TPTP Problem File: ITP126^1.p
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%------------------------------------------------------------------------------
% File : ITP126^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Monitor problem prob_770__6461298_1
% Version : Especial.
% English :
% Refs : [BH+15] Blanchette et al. (2015), Mining the Archive of Formal
% : [Des21] Desharnais (2021), Email to Geoff Sutcliffe
% Source : [Des21]
% Names : Monitor/prob_770__6461298_1 [Des21]
% Status : Theorem
% Rating : 0.90 v8.2.0, 1.00 v8.1.0, 0.91 v7.5.0
% Syntax : Number of formulae : 380 ( 210 unt; 51 typ; 0 def)
% Number of atoms : 729 ( 452 equ; 0 cnn)
% Maximal formula atoms : 4 ( 2 avg)
% Number of connectives : 2042 ( 105 ~; 15 |; 53 &;1643 @)
% ( 0 <=>; 226 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 5 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 123 ( 123 >; 0 *; 0 +; 0 <<)
% Number of symbols : 45 ( 42 usr; 11 con; 0-3 aty)
% Number of variables : 780 ( 40 ^; 698 !; 42 ?; 780 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:33:17.167
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_n_t__Option__Ooption_It__Option__Ooption_Itf__b_J_J,type,
option_option_b: $tType ).
thf(ty_n_t__List__Olist_It__Option__Ooption_Itf__b_J_J,type,
list_option_b: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Option__Ooption_Itf__b_J,type,
option_b: $tType ).
thf(ty_n_t__MFOTL__Oformula_Itf__a_J,type,
formula_a: $tType ).
thf(ty_n_t__List__Olist_Itf__b_J,type,
list_b: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__b,type,
b: $tType ).
% Explicit typings (42)
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_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__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_List_Odrop_001t__Option__Ooption_Itf__b_J,type,
drop_option_b: nat > list_option_b > list_option_b ).
thf(sy_c_List_Odrop_001tf__b,type,
drop_b: nat > list_b > list_b ).
thf(sy_c_List_Ofind_001t__Option__Ooption_Itf__b_J,type,
find_option_b: ( option_b > $o ) > list_option_b > option_option_b ).
thf(sy_c_List_Ofind_001tf__b,type,
find_b: ( b > $o ) > list_b > option_b ).
thf(sy_c_List_Ogen__length_001t__Option__Ooption_Itf__b_J,type,
gen_length_option_b: nat > list_option_b > nat ).
thf(sy_c_List_Ogen__length_001tf__b,type,
gen_length_b: nat > list_b > nat ).
thf(sy_c_List_Olist_OCons_001t__Option__Ooption_Itf__b_J,type,
cons_option_b: option_b > list_option_b > list_option_b ).
thf(sy_c_List_Olist_OCons_001tf__b,type,
cons_b: b > list_b > list_b ).
thf(sy_c_List_Olist__ex_001t__Option__Ooption_Itf__b_J,type,
list_ex_option_b: ( option_b > $o ) > list_option_b > $o ).
thf(sy_c_List_Olist__ex_001tf__b,type,
list_ex_b: ( b > $o ) > list_b > $o ).
thf(sy_c_List_Onth_001t__Option__Ooption_Itf__b_J,type,
nth_option_b: list_option_b > nat > option_b ).
thf(sy_c_List_Onth_001tf__b,type,
nth_b: list_b > nat > b ).
thf(sy_c_MFOTL_Ofvi_001tf__a,type,
fvi_a: nat > formula_a > set_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri2019852685at_int: nat > int ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat,type,
semiri1382578993at_nat: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Option__Ooption_Itf__b_J_J,type,
size_s1671393719tion_b: list_option_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__b_J,type,
size_size_list_b: list_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Option__Ooption_Itf__b_J_J,type,
size_s684879735tion_b: option_option_b > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_Itf__b_J,type,
size_size_option_b: option_b > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_Option_Ooption_ONone_001t__Option__Ooption_Itf__b_J,type,
none_option_b: option_option_b ).
thf(sy_c_Option_Ooption_ONone_001tf__b,type,
none_b: option_b ).
thf(sy_c_Option_Ooption_OSome_001t__Option__Ooption_Itf__b_J,type,
some_option_b: option_b > option_option_b ).
thf(sy_c_Option_Ooption_OSome_001tf__b,type,
some_b: b > option_b ).
thf(sy_c_Option_Ooption_Osize__option_001t__Option__Ooption_Itf__b_J,type,
size_option_option_b: ( option_b > nat ) > option_option_b > nat ).
thf(sy_c_Option_Ooption_Osize__option_001tf__b,type,
size_option_b: ( b > nat ) > option_b > nat ).
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__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_v__092_060phi_062,type,
phi: formula_a ).
thf(sy_v_b,type,
b2: nat ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_v,type,
v: list_option_b ).
thf(sy_v_x,type,
x: b ).
% Relevant facts (323)
thf(fact_0_nth__Cons__Suc,axiom,
! [X: b,Xs: list_b,N: nat] :
( ( nth_b @ ( cons_b @ X @ Xs ) @ ( suc @ N ) )
= ( nth_b @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_1_nth__Cons__Suc,axiom,
! [X: option_b,Xs: list_option_b,N: nat] :
( ( nth_option_b @ ( cons_option_b @ X @ Xs ) @ ( suc @ N ) )
= ( nth_option_b @ Xs @ N ) ) ).
% nth_Cons_Suc
thf(fact_2_nth__Cons__0,axiom,
! [X: b,Xs: list_b] :
( ( nth_b @ ( cons_b @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_3_nth__Cons__0,axiom,
! [X: option_b,Xs: list_option_b] :
( ( nth_option_b @ ( cons_option_b @ X @ Xs ) @ zero_zero_nat )
= X ) ).
% nth_Cons_0
thf(fact_4_less__Suc0,axiom,
! [N: nat] :
( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
= ( N = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_5_zero__less__Suc,axiom,
! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_6_not__None__eq,axiom,
! [X: option_option_b] :
( ( X != none_option_b )
= ( ? [Y: option_b] :
( X
= ( some_option_b @ Y ) ) ) ) ).
% not_None_eq
thf(fact_7_not__None__eq,axiom,
! [X: option_b] :
( ( X != none_b )
= ( ? [Y: b] :
( X
= ( some_b @ Y ) ) ) ) ).
% not_None_eq
thf(fact_8_not__Some__eq,axiom,
! [X: option_option_b] :
( ( ! [Y: option_b] :
( X
!= ( some_option_b @ Y ) ) )
= ( X = none_option_b ) ) ).
% not_Some_eq
thf(fact_9_not__Some__eq,axiom,
! [X: option_b] :
( ( ! [Y: b] :
( X
!= ( some_b @ Y ) ) )
= ( X = none_b ) ) ).
% not_Some_eq
thf(fact_10_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_11_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_12_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_13_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% neq0_conv
thf(fact_14_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_15_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_16_list_Oinject,axiom,
! [X21: b,X22: list_b,Y21: b,Y22: list_b] :
( ( ( cons_b @ X21 @ X22 )
= ( cons_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_17_list_Oinject,axiom,
! [X21: option_b,X22: list_option_b,Y21: option_b,Y22: list_option_b] :
( ( ( cons_option_b @ X21 @ X22 )
= ( cons_option_b @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_18_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_19_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_20_option_Oinject,axiom,
! [X2: option_b,Y2: option_b] :
( ( ( some_option_b @ X2 )
= ( some_option_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_21_option_Oinject,axiom,
! [X2: b,Y2: b] :
( ( ( some_b @ X2 )
= ( some_b @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_22_not__Cons__self2,axiom,
! [X: b,Xs: list_b] :
( ( cons_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_23_not__Cons__self2,axiom,
! [X: option_b,Xs: list_option_b] :
( ( cons_option_b @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_24_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_25_Suc__inject,axiom,
! [X: nat,Y3: nat] :
( ( ( suc @ X )
= ( suc @ Y3 ) )
=> ( X = Y3 ) ) ).
% Suc_inject
thf(fact_26_linorder__neqE__nat,axiom,
! [X: nat,Y3: nat] :
( ( X != Y3 )
=> ( ~ ( ord_less_nat @ X @ Y3 )
=> ( ord_less_nat @ Y3 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_27_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_28_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_29_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_30_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_31_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_32_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_33_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_34_size__neq__size__imp__neq,axiom,
! [X: option_b,Y3: option_b] :
( ( ( size_size_option_b @ X )
!= ( size_size_option_b @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_35_size__neq__size__imp__neq,axiom,
! [X: list_b,Y3: list_b] :
( ( ( size_size_list_b @ X )
!= ( size_size_list_b @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_36_size__neq__size__imp__neq,axiom,
! [X: list_option_b,Y3: list_option_b] :
( ( ( size_s1671393719tion_b @ X )
!= ( size_s1671393719tion_b @ Y3 ) )
=> ( X != Y3 ) ) ).
% size_neq_size_imp_neq
thf(fact_37_neq__if__length__neq,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
!= ( size_size_list_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_38_neq__if__length__neq,axiom,
! [Xs: list_option_b,Ys: list_option_b] :
( ( ( size_s1671393719tion_b @ Xs )
!= ( size_s1671393719tion_b @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_39_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_b] :
( ( size_size_list_b @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_40_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_option_b] :
( ( size_s1671393719tion_b @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_41_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_42_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_43_old_Onat_Oexhaust,axiom,
! [Y3: nat] :
( ( Y3 != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y3
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_44_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_45_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_46_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_47_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_48_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X3: nat,Y4: nat] :
( ( P @ X3 @ Y4 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
=> ( P @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_49_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_50_mem__Collect__eq,axiom,
! [A: nat,P: nat > $o] :
( ( member_nat @ A @ ( collect_nat @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__cong,axiom,
! [P: nat > $o,Q: nat > $o] :
( ! [X3: nat] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect_nat @ P )
= ( collect_nat @ Q ) ) ) ).
% Collect_cong
thf(fact_53_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_54_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_55_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_56_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_57_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_58_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_59_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_60_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% less_zeroE
thf(fact_61_not__less0,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less0
thf(fact_62_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr0
thf(fact_63_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr0I
thf(fact_64_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_65_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_66_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_67_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_68_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_69_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_70_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_71_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ N )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_72_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_73_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_74_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ N )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_75_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_76_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_77_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_78_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_79_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_80_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_81_length__induct,axiom,
! [P: list_b > $o,Xs: list_b] :
( ! [Xs2: list_b] :
( ! [Ys2: list_b] :
( ( ord_less_nat @ ( size_size_list_b @ Ys2 ) @ ( size_size_list_b @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_82_length__induct,axiom,
! [P: list_option_b > $o,Xs: list_option_b] :
( ! [Xs2: list_option_b] :
( ! [Ys2: list_option_b] :
( ( ord_less_nat @ ( size_s1671393719tion_b @ Ys2 ) @ ( size_s1671393719tion_b @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_83_combine__options__cases,axiom,
! [X: option_b,P: option_b > option_option_b > $o,Y3: option_option_b] :
( ( ( X = none_b )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_option_b )
=> ( P @ X @ Y3 ) )
=> ( ! [A3: b,B: option_b] :
( ( X
= ( some_b @ A3 ) )
=> ( ( Y3
= ( some_option_b @ B ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_84_combine__options__cases,axiom,
! [X: option_option_b,P: option_option_b > option_b > $o,Y3: option_b] :
( ( ( X = none_option_b )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_b )
=> ( P @ X @ Y3 ) )
=> ( ! [A3: option_b,B: b] :
( ( X
= ( some_option_b @ A3 ) )
=> ( ( Y3
= ( some_b @ B ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_85_combine__options__cases,axiom,
! [X: option_option_b,P: option_option_b > option_option_b > $o,Y3: option_option_b] :
( ( ( X = none_option_b )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_option_b )
=> ( P @ X @ Y3 ) )
=> ( ! [A3: option_b,B: option_b] :
( ( X
= ( some_option_b @ A3 ) )
=> ( ( Y3
= ( some_option_b @ B ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_86_combine__options__cases,axiom,
! [X: option_b,P: option_b > option_b > $o,Y3: option_b] :
( ( ( X = none_b )
=> ( P @ X @ Y3 ) )
=> ( ( ( Y3 = none_b )
=> ( P @ X @ Y3 ) )
=> ( ! [A3: b,B: b] :
( ( X
= ( some_b @ A3 ) )
=> ( ( Y3
= ( some_b @ B ) )
=> ( P @ X @ Y3 ) ) )
=> ( P @ X @ Y3 ) ) ) ) ).
% combine_options_cases
thf(fact_87_split__option__all,axiom,
( ( ^ [P2: option_option_b > $o] :
! [X5: option_option_b] : ( P2 @ X5 ) )
= ( ^ [P3: option_option_b > $o] :
( ( P3 @ none_option_b )
& ! [X4: option_b] : ( P3 @ ( some_option_b @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_88_split__option__all,axiom,
( ( ^ [P2: option_b > $o] :
! [X5: option_b] : ( P2 @ X5 ) )
= ( ^ [P3: option_b > $o] :
( ( P3 @ none_b )
& ! [X4: b] : ( P3 @ ( some_b @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_89_split__option__ex,axiom,
( ( ^ [P2: option_option_b > $o] :
? [X5: option_option_b] : ( P2 @ X5 ) )
= ( ^ [P3: option_option_b > $o] :
( ( P3 @ none_option_b )
| ? [X4: option_b] : ( P3 @ ( some_option_b @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_90_split__option__ex,axiom,
( ( ^ [P2: option_b > $o] :
? [X5: option_b] : ( P2 @ X5 ) )
= ( ^ [P3: option_b > $o] :
( ( P3 @ none_b )
| ? [X4: b] : ( P3 @ ( some_b @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_91_option_Oinducts,axiom,
! [P: option_option_b > $o,Option: option_option_b] :
( ( P @ none_option_b )
=> ( ! [X3: option_b] : ( P @ ( some_option_b @ X3 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_92_option_Oinducts,axiom,
! [P: option_b > $o,Option: option_b] :
( ( P @ none_b )
=> ( ! [X3: b] : ( P @ ( some_b @ X3 ) )
=> ( P @ Option ) ) ) ).
% option.inducts
thf(fact_93_option_Oexhaust,axiom,
! [Y3: option_option_b] :
( ( Y3 != none_option_b )
=> ~ ! [X23: option_b] :
( Y3
!= ( some_option_b @ X23 ) ) ) ).
% option.exhaust
thf(fact_94_option_Oexhaust,axiom,
! [Y3: option_b] :
( ( Y3 != none_b )
=> ~ ! [X23: b] :
( Y3
!= ( some_b @ X23 ) ) ) ).
% option.exhaust
thf(fact_95_option_OdiscI,axiom,
! [Option: option_option_b,X2: option_b] :
( ( Option
= ( some_option_b @ X2 ) )
=> ( Option != none_option_b ) ) ).
% option.discI
thf(fact_96_option_OdiscI,axiom,
! [Option: option_b,X2: b] :
( ( Option
= ( some_b @ X2 ) )
=> ( Option != none_b ) ) ).
% option.discI
thf(fact_97_option_Odistinct_I1_J,axiom,
! [X2: option_b] :
( none_option_b
!= ( some_option_b @ X2 ) ) ).
% option.distinct(1)
thf(fact_98_option_Odistinct_I1_J,axiom,
! [X2: b] :
( none_b
!= ( some_b @ X2 ) ) ).
% option.distinct(1)
thf(fact_99_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_100_lift__Suc__mono__less__iff,axiom,
! [F: nat > int,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_int @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_101_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_102_lift__Suc__mono__less,axiom,
! [F: nat > int,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_int @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N3 )
=> ( ord_less_int @ ( F @ N ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_103_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_104_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_105_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_106_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_107_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_108_length__Suc__conv,axiom,
! [Xs: list_b,N: nat] :
( ( ( size_size_list_b @ Xs )
= ( suc @ N ) )
= ( ? [Y: b,Ys3: list_b] :
( ( Xs
= ( cons_b @ Y @ Ys3 ) )
& ( ( size_size_list_b @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_109_length__Suc__conv,axiom,
! [Xs: list_option_b,N: nat] :
( ( ( size_s1671393719tion_b @ Xs )
= ( suc @ N ) )
= ( ? [Y: option_b,Ys3: list_option_b] :
( ( Xs
= ( cons_option_b @ Y @ Ys3 ) )
& ( ( size_s1671393719tion_b @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_110_Suc__length__conv,axiom,
! [N: nat,Xs: list_b] :
( ( ( suc @ N )
= ( size_size_list_b @ Xs ) )
= ( ? [Y: b,Ys3: list_b] :
( ( Xs
= ( cons_b @ Y @ Ys3 ) )
& ( ( size_size_list_b @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_111_Suc__length__conv,axiom,
! [N: nat,Xs: list_option_b] :
( ( ( suc @ N )
= ( size_s1671393719tion_b @ Xs ) )
= ( ? [Y: option_b,Ys3: list_option_b] :
( ( Xs
= ( cons_option_b @ Y @ Ys3 ) )
& ( ( size_s1671393719tion_b @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_112_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_b,Z: list_b] : Y5 = Z )
= ( ^ [Xs3: list_b,Ys3: list_b] :
( ( ( size_size_list_b @ Xs3 )
= ( size_size_list_b @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs3 ) )
=> ( ( nth_b @ Xs3 @ I3 )
= ( nth_b @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_113_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_option_b,Z: list_option_b] : Y5 = Z )
= ( ^ [Xs3: list_option_b,Ys3: list_option_b] :
( ( ( size_s1671393719tion_b @ Xs3 )
= ( size_s1671393719tion_b @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1671393719tion_b @ Xs3 ) )
=> ( ( nth_option_b @ Xs3 @ I3 )
= ( nth_option_b @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_114_Skolem__list__nth,axiom,
! [K: nat,P: nat > b > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: b] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_b] :
( ( ( size_size_list_b @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_b @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_115_Skolem__list__nth,axiom,
! [K: nat,P: nat > option_b > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X5: option_b] : ( P @ I3 @ X5 ) ) )
= ( ? [Xs3: list_option_b] :
( ( ( size_s1671393719tion_b @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_option_b @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_116_nth__equalityI,axiom,
! [Xs: list_b,Ys: list_b] :
( ( ( size_size_list_b @ Xs )
= ( size_size_list_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ Xs @ I2 )
= ( nth_b @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_117_nth__equalityI,axiom,
! [Xs: list_option_b,Ys: list_option_b] :
( ( ( size_s1671393719tion_b @ Xs )
= ( size_s1671393719tion_b @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1671393719tion_b @ Xs ) )
=> ( ( nth_option_b @ Xs @ I2 )
= ( nth_option_b @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_118_not__gr__zero,axiom,
! [N: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
= ( N = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_119_fvi__Suc__bound,axiom,
! [B2: nat,Phi: formula_a,N: nat] :
( ! [X3: nat] :
( ( member_nat @ X3 @ ( fvi_a @ ( suc @ B2 ) @ Phi ) )
=> ( ord_less_nat @ X3 @ N ) )
=> ! [X6: nat] :
( ( member_nat @ X6 @ ( fvi_a @ B2 @ Phi ) )
=> ( ord_less_nat @ X6 @ ( suc @ N ) ) ) ) ).
% fvi_Suc_bound
thf(fact_120_length__Cons,axiom,
! [X: b,Xs: list_b] :
( ( size_size_list_b @ ( cons_b @ X @ Xs ) )
= ( suc @ ( size_size_list_b @ Xs ) ) ) ).
% length_Cons
thf(fact_121_length__Cons,axiom,
! [X: option_b,Xs: list_option_b] :
( ( size_s1671393719tion_b @ ( cons_option_b @ X @ Xs ) )
= ( suc @ ( size_s1671393719tion_b @ Xs ) ) ) ).
% length_Cons
thf(fact_122_option_Osize__gen_I1_J,axiom,
! [X: b > nat] :
( ( size_option_b @ X @ none_b )
= ( suc @ zero_zero_nat ) ) ).
% option.size_gen(1)
thf(fact_123_fvi__Suc,axiom,
! [X: nat,B2: nat,Phi: formula_a] :
( ( member_nat @ X @ ( fvi_a @ ( suc @ B2 ) @ Phi ) )
= ( member_nat @ ( suc @ X ) @ ( fvi_a @ B2 @ Phi ) ) ) ).
% fvi_Suc
thf(fact_124_option_Osize_I4_J,axiom,
! [X2: option_b] :
( ( size_s684879735tion_b @ ( some_option_b @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_125_option_Osize_I4_J,axiom,
! [X2: b] :
( ( size_size_option_b @ ( some_b @ X2 ) )
= ( suc @ zero_zero_nat ) ) ).
% option.size(4)
thf(fact_126_option_Osize_I3_J,axiom,
( ( size_size_option_b @ none_b )
= ( suc @ zero_zero_nat ) ) ).
% option.size(3)
thf(fact_127_find__Some__iff,axiom,
! [P: b > $o,Xs: list_b,X: b] :
( ( ( find_b @ P @ Xs )
= ( some_b @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_b @ Xs ) )
& ( P @ ( nth_b @ Xs @ I3 ) )
& ( X
= ( nth_b @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_b @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_128_find__Some__iff,axiom,
! [P: option_b > $o,Xs: list_option_b,X: option_b] :
( ( ( find_option_b @ P @ Xs )
= ( some_option_b @ X ) )
= ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1671393719tion_b @ Xs ) )
& ( P @ ( nth_option_b @ Xs @ I3 ) )
& ( X
= ( nth_option_b @ Xs @ I3 ) )
& ! [J3: nat] :
( ( ord_less_nat @ J3 @ I3 )
=> ~ ( P @ ( nth_option_b @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_129_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ zero_zero_nat )
=> ( ? [X_1: nat] : ( P @ X_1 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_130_option_Osize__neq,axiom,
! [X: option_b] :
( ( size_size_option_b @ X )
!= zero_zero_nat ) ).
% option.size_neq
thf(fact_131_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_132_zero__reorient,axiom,
! [X: int] :
( ( zero_zero_int = X )
= ( X = zero_zero_int ) ) ).
% zero_reorient
thf(fact_133_find_Osimps_I2_J,axiom,
! [P: option_b > $o,X: option_b,Xs: list_option_b] :
( ( ( P @ X )
=> ( ( find_option_b @ P @ ( cons_option_b @ X @ Xs ) )
= ( some_option_b @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_option_b @ P @ ( cons_option_b @ X @ Xs ) )
= ( find_option_b @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_134_find_Osimps_I2_J,axiom,
! [P: b > $o,X: b,Xs: list_b] :
( ( ( P @ X )
=> ( ( find_b @ P @ ( cons_b @ X @ Xs ) )
= ( some_b @ X ) ) )
& ( ~ ( P @ X )
=> ( ( find_b @ P @ ( cons_b @ X @ Xs ) )
= ( find_b @ P @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_135_gr__zeroI,axiom,
! [N: nat] :
( ( N != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% gr_zeroI
thf(fact_136_not__less__zero,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ zero_zero_nat ) ).
% not_less_zero
thf(fact_137_gr__implies__not__zero,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( N != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_138_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_139_list__decode_Ocases,axiom,
! [X: nat] :
( ( X != zero_zero_nat )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_140_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_141_less__numeral__extra_I3_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_numeral_extra(3)
thf(fact_142_length__code,axiom,
( size_size_list_b
= ( gen_length_b @ zero_zero_nat ) ) ).
% length_code
thf(fact_143_length__code,axiom,
( size_s1671393719tion_b
= ( gen_length_option_b @ zero_zero_nat ) ) ).
% length_code
thf(fact_144_gen__length__code_I2_J,axiom,
! [N: nat,X: b,Xs: list_b] :
( ( gen_length_b @ N @ ( cons_b @ X @ Xs ) )
= ( gen_length_b @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_145_gen__length__code_I2_J,axiom,
! [N: nat,X: option_b,Xs: list_option_b] :
( ( gen_length_option_b @ N @ ( cons_option_b @ X @ Xs ) )
= ( gen_length_option_b @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_146_option_Osize__gen_I2_J,axiom,
! [X: option_b > nat,X2: option_b] :
( ( size_option_option_b @ X @ ( some_option_b @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_147_option_Osize__gen_I2_J,axiom,
! [X: b > nat,X2: b] :
( ( size_option_b @ X @ ( some_b @ X2 ) )
= ( plus_plus_nat @ ( X @ X2 ) @ ( suc @ zero_zero_nat ) ) ) ).
% option.size_gen(2)
thf(fact_148_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_b] :
( ( ord_less_nat @ I @ ( size_size_list_b @ Xs ) )
=> ( ( cons_b @ ( nth_b @ Xs @ I ) @ ( drop_b @ ( suc @ I ) @ Xs ) )
= ( drop_b @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_149_Cons__nth__drop__Suc,axiom,
! [I: nat,Xs: list_option_b] :
( ( ord_less_nat @ I @ ( size_s1671393719tion_b @ Xs ) )
=> ( ( cons_option_b @ ( nth_option_b @ Xs @ I ) @ ( drop_option_b @ ( suc @ I ) @ Xs ) )
= ( drop_option_b @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_150_list__ex__length,axiom,
( list_ex_b
= ( ^ [P3: b > $o,Xs3: list_b] :
? [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_size_list_b @ Xs3 ) )
& ( P3 @ ( nth_b @ Xs3 @ N4 ) ) ) ) ) ).
% list_ex_length
thf(fact_151_list__ex__length,axiom,
( list_ex_option_b
= ( ^ [P3: option_b > $o,Xs3: list_option_b] :
? [N4: nat] :
( ( ord_less_nat @ N4 @ ( size_s1671393719tion_b @ Xs3 ) )
& ( P3 @ ( nth_option_b @ Xs3 @ N4 ) ) ) ) ) ).
% list_ex_length
thf(fact_152_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_153_of__nat__0__less__iff,axiom,
! [N: nat] :
( ( ord_less_int @ zero_zero_int @ ( semiri2019852685at_int @ N ) )
= ( ord_less_nat @ zero_zero_nat @ N ) ) ).
% of_nat_0_less_iff
thf(fact_154_add__left__cancel,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_155_add__left__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add_left_cancel
thf(fact_156_add__right__cancel,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_157_add__right__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add_right_cancel
thf(fact_158_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri1382578993at_nat @ M )
= ( semiri1382578993at_nat @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_159_of__nat__eq__iff,axiom,
! [M: nat,N: nat] :
( ( ( semiri2019852685at_int @ M )
= ( semiri2019852685at_int @ N ) )
= ( M = N ) ) ).
% of_nat_eq_iff
thf(fact_160_zero__eq__add__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( zero_zero_nat
= ( plus_plus_nat @ X @ Y3 ) )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_161_add__eq__0__iff__both__eq__0,axiom,
! [X: nat,Y3: nat] :
( ( ( plus_plus_nat @ X @ Y3 )
= zero_zero_nat )
= ( ( X = zero_zero_nat )
& ( Y3 = zero_zero_nat ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_162_add__cancel__right__right,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ A @ B2 ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_right
thf(fact_163_add__cancel__right__right,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ A @ B2 ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_right
thf(fact_164_add__cancel__right__left,axiom,
! [A: nat,B2: nat] :
( ( A
= ( plus_plus_nat @ B2 @ A ) )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_right_left
thf(fact_165_add__cancel__right__left,axiom,
! [A: int,B2: int] :
( ( A
= ( plus_plus_int @ B2 @ A ) )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_right_left
thf(fact_166_add__cancel__left__right,axiom,
! [A: nat,B2: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_right
thf(fact_167_add__cancel__left__right,axiom,
! [A: int,B2: int] :
( ( ( plus_plus_int @ A @ B2 )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_right
thf(fact_168_add__cancel__left__left,axiom,
! [B2: nat,A: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= A )
= ( B2 = zero_zero_nat ) ) ).
% add_cancel_left_left
thf(fact_169_add__cancel__left__left,axiom,
! [B2: int,A: int] :
( ( ( plus_plus_int @ B2 @ A )
= A )
= ( B2 = zero_zero_int ) ) ).
% add_cancel_left_left
thf(fact_170_double__zero__sym,axiom,
! [A: int] :
( ( zero_zero_int
= ( plus_plus_int @ A @ A ) )
= ( A = zero_zero_int ) ) ).
% double_zero_sym
thf(fact_171_double__zero,axiom,
! [A: int] :
( ( ( plus_plus_int @ A @ A )
= zero_zero_int )
= ( A = zero_zero_int ) ) ).
% double_zero
thf(fact_172_add_Oright__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.right_neutral
thf(fact_173_add_Oright__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.right_neutral
thf(fact_174_add_Oleft__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ zero_zero_nat @ A )
= A ) ).
% add.left_neutral
thf(fact_175_add_Oleft__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.left_neutral
thf(fact_176_add__less__cancel__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_177_add__less__cancel__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_right
thf(fact_178_add__less__cancel__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
= ( ord_less_nat @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_179_add__less__cancel__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
= ( ord_less_int @ A @ B2 ) ) ).
% add_less_cancel_left
thf(fact_180_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_181_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus_nat @ M @ zero_zero_nat )
= M ) ).
% Nat.add_0_right
thf(fact_182_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_183_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_184_drop0,axiom,
( ( drop_option_b @ zero_zero_nat )
= ( ^ [X4: list_option_b] : X4 ) ) ).
% drop0
thf(fact_185_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_option_b] :
( ( drop_option_b @ N @ ( drop_option_b @ M @ Xs ) )
= ( drop_option_b @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_186_list__ex__simps_I1_J,axiom,
! [P: b > $o,X: b,Xs: list_b] :
( ( list_ex_b @ P @ ( cons_b @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_b @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_187_list__ex__simps_I1_J,axiom,
! [P: option_b > $o,X: option_b,Xs: list_option_b] :
( ( list_ex_option_b @ P @ ( cons_option_b @ X @ Xs ) )
= ( ( P @ X )
| ( list_ex_option_b @ P @ Xs ) ) ) ).
% list_ex_simps(1)
thf(fact_188_add__less__same__cancel1,axiom,
! [B2: nat,A: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel1
thf(fact_189_add__less__same__cancel1,axiom,
! [B2: int,A: int] :
( ( ord_less_int @ ( plus_plus_int @ B2 @ A ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel1
thf(fact_190_add__less__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ B2 )
= ( ord_less_nat @ A @ zero_zero_nat ) ) ).
% add_less_same_cancel2
thf(fact_191_add__less__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ B2 )
= ( ord_less_int @ A @ zero_zero_int ) ) ).
% add_less_same_cancel2
thf(fact_192_less__add__same__cancel1,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B2 ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_193_less__add__same__cancel1,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ A @ B2 ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel1
thf(fact_194_less__add__same__cancel2,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ ( plus_plus_nat @ B2 @ A ) )
= ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_195_less__add__same__cancel2,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ ( plus_plus_int @ B2 @ A ) )
= ( ord_less_int @ zero_zero_int @ B2 ) ) ).
% less_add_same_cancel2
thf(fact_196_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_197_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_198_of__nat__0,axiom,
( ( semiri1382578993at_nat @ zero_zero_nat )
= zero_zero_nat ) ).
% of_nat_0
thf(fact_199_of__nat__0,axiom,
( ( semiri2019852685at_int @ zero_zero_nat )
= zero_zero_int ) ).
% of_nat_0
thf(fact_200_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_nat
= ( semiri1382578993at_nat @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_201_of__nat__0__eq__iff,axiom,
! [N: nat] :
( ( zero_zero_int
= ( semiri2019852685at_int @ N ) )
= ( zero_zero_nat = N ) ) ).
% of_nat_0_eq_iff
thf(fact_202_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri1382578993at_nat @ M )
= zero_zero_nat )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_203_of__nat__eq__0__iff,axiom,
! [M: nat] :
( ( ( semiri2019852685at_int @ M )
= zero_zero_int )
= ( M = zero_zero_nat ) ) ).
% of_nat_eq_0_iff
thf(fact_204_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_205_of__nat__less__iff,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_iff
thf(fact_206_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri1382578993at_nat @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% of_nat_add
thf(fact_207_of__nat__add,axiom,
! [M: nat,N: nat] :
( ( semiri2019852685at_int @ ( plus_plus_nat @ M @ N ) )
= ( plus_plus_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% of_nat_add
thf(fact_208_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_209_drop__Suc__Cons,axiom,
! [N: nat,X: b,Xs: list_b] :
( ( drop_b @ ( suc @ N ) @ ( cons_b @ X @ Xs ) )
= ( drop_b @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_210_drop__Suc__Cons,axiom,
! [N: nat,X: option_b,Xs: list_option_b] :
( ( drop_option_b @ ( suc @ N ) @ ( cons_option_b @ X @ Xs ) )
= ( drop_option_b @ N @ Xs ) ) ).
% drop_Suc_Cons
thf(fact_211_is__num__normalize_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% is_num_normalize(1)
thf(fact_212_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_213_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_214_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_215_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_216_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_217_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_218_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_219_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_220_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus_nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus_nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_221_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_222_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_223_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_224_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N )
= N ) ).
% plus_nat.add_0
thf(fact_225_drop__0,axiom,
! [Xs: list_option_b] :
( ( drop_option_b @ zero_zero_nat @ Xs )
= Xs ) ).
% drop_0
thf(fact_226_add__less__imp__less__right,axiom,
! [A: nat,C: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_227_add__less__imp__less__right,axiom,
! [A: int,C: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_right
thf(fact_228_add__less__imp__less__left,axiom,
! [C: nat,A: nat,B2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) )
=> ( ord_less_nat @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_229_add__less__imp__less__left,axiom,
! [C: int,A: int,B2: int] :
( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) )
=> ( ord_less_int @ A @ B2 ) ) ).
% add_less_imp_less_left
thf(fact_230_add__strict__right__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_231_add__strict__right__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add_strict_right_mono
thf(fact_232_add__strict__left__mono,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_233_add__strict__left__mono,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ A @ B2 )
=> ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B2 ) ) ) ).
% add_strict_left_mono
thf(fact_234_add__strict__mono,axiom,
! [A: nat,B2: nat,C: nat,D: nat] :
( ( ord_less_nat @ A @ B2 )
=> ( ( ord_less_nat @ C @ D )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_235_add__strict__mono,axiom,
! [A: int,B2: int,C: int,D: int] :
( ( ord_less_int @ A @ B2 )
=> ( ( ord_less_int @ C @ D )
=> ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B2 @ D ) ) ) ) ).
% add_strict_mono
thf(fact_236_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_237_add__mono__thms__linordered__field_I1_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( K = L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_238_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_239_add__mono__thms__linordered__field_I2_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_240_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_241_add__mono__thms__linordered__field_I5_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( ord_less_int @ I @ J )
& ( ord_less_int @ K @ L ) )
=> ( ord_less_int @ ( plus_plus_int @ I @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_242_add_Ogroup__left__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ zero_zero_int @ A )
= A ) ).
% add.group_left_neutral
thf(fact_243_add_Ocomm__neutral,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% add.comm_neutral
thf(fact_244_add_Ocomm__neutral,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% add.comm_neutral
thf(fact_245_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_246_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_247_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_248_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_249_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_250_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: int,J: int,K: int,L: int] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_int @ I @ K )
= ( plus_plus_int @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_251_group__cancel_Oadd1,axiom,
! [A2: nat,K: nat,A: nat,B2: nat] :
( ( A2
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A2 @ B2 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_252_group__cancel_Oadd1,axiom,
! [A2: int,K: int,A: int,B2: int] :
( ( A2
= ( plus_plus_int @ K @ A ) )
=> ( ( plus_plus_int @ A2 @ B2 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add1
thf(fact_253_group__cancel_Oadd2,axiom,
! [B3: nat,K: nat,B2: nat,A: nat] :
( ( B3
= ( plus_plus_nat @ K @ B2 ) )
=> ( ( plus_plus_nat @ A @ B3 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_254_group__cancel_Oadd2,axiom,
! [B3: int,K: int,B2: int,A: int] :
( ( B3
= ( plus_plus_int @ K @ B2 ) )
=> ( ( plus_plus_int @ A @ B3 )
= ( plus_plus_int @ K @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% group_cancel.add2
thf(fact_255_add_Oassoc,axiom,
! [A: nat,B2: nat,C: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B2 ) @ C )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_256_add_Oassoc,axiom,
! [A: int,B2: int,C: int] :
( ( plus_plus_int @ ( plus_plus_int @ A @ B2 ) @ C )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.assoc
thf(fact_257_add_Oleft__cancel,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
= ( B2 = C ) ) ).
% add.left_cancel
thf(fact_258_add_Oright__cancel,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
= ( B2 = C ) ) ).
% add.right_cancel
thf(fact_259_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A4: nat,B4: nat] : ( plus_plus_nat @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_260_add_Ocommute,axiom,
( plus_plus_int
= ( ^ [A4: int,B4: int] : ( plus_plus_int @ B4 @ A4 ) ) ) ).
% add.commute
thf(fact_261_add_Oleft__commute,axiom,
! [B2: nat,A: nat,C: nat] :
( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A @ C ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_262_add_Oleft__commute,axiom,
! [B2: int,A: int,C: int] :
( ( plus_plus_int @ B2 @ ( plus_plus_int @ A @ C ) )
= ( plus_plus_int @ A @ ( plus_plus_int @ B2 @ C ) ) ) ).
% add.left_commute
thf(fact_263_add__left__imp__eq,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ( plus_plus_nat @ A @ B2 )
= ( plus_plus_nat @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_264_add__left__imp__eq,axiom,
! [A: int,B2: int,C: int] :
( ( ( plus_plus_int @ A @ B2 )
= ( plus_plus_int @ A @ C ) )
=> ( B2 = C ) ) ).
% add_left_imp_eq
thf(fact_265_add__right__imp__eq,axiom,
! [B2: nat,A: nat,C: nat] :
( ( ( plus_plus_nat @ B2 @ A )
= ( plus_plus_nat @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_266_add__right__imp__eq,axiom,
! [B2: int,A: int,C: int] :
( ( ( plus_plus_int @ B2 @ A )
= ( plus_plus_int @ C @ A ) )
=> ( B2 = C ) ) ).
% add_right_imp_eq
thf(fact_267_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ zero_zero_nat ) ).
% of_nat_less_0_iff
thf(fact_268_of__nat__less__0__iff,axiom,
! [M: nat] :
~ ( ord_less_int @ ( semiri2019852685at_int @ M ) @ zero_zero_int ) ).
% of_nat_less_0_iff
thf(fact_269_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri1382578993at_nat @ ( suc @ N ) )
!= zero_zero_nat ) ).
% of_nat_neq_0
thf(fact_270_of__nat__neq__0,axiom,
! [N: nat] :
( ( semiri2019852685at_int @ ( suc @ N ) )
!= zero_zero_int ) ).
% of_nat_neq_0
thf(fact_271_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_272_less__imp__of__nat__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) ) ) ).
% less_imp_of_nat_less
thf(fact_273_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( semiri1382578993at_nat @ M ) @ ( semiri1382578993at_nat @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_274_of__nat__less__imp__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_int @ ( semiri2019852685at_int @ M ) @ ( semiri2019852685at_int @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% of_nat_less_imp_less
thf(fact_275_add__neg__neg,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ zero_zero_nat )
=> ( ( ord_less_nat @ B2 @ zero_zero_nat )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ B2 ) @ zero_zero_nat ) ) ) ).
% add_neg_neg
thf(fact_276_add__neg__neg,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ A @ zero_zero_int )
=> ( ( ord_less_int @ B2 @ zero_zero_int )
=> ( ord_less_int @ ( plus_plus_int @ A @ B2 ) @ zero_zero_int ) ) ) ).
% add_neg_neg
thf(fact_277_add__pos__pos,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ zero_zero_nat @ B2 )
=> ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_278_add__pos__pos,axiom,
! [A: int,B2: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ zero_zero_int @ B2 )
=> ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B2 ) ) ) ) ).
% add_pos_pos
thf(fact_279_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: nat,B2: nat] :
( ( ord_less_nat @ A @ B2 )
=> ~ ! [C2: nat] :
( ( B2
= ( plus_plus_nat @ A @ C2 ) )
=> ( C2 = zero_zero_nat ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_280_pos__add__strict,axiom,
! [A: nat,B2: nat,C: nat] :
( ( ord_less_nat @ zero_zero_nat @ A )
=> ( ( ord_less_nat @ B2 @ C )
=> ( ord_less_nat @ B2 @ ( plus_plus_nat @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_281_pos__add__strict,axiom,
! [A: int,B2: int,C: int] :
( ( ord_less_int @ zero_zero_int @ A )
=> ( ( ord_less_int @ B2 @ C )
=> ( ord_less_int @ B2 @ ( plus_plus_int @ A @ C ) ) ) ) ).
% pos_add_strict
thf(fact_282_nth__via__drop,axiom,
! [N: nat,Xs: list_b,Y3: b,Ys: list_b] :
( ( ( drop_b @ N @ Xs )
= ( cons_b @ Y3 @ Ys ) )
=> ( ( nth_b @ Xs @ N )
= Y3 ) ) ).
% nth_via_drop
thf(fact_283_nth__via__drop,axiom,
! [N: nat,Xs: list_option_b,Y3: option_b,Ys: list_option_b] :
( ( ( drop_option_b @ N @ Xs )
= ( cons_option_b @ Y3 @ Ys ) )
=> ( ( nth_option_b @ Xs @ N )
= Y3 ) ) ).
% nth_via_drop
thf(fact_284_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_285_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_286_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K2: nat] :
( ( ord_less_nat @ zero_zero_nat @ K2 )
& ( ( plus_plus_nat @ I @ K2 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_287_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ~ ! [Q2: nat] :
( N
!= ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).
% less_natE
thf(fact_288_less__add__Suc1,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).
% less_add_Suc1
thf(fact_289_less__add__Suc2,axiom,
! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).
% less_add_Suc2
thf(fact_290_less__iff__Suc__add,axiom,
( ord_less_nat
= ( ^ [M5: nat,N4: nat] :
? [K3: nat] :
( N4
= ( suc @ ( plus_plus_nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_291_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_292_gen__length__def,axiom,
( gen_length_b
= ( ^ [N4: nat,Xs3: list_b] : ( plus_plus_nat @ N4 @ ( size_size_list_b @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_293_gen__length__def,axiom,
( gen_length_option_b
= ( ^ [N4: nat,Xs3: list_option_b] : ( plus_plus_nat @ N4 @ ( size_s1671393719tion_b @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_294_list_Osize_I4_J,axiom,
! [X21: b,X22: list_b] :
( ( size_size_list_b @ ( cons_b @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_size_list_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_295_list_Osize_I4_J,axiom,
! [X21: option_b,X22: list_option_b] :
( ( size_s1671393719tion_b @ ( cons_option_b @ X21 @ X22 ) )
= ( plus_plus_nat @ ( size_s1671393719tion_b @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).
% list.size(4)
thf(fact_296_add__less__zeroD,axiom,
! [X: int,Y3: int] :
( ( ord_less_int @ ( plus_plus_int @ X @ Y3 ) @ zero_zero_int )
=> ( ( ord_less_int @ X @ zero_zero_int )
| ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).
% add_less_zeroD
thf(fact_297_Euclid__induct,axiom,
! [P: nat > nat > $o,A: nat,B2: nat] :
( ! [A3: nat,B: nat] :
( ( P @ A3 @ B )
= ( P @ B @ A3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ zero_zero_nat )
=> ( ! [A3: nat,B: nat] :
( ( P @ A3 @ B )
=> ( P @ A3 @ ( plus_plus_nat @ A3 @ B ) ) )
=> ( P @ A @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_298_pos__int__cases,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiri2019852685at_int @ N2 ) )
=> ~ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% pos_int_cases
thf(fact_299_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_int @ zero_zero_int @ K )
=> ? [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
& ( K
= ( semiri2019852685at_int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_300_zadd__int__left,axiom,
! [M: nat,N: nat,Z2: int] :
( ( plus_plus_int @ ( semiri2019852685at_int @ M ) @ ( plus_plus_int @ ( semiri2019852685at_int @ N ) @ Z2 ) )
= ( plus_plus_int @ ( semiri2019852685at_int @ ( plus_plus_nat @ M @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_301_int__int__eq,axiom,
! [M: nat,N: nat] :
( ( ( semiri2019852685at_int @ M )
= ( semiri2019852685at_int @ N ) )
= ( M = N ) ) ).
% int_int_eq
thf(fact_302_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus_int @ zero_zero_int @ L )
= L ) ).
% plus_int_code(2)
thf(fact_303_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus_int @ K @ zero_zero_int )
= K ) ).
% plus_int_code(1)
thf(fact_304_zless__iff__Suc__zadd,axiom,
( ord_less_int
= ( ^ [W: int,Z3: int] :
? [N4: nat] :
( Z3
= ( plus_plus_int @ W @ ( semiri2019852685at_int @ ( suc @ N4 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_305_less__int__code_I1_J,axiom,
~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).
% less_int_code(1)
thf(fact_306_linorder__neqE__linordered__idom,axiom,
! [X: int,Y3: int] :
( ( X != Y3 )
=> ( ~ ( ord_less_int @ X @ Y3 )
=> ( ord_less_int @ Y3 @ X ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_307_nat__int__comparison_I2_J,axiom,
( ord_less_nat
= ( ^ [A4: nat,B4: nat] : ( ord_less_int @ ( semiri2019852685at_int @ A4 ) @ ( semiri2019852685at_int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_308_int__ops_I1_J,axiom,
( ( semiri2019852685at_int @ zero_zero_nat )
= zero_zero_int ) ).
% int_ops(1)
thf(fact_309_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_310_verit__comp__simplify1_I1_J,axiom,
! [A: int] :
~ ( ord_less_int @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_311_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z: nat] : Y5 = Z )
= ( ^ [A4: nat,B4: nat] :
( ( semiri2019852685at_int @ A4 )
= ( semiri2019852685at_int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_312_int__if,axiom,
! [P: $o,A: nat,B2: nat] :
( ( P
=> ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri2019852685at_int @ A ) ) )
& ( ~ P
=> ( ( semiri2019852685at_int @ ( if_nat @ P @ A @ B2 ) )
= ( semiri2019852685at_int @ B2 ) ) ) ) ).
% int_if
thf(fact_313_verit__sum__simplify,axiom,
! [A: nat] :
( ( plus_plus_nat @ A @ zero_zero_nat )
= A ) ).
% verit_sum_simplify
thf(fact_314_verit__sum__simplify,axiom,
! [A: int] :
( ( plus_plus_int @ A @ zero_zero_int )
= A ) ).
% verit_sum_simplify
thf(fact_315_int__ops_I5_J,axiom,
! [A: nat,B2: nat] :
( ( semiri2019852685at_int @ ( plus_plus_nat @ A @ B2 ) )
= ( plus_plus_int @ ( semiri2019852685at_int @ A ) @ ( semiri2019852685at_int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_316_int__plus,axiom,
! [N: nat,M: nat] :
( ( semiri2019852685at_int @ ( plus_plus_nat @ N @ M ) )
= ( plus_plus_int @ ( semiri2019852685at_int @ N ) @ ( semiri2019852685at_int @ M ) ) ) ).
% int_plus
thf(fact_317_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus_nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_318_add__0__iff,axiom,
! [B2: nat,A: nat] :
( ( B2
= ( plus_plus_nat @ B2 @ A ) )
= ( A = zero_zero_nat ) ) ).
% add_0_iff
thf(fact_319_add__0__iff,axiom,
! [B2: int,A: int] :
( ( B2
= ( plus_plus_int @ B2 @ A ) )
= ( A = zero_zero_int ) ) ).
% add_0_iff
thf(fact_320_triangle__0,axiom,
( ( nat_triangle @ zero_zero_nat )
= zero_zero_nat ) ).
% triangle_0
thf(fact_321_nth__drop,axiom,
! [N: nat,Xs: list_b,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_b @ Xs ) )
=> ( ( nth_b @ ( drop_b @ N @ Xs ) @ I )
= ( nth_b @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_322_nth__drop,axiom,
! [N: nat,Xs: list_option_b,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_s1671393719tion_b @ Xs ) )
=> ( ( nth_option_b @ ( drop_option_b @ N @ Xs ) @ I )
= ( nth_option_b @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
% Helper facts (3)
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,Y3: nat] :
( ( if_nat @ $false @ X @ Y3 )
= Y3 ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y3: nat] :
( ( if_nat @ $true @ X @ Y3 )
= X ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
member_nat @ zero_zero_nat @ ( fvi_a @ b2 @ phi ) ).
thf(conj_1,hypothesis,
( ( ( size_s1671393719tion_b @ v )
= n )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ n )
=> ( ( ( nth_option_b @ v @ I4 )
= none_b )
= ( ~ ( member_nat @ I4 @ ( fvi_a @ ( suc @ b2 ) @ phi ) ) ) ) ) ) ).
thf(conj_2,conjecture,
( ( ( size_s1671393719tion_b @ ( cons_option_b @ ( some_b @ x ) @ v ) )
= ( suc @ n ) )
& ! [I2: nat] :
( ~ ( ord_less_nat @ I2 @ ( suc @ n ) )
| ( ( ( nth_option_b @ ( cons_option_b @ ( some_b @ x ) @ v ) @ I2 )
= none_b )
= ( ~ ( member_nat @ I2 @ ( fvi_a @ b2 @ phi ) ) ) ) ) ) ).
%------------------------------------------------------------------------------