TPTP Problem File: DAT140^1.p
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%------------------------------------------------------------------------------
% File : DAT140^1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Data Structures
% Problem : Coinductive list prefix 36
% Version : [Bla16] axioms : Especial.
% English :
% Refs : [Loc10] Lochbihler (2010), Coinductive
% : [RB15] Reynolds & Blanchette (2015), A Decision Procedure for
% : [Bla16] Blanchette (2016), Email to Geoff Sutcliffe
% Source : [Bla16]
% Names : coinductive_list_prefix__36.p [Bla16]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.3.0, 0.00 v7.1.0
% Syntax : Number of formulae : 309 ( 76 unt; 35 typ; 0 def)
% Number of atoms : 929 ( 201 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3616 ( 98 ~; 36 |; 69 &;2907 @)
% ( 0 <=>; 506 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 146 ( 146 >; 0 *; 0 +; 0 <<)
% Number of symbols : 34 ( 33 usr; 0 con; 1-4 aty)
% Number of variables : 1045 ( 40 ^; 910 !; 60 ?;1045 :)
% ( 35 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2016-07-13 14:40:41.558
%------------------------------------------------------------------------------
%----Could-be-implicit typings (4)
thf(ty_t_Coinductive__List_Ollist,type,
coinductive_llist: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
thf(ty_tf_a,type,
a: $tType ).
%----Explicit typings (31)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Complete__Lattices_Ocomplete__lattice,type,
comple187826305attice:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : ( ( itself @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ofinite__lprefix,type,
coindu328551480prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olappend,type,
coinductive_lappend:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Olfinite,type,
coinductive_lfinite:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Ollast,type,
coinductive_llast:
!>[A: $tType] : ( ( coinductive_llist @ A ) > A ) ).
thf(sy_c_Coinductive__List_Ollist_OLCons,type,
coinductive_LCons:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > ( coinductive_llist @ A ) ) ).
thf(sy_c_Coinductive__List_Ollist_OLNil,type,
coinductive_LNil:
!>[A: $tType] : ( coinductive_llist @ A ) ).
thf(sy_c_Coinductive__List_Ollist_Olnull,type,
coinductive_lnull:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olmember,type,
coinductive_lmember:
!>[A: $tType] : ( A > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olprefix,type,
coinductive_lprefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Olstrict__prefix,type,
coindu1478340336prefix:
!>[A: $tType] : ( ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Coinductive__List_Oord__class_Olsorted,type,
coindu63249387sorted:
!>[A: $tType] : ( ( coinductive_llist @ A ) > $o ) ).
thf(sy_c_Hilbert__Choice_OGreatestM,type,
hilbert_GreatestM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
thf(sy_c_Hilbert__Choice_OLeastM,type,
hilbert_LeastM:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > $o ) > A ) ).
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_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
%----Relevant facts (254)
thf(fact_0_less__eq__llist__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( coinductive_llist @ A ) )
= ( coinductive_lprefix @ A ) ) ).
% less_eq_llist_def
thf(fact_1_lprefix__refl,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ Xs ) ).
% lprefix_refl
thf(fact_2_llist_Oleq__refl,axiom,
! [A: $tType,X: coinductive_llist @ A] : ( coinductive_lprefix @ A @ X @ X ) ).
% llist.leq_refl
thf(fact_3_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_4_lprefix__trans,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( coinductive_lprefix @ A @ Xs @ Zs ) ) ) ).
% lprefix_trans
thf(fact_5_llist_Oleq__trans,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A,Z: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ Z )
=> ( coinductive_lprefix @ A @ X @ Z ) ) ) ).
% llist.leq_trans
thf(fact_6_lprefix__antisym,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lprefix @ A @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ).
% lprefix_antisym
thf(fact_7_llist_Oleq__antisym,axiom,
! [A: $tType,X: coinductive_llist @ A,Y: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ X @ Y )
=> ( ( coinductive_lprefix @ A @ Y @ X )
=> ( X = Y ) ) ) ).
% llist.leq_antisym
thf(fact_8_lprefix__down__linear,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Zs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Zs )
=> ( ( coinductive_lprefix @ A @ Ys @ Zs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ) ).
% lprefix_down_linear
thf(fact_9_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_10_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_11_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [F: A > B,G: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F @ X2 ) @ ( G @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G ) ) ) ).
% le_funI
thf(fact_12_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
! [X3: A] : ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( G2 @ X3 ) ) ) ) ) ).
% le_fun_def
thf(fact_13_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_14_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less_eq @ A @ C @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_15_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: A,B3: A] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_16_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_17_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_18_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_19_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_le_eq_trans
thf(fact_20_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% ord_eq_le_trans
thf(fact_21_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_22_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_23_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less_eq @ A @ A2 @ C ) ) ) ) ).
% order.trans
thf(fact_24_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_25_wlog__linorder__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > A > $o,B2: A,A2: A] :
( ! [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ( ( P @ B2 @ A2 )
=> ( P @ A2 @ B2 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% wlog_linorder_le
thf(fact_26_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_27_linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_28_antisym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_29_eq__iff,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ^ [Y2: A,Z2: A] : Y2 = Z2 )
= ( ^ [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
& ( ord_less_eq @ A @ Y3 @ X3 ) ) ) ) ) ).
% eq_iff
thf(fact_30_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_31_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_32_order__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_subst2
thf(fact_33_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_subst1
thf(fact_34_lfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% lfp.leq_antisym
thf(fact_35_gfp_Oleq__antisym,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( X = Y ) ) ) ) ).
% gfp.leq_antisym
thf(fact_36_lfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% lfp.leq_trans
thf(fact_37_gfp_Oleq__trans,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ Y )
=> ( ord_less_eq @ A @ Z @ X ) ) ) ) ).
% gfp.leq_trans
thf(fact_38_lfp_Oleq__refl,axiom,
! [A: $tType] :
( ( comple187826305attice @ A @ ( type2 @ A ) )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% lfp.leq_refl
thf(fact_39_Coinductive__List_Ofinite__lprefix__def,axiom,
! [A: $tType] :
( ( coindu328551480prefix @ A )
= ( coinductive_lprefix @ A ) ) ).
% Coinductive_List.finite_lprefix_def
thf(fact_40_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less_eq @ B @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_41_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F2: A > B] :
! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ Y3 ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% antimono_def
thf(fact_42_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ Y4 ) @ ( F @ X2 ) ) )
=> ( order_antimono @ A @ B @ F ) ) ) ).
% antimonoI
thf(fact_43_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoE
thf(fact_44_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F @ Y ) @ ( F @ X ) ) ) ) ) ).
% antimonoD
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G: A > B] :
( ! [X2: A] :
( ( F @ X2 )
= ( G @ X2 ) )
=> ( F = G ) ) ).
% ext
thf(fact_46_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_47_decseqD,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [F: nat > A,I: nat,J: nat] :
( ( order_antimono @ nat @ A @ F )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ A @ ( F @ J ) @ ( F @ I ) ) ) ) ) ).
% decseqD
thf(fact_48_decseq__def,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X4: nat > A] :
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ A @ ( X4 @ N ) @ ( X4 @ M ) ) ) ) ) ) ).
% decseq_def
thf(fact_49_Coinductive__List_Olprefix__nitpick__simps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( ( coinductive_lfinite @ A @ Xs2 )
=> ( coindu328551480prefix @ A @ Xs2 @ Ys2 ) )
& ( ~ ( coinductive_lfinite @ A @ Xs2 )
=> ( Xs2 = Ys2 ) ) ) ) ) ).
% Coinductive_List.lprefix_nitpick_simps
thf(fact_50_lstrict__prefix__def,axiom,
! [A: $tType] :
( ( coindu1478340336prefix @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs2 @ Ys2 )
& ( Xs2 != Ys2 ) ) ) ) ).
% lstrict_prefix_def
thf(fact_51_Coinductive__List_Ofinite__lprefix__nitpick__simps_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coindu328551480prefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(2)
thf(fact_52_Coinductive__List_Ofinite__lprefix__nitpick__simps_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( Xs
= ( coinductive_LNil @ A ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(1)
thf(fact_53_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_54_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B] :
( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( order_strict_mono @ A @ B @ F ) ) ) ).
% strict_monoI
thf(fact_55_lprefix__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Ys ) ).
% lprefix_code(1)
thf(fact_56_lfinite__code_I1_J,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_code(1)
thf(fact_57_lstrict__prefix__code_I1_J,axiom,
! [A: $tType] :
~ ( coindu1478340336prefix @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) ) ).
% lstrict_prefix_code(1)
thf(fact_58_less__llist__def,axiom,
! [A: $tType] :
( ( ord_less @ ( coinductive_llist @ A ) )
= ( coindu1478340336prefix @ A ) ) ).
% less_llist_def
thf(fact_59_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_60_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_61_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_62_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_63_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_64_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans
thf(fact_65_lstrict__prefix__lfinite1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lstrict_prefix_lfinite1
thf(fact_66_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_67_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_68_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_69_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_70_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_71_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X2 )
=> ( P @ Y5 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_72_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_73_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_74_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_75_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_less_eq_trans
thf(fact_76_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% ord_eq_less_trans
thf(fact_77_lfinite__LNil,axiom,
! [A: $tType] : ( coinductive_lfinite @ A @ ( coinductive_LNil @ A ) ) ).
% lfinite_LNil
thf(fact_78_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_79_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_80_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_trans
thf(fact_81_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_82_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_83_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_84_llist__less__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ! [Xs3: coinductive_llist @ A] :
( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs3 )
=> ( P @ Ys3 ) )
=> ( P @ Xs3 ) )
=> ( P @ Xs ) ) ).
% llist_less_induct
thf(fact_85_dense,axiom,
! [A: $tType] :
( ( dense_order @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_86_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_87_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_88_neqE,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_89_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [X1: A] : ( ord_less @ A @ X @ X1 ) ) ).
% gt_ex
thf(fact_90_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A @ ( type2 @ A ) )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_91_order__less__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_subst2
thf(fact_92_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_93_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > B,C: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ B @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A2 ) @ C ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_94_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B @ ( type2 @ B ) )
& ( ord @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_95_not__lfinite__lprefix__conv__eq,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( Xs = Ys ) ) ) ).
% not_lfinite_lprefix_conv_eq
thf(fact_96_lprefix__lfiniteD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lfinite @ A @ Ys )
=> ( coinductive_lfinite @ A @ Xs ) ) ) ).
% lprefix_lfiniteD
thf(fact_97_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_98_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_99_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less_eq @ A @ B4 @ A4 )
& ( A4 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_100_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A4: A] :
( ( ord_less @ A @ B4 @ A4 )
| ( A4 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_101_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_102_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_103_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_104_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_105_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [B2: A,A2: A,C: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C @ B2 )
=> ( ord_less @ A @ C @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_106_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_107_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ( ( ord_less_eq @ A )
= ( ^ [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_108_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans2
thf(fact_109_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,C: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C )
=> ( ord_less @ A @ A2 @ C ) ) ) ) ).
% order.strict_trans1
thf(fact_110_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_111_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ 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 @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_113_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_114_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,Z: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_115_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A @ ( type2 @ A ) )
=> ! [Z: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_116_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% less_le_trans
thf(fact_117_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% le_less_trans
thf(fact_118_antisym__conv2,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_119_antisym__conv1,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_120_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_121_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_122_not__less,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_123_not__le,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_124_order__less__le__subst2,axiom,
! [A: $tType,C2: $tType] :
( ( ( order @ C2 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less @ A @ X2 @ Y4 )
=> ( ord_less @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_125_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less_eq @ B @ X2 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X2 ) @ ( 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 @ ( type2 @ C2 ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,B2: A,F: A > C2,C: C2] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C2 @ ( F @ B2 ) @ C )
=> ( ! [X2: A,Y4: A] :
( ( ord_less_eq @ A @ X2 @ Y4 )
=> ( ord_less_eq @ C2 @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C2 @ ( F @ A2 ) @ C ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_127_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B @ ( type2 @ B ) )
& ( order @ A @ ( type2 @ A ) ) )
=> ! [A2: A,F: B > A,B2: B,C: B] :
( ( ord_less_eq @ A @ A2 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C )
=> ( ! [X2: B,Y4: B] :
( ( ord_less @ B @ X2 @ Y4 )
=> ( ord_less @ A @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A2 @ ( F @ C ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_128_less__le,axiom,
! [A: $tType] :
( ( order @ A @ ( type2 @ 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 @ ( type2 @ 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 @ ( type2 @ A ) )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_131_leD,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_132_LNil__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A] : ( coinductive_lprefix @ A @ ( coinductive_LNil @ A ) @ Xs ) ).
% LNil_lprefix
thf(fact_133_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F2: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F2 ) ) ) ) ) ).
% less_fun_def
thf(fact_134_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ! [F: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F )
=> ( ( ord_less @ B @ ( F @ X ) @ ( F @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_135_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A @ ( type2 @ A ) )
& ( order @ B @ ( type2 @ B ) ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F2: A > B] :
! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_136_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% minf(8)
thf(fact_137_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% minf(6)
thf(fact_138_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ T @ X5 ) ) ) ).
% pinf(8)
thf(fact_139_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less_eq @ A @ X5 @ T ) ) ) ).
% pinf(6)
thf(fact_140_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A @ ( type2 @ A ) )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ C3 ) )
=> ( P @ X5 ) )
& ! [D: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_141_lmember__code_I1_J,axiom,
! [A: $tType,X: A] :
~ ( coinductive_lmember @ A @ X @ ( coinductive_LNil @ A ) ) ).
% lmember_code(1)
thf(fact_142_llimit__induct,axiom,
! [A: $tType,P: ( coinductive_llist @ A ) > $o,Xs: coinductive_llist @ A] :
( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( ( ! [Ys3: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Ys3 @ Xs )
=> ( P @ Ys3 ) )
=> ( P @ Xs ) )
=> ( P @ Xs ) ) ) ) ).
% llimit_induct
thf(fact_143_llist_Oinject,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A,Y21: A,Y22: coinductive_llist @ A] :
( ( ( coinductive_LCons @ A @ X21 @ X22 )
= ( coinductive_LCons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% llist.inject
thf(fact_144_LCons__lprefix__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
& ( coinductive_lprefix @ A @ Xs @ Ys ) ) ) ).
% LCons_lprefix_LCons
thf(fact_145_lfinite__code_I2_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
( ( coinductive_lfinite @ B @ ( coinductive_LCons @ B @ X @ Xs ) )
= ( coinductive_lfinite @ B @ Xs ) ) ).
% lfinite_code(2)
thf(fact_146_lfinite__LCons,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_lfinite @ A @ Xs ) ) ).
% lfinite_LCons
thf(fact_147_lstrict__prefix__code_I4_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B,Y: B,Ys: coinductive_llist @ B] :
( ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LCons @ B @ Y @ Ys ) )
= ( ( X = Y )
& ( coindu1478340336prefix @ B @ Xs @ Ys ) ) ) ).
% lstrict_prefix_code(4)
thf(fact_148_lstrict__prefix__code_I3_J,axiom,
! [B: $tType,X: B,Xs: coinductive_llist @ B] :
~ ( coindu1478340336prefix @ B @ ( coinductive_LCons @ B @ X @ Xs ) @ ( coinductive_LNil @ B ) ) ).
% lstrict_prefix_code(3)
thf(fact_149_lstrict__prefix__code_I2_J,axiom,
! [B: $tType,Y: B,Ys: coinductive_llist @ B] : ( coindu1478340336prefix @ B @ ( coinductive_LNil @ B ) @ ( coinductive_LCons @ B @ Y @ Ys ) ) ).
% lstrict_prefix_code(2)
thf(fact_150_lmember__code_I2_J,axiom,
! [A: $tType,X: A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lmember @ A @ X @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( X = Y )
| ( coinductive_lmember @ A @ X @ Ys ) ) ) ).
% lmember_code(2)
thf(fact_151_Le__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,X: A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LCons @ A @ X @ Ys ) ) ) ).
% Le_LCons
thf(fact_152_LCons__lprefix__conv,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ Ys )
= ( ? [Ys4: coinductive_llist @ A] :
( ( Ys
= ( coinductive_LCons @ A @ X @ Ys4 ) )
& ( coinductive_lprefix @ A @ Xs @ Ys4 ) ) ) ) ).
% LCons_lprefix_conv
thf(fact_153_llist_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
( ( coinductive_LNil @ A )
!= ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.distinct(1)
thf(fact_154_llist_Oexhaust,axiom,
! [A: $tType,Y: coinductive_llist @ A] :
( ( Y
!= ( coinductive_LNil @ A ) )
=> ~ ! [X212: A,X222: coinductive_llist @ A] :
( Y
!= ( coinductive_LCons @ A @ X212 @ X222 ) ) ) ).
% llist.exhaust
thf(fact_155_neq__LNil__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( Xs
!= ( coinductive_LNil @ A ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% neq_LNil_conv
thf(fact_156_lfinite__LConsI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lfinite @ A @ ( coinductive_LCons @ A @ X @ Xs ) ) ) ).
% lfinite_LConsI
thf(fact_157_lprefix__code_I2_J,axiom,
! [A: $tType,X: A,Xs: coinductive_llist @ A] :
~ ( coinductive_lprefix @ A @ ( coinductive_LCons @ A @ X @ Xs ) @ ( coinductive_LNil @ A ) ) ).
% lprefix_code(2)
thf(fact_158_lprefix_Ocases,axiom,
! [A: $tType,A1: coinductive_llist @ A,A22: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ A1 @ A22 )
=> ( ( ( A1
= ( coinductive_LNil @ A ) )
=> ! [Xs3: coinductive_llist @ A] : A22 != Xs3 )
=> ~ ! [Xs3: coinductive_llist @ A,Ys5: coinductive_llist @ A,X2: A] :
( ( A1
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
=> ( ( A22
= ( coinductive_LCons @ A @ X2 @ Ys5 ) )
=> ~ ( coinductive_lprefix @ A @ Xs3 @ Ys5 ) ) ) ) ) ).
% lprefix.cases
thf(fact_159_lprefix_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [A12: coinductive_llist @ A,A23: coinductive_llist @ A] :
( ? [Xs2: coinductive_llist @ A] :
( ( A12
= ( coinductive_LNil @ A ) )
& ( A23 = Xs2 ) )
| ? [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A,X3: A] :
( ( A12
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
& ( A23
= ( coinductive_LCons @ A @ X3 @ Ys2 ) )
& ( coinductive_lprefix @ A @ Xs2 @ Ys2 ) ) ) ) ) ).
% lprefix.simps
thf(fact_160_lprefix_Ocoinduct,axiom,
! [A: $tType,X6: ( coinductive_llist @ A ) > ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A,Xa: coinductive_llist @ A] :
( ( X6 @ X @ Xa )
=> ( ! [X2: coinductive_llist @ A,Xa2: coinductive_llist @ A] :
( ( X6 @ X2 @ Xa2 )
=> ( ? [Xs5: coinductive_llist @ A] :
( ( X2
= ( coinductive_LNil @ A ) )
& ( Xa2 = Xs5 ) )
| ? [Xs5: coinductive_llist @ A,Ys3: coinductive_llist @ A,Xb: A] :
( ( X2
= ( coinductive_LCons @ A @ Xb @ Xs5 ) )
& ( Xa2
= ( coinductive_LCons @ A @ Xb @ Ys3 ) )
& ( ( X6 @ Xs5 @ Ys3 )
| ( coinductive_lprefix @ A @ Xs5 @ Ys3 ) ) ) ) )
=> ( coinductive_lprefix @ A @ X @ Xa ) ) ) ).
% lprefix.coinduct
thf(fact_161_lprefix__LCons__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs4 ) )
& ( coinductive_lprefix @ A @ Xs4 @ Ys ) ) ) ) ).
% lprefix_LCons_conv
thf(fact_162_lfinite_Ocases,axiom,
! [A: $tType,A2: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ~ ! [Xs3: coinductive_llist @ A] :
( ? [X2: A] :
( A2
= ( coinductive_LCons @ A @ X2 @ Xs3 ) )
=> ~ ( coinductive_lfinite @ A @ Xs3 ) ) ) ) ).
% lfinite.cases
thf(fact_163_lfinite_Osimps,axiom,
! [A: $tType] :
( ( coinductive_lfinite @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [Xs2: coinductive_llist @ A,X3: A] :
( ( A4
= ( coinductive_LCons @ A @ X3 @ Xs2 ) )
& ( coinductive_lfinite @ A @ Xs2 ) ) ) ) ) ).
% lfinite.simps
thf(fact_164_lfinite_Oinducts,axiom,
! [A: $tType,X: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ X )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [Xs3: coinductive_llist @ A,X2: A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_LCons @ A @ X2 @ Xs3 ) ) ) )
=> ( P @ X ) ) ) ) ).
% lfinite.inducts
thf(fact_165_Coinductive__List_Ofinite__lprefix__nitpick__simps_I3_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coindu328551480prefix @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
| ? [Xs4: coinductive_llist @ A] :
( ( Xs
= ( coinductive_LCons @ A @ Y @ Xs4 ) )
& ( coindu328551480prefix @ A @ Xs4 @ Ys ) ) ) ) ).
% Coinductive_List.finite_lprefix_nitpick_simps(3)
thf(fact_166_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A @ ( type2 @ A ) )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_167_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_168_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_169_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(3)
thf(fact_170_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( X5 != T ) ) ) ).
% pinf(4)
thf(fact_171_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ~ ( ord_less @ A @ X5 @ T ) ) ) ).
% pinf(5)
thf(fact_172_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less @ A @ T @ X5 ) ) ) ).
% pinf(7)
thf(fact_173_pinf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ Z3 @ X5 )
=> ( F3 = F3 ) ) ) ).
% pinf(11)
thf(fact_174_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
& ( Q @ X5 ) )
= ( ( P2 @ X5 )
& ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_175_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z4: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z4 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ( ( P @ X5 )
| ( Q @ X5 ) )
= ( ( P2 @ X5 )
| ( Q2 @ X5 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_176_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(3)
thf(fact_177_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( X5 != T ) ) ) ).
% minf(4)
thf(fact_178_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ( ord_less @ A @ X5 @ T ) ) ) ).
% minf(5)
thf(fact_179_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A @ ( type2 @ A ) )
=> ! [T: A] :
? [Z3: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z3 )
=> ~ ( ord_less @ A @ T @ X5 ) ) ) ).
% minf(7)
thf(fact_180_minf_I11_J,axiom,
! [C2: $tType,D2: $tType] :
( ( ord @ C2 @ ( type2 @ C2 ) )
=> ! [F3: D2] :
? [Z3: C2] :
! [X5: C2] :
( ( ord_less @ C2 @ X5 @ Z3 )
=> ( F3 = F3 ) ) ) ).
% minf(11)
thf(fact_181_llast__singleton,axiom,
! [A: $tType,X: A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) )
= X ) ).
% llast_singleton
thf(fact_182_lsorted_Ocases,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [A2: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ A2 )
=> ( ( A2
!= ( coinductive_LNil @ A ) )
=> ( ! [X2: A] :
( A2
!= ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) )
=> ~ ! [X2: A,Y4: A,Xs3: coinductive_llist @ A] :
( ( A2
= ( coinductive_LCons @ A @ X2 @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) )
=> ( ( ord_less_eq @ A @ X2 @ Y4 )
=> ~ ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y4 @ Xs3 ) ) ) ) ) ) ) ) ).
% lsorted.cases
thf(fact_183_lsorted_Osimps,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( ( coindu63249387sorted @ A )
= ( ^ [A4: coinductive_llist @ A] :
( ( A4
= ( coinductive_LNil @ A ) )
| ? [X3: A] :
( A4
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LNil @ A ) ) )
| ? [X3: A,Y3: A,Xs2: coinductive_llist @ A] :
( ( A4
= ( coinductive_LCons @ A @ X3 @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) )
& ( ord_less_eq @ A @ X3 @ Y3 )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ).
% lsorted.simps
thf(fact_184_lsorted__code_I1_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% lsorted_code(1)
thf(fact_185_llast__LCons2,axiom,
! [A: $tType,X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ).
% llast_LCons2
thf(fact_186_lsorted__LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ).
% lsorted_LCons_LCons
thf(fact_187_lsorted__code_I2_J,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% lsorted_code(2)
thf(fact_188_lsorted__lprefixD,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coindu63249387sorted @ A @ Ys )
=> ( coindu63249387sorted @ A @ Xs ) ) ) ) ).
% lsorted_lprefixD
thf(fact_189_LNil,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LNil @ A ) ) ) ).
% LNil
thf(fact_190_LCons__LCons,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A,Y: A,Xs: coinductive_llist @ A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y @ Xs ) )
=> ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LCons @ A @ Y @ Xs ) ) ) ) ) ) ).
% LCons_LCons
thf(fact_191_Singleton,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X: A] : ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ X @ ( coinductive_LNil @ A ) ) ) ) ).
% Singleton
thf(fact_192_lsorted_Ocoinduct,axiom,
! [A: $tType] :
( ( ord @ A @ ( type2 @ A ) )
=> ! [X6: ( coinductive_llist @ A ) > $o,X: coinductive_llist @ A] :
( ( X6 @ X )
=> ( ! [X2: coinductive_llist @ A] :
( ( X6 @ X2 )
=> ( ( X2
= ( coinductive_LNil @ A ) )
| ? [Xa3: A] :
( X2
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LNil @ A ) ) )
| ? [Xa3: A,Y5: A,Xs5: coinductive_llist @ A] :
( ( X2
= ( coinductive_LCons @ A @ Xa3 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) )
& ( ord_less_eq @ A @ Xa3 @ Y5 )
& ( ( X6 @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) )
| ( coindu63249387sorted @ A @ ( coinductive_LCons @ A @ Y5 @ Xs5 ) ) ) ) ) )
=> ( coindu63249387sorted @ A @ X ) ) ) ) ).
% lsorted.coinduct
thf(fact_193_llast__lappend__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) )
= ( coinductive_llast @ A @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ) ).
% llast_lappend_LCons
thf(fact_194_ex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) ) ) ) ) ).
% ex_has_greatest_nat
thf(fact_195_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_196_lappend__code_I2_J,axiom,
! [A: $tType,Xa: A,X: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LCons @ A @ Xa @ X ) @ Ys )
= ( coinductive_LCons @ A @ Xa @ ( coinductive_lappend @ A @ X @ Ys ) ) ) ).
% lappend_code(2)
thf(fact_197_lappend__LNil2,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ Xs @ ( coinductive_LNil @ A ) )
= Xs ) ).
% lappend_LNil2
thf(fact_198_lappend__code_I1_J,axiom,
! [A: $tType,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ Ys )
= Ys ) ).
% lappend_code(1)
thf(fact_199_lfinite__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ( coinductive_lfinite @ A @ Ys ) ) ) ).
% lfinite_lappend
thf(fact_200_lprefix__lappend__same,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ ( coinductive_lappend @ A @ Xs @ Zs ) )
= ( ( coinductive_lfinite @ A @ Xs )
=> ( coinductive_lprefix @ A @ Ys @ Zs ) ) ) ).
% lprefix_lappend_same
thf(fact_201_lappend__snocL1__conv__LCons2,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Y: A,Ys: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ ( coinductive_LNil @ A ) ) ) @ Ys )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_LCons @ A @ Y @ Ys ) ) ) ).
% lappend_snocL1_conv_LCons2
thf(fact_202_lappend__assoc,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lappend @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
= ( coinductive_lappend @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) ) ) ).
% lappend_assoc
thf(fact_203_lappend__inf,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ~ ( coinductive_lfinite @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_inf
thf(fact_204_lappend__eq__LNil__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend_eq_LNil_iff
thf(fact_205_LNil__eq__lappend__iff,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_LNil @ A )
= ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( Xs
= ( coinductive_LNil @ A ) )
& ( Ys
= ( coinductive_LNil @ A ) ) ) ) ).
% LNil_eq_lappend_iff
thf(fact_206_lappend__LNil__LNil,axiom,
! [A: $tType] :
( ( coinductive_lappend @ A @ ( coinductive_LNil @ A ) @ ( coinductive_LNil @ A ) )
= ( coinductive_LNil @ A ) ) ).
% lappend_LNil_LNil
thf(fact_207_lprefix__lappend__sameI,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Zs @ Xs ) @ ( coinductive_lappend @ A @ Zs @ Ys ) ) ) ).
% lprefix_lappend_sameI
thf(fact_208_lprefix__conv__lappend,axiom,
! [A: $tType] :
( ( coinductive_lprefix @ A )
= ( ^ [Xs2: coinductive_llist @ A,Ys2: coinductive_llist @ A] :
? [Zs2: coinductive_llist @ A] :
( Ys2
= ( coinductive_lappend @ A @ Xs2 @ Zs2 ) ) ) ) ).
% lprefix_conv_lappend
thf(fact_209_lprefix__lappendD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Ys @ Zs ) )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
| ( coinductive_lprefix @ A @ Ys @ Xs ) ) ) ).
% lprefix_lappendD
thf(fact_210_lappend__lprefixE,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A,Zs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) @ Zs )
=> ~ ! [Zs3: coinductive_llist @ A] :
( Zs
!= ( coinductive_lappend @ A @ Xs @ Zs3 ) ) ) ).
% lappend_lprefixE
thf(fact_211_lprefix__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] : ( coinductive_lprefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ).
% lprefix_lappend
thf(fact_212_lfinite__rev__induct,axiom,
! [A: $tType,Xs: coinductive_llist @ A,P: ( coinductive_llist @ A ) > $o] :
( ( coinductive_lfinite @ A @ Xs )
=> ( ( P @ ( coinductive_LNil @ A ) )
=> ( ! [X2: A,Xs3: coinductive_llist @ A] :
( ( coinductive_lfinite @ A @ Xs3 )
=> ( ( P @ Xs3 )
=> ( P @ ( coinductive_lappend @ A @ Xs3 @ ( coinductive_LCons @ A @ X2 @ ( coinductive_LNil @ A ) ) ) ) ) )
=> ( P @ Xs ) ) ) ) ).
% lfinite_rev_induct
thf(fact_213_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
& ( M != N ) ) ) ) ).
% nat_less_le
thf(fact_214_less__imp__le__nat,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_imp_le_nat
thf(fact_215_nat__le__linear,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
| ( ord_less_eq @ nat @ N2 @ M2 ) ) ).
% nat_le_linear
thf(fact_216_le__antisym,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( ord_less_eq @ nat @ N2 @ M2 )
=> ( M2 = N2 ) ) ) ).
% le_antisym
thf(fact_217_eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( M2 = N2 )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% eq_imp_le
thf(fact_218_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_219_le__refl,axiom,
! [N2: nat] : ( ord_less_eq @ nat @ N2 @ N2 ) ).
% le_refl
thf(fact_220_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_221_le__neq__implies__less,axiom,
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( M2 != N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_222_less__or__eq__imp__le,axiom,
! [M2: nat,N2: nat] :
( ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) )
=> ( ord_less_eq @ nat @ M2 @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_223_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
| ( M = N ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_224_GreatestM__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M2: A > nat,B2: nat] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( M2 @ X ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ).
% GreatestM_nat_le
thf(fact_225_GreatestM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less @ nat @ ( M2 @ Y4 ) @ B2 ) )
=> ( ( P @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ Y5 ) @ ( M2 @ ( hilbert_GreatestM @ A @ nat @ M2 @ P ) ) ) ) ) ) ) ).
% GreatestM_nat_lemma
thf(fact_226_GreatestMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M2 @ Y4 ) @ ( M2 @ X ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ Y5 ) @ ( M2 @ X2 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( hilbert_GreatestM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% GreatestMI2
thf(fact_227_LeastMI2,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B @ ( type2 @ B ) )
=> ! [P: A > $o,X: A,M2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ B @ ( M2 @ X ) @ ( M2 @ Y4 ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ B @ ( M2 @ X2 ) @ ( M2 @ Y5 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( hilbert_LeastM @ A @ B @ M2 @ P ) ) ) ) ) ) ).
% LeastMI2
thf(fact_228_lstrict__prefix__lappend__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coindu1478340336prefix @ A @ Xs @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lfinite @ A @ Xs )
& ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lstrict_prefix_lappend_conv
thf(fact_229_lappend_Odisc__iff_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) )
= ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.disc_iff(2)
thf(fact_230_lnull__lappend,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) )
= ( ( coinductive_lnull @ A @ Xs )
& ( coinductive_lnull @ A @ Ys ) ) ) ).
% lnull_lappend
thf(fact_231_lprefix__LNil,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ ( coinductive_LNil @ A ) )
= ( coinductive_lnull @ A @ Xs ) ) ).
% lprefix_LNil
thf(fact_232_lappend_Octr_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= ( coinductive_LNil @ A ) ) ) ) ).
% lappend.ctr(1)
thf(fact_233_lappend_Odisc_I2_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) )
=> ~ ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ).
% lappend.disc(2)
thf(fact_234_lappend_Odisc_I1_J,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ ( coinductive_lappend @ A @ Xs @ Ys ) ) ) ) ).
% lappend.disc(1)
thf(fact_235_lappend__lnull1,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Ys ) ) ).
% lappend_lnull1
thf(fact_236_lappend__lnull2,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lappend @ A @ Xs @ Ys )
= Xs ) ) ).
% lappend_lnull2
thf(fact_237_LeastM__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,M2: A > nat] :
( ( P @ K )
=> ( ( P @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) )
& ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ Y5 ) ) ) ) ) ).
% LeastM_nat_lemma
thf(fact_238_LeastM__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,M2: A > nat] :
( ( P @ X )
=> ( ord_less_eq @ nat @ ( M2 @ ( hilbert_LeastM @ A @ nat @ M2 @ P ) ) @ ( M2 @ X ) ) ) ).
% LeastM_nat_le
thf(fact_239_llast__LCons,axiom,
! [A: $tType,Xs: coinductive_llist @ A,X: A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= X ) )
& ( ~ ( coinductive_lnull @ A @ Xs )
=> ( ( coinductive_llast @ A @ ( coinductive_LCons @ A @ X @ Xs ) )
= ( coinductive_llast @ A @ Xs ) ) ) ) ).
% llast_LCons
thf(fact_240_llist_Odisc_I2_J,axiom,
! [A: $tType,X21: A,X22: coinductive_llist @ A] :
~ ( coinductive_lnull @ A @ ( coinductive_LCons @ A @ X21 @ X22 ) ) ).
% llist.disc(2)
thf(fact_241_llist_OdiscI_I2_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A,X21: A,X22: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LCons @ A @ X21 @ X22 ) )
=> ~ ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(2)
thf(fact_242_not__lnull__conv,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( ~ ( coinductive_lnull @ A @ Xs ) )
= ( ? [X3: A,Xs4: coinductive_llist @ A] :
( Xs
= ( coinductive_LCons @ A @ X3 @ Xs4 ) ) ) ) ).
% not_lnull_conv
thf(fact_243_lnull__lprefix,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lprefix @ A @ Xs @ Ys ) ) ).
% lnull_lprefix
thf(fact_244_lprefix__lnull,axiom,
! [A: $tType,Ys: coinductive_llist @ A,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Ys )
=> ( ( coinductive_lprefix @ A @ Xs @ Ys )
= ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnull
thf(fact_245_lprefix__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ( coinductive_lnull @ A @ Ys )
=> ( coinductive_lnull @ A @ Xs ) ) ) ).
% lprefix_lnullD
thf(fact_246_lprefix__not__lnullD,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( coinductive_lprefix @ A @ Xs @ Ys )
=> ( ~ ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lprefix_not_lnullD
thf(fact_247_llist_Odisc_I1_J,axiom,
! [A: $tType] : ( coinductive_lnull @ A @ ( coinductive_LNil @ A ) ) ).
% llist.disc(1)
thf(fact_248_llist_OdiscI_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( Llist
= ( coinductive_LNil @ A ) )
=> ( coinductive_lnull @ A @ Llist ) ) ).
% llist.discI(1)
thf(fact_249_llist_Ocollapse_I1_J,axiom,
! [A: $tType,Llist: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Llist )
=> ( Llist
= ( coinductive_LNil @ A ) ) ) ).
% llist.collapse(1)
thf(fact_250_lnull__def,axiom,
! [A: $tType] :
( ( coinductive_lnull @ A )
= ( ^ [Llist2: coinductive_llist @ A] :
( Llist2
= ( coinductive_LNil @ A ) ) ) ) ).
% lnull_def
thf(fact_251_lnull__imp__lfinite,axiom,
! [A: $tType,Xs: coinductive_llist @ A] :
( ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lfinite @ A @ Xs ) ) ).
% lnull_imp_lfinite
thf(fact_252_lappend_Oexhaust,axiom,
! [A: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ A] :
( ( ( coinductive_lnull @ A @ Xs )
=> ~ ( coinductive_lnull @ A @ Ys ) )
=> ( ~ ( coinductive_lnull @ A @ Xs )
| ~ ( coinductive_lnull @ A @ Ys ) ) ) ).
% lappend.exhaust
thf(fact_253_lzip_Oexhaust,axiom,
! [A: $tType,B: $tType,Xs: coinductive_llist @ A,Ys: coinductive_llist @ B] :
( ~ ( ( coinductive_lnull @ A @ Xs )
| ( coinductive_lnull @ B @ Ys ) )
=> ~ ( ~ ( coinductive_lnull @ A @ Xs )
=> ( coinductive_lnull @ B @ Ys ) ) ) ).
% lzip.exhaust
%----Type constructors (19)
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A5: $tType,A6: $tType] :
( ( comple187826305attice @ A6 @ ( type2 @ A6 ) )
=> ( comple187826305attice @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A5: $tType,A6: $tType] :
( ( preorder @ A6 @ ( type2 @ A6 ) )
=> ( preorder @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A5: $tType,A6: $tType] :
( ( order @ A6 @ ( type2 @ A6 ) )
=> ( order @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A5: $tType,A6: $tType] :
( ( ord @ A6 @ ( type2 @ A6 ) )
=> ( ord @ ( A5 > A6 ) @ ( type2 @ ( A5 > A6 ) ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat @ ( type2 @ nat ) ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat @ ( type2 @ nat ) ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_4,axiom,
comple187826305attice @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Opreorder_5,axiom,
preorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_6,axiom,
linorder @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oorder_7,axiom,
order @ $o @ ( type2 @ $o ) ).
thf(tcon_HOL_Obool___Orderings_Oord_8,axiom,
ord @ $o @ ( type2 @ $o ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Opreorder_9,axiom,
! [A5: $tType] : ( preorder @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oorder_10,axiom,
! [A5: $tType] : ( order @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
thf(tcon_Coinductive__List_Ollist___Orderings_Oord_11,axiom,
! [A5: $tType] : ( ord @ ( coinductive_llist @ A5 ) @ ( type2 @ ( coinductive_llist @ A5 ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( ord_less_eq @ ( coinductive_llist @ a ) )
= ( coinductive_lprefix @ a ) ) ).
%------------------------------------------------------------------------------