TPTP Problem File: ITP099^1.p
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%------------------------------------------------------------------------------
% File : ITP099^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_412__5414926_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 : ListInf/prob_412__5414926_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 286 ( 89 unt; 33 typ; 0 def)
% Number of atoms : 736 ( 272 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2212 ( 82 ~; 25 |; 37 &;1727 @)
% ( 0 <=>; 341 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 7 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 134 ( 134 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 29 usr; 4 con; 0-3 aty)
% Number of variables : 781 ( 70 ^; 671 !; 40 ?; 781 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:11.229
%------------------------------------------------------------------------------
% Could-be-implicit typings (4)
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (29)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_If_001tf__a,type,
if_a: $o > a > a > a ).
thf(sy_c_List2_Olist__asc_001t__Nat__Onat,type,
list_asc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__desc_001t__Nat__Onat,type,
list_desc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__strict__asc_001t__Nat__Onat,type,
list_strict_asc_nat: list_nat > $o ).
thf(sy_c_List2_Olist__strict__desc_001t__Nat__Onat,type,
list_strict_desc_nat: list_nat > $o ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001t__Nat__Onat,type,
listIn923761578nd_nat: list_nat > ( nat > nat ) > nat > nat ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__append_001tf__a,type,
listIn1312259492pend_a: list_a > ( nat > a ) > nat > a ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001t__Nat__Onat,type,
listIn870223551op_nat: nat > ( nat > nat ) > nat > nat ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__drop_001tf__a,type,
listIn1417627087drop_a: nat > ( nat > a ) > nat > a ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001t__Nat__Onat,type,
listIn1299132951ke_nat: nat > ( nat > nat ) > list_nat ).
thf(sy_c_ListInf__Mirabelle__akbajwqfbr_Oi__take_001tf__a,type,
listIn84854903take_a: nat > ( nat > a ) > list_a ).
thf(sy_c_List_Oappend_001t__Nat__Onat,type,
append_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Oappend_001tf__a,type,
append_a: list_a > list_a > list_a ).
thf(sy_c_List_Odrop_001t__Nat__Onat,type,
drop_nat: nat > list_nat > list_nat ).
thf(sy_c_List_Odrop_001tf__a,type,
drop_a: nat > list_a > list_a ).
thf(sy_c_List_Olast_001t__Nat__Onat,type,
last_nat: list_nat > nat ).
thf(sy_c_List_Olast_001tf__a,type,
last_a: list_a > a ).
thf(sy_c_List_Onth_001t__Nat__Onat,type,
nth_nat: list_nat > nat > nat ).
thf(sy_c_List_Onth_001tf__a,type,
nth_a: list_a > nat > a ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_Itf__a_J,type,
size_size_list_a: list_a > nat ).
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_v_f,type,
f: nat > a ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_xs,type,
xs: list_a ).
% Relevant facts (247)
thf(fact_0_same__i__append__eq,axiom,
! [Xs: list_a,F: nat > a,G: nat > a] :
( ( ( listIn1312259492pend_a @ Xs @ F )
= ( listIn1312259492pend_a @ Xs @ G ) )
= ( F = G ) ) ).
% same_i_append_eq
thf(fact_1_i__append__eq__i__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ( ( listIn923761578nd_nat @ Xs @ F )
= ( listIn923761578nd_nat @ Ys @ G ) )
= ( ( Xs = Ys )
& ( F = G ) ) ) ) ).
% i_append_eq_i_append_conv
thf(fact_2_i__append__eq__i__append__conv,axiom,
! [Xs: list_a,Ys: list_a,F: nat > a,G: nat > a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ( ( listIn1312259492pend_a @ Xs @ F )
= ( listIn1312259492pend_a @ Ys @ G ) )
= ( ( Xs = Ys )
& ( F = G ) ) ) ) ).
% i_append_eq_i_append_conv
thf(fact_3_length__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).
% length_drop
thf(fact_4_length__drop,axiom,
! [N: nat,Xs: list_a] :
( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
= ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).
% length_drop
thf(fact_5_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_6_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_7_less__diff__imp__less,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
=> ( ord_less_nat @ I @ J ) ) ).
% less_diff_imp_less
thf(fact_8_nat__diff__left__cancel__eq1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( minus_minus_nat @ K @ M )
= ( minus_minus_nat @ K @ N ) )
=> ( ( ord_less_nat @ M @ K )
=> ( M = N ) ) ) ).
% nat_diff_left_cancel_eq1
thf(fact_9_nat__diff__left__cancel__eq2,axiom,
! [K: nat,M: nat,N: nat] :
( ( ( minus_minus_nat @ K @ M )
= ( minus_minus_nat @ K @ N ) )
=> ( ( ord_less_nat @ N @ K )
=> ( M = N ) ) ) ).
% nat_diff_left_cancel_eq2
thf(fact_10_nat__diff__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
=> ( ord_less_nat @ N @ M ) ) ).
% nat_diff_left_cancel_less
thf(fact_11_nat__diff__right__cancel__eq1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
=> ( ( ord_less_nat @ K @ M )
=> ( M = N ) ) ) ).
% nat_diff_right_cancel_eq1
thf(fact_12_nat__diff__right__cancel__eq2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
=> ( ( ord_less_nat @ K @ N )
=> ( M = N ) ) ) ).
% nat_diff_right_cancel_eq2
thf(fact_13_nat__diff__right__cancel__less,axiom,
! [N: nat,K: nat,M: nat] :
( ( ord_less_nat @ ( minus_minus_nat @ N @ K ) @ ( minus_minus_nat @ M @ K ) )
=> ( ord_less_nat @ N @ M ) ) ).
% nat_diff_right_cancel_less
thf(fact_14_length__induct,axiom,
! [P: list_nat > $o,Xs: list_nat] :
( ! [Xs2: list_nat] :
( ! [Ys2: list_nat] :
( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_15_length__induct,axiom,
! [P: list_a > $o,Xs: list_a] :
( ! [Xs2: list_a] :
( ! [Ys2: list_a] :
( ( ord_less_nat @ ( size_size_list_a @ Ys2 ) @ ( size_size_list_a @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_16_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_17_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_18_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_19_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_20_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_21_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_22_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_23_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_24_size__neq__size__imp__neq,axiom,
! [X: list_nat,Y: list_nat] :
( ( ( size_size_list_nat @ X )
!= ( size_size_list_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_25_size__neq__size__imp__neq,axiom,
! [X: list_a,Y: list_a] :
( ( ( size_size_list_a @ X )
!= ( size_size_list_a @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_26_neq__if__length__neq,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
!= ( size_size_list_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_27_neq__if__length__neq,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
!= ( size_size_list_a @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_28_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_nat] :
( ( size_size_list_nat @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_29_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs2: list_a] :
( ( size_size_list_a @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_30_eq__imp__diff__eq,axiom,
! [M: nat,N: nat,K: nat] :
( ( M = N )
=> ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) ) ) ).
% eq_imp_diff_eq
thf(fact_31_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_32_i__append__i__drop__eq1,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
= ( listIn923761578nd_nat @ ( drop_nat @ N @ Xs ) @ F ) ) ) ).
% i_append_i_drop_eq1
thf(fact_33_i__append__i__drop__eq1,axiom,
! [N: nat,Xs: list_a,F: nat > a] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
= ( listIn1312259492pend_a @ ( drop_a @ N @ Xs ) @ F ) ) ) ).
% i_append_i_drop_eq1
thf(fact_34_i__append__i__drop__eq2,axiom,
! [Xs: list_nat,N: nat,F: nat > nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( listIn870223551op_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
= ( listIn870223551op_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ).
% i_append_i_drop_eq2
thf(fact_35_i__append__i__drop__eq2,axiom,
! [Xs: list_a,N: nat,F: nat > a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( listIn1417627087drop_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ).
% i_append_i_drop_eq2
thf(fact_36_last__drop,axiom,
! [N: nat,Xs: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( last_nat @ ( drop_nat @ N @ Xs ) )
= ( last_nat @ Xs ) ) ) ).
% last_drop
thf(fact_37_last__drop,axiom,
! [N: nat,Xs: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( last_a @ ( drop_a @ N @ Xs ) )
= ( last_a @ Xs ) ) ) ).
% last_drop
thf(fact_38_i__append__nth2,axiom,
! [Xs: list_nat,N: nat,F: nat > nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( listIn923761578nd_nat @ Xs @ F @ N )
= ( F @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% i_append_nth2
thf(fact_39_i__append__nth2,axiom,
! [Xs: list_a,N: nat,F: nat > a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( listIn1312259492pend_a @ Xs @ F @ N )
= ( F @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% i_append_nth2
thf(fact_40_i__append__nth,axiom,
( listIn923761578nd_nat
= ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% i_append_nth
thf(fact_41_i__append__nth,axiom,
( listIn1312259492pend_a
= ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% i_append_nth
thf(fact_42_i__append__def,axiom,
( listIn923761578nd_nat
= ( ^ [Xs3: list_nat,F2: nat > nat,N3: nat] : ( if_nat @ ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) @ ( nth_nat @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).
% i_append_def
thf(fact_43_i__append__def,axiom,
( listIn1312259492pend_a
= ( ^ [Xs3: list_a,F2: nat > a,N3: nat] : ( if_a @ ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) @ ( nth_a @ Xs3 @ N3 ) @ ( F2 @ ( minus_minus_nat @ N3 @ ( size_size_list_a @ Xs3 ) ) ) ) ) ) ).
% i_append_def
thf(fact_44_drop__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
= ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_45_drop__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
= ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_46_append_Oassoc,axiom,
! [A: list_a,B: list_a,C: list_a] :
( ( append_a @ ( append_a @ A @ B ) @ C )
= ( append_a @ A @ ( append_a @ B @ C ) ) ) ).
% append.assoc
thf(fact_47_append__assoc,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( append_a @ ( append_a @ Xs @ Ys ) @ Zs )
= ( append_a @ Xs @ ( append_a @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_48_append__same__eq,axiom,
! [Ys: list_a,Xs: list_a,Zs: list_a] :
( ( ( append_a @ Ys @ Xs )
= ( append_a @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_49_same__append__eq,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_50_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_51_append__eq__append__conv,axiom,
! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
( ( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
| ( ( size_size_list_nat @ Us )
= ( size_size_list_nat @ Vs ) ) )
=> ( ( ( append_nat @ Xs @ Us )
= ( append_nat @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_52_append__eq__append__conv,axiom,
! [Xs: list_a,Ys: list_a,Us: list_a,Vs: list_a] :
( ( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
| ( ( size_size_list_a @ Us )
= ( size_size_list_a @ Vs ) ) )
=> ( ( ( append_a @ Xs @ Us )
= ( append_a @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_53_i__append__assoc,axiom,
! [Xs: list_a,Ys: list_a,F: nat > a] :
( ( listIn1312259492pend_a @ Xs @ ( listIn1312259492pend_a @ Ys @ F ) )
= ( listIn1312259492pend_a @ ( append_a @ Xs @ Ys ) @ F ) ) ).
% i_append_assoc
thf(fact_54_i__append__nth1,axiom,
! [N: nat,Xs: list_nat,F: nat > nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( listIn923761578nd_nat @ Xs @ F @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% i_append_nth1
thf(fact_55_i__append__nth1,axiom,
! [N: nat,Xs: list_a,F: nat > a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( listIn1312259492pend_a @ Xs @ F @ N )
= ( nth_a @ Xs @ N ) ) ) ).
% i_append_nth1
thf(fact_56_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_57_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_58_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_59_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_60_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_61_append__eq__appendI,axiom,
! [Xs: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
( ( ( append_a @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append_a @ Xs1 @ Us ) )
=> ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_62_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_63_append__eq__append__conv2,axiom,
! [Xs: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
( ( ( append_a @ Xs @ Ys )
= ( append_a @ Zs @ Ts ) )
= ( ? [Us2: list_a] :
( ( ( Xs
= ( append_a @ Zs @ Us2 ) )
& ( ( append_a @ Us2 @ Ys )
= Ts ) )
| ( ( ( append_a @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_64_i__append__eq__i__append__conv2__aux,axiom,
! [Xs: list_nat,F: nat > nat,Ys: list_nat,G: nat > nat] :
( ( ( listIn923761578nd_nat @ Xs @ F )
= ( listIn923761578nd_nat @ Ys @ G ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
=> ? [Zs2: list_nat] :
( ( ( append_nat @ Xs @ Zs2 )
= Ys )
& ( F
= ( listIn923761578nd_nat @ Zs2 @ G ) ) ) ) ) ).
% i_append_eq_i_append_conv2_aux
thf(fact_65_i__append__eq__i__append__conv2__aux,axiom,
! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
( ( ( listIn1312259492pend_a @ Xs @ F )
= ( listIn1312259492pend_a @ Ys @ G ) )
=> ( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) )
=> ? [Zs2: list_a] :
( ( ( append_a @ Xs @ Zs2 )
= Ys )
& ( F
= ( listIn1312259492pend_a @ Zs2 @ G ) ) ) ) ) ).
% i_append_eq_i_append_conv2_aux
thf(fact_66_ge__less__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,N3: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ A2 )
=> ( N3 != X3 ) ) ) ) ).
% ge_less_neq_conv
thf(fact_67_less__ge__neq__conv,axiom,
( ord_less_nat
= ( ^ [N3: nat,A2: nat] :
! [X3: nat] :
( ( ord_less_eq_nat @ A2 @ X3 )
=> ( N3 != X3 ) ) ) ) ).
% less_ge_neq_conv
thf(fact_68_greater__le__neq__conv,axiom,
( ord_less_nat
= ( ^ [A2: nat,N3: nat] :
! [X3: nat] :
( ( ord_less_eq_nat @ X3 @ A2 )
=> ( N3 != X3 ) ) ) ) ).
% greater_le_neq_conv
thf(fact_69_le__greater__neq__conv,axiom,
( ord_less_eq_nat
= ( ^ [N3: nat,A2: nat] :
! [X3: nat] :
( ( ord_less_nat @ A2 @ X3 )
=> ( N3 != X3 ) ) ) ) ).
% le_greater_neq_conv
thf(fact_70_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_71_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_72_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_73_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_74_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_75_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_76_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_77_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_78_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_79_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_80_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_81_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_82_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_83_le__diff__swap,axiom,
! [I: nat,K: nat,J: nat] :
( ( ord_less_eq_nat @ I @ K )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ J ) @ I )
= ( ord_less_eq_nat @ ( minus_minus_nat @ K @ I ) @ J ) ) ) ) ).
% le_diff_swap
thf(fact_84_le__diff__imp__le,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
=> ( ord_less_eq_nat @ I @ J ) ) ).
% le_diff_imp_le
thf(fact_85_le__imp__diff__le,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% le_imp_diff_le
thf(fact_86_eq__diff__left__iff,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ M @ K )
=> ( ( ord_less_eq_nat @ N @ K )
=> ( ( ( minus_minus_nat @ K @ M )
= ( minus_minus_nat @ K @ N ) )
= ( M = N ) ) ) ) ).
% eq_diff_left_iff
thf(fact_87_le__diff__le__imp__le,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ M ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% le_diff_le_imp_le
thf(fact_88_nat__diff__left__cancel__le2,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
=> ( ( ord_less_eq_nat @ N @ K )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% nat_diff_left_cancel_le2
thf(fact_89_nat__diff__right__cancel__le2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_diff_right_cancel_le2
thf(fact_90_i__append__eq__i__appendI,axiom,
! [Xs: list_a,Xs4: list_a,Ys: list_a,F: nat > a,G: nat > a] :
( ( ( append_a @ Xs @ Xs4 )
= Ys )
=> ( ( F
= ( listIn1312259492pend_a @ Xs4 @ G ) )
=> ( ( listIn1312259492pend_a @ Xs @ F )
= ( listIn1312259492pend_a @ Ys @ G ) ) ) ) ).
% i_append_eq_i_appendI
thf(fact_91_i__append__eq__i__append__conv2,axiom,
! [Xs: list_a,F: nat > a,Ys: list_a,G: nat > a] :
( ( ( listIn1312259492pend_a @ Xs @ F )
= ( listIn1312259492pend_a @ Ys @ G ) )
= ( ? [Zs3: list_a] :
( ( ( Xs
= ( append_a @ Ys @ Zs3 ) )
& ( ( listIn1312259492pend_a @ Zs3 @ F )
= G ) )
| ( ( ( append_a @ Xs @ Zs3 )
= Ys )
& ( F
= ( listIn1312259492pend_a @ Zs3 @ G ) ) ) ) ) ) ).
% i_append_eq_i_append_conv2
thf(fact_92_nth__append,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_93_nth__append,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) )
& ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_94_nth__equalityI,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( ( size_size_list_nat @ Xs )
= ( size_size_list_nat @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ Xs @ I2 )
= ( nth_nat @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_95_nth__equalityI,axiom,
! [Xs: list_a,Ys: list_a] :
( ( ( size_size_list_a @ Xs )
= ( size_size_list_a @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ Xs @ I2 )
= ( nth_a @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_96_Skolem__list__nth,axiom,
! [K: nat,P: nat > nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: nat] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_97_Skolem__list__nth,axiom,
! [K: nat,P: nat > a > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: a] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_a @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_98_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_nat,Z: list_nat] : Y4 = Z )
= ( ^ [Xs3: list_nat,Ys3: list_nat] :
( ( ( size_size_list_nat @ Xs3 )
= ( size_size_list_nat @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
=> ( ( nth_nat @ Xs3 @ I3 )
= ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_99_list__eq__iff__nth__eq,axiom,
( ( ^ [Y4: list_a,Z: list_a] : Y4 = Z )
= ( ^ [Xs3: list_a,Ys3: list_a] :
( ( ( size_size_list_a @ Xs3 )
= ( size_size_list_a @ Ys3 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs3 ) )
=> ( ( nth_a @ Xs3 @ I3 )
= ( nth_a @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_100_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_101_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_102_less__diff__le__imp__less,axiom,
! [I: nat,J: nat,M: nat,N: nat] :
( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ M ) )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( ord_less_nat @ I @ ( minus_minus_nat @ J @ N ) ) ) ) ).
% less_diff_le_imp_less
thf(fact_103_nat__diff__left__cancel__le1,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ K @ M ) @ ( minus_minus_nat @ K @ N ) )
=> ( ( ord_less_nat @ M @ K )
=> ( ord_less_eq_nat @ N @ M ) ) ) ).
% nat_diff_left_cancel_le1
thf(fact_104_nat__diff__right__cancel__le1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
=> ( ( ord_less_nat @ K @ M )
=> ( ord_less_eq_nat @ M @ N ) ) ) ).
% nat_diff_right_cancel_le1
thf(fact_105_i__drop__nth__sub,axiom,
! [N: nat,X: nat,S: nat > a] :
( ( ord_less_eq_nat @ N @ X )
=> ( ( listIn1417627087drop_a @ N @ S @ ( minus_minus_nat @ X @ N ) )
= ( S @ X ) ) ) ).
% i_drop_nth_sub
thf(fact_106_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_107_nth__append2,axiom,
! [Xs: list_nat,N: nat,Ys: list_nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ).
% nth_append2
thf(fact_108_nth__append2,axiom,
! [Xs: list_a,N: nat,Ys: list_a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ).
% nth_append2
thf(fact_109_nth__append1,axiom,
! [N: nat,Xs: list_nat,Ys: list_nat] :
( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
= ( nth_nat @ Xs @ N ) ) ) ).
% nth_append1
thf(fact_110_nth__append1,axiom,
! [N: nat,Xs: list_a,Ys: list_a] :
( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
= ( nth_a @ Xs @ N ) ) ) ).
% nth_append1
thf(fact_111_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_112_list__desc__trans__le,axiom,
( list_desc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% list_desc_trans_le
thf(fact_113_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_114_list__asc__trans__le,axiom,
( list_asc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_eq_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% list_asc_trans_le
thf(fact_115_list__strict__asc__trans__le,axiom,
! [Xs: list_nat] :
( ( list_strict_asc_nat @ Xs )
=> ! [J4: nat] :
( ( ord_less_nat @ J4 @ ( size_size_list_nat @ Xs ) )
=> ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ J4 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs @ I4 ) @ ( nth_nat @ Xs @ J4 ) ) ) ) ) ).
% list_strict_asc_trans_le
thf(fact_116_list__strict__asc__imp__list__asc,axiom,
! [Xs: list_nat] :
( ( list_strict_asc_nat @ Xs )
=> ( list_asc_nat @ Xs ) ) ).
% list_strict_asc_imp_list_asc
thf(fact_117_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_118_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_119_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_120_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_121_eq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_122_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_123_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_124_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_125_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_126_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_127_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_128_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_129_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_130_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_131_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_132_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_133_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_134_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_135_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_136_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_137_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_138_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_139_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_140_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_141_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_142_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_143_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_144_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X4: nat] : ( P2 @ X4 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P3 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_145_less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_146_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_147_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_148_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_149_less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_150_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_151_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_152_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_153_less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_not_sym
thf(fact_154_less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_155_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_156_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_157_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_158_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_159_less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% less_linear
thf(fact_160_less__trans,axiom,
! [X: nat,Y: nat,Z2: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z2 )
=> ( ord_less_nat @ X @ Z2 ) ) ) ).
% less_trans
thf(fact_161_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_162_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_163_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_164_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_165_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_166_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_167_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_168_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_169_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_170_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_171_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_172_list__drop__eq__conv,axiom,
( ( ^ [Y4: list_a,Z: list_a] : Y4 = Z )
= ( ^ [Xs3: list_a,Ys3: list_a] :
! [N3: nat] :
( ( drop_a @ N3 @ Xs3 )
= ( drop_a @ N3 @ Ys3 ) ) ) ) ).
% list_drop_eq_conv
thf(fact_173_list__asc__trans,axiom,
( list_asc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_174_list__strict__asc__trans,axiom,
( list_strict_asc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Xs3 @ J3 ) ) ) ) ) ) ).
% list_strict_asc_trans
thf(fact_175_list__desc__trans,axiom,
( list_desc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_eq_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_176_order_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_177_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_178_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_179_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_180_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_181_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_182_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_183_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_184_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_185_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_186_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_187_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_188_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_189_le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_190_le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% le_less_linear
thf(fact_191_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 ) ) ) ).
% less_le_trans
thf(fact_192_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 ) ) ) ).
% le_less_trans
thf(fact_193_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_194_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_195_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_196_le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_197_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_198_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_199_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_200_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_201_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_202_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 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_203_less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_eq_nat @ X3 @ Y5 )
& ( X3 != Y5 ) ) ) ) ).
% less_le
thf(fact_204_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y5: nat] :
( ( ord_less_nat @ X3 @ Y5 )
| ( X3 = Y5 ) ) ) ) ).
% le_less
thf(fact_205_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_206_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_207_list__strict__desc__trans,axiom,
( list_strict_desc_nat
= ( ^ [Xs3: list_nat] :
! [J3: nat] :
( ( ord_less_nat @ J3 @ ( size_size_list_nat @ Xs3 ) )
=> ! [I3: nat] :
( ( ord_less_nat @ I3 @ J3 )
=> ( ord_less_nat @ ( nth_nat @ Xs3 @ J3 ) @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).
% list_strict_desc_trans
thf(fact_208_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 )
& ! [X5: nat] :
( ( ( ord_less_eq_nat @ A @ X5 )
& ( ord_less_nat @ X5 @ C2 ) )
=> ( P @ X5 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C2 ) ) ) ) ) ) ).
% complete_interval
thf(fact_209_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_210_pinf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_eq_nat @ X5 @ T ) ) ).
% pinf(6)
thf(fact_211_pinf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_eq_nat @ T @ X5 ) ) ).
% pinf(8)
thf(fact_212_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_213_minf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_nat @ T @ X5 ) ) ).
% minf(7)
thf(fact_214_minf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_nat @ X5 @ T ) ) ).
% minf(5)
thf(fact_215_minf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(4)
thf(fact_216_minf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( X5 != T ) ) ).
% minf(3)
thf(fact_217_minf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(2)
thf(fact_218_minf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% minf(1)
thf(fact_219_pinf_I7_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ord_less_nat @ T @ X5 ) ) ).
% pinf(7)
thf(fact_220_pinf_I5_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ~ ( ord_less_nat @ X5 @ T ) ) ).
% pinf(5)
thf(fact_221_pinf_I4_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(4)
thf(fact_222_pinf_I3_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( X5 != T ) ) ).
% pinf(3)
thf(fact_223_pinf_I2_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_224_pinf_I1_J,axiom,
! [P: nat > $o,P4: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P4 @ X2 ) ) )
=> ( ? [Z4: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_225_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_226_list__strict__desc__imp__list__desc,axiom,
! [Xs: list_nat] :
( ( list_strict_desc_nat @ Xs )
=> ( list_desc_nat @ Xs ) ) ).
% list_strict_desc_imp_list_desc
thf(fact_227_minf_I8_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ~ ( ord_less_eq_nat @ T @ X5 ) ) ).
% minf(8)
thf(fact_228_minf_I6_J,axiom,
! [T: nat] :
? [Z3: nat] :
! [X5: nat] :
( ( ord_less_nat @ X5 @ Z3 )
=> ( ord_less_eq_nat @ X5 @ T ) ) ).
% minf(6)
thf(fact_229_i__append__i__take__eq2,axiom,
! [Xs: list_nat,N: nat,F: nat > nat] :
( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ N )
=> ( ( listIn1299132951ke_nat @ N @ ( listIn923761578nd_nat @ Xs @ F ) )
= ( append_nat @ Xs @ ( listIn1299132951ke_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ F ) ) ) ) ).
% i_append_i_take_eq2
thf(fact_230_i__append__i__take__eq2,axiom,
! [Xs: list_a,N: nat,F: nat > a] :
( ( ord_less_eq_nat @ ( size_size_list_a @ Xs ) @ N )
=> ( ( listIn84854903take_a @ N @ ( listIn1312259492pend_a @ Xs @ F ) )
= ( append_a @ Xs @ ( listIn84854903take_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ F ) ) ) ) ).
% i_append_i_take_eq2
thf(fact_231_nth__drop,axiom,
! [N: nat,Xs: list_nat,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_nat @ Xs ) )
=> ( ( nth_nat @ ( drop_nat @ N @ Xs ) @ I )
= ( nth_nat @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_232_nth__drop,axiom,
! [N: nat,Xs: list_a,I: nat] :
( ( ord_less_eq_nat @ N @ ( size_size_list_a @ Xs ) )
=> ( ( nth_a @ ( drop_a @ N @ Xs ) @ I )
= ( nth_a @ Xs @ ( plus_plus_nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_233_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_234_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_235_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_236_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_237_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_238_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_239_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_240_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_241_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_242_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_243_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_244_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_245_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_246_drop__drop,axiom,
! [N: nat,M: nat,Xs: list_a] :
( ( drop_a @ N @ ( drop_a @ M @ Xs ) )
= ( drop_a @ ( plus_plus_nat @ N @ M ) @ Xs ) ) ).
% drop_drop
% Helper facts (5)
thf(help_If_2_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001tf__a_T,axiom,
! [X: a,Y: a] :
( ( if_a @ $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 (1)
thf(conj_0,conjecture,
( ( ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1312259492pend_a @ ( drop_a @ n @ xs ) @ f ) ) )
& ( ~ ( ord_less_nat @ n @ ( size_size_list_a @ xs ) )
=> ( ( listIn1417627087drop_a @ n @ ( listIn1312259492pend_a @ xs @ f ) )
= ( listIn1417627087drop_a @ ( minus_minus_nat @ n @ ( size_size_list_a @ xs ) ) @ f ) ) ) ) ).
%------------------------------------------------------------------------------