811_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A5: real,B: real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_812_mult__less__cancel__left__disj,axiom,
! [C: int,A5: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A5 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A5 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_813_mult__less__cancel__left__disj,axiom,
! [C: real,A5: real,B: real] :
( ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A5 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A5 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_814_mult__strict__right__mono__neg,axiom,
! [B: int,A5: int,C: int] :
( ( ord_less_int @ B @ A5 )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A5 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_815_mult__strict__right__mono__neg,axiom,
! [B: real,A5: real,C: real] :
( ( ord_less_real @ B @ A5 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A5 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_816_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A5: int,B: int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A5 @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_817_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A5: nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A5 @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_818_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A5: real,B: real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A5 @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_819_mult__less__cancel__right__disj,axiom,
! [A5: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A5 @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A5 @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A5 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_820_mult__less__cancel__right__disj,axiom,
! [A5: real,C: real,B: real] :
( ( ord_less_real @ ( times_times_real @ A5 @ C ) @ ( times_times_real @ B @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A5 @ B ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B @ A5 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_821_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A5: int,B: int,C: int] :
( ( ord_less_int @ A5 @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A5 ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_822_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A5: nat,B: nat,C: nat] :
( ( ord_less_nat @ A5 @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A5 ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_823_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A5: real,B: real,C: real] :
( ( ord_less_real @ A5 @ B )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A5 ) @ ( times_times_real @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_824_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_825_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_826_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_827_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_828_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A5 @ B ) @ C )
= ( times_5121417576591743744ring_a @ A5 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_829_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A5: kyber_qr_a,B: kyber_qr_a,C: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ ( times_2095635435063429214r_qr_a @ A5 @ B ) @ C )
= ( times_2095635435063429214r_qr_a @ A5 @ ( times_2095635435063429214r_qr_a @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_830_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A5: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A5 @ B ) @ C )
= ( times_times_int @ A5 @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_831_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A5: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A5 @ B ) @ C )
= ( times_times_nat @ A5 @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_832_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A5: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A5 @ B ) @ C )
= ( times_times_real @ A5 @ ( times_times_real @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_833_mult_Oassoc,axiom,
! [A5: finite_mod_ring_a,B: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ ( times_5121417576591743744ring_a @ A5 @ B ) @ C )
= ( times_5121417576591743744ring_a @ A5 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_834_mult_Oassoc,axiom,
! [A5: kyber_qr_a,B: kyber_qr_a,C: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ ( times_2095635435063429214r_qr_a @ A5 @ B ) @ C )
= ( times_2095635435063429214r_qr_a @ A5 @ ( times_2095635435063429214r_qr_a @ B @ C ) ) ) ).
% mult.assoc
thf(fact_835_mult_Oassoc,axiom,
! [A5: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A5 @ B ) @ C )
= ( times_times_int @ A5 @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_836_mult_Oassoc,axiom,
! [A5: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A5 @ B ) @ C )
= ( times_times_nat @ A5 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_837_mult_Oassoc,axiom,
! [A5: real,B: real,C: real] :
( ( times_times_real @ ( times_times_real @ A5 @ B ) @ C )
= ( times_times_real @ A5 @ ( times_times_real @ B @ C ) ) ) ).
% mult.assoc
thf(fact_838_mult_Ocommute,axiom,
( times_5121417576591743744ring_a
= ( ^ [A3: finite_mod_ring_a,B4: finite_mod_ring_a] : ( times_5121417576591743744ring_a @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_839_mult_Ocommute,axiom,
( times_2095635435063429214r_qr_a
= ( ^ [A3: kyber_qr_a,B4: kyber_qr_a] : ( times_2095635435063429214r_qr_a @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_840_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B4: int] : ( times_times_int @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_841_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B4: nat] : ( times_times_nat @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_842_mult_Ocommute,axiom,
( times_times_real
= ( ^ [A3: real,B4: real] : ( times_times_real @ B4 @ A3 ) ) ) ).
% mult.commute
thf(fact_843_mult_Oleft__commute,axiom,
! [B: finite_mod_ring_a,A5: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ B @ ( times_5121417576591743744ring_a @ A5 @ C ) )
= ( times_5121417576591743744ring_a @ A5 @ ( times_5121417576591743744ring_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_844_mult_Oleft__commute,axiom,
! [B: kyber_qr_a,A5: kyber_qr_a,C: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ B @ ( times_2095635435063429214r_qr_a @ A5 @ C ) )
= ( times_2095635435063429214r_qr_a @ A5 @ ( times_2095635435063429214r_qr_a @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_845_mult_Oleft__commute,axiom,
! [B: int,A5: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A5 @ C ) )
= ( times_times_int @ A5 @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_846_mult_Oleft__commute,axiom,
! [B: nat,A5: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A5 @ C ) )
= ( times_times_nat @ A5 @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_847_mult_Oleft__commute,axiom,
! [B: real,A5: real,C: real] :
( ( times_times_real @ B @ ( times_times_real @ A5 @ C ) )
= ( times_times_real @ A5 @ ( times_times_real @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_848_kyber__spec_Oq__gt__zero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% kyber_spec.q_gt_zero
thf(fact_849_kyber__spec_Oq__gt__zero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ zero_zero_int @ Q2 ) ) ).
% kyber_spec.q_gt_zero
thf(fact_850_kyber__spec_On__gt__zero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ zero_zero_int @ N2 ) ) ).
% kyber_spec.n_gt_zero
thf(fact_851_kyber__spec_On__gt__zero,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ zero_zero_int @ N2 ) ) ).
% kyber_spec.n_gt_zero
thf(fact_852_finite__interval__int4,axiom,
! [A5: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A5 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int4
thf(fact_853_prime__gt__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ zero_zero_int @ P2 ) ) ).
% prime_gt_0_int
thf(fact_854_n__gt__1,axiom,
ord_less_int @ one_one_int @ n ).
% n_gt_1
thf(fact_855_n_H__gr__0,axiom,
ord_less_nat @ zero_zero_nat @ n2 ).
% n'_gr_0
thf(fact_856_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_nat @ N4 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_857_mult__1,axiom,
! [A5: complex] :
( ( times_times_complex @ one_one_complex @ A5 )
= A5 ) ).
% mult_1
thf(fact_858_mult__1,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A5 )
= A5 ) ).
% mult_1
thf(fact_859_mult__1,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ one_one_Kyber_qr_a @ A5 )
= A5 ) ).
% mult_1
thf(fact_860_mult__1,axiom,
! [A5: int] :
( ( times_times_int @ one_one_int @ A5 )
= A5 ) ).
% mult_1
thf(fact_861_mult__1,axiom,
! [A5: nat] :
( ( times_times_nat @ one_one_nat @ A5 )
= A5 ) ).
% mult_1
thf(fact_862_mult__1,axiom,
! [A5: real] :
( ( times_times_real @ one_one_real @ A5 )
= A5 ) ).
% mult_1
thf(fact_863_mult_Oright__neutral,axiom,
! [A5: complex] :
( ( times_times_complex @ A5 @ one_one_complex )
= A5 ) ).
% mult.right_neutral
thf(fact_864_mult_Oright__neutral,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A5 @ one_on2109788427901206336ring_a )
= A5 ) ).
% mult.right_neutral
thf(fact_865_mult_Oright__neutral,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ A5 @ one_one_Kyber_qr_a )
= A5 ) ).
% mult.right_neutral
thf(fact_866_mult_Oright__neutral,axiom,
! [A5: int] :
( ( times_times_int @ A5 @ one_one_int )
= A5 ) ).
% mult.right_neutral
thf(fact_867_mult_Oright__neutral,axiom,
! [A5: nat] :
( ( times_times_nat @ A5 @ one_one_nat )
= A5 ) ).
% mult.right_neutral
thf(fact_868_mult_Oright__neutral,axiom,
! [A5: real] :
( ( times_times_real @ A5 @ one_one_real )
= A5 ) ).
% mult.right_neutral
thf(fact_869_mult__cancel__right2,axiom,
! [A5: complex,C: complex] :
( ( ( times_times_complex @ A5 @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A5 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_870_mult__cancel__right2,axiom,
! [A5: finite_mod_ring_a,C: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ A5 @ C )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A5 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right2
thf(fact_871_mult__cancel__right2,axiom,
! [A5: int,C: int] :
( ( ( times_times_int @ A5 @ C )
= C )
= ( ( C = zero_zero_int )
| ( A5 = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_872_mult__cancel__right2,axiom,
! [A5: real,C: real] :
( ( ( times_times_real @ A5 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A5 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_873_mult__cancel__right1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ B @ C ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_874_mult__cancel__right1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ B @ C ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_right1
thf(fact_875_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_876_mult__cancel__right1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ B @ C ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_877_mult__cancel__left2,axiom,
! [C: complex,A5: complex] :
( ( ( times_times_complex @ C @ A5 )
= C )
= ( ( C = zero_zero_complex )
| ( A5 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_878_mult__cancel__left2,axiom,
! [C: finite_mod_ring_a,A5: finite_mod_ring_a] :
( ( ( times_5121417576591743744ring_a @ C @ A5 )
= C )
= ( ( C = zero_z7902377541816115708ring_a )
| ( A5 = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left2
thf(fact_879_mult__cancel__left2,axiom,
! [C: int,A5: int] :
( ( ( times_times_int @ C @ A5 )
= C )
= ( ( C = zero_zero_int )
| ( A5 = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_880_mult__cancel__left2,axiom,
! [C: real,A5: real] :
( ( ( times_times_real @ C @ A5 )
= C )
= ( ( C = zero_zero_real )
| ( A5 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_881_mult__cancel__left1,axiom,
! [C: complex,B: complex] :
( ( C
= ( times_times_complex @ C @ B ) )
= ( ( C = zero_zero_complex )
| ( B = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_882_mult__cancel__left1,axiom,
! [C: finite_mod_ring_a,B: finite_mod_ring_a] :
( ( C
= ( times_5121417576591743744ring_a @ C @ B ) )
= ( ( C = zero_z7902377541816115708ring_a )
| ( B = one_on2109788427901206336ring_a ) ) ) ).
% mult_cancel_left1
thf(fact_883_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_884_mult__cancel__left1,axiom,
! [C: real,B: real] :
( ( C
= ( times_times_real @ C @ B ) )
= ( ( C = zero_zero_real )
| ( B = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_885_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_886_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_887_one__reorient,axiom,
! [X: complex] :
( ( one_one_complex = X )
= ( X = one_one_complex ) ) ).
% one_reorient
thf(fact_888_one__reorient,axiom,
! [X: real] :
( ( one_one_real = X )
= ( X = one_one_real ) ) ).
% one_reorient
thf(fact_889_finite__psubset__induct,axiom,
! [A2: set_int,P: set_int > $o] :
( ( finite_finite_int @ A2 )
=> ( ! [A6: set_int] :
( ( finite_finite_int @ A6 )
=> ( ! [B5: set_int] :
( ( ord_less_set_int @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_890_finite__psubset__induct,axiom,
! [A2: set_nat,P: set_nat > $o] :
( ( finite_finite_nat @ A2 )
=> ( ! [A6: set_nat] :
( ( finite_finite_nat @ A6 )
=> ( ! [B5: set_nat] :
( ( ord_less_set_nat @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_891_finite__psubset__induct,axiom,
! [A2: set_complex,P: set_complex > $o] :
( ( finite3207457112153483333omplex @ A2 )
=> ( ! [A6: set_complex] :
( ( finite3207457112153483333omplex @ A6 )
=> ( ! [B5: set_complex] :
( ( ord_less_set_complex @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_892_finite__psubset__induct,axiom,
! [A2: set_Sum_sum_a_int,P: set_Sum_sum_a_int > $o] :
( ( finite5547626034989006084_a_int @ A2 )
=> ( ! [A6: set_Sum_sum_a_int] :
( ( finite5547626034989006084_a_int @ A6 )
=> ( ! [B5: set_Sum_sum_a_int] :
( ( ord_le2148885262193138791_a_int @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_893_finite__psubset__induct,axiom,
! [A2: set_Sum_sum_a_k,P: set_Sum_sum_a_k > $o] :
( ( finite51705190296372934um_a_k @ A2 )
=> ( ! [A6: set_Sum_sum_a_k] :
( ( finite51705190296372934um_a_k @ A6 )
=> ( ! [B5: set_Sum_sum_a_k] :
( ( ord_le1054565356735369145um_a_k @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_894_finite__psubset__induct,axiom,
! [A2: set_Sum_sum_a_a,P: set_Sum_sum_a_a > $o] :
( ( finite51705147264084924um_a_a @ A2 )
=> ( ! [A6: set_Sum_sum_a_a] :
( ( finite51705147264084924um_a_a @ A6 )
=> ( ! [B5: set_Sum_sum_a_a] :
( ( ord_le344224956271909295um_a_a @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_895_finite__psubset__induct,axiom,
! [A2: set_k,P: set_k > $o] :
( ( finite_finite_k @ A2 )
=> ( ! [A6: set_k] :
( ( finite_finite_k @ A6 )
=> ( ! [B5: set_k] :
( ( ord_less_set_k @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_896_finite__psubset__induct,axiom,
! [A2: set_a,P: set_a > $o] :
( ( finite_finite_a @ A2 )
=> ( ! [A6: set_a] :
( ( finite_finite_a @ A6 )
=> ( ! [B5: set_a] :
( ( ord_less_set_a @ B5 @ A6 )
=> ( P @ B5 ) )
=> ( P @ A6 ) ) )
=> ( P @ A2 ) ) ) ).
% finite_psubset_induct
thf(fact_897_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_898_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_899_zero__neq__one,axiom,
zero_zero_complex != one_one_complex ).
% zero_neq_one
thf(fact_900_zero__neq__one,axiom,
zero_zero_real != one_one_real ).
% zero_neq_one
thf(fact_901_mult_Ocomm__neutral,axiom,
! [A5: complex] :
( ( times_times_complex @ A5 @ one_one_complex )
= A5 ) ).
% mult.comm_neutral
thf(fact_902_mult_Ocomm__neutral,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ A5 @ one_on2109788427901206336ring_a )
= A5 ) ).
% mult.comm_neutral
thf(fact_903_mult_Ocomm__neutral,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ A5 @ one_one_Kyber_qr_a )
= A5 ) ).
% mult.comm_neutral
thf(fact_904_mult_Ocomm__neutral,axiom,
! [A5: int] :
( ( times_times_int @ A5 @ one_one_int )
= A5 ) ).
% mult.comm_neutral
thf(fact_905_mult_Ocomm__neutral,axiom,
! [A5: nat] :
( ( times_times_nat @ A5 @ one_one_nat )
= A5 ) ).
% mult.comm_neutral
thf(fact_906_mult_Ocomm__neutral,axiom,
! [A5: real] :
( ( times_times_real @ A5 @ one_one_real )
= A5 ) ).
% mult.comm_neutral
thf(fact_907_comm__monoid__mult__class_Omult__1,axiom,
! [A5: complex] :
( ( times_times_complex @ one_one_complex @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_908_comm__monoid__mult__class_Omult__1,axiom,
! [A5: finite_mod_ring_a] :
( ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_909_comm__monoid__mult__class_Omult__1,axiom,
! [A5: kyber_qr_a] :
( ( times_2095635435063429214r_qr_a @ one_one_Kyber_qr_a @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_910_comm__monoid__mult__class_Omult__1,axiom,
! [A5: int] :
( ( times_times_int @ one_one_int @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_911_comm__monoid__mult__class_Omult__1,axiom,
! [A5: nat] :
( ( times_times_nat @ one_one_nat @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_912_comm__monoid__mult__class_Omult__1,axiom,
! [A5: real] :
( ( times_times_real @ one_one_real @ A5 )
= A5 ) ).
% comm_monoid_mult_class.mult_1
thf(fact_913_not__prime__1,axiom,
~ ( factor5938532291743052070omplex @ one_one_complex ) ).
% not_prime_1
thf(fact_914_not__prime__1,axiom,
~ ( factor1868233065478773540e_real @ one_one_real ) ).
% not_prime_1
thf(fact_915_not__prime__1,axiom,
~ ( factor1798656936486255268me_int @ one_one_int ) ).
% not_prime_1
thf(fact_916_not__prime__1,axiom,
~ ( factor1801147406995305544me_nat @ one_one_nat ) ).
% not_prime_1
thf(fact_917_infinite__nat__iff__unbounded,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_918_unbounded__k__infinite,axiom,
! [K: nat,S: set_nat] :
( ! [M4: nat] :
( ( ord_less_nat @ K @ M4 )
=> ? [N6: nat] :
( ( ord_less_nat @ M4 @ N6 )
& ( member_nat @ N6 @ S ) ) )
=> ~ ( finite_finite_nat @ S ) ) ).
% unbounded_k_infinite
thf(fact_919_pos__zmult__eq__1__iff,axiom,
! [M3: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M3 )
=> ( ( ( times_times_int @ M3 @ N2 )
= one_one_int )
= ( ( M3 = one_one_int )
& ( N2 = one_one_int ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_920_prime__gt__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_int @ one_one_int @ P2 ) ) ).
% prime_gt_1_int
thf(fact_921_lambda__one,axiom,
( ( ^ [X2: complex] : X2 )
= ( times_times_complex @ one_one_complex ) ) ).
% lambda_one
thf(fact_922_lambda__one,axiom,
( ( ^ [X2: finite_mod_ring_a] : X2 )
= ( times_5121417576591743744ring_a @ one_on2109788427901206336ring_a ) ) ).
% lambda_one
thf(fact_923_lambda__one,axiom,
( ( ^ [X2: kyber_qr_a] : X2 )
= ( times_2095635435063429214r_qr_a @ one_one_Kyber_qr_a ) ) ).
% lambda_one
thf(fact_924_lambda__one,axiom,
( ( ^ [X2: int] : X2 )
= ( times_times_int @ one_one_int ) ) ).
% lambda_one
thf(fact_925_lambda__one,axiom,
( ( ^ [X2: nat] : X2 )
= ( times_times_nat @ one_one_nat ) ) ).
% lambda_one
thf(fact_926_lambda__one,axiom,
( ( ^ [X2: real] : X2 )
= ( times_times_real @ one_one_real ) ) ).
% lambda_one
thf(fact_927_zero__less__one__class_Ozero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_less_one
thf(fact_928_zero__less__one__class_Ozero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_less_one
thf(fact_929_zero__less__one__class_Ozero__less__one,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_less_one
thf(fact_930_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_931_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_932_not__one__less__zero,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% not_one_less_zero
thf(fact_933_less__1__mult,axiom,
! [M3: int,N2: int] :
( ( ord_less_int @ one_one_int @ M3 )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M3 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_934_less__1__mult,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M3 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_935_less__1__mult,axiom,
! [M3: real,N2: real] :
( ( ord_less_real @ one_one_real @ M3 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M3 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_936_kyber__spec_On_H__gr__0,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% kyber_spec.n'_gr_0
thf(fact_937_kyber__spec_On_H__gr__0,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_nat @ zero_zero_nat @ N3 ) ) ).
% kyber_spec.n'_gr_0
thf(fact_938_finite__conv__nat__seg__image,axiom,
( finite_finite_int
= ( ^ [A: set_int] :
? [N4: nat,F3: nat > int] :
( A
= ( image_nat_int @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_939_finite__conv__nat__seg__image,axiom,
( finite_finite_nat
= ( ^ [A: set_nat] :
? [N4: nat,F3: nat > nat] :
( A
= ( image_nat_nat @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_940_finite__conv__nat__seg__image,axiom,
( finite3207457112153483333omplex
= ( ^ [A: set_complex] :
? [N4: nat,F3: nat > complex] :
( A
= ( image_nat_complex @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_941_finite__conv__nat__seg__image,axiom,
( finite5547626034989006084_a_int
= ( ^ [A: set_Sum_sum_a_int] :
? [N4: nat,F3: nat > sum_sum_a_int] :
( A
= ( image_3115417691219061956_a_int @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_942_finite__conv__nat__seg__image,axiom,
( finite51705190296372934um_a_k
= ( ^ [A: set_Sum_sum_a_k] :
? [N4: nat,F3: nat > sum_sum_a_k] :
( A
= ( image_95481375269537542um_a_k @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_943_finite__conv__nat__seg__image,axiom,
( finite51705147264084924um_a_a
= ( ^ [A: set_Sum_sum_a_a] :
? [N4: nat,F3: nat > sum_sum_a_a] :
( A
= ( image_95481332237249532um_a_a @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_944_finite__conv__nat__seg__image,axiom,
( finite_finite_k
= ( ^ [A: set_k] :
? [N4: nat,F3: nat > k] :
( A
= ( image_nat_k @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_945_finite__conv__nat__seg__image,axiom,
( finite_finite_a
= ( ^ [A: set_a] :
? [N4: nat,F3: nat > a] :
( A
= ( image_nat_a @ F3
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N4 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_946_nat__seg__image__imp__finite,axiom,
! [A2: set_int,F2: nat > int,N2: nat] :
( ( A2
= ( image_nat_int @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_int @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_947_nat__seg__image__imp__finite,axiom,
! [A2: set_nat,F2: nat > nat,N2: nat] :
( ( A2
= ( image_nat_nat @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_nat @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_948_nat__seg__image__imp__finite,axiom,
! [A2: set_complex,F2: nat > complex,N2: nat] :
( ( A2
= ( image_nat_complex @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite3207457112153483333omplex @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_949_nat__seg__image__imp__finite,axiom,
! [A2: set_Sum_sum_a_int,F2: nat > sum_sum_a_int,N2: nat] :
( ( A2
= ( image_3115417691219061956_a_int @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite5547626034989006084_a_int @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_950_nat__seg__image__imp__finite,axiom,
! [A2: set_Sum_sum_a_k,F2: nat > sum_sum_a_k,N2: nat] :
( ( A2
= ( image_95481375269537542um_a_k @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite51705190296372934um_a_k @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_951_nat__seg__image__imp__finite,axiom,
! [A2: set_Sum_sum_a_a,F2: nat > sum_sum_a_a,N2: nat] :
( ( A2
= ( image_95481332237249532um_a_a @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite51705147264084924um_a_a @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_952_nat__seg__image__imp__finite,axiom,
! [A2: set_k,F2: nat > k,N2: nat] :
( ( A2
= ( image_nat_k @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_k @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_953_nat__seg__image__imp__finite,axiom,
! [A2: set_a,F2: nat > a,N2: nat] :
( ( A2
= ( image_nat_a @ F2
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) ) )
=> ( finite_finite_a @ A2 ) ) ).
% nat_seg_image_imp_finite
thf(fact_954_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times_int @ K @ zero_zero_int )
= zero_zero_int ) ).
% times_int_code(1)
thf(fact_955_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times_int @ zero_zero_int @ L )
= zero_zero_int ) ).
% times_int_code(2)
thf(fact_956_kyber__spec_On__gt__1,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_a @ type_a @ type_a @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ one_one_int @ N2 ) ) ).
% kyber_spec.n_gt_1
thf(fact_957_kyber__spec_On__gt__1,axiom,
! [N2: int,Q2: int,K: nat,N3: nat] :
( ( kyber_kyber_spec_a_k @ type_a @ type_k @ N2 @ Q2 @ K @ N3 )
=> ( ord_less_int @ one_one_int @ N2 ) ) ).
% kyber_spec.n_gt_1
thf(fact_958_infinite__UNIV__int,axiom,
~ ( finite_finite_int @ top_top_set_int ) ).
% infinite_UNIV_int
thf(fact_959_primes__infinite,axiom,
~ ( finite_finite_nat @ ( collect_nat @ factor1801147406995305544me_nat ) ) ).
% primes_infinite
thf(fact_960_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_961_zmult__zless__mono2,axiom,
! [I2: int,J: int,K: int] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_int @ zero_zero_int @ K )
=> ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_962_prod_Ofinite__Collect__op,axiom,
! [I3: set_Product_unit,X: product_unit > complex,Y3: product_unit > complex] :
( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_963_prod_Ofinite__Collect__op,axiom,
! [I3: set_int,X: int > complex,Y3: int > complex] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_964_prod_Ofinite__Collect__op,axiom,
! [I3: set_nat,X: nat > complex,Y3: nat > complex] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_965_prod_Ofinite__Collect__op,axiom,
! [I3: set_complex,X: complex > complex,Y3: complex > complex] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_966_prod_Ofinite__Collect__op,axiom,
! [I3: set_k,X: k > complex,Y3: k > complex] :
( ( finite_finite_k
@ ( collect_k
@ ^ [I: k] :
( ( member_k @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_k
@ ( collect_k
@ ^ [I: k] :
( ( member_k @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite_finite_k
@ ( collect_k
@ ^ [I: k] :
( ( member_k @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_967_prod_Ofinite__Collect__op,axiom,
! [I3: set_a,X: a > complex,Y3: a > complex] :
( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( X @ I )
!= one_one_complex ) ) ) )
=> ( ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_complex ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [I: a] :
( ( member_a @ I @ I3 )
& ( ( times_times_complex @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_complex ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_968_prod_Ofinite__Collect__op,axiom,
! [I3: set_Product_unit,X: product_unit > int,Y3: product_unit > int] :
( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( X @ I )
!= one_one_int ) ) ) )
=> ( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_int ) ) ) )
=> ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [I: product_unit] :
( ( member_Product_unit @ I @ I3 )
& ( ( times_times_int @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_969_prod_Ofinite__Collect__op,axiom,
! [I3: set_int,X: int > int,Y3: int > int] :
( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( X @ I )
!= one_one_int ) ) ) )
=> ( ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_int ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( member_int @ I @ I3 )
& ( ( times_times_int @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_970_prod_Ofinite__Collect__op,axiom,
! [I3: set_nat,X: nat > int,Y3: nat > int] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( X @ I )
!= one_one_int ) ) ) )
=> ( ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_int ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [I: nat] :
( ( member_nat @ I @ I3 )
& ( ( times_times_int @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_971_prod_Ofinite__Collect__op,axiom,
! [I3: set_complex,X: complex > int,Y3: complex > int] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( X @ I )
!= one_one_int ) ) ) )
=> ( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( Y3 @ I )
!= one_one_int ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [I: complex] :
( ( member_complex @ I @ I3 )
& ( ( times_times_int @ ( X @ I ) @ ( Y3 @ I ) )
!= one_one_int ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_972_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_973_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_974_less__numeral__extra_I1_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% less_numeral_extra(1)
thf(fact_975_mult__less__iff1,axiom,
! [Z: int,X: int,Y3: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z ) @ ( times_times_int @ Y3 @ Z ) )
= ( ord_less_int @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_976_mult__less__iff1,axiom,
! [Z: real,X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y3 @ Z ) )
= ( ord_less_real @ X @ Y3 ) ) ) ).
% mult_less_iff1
thf(fact_977_fps__tan__0,axiom,
( ( formal6482914284900457064omplex @ zero_zero_complex )
= zero_z1877163951443063103omplex ) ).
% fps_tan_0
thf(fact_978_fps__tan__0,axiom,
( ( formal3683295897622742886n_real @ zero_zero_real )
= zero_z7760665558314615101s_real ) ).
% fps_tan_0
thf(fact_979_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_980_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_981_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_982_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_983_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_984_less__numeral__extra_I3_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% less_numeral_extra(3)
thf(fact_985_field__lbound__gt__zero,axiom,
! [D1: real,D22: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D22 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D22 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_986_Primes_Oprime__int__def,axiom,
prime_int = factor1798656936486255268me_int ).
% Primes.prime_int_def
thf(fact_987_of__qr__1,axiom,
( ( kyber_of_qr_a @ one_one_Kyber_qr_a )
= one_on3394844594818161742ring_a ) ).
% of_qr_1
thf(fact_988_prime__gt__0__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ zero_zero_nat @ P2 ) ) ).
% prime_gt_0_nat
thf(fact_989_psubsetD,axiom,
! [A2: set_nat,B2: set_nat,C: nat] :
( ( ord_less_set_nat @ A2 @ B2 )
=> ( ( member_nat @ C @ A2 )
=> ( member_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_990_less__set__def,axiom,
( ord_less_set_nat
= ( ^ [A: set_nat,B6: set_nat] :
( ord_less_nat_o
@ ^ [X2: nat] : ( member_nat @ X2 @ A )
@ ^ [X2: nat] : ( member_nat @ X2 @ B6 ) ) ) ) ).
% less_set_def
thf(fact_991_bigger__prime,axiom,
! [N2: nat] :
? [P6: nat] :
( ( factor1801147406995305544me_nat @ P6 )
& ( ord_less_nat @ N2 @ P6 ) ) ).
% bigger_prime
thf(fact_992_prime__gt__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_nat @ one_one_nat @ P2 ) ) ).
% prime_gt_1_nat
thf(fact_993_not__prime__eq__prod__nat,axiom,
! [M3: nat] :
( ( ord_less_nat @ one_one_nat @ M3 )
=> ( ~ ( factor1801147406995305544me_nat @ M3 )
=> ? [N5: nat,K2: nat] :
( ( N5
= ( times_times_nat @ M3 @ K2 ) )
& ( ord_less_nat @ one_one_nat @ M3 )
& ( ord_less_nat @ M3 @ N5 )
& ( ord_less_nat @ one_one_nat @ K2 )
& ( ord_less_nat @ K2 @ N5 ) ) ) ) ).
% not_prime_eq_prod_nat
thf(fact_994_prime__product,axiom,
! [P2: nat,Q2: nat] :
( ( factor1801147406995305544me_nat @ ( times_times_nat @ P2 @ Q2 ) )
=> ( ( P2 = one_one_nat )
| ( Q2 = one_one_nat ) ) ) ).
% prime_product
thf(fact_995_mult__less__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_996_nat__0__less__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M3 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M3 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_997_nat__mult__less__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M3 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_998_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_999_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1000_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1001_bot__nat__0_Onot__eq__extremum,axiom,
! [A5: nat] :
( ( A5 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A5 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1002_nat__mult__eq__1__iff,axiom,
! [M3: nat,N2: nat] :
( ( ( times_times_nat @ M3 @ N2 )
= one_one_nat )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1003_nat__1__eq__mult__iff,axiom,
! [M3: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M3 @ N2 ) )
= ( ( M3 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1004_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1005_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1006_mult__eq__self__implies__10,axiom,
! [M3: nat,N2: nat] :
( ( M3
= ( times_times_nat @ M3 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M3 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1007_nat__neq__iff,axiom,
! [M3: nat,N2: nat] :
( ( M3 != N2 )
= ( ( ord_less_nat @ M3 @ N2 )
| ( ord_less_nat @ N2 @ M3 ) ) ) ).
% nat_neq_iff
thf(fact_1008_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_1009_less__not__refl2,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ N2 @ M3 )
=> ( M3 != N2 ) ) ).
% less_not_refl2
thf(fact_1010_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less_nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_1011_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_1012_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ! [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
=> ( P @ M5 ) )
=> ( P @ N5 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_1013_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N5: nat] :
( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_1014_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_1015_bot__nat__0_Oextremum__strict,axiom,
! [A5: nat] :
~ ( ord_less_nat @ A5 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_1016_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_1017_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_1018_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_1019_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_1020_gr__implies__not0,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_1021_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
=> ( ~ ( P @ N5 )
=> ? [M5: nat] :
( ( ord_less_nat @ M5 @ N5 )
& ~ ( P @ M5 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_1022_nat__mult__less__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1023_nat__mult__eq__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M3 )
= ( times_times_nat @ K @ N2 ) )
= ( M3 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1024_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1025_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1026_filter__preserves__multiset,axiom,
! [M: product_unit > nat,P: product_unit > $o] :
( ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X2: product_unit] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite4290736615968046902t_unit
@ ( collect_Product_unit
@ ^ [X2: product_unit] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1027_filter__preserves__multiset,axiom,
! [M: int > nat,P: int > $o] :
( ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite_finite_int
@ ( collect_int
@ ^ [X2: int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1028_filter__preserves__multiset,axiom,
! [M: nat > nat,P: nat > $o] :
( ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [X2: nat] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1029_filter__preserves__multiset,axiom,
! [M: complex > nat,P: complex > $o] :
( ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [X2: complex] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1030_filter__preserves__multiset,axiom,
! [M: sum_sum_a_int > nat,P: sum_sum_a_int > $o] :
( ( finite5547626034989006084_a_int
@ ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite5547626034989006084_a_int
@ ( collec2895206842034026310_a_int
@ ^ [X2: sum_sum_a_int] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1031_filter__preserves__multiset,axiom,
! [M: sum_sum_a_k > nat,P: sum_sum_a_k > $o] :
( ( finite51705190296372934um_a_k
@ ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite51705190296372934um_a_k
@ ( collect_Sum_sum_a_k
@ ^ [X2: sum_sum_a_k] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1032_filter__preserves__multiset,axiom,
! [M: sum_sum_a_a > nat,P: sum_sum_a_a > $o] :
( ( finite51705147264084924um_a_a
@ ( collect_Sum_sum_a_a
@ ^ [X2: sum_sum_a_a] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite51705147264084924um_a_a
@ ( collect_Sum_sum_a_a
@ ^ [X2: sum_sum_a_a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1033_filter__preserves__multiset,axiom,
! [M: k > nat,P: k > $o] :
( ( finite_finite_k
@ ( collect_k
@ ^ [X2: k] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite_finite_k
@ ( collect_k
@ ^ [X2: k] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1034_filter__preserves__multiset,axiom,
! [M: a > nat,P: a > $o] :
( ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] : ( ord_less_nat @ zero_zero_nat @ ( M @ X2 ) ) ) )
=> ( finite_finite_a
@ ( collect_a
@ ^ [X2: a] : ( ord_less_nat @ zero_zero_nat @ ( if_nat @ ( P @ X2 ) @ ( M @ X2 ) @ zero_zero_nat ) ) ) ) ) ).
% filter_preserves_multiset
thf(fact_1035_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I2: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less_nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1036_finite__nat__set__iff__bounded,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_1037_bounded__nat__set__is__finite,axiom,
! [N: set_nat,N2: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ N )
=> ( ord_less_nat @ X3 @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% bounded_nat_set_is_finite
thf(fact_1038_coeff__poly__cutoff,axiom,
! [K: nat,N2: nat,P2: poly_F3299452240248304339ring_a] :
( ( ( ord_less_nat @ K @ N2 )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N2 @ P2 ) @ K )
= ( coeff_1607515655354303335ring_a @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N2 )
=> ( ( coeff_1607515655354303335ring_a @ ( poly_c8149583573515411563ring_a @ N2 @ P2 ) @ K )
= zero_z7902377541816115708ring_a ) ) ) ).
% coeff_poly_cutoff
thf(fact_1039_coeff__poly__cutoff,axiom,
! [K: nat,N2: nat,P2: poly_int] :
( ( ( ord_less_nat @ K @ N2 )
=> ( ( coeff_int @ ( poly_cutoff_int @ N2 @ P2 ) @ K )
= ( coeff_int @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N2 )
=> ( ( coeff_int @ ( poly_cutoff_int @ N2 @ P2 ) @ K )
= zero_zero_int ) ) ) ).
% coeff_poly_cutoff
thf(fact_1040_coeff__poly__cutoff,axiom,
! [K: nat,N2: nat,P2: poly_nat] :
( ( ( ord_less_nat @ K @ N2 )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N2 @ P2 ) @ K )
= ( coeff_nat @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N2 )
=> ( ( coeff_nat @ ( poly_cutoff_nat @ N2 @ P2 ) @ K )
= zero_zero_nat ) ) ) ).
% coeff_poly_cutoff
thf(fact_1041_coeff__poly__cutoff,axiom,
! [K: nat,N2: nat,P2: poly_complex] :
( ( ( ord_less_nat @ K @ N2 )
=> ( ( coeff_complex @ ( poly_cutoff_complex @ N2 @ P2 ) @ K )
= ( coeff_complex @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N2 )
=> ( ( coeff_complex @ ( poly_cutoff_complex @ N2 @ P2 ) @ K )
= zero_zero_complex ) ) ) ).
% coeff_poly_cutoff
thf(fact_1042_coeff__poly__cutoff,axiom,
! [K: nat,N2: nat,P2: poly_real] :
( ( ( ord_less_nat @ K @ N2 )
=> ( ( coeff_real @ ( poly_cutoff_real @ N2 @ P2 ) @ K )
= ( coeff_real @ P2 @ K ) ) )
& ( ~ ( ord_less_nat @ K @ N2 )
=> ( ( coeff_real @ ( poly_cutoff_real @ N2 @ P2 ) @ K )
= zero_zero_real ) ) ) ).
% coeff_poly_cutoff
thf(fact_1043_euclidean__size__field__def,axiom,
( field_5488087805207052648omplex
= ( ^ [X2: complex] : ( if_nat @ ( X2 = zero_zero_complex ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_1044_euclidean__size__field__def,axiom,
( field_5283244131969691238d_real
= ( ^ [X2: real] : ( if_nat @ ( X2 = zero_zero_real ) @ zero_zero_nat @ one_one_nat ) ) ) ).
% euclidean_size_field_def
thf(fact_1045_normalize__field__def,axiom,
( field_8342942429055804762omplex
= ( ^ [X2: complex] : ( if_complex @ ( X2 = zero_zero_complex ) @ zero_zero_complex @ one_one_complex ) ) ) ).
% normalize_field_def
thf(fact_1046_normalize__field__def,axiom,
( field_8354674766439439704d_real
= ( ^ [X2: real] : ( if_real @ ( X2 = zero_zero_real ) @ zero_zero_real @ one_one_real ) ) ) ).
% normalize_field_def
thf(fact_1047_mod__field__def,axiom,
( field_4028222145872571717omplex
= ( ^ [X2: complex,Y2: complex] : ( if_complex @ ( Y2 = zero_zero_complex ) @ X2 @ zero_zero_complex ) ) ) ).
% mod_field_def
thf(fact_1048_mod__field__def,axiom,
( field_341224784244110787d_real
= ( ^ [X2: real,Y2: real] : ( if_real @ ( Y2 = zero_zero_real ) @ X2 @ zero_zero_real ) ) ) ).
% mod_field_def
thf(fact_1049_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_1050_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
= one_one_complex ) ).
% dbl_inc_simps(2)
thf(fact_1051_dbl__inc__simps_I2_J,axiom,
( ( neg_nu8295874005876285629c_real @ zero_zero_real )
= one_one_real ) ).
% dbl_inc_simps(2)
thf(fact_1052_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ ( reflec4498816349307343611ring_a @ P2 ) @ zero_zero_nat )
= zero_z7902377541816115708ring_a )
= ( P2 = zero_z1830546546923837194ring_a ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_1053_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ ( reflect_poly_int @ P2 ) @ zero_zero_nat )
= zero_zero_int )
= ( P2 = zero_zero_poly_int ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_1054_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ ( reflect_poly_nat @ P2 ) @ zero_zero_nat )
= zero_zero_nat )
= ( P2 = zero_zero_poly_nat ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_1055_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_complex] :
( ( ( coeff_complex @ ( reflect_poly_complex @ P2 ) @ zero_zero_nat )
= zero_zero_complex )
= ( P2 = zero_z2709840015065127615omplex ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_1056_coeff__0__reflect__poly__0__iff,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ ( reflect_poly_real @ P2 ) @ zero_zero_nat )
= zero_zero_real )
= ( P2 = zero_zero_poly_real ) ) ).
% coeff_0_reflect_poly_0_iff
thf(fact_1057_deg__qr__pos,axiom,
ord_less_nat @ zero_zero_nat @ ( kyber_5808863167042391122g_qr_a @ type_a ) ).
% deg_qr_pos
thf(fact_1058_poly__cutoff__def,axiom,
( poly_c8149583573515411563ring_a
= ( ^ [N4: nat,P3: poly_F3299452240248304339ring_a] :
( abs_po1984167875446606498ring_a
@ ^ [K3: nat] : ( if_Finite_mod_ring_a @ ( ord_less_nat @ K3 @ N4 ) @ ( coeff_1607515655354303335ring_a @ P3 @ K3 ) @ zero_z7902377541816115708ring_a ) ) ) ) ).
% poly_cutoff_def
thf(fact_1059_poly__cutoff__def,axiom,
( poly_cutoff_int
= ( ^ [N4: nat,P3: poly_int] :
( abs_poly_int
@ ^ [K3: nat] : ( if_int @ ( ord_less_nat @ K3 @ N4 ) @ ( coeff_int @ P3 @ K3 ) @ zero_zero_int ) ) ) ) ).
% poly_cutoff_def
thf(fact_1060_poly__cutoff__def,axiom,
( poly_cutoff_nat
= ( ^ [N4: nat,P3: poly_nat] :
( abs_poly_nat
@ ^ [K3: nat] : ( if_nat @ ( ord_less_nat @ K3 @ N4 ) @ ( coeff_nat @ P3 @ K3 ) @ zero_zero_nat ) ) ) ) ).
% poly_cutoff_def
thf(fact_1061_poly__cutoff__def,axiom,
( poly_cutoff_complex
= ( ^ [N4: nat,P3: poly_complex] :
( abs_poly_complex
@ ^ [K3: nat] : ( if_complex @ ( ord_less_nat @ K3 @ N4 ) @ ( coeff_complex @ P3 @ K3 ) @ zero_zero_complex ) ) ) ) ).
% poly_cutoff_def
thf(fact_1062_poly__cutoff__def,axiom,
( poly_cutoff_real
= ( ^ [N4: nat,P3: poly_real] :
( abs_poly_real
@ ^ [K3: nat] : ( if_real @ ( ord_less_nat @ K3 @ N4 ) @ ( coeff_real @ P3 @ K3 ) @ zero_zero_real ) ) ) ) ).
% poly_cutoff_def
thf(fact_1063_reflect__poly__reflect__poly,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( reflec4498816349307343611ring_a @ ( reflec4498816349307343611ring_a @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_1064_reflect__poly__reflect__poly,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ zero_zero_nat )
!= zero_zero_int )
=> ( ( reflect_poly_int @ ( reflect_poly_int @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_1065_reflect__poly__reflect__poly,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( reflect_poly_nat @ ( reflect_poly_nat @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_1066_reflect__poly__reflect__poly,axiom,
! [P2: poly_complex] :
( ( ( coeff_complex @ P2 @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( reflect_poly_complex @ ( reflect_poly_complex @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_1067_reflect__poly__reflect__poly,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( reflect_poly_real @ ( reflect_poly_real @ P2 ) )
= P2 ) ) ).
% reflect_poly_reflect_poly
thf(fact_1068_coeff__inverse,axiom,
! [X: poly_F3299452240248304339ring_a] :
( ( abs_po1984167875446606498ring_a @ ( coeff_1607515655354303335ring_a @ X ) )
= X ) ).
% coeff_inverse
thf(fact_1069_zero__poly__def,axiom,
( zero_zero_poly_int
= ( abs_poly_int
@ ^ [Uu: nat] : zero_zero_int ) ) ).
% zero_poly_def
thf(fact_1070_zero__poly__def,axiom,
( zero_zero_poly_nat
= ( abs_poly_nat
@ ^ [Uu: nat] : zero_zero_nat ) ) ).
% zero_poly_def
thf(fact_1071_zero__poly__def,axiom,
( zero_z2709840015065127615omplex
= ( abs_poly_complex
@ ^ [Uu: nat] : zero_zero_complex ) ) ).
% zero_poly_def
thf(fact_1072_zero__poly__def,axiom,
( zero_zero_poly_real
= ( abs_poly_real
@ ^ [Uu: nat] : zero_zero_real ) ) ).
% zero_poly_def
thf(fact_1073_deg__qr__n,axiom,
( ( semiri1314217659103216013at_int @ ( kyber_5808863167042391122g_qr_a @ type_a ) )
= n ) ).
% deg_qr_n
thf(fact_1074_Primes_Oprime__nat__def,axiom,
prime_nat = factor1801147406995305544me_nat ).
% Primes.prime_nat_def
thf(fact_1075_radical__0,axiom,
! [N2: nat,R2: nat > complex > complex,A5: formal670952693614245302omplex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( formal3405225654480437512omplex @ R2 @ zero_zero_nat @ A5 @ N2 )
= zero_zero_complex ) ) ).
% radical_0
thf(fact_1076_radical__0,axiom,
! [N2: nat,R2: nat > real > real,A5: formal3361831859752904756s_real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( formal8005797870169972230l_real @ R2 @ zero_zero_nat @ A5 @ N2 )
= zero_zero_real ) ) ).
% radical_0
thf(fact_1077_degree__reflect__poly__eq,axiom,
! [P2: poly_F3299452240248304339ring_a] :
( ( ( coeff_1607515655354303335ring_a @ P2 @ zero_zero_nat )
!= zero_z7902377541816115708ring_a )
=> ( ( degree4881254707062955960ring_a @ ( reflec4498816349307343611ring_a @ P2 ) )
= ( degree4881254707062955960ring_a @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_1078_degree__reflect__poly__eq,axiom,
! [P2: poly_int] :
( ( ( coeff_int @ P2 @ zero_zero_nat )
!= zero_zero_int )
=> ( ( degree_int @ ( reflect_poly_int @ P2 ) )
= ( degree_int @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_1079_degree__reflect__poly__eq,axiom,
! [P2: poly_nat] :
( ( ( coeff_nat @ P2 @ zero_zero_nat )
!= zero_zero_nat )
=> ( ( degree_nat @ ( reflect_poly_nat @ P2 ) )
= ( degree_nat @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_1080_degree__reflect__poly__eq,axiom,
! [P2: poly_complex] :
( ( ( coeff_complex @ P2 @ zero_zero_nat )
!= zero_zero_complex )
=> ( ( degree_complex @ ( reflect_poly_complex @ P2 ) )
= ( degree_complex @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_1081_degree__reflect__poly__eq,axiom,
! [P2: poly_real] :
( ( ( coeff_real @ P2 @ zero_zero_nat )
!= zero_zero_real )
=> ( ( degree_real @ ( reflect_poly_real @ P2 ) )
= ( degree_real @ P2 ) ) ) ).
% degree_reflect_poly_eq
thf(fact_1082_of__nat__0,axiom,
( ( semiri1316708129612266289at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_1083_of__nat__0,axiom,
( ( semiri8010041392384452111omplex @ zero_zero_nat )
= zero_zero_complex ) ).
% of_nat_0
thf(fact_1084_of__nat__0,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_1085_of__nat__0,axiom,
( ( semiri5074537144036343181t_real @ zero_zero_nat )
= zero_zero_real ) ).
% of_nat_0
thf(fact_1086_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1087_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1088_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1089_of__nat__0__eq__iff,axiom,
! [N2: nat] :
( ( zero_zero_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( zero_zero_nat = N2 ) ) ).
% of_nat_0_eq_iff
thf(fact_1090_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1316708129612266289at_nat @ M3 )
= zero_zero_nat )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1091_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri8010041392384452111omplex @ M3 )
= zero_zero_complex )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1092_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= zero_zero_int )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1093_of__nat__eq__0__iff,axiom,
! [M3: nat] :
( ( ( semiri5074537144036343181t_real @ M3 )
= zero_zero_real )
= ( M3 = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1094_of__nat__less__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M3 ) @ ( semiri1316708129612266289at_nat @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1095_of__nat__less__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1096_of__nat__less__iff,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_real @ ( semiri5074537144036343181t_real @ M3 ) @ ( semiri5074537144036343181t_real @ N2 ) )
= ( ord_less_nat @ M3 @ N2 ) ) ).
% of_nat_less_iff
thf(fact_1097_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1316708129612266289at_nat @ N2 )
= one_one_nat )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_1098_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri8010041392384452111omplex @ N2 )
= one_one_complex )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_1099_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri1314217659103216013at_int @ N2 )
= one_one_int )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_1100_semiring__char__0__class_Oof__nat__eq__1__iff,axiom,
! [N2: nat] :
( ( ( semiri5074537144036343181t_real @ N2 )
= one_one_real )
= ( N2 = one_one_nat ) ) ).
% semiring_char_0_class.of_nat_eq_1_iff
thf(fact_1101_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_nat
= ( semiri1316708129612266289at_nat @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1102_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_complex
= ( semiri8010041392384452111omplex @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1103_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_int
= ( semiri1314217659103216013at_int @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1104_of__nat__1__eq__iff,axiom,
! [N2: nat] :
( ( one_one_real
= ( semiri5074537144036343181t_real @ N2 ) )
= ( N2 = one_one_nat ) ) ).
% of_nat_1_eq_iff
thf(fact_1105_of__nat__1,axiom,
( ( semiri8010041392384452111omplex @ one_one_nat )
= one_one_complex ) ).
% of_nat_1
thf(fact_1106_of__nat__1,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% of_nat_1
thf(fact_1107_of__nat__1,axiom,
( ( semiri5074537144036343181t_real @ one_one_nat )
= one_one_real ) ).
% of_nat_1
thf(fact_1108_prime__nat__int__transfer,axiom,
! [N2: nat] :
( ( factor1798656936486255268me_int @ ( semiri1314217659103216013at_int @ N2 ) )
= ( factor1801147406995305544me_nat @ N2 ) ) ).
% prime_nat_int_transfer
thf(fact_1109_int__int__eq,axiom,
! [M3: nat,N2: nat] :
( ( ( semiri1314217659103216013at_int @ M3 )
= ( semiri1314217659103216013at_int @ N2 ) )
= ( M3 = N2 ) ) ).
% int_int_eq
thf(fact_1110_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( ( ord_less_nat @ zero_zero_nat @ N5 )
& ( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1111_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N5 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N5 ) ) ) ).
% pos_int_cases
thf(fact_1112_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1113_nat__less__as__int,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_1114_CARD__k,axiom,
( ( semiri1314217659103216013at_int @ ( finite_card_k @ top_top_set_k ) )
= ( semiri1314217659103216013at_int @ k2 ) ) ).
% CARD_k
thf(fact_1115_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1116_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B4: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1117_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1118_CARD__a,axiom,
( ( semiri1314217659103216013at_int @ ( finite_card_a @ top_top_set_a ) )
= q ) ).
% CARD_a
thf(fact_1119_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1120_card__Collect__less__nat,axiom,
! [N2: nat] :
( ( finite_card_nat
@ ( collect_nat
@ ^ [I: nat] : ( ord_less_nat @ I @ N2 ) ) )
= N2 ) ).
% card_Collect_less_nat
thf(fact_1121_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1122_nat__n,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ n ) )
= n ) ).
% nat_n
thf(fact_1123_nat__q,axiom,
( ( semiri1314217659103216013at_int @ ( nat2 @ q ) )
= q ) ).
% nat_q
thf(fact_1124_nat__dvd__1__iff__1,axiom,
! [M3: nat] :
( ( dvd_dvd_nat @ M3 @ one_one_nat )
= ( M3 = one_one_nat ) ) ).
% nat_dvd_1_iff_1
thf(fact_1125_nat__int,axiom,
! [N2: nat] :
( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
= N2 ) ).
% nat_int
thf(fact_1126_prime__nat__iff__prime,axiom,
! [K: int] :
( ( factor1801147406995305544me_nat @ ( nat2 @ K ) )
= ( factor1798656936486255268me_int @ K ) ) ).
% prime_nat_iff_prime
thf(fact_1127_zless__nat__conj,axiom,
! [W: int,Z: int] :
( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ( ord_less_int @ zero_zero_int @ Z )
& ( ord_less_int @ W @ Z ) ) ) ).
% zless_nat_conj
thf(fact_1128_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% zero_less_nat_eq
thf(fact_1129_nat__zero__as__int,axiom,
( zero_zero_nat
= ( nat2 @ zero_zero_int ) ) ).
% nat_zero_as_int
thf(fact_1130_nat__one__as__int,axiom,
( one_one_nat
= ( nat2 @ one_one_int ) ) ).
% nat_one_as_int
thf(fact_1131_nat__dvd__not__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( ord_less_nat @ M3 @ N2 )
=> ~ ( dvd_dvd_nat @ N2 @ M3 ) ) ) ).
% nat_dvd_not_less
thf(fact_1132_prime__factor__nat,axiom,
! [N2: nat] :
( ( N2 != one_one_nat )
=> ? [P6: nat] :
( ( factor1801147406995305544me_nat @ P6 )
& ( dvd_dvd_nat @ P6 @ N2 ) ) ) ).
% prime_factor_nat
thf(fact_1133_prime__dvd__mult__eq__nat,axiom,
! [P2: nat,A5: nat,B: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ P2 @ ( times_times_nat @ A5 @ B ) )
= ( ( dvd_dvd_nat @ P2 @ A5 )
| ( dvd_dvd_nat @ P2 @ B ) ) ) ) ).
% prime_dvd_mult_eq_nat
thf(fact_1134_nat__mono__iff,axiom,
! [Z: int,W: int] :
( ( ord_less_int @ zero_zero_int @ Z )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_mono_iff
thf(fact_1135_zless__nat__eq__int__zless,axiom,
! [M3: nat,Z: int] :
( ( ord_less_nat @ M3 @ ( nat2 @ Z ) )
= ( ord_less_int @ ( semiri1314217659103216013at_int @ M3 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_1136_dvd__mult__cancel,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( dvd_dvd_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( dvd_dvd_nat @ M3 @ N2 ) ) ) ).
% dvd_mult_cancel
thf(fact_1137_nat__mult__dvd__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( dvd_dvd_nat @ M3 @ N2 ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1138_prime__nat__iff,axiom,
( factor1801147406995305544me_nat
= ( ^ [N4: nat] :
( ( ord_less_nat @ one_one_nat @ N4 )
& ! [M2: nat] :
( ( dvd_dvd_nat @ M2 @ N4 )
=> ( ( M2 = one_one_nat )
| ( M2 = N4 ) ) ) ) ) ) ).
% prime_nat_iff
thf(fact_1139_prime__nat__not__dvd,axiom,
! [P2: nat,N2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ N2 @ P2 )
=> ( ( N2 != one_one_nat )
=> ~ ( dvd_dvd_nat @ N2 @ P2 ) ) ) ) ).
% prime_nat_not_dvd
thf(fact_1140_nat__times__as__int,axiom,
( times_times_nat
= ( ^ [A3: nat,B4: nat] : ( nat2 @ ( times_times_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_1141_split__nat,axiom,
! [P: nat > $o,I2: int] :
( ( P @ ( nat2 @ I2 ) )
= ( ! [N4: nat] :
( ( I2
= ( semiri1314217659103216013at_int @ N4 ) )
=> ( P @ N4 ) )
& ( ( ord_less_int @ I2 @ zero_zero_int )
=> ( P @ zero_zero_nat ) ) ) ) ).
% split_nat
thf(fact_1142_dvd__mult__cancel1,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ M3 @ N2 ) @ M3 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel1
thf(fact_1143_dvd__mult__cancel2,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M3 ) @ M3 )
= ( N2 = one_one_nat ) ) ) ).
% dvd_mult_cancel2
thf(fact_1144_int__dvd__int__iff,axiom,
! [M3: nat,N2: nat] :
( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( dvd_dvd_nat @ M3 @ N2 ) ) ).
% int_dvd_int_iff
thf(fact_1145_zdvd__not__zless,axiom,
! [M3: int,N2: int] :
( ( ord_less_int @ zero_zero_int @ M3 )
=> ( ( ord_less_int @ M3 @ N2 )
=> ~ ( dvd_dvd_int @ N2 @ M3 ) ) ) ).
% zdvd_not_zless
thf(fact_1146_zdvd__mult__cancel,axiom,
! [K: int,M3: int,N2: int] :
( ( dvd_dvd_int @ ( times_times_int @ K @ M3 ) @ ( times_times_int @ K @ N2 ) )
=> ( ( K != zero_zero_int )
=> ( dvd_dvd_int @ M3 @ N2 ) ) ) ).
% zdvd_mult_cancel
thf(fact_1147_prime__dvd__mult__eq__int,axiom,
! [P2: int,A5: int,B: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( dvd_dvd_int @ P2 @ ( times_times_int @ A5 @ B ) )
= ( ( dvd_dvd_int @ P2 @ A5 )
| ( dvd_dvd_int @ P2 @ B ) ) ) ) ).
% prime_dvd_mult_eq_int
thf(fact_1148_prime__int__not__dvd,axiom,
! [P2: int,N2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ord_less_int @ N2 @ P2 )
=> ( ( ord_less_int @ one_one_int @ N2 )
=> ~ ( dvd_dvd_int @ N2 @ P2 ) ) ) ) ).
% prime_int_not_dvd
thf(fact_1149_finite__divisors__nat,axiom,
! [M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ M3 ) ) ) ) ).
% finite_divisors_nat
thf(fact_1150_finite__divisors__int,axiom,
! [I2: int] :
( ( I2 != zero_zero_int )
=> ( finite_finite_int
@ ( collect_int
@ ^ [D3: int] : ( dvd_dvd_int @ D3 @ I2 ) ) ) ) ).
% finite_divisors_int
thf(fact_1151_dvd__pos__nat,axiom,
! [N2: nat,M3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ M3 @ N2 )
=> ( ord_less_nat @ zero_zero_nat @ M3 ) ) ) ).
% dvd_pos_nat
thf(fact_1152_zdvd__mono,axiom,
! [K: int,M3: int,T: int] :
( ( K != zero_zero_int )
=> ( ( dvd_dvd_int @ M3 @ T )
= ( dvd_dvd_int @ ( times_times_int @ K @ M3 ) @ ( times_times_int @ K @ T ) ) ) ) ).
% zdvd_mono
thf(fact_1153_prime__power__exp__nat,axiom,
! [P2: nat,N2: nat,X: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( N2 != zero_zero_nat )
=> ( ( ( power_power_nat @ X @ N2 )
= ( power_power_nat @ P2 @ K ) )
=> ? [I4: nat] :
( X
= ( power_power_nat @ P2 @ I4 ) ) ) ) ) ).
% prime_power_exp_nat
thf(fact_1154_prime__power__mult__nat,axiom,
! [P2: nat,X: nat,Y3: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ( times_times_nat @ X @ Y3 )
= ( power_power_nat @ P2 @ K ) )
=> ? [I4: nat,J2: nat] :
( ( X
= ( power_power_nat @ P2 @ I4 ) )
& ( Y3
= ( power_power_nat @ P2 @ J2 ) ) ) ) ) ).
% prime_power_mult_nat
thf(fact_1155_prime__dvd__power__nat,axiom,
! [P2: nat,X: nat,N2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ P2 @ ( power_power_nat @ X @ N2 ) )
=> ( dvd_dvd_nat @ P2 @ X ) ) ) ).
% prime_dvd_power_nat
thf(fact_1156_prime__dvd__power__int,axiom,
! [P2: int,X: int,N2: nat] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( dvd_dvd_int @ P2 @ ( power_power_int @ X @ N2 ) )
=> ( dvd_dvd_int @ P2 @ X ) ) ) ).
% prime_dvd_power_int
thf(fact_1157_prime__dvd__power__nat__iff,axiom,
! [P2: nat,N2: nat,X: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_nat @ P2 @ ( power_power_nat @ X @ N2 ) )
= ( dvd_dvd_nat @ P2 @ X ) ) ) ) ).
% prime_dvd_power_nat_iff
thf(fact_1158_prime__dvd__power__int__iff,axiom,
! [P2: int,N2: nat,X: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( dvd_dvd_int @ P2 @ ( power_power_int @ X @ N2 ) )
= ( dvd_dvd_int @ P2 @ X ) ) ) ) ).
% prime_dvd_power_int_iff
thf(fact_1159_prime__power__canonical,axiom,
! [P2: nat,M3: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ? [K2: nat,N5: nat] :
( ~ ( dvd_dvd_nat @ P2 @ N5 )
& ( M3
= ( times_times_nat @ N5 @ ( power_power_nat @ P2 @ K2 ) ) ) ) ) ) ).
% prime_power_canonical
thf(fact_1160_nat__zero__less__power__iff,axiom,
! [X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N2 = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1161_nat__power__less__imp__less,axiom,
! [I2: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M3 ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% nat_power_less_imp_less
thf(fact_1162_finite__nth__roots,axiom,
! [N2: nat,C: complex] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( finite3207457112153483333omplex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N2 )
= C ) ) ) ) ).
% finite_nth_roots
thf(fact_1163_card__roots__unity__eq,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N2 )
= one_one_complex ) ) )
= N2 ) ) ).
% card_roots_unity_eq
thf(fact_1164_card__nth__roots,axiom,
! [C: complex,N2: nat] :
( ( C != zero_zero_complex )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( finite_card_complex
@ ( collect_complex
@ ^ [Z4: complex] :
( ( power_power_complex @ Z4 @ N2 )
= C ) ) )
= N2 ) ) ) ).
% card_nth_roots
thf(fact_1165_realpow__pos__nth__unique,axiom,
! [N2: nat,A5: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A5 )
=> ? [X3: real] :
( ( ord_less_real @ zero_zero_real @ X3 )
& ( ( power_power_real @ X3 @ N2 )
= A5 )
& ! [Y6: real] :
( ( ( ord_less_real @ zero_zero_real @ Y6 )
& ( ( power_power_real @ Y6 @ N2 )
= A5 ) )
=> ( Y6 = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1166_realpow__pos__nth,axiom,
! [N2: nat,A5: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ A5 )
=> ? [R3: real] :
( ( ord_less_real @ zero_zero_real @ R3 )
& ( ( power_power_real @ R3 @ N2 )
= A5 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1167_nat__dvd__iff,axiom,
! [Z: int,M3: nat] :
( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_dvd_iff
thf(fact_1168_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X @ X ) ) )
= ( X = zero_zero_real ) ) ).
% not_real_square_gt_zero
thf(fact_1169_finite__interval__int1,axiom,
! [A5: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A5 @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int1
thf(fact_1170_finite__interval__int2,axiom,
! [A5: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_eq_int @ A5 @ I )
& ( ord_less_int @ I @ B ) ) ) ) ).
% finite_interval_int2
thf(fact_1171_finite__interval__int3,axiom,
! [A5: int,B: int] :
( finite_finite_int
@ ( collect_int
@ ^ [I: int] :
( ( ord_less_int @ A5 @ I )
& ( ord_less_eq_int @ I @ B ) ) ) ) ).
% finite_interval_int3
thf(fact_1172_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq_int @ Z @ zero_zero_int )
=> ( ( nat2 @ Z )
= zero_zero_nat ) ) ).
% nat_le_0
thf(fact_1173_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= zero_zero_nat )
= ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).
% nat_0_iff
thf(fact_1174_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= zero_zero_int ) ) ) ).
% int_nat_eq
thf(fact_1175_real__arch__pow__inv,axiom,
! [Y3: real,X: real] :
( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ X @ one_one_real )
=> ? [N5: nat] : ( ord_less_real @ ( power_power_real @ X @ N5 ) @ Y3 ) ) ) ).
% real_arch_pow_inv
thf(fact_1176_real__arch__pow,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ one_one_real @ X )
=> ? [N5: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X @ N5 ) ) ) ).
% real_arch_pow
thf(fact_1177_imp__le__cong,axiom,
! [X: int,X7: int,P: $o,P7: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P7 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
=> P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> P7 ) ) ) ) ).
% imp_le_cong
thf(fact_1178_conj__le__cong,axiom,
! [X: int,X7: int,P: $o,P7: $o] :
( ( X = X7 )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
=> ( P = P7 ) )
=> ( ( ( ord_less_eq_int @ zero_zero_int @ X )
& P )
= ( ( ord_less_eq_int @ zero_zero_int @ X7 )
& P7 ) ) ) ) ).
% conj_le_cong
thf(fact_1179_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ! [Y6: real] :
? [N5: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N5 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1180_less__eq__int__code_I1_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% less_eq_int_code(1)
thf(fact_1181_real__sup__exists,axiom,
! [P: real > $o] :
( ? [X_12: real] : ( P @ X_12 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( P @ X3 )
=> ( ord_less_real @ X3 @ Z5 ) )
=> ? [S3: real] :
! [Y6: real] :
( ( ? [X2: real] :
( ( P @ X2 )
& ( ord_less_real @ Y6 @ X2 ) ) )
= ( ord_less_real @ Y6 @ S3 ) ) ) ) ).
% real_sup_exists
thf(fact_1182_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ~ ! [N5: nat] :
( K
!= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% nonneg_int_cases
thf(fact_1183_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ? [N5: nat] :
( K
= ( semiri1314217659103216013at_int @ N5 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_1184_ex__nat,axiom,
( ( ^ [P4: nat > $o] :
? [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
? [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
& ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_1185_all__nat,axiom,
( ( ^ [P4: nat > $o] :
! [X6: nat] : ( P4 @ X6 ) )
= ( ^ [P5: nat > $o] :
! [X2: int] :
( ( ord_less_eq_int @ zero_zero_int @ X2 )
=> ( P5 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_1186_eq__nat__nat__iff,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z6 ) )
= ( Z = Z6 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_1187_zdvd__antisym__nonneg,axiom,
! [M3: int,N2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M3 )
=> ( ( ord_less_eq_int @ zero_zero_int @ N2 )
=> ( ( dvd_dvd_int @ M3 @ N2 )
=> ( ( dvd_dvd_int @ N2 @ M3 )
=> ( M3 = N2 ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_1188_prime__ge__0__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ zero_zero_int @ P2 ) ) ).
% prime_ge_0_int
thf(fact_1189_prime__ge__1__int,axiom,
! [P2: int] :
( ( factor1798656936486255268me_int @ P2 )
=> ( ord_less_eq_int @ one_one_int @ P2 ) ) ).
% prime_ge_1_int
thf(fact_1190_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq_int @ one_one_int @ Z )
= ( ord_less_int @ zero_zero_int @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_1191_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_1192_int__eq__iff,axiom,
! [M3: nat,Z: int] :
( ( ( semiri1314217659103216013at_int @ M3 )
= Z )
= ( ( M3
= ( nat2 @ Z ) )
& ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).
% int_eq_iff
thf(fact_1193_zdvd__imp__le,axiom,
! [Z: int,N2: int] :
( ( dvd_dvd_int @ Z @ N2 )
=> ( ( ord_less_int @ zero_zero_int @ N2 )
=> ( ord_less_eq_int @ Z @ N2 ) ) ) ).
% zdvd_imp_le
thf(fact_1194_nat__eq__iff,axiom,
! [W: int,M3: nat] :
( ( ( nat2 @ W )
= M3 )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff
thf(fact_1195_nat__eq__iff2,axiom,
! [M3: nat,W: int] :
( ( M3
= ( nat2 @ W ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( W
= ( semiri1314217659103216013at_int @ M3 ) ) )
& ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
=> ( M3 = zero_zero_nat ) ) ) ) ).
% nat_eq_iff2
thf(fact_1196_nat__less__eq__zless,axiom,
! [W: int,Z: int] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_int @ W @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_1197_nat__mult__distrib,axiom,
! [Z: int,Z6: int] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
= ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1198_nat__power__eq,axiom,
! [Z: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ Z )
=> ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
= ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).
% nat_power_eq
thf(fact_1199_prime__int__nat__transfer,axiom,
( factor1798656936486255268me_int
= ( ^ [K3: int] :
( ( ord_less_eq_int @ zero_zero_int @ K3 )
& ( factor1801147406995305544me_nat @ ( nat2 @ K3 ) ) ) ) ) ).
% prime_int_nat_transfer
thf(fact_1200_nat__less__iff,axiom,
! [W: int,M3: nat] :
( ( ord_less_eq_int @ zero_zero_int @ W )
=> ( ( ord_less_nat @ ( nat2 @ W ) @ M3 )
= ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M3 ) ) ) ) ).
% nat_less_iff
thf(fact_1201_prime__int__altdef,axiom,
( factor1798656936486255268me_int
= ( ^ [P3: int] :
( ( ord_less_int @ one_one_int @ P3 )
& ! [M2: int] :
( ( ord_less_eq_int @ zero_zero_int @ M2 )
=> ( ( dvd_dvd_int @ M2 @ P3 )
=> ( ( M2 = one_one_int )
| ( M2 = P3 ) ) ) ) ) ) ) ).
% prime_int_altdef
thf(fact_1202_Primes_Oprime__int__iff,axiom,
( factor1798656936486255268me_int
= ( ^ [N4: int] :
( ( ord_less_int @ one_one_int @ N4 )
& ! [M2: int] :
( ( ( ord_less_eq_int @ zero_zero_int @ M2 )
& ( dvd_dvd_int @ M2 @ N4 ) )
=> ( ( M2 = one_one_int )
| ( M2 = N4 ) ) ) ) ) ) ).
% Primes.prime_int_iff
thf(fact_1203_Max__divisors__self__nat,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [D3: nat] : ( dvd_dvd_nat @ D3 @ N2 ) ) )
= N2 ) ) ).
% Max_divisors_self_nat
thf(fact_1204_finite__Collect__le__nat,axiom,
! [K: nat] :
( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ N4 @ K ) ) ) ).
% finite_Collect_le_nat
thf(fact_1205_nat__mult__le__cancel__disj,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1206_mult__le__cancel2,axiom,
! [M3: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M3 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1207_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N7: set_nat] :
? [M2: nat] :
! [X2: nat] :
( ( member_nat @ X2 @ N7 )
=> ( ord_less_eq_nat @ X2 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_1208_infinite__nat__iff__unbounded__le,axiom,
! [S: set_nat] :
( ( ~ ( finite_finite_nat @ S ) )
= ( ! [M2: nat] :
? [N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( member_nat @ N4 @ S ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_1209_finite__less__ub,axiom,
! [F2: nat > nat,U: nat] :
( ! [N5: nat] : ( ord_less_eq_nat @ N5 @ ( F2 @ N5 ) )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [N4: nat] : ( ord_less_eq_nat @ ( F2 @ N4 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1210_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I2: nat,J: nat] :
( ! [I4: nat,J2: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ord_less_nat @ ( F2 @ I4 ) @ ( F2 @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F2 @ I2 ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_1211_le__neq__implies__less,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_nat @ M3 @ N2 )
=> ( ( M3 != N2 )
=> ( ord_less_nat @ M3 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_1212_less__or__eq__imp__le,axiom,
! [M3: nat,N2: nat] :
( ( ( ord_less_nat @ M3 @ N2 )
| ( M3 = N2 ) )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_1213_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_1214_less__imp__le__nat,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_nat @ M3 @ N2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_1215_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_1216_less__eq__real__def,axiom,
( ord_less_eq_real
= ( ^ [X2: real,Y2: real] :
( ( ord_less_real @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% less_eq_real_def
thf(fact_1217_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K2 )
=> ~ ( P @ I5 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1218_zle__int,axiom,
! [M3: nat,N2: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ).
% zle_int
thf(fact_1219_nat__mono,axiom,
! [X: int,Y3: int] :
( ( ord_less_eq_int @ X @ Y3 )
=> ( ord_less_eq_nat @ ( nat2 @ X ) @ ( nat2 @ Y3 ) ) ) ).
% nat_mono
thf(fact_1220_prime__ge__1__nat,axiom,
! [P2: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ord_less_eq_nat @ one_one_nat @ P2 ) ) ).
% prime_ge_1_nat
thf(fact_1221_nat__leq__as__int,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B4: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_1222_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ! [M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ M4 )
=> ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M4 ) @ X ) @ C ) )
=> ( X = zero_zero_real ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1223_nat__mult__le__cancel1,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M3 ) @ ( times_times_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1224_dvd__imp__le,axiom,
! [K: nat,N2: nat] :
( ( dvd_dvd_nat @ K @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% dvd_imp_le
thf(fact_1225_nat__le__iff,axiom,
! [X: int,N2: nat] :
( ( ord_less_eq_nat @ ( nat2 @ X ) @ N2 )
= ( ord_less_eq_int @ X @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).
% nat_le_iff
thf(fact_1226_divides__primepow__nat,axiom,
! [P2: nat,D: nat,K: nat] :
( ( factor1801147406995305544me_nat @ P2 )
=> ( ( dvd_dvd_nat @ D @ ( power_power_nat @ P2 @ K ) )
= ( ? [I: nat] :
( ( ord_less_eq_nat @ I @ K )
& ( D
= ( power_power_nat @ P2 @ I ) ) ) ) ) ) ).
% divides_primepow_nat
thf(fact_1227_power__dvd__imp__le,axiom,
! [I2: nat,M3: nat,N2: nat] :
( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M3 ) @ ( power_power_nat @ I2 @ N2 ) )
=> ( ( ord_less_nat @ one_one_nat @ I2 )
=> ( ord_less_eq_nat @ M3 @ N2 ) ) ) ).
% power_dvd_imp_le
thf(fact_1228_le__nat__iff,axiom,
! [K: int,N2: nat] :
( ( ord_less_eq_int @ zero_zero_int @ K )
=> ( ( ord_less_eq_nat @ N2 @ ( nat2 @ K ) )
= ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).
% le_nat_iff
thf(fact_1229_nat__le__eq__zle,axiom,
! [W: int,Z: int] :
( ( ( ord_less_int @ zero_zero_int @ W )
| ( ord_less_eq_int @ zero_zero_int @ Z ) )
=> ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
= ( ord_less_eq_int @ W @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_1230_nth__root__nat__aux2_I1_J,axiom,
! [K: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( finite_finite_nat
@ ( collect_nat
@ ^ [M2: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M2 @ K ) @ N2 ) ) ) ) ).
% nth_root_nat_aux2(1)
thf(fact_1231_dvd__nat__bounds,axiom,
! [P2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ P2 )
=> ( ( dvd_dvd_nat @ N2 @ P2 )
=> ( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( ord_less_eq_nat @ N2 @ P2 ) ) ) ) ).
% dvd_nat_bounds
thf(fact_1232_ln__less__cancel__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
= ( ord_less_real @ X @ Y3 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1233_ln__inj__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ( ln_ln_real @ X )
= ( ln_ln_real @ Y3 ) )
= ( X = Y3 ) ) ) ) ).
% ln_inj_iff
thf(fact_1234_ln__le__cancel__iff,axiom,
! [X: real,Y3: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ Y3 )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ ( ln_ln_real @ Y3 ) )
= ( ord_less_eq_real @ X @ Y3 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1235_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% ln_less_zero_iff
thf(fact_1236_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_1237_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ( ln_ln_real @ X )
= zero_zero_real )
= ( X = one_one_real ) ) ) ).
% ln_eq_zero_iff
thf(fact_1238_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ ( ln_ln_real @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% ln_le_zero_iff
thf(fact_1239_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
= ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_1240_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_ge_zero
thf(fact_1241_ln__bound,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_bound
thf(fact_1242_ln__less__self,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_1243_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) ) ) ).
% ln_gt_zero
thf(fact_1244_ln__less__zero,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( ln_ln_real @ X ) @ zero_zero_real ) ) ) ).
% ln_less_zero
thf(fact_1245_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ one_one_real @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_1246_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X ) )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ one_one_real @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_1247_ln__realpow,axiom,
! [X: real,N2: nat] :
( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ln_ln_real @ ( power_power_real @ X @ N2 ) )
= ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X ) ) ) ) ).
% ln_realpow
thf(fact_1248_nth__root__nat__def,axiom,
( nth_nth_root_nat
= ( ^ [K3: nat,N4: nat] :
( if_nat @ ( K3 = zero_zero_nat ) @ zero_zero_nat
@ ( lattic8265883725875713057ax_nat
@ ( collect_nat
@ ^ [M2: nat] : ( ord_less_eq_nat @ ( power_power_nat @ M2 @ K3 ) @ N4 ) ) ) ) ) ) ).
% nth_root_nat_def
thf(fact_1249_to__int__mod__ring__range,axiom,
( ( image_4238506139956901036_a_int @ finite1095367895020317408ring_a @ top_to2069866484006881781ring_a )
= ( set_or4662586982721622107an_int @ zero_zero_int @ q ) ) ).
% to_int_mod_ring_range
thf(fact_1250_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_1251_first__root__nat,axiom,
! [N2: nat] :
( ( nth_nth_root_nat @ one_one_nat @ N2 )
= N2 ) ).
% first_root_nat
thf(fact_1252_nth__root__nat__1,axiom,
! [K: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( nth_nth_root_nat @ K @ one_one_nat )
= one_one_nat ) ) ).
% nth_root_nat_1
thf(fact_1253_nth__root__nat__nth__power,axiom,
! [K: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( nth_nth_root_nat @ K @ ( power_power_nat @ N2 @ K ) )
= N2 ) ) ).
% nth_root_nat_nth_power
thf(fact_1254_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_1255_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_1256_nth__root__nat__less,axiom,
! [K: nat,N2: nat,X: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ N2 @ ( power_power_nat @ X @ K ) )
=> ( ord_less_nat @ ( nth_nth_root_nat @ K @ N2 ) @ X ) ) ) ).
% nth_root_nat_less
thf(fact_1257_nth__root__nat__ge,axiom,
! [K: nat,X: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ X @ K ) @ N2 )
=> ( ord_less_eq_nat @ X @ ( nth_nth_root_nat @ K @ N2 ) ) ) ) ).
% nth_root_nat_ge
thf(fact_1258_nth__root__nat__power__le,axiom,
! [K: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ ( power_power_nat @ ( nth_nth_root_nat @ K @ N2 ) @ K ) @ N2 ) ) ).
% nth_root_nat_power_le
thf(fact_1259_nth__root__nat__aux__correct,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ K @ M3 ) @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ( nth_nth_root_nat_aux @ M3 @ K @ ( power_power_nat @ K @ M3 ) @ N2 )
= ( nth_nth_root_nat @ M3 @ N2 ) ) ) ) ).
% nth_root_nat_aux_correct
thf(fact_1260_nth__root__nat__aux__le,axiom,
! [K: nat,M3: nat,N2: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ K @ M3 ) @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M3 )
=> ( ord_less_eq_nat @ ( power_power_nat @ ( nth_nth_root_nat_aux @ M3 @ K @ ( power_power_nat @ K @ M3 ) @ N2 ) @ M3 ) @ N2 ) ) ) ).
% nth_root_nat_aux_le
thf(fact_1261_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_1262_subset__eq__atLeast0__lessThan__finite,axiom,
! [N: set_nat,N2: nat] :
( ( ord_less_eq_set_nat @ N @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( finite_finite_nat @ N ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_1263_image__int__atLeastLessThan,axiom,
! [A5: nat,B: nat] :
( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A5 @ B ) )
= ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A5 ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% image_int_atLeastLessThan
thf(fact_1264_nth__root__nat__naive__code,axiom,
( nth_nth_root_nat
= ( ^ [M2: nat,N4: nat] :
( if_nat
@ ( ( M2 = zero_zero_nat )
| ( N4 = zero_zero_nat ) )
@ zero_zero_nat
@ ( if_nat
@ ( ( M2 = one_one_nat )
| ( N4 = one_one_nat ) )
@ N4
@ ( nth_nth_root_nat_aux @ M2 @ one_one_nat @ one_one_nat @ N4 ) ) ) ) ) ).
% nth_root_nat_naive_code
thf(fact_1265_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M3: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N2 @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K2 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K2 ) ) )
=> ( P @ M3 ) ) ) ).
% nat_descend_induct
thf(fact_1266_inf__pigeonhole__principle,axiom,
! [N2: nat,F2: nat > nat > $o] :
( ! [K2: nat] :
? [I5: nat] :
( ( ord_less_nat @ I5 @ N2 )
& ( F2 @ K2 @ I5 ) )
=> ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F2 @ K5 @ I4 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_1267_ex__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ( P @ M2 ) ) )
= ( ? [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_1268_all__nat__less__eq,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) ) )
= ( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
% Helper facts (13)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y3: int] :
( ( if_int @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y3: real] :
( ( if_real @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y3: complex] :
( ( if_complex @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y3: complex] :
( ( if_complex @ $true @ X @ Y3 )
= X ) ).
thf(help_If_2_1_If_001t__Kyber____spec__Oqr_Itf__a_J_T,axiom,
! [X: kyber_qr_a,Y3: kyber_qr_a] :
( ( if_Kyber_qr_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Kyber____spec__Oqr_Itf__a_J_T,axiom,
! [X: kyber_qr_a,Y3: kyber_qr_a] :
( ( if_Kyber_qr_a @ $true @ X @ Y3 )
= X ) ).
thf(help_If_3_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y3: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Finite____Field__Omod____ring_Itf__a_J_T,axiom,
! [X: finite_mod_ring_a,Y3: finite_mod_ring_a] :
( ( if_Finite_mod_ring_a @ $true @ X @ Y3 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( finite_finite_int
@ ( image_nat_int
@ ^ [Xa: nat] : ( abs_ky7385543178848499077ty_q_a @ q @ ( times_5121417576591743744ring_a @ ( finite8272632373135393572ring_a @ s ) @ ( coeff_1607515655354303335ring_a @ ( kyber_of_qr_a @ x ) @ Xa ) ) )
@ top_top_set_nat ) ) ).
%------------------------------------------------------------------------------