TPTP Problem File: ITP102^2.p
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%------------------------------------------------------------------------------
% File : ITP102^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer ListInf problem prob_77__5408624_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_77__5408624_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 308 ( 58 unt; 34 typ; 0 def)
% Number of atoms : 925 ( 201 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3253 ( 93 ~; 25 |; 45 &;2535 @)
% ( 0 <=>; 555 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 8 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 125 ( 125 >; 0 *; 0 +; 0 <<)
% Number of symbols : 33 ( 32 usr; 2 con; 0-4 aty)
% Number of variables : 992 ( 60 ^; 860 !; 46 ?; 992 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:15:42.791
%------------------------------------------------------------------------------
% Could-be-implicit typings (3)
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (31)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_List2_Olist__asc,type,
list_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__desc,type,
list_desc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__strict__asc,type,
list_strict_asc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List2_Olist__strict__desc,type,
list_strict_desc:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_ListInf__Mirabelle__nfmaokebij_Oi__append,type,
listIn521021761append:
!>[A: $tType] : ( ( list @ A ) > ( nat > A ) > nat > A ) ).
thf(sy_c_List_Olinorder__class_Osorted,type,
linorder_sorted:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_v_f,type,
f: nat > a ).
thf(sy_v_g,type,
g: nat > a ).
thf(sy_v_xs,type,
xs: list @ a ).
thf(sy_v_ys,type,
ys: list @ a ).
% Relevant facts (255)
thf(fact_0_i__append__eq__i__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,F: nat > A,G: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( listIn521021761append @ A @ Xs @ F )
= ( listIn521021761append @ A @ Ys @ G ) )
= ( ( Xs = Ys )
& ( F = G ) ) ) ) ).
% i_append_eq_i_append_conv
thf(fact_1_i__append__nth1,axiom,
! [A: $tType,N: nat,Xs: list @ A,F: nat > A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( listIn521021761append @ A @ Xs @ F @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% i_append_nth1
thf(fact_2_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Ys @ I ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_3_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X: A] : ( P @ I2 @ X ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_4_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y: list @ A,Z: list @ A] : Y = Z )
= ( ^ [Xs2: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I2 )
= ( nth @ A @ Ys2 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_5_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] : ( ord_less_eq @ A @ X2 @ X2 ) ) ).
% order_refl
thf(fact_6_list__desc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_desc_trans_le
thf(fact_7_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys3 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_8_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N2: nat] :
( ( ord_less_eq @ nat @ M @ N2 )
& ( M != N2 ) ) ) ) ).
% nat_less_le
thf(fact_9_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_10_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N2: nat] :
( ( ord_less @ nat @ M @ N2 )
| ( M = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_11_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_12_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_13_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I3: nat,J2: nat] :
( ! [I: nat,J3: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( ord_less @ nat @ ( F @ I ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ord_less_eq @ nat @ ( F @ I3 ) @ ( F @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_14_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( A2 = B ) ) ) ) ).
% dual_order.antisym
thf(fact_15_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_16_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_17_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ).
% linorder_wlog
thf(fact_18_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_19_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less_eq @ A @ X2 @ Z2 ) ) ) ) ).
% order_trans
thf(fact_20_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ A2 )
=> ( A2 = B ) ) ) ) ).
% order_class.order.antisym
thf(fact_21_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_22_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_23_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_24_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv
thf(fact_25_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ( ord_less_eq @ A @ X2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X2 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X2 ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_26_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_27_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% le_cases
thf(fact_28_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 = Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% eq_refl
thf(fact_29_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% linear
thf(fact_30_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X2 )
=> ( X2 = Y2 ) ) ) ) ).
% antisym
thf(fact_31_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y: A,Z: A] : Y = Z )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_32_ord__le__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B4,C: B4] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_33_ord__eq__le__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_34_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_35_order__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_36_le__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ( ( ord_less_eq @ ( A > B4 ) )
= ( ^ [F2: A > B4,G2: A > B4] :
! [X3: A] : ( ord_less_eq @ B4 @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_37_le__funI,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4] :
( ! [X4: A] : ( ord_less_eq @ B4 @ ( F @ X4 ) @ ( G @ X4 ) )
=> ( ord_less_eq @ ( A > B4 ) @ F @ G ) ) ) ).
% le_funI
thf(fact_38_le__funE,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4,X2: A] :
( ( ord_less_eq @ ( A > B4 ) @ F @ G )
=> ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funE
thf(fact_39_le__funD,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ! [F: A > B4,G: A > B4,X2: A] :
( ( ord_less_eq @ ( A > B4 ) @ F @ G )
=> ( ord_less_eq @ B4 @ ( F @ X2 ) @ ( G @ X2 ) ) ) ) ).
% le_funD
thf(fact_40_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( A2 != B ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_41_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( A2 != B ) ) ) ).
% order.strict_implies_not_eq
thf(fact_42_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X2 )
| ( X2 = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_43_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_44_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B: A] :
( ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B3: A] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A2 @ B ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_45_ext,axiom,
! [B4: $tType,A: $tType,F: A > B4,G: A > B4] :
( ! [X4: A] :
( ( F @ X4 )
= ( G @ X4 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X: A] : ( P2 @ X ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M: A] :
( ( ord_less @ A @ M @ N2 )
=> ~ ( P3 @ M ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_47_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_imp_not_less
thf(fact_48_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_49_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_50_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ( X2 != Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% linorder_cases
thf(fact_51_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,P: $o] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X2 )
=> P ) ) ) ).
% less_imp_triv
thf(fact_52_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( Y2 != X2 ) ) ) ).
% less_imp_not_eq2
thf(fact_53_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X2: A] :
( ~ ( ord_less @ A @ Y2 @ X2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_54_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_55_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_not_sym
thf(fact_56_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ) ).
% less_imp_not_eq
thf(fact_57_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ~ ( ord_less @ A @ A2 @ B ) ) ) ).
% dual_order.asym
thf(fact_58_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( B = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_59_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B: A,C: A] :
( ( A2 = B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_60_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A] :
~ ( ord_less @ A @ X2 @ X2 ) ) ).
% less_irrefl
thf(fact_61_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_linear
thf(fact_62_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% less_trans
thf(fact_63_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% less_asym'
thf(fact_64_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% less_asym
thf(fact_65_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( X2 != Y2 ) ) ) ).
% less_imp_neq
thf(fact_66_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ? [Z3: A] :
( ( ord_less @ A @ X2 @ Z3 )
& ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% dense
thf(fact_67_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ~ ( ord_less @ A @ B @ A2 ) ) ) ).
% order.asym
thf(fact_68_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
= ( ( ord_less @ A @ X2 @ Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% neq_iff
thf(fact_69_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less @ A @ Y2 @ X2 ) ) ) ) ).
% neqE
thf(fact_70_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X2: A] :
? [X_1: A] : ( ord_less @ A @ X2 @ X_1 ) ) ).
% gt_ex
thf(fact_71_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X2: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X2 ) ) ).
% lt_ex
thf(fact_72_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_73_order__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_74_ord__less__eq__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,B: A,F: A > B4,C: B4] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B4 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B4 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_75_ord__eq__less__subst,axiom,
! [A: $tType,B4: $tType] :
( ( ( ord @ B4 )
& ( ord @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( A2
= ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_76_measure__induct__rule,axiom,
! [B4: $tType,A: $tType] :
( ( wellorder @ B4 )
=> ! [F: A > B4,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_77_measure__induct,axiom,
! [B4: $tType,A: $tType] :
( ( wellorder @ B4 )
=> ! [F: A > B4,P: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ B4 @ ( F @ Y5 ) @ ( F @ X4 ) )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_78_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X2: A] :
( ! [X4: A] :
( ~ ( P @ X4 )
=> ? [Y5: A] :
( ( ord_less @ nat @ ( V @ Y5 ) @ ( V @ X4 ) )
& ~ ( P @ Y5 ) ) )
=> ( P @ X2 ) ) ).
% infinite_descent_measure
thf(fact_79_linorder__neqE__nat,axiom,
! [X2: nat,Y2: nat] :
( ( X2 != Y2 )
=> ( ~ ( ord_less @ nat @ X2 @ Y2 )
=> ( ord_less @ nat @ Y2 @ X2 ) ) ) ).
% linorder_neqE_nat
thf(fact_80_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P @ M3 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_81_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_82_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_83_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less @ nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_84_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_85_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_86_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_87_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y5: nat] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ Y5 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_88_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
| ( ord_less_eq @ nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_89_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_90_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_91_le__trans,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less_eq @ nat @ J2 @ K )
=> ( ord_less_eq @ nat @ I3 @ K ) ) ) ).
% le_trans
thf(fact_92_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_93_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X2: A,Y2: A] :
( ( ( size_size @ A @ X2 )
!= ( size_size @ A @ Y2 ) )
=> ( X2 != Y2 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_94_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_95_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_96_list__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_desc_trans
thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( A2 != B )
=> ( ( ord_less_eq @ A @ A2 @ B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_98_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_99_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_100_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_101_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ord_less_eq @ A @ A2 @ B ) ) ) ).
% order.strict_implies_order
thf(fact_102_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ! [W: A] :
( ( ord_less @ A @ X2 @ W )
=> ( ( ord_less @ A @ W @ Y2 )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_103_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X2: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X2 )
=> ( ord_less_eq @ A @ Y2 @ W ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_104_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less @ A @ B @ A2 )
=> ( ( ord_less_eq @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_105_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B @ A2 )
=> ( ( ord_less @ A @ C @ B )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_106_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_107_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_108_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_109_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ A @ B @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_110_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X2: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X2 )
=> ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_111_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ~ ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% less_le_not_le
thf(fact_112_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_113_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
| ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% le_less_linear
thf(fact_114_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z2: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y2 )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_115_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y2: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ Y2 @ X4 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_116_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_117_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X2 @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_118_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ X2 @ Y2 ) ) ) ).
% less_imp_le
thf(fact_119_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
=> ( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_120_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ( ord_less_eq @ A @ X2 @ Y2 )
= ( X2 = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_121_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B: A] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( A2 != B )
=> ( ord_less @ A @ A2 @ B ) ) ) ) ).
% le_neq_trans
thf(fact_122_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less @ A @ X2 @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% not_less
thf(fact_123_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X2 @ Y2 ) )
= ( ord_less @ A @ Y2 @ X2 ) ) ) ).
% not_le
thf(fact_124_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B )
=> ( ( ord_less_eq @ C2 @ ( F @ B ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_125_order__less__le__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( ord_less @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less_eq @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less_eq @ B4 @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_126_order__le__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 )
& ( order @ A ) )
=> ! [A2: A,B: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B )
=> ( ( ord_less @ C2 @ ( F @ B ) @ C )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_127_order__le__less__subst1,axiom,
! [A: $tType,B4: $tType] :
( ( ( order @ B4 )
& ( order @ A ) )
=> ! [A2: A,F: B4 > A,B: B4,C: B4] :
( ( ord_less_eq @ A @ A2 @ ( F @ B ) )
=> ( ( ord_less @ B4 @ B @ C )
=> ( ! [X4: B4,Y4: B4] :
( ( ord_less @ B4 @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_128_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( X3 != Y3 ) ) ) ) ) ).
% less_le
thf(fact_129_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
| ( X3 = Y3 ) ) ) ) ) ).
% le_less
thf(fact_130_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X2: A,Y2: A] :
( ~ ( ord_less @ A @ X2 @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ).
% leI
thf(fact_131_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X2: A] :
( ( ord_less_eq @ A @ Y2 @ X2 )
=> ~ ( ord_less @ A @ X2 @ Y2 ) ) ) ).
% leD
thf(fact_132_list__strict__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ! [J4: nat] :
( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ J4 )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% list_strict_asc_trans_le
thf(fact_133_list__asc__trans__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% list_asc_trans_le
thf(fact_134_list__strict__desc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_desc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ A @ ( nth @ A @ Xs2 @ J ) @ ( nth @ A @ Xs2 @ I2 ) ) ) ) ) ) ) ).
% list_strict_desc_trans
thf(fact_135_list__strict__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_strict_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% list_strict_asc_trans
thf(fact_136_list__asc__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( list_asc @ A )
= ( ^ [Xs2: list @ A] :
! [J: nat] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% list_asc_trans
thf(fact_137_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F: A > nat,B: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( F @ Y4 ) @ B ) )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( F @ Y5 ) @ ( F @ X4 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_138_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: 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 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_139_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B )
=> ( ( P @ A2 )
=> ( ~ ( P @ B )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D ) )
=> ( P @ X4 ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_140_le__greater__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [N2: A,A3: A] :
! [X3: A] :
( ( ord_less @ A @ A3 @ X3 )
=> ( N2 != X3 ) ) ) ) ) ).
% le_greater_neq_conv
thf(fact_141_greater__le__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,N2: A] :
! [X3: A] :
( ( ord_less_eq @ A @ X3 @ A3 )
=> ( N2 != X3 ) ) ) ) ) ).
% greater_le_neq_conv
thf(fact_142_less__ge__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [N2: A,A3: A] :
! [X3: A] :
( ( ord_less_eq @ A @ A3 @ X3 )
=> ( N2 != X3 ) ) ) ) ) ).
% less_ge_neq_conv
thf(fact_143_less__fun__def,axiom,
! [B4: $tType,A: $tType] :
( ( ord @ B4 )
=> ( ( ord_less @ ( A > B4 ) )
= ( ^ [F2: A > B4,G2: A > B4] :
( ( ord_less_eq @ ( A > B4 ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B4 ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_144_list__strict__asc__imp__list__asc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_asc @ A @ Xs )
=> ( list_asc @ A @ Xs ) ) ) ).
% list_strict_asc_imp_list_asc
thf(fact_145_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_146_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X4: A] :
( ( P @ X4 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ X4 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_147_list__strict__desc__imp__list__desc,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [Xs: list @ A] :
( ( list_strict_desc @ A @ Xs )
=> ( list_desc @ A @ Xs ) ) ) ).
% list_strict_desc_imp_list_desc
thf(fact_148_ge__less__neq__conv,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,N2: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ A3 )
=> ( N2 != X3 ) ) ) ) ) ).
% ge_less_neq_conv
thf(fact_149_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_150_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_151_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_152_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_153_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_154_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_155_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_156_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_157_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_158_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_159_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P4 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_160_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P4: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P @ X4 )
= ( P4 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q2 @ X4 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P4 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_161_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_162_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_163_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_164_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_165_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_166_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_167_verit__comp__simplify1_I3_J,axiom,
! [B4: $tType] :
( ( linorder @ B4 )
=> ! [B5: B4,A5: B4] :
( ( ~ ( ord_less_eq @ B4 @ B5 @ A5 ) )
= ( ord_less @ B4 @ A5 @ B5 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_168_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sorted @ A )
= ( ^ [Xs2: list @ A] :
! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_169_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I3: nat,J2: nat] :
( ( linorder_sorted @ A @ Xs )
=> ( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_170_list__ord__le__sorted__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( list_asc @ A )
= ( linorder_sorted @ A ) ) ) ).
% list_ord_le_sorted_eq
thf(fact_171_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B: A] :
( ( A2 = B )
| ~ ( ord_less_eq @ A @ A2 @ B )
| ~ ( ord_less_eq @ A @ B @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_172_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_173_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sorted @ A )
= ( ^ [Xs2: list @ A] :
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ J ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_174_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I3: nat,J2: nat] :
( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I3 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_175_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
= ( ! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_176_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linorder_sorted @ A )
= ( ^ [Xs2: list @ A] :
! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs2 @ I2 ) @ ( nth @ A @ Xs2 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_177_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_178_nat_Oinject,axiom,
! [X22: nat,Y22: nat] :
( ( ( suc @ X22 )
= ( suc @ Y22 ) )
= ( X22 = Y22 ) ) ).
% nat.inject
thf(fact_179_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_180_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_181_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_182_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_183_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_184_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_185_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_186_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_187_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_188_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_189_Suc__le__D,axiom,
! [N: nat,M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M4 )
=> ? [M5: nat] :
( M4
= ( suc @ M5 ) ) ) ).
% Suc_le_D
thf(fact_190_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_191_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_192_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M2 @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_193_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P @ M3 ) )
=> ( P @ N3 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_194_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_195_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y4: nat,Z3: nat] :
( ( R @ X4 @ Y4 )
=> ( ( R @ Y4 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_196_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_197_Suc__inject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
=> ( X2 = Y2 ) ) ).
% Suc_inject
thf(fact_198_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_199_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_200_strict__inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ! [I: nat] :
( ( J2
= ( suc @ I ) )
=> ( P @ I ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( P @ ( suc @ I ) )
=> ( P @ I ) ) )
=> ( P @ I3 ) ) ) ) ).
% strict_inc_induct
thf(fact_201_less__Suc__induct,axiom,
! [I3: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ! [I: nat] : ( P @ I @ ( suc @ I ) )
=> ( ! [I: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ( P @ I @ J3 )
=> ( ( P @ J3 @ K2 )
=> ( P @ I @ K2 ) ) ) ) )
=> ( P @ I3 @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_202_less__trans__Suc,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( suc @ I3 ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_203_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_204_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_205_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_206_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_207_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_208_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_209_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_210_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_211_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_212_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_213_Suc__lessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I3 ) @ K )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ).
% Suc_lessE
thf(fact_214_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_215_Nat_OlessE,axiom,
! [I3: nat,K: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( K
!= ( suc @ I3 ) )
=> ~ ! [J3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( K
!= ( suc @ J3 ) ) ) ) ) ).
% Nat.lessE
thf(fact_216_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F @ N ) @ ( F @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_217_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ord_less @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_218_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N4 ) @ ( F @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_219_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F: nat > A,N: nat,N4: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ord_less_eq @ A @ ( F @ N ) @ ( F @ N4 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_220_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_221_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_222_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_223_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_224_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_225_inc__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( P @ J2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J2 )
=> ( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) ) ) )
=> ( P @ I3 ) ) ) ) ).
% inc_induct
thf(fact_226_dec__induct,axiom,
! [I3: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( P @ I3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I3 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J2 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_227_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_228_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_229_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sorted @ A @ ( rev @ A @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_230_nat__induct_H,axiom,
! [P: nat > $o,N0: nat,N: nat] :
( ( P @ N0 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N0 @ N3 )
=> ( ( P @ N3 )
=> ( P @ ( suc @ N3 ) ) ) )
=> ( ( ord_less_eq @ nat @ N0 @ N )
=> ( P @ N ) ) ) ) ).
% nat_induct'
thf(fact_231_rev__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rev @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( suc @ N ) ) ) ) ) ).
% rev_nth
thf(fact_232_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P3: A > $o,Xs2: list @ A] :
? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs2 ) )
& ( P3 @ ( nth @ A @ Xs2 @ N2 ) ) ) ) ) ).
% list_ex_length
thf(fact_233_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_234_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_235_diff__diff__cancel,axiom,
! [I3: nat,N: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I3 ) )
= I3 ) ) ).
% diff_diff_cancel
thf(fact_236_list__ex__rev,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( list_ex @ A @ P @ ( rev @ A @ Xs ) )
= ( list_ex @ A @ P @ Xs ) ) ).
% list_ex_rev
thf(fact_237_i__append__nth2,axiom,
! [A: $tType,Xs: list @ A,N: nat,F: nat > A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N )
=> ( ( listIn521021761append @ A @ Xs @ F @ N )
= ( F @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% i_append_nth2
thf(fact_238_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I3: nat] :
( ( P @ K )
=> ( ! [N3: nat] :
( ( P @ ( suc @ N3 ) )
=> ( P @ N3 ) )
=> ( P @ ( minus_minus @ nat @ K @ I3 ) ) ) ) ).
% zero_induct_lemma
thf(fact_239_less__diff__imp__less,axiom,
! [I3: nat,J2: nat,M2: nat] :
( ( ord_less @ nat @ I3 @ ( minus_minus @ nat @ J2 @ M2 ) )
=> ( ord_less @ nat @ I3 @ J2 ) ) ).
% less_diff_imp_less
thf(fact_240_nat__diff__left__cancel__eq1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( minus_minus @ nat @ K @ M2 )
= ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ M2 @ K )
=> ( M2 = N ) ) ) ).
% nat_diff_left_cancel_eq1
thf(fact_241_nat__diff__left__cancel__eq2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( minus_minus @ nat @ K @ M2 )
= ( minus_minus @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ N @ K )
=> ( M2 = N ) ) ) ).
% nat_diff_left_cancel_eq2
thf(fact_242_nat__diff__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ K @ M2 ) @ ( minus_minus @ nat @ K @ N ) )
=> ( ord_less @ nat @ N @ M2 ) ) ).
% nat_diff_left_cancel_less
thf(fact_243_nat__diff__right__cancel__eq1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ K )
= ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less @ nat @ K @ M2 )
=> ( M2 = N ) ) ) ).
% nat_diff_right_cancel_eq1
thf(fact_244_nat__diff__right__cancel__eq2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ K )
= ( minus_minus @ nat @ N @ K ) )
=> ( ( ord_less @ nat @ K @ N )
=> ( M2 = N ) ) ) ).
% nat_diff_right_cancel_eq2
thf(fact_245_nat__diff__right__cancel__less,axiom,
! [N: nat,K: nat,M2: nat] :
( ( ord_less @ nat @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M2 @ K ) )
=> ( ord_less @ nat @ N @ M2 ) ) ).
% nat_diff_right_cancel_less
thf(fact_246_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_247_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_248_diff__commute,axiom,
! [I3: nat,J2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ J2 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I3 @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_249_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_250_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B: nat] :
( ( ord_less_eq @ nat @ A2 @ C )
=> ( ( ord_less_eq @ nat @ B @ C )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C @ A2 ) @ ( minus_minus @ nat @ C @ B ) )
= ( ord_less_eq @ nat @ B @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_251_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_252_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_253_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_254_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
% Type constructors (16)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A6: $tType,A7: $tType] :
( ( preorder @ A7 )
=> ( preorder @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A6: $tType,A7: $tType] :
( ( order @ A7 )
=> ( order @ ( A6 > A7 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A6: $tType,A7: $tType] :
( ( ord @ A7 )
=> ( ord @ ( A6 > A7 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_8,axiom,
! [A6: $tType] : ( size @ ( list @ A6 ) ) ).
% Conjectures (3)
thf(conj_0,hypothesis,
! [X5: nat] :
( ( listIn521021761append @ a @ xs @ f @ X5 )
= ( listIn521021761append @ a @ ys @ g @ X5 ) ) ).
thf(conj_1,hypothesis,
ord_less_eq @ nat @ ( size_size @ ( list @ a ) @ xs ) @ ( size_size @ ( list @ a ) @ ys ) ).
thf(conj_2,conjecture,
! [I: nat] :
( ~ ( ord_less @ nat @ I @ ( size_size @ ( list @ a ) @ xs ) )
| ( ( nth @ a @ xs @ I )
= ( nth @ a @ ys @ I ) ) ) ).
%------------------------------------------------------------------------------