TPTP Problem File: ITP058^2.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : ITP058^2 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer FLPTheorem problem prob_419__3294834_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 : FLPTheorem/prob_419__3294834_1 [Des21]
% Status : Theorem
% Rating : 0.33 v8.1.0, 0.25 v7.5.0
% Syntax : Number of formulae : 343 ( 122 unt; 64 typ; 0 def)
% Number of atoms : 686 ( 312 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 5200 ( 160 ~; 20 |; 52 &;4519 @)
% ( 0 <=>; 449 =>; 0 <=; 0 <~>)
% Maximal formula depth : 28 ( 8 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 550 ( 550 >; 0 *; 0 +; 0 <<)
% Number of symbols : 60 ( 59 usr; 1 con; 0-7 aty)
% Number of variables : 1173 ( 24 ^;1023 !; 64 ?;1173 :)
% ( 62 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:16:10.530
%------------------------------------------------------------------------------
% Could-be-implicit typings (10)
thf(ty_t_AsynchronousSystem_Oconfiguration_Oconfiguration__ext,type,
configuration_ext: $tType > $tType > $tType > $tType > $tType ).
thf(ty_t_AsynchronousSystem_OmessageValue,type,
messageValue: $tType > $tType ).
thf(ty_t_AsynchronousSystem_Omessage,type,
message: $tType > $tType > $tType ).
thf(ty_t_Product__Type_Ounit,type,
product_unit: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Set_Oset,type,
set: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_v,type,
v: $tType ).
thf(ty_tf_s,type,
s: $tType ).
thf(ty_tf_p,type,
p: $tType ).
% Explicit typings (54)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osize,type,
size:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[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_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__nonzero__semiring,type,
linord1659791738miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_c_FLPSystem_OflpSystem,type,
flpSystem:
!>[P: $tType,S: $tType,V: $tType] : ( ( P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat ) > $o ) ).
thf(sy_c_FLPTheorem__Mirabelle__yavhmxymmt_OflpPseudoConsensus,type,
fLPThe1922692578sensus:
!>[P: $tType,S: $tType,V: $tType] : ( ( P > S > ( messageValue @ V ) > S ) > ( P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat ) > ( P > S ) > $o ) ).
thf(sy_c_FLPTheorem__Mirabelle__yavhmxymmt_OflpPseudoConsensus_OinfiniteExecutionCfg,type,
fLPThe137922386ionCfg:
!>[P: $tType,V: $tType,S: $tType] : ( ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) ).
thf(sy_c_FLPTheorem__Mirabelle__yavhmxymmt_OflpPseudoConsensus_OinfiniteExecutionMsg,type,
fLPThe221390223ionMsg:
!>[P: $tType,V: $tType,S: $tType] : ( ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > ( list @ ( message @ P @ V ) ) ) ).
thf(sy_c_FLPTheorem__Mirabelle__yavhmxymmt_OflpPseudoConsensus__axioms,type,
fLPThe128449925axioms:
!>[P: $tType,S: $tType,V: $tType] : ( ( P > S > ( messageValue @ V ) > S ) > ( P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat ) > ( P > S ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist__ex1,type,
list_ex1:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Omap__tailrec__rev,type,
map_tailrec_rev:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Omember,type,
member:
!>[A: $tType] : ( ( list @ A ) > A > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord_Olexordp__eq,type,
lexordp_eq:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_member,type,
member2:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_cfg,type,
cfg: configuration_ext @ p @ v @ s @ product_unit ).
thf(sy_v_fStepCfg____,type,
fStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
thf(sy_v_fStepMsg____,type,
fStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ).
thf(sy_v_fe____,type,
fe: nat > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ).
thf(sy_v_ft____,type,
ft: nat > ( list @ ( message @ p @ v ) ) ).
% Relevant facts (252)
thf(fact_0_ft__def,axiom,
( ft
= ( fLPThe221390223ionMsg @ p @ v @ s @ cfg @ fStepCfg @ fStepMsg ) ) ).
% ft_def
thf(fact_1_fe__def,axiom,
( fe
= ( fLPThe137922386ionCfg @ p @ v @ s @ cfg @ fStepCfg @ fStepMsg ) ) ).
% fe_def
thf(fact_2_infiniteExecutionCfg_Osimps_I1_J,axiom,
! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] :
( ( fLPThe137922386ionCfg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
= ( cons @ ( configuration_ext @ p @ v @ s @ product_unit ) @ Cfg @ ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ) ).
% infiniteExecutionCfg.simps(1)
thf(fact_3_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_4_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_5_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_6_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_7_list_Oinducts,axiom,
! [A: $tType,P2: ( list @ A ) > $o,List: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X1: A,X2: list @ A] :
( ( P2 @ X2 )
=> ( P2 @ ( cons @ A @ X1 @ X2 ) ) )
=> ( P2 @ List ) ) ) ).
% list.inducts
thf(fact_8_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y2: A,Ys: list @ A] :
( Xs
= ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% neq_Nil_conv
thf(fact_9_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys2: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A] : ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys3: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ) ).
% list_induct2'
thf(fact_10_splice_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( P2 @ Ys3 @ Xs2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ Ys3 ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% splice.induct
thf(fact_11_induct__list012,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Zs: list @ A] :
( ( P2 @ Zs )
=> ( ( P2 @ ( cons @ A @ Y3 @ Zs ) )
=> ( P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% induct_list012
thf(fact_12_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: list @ A] :
( ! [X: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ Xs2 ) )
=> ( X3
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_13_min__list_Oinduct,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ! [X: A,Xs2: list @ A] :
( ! [X213: A,X223: list @ A] :
( ( Xs2
= ( cons @ A @ X213 @ X223 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( P2 @ ( nil @ A ) )
=> ( P2 @ A0 ) ) ) ) ).
% min_list.induct
thf(fact_14_infiniteExecutionMsg_Osimps_I1_J,axiom,
! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] :
( ( fLPThe221390223ionMsg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
= ( nil @ ( message @ p @ v ) ) ) ).
% infiniteExecutionMsg.simps(1)
thf(fact_15_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X3: A] :
( ( ( zero_zero @ A )
= X3 )
= ( X3
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_16_not__Cons__self2,axiom,
! [A: $tType,X3: A,Xs: list @ A] :
( ( cons @ A @ X3 @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_17_strict__sorted_Oinduct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A,Ys3: list @ A] :
( ( P2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% strict_sorted.induct
thf(fact_18_strict__sorted_Ocases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ~ ! [X: A,Ys3: list @ A] :
( X3
!= ( cons @ A @ X @ Ys3 ) ) ) ) ).
% strict_sorted.cases
thf(fact_19_map__tailrec__rev_Oinduct,axiom,
! [A: $tType,B: $tType,P2: ( A > B ) > ( list @ A ) > ( list @ B ) > $o,A0: A > B,A1: list @ A,A2: list @ B] :
( ! [F: A > B,X_1: list @ B] : ( P2 @ F @ ( nil @ A ) @ X_1 )
=> ( ! [F: A > B,A3: A,As: list @ A,Bs: list @ B] :
( ( P2 @ F @ As @ ( cons @ B @ ( F @ A3 ) @ Bs ) )
=> ( P2 @ F @ ( cons @ A @ A3 @ As ) @ Bs ) )
=> ( P2 @ A0 @ A1 @ A2 ) ) ) ).
% map_tailrec_rev.induct
thf(fact_20_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_21_successively_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P3: A > A > $o] : ( P2 @ P3 @ ( nil @ A ) )
=> ( ! [P3: A > A > $o,X: A] : ( P2 @ P3 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [P3: A > A > $o,X: A,Y3: A,Xs2: list @ A] :
( ( P2 @ P3 @ ( cons @ A @ Y3 @ Xs2 ) )
=> ( P2 @ P3 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% successively.induct
thf(fact_22_arg__min__list_Oinduct,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [P2: ( A > B ) > ( list @ A ) > $o,A0: A > B,A1: list @ A] :
( ! [F: A > B,X: A] : ( P2 @ F @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [F: A > B,X: A,Y3: A,Zs: list @ A] :
( ( P2 @ F @ ( cons @ A @ Y3 @ Zs ) )
=> ( P2 @ F @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Zs ) ) ) )
=> ( ! [A3: A > B] : ( P2 @ A3 @ ( nil @ A ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ) ).
% arg_min_list.induct
thf(fact_23_remdups__adj_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,A0: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X: A] : ( P2 @ ( cons @ A @ X @ ( nil @ A ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A] :
( ( ( X = Y3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) ) )
=> ( ( ( X != Y3 )
=> ( P2 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( P2 @ ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) )
=> ( P2 @ A0 ) ) ) ) ).
% remdups_adj.induct
thf(fact_24_sorted__wrt_Oinduct,axiom,
! [A: $tType,P2: ( A > A > $o ) > ( list @ A ) > $o,A0: A > A > $o,A1: list @ A] :
( ! [P3: A > A > $o] : ( P2 @ P3 @ ( nil @ A ) )
=> ( ! [P3: A > A > $o,X: A,Ys3: list @ A] :
( ( P2 @ P3 @ Ys3 )
=> ( P2 @ P3 @ ( cons @ A @ X @ Ys3 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% sorted_wrt.induct
thf(fact_25_remdups__adj_Ocases,axiom,
! [A: $tType,X3: list @ A] :
( ( X3
!= ( nil @ A ) )
=> ( ! [X: A] :
( X3
!= ( cons @ A @ X @ ( nil @ A ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( X3
!= ( cons @ A @ X @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_26_transpose_Ocases,axiom,
! [A: $tType,X3: list @ ( list @ A )] :
( ( X3
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ~ ! [X: A,Xs2: list @ A,Xss: list @ ( list @ A )] :
( X3
!= ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs2 ) @ Xss ) ) ) ) ).
% transpose.cases
thf(fact_27_shuffles_Oinduct,axiom,
! [A: $tType,P2: ( list @ A ) > ( list @ A ) > $o,A0: list @ A,A1: list @ A] :
( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [Xs2: list @ A] : ( P2 @ Xs2 @ ( nil @ A ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( P2 @ Xs2 @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ( P2 @ ( cons @ A @ X @ Xs2 ) @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% shuffles.induct
thf(fact_28_infiniteExecutionCfg_Oelims,axiom,
! [X3: configuration_ext @ p @ v @ s @ product_unit,Xa: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),Xb: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),Xc: nat,Y: list @ ( configuration_ext @ p @ v @ s @ product_unit )] :
( ( ( fLPThe137922386ionCfg @ p @ v @ s @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc
= ( zero_zero @ nat ) )
=> ( Y
!= ( cons @ ( configuration_ext @ p @ v @ s @ product_unit ) @ X3 @ ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ) )
=> ~ ! [N: nat] :
( ( Xc
= ( suc @ N ) )
=> ( Y
!= ( Xa @ ( fLPThe137922386ionCfg @ p @ v @ s @ X3 @ Xa @ Xb @ N ) @ ( fLPThe221390223ionMsg @ p @ v @ s @ X3 @ Xa @ Xb @ N ) ) ) ) ) ) ).
% infiniteExecutionCfg.elims
thf(fact_29_infiniteExecutionMsg_Oelims,axiom,
! [X3: configuration_ext @ p @ v @ s @ product_unit,Xa: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),Xb: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),Xc: nat,Y: list @ ( message @ p @ v )] :
( ( ( fLPThe221390223ionMsg @ p @ v @ s @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ ( message @ p @ v ) ) ) )
=> ~ ! [N: nat] :
( ( Xc
= ( suc @ N ) )
=> ( Y
!= ( Xb @ ( fLPThe137922386ionCfg @ p @ v @ s @ X3 @ Xa @ Xb @ N ) @ ( fLPThe221390223ionMsg @ p @ v @ s @ X3 @ Xa @ Xb @ N ) ) ) ) ) ) ).
% infiniteExecutionMsg.elims
thf(fact_30_infiniteExecutionCfg_Osimps_I2_J,axiom,
! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N2: nat] :
( ( fLPThe137922386ionCfg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N2 ) )
= ( FStepCfg @ ( fLPThe137922386ionCfg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ N2 ) @ ( fLPThe221390223ionMsg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ N2 ) ) ) ).
% infiniteExecutionCfg.simps(2)
thf(fact_31_infiniteExecutionMsg_Osimps_I2_J,axiom,
! [Cfg: configuration_ext @ p @ v @ s @ product_unit,FStepCfg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N2: nat] :
( ( fLPThe221390223ionMsg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N2 ) )
= ( FStepMsg @ ( fLPThe137922386ionCfg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ N2 ) @ ( fLPThe221390223ionMsg @ p @ v @ s @ Cfg @ FStepCfg @ FStepMsg @ N2 ) ) ) ).
% infiniteExecutionMsg.simps(2)
thf(fact_32_flpPseudoConsensus_OinfiniteExecutionCfg_Osimps_I1_J,axiom,
! [S: $tType,V: $tType,P: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,Cfg: configuration_ext @ P @ V @ S @ product_unit,FStepCfg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( fLPThe137922386ionCfg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
= ( cons @ ( configuration_ext @ P @ V @ S @ product_unit ) @ Cfg @ ( nil @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionCfg.simps(1)
thf(fact_33_n__lists__Nil,axiom,
! [A: $tType,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( ( n_lists @ A @ N2 @ ( nil @ A ) )
= ( nil @ ( list @ A ) ) ) ) ) ).
% n_lists_Nil
thf(fact_34_insert__Nil,axiom,
! [A: $tType,X3: A] :
( ( insert @ A @ X3 @ ( nil @ A ) )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) ).
% insert_Nil
thf(fact_35_nths__singleton,axiom,
! [A: $tType,A4: set @ nat,X3: A] :
( ( ( member2 @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X3 @ ( nil @ A ) ) ) )
& ( ~ ( member2 @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_36_list__ex1__simps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
~ ( list_ex1 @ A @ P2 @ ( nil @ A ) ) ).
% list_ex1_simps(1)
thf(fact_37_n__lists_Osimps_I1_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( n_lists @ A @ ( zero_zero @ nat ) @ Xs )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% n_lists.simps(1)
thf(fact_38_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I2_J,axiom,
! [P2: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,Q: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,A42: configuration_ext @ p @ v @ s @ product_unit,A5: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),A6: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),A7: nat] :
( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( Q @ A42 @ A5 @ A6 @ A7 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(2)
thf(fact_39_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I1_J,axiom,
! [P2: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,Q: ( configuration_ext @ p @ v @ s @ product_unit ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ) ) > nat > $o,A0: configuration_ext @ p @ v @ s @ product_unit,A1: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),A2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),A32: nat] :
( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) )] : ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ p @ v @ s @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ p @ v @ s @ product_unit ) ) > ( list @ ( message @ p @ v ) ) > ( list @ ( message @ p @ v ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( P2 @ A0 @ A1 @ A2 @ A32 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(1)
thf(fact_40_nths__nil,axiom,
! [A: $tType,A4: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A4 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_41_flpPseudoConsensus_OinfiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I2_J,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,P2: ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > $o,Q: ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > $o,A42: configuration_ext @ P @ V @ S @ product_unit,A5: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),A6: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),A7: nat] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] : ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] : ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( Q @ A42 @ A5 @ A6 @ A7 ) ) ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionCfg_infiniteExecutionMsg.induct(2)
thf(fact_42_flpPseudoConsensus_OinfiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I1_J,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,P2: ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > $o,Q: ( configuration_ext @ P @ V @ S @ product_unit ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) > ( ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ) ) > nat > $o,A0: configuration_ext @ P @ V @ S @ product_unit,A1: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),A2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),A32: nat] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] : ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] : ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( zero_zero @ nat ) )
=> ( ! [Cfg2: configuration_ext @ P @ V @ S @ product_unit,FStepCfg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg2: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N: nat] :
( ( P2 @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ N )
=> ( Q @ Cfg2 @ FStepCfg2 @ FStepMsg2 @ ( suc @ N ) ) ) )
=> ( P2 @ A0 @ A1 @ A2 @ A32 ) ) ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionCfg_infiniteExecutionMsg.induct(1)
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A8: A,P2: A > $o] :
( ( member2 @ A @ A8 @ ( collect @ A @ P2 ) )
= ( P2 @ A8 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member2 @ A @ X4 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X: A] :
( ( P2 @ X )
= ( Q @ X ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X: A] :
( ( F2 @ X )
= ( G @ X ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_47_flpPseudoConsensus_OinfiniteExecutionMsg_Osimps_I2_J,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,Cfg: configuration_ext @ P @ V @ S @ product_unit,FStepCfg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N2: nat] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( fLPThe221390223ionMsg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N2 ) )
= ( FStepMsg @ ( fLPThe137922386ionCfg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ N2 ) @ ( fLPThe221390223ionMsg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ N2 ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionMsg.simps(2)
thf(fact_48_flpPseudoConsensus_OinfiniteExecutionCfg_Osimps_I2_J,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,Cfg: configuration_ext @ P @ V @ S @ product_unit,FStepCfg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),N2: nat] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( fLPThe137922386ionCfg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N2 ) )
= ( FStepCfg @ ( fLPThe137922386ionCfg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ N2 ) @ ( fLPThe221390223ionMsg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ N2 ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionCfg.simps(2)
thf(fact_49_flpPseudoConsensus_OinfiniteExecutionMsg_Oelims,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,X3: configuration_ext @ P @ V @ S @ product_unit,Xa: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),Xb: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),Xc: nat,Y: list @ ( message @ P @ V )] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( ( fLPThe221390223ionMsg @ P @ V @ S @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc
= ( zero_zero @ nat ) )
=> ( Y
!= ( nil @ ( message @ P @ V ) ) ) )
=> ~ ! [N: nat] :
( ( Xc
= ( suc @ N ) )
=> ( Y
!= ( Xb @ ( fLPThe137922386ionCfg @ P @ V @ S @ X3 @ Xa @ Xb @ N ) @ ( fLPThe221390223ionMsg @ P @ V @ S @ X3 @ Xa @ Xb @ N ) ) ) ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionMsg.elims
thf(fact_50_flpPseudoConsensus_OinfiniteExecutionMsg_Osimps_I1_J,axiom,
! [S: $tType,V: $tType,P: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,Cfg: configuration_ext @ P @ V @ S @ product_unit,FStepCfg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),FStepMsg: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) )] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( fLPThe221390223ionMsg @ P @ V @ S @ Cfg @ FStepCfg @ FStepMsg @ ( zero_zero @ nat ) )
= ( nil @ ( message @ P @ V ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionMsg.simps(1)
thf(fact_51_flpPseudoConsensus_OinfiniteExecutionCfg_Oelims,axiom,
! [P: $tType,V: $tType,S: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S,X3: configuration_ext @ P @ V @ S @ product_unit,Xa: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ),Xb: ( list @ ( configuration_ext @ P @ V @ S @ product_unit ) ) > ( list @ ( message @ P @ V ) ) > ( list @ ( message @ P @ V ) ),Xc: nat,Y: list @ ( configuration_ext @ P @ V @ S @ product_unit )] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( ( ( fLPThe137922386ionCfg @ P @ V @ S @ X3 @ Xa @ Xb @ Xc )
= Y )
=> ( ( ( Xc
= ( zero_zero @ nat ) )
=> ( Y
!= ( cons @ ( configuration_ext @ P @ V @ S @ product_unit ) @ X3 @ ( nil @ ( configuration_ext @ P @ V @ S @ product_unit ) ) ) ) )
=> ~ ! [N: nat] :
( ( Xc
= ( suc @ N ) )
=> ( Y
!= ( Xa @ ( fLPThe137922386ionCfg @ P @ V @ S @ X3 @ Xa @ Xb @ N ) @ ( fLPThe221390223ionMsg @ P @ V @ S @ X3 @ Xa @ Xb @ N ) ) ) ) ) ) ) ).
% flpPseudoConsensus.infiniteExecutionCfg.elims
thf(fact_52_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_53_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_54_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_55_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_56_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_57_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_58_nat__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( P2 @ N )
=> ( P2 @ ( suc @ N ) ) )
=> ( P2 @ N2 ) ) ) ).
% nat_induct
thf(fact_59_Suc__inject,axiom,
! [X3: nat,Y: nat] :
( ( ( suc @ X3 )
= ( suc @ Y ) )
=> ( X3 = Y ) ) ).
% Suc_inject
thf(fact_60_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_61_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_62_old_Onat_Oinducts,axiom,
! [P2: nat > $o,Nat: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [Nat4: nat] :
( ( P2 @ Nat4 )
=> ( P2 @ ( suc @ Nat4 ) ) )
=> ( P2 @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_63_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_64_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_65_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_66_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_67_zero__induct,axiom,
! [P2: nat > $o,K: nat] :
( ( P2 @ K )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_68_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N2: nat] :
( ! [X: nat] : ( P2 @ X @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X: nat,Y3: nat] :
( ( P2 @ X @ Y3 )
=> ( P2 @ ( suc @ X ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_69_dependent__nat__choice,axiom,
! [A: $tType,P2: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P2 @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X: A,N: nat] :
( ( P2 @ N @ X )
=> ? [Y4: A] :
( ( P2 @ ( suc @ N ) @ Y4 )
& ( Q @ N @ X @ Y4 ) ) )
=> ? [F: nat > A] :
! [N3: nat] :
( ( P2 @ N3 @ ( F @ N3 ) )
& ( Q @ N3 @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_70_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_71_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_72_gen__length__code_I2_J,axiom,
! [B: $tType,N2: nat,X3: B,Xs: list @ B] :
( ( gen_length @ B @ N2 @ ( cons @ B @ X3 @ Xs ) )
= ( gen_length @ B @ ( suc @ N2 ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_73_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_74_map__tailrec__rev_Oelims,axiom,
! [A: $tType,B: $tType,X3: A > B,Xa: list @ A,Xb: list @ B,Y: list @ B] :
( ( ( map_tailrec_rev @ A @ B @ X3 @ Xa @ Xb )
= Y )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y != Xb ) )
=> ~ ! [A3: A,As: list @ A] :
( ( Xa
= ( cons @ A @ A3 @ As ) )
=> ( Y
!= ( map_tailrec_rev @ A @ B @ X3 @ As @ ( cons @ B @ ( X3 @ A3 ) @ Xb ) ) ) ) ) ) ).
% map_tailrec_rev.elims
thf(fact_75_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F2: A > B,X3: A] :
( ( arg_min_list @ A @ B @ F2 @ ( cons @ A @ X3 @ ( nil @ A ) ) )
= X3 ) ) ).
% arg_min_list.simps(1)
thf(fact_76_map__tailrec__rev_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,F2: A > B,A8: A,As2: list @ A,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( cons @ A @ A8 @ As2 ) @ Bs2 )
= ( map_tailrec_rev @ A @ B @ F2 @ As2 @ ( cons @ B @ ( F2 @ A8 ) @ Bs2 ) ) ) ).
% map_tailrec_rev.simps(2)
thf(fact_77_map__tailrec__rev_Osimps_I1_J,axiom,
! [A: $tType,B: $tType,F2: A > B,Bs2: list @ B] :
( ( map_tailrec_rev @ A @ B @ F2 @ ( nil @ A ) @ Bs2 )
= Bs2 ) ).
% map_tailrec_rev.simps(1)
thf(fact_78_gen__length__code_I1_J,axiom,
! [A: $tType,N2: nat] :
( ( gen_length @ A @ N2 @ ( nil @ A ) )
= N2 ) ).
% gen_length_code(1)
thf(fact_79_ord_Olexordp__eq__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A] :
~ ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ).
% ord.lexordp_eq_simps(3)
thf(fact_80_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F2: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F2 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_81_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A1: list @ A,A2: list @ B] :
( ( listrelp @ A @ B @ R @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A2
!= ( nil @ B ) ) )
=> ~ ! [X: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ B] :
( ( A2
= ( cons @ B @ Y3 @ Ys3 ) )
=> ( ( R @ X @ Y3 )
=> ~ ( listrelp @ A @ B @ R @ Xs2 @ Ys3 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_82_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R2: A > B > $o,A12: list @ A,A22: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X4: A,Y2: B,Xs3: list @ A,Ys: list @ B] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y2 @ Ys ) )
& ( R2 @ X4 @ Y2 )
& ( listrelp @ A @ B @ R2 @ Xs3 @ Ys ) ) ) ) ) ).
% listrelp.simps
thf(fact_83_listrelp_Oinducts,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X12: list @ A,X23: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( listrelp @ A @ B @ R @ X12 @ X23 )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Y3: B,Xs2: list @ A,Ys3: list @ B] :
( ( R @ X @ Y3 )
=> ( ( listrelp @ A @ B @ R @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) ) )
=> ( P2 @ X12 @ X23 ) ) ) ) ).
% listrelp.inducts
thf(fact_84_flpPseudoConsensus_Oaxioms_I2_J,axiom,
! [V: $tType,S: $tType,P: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( fLPThe128449925axioms @ P @ S @ V @ Trans @ Sends @ Start ) ) ).
% flpPseudoConsensus.axioms(2)
thf(fact_85_ord_Olexordp__eq__simps_I4_J,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( Less @ X3 @ Y )
| ( ~ ( Less @ Y @ X3 )
& ( lexordp_eq @ A @ Less @ Xs @ Ys2 ) ) ) ) ).
% ord.lexordp_eq_simps(4)
thf(fact_86_ord_Olexordp__eq__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq_simps(1)
thf(fact_87_ord_Olexordp__eq__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ( lexordp_eq @ A @ Less @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% ord.lexordp_eq_simps(2)
thf(fact_88_ord_Olexordp__eq__refl,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] : ( lexordp_eq @ A @ Less @ Xs @ Xs ) ).
% ord.lexordp_eq_refl
thf(fact_89_ord_Olexordp__eq_Ocong,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( lexordp_eq @ A ) ) ).
% ord.lexordp_eq.cong
thf(fact_90_ord_Olexordp__eq_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( Less @ X3 @ Y )
=> ( ~ ( Less @ Y @ X3 )
=> ( ( lexordp_eq @ A @ Less @ Xs @ Ys2 )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ).
% ord.lexordp_eq.Cons_eq
thf(fact_91_ord_Olexordp__eq_OCons,axiom,
! [A: $tType,Less: A > A > $o,X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( Less @ X3 @ Y )
=> ( lexordp_eq @ A @ Less @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% ord.lexordp_eq.Cons
thf(fact_92_ord_Olexordp__eq_ONil,axiom,
! [A: $tType,Less: A > A > $o,Ys2: list @ A] : ( lexordp_eq @ A @ Less @ ( nil @ A ) @ Ys2 ) ).
% ord.lexordp_eq.Nil
thf(fact_93_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X3: A,Y: B,Xs: list @ A,Ys2: list @ B] :
( ( R @ X3 @ Y )
=> ( ( listrelp @ A @ B @ R @ Xs @ Ys2 )
=> ( listrelp @ A @ B @ R @ ( cons @ A @ X3 @ Xs ) @ ( cons @ B @ Y @ Ys2 ) ) ) ) ).
% listrelp.Cons
thf(fact_94_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( listrelp @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_95_ord_Olexordp__eq_Oinducts,axiom,
! [A: $tType,Less: A > A > $o,X12: list @ A,X23: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( lexordp_eq @ A @ Less @ X12 @ X23 )
=> ( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( Less @ X @ Y3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ( ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X12 @ X23 ) ) ) ) ) ).
% ord.lexordp_eq.inducts
thf(fact_96_ord_Olexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( lexordp_eq @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A22: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( Less2 @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( Less2 @ X4 @ Y2 )
& ~ ( Less2 @ Y2 @ X4 )
& ( lexordp_eq @ A @ Less2 @ Xs3 @ Ys ) ) ) ) ) ).
% ord.lexordp_eq.simps
thf(fact_97_ord_Olexordp__eq_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A2: list @ A] :
( ( lexordp_eq @ A @ Less @ A1 @ A2 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( Less @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( Less @ X @ Y3 )
=> ( ~ ( Less @ Y3 @ X )
=> ~ ( lexordp_eq @ A @ Less @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ).
% ord.lexordp_eq.cases
thf(fact_98_flpPseudoConsensus__def,axiom,
! [V: $tType,S: $tType,P: $tType] :
( ( fLPThe1922692578sensus @ P @ S @ V )
= ( ^ [Trans2: P > S > ( messageValue @ V ) > S,Sends2: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start2: P > S] :
( ( flpSystem @ P @ S @ V @ Sends2 )
& ( fLPThe128449925axioms @ P @ S @ V @ Trans2 @ Sends2 @ Start2 ) ) ) ) ).
% flpPseudoConsensus_def
thf(fact_99_flpPseudoConsensus_Ointro,axiom,
! [V: $tType,S: $tType,P: $tType,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Trans: P > S > ( messageValue @ V ) > S,Start: P > S] :
( ( flpSystem @ P @ S @ V @ Sends )
=> ( ( fLPThe128449925axioms @ P @ S @ V @ Trans @ Sends @ Start )
=> ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start ) ) ) ).
% flpPseudoConsensus.intro
thf(fact_100_lexordp__eq__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A] :
~ ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( nil @ A ) ) ) ).
% lexordp_eq_simps(3)
thf(fact_101_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_102_member__rec_I2_J,axiom,
! [A: $tType,Y: A] :
~ ( member @ A @ ( nil @ A ) @ Y ) ).
% member_rec(2)
thf(fact_103_member__rec_I1_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Y: A] :
( ( member @ A @ ( cons @ A @ X3 @ Xs ) @ Y )
= ( ( X3 = Y )
| ( member @ A @ Xs @ Y ) ) ) ).
% member_rec(1)
thf(fact_104_lexordp__eq__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ) ).
% lexordp_eq_simps(2)
thf(fact_105_lexordp__eq__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq_simps(1)
thf(fact_106_lexordp__eq__refl,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] : ( ord_lexordp_eq @ A @ Xs @ Xs ) ) ).
% lexordp_eq_refl
thf(fact_107_lexordp__eq__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp_eq @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_eq_trans
thf(fact_108_lexordp__eq__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
| ( ord_lexordp_eq @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_eq_linear
thf(fact_109_lexordp__eq__antisym,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp_eq @ A @ Ys2 @ Xs )
=> ( Xs = Ys2 ) ) ) ) ).
% lexordp_eq_antisym
thf(fact_110_lexordp__eq_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] : ( ord_lexordp_eq @ A @ ( nil @ A ) @ Ys2 ) ) ).
% lexordp_eq.Nil
thf(fact_111_flpPseudoConsensus_Oaxioms_I1_J,axiom,
! [V: $tType,S: $tType,P: $tType,Trans: P > S > ( messageValue @ V ) > S,Sends: P > S > ( messageValue @ V ) > ( message @ P @ V ) > nat,Start: P > S] :
( ( fLPThe1922692578sensus @ P @ S @ V @ Trans @ Sends @ Start )
=> ( flpSystem @ P @ S @ V @ Sends ) ) ).
% flpPseudoConsensus.axioms(1)
thf(fact_112_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A2 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_113_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22 = Ys ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_114_lexordp__eq_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X12: list @ A,X23: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp_eq @ A @ X12 @ X23 )
=> ( ! [X_1: list @ A] : ( P2 @ ( nil @ A ) @ X_1 )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X12 @ X23 ) ) ) ) ) ) ).
% lexordp_eq.inducts
thf(fact_115_splice_Oelims,axiom,
! [A: $tType,X3: list @ A,Xa: list @ A,Y: list @ A] :
( ( ( splice @ A @ X3 @ Xa )
= Y )
=> ( ( ( X3
= ( nil @ A ) )
=> ( Y != Xa ) )
=> ~ ! [X: A,Xs2: list @ A] :
( ( X3
= ( cons @ A @ X @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X @ ( splice @ A @ Xa @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_116_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N2: nat,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( suc @ N2 ) @ I )
= ( semiri532925092at_aux @ A @ Inc @ N2 @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_117_bot__nat__0_Onot__eq__extremum,axiom,
! [A8: nat] :
( ( A8
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A8 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_118_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_119_neq0__conv,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% neq0_conv
thf(fact_120_lessI,axiom,
! [N2: nat] : ( ord_less @ nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_121_Suc__mono,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_122_Suc__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_eq
thf(fact_123_not__gr__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) )
= ( N2
= ( zero_zero @ A ) ) ) ) ).
% not_gr_zero
thf(fact_124_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_125_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_126_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( splice @ A @ Xs @ Ys2 )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys2
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_127_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_128_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_129_gr__zeroI,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( N2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ N2 ) ) ) ).
% gr_zeroI
thf(fact_130_not__less__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
~ ( ord_less @ A @ N2 @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_131_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [M2: A,N2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_132_zero__less__iff__neq__zero,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [N2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ N2 )
= ( N2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_iff_neq_zero
thf(fact_133_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P2: A > $o,X3: A] :
( ! [X: A] :
( ( ( V2 @ X )
= ( zero_zero @ nat ) )
=> ( P2 @ X ) )
=> ( ! [X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X ) )
=> ( ~ ( P2 @ X )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X ) )
& ~ ( P2 @ Y4 ) ) ) )
=> ( P2 @ X3 ) ) ) ).
% infinite_descent0_measure
thf(fact_134_bot__nat__0_Oextremum__strict,axiom,
! [A8: nat] :
~ ( ord_less @ nat @ A8 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_135_infinite__descent0,axiom,
! [P2: nat > $o,N2: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( P2 @ N )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ~ ( P2 @ M3 ) ) ) )
=> ( P2 @ N2 ) ) ) ).
% infinite_descent0
thf(fact_136_gr__implies__not0,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( N2
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_137_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_138_not__less0,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_139_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
= ( N2
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_140_gr0I,axiom,
! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ).
% gr0I
thf(fact_141_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_142_Suc__lessD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N2 )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_lessD
thf(fact_143_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_144_Suc__lessI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ( suc @ M2 )
!= N2 )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_145_less__SucE,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
=> ( ~ ( ord_less @ nat @ M2 @ N2 )
=> ( M2 = N2 ) ) ) ).
% less_SucE
thf(fact_146_less__SucI,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ nat @ M2 @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_147_Ex__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ N2 )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N2 )
& ( P2 @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_148_less__Suc__eq,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ).
% less_Suc_eq
thf(fact_149_not__less__eq,axiom,
! [M2: nat,N2: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N2 ) )
= ( ord_less @ nat @ N2 @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_150_All__less__Suc,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ N2 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N2 )
=> ( P2 @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_151_Suc__less__eq2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N2 ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_152_less__antisym,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
=> ( M2 = N2 ) ) ) ).
% less_antisym
thf(fact_153_Suc__less__SucD,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ).
% Suc_less_SucD
thf(fact_154_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_155_less__Suc__induct,axiom,
! [I: nat,J2: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ K2 )
=> ( ( P2 @ I3 @ J )
=> ( ( P2 @ J @ K2 )
=> ( P2 @ I3 @ K2 ) ) ) ) )
=> ( P2 @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_156_strict__inc__induct,axiom,
! [I: nat,J2: nat,P2: nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_157_not__less__less__Suc__eq,axiom,
! [N2: nat,M2: nat] :
( ~ ( ord_less @ nat @ N2 @ M2 )
=> ( ( ord_less @ nat @ N2 @ ( suc @ M2 ) )
= ( N2 = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_158_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P2: A > $o,A8: A] :
( ! [X: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X ) )
=> ( P2 @ A8 ) ) ) ).
% measure_induct_rule
thf(fact_159_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P2: A > $o,A8: A] :
( ! [X: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F2 @ Y4 ) @ ( F2 @ X ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X ) )
=> ( P2 @ A8 ) ) ) ).
% measure_induct
thf(fact_160_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,N4: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less @ nat @ N2 @ N4 )
=> ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ N4 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_161_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N2: nat,M2: nat] :
( ! [N: nat] : ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ ( suc @ N ) ) )
=> ( ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ M2 ) )
= ( ord_less @ nat @ N2 @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_162_splice_Osimps_I2_J,axiom,
! [A: $tType,X3: A,Xs: list @ A,Ys2: list @ A] :
( ( splice @ A @ ( cons @ A @ X3 @ Xs ) @ Ys2 )
= ( cons @ A @ X3 @ ( splice @ A @ Ys2 @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_163_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys2: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys2 )
= Ys2 ) ).
% splice.simps(1)
thf(fact_164_less__Suc__eq__0__disj,axiom,
! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N2 ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_165_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ? [M: nat] :
( N2
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_166_All__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
=> ( P2 @ I2 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N2 )
=> ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_167_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_168_Ex__less__Suc2,axiom,
! [N2: nat,P2: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N2 ) )
& ( P2 @ I2 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N2 )
& ( P2 @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_169_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_170_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_171_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri532925092at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_172_NatPredicateTippingPoint,axiom,
! [N22: nat,Pr: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N22 )
=> ( ( Pr @ ( zero_zero @ nat ) )
=> ( ~ ( Pr @ N22 )
=> ? [N: nat] :
( ( ord_less @ nat @ N @ N22 )
& ( Pr @ N )
& ~ ( Pr @ ( suc @ N ) ) ) ) ) ) ).
% NatPredicateTippingPoint
thf(fact_173_less__numeral__extra_I3_J,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ~ ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_zero @ A ) ) ) ).
% less_numeral_extra(3)
thf(fact_174_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A2: list @ A] :
( ( ord_lexordp @ A @ A1 @ A2 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys3: list @ A] :
( A2
!= ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys3: list @ A] :
( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X @ Xs2 ) )
=> ! [Ys3: list @ A] :
( ( A2
= ( cons @ A @ Y3 @ Ys3 ) )
=> ( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys3 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_175_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A22: list @ A] :
( ? [Y2: A,Ys: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ( ord_less @ A @ X4 @ Y2 ) )
| ? [X4: A,Y2: A,Xs3: list @ A,Ys: list @ A] :
( ( A12
= ( cons @ A @ X4 @ Xs3 ) )
& ( A22
= ( cons @ A @ Y2 @ Ys ) )
& ~ ( ord_less @ A @ X4 @ Y2 )
& ~ ( ord_less @ A @ Y2 @ X4 )
& ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_176_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_177_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys2: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys2 )
= ( Ys2
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_178_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Xs: list @ A,Y: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) )
= ( ( ord_less @ A @ X3 @ Y )
| ( ~ ( ord_less @ A @ Y @ X3 )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_179_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X: A] :
~ ( ord_less @ A @ X @ X )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_180_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V2: A > nat,X3: A] :
( ! [X: A] :
( ~ ( P2 @ X )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X ) )
& ~ ( P2 @ Y4 ) ) )
=> ( P2 @ X3 ) ) ).
% infinite_descent_measure
thf(fact_181_linorder__neqE__nat,axiom,
! [X3: nat,Y: nat] :
( ( X3 != Y )
=> ( ~ ( ord_less @ nat @ X3 @ Y )
=> ( ord_less @ nat @ Y @ X3 ) ) ) ).
% linorder_neqE_nat
thf(fact_182_infinite__descent,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P2 @ N )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ~ ( P2 @ M3 ) ) )
=> ( P2 @ N2 ) ) ).
% infinite_descent
thf(fact_183_nat__less__induct,axiom,
! [P2: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P2 @ M3 ) )
=> ( P2 @ N ) )
=> ( P2 @ N2 ) ) ).
% nat_less_induct
thf(fact_184_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_185_less__not__refl3,axiom,
! [S2: nat,T: nat] :
( ( ord_less @ nat @ S2 @ T )
=> ( S2 != T ) ) ).
% less_not_refl3
thf(fact_186_less__not__refl2,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( M2 != N2 ) ) ).
% less_not_refl2
thf(fact_187_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less @ nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_188_nat__neq__iff,axiom,
! [M2: nat,N2: nat] :
( ( M2 != N2 )
= ( ( ord_less @ nat @ M2 @ N2 )
| ( ord_less @ nat @ N2 @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_189_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ~ ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_190_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,Zs2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ord_lexordp @ A @ Ys2 @ Zs2 )
=> ( ord_lexordp @ A @ Xs @ Zs2 ) ) ) ) ).
% lexordp_trans
thf(fact_191_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
| ( Xs = Ys2 )
| ( ord_lexordp @ A @ Ys2 @ Xs ) ) ) ).
% lexordp_linear
thf(fact_192_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_193_lexordp__into__lexordp__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ).
% lexordp_into_lexordp_eq
thf(fact_194_lexordp__eq__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( Xs3 = Ys )
| ( ord_lexordp @ A @ Xs3 @ Ys ) ) ) ) ) ).
% lexordp_eq_conv_lexord
thf(fact_195_lexordp__conv__lexordp__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys: list @ A] :
( ( ord_lexordp_eq @ A @ Xs3 @ Ys )
& ~ ( ord_lexordp_eq @ A @ Ys @ Xs3 ) ) ) ) ) ).
% lexordp_conv_lexordp_eq
thf(fact_196_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X3 @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_197_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X3: A,Y: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X3 @ Y )
=> ( ~ ( ord_less @ A @ Y @ X3 )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X3 @ Xs ) @ ( cons @ A @ Y @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_198_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys2: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys2 ) ) ) ).
% lexordp.Nil
thf(fact_199_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P2 @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ X @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_200_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( Ys2
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( Ys2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ord_less @ A @ X @ Y3 ) ) )
=> ~ ! [X: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X @ Xs4 ) )
=> ! [Ys4: list @ A] :
( ( Ys2
= ( cons @ A @ X @ Ys4 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys4 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_201_lexordp_Oinducts,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X12: list @ A,X23: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ X12 @ X23 )
=> ( ! [Y3: A,Ys3: list @ A] : ( P2 @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys3 ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ A @ X @ Y3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) )
=> ( ! [X: A,Y3: A,Xs2: list @ A,Ys3: list @ A] :
( ~ ( ord_less @ A @ X @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X )
=> ( ( ord_lexordp @ A @ Xs2 @ Ys3 )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ A @ Y3 @ Ys3 ) ) ) ) ) )
=> ( P2 @ X12 @ X23 ) ) ) ) ) ) ).
% lexordp.inducts
thf(fact_202_of__nat__0__less__iff,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [N2: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% of_nat_0_less_iff
thf(fact_203_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( suc @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_204_length__greater__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( Xs
!= ( nil @ A ) ) ) ).
% length_greater_0_conv
thf(fact_205_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat,N2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( M2 = N2 ) ) ) ).
% of_nat_eq_iff
thf(fact_206_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ A8 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_207_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ ( zero_zero @ A ) )
= A8 ) ) ).
% diff_0_right
thf(fact_208_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A8 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_209_diff__zero,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ ( zero_zero @ A ) )
= A8 ) ) ).
% diff_zero
thf(fact_210_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1352612707id_add @ A )
=> ! [A8: A] :
( ( minus_minus @ A @ A8 @ A8 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_211_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N2 )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_212_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_213_Suc__diff__diff,axiom,
! [M2: nat,N2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_214_diff__Suc__Suc,axiom,
! [M2: nat,N2: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_215_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A8 @ B2 ) )
= ( ord_less @ A @ B2 @ A8 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_216_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_217_of__nat__0__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( ( zero_zero @ A )
= ( semiring_1_of_nat @ A @ N2 ) )
= ( ( zero_zero @ nat )
= N2 ) ) ) ).
% of_nat_0_eq_iff
thf(fact_218_of__nat__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_0
thf(fact_219_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_iff
thf(fact_220_length__0__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( zero_zero @ nat ) )
= ( Xs
= ( nil @ A ) ) ) ).
% length_0_conv
thf(fact_221_zero__less__diff,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N2 @ M2 ) )
= ( ord_less @ nat @ M2 @ N2 ) ) ).
% zero_less_diff
thf(fact_222_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B2: A,C: A] :
( ( ord_less @ A @ A8 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B2 @ C ) ) ) ) ).
% diff_strict_right_mono
thf(fact_223_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A8: A,C: A] :
( ( ord_less @ A @ B2 @ A8 )
=> ( ord_less @ A @ ( minus_minus @ A @ C @ A8 ) @ ( minus_minus @ A @ C @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_224_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A8 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( ord_less @ A @ A8 @ B2 )
= ( ord_less @ A @ C @ D ) ) ) ) ).
% diff_eq_diff_less
thf(fact_225_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A8: A,B2: A,D: A,C: A] :
( ( ord_less @ A @ A8 @ B2 )
=> ( ( ord_less @ A @ D @ C )
=> ( ord_less @ A @ ( minus_minus @ A @ A8 @ C ) @ ( minus_minus @ A @ B2 @ D ) ) ) ) ) ).
% diff_strict_mono
thf(fact_226_length__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys5: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys5 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ Ys5 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_227_diff__less,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ M2 ) ) ) ).
% diff_less
thf(fact_228_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( ord_less @ nat @ M2 @ N2 ) ) ) ).
% of_nat_less_imp_less
thf(fact_229_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_230_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N2: nat] :
( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_231_diff__less__mono2,axiom,
! [M2: nat,N2: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N2 ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_232_Suc__diff__Suc,axiom,
! [N2: nat,M2: nat] :
( ( ord_less @ nat @ N2 @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N2 ) ) )
= ( minus_minus @ nat @ M2 @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_233_diff__less__Suc,axiom,
! [M2: nat,N2: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_234_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord1659791738miring @ A )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_235_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A9: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A9 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_236_of__nat__neq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ N2 ) )
!= ( zero_zero @ A ) ) ) ).
% of_nat_neq_0
thf(fact_237_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N2 ) )
= ( ? [Y2: A,Ys: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ) ) ).
% length_Suc_conv
thf(fact_238_Suc__length__conv,axiom,
! [A: $tType,N2: nat,Xs: list @ A] :
( ( ( suc @ N2 )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y2: A,Ys: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Ys )
= N2 ) ) ) ) ).
% Suc_length_conv
thf(fact_239_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_240_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( P2 @ Xs2 @ Ys3 )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) ) ) )
=> ( P2 @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_241_list__induct3,axiom,
! [B: $tType,A: $tType,C2: $tType,Xs: list @ A,Ys2: list @ B,Zs2: list @ C2,P2: ( list @ A ) > ( list @ B ) > ( list @ C2 ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C2 ) @ Zs2 ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C2 ) )
=> ( ! [X: A,Xs2: list @ A,Y3: B,Ys3: list @ B,Z: C2,Zs: list @ C2] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys3 )
= ( size_size @ ( list @ C2 ) @ Zs ) )
=> ( ( P2 @ Xs2 @ Ys3 @ Zs )
=> ( P2 @ ( cons @ A @ X @ Xs2 ) @ ( cons @ B @ Y3 @ Ys3 ) @ ( cons @ C2 @ Z @ Zs ) ) ) ) )
=> ( P2 @ Xs @ Ys2 @ Zs2 ) ) ) ) ) ).
% list_induct3
thf(fact_242_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A9: A,B3: A] :
( ( minus_minus @ A @ A9 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_243_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_244_diffs0__imp__equal,axiom,
! [M2: nat,N2: nat] :
( ( ( minus_minus @ nat @ M2 @ N2 )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N2 @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_245_zero__induct__lemma,axiom,
! [P2: nat > $o,K: nat,I: nat] :
( ( P2 @ K )
=> ( ! [N: nat] :
( ( P2 @ ( suc @ N ) )
=> ( P2 @ N ) )
=> ( P2 @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_246_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X3: A,Y: A] :
( ( ( size_size @ A @ X3 )
!= ( size_size @ A @ Y ) )
=> ( X3 != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_247_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel146912293up_add @ A )
=> ! [A8: A,C: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A8 @ C ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A8 @ B2 ) @ C ) ) ) ).
% diff_right_commute
thf(fact_248_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A8: A,B2: A,C: A,D: A] :
( ( ( minus_minus @ A @ A8 @ B2 )
= ( minus_minus @ A @ C @ D ) )
=> ( ( A8 = B2 )
= ( C = D ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_249_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_250_Ex__list__of__length,axiom,
! [A: $tType,N2: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_251_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J2 ) ) ).
% diff_commute
% Type constructors (26)
thf(tcon_fun___Orderings_Oorder,axiom,
! [A10: $tType,A11: $tType] :
( ( order @ A11 )
=> ( order @ ( A10 > A11 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A10: $tType,A11: $tType] :
( ( ord @ A11 )
=> ( ord @ ( A10 > A11 ) ) ) ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring,axiom,
linord1659791738miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__diff,axiom,
comm_monoid_diff @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_1,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_2,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Set_Oset___Orderings_Oorder_3,axiom,
! [A10: $tType] : ( order @ ( set @ A10 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_4,axiom,
! [A10: $tType] : ( ord @ ( set @ A10 ) ) ).
thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o ).
thf(tcon_List_Olist___Nat_Osize_8,axiom,
! [A10: $tType] : ( size @ ( list @ A10 ) ) ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_9,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_10,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_11,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_12,axiom,
ord @ product_unit ).
thf(tcon_AsynchronousSystem_Omessage___Nat_Osize_13,axiom,
! [A10: $tType,A11: $tType] : ( size @ ( message @ A10 @ A11 ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( fe @ ( zero_zero @ nat ) )
= ( cons @ ( configuration_ext @ p @ v @ s @ product_unit ) @ cfg @ ( nil @ ( configuration_ext @ p @ v @ s @ product_unit ) ) ) ) ).
%------------------------------------------------------------------------------