TPTP Problem File: ITP127^2.p
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%------------------------------------------------------------------------------
% File : ITP127^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Monomorphic_Monad problem prob_2789__7122292_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 : Monomorphic_Monad/prob_2789__7122292_1 [Des21]
% Status : Theorem
% Rating : 0.00 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 326 ( 78 unt; 47 typ; 0 def)
% Number of atoms : 843 ( 228 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3576 ( 87 ~; 17 |; 40 &;3009 @)
% ( 0 <=>; 423 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 9 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 427 ( 427 >; 0 *; 0 +; 0 <<)
% Number of symbols : 47 ( 44 usr; 3 con; 0-9 aty)
% Number of variables : 1233 ( 74 ^;1078 !; 13 ?;1233 :)
% ( 68 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:22:28.107
%------------------------------------------------------------------------------
% Could-be-implicit typings (5)
thf(ty_t_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT,type,
monomo1453983439ondetT: $tType > $tType > $tType ).
thf(ty_t_Countable__Set__Type_Ocset,type,
countable_Set_cset: $tType > $tType ).
thf(ty_tf_m,type,
m: $tType ).
thf(ty_tf_c,type,
c: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (42)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[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_Oorder__bot,type,
order_bot:
!>[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_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_c_Countable__Set__Type_Ocin,type,
countable_Set_cin:
!>[A: $tType] : ( A > ( countable_Set_cset @ A ) > $o ) ).
thf(sy_c_Countable__Set__Type_Ocinsert,type,
counta2111716221insert:
!>[A: $tType] : ( A > ( countable_Set_cset @ A ) > ( countable_Set_cset @ A ) ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM,type,
monomo2011693592ondetM:
!>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__axioms,type,
monomo759963067axioms:
!>[A: $tType,M: $tType,C: $tType] : ( ( ( countable_Set_cset @ A ) > M ) > ( M > ( ( countable_Set_cset @ A ) > M ) > M ) > ( ( countable_Set_cset @ A ) > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Ocset__nondetM__base_Oaltc__nondet,type,
monomo955616121nondet:
!>[C: $tType,M: $tType,A: $tType] : ( ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > ( countable_Set_cset @ C ) > ( C > ( monomo1453983439ondetT @ A @ M ) ) > ( monomo1453983439ondetT @ A @ M ) ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc,type,
monomo439771545d_altc:
!>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc3,type,
monomo1425736922_altc3:
!>[A: $tType,M: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__altc__axioms,type,
monomo555039548axioms:
!>[M: $tType,A: $tType,C: $tType] : ( ( M > ( A > M ) > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state,type,
monomo109450930_state:
!>[A: $tType,M: $tType,S: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc,type,
monomo1036387116e_altc:
!>[A: $tType,M: $tType,S: $tType,C: $tType] : ( ( A > M ) > ( M > ( A > M ) > M ) > ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_Omonad__state__altc__axioms,type,
monomo1324021455axioms:
!>[S: $tType,M: $tType,C: $tType] : ( ( ( S > M ) > M ) > ( S > M > M ) > ( ( countable_Set_cset @ C ) > ( C > M ) > M ) > $o ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetM__base_Oput__nondet,type,
monomo410456285nondet:
!>[State: $tType,M: $tType,A: $tType] : ( ( State > M > M ) > State > ( monomo1453983439ondetT @ A @ M ) > ( monomo1453983439ondetT @ A @ M ) ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_ONondetT,type,
monomo1975189248ondetT:
!>[M: $tType,A: $tType] : ( M > ( monomo1453983439ondetT @ A @ M ) ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Ocase__nondetT,type,
monomo1564381680ondetT:
!>[M: $tType,B: $tType,A: $tType] : ( ( M > B ) > ( monomo1453983439ondetT @ A @ M ) > B ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orec__nondetT,type,
monomo1291740684ondetT:
!>[M: $tType,D: $tType,A: $tType] : ( ( M > D ) > ( monomo1453983439ondetT @ A @ M ) > D ) ).
thf(sy_c_Monomorphic__Monad__Mirabelle__jpzjukxpzs_OnondetT_Orun__nondet,type,
monomo603716163nondet:
!>[A: $tType,M: $tType] : ( ( monomo1453983439ondetT @ A @ M ) > M ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : 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_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_v_f____,type,
f: c > ( monomo1453983439ondetT @ a @ m ) ).
thf(sy_v_mergec,type,
mergec: ( countable_Set_cset @ c ) > ( c > m ) > m ).
thf(sy_v_x____,type,
x: c ).
% Relevant facts (256)
thf(fact_0_mergec__single,axiom,
! [X: c,F: c > m] :
( ( mergec @ ( counta2111716221insert @ c @ X @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ F )
= ( F @ X ) ) ).
% mergec_single
thf(fact_1_cset__nondetM__base_Oaltc__nondet_Ocong,axiom,
! [A: $tType,M: $tType,C: $tType] :
( ( monomo955616121nondet @ C @ M @ A )
= ( monomo955616121nondet @ C @ M @ A ) ) ).
% cset_nondetM_base.altc_nondet.cong
thf(fact_2_cinsert__absorb2,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A] :
( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ X @ A2 ) )
= ( counta2111716221insert @ A @ X @ A2 ) ) ).
% cinsert_absorb2
thf(fact_3_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X2: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_4_cdoubleton__eq__iff,axiom,
! [A: $tType,A3: A,B2: A,C2: A,D2: A] :
( ( ( counta2111716221insert @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
= ( counta2111716221insert @ A @ C2 @ ( counta2111716221insert @ A @ D2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
= ( ( ( A3 = C2 )
& ( B2 = D2 ) )
| ( ( A3 = D2 )
& ( B2 = C2 ) ) ) ) ).
% cdoubleton_eq_iff
thf(fact_5_csingleton__inject,axiom,
! [A: $tType,A3: A,B2: A] :
( ( ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
= ( counta2111716221insert @ A @ B2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
=> ( A3 = B2 ) ) ).
% csingleton_inject
thf(fact_6_cinsert__not__cempty,axiom,
! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
( ( counta2111716221insert @ A @ A3 @ A2 )
!= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% cinsert_not_cempty
thf(fact_7_cinsert__commute,axiom,
! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
( ( counta2111716221insert @ A @ X @ ( counta2111716221insert @ A @ Y @ A2 ) )
= ( counta2111716221insert @ A @ Y @ ( counta2111716221insert @ A @ X @ A2 ) ) ) ).
% cinsert_commute
thf(fact_8_cset__nondetM_Omerge__single,axiom,
! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: A,F: A > M] :
( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
=> ( ( Merge @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) @ F )
= ( F @ X ) ) ) ).
% cset_nondetM.merge_single
thf(fact_9_cset__nondetM_Omergec__single,axiom,
! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
=> ( ( Mergec @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
= ( F @ X ) ) ) ).
% cset_nondetM.mergec_single
thf(fact_10_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X2: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_11_monad__altc_Oaltc__single,axiom,
! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M,X: C,F: C > M] :
( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
=> ( ( Altc @ ( counta2111716221insert @ C @ X @ ( bot_bot @ ( countable_Set_cset @ C ) ) ) @ F )
= ( F @ X ) ) ) ).
% monad_altc.altc_single
thf(fact_12_run__altc__nondet,axiom,
! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
( ( monomo603716163nondet @ a @ m @ ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F ) )
= ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ).
% run_altc_nondet
thf(fact_13_cset__nondetM_Omerge__empty,axiom,
! [C: $tType,M: $tType,A: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,F: A > M] :
( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
=> ( ( Merge @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ F )
= ( Return @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% cset_nondetM.merge_empty
thf(fact_14_nondetT_Oexpand,axiom,
! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M,NondetT2: monomo1453983439ondetT @ A @ M] :
( ( ( monomo603716163nondet @ A @ M @ NondetT )
= ( monomo603716163nondet @ A @ M @ NondetT2 ) )
=> ( NondetT = NondetT2 ) ) ).
% nondetT.expand
thf(fact_15_cset__nondetM__base_Orun__altc__nondet,axiom,
! [M: $tType,A: $tType,C: $tType,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M,A2: countable_Set_cset @ C,F: C > ( monomo1453983439ondetT @ A @ M )] :
( ( monomo603716163nondet @ A @ M @ ( monomo955616121nondet @ C @ M @ A @ Mergec @ A2 @ F ) )
= ( Mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F ) ) ) ).
% cset_nondetM_base.run_altc_nondet
thf(fact_16_altc__nondet__def,axiom,
! [A2: countable_Set_cset @ c,F: c > ( monomo1453983439ondetT @ a @ m )] :
( ( monomo955616121nondet @ c @ m @ a @ mergec @ A2 @ F )
= ( monomo1975189248ondetT @ m @ a @ ( mergec @ A2 @ ( comp @ ( monomo1453983439ondetT @ a @ m ) @ m @ c @ ( monomo603716163nondet @ a @ m ) @ F ) ) ) ) ).
% altc_nondet_def
thf(fact_17_comp__apply,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comp @ B @ A @ C )
= ( ^ [F2: B > A,G: C > B,X2: C] : ( F2 @ ( G @ X2 ) ) ) ) ).
% comp_apply
thf(fact_18_cset__nondetM__base_Oaltc__nondet__def,axiom,
! [A: $tType,M: $tType,C: $tType] :
( ( monomo955616121nondet @ C @ M @ A )
= ( ^ [Mergec2: ( countable_Set_cset @ C ) > ( C > M ) > M,A4: countable_Set_cset @ C,F2: C > ( monomo1453983439ondetT @ A @ M )] : ( monomo1975189248ondetT @ M @ A @ ( Mergec2 @ A4 @ ( comp @ ( monomo1453983439ondetT @ A @ M ) @ M @ C @ ( monomo603716163nondet @ A @ M ) @ F2 ) ) ) ) ) ).
% cset_nondetM_base.altc_nondet_def
thf(fact_19_cset__nondetM_Oaxioms_I3_J,axiom,
! [A: $tType,M: $tType,C: $tType,Return: ( countable_Set_cset @ A ) > M,Bind: M > ( ( countable_Set_cset @ A ) > M ) > M,Merge: ( countable_Set_cset @ A ) > ( A > M ) > M,Mergec: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo2011693592ondetM @ A @ M @ C @ Return @ Bind @ Merge @ Mergec )
=> ( monomo759963067axioms @ A @ M @ C @ Return @ Bind @ Merge @ Mergec ) ) ).
% cset_nondetM.axioms(3)
thf(fact_20_comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comp @ B @ C @ A )
= ( ^ [F2: B > C,G: A > B,X2: A] : ( F2 @ ( G @ X2 ) ) ) ) ).
% comp_def
thf(fact_21_comp__assoc,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F: D > B,G2: C > D,H: A > C] :
( ( comp @ C @ B @ A @ ( comp @ D @ B @ C @ F @ G2 ) @ H )
= ( comp @ D @ B @ A @ F @ ( comp @ C @ D @ A @ G2 @ H ) ) ) ).
% comp_assoc
thf(fact_22_comp__eq__dest,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ ( D2 @ V ) ) ) ) ).
% comp_eq_dest
thf(fact_23_comp__eq__elim,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,A3: C > B,B2: A > C,C2: D > B,D2: A > D] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= ( comp @ D @ B @ A @ C2 @ D2 ) )
=> ! [V2: A] :
( ( A3 @ ( B2 @ V2 ) )
= ( C2 @ ( D2 @ V2 ) ) ) ) ).
% comp_eq_elim
thf(fact_24_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F: B > A,G2: C > B,X: C,F3: D > A,G3: E > D,X3: E] :
( ( ( F @ ( G2 @ X ) )
= ( F3 @ ( G3 @ X3 ) ) )
=> ( ( comp @ B @ A @ C @ F @ G2 @ X )
= ( comp @ D @ A @ E @ F3 @ G3 @ X3 ) ) ) ).
% comp_cong
thf(fact_25_comp__eq__dest__lhs,axiom,
! [C: $tType,B: $tType,A: $tType,A3: C > B,B2: A > C,C2: A > B,V: A] :
( ( ( comp @ C @ B @ A @ A3 @ B2 )
= C2 )
=> ( ( A3 @ ( B2 @ V ) )
= ( C2 @ V ) ) ) ).
% comp_eq_dest_lhs
thf(fact_26_comp__apply__eq,axiom,
! [B: $tType,D: $tType,A: $tType,C: $tType,F: B > A,G2: C > B,X: C,H: D > A,K: C > D] :
( ( ( F @ ( G2 @ X ) )
= ( H @ ( K @ X ) ) )
=> ( ( comp @ B @ A @ C @ F @ G2 @ X )
= ( comp @ D @ A @ C @ H @ K @ X ) ) ) ).
% comp_apply_eq
thf(fact_27_nondetT_Oinject,axiom,
! [A: $tType,M: $tType,X: M,Ya: M] :
( ( ( monomo1975189248ondetT @ M @ A @ X )
= ( monomo1975189248ondetT @ M @ A @ Ya ) )
= ( X = Ya ) ) ).
% nondetT.inject
thf(fact_28_nondetT_Ocollapse,axiom,
! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
( ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) )
= NondetT ) ).
% nondetT.collapse
thf(fact_29_nondetT_Oexhaust,axiom,
! [A: $tType,M: $tType,Y: monomo1453983439ondetT @ A @ M] :
~ ! [X4: M] :
( Y
!= ( monomo1975189248ondetT @ M @ A @ X4 ) ) ).
% nondetT.exhaust
thf(fact_30_nondetT_Oinduct,axiom,
! [M: $tType,A: $tType,P: ( monomo1453983439ondetT @ A @ M ) > $o,NondetT: monomo1453983439ondetT @ A @ M] :
( ! [Xa: M] : ( P @ ( monomo1975189248ondetT @ M @ A @ Xa ) )
=> ( P @ NondetT ) ) ).
% nondetT.induct
thf(fact_31_nondetT_Oexhaust__sel,axiom,
! [M: $tType,A: $tType,NondetT: monomo1453983439ondetT @ A @ M] :
( NondetT
= ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ).
% nondetT.exhaust_sel
thf(fact_32_nondetT_Osel,axiom,
! [Aa: $tType,A: $tType,X: A] :
( ( monomo603716163nondet @ Aa @ A @ ( monomo1975189248ondetT @ A @ Aa @ X ) )
= X ) ).
% nondetT.sel
thf(fact_33_put__nondet__def,axiom,
! [State: $tType] :
( ( monomo410456285nondet @ State @ m @ a )
= ( ^ [Put: State > m > m,S2: State,M2: monomo1453983439ondetT @ a @ m] : ( monomo1975189248ondetT @ m @ a @ ( Put @ S2 @ ( monomo603716163nondet @ a @ m @ M2 ) ) ) ) ) ).
% put_nondet_def
thf(fact_34_nondetM__base_Oput__nondet__def,axiom,
! [A: $tType,M: $tType,State: $tType] :
( ( monomo410456285nondet @ State @ M @ A )
= ( ^ [Put: State > M > M,S2: State,M2: monomo1453983439ondetT @ A @ M] : ( monomo1975189248ondetT @ M @ A @ ( Put @ S2 @ ( monomo603716163nondet @ A @ M @ M2 ) ) ) ) ) ).
% nondetM_base.put_nondet_def
thf(fact_35_nondetT_Osplit__sel__asm,axiom,
! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
= ( ~ ( ( NondetT
= ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
& ~ ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ) ).
% nondetT.split_sel_asm
thf(fact_36_nondetT_Osplit__sel,axiom,
! [B: $tType,M: $tType,A: $tType,P: B > $o,F: M > B,NondetT: monomo1453983439ondetT @ A @ M] :
( ( P @ ( monomo1564381680ondetT @ M @ B @ A @ F @ NondetT ) )
= ( ( NondetT
= ( monomo1975189248ondetT @ M @ A @ ( monomo603716163nondet @ A @ M @ NondetT ) ) )
=> ( P @ ( F @ ( monomo603716163nondet @ A @ M @ NondetT ) ) ) ) ) ).
% nondetT.split_sel
thf(fact_37_rewriteR__comp__comp2,axiom,
! [C: $tType,B: $tType,E: $tType,D: $tType,A: $tType,G2: C > B,H: A > C,R1: D > B,R2: A > D,F: B > E,L: D > E] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= ( comp @ D @ B @ A @ R1 @ R2 ) )
=> ( ( ( comp @ B @ E @ D @ F @ R1 )
= L )
=> ( ( comp @ C @ E @ A @ ( comp @ B @ E @ C @ F @ G2 ) @ H )
= ( comp @ D @ E @ A @ L @ R2 ) ) ) ) ).
% rewriteR_comp_comp2
thf(fact_38_rewriteL__comp__comp2,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,E: $tType,F: C > B,G2: A > C,L1: D > B,L2: A > D,H: E > A,R: E > D] :
( ( ( comp @ C @ B @ A @ F @ G2 )
= ( comp @ D @ B @ A @ L1 @ L2 ) )
=> ( ( ( comp @ A @ D @ E @ L2 @ H )
= R )
=> ( ( comp @ C @ B @ E @ F @ ( comp @ A @ C @ E @ G2 @ H ) )
= ( comp @ D @ B @ E @ L1 @ R ) ) ) ) ).
% rewriteL_comp_comp2
thf(fact_39_rewriteR__comp__comp,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,G2: C > B,H: A > C,R: A > B,F: B > D] :
( ( ( comp @ C @ B @ A @ G2 @ H )
= R )
=> ( ( comp @ C @ D @ A @ ( comp @ B @ D @ C @ F @ G2 ) @ H )
= ( comp @ B @ D @ A @ F @ R ) ) ) ).
% rewriteR_comp_comp
thf(fact_40_rewriteL__comp__comp,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType,F: C > B,G2: A > C,L: A > B,H: D > A] :
( ( ( comp @ C @ B @ A @ F @ G2 )
= L )
=> ( ( comp @ C @ B @ D @ F @ ( comp @ A @ C @ D @ G2 @ H ) )
= ( comp @ A @ B @ D @ L @ H ) ) ) ).
% rewriteL_comp_comp
thf(fact_41_type__copy__map__cong0,axiom,
! [B: $tType,D: $tType,E: $tType,A: $tType,C: $tType,M3: B > A,G2: C > B,X: C,N: D > A,H: C > D,F: A > E] :
( ( ( M3 @ ( G2 @ X ) )
= ( N @ ( H @ X ) ) )
=> ( ( comp @ B @ E @ C @ ( comp @ A @ E @ B @ F @ M3 ) @ G2 @ X )
= ( comp @ D @ E @ C @ ( comp @ A @ E @ D @ F @ N ) @ H @ X ) ) ) ).
% type_copy_map_cong0
thf(fact_42_run__put__nondet,axiom,
! [B: $tType,Put2: B > m > m,S3: B,M4: monomo1453983439ondetT @ a @ m] :
( ( monomo603716163nondet @ a @ m @ ( monomo410456285nondet @ B @ m @ a @ Put2 @ S3 @ M4 ) )
= ( Put2 @ S3 @ ( monomo603716163nondet @ a @ m @ M4 ) ) ) ).
% run_put_nondet
thf(fact_43_nondetT_Ocase__eq__if,axiom,
! [A: $tType,B: $tType,M: $tType] :
( ( monomo1564381680ondetT @ M @ B @ A )
= ( ^ [F2: M > B,NondetT3: monomo1453983439ondetT @ A @ M] : ( F2 @ ( monomo603716163nondet @ A @ M @ NondetT3 ) ) ) ) ).
% nondetT.case_eq_if
thf(fact_44_nondetT_Ocase,axiom,
! [A: $tType,B: $tType,M: $tType,F: M > B,X: M] :
( ( monomo1564381680ondetT @ M @ B @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
= ( F @ X ) ) ).
% nondetT.case
thf(fact_45_ext,axiom,
! [B: $tType,A: $tType,F: A > B,G2: A > B] :
( ! [X4: A] :
( ( F @ X4 )
= ( G2 @ X4 ) )
=> ( F = G2 ) ) ).
% ext
thf(fact_46_nondetM__base_Orun__put__nondet,axiom,
! [B: $tType,M: $tType,A: $tType,Put2: B > M > M,S3: B,M4: monomo1453983439ondetT @ A @ M] :
( ( monomo603716163nondet @ A @ M @ ( monomo410456285nondet @ B @ M @ A @ Put2 @ S3 @ M4 ) )
= ( Put2 @ S3 @ ( monomo603716163nondet @ A @ M @ M4 ) ) ) ).
% nondetM_base.run_put_nondet
thf(fact_47_fun_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,D: $tType,G2: B > C,F: A > B,V: D > A] :
( ( comp @ B @ C @ D @ G2 @ ( comp @ A @ B @ D @ F @ V ) )
= ( comp @ A @ C @ D @ ( comp @ B @ C @ A @ G2 @ F ) @ V ) ) ).
% fun.map_comp
thf(fact_48_function__factors__left,axiom,
! [A: $tType,C: $tType,B: $tType,G2: A > B,F: A > C] :
( ( ! [X2: A,Y2: A] :
( ( ( G2 @ X2 )
= ( G2 @ Y2 ) )
=> ( ( F @ X2 )
= ( F @ Y2 ) ) ) )
= ( ? [H2: B > C] :
( F
= ( comp @ B @ C @ A @ H2 @ G2 ) ) ) ) ).
% function_factors_left
thf(fact_49_function__factors__right,axiom,
! [C: $tType,B: $tType,A: $tType,G2: B > C,F: A > C] :
( ( ! [X2: A] :
? [Y2: B] :
( ( G2 @ Y2 )
= ( F @ X2 ) ) )
= ( ? [H2: A > B] :
( F
= ( comp @ B @ C @ A @ G2 @ H2 ) ) ) ) ).
% function_factors_right
thf(fact_50_monad__altc_Oaxioms_I2_J,axiom,
! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
=> ( monomo555039548axioms @ M @ A @ C @ Bind @ Altc ) ) ).
% monad_altc.axioms(2)
thf(fact_51_cinsert__cDiff__single,axiom,
! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
= ( counta2111716221insert @ A @ A3 @ A2 ) ) ).
% cinsert_cDiff_single
thf(fact_52_monad__altc3_Oaxioms_I1_J,axiom,
! [A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo1425736922_altc3 @ A @ M @ C @ Return @ Bind @ Altc )
=> ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% monad_altc3.axioms(1)
thf(fact_53_nondetT_Orec,axiom,
! [A: $tType,D: $tType,M: $tType,F: M > D,X: M] :
( ( monomo1291740684ondetT @ M @ D @ A @ F @ ( monomo1975189248ondetT @ M @ A @ X ) )
= ( F @ X ) ) ).
% nondetT.rec
thf(fact_54_monad__state__altc_Oaxioms_I2_J,axiom,
! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
=> ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc ) ) ).
% monad_state_altc.axioms(2)
thf(fact_55_cDiff__idemp,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ B3 )
= ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% cDiff_idemp
thf(fact_56_cDiff__cancel,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ A2 )
= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% cDiff_cancel
thf(fact_57_cDiff__cempty,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
= A2 ) ).
% cDiff_cempty
thf(fact_58_cempty__cDiff,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ A2 )
= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% cempty_cDiff
thf(fact_59_cDiff__cinsert,axiom,
! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
= ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ).
% cDiff_cinsert
thf(fact_60_cDiff__cinsert2,axiom,
! [A: $tType,A2: countable_Set_cset @ A,A3: A,B3: countable_Set_cset @ A] :
( ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ B3 ) )
= ( minus_minus @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) ) ).
% cDiff_cinsert2
thf(fact_61_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_62_cDiff__single__cinsert,axiom,
! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) ) ) ).
% cDiff_single_cinsert
thf(fact_63_monad__state__altc_Oaxioms_I3_J,axiom,
! [A: $tType,S: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
=> ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc ) ) ).
% monad_state_altc.axioms(3)
thf(fact_64_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_65_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A3 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_66_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A4: A > B,B4: A > B,X2: A] : ( minus_minus @ B @ ( A4 @ X2 ) @ ( B4 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_67_cinsert__cDiff,axiom,
! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ A2 )
=> ( ( counta2111716221insert @ A @ A3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) )
= A2 ) ) ).
% cinsert_cDiff
thf(fact_68_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_69_csubset__antisym,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% csubset_antisym
thf(fact_70_all__not__cin__conv,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( ! [X2: A] :
~ ( countable_Set_cin @ A @ X2 @ A2 ) )
= ( A2
= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% all_not_cin_conv
thf(fact_71_cempty__iff,axiom,
! [A: $tType,C2: A] :
~ ( countable_Set_cin @ A @ C2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% cempty_iff
thf(fact_72_cinsert__iff,axiom,
! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
= ( ( A3 = B2 )
| ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% cinsert_iff
thf(fact_73_cinsertCI,axiom,
! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
( ( ~ ( countable_Set_cin @ A @ A3 @ B3 )
=> ( A3 = B2 ) )
=> ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% cinsertCI
thf(fact_74_csubsetI,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ! [X4: A] :
( ( countable_Set_cin @ A @ X4 @ A2 )
=> ( countable_Set_cin @ A @ X4 @ B3 ) )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% csubsetI
thf(fact_75_cempty__fsubsetI,axiom,
! [A: $tType,X: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( bot_bot @ ( countable_Set_cset @ A ) ) @ X ) ).
% cempty_fsubsetI
thf(fact_76_csubset__cempty,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) )
= ( A2
= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% csubset_cempty
thf(fact_77_cDiff__iff,axiom,
! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
= ( ( countable_Set_cin @ A @ C2 @ A2 )
& ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% cDiff_iff
thf(fact_78_cDiffI,axiom,
! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ C2 @ A2 )
=> ( ~ ( countable_Set_cin @ A @ C2 @ B3 )
=> ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% cDiffI
thf(fact_79_cinsert__csubset,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
= ( ( countable_Set_cin @ A @ X @ B3 )
& ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% cinsert_csubset
thf(fact_80_cinsert__cDiff1,axiom,
! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ X @ B3 )
=> ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
= ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% cinsert_cDiff1
thf(fact_81_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_82_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_83_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_84_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_85_csubset__cinsert,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ~ ( countable_Set_cin @ A @ X @ A2 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
= ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ).
% csubset_cinsert
thf(fact_86_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_87_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( ord_less_eq @ A @ A5 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_88_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.trans
thf(fact_89_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less_eq @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_90_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% dual_order.refl
thf(fact_91_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% order_trans
thf(fact_92_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_93_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_94_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_95_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( ord_less_eq @ A @ B5 @ A5 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_96_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_97_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z2 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z2 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z2 )
=> ~ ( ord_less_eq @ A @ Z2 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_98_eqcelem__imp__iff,axiom,
! [A: $tType,X: A,Y: A,A2: countable_Set_cset @ A] :
( ( X = Y )
=> ( ( countable_Set_cin @ A @ X @ A2 )
= ( countable_Set_cin @ A @ Y @ A2 ) ) ) ).
% eqcelem_imp_iff
thf(fact_99_cset__eq__csubset,axiom,
! [A: $tType] :
( ( ^ [Y3: countable_Set_cset @ A,Z: countable_Set_cset @ A] : Y3 = Z )
= ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
& ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% cset_eq_csubset
thf(fact_100_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% order.trans
thf(fact_101_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_102_eqcset__imp__iff,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
( ( A2 = B3 )
=> ( ( countable_Set_cin @ A @ X @ A2 )
= ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% eqcset_imp_iff
thf(fact_103_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_104_if__split__cin2,axiom,
! [A: $tType,A3: A,Q: $o,X: countable_Set_cset @ A,Y: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ ( if @ ( countable_Set_cset @ A ) @ Q @ X @ Y ) )
= ( ( Q
=> ( countable_Set_cin @ A @ A3 @ X ) )
& ( ~ Q
=> ( countable_Set_cin @ A @ A3 @ Y ) ) ) ) ).
% if_split_cin2
thf(fact_105_if__split__cin1,axiom,
! [A: $tType,Q: $o,X: A,Y: A,B2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ ( if @ A @ Q @ X @ Y ) @ B2 )
= ( ( Q
=> ( countable_Set_cin @ A @ X @ B2 ) )
& ( ~ Q
=> ( countable_Set_cin @ A @ Y @ B2 ) ) ) ) ).
% if_split_cin1
thf(fact_106_eq__cmem__trans,axiom,
! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
( ( A3 = B2 )
=> ( ( countable_Set_cin @ A @ B2 @ A2 )
=> ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% eq_cmem_trans
thf(fact_107_csubset__trans,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% csubset_trans
thf(fact_108_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_109_csubset__refl,axiom,
! [A: $tType,A2: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ A2 ) ).
% csubset_refl
thf(fact_110_cequalityD2,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ).
% cequalityD2
thf(fact_111_cequalityD1,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( A2 = B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% cequalityD1
thf(fact_112_cequalityCE,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
( ( A2 = B3 )
=> ( ( ( countable_Set_cin @ A @ C2 @ A2 )
=> ~ ( countable_Set_cin @ A @ C2 @ B3 ) )
=> ~ ( ~ ( countable_Set_cin @ A @ C2 @ A2 )
=> ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ) ).
% cequalityCE
thf(fact_113_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_114_cequalityE,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% cequalityE
thf(fact_115_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y3: A,Z: A] : Y3 = Z )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% eq_iff
thf(fact_116_csubsetD,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( countable_Set_cin @ A @ C2 @ A2 )
=> ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% csubsetD
thf(fact_117_cset__eqI,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ! [X4: A] :
( ( countable_Set_cin @ A @ X4 @ A2 )
= ( countable_Set_cin @ A @ X4 @ B3 ) )
=> ( A2 = B3 ) ) ).
% cset_eqI
thf(fact_118_cin__mono,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,X: A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( countable_Set_cin @ A @ X @ A2 )
=> ( countable_Set_cin @ A @ X @ B3 ) ) ) ).
% cin_mono
thf(fact_119_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C2: B] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_120_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_121_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_122_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_123_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F2: A > B,G: A > B] :
! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ).
% le_fun_def
thf(fact_124_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F @ X4 ) @ ( G2 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F @ G2 ) ) ) ).
% le_funI
thf(fact_125_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_126_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F @ G2 )
=> ( ord_less_eq @ B @ ( F @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_127_equals__cemptyI,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ! [Y4: A] :
~ ( countable_Set_cin @ A @ Y4 @ A2 )
=> ( A2
= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% equals_cemptyI
thf(fact_128_equals__cemptyD,axiom,
! [A: $tType,A2: countable_Set_cset @ A,A3: A] :
( ( A2
= ( bot_bot @ ( countable_Set_cset @ A ) ) )
=> ~ ( countable_Set_cin @ A @ A3 @ A2 ) ) ).
% equals_cemptyD
thf(fact_129_ex__cin__conv,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
( ( ? [X2: A] : ( countable_Set_cin @ A @ X2 @ A2 ) )
= ( A2
!= ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ).
% ex_cin_conv
thf(fact_130_cemptyE,axiom,
! [A: $tType,A3: A] :
~ ( countable_Set_cin @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% cemptyE
thf(fact_131_mk__disjoint__cinsert,axiom,
! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ A2 )
=> ? [B7: countable_Set_cset @ A] :
( ( A2
= ( counta2111716221insert @ A @ A3 @ B7 ) )
& ~ ( countable_Set_cin @ A @ A3 @ B7 ) ) ) ).
% mk_disjoint_cinsert
thf(fact_132_cinsert__absorb,axiom,
! [A: $tType,A3: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ A2 )
=> ( ( counta2111716221insert @ A @ A3 @ A2 )
= A2 ) ) ).
% cinsert_absorb
thf(fact_133_cinsert__ident,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ~ ( countable_Set_cin @ A @ X @ A2 )
=> ( ~ ( countable_Set_cin @ A @ X @ B3 )
=> ( ( ( counta2111716221insert @ A @ X @ A2 )
= ( counta2111716221insert @ A @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% cinsert_ident
thf(fact_134_set__cinsert,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ X @ A2 )
=> ~ ! [B7: countable_Set_cset @ A] :
( ( A2
= ( counta2111716221insert @ A @ X @ B7 ) )
=> ( countable_Set_cin @ A @ X @ B7 ) ) ) ).
% set_cinsert
thf(fact_135_cinsertI2,axiom,
! [A: $tType,A3: A,B3: countable_Set_cset @ A,B2: A] :
( ( countable_Set_cin @ A @ A3 @ B3 )
=> ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% cinsertI2
thf(fact_136_cinsertI1,axiom,
! [A: $tType,A3: A,B3: countable_Set_cset @ A] : ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% cinsertI1
thf(fact_137_cinsertE,axiom,
! [A: $tType,A3: A,B2: A,A2: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ A3 @ ( counta2111716221insert @ A @ B2 @ A2 ) )
=> ( ( A3 != B2 )
=> ( countable_Set_cin @ A @ A3 @ A2 ) ) ) ).
% cinsertE
thf(fact_138_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
=> ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_139_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( bot_bot @ A ) )
= ( A3
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_140_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A3 ) ) ).
% bot.extremum
thf(fact_141_cDiffD2,axiom,
! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
=> ~ ( countable_Set_cin @ A @ C2 @ B3 ) ) ).
% cDiffD2
thf(fact_142_cDiffD1,axiom,
! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
=> ( countable_Set_cin @ A @ C2 @ A2 ) ) ).
% cDiffD1
thf(fact_143_cDiffE,axiom,
! [A: $tType,C2: A,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( countable_Set_cin @ A @ C2 @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
=> ~ ( ( countable_Set_cin @ A @ C2 @ A2 )
=> ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% cDiffE
thf(fact_144_csubset__cinsertI2,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,B2: A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ B2 @ B3 ) ) ) ).
% csubset_cinsertI2
thf(fact_145_csubset__cinsertI,axiom,
! [A: $tType,B3: countable_Set_cset @ A,A3: A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ ( counta2111716221insert @ A @ A3 @ B3 ) ) ).
% csubset_cinsertI
thf(fact_146_cinsert__mono,axiom,
! [A: $tType,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,A3: A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ C3 @ D3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ A3 @ C3 ) @ ( counta2111716221insert @ A @ A3 @ D3 ) ) ) ).
% cinsert_mono
thf(fact_147_cDiff__csubset,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] : ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ A2 ) ).
% cDiff_csubset
thf(fact_148_double__cDiff,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
=> ( ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_cDiff
thf(fact_149_cminus__mono,axiom,
! [A: $tType,A2: countable_Set_cset @ A,C3: countable_Set_cset @ A,D3: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ C3 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ D3 @ B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ C3 @ D3 ) ) ) ) ).
% cminus_mono
thf(fact_150_csubset__cinsert__iff,axiom,
! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
= ( ( ( countable_Set_cin @ A @ X @ A2 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
& ( ~ ( countable_Set_cin @ A @ X @ A2 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ).
% csubset_cinsert_iff
thf(fact_151_csingleton__iff,axiom,
! [A: $tType,B2: A,A3: A] :
( ( countable_Set_cin @ A @ B2 @ ( counta2111716221insert @ A @ A3 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
= ( B2 = A3 ) ) ).
% csingleton_iff
thf(fact_152_cinsert__cDiff__if,axiom,
! [A: $tType,X: A,B3: countable_Set_cset @ A,A2: countable_Set_cset @ A] :
( ( ( countable_Set_cin @ A @ X @ B3 )
=> ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
= ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) )
& ( ~ ( countable_Set_cin @ A @ X @ B3 )
=> ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ B3 )
= ( counta2111716221insert @ A @ X @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ).
% cinsert_cDiff_if
thf(fact_153_csubset__csingletonD,axiom,
! [A: $tType,A2: countable_Set_cset @ A,X: A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
=> ( ( A2
= ( bot_bot @ ( countable_Set_cset @ A ) ) )
| ( A2
= ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) ) ) ).
% csubset_csingletonD
thf(fact_154_cDiff__cinsert__absorb,axiom,
! [A: $tType,X: A,A2: countable_Set_cset @ A] :
( ~ ( countable_Set_cin @ A @ X @ A2 )
=> ( ( minus_minus @ ( countable_Set_cset @ A ) @ ( counta2111716221insert @ A @ X @ A2 ) @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) )
= A2 ) ) ).
% cDiff_cinsert_absorb
thf(fact_155_cpsubset__finsert__iff,axiom,
! [A: $tType,A2: countable_Set_cset @ A,X: A,B3: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ B3 ) )
= ( ( ( countable_Set_cin @ A @ X @ B3 )
=> ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 ) )
& ( ~ ( countable_Set_cin @ A @ X @ B3 )
=> ( ( ( countable_Set_cin @ A @ X @ A2 )
=> ( ord_less @ ( countable_Set_cset @ A ) @ ( minus_minus @ ( countable_Set_cset @ A ) @ A2 @ ( counta2111716221insert @ A @ X @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ) @ B3 ) )
& ( ~ ( countable_Set_cin @ A @ X @ A2 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ) ) ) ) ).
% cpsubset_finsert_iff
thf(fact_156_monad__state__altc_Ointro,axiom,
! [S: $tType,A: $tType,M: $tType,C: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
=> ( ( monomo439771545d_altc @ A @ M @ C @ Return @ Bind @ Altc )
=> ( ( monomo1324021455axioms @ S @ M @ C @ Get @ Put2 @ Altc )
=> ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc ) ) ) ) ).
% monad_state_altc.intro
thf(fact_157_monad__state__altc__def,axiom,
! [C: $tType,S: $tType,M: $tType,A: $tType] :
( ( monomo1036387116e_altc @ A @ M @ S @ C )
= ( ^ [Return2: A > M,Bind2: M > ( A > M ) > M,Get2: ( S > M ) > M,Put: S > M > M,Altc2: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo109450930_state @ A @ M @ S @ Return2 @ Bind2 @ Get2 @ Put )
& ( monomo439771545d_altc @ A @ M @ C @ Return2 @ Bind2 @ Altc2 )
& ( monomo1324021455axioms @ S @ M @ C @ Get2 @ Put @ Altc2 ) ) ) ) ).
% monad_state_altc_def
thf(fact_158_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ( order_Greatest @ A @ P )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_159_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P: A > $o,X: A,Q: A > $o] :
( ( P @ X )
=> ( ! [Y4: A] :
( ( P @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X ) )
=> ( ! [X4: A] :
( ( P @ X4 )
=> ( ! [Y5: A] :
( ( P @ Y5 )
=> ( ord_less_eq @ A @ Y5 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_160_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
| ~ ( ord_less_eq @ A @ A3 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_161_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X5: $o > A,Y6: $o > A] :
( ( ord_less_eq @ A @ ( X5 @ $false ) @ ( Y6 @ $false ) )
& ( ord_less_eq @ A @ ( X5 @ $true ) @ ( Y6 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_162_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B8: B,A7: B] :
( ( ~ ( ord_less_eq @ B @ B8 @ A7 ) )
= ( ord_less @ B @ A7 @ B8 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_163_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( A3
= ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_164_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F @ B2 )
= C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ B @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ B @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_165_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_166_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_167_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y4: A] : ( ord_less @ A @ Y4 @ X ) ) ).
% lt_ex
thf(fact_168_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_169_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_170_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_171_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_172_cpsubset__trans,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
=> ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% cpsubset_trans
thf(fact_173_dense,axiom,
! [A: $tType] :
( ( dense_order @ 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_174_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_175_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_176_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% less_asym'
thf(fact_177_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_trans
thf(fact_178_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_179_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_180_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_181_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_182_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_183_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_184_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_185_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X4: A] :
( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X4 )
=> ( P @ Y5 ) )
=> ( P @ X4 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_186_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_187_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_188_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_189_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_190_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_191_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_192_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_193_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X6: A] : ( P2 @ X6 ) )
= ( ^ [P3: A > $o] :
? [N2: A] :
( ( P3 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P3 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_194_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A6: A,B6: A] :
( ( ord_less @ A @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B6: A] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_195_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_196_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ 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_197_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_198_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_199_monad__state_Obind__put,axiom,
! [S: $tType,M: $tType,A: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,M4: M,F: A > M] :
( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
=> ( ( Bind @ ( Put2 @ S3 @ M4 ) @ F )
= ( Put2 @ S3 @ ( Bind @ M4 @ F ) ) ) ) ).
% monad_state.bind_put
thf(fact_200_monad__state_Oput__put,axiom,
! [A: $tType,S: $tType,M: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,S4: S,M4: M] :
( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
=> ( ( Put2 @ S3 @ ( Put2 @ S4 @ M4 ) )
= ( Put2 @ S4 @ M4 ) ) ) ).
% monad_state.put_put
thf(fact_201_monad__state_Oput__get,axiom,
! [A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,S3: S,F: S > M] :
( ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 )
=> ( ( Put2 @ S3 @ ( Get @ F ) )
= ( Put2 @ S3 @ ( F @ S3 ) ) ) ) ).
% monad_state.put_get
thf(fact_202_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_203_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_204_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less @ A @ X2 @ Y2 )
| ( X2 = Y2 ) ) ) ) ) ).
% le_less
thf(fact_205_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ( X2 != Y2 ) ) ) ) ) ).
% less_le
thf(fact_206_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less @ B @ X4 @ Y4 )
=> ( ord_less @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_207_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less_eq @ A @ X4 @ Y4 )
=> ( ord_less_eq @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_208_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X4: B,Y4: B] :
( ( ord_less_eq @ B @ X4 @ Y4 )
=> ( ord_less_eq @ A @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ A @ A3 @ ( F @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_209_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F @ B2 ) @ C2 )
=> ( ! [X4: A,Y4: A] :
( ( ord_less @ A @ X4 @ Y4 )
=> ( ord_less @ C @ ( F @ X4 ) @ ( F @ Y4 ) ) )
=> ( ord_less @ C @ ( F @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_210_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_211_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_212_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_213_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_214_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_215_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_216_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% le_less_trans
thf(fact_217_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% less_le_trans
thf(fact_218_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z2 @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_ge
thf(fact_219_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z2: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z2 ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ).
% dense_le
thf(fact_220_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_221_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ 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_222_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y2: A] :
( ( ord_less_eq @ A @ X2 @ Y2 )
& ~ ( ord_less_eq @ A @ Y2 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_223_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_224_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_225_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_226_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less @ A @ A5 @ B5 )
| ( A5 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_227_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B5: A] :
( ( ord_less_eq @ A @ A5 @ B5 )
& ( A5 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_228_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_229_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_230_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z2 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_231_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z2: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_232_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_233_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less @ A @ B5 @ A5 )
| ( A5 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_234_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A5: A] :
( ( ord_less_eq @ A @ B5 @ A5 )
& ( A5 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_235_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_236_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_237_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_238_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_239_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_240_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_241_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_242_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_243_cpsubsetD,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C2: A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( countable_Set_cin @ A @ C2 @ A2 )
=> ( countable_Set_cin @ A @ C2 @ B3 ) ) ) ).
% cpsubsetD
thf(fact_244_not__cpsubset__cempty,axiom,
! [A: $tType,A2: countable_Set_cset @ A] :
~ ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ ( bot_bot @ ( countable_Set_cset @ A ) ) ) ).
% not_cpsubset_cempty
thf(fact_245_cpsubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( countable_Set_cset @ A ) )
= ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
& ( A4 != B4 ) ) ) ) ).
% cpsubset_eq
thf(fact_246_less__cset__def,axiom,
! [A: $tType] :
( ( ord_less @ ( countable_Set_cset @ A ) )
= ( ^ [Xs: countable_Set_cset @ A,Ys: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ Xs @ Ys )
& ( Xs != Ys ) ) ) ) ).
% less_cset_def
thf(fact_247_cpsubset__imp__fsubset,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 ) ) ).
% cpsubset_imp_fsubset
thf(fact_248_cpsubset__csubset__trans,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less_eq @ ( countable_Set_cset @ A ) @ B3 @ C3 )
=> ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% cpsubset_csubset_trans
thf(fact_249_csubset__cpsubset__trans,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A,C3: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ( ( ord_less @ ( countable_Set_cset @ A ) @ B3 @ C3 )
=> ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ C3 ) ) ) ).
% csubset_cpsubset_trans
thf(fact_250_csubset__not__fsubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( countable_Set_cset @ A ) )
= ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) @ A4 @ B4 )
& ~ ( ord_less_eq @ ( countable_Set_cset @ A ) @ B4 @ A4 ) ) ) ) ).
% csubset_not_fsubset_eq
thf(fact_251_csubset__iff__pfsubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( countable_Set_cset @ A ) )
= ( ^ [A4: countable_Set_cset @ A,B4: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A4 @ B4 )
| ( A4 = B4 ) ) ) ) ).
% csubset_iff_pfsubset_eq
thf(fact_252_monad__state__altc_Oaxioms_I1_J,axiom,
! [C: $tType,A: $tType,M: $tType,S: $tType,Return: A > M,Bind: M > ( A > M ) > M,Get: ( S > M ) > M,Put2: S > M > M,Altc: ( countable_Set_cset @ C ) > ( C > M ) > M] :
( ( monomo1036387116e_altc @ A @ M @ S @ C @ Return @ Bind @ Get @ Put2 @ Altc )
=> ( monomo109450930_state @ A @ M @ S @ Return @ Bind @ Get @ Put2 ) ) ).
% monad_state_altc.axioms(1)
thf(fact_253_cpsubset__imp__ex__fmem,axiom,
! [A: $tType,A2: countable_Set_cset @ A,B3: countable_Set_cset @ A] :
( ( ord_less @ ( countable_Set_cset @ A ) @ A2 @ B3 )
=> ? [B6: A] : ( countable_Set_cin @ A @ B6 @ ( minus_minus @ ( countable_Set_cset @ A ) @ B3 @ A2 ) ) ) ).
% cpsubset_imp_ex_fmem
thf(fact_254_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ~ ( ord_less_eq @ A @ T @ X7 ) ) ) ).
% minf(8)
thf(fact_255_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z3: A] :
! [X7: A] :
( ( ord_less @ A @ X7 @ Z3 )
=> ( ord_less_eq @ A @ X7 @ T ) ) ) ).
% minf(6)
% Type constructors (19)
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A8: $tType,A9: $tType] :
( ( order_bot @ A9 )
=> ( order_bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A8: $tType,A9: $tType] :
( ( preorder @ A9 )
=> ( preorder @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A8: $tType,A9: $tType] :
( ( order @ A9 )
=> ( order @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A8: $tType,A9: $tType] :
( ( ord @ A9 )
=> ( ord @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A8: $tType,A9: $tType] :
( ( bot @ A9 )
=> ( bot @ ( A8 > A9 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A8: $tType,A9: $tType] :
( ( minus @ A9 )
=> ( minus @ ( A8 > A9 ) ) ) ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_1,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_2,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_3,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_4,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_5,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_6,axiom,
minus @ $o ).
thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder__bot_7,axiom,
! [A8: $tType] : ( order_bot @ ( countable_Set_cset @ A8 ) ) ).
thf(tcon_Countable__Set__Type_Ocset___Orderings_Opreorder_8,axiom,
! [A8: $tType] : ( preorder @ ( countable_Set_cset @ A8 ) ) ).
thf(tcon_Countable__Set__Type_Ocset___Orderings_Oorder_9,axiom,
! [A8: $tType] : ( order @ ( countable_Set_cset @ A8 ) ) ).
thf(tcon_Countable__Set__Type_Ocset___Orderings_Oord_10,axiom,
! [A8: $tType] : ( ord @ ( countable_Set_cset @ A8 ) ) ).
thf(tcon_Countable__Set__Type_Ocset___Orderings_Obot_11,axiom,
! [A8: $tType] : ( bot @ ( countable_Set_cset @ A8 ) ) ).
thf(tcon_Countable__Set__Type_Ocset___Groups_Ominus_12,axiom,
! [A8: $tType] : ( minus @ ( countable_Set_cset @ A8 ) ) ).
% Helper facts (3)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( monomo955616121nondet @ c @ m @ a @ mergec @ ( counta2111716221insert @ c @ x @ ( bot_bot @ ( countable_Set_cset @ c ) ) ) @ f )
= ( f @ x ) ) ).
%------------------------------------------------------------------------------