TPTP Problem File: SLH0777^1.p
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%------------------------------------------------------------------------------
% 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 : Commuting_Hermitian/0002_Commuting_Hermitian/prob_02640_103444__19642114_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1315 ( 415 unt; 233 typ; 0 def)
% Number of atoms : 3538 (1157 equ; 0 cnn)
% Maximal formula atoms : 18 ( 3 avg)
% Number of connectives : 10768 ( 342 ~; 103 |; 252 &;8347 @)
% ( 0 <=>;1724 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 7 avg)
% Number of types : 43 ( 42 usr)
% Number of type conns : 645 ( 645 >; 0 *; 0 +; 0 <<)
% Number of symbols : 194 ( 191 usr; 31 con; 0-4 aty)
% Number of variables : 3273 ( 184 ^;3002 !; 87 ?;3273 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:38:28.006
%------------------------------------------------------------------------------
% Could-be-implicit typings (42)
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% Explicit typings (191)
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thf(sy_c_Column__Operations_Omat__swapcols_001t__Complex__Ocomplex,type,
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thf(sy_c_Commuting__Hermitian_Oper__diag_001t__Complex__Ocomplex,type,
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thf(sy_c_Determinant_Opermutation__delete,type,
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thf(sy_c_Determinant_Opermutation__insert_001t__Nat__Onat,type,
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member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
member8206827879206165904at_nat: produc859450856879609959at_nat > set_Pr8693737435421807431at_nat > $o ).
thf(sy_c_member_001t__Real__Oreal,type,
member_real: real > set_real > $o ).
thf(sy_v_B,type,
b: mat_complex ).
thf(sy_v_f,type,
f: nat > nat ).
thf(sy_v_n,type,
n: nat ).
% Relevant facts (1072)
thf(fact_0_assms_I2_J,axiom,
! [I: nat] :
( ( ord_less_nat @ I @ n )
=> ( member_complex @ ( index_mat_complex @ b @ ( product_Pair_nat_nat @ I @ I ) ) @ real_V2521375963428798218omplex ) ) ).
% assms(2)
thf(fact_1_assms_I1_J,axiom,
member_mat_complex @ b @ ( carrier_mat_complex @ n @ n ) ).
% assms(1)
thf(fact_2_assms_I3_J,axiom,
bij_betw_nat_nat @ f @ ( set_ord_lessThan_nat @ n ) @ ( set_ord_lessThan_nat @ n ) ).
% assms(3)
thf(fact_3_ev__blocks__part__def,axiom,
( jordan4637981584770492064omplex
= ( ^ [M: nat,A: mat_complex] :
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ( ord_less_nat @ K @ M )
=> ( ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ K @ K ) )
= ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) ) )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ J @ J ) )
= ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) ) ) ) ) ) ) ) ) ).
% ev_blocks_part_def
thf(fact_4_same__diag__def,axiom,
( jordan2620430285385836103omplex
= ( ^ [N: nat,A: mat_complex,B: mat_complex] :
! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) )
= ( index_mat_complex @ B @ ( product_Pair_nat_nat @ I2 @ I2 ) ) ) ) ) ) ).
% same_diag_def
thf(fact_5_ev__blockD,axiom,
! [N2: nat,A2: mat_complex,I3: nat,J2: nat] :
( ( jordan8042990603089931364omplex @ N2 @ A2 )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ I3 ) )
= ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) ) ) ) ) ).
% ev_blockD
thf(fact_6_ev__block__def,axiom,
( jordan8042990603089931364omplex
= ( ^ [N: nat,A: mat_complex] :
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( ( ord_less_nat @ J @ N )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) )
= ( index_mat_complex @ A @ ( product_Pair_nat_nat @ J @ J ) ) ) ) ) ) ) ).
% ev_block_def
thf(fact_7_bezw_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [X2: nat,Y: nat] :
( X
!= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% bezw.cases
thf(fact_8_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less_nat @ M2 @ N2 )
| ( ord_less_nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_9_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_10_same__diag__ev__block,axiom,
! [N2: nat,A2: mat_complex,B2: mat_complex] :
( ( jordan2620430285385836103omplex @ N2 @ A2 @ B2 )
=> ( ( jordan8042990603089931364omplex @ N2 @ A2 )
=> ( jordan8042990603089931364omplex @ N2 @ B2 ) ) ) ).
% same_diag_ev_block
thf(fact_11_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_12_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_13_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_14_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_15_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_16_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_17_lessThan__strict__subset__iff,axiom,
! [M2: real,N2: real] :
( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M2 ) @ ( set_or5984915006950818249n_real @ N2 ) )
= ( ord_less_real @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_18_lessThan__strict__subset__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% lessThan_strict_subset_iff
thf(fact_19_lessThan__iff,axiom,
! [I3: complex,K2: complex] :
( ( member_complex @ I3 @ ( set_or7194820819169546315omplex @ K2 ) )
= ( ord_less_complex @ I3 @ K2 ) ) ).
% lessThan_iff
thf(fact_20_lessThan__iff,axiom,
! [I3: mat_complex,K2: mat_complex] :
( ( member_mat_complex @ I3 @ ( set_or6592934062653983178omplex @ K2 ) )
= ( ord_less_mat_complex @ I3 @ K2 ) ) ).
% lessThan_iff
thf(fact_21_lessThan__iff,axiom,
! [I3: produc859450856879609959at_nat,K2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ I3 @ ( set_or6242363224033793546at_nat @ K2 ) )
= ( ord_le9033551061567896339at_nat @ I3 @ K2 ) ) ).
% lessThan_iff
thf(fact_22_lessThan__iff,axiom,
! [I3: real,K2: real] :
( ( member_real @ I3 @ ( set_or5984915006950818249n_real @ K2 ) )
= ( ord_less_real @ I3 @ K2 ) ) ).
% lessThan_iff
thf(fact_23_lessThan__iff,axiom,
! [I3: nat,K2: nat] :
( ( member_nat @ I3 @ ( set_ord_lessThan_nat @ K2 ) )
= ( ord_less_nat @ I3 @ K2 ) ) ).
% lessThan_iff
thf(fact_24_ev__blocks__def,axiom,
jordan4650062548456832493omplex = jordan4637981584770492064omplex ).
% ev_blocks_def
thf(fact_25_per__diag__diagonal,axiom,
! [D: mat_complex,N2: nat,F: nat > nat] :
( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( diagonal_mat_complex @ D )
=> ( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ N2 ) )
=> ( diagonal_mat_complex @ ( commut4119912100034661455omplex @ D @ F ) ) ) ) ) ).
% per_diag_diagonal
thf(fact_26_idty__index,axiom,
! [F: nat > nat,N2: nat,I3: nat,J2: nat] :
( ( bij_betw_nat_nat @ F @ ( set_ord_lessThan_nat @ N2 ) @ ( set_ord_lessThan_nat @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_complex @ ( one_mat_complex @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( index_mat_complex @ ( one_mat_complex @ N2 ) @ ( product_Pair_nat_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ) ) ) ).
% idty_index
thf(fact_27_elements__matI,axiom,
! [A2: mat_mat_complex,Nr: nat,Nc: nat,I3: nat,J2: nat,A3: mat_complex] :
( ( member7752848204589936667omplex @ A2 @ ( carrie8442657464762054641omplex @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( A3
= ( index_7093623372566408491omplex @ A2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) )
=> ( member_mat_complex @ A3 @ ( elemen3580889201824698026omplex @ A2 ) ) ) ) ) ) ).
% elements_matI
thf(fact_28_elements__matI,axiom,
! [A2: mat_Pr7664800203558610372at_nat,Nr: nat,Nc: nat,I3: nat,J2: nat,A3: produc859450856879609959at_nat] :
( ( member1116373046597317467at_nat @ A2 @ ( carrie8012128234321260977at_nat @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( A3
= ( index_4502547381347762155at_nat @ A2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) )
=> ( member8206827879206165904at_nat @ A3 @ ( elemen449274506822781674at_nat @ A2 ) ) ) ) ) ) ).
% elements_matI
thf(fact_29_elements__matI,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,I3: nat,J2: nat,A3: complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ord_less_nat @ I3 @ Nr )
=> ( ( ord_less_nat @ J2 @ Nc )
=> ( ( A3
= ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) )
=> ( member_complex @ A3 @ ( elements_mat_complex @ A2 ) ) ) ) ) ) ).
% elements_matI
thf(fact_30_bij__betwE,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A2 )
=> ( member_nat @ ( F @ X3 ) @ B2 ) ) ) ).
% bij_betwE
thf(fact_31_bij__betw__inv,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ? [G: nat > nat] : ( bij_betw_nat_nat @ G @ B2 @ A2 ) ) ).
% bij_betw_inv
thf(fact_32_bij__betw__ball,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,Phi: nat > $o] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( ! [X4: nat] :
( ( member_nat @ X4 @ B2 )
=> ( Phi @ X4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A2 )
=> ( Phi @ ( F @ X4 ) ) ) ) ) ) ).
% bij_betw_ball
thf(fact_33_bij__betw__cong,axiom,
! [A2: set_nat,F: nat > nat,G2: nat > nat,A4: set_nat] :
( ! [A5: nat] :
( ( member_nat @ A5 @ A2 )
=> ( ( F @ A5 )
= ( G2 @ A5 ) ) )
=> ( ( bij_betw_nat_nat @ F @ A2 @ A4 )
= ( bij_betw_nat_nat @ G2 @ A2 @ A4 ) ) ) ).
% bij_betw_cong
thf(fact_34_bij__betw__apply,axiom,
! [F: complex > complex,A2: set_complex,B2: set_complex,A3: complex] :
( ( bij_be1856998921033663316omplex @ F @ A2 @ B2 )
=> ( ( member_complex @ A3 @ A2 )
=> ( member_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_35_bij__betw__apply,axiom,
! [F: complex > mat_complex,A2: set_complex,B2: set_mat_complex,A3: complex] :
( ( bij_be1079348409077485953omplex @ F @ A2 @ B2 )
=> ( ( member_complex @ A3 @ A2 )
=> ( member_mat_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_36_bij__betw__apply,axiom,
! [F: complex > produc859450856879609959at_nat,A2: set_complex,B2: set_Pr8693737435421807431at_nat,A3: complex] :
( ( bij_be8786120691309409089at_nat @ F @ A2 @ B2 )
=> ( ( member_complex @ A3 @ A2 )
=> ( member8206827879206165904at_nat @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_37_bij__betw__apply,axiom,
! [F: mat_complex > complex,A2: set_mat_complex,B2: set_complex,A3: mat_complex] :
( ( bij_be8380097718144844511omplex @ F @ A2 @ B2 )
=> ( ( member_mat_complex @ A3 @ A2 )
=> ( member_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_38_bij__betw__apply,axiom,
! [F: mat_complex > mat_complex,A2: set_mat_complex,B2: set_mat_complex,A3: mat_complex] :
( ( bij_be1442909924300985910omplex @ F @ A2 @ B2 )
=> ( ( member_mat_complex @ A3 @ A2 )
=> ( member_mat_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_39_bij__betw__apply,axiom,
! [F: mat_complex > produc859450856879609959at_nat,A2: set_mat_complex,B2: set_Pr8693737435421807431at_nat,A3: mat_complex] :
( ( bij_be6229314974531182710at_nat @ F @ A2 @ B2 )
=> ( ( member_mat_complex @ A3 @ A2 )
=> ( member8206827879206165904at_nat @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_40_bij__betw__apply,axiom,
! [F: produc859450856879609959at_nat > complex,A2: set_Pr8693737435421807431at_nat,B2: set_complex,A3: produc859450856879609959at_nat] :
( ( bij_be2724670508752533023omplex @ F @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ A3 @ A2 )
=> ( member_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_41_bij__betw__apply,axiom,
! [F: produc859450856879609959at_nat > mat_complex,A2: set_Pr8693737435421807431at_nat,B2: set_mat_complex,A3: produc859450856879609959at_nat] :
( ( bij_be4765843641008149238omplex @ F @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ A3 @ A2 )
=> ( member_mat_complex @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_42_bij__betw__apply,axiom,
! [F: produc859450856879609959at_nat > produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,A3: produc859450856879609959at_nat] :
( ( bij_be4529855521105804598at_nat @ F @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ A3 @ A2 )
=> ( member8206827879206165904at_nat @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_43_bij__betw__apply,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,A3: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( member_nat @ ( F @ A3 ) @ B2 ) ) ) ).
% bij_betw_apply
thf(fact_44_bij__betw__iff__bijections,axiom,
( bij_be1856998921033663316omplex
= ( ^ [F2: complex > complex,A: set_complex,B: set_complex] :
? [G3: complex > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ( ( member_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: complex] :
( ( member_complex @ X4 @ B )
=> ( ( member_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_45_bij__betw__iff__bijections,axiom,
( bij_be8380097718144844511omplex
= ( ^ [F2: mat_complex > complex,A: set_mat_complex,B: set_complex] :
? [G3: complex > mat_complex] :
( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A )
=> ( ( member_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: complex] :
( ( member_complex @ X4 @ B )
=> ( ( member_mat_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_46_bij__betw__iff__bijections,axiom,
( bij_be2724670508752533023omplex
= ( ^ [F2: produc859450856879609959at_nat > complex,A: set_Pr8693737435421807431at_nat,B: set_complex] :
? [G3: complex > produc859450856879609959at_nat] :
( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A )
=> ( ( member_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: complex] :
( ( member_complex @ X4 @ B )
=> ( ( member8206827879206165904at_nat @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_47_bij__betw__iff__bijections,axiom,
( bij_be1079348409077485953omplex
= ( ^ [F2: complex > mat_complex,A: set_complex,B: set_mat_complex] :
? [G3: mat_complex > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ( ( member_mat_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ B )
=> ( ( member_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_48_bij__betw__iff__bijections,axiom,
( bij_be1442909924300985910omplex
= ( ^ [F2: mat_complex > mat_complex,A: set_mat_complex,B: set_mat_complex] :
? [G3: mat_complex > mat_complex] :
( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A )
=> ( ( member_mat_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ B )
=> ( ( member_mat_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_49_bij__betw__iff__bijections,axiom,
( bij_be4765843641008149238omplex
= ( ^ [F2: produc859450856879609959at_nat > mat_complex,A: set_Pr8693737435421807431at_nat,B: set_mat_complex] :
? [G3: mat_complex > produc859450856879609959at_nat] :
( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A )
=> ( ( member_mat_complex @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ B )
=> ( ( member8206827879206165904at_nat @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_50_bij__betw__iff__bijections,axiom,
( bij_be8786120691309409089at_nat
= ( ^ [F2: complex > produc859450856879609959at_nat,A: set_complex,B: set_Pr8693737435421807431at_nat] :
? [G3: produc859450856879609959at_nat > complex] :
( ! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ( ( member8206827879206165904at_nat @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ B )
=> ( ( member_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_51_bij__betw__iff__bijections,axiom,
( bij_be6229314974531182710at_nat
= ( ^ [F2: mat_complex > produc859450856879609959at_nat,A: set_mat_complex,B: set_Pr8693737435421807431at_nat] :
? [G3: produc859450856879609959at_nat > mat_complex] :
( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A )
=> ( ( member8206827879206165904at_nat @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ B )
=> ( ( member_mat_complex @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_52_bij__betw__iff__bijections,axiom,
( bij_be4529855521105804598at_nat
= ( ^ [F2: produc859450856879609959at_nat > produc859450856879609959at_nat,A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
? [G3: produc859450856879609959at_nat > produc859450856879609959at_nat] :
( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A )
=> ( ( member8206827879206165904at_nat @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ B )
=> ( ( member8206827879206165904at_nat @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_53_bij__betw__iff__bijections,axiom,
( bij_betw_nat_nat
= ( ^ [F2: nat > nat,A: set_nat,B: set_nat] :
? [G3: nat > nat] :
( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( ( member_nat @ ( F2 @ X4 ) @ B )
& ( ( G3 @ ( F2 @ X4 ) )
= X4 ) ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( ( member_nat @ ( G3 @ X4 ) @ A )
& ( ( F2 @ ( G3 @ X4 ) )
= X4 ) ) ) ) ) ) ).
% bij_betw_iff_bijections
thf(fact_54_one__carrier__mat,axiom,
! [N2: nat] : ( member_mat_complex @ ( one_mat_complex @ N2 ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% one_carrier_mat
thf(fact_55_lessThan__eq__iff,axiom,
! [X: nat,Y2: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y2 ) )
= ( X = Y2 ) ) ).
% lessThan_eq_iff
thf(fact_56_psubsetD,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_57_psubsetD,axiom,
! [A2: set_mat_complex,B2: set_mat_complex,C: mat_complex] :
( ( ord_le5598786136212072115omplex @ A2 @ B2 )
=> ( ( member_mat_complex @ C @ A2 )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_58_psubsetD,axiom,
! [A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
( ( ord_le6428140832669894131at_nat @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ C @ A2 )
=> ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% psubsetD
thf(fact_59_bij__betw__singleton__eq,axiom,
! [F: nat > nat,A2: set_nat,B2: set_nat,G2: nat > nat,A3: nat] :
( ( bij_betw_nat_nat @ F @ A2 @ B2 )
=> ( ( bij_betw_nat_nat @ G2 @ A2 @ B2 )
=> ( ( member_nat @ A3 @ A2 )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A2 )
=> ( ( X2 != A3 )
=> ( ( F @ X2 )
= ( G2 @ X2 ) ) ) )
=> ( ( F @ A3 )
= ( G2 @ A3 ) ) ) ) ) ) ).
% bij_betw_singleton_eq
thf(fact_60_bij__betwI_H,axiom,
! [X5: set_complex,F: complex > complex,Y3: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ! [Y: complex] :
( ( member_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ( member_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: complex] :
( ( member_complex @ Y @ Y3 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be1856998921033663316omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_61_bij__betwI_H,axiom,
! [X5: set_complex,F: complex > mat_complex,Y3: set_mat_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ! [Y: complex] :
( ( member_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ( member_mat_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ Y3 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be1079348409077485953omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_62_bij__betwI_H,axiom,
! [X5: set_complex,F: complex > produc859450856879609959at_nat,Y3: set_Pr8693737435421807431at_nat] :
( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ! [Y: complex] :
( ( member_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: complex] :
( ( member_complex @ X2 @ X5 )
=> ( member8206827879206165904at_nat @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ Y3 )
=> ? [X3: complex] :
( ( member_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be8786120691309409089at_nat @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_63_bij__betwI_H,axiom,
! [X5: set_mat_complex,F: mat_complex > complex,Y3: set_complex] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ( member_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: complex] :
( ( member_complex @ Y @ Y3 )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be8380097718144844511omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_64_bij__betwI_H,axiom,
! [X5: set_mat_complex,F: mat_complex > mat_complex,Y3: set_mat_complex] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ( member_mat_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ Y3 )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be1442909924300985910omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_65_bij__betwI_H,axiom,
! [X5: set_mat_complex,F: mat_complex > produc859450856879609959at_nat,Y3: set_Pr8693737435421807431at_nat] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ X5 )
=> ( member8206827879206165904at_nat @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ Y3 )
=> ? [X3: mat_complex] :
( ( member_mat_complex @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be6229314974531182710at_nat @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_66_bij__betwI_H,axiom,
! [X5: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > complex,Y3: set_complex] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ( member_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: complex] :
( ( member_complex @ Y @ Y3 )
=> ? [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be2724670508752533023omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_67_bij__betwI_H,axiom,
! [X5: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > mat_complex,Y3: set_mat_complex] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ( member_mat_complex @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: mat_complex] :
( ( member_mat_complex @ Y @ Y3 )
=> ? [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be4765843641008149238omplex @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_68_bij__betwI_H,axiom,
! [X5: set_Pr8693737435421807431at_nat,F: produc859450856879609959at_nat > produc859450856879609959at_nat,Y3: set_Pr8693737435421807431at_nat] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ X5 )
=> ( member8206827879206165904at_nat @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ Y @ Y3 )
=> ? [X3: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_be4529855521105804598at_nat @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_69_bij__betwI_H,axiom,
! [X5: set_nat,F: nat > nat,Y3: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ! [Y: nat] :
( ( member_nat @ Y @ X5 )
=> ( ( ( F @ X2 )
= ( F @ Y ) )
= ( X2 = Y ) ) ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ X5 )
=> ( member_nat @ ( F @ X2 ) @ Y3 ) )
=> ( ! [Y: nat] :
( ( member_nat @ Y @ Y3 )
=> ? [X3: nat] :
( ( member_nat @ X3 @ X5 )
& ( Y
= ( F @ X3 ) ) ) )
=> ( bij_betw_nat_nat @ F @ X5 @ Y3 ) ) ) ) ).
% bij_betwI'
thf(fact_70_pivot__positions__main__gen_Oinduct,axiom,
! [Nr: nat,Nc: nat,A2: mat_complex,Zero: complex,P: nat > nat > $o,A0: nat,A1: nat] :
( ! [I4: nat,J3: nat] :
( ( ( ord_less_nat @ I4 @ Nr )
=> ( ( ord_less_nat @ J3 @ Nc )
=> ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
= Zero )
=> ( P @ I4 @ ( suc @ J3 ) ) ) ) )
=> ( ( ( ord_less_nat @ I4 @ Nr )
=> ( ( ord_less_nat @ J3 @ Nc )
=> ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I4 @ J3 ) )
!= Zero )
=> ( P @ ( suc @ I4 ) @ ( suc @ J3 ) ) ) ) )
=> ( P @ I4 @ J3 ) ) )
=> ( P @ A0 @ A1 ) ) ).
% pivot_positions_main_gen.induct
thf(fact_71_mem__Collect__eq,axiom,
! [A3: complex,P: complex > $o] :
( ( member_complex @ A3 @ ( collect_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
! [A3: mat_complex,P: mat_complex > $o] :
( ( member_mat_complex @ A3 @ ( collect_mat_complex @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
! [A3: produc859450856879609959at_nat,P: produc859450856879609959at_nat > $o] :
( ( member8206827879206165904at_nat @ A3 @ ( collec7088162979684241874at_nat @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_74_Collect__mem__eq,axiom,
! [A2: set_complex] :
( ( collect_complex
@ ^ [X4: complex] : ( member_complex @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
! [A2: set_mat_complex] :
( ( collect_mat_complex
@ ^ [X4: mat_complex] : ( member_mat_complex @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
! [A2: set_Pr8693737435421807431at_nat] :
( ( collec7088162979684241874at_nat
@ ^ [X4: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_77_prod_Osimps_I1_J,axiom,
! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
( ( ( product_Pair_nat_nat @ X1 @ X22 )
= ( product_Pair_nat_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.simps(1)
thf(fact_78_prod_Osimps_I1_J,axiom,
! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
( ( ( produc72220940542539688at_nat @ X1 @ X22 )
= ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.simps(1)
thf(fact_79_prod_Osimps_I1_J,axiom,
! [X1: product_prod_nat_nat,X22: product_prod_nat_nat,Y1: product_prod_nat_nat,Y22: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ X1 @ X22 )
= ( produc6161850002892822231at_nat @ Y1 @ Y22 ) )
= ( ( X1 = Y1 )
& ( X22 = Y22 ) ) ) ).
% prod.simps(1)
thf(fact_80_old_Oprod_Osimps_I1_J,axiom,
! [A3: nat,B3: nat,A6: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B3 )
= ( product_Pair_nat_nat @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.simps(1)
thf(fact_81_old_Oprod_Osimps_I1_J,axiom,
! [A3: nat > nat,B3: nat,A6: nat > nat,B4: nat] :
( ( ( produc72220940542539688at_nat @ A3 @ B3 )
= ( produc72220940542539688at_nat @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.simps(1)
thf(fact_82_old_Oprod_Osimps_I1_J,axiom,
! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,A6: product_prod_nat_nat,B4: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A3 @ B3 )
= ( produc6161850002892822231at_nat @ A6 @ B4 ) )
= ( ( A3 = A6 )
& ( B3 = B4 ) ) ) ).
% old.prod.simps(1)
thf(fact_83_prod_Oinduct,axiom,
! [P: product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_84_prod_Oinduct,axiom,
! [P: produc8199716216217303280at_nat > $o,Prod: produc8199716216217303280at_nat] :
( ! [A5: nat > nat,B5: nat] : ( P @ ( produc72220940542539688at_nat @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_85_prod_Oinduct,axiom,
! [P: produc859450856879609959at_nat > $o,Prod: produc859450856879609959at_nat] :
( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B5 ) )
=> ( P @ Prod ) ) ).
% prod.induct
thf(fact_86_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_87_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_88_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_89_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_90_fold__atLeastAtMost__nat_Ocases,axiom,
! [X: produc4471711990508489141at_nat] :
~ ! [F3: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
( X
!= ( produc3209952032786966637at_nat @ F3 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_91_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_92_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I4: nat] :
( ( J2
= ( suc @ I4 ) )
=> ( P @ I4 ) )
=> ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ J2 )
=> ( ( P @ ( suc @ I4 ) )
=> ( P @ I4 ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_93_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
=> ( ! [I4: nat,J3: nat,K3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ( ord_less_nat @ J3 @ K3 )
=> ( ( P @ I4 @ J3 )
=> ( ( P @ J3 @ K3 )
=> ( P @ I4 @ K3 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_94_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( suc @ I3 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_95_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_96_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less_nat @ N2 @ M2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_97_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_98_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_99_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M2 @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_100_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_101_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_102_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ N2 )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_103_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_104_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_105_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_106_Suc__lessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ ( suc @ I3 ) @ K2 )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_107_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_108_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_109_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_110_Nat_OlessE,axiom,
! [I3: nat,K2: nat] :
( ( ord_less_nat @ I3 @ K2 )
=> ( ( K2
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( K2
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_111_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_112_lift__Suc__mono__less__iff,axiom,
! [F: nat > real,N2: nat,M2: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M2 ) )
= ( ord_less_nat @ N2 @ M2 ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_113_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_114_lift__Suc__mono__less,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N2 @ N4 )
=> ( ord_less_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_115_basic__trans__rules_I1_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_116_basic__trans__rules_I1_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_117_basic__trans__rules_I1_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_118_basic__trans__rules_I1_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(1)
thf(fact_119_basic__trans__rules_I2_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_120_basic__trans__rules_I2_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_121_basic__trans__rules_I2_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_122_basic__trans__rules_I2_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(2)
thf(fact_123_basic__trans__rules_I11_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_124_basic__trans__rules_I11_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_125_basic__trans__rules_I11_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_126_basic__trans__rules_I11_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(11)
thf(fact_127_basic__trans__rules_I12_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_128_basic__trans__rules_I12_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_129_basic__trans__rules_I12_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_130_basic__trans__rules_I12_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(12)
thf(fact_131_basic__trans__rules_I19_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(19)
thf(fact_132_basic__trans__rules_I19_J,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% basic_trans_rules(19)
thf(fact_133_basic__trans__rules_I20_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% basic_trans_rules(20)
thf(fact_134_basic__trans__rules_I20_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% basic_trans_rules(20)
thf(fact_135_basic__trans__rules_I27_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(27)
thf(fact_136_basic__trans__rules_I27_J,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% basic_trans_rules(27)
thf(fact_137_basic__trans__rules_I28_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(28)
thf(fact_138_basic__trans__rules_I28_J,axiom,
! [A3: real,B3: real,C: real] :
( ( A3 = B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% basic_trans_rules(28)
thf(fact_139_ssubst__Pair__rhs,axiom,
! [R: nat,S: nat,R2: set_Pr1261947904930325089at_nat,S2: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_140_ssubst__Pair__rhs,axiom,
! [R: nat > nat,S: nat,R2: set_Pr9093778441882193744at_nat,S2: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_141_ssubst__Pair__rhs,axiom,
! [R: product_prod_nat_nat,S: product_prod_nat_nat,R2: set_Pr8693737435421807431at_nat,S2: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S ) @ R2 )
=> ( ( S2 = S )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ R @ S2 ) @ R2 ) ) ) ).
% ssubst_Pair_rhs
thf(fact_142_prod__induct3,axiom,
! [P: produc859450856879609959at_nat > $o,X: produc859450856879609959at_nat] :
( ! [A5: product_prod_nat_nat,B5: nat,C2: nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C2 ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_143_prod__cases3,axiom,
! [Y2: produc859450856879609959at_nat] :
~ ! [A5: product_prod_nat_nat,B5: nat,C2: nat] :
( Y2
!= ( produc6161850002892822231at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ C2 ) ) ) ).
% prod_cases3
thf(fact_144_Pair__inject,axiom,
! [A3: nat,B3: nat,A6: nat,B4: nat] :
( ( ( product_Pair_nat_nat @ A3 @ B3 )
= ( product_Pair_nat_nat @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_145_Pair__inject,axiom,
! [A3: nat > nat,B3: nat,A6: nat > nat,B4: nat] :
( ( ( produc72220940542539688at_nat @ A3 @ B3 )
= ( produc72220940542539688at_nat @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_146_Pair__inject,axiom,
! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,A6: product_prod_nat_nat,B4: product_prod_nat_nat] :
( ( ( produc6161850002892822231at_nat @ A3 @ B3 )
= ( produc6161850002892822231at_nat @ A6 @ B4 ) )
=> ~ ( ( A3 = A6 )
=> ( B3 != B4 ) ) ) ).
% Pair_inject
thf(fact_147_prod__cases,axiom,
! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
( ! [A5: nat,B5: nat] : ( P @ ( product_Pair_nat_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_148_prod__cases,axiom,
! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
( ! [A5: nat > nat,B5: nat] : ( P @ ( produc72220940542539688at_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_149_prod__cases,axiom,
! [P: produc859450856879609959at_nat > $o,P2: produc859450856879609959at_nat] :
( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] : ( P @ ( produc6161850002892822231at_nat @ A5 @ B5 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_150_surj__pair,axiom,
! [P2: product_prod_nat_nat] :
? [X2: nat,Y: nat] :
( P2
= ( product_Pair_nat_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_151_surj__pair,axiom,
! [P2: produc8199716216217303280at_nat] :
? [X2: nat > nat,Y: nat] :
( P2
= ( produc72220940542539688at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_152_surj__pair,axiom,
! [P2: produc859450856879609959at_nat] :
? [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
( P2
= ( produc6161850002892822231at_nat @ X2 @ Y ) ) ).
% surj_pair
thf(fact_153_old_Oprod_Oexhaust,axiom,
! [Y2: product_prod_nat_nat] :
~ ! [A5: nat,B5: nat] :
( Y2
!= ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_154_old_Oprod_Oexhaust,axiom,
! [Y2: produc8199716216217303280at_nat] :
~ ! [A5: nat > nat,B5: nat] :
( Y2
!= ( produc72220940542539688at_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_155_old_Oprod_Oexhaust,axiom,
! [Y2: produc859450856879609959at_nat] :
~ ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
( Y2
!= ( produc6161850002892822231at_nat @ A5 @ B5 ) ) ).
% old.prod.exhaust
thf(fact_156_lookup__other__ev_Oinduct,axiom,
! [P: complex > nat > mat_complex > $o,A0: complex,A1: nat,A22: mat_complex] :
( ! [Ev: complex,X_1: mat_complex] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: complex,I4: nat,A7: mat_complex] :
( ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
= Ev )
=> ( P @ Ev @ I4 @ A7 ) )
=> ( P @ Ev @ ( suc @ I4 ) @ A7 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_other_ev.induct
thf(fact_157_lookup__ev_Oinduct,axiom,
! [P: complex > nat > mat_complex > $o,A0: complex,A1: nat,A22: mat_complex] :
( ! [Ev: complex,X_1: mat_complex] : ( P @ Ev @ zero_zero_nat @ X_1 )
=> ( ! [Ev: complex,I4: nat,A7: mat_complex] :
( ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ I4 ) )
!= Ev )
=> ( P @ Ev @ I4 @ A7 ) )
=> ( P @ Ev @ ( suc @ I4 ) @ A7 ) )
=> ( P @ A0 @ A1 @ A22 ) ) ) ).
% lookup_ev.induct
thf(fact_158_gcd_Ocases,axiom,
! [X: product_prod_nat_nat] :
~ ! [A5: nat,B5: nat] :
( X
!= ( product_Pair_nat_nat @ A5 @ B5 ) ) ).
% gcd.cases
thf(fact_159_gauss__jordan__compute__inverse_I3_J,axiom,
! [A2: mat_complex,N2: nat,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ ( one_mat_complex @ N2 ) )
= ( produc3658446505030690647omplex @ ( one_mat_complex @ N2 ) @ B6 ) )
=> ( member_mat_complex @ B6 @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ) ).
% gauss_jordan_compute_inverse(3)
thf(fact_160_mat__delete__index,axiom,
! [A2: mat_complex,N2: nat,I3: nat,J2: nat,I5: nat,J4: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ ( suc @ N2 ) @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( ( ord_less_nat @ J2 @ ( suc @ N2 ) )
=> ( ( ord_less_nat @ I5 @ N2 )
=> ( ( ord_less_nat @ J4 @ N2 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ ( insert_index @ I3 @ I5 ) @ ( insert_index @ J2 @ J4 ) ) )
= ( index_mat_complex @ ( mat_delete_complex @ A2 @ I3 @ J2 ) @ ( product_Pair_nat_nat @ I5 @ J4 ) ) ) ) ) ) ) ) ).
% mat_delete_index
thf(fact_161_order__less__imp__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_162_order__less__imp__not__less,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% order_less_imp_not_less
thf(fact_163_order__less__imp__not__eq2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_164_order__less__imp__not__eq2,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( Y2 != X ) ) ).
% order_less_imp_not_eq2
thf(fact_165_insert__index__exclude,axiom,
! [I3: nat,I5: nat] :
( ( insert_index @ I3 @ I5 )
!= I3 ) ).
% insert_index_exclude
thf(fact_166_inf__concat__simple_Oinduct,axiom,
! [P: ( nat > nat ) > nat > $o,A0: nat > nat,A1: nat] :
( ! [F3: nat > nat] : ( P @ F3 @ zero_zero_nat )
=> ( ! [F3: nat > nat,N3: nat] :
( ( P @ F3 @ N3 )
=> ( P @ F3 @ ( suc @ N3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% inf_concat_simple.induct
thf(fact_167_inf__concat__simple_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
( ! [F3: nat > nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
=> ~ ! [F3: nat > nat,N3: nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ ( suc @ N3 ) ) ) ) ).
% inf_concat_simple.cases
thf(fact_168_insert__index_I1_J,axiom,
! [I5: nat,I3: nat] :
( ( ord_less_nat @ I5 @ I3 )
=> ( ( insert_index @ I3 @ I5 )
= I5 ) ) ).
% insert_index(1)
thf(fact_169_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_170_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_171_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_172_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_173_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_174_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_175_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_176_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_177_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_178_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_179_nat_Osimps_I3_J,axiom,
! [X22: nat] :
( ( suc @ X22 )
!= zero_zero_nat ) ).
% nat.simps(3)
thf(fact_180_old_Onat_Osimps_I3_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.simps(3)
thf(fact_181_old_Onat_Osimps_I2_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.simps(2)
thf(fact_182_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_183_nat_Oinduct,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% nat.induct
thf(fact_184_nat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2 != zero_zero_nat )
=> ~ ! [X23: nat] :
( Y2
!= ( suc @ X23 ) ) ) ).
% nat.exhaust
thf(fact_185_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_186_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_187_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y: nat] : ( P @ zero_zero_nat @ ( suc @ Y ) )
=> ( ! [X2: nat,Y: nat] :
( ( P @ X2 @ Y )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y ) ) )
=> ( P @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_188_zero__induct,axiom,
! [P: nat > $o,K2: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_189_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_190_Suc__not__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= zero_zero_nat ) ).
% Suc_not_Zero
thf(fact_191_Zero__neq__Suc,axiom,
! [M2: nat] :
( zero_zero_nat
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_192_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_193_unit__vecs__last_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [N3: nat] : ( P @ N3 @ zero_zero_nat )
=> ( ! [N3: nat,I4: nat] :
( ( P @ N3 @ I4 )
=> ( P @ N3 @ ( suc @ I4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% unit_vecs_last.induct
thf(fact_194_gauss__jordan__carrier_I1_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,Nc2: nat,A4: mat_complex,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc2 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ A4 @ B6 ) )
=> ( member_mat_complex @ A4 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ) ).
% gauss_jordan_carrier(1)
thf(fact_195_gauss__jordan__carrier_I2_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,Nc2: nat,A4: mat_complex,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc2 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ A4 @ B6 ) )
=> ( member_mat_complex @ B6 @ ( carrier_mat_complex @ Nr @ Nc2 ) ) ) ) ) ).
% gauss_jordan_carrier(2)
thf(fact_196_gauss__jordan_I3_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,Nc22: nat,C3: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc22 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ C3 @ D ) )
=> ( member_mat_complex @ C3 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ) ).
% gauss_jordan(3)
thf(fact_197_gauss__jordan_I4_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,Nc22: nat,C3: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc22 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ C3 @ D ) )
=> ( member_mat_complex @ D @ ( carrier_mat_complex @ Nr @ Nc22 ) ) ) ) ) ).
% gauss_jordan(4)
thf(fact_198_insert__index__def,axiom,
( insert_index
= ( ^ [I2: nat,I6: nat] : ( if_nat @ ( ord_less_nat @ I6 @ I2 ) @ I6 @ ( suc @ I6 ) ) ) ) ).
% insert_index_def
thf(fact_199_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ( M2 = zero_zero_nat )
| ? [J: nat] :
( ( M2
= ( suc @ J ) )
& ( ord_less_nat @ J @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_200_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_201_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_202_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_203_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M: nat] :
( N2
= ( suc @ M ) ) ) ) ).
% gr0_conv_Suc
thf(fact_204_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
& ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N2 )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_205_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_206_unit__vecs__first_Ocases,axiom,
! [X: product_prod_nat_nat] :
( ! [N3: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ zero_zero_nat ) )
=> ~ ! [N3: nat,I4: nat] :
( X
!= ( product_Pair_nat_nat @ N3 @ ( suc @ I4 ) ) ) ) ).
% unit_vecs_first.cases
thf(fact_207_lt__ex,axiom,
! [X: real] :
? [Y: real] : ( ord_less_real @ Y @ X ) ).
% lt_ex
thf(fact_208_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_209_gt__ex,axiom,
! [X: real] :
? [X_1: real] : ( ord_less_real @ X @ X_1 ) ).
% gt_ex
thf(fact_210_neqE,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% neqE
thf(fact_211_neqE,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% neqE
thf(fact_212_neq__iff,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
= ( ( ord_less_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ) ).
% neq_iff
thf(fact_213_neq__iff,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
= ( ( ord_less_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ) ).
% neq_iff
thf(fact_214_dense,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ? [Z2: real] :
( ( ord_less_real @ X @ Z2 )
& ( ord_less_real @ Z2 @ Y2 ) ) ) ).
% dense
thf(fact_215_less__imp__neq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_216_less__imp__neq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_neq
thf(fact_217_less__asym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% less_asym
thf(fact_218_less__asym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% less_asym
thf(fact_219_order_Oasym,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ~ ( ord_less_nat @ B3 @ A3 ) ) ).
% order.asym
thf(fact_220_order_Oasym,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ~ ( ord_less_real @ B3 @ A3 ) ) ).
% order.asym
thf(fact_221_less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% less_linear
thf(fact_222_less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
| ( X = Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% less_linear
thf(fact_223_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_224_less__irrefl,axiom,
! [X: real] :
~ ( ord_less_real @ X @ X ) ).
% less_irrefl
thf(fact_225_less__imp__not__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_not_eq
thf(fact_226_less__imp__not__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( X != Y2 ) ) ).
% less_imp_not_eq
thf(fact_227_less__not__sym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ~ ( ord_less_nat @ Y2 @ X ) ) ).
% less_not_sym
thf(fact_228_less__not__sym,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ~ ( ord_less_real @ Y2 @ X ) ) ).
% less_not_sym
thf(fact_229_order_Oirrefl,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% order.irrefl
thf(fact_230_order_Oirrefl,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% order.irrefl
thf(fact_231_less__induct,axiom,
! [P: nat > $o,A3: nat] :
( ! [X2: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X2 )
=> ( P @ Y4 ) )
=> ( P @ X2 ) )
=> ( P @ A3 ) ) ).
% less_induct
thf(fact_232_antisym__conv3,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_nat @ Y2 @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_233_antisym__conv3,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_real @ Y2 @ X )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv3
thf(fact_234_less__imp__triv,axiom,
! [X: nat,Y2: nat,P: $o] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_235_less__imp__triv,axiom,
! [X: real,Y2: real,P: $o] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_236_linorder__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_nat @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_237_linorder__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_cases
thf(fact_238_dual__order_Oasym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ~ ( ord_less_nat @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_239_dual__order_Oasym,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ~ ( ord_less_real @ A3 @ B3 ) ) ).
% dual_order.asym
thf(fact_240_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [N: nat] :
( ( P4 @ N )
& ! [M: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ( P4 @ M ) ) ) ) ) ).
% exists_least_iff
thf(fact_241_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat] : ( P @ A5 @ A5 )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_242_linorder__less__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A5: real,B5: real] :
( ( ord_less_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real] : ( P @ A5 @ A5 )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% linorder_less_wlog
thf(fact_243_order_Ostrict__trans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_244_order_Ostrict__trans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans
thf(fact_245_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ( ord_less_nat @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_246_not__less__iff__gr__or__eq,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ( ord_less_real @ Y2 @ X )
| ( X = Y2 ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_247_dual__order_Ostrict__trans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_248_dual__order_Ostrict__trans,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans
thf(fact_249_order_Ostrict__implies__not__eq,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_250_order_Ostrict__implies__not__eq,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( A3 != B3 ) ) ).
% order.strict_implies_not_eq
thf(fact_251_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_252_dual__order_Ostrict__implies__not__eq,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( A3 != B3 ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_253_all__less__two,axiom,
! [P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ ( suc @ zero_zero_nat ) ) )
=> ( P @ I2 ) ) )
= ( ( P @ zero_zero_nat )
& ( P @ ( suc @ zero_zero_nat ) ) ) ) ).
% all_less_two
thf(fact_254_diff__ev__def,axiom,
( jordan8934236962569034858v_real
= ( ^ [A: mat_real,I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( index_mat_real @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) )
!= ( index_mat_real @ A @ ( product_Pair_nat_nat @ J @ J ) ) )
=> ( ( index_mat_real @ A @ ( product_Pair_nat_nat @ I2 @ J ) )
= zero_zero_real ) ) ) ) ) ).
% diff_ev_def
thf(fact_255_diff__ev__def,axiom,
( jordan8650160714669549932omplex
= ( ^ [A: mat_complex,I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ I2 ) )
!= ( index_mat_complex @ A @ ( product_Pair_nat_nat @ J @ J ) ) )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ J ) )
= zero_zero_complex ) ) ) ) ) ).
% diff_ev_def
thf(fact_256_swap__cols__rows__carrier_I3_J,axiom,
! [A2: mat_complex,N2: nat,K2: nat,L: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( member_mat_complex @ ( column7161609239796038556omplex @ K2 @ L @ A2 ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ) ).
% swap_cols_rows_carrier(3)
thf(fact_257_uppert__def,axiom,
( jordan3508124462612338182t_real
= ( ^ [A: mat_real,I2: nat,J: nat] :
( ( ord_less_nat @ J @ I2 )
=> ( ( index_mat_real @ A @ ( product_Pair_nat_nat @ I2 @ J ) )
= zero_zero_real ) ) ) ) ).
% uppert_def
thf(fact_258_uppert__def,axiom,
( jordan3528196489273997576omplex
= ( ^ [A: mat_complex,I2: nat,J: nat] :
( ( ord_less_nat @ J @ I2 )
=> ( ( index_mat_complex @ A @ ( product_Pair_nat_nat @ I2 @ J ) )
= zero_zero_complex ) ) ) ) ).
% uppert_def
thf(fact_259_Reals__0,axiom,
member_real @ zero_zero_real @ real_V470468836141973256s_real ).
% Reals_0
thf(fact_260_Reals__0,axiom,
member_complex @ zero_zero_complex @ real_V2521375963428798218omplex ).
% Reals_0
thf(fact_261_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N3: nat] :
( ~ ( P @ N3 )
& ( P @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_262_zero__prod__def,axiom,
( zero_z8332228408419305374at_nat
= ( produc6161850002892822231at_nat @ zero_z3979849011205770936at_nat @ zero_z3979849011205770936at_nat ) ) ).
% zero_prod_def
thf(fact_263_zero__prod__def,axiom,
( zero_z3979849011205770936at_nat
= ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_264_zero__prod__def,axiom,
( zero_z738777567634093332t_real
= ( produc7837566107596912789t_real @ zero_zero_nat @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_265_zero__prod__def,axiom,
( zero_z631996502013145750omplex
= ( produc6973218034000581911omplex @ zero_zero_nat @ zero_zero_complex ) ) ).
% zero_prod_def
thf(fact_266_zero__prod__def,axiom,
( zero_z5987101913011988884al_nat
= ( produc3181502643871035669al_nat @ zero_zero_real @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_267_zero__prod__def,axiom,
( zero_z1365759597461889520l_real
= ( produc4511245868158468465l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_268_zero__prod__def,axiom,
( zero_z5510983778958038386omplex
= ( produc1693001998875562995omplex @ zero_zero_real @ zero_zero_complex ) ) ).
% zero_prod_def
thf(fact_269_zero__prod__def,axiom,
( zero_z6791906118007317398ex_nat
= ( produc1369629321580543767ex_nat @ zero_zero_complex @ zero_zero_nat ) ) ).
% zero_prod_def
thf(fact_270_zero__prod__def,axiom,
( zero_z7423682798772796146x_real
= ( produc1746590499379883635x_real @ zero_zero_complex @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_271_zero__prod__def,axiom,
( zero_z1220838019464432500omplex
= ( produc101793102246108661omplex @ zero_zero_complex @ zero_zero_complex ) ) ).
% zero_prod_def
thf(fact_272_rel__simps_I70_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% rel_simps(70)
thf(fact_273_rel__simps_I70_J,axiom,
~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).
% rel_simps(70)
thf(fact_274_zero__order_I5_J,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% zero_order(5)
thf(fact_275_zero__order_I4_J,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_order(4)
thf(fact_276_zero__order_I3_J,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% zero_order(3)
thf(fact_277_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_278_zero__reorient,axiom,
! [X: real] :
( ( zero_zero_real = X )
= ( X = zero_zero_real ) ) ).
% zero_reorient
thf(fact_279_zero__reorient,axiom,
! [X: complex] :
( ( zero_zero_complex = X )
= ( X = zero_zero_complex ) ) ).
% zero_reorient
thf(fact_280_gr__implies__not__zero,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_281_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_282_field__lbound__gt__zero,axiom,
! [D1: real,D2: real] :
( ( ord_less_real @ zero_zero_real @ D1 )
=> ( ( ord_less_real @ zero_zero_real @ D2 )
=> ? [E: real] :
( ( ord_less_real @ zero_zero_real @ E )
& ( ord_less_real @ E @ D1 )
& ( ord_less_real @ E @ D2 ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_283_commute__diag__mat__zero__comp,axiom,
! [D: mat_real,N2: nat,B2: mat_real,I3: nat,J2: nat] :
( ( diagonal_mat_real @ D )
=> ( ( member_mat_real @ D @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( ( times_times_mat_real @ B2 @ D )
= ( times_times_mat_real @ D @ B2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( index_mat_real @ D @ ( product_Pair_nat_nat @ I3 @ I3 ) )
!= ( index_mat_real @ D @ ( product_Pair_nat_nat @ J2 @ J2 ) ) )
=> ( ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_real ) ) ) ) ) ) ) ) ).
% commute_diag_mat_zero_comp
thf(fact_284_commute__diag__mat__zero__comp,axiom,
! [D: mat_complex,N2: nat,B2: mat_complex,I3: nat,J2: nat] :
( ( diagonal_mat_complex @ D )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( times_8009071140041733218omplex @ B2 @ D )
= ( times_8009071140041733218omplex @ D @ B2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( index_mat_complex @ D @ ( product_Pair_nat_nat @ I3 @ I3 ) )
!= ( index_mat_complex @ D @ ( product_Pair_nat_nat @ J2 @ J2 ) ) )
=> ( ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_complex ) ) ) ) ) ) ) ) ).
% commute_diag_mat_zero_comp
thf(fact_285_index__one__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_nat @ ( one_mat_nat @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_nat ) )
& ( ( I3 != J2 )
=> ( ( index_mat_nat @ ( one_mat_nat @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_nat ) ) ) ) ) ).
% index_one_mat(1)
thf(fact_286_index__one__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_real @ ( one_mat_real @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_real ) )
& ( ( I3 != J2 )
=> ( ( index_mat_real @ ( one_mat_real @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_real ) ) ) ) ) ).
% index_one_mat(1)
thf(fact_287_index__one__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_complex @ ( one_mat_complex @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_complex ) )
& ( ( I3 != J2 )
=> ( ( index_mat_complex @ ( one_mat_complex @ N2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_complex ) ) ) ) ) ).
% index_one_mat(1)
thf(fact_288_gauss__jordan__row__echelon,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,A4: mat_complex,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ A4 @ B6 ) )
=> ( gauss_194721375535881179omplex @ A4 ) ) ) ).
% gauss_jordan_row_echelon
thf(fact_289_synthetic__divmod__0,axiom,
! [C: nat] :
( ( synthetic_divmod_nat @ zero_zero_poly_nat @ C )
= ( produc8516950543009317441at_nat @ zero_zero_poly_nat @ zero_zero_nat ) ) ).
% synthetic_divmod_0
thf(fact_290_synthetic__divmod__0,axiom,
! [C: real] :
( ( synthe4136317906449589571d_real @ zero_zero_poly_real @ C )
= ( produc8317220421998033145l_real @ zero_zero_poly_real @ zero_zero_real ) ) ).
% synthetic_divmod_0
thf(fact_291_synthetic__divmod__0,axiom,
! [C: complex] :
( ( synthe6525044291898662085omplex @ zero_z2709840015065127615omplex @ C )
= ( produc7503770182060870909omplex @ zero_z2709840015065127615omplex @ zero_zero_complex ) ) ).
% synthetic_divmod_0
thf(fact_292_mult__delta__left,axiom,
! [B3: $o,X: complex,Y2: complex] :
( ( B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X @ zero_zero_complex ) @ Y2 )
= ( times_times_complex @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_complex @ ( if_complex @ B3 @ X @ zero_zero_complex ) @ Y2 )
= zero_zero_complex ) ) ) ).
% mult_delta_left
thf(fact_293_mult__delta__left,axiom,
! [B3: $o,X: nat,Y2: nat] :
( ( B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X @ zero_zero_nat ) @ Y2 )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_nat @ ( if_nat @ B3 @ X @ zero_zero_nat ) @ Y2 )
= zero_zero_nat ) ) ) ).
% mult_delta_left
thf(fact_294_mult__delta__left,axiom,
! [B3: $o,X: real,Y2: real] :
( ( B3
=> ( ( times_times_real @ ( if_real @ B3 @ X @ zero_zero_real ) @ Y2 )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_real @ ( if_real @ B3 @ X @ zero_zero_real ) @ Y2 )
= zero_zero_real ) ) ) ).
% mult_delta_left
thf(fact_295_mult__delta__right,axiom,
! [B3: $o,X: complex,Y2: complex] :
( ( B3
=> ( ( times_times_complex @ X @ ( if_complex @ B3 @ Y2 @ zero_zero_complex ) )
= ( times_times_complex @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_complex @ X @ ( if_complex @ B3 @ Y2 @ zero_zero_complex ) )
= zero_zero_complex ) ) ) ).
% mult_delta_right
thf(fact_296_mult__delta__right,axiom,
! [B3: $o,X: nat,Y2: nat] :
( ( B3
=> ( ( times_times_nat @ X @ ( if_nat @ B3 @ Y2 @ zero_zero_nat ) )
= ( times_times_nat @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_nat @ X @ ( if_nat @ B3 @ Y2 @ zero_zero_nat ) )
= zero_zero_nat ) ) ) ).
% mult_delta_right
thf(fact_297_mult__delta__right,axiom,
! [B3: $o,X: real,Y2: real] :
( ( B3
=> ( ( times_times_real @ X @ ( if_real @ B3 @ Y2 @ zero_zero_real ) )
= ( times_times_real @ X @ Y2 ) ) )
& ( ~ B3
=> ( ( times_times_real @ X @ ( if_real @ B3 @ Y2 @ zero_zero_real ) )
= zero_zero_real ) ) ) ).
% mult_delta_right
thf(fact_298_semiring__norm_I63_J,axiom,
! [A3: complex] :
( ( times_times_complex @ zero_zero_complex @ A3 )
= zero_zero_complex ) ).
% semiring_norm(63)
thf(fact_299_semiring__norm_I63_J,axiom,
! [A3: nat] :
( ( times_times_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% semiring_norm(63)
thf(fact_300_semiring__norm_I63_J,axiom,
! [A3: real] :
( ( times_times_real @ zero_zero_real @ A3 )
= zero_zero_real ) ).
% semiring_norm(63)
thf(fact_301_semiring__norm_I64_J,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% semiring_norm(64)
thf(fact_302_semiring__norm_I64_J,axiom,
! [A3: nat] :
( ( times_times_nat @ A3 @ zero_zero_nat )
= zero_zero_nat ) ).
% semiring_norm(64)
thf(fact_303_semiring__norm_I64_J,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% semiring_norm(64)
thf(fact_304_rel__simps_I93_J,axiom,
one_one_nat != zero_zero_nat ).
% rel_simps(93)
thf(fact_305_rel__simps_I93_J,axiom,
one_one_real != zero_zero_real ).
% rel_simps(93)
thf(fact_306_rel__simps_I93_J,axiom,
one_one_complex != zero_zero_complex ).
% rel_simps(93)
thf(fact_307_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_308_less__numeral__extra_I4_J,axiom,
~ ( ord_less_real @ one_one_real @ one_one_real ) ).
% less_numeral_extra(4)
thf(fact_309_mult__carrier__mat,axiom,
! [A2: mat_complex,Nr: nat,N2: nat,B2: mat_complex,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( member_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% mult_carrier_mat
thf(fact_310_assoc__mult__mat,axiom,
! [A2: mat_complex,N_1: nat,N_2: nat,B2: mat_complex,N_3: nat,C3: mat_complex,N_4: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N_1 @ N_2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N_2 @ N_3 ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N_3 @ N_4 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ C3 )
= ( times_8009071140041733218omplex @ A2 @ ( times_8009071140041733218omplex @ B2 @ C3 ) ) ) ) ) ) ).
% assoc_mult_mat
thf(fact_311_Reals__mult,axiom,
! [A3: real,B3: real] :
( ( member_real @ A3 @ real_V470468836141973256s_real )
=> ( ( member_real @ B3 @ real_V470468836141973256s_real )
=> ( member_real @ ( times_times_real @ A3 @ B3 ) @ real_V470468836141973256s_real ) ) ) ).
% Reals_mult
thf(fact_312_Reals__mult,axiom,
! [A3: complex,B3: complex] :
( ( member_complex @ A3 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ B3 @ real_V2521375963428798218omplex )
=> ( member_complex @ ( times_times_complex @ A3 @ B3 ) @ real_V2521375963428798218omplex ) ) ) ).
% Reals_mult
thf(fact_313_Reals__1,axiom,
member_real @ one_one_real @ real_V470468836141973256s_real ).
% Reals_1
thf(fact_314_Reals__1,axiom,
member_complex @ one_one_complex @ real_V2521375963428798218omplex ).
% Reals_1
thf(fact_315_rel__simps_I69_J,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% rel_simps(69)
thf(fact_316_rel__simps_I69_J,axiom,
~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).
% rel_simps(69)
thf(fact_317_rel__simps_I68_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% rel_simps(68)
thf(fact_318_rel__simps_I68_J,axiom,
ord_less_real @ zero_zero_real @ one_one_real ).
% rel_simps(68)
thf(fact_319_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_320_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_321_mat__mult__left__right__inverse,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( times_8009071140041733218omplex @ A2 @ B2 )
= ( one_mat_complex @ N2 ) )
=> ( ( times_8009071140041733218omplex @ B2 @ A2 )
= ( one_mat_complex @ N2 ) ) ) ) ) ).
% mat_mult_left_right_inverse
thf(fact_322_left__mult__one__mat,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( one_mat_complex @ Nr ) @ A2 )
= A2 ) ) ).
% left_mult_one_mat
thf(fact_323_right__mult__one__mat,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A2 @ ( one_mat_complex @ Nc ) )
= A2 ) ) ).
% right_mult_one_mat
thf(fact_324_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_325_gauss__jordan_I2_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex,Nc22: nat,C3: mat_complex,D: mat_complex,X5: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc22 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ C3 @ D ) )
=> ( ( member_mat_complex @ X5 @ ( carrier_mat_complex @ Nc @ Nc22 ) )
=> ( ( ( times_8009071140041733218omplex @ A2 @ X5 )
= B2 )
= ( ( times_8009071140041733218omplex @ C3 @ X5 )
= D ) ) ) ) ) ) ).
% gauss_jordan(2)
thf(fact_326_identify__block_Oinduct,axiom,
! [P: mat_complex > nat > $o,A0: mat_complex,A1: nat] :
( ! [A7: mat_complex] : ( P @ A7 @ zero_zero_nat )
=> ( ! [A7: mat_complex,I4: nat] :
( ( ( ( index_mat_complex @ A7 @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_complex )
=> ( P @ A7 @ I4 ) )
=> ( P @ A7 @ ( suc @ I4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_327_identify__block_Oinduct,axiom,
! [P: mat_nat > nat > $o,A0: mat_nat,A1: nat] :
( ! [A7: mat_nat] : ( P @ A7 @ zero_zero_nat )
=> ( ! [A7: mat_nat,I4: nat] :
( ( ( ( index_mat_nat @ A7 @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_nat )
=> ( P @ A7 @ I4 ) )
=> ( P @ A7 @ ( suc @ I4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_328_identify__block_Oinduct,axiom,
! [P: mat_real > nat > $o,A0: mat_real,A1: nat] :
( ! [A7: mat_real] : ( P @ A7 @ zero_zero_nat )
=> ( ! [A7: mat_real,I4: nat] :
( ( ( ( index_mat_real @ A7 @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_real )
=> ( P @ A7 @ I4 ) )
=> ( P @ A7 @ ( suc @ I4 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% identify_block.induct
thf(fact_329_gauss__jordan__compute__inverse_I2_J,axiom,
! [A2: mat_complex,N2: nat,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ ( one_mat_complex @ N2 ) )
= ( produc3658446505030690647omplex @ ( one_mat_complex @ N2 ) @ B6 ) )
=> ( ( times_8009071140041733218omplex @ B6 @ A2 )
= ( one_mat_complex @ N2 ) ) ) ) ).
% gauss_jordan_compute_inverse(2)
thf(fact_330_gauss__jordan__compute__inverse_I1_J,axiom,
! [A2: mat_complex,N2: nat,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ ( one_mat_complex @ N2 ) )
= ( produc3658446505030690647omplex @ ( one_mat_complex @ N2 ) @ B6 ) )
=> ( ( times_8009071140041733218omplex @ A2 @ B6 )
= ( one_mat_complex @ N2 ) ) ) ) ).
% gauss_jordan_compute_inverse(1)
thf(fact_331_gauss__jordan__inverse__one__direction_I2_J,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,Nc: nat,B6: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( ( ( gauss_1847731261493334972omplex @ A2 @ B2 )
= ( produc3658446505030690647omplex @ ( one_mat_complex @ N2 ) @ B6 ) )
=> ( ( B2
= ( one_mat_complex @ N2 ) )
=> ( ( ( times_8009071140041733218omplex @ A2 @ B6 )
= ( one_mat_complex @ N2 ) )
& ( ( times_8009071140041733218omplex @ B6 @ A2 )
= ( one_mat_complex @ N2 ) ) ) ) ) ) ) ).
% gauss_jordan_inverse_one_direction(2)
thf(fact_332_diagonal__mat__sq__index,axiom,
! [B2: mat_real,N2: nat,I3: nat,J2: nat] :
( ( diagonal_mat_real @ B2 )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_real @ ( times_times_mat_real @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_real @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I3 @ I3 ) ) @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ J2 @ I3 ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index
thf(fact_333_diagonal__mat__sq__index,axiom,
! [B2: mat_complex,N2: nat,I3: nat,J2: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I3 @ I3 ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ J2 @ I3 ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index
thf(fact_334_diagonal__mat__sq__index_H,axiom,
! [B2: mat_real,N2: nat,I3: nat,J2: nat] :
( ( diagonal_mat_real @ B2 )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_real @ ( times_times_mat_real @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_real @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index'
thf(fact_335_diagonal__mat__sq__index_H,axiom,
! [B2: mat_complex,N2: nat,I3: nat,J2: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ).
% diagonal_mat_sq_index'
thf(fact_336_diagonal__mat__mult__index,axiom,
! [A2: mat_real,N2: nat,B2: mat_real,I3: nat,J2: nat] :
( ( diagonal_mat_real @ A2 )
=> ( ( member_mat_real @ A2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_real @ ( times_times_mat_real @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_real @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I3 @ I3 ) ) @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index
thf(fact_337_diagonal__mat__mult__index,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,I3: nat,J2: nat] :
( ( diagonal_mat_complex @ A2 )
=> ( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_complex @ ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ I3 ) ) @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index
thf(fact_338_diagonal__mat__mult__index_H,axiom,
! [A2: mat_real,N2: nat,B2: mat_real,J2: nat,I3: nat] :
( ( member_mat_real @ A2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( member_mat_real @ B2 @ ( carrier_mat_real @ N2 @ N2 ) )
=> ( ( diagonal_mat_real @ B2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( index_mat_real @ ( times_times_mat_real @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_real @ ( index_mat_real @ B2 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) @ ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index'
thf(fact_339_diagonal__mat__mult__index_H,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,J2: nat,I3: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ord_less_nat @ I3 @ N2 )
=> ( ( index_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= ( times_times_complex @ ( index_mat_complex @ B2 @ ( product_Pair_nat_nat @ J2 @ J2 ) ) @ ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ J2 ) ) ) ) ) ) ) ) ) ).
% diagonal_mat_mult_index'
thf(fact_340_diagonal__mat__commute,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( diagonal_mat_complex @ A2 )
=> ( ( diagonal_mat_complex @ B2 )
=> ( ( times_8009071140041733218omplex @ A2 @ B2 )
= ( times_8009071140041733218omplex @ B2 @ A2 ) ) ) ) ) ) ).
% diagonal_mat_commute
thf(fact_341_diagonal__mat__sq__diag,axiom,
! [B2: mat_complex,N2: nat] :
( ( diagonal_mat_complex @ B2 )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ B2 @ B2 ) ) ) ) ).
% diagonal_mat_sq_diag
thf(fact_342_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_343_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_344_nat__1__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_345_nat__mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= one_one_nat )
= ( ( M2 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_346_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( ( K2 = zero_zero_nat )
| ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_347_mult__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ K2 )
= ( times_times_nat @ N2 @ K2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_348_mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( ( M2 = N2 )
| ( K2 = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_349_mult__0__right,axiom,
! [M2: nat] :
( ( times_times_nat @ M2 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_350_mult__is__0,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ( M2 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_351_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_352_mult__eq__self__implies__10,axiom,
! [M2: nat,N2: nat] :
( ( M2
= ( times_times_nat @ M2 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M2 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_353_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K2 ) @ M2 )
= ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( M2 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_354_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_355_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ).
% nat_mult_less_cancel1
thf(fact_356_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ( times_times_nat @ K2 @ M2 )
= ( times_times_nat @ K2 @ N2 ) )
= ( M2 = N2 ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_357_mult__less__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_358_mult__less__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_359_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ord_less_nat @ M2 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_360_nat__0__less__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M2 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_361_one__eq__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M2 @ N2 ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_362_mult__eq__1__iff,axiom,
! [M2: nat,N2: nat] :
( ( ( times_times_nat @ M2 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M2
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_363_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_364_one__less__mult,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% one_less_mult
thf(fact_365_n__less__m__mult__n,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% n_less_m_mult_n
thf(fact_366_n__less__n__mult__m,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M2 )
=> ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_367_diagonal__mat__times__diag,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( diagonal_mat_complex @ A2 )
=> ( ( diagonal_mat_complex @ B2 )
=> ( diagonal_mat_complex @ ( times_8009071140041733218omplex @ A2 @ B2 ) ) ) ) ) ) ).
% diagonal_mat_times_diag
thf(fact_368_less__1__mult,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ one_one_nat @ M2 )
=> ( ( ord_less_nat @ one_one_nat @ N2 )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_369_less__1__mult,axiom,
! [M2: real,N2: real] :
( ( ord_less_real @ one_one_real @ M2 )
=> ( ( ord_less_real @ one_one_real @ N2 )
=> ( ord_less_real @ one_one_real @ ( times_times_real @ M2 @ N2 ) ) ) ) ).
% less_1_mult
thf(fact_370_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_371_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_372_mult__if__delta,axiom,
! [P: $o,Q: complex] :
( ( P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_complex @ ( if_complex @ P @ one_one_complex @ zero_zero_complex ) @ Q )
= zero_zero_complex ) ) ) ).
% mult_if_delta
thf(fact_373_mult__if__delta,axiom,
! [P: $o,Q: nat] :
( ( P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_nat @ ( if_nat @ P @ one_one_nat @ zero_zero_nat ) @ Q )
= zero_zero_nat ) ) ) ).
% mult_if_delta
thf(fact_374_mult__if__delta,axiom,
! [P: $o,Q: real] :
( ( P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= Q ) )
& ( ~ P
=> ( ( times_times_real @ ( if_real @ P @ one_one_real @ zero_zero_real ) @ Q )
= zero_zero_real ) ) ) ).
% mult_if_delta
thf(fact_375_verit__comp__simplify_I1_J,axiom,
! [A3: nat] :
~ ( ord_less_nat @ A3 @ A3 ) ).
% verit_comp_simplify(1)
thf(fact_376_verit__comp__simplify_I1_J,axiom,
! [A3: real] :
~ ( ord_less_real @ A3 @ A3 ) ).
% verit_comp_simplify(1)
thf(fact_377_linorder__neqE__linordered__idom,axiom,
! [X: real,Y2: real] :
( ( X != Y2 )
=> ( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ Y2 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_378_mult__not__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
& ( B3 != zero_zero_complex ) ) ) ).
% mult_not_zero
thf(fact_379_mult__not__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat )
=> ( ( A3 != zero_zero_nat )
& ( B3 != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_380_mult__not__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real )
=> ( ( A3 != zero_zero_real )
& ( B3 != zero_zero_real ) ) ) ).
% mult_not_zero
thf(fact_381_divisors__zero,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
=> ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% divisors_zero
thf(fact_382_divisors__zero,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
=> ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_383_divisors__zero,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
=> ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% divisors_zero
thf(fact_384_mult__eq__0__iff,axiom,
! [A3: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ B3 )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( B3 = zero_zero_complex ) ) ) ).
% mult_eq_0_iff
thf(fact_385_mult__eq__0__iff,axiom,
! [A3: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ B3 )
= zero_zero_nat )
= ( ( A3 = zero_zero_nat )
| ( B3 = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_386_mult__eq__0__iff,axiom,
! [A3: real,B3: real] :
( ( ( times_times_real @ A3 @ B3 )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( B3 = zero_zero_real ) ) ) ).
% mult_eq_0_iff
thf(fact_387_no__zero__divisors,axiom,
! [A3: complex,B3: complex] :
( ( A3 != zero_zero_complex )
=> ( ( B3 != zero_zero_complex )
=> ( ( times_times_complex @ A3 @ B3 )
!= zero_zero_complex ) ) ) ).
% no_zero_divisors
thf(fact_388_no__zero__divisors,axiom,
! [A3: nat,B3: nat] :
( ( A3 != zero_zero_nat )
=> ( ( B3 != zero_zero_nat )
=> ( ( times_times_nat @ A3 @ B3 )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_389_no__zero__divisors,axiom,
! [A3: real,B3: real] :
( ( A3 != zero_zero_real )
=> ( ( B3 != zero_zero_real )
=> ( ( times_times_real @ A3 @ B3 )
!= zero_zero_real ) ) ) ).
% no_zero_divisors
thf(fact_390_mult__cancel__left,axiom,
! [C: complex,A3: complex,B3: complex] :
( ( ( times_times_complex @ C @ A3 )
= ( times_times_complex @ C @ B3 ) )
= ( ( C = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_391_mult__cancel__left,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B3 ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_392_mult__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_left
thf(fact_393_mult__left__cancel,axiom,
! [C: complex,A3: complex,B3: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ C @ A3 )
= ( times_times_complex @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_394_mult__left__cancel,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A3 )
= ( times_times_nat @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_395_mult__left__cancel,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ C @ A3 )
= ( times_times_real @ C @ B3 ) )
= ( A3 = B3 ) ) ) ).
% mult_left_cancel
thf(fact_396_mult__cancel__right,axiom,
! [A3: complex,C: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ C )
= ( times_times_complex @ B3 @ C ) )
= ( ( C = zero_zero_complex )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_397_mult__cancel__right,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( ( C = zero_zero_nat )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_398_mult__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( A3 = B3 ) ) ) ).
% mult_cancel_right
thf(fact_399_mult__right__cancel,axiom,
! [C: complex,A3: complex,B3: complex] :
( ( C != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ C )
= ( times_times_complex @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_400_mult__right__cancel,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A3 @ C )
= ( times_times_nat @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_401_mult__right__cancel,axiom,
! [C: real,A3: real,B3: real] :
( ( C != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ C )
= ( times_times_real @ B3 @ C ) )
= ( A3 = B3 ) ) ) ).
% mult_right_cancel
thf(fact_402_mult__sign__intros_I8_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(8)
thf(fact_403_mult__sign__intros_I7_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(7)
thf(fact_404_mult__sign__intros_I7_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ zero_zero_real )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(7)
thf(fact_405_mult__sign__intros_I6_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(6)
thf(fact_406_mult__sign__intros_I6_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(6)
thf(fact_407_mult__sign__intros_I5_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_408_mult__sign__intros_I5_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(5)
thf(fact_409_not__square__less__zero,axiom,
! [A3: real] :
~ ( ord_less_real @ ( times_times_real @ A3 @ A3 ) @ zero_zero_real ) ).
% not_square_less_zero
thf(fact_410_mult__less__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_less_0_iff
thf(fact_411_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ B3 @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_412_linordered__semiring__strict__class_Omult__pos__neg2,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ B3 @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% linordered_semiring_strict_class.mult_pos_neg2
thf(fact_413_zero__less__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ A3 )
& ( ord_less_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_real @ A3 @ zero_zero_real )
& ( ord_less_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_less_mult_iff
thf(fact_414_zero__less__mult__pos,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_415_zero__less__mult__pos,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos
thf(fact_416_zero__less__mult__pos2,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B3 @ A3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ord_less_nat @ zero_zero_nat @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_417_zero__less__mult__pos2,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B3 @ A3 ) )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ord_less_real @ zero_zero_real @ B3 ) ) ) ).
% zero_less_mult_pos2
thf(fact_418_mult__less__cancel__left__neg,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_419_mult__less__cancel__left__pos,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_420_mult__strict__left__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_421_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_422_linordered__semiring__strict__class_Omult__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_left_mono
thf(fact_423_mult__less__cancel__left__disj,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_424_mult__strict__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_425_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_426_linordered__semiring__strict__class_Omult__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_right_mono
thf(fact_427_mult__less__cancel__right__disj,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
& ( ord_less_real @ A3 @ B3 ) )
| ( ( ord_less_real @ C @ zero_zero_real )
& ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_428_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_429_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_430_mult__cancel__left1,axiom,
! [C: complex,B3: complex] :
( ( C
= ( times_times_complex @ C @ B3 ) )
= ( ( C = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_left1
thf(fact_431_mult__cancel__left1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ C @ B3 ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_left1
thf(fact_432_mult__cancel__left2,axiom,
! [C: complex,A3: complex] :
( ( ( times_times_complex @ C @ A3 )
= C )
= ( ( C = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_left2
thf(fact_433_mult__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ( times_times_real @ C @ A3 )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_left2
thf(fact_434_mult__cancel__right1,axiom,
! [C: complex,B3: complex] :
( ( C
= ( times_times_complex @ B3 @ C ) )
= ( ( C = zero_zero_complex )
| ( B3 = one_one_complex ) ) ) ).
% mult_cancel_right1
thf(fact_435_mult__cancel__right1,axiom,
! [C: real,B3: real] :
( ( C
= ( times_times_real @ B3 @ C ) )
= ( ( C = zero_zero_real )
| ( B3 = one_one_real ) ) ) ).
% mult_cancel_right1
thf(fact_436_mult__cancel__right2,axiom,
! [A3: complex,C: complex] :
( ( ( times_times_complex @ A3 @ C )
= C )
= ( ( C = zero_zero_complex )
| ( A3 = one_one_complex ) ) ) ).
% mult_cancel_right2
thf(fact_437_mult__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ( times_times_real @ A3 @ C )
= C )
= ( ( C = zero_zero_real )
| ( A3 = one_one_real ) ) ) ).
% mult_cancel_right2
thf(fact_438_mult__less__iff1,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ Z ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% mult_less_iff1
thf(fact_439_gauss__jordan__single_I4_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,C3: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A2 )
= C3 )
=> ? [P5: mat_complex,Q2: mat_complex] :
( ( C3
= ( times_8009071140041733218omplex @ P5 @ A2 ) )
& ( member_mat_complex @ P5 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( member_mat_complex @ Q2 @ ( carrier_mat_complex @ Nr @ Nr ) )
& ( ( times_8009071140041733218omplex @ P5 @ Q2 )
= ( one_mat_complex @ Nr ) )
& ( ( times_8009071140041733218omplex @ Q2 @ P5 )
= ( one_mat_complex @ Nr ) ) ) ) ) ).
% gauss_jordan_single(4)
thf(fact_440_index__mat__multrow__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,A3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= A3 ) )
& ( ~ ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_nat ) )
& ( ( I3 != J2 )
=> ( ( index_mat_nat @ ( gauss_3195076542185637913at_nat @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_nat ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_441_index__mat__multrow__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,A3: real] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= A3 ) )
& ( ~ ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_real ) )
& ( ( I3 != J2 )
=> ( ( index_mat_real @ ( gauss_7241202418770761333t_real @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_real ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_442_index__mat__multrow__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,A3: complex] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( index_mat_complex @ ( gauss_6868829418328711927omplex @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= A3 ) )
& ( ~ ( ( K2 = I3 )
& ( K2 = J2 ) )
=> ( ( ( I3 = J2 )
=> ( ( index_mat_complex @ ( gauss_6868829418328711927omplex @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_complex ) )
& ( ( I3 != J2 )
=> ( ( index_mat_complex @ ( gauss_6868829418328711927omplex @ N2 @ K2 @ A3 ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_complex ) ) ) ) ) ) ) ).
% index_mat_multrow_mat(1)
thf(fact_443_multrow__mat__carrier,axiom,
! [N2: nat,K2: nat,A3: complex] : ( member_mat_complex @ ( gauss_6868829418328711927omplex @ N2 @ K2 @ A3 ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% multrow_mat_carrier
thf(fact_444_gauss__jordan__single_I2_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,C3: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A2 )
= C3 )
=> ( member_mat_complex @ C3 @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% gauss_jordan_single(2)
thf(fact_445_gauss__jordan__single_I3_J,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,C3: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( ( gauss_4244865067341541924omplex @ A2 )
= C3 )
=> ( gauss_194721375535881179omplex @ C3 ) ) ) ).
% gauss_jordan_single(3)
thf(fact_446_index__mat__swaprows__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_nat @ ( gauss_4919907329869174035at_nat @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_nat ) )
& ( ~ ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_nat @ ( gauss_4919907329869174035at_nat @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_nat ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_447_index__mat__swaprows__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_real @ ( gauss_1271566072679876207t_real @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_real ) )
& ( ~ ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_real @ ( gauss_1271566072679876207t_real @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_real ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_448_index__mat__swaprows__mat_I1_J,axiom,
! [I3: nat,N2: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( ( ord_less_nat @ J2 @ N2 )
=> ( ( ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_complex @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= one_one_complex ) )
& ( ~ ( ( ( K2 = I3 )
& ( L = J2 ) )
| ( ( K2 = J2 )
& ( L = I3 ) )
| ( ( I3 = J2 )
& ( I3 != K2 )
& ( I3 != L ) ) )
=> ( ( index_mat_complex @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) @ ( product_Pair_nat_nat @ I3 @ J2 ) )
= zero_zero_complex ) ) ) ) ) ).
% index_mat_swaprows_mat(1)
thf(fact_449_multcol__mat,axiom,
! [A2: mat_complex,Nr: nat,N2: nat,K2: nat,A3: complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( column4410001698458707789omplex @ K2 @ A3 @ A2 )
= ( times_8009071140041733218omplex @ A2 @ ( gauss_6868829418328711927omplex @ N2 @ K2 @ A3 ) ) ) ) ).
% multcol_mat
thf(fact_450_class__field_Oone__not__zero,axiom,
one_one_real != zero_zero_real ).
% class_field.one_not_zero
thf(fact_451_class__field_Oone__not__zero,axiom,
one_one_complex != zero_zero_complex ).
% class_field.one_not_zero
thf(fact_452_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: complex,A3: complex,B3: complex] :
( ( X != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_453_vector__space__over__itself_Oscale__right__imp__eq,axiom,
! [X: real,A3: real,B3: real] :
( ( X != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B3 @ X ) )
=> ( A3 = B3 ) ) ) ).
% vector_space_over_itself.scale_right_imp_eq
thf(fact_454_swaprows__mat__carrier,axiom,
! [N2: nat,K2: nat,L: nat] : ( member_mat_complex @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) @ ( carrier_mat_complex @ N2 @ N2 ) ) ).
% swaprows_mat_carrier
thf(fact_455_swaprows__mat__inv,axiom,
! [K2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ L @ N2 )
=> ( ( times_8009071140041733218omplex @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) )
= ( one_mat_complex @ N2 ) ) ) ) ).
% swaprows_mat_inv
thf(fact_456_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: complex,X: complex] :
( ( ( times_times_complex @ A3 @ X )
= zero_zero_complex )
= ( ( A3 = zero_zero_complex )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_457_vector__space__over__itself_Oscale__eq__0__iff,axiom,
! [A3: real,X: real] :
( ( ( times_times_real @ A3 @ X )
= zero_zero_real )
= ( ( A3 = zero_zero_real )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_eq_0_iff
thf(fact_458_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: complex] :
( ( times_times_complex @ zero_zero_complex @ X )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_459_vector__space__over__itself_Oscale__zero__left,axiom,
! [X: real] :
( ( times_times_real @ zero_zero_real @ X )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_left
thf(fact_460_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: complex] :
( ( times_times_complex @ A3 @ zero_zero_complex )
= zero_zero_complex ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_461_vector__space__over__itself_Oscale__zero__right,axiom,
! [A3: real] :
( ( times_times_real @ A3 @ zero_zero_real )
= zero_zero_real ) ).
% vector_space_over_itself.scale_zero_right
thf(fact_462_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: complex,X: complex,Y2: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y2 ) )
= ( ( X = Y2 )
| ( A3 = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_463_vector__space__over__itself_Oscale__cancel__left,axiom,
! [A3: real,X: real,Y2: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
= ( ( X = Y2 )
| ( A3 = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_left
thf(fact_464_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: complex,X: complex,Y2: complex] :
( ( A3 != zero_zero_complex )
=> ( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ A3 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_465_vector__space__over__itself_Oscale__left__imp__eq,axiom,
! [A3: real,X: real,Y2: real] :
( ( A3 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ A3 @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% vector_space_over_itself.scale_left_imp_eq
thf(fact_466_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: complex,X: complex,B3: complex] :
( ( ( times_times_complex @ A3 @ X )
= ( times_times_complex @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X = zero_zero_complex ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_467_vector__space__over__itself_Oscale__cancel__right,axiom,
! [A3: real,X: real,B3: real] :
( ( ( times_times_real @ A3 @ X )
= ( times_times_real @ B3 @ X ) )
= ( ( A3 = B3 )
| ( X = zero_zero_real ) ) ) ).
% vector_space_over_itself.scale_cancel_right
thf(fact_468_swapcols__mat,axiom,
! [A2: mat_complex,Nr: nat,N2: nat,K2: nat,L: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( ord_less_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ L @ N2 )
=> ( ( column4357519492343924999omplex @ K2 @ L @ A2 )
= ( times_8009071140041733218omplex @ A2 @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) ) ) ) ) ) ).
% swapcols_mat
thf(fact_469_mult__hom_Ohom__zero,axiom,
! [C: complex] :
( ( times_times_complex @ C @ zero_zero_complex )
= zero_zero_complex ) ).
% mult_hom.hom_zero
thf(fact_470_mult__hom_Ohom__zero,axiom,
! [C: nat] :
( ( times_times_nat @ C @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_hom.hom_zero
thf(fact_471_mult__hom_Ohom__zero,axiom,
! [C: real] :
( ( times_times_real @ C @ zero_zero_real )
= zero_zero_real ) ).
% mult_hom.hom_zero
thf(fact_472_identify__block_Oelims,axiom,
! [X: mat_complex,Xa: nat,Y2: nat] :
( ( ( jordan3525277539992963945omplex @ X @ Xa )
= Y2 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [I4: nat] :
( ( Xa
= ( suc @ I4 ) )
=> ~ ( ( ( ( index_mat_complex @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_complex )
=> ( Y2
= ( jordan3525277539992963945omplex @ X @ I4 ) ) )
& ( ( ( index_mat_complex @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
!= one_one_complex )
=> ( Y2
= ( suc @ I4 ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_473_identify__block_Oelims,axiom,
! [X: mat_nat,Xa: nat,Y2: nat] :
( ( ( jordan8923406848002823307ck_nat @ X @ Xa )
= Y2 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [I4: nat] :
( ( Xa
= ( suc @ I4 ) )
=> ~ ( ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_nat )
=> ( Y2
= ( jordan8923406848002823307ck_nat @ X @ I4 ) ) )
& ( ( ( index_mat_nat @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
!= one_one_nat )
=> ( Y2
= ( suc @ I4 ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_474_identify__block_Oelims,axiom,
! [X: mat_real,Xa: nat,Y2: nat] :
( ( ( jordan6672758942465739239k_real @ X @ Xa )
= Y2 )
=> ( ( ( Xa = zero_zero_nat )
=> ( Y2 != zero_zero_nat ) )
=> ~ ! [I4: nat] :
( ( Xa
= ( suc @ I4 ) )
=> ~ ( ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
= one_one_real )
=> ( Y2
= ( jordan6672758942465739239k_real @ X @ I4 ) ) )
& ( ( ( index_mat_real @ X @ ( product_Pair_nat_nat @ I4 @ ( suc @ I4 ) ) )
!= one_one_real )
=> ( Y2
= ( suc @ I4 ) ) ) ) ) ) ) ).
% identify_block.elims
thf(fact_475_swapcols__carrier,axiom,
! [L: nat,K2: nat,A2: mat_complex,N2: nat,M2: nat] :
( ( member_mat_complex @ ( column4357519492343924999omplex @ L @ K2 @ A2 ) @ ( carrier_mat_complex @ N2 @ M2 ) )
= ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ M2 ) ) ) ).
% swapcols_carrier
thf(fact_476_identify__block_Osimps_I2_J,axiom,
! [A2: mat_complex,I3: nat] :
( ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
= one_one_complex )
=> ( ( jordan3525277539992963945omplex @ A2 @ ( suc @ I3 ) )
= ( jordan3525277539992963945omplex @ A2 @ I3 ) ) )
& ( ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
!= one_one_complex )
=> ( ( jordan3525277539992963945omplex @ A2 @ ( suc @ I3 ) )
= ( suc @ I3 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_477_identify__block_Osimps_I2_J,axiom,
! [A2: mat_nat,I3: nat] :
( ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A2 @ ( suc @ I3 ) )
= ( jordan8923406848002823307ck_nat @ A2 @ I3 ) ) )
& ( ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
!= one_one_nat )
=> ( ( jordan8923406848002823307ck_nat @ A2 @ ( suc @ I3 ) )
= ( suc @ I3 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_478_identify__block_Osimps_I2_J,axiom,
! [A2: mat_real,I3: nat] :
( ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A2 @ ( suc @ I3 ) )
= ( jordan6672758942465739239k_real @ A2 @ I3 ) ) )
& ( ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ I3 @ ( suc @ I3 ) ) )
!= one_one_real )
=> ( ( jordan6672758942465739239k_real @ A2 @ ( suc @ I3 ) )
= ( suc @ I3 ) ) ) ) ).
% identify_block.simps(2)
thf(fact_479_identify__block_I3_J,axiom,
! [A2: mat_complex,J2: nat,I3: nat,K2: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ K2 )
=> ( ( ord_less_nat @ K2 @ J2 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ K2 @ ( suc @ K2 ) ) )
= one_one_complex ) ) ) ) ).
% identify_block(3)
thf(fact_480_identify__block_I3_J,axiom,
! [A2: mat_nat,J2: nat,I3: nat,K2: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ K2 )
=> ( ( ord_less_nat @ K2 @ J2 )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ K2 @ ( suc @ K2 ) ) )
= one_one_nat ) ) ) ) ).
% identify_block(3)
thf(fact_481_identify__block_I3_J,axiom,
! [A2: mat_real,J2: nat,I3: nat,K2: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ K2 )
=> ( ( ord_less_nat @ K2 @ J2 )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ K2 @ ( suc @ K2 ) ) )
= one_one_real ) ) ) ) ).
% identify_block(3)
thf(fact_482_identify__block_I2_J,axiom,
! [A2: mat_complex,J2: nat,I3: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J2 )
= I3 )
=> ( ( I3 = zero_zero_nat )
| ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_complex ) ) ) ).
% identify_block(2)
thf(fact_483_identify__block_I2_J,axiom,
! [A2: mat_nat,J2: nat,I3: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J2 )
= I3 )
=> ( ( I3 = zero_zero_nat )
| ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_nat ) ) ) ).
% identify_block(2)
thf(fact_484_identify__block_I2_J,axiom,
! [A2: mat_real,J2: nat,I3: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J2 )
= I3 )
=> ( ( I3 = zero_zero_nat )
| ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_real ) ) ) ).
% identify_block(2)
thf(fact_485_swaprows__mat,axiom,
! [A2: mat_complex,N2: nat,Nc: nat,K2: nat,L: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( ( ord_less_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ L @ N2 )
=> ( ( gauss_1020679828357514249omplex @ K2 @ L @ A2 )
= ( times_8009071140041733218omplex @ ( gauss_8970452565587180529omplex @ N2 @ K2 @ L ) @ A2 ) ) ) ) ) ).
% swaprows_mat
thf(fact_486_arithmetic__simps_I57_J,axiom,
! [A3: complex] :
( ( minus_minus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% arithmetic_simps(57)
thf(fact_487_arithmetic__simps_I57_J,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% arithmetic_simps(57)
thf(fact_488_ge__iff__diff__ge__0,axiom,
( ord_less_eq_real
= ( ^ [B7: real,A8: real] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A8 @ B7 ) ) ) ) ).
% ge_iff_diff_ge_0
thf(fact_489_diff__self,axiom,
! [A3: complex] :
( ( minus_minus_complex @ A3 @ A3 )
= zero_zero_complex ) ).
% diff_self
thf(fact_490_diff__self,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% diff_self
thf(fact_491_right__minus__eq,axiom,
! [A3: complex,B3: complex] :
( ( ( minus_minus_complex @ A3 @ B3 )
= zero_zero_complex )
= ( A3 = B3 ) ) ).
% right_minus_eq
thf(fact_492_right__minus__eq,axiom,
! [A3: real,B3: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= zero_zero_real )
= ( A3 = B3 ) ) ).
% right_minus_eq
thf(fact_493_zero__diff,axiom,
! [A3: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A3 )
= zero_zero_nat ) ).
% zero_diff
thf(fact_494_diff__zero,axiom,
! [A3: complex] :
( ( minus_minus_complex @ A3 @ zero_zero_complex )
= A3 ) ).
% diff_zero
thf(fact_495_diff__zero,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ zero_zero_nat )
= A3 ) ).
% diff_zero
thf(fact_496_diff__zero,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ zero_zero_real )
= A3 ) ).
% diff_zero
thf(fact_497_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: complex] :
( ( minus_minus_complex @ A3 @ A3 )
= zero_zero_complex ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_498_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: nat] :
( ( minus_minus_nat @ A3 @ A3 )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_499_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A3: real] :
( ( minus_minus_real @ A3 @ A3 )
= zero_zero_real ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_500_diff__strict__mono,axiom,
! [A3: real,B3: real,D3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ D3 @ C )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D3 ) ) ) ) ).
% diff_strict_mono
thf(fact_501_diff__eq__diff__less,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ C @ D3 ) )
=> ( ( ord_less_real @ A3 @ B3 )
= ( ord_less_real @ C @ D3 ) ) ) ).
% diff_eq_diff_less
thf(fact_502_diff__strict__left__mono,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_real @ ( minus_minus_real @ C @ A3 ) @ ( minus_minus_real @ C @ B3 ) ) ) ).
% diff_strict_left_mono
thf(fact_503_diff__strict__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_504_rel__simps_I46_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% rel_simps(46)
thf(fact_505_rel__simps_I46_J,axiom,
ord_less_eq_real @ zero_zero_real @ zero_zero_real ).
% rel_simps(46)
thf(fact_506_zero__order_I2_J,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% zero_order(2)
thf(fact_507_zero__order_I1_J,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_order(1)
thf(fact_508_basic__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_509_basic__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_510_basic__trans__rules_I3_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_511_basic__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_512_basic__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_513_basic__trans__rules_I3_J,axiom,
! [A3: real,B3: real,F: real > complex,C: complex] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_514_basic__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_515_basic__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > real,C: real] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_real @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_516_basic__trans__rules_I3_J,axiom,
! [A3: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(3)
thf(fact_517_basic__trans__rules_I4_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_518_basic__trans__rules_I4_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_519_basic__trans__rules_I4_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_520_basic__trans__rules_I4_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_521_basic__trans__rules_I4_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_522_basic__trans__rules_I4_J,axiom,
! [A3: complex,F: real > complex,B3: real,C: real] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(4)
thf(fact_523_basic__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_524_basic__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_525_basic__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_526_basic__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_527_basic__trans__rules_I5_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_nat @ X2 @ Y )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_528_basic__trans__rules_I5_J,axiom,
! [A3: real,B3: real,F: real > complex,C: complex] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_real @ X2 @ Y )
=> ( ord_less_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(5)
thf(fact_529_basic__trans__rules_I6_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_530_basic__trans__rules_I6_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_531_basic__trans__rules_I6_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_532_basic__trans__rules_I6_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_533_basic__trans__rules_I6_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_534_basic__trans__rules_I6_J,axiom,
! [A3: complex,F: real > complex,B3: real,C: real] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_535_basic__trans__rules_I6_J,axiom,
! [A3: nat,F: complex > nat,B3: complex,C: complex] :
( ( ord_less_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_536_basic__trans__rules_I6_J,axiom,
! [A3: real,F: complex > real,B3: complex,C: complex] :
( ( ord_less_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_537_basic__trans__rules_I6_J,axiom,
! [A3: complex,F: complex > complex,B3: complex,C: complex] :
( ( ord_less_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(6)
thf(fact_538_basic__trans__rules_I17_J,axiom,
! [A3: nat,B3: nat] :
( ( A3 != B3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_539_basic__trans__rules_I17_J,axiom,
! [A3: real,B3: real] :
( ( A3 != B3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_540_basic__trans__rules_I17_J,axiom,
! [A3: complex,B3: complex] :
( ( A3 != B3 )
=> ( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ord_less_complex @ A3 @ B3 ) ) ) ).
% basic_trans_rules(17)
thf(fact_541_basic__trans__rules_I18_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_542_basic__trans__rules_I18_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_543_basic__trans__rules_I18_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( A3 != B3 )
=> ( ord_less_complex @ A3 @ B3 ) ) ) ).
% basic_trans_rules(18)
thf(fact_544_basic__trans__rules_I21_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(21)
thf(fact_545_basic__trans__rules_I21_J,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% basic_trans_rules(21)
thf(fact_546_basic__trans__rules_I21_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_complex @ Y2 @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% basic_trans_rules(21)
thf(fact_547_basic__trans__rules_I22_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_nat @ X @ Z ) ) ) ).
% basic_trans_rules(22)
thf(fact_548_basic__trans__rules_I22_J,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( ord_less_real @ X @ Z ) ) ) ).
% basic_trans_rules(22)
thf(fact_549_basic__trans__rules_I22_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ Z )
=> ( ord_less_complex @ X @ Z ) ) ) ).
% basic_trans_rules(22)
thf(fact_550_order__le__imp__less__or__eq,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_nat @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_551_order__le__imp__less__or__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_real @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_552_order__le__imp__less__or__eq,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_complex @ X @ Y2 )
| ( X = Y2 ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_553_linorder__le__less__linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_554_linorder__le__less__linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_real @ Y2 @ X ) ) ).
% linorder_le_less_linear
thf(fact_555_order__less__imp__le,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_556_order__less__imp__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_557_order__less__imp__le,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_complex @ X @ Y2 )
=> ( ord_less_eq_complex @ X @ Y2 ) ) ).
% order_less_imp_le
thf(fact_558_linorder__not__less,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_559_linorder__not__less,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( ord_less_eq_real @ Y2 @ X ) ) ).
% linorder_not_less
thf(fact_560_linorder__not__le,axiom,
! [X: nat,Y2: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y2 ) )
= ( ord_less_nat @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_561_linorder__not__le,axiom,
! [X: real,Y2: real] :
( ( ~ ( ord_less_eq_real @ X @ Y2 ) )
= ( ord_less_real @ Y2 @ X ) ) ).
% linorder_not_le
thf(fact_562_order__less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_563_order__less__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_564_order__less__le,axiom,
( ord_less_complex
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_eq_complex @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% order_less_le
thf(fact_565_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_566_order__le__less,axiom,
( ord_less_eq_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_real @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_567_order__le__less,axiom,
( ord_less_eq_complex
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_complex @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% order_le_less
thf(fact_568_dual__order_Ostrict__implies__order,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ord_less_eq_nat @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_569_dual__order_Ostrict__implies__order,axiom,
! [B3: real,A3: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_570_dual__order_Ostrict__implies__order,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_complex @ B3 @ A3 )
=> ( ord_less_eq_complex @ B3 @ A3 ) ) ).
% dual_order.strict_implies_order
thf(fact_571_order_Ostrict__implies__order,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_572_order_Ostrict__implies__order,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_573_order_Ostrict__implies__order,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ord_less_eq_complex @ A3 @ B3 ) ) ).
% order.strict_implies_order
thf(fact_574_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B7: nat,A8: nat] :
( ( ord_less_eq_nat @ B7 @ A8 )
& ~ ( ord_less_eq_nat @ A8 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_575_dual__order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [B7: real,A8: real] :
( ( ord_less_eq_real @ B7 @ A8 )
& ~ ( ord_less_eq_real @ A8 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_576_dual__order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [B7: complex,A8: complex] :
( ( ord_less_eq_complex @ B7 @ A8 )
& ~ ( ord_less_eq_complex @ A8 @ B7 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_577_dual__order_Ostrict__trans2,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_578_dual__order_Ostrict__trans2,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_579_dual__order_Ostrict__trans2,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ord_less_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C @ B3 )
=> ( ord_less_complex @ C @ A3 ) ) ) ).
% dual_order.strict_trans2
thf(fact_580_dual__order_Ostrict__trans1,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_nat @ C @ B3 )
=> ( ord_less_nat @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_581_dual__order_Ostrict__trans1,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_real @ C @ B3 )
=> ( ord_less_real @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_582_dual__order_Ostrict__trans1,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_complex @ C @ B3 )
=> ( ord_less_complex @ C @ A3 ) ) ) ).
% dual_order.strict_trans1
thf(fact_583_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B7: nat,A8: nat] :
( ( ord_less_eq_nat @ B7 @ A8 )
& ( A8 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_584_dual__order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [B7: real,A8: real] :
( ( ord_less_eq_real @ B7 @ A8 )
& ( A8 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_585_dual__order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [B7: complex,A8: complex] :
( ( ord_less_eq_complex @ B7 @ A8 )
& ( A8 != B7 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_586_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B7: nat,A8: nat] :
( ( ord_less_nat @ B7 @ A8 )
| ( A8 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_587_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [B7: real,A8: real] :
( ( ord_less_real @ B7 @ A8 )
| ( A8 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_588_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [B7: complex,A8: complex] :
( ( ord_less_complex @ B7 @ A8 )
| ( A8 = B7 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_589_dense__le__bounded,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_real @ X @ Y2 )
=> ( ! [W: real] :
( ( ord_less_real @ X @ W )
=> ( ( ord_less_real @ W @ Y2 )
=> ( ord_less_eq_real @ W @ Z ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_le_bounded
thf(fact_590_dense__ge__bounded,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ Z @ X )
=> ( ! [W: real] :
( ( ord_less_real @ Z @ W )
=> ( ( ord_less_real @ W @ X )
=> ( ord_less_eq_real @ Y2 @ W ) ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ) ).
% dense_ge_bounded
thf(fact_591_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A8: nat,B7: nat] :
( ( ord_less_eq_nat @ A8 @ B7 )
& ~ ( ord_less_eq_nat @ B7 @ A8 ) ) ) ) ).
% order.strict_iff_not
thf(fact_592_order_Ostrict__iff__not,axiom,
( ord_less_real
= ( ^ [A8: real,B7: real] :
( ( ord_less_eq_real @ A8 @ B7 )
& ~ ( ord_less_eq_real @ B7 @ A8 ) ) ) ) ).
% order.strict_iff_not
thf(fact_593_order_Ostrict__iff__not,axiom,
( ord_less_complex
= ( ^ [A8: complex,B7: complex] :
( ( ord_less_eq_complex @ A8 @ B7 )
& ~ ( ord_less_eq_complex @ B7 @ A8 ) ) ) ) ).
% order.strict_iff_not
thf(fact_594_order_Ostrict__trans2,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_595_order_Ostrict__trans2,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_596_order_Ostrict__trans2,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_complex @ A3 @ C ) ) ) ).
% order.strict_trans2
thf(fact_597_order_Ostrict__trans1,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ B3 @ C )
=> ( ord_less_nat @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_598_order_Ostrict__trans1,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ B3 @ C )
=> ( ord_less_real @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_599_order_Ostrict__trans1,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_complex @ B3 @ C )
=> ( ord_less_complex @ A3 @ C ) ) ) ).
% order.strict_trans1
thf(fact_600_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A8: nat,B7: nat] :
( ( ord_less_eq_nat @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_601_order_Ostrict__iff__order,axiom,
( ord_less_real
= ( ^ [A8: real,B7: real] :
( ( ord_less_eq_real @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_602_order_Ostrict__iff__order,axiom,
( ord_less_complex
= ( ^ [A8: complex,B7: complex] :
( ( ord_less_eq_complex @ A8 @ B7 )
& ( A8 != B7 ) ) ) ) ).
% order.strict_iff_order
thf(fact_603_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A8: nat,B7: nat] :
( ( ord_less_nat @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_604_order_Oorder__iff__strict,axiom,
( ord_less_eq_real
= ( ^ [A8: real,B7: real] :
( ( ord_less_real @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_605_order_Oorder__iff__strict,axiom,
( ord_less_eq_complex
= ( ^ [A8: complex,B7: complex] :
( ( ord_less_complex @ A8 @ B7 )
| ( A8 = B7 ) ) ) ) ).
% order.order_iff_strict
thf(fact_606_not__le__imp__less,axiom,
! [Y2: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y2 @ X )
=> ( ord_less_nat @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_607_not__le__imp__less,axiom,
! [Y2: real,X: real] :
( ~ ( ord_less_eq_real @ Y2 @ X )
=> ( ord_less_real @ X @ Y2 ) ) ).
% not_le_imp_less
thf(fact_608_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_609_less__le__not__le,axiom,
( ord_less_real
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ~ ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_610_less__le__not__le,axiom,
( ord_less_complex
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_eq_complex @ X4 @ Y5 )
& ~ ( ord_less_eq_complex @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_611_dense__le,axiom,
! [Y2: real,Z: real] :
( ! [X2: real] :
( ( ord_less_real @ X2 @ Y2 )
=> ( ord_less_eq_real @ X2 @ Z ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_le
thf(fact_612_dense__ge,axiom,
! [Z: real,Y2: real] :
( ! [X2: real] :
( ( ord_less_real @ Z @ X2 )
=> ( ord_less_eq_real @ Y2 @ X2 ) )
=> ( ord_less_eq_real @ Y2 @ Z ) ) ).
% dense_ge
thf(fact_613_antisym__conv2,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ~ ( ord_less_nat @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_614_antisym__conv2,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ~ ( ord_less_real @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_615_antisym__conv2,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ~ ( ord_less_complex @ X @ Y2 ) )
= ( X = Y2 ) ) ) ).
% antisym_conv2
thf(fact_616_antisym__conv1,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_617_antisym__conv1,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_618_antisym__conv1,axiom,
! [X: complex,Y2: complex] :
( ~ ( ord_less_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv1
thf(fact_619_nless__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_nat @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_620_nless__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_real @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_real @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_621_nless__le,axiom,
! [A3: complex,B3: complex] :
( ( ~ ( ord_less_complex @ A3 @ B3 ) )
= ( ~ ( ord_less_eq_complex @ A3 @ B3 )
| ( A3 = B3 ) ) ) ).
% nless_le
thf(fact_622_leI,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% leI
thf(fact_623_leI,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% leI
thf(fact_624_leD,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_nat @ X @ Y2 ) ) ).
% leD
thf(fact_625_leD,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_real @ X @ Y2 ) ) ).
% leD
thf(fact_626_leD,axiom,
! [Y2: complex,X: complex] :
( ( ord_less_eq_complex @ Y2 @ X )
=> ~ ( ord_less_complex @ X @ Y2 ) ) ).
% leD
thf(fact_627_verit__comp__simplify_I3_J,axiom,
! [B4: nat,A6: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A6 ) )
= ( ord_less_nat @ A6 @ B4 ) ) ).
% verit_comp_simplify(3)
thf(fact_628_verit__comp__simplify_I3_J,axiom,
! [B4: real,A6: real] :
( ( ~ ( ord_less_eq_real @ B4 @ A6 ) )
= ( ord_less_real @ A6 @ B4 ) ) ).
% verit_comp_simplify(3)
thf(fact_629_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M2 )
= zero_zero_nat )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_630_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ M2 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_631_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_632_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus_nat @ M2 @ zero_zero_nat )
= M2 ) ).
% minus_nat.diff_0
thf(fact_633_less__imp__diff__less,axiom,
! [J2: nat,K2: nat,N2: nat] :
( ( ord_less_nat @ J2 @ K2 )
=> ( ord_less_nat @ ( minus_minus_nat @ J2 @ N2 ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_634_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ M2 @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_635_zero__induct__lemma,axiom,
! [P: nat > $o,K2: nat,I3: nat] :
( ( P @ K2 )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K2 @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_636_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_637_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_638_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_639_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_640_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_641_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
= ( A3 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_642_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).
% bot_nat_0.extremum
thf(fact_643_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_644_inf__pigeonhole__principle,axiom,
! [N2: nat,F: nat > nat > $o] :
( ! [K3: nat] :
? [I: nat] :
( ( ord_less_nat @ I @ N2 )
& ( F @ K3 @ I ) )
=> ? [I4: nat] :
( ( ord_less_nat @ I4 @ N2 )
& ! [K4: nat] :
? [K5: nat] :
( ( ord_less_eq_nat @ K4 @ K5 )
& ( F @ K5 @ I4 ) ) ) ) ).
% inf_pigeonhole_principle
thf(fact_645_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I4: nat,J3: nat] :
( ( ord_less_nat @ I4 @ J3 )
=> ( ord_less_nat @ ( F @ I4 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_646_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_647_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less_nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_648_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_649_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_650_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_651_transitive__stepwise__le,axiom,
! [M2: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y: nat,Z2: nat] :
( ( R2 @ X2 @ Y )
=> ( ( R2 @ Y @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [N3: nat] : ( R2 @ N3 @ ( suc @ N3 ) )
=> ( R2 @ M2 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_652_nat__induct__at__least,axiom,
! [M2: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_653_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_654_not__less__eq__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M2 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_655_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_656_Suc__le__mono,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M2 ) )
= ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% Suc_le_mono
thf(fact_657_le__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M2 @ N2 )
| ( M2
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_658_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M5: nat] :
( M6
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_659_le__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_660_le__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M2 @ N2 )
=> ( M2
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_661_Suc__leD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_leD
thf(fact_662_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( times_times_nat @ K2 @ ( minus_minus_nat @ M2 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_663_diff__mult__distrib,axiom,
! [M2: nat,N2: nat,K2: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K2 )
= ( minus_minus_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_664_mult__le__mono2,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ K2 @ I3 ) @ ( times_times_nat @ K2 @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_665_mult__le__mono1,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_666_mult__le__mono,axiom,
! [I3: nat,J2: nat,K2: nat,L: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ K2 @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I3 @ K2 ) @ ( times_times_nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_667_le__square,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_668_le__cube,axiom,
! [M2: nat] : ( ord_less_eq_nat @ M2 @ ( times_times_nat @ M2 @ ( times_times_nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_669_basic__trans__rules_I26_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( A3 = B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_670_basic__trans__rules_I26_J,axiom,
! [A3: real,B3: real,C: real] :
( ( A3 = B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_671_basic__trans__rules_I26_J,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( A3 = B3 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_eq_complex @ A3 @ C ) ) ) ).
% basic_trans_rules(26)
thf(fact_672_basic__trans__rules_I25_J,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_673_basic__trans__rules_I25_J,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_674_basic__trans__rules_I25_J,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( B3 = C )
=> ( ord_less_eq_complex @ A3 @ C ) ) ) ).
% basic_trans_rules(25)
thf(fact_675_basic__trans__rules_I24_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_676_basic__trans__rules_I24_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_677_basic__trans__rules_I24_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ B3 @ A3 )
=> ( A3 = B3 ) ) ) ).
% basic_trans_rules(24)
thf(fact_678_basic__trans__rules_I23_J,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_679_basic__trans__rules_I23_J,axiom,
! [X: real,Y2: real,Z: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ Z )
=> ( ord_less_eq_real @ X @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_680_basic__trans__rules_I23_J,axiom,
! [X: complex,Y2: complex,Z: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ Z )
=> ( ord_less_eq_complex @ X @ Z ) ) ) ).
% basic_trans_rules(23)
thf(fact_681_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_682_basic__trans__rules_I10_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_683_basic__trans__rules_I10_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C: nat] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_684_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_685_basic__trans__rules_I10_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_686_basic__trans__rules_I10_J,axiom,
! [A3: complex,F: real > complex,B3: real,C: real] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_687_basic__trans__rules_I10_J,axiom,
! [A3: nat,F: complex > nat,B3: complex,C: complex] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_688_basic__trans__rules_I10_J,axiom,
! [A3: real,F: complex > real,B3: complex,C: complex] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_689_basic__trans__rules_I10_J,axiom,
! [A3: complex,F: complex > complex,B3: complex,C: complex] :
( ( A3
= ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(10)
thf(fact_690_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_691_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_692_basic__trans__rules_I9_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_693_basic__trans__rules_I9_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_694_basic__trans__rules_I9_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_695_basic__trans__rules_I9_J,axiom,
! [A3: real,B3: real,F: real > complex,C: complex] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_696_basic__trans__rules_I9_J,axiom,
! [A3: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_697_basic__trans__rules_I9_J,axiom,
! [A3: complex,B3: complex,F: complex > real,C: real] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_698_basic__trans__rules_I9_J,axiom,
! [A3: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ( F @ B3 )
= C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(9)
thf(fact_699_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: nat > nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_700_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: real > nat,B3: real,C: real] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_701_basic__trans__rules_I8_J,axiom,
! [A3: nat,F: complex > nat,B3: complex,C: complex] :
( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_702_basic__trans__rules_I8_J,axiom,
! [A3: real,F: nat > real,B3: nat,C: nat] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_703_basic__trans__rules_I8_J,axiom,
! [A3: real,F: real > real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_704_basic__trans__rules_I8_J,axiom,
! [A3: real,F: complex > real,B3: complex,C: complex] :
( ( ord_less_eq_real @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_705_basic__trans__rules_I8_J,axiom,
! [A3: complex,F: nat > complex,B3: nat,C: nat] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_706_basic__trans__rules_I8_J,axiom,
! [A3: complex,F: real > complex,B3: real,C: real] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_707_basic__trans__rules_I8_J,axiom,
! [A3: complex,F: complex > complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ ( F @ B3 ) )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ A3 @ ( F @ C ) ) ) ) ) ).
% basic_trans_rules(8)
thf(fact_708_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_709_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > real,C: real] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_710_basic__trans__rules_I7_J,axiom,
! [A3: nat,B3: nat,F: nat > complex,C: complex] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: nat,Y: nat] :
( ( ord_less_eq_nat @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_711_basic__trans__rules_I7_J,axiom,
! [A3: real,B3: real,F: real > nat,C: nat] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_712_basic__trans__rules_I7_J,axiom,
! [A3: real,B3: real,F: real > real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_713_basic__trans__rules_I7_J,axiom,
! [A3: real,B3: real,F: real > complex,C: complex] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: real,Y: real] :
( ( ord_less_eq_real @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_714_basic__trans__rules_I7_J,axiom,
! [A3: complex,B3: complex,F: complex > nat,C: nat] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_nat @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_715_basic__trans__rules_I7_J,axiom,
! [A3: complex,B3: complex,F: complex > real,C: real] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_real @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_real @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_real @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_716_basic__trans__rules_I7_J,axiom,
! [A3: complex,B3: complex,F: complex > complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ ( F @ B3 ) @ C )
=> ( ! [X2: complex,Y: complex] :
( ( ord_less_eq_complex @ X2 @ Y )
=> ( ord_less_eq_complex @ ( F @ X2 ) @ ( F @ Y ) ) )
=> ( ord_less_eq_complex @ ( F @ A3 ) @ C ) ) ) ) ).
% basic_trans_rules(7)
thf(fact_717_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_718_le__trans,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( ord_less_eq_nat @ J2 @ K2 )
=> ( ord_less_eq_nat @ I3 @ K2 ) ) ) ).
% le_trans
thf(fact_719_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_720_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_721_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ( minus_minus_nat @ M2 @ K2 )
= ( minus_minus_nat @ N2 @ K2 ) )
= ( M2 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_722_le__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_723_diff__commute,axiom,
! [I3: nat,J2: nat,K2: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K2 )
= ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K2 ) @ J2 ) ) ).
% diff_commute
thf(fact_724_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_725_diff__le__mono,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_726_diff__le__self,axiom,
! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).
% diff_le_self
thf(fact_727_le__diff__iff_H,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ C )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A3 ) @ ( minus_minus_nat @ C @ B3 ) )
= ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_728_diff__le__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_729_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
| ( ord_less_eq_nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_730_diff__diff__cancel,axiom,
! [I3: nat,N2: nat] :
( ( ord_less_eq_nat @ I3 @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_731_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K2: nat,B3: nat] :
( ( P @ K2 )
=> ( ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq_nat @ Y @ B3 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_732_diff__left__imp__eq,axiom,
! [A3: real,B3: real,C: real] :
( ( ( minus_minus_real @ A3 @ B3 )
= ( minus_minus_real @ A3 @ C ) )
=> ( B3 = C ) ) ).
% diff_left_imp_eq
thf(fact_733_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_734_lessThan__subset__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X ) @ ( set_or5984915006950818249n_real @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_735_lessThan__subset__iff,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y2 ) )
= ( ord_less_eq_nat @ X @ Y2 ) ) ).
% lessThan_subset_iff
thf(fact_736_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_737_lift__Suc__antimono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_738_lift__Suc__antimono__le,axiom,
! [F: nat > complex,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_complex @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_complex @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_739_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_740_lift__Suc__mono__le,axiom,
! [F: nat > real,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_real @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_741_lift__Suc__mono__le,axiom,
! [F: nat > complex,N2: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq_complex @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N4 )
=> ( ord_less_eq_complex @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_742_Suc__diff__le,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_eq_nat @ N2 @ M2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_743_prod__decode__aux_Oinduct,axiom,
! [P: nat > nat > $o,A0: nat,A1: nat] :
( ! [K3: nat,M5: nat] :
( ( ~ ( ord_less_eq_nat @ M5 @ K3 )
=> ( P @ ( suc @ K3 ) @ ( minus_minus_nat @ M5 @ ( suc @ K3 ) ) ) )
=> ( P @ K3 @ M5 ) )
=> ( P @ A0 @ A1 ) ) ).
% prod_decode_aux.induct
thf(fact_744_less__diff__iff,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ K2 @ M2 )
=> ( ( ord_less_eq_nat @ K2 @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K2 ) @ ( minus_minus_nat @ N2 @ K2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_745_diff__less__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ A3 )
=> ( ord_less_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ C ) ) ) ) ).
% diff_less_mono
thf(fact_746_linear,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
| ( ord_less_eq_nat @ Y2 @ X ) ) ).
% linear
thf(fact_747_linear,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
| ( ord_less_eq_real @ Y2 @ X ) ) ).
% linear
thf(fact_748_nle__le,axiom,
! [A3: nat,B3: nat] :
( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
= ( ( ord_less_eq_nat @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_749_nle__le,axiom,
! [A3: real,B3: real] :
( ( ~ ( ord_less_eq_real @ A3 @ B3 ) )
= ( ( ord_less_eq_real @ B3 @ A3 )
& ( B3 != A3 ) ) ) ).
% nle_le
thf(fact_750_eq__refl,axiom,
! [X: nat,Y2: nat] :
( ( X = Y2 )
=> ( ord_less_eq_nat @ X @ Y2 ) ) ).
% eq_refl
thf(fact_751_eq__refl,axiom,
! [X: real,Y2: real] :
( ( X = Y2 )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% eq_refl
thf(fact_752_eq__refl,axiom,
! [X: complex,Y2: complex] :
( ( X = Y2 )
=> ( ord_less_eq_complex @ X @ Y2 ) ) ).
% eq_refl
thf(fact_753_le__cases,axiom,
! [X: nat,Y2: nat] :
( ~ ( ord_less_eq_nat @ X @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ X ) ) ).
% le_cases
thf(fact_754_le__cases,axiom,
! [X: real,Y2: real] :
( ~ ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ Y2 @ X ) ) ).
% le_cases
thf(fact_755_le__cases3,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y2 )
=> ~ ( ord_less_eq_nat @ Y2 @ X ) )
=> ( ( ( ord_less_eq_nat @ Y2 @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_756_le__cases3,axiom,
! [X: real,Y2: real,Z: real] :
( ( ( ord_less_eq_real @ X @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ Z ) )
=> ( ( ( ord_less_eq_real @ Y2 @ X )
=> ~ ( ord_less_eq_real @ X @ Z ) )
=> ( ( ( ord_less_eq_real @ X @ Z )
=> ~ ( ord_less_eq_real @ Z @ Y2 ) )
=> ( ( ( ord_less_eq_real @ Z @ Y2 )
=> ~ ( ord_less_eq_real @ Y2 @ X ) )
=> ( ( ( ord_less_eq_real @ Y2 @ Z )
=> ~ ( ord_less_eq_real @ Z @ X ) )
=> ~ ( ( ord_less_eq_real @ Z @ X )
=> ~ ( ord_less_eq_real @ X @ Y2 ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_757_antisym__conv,axiom,
! [Y2: nat,X: nat] :
( ( ord_less_eq_nat @ Y2 @ X )
=> ( ( ord_less_eq_nat @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_758_antisym__conv,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq_real @ Y2 @ X )
=> ( ( ord_less_eq_real @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_759_antisym__conv,axiom,
! [Y2: complex,X: complex] :
( ( ord_less_eq_complex @ Y2 @ X )
=> ( ( ord_less_eq_complex @ X @ Y2 )
= ( X = Y2 ) ) ) ).
% antisym_conv
thf(fact_760_order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A8: nat,B7: nat] :
( ( ord_less_eq_nat @ A8 @ B7 )
& ( ord_less_eq_nat @ B7 @ A8 ) ) ) ) ).
% order.eq_iff
thf(fact_761_order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A8: real,B7: real] :
( ( ord_less_eq_real @ A8 @ B7 )
& ( ord_less_eq_real @ B7 @ A8 ) ) ) ) ).
% order.eq_iff
thf(fact_762_order_Oeq__iff,axiom,
( ( ^ [Y6: complex,Z3: complex] : ( Y6 = Z3 ) )
= ( ^ [A8: complex,B7: complex] :
( ( ord_less_eq_complex @ A8 @ B7 )
& ( ord_less_eq_complex @ B7 @ A8 ) ) ) ) ).
% order.eq_iff
thf(fact_763_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_764_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [X4: real,Y5: real] :
( ( ord_less_eq_real @ X4 @ Y5 )
& ( ord_less_eq_real @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_765_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: complex,Z3: complex] : ( Y6 = Z3 ) )
= ( ^ [X4: complex,Y5: complex] :
( ( ord_less_eq_complex @ X4 @ Y5 )
& ( ord_less_eq_complex @ Y5 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_766_order__antisym,axiom,
! [X: nat,Y2: nat] :
( ( ord_less_eq_nat @ X @ Y2 )
=> ( ( ord_less_eq_nat @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_767_order__antisym,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ X @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_768_order__antisym,axiom,
! [X: complex,Y2: complex] :
( ( ord_less_eq_complex @ X @ Y2 )
=> ( ( ord_less_eq_complex @ Y2 @ X )
=> ( X = Y2 ) ) ) ).
% order_antisym
thf(fact_769_order_Orefl,axiom,
! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).
% order.refl
thf(fact_770_order_Orefl,axiom,
! [A3: real] : ( ord_less_eq_real @ A3 @ A3 ) ).
% order.refl
thf(fact_771_order_Orefl,axiom,
! [A3: complex] : ( ord_less_eq_complex @ A3 @ A3 ) ).
% order.refl
thf(fact_772_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_773_order__refl,axiom,
! [X: real] : ( ord_less_eq_real @ X @ X ) ).
% order_refl
thf(fact_774_order__refl,axiom,
! [X: complex] : ( ord_less_eq_complex @ X @ X ) ).
% order_refl
thf(fact_775_order_Otrans,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ C )
=> ( ord_less_eq_nat @ A3 @ C ) ) ) ).
% order.trans
thf(fact_776_order_Otrans,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ B3 @ C )
=> ( ord_less_eq_real @ A3 @ C ) ) ) ).
% order.trans
thf(fact_777_order_Otrans,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ B3 @ C )
=> ( ord_less_eq_complex @ A3 @ C ) ) ) ).
% order.trans
thf(fact_778_linorder__wlog,axiom,
! [P: nat > nat > $o,A3: nat,B3: nat] :
( ! [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: nat,B5: nat] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_779_linorder__wlog,axiom,
! [P: real > real > $o,A3: real,B3: real] :
( ! [A5: real,B5: real] :
( ( ord_less_eq_real @ A5 @ B5 )
=> ( P @ A5 @ B5 ) )
=> ( ! [A5: real,B5: real] :
( ( P @ B5 @ A5 )
=> ( P @ A5 @ B5 ) )
=> ( P @ A3 @ B3 ) ) ) ).
% linorder_wlog
thf(fact_780_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A8: nat,B7: nat] :
( ( ord_less_eq_nat @ B7 @ A8 )
& ( ord_less_eq_nat @ A8 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_781_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: real,Z3: real] : ( Y6 = Z3 ) )
= ( ^ [A8: real,B7: real] :
( ( ord_less_eq_real @ B7 @ A8 )
& ( ord_less_eq_real @ A8 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_782_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: complex,Z3: complex] : ( Y6 = Z3 ) )
= ( ^ [A8: complex,B7: complex] :
( ( ord_less_eq_complex @ B7 @ A8 )
& ( ord_less_eq_complex @ A8 @ B7 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_783_dual__order_Oantisym,axiom,
! [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_784_dual__order_Oantisym,axiom,
! [B3: real,A3: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_785_dual__order_Oantisym,axiom,
! [B3: complex,A3: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ A3 @ B3 )
=> ( A3 = B3 ) ) ) ).
% dual_order.antisym
thf(fact_786_dual__order_Otrans,axiom,
! [B3: nat,A3: nat,C: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
=> ( ( ord_less_eq_nat @ C @ B3 )
=> ( ord_less_eq_nat @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_787_dual__order_Otrans,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ B3 )
=> ( ord_less_eq_real @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_788_dual__order_Otrans,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C @ B3 )
=> ( ord_less_eq_complex @ C @ A3 ) ) ) ).
% dual_order.trans
thf(fact_789_diff__is__0__eq,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_790_diff__is__0__eq_H,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( minus_minus_nat @ M2 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_791_le__iff__diff__le__0,axiom,
( ord_less_eq_real
= ( ^ [A8: real,B7: real] : ( ord_less_eq_real @ ( minus_minus_real @ A8 @ B7 ) @ zero_zero_real ) ) ) ).
% le_iff_diff_le_0
thf(fact_792_le__iff__diff__le__0,axiom,
( ord_less_eq_complex
= ( ^ [A8: complex,B7: complex] : ( ord_less_eq_complex @ ( minus_minus_complex @ A8 @ B7 ) @ zero_zero_complex ) ) ) ).
% le_iff_diff_le_0
thf(fact_793_diff__ge__0__iff__ge,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_794_diff__ge__0__iff__ge,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ ( minus_minus_complex @ A3 @ B3 ) )
= ( ord_less_eq_complex @ B3 @ A3 ) ) ).
% diff_ge_0_iff_ge
thf(fact_795_diff__Pair,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( minus_4365393887724441320at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ C @ D3 ) )
= ( product_Pair_nat_nat @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_796_diff__Pair,axiom,
! [A3: nat,B3: real,C: nat,D3: real] :
( ( minus_5557628854490389828t_real @ ( produc7837566107596912789t_real @ A3 @ B3 ) @ ( produc7837566107596912789t_real @ C @ D3 ) )
= ( produc7837566107596912789t_real @ ( minus_minus_nat @ A3 @ C ) @ ( minus_minus_real @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_797_diff__Pair,axiom,
! [A3: real,B3: nat,C: real,D3: nat] :
( ( minus_1582581163013509572al_nat @ ( produc3181502643871035669al_nat @ A3 @ B3 ) @ ( produc3181502643871035669al_nat @ C @ D3 ) )
= ( produc3181502643871035669al_nat @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_nat @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_798_diff__Pair,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( minus_885040589139849760l_real @ ( produc4511245868158468465l_real @ A3 @ B3 ) @ ( produc4511245868158468465l_real @ C @ D3 ) )
= ( produc4511245868158468465l_real @ ( minus_minus_real @ A3 @ C ) @ ( minus_minus_real @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_799_diff__Pair,axiom,
! [A3: nat,B3: mat_complex,C: nat,D3: mat_complex] :
( ( minus_9125208095613564965omplex @ ( produc4998868960714853886omplex @ A3 @ B3 ) @ ( produc4998868960714853886omplex @ C @ D3 ) )
= ( produc4998868960714853886omplex @ ( minus_minus_nat @ A3 @ C ) @ ( minus_2412168080157227406omplex @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_800_diff__Pair,axiom,
! [A3: mat_complex,B3: nat,C: mat_complex,D3: nat] :
( ( minus_1583438508407137535ex_nat @ ( produc3916067632315525152ex_nat @ A3 @ B3 ) @ ( produc3916067632315525152ex_nat @ C @ D3 ) )
= ( produc3916067632315525152ex_nat @ ( minus_2412168080157227406omplex @ A3 @ C ) @ ( minus_minus_nat @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_801_diff__Pair,axiom,
! [A3: mat_complex,B3: real,C: mat_complex,D3: real] :
( ( minus_5460563211077212123x_real @ ( produc8172131712939336444x_real @ A3 @ B3 ) @ ( produc8172131712939336444x_real @ C @ D3 ) )
= ( produc8172131712939336444x_real @ ( minus_2412168080157227406omplex @ A3 @ C ) @ ( minus_minus_real @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_802_diff__Pair,axiom,
! [A3: real,B3: mat_complex,C: real,D3: mat_complex] :
( ( minus_7607308147470025289omplex @ ( produc3178892618276054050omplex @ A3 @ B3 ) @ ( produc3178892618276054050omplex @ C @ D3 ) )
= ( produc3178892618276054050omplex @ ( minus_minus_real @ A3 @ C ) @ ( minus_2412168080157227406omplex @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_803_diff__Pair,axiom,
! [A3: nat > nat,B3: nat,C: nat > nat,D3: nat] :
( ( minus_9067931446424981591at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ C @ D3 ) )
= ( produc72220940542539688at_nat @ ( minus_minus_nat_nat @ A3 @ C ) @ ( minus_minus_nat @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_804_diff__Pair,axiom,
! [A3: mat_complex,B3: mat_complex,C: mat_complex,D3: mat_complex] :
( ( minus_2734116836287720782omplex @ ( produc3658446505030690647omplex @ A3 @ B3 ) @ ( produc3658446505030690647omplex @ C @ D3 ) )
= ( produc3658446505030690647omplex @ ( minus_2412168080157227406omplex @ A3 @ C ) @ ( minus_2412168080157227406omplex @ B3 @ D3 ) ) ) ).
% diff_Pair
thf(fact_805_Reals__diff,axiom,
! [A3: real,B3: real] :
( ( member_real @ A3 @ real_V470468836141973256s_real )
=> ( ( member_real @ B3 @ real_V470468836141973256s_real )
=> ( member_real @ ( minus_minus_real @ A3 @ B3 ) @ real_V470468836141973256s_real ) ) ) ).
% Reals_diff
thf(fact_806_Reals__diff,axiom,
! [A3: complex,B3: complex] :
( ( member_complex @ A3 @ real_V2521375963428798218omplex )
=> ( ( member_complex @ B3 @ real_V2521375963428798218omplex )
=> ( member_complex @ ( minus_minus_complex @ A3 @ B3 ) @ real_V2521375963428798218omplex ) ) ) ).
% Reals_diff
thf(fact_807_swaprows__carrier,axiom,
! [K2: nat,L: nat,A2: mat_complex,N2: nat,Nc: nat] :
( ( member_mat_complex @ ( gauss_1020679828357514249omplex @ K2 @ L @ A2 ) @ ( carrier_mat_complex @ N2 @ Nc ) )
= ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ Nc ) ) ) ).
% swaprows_carrier
thf(fact_808_diff__gt__0__iff__gt,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ zero_zero_complex @ ( minus_minus_complex @ A3 @ B3 ) )
= ( ord_less_complex @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_809_diff__gt__0__iff__gt,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A3 @ B3 ) )
= ( ord_less_real @ B3 @ A3 ) ) ).
% diff_gt_0_iff_gt
thf(fact_810_diff__less__0__iff__less,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_complex @ ( minus_minus_complex @ A3 @ B3 ) @ zero_zero_complex )
= ( ord_less_complex @ A3 @ B3 ) ) ).
% diff_less_0_iff_less
thf(fact_811_diff__less__0__iff__less,axiom,
! [A3: real,B3: real] :
( ( ord_less_real @ ( minus_minus_real @ A3 @ B3 ) @ zero_zero_real )
= ( ord_less_real @ A3 @ B3 ) ) ).
% diff_less_0_iff_less
thf(fact_812_arith__special_I21_J,axiom,
( ( minus_minus_complex @ one_one_complex @ one_one_complex )
= zero_zero_complex ) ).
% arith_special(21)
thf(fact_813_arith__special_I21_J,axiom,
( ( minus_minus_real @ one_one_real @ one_one_real )
= zero_zero_real ) ).
% arith_special(21)
thf(fact_814_mult__sign__intros_I4_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_815_mult__sign__intros_I4_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(4)
thf(fact_816_mult__sign__intros_I3_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(3)
thf(fact_817_mult__sign__intros_I3_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ zero_zero_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(3)
thf(fact_818_mult__sign__intros_I3_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(3)
thf(fact_819_mult__sign__intros_I2_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ) ).
% mult_sign_intros(2)
thf(fact_820_mult__sign__intros_I2_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ) ).
% mult_sign_intros(2)
thf(fact_821_mult__sign__intros_I2_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ) ).
% mult_sign_intros(2)
thf(fact_822_mult__sign__intros_I1_J,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_823_mult__sign__intros_I1_J,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_824_mult__sign__intros_I1_J,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ) ).
% mult_sign_intros(1)
thf(fact_825_mult__mono,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_826_mult__mono,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_827_mult__mono,axiom,
! [A3: complex,B3: complex,C: complex,D3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C ) @ ( times_times_complex @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono
thf(fact_828_mult__mono_H,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_829_mult__mono_H,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_830_mult__mono_H,axiom,
! [A3: complex,B3: complex,C: complex,D3: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ C @ D3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C ) @ ( times_times_complex @ B3 @ D3 ) ) ) ) ) ) ).
% mult_mono'
thf(fact_831_zero__le__square,axiom,
! [A3: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ A3 ) ) ).
% zero_le_square
thf(fact_832_split__mult__pos__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) )
=> ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_833_split__mult__pos__le,axiom,
! [A3: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) ) )
=> ( ord_less_eq_complex @ zero_zero_complex @ ( times_times_complex @ A3 @ B3 ) ) ) ).
% split_mult_pos_le
thf(fact_834_mult__left__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_835_mult__left__mono__neg,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A3 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% mult_left_mono_neg
thf(fact_836_mult__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_837_mult__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_838_mult__left__mono,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A3 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% mult_left_mono
thf(fact_839_mult__right__mono__neg,axiom,
! [B3: real,A3: real,C: real] :
( ( ord_less_eq_real @ B3 @ A3 )
=> ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_840_mult__right__mono__neg,axiom,
! [B3: complex,A3: complex,C: complex] :
( ( ord_less_eq_complex @ B3 @ A3 )
=> ( ( ord_less_eq_complex @ C @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_841_mult__right__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_842_mult__right__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_843_mult__right__mono,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ C ) @ ( times_times_complex @ B3 @ C ) ) ) ) ).
% mult_right_mono
thf(fact_844_mult__le__0__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) ) ) ).
% mult_le_0_iff
thf(fact_845_split__mult__neg__le,axiom,
! [A3: nat,B3: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
& ( ord_less_eq_nat @ B3 @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B3 ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_846_split__mult__neg__le,axiom,
! [A3: real,B3: real] :
( ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) ) )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ zero_zero_real ) ) ).
% split_mult_neg_le
thf(fact_847_split__mult__neg__le,axiom,
! [A3: complex,B3: complex] :
( ( ( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
& ( ord_less_eq_complex @ B3 @ zero_zero_complex ) )
| ( ( ord_less_eq_complex @ A3 @ zero_zero_complex )
& ( ord_less_eq_complex @ zero_zero_complex @ B3 ) ) )
=> ( ord_less_eq_complex @ ( times_times_complex @ A3 @ B3 ) @ zero_zero_complex ) ) ).
% split_mult_neg_le
thf(fact_848_mult__nonneg__nonpos2,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ B3 @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B3 @ A3 ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_849_mult__nonneg__nonpos2,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ B3 @ zero_zero_real )
=> ( ord_less_eq_real @ ( times_times_real @ B3 @ A3 ) @ zero_zero_real ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_850_mult__nonneg__nonpos2,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ B3 @ zero_zero_complex )
=> ( ord_less_eq_complex @ ( times_times_complex @ B3 @ A3 ) @ zero_zero_complex ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_851_zero__le__mult__iff,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A3 @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ A3 )
& ( ord_less_eq_real @ zero_zero_real @ B3 ) )
| ( ( ord_less_eq_real @ A3 @ zero_zero_real )
& ( ord_less_eq_real @ B3 @ zero_zero_real ) ) ) ) ).
% zero_le_mult_iff
thf(fact_852_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: nat,B3: nat,C: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_853_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: real,B3: real,C: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_854_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A3: complex,B3: complex,C: complex] :
( ( ord_less_eq_complex @ A3 @ B3 )
=> ( ( ord_less_eq_complex @ zero_zero_complex @ C )
=> ( ord_less_eq_complex @ ( times_times_complex @ C @ A3 ) @ ( times_times_complex @ C @ B3 ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_855_verit__comp__simplify_I29_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% verit_comp_simplify(29)
thf(fact_856_verit__comp__simplify_I29_J,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% verit_comp_simplify(29)
thf(fact_857_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one_class.zero_le_one
thf(fact_858_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_real @ zero_zero_real @ one_one_real ).
% zero_less_one_class.zero_le_one
thf(fact_859_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_860_not__one__le__zero,axiom,
~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).
% not_one_le_zero
thf(fact_861_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_862_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M2 )
=> ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_863_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_864_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_865_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_866_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_867_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I: nat] :
( ( ord_less_nat @ I @ K3 )
=> ~ ( P @ I ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_868_le__imp__less__Suc,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_869_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).
% less_eq_Suc_le
thf(fact_870_less__Suc__eq__le,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_871_le__less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ M2 @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_872_Suc__le__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_lessD
thf(fact_873_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_874_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I3 @ N3 )
=> ( ( ord_less_nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_875_Suc__le__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 )
= ( ord_less_nat @ M2 @ N2 ) ) ).
% Suc_le_eq
thf(fact_876_Suc__leI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M2 ) @ N2 ) ) ).
% Suc_leI
thf(fact_877_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K2 ) @ M2 ) @ ( times_times_nat @ ( suc @ K2 ) @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_878_insert__index_I2_J,axiom,
! [I3: nat,I5: nat] :
( ( ord_less_eq_nat @ I3 @ I5 )
=> ( ( insert_index @ I3 @ I5 )
= ( suc @ I5 ) ) ) ).
% insert_index(2)
thf(fact_879_mult__le__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_left
thf(fact_880_mult__le__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ A3 ) ) ) ) ).
% mult_le_cancel_right
thf(fact_881_mult__left__less__imp__less,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_882_mult__left__less__imp__less,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_left_less_imp_less
thf(fact_883_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_884_linordered__semiring__strict__class_Omult__strict__mono,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono
thf(fact_885_mult__less__cancel__left,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_left
thf(fact_886_mult__right__less__imp__less,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_887_mult__right__less__imp__less,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) ) ) ).
% mult_right_less_imp_less
thf(fact_888_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_889_linordered__semiring__strict__class_Omult__strict__mono_H,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_strict_mono'
thf(fact_890_mult__less__cancel__right,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ A3 ) ) ) ) ).
% mult_less_cancel_right
thf(fact_891_mult__le__cancel__left__neg,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ C @ zero_zero_real )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_eq_real @ B3 @ A3 ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_892_mult__le__cancel__left__pos,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_real @ zero_zero_real @ C )
=> ( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
= ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_893_mult__left__le__imp__le,axiom,
! [C: nat,A3: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A3 ) @ ( times_times_nat @ C @ B3 ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_894_mult__left__le__imp__le,axiom,
! [C: real,A3: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ ( times_times_real @ C @ B3 ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_left_le_imp_le
thf(fact_895_mult__right__le__imp__le,axiom,
! [A3: nat,C: nat,B3: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_896_mult__right__le__imp__le,axiom,
! [A3: real,C: real,B3: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ C ) )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ B3 ) ) ) ).
% mult_right_le_imp_le
thf(fact_897_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ C @ D3 )
=> ( ( ord_less_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_898_linordered__semiring__strict__class_Omult__le__less__imp__less,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ( ord_less_real @ C @ D3 )
=> ( ( ord_less_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_le_less_imp_less
thf(fact_899_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ C @ D3 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A3 @ C ) @ ( times_times_nat @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_900_linordered__semiring__strict__class_Omult__less__le__imp__less,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_less_real @ A3 @ B3 )
=> ( ( ord_less_eq_real @ C @ D3 )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_real @ ( times_times_real @ A3 @ C ) @ ( times_times_real @ B3 @ D3 ) ) ) ) ) ) ).
% linordered_semiring_strict_class.mult_less_le_imp_less
thf(fact_901_mult__le__cancel__iff1,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ X @ Z ) @ ( times_times_real @ Y2 @ Z ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff1
thf(fact_902_mult__le__cancel__iff2,axiom,
! [Z: real,X: real,Y2: real] :
( ( ord_less_real @ zero_zero_real @ Z )
=> ( ( ord_less_eq_real @ ( times_times_real @ Z @ X ) @ ( times_times_real @ Z @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% mult_le_cancel_iff2
thf(fact_903_mult__left__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Y2 @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_904_mult__right__le__one__le,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ zero_zero_real @ Y2 )
=> ( ( ord_less_eq_real @ Y2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ X @ Y2 ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_905_mult__le__one,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B3 )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ B3 ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_906_mult__le__one,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ B3 )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ B3 ) @ one_one_real ) ) ) ) ).
% mult_le_one
thf(fact_907_mult__left__le,axiom,
! [C: nat,A3: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ord_less_eq_nat @ ( times_times_nat @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_908_mult__left__le,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ C @ one_one_real )
=> ( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ A3 ) ) ) ).
% mult_left_le
thf(fact_909_diff__Suc__less,axiom,
! [N2: nat,I3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_910_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_911_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N2 )
& ! [I: nat] :
( ( ord_less_eq_nat @ I @ K3 )
=> ~ ( P @ I ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_912_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M2 @ K2 ) @ ( times_times_nat @ N2 @ K2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_913_mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% mult_le_cancel1
thf(fact_914_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K2 @ M2 ) @ ( times_times_nat @ K2 @ N2 ) )
= ( ord_less_eq_nat @ M2 @ N2 ) ) ) ).
% nat_mult_le_cancel1
thf(fact_915_one__le__mult__iff,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M2 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M2 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_916_mat__delete__carrier,axiom,
! [A2: mat_complex,M2: nat,N2: nat,I3: nat,J2: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ M2 @ N2 ) )
=> ( member_mat_complex @ ( mat_delete_complex @ A2 @ I3 @ J2 ) @ ( carrier_mat_complex @ ( minus_minus_nat @ M2 @ one_one_nat ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% mat_delete_carrier
thf(fact_917_mult__less__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_918_mult__less__cancel__right1,axiom,
! [C: real,B3: real] :
( ( ord_less_real @ C @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_919_mult__less__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ A3 @ one_one_real ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ one_one_real @ A3 ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_920_mult__less__cancel__left1,axiom,
! [C: real,B3: real] :
( ( ord_less_real @ C @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_eq_real @ zero_zero_real @ C )
=> ( ord_less_real @ one_one_real @ B3 ) )
& ( ( ord_less_eq_real @ C @ zero_zero_real )
=> ( ord_less_real @ B3 @ one_one_real ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_921_mult__le__cancel__right2,axiom,
! [A3: real,C: real] :
( ( ord_less_eq_real @ ( times_times_real @ A3 @ C ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_922_mult__le__cancel__right1,axiom,
! [C: real,B3: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ B3 @ C ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_923_mult__le__cancel__left2,axiom,
! [C: real,A3: real] :
( ( ord_less_eq_real @ ( times_times_real @ C @ A3 ) @ C )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ A3 @ one_one_real ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ one_one_real @ A3 ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_924_mult__le__cancel__left1,axiom,
! [C: real,B3: real] :
( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B3 ) )
= ( ( ( ord_less_real @ zero_zero_real @ C )
=> ( ord_less_eq_real @ one_one_real @ B3 ) )
& ( ( ord_less_real @ C @ zero_zero_real )
=> ( ord_less_eq_real @ B3 @ one_one_real ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_925_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
= ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_926_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_927_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_928_identify__block__main,axiom,
! [A2: mat_complex,J2: nat,I3: nat] :
( ( ( jordan3525277539992963945omplex @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ( I3 = zero_zero_nat )
| ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_complex ) )
& ! [K4: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ( ( ord_less_nat @ K4 @ J2 )
=> ( ( index_mat_complex @ A2 @ ( product_Pair_nat_nat @ K4 @ ( suc @ K4 ) ) )
= one_one_complex ) ) ) ) ) ).
% identify_block_main
thf(fact_929_identify__block__main,axiom,
! [A2: mat_nat,J2: nat,I3: nat] :
( ( ( jordan8923406848002823307ck_nat @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ( I3 = zero_zero_nat )
| ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_nat ) )
& ! [K4: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ( ( ord_less_nat @ K4 @ J2 )
=> ( ( index_mat_nat @ A2 @ ( product_Pair_nat_nat @ K4 @ ( suc @ K4 ) ) )
= one_one_nat ) ) ) ) ) ).
% identify_block_main
thf(fact_930_identify__block__main,axiom,
! [A2: mat_real,J2: nat,I3: nat] :
( ( ( jordan6672758942465739239k_real @ A2 @ J2 )
= I3 )
=> ( ( ord_less_eq_nat @ I3 @ J2 )
& ( ( I3 = zero_zero_nat )
| ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ ( minus_minus_nat @ I3 @ one_one_nat ) @ I3 ) )
!= one_one_real ) )
& ! [K4: nat] :
( ( ord_less_eq_nat @ I3 @ K4 )
=> ( ( ord_less_nat @ K4 @ J2 )
=> ( ( index_mat_real @ A2 @ ( product_Pair_nat_nat @ K4 @ ( suc @ K4 ) ) )
= one_one_real ) ) ) ) ) ).
% identify_block_main
thf(fact_931_permutation__insert__expand,axiom,
( permut3695043542826343943rt_nat
= ( ^ [I2: nat,J: nat,P6: nat > nat,I6: nat] : ( if_nat @ ( ord_less_nat @ I6 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P6 @ I6 ) @ J ) @ ( P6 @ I6 ) @ ( suc @ ( P6 @ I6 ) ) ) @ ( if_nat @ ( I6 = I2 ) @ J @ ( if_nat @ ( ord_less_nat @ ( P6 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) @ J ) @ ( P6 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) @ ( suc @ ( P6 @ ( minus_minus_nat @ I6 @ one_one_nat ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_932_permutation__insert__expand,axiom,
( permut4060954620988167523t_real
= ( ^ [I2: real,J: nat,P6: real > nat,I6: real] : ( if_nat @ ( ord_less_real @ I6 @ I2 ) @ ( if_nat @ ( ord_less_nat @ ( P6 @ I6 ) @ J ) @ ( P6 @ I6 ) @ ( suc @ ( P6 @ I6 ) ) ) @ ( if_nat @ ( I6 = I2 ) @ J @ ( if_nat @ ( ord_less_nat @ ( P6 @ ( minus_minus_real @ I6 @ one_one_real ) ) @ J ) @ ( P6 @ ( minus_minus_real @ I6 @ one_one_real ) ) @ ( suc @ ( P6 @ ( minus_minus_real @ I6 @ one_one_real ) ) ) ) ) ) ) ) ).
% permutation_insert_expand
thf(fact_933_field__le__mult__one__interval,axiom,
! [X: real,Y2: real] :
( ! [Z2: real] :
( ( ord_less_real @ zero_zero_real @ Z2 )
=> ( ( ord_less_real @ Z2 @ one_one_real )
=> ( ord_less_eq_real @ ( times_times_real @ Z2 @ X ) @ Y2 ) ) )
=> ( ord_less_eq_real @ X @ Y2 ) ) ).
% field_le_mult_one_interval
thf(fact_934_mult__eq__1,axiom,
! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A3 )
=> ( ( ord_less_eq_nat @ A3 @ one_one_nat )
=> ( ( ord_less_eq_nat @ B3 @ one_one_nat )
=> ( ( ( times_times_nat @ A3 @ B3 )
= one_one_nat )
= ( ( A3 = one_one_nat )
& ( B3 = one_one_nat ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_935_mult__eq__1,axiom,
! [A3: real,B3: real] :
( ( ord_less_eq_real @ zero_zero_real @ A3 )
=> ( ( ord_less_eq_real @ A3 @ one_one_real )
=> ( ( ord_less_eq_real @ B3 @ one_one_real )
=> ( ( ( times_times_real @ A3 @ B3 )
= one_one_real )
= ( ( A3 = one_one_real )
& ( B3 = one_one_real ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_936_mult__eq__1,axiom,
! [A3: complex,B3: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ A3 )
=> ( ( ord_less_eq_complex @ A3 @ one_one_complex )
=> ( ( ord_less_eq_complex @ B3 @ one_one_complex )
=> ( ( ( times_times_complex @ A3 @ B3 )
= one_one_complex )
= ( ( A3 = one_one_complex )
& ( B3 = one_one_complex ) ) ) ) ) ) ).
% mult_eq_1
thf(fact_937_minus__carrier__mat,axiom,
! [B2: mat_complex,Nr: nat,Nc: nat,A2: mat_complex] :
( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ).
% minus_carrier_mat
thf(fact_938_Set_Obasic__monos_I7_J,axiom,
! [A2: set_complex,B2: set_complex,X: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ X @ A2 )
=> ( member_complex @ X @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_939_Set_Obasic__monos_I7_J,axiom,
! [A2: set_mat_complex,B2: set_mat_complex,X: mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B2 )
=> ( ( member_mat_complex @ X @ A2 )
=> ( member_mat_complex @ X @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_940_Set_Obasic__monos_I7_J,axiom,
! [A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,X: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ X @ A2 )
=> ( member8206827879206165904at_nat @ X @ B2 ) ) ) ).
% Set.basic_monos(7)
thf(fact_941_basic__trans__rules_I31_J,axiom,
! [A2: set_complex,B2: set_complex,C: complex] :
( ( ord_le211207098394363844omplex @ A2 @ B2 )
=> ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_942_basic__trans__rules_I31_J,axiom,
! [A2: set_mat_complex,B2: set_mat_complex,C: mat_complex] :
( ( ord_le3632134057777142183omplex @ A2 @ B2 )
=> ( ( member_mat_complex @ C @ A2 )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_943_basic__trans__rules_I31_J,axiom,
! [A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat,C: produc859450856879609959at_nat] :
( ( ord_le3000389064537975527at_nat @ A2 @ B2 )
=> ( ( member8206827879206165904at_nat @ C @ A2 )
=> ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% basic_trans_rules(31)
thf(fact_944_subsetI,axiom,
! [A2: set_complex,B2: set_complex] :
( ! [X2: complex] :
( ( member_complex @ X2 @ A2 )
=> ( member_complex @ X2 @ B2 ) )
=> ( ord_le211207098394363844omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_945_subsetI,axiom,
! [A2: set_mat_complex,B2: set_mat_complex] :
( ! [X2: mat_complex] :
( ( member_mat_complex @ X2 @ A2 )
=> ( member_mat_complex @ X2 @ B2 ) )
=> ( ord_le3632134057777142183omplex @ A2 @ B2 ) ) ).
% subsetI
thf(fact_946_subsetI,axiom,
! [A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ! [X2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X2 @ A2 )
=> ( member8206827879206165904at_nat @ X2 @ B2 ) )
=> ( ord_le3000389064537975527at_nat @ A2 @ B2 ) ) ).
% subsetI
thf(fact_947_subset__eq,axiom,
( ord_le211207098394363844omplex
= ( ^ [A: set_complex,B: set_complex] :
! [X4: complex] :
( ( member_complex @ X4 @ A )
=> ( member_complex @ X4 @ B ) ) ) ) ).
% subset_eq
thf(fact_948_subset__eq,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A: set_mat_complex,B: set_mat_complex] :
! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ A )
=> ( member_mat_complex @ X4 @ B ) ) ) ) ).
% subset_eq
thf(fact_949_subset__eq,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ A )
=> ( member8206827879206165904at_nat @ X4 @ B ) ) ) ) ).
% subset_eq
thf(fact_950_subset__iff,axiom,
( ord_le211207098394363844omplex
= ( ^ [A: set_complex,B: set_complex] :
! [T2: complex] :
( ( member_complex @ T2 @ A )
=> ( member_complex @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_951_subset__iff,axiom,
( ord_le3632134057777142183omplex
= ( ^ [A: set_mat_complex,B: set_mat_complex] :
! [T2: mat_complex] :
( ( member_mat_complex @ T2 @ A )
=> ( member_mat_complex @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_952_subset__iff,axiom,
( ord_le3000389064537975527at_nat
= ( ^ [A: set_Pr8693737435421807431at_nat,B: set_Pr8693737435421807431at_nat] :
! [T2: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ T2 @ A )
=> ( member8206827879206165904at_nat @ T2 @ B ) ) ) ) ).
% subset_iff
thf(fact_953_psubset__imp__ex__mem,axiom,
! [A2: set_complex,B2: set_complex] :
( ( ord_less_set_complex @ A2 @ B2 )
=> ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_954_psubset__imp__ex__mem,axiom,
! [A2: set_mat_complex,B2: set_mat_complex] :
( ( ord_le5598786136212072115omplex @ A2 @ B2 )
=> ? [B5: mat_complex] : ( member_mat_complex @ B5 @ ( minus_8760755521168068590omplex @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_955_psubset__imp__ex__mem,axiom,
! [A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( ord_le6428140832669894131at_nat @ A2 @ B2 )
=> ? [B5: produc859450856879609959at_nat] : ( member8206827879206165904at_nat @ B5 @ ( minus_8321449233255521966at_nat @ B2 @ A2 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_956_minus__mult__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N2: nat,B2: mat_complex,C3: mat_complex,Nc: nat] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( ( times_8009071140041733218omplex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ C3 )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A2 @ C3 ) @ ( times_8009071140041733218omplex @ B2 @ C3 ) ) ) ) ) ) ).
% minus_mult_distrib_mat
thf(fact_957_mult__minus__distrib__mat,axiom,
! [A2: mat_complex,Nr: nat,N2: nat,B2: mat_complex,Nc: nat,C3: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N2 @ Nc ) )
=> ( ( times_8009071140041733218omplex @ A2 @ ( minus_2412168080157227406omplex @ B2 @ C3 ) )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( times_8009071140041733218omplex @ A2 @ C3 ) ) ) ) ) ) ).
% mult_minus_distrib_mat
thf(fact_958_linordered__field__no__ub,axiom,
! [X3: real] :
? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).
% linordered_field_no_ub
thf(fact_959_linordered__field__no__lb,axiom,
! [X3: real] :
? [Y: real] : ( ord_less_real @ Y @ X3 ) ).
% linordered_field_no_lb
thf(fact_960_kuhn__labelling__lemma_H,axiom,
! [P: ( nat > real ) > $o,F: ( nat > real ) > nat > real,Q3: nat > $o] :
( ! [X2: nat > real] :
( ( P @ X2 )
=> ( P @ ( F @ X2 ) ) )
=> ( ! [X2: nat > real] :
( ( P @ X2 )
=> ! [I4: nat] :
( ( Q3 @ I4 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( X2 @ I4 ) )
& ( ord_less_eq_real @ ( X2 @ I4 ) @ one_one_real ) ) ) )
=> ? [L2: ( nat > real ) > nat > nat] :
( ! [X3: nat > real,I: nat] : ( ord_less_eq_nat @ ( L2 @ X3 @ I ) @ one_one_nat )
& ! [X3: nat > real,I: nat] :
( ( ( P @ X3 )
& ( Q3 @ I )
& ( ( X3 @ I )
= zero_zero_real ) )
=> ( ( L2 @ X3 @ I )
= zero_zero_nat ) )
& ! [X3: nat > real,I: nat] :
( ( ( P @ X3 )
& ( Q3 @ I )
& ( ( X3 @ I )
= one_one_real ) )
=> ( ( L2 @ X3 @ I )
= one_one_nat ) )
& ! [X3: nat > real,I: nat] :
( ( ( P @ X3 )
& ( Q3 @ I )
& ( ( L2 @ X3 @ I )
= zero_zero_nat ) )
=> ( ord_less_eq_real @ ( X3 @ I ) @ ( F @ X3 @ I ) ) )
& ! [X3: nat > real,I: nat] :
( ( ( P @ X3 )
& ( Q3 @ I )
& ( ( L2 @ X3 @ I )
= one_one_nat ) )
=> ( ord_less_eq_real @ ( F @ X3 @ I ) @ ( X3 @ I ) ) ) ) ) ) ).
% kuhn_labelling_lemma'
thf(fact_961_poly__cancel__eq__conv,axiom,
! [X: complex,A3: complex,Y2: complex,B3: complex] :
( ( X = zero_zero_complex )
=> ( ( A3 != zero_zero_complex )
=> ( ( Y2 = zero_zero_complex )
= ( ( minus_minus_complex @ ( times_times_complex @ A3 @ Y2 ) @ ( times_times_complex @ B3 @ X ) )
= zero_zero_complex ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_962_poly__cancel__eq__conv,axiom,
! [X: real,A3: real,Y2: real,B3: real] :
( ( X = zero_zero_real )
=> ( ( A3 != zero_zero_real )
=> ( ( Y2 = zero_zero_real )
= ( ( minus_minus_real @ ( times_times_real @ A3 @ Y2 ) @ ( times_times_real @ B3 @ X ) )
= zero_zero_real ) ) ) ) ).
% poly_cancel_eq_conv
thf(fact_963_DiffE,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ~ ( ( member_complex @ C @ A2 )
=> ( member_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_964_DiffE,axiom,
! [C: mat_complex,A2: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A2 @ B2 ) )
=> ~ ( ( member_mat_complex @ C @ A2 )
=> ( member_mat_complex @ C @ B2 ) ) ) ).
% DiffE
thf(fact_965_DiffE,axiom,
! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B2 ) )
=> ~ ( ( member8206827879206165904at_nat @ C @ A2 )
=> ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% DiffE
thf(fact_966_DiffI,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ A2 )
=> ( ~ ( member_complex @ C @ B2 )
=> ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_967_DiffI,axiom,
! [C: mat_complex,A2: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ A2 )
=> ( ~ ( member_mat_complex @ C @ B2 )
=> ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_968_DiffI,axiom,
! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ A2 )
=> ( ~ ( member8206827879206165904at_nat @ C @ B2 )
=> ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B2 ) ) ) ) ).
% DiffI
thf(fact_969_DiffD1,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ( member_complex @ C @ A2 ) ) ).
% DiffD1
thf(fact_970_DiffD1,axiom,
! [C: mat_complex,A2: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A2 @ B2 ) )
=> ( member_mat_complex @ C @ A2 ) ) ).
% DiffD1
thf(fact_971_DiffD1,axiom,
! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B2 ) )
=> ( member8206827879206165904at_nat @ C @ A2 ) ) ).
% DiffD1
thf(fact_972_DiffD2,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
=> ~ ( member_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_973_DiffD2,axiom,
! [C: mat_complex,A2: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A2 @ B2 ) )
=> ~ ( member_mat_complex @ C @ B2 ) ) ).
% DiffD2
thf(fact_974_DiffD2,axiom,
! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B2 ) )
=> ~ ( member8206827879206165904at_nat @ C @ B2 ) ) ).
% DiffD2
thf(fact_975_Diff__iff,axiom,
! [C: complex,A2: set_complex,B2: set_complex] :
( ( member_complex @ C @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
= ( ( member_complex @ C @ A2 )
& ~ ( member_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_976_Diff__iff,axiom,
! [C: mat_complex,A2: set_mat_complex,B2: set_mat_complex] :
( ( member_mat_complex @ C @ ( minus_8760755521168068590omplex @ A2 @ B2 ) )
= ( ( member_mat_complex @ C @ A2 )
& ~ ( member_mat_complex @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_977_Diff__iff,axiom,
! [C: produc859450856879609959at_nat,A2: set_Pr8693737435421807431at_nat,B2: set_Pr8693737435421807431at_nat] :
( ( member8206827879206165904at_nat @ C @ ( minus_8321449233255521966at_nat @ A2 @ B2 ) )
= ( ( member8206827879206165904at_nat @ C @ A2 )
& ~ ( member8206827879206165904at_nat @ C @ B2 ) ) ) ).
% Diff_iff
thf(fact_978_less__eq__fract__respect,axiom,
! [B3: real,B4: real,D3: real,D4: real,A3: real,A6: real,C: real,C4: real] :
( ( B3 != zero_zero_real )
=> ( ( B4 != zero_zero_real )
=> ( ( D3 != zero_zero_real )
=> ( ( D4 != zero_zero_real )
=> ( ( ( times_times_real @ A3 @ B4 )
= ( times_times_real @ A6 @ B3 ) )
=> ( ( ( times_times_real @ C @ D4 )
= ( times_times_real @ C4 @ D3 ) )
=> ( ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A3 @ D3 ) @ ( times_times_real @ B3 @ D3 ) ) @ ( times_times_real @ ( times_times_real @ C @ B3 ) @ ( times_times_real @ B3 @ D3 ) ) )
= ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ A6 @ D4 ) @ ( times_times_real @ B4 @ D4 ) ) @ ( times_times_real @ ( times_times_real @ C4 @ B4 ) @ ( times_times_real @ B4 @ D4 ) ) ) ) ) ) ) ) ) ) ).
% less_eq_fract_respect
thf(fact_979_delete__index__def,axiom,
( delete_index
= ( ^ [I2: nat,I6: nat] : ( if_nat @ ( ord_less_nat @ I6 @ I2 ) @ I6 @ ( minus_minus_nat @ I6 @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% delete_index_def
thf(fact_980_delete__insert__index,axiom,
! [I3: nat,I5: nat] :
( ( delete_index @ I3 @ ( insert_index @ I3 @ I5 ) )
= I5 ) ).
% delete_insert_index
thf(fact_981_insert__delete__index,axiom,
! [I5: nat,I3: nat] :
( ( I5 != I3 )
=> ( ( insert_index @ I3 @ ( delete_index @ I3 @ I5 ) )
= I5 ) ) ).
% insert_delete_index
thf(fact_982_permutation__delete__expand,axiom,
( permutation_delete
= ( ^ [P6: nat > nat,I2: nat,J: nat] : ( if_nat @ ( ord_less_nat @ ( P6 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( P6 @ I2 ) ) @ ( P6 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( minus_minus_nat @ ( P6 @ ( if_nat @ ( ord_less_nat @ J @ I2 ) @ J @ ( suc @ J ) ) ) @ ( suc @ zero_zero_nat ) ) ) ) ) ).
% permutation_delete_expand
thf(fact_983_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K: nat,M: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M @ K ) @ ( product_Pair_nat_nat @ M @ ( minus_minus_nat @ K @ M ) ) @ ( nat_prod_decode_aux @ ( suc @ K ) @ ( minus_minus_nat @ M @ ( suc @ K ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_984_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y2: product_prod_nat_nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y2 )
=> ( ( ( ord_less_eq_nat @ Xa @ X )
=> ( Y2
= ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq_nat @ Xa @ X )
=> ( Y2
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus_nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_985_minus__carrier__mat_H,axiom,
! [A2: mat_complex,Nr: nat,Nc: nat,B2: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ Nr @ Nc ) )
=> ( member_mat_complex @ ( minus_2412168080157227406omplex @ A2 @ B2 ) @ ( carrier_mat_complex @ Nr @ Nc ) ) ) ) ).
% minus_carrier_mat'
thf(fact_986_mat__assoc__test_I1_J,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,C3: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ ( times_8009071140041733218omplex @ C3 @ D ) )
= ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ C3 ) @ D ) ) ) ) ) ) ).
% mat_assoc_test(1)
thf(fact_987_mat__assoc__test_I3_J,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,C3: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ ( one_mat_complex @ N2 ) ) @ ( one_mat_complex @ N2 ) ) @ B2 ) @ ( one_mat_complex @ N2 ) )
= ( times_8009071140041733218omplex @ A2 @ B2 ) ) ) ) ) ) ).
% mat_assoc_test(3)
thf(fact_988_mat__assoc__test_I9_J,axiom,
! [A2: mat_complex,N2: nat,B2: mat_complex,C3: mat_complex,D: mat_complex] :
( ( member_mat_complex @ A2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ B2 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ C3 @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( member_mat_complex @ D @ ( carrier_mat_complex @ N2 @ N2 ) )
=> ( ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ ( minus_2412168080157227406omplex @ B2 @ C3 ) ) @ D )
= ( minus_2412168080157227406omplex @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ B2 ) @ D ) @ ( times_8009071140041733218omplex @ ( times_8009071140041733218omplex @ A2 @ C3 ) @ D ) ) ) ) ) ) ) ).
% mat_assoc_test(9)
thf(fact_989_nat__descend__induct,axiom,
! [N2: nat,P: nat > $o,M2: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N2 @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
=> ( ! [I: nat] :
( ( ord_less_nat @ K3 @ I )
=> ( P @ I ) )
=> ( P @ K3 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_990_Bolzano,axiom,
! [A3: real,B3: real,P: real > real > $o] :
( ( ord_less_eq_real @ A3 @ B3 )
=> ( ! [A5: real,B5: real,C2: real] :
( ( P @ A5 @ B5 )
=> ( ( P @ B5 @ C2 )
=> ( ( ord_less_eq_real @ A5 @ B5 )
=> ( ( ord_less_eq_real @ B5 @ C2 )
=> ( P @ A5 @ C2 ) ) ) ) )
=> ( ! [X2: real] :
( ( ord_less_eq_real @ A3 @ X2 )
=> ( ( ord_less_eq_real @ X2 @ B3 )
=> ? [D5: real] :
( ( ord_less_real @ zero_zero_real @ D5 )
& ! [A5: real,B5: real] :
( ( ( ord_less_eq_real @ A5 @ X2 )
& ( ord_less_eq_real @ X2 @ B5 )
& ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D5 ) )
=> ( P @ A5 @ B5 ) ) ) ) )
=> ( P @ A3 @ B3 ) ) ) ) ).
% Bolzano
thf(fact_991_bgauge__existence__lemma,axiom,
! [S: set_complex,Q: real > complex > $o] :
( ( ! [X4: complex] :
( ( member_complex @ X4 @ S )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( Q @ D6 @ X4 ) ) ) )
= ( ! [X4: complex] :
? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( ( member_complex @ X4 @ S )
=> ( Q @ D6 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_992_bgauge__existence__lemma,axiom,
! [S: set_mat_complex,Q: real > mat_complex > $o] :
( ( ! [X4: mat_complex] :
( ( member_mat_complex @ X4 @ S )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( Q @ D6 @ X4 ) ) ) )
= ( ! [X4: mat_complex] :
? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( ( member_mat_complex @ X4 @ S )
=> ( Q @ D6 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_993_bgauge__existence__lemma,axiom,
! [S: set_Pr8693737435421807431at_nat,Q: real > produc859450856879609959at_nat > $o] :
( ( ! [X4: produc859450856879609959at_nat] :
( ( member8206827879206165904at_nat @ X4 @ S )
=> ? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( Q @ D6 @ X4 ) ) ) )
= ( ! [X4: produc859450856879609959at_nat] :
? [D6: real] :
( ( ord_less_real @ zero_zero_real @ D6 )
& ( ( member8206827879206165904at_nat @ X4 @ S )
=> ( Q @ D6 @ X4 ) ) ) ) ) ).
% bgauge_existence_lemma
thf(fact_994_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_995_Abstract__Rewriting_Ochain__mono,axiom,
! [R3: set_Pr1261947904930325089at_nat,R2: set_Pr1261947904930325089at_nat,Seq: nat > nat] :
( ( ord_le3146513528884898305at_nat @ R3 @ R2 )
=> ( ! [I4: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R3 )
=> ! [I: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_996_Abstract__Rewriting_Ochain__mono,axiom,
! [R3: set_Pr8693737435421807431at_nat,R2: set_Pr8693737435421807431at_nat,Seq: nat > product_prod_nat_nat] :
( ( ord_le3000389064537975527at_nat @ R3 @ R2 )
=> ( ! [I4: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( Seq @ I4 ) @ ( Seq @ ( suc @ I4 ) ) ) @ R3 )
=> ! [I: nat] : ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R2 ) ) ) ).
% Abstract_Rewriting.chain_mono
thf(fact_997_shift_Ocases,axiom,
! [X: produc8199716216217303280at_nat] :
~ ! [F3: nat > nat,J3: nat] :
( X
!= ( produc72220940542539688at_nat @ F3 @ J3 ) ) ).
% shift.cases
thf(fact_998_subrelI,axiom,
! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
( ! [X2: nat,Y: nat] :
( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ R )
=> ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ S ) )
=> ( ord_le3146513528884898305at_nat @ R @ S ) ) ).
% subrelI
thf(fact_999_subrelI,axiom,
! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
( ! [X2: nat > nat,Y: nat] :
( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y ) @ R )
=> ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X2 @ Y ) @ S ) )
=> ( ord_le3678578370064672496at_nat @ R @ S ) ) ).
% subrelI
thf(fact_1000_subrelI,axiom,
! [R: set_Pr8693737435421807431at_nat,S: set_Pr8693737435421807431at_nat] :
( ! [X2: product_prod_nat_nat,Y: product_prod_nat_nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y ) @ R )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X2 @ Y ) @ S ) )
=> ( ord_le3000389064537975527at_nat @ R @ S ) ) ).
% subrelI
thf(fact_1001_Pair__le,axiom,
! [A3: nat,B3: nat,C: nat,D3: nat] :
( ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ C @ D3 ) )
= ( ( ord_less_eq_nat @ A3 @ C )
& ( ord_less_eq_nat @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1002_Pair__le,axiom,
! [A3: nat,B3: real,C: nat,D3: real] :
( ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ A3 @ B3 ) @ ( produc7837566107596912789t_real @ C @ D3 ) )
= ( ( ord_less_eq_nat @ A3 @ C )
& ( ord_less_eq_real @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1003_Pair__le,axiom,
! [A3: nat,B3: complex,C: nat,D3: complex] :
( ( ord_le5145267895229885055omplex @ ( produc6973218034000581911omplex @ A3 @ B3 ) @ ( produc6973218034000581911omplex @ C @ D3 ) )
= ( ( ord_less_eq_nat @ A3 @ C )
& ( ord_less_eq_complex @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1004_Pair__le,axiom,
! [A3: real,B3: nat,C: real,D3: nat] :
( ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ A3 @ B3 ) @ ( produc3181502643871035669al_nat @ C @ D3 ) )
= ( ( ord_less_eq_real @ A3 @ C )
& ( ord_less_eq_nat @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1005_Pair__le,axiom,
! [A3: real,B3: real,C: real,D3: real] :
( ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ A3 @ B3 ) @ ( produc4511245868158468465l_real @ C @ D3 ) )
= ( ( ord_less_eq_real @ A3 @ C )
& ( ord_less_eq_real @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1006_Pair__le,axiom,
! [A3: real,B3: complex,C: real,D3: complex] :
( ( ord_le6068715119631280347omplex @ ( produc1693001998875562995omplex @ A3 @ B3 ) @ ( produc1693001998875562995omplex @ C @ D3 ) )
= ( ( ord_less_eq_real @ A3 @ C )
& ( ord_less_eq_complex @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1007_Pair__le,axiom,
! [A3: complex,B3: nat,C: complex,D3: nat] :
( ( ord_le2081805474369280895ex_nat @ ( produc1369629321580543767ex_nat @ A3 @ B3 ) @ ( produc1369629321580543767ex_nat @ C @ D3 ) )
= ( ( ord_less_eq_complex @ A3 @ C )
& ( ord_less_eq_nat @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1008_Pair__le,axiom,
! [A3: complex,B3: real,C: complex,D3: real] :
( ( ord_le7981414139446038107x_real @ ( produc1746590499379883635x_real @ A3 @ B3 ) @ ( produc1746590499379883635x_real @ C @ D3 ) )
= ( ( ord_less_eq_complex @ A3 @ C )
& ( ord_less_eq_real @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1009_Pair__le,axiom,
! [A3: complex,B3: complex,C: complex,D3: complex] :
( ( ord_le6295960533335388509omplex @ ( produc101793102246108661omplex @ A3 @ B3 ) @ ( produc101793102246108661omplex @ C @ D3 ) )
= ( ( ord_less_eq_complex @ A3 @ C )
& ( ord_less_eq_complex @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1010_Pair__le,axiom,
! [A3: nat > nat,B3: nat,C: nat > nat,D3: nat] :
( ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ A3 @ B3 ) @ ( produc72220940542539688at_nat @ C @ D3 ) )
= ( ( ord_less_eq_nat_nat @ A3 @ C )
& ( ord_less_eq_nat @ B3 @ D3 ) ) ) ).
% Pair_le
thf(fact_1011_Pair__mono,axiom,
! [X: nat,X7: nat,Y2: nat,Y7: nat] :
( ( ord_less_eq_nat @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y2 @ Y7 )
=> ( ord_le8460144461188290721at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( product_Pair_nat_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1012_Pair__mono,axiom,
! [X: nat,X7: nat,Y2: real,Y7: real] :
( ( ord_less_eq_nat @ X @ X7 )
=> ( ( ord_less_eq_real @ Y2 @ Y7 )
=> ( ord_le8710666929947597437t_real @ ( produc7837566107596912789t_real @ X @ Y2 ) @ ( produc7837566107596912789t_real @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1013_Pair__mono,axiom,
! [X: nat,X7: nat,Y2: complex,Y7: complex] :
( ( ord_less_eq_nat @ X @ X7 )
=> ( ( ord_less_eq_complex @ Y2 @ Y7 )
=> ( ord_le5145267895229885055omplex @ ( produc6973218034000581911omplex @ X @ Y2 ) @ ( produc6973218034000581911omplex @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1014_Pair__mono,axiom,
! [X: real,X7: real,Y2: nat,Y7: nat] :
( ( ord_less_eq_real @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y2 @ Y7 )
=> ( ord_le4735619238470717181al_nat @ ( produc3181502643871035669al_nat @ X @ Y2 ) @ ( produc3181502643871035669al_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1015_Pair__mono,axiom,
! [X: real,X7: real,Y2: real,Y7: real] :
( ( ord_less_eq_real @ X @ X7 )
=> ( ( ord_less_eq_real @ Y2 @ Y7 )
=> ( ord_le1075799226346578649l_real @ ( produc4511245868158468465l_real @ X @ Y2 ) @ ( produc4511245868158468465l_real @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1016_Pair__mono,axiom,
! [X: real,X7: real,Y2: complex,Y7: complex] :
( ( ord_less_eq_real @ X @ X7 )
=> ( ( ord_less_eq_complex @ Y2 @ Y7 )
=> ( ord_le6068715119631280347omplex @ ( produc1693001998875562995omplex @ X @ Y2 ) @ ( produc1693001998875562995omplex @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1017_Pair__mono,axiom,
! [X: complex,X7: complex,Y2: nat,Y7: nat] :
( ( ord_less_eq_complex @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y2 @ Y7 )
=> ( ord_le2081805474369280895ex_nat @ ( produc1369629321580543767ex_nat @ X @ Y2 ) @ ( produc1369629321580543767ex_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1018_Pair__mono,axiom,
! [X: complex,X7: complex,Y2: real,Y7: real] :
( ( ord_less_eq_complex @ X @ X7 )
=> ( ( ord_less_eq_real @ Y2 @ Y7 )
=> ( ord_le7981414139446038107x_real @ ( produc1746590499379883635x_real @ X @ Y2 ) @ ( produc1746590499379883635x_real @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1019_Pair__mono,axiom,
! [X: complex,X7: complex,Y2: complex,Y7: complex] :
( ( ord_less_eq_complex @ X @ X7 )
=> ( ( ord_less_eq_complex @ Y2 @ Y7 )
=> ( ord_le6295960533335388509omplex @ ( produc101793102246108661omplex @ X @ Y2 ) @ ( produc101793102246108661omplex @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1020_Pair__mono,axiom,
! [X: nat > nat,X7: nat > nat,Y2: nat,Y7: nat] :
( ( ord_less_eq_nat_nat @ X @ X7 )
=> ( ( ord_less_eq_nat @ Y2 @ Y7 )
=> ( ord_le2819838839419867280at_nat @ ( produc72220940542539688at_nat @ X @ Y2 ) @ ( produc72220940542539688at_nat @ X7 @ Y7 ) ) ) ) ).
% Pair_mono
thf(fact_1021_minf_I8_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_1022_minf_I8_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ~ ( ord_less_eq_real @ T @ X3 ) ) ).
% minf(8)
thf(fact_1023_minf_I6_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_1024_minf_I6_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ord_less_eq_real @ X3 @ T ) ) ).
% minf(6)
thf(fact_1025_minf_I7_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_1026_minf_I7_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ~ ( ord_less_real @ T @ X3 ) ) ).
% minf(7)
thf(fact_1027_minf_I5_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_1028_minf_I5_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ord_less_real @ X3 @ T ) ) ).
% minf(5)
thf(fact_1029_minf_I4_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1030_minf_I4_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_1031_minf_I3_J,axiom,
! [T: nat] :
? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1032_minf_I3_J,axiom,
! [T: real] :
? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_1033_minf_I2_J,axiom,
! [P: nat > $o,P7: nat > $o,Q3: nat > $o,Q4: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P7 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1034_minf_I2_J,axiom,
! [P: real > $o,P7: real > $o,Q3: real > $o,Q4: real > $o] :
( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P7 @ X2 ) ) )
=> ( ? [Z4: real] :
! [X2: real] :
( ( ord_less_real @ X2 @ Z4 )
=> ( ( Q3 @ X2 )
= ( Q4 @ X2 ) ) )
=> ? [Z2: real] :
! [X3: real] :
( ( ord_less_real @ X3 @ Z2 )
=> ( ( ( P @ X3 )
| ( Q3 @ X3 ) )
= ( ( P7 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_1035_bell00__index_I2_J,axiom,
( ( index_mat_complex @ bell00 @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) )
= zero_zero_complex ) ).
% bell00_index(2)
thf(fact_1036_bell10__index_I2_J,axiom,
( ( index_mat_complex @ bell10 @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) )
= zero_zero_complex ) ).
% bell10_index(2)
thf(fact_1037_bell__11__index_I1_J,axiom,
( ( index_mat_complex @ bell11 @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
= zero_zero_complex ) ).
% bell_11_index(1)
thf(fact_1038_bell01__index_I1_J,axiom,
( ( index_mat_complex @ bell01 @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
= zero_zero_complex ) ).
% bell01_index(1)
thf(fact_1039_complex__is__real__iff__compare0,axiom,
! [X: complex] :
( ( member_complex @ X @ real_V2521375963428798218omplex )
= ( ( ord_less_eq_complex @ X @ zero_zero_complex )
| ( ord_less_eq_complex @ zero_zero_complex @ X ) ) ) ).
% complex_is_real_iff_compare0
thf(fact_1040_nonnegative__complex__is__real,axiom,
! [X: complex] :
( ( ord_less_eq_complex @ zero_zero_complex @ X )
=> ( member_complex @ X @ real_V2521375963428798218omplex ) ) ).
% nonnegative_complex_is_real
thf(fact_1041_real__root__increasing,axiom,
! [N2: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ( ord_less_eq_real @ X @ one_one_real )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_1042_real__root__ge__zero,axiom,
! [X: real,N2: nat] :
( ( ord_less_eq_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ).
% real_root_ge_zero
thf(fact_1043_real__root__le__mono,axiom,
! [N2: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ X @ Y2 )
=> ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_1044_real__root__le__iff,axiom,
! [N2: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y2 ) )
= ( ord_less_eq_real @ X @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_1045_root__0,axiom,
! [X: real] :
( ( root @ zero_zero_nat @ X )
= zero_zero_real ) ).
% root_0
thf(fact_1046_real__root__zero,axiom,
! [N2: nat] :
( ( root @ N2 @ zero_zero_real )
= zero_zero_real ) ).
% real_root_zero
thf(fact_1047_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ zero_zero_nat ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_1048_real__root__eq__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= zero_zero_real )
= ( X = zero_zero_real ) ) ) ).
% real_root_eq_0_iff
thf(fact_1049_real__root__less__mono,axiom,
! [N2: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ X @ Y2 )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_1050_real__root__less__iff,axiom,
! [N2: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N2 @ Y2 ) )
= ( ord_less_real @ X @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_1051_real__root__eq__iff,axiom,
! [N2: nat,X: real,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= ( root @ N2 @ Y2 ) )
= ( X = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_1052_real__root__eq__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ( root @ N2 @ X )
= one_one_real )
= ( X = one_one_real ) ) ) ).
% real_root_eq_1_iff
thf(fact_1053_real__root__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( root @ N2 @ one_one_real )
= one_one_real ) ) ).
% real_root_one
thf(fact_1054_real__root__ge__0__iff,axiom,
! [N2: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
= ( ord_less_eq_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_1055_real__root__le__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_eq_real @ X @ zero_zero_real ) ) ) ).
% real_root_le_0_iff
thf(fact_1056_real__root__lt__0__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ zero_zero_real )
= ( ord_less_real @ X @ zero_zero_real ) ) ) ).
% real_root_lt_0_iff
thf(fact_1057_real__root__gt__0__iff,axiom,
! [N2: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y2 ) )
= ( ord_less_real @ zero_zero_real @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_1058_real__root__gt__zero,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_1059_real__root__le__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_eq_real @ X @ one_one_real ) ) ) ).
% real_root_le_1_iff
thf(fact_1060_real__root__ge__1__iff,axiom,
! [N2: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y2 ) )
= ( ord_less_eq_real @ one_one_real @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_1061_real__root__strict__decreasing,axiom,
! [N2: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_real @ one_one_real @ X )
=> ( ord_less_real @ ( root @ N5 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_1062_real__root__lt__1__iff,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ ( root @ N2 @ X ) @ one_one_real )
= ( ord_less_real @ X @ one_one_real ) ) ) ).
% real_root_lt_1_iff
thf(fact_1063_real__root__gt__1__iff,axiom,
! [N2: nat,Y2: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y2 ) )
= ( ord_less_real @ one_one_real @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_1064_real__root__pos__pos,axiom,
! [N2: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_1065_real__root__strict__increasing,axiom,
! [N2: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ N2 @ N5 )
=> ( ( ord_less_real @ zero_zero_real @ X )
=> ( ( ord_less_real @ X @ one_one_real )
=> ( ord_less_real @ ( root @ N2 @ X ) @ ( root @ N5 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_1066_real__root__decreasing,axiom,
! [N2: nat,N5: nat,X: real] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ N5 )
=> ( ( ord_less_eq_real @ one_one_real @ X )
=> ( ord_less_eq_real @ ( root @ N5 @ X ) @ ( root @ N2 @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_1067_pair__lessI2,axiom,
! [A3: nat,B3: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_1068_pair__less__iff1,axiom,
! [X: nat,Y2: nat,Z: nat] :
( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( product_Pair_nat_nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less_nat @ Y2 @ Z ) ) ).
% pair_less_iff1
thf(fact_1069_pair__lessI1,axiom,
! [A3: nat,B3: nat,S: nat,T: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_1070_pair__leqI2,axiom,
! [A3: nat,B3: nat,S: nat,T: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( ( ord_less_eq_nat @ S @ T )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_1071_pair__leqI1,axiom,
! [A3: nat,B3: nat,S: nat,T: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ S ) @ ( product_Pair_nat_nat @ B3 @ T ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
% Helper facts (9)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y2: nat] :
( ( if_nat @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
! [X: real,Y2: real] :
( ( if_real @ $true @ X @ Y2 )
= X ) ).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
! [X: complex,Y2: complex] :
( ( if_complex @ $true @ X @ Y2 )
= X ) ).
thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
! [X: product_prod_nat_nat,Y2: product_prod_nat_nat] :
( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y2 )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
! [J3: nat] :
( ~ ( ord_less_nat @ J3 @ n )
| ( member_complex @ ( index_mat_complex @ ( commut4119912100034661455omplex @ b @ f ) @ ( product_Pair_nat_nat @ J3 @ J3 ) ) @ real_V2521375963428798218omplex ) ) ).
%------------------------------------------------------------------------------