TPTP Problem File: SLH0491^1.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 : ResiduatedTransitionSystem/0001_LambdaCalculus/prob_00644_026273__14356420_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1339 ( 594 unt; 60 typ; 0 def)
% Number of atoms : 3556 (1344 equ; 0 cnn)
% Maximal formula atoms : 23 ( 2 avg)
% Number of connectives : 11170 ( 406 ~; 95 |; 189 &;8981 @)
% ( 0 <=>;1499 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 6 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 215 ( 215 >; 0 *; 0 +; 0 <<)
% Number of symbols : 58 ( 55 usr; 13 con; 0-3 aty)
% Number of variables : 3457 ( 113 ^;3248 !; 96 ?;3457 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 11:48:34.340
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_n_t__LambdaCalculus__Olambda____calculus__Olambda,type,
lambda_lambda: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__String__Ochar,type,
char: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
% Explicit typings (55)
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
gbinomial_int: int > nat > int ).
thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
gbinomial_nat: nat > nat > nat ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
one_one_int: int ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint,type,
zero_zero_int: int ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Int__Oint,type,
if_int: $o > int > int > int ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OArr,type,
lambda_Arr: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OArr__rel,type,
lambda_Arr_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OFV,type,
lambda_FV: lambda_lambda > set_nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OIde,type,
lambda_Ide: lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OIde__rel,type,
lambda_Ide_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_ORaise,type,
lambda_Raise: nat > nat > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_OSubst,type,
lambda_Subst: nat > lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OApp,type,
lambda_App: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OBeta,type,
lambda_Beta: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OLam,type,
lambda_Lam: lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_ONil,type,
lambda_Nil: lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_OVar,type,
lambda_Var: nat > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Olambda_Osize__lambda,type,
lambda_size_lambda: lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Oresid,type,
lambda_resid: lambda_lambda > lambda_lambda > lambda_lambda ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osize,type,
lambda_size: lambda_lambda > nat ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osize__rel,type,
lambda_size_rel: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_LambdaCalculus_Olambda__calculus_Osubterm,type,
lambda_subterm: lambda_lambda > lambda_lambda > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1316708129612266289at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
size_s1768714712973771222lambda: lambda_lambda > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
size_size_char: char > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Int__Oint,type,
neg_nu3811975205180677377ec_int: int > int ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Int__Oint,type,
neg_nu5851722552734809277nc_int: int > int ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint,type,
ord_less_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
power_power_int: int > nat > int ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_Wellfounded_Oaccp_001t__LambdaCalculus__Olambda____calculus__Olambda,type,
accp_lambda_lambda: ( lambda_lambda > lambda_lambda > $o ) > lambda_lambda > $o ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v_t1____,type,
t1: lambda_lambda ).
thf(sy_v_t2____,type,
t2: lambda_lambda ).
thf(sy_v_ua____,type,
ua: lambda_lambda ).
thf(sy_v_x3,type,
x3: lambda_lambda ).
thf(sy_v_x51,type,
x51: lambda_lambda ).
thf(sy_v_x52,type,
x52: lambda_lambda ).
% Relevant facts (1265)
thf(fact_0_Arr__not__Nil,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( T != lambda_Nil ) ) ).
% Arr_not_Nil
thf(fact_1_Subst__Nil,axiom,
! [N: nat,V: lambda_lambda] :
( ( lambda_Subst @ N @ V @ lambda_Nil )
= lambda_Nil ) ).
% Subst_Nil
thf(fact_2_Subst__not__Nil,axiom,
! [V: lambda_lambda,T: lambda_lambda,N: nat] :
( ( V != lambda_Nil )
=> ( ( T != lambda_Nil )
=> ( ( T != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ T )
!= lambda_Nil ) ) ) ) ).
% Subst_not_Nil
thf(fact_3_Arr__Subst,axiom,
! [V: lambda_lambda,T: lambda_lambda,N: nat] :
( ( lambda_Arr @ V )
=> ( ( lambda_Arr @ T )
=> ( lambda_Arr @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% Arr_Subst
thf(fact_4_Con__implies__Arr1,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ T ) ) ).
% Con_implies_Arr1
thf(fact_5_Con__implies__Arr2,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_resid @ T @ U )
!= lambda_Nil )
=> ( lambda_Arr @ U ) ) ).
% Con_implies_Arr2
thf(fact_6_lambda__calculus_Olambda_Oinject_I4_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda,Y51: lambda_lambda,Y52: lambda_lambda] :
( ( ( lambda_Beta @ X51 @ X52 )
= ( lambda_Beta @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% lambda_calculus.lambda.inject(4)
thf(fact_7_lambda__calculus_Olambda_Oinject_I2_J,axiom,
! [X3: lambda_lambda,Y3: lambda_lambda] :
( ( ( lambda_Lam @ X3 )
= ( lambda_Lam @ Y3 ) )
= ( X3 = Y3 ) ) ).
% lambda_calculus.lambda.inject(2)
thf(fact_8_ind1,axiom,
! [U: lambda_lambda] :
( ( ( lambda_resid @ t1 @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ t1 @ U ) ) ) ).
% ind1
thf(fact_9_ind2,axiom,
! [U: lambda_lambda] :
( ( ( lambda_resid @ t2 @ U )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ t2 @ U ) ) ) ).
% ind2
thf(fact_10_lambda__calculus_OConD_I2_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(2)
thf(fact_11_lambda__calculus_OArr_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Beta @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(5)
thf(fact_12_lambda__calculus_OArr_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ ( lambda_Lam @ T ) )
= ( lambda_Arr @ T ) ) ).
% lambda_calculus.Arr.simps(3)
thf(fact_13_lambda__calculus_OArr_Osimps_I1_J,axiom,
~ ( lambda_Arr @ lambda_Nil ) ).
% lambda_calculus.Arr.simps(1)
thf(fact_14_lambda__calculus_OSubst_Osimps_I1_J,axiom,
! [Uu: nat,Uv: lambda_lambda] :
( ( lambda_Subst @ Uu @ Uv @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Subst.simps(1)
thf(fact_15_lambda__calculus_Oresid_Osimps_I40_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(40)
thf(fact_16_lambda__calculus_Oresid_Osimps_I38_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(38)
thf(fact_17_lambda__calculus_Oresid_Osimps_I30_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(30)
thf(fact_18_lambda__calculus_Oresid_Osimps_I27_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(27)
thf(fact_19_lambda__calculus_Oresid_Osimps_I26_J,axiom,
! [Uu: lambda_lambda] :
( ( lambda_resid @ Uu @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(26)
thf(fact_20_lambda__calculus_Oresid_Osimps_I21_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(21)
thf(fact_21_lambda__calculus_Oresid_Osimps_I19_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(19)
thf(fact_22_lambda__calculus_Oresid_Osimps_I11_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Beta @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(11)
thf(fact_23_lambda__calculus_Oresid_Osimps_I8_J,axiom,
! [V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(8)
thf(fact_24_lambda__calculus_Oresid_Osimps_I7_J,axiom,
! [Uv: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ Uv )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(7)
thf(fact_25_lambda__calculus_Oresid_Osimps_I4_J,axiom,
! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U2 ) @ ( lambda_resid @ T @ T2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(4)
thf(fact_26_lambda__calculus_Oresid_Osimps_I2_J,axiom,
! [T: lambda_lambda,T2: lambda_lambda] :
( ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T2 ) )
= lambda_Nil ) )
& ( ( ( lambda_resid @ T @ T2 )
!= lambda_Nil )
=> ( ( lambda_resid @ ( lambda_Lam @ T ) @ ( lambda_Lam @ T2 ) )
= ( lambda_Lam @ ( lambda_resid @ T @ T2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(2)
thf(fact_27_lambda__calculus_Olambda_Odistinct_I17_J,axiom,
! [X3: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(17)
thf(fact_28_lambda__calculus_Olambda_Odistinct_I7_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( lambda_Nil
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(7)
thf(fact_29_lambda__calculus_Olambda_Odistinct_I3_J,axiom,
! [X3: lambda_lambda] :
( lambda_Nil
!= ( lambda_Lam @ X3 ) ) ).
% lambda_calculus.lambda.distinct(3)
thf(fact_30_substitution__lemma,axiom,
! [V: lambda_lambda,W: lambda_lambda,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Subst @ zero_zero_nat @ W @ T ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Subst @ N @ V @ W ) @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) ) ) ) ) ).
% substitution_lemma
thf(fact_31_lambda__calculus_Oresid_Osimps_I5_J,axiom,
! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) @ ( lambda_Beta @ T2 @ U2 ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ U @ U2 ) @ ( lambda_resid @ T @ T2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(5)
thf(fact_32_Ide__Subst,axiom,
! [T: lambda_lambda,V: lambda_lambda,N: nat] :
( ( lambda_Ide @ T )
=> ( ( lambda_Ide @ V )
=> ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) ) ) ) ).
% Ide_Subst
thf(fact_33_Ide__implies__Arr,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ T )
=> ( lambda_Arr @ T ) ) ).
% Ide_implies_Arr
thf(fact_34_lambda__calculus_OArr_Oelims_I3_J,axiom,
! [X: lambda_lambda] :
( ~ ( lambda_Arr @ X )
=> ( ( X != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(3)
thf(fact_35_lambda__calculus_OConD_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_Beta @ V2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ ( lambda_Lam @ V2 ) )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(4)
thf(fact_36_lambda__calculus_OConD_I3_J,axiom,
! [V: lambda_lambda,U: lambda_lambda,T2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Beta @ V @ U ) @ ( lambda_App @ T2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ ( lambda_Lam @ V ) @ T2 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(3)
thf(fact_37_lambda__calculus_Oresid_Osimps_I6_J,axiom,
! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_Beta @ T @ U ) @ ( lambda_App @ ( lambda_Lam @ T2 ) @ U2 ) )
= ( lambda_Beta @ ( lambda_resid @ T @ T2 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(6)
thf(fact_38_Arr__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Arr
= ( ^ [T4: lambda_lambda] : ( lambda_Arr @ ( lambda_Raise @ D @ N @ T4 ) ) ) ) ).
% Arr_Raise
thf(fact_39_lambda__calculus_Olambda_Osize__gen_I1_J,axiom,
( ( lambda_size_lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size_gen(1)
thf(fact_40_Raise__not__Nil,axiom,
! [T: lambda_lambda,D: nat,N: nat] :
( ( T != lambda_Nil )
=> ( ( lambda_Raise @ D @ N @ T )
!= lambda_Nil ) ) ).
% Raise_not_Nil
thf(fact_41_Raise__inj,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( ( lambda_Raise @ D @ N @ T )
= ( lambda_Raise @ D @ N @ U ) )
=> ( T = U ) ) ).
% Raise_inj
thf(fact_42_Raise__0,axiom,
! [D: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ zero_zero_nat @ T )
= T ) ).
% Raise_0
thf(fact_43_Ide__Raise,axiom,
! [D: nat,N: nat] :
( lambda_Ide
= ( ^ [T4: lambda_lambda] : ( lambda_Ide @ ( lambda_Raise @ D @ N @ T4 ) ) ) ) ).
% Ide_Raise
thf(fact_44_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X2: nat] : ( member_nat @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_46_Raise__subst,axiom,
! [N: nat,K: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Raise @ N @ K @ ( lambda_Subst @ zero_zero_nat @ V @ T ) )
= ( lambda_Subst @ zero_zero_nat @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ N ) @ K @ T ) ) ) ).
% Raise_subst
thf(fact_47_lambda__calculus_Olambda_Oinject_I3_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda,Y41: lambda_lambda,Y42: lambda_lambda] :
( ( ( lambda_App @ X41 @ X42 )
= ( lambda_App @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% lambda_calculus.lambda.inject(3)
thf(fact_48_lambda__calculus_OIde_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Ide @ ( lambda_App @ T @ U ) )
= ( ( lambda_Ide @ T )
& ( lambda_Ide @ U ) ) ) ).
% lambda_calculus.Ide.simps(4)
thf(fact_49_lambda__calculus_ORaise_Osimps_I4_J,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Raise @ D @ N @ T ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(4)
thf(fact_50_lambda__calculus_ORaise_Osimps_I5_J,axiom,
! [D: nat,N: nat,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Raise @ ( suc @ D ) @ N @ T ) @ ( lambda_Raise @ D @ N @ U ) ) ) ).
% lambda_calculus.Raise.simps(5)
thf(fact_51_lambda__calculus_ORaise_Osimps_I3_J,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ N @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Raise @ ( suc @ D ) @ N @ T ) ) ) ).
% lambda_calculus.Raise.simps(3)
thf(fact_52_lambda__calculus_Olambda_Odistinct_I5_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( lambda_Nil
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(5)
thf(fact_53_lambda__calculus_Olambda_Odistinct_I19_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_App @ X41 @ X42 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(19)
thf(fact_54_lambda__calculus_Olambda_Odistinct_I15_J,axiom,
! [X3: lambda_lambda,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Lam @ X3 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(15)
thf(fact_55_lambda__calculus_OArr_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Arr @ ( lambda_App @ T @ U ) )
= ( ( lambda_Arr @ T )
& ( lambda_Arr @ U ) ) ) ).
% lambda_calculus.Arr.simps(4)
thf(fact_56_lambda__calculus_OSubst_Osimps_I4_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_App @ T @ U ) )
= ( lambda_App @ ( lambda_Subst @ N @ V @ T ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(4)
thf(fact_57_lambda__calculus_ORaise_Osimps_I1_J,axiom,
! [Uu: nat,Uv: nat] :
( ( lambda_Raise @ Uu @ Uv @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.Raise.simps(1)
thf(fact_58_lambda__calculus_OIde_Osimps_I1_J,axiom,
~ ( lambda_Ide @ lambda_Nil ) ).
% lambda_calculus.Ide.simps(1)
thf(fact_59_lambda__calculus_OIde_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
~ ( lambda_Ide @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.Ide.simps(5)
thf(fact_60_lambda__calculus_OIde_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Lam @ T ) )
= ( lambda_Ide @ T ) ) ).
% lambda_calculus.Ide.simps(3)
thf(fact_61_lambda__calculus_OIde_Oelims_I3_J,axiom,
! [X: lambda_lambda] :
( ~ ( lambda_Ide @ X )
=> ( ( X != lambda_Nil )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( X
!= ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(3)
thf(fact_62_lambda__calculus_Oresid_Osimps_I3_J,axiom,
! [T: lambda_lambda,T2: lambda_lambda,U: lambda_lambda,U2: lambda_lambda] :
( ( ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T2 @ U2 ) )
= lambda_Nil ) )
& ( ~ ( ( ( lambda_resid @ T @ T2 )
= lambda_Nil )
| ( ( lambda_resid @ U @ U2 )
= lambda_Nil ) )
=> ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T2 @ U2 ) )
= ( lambda_App @ ( lambda_resid @ T @ T2 ) @ ( lambda_resid @ U @ U2 ) ) ) ) ) ).
% lambda_calculus.resid.simps(3)
thf(fact_63_lambda__calculus_Oresid_Osimps_I12_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ lambda_Nil )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(12)
thf(fact_64_lambda__calculus_Oresid_Osimps_I31_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ lambda_Nil @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(31)
thf(fact_65_lambda__calculus_OConD_I1_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,T2: lambda_lambda,U2: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_App @ T @ U ) @ ( lambda_App @ T2 @ U2 ) )
!= lambda_Nil )
=> ( ( ( lambda_resid @ T @ T2 )
!= lambda_Nil )
& ( ( lambda_resid @ U @ U2 )
!= lambda_Nil ) ) ) ).
% lambda_calculus.ConD(1)
thf(fact_66_lambda__calculus_OSubst_Osimps_I5_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda,U: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Beta @ T @ U ) )
= ( lambda_Beta @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) @ ( lambda_Subst @ N @ V @ U ) ) ) ).
% lambda_calculus.Subst.simps(5)
thf(fact_67_lambda__calculus_OSubst_Osimps_I3_J,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Subst @ N @ V @ ( lambda_Lam @ T ) )
= ( lambda_Lam @ ( lambda_Subst @ ( suc @ N ) @ V @ T ) ) ) ).
% lambda_calculus.Subst.simps(3)
thf(fact_68_lambda__calculus_Oresid_Osimps_I15_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(15)
thf(fact_69_lambda__calculus_Oresid_Osimps_I17_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(17)
thf(fact_70_lambda__calculus_Oresid_Osimps_I18_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(18)
thf(fact_71_lambda__calculus_Oresid_Osimps_I22_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ lambda_Nil @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(22)
thf(fact_72_lambda__calculus_Oresid_Osimps_I24_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(24)
thf(fact_73_lambda__calculus_Oresid_Osimps_I25_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(25)
thf(fact_74_lambda__calculus_Oresid_Osimps_I34_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ lambda_Nil @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(34)
thf(fact_75_lambda__calculus_Oresid_Osimps_I36_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(36)
thf(fact_76_lambda__calculus_Oresid_Osimps_I37_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: lambda_lambda,Ve: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(37)
thf(fact_77_lambda__calculus_Oresid_Osimps_I41_J,axiom,
! [Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ lambda_Nil @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(41)
thf(fact_78_lambda__calculus_Oresid_Osimps_I43_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_App @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(43)
thf(fact_79_lambda__calculus_Oresid_Osimps_I44_J,axiom,
! [Vd: lambda_lambda,Ve: lambda_lambda,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Beta @ Vd @ Ve ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(44)
thf(fact_80_lambda__calculus_Oresid_Osimps_I10_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_App @ Va @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(10)
thf(fact_81_lambda__calculus_Oresid_Osimps_I14_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Lam @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(14)
thf(fact_82_lambda__calculus_Oresid_Osimps_I29_J,axiom,
! [Va: lambda_lambda,Vb: lambda_lambda,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ Va @ Vb ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(29)
thf(fact_83_lambda__calculus_Oresid_Osimps_I33_J,axiom,
! [Vb: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Lam @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(33)
thf(fact_84_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_85_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_86_Subst__raise,axiom,
! [V: lambda_lambda,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( lambda_Subst @ M @ V @ ( lambda_Raise @ zero_zero_nat @ ( suc @ N ) @ T ) )
= ( lambda_Raise @ zero_zero_nat @ N @ T ) ) ) ) ).
% Subst_raise
thf(fact_87_raise__Subst,axiom,
! [T: lambda_lambda,V: lambda_lambda,P2: nat,N: nat] :
( ( T != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Subst @ N @ V @ T ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ zero_zero_nat @ P2 @ T ) ) ) ) ) ).
% raise_Subst
thf(fact_88_Ide__Subst__iff,axiom,
! [N: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Ide @ ( lambda_Subst @ N @ V @ T ) )
= ( ( lambda_Ide @ T )
& ( ( member_nat @ N @ ( lambda_FV @ T ) )
=> ( lambda_Ide @ V ) ) ) ) ).
% Ide_Subst_iff
thf(fact_89_ArrE,axiom,
! [T: lambda_lambda] :
( ( lambda_Arr @ T )
=> ( ! [I: nat] :
( T
!= ( lambda_Var @ I ) )
=> ( ! [U3: lambda_lambda] :
( T
!= ( lambda_Lam @ U3 ) )
=> ( ! [U3: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_App @ U3 @ V3 ) )
=> ~ ! [U3: lambda_lambda,V3: lambda_lambda] :
( T
!= ( lambda_Beta @ U3 @ V3 ) ) ) ) ) ) ).
% ArrE
thf(fact_90_Raise__Subst,axiom,
! [P2: nat,N: nat,K: nat,V: lambda_lambda,T: lambda_lambda] :
( ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Subst @ P2 @ V @ T ) )
= ( lambda_Subst @ P2 @ ( lambda_Raise @ N @ K @ V ) @ ( lambda_Raise @ ( suc @ ( plus_plus_nat @ P2 @ N ) ) @ K @ T ) ) ) ).
% Raise_Subst
thf(fact_91_Subst__Subst,axiom,
! [V: lambda_lambda,W: lambda_lambda,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( W != lambda_Nil )
=> ( ( lambda_Subst @ ( plus_plus_nat @ M @ N ) @ W @ ( lambda_Subst @ M @ V @ T ) )
= ( lambda_Subst @ M @ ( lambda_Subst @ N @ W @ V ) @ ( lambda_Subst @ ( suc @ ( plus_plus_nat @ M @ N ) ) @ W @ T ) ) ) ) ) ).
% Subst_Subst
thf(fact_92_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_93_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_94_size__Raise,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_size @ ( lambda_Raise @ D @ N @ T ) )
= ( lambda_size @ T ) ) ).
% size_Raise
thf(fact_95_Raise__Suc,axiom,
! [D: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ ( suc @ N ) @ T )
= ( lambda_Raise @ D @ one_one_nat @ ( lambda_Raise @ D @ N @ T ) ) ) ).
% Raise_Suc
thf(fact_96_Raise__plus,axiom,
! [D: nat,M: nat,N: nat,T: lambda_lambda] :
( ( lambda_Raise @ D @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ ( plus_plus_nat @ D @ M ) @ N @ ( lambda_Raise @ D @ M @ T ) ) ) ).
% Raise_plus
thf(fact_97_raise__Raise,axiom,
! [P2: nat,N: nat,K: nat,T: lambda_lambda] :
( ( lambda_Raise @ zero_zero_nat @ P2 @ ( lambda_Raise @ N @ K @ T ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ zero_zero_nat @ P2 @ T ) ) ) ).
% raise_Raise
thf(fact_98_Raise__plus_H,axiom,
! [D2: nat,D: nat,N: nat,M: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ D2 @ ( plus_plus_nat @ D @ N ) )
=> ( ( ord_less_eq_nat @ D @ D2 )
=> ( ( lambda_Raise @ D @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ D2 @ M @ ( lambda_Raise @ D @ N @ T ) ) ) ) ) ).
% Raise_plus'
thf(fact_99_Raise__Raise,axiom,
! [I2: nat,N: nat,P2: nat,K: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( lambda_Raise @ I2 @ P2 @ ( lambda_Raise @ N @ K @ T ) )
= ( lambda_Raise @ ( plus_plus_nat @ P2 @ N ) @ K @ ( lambda_Raise @ I2 @ P2 @ T ) ) ) ) ).
% Raise_Raise
thf(fact_100_raise__plus,axiom,
! [D: nat,N: nat,M: nat,T: lambda_lambda] :
( ( ord_less_eq_nat @ D @ N )
=> ( ( lambda_Raise @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) @ T )
= ( lambda_Raise @ D @ M @ ( lambda_Raise @ zero_zero_nat @ N @ T ) ) ) ) ).
% raise_plus
thf(fact_101_add__left__cancel,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_102_add__left__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add_left_cancel
thf(fact_103_add__right__cancel,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_104_add__right__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add_right_cancel
thf(fact_105_lambda__calculus_Olambda_Oinject_I1_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( lambda_Var @ X22 )
= ( lambda_Var @ Y2 ) )
= ( X22 = Y2 ) ) ).
% lambda_calculus.lambda.inject(1)
thf(fact_106_Raise__Subst_H,axiom,
! [T: lambda_lambda,V: lambda_lambda,K: nat,N: nat,P2: nat] :
( ( T != lambda_Nil )
=> ( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( lambda_Raise @ K @ P2 @ ( lambda_Subst @ N @ V @ T ) )
= ( lambda_Subst @ ( plus_plus_nat @ P2 @ N ) @ V @ ( lambda_Raise @ K @ P2 @ T ) ) ) ) ) ) ).
% Raise_Subst'
thf(fact_107_vacuous__Subst,axiom,
! [V: lambda_lambda,I2: nat,T: lambda_lambda] :
( ( lambda_Arr @ V )
=> ( ~ ( member_nat @ I2 @ ( lambda_FV @ T ) )
=> ( ( lambda_Raise @ I2 @ one_one_nat @ ( lambda_Subst @ I2 @ V @ T ) )
= T ) ) ) ).
% vacuous_Subst
thf(fact_108_Subst__Raise,axiom,
! [V: lambda_lambda,D: nat,M: nat,N: nat,T: lambda_lambda] :
( ( V != lambda_Nil )
=> ( ( ord_less_eq_nat @ D @ M )
=> ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N @ D ) )
=> ( ( lambda_Subst @ M @ V @ ( lambda_Raise @ D @ ( suc @ N ) @ T ) )
= ( lambda_Raise @ D @ N @ T ) ) ) ) ) ).
% Subst_Raise
thf(fact_109_le__zero__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_110_add__le__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_111_add__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_112_add__le__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_113_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_eq_int @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_114_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_115_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_116_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_117_add__cancel__left__left,axiom,
! [B: nat,A: nat] :
( ( ( plus_plus_nat @ B @ A )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_118_add__cancel__left__left,axiom,
! [B: int,A: int] :
( ( ( plus_plus_int @ B @ A )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_119_add__cancel__left__right,axiom,
! [A: nat,B: nat] :
( ( ( plus_plus_nat @ A @ B )
= A )
= ( B = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_120_add__cancel__left__right,axiom,
! [A: int,B: int] :
( ( ( plus_plus_int @ A @ B )
= A )
= ( B = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_121_add__cancel__right__left,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ B @ A ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_122_add__cancel__right__left,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ B @ A ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_123_add__cancel__right__right,axiom,
! [A: nat,B: nat] :
( ( A
= ( plus_plus_nat @ A @ B ) )
= ( B = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_124_add__cancel__right__right,axiom,
! [A: int,B: int] :
( ( A
= ( plus_plus_int @ A @ B ) )
= ( B = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_125_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_126_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y ) )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_127_add__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add_0
thf(fact_128_add__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add_0
thf(fact_129_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_130_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_131_le0,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% le0
thf(fact_132_add__Suc__right,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ M @ ( suc @ N ) )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc_right
thf(fact_133_add__is__0,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
& ( N = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_134_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_135_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_136_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ A ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_137_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_138_le__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel2
thf(fact_139_le__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel2
thf(fact_140_le__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).
% le_add_same_cancel1
thf(fact_141_le__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_eq_int @ zero_zero_int @ B ) ) ).
% le_add_same_cancel1
thf(fact_142_add__le__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel2
thf(fact_143_add__le__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel2
thf(fact_144_add__le__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).
% add_le_same_cancel1
thf(fact_145_add__le__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_eq_int @ A @ zero_zero_int ) ) ).
% add_le_same_cancel1
thf(fact_146_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_147_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_148_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I2 @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_149_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I2 @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_150_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_151_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_152_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_153_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_154_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_155_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_156_lambda__calculus_Osize_Osimps_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_App @ T @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(4)
thf(fact_157_lambda__calculus_ORaise_Osimps_I2_J,axiom,
! [D: nat,I2: nat,N: nat] :
( ( ( ord_less_eq_nat @ D @ I2 )
=> ( ( lambda_Raise @ D @ N @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ ( plus_plus_nat @ I2 @ N ) ) ) )
& ( ~ ( ord_less_eq_nat @ D @ I2 )
=> ( ( lambda_Raise @ D @ N @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ I2 ) ) ) ) ).
% lambda_calculus.Raise.simps(2)
thf(fact_158_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_159_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_160_group__cancel_Oadd2,axiom,
! [B2: nat,K: nat,B: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_161_group__cancel_Oadd2,axiom,
! [B2: int,K: int,B: int,A: int] :
( ( B2
= ( plus_plus_int @ K @ B ) )
=> ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_162_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_163_lift__Suc__mono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_164_add_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.assoc
thf(fact_165_add_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.assoc
thf(fact_166_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_167_lift__Suc__antimono__le,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq_nat @ N @ N2 )
=> ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_168_add_Oleft__cancel,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
= ( B = C ) ) ).
% add.left_cancel
thf(fact_169_add_Oright__cancel,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
= ( B = C ) ) ).
% add.right_cancel
thf(fact_170_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_171_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).
% add.commute
thf(fact_172_add_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% add.left_commute
thf(fact_173_add_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% add.left_commute
thf(fact_174_add__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_mono
thf(fact_175_add__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_mono
thf(fact_176_add__left__imp__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_177_add__left__imp__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( plus_plus_int @ A @ B )
= ( plus_plus_int @ A @ C ) )
=> ( B = C ) ) ).
% add_left_imp_eq
thf(fact_178_add__right__imp__eq,axiom,
! [B: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_179_add__right__imp__eq,axiom,
! [B: int,A: int,C: int] :
( ( ( plus_plus_int @ B @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B = C ) ) ).
% add_right_imp_eq
thf(fact_180_add__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_left_mono
thf(fact_181_add__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_left_mono
thf(fact_182_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C2: nat] :
( B
!= ( plus_plus_nat @ A @ C2 ) ) ) ).
% less_eqE
thf(fact_183_add__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_right_mono
thf(fact_184_add__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_right_mono
thf(fact_185_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
? [C3: nat] :
( B3
= ( plus_plus_nat @ A3 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_186_add__le__imp__le__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_187_add__le__imp__le__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_188_add__le__imp__le__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_189_add__le__imp__le__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_eq_int @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_190_Suc__eq__plus1,axiom,
( suc
= ( ^ [N4: nat] : ( plus_plus_nat @ N4 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_191_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_192_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_193_one__reorient,axiom,
! [X: int] :
( ( one_one_int = X )
= ( X = one_one_int ) ) ).
% one_reorient
thf(fact_194_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_195_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_196_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_197_trans__le__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_198_trans__le__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_199_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_200_add__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_201_add__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_202_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_203_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N3: nat] :
( L
= ( plus_plus_nat @ K @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_204_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_205_le__trans,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I2 @ K ) ) ) ).
% le_trans
thf(fact_206_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_207_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_208_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_209_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_210_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_211_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_212_lambda__calculus_Osize_Osimps_I2_J,axiom,
! [Uu: nat] :
( ( lambda_size @ ( lambda_Var @ Uu ) )
= one_one_nat ) ).
% lambda_calculus.size.simps(2)
thf(fact_213_lambda__calculus_Osize_Osimps_I3_J,axiom,
! [T: lambda_lambda] :
( ( lambda_size @ ( lambda_Lam @ T ) )
= ( plus_plus_nat @ ( lambda_size @ T ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(3)
thf(fact_214_lambda__calculus_Osize_Osimps_I5_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_size @ ( lambda_Beta @ T @ U ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T ) @ one_one_nat ) @ ( lambda_size @ U ) ) @ one_one_nat ) ) ).
% lambda_calculus.size.simps(5)
thf(fact_215_add__nonpos__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_216_add__nonpos__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ zero_zero_int )
=> ( ( ord_less_eq_int @ Y @ zero_zero_int )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_217_add__nonneg__eq__0__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ X )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
=> ( ( ( plus_plus_nat @ X @ Y )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y = zero_zero_nat ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_218_add__nonneg__eq__0__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ( plus_plus_int @ X @ Y )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_219_add__nonpos__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_nonpos
thf(fact_220_add__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_nonpos
thf(fact_221_add__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_222_add__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_223_add__increasing2,axiom,
! [C: nat,B: nat,A: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_224_add__increasing2,axiom,
! [C: int,B: int,A: int] :
( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing2
thf(fact_225_add__decreasing2,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ C @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_226_add__decreasing2,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing2
thf(fact_227_add__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_228_add__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_increasing
thf(fact_229_add__decreasing,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_230_add__decreasing,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).
% add_decreasing
thf(fact_231_add__Suc__shift,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ M @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_232_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_233_nat__arith_Osuc1,axiom,
! [A2: nat,K: nat,A: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( suc @ A2 )
= ( plus_plus_nat @ K @ ( suc @ A ) ) ) ) ).
% nat_arith.suc1
thf(fact_234_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_235_add__eq__self__zero,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= M )
=> ( N = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_236_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z )
=> ( R @ X4 @ Z ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_237_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ M @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_238_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_239_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_240_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_241_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_242_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_243_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_244_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_245_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_246_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).
% less_eq_nat.simps(1)
thf(fact_247_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_248_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_249_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq_nat @ N @ zero_zero_nat )
= ( N = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_250_comm__monoid__add__class_Oadd__0,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_251_comm__monoid__add__class_Oadd__0,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% comm_monoid_add_class.add_0
thf(fact_252_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_253_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_254_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_255_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_256_lambda__calculus_Olambda_Odistinct_I1_J,axiom,
! [X22: nat] :
( lambda_Nil
!= ( lambda_Var @ X22 ) ) ).
% lambda_calculus.lambda.distinct(1)
thf(fact_257_lambda__calculus_Olambda_Odistinct_I11_J,axiom,
! [X22: nat,X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_App @ X41 @ X42 ) ) ).
% lambda_calculus.lambda.distinct(11)
thf(fact_258_lambda__calculus_Olambda_Odistinct_I13_J,axiom,
! [X22: nat,X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_Beta @ X51 @ X52 ) ) ).
% lambda_calculus.lambda.distinct(13)
thf(fact_259_lambda__calculus_Olambda_Odistinct_I9_J,axiom,
! [X22: nat,X3: lambda_lambda] :
( ( lambda_Var @ X22 )
!= ( lambda_Lam @ X3 ) ) ).
% lambda_calculus.lambda.distinct(9)
thf(fact_260_lambda__calculus_OArr_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Arr @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Arr.simps(2)
thf(fact_261_lambda__calculus_OIde_Osimps_I2_J,axiom,
! [Uu: nat] : ( lambda_Ide @ ( lambda_Var @ Uu ) ) ).
% lambda_calculus.Ide.simps(2)
thf(fact_262_lambda__calculus_Osize_Oelims,axiom,
! [X: lambda_lambda,Y: nat] :
( ( ( lambda_size @ X )
= Y )
=> ( ( ( X = lambda_Nil )
=> ( Y != zero_zero_nat ) )
=> ( ( ? [Uu2: nat] :
( X
= ( lambda_Var @ Uu2 ) )
=> ( Y != one_one_nat ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( Y
!= ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( Y
!= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( Y
!= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.elims
thf(fact_263_one__is__add,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M @ N ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_264_add__is__1,axiom,
! [M: nat,N: nat] :
( ( ( plus_plus_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( ( M
= ( suc @ zero_zero_nat ) )
& ( N = zero_zero_nat ) )
| ( ( M = zero_zero_nat )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_265_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_266_lambda__calculus_Oresid_Osimps_I1_J,axiom,
! [I2: nat,I3: nat] :
( ( ( I2 = I3 )
=> ( ( lambda_resid @ ( lambda_Var @ I2 ) @ ( lambda_Var @ I3 ) )
= ( lambda_Var @ I2 ) ) )
& ( ( I2 != I3 )
=> ( ( lambda_resid @ ( lambda_Var @ I2 ) @ ( lambda_Var @ I3 ) )
= lambda_Nil ) ) ) ).
% lambda_calculus.resid.simps(1)
thf(fact_267_lambda__calculus_Olambda_Osize__gen_I2_J,axiom,
! [X22: nat] :
( ( lambda_size_lambda @ ( lambda_Var @ X22 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size_gen(2)
thf(fact_268_lambda__calculus_Osize_Osimps_I1_J,axiom,
( ( lambda_size @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.size.simps(1)
thf(fact_269_lambda__calculus_Oresid_Osimps_I32_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_App @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(32)
thf(fact_270_lambda__calculus_Oresid_Osimps_I13_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_App @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(13)
thf(fact_271_lambda__calculus_Oresid_Osimps_I39_J,axiom,
! [Vb: nat,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Vb ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(39)
thf(fact_272_lambda__calculus_Oresid_Osimps_I20_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vb: nat] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_Var @ Vb ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(20)
thf(fact_273_lambda__calculus_Oresid_Osimps_I28_J,axiom,
! [Va: nat,V: lambda_lambda] :
( ( lambda_resid @ ( lambda_Var @ Va ) @ ( lambda_Lam @ V ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(28)
thf(fact_274_lambda__calculus_Oresid_Osimps_I9_J,axiom,
! [V: lambda_lambda,Va: nat] :
( ( lambda_resid @ ( lambda_Lam @ V ) @ ( lambda_Var @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(9)
thf(fact_275_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_276_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_277_lambda__calculus_OIde_Oelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ! [Uu2: nat] :
( X
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(2)
thf(fact_278_lambda__calculus_Osize_Ocases,axiom,
! [X: lambda_lambda] :
( ( X != lambda_Nil )
=> ( ! [Uu2: nat] :
( X
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( X
!= ( lambda_Lam @ T3 ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( X
!= ( lambda_App @ T3 @ U3 ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( X
!= ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ).
% lambda_calculus.size.cases
thf(fact_279_lambda__calculus_Olambda_Oexhaust,axiom,
! [Y: lambda_lambda] :
( ( Y != lambda_Nil )
=> ( ! [X23: nat] :
( Y
!= ( lambda_Var @ X23 ) )
=> ( ! [X32: lambda_lambda] :
( Y
!= ( lambda_Lam @ X32 ) )
=> ( ! [X412: lambda_lambda,X422: lambda_lambda] :
( Y
!= ( lambda_App @ X412 @ X422 ) )
=> ~ ! [X512: lambda_lambda,X522: lambda_lambda] :
( Y
!= ( lambda_Beta @ X512 @ X522 ) ) ) ) ) ) ).
% lambda_calculus.lambda.exhaust
thf(fact_280_lambda__calculus_Oresid_Osimps_I42_J,axiom,
! [Vd: nat,Vc: lambda_lambda,V: lambda_lambda,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) @ ( lambda_Beta @ V @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(42)
thf(fact_281_lambda__calculus_Oresid_Osimps_I35_J,axiom,
! [Vb: lambda_lambda,Vc: lambda_lambda,Vd: nat,Va: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ Vb @ Vc ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(35)
thf(fact_282_lambda__calculus_Oresid_Osimps_I23_J,axiom,
! [V: lambda_lambda,Va: lambda_lambda,Vd: nat,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_Beta @ V @ Va ) @ ( lambda_App @ ( lambda_Var @ Vd ) @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(23)
thf(fact_283_lambda__calculus_Oresid_Osimps_I16_J,axiom,
! [Vd: nat,Va: lambda_lambda,Vb: lambda_lambda,Vc: lambda_lambda] :
( ( lambda_resid @ ( lambda_App @ ( lambda_Var @ Vd ) @ Va ) @ ( lambda_Beta @ Vb @ Vc ) )
= lambda_Nil ) ).
% lambda_calculus.resid.simps(16)
thf(fact_284_lambda__calculus_OArr_Oelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda_Arr @ X )
=> ( ! [Uu2: nat] :
( X
!= ( lambda_Var @ Uu2 ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(2)
thf(fact_285_lambda__calculus_Olambda_Osize__gen_I4_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_App @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size_lambda @ X41 ) @ ( lambda_size_lambda @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(4)
thf(fact_286_lambda__calculus_Olambda_Osize__gen_I5_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_Beta @ X51 @ X52 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size_lambda @ X51 ) @ ( lambda_size_lambda @ X52 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(5)
thf(fact_287_lambda__calculus_Olambda_Osize__gen_I3_J,axiom,
! [X3: lambda_lambda] :
( ( lambda_size_lambda @ ( lambda_Lam @ X3 ) )
= ( plus_plus_nat @ ( lambda_size_lambda @ X3 ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size_gen(3)
thf(fact_288_lambda__calculus_OArr_Oelims_I1_J,axiom,
! [X: lambda_lambda,Y: $o] :
( ( ( lambda_Arr @ X )
= Y )
=> ( ( ( X = lambda_Nil )
=> Y )
=> ( ( ? [Uu2: nat] :
( X
= ( lambda_Var @ Uu2 ) )
=> ~ Y )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( Y
= ( ~ ( lambda_Arr @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( Y
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( Y
= ( ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.elims(1)
thf(fact_289_lambda__calculus_OIde_Oelims_I1_J,axiom,
! [X: lambda_lambda,Y: $o] :
( ( ( lambda_Ide @ X )
= Y )
=> ( ( ( X = lambda_Nil )
=> Y )
=> ( ( ? [Uu2: nat] :
( X
= ( lambda_Var @ Uu2 ) )
=> ~ Y )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( Y
= ( ~ ( lambda_Ide @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( Y
= ( ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) ) ) )
=> ~ ( ? [T3: lambda_lambda,U3: lambda_lambda] :
( X
= ( lambda_Beta @ T3 @ U3 ) )
=> Y ) ) ) ) ) ) ).
% lambda_calculus.Ide.elims(1)
thf(fact_290_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_291_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_292_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_293_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_294_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_295_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N3: nat] :
( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_296_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X4: nat,Y4: nat] :
( ( P @ X4 @ Y4 )
=> ( P @ ( suc @ X4 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_297_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_298_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_299_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_300_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_301_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% not0_implies_Suc
thf(fact_302_double__eq__0__iff,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_eq_0_iff
thf(fact_303_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_304_zero__less__one__class_Ozero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_less_one_class.zero_le_one
thf(fact_305_zero__le__one,axiom,
ord_less_eq_nat @ zero_zero_nat @ one_one_nat ).
% zero_le_one
thf(fact_306_zero__le__one,axiom,
ord_less_eq_int @ zero_zero_int @ one_one_int ).
% zero_le_one
thf(fact_307_not__one__le__zero,axiom,
~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_le_zero
thf(fact_308_not__one__le__zero,axiom,
~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).
% not_one_le_zero
thf(fact_309_Raise__Var,axiom,
! [D: nat,N: nat,I2: nat] :
( ( lambda_Raise @ D @ N @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ ( if_nat @ ( ord_less_nat @ I2 @ D ) @ I2 @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).
% Raise_Var
thf(fact_310_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_311_dual__order_Orefl,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% dual_order.refl
thf(fact_312_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_313_order__refl,axiom,
! [X: int] : ( ord_less_eq_int @ X @ X ) ).
% order_refl
thf(fact_314_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
= ( P @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
=> ( ! [A4: nat,B4: nat] :
( ( P @ A4 @ B4 )
=> ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
=> ( P @ A @ B ) ) ) ) ).
% Euclid_induct
thf(fact_315_lambda__calculus_Olambda_Osize_I8_J,axiom,
! [X3: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_Lam @ X3 ) )
= ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X3 ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(8)
thf(fact_316_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_317_add__less__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_318_add__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_319_add__less__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_320_add__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_321_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_322_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_323_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_324_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_325_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_326_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_327_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_328_add__less__same__cancel1,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_329_add__less__same__cancel1,axiom,
! [B: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_330_add__less__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_331_add__less__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_332_less__add__same__cancel1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel1
thf(fact_333_less__add__same__cancel1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel1
thf(fact_334_less__add__same__cancel2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B ) ) ).
% less_add_same_cancel2
thf(fact_335_less__add__same__cancel2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
= ( ord_less_int @ zero_zero_int @ B ) ) ).
% less_add_same_cancel2
thf(fact_336_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ A ) @ zero_zero_int )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_337_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ A ) )
= ( ord_less_int @ zero_zero_int @ A ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_338_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_339_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_340_add__gr__0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
| ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% add_gr_0
thf(fact_341_less__one,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ one_one_nat )
= ( N = zero_zero_nat ) ) ).
% less_one
thf(fact_342_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_343_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N2: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ N @ N2 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_344_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_345_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_346_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_347_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_348_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_349_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_350_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_351_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_352_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_353_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_354_lt__ex,axiom,
! [X: int] :
? [Y4: int] : ( ord_less_int @ Y4 @ X ) ).
% lt_ex
thf(fact_355_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_356_gt__ex,axiom,
! [X: int] :
? [X_1: int] : ( ord_less_int @ X @ X_1 ) ).
% gt_ex
thf(fact_357_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_358_less__imp__neq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_359_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_360_order_Oasym,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order.asym
thf(fact_361_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_362_ord__eq__less__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_363_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_364_ord__less__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_int @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_365_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y5: nat] :
( ( ord_less_nat @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_366_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_367_antisym__conv3,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_int @ Y @ X )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_368_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_369_linorder__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_370_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_371_dual__order_Oasym,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ~ ( ord_less_int @ A @ B ) ) ).
% dual_order.asym
thf(fact_372_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_373_dual__order_Oirrefl,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% dual_order.irrefl
thf(fact_374_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N4: nat] :
( ( P4 @ N4 )
& ! [M2: nat] :
( ( ord_less_nat @ M2 @ N4 )
=> ~ ( P4 @ M2 ) ) ) ) ) ).
% exists_least_iff
thf(fact_375_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_376_linorder__less__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int] : ( P @ A4 @ A4 )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_377_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_378_order_Ostrict__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_379_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_380_not__less__iff__gr__or__eq,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ( ord_less_int @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_381_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_382_dual__order_Ostrict__trans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_383_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_384_order_Ostrict__implies__not__eq,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_385_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_386_dual__order_Ostrict__implies__not__eq,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_387_linorder__neqE__linordered__idom,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_388_linorder__neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_389_linorder__neqE,axiom,
! [X: int,Y: int] :
( ( X != Y )
=> ( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neqE
thf(fact_390_order__less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_asym
thf(fact_391_order__less__asym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_asym
thf(fact_392_linorder__neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_393_linorder__neq__iff,axiom,
! [X: int,Y: int] :
( ( X != Y )
= ( ( ord_less_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ) ).
% linorder_neq_iff
thf(fact_394_order__less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order_less_asym'
thf(fact_395_order__less__asym_H,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ~ ( ord_less_int @ B @ A ) ) ).
% order_less_asym'
thf(fact_396_order__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_397_order__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_trans
thf(fact_398_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_399_ord__eq__less__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_400_ord__eq__less__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_401_ord__eq__less__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_402_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_403_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_404_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_405_ord__less__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_406_order__less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% order_less_irrefl
thf(fact_407_order__less__irrefl,axiom,
! [X: int] :
~ ( ord_less_int @ X @ X ) ).
% order_less_irrefl
thf(fact_408_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_409_order__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_410_order__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_411_order__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_412_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_413_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_414_order__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_415_order__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_416_order__less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_417_order__less__not__sym,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_not_sym
thf(fact_418_order__less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_419_order__less__imp__triv,axiom,
! [X: int,Y: int,P: $o] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_int @ Y @ X )
=> P ) ) ).
% order_less_imp_triv
thf(fact_420_linorder__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_421_linorder__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
| ( X = Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_less_linear
thf(fact_422_order__less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_423_order__less__imp__not__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( X != Y ) ) ).
% order_less_imp_not_eq
thf(fact_424_order__less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_425_order__less__imp__not__eq2,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( Y != X ) ) ).
% order_less_imp_not_eq2
thf(fact_426_order__less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_427_order__less__imp__not__less,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ~ ( ord_less_int @ Y @ X ) ) ).
% order_less_imp_not_less
thf(fact_428_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_429_leD,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_int @ X @ Y ) ) ).
% leD
thf(fact_430_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_431_leI,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% leI
thf(fact_432_nless__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_nat @ A @ B ) )
= ( ~ ( ord_less_eq_nat @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_433_nless__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_int @ A @ B ) )
= ( ~ ( ord_less_eq_int @ A @ B )
| ( A = B ) ) ) ).
% nless_le
thf(fact_434_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_435_antisym__conv1,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_436_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_437_antisym__conv2,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ~ ( ord_less_int @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_438_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_439_less__le__not__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ~ ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% less_le_not_le
thf(fact_440_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_441_not__le__imp__less,axiom,
! [Y: int,X: int] :
( ~ ( ord_less_eq_int @ Y @ X )
=> ( ord_less_int @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_442_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_443_order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_int @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ).
% order.order_iff_strict
thf(fact_444_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_445_order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ).
% order.strict_iff_order
thf(fact_446_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_447_order_Ostrict__trans1,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_448_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_449_order_Ostrict__trans2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_450_order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ~ ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_451_order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ~ ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order.strict_iff_not
thf(fact_452_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_nat @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_453_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_int @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_454_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_455_dual__order_Ostrict__iff__order,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_456_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_457_dual__order_Ostrict__trans1,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_458_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_459_dual__order_Ostrict__trans2,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_int @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_460_dual__order_Ostrict__iff__not,axiom,
( ord_less_nat
= ( ^ [B3: nat,A3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ~ ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_461_dual__order_Ostrict__iff__not,axiom,
( ord_less_int
= ( ^ [B3: int,A3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ~ ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_462_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_463_order_Ostrict__implies__order,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_eq_int @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_464_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_465_dual__order_Ostrict__implies__order,axiom,
! [B: int,A: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_eq_int @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_466_order__le__less,axiom,
( ord_less_eq_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_nat @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_467_order__le__less,axiom,
( ord_less_eq_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_int @ X2 @ Y6 )
| ( X2 = Y6 ) ) ) ) ).
% order_le_less
thf(fact_468_order__less__le,axiom,
( ord_less_nat
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_469_order__less__le,axiom,
( ord_less_int
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( X2 != Y6 ) ) ) ) ).
% order_less_le
thf(fact_470_linorder__not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% linorder_not_le
thf(fact_471_linorder__not__le,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_eq_int @ X @ Y ) )
= ( ord_less_int @ Y @ X ) ) ).
% linorder_not_le
thf(fact_472_linorder__not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_not_less
thf(fact_473_linorder__not__less,axiom,
! [X: int,Y: int] :
( ( ~ ( ord_less_int @ X @ Y ) )
= ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_not_less
thf(fact_474_order__less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_475_order__less__imp__le,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ X @ Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_less_imp_le
thf(fact_476_order__le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_477_order__le__neq__trans,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( A != B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_le_neq_trans
thf(fact_478_order__neq__le__trans,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_479_order__neq__le__trans,axiom,
! [A: int,B: int] :
( ( A != B )
=> ( ( ord_less_eq_int @ A @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% order_neq_le_trans
thf(fact_480_order__le__less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_481_order__le__less__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_le_less_trans
thf(fact_482_order__less__le__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_483_order__less__le__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_int @ X @ Z2 ) ) ) ).
% order_less_le_trans
thf(fact_484_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_485_order__le__less__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_486_order__le__less__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_487_order__le__less__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_488_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_489_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_490_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_491_order__le__less__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_492_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_493_order__less__le__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_494_order__less__le__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_495_order__less__le__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_496_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_497_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_498_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_nat @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_499_order__less__le__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_int @ X4 @ Y4 )
=> ( ord_less_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_500_linorder__le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_501_linorder__le__less__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_int @ Y @ X ) ) ).
% linorder_le_less_linear
thf(fact_502_order__le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_503_order__le__imp__less__or__eq,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_int @ X @ Y )
| ( X = Y ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_504_size__neq__size__imp__neq,axiom,
! [X: lambda_lambda,Y: lambda_lambda] :
( ( ( size_s1768714712973771222lambda @ X )
!= ( size_s1768714712973771222lambda @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_505_size__neq__size__imp__neq,axiom,
! [X: char,Y: char] :
( ( ( size_size_char @ X )
!= ( size_size_char @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_506_zero__less__iff__neq__zero,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( N != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_507_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_508_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_509_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_510_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_511_add__less__imp__less__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_512_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_513_add__less__imp__less__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
=> ( ord_less_int @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_514_add__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_515_add__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).
% add_strict_right_mono
thf(fact_516_add__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_517_add__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).
% add_strict_left_mono
thf(fact_518_add__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_519_add__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_strict_mono
thf(fact_520_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_521_add__mono__thms__linordered__field_I1_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_522_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( I2 = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_523_add__mono__thms__linordered__field_I2_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( I2 = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_524_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_525_add__mono__thms__linordered__field_I5_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_526_lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( ( K
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% lessE
thf(fact_527_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_528_Suc__lessE,axiom,
! [I2: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I2 ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_529_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_530_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_531_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_532_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ N )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_533_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_534_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_535_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ N )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_536_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M6: nat] :
( ( M
= ( suc @ M6 ) )
& ( ord_less_nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_537_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_538_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_539_less__trans__Suc,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_540_less__Suc__induct,axiom,
! [I2: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J2: nat,K3: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ( ord_less_nat @ J2 @ K3 )
=> ( ( P @ I @ J2 )
=> ( ( P @ J2 @ K3 )
=> ( P @ I @ K3 ) ) ) ) )
=> ( P @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_541_strict__inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I2 @ J )
=> ( ! [I: nat] :
( ( J
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_542_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_543_infinite__descent0,axiom,
! [P: nat > $o,N: 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 @ N ) ) ) ).
% infinite_descent0
thf(fact_544_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_545_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_546_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_547_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_548_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_549_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_550_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_eq_nat @ M2 @ N4 )
& ( M2 != N4 ) ) ) ) ).
% nat_less_le
thf(fact_551_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_552_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M2: nat,N4: nat] :
( ( ord_less_nat @ M2 @ N4 )
| ( M2 = N4 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_553_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_554_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_555_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I2: nat,J: nat] :
( ! [I: nat,J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( ord_less_nat @ ( F @ I ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_556_add__lessD1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
=> ( ord_less_nat @ I2 @ K ) ) ).
% add_lessD1
thf(fact_557_add__less__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_558_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_559_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_560_add__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_561_trans__less__add1,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_562_trans__less__add2,axiom,
! [I2: nat,J: nat,M: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_563_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_564_add__less__zeroD,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_565_not__one__less__zero,axiom,
~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).
% not_one_less_zero
thf(fact_566_not__one__less__zero,axiom,
~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).
% not_one_less_zero
thf(fact_567_zero__less__one,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% zero_less_one
thf(fact_568_zero__less__one,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% zero_less_one
thf(fact_569_less__add__one,axiom,
! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).
% less_add_one
thf(fact_570_less__add__one,axiom,
! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).
% less_add_one
thf(fact_571_add__mono1,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).
% add_mono1
thf(fact_572_add__mono1,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).
% add_mono1
thf(fact_573_zero__less__two,axiom,
ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).
% zero_less_two
thf(fact_574_zero__less__two,axiom,
ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).
% zero_less_two
thf(fact_575_add__less__le__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_576_add__less__le__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_less_le_mono
thf(fact_577_add__le__less__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_578_add__le__less__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).
% add_le_less_mono
thf(fact_579_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I2 @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_580_add__mono__thms__linordered__field_I3_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I2 @ J )
& ( ord_less_eq_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_581_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I2 @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_582_add__mono__thms__linordered__field_I4_J,axiom,
! [I2: int,J: int,K: int,L: int] :
( ( ( ord_less_eq_int @ I2 @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_583_add__neg__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_584_add__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_585_add__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_586_add__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_pos
thf(fact_587_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ! [C2: nat] :
( ( B
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_588_pos__add__strict,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_589_pos__add__strict,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_590_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
& ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I4: nat] :
( ( ord_less_nat @ I4 @ N )
& ( P @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_591_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_592_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less_nat @ I4 @ ( suc @ N ) )
=> ( P @ I4 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ N )
=> ( P @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_593_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ).
% gr0_implies_Suc
thf(fact_594_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_595_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_596_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_597_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_598_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_599_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_600_inc__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I2 ) ) ) ) ).
% inc_induct
thf(fact_601_dec__induct,axiom,
! [I2: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( P @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq_nat @ I2 @ N3 )
=> ( ( ord_less_nat @ N3 @ J )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_602_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_603_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_604_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_605_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K3: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K3 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_606_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M2: nat,N4: nat] :
? [K2: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_607_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_608_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_609_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q ) ) ) ) ).
% less_natE
thf(fact_610_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less_nat @ I2 @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I2 @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_611_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M5: nat,N3: nat] :
( ( ord_less_nat @ M5 @ N3 )
=> ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_612_add__strict__increasing2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_613_add__strict__increasing2,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing2
thf(fact_614_add__strict__increasing,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_615_add__strict__increasing,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).
% add_strict_increasing
thf(fact_616_add__pos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_617_add__pos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_pos_nonneg
thf(fact_618_add__nonpos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_nonpos_neg
thf(fact_619_add__nonpos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_nonpos_neg
thf(fact_620_add__nonneg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_621_add__nonneg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).
% add_nonneg_pos
thf(fact_622_add__neg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% add_neg_nonpos
thf(fact_623_add__neg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).
% add_neg_nonpos
thf(fact_624_nle__le,axiom,
! [A: nat,B: nat] :
( ( ~ ( ord_less_eq_nat @ A @ B ) )
= ( ( ord_less_eq_nat @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_625_nle__le,axiom,
! [A: int,B: int] :
( ( ~ ( ord_less_eq_int @ A @ B ) )
= ( ( ord_less_eq_int @ B @ A )
& ( B != A ) ) ) ).
% nle_le
thf(fact_626_le__cases3,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z2 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z2 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z2 )
=> ~ ( ord_less_eq_nat @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z2 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_627_le__cases3,axiom,
! [X: int,Y: int,Z2: int] :
( ( ( ord_less_eq_int @ X @ Y )
=> ~ ( ord_less_eq_int @ Y @ Z2 ) )
=> ( ( ( ord_less_eq_int @ Y @ X )
=> ~ ( ord_less_eq_int @ X @ Z2 ) )
=> ( ( ( ord_less_eq_int @ X @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ Y ) )
=> ( ( ( ord_less_eq_int @ Z2 @ Y )
=> ~ ( ord_less_eq_int @ Y @ X ) )
=> ( ( ( ord_less_eq_int @ Y @ Z2 )
=> ~ ( ord_less_eq_int @ Z2 @ X ) )
=> ~ ( ( ord_less_eq_int @ Z2 @ X )
=> ~ ( ord_less_eq_int @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_628_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 ) )
= ( ^ [X2: nat,Y6: nat] :
( ( ord_less_eq_nat @ X2 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_629_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y7: int,Z3: int] : ( Y7 = Z3 ) )
= ( ^ [X2: int,Y6: int] :
( ( ord_less_eq_int @ X2 @ Y6 )
& ( ord_less_eq_int @ Y6 @ X2 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_630_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_631_ord__eq__le__trans,axiom,
! [A: int,B: int,C: int] :
( ( A = B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_632_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_633_ord__le__eq__trans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_634_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_635_order__antisym,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_636_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_637_order_Otrans,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ord_less_eq_int @ A @ C ) ) ) ).
% order.trans
thf(fact_638_order__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z2 )
=> ( ord_less_eq_nat @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_639_order__trans,axiom,
! [X: int,Y: int,Z2: int] :
( ( ord_less_eq_int @ X @ Y )
=> ( ( ord_less_eq_int @ Y @ Z2 )
=> ( ord_less_eq_int @ X @ Z2 ) ) ) ).
% order_trans
thf(fact_640_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B4: nat] :
( ( ord_less_eq_nat @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: nat,B4: nat] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_641_linorder__wlog,axiom,
! [P: int > int > $o,A: int,B: int] :
( ! [A4: int,B4: int] :
( ( ord_less_eq_int @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: int,B4: int] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_642_dual__order_Oeq__iff,axiom,
( ( ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ B3 @ A3 )
& ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_643_dual__order_Oeq__iff,axiom,
( ( ^ [Y7: int,Z3: int] : ( Y7 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ B3 @ A3 )
& ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_644_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_645_dual__order_Oantisym,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_646_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_647_dual__order_Otrans,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ B )
=> ( ord_less_eq_int @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_648_antisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_649_antisym,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ B @ A )
=> ( A = B ) ) ) ).
% antisym
thf(fact_650_order__eq__iff,axiom,
( ( ^ [Y7: nat,Z3: nat] : ( Y7 = Z3 ) )
= ( ^ [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
& ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).
% order_eq_iff
thf(fact_651_order__eq__iff,axiom,
( ( ^ [Y7: int,Z3: int] : ( Y7 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( ord_less_eq_int @ A3 @ B3 )
& ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).
% order_eq_iff
thf(fact_652_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_653_order__subst1,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_654_order__subst1,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_655_order__subst1,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( ord_less_eq_int @ A @ ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_656_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_657_order__subst2,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_658_order__subst2,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_659_order__subst2,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ ( F @ B ) @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_660_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_661_order__eq__refl,axiom,
! [X: int,Y: int] :
( ( X = Y )
=> ( ord_less_eq_int @ X @ Y ) ) ).
% order_eq_refl
thf(fact_662_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_663_linorder__linear,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ X @ Y )
| ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_linear
thf(fact_664_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_665_ord__eq__le__subst,axiom,
! [A: int,F: nat > int,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_666_ord__eq__le__subst,axiom,
! [A: nat,F: int > nat,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_667_ord__eq__le__subst,axiom,
! [A: int,F: int > int,B: int,C: int] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_int @ B @ C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_668_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_669_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > int,C: int] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y4: nat] :
( ( ord_less_eq_nat @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_670_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > nat,C: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_671_ord__le__eq__subst,axiom,
! [A: int,B: int,F: int > int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: int,Y4: int] :
( ( ord_less_eq_int @ X4 @ Y4 )
=> ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_672_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_673_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_674_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_675_order__antisym__conv,axiom,
! [Y: int,X: int] :
( ( ord_less_eq_int @ Y @ X )
=> ( ( ord_less_eq_int @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_676_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_nat @ K3 @ N )
& ! [I5: nat] :
( ( ord_less_eq_nat @ I5 @ K3 )
=> ~ ( P @ I5 ) )
& ( P @ ( suc @ K3 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_677_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ one_one_nat )
=> ( ! [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_678_lambda__calculus_Olambda_Osize_I6_J,axiom,
( ( size_s1768714712973771222lambda @ lambda_Nil )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(6)
thf(fact_679_lambda__calculus_Olambda_Osize_I7_J,axiom,
! [X22: nat] :
( ( size_s1768714712973771222lambda @ ( lambda_Var @ X22 ) )
= zero_zero_nat ) ).
% lambda_calculus.lambda.size(7)
thf(fact_680_zero__neq__one,axiom,
zero_zero_nat != one_one_nat ).
% zero_neq_one
thf(fact_681_zero__neq__one,axiom,
zero_zero_int != one_one_int ).
% zero_neq_one
thf(fact_682_lambda__calculus_Olambda_Osize_I9_J,axiom,
! [X41: lambda_lambda,X42: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_App @ X41 @ X42 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X41 ) @ ( size_s1768714712973771222lambda @ X42 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(9)
thf(fact_683_lambda__calculus_Olambda_Osize_I10_J,axiom,
! [X51: lambda_lambda,X52: lambda_lambda] :
( ( size_s1768714712973771222lambda @ ( lambda_Beta @ X51 @ X52 ) )
= ( plus_plus_nat @ ( plus_plus_nat @ ( size_s1768714712973771222lambda @ X51 ) @ ( size_s1768714712973771222lambda @ X52 ) ) @ ( suc @ zero_zero_nat ) ) ) ).
% lambda_calculus.lambda.size(10)
thf(fact_684_subterm__implies__smaller,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ U )
=> ( ord_less_nat @ ( lambda_size @ T ) @ ( lambda_size @ U ) ) ) ).
% subterm_implies_smaller
thf(fact_685_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_686_less__numeral__extra_I1_J,axiom,
ord_less_int @ zero_zero_int @ one_one_int ).
% less_numeral_extra(1)
thf(fact_687_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K3: nat] :
( ( ord_less_nat @ N @ K3 )
=> ( P @ K3 ) )
=> ( ! [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N )
=> ( ! [I5: nat] :
( ( ord_less_nat @ K3 @ I5 )
=> ( P @ I5 ) )
=> ( P @ K3 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_688_lambda__calculus_Osubterm_Ointros_I6_J,axiom,
! [T: lambda_lambda,U: lambda_lambda,V: lambda_lambda] :
( ( lambda_subterm @ T @ U )
=> ( ( lambda_subterm @ U @ V )
=> ( lambda_subterm @ T @ V ) ) ) ).
% lambda_calculus.subterm.intros(6)
thf(fact_689_lambda__calculus_Osubterm__lemmas_I3_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_App @ T @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ T @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(3)
thf(fact_690_lambda__calculus_Osubterm_Ointros_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_App @ T @ U ) ) ).
% lambda_calculus.subterm.intros(2)
thf(fact_691_lambda__calculus_Osubterm_Ointros_I3_J,axiom,
! [U: lambda_lambda,T: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_App @ T @ U ) ) ).
% lambda_calculus.subterm.intros(3)
thf(fact_692_lambda__calculus_Osubterm_Ointros_I5_J,axiom,
! [U: lambda_lambda,T: lambda_lambda] : ( lambda_subterm @ U @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.subterm.intros(5)
thf(fact_693_lambda__calculus_Osubterm_Ointros_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_Beta @ T @ U ) ) ).
% lambda_calculus.subterm.intros(4)
thf(fact_694_lambda__calculus_Osubterm__lemmas_I4_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_Beta @ T @ U ) )
& ( lambda_subterm @ U @ ( lambda_Beta @ T @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(4)
thf(fact_695_lambda__calculus_Osubterm__lemmas_I1_J,axiom,
! [T: lambda_lambda] : ( lambda_subterm @ T @ ( lambda_Lam @ T ) ) ).
% lambda_calculus.subterm_lemmas(1)
thf(fact_696_lambda__calculus_Osubterm__lemmas_I2_J,axiom,
! [T: lambda_lambda,U: lambda_lambda] :
( ( lambda_subterm @ T @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) )
& ( lambda_subterm @ U @ ( lambda_App @ ( lambda_Lam @ T ) @ U ) ) ) ).
% lambda_calculus.subterm_lemmas(2)
thf(fact_697_is__num__normalize_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_698_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M7: nat] :
( ( P @ X )
=> ( ! [X4: nat] :
( ( P @ X4 )
=> ( ord_less_eq_nat @ X4 @ M7 ) )
=> ~ ! [M5: nat] :
( ( P @ M5 )
=> ~ ! [X6: nat] :
( ( P @ X6 )
=> ( ord_less_eq_nat @ X6 @ M5 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_699_lambda__calculus_Osubterm_Ocases,axiom,
! [A1: lambda_lambda,A22: lambda_lambda] :
( ( lambda_subterm @ A1 @ A22 )
=> ( ( A22
!= ( lambda_Lam @ A1 ) )
=> ( ! [U3: lambda_lambda] :
( A22
!= ( lambda_App @ A1 @ U3 ) )
=> ( ! [T3: lambda_lambda] :
( A22
!= ( lambda_App @ T3 @ A1 ) )
=> ( ! [U3: lambda_lambda] :
( A22
!= ( lambda_Beta @ A1 @ U3 ) )
=> ( ! [T3: lambda_lambda] :
( A22
!= ( lambda_Beta @ T3 @ A1 ) )
=> ~ ! [U3: lambda_lambda] :
( ( lambda_subterm @ A1 @ U3 )
=> ~ ( lambda_subterm @ U3 @ A22 ) ) ) ) ) ) ) ) ).
% lambda_calculus.subterm.cases
thf(fact_700_lambda__calculus_Osubterm_Osimps,axiom,
( lambda_subterm
= ( ^ [A12: lambda_lambda,A23: lambda_lambda] :
( ? [T4: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_Lam @ T4 ) ) )
| ? [T4: lambda_lambda,U4: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_App @ T4 @ U4 ) ) )
| ? [T4: lambda_lambda,U4: lambda_lambda] :
( ( A12 = U4 )
& ( A23
= ( lambda_App @ T4 @ U4 ) ) )
| ? [T4: lambda_lambda,U4: lambda_lambda] :
( ( A12 = T4 )
& ( A23
= ( lambda_Beta @ T4 @ U4 ) ) )
| ? [T4: lambda_lambda,U4: lambda_lambda] :
( ( A12 = U4 )
& ( A23
= ( lambda_Beta @ T4 @ U4 ) ) )
| ? [T4: lambda_lambda,U4: lambda_lambda,V4: lambda_lambda] :
( ( A12 = T4 )
& ( A23 = V4 )
& ( lambda_subterm @ T4 @ U4 )
& ( lambda_subterm @ U4 @ V4 ) ) ) ) ) ).
% lambda_calculus.subterm.simps
thf(fact_701_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_702_le__numeral__extra_I3_J,axiom,
ord_less_eq_int @ zero_zero_int @ zero_zero_int ).
% le_numeral_extra(3)
thf(fact_703_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_704_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_705_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_706_le__numeral__extra_I4_J,axiom,
ord_less_eq_int @ one_one_int @ one_one_int ).
% le_numeral_extra(4)
thf(fact_707_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_708_less__numeral__extra_I4_J,axiom,
~ ( ord_less_int @ one_one_int @ one_one_int ) ).
% less_numeral_extra(4)
thf(fact_709_size__char__eq__0,axiom,
( size_size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size_char_eq_0
thf(fact_710_add__0__iff,axiom,
! [B: nat,A: nat] :
( ( B
= ( plus_plus_nat @ B @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_711_add__0__iff,axiom,
! [B: int,A: int] :
( ( B
= ( plus_plus_int @ B @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_712_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_713_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_714_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: nat] :
( ( ord_less_eq_nat @ A @ C2 )
& ( ord_less_eq_nat @ C2 @ B )
& ! [X6: nat] :
( ( ( ord_less_eq_nat @ A @ X6 )
& ( ord_less_nat @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: nat] :
( ! [X4: nat] :
( ( ( ord_less_eq_nat @ A @ X4 )
& ( ord_less_nat @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_715_complete__interval,axiom,
! [A: int,B: int,P: int > $o] :
( ( ord_less_int @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C2: int] :
( ( ord_less_eq_int @ A @ C2 )
& ( ord_less_eq_int @ C2 @ B )
& ! [X6: int] :
( ( ( ord_less_eq_int @ A @ X6 )
& ( ord_less_int @ X6 @ C2 ) )
=> ( P @ X6 ) )
& ! [D3: int] :
( ! [X4: int] :
( ( ( ord_less_eq_int @ A @ X4 )
& ( ord_less_int @ X4 @ D3 ) )
=> ( P @ X4 ) )
=> ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_716_verit__comp__simplify1_I2_J,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_717_verit__comp__simplify1_I2_J,axiom,
! [A: int] : ( ord_less_eq_int @ A @ A ) ).
% verit_comp_simplify1(2)
thf(fact_718_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_719_verit__la__disequality,axiom,
! [A: int,B: int] :
( ( A = B )
| ~ ( ord_less_eq_int @ A @ B )
| ~ ( ord_less_eq_int @ B @ A ) ) ).
% verit_la_disequality
thf(fact_720_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_721_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_722_verit__comp__simplify1_I3_J,axiom,
! [B5: nat,A5: nat] :
( ( ~ ( ord_less_eq_nat @ B5 @ A5 ) )
= ( ord_less_nat @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_723_verit__comp__simplify1_I3_J,axiom,
! [B5: int,A5: int] :
( ( ~ ( ord_less_eq_int @ B5 @ A5 ) )
= ( ord_less_int @ A5 @ B5 ) ) ).
% verit_comp_simplify1(3)
thf(fact_724_size_H__char__eq__0,axiom,
( size_char
= ( ^ [C3: char] : zero_zero_nat ) ) ).
% size'_char_eq_0
thf(fact_725_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X6 ) ) ).
% minf(8)
thf(fact_726_minf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ~ ( ord_less_eq_int @ T @ X6 ) ) ).
% minf(8)
thf(fact_727_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_eq_nat @ X6 @ T ) ) ).
% minf(6)
thf(fact_728_minf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ord_less_eq_int @ X6 @ T ) ) ).
% minf(6)
thf(fact_729_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_eq_nat @ T @ X6 ) ) ).
% pinf(8)
thf(fact_730_pinf_I8_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ord_less_eq_int @ T @ X6 ) ) ).
% pinf(8)
thf(fact_731_pinf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_732_pinf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_733_pinf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_734_pinf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ Z4 @ X4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_735_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_736_pinf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(3)
thf(fact_737_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_738_pinf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( X6 != T ) ) ).
% pinf(4)
thf(fact_739_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_nat @ X6 @ T ) ) ).
% pinf(5)
thf(fact_740_pinf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ~ ( ord_less_int @ X6 @ T ) ) ).
% pinf(5)
thf(fact_741_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ( ord_less_nat @ T @ X6 ) ) ).
% pinf(7)
thf(fact_742_pinf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ( ord_less_int @ T @ X6 ) ) ).
% pinf(7)
thf(fact_743_minf_I1_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_744_minf_I1_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ( ( P @ X6 )
& ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
& ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(1)
thf(fact_745_minf_I2_J,axiom,
! [P: nat > $o,P5: nat > $o,Q2: nat > $o,Q3: nat > $o] :
( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: nat] :
! [X4: nat] :
( ( ord_less_nat @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_746_minf_I2_J,axiom,
! [P: int > $o,P5: int > $o,Q2: int > $o,Q3: int > $o] :
( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less_int @ X4 @ Z4 )
=> ( ( Q2 @ X4 )
= ( Q3 @ X4 ) ) )
=> ? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ( ( P @ X6 )
| ( Q2 @ X6 ) )
= ( ( P5 @ X6 )
| ( Q3 @ X6 ) ) ) ) ) ) ).
% minf(2)
thf(fact_747_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_748_minf_I3_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(3)
thf(fact_749_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_750_minf_I4_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( X6 != T ) ) ).
% minf(4)
thf(fact_751_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ( ord_less_nat @ X6 @ T ) ) ).
% minf(5)
thf(fact_752_minf_I5_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ( ord_less_int @ X6 @ T ) ) ).
% minf(5)
thf(fact_753_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ X6 @ Z )
=> ~ ( ord_less_nat @ T @ X6 ) ) ).
% minf(7)
thf(fact_754_minf_I7_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ X6 @ Z )
=> ~ ( ord_less_int @ T @ X6 ) ) ).
% minf(7)
thf(fact_755_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X6: nat] :
( ( ord_less_nat @ Z @ X6 )
=> ~ ( ord_less_eq_nat @ X6 @ T ) ) ).
% pinf(6)
thf(fact_756_pinf_I6_J,axiom,
! [T: int] :
? [Z: int] :
! [X6: int] :
( ( ord_less_int @ Z @ X6 )
=> ~ ( ord_less_eq_int @ X6 @ T ) ) ).
% pinf(6)
thf(fact_757_lambda__calculus_OSubst_Osimps_I2_J,axiom,
! [N: nat,I2: nat,V: lambda_lambda] :
( ( ( ord_less_nat @ N @ I2 )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ ( minus_minus_nat @ I2 @ one_one_nat ) ) ) )
& ( ~ ( ord_less_nat @ N @ I2 )
=> ( ( ( N = I2 )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I2 ) )
= ( lambda_Raise @ zero_zero_nat @ N @ V ) ) )
& ( ( N != I2 )
=> ( ( lambda_Subst @ N @ V @ ( lambda_Var @ I2 ) )
= ( lambda_Var @ I2 ) ) ) ) ) ) ).
% lambda_calculus.Subst.simps(2)
thf(fact_758_lambda__calculus_Osize_Opelims,axiom,
! [X: lambda_lambda,Y: nat] :
( ( ( lambda_size @ X )
= Y )
=> ( ( accp_lambda_lambda @ lambda_size_rel @ X )
=> ( ( ( X = lambda_Nil )
=> ( ( Y = zero_zero_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X
= ( lambda_Var @ Uu2 ) )
=> ( ( Y = one_one_nat )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( Y
= ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( Y
= ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_App @ T3 @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ( Y
= ( plus_plus_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( lambda_size @ T3 ) @ one_one_nat ) @ ( lambda_size @ U3 ) ) @ one_one_nat ) )
=> ~ ( accp_lambda_lambda @ lambda_size_rel @ ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.size.pelims
thf(fact_759_dbl__inc__simps_I2_J,axiom,
( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
= one_one_int ) ).
% dbl_inc_simps(2)
thf(fact_760_Suc__diff__1,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
= N ) ) ).
% Suc_diff_1
thf(fact_761_diff__self,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% diff_self
thf(fact_762_diff__0__right,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_0_right
thf(fact_763_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_764_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_765_diff__zero,axiom,
! [A: int] :
( ( minus_minus_int @ A @ zero_zero_int )
= A ) ).
% diff_zero
thf(fact_766_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_767_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: int] :
( ( minus_minus_int @ A @ A )
= zero_zero_int ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_768_add__diff__cancel,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel
thf(fact_769_diff__add__cancel,axiom,
! [A: int,B: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ).
% diff_add_cancel
thf(fact_770_add__diff__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_771_add__diff__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_left
thf(fact_772_add__diff__cancel__left_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_773_add__diff__cancel__left_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
= B ) ).
% add_diff_cancel_left'
thf(fact_774_add__diff__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
= ( minus_minus_nat @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_775_add__diff__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ A @ B ) ) ).
% add_diff_cancel_right
thf(fact_776_add__diff__cancel__right_H,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_777_add__diff__cancel__right_H,axiom,
! [A: int,B: int] :
( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
= A ) ).
% add_diff_cancel_right'
thf(fact_778_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_779_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_780_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_781_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_782_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_783_diff__diff__left,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_784_diff__ge__0__iff__ge,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ).
% diff_ge_0_iff_ge
thf(fact_785_diff__gt__0__iff__gt,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
= ( ord_less_int @ B @ A ) ) ).
% diff_gt_0_iff_gt
thf(fact_786_le__add__diff__inverse,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_787_le__add__diff__inverse,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% le_add_diff_inverse
thf(fact_788_le__add__diff__inverse2,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_789_le__add__diff__inverse2,axiom,
! [B: int,A: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
= A ) ) ).
% le_add_diff_inverse2
thf(fact_790_diff__add__zero,axiom,
! [A: nat,B: nat] :
( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
= zero_zero_nat ) ).
% diff_add_zero
thf(fact_791_diff__numeral__special_I9_J,axiom,
( ( minus_minus_int @ one_one_int @ one_one_int )
= zero_zero_int ) ).
% diff_numeral_special(9)
thf(fact_792_zero__less__diff,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M ) )
= ( ord_less_nat @ M @ N ) ) ).
% zero_less_diff
thf(fact_793_diff__is__0__eq_H,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( minus_minus_nat @ M @ N )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_794_diff__is__0__eq,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N ) ) ).
% diff_is_0_eq
thf(fact_795_add__diff__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).
% add_diff_assoc
thf(fact_796_add__diff__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).
% add_diff_assoc2
thf(fact_797_diff__diff__right,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% diff_diff_right
thf(fact_798_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_799_Suc__pred,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
= N ) ) ).
% Suc_pred
thf(fact_800_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_801_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_802_diff__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ D @ C )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_mono
thf(fact_803_diff__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_left_mono
thf(fact_804_diff__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_right_mono
thf(fact_805_diff__eq__diff__less__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_eq_int @ A @ B )
= ( ord_less_eq_int @ C @ D ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_806_eq__iff__diff__eq__0,axiom,
( ( ^ [Y7: int,Z3: int] : ( Y7 = Z3 ) )
= ( ^ [A3: int,B3: int] :
( ( minus_minus_int @ A3 @ B3 )
= zero_zero_int ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_807_group__cancel_Osub1,axiom,
! [A2: int,K: int,A: int,B: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( minus_minus_int @ A2 @ B )
= ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).
% group_cancel.sub1
thf(fact_808_diff__eq__eq,axiom,
! [A: int,B: int,C: int] :
( ( ( minus_minus_int @ A @ B )
= C )
= ( A
= ( plus_plus_int @ C @ B ) ) ) ).
% diff_eq_eq
thf(fact_809_eq__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( A
= ( minus_minus_int @ C @ B ) )
= ( ( plus_plus_int @ A @ B )
= C ) ) ).
% eq_diff_eq
thf(fact_810_add__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% add_diff_eq
thf(fact_811_diff__diff__eq2,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_diff_eq2
thf(fact_812_diff__add__eq,axiom,
! [A: int,B: int,C: int] :
( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).
% diff_add_eq
thf(fact_813_diff__add__eq__diff__diff__swap,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
= ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_814_add__implies__diff,axiom,
! [C: nat,B: nat,A: nat] :
( ( ( plus_plus_nat @ C @ B )
= A )
=> ( C
= ( minus_minus_nat @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_815_add__implies__diff,axiom,
! [C: int,B: int,A: int] :
( ( ( plus_plus_int @ C @ B )
= A )
=> ( C
= ( minus_minus_int @ A @ B ) ) ) ).
% add_implies_diff
thf(fact_816_diff__diff__eq,axiom,
! [A: nat,B: nat,C: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
= ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_817_diff__diff__eq,axiom,
! [A: int,B: int,C: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).
% diff_diff_eq
thf(fact_818_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I2: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_819_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_820_diffs0__imp__equal,axiom,
! [M: nat,N: nat] :
( ( ( minus_minus_nat @ M @ N )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N @ M )
= zero_zero_nat )
=> ( M = N ) ) ) ).
% diffs0_imp_equal
thf(fact_821_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_822_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_823_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_824_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_825_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_826_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_827_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_828_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_829_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_830_diff__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_831_diff__cancel,axiom,
! [K: nat,M: nat,N: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_cancel
thf(fact_832_diff__commute,axiom,
! [I2: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).
% diff_commute
thf(fact_833_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_834_diff__right__commute,axiom,
! [A: int,C: int,B: int] :
( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
= ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_835_diff__eq__diff__eq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( A = B )
= ( C = D ) ) ) ).
% diff_eq_diff_eq
thf(fact_836_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_837_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_838_diff__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).
% diff_strict_right_mono
thf(fact_839_diff__strict__left__mono,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).
% diff_strict_left_mono
thf(fact_840_diff__eq__diff__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( minus_minus_int @ A @ B )
= ( minus_minus_int @ C @ D ) )
=> ( ( ord_less_int @ A @ B )
= ( ord_less_int @ C @ D ) ) ) ).
% diff_eq_diff_less
thf(fact_841_diff__strict__mono,axiom,
! [A: int,B: int,D: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ D @ C )
=> ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).
% diff_strict_mono
thf(fact_842_le__iff__diff__le__0,axiom,
( ord_less_eq_int
= ( ^ [A3: int,B3: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% le_iff_diff_le_0
thf(fact_843_less__iff__diff__less__0,axiom,
( ord_less_int
= ( ^ [A3: int,B3: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B3 ) @ zero_zero_int ) ) ) ).
% less_iff_diff_less_0
thf(fact_844_add__le__imp__le__diff,axiom,
! [I2: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_845_add__le__imp__le__diff,axiom,
! [I2: int,K: int,N: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N @ K ) ) ) ).
% add_le_imp_le_diff
thf(fact_846_add__le__add__imp__diff__le,axiom,
! [I2: nat,K: nat,N: nat,J: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N )
=> ( ( ord_less_eq_nat @ N @ ( plus_plus_nat @ J @ K ) )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_847_add__le__add__imp__diff__le,axiom,
! [I2: int,K: int,N: int,J: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N )
=> ( ( ord_less_eq_int @ N @ ( plus_plus_int @ J @ K ) )
=> ( ord_less_eq_int @ ( minus_minus_int @ N @ K ) @ J ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_848_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ( ( minus_minus_nat @ B @ A )
= C )
= ( B
= ( plus_plus_nat @ C @ A ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_849_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
= B ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_850_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_851_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
= ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_852_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
= ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_853_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
= ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_854_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_855_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_856_le__add__diff,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).
% le_add_diff
thf(fact_857_diff__add,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
= B ) ) ).
% diff_add
thf(fact_858_le__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% le_diff_eq
thf(fact_859_diff__le__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_le_eq
thf(fact_860_diff__less__eq,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
= ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).
% diff_less_eq
thf(fact_861_less__diff__eq,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
= ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).
% less_diff_eq
thf(fact_862_add__diff__inverse,axiom,
! [A: nat,B: nat] :
( ~ ( ord_less_nat @ A @ B )
=> ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
= A ) ) ).
% add_diff_inverse
thf(fact_863_add__diff__inverse,axiom,
! [A: int,B: int] :
( ~ ( ord_less_int @ A @ B )
=> ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
= A ) ) ).
% add_diff_inverse
thf(fact_864_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_865_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_866_diff__less,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ M ) ) ) ).
% diff_less
thf(fact_867_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_868_diff__add__0,axiom,
! [N: nat,M: nat] :
( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_869_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_870_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_871_add__diff__inverse__nat,axiom,
! [M: nat,N: nat] :
( ~ ( ord_less_nat @ M @ N )
=> ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M @ N ) )
= M ) ) ).
% add_diff_inverse_nat
thf(fact_872_less__diff__conv,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).
% less_diff_conv
thf(fact_873_le__diff__conv,axiom,
! [J: nat,K: nat,I2: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).
% le_diff_conv
thf(fact_874_le__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).
% le_diff_conv2
thf(fact_875_diff__add__assoc,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
= ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% diff_add_assoc
thf(fact_876_diff__add__assoc2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).
% diff_add_assoc2
thf(fact_877_le__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( minus_minus_nat @ J @ I2 )
= K )
= ( J
= ( plus_plus_nat @ K @ I2 ) ) ) ) ).
% le_imp_diff_is_add
thf(fact_878_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_879_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_880_nat__diff__split__asm,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ~ ( ( ( ord_less_nat @ A @ B )
& ~ ( P @ zero_zero_nat ) )
| ? [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_881_nat__diff__split,axiom,
! [P: nat > $o,A: nat,B: nat] :
( ( P @ ( minus_minus_nat @ A @ B ) )
= ( ( ( ord_less_nat @ A @ B )
=> ( P @ zero_zero_nat ) )
& ! [D4: nat] :
( ( A
= ( plus_plus_nat @ B @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_882_less__diff__conv2,axiom,
! [K: nat,J: nat,I2: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
= ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_883_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_884_Suc__pred_H,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( N
= ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_885_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N4 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% add_eq_if
thf(fact_886_dbl__inc__def,axiom,
( neg_nu5851722552734809277nc_int
= ( ^ [X2: int] : ( plus_plus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_inc_def
thf(fact_887_lambda__calculus_OIde_Opelims_I1_J,axiom,
! [X: lambda_lambda,Y: $o] :
( ( ( lambda_Ide @ X )
= Y )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X )
=> ( ( ( X = lambda_Nil )
=> ( ~ Y
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X
= ( lambda_Var @ Uu2 ) )
=> ( Y
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( Y
= ( lambda_Ide @ T3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( Y
= ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ~ Y
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(1)
thf(fact_888_lambda__calculus_OArr_Opelims_I1_J,axiom,
! [X: lambda_lambda,Y: $o] :
( ( ( lambda_Arr @ X )
= Y )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X )
=> ( ( ( X = lambda_Nil )
=> ( ~ Y
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) ) )
=> ( ! [Uu2: nat] :
( ( X
= ( lambda_Var @ Uu2 ) )
=> ( Y
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( Y
= ( lambda_Arr @ T3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( Y
= ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ( Y
= ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(1)
thf(fact_889_lambda__calculus_OIde_Opelims_I3_J,axiom,
! [X: lambda_lambda] :
( ~ ( lambda_Ide @ X )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X )
=> ( ( ( X = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ lambda_Nil ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) )
=> ( lambda_Ide @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U3 ) )
=> ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Beta @ T3 @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(3)
thf(fact_890_lambda__calculus_OArr_Opelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda_Arr @ X )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X )
=> ( ! [Uu2: nat] :
( ( X
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Arr @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U3 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U3 ) )
=> ~ ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(2)
thf(fact_891_lambda__calculus_OIde_Opelims_I2_J,axiom,
! [X: lambda_lambda] :
( ( lambda_Ide @ X )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ X )
=> ( ! [Uu2: nat] :
( ( X
= ( lambda_Var @ Uu2 ) )
=> ~ ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Var @ Uu2 ) ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_Lam @ T3 ) )
=> ~ ( lambda_Ide @ T3 ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Ide_rel @ ( lambda_App @ T3 @ U3 ) )
=> ~ ( ( lambda_Ide @ T3 )
& ( lambda_Ide @ U3 ) ) ) ) ) ) ) ) ).
% lambda_calculus.Ide.pelims(2)
thf(fact_892_lambda__calculus_OArr_Opelims_I3_J,axiom,
! [X: lambda_lambda] :
( ~ ( lambda_Arr @ X )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ X )
=> ( ( ( X = lambda_Nil )
=> ~ ( accp_lambda_lambda @ lambda_Arr_rel @ lambda_Nil ) )
=> ( ! [T3: lambda_lambda] :
( ( X
= ( lambda_Lam @ T3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Lam @ T3 ) )
=> ( lambda_Arr @ T3 ) ) )
=> ( ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_App @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_App @ T3 @ U3 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) )
=> ~ ! [T3: lambda_lambda,U3: lambda_lambda] :
( ( X
= ( lambda_Beta @ T3 @ U3 ) )
=> ( ( accp_lambda_lambda @ lambda_Arr_rel @ ( lambda_Beta @ T3 @ U3 ) )
=> ( ( lambda_Arr @ T3 )
& ( lambda_Arr @ U3 ) ) ) ) ) ) ) ) ) ).
% lambda_calculus.Arr.pelims(3)
thf(fact_893_dbl__dec__def,axiom,
( neg_nu3811975205180677377ec_int
= ( ^ [X2: int] : ( minus_minus_int @ ( plus_plus_int @ X2 @ X2 ) @ one_one_int ) ) ) ).
% dbl_dec_def
thf(fact_894_convex__bound__lt,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_int @ X @ A )
=> ( ( ord_less_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_895_mult__is__0,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= zero_zero_nat )
= ( ( M = zero_zero_nat )
| ( N = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_896_mult__0__right,axiom,
! [M: nat] :
( ( times_times_nat @ M @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_897_mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_898_mult__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( times_times_nat @ M @ K )
= ( times_times_nat @ N @ K ) )
= ( ( M = N )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_899_nat__mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= one_one_nat )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_900_nat__1__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( one_one_nat
= ( times_times_nat @ M @ N ) )
= ( ( M = one_one_nat )
& ( N = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_901_mult__zero__left,axiom,
! [A: nat] :
( ( times_times_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% mult_zero_left
thf(fact_902_mult__zero__left,axiom,
! [A: int] :
( ( times_times_int @ zero_zero_int @ A )
= zero_zero_int ) ).
% mult_zero_left
thf(fact_903_mult__zero__right,axiom,
! [A: nat] :
( ( times_times_nat @ A @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_zero_right
thf(fact_904_mult__zero__right,axiom,
! [A: int] :
( ( times_times_int @ A @ zero_zero_int )
= zero_zero_int ) ).
% mult_zero_right
thf(fact_905_mult__eq__0__iff,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% mult_eq_0_iff
thf(fact_906_mult__eq__0__iff,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
= ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% mult_eq_0_iff
thf(fact_907_mult__cancel__left,axiom,
! [C: nat,A: nat,B: nat] :
( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_908_mult__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_left
thf(fact_909_mult__cancel__right,axiom,
! [A: nat,C: nat,B: nat] :
( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( ( C = zero_zero_nat )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_910_mult__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( A = B ) ) ) ).
% mult_cancel_right
thf(fact_911_mult_Oright__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.right_neutral
thf(fact_912_mult_Oright__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.right_neutral
thf(fact_913_mult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% mult_1
thf(fact_914_mult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% mult_1
thf(fact_915_mult__eq__1__iff,axiom,
! [M: nat,N: nat] :
( ( ( times_times_nat @ M @ N )
= ( suc @ zero_zero_nat ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_916_one__eq__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M @ N ) )
= ( ( M
= ( suc @ zero_zero_nat ) )
& ( N
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_917_nat__0__less__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ M )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_918_mult__less__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_919_mult__Suc__right,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ M @ ( suc @ N ) )
= ( plus_plus_nat @ M @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc_right
thf(fact_920_dbl__dec__simps_I3_J,axiom,
( ( neg_nu3811975205180677377ec_int @ one_one_int )
= one_one_int ) ).
% dbl_dec_simps(3)
thf(fact_921_mult__cancel__right2,axiom,
! [A: int,C: int] :
( ( ( times_times_int @ A @ C )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_right2
thf(fact_922_mult__cancel__right1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ B @ C ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_right1
thf(fact_923_mult__cancel__left2,axiom,
! [C: int,A: int] :
( ( ( times_times_int @ C @ A )
= C )
= ( ( C = zero_zero_int )
| ( A = one_one_int ) ) ) ).
% mult_cancel_left2
thf(fact_924_mult__cancel__left1,axiom,
! [C: int,B: int] :
( ( C
= ( times_times_int @ C @ B ) )
= ( ( C = zero_zero_int )
| ( B = one_one_int ) ) ) ).
% mult_cancel_left1
thf(fact_925_one__le__mult__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_926_mult__le__cancel2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_927_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% gbinomial_0(2)
thf(fact_928_gbinomial__0_I2_J,axiom,
! [K: nat] :
( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
= zero_zero_int ) ).
% gbinomial_0(2)
thf(fact_929_gbinomial__0_I1_J,axiom,
! [A: nat] :
( ( gbinomial_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% gbinomial_0(1)
thf(fact_930_gbinomial__0_I1_J,axiom,
! [A: int] :
( ( gbinomial_int @ A @ zero_zero_nat )
= one_one_int ) ).
% gbinomial_0(1)
thf(fact_931_diff__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_932_diff__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M @ N ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_933_right__diff__distrib_H,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
= ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_934_right__diff__distrib_H,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib'
thf(fact_935_left__diff__distrib_H,axiom,
! [B: nat,C: nat,A: nat] :
( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
= ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_936_left__diff__distrib_H,axiom,
! [B: int,C: int,A: int] :
( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
= ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).
% left_diff_distrib'
thf(fact_937_right__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
= ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% right_diff_distrib
thf(fact_938_left__diff__distrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
= ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% left_diff_distrib
thf(fact_939_mult__not__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
!= zero_zero_nat )
=> ( ( A != zero_zero_nat )
& ( B != zero_zero_nat ) ) ) ).
% mult_not_zero
thf(fact_940_mult__not__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
!= zero_zero_int )
=> ( ( A != zero_zero_int )
& ( B != zero_zero_int ) ) ) ).
% mult_not_zero
thf(fact_941_divisors__zero,axiom,
! [A: nat,B: nat] :
( ( ( times_times_nat @ A @ B )
= zero_zero_nat )
=> ( ( A = zero_zero_nat )
| ( B = zero_zero_nat ) ) ) ).
% divisors_zero
thf(fact_942_divisors__zero,axiom,
! [A: int,B: int] :
( ( ( times_times_int @ A @ B )
= zero_zero_int )
=> ( ( A = zero_zero_int )
| ( B = zero_zero_int ) ) ) ).
% divisors_zero
thf(fact_943_no__zero__divisors,axiom,
! [A: nat,B: nat] :
( ( A != zero_zero_nat )
=> ( ( B != zero_zero_nat )
=> ( ( times_times_nat @ A @ B )
!= zero_zero_nat ) ) ) ).
% no_zero_divisors
thf(fact_944_no__zero__divisors,axiom,
! [A: int,B: int] :
( ( A != zero_zero_int )
=> ( ( B != zero_zero_int )
=> ( ( times_times_int @ A @ B )
!= zero_zero_int ) ) ) ).
% no_zero_divisors
thf(fact_945_mult__left__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ C @ A )
= ( times_times_nat @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_946_mult__left__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ C @ A )
= ( times_times_int @ C @ B ) )
= ( A = B ) ) ) ).
% mult_left_cancel
thf(fact_947_mult__right__cancel,axiom,
! [C: nat,A: nat,B: nat] :
( ( C != zero_zero_nat )
=> ( ( ( times_times_nat @ A @ C )
= ( times_times_nat @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_948_mult__right__cancel,axiom,
! [C: int,A: int,B: int] :
( ( C != zero_zero_int )
=> ( ( ( times_times_int @ A @ C )
= ( times_times_int @ B @ C ) )
= ( A = B ) ) ) ).
% mult_right_cancel
thf(fact_949_combine__common__factor,axiom,
! [A: nat,E: nat,B: nat,C: nat] :
( ( plus_plus_nat @ ( times_times_nat @ A @ E ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E ) @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_950_combine__common__factor,axiom,
! [A: int,E: int,B: int,C: int] :
( ( plus_plus_int @ ( times_times_int @ A @ E ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ C ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E ) @ C ) ) ).
% combine_common_factor
thf(fact_951_distrib__right,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% distrib_right
thf(fact_952_distrib__right,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% distrib_right
thf(fact_953_distrib__left,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
= ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).
% distrib_left
thf(fact_954_distrib__left,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% distrib_left
thf(fact_955_comm__semiring__class_Odistrib,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
= ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_956_comm__semiring__class_Odistrib,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% comm_semiring_class.distrib
thf(fact_957_ring__class_Oring__distribs_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
= ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_958_ring__class_Oring__distribs_I2_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
= ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_959_comm__monoid__mult__class_Omult__1,axiom,
! [A: nat] :
( ( times_times_nat @ one_one_nat @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_960_comm__monoid__mult__class_Omult__1,axiom,
! [A: int] :
( ( times_times_int @ one_one_int @ A )
= A ) ).
% comm_monoid_mult_class.mult_1
thf(fact_961_mult_Ocomm__neutral,axiom,
! [A: nat] :
( ( times_times_nat @ A @ one_one_nat )
= A ) ).
% mult.comm_neutral
thf(fact_962_mult_Ocomm__neutral,axiom,
! [A: int] :
( ( times_times_int @ A @ one_one_int )
= A ) ).
% mult.comm_neutral
thf(fact_963_Suc__mult__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M )
= ( times_times_nat @ ( suc @ K ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_964_mult__0,axiom,
! [N: nat] :
( ( times_times_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ).
% mult_0
thf(fact_965_mult__le__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_966_mult__le__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_967_mult__le__mono,axiom,
! [I2: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_968_le__square,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).
% le_square
thf(fact_969_le__cube,axiom,
! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).
% le_cube
thf(fact_970_add__mult__distrib2,axiom,
! [K: nat,M: nat,N: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_971_add__mult__distrib,axiom,
! [M: nat,N: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M @ N ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_972_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times_nat @ N @ one_one_nat )
= N ) ).
% nat_mult_1_right
thf(fact_973_nat__mult__1,axiom,
! [N: nat] :
( ( times_times_nat @ one_one_nat @ N )
= N ) ).
% nat_mult_1
thf(fact_974_mult_Oleft__commute,axiom,
! [B: nat,A: nat,C: nat] :
( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_975_mult_Oleft__commute,axiom,
! [B: int,A: int,C: int] :
( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.left_commute
thf(fact_976_mult_Ocommute,axiom,
( times_times_nat
= ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_977_mult_Ocommute,axiom,
( times_times_int
= ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).
% mult.commute
thf(fact_978_mult_Oassoc,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% mult.assoc
thf(fact_979_mult_Oassoc,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% mult.assoc
thf(fact_980_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: nat,B: nat,C: nat] :
( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
= ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_981_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: int,B: int,C: int] :
( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
= ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_982_crossproduct__eq,axiom,
! [W: nat,Y: nat,X: nat,Z2: nat] :
( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X @ Z2 ) )
= ( plus_plus_nat @ ( times_times_nat @ W @ Z2 ) @ ( times_times_nat @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_983_crossproduct__eq,axiom,
! [W: int,Y: int,X: int,Z2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z2 ) @ ( times_times_int @ X @ Y ) ) )
= ( ( W = X )
| ( Y = Z2 ) ) ) ).
% crossproduct_eq
thf(fact_984_crossproduct__noteq,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
!= ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_985_crossproduct__noteq,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ( A != B )
& ( C != D ) )
= ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
!= ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).
% crossproduct_noteq
thf(fact_986_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_987_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_988_zero__le__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).
% zero_le_mult_iff
thf(fact_989_mult__nonneg__nonpos2,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_990_mult__nonneg__nonpos2,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_991_mult__nonpos__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonpos_nonneg
thf(fact_992_mult__nonpos__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonpos_nonneg
thf(fact_993_mult__nonneg__nonpos,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ B @ zero_zero_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_nonneg_nonpos
thf(fact_994_mult__nonneg__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_nonneg_nonpos
thf(fact_995_mult__nonneg__nonneg,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_996_mult__nonneg__nonneg,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_997_split__mult__neg__le,axiom,
! [A: nat,B: nat] :
( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
& ( ord_less_eq_nat @ B @ zero_zero_nat ) )
| ( ( ord_less_eq_nat @ A @ zero_zero_nat )
& ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).
% split_mult_neg_le
thf(fact_998_split__mult__neg__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).
% split_mult_neg_le
thf(fact_999_mult__le__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ B @ zero_zero_int ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).
% mult_le_0_iff
thf(fact_1000_mult__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1001_mult__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono
thf(fact_1002_mult__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1003_mult__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1004_mult__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono
thf(fact_1005_mult__nonpos__nonpos,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ zero_zero_int )
=> ( ( ord_less_eq_int @ B @ zero_zero_int )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1006_mult__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_eq_int @ B @ A )
=> ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1007_split__mult__pos__le,axiom,
! [A: int,B: int] :
( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
& ( ord_less_eq_int @ zero_zero_int @ B ) )
| ( ( ord_less_eq_int @ A @ zero_zero_int )
& ( ord_less_eq_int @ B @ zero_zero_int ) ) )
=> ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).
% split_mult_pos_le
thf(fact_1008_zero__le__square,axiom,
! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).
% zero_le_square
thf(fact_1009_mult__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1010_mult__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1011_mult__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1012_mult__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_mono
thf(fact_1013_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1014_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1015_mult__less__cancel__right__disj,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1016_mult__strict__right__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1017_mult__strict__right__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1018_mult__strict__right__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1019_mult__less__cancel__left__disj,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
& ( ord_less_int @ A @ B ) )
| ( ( ord_less_int @ C @ zero_zero_int )
& ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1020_mult__strict__left__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1021_mult__strict__left__mono,axiom,
! [A: int,B: int,C: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1022_mult__strict__left__mono__neg,axiom,
! [B: int,A: int,C: int] :
( ( ord_less_int @ B @ A )
=> ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1023_mult__less__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ A @ B ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1024_mult__less__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_int @ B @ A ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1025_zero__less__mult__pos2,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1026_zero__less__mult__pos2,axiom,
! [B: int,A: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos2
thf(fact_1027_zero__less__mult__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1028_zero__less__mult__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ B ) ) ) ).
% zero_less_mult_pos
thf(fact_1029_zero__less__mult__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ zero_zero_int @ B ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1030_mult__pos__neg2,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg2
thf(fact_1031_mult__pos__neg2,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).
% mult_pos_neg2
thf(fact_1032_mult__pos__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1033_mult__pos__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_pos_pos
thf(fact_1034_mult__pos__neg,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B @ zero_zero_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_pos_neg
thf(fact_1035_mult__pos__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_pos_neg
thf(fact_1036_mult__neg__pos,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).
% mult_neg_pos
thf(fact_1037_mult__neg__pos,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).
% mult_neg_pos
thf(fact_1038_mult__less__0__iff,axiom,
! [A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
= ( ( ( ord_less_int @ zero_zero_int @ A )
& ( ord_less_int @ B @ zero_zero_int ) )
| ( ( ord_less_int @ A @ zero_zero_int )
& ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).
% mult_less_0_iff
thf(fact_1039_not__square__less__zero,axiom,
! [A: int] :
~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).
% not_square_less_zero
thf(fact_1040_mult__neg__neg,axiom,
! [A: int,B: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B @ zero_zero_int )
=> ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).
% mult_neg_neg
thf(fact_1041_add__scale__eq__noteq,axiom,
! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
( ( R2 != zero_zero_nat )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
!= ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1042_add__scale__eq__noteq,axiom,
! [R2: int,A: int,B: int,C: int,D: int] :
( ( R2 != zero_zero_int )
=> ( ( ( A = B )
& ( C != D ) )
=> ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
!= ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1043_less__1__mult,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ M )
=> ( ( ord_less_nat @ one_one_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1044_less__1__mult,axiom,
! [M: int,N: int] :
( ( ord_less_int @ one_one_int @ M )
=> ( ( ord_less_int @ one_one_int @ N )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N ) ) ) ) ).
% less_1_mult
thf(fact_1045_eq__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C )
= D ) ) ).
% eq_add_iff1
thf(fact_1046_eq__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ( plus_plus_int @ ( times_times_int @ A @ E ) @ C )
= ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( C
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% eq_add_iff2
thf(fact_1047_square__diff__square__factored,axiom,
! [X: int,Y: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= ( times_times_int @ ( plus_plus_int @ X @ Y ) @ ( minus_minus_int @ X @ Y ) ) ) ).
% square_diff_square_factored
thf(fact_1048_Suc__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1049_mult__less__mono1,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1050_mult__less__mono2,axiom,
! [I2: nat,J: nat,K: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1051_Suc__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1052_mult__Suc,axiom,
! [M: nat,N: nat] :
( ( times_times_nat @ ( suc @ M ) @ N )
= ( plus_plus_nat @ N @ ( times_times_nat @ M @ N ) ) ) ).
% mult_Suc
thf(fact_1053_mult__eq__self__implies__10,axiom,
! [M: nat,N: nat] :
( ( M
= ( times_times_nat @ M @ N ) )
=> ( ( N = one_one_nat )
| ( M = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1054_mult__less__le__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1055_mult__less__le__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1056_mult__le__less__imp__less,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1057_mult__le__less__imp__less,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1058_mult__right__le__imp__le,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1059_mult__right__le__imp__le,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_right_le_imp_le
thf(fact_1060_mult__left__le__imp__le,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_nat @ zero_zero_nat @ C )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1061_mult__left__le__imp__le,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_left_le_imp_le
thf(fact_1062_mult__le__cancel__left__pos,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ zero_zero_int @ C )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ A @ B ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1063_mult__le__cancel__left__neg,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ C @ zero_zero_int )
=> ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ord_less_eq_int @ B @ A ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1064_mult__less__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1065_mult__strict__mono_H,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1066_mult__strict__mono_H,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1067_mult__right__less__imp__less,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1068_mult__right__less__imp__less,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_right_less_imp_less
thf(fact_1069_mult__less__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ A ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1070_mult__strict__mono,axiom,
! [A: nat,B: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C @ D )
=> ( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1071_mult__strict__mono,axiom,
! [A: int,B: int,C: int,D: int] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_int @ C @ D )
=> ( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1072_mult__left__less__imp__less,axiom,
! [C: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ C )
=> ( ord_less_nat @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1073_mult__left__less__imp__less,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ B ) ) ) ).
% mult_left_less_imp_less
thf(fact_1074_mult__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1075_mult__le__cancel__left,axiom,
! [C: int,A: int,B: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ A ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1076_sum__squares__ge__zero,axiom,
! [X: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) ) ).
% sum_squares_ge_zero
thf(fact_1077_mult__left__le,axiom,
! [C: nat,A: nat] :
( ( ord_less_eq_nat @ C @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1078_mult__left__le,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ C @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ A ) ) ) ).
% mult_left_le
thf(fact_1079_mult__le__one,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_eq_nat @ B @ one_one_nat )
=> ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).
% mult_le_one
thf(fact_1080_mult__le__one,axiom,
! [A: int,B: int] :
( ( ord_less_eq_int @ A @ one_one_int )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_eq_int @ B @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).
% mult_le_one
thf(fact_1081_mult__right__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ X @ Y ) @ X ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1082_mult__left__le__one__le,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ zero_zero_int @ X )
=> ( ( ord_less_eq_int @ zero_zero_int @ Y )
=> ( ( ord_less_eq_int @ Y @ one_one_int )
=> ( ord_less_eq_int @ ( times_times_int @ Y @ X ) @ X ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1083_not__sum__squares__lt__zero,axiom,
! [X: int,Y: int] :
~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).
% not_sum_squares_lt_zero
thf(fact_1084_le__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% le_add_iff2
thf(fact_1085_le__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% le_add_iff1
thf(fact_1086_less__add__iff2,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E ) @ D ) ) ) ).
% less_add_iff2
thf(fact_1087_less__add__iff1,axiom,
! [A: int,E: int,C: int,B: int,D: int] :
( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E ) @ D ) )
= ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E ) @ C ) @ D ) ) ).
% less_add_iff1
thf(fact_1088_square__diff__one__factored,axiom,
! [X: int] :
( ( minus_minus_int @ ( times_times_int @ X @ X ) @ one_one_int )
= ( times_times_int @ ( plus_plus_int @ X @ one_one_int ) @ ( minus_minus_int @ X @ one_one_int ) ) ) ).
% square_diff_one_factored
thf(fact_1089_one__less__mult,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N ) ) ) ) ).
% one_less_mult
thf(fact_1090_n__less__m__mult__n,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ M @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1091_n__less__n__mult__m,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
=> ( ord_less_nat @ N @ ( times_times_nat @ N @ M ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1092_mult__less__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1093_mult__less__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1094_mult__less__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ A @ one_one_int ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ one_one_int @ A ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1095_mult__less__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_eq_int @ zero_zero_int @ C )
=> ( ord_less_int @ one_one_int @ B ) )
& ( ( ord_less_eq_int @ C @ zero_zero_int )
=> ( ord_less_int @ B @ one_one_int ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1096_mult__le__cancel__right2,axiom,
! [A: int,C: int] :
( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1097_mult__le__cancel__right1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1098_mult__le__cancel__left2,axiom,
! [C: int,A: int] :
( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ C )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ A @ one_one_int ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ one_one_int @ A ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1099_mult__le__cancel__left1,axiom,
! [C: int,B: int] :
( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
= ( ( ( ord_less_int @ zero_zero_int @ C )
=> ( ord_less_eq_int @ one_one_int @ B ) )
& ( ( ord_less_int @ C @ zero_zero_int )
=> ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1100_convex__bound__le,axiom,
! [X: int,A: int,Y: int,U: int,V: int] :
( ( ord_less_eq_int @ X @ A )
=> ( ( ord_less_eq_int @ Y @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ U )
=> ( ( ord_less_eq_int @ zero_zero_int @ V )
=> ( ( ( plus_plus_int @ U @ V )
= one_one_int )
=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1101_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M2: nat,N4: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N4 @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N4 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1102_nat__mult__le__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1103_nat__mult__less__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1104_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1105_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( ( K = zero_zero_nat )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1106_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K: nat] :
( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
= ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).
% left_add_mult_distrib
thf(fact_1107_nat__mult__eq__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ( times_times_nat @ K @ M )
= ( times_times_nat @ K @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1108_nat__mult__less__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1109_nat__mult__le__cancel1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1110_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1111_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1112_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1113_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1114_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I2 @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( M
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1115_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1116_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ J @ I2 )
=> ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N ) )
= ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1117_sum__squares__gt__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) )
= ( ( X != zero_zero_int )
| ( Y != zero_zero_int ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1118_sum__squares__le__zero__iff,axiom,
! [X: int,Y: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1119_sum__squares__eq__zero__iff,axiom,
! [X: int,Y: int] :
( ( ( plus_plus_int @ ( times_times_int @ X @ X ) @ ( times_times_int @ Y @ Y ) )
= zero_zero_int )
= ( ( X = zero_zero_int )
& ( Y = zero_zero_int ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_1120_mult__le__cancel__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1121_mult__le__cancel__iff2,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X ) @ ( times_times_int @ Z2 @ Y ) )
= ( ord_less_eq_int @ X @ Y ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1122_mult__less__iff1,axiom,
! [Z2: int,X: int,Y: int] :
( ( ord_less_int @ zero_zero_int @ Z2 )
=> ( ( ord_less_int @ ( times_times_int @ X @ Z2 ) @ ( times_times_int @ Y @ Z2 ) )
= ( ord_less_int @ X @ Y ) ) ) ).
% mult_less_iff1
thf(fact_1123_power__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1124_power__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_nat @ N @ M ) ) ) ) ).
% power_decreasing_iff
thf(fact_1125_power__Suc__0,axiom,
! [N: nat] :
( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N )
= ( suc @ zero_zero_nat ) ) ).
% power_Suc_0
thf(fact_1126_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M: nat] :
( ( ( power_power_nat @ X @ M )
= ( suc @ zero_zero_nat ) )
= ( ( M = zero_zero_nat )
| ( X
= ( suc @ zero_zero_nat ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_1127_power__one,axiom,
! [N: nat] :
( ( power_power_int @ one_one_int @ N )
= one_one_int ) ).
% power_one
thf(fact_1128_power__one,axiom,
! [N: nat] :
( ( power_power_nat @ one_one_nat @ N )
= one_one_nat ) ).
% power_one
thf(fact_1129_nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% nat_zero_less_power_iff
thf(fact_1130_power__one__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ one_one_nat )
= A ) ).
% power_one_right
thf(fact_1131_power__inject__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ( power_power_nat @ A @ M )
= ( power_power_nat @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1132_power__inject__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ( power_power_int @ A @ M )
= ( power_power_int @ A @ N ) )
= ( M = N ) ) ) ).
% power_inject_exp
thf(fact_1133_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_int @ zero_zero_int @ ( suc @ N ) )
= zero_zero_int ) ).
% power_0_Suc
thf(fact_1134_power__0__Suc,axiom,
! [N: nat] :
( ( power_power_nat @ zero_zero_nat @ ( suc @ N ) )
= zero_zero_nat ) ).
% power_0_Suc
thf(fact_1135_power__Suc0__right,axiom,
! [A: nat] :
( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
= A ) ).
% power_Suc0_right
thf(fact_1136_power__strict__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1137_power__strict__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_nat @ X @ Y ) ) ) ).
% power_strict_increasing_iff
thf(fact_1138_power__eq__0__iff,axiom,
! [A: int,N: nat] :
( ( ( power_power_int @ A @ N )
= zero_zero_int )
= ( ( A = zero_zero_int )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1139_power__eq__0__iff,axiom,
! [A: nat,N: nat] :
( ( ( power_power_nat @ A @ N )
= zero_zero_nat )
= ( ( A = zero_zero_nat )
& ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).
% power_eq_0_iff
thf(fact_1140_power__strict__decreasing__iff,axiom,
! [B: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ B @ one_one_nat )
=> ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1141_power__strict__decreasing__iff,axiom,
! [B: int,M: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ B )
=> ( ( ord_less_int @ B @ one_one_int )
=> ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N ) )
= ( ord_less_nat @ N @ M ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_1142_power__increasing__iff,axiom,
! [B: nat,X: nat,Y: nat] :
( ( ord_less_nat @ one_one_nat @ B )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X ) @ ( power_power_nat @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1143_power__increasing__iff,axiom,
! [B: int,X: nat,Y: nat] :
( ( ord_less_int @ one_one_int @ B )
=> ( ( ord_less_eq_int @ ( power_power_int @ B @ X ) @ ( power_power_int @ B @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ) ).
% power_increasing_iff
thf(fact_1144_power__mono__iff,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
= ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1145_power__mono__iff,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
= ( ord_less_eq_int @ A @ B ) ) ) ) ) ).
% power_mono_iff
thf(fact_1146_zero__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1147_zero__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_less_power
thf(fact_1148_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ I2 )
=> ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_1149_power__not__zero,axiom,
! [A: int,N: nat] :
( ( A != zero_zero_int )
=> ( ( power_power_int @ A @ N )
!= zero_zero_int ) ) ).
% power_not_zero
thf(fact_1150_power__not__zero,axiom,
! [A: nat,N: nat] :
( ( A != zero_zero_nat )
=> ( ( power_power_nat @ A @ N )
!= zero_zero_nat ) ) ).
% power_not_zero
thf(fact_1151_power__0,axiom,
! [A: int] :
( ( power_power_int @ A @ zero_zero_nat )
= one_one_int ) ).
% power_0
thf(fact_1152_power__0,axiom,
! [A: nat] :
( ( power_power_nat @ A @ zero_zero_nat )
= one_one_nat ) ).
% power_0
thf(fact_1153_one__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ).
% one_le_power
thf(fact_1154_one__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ).
% one_le_power
thf(fact_1155_zero__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ zero_zero_nat @ ( power_power_nat @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1156_zero__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ N ) ) ) ).
% zero_le_power
thf(fact_1157_power__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1158_power__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_eq_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ).
% power_mono
thf(fact_1159_left__right__inverse__power,axiom,
! [X: nat,Y: nat,N: nat] :
( ( ( times_times_nat @ X @ Y )
= one_one_nat )
=> ( ( times_times_nat @ ( power_power_nat @ X @ N ) @ ( power_power_nat @ Y @ N ) )
= one_one_nat ) ) ).
% left_right_inverse_power
thf(fact_1160_left__right__inverse__power,axiom,
! [X: int,Y: int,N: nat] :
( ( ( times_times_int @ X @ Y )
= one_one_int )
=> ( ( times_times_int @ ( power_power_int @ X @ N ) @ ( power_power_int @ Y @ N ) )
= one_one_int ) ) ).
% left_right_inverse_power
thf(fact_1161_power__add,axiom,
! [A: nat,M: nat,N: nat] :
( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) ) ) ).
% power_add
thf(fact_1162_power__add,axiom,
! [A: int,M: nat,N: nat] :
( ( power_power_int @ A @ ( plus_plus_nat @ M @ N ) )
= ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) ) ) ).
% power_add
thf(fact_1163_power__Suc,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ).
% power_Suc
thf(fact_1164_power__Suc,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ).
% power_Suc
thf(fact_1165_power__Suc2,axiom,
! [A: nat,N: nat] :
( ( power_power_nat @ A @ ( suc @ N ) )
= ( times_times_nat @ ( power_power_nat @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_1166_power__Suc2,axiom,
! [A: int,N: nat] :
( ( power_power_int @ A @ ( suc @ N ) )
= ( times_times_int @ ( power_power_int @ A @ N ) @ A ) ) ).
% power_Suc2
thf(fact_1167_power__less__imp__less__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1168_power__less__imp__less__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_int @ A @ B ) ) ) ).
% power_less_imp_less_base
thf(fact_1169_power__le__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ one_one_nat ) ) ) ).
% power_le_one
thf(fact_1170_power__le__one,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ one_one_int ) ) ) ).
% power_le_one
thf(fact_1171_power__le__imp__le__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1172_power__le__imp__le__base,axiom,
! [A: int,N: nat,B: int] :
( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ord_less_eq_int @ A @ B ) ) ) ).
% power_le_imp_le_base
thf(fact_1173_power__inject__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ ( suc @ N ) )
= ( power_power_nat @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1174_power__inject__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ ( suc @ N ) )
= ( power_power_int @ B @ ( suc @ N ) ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( A = B ) ) ) ) ).
% power_inject_base
thf(fact_1175_power__gt1__lemma,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1176_power__gt1__lemma,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_gt1_lemma
thf(fact_1177_power__less__power__Suc,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1178_power__less__power__Suc,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% power_less_power_Suc
thf(fact_1179_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= one_one_int ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ) ).
% power_0_left
thf(fact_1180_power__0__left,axiom,
! [N: nat] :
( ( ( N = zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= one_one_nat ) )
& ( ( N != zero_zero_nat )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ) ).
% power_0_left
thf(fact_1181_power__gt1,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1182_power__gt1,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ ( suc @ N ) ) ) ) ).
% power_gt1
thf(fact_1183_power__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1184_power__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ one_one_int @ A )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_increasing
thf(fact_1185_power__less__imp__less__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1186_power__less__imp__less__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% power_less_imp_less_exp
thf(fact_1187_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ one_one_nat @ A )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1188_power__strict__increasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ one_one_int @ A )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ A @ N5 ) ) ) ) ).
% power_strict_increasing
thf(fact_1189_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_int @ zero_zero_int @ N )
= zero_zero_int ) ) ).
% zero_power
thf(fact_1190_zero__power,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( power_power_nat @ zero_zero_nat @ N )
= zero_zero_nat ) ) ).
% zero_power
thf(fact_1191_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N )
=> ( ord_less_nat @ K @ ( power_power_nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_1192_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_1193_power__Suc__less,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( times_times_nat @ A @ ( power_power_nat @ A @ N ) ) @ ( power_power_nat @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1194_power__Suc__less,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( times_times_int @ A @ ( power_power_int @ A @ N ) ) @ ( power_power_int @ A @ N ) ) ) ) ).
% power_Suc_less
thf(fact_1195_power__Suc__le__self,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1196_power__Suc__le__self,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N ) ) @ A ) ) ) ).
% power_Suc_le_self
thf(fact_1197_power__Suc__less__one,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ ( suc @ N ) ) @ one_one_nat ) ) ) ).
% power_Suc_less_one
thf(fact_1198_power__Suc__less__one,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ ( suc @ N ) ) @ one_one_int ) ) ) ).
% power_Suc_less_one
thf(fact_1199_power__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ A @ one_one_nat )
=> ( ord_less_eq_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1200_power__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_eq_nat @ N @ N5 )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ A @ one_one_int )
=> ( ord_less_eq_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_decreasing
thf(fact_1201_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: nat] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ A @ one_one_nat )
=> ( ord_less_nat @ ( power_power_nat @ A @ N5 ) @ ( power_power_nat @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1202_power__strict__decreasing,axiom,
! [N: nat,N5: nat,A: int] :
( ( ord_less_nat @ N @ N5 )
=> ( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ A @ one_one_int )
=> ( ord_less_int @ ( power_power_int @ A @ N5 ) @ ( power_power_int @ A @ N ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1203_power__le__imp__le__exp,axiom,
! [A: nat,M: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1204_power__le__imp__le__exp,axiom,
! [A: int,M: nat,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% power_le_imp_le_exp
thf(fact_1205_power__eq__iff__eq__base,axiom,
! [N: nat,A: nat,B: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1206_power__eq__iff__eq__base,axiom,
! [N: nat,A: int,B: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
= ( A = B ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_1207_power__eq__imp__eq__base,axiom,
! [A: nat,N: nat,B: nat] :
( ( ( power_power_nat @ A @ N )
= ( power_power_nat @ B @ N ) )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1208_power__eq__imp__eq__base,axiom,
! [A: int,N: nat,B: int] :
( ( ( power_power_int @ A @ N )
= ( power_power_int @ B @ N ) )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_eq_int @ zero_zero_int @ B )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( A = B ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_1209_self__le__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_eq_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_nat @ A @ ( power_power_nat @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1210_self__le__power,axiom,
! [A: int,N: nat] :
( ( ord_less_eq_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_eq_int @ A @ ( power_power_int @ A @ N ) ) ) ) ).
% self_le_power
thf(fact_1211_one__less__power,axiom,
! [A: nat,N: nat] :
( ( ord_less_nat @ one_one_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ one_one_nat @ ( power_power_nat @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1212_one__less__power,axiom,
! [A: int,N: nat] :
( ( ord_less_int @ one_one_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ one_one_int @ ( power_power_int @ A @ N ) ) ) ) ).
% one_less_power
thf(fact_1213_power__strict__mono,axiom,
! [A: nat,B: nat,N: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_nat @ ( power_power_nat @ A @ N ) @ ( power_power_nat @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1214_power__strict__mono,axiom,
! [A: int,B: int,N: nat] :
( ( ord_less_int @ A @ B )
=> ( ( ord_less_eq_int @ zero_zero_int @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ord_less_int @ ( power_power_int @ A @ N ) @ ( power_power_int @ B @ N ) ) ) ) ) ).
% power_strict_mono
thf(fact_1215_power__eq__if,axiom,
( power_power_nat
= ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1216_power__eq__if,axiom,
( power_power_int
= ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1217_power__minus__mult,axiom,
! [N: nat,A: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_nat @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1218_power__minus__mult,axiom,
! [N: nat,A: int] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( times_times_int @ ( power_power_int @ A @ ( minus_minus_nat @ N @ one_one_nat ) ) @ A )
= ( power_power_int @ A @ N ) ) ) ).
% power_minus_mult
thf(fact_1219_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1220_of__nat__zero__less__power__iff,axiom,
! [X: nat,N: nat] :
( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X ) @ N ) )
= ( ( ord_less_nat @ zero_zero_nat @ X )
| ( N = zero_zero_nat ) ) ) ).
% of_nat_zero_less_power_iff
thf(fact_1221_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_1222_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= ( semiri1314217659103216013at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_1223_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_1224_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= one_one_nat ) ).
% binomial_n_n
thf(fact_1225_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1316708129612266289at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1226_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1314217659103216013at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_1227_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri1314217659103216013at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_1228_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ zero_zero_nat @ ( suc @ K ) )
= zero_zero_nat ) ).
% binomial_0_Suc
thf(fact_1229_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ zero_zero_nat ) )
= N ) ).
% binomial_1
thf(fact_1230_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= zero_zero_nat )
= ( ord_less_nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_1231_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus_nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_1232_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ zero_zero_nat )
= one_one_nat ) ).
% binomial_n_0
thf(fact_1233_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) )
= ( ord_less_eq_nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_1234_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq_nat @ R2 @ N )
=> ( ord_less_eq_nat @ ( binomial @ N @ R2 ) @ ( power_power_nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_1235_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus_nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_1236_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ one_one_nat )
= N ) ).
% choose_one
thf(fact_1237_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ N @ K )
=> ( ( binomial @ N @ K )
= zero_zero_nat ) ) ).
% binomial_eq_0
thf(fact_1238_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W2: int,Z5: int] :
? [N4: nat] :
( Z5
= ( plus_plus_int @ W2 @ ( semiri1314217659103216013at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1239_nat__int__comparison_I3_J,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_1240_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A3: nat,B3: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_1241_int__ops_I1_J,axiom,
( ( semiri1314217659103216013at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_1242_int__ops_I5_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(5)
thf(fact_1243_int__ops_I2_J,axiom,
( ( semiri1314217659103216013at_int @ one_one_nat )
= one_one_int ) ).
% int_ops(2)
thf(fact_1244_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).
% int_plus
thf(fact_1245_int__Suc,axiom,
! [N: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ N ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ one_one_int ) ) ).
% int_Suc
thf(fact_1246_int__ops_I4_J,axiom,
! [A: nat] :
( ( semiri1314217659103216013at_int @ ( suc @ A ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ one_one_int ) ) ).
% int_ops(4)
thf(fact_1247_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1248_zle__int,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% zle_int
thf(fact_1249_int__ops_I6_J,axiom,
! [A: nat,B: nat] :
( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= zero_zero_int ) )
& ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
=> ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
= ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).
% int_ops(6)
thf(fact_1250_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K: nat] :
( ( ord_less_int @ I2 @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1251_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N3: nat] :
( ( ord_less_nat @ zero_zero_nat @ N3 )
& ( K
= ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_1252_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N3: nat] :
( ( K
= ( semiri1314217659103216013at_int @ N3 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N3 ) ) ) ).
% pos_int_cases
thf(fact_1253_int__ops_I7_J,axiom,
! [A: nat,B: nat] :
( ( semiri1314217659103216013at_int @ ( times_times_nat @ A @ B ) )
= ( times_times_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).
% int_ops(7)
thf(fact_1254_choose__mult__lemma,axiom,
! [M: nat,R2: nat,K: nat] :
( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
= ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_1255_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_1256_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_1257_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_nat @ zero_zero_nat @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_1258_Suc__times__binomial__add,axiom,
! [A: nat,B: nat] :
( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
= ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).
% Suc_times_binomial_add
thf(fact_1259_choose__mult,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ M @ N )
=> ( ( times_times_nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K ) )
= ( times_times_nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_1260_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_1261_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y: nat] :
( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X @ Y ) ) )
= ( ( ( ord_less_eq_nat @ Y @ X )
=> ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
& ( ( ord_less_nat @ X @ Y )
=> ( P @ zero_zero_int ) ) ) ) ).
% zdiff_int_split
thf(fact_1262_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times_nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ).
% binomial_absorption
thf(fact_1263_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus_nat @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_1264_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ( times_times_nat @ K @ ( binomial @ N @ K ) )
= ( times_times_nat @ N @ ( binomial @ ( minus_minus_nat @ N @ one_one_nat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).
% times_binomial_minus1_eq
% Helper facts (5)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
! [X: int,Y: int] :
( ( if_int @ $true @ X @ Y )
= X ) ).
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (9)
thf(conj_0,hypothesis,
( ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ t2 @ x52 ) @ ( lambda_resid @ x3 @ x51 ) )
!= lambda_Nil ) ).
thf(conj_1,hypothesis,
( ua
= ( lambda_Beta @ x51 @ x52 ) ) ).
thf(conj_2,hypothesis,
! [U5: lambda_lambda] :
( ( ( lambda_resid @ ( lambda_Lam @ x3 ) @ U5 )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ ( lambda_Lam @ x3 ) @ U5 ) ) ) ).
thf(conj_3,hypothesis,
! [U5: lambda_lambda] :
( ( ( lambda_resid @ t2 @ U5 )
!= lambda_Nil )
=> ( lambda_Arr @ ( lambda_resid @ t2 @ U5 ) ) ) ).
thf(conj_4,hypothesis,
! [V5: lambda_lambda,T5: lambda_lambda,N6: nat] :
( ( lambda_Arr @ V5 )
=> ( ( lambda_Arr @ T5 )
=> ( lambda_Arr @ ( lambda_Subst @ N6 @ V5 @ T5 ) ) ) ) ).
thf(conj_5,hypothesis,
( t1
= ( lambda_Lam @ x3 ) ) ).
thf(conj_6,hypothesis,
( ( lambda_resid @ x3 @ x51 )
!= lambda_Nil ) ).
thf(conj_7,hypothesis,
( ( lambda_resid @ t2 @ x52 )
!= lambda_Nil ) ).
thf(conj_8,conjecture,
lambda_Arr @ ( lambda_Subst @ zero_zero_nat @ ( lambda_resid @ t2 @ x52 ) @ ( lambda_resid @ x3 @ x51 ) ) ).
%------------------------------------------------------------------------------