TPTP Problem File: ITP061^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP061^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer FLPTheorem problem prob_979__3303242_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_979__3303242_1 [Des21]
% Status : Theorem
% Rating : 0.50 v8.2.0, 0.38 v8.1.0, 0.36 v7.5.0
% Syntax : Number of formulae : 274 ( 94 unt; 38 typ; 0 def)
% Number of atoms : 616 ( 172 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 1864 ( 84 ~; 18 |; 27 &;1410 @)
% ( 0 <=>; 325 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 7 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% Number of symbols : 32 ( 31 usr; 12 con; 0-4 aty)
% Number of variables : 600 ( 42 ^; 537 !; 21 ?; 600 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:50.858
%------------------------------------------------------------------------------
% Could-be-implicit typings (7)
thf(ty_n_t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
list_c1059388851t_unit: $tType ).
thf(ty_n_t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
config256849571t_unit: $tType ).
thf(ty_n_t__List__Olist_It__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J_J,type,
list_message_p_v: $tType ).
thf(ty_n_t__Set__Oset_It__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J_J,type,
set_message_p_v: $tType ).
thf(ty_n_t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
message_p_v: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__p,type,
p: $tType ).
% Explicit typings (31)
thf(sy_c_AsynchronousSystem_Oenabled_001tf__p_001tf__v_001tf__s,type,
enabled_p_v_s: config256849571t_unit > message_p_v > $o ).
thf(sy_c_AsynchronousSystem_OisReceiverOf_001tf__p_001tf__v,type,
isReceiverOf_p_v: p > message_p_v > $o ).
thf(sy_c_Execution_Oexecution_OfirstOccurrence_001tf__p_001tf__v_001tf__s,type,
firstO1414030372_p_v_s: list_c1059388851t_unit > list_message_p_v > message_p_v > nat > $o ).
thf(sy_c_Execution_Oexecution_OminimalEnabled_001tf__p_001tf__v_001tf__s,type,
minimalEnabled_p_v_s: list_c1059388851t_unit > list_message_p_v > message_p_v > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_ListUtilities_OprefixList_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
prefix1615116500t_unit: list_c1059388851t_unit > list_c1059388851t_unit > $o ).
thf(sy_c_ListUtilities_OprefixList_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
prefix47729710ge_p_v: list_message_p_v > list_message_p_v > $o ).
thf(sy_c_List_Olist_ONil_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
nil_co1338500125t_unit: list_c1059388851t_unit ).
thf(sy_c_List_Onth_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
nth_co1649820636t_unit: list_c1059388851t_unit > nat > config256849571t_unit ).
thf(sy_c_List_Onth_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
nth_message_p_v: list_message_p_v > nat > message_p_v ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J_J,type,
size_s1406904903t_unit: list_c1059388851t_unit > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J_J,type,
size_s1168481041ge_p_v: list_message_p_v > nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Set_OCollect_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
collect_message_p_v: ( message_p_v > $o ) > set_message_p_v ).
thf(sy_c_member_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
member_message_p_v: message_p_v > set_message_p_v > $o ).
thf(sy_v_OccM_H____,type,
occM: nat ).
thf(sy_v_OccM____,type,
occM2: nat ).
thf(sy_v_consumedMsg____,type,
consumedMsg: message_p_v ).
thf(sy_v_fe____,type,
fe: nat > list_c1059388851t_unit ).
thf(sy_v_firstOccSet____,type,
firstOccSet: nat > set_message_p_v ).
thf(sy_v_ft____,type,
ft: nat > list_message_p_v ).
thf(sy_v_index____,type,
index: nat ).
thf(sy_v_msg_H____,type,
msg: message_p_v ).
thf(sy_v_msg____,type,
msg2: message_p_v ).
thf(sy_v_msga____,type,
msga: message_p_v ).
thf(sy_v_n0____,type,
n0: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p2: p ).
% Relevant facts (235)
thf(fact_0_AssumpOccMFirstOccurrence_I3_J,axiom,
ord_less_nat @ occM2 @ ( size_s1406904903t_unit @ ( fe @ index ) ) ).
% AssumpOccMFirstOccurrence(3)
thf(fact_1__092_060open_062length_A_Ife_Aindex_J_A_N_A1_A_092_060le_062_AOccM_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ one_one_nat ) @ occM2 ).
% \<open>length (fe index) - 1 \<le> OccM\<close>
thf(fact_2_OccM_H_I3_J,axiom,
ord_less_nat @ occM @ ( size_s1406904903t_unit @ ( fe @ index ) ) ).
% OccM'(3)
thf(fact_3__092_060open_062length_A_Ife_Aindex_J_A_N_A1_A_092_060le_062_Alength_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_092_060close_062,axiom,
ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ one_one_nat ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) @ one_one_nat ) ).
% \<open>length (fe index) - 1 \<le> length (fe (Suc index)) - 1\<close>
thf(fact_4_NotEmpty_I2_J,axiom,
( ( fe @ index )
!= nil_co1338500125t_unit ) ).
% NotEmpty(2)
thf(fact_5_AssumptionFair_I2_J,axiom,
ord_less_nat @ n0 @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% AssumptionFair(2)
thf(fact_6_one__natural_Orsp,axiom,
one_one_nat = one_one_nat ).
% one_natural.rsp
thf(fact_7_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_8_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_c1059388851t_unit] :
( ( size_s1406904903t_unit @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_9_Ex__list__of__length,axiom,
! [N: nat] :
? [Xs: list_message_p_v] :
( ( size_s1168481041ge_p_v @ Xs )
= N ) ).
% Ex_list_of_length
thf(fact_10_neq__if__length__neq,axiom,
! [Xs2: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs2 )
!= ( size_s1406904903t_unit @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_11_neq__if__length__neq,axiom,
! [Xs2: list_message_p_v,Ys: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs2 )
!= ( size_s1168481041ge_p_v @ Ys ) )
=> ( Xs2 != Ys ) ) ).
% neq_if_length_neq
thf(fact_12_size__neq__size__imp__neq,axiom,
! [X: list_c1059388851t_unit,Y: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ X )
!= ( size_s1406904903t_unit @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_13_size__neq__size__imp__neq,axiom,
! [X: list_message_p_v,Y: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ X )
!= ( size_s1168481041ge_p_v @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_14_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_15_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_16_NotEmpty_I1_J,axiom,
( ( fe @ ( suc @ index ) )
!= nil_co1338500125t_unit ) ).
% NotEmpty(1)
thf(fact_17_lessI,axiom,
! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_18_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_19_Suc__less__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_less_eq
thf(fact_20_Suc__le__mono,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N @ M ) ) ).
% Suc_le_mono
thf(fact_21_Suc__diff__diff,axiom,
! [M: nat,N: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_22_diff__Suc__Suc,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N ) )
= ( minus_minus_nat @ M @ N ) ) ).
% diff_Suc_Suc
thf(fact_23_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_24_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus_nat @ ( suc @ N ) @ one_one_nat )
= N ) ).
% diff_Suc_1
thf(fact_25_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_26_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% Suc_leD
thf(fact_27_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_28_le__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_eq_nat @ M @ N )
=> ( M
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_29_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_30_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_31_Suc__le__D,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq_nat @ ( suc @ N ) @ M2 )
=> ? [M3: nat] :
( M2
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_32_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_33_Suc__le__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
= ( ord_less_nat @ M @ N ) ) ).
% Suc_le_eq
thf(fact_34_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_35_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less_nat @ ( suc @ I ) @ K )
=> ~ ! [J2: nat] :
( ( ord_less_nat @ I @ J2 )
=> ( K
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_36_Suc__lessI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ( suc @ M )
!= N )
=> ( ord_less_nat @ ( suc @ M ) @ N ) ) ) ).
% Suc_lessI
thf(fact_37_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_38_le__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N ) )
= ( ( ord_less_eq_nat @ M @ N )
| ( M
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_39_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less_nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_40_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_41_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_42_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_43_mem__Collect__eq,axiom,
! [A: message_p_v,P: message_p_v > $o] :
( ( member_message_p_v @ A @ ( collect_message_p_v @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A2: set_message_p_v] :
( ( collect_message_p_v
@ ^ [X3: message_p_v] : ( member_message_p_v @ X3 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [P: message_p_v > $o,Q: message_p_v > $o] :
( ! [X4: message_p_v] :
( ( P @ X4 )
= ( Q @ X4 ) )
=> ( ( collect_message_p_v @ P )
= ( collect_message_p_v @ Q ) ) ) ).
% Collect_cong
thf(fact_46_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( ord_less_nat @ N2 @ J )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_47_le__antisym,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_eq_nat @ N @ M )
=> ( M = N ) ) ) ).
% le_antisym
thf(fact_48_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less_nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_49_less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ( ord_less_nat @ M @ N )
| ( M = N ) ) ) ).
% less_Suc_eq
thf(fact_50_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_51_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_eq_nat @ M4 @ N3 )
& ( M4 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_52_nat__neq__iff,axiom,
! [M: nat,N: nat] :
( ( M != N )
= ( ( ord_less_nat @ M @ N )
| ( ord_less_nat @ N @ M ) ) ) ).
% nat_neq_iff
thf(fact_53_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_nat @ M @ N ) )
= ( ord_less_nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_54_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_55_Suc__le__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_le_lessD
thf(fact_56_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less_nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_57_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_58_Suc__less__SucD,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N ) )
=> ( ord_less_nat @ M @ N ) ) ).
% Suc_less_SucD
thf(fact_59_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_60_nat__le__linear,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
| ( ord_less_eq_nat @ N @ M ) ) ).
% nat_le_linear
thf(fact_61_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_62_le__less__Suc__eq,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% le_less_Suc_eq
thf(fact_63_less__Suc__eq__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ ( suc @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% less_Suc_eq_le
thf(fact_64_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_65_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_66_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_67_less__trans__Suc,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_68_not__less__eq__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N ) )
= ( ord_less_eq_nat @ ( suc @ N ) @ M ) ) ).
% not_less_eq_eq
thf(fact_69_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M6: nat] :
( ( ord_less_eq_nat @ ( suc @ M6 ) @ N2 )
=> ( P @ M6 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_70_le__imp__less__Suc,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_nat @ M @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_71_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I3 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_72_less__imp__le__nat,axiom,
! [M: nat,N: nat] :
( ( ord_less_nat @ M @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_imp_le_nat
thf(fact_73_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_74_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
=> ( P @ M6 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_75_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M6: nat] :
( ( ord_less_nat @ M6 @ N2 )
& ~ ( P @ M6 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_76_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M4: nat,N3: nat] :
( ( ord_less_nat @ M4 @ N3 )
| ( M4 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_77_less__or__eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( ( ord_less_nat @ M @ N )
| ( M = N ) )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% less_or_eq_imp_le
thf(fact_78_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_79_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ J )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_80_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ B ) )
=> ? [X4: nat] :
( ( P @ X4 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_81_le__neq__implies__less,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( M != N )
=> ( ord_less_nat @ M @ N ) ) ) ).
% le_neq_implies_less
thf(fact_82_nat__induct__at__least,axiom,
! [M: nat,N: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ M )
=> ( ! [N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_83_not__less__less__Suc__eq,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less_nat @ N @ M )
=> ( ( ord_less_nat @ N @ ( suc @ M ) )
= ( N = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_84_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_nat @ I3 @ J2 )
=> ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_85_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X4 @ Z ) ) )
=> ( ! [N2: nat] : ( R @ N2 @ ( suc @ N2 ) )
=> ( R @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_86_lift__Suc__mono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_87_lift__Suc__mono__less,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ N @ N4 )
=> ( ord_less_nat @ ( F @ N ) @ ( F @ N4 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_88_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N: nat,N4: nat] :
( ! [N2: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N2 ) ) @ ( F @ N2 ) )
=> ( ( ord_less_eq_nat @ N @ N4 )
=> ( ord_less_eq_nat @ ( F @ N4 ) @ ( F @ N ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_89_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N: nat,M: nat] :
( ! [N2: nat] : ( ord_less_nat @ ( F @ N2 ) @ ( F @ ( suc @ N2 ) ) )
=> ( ( ord_less_nat @ ( F @ N ) @ ( F @ M ) )
= ( ord_less_nat @ N @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_90_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_91_Suc__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N ) ) )
= ( minus_minus_nat @ M @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_92_Suc__diff__le,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq_nat @ N @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N )
= ( suc @ ( minus_minus_nat @ M @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_93_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_94_diff__less__mono,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C @ A )
=> ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).
% diff_less_mono
thf(fact_95_less__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_96_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N: nat] :
( ( minus_minus_nat @ M @ ( suc @ N ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_97_length__induct,axiom,
! [P: list_c1059388851t_unit > $o,Xs2: list_c1059388851t_unit] :
( ! [Xs: list_c1059388851t_unit] :
( ! [Ys2: list_c1059388851t_unit] :
( ( ord_less_nat @ ( size_s1406904903t_unit @ Ys2 ) @ ( size_s1406904903t_unit @ Xs ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_98_length__induct,axiom,
! [P: list_message_p_v > $o,Xs2: list_message_p_v] :
( ! [Xs: list_message_p_v] :
( ! [Ys2: list_message_p_v] :
( ( ord_less_nat @ ( size_s1168481041ge_p_v @ Ys2 ) @ ( size_s1168481041ge_p_v @ Xs ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs ) )
=> ( P @ Xs2 ) ) ).
% length_induct
thf(fact_99_less__imp__diff__less,axiom,
! [J: nat,K: nat,N: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_100_diff__less__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_nat @ M @ N )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_101_diff__le__mono2,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_102_le__diff__iff_H,axiom,
! [A: nat,C: nat,B: nat] :
( ( ord_less_eq_nat @ A @ C )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
= ( ord_less_eq_nat @ B @ A ) ) ) ) ).
% le_diff_iff'
thf(fact_103_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_104_diff__le__mono,axiom,
! [M: nat,N: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_105_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( minus_minus_nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_106_le__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N @ K ) )
= ( ord_less_eq_nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_107_eq__diff__iff,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N @ K ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_108_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_109_diff__right__commute,axiom,
! [A: nat,C: nat,B: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
= ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).
% diff_right_commute
thf(fact_110_Subset,axiom,
! [MsgInSet: message_p_v] :
( ( member_message_p_v @ MsgInSet @ ( firstOccSet @ ( suc @ index ) ) )
=> ( member_message_p_v @ MsgInSet @ ( firstOccSet @ index ) ) ) ).
% Subset
thf(fact_111_SameCfgOnLow,axiom,
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1406904903t_unit @ ( fe @ index ) ) )
=> ( ( nth_co1649820636t_unit @ ( fe @ index ) @ I4 )
= ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ I4 ) ) ) ).
% SameCfgOnLow
thf(fact_112__092_060open_062_092_060not_062_AOccM_A_060_Alength_A_Ift_Aindex_J_092_060close_062,axiom,
~ ( ord_less_nat @ occM2 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) ) ).
% \<open>\<not> OccM < length (ft index)\<close>
thf(fact_113_IPrefixListEx,axiom,
! [I4: nat] : ( prefix1615116500t_unit @ ( fe @ I4 ) @ ( fe @ ( suc @ I4 ) ) ) ).
% IPrefixListEx
thf(fact_114_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_115_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M: nat] :
( ! [K2: nat] :
( ( ord_less_nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less_nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_116_KeepProperty,axiom,
! [Low: nat,P: nat > $o,Q: nat > $o] :
( ! [I3: nat] :
( ( ord_less_eq_nat @ Low @ I3 )
=> ( ( P @ I3 )
=> ( ( P @ ( suc @ I3 ) )
& ( Q @ I3 ) ) ) )
=> ( ( P @ Low )
=> ! [I4: nat] :
( ( ord_less_eq_nat @ Low @ I4 )
=> ( Q @ I4 ) ) ) ) ).
% KeepProperty
thf(fact_117_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_118_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_119_LengthStep,axiom,
ord_less_nat @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) ).
% LengthStep
thf(fact_120_ConsumedMsg_I1_J,axiom,
minimalEnabled_p_v_s @ ( fe @ index ) @ ( ft @ index ) @ consumedMsg ).
% ConsumedMsg(1)
thf(fact_121_PrefixListTransitive,axiom,
! [L1: list_c1059388851t_unit,L2: list_c1059388851t_unit,L3: list_c1059388851t_unit] :
( ( prefix1615116500t_unit @ L1 @ L2 )
=> ( ( prefix1615116500t_unit @ L2 @ L3 )
=> ( prefix1615116500t_unit @ L1 @ L3 ) ) ) ).
% PrefixListTransitive
thf(fact_122_PrefixListTransitive,axiom,
! [L1: list_message_p_v,L2: list_message_p_v,L3: list_message_p_v] :
( ( prefix47729710ge_p_v @ L1 @ L2 )
=> ( ( prefix47729710ge_p_v @ L2 @ L3 )
=> ( prefix47729710ge_p_v @ L1 @ L3 ) ) ) ).
% PrefixListTransitive
thf(fact_123_PrefixSameOnLow,axiom,
! [L1: list_c1059388851t_unit,L2: list_c1059388851t_unit] :
( ( prefix1615116500t_unit @ L1 @ L2 )
=> ! [Index: nat] :
( ( ord_less_nat @ Index @ ( size_s1406904903t_unit @ L1 ) )
=> ( ( nth_co1649820636t_unit @ L1 @ Index )
= ( nth_co1649820636t_unit @ L2 @ Index ) ) ) ) ).
% PrefixSameOnLow
thf(fact_124_PrefixSameOnLow,axiom,
! [L1: list_message_p_v,L2: list_message_p_v] :
( ( prefix47729710ge_p_v @ L1 @ L2 )
=> ! [Index: nat] :
( ( ord_less_nat @ Index @ ( size_s1168481041ge_p_v @ L1 ) )
=> ( ( nth_message_p_v @ L1 @ Index )
= ( nth_message_p_v @ L2 @ Index ) ) ) ) ).
% PrefixSameOnLow
thf(fact_125_PrefixListMonotonicity,axiom,
! [L1: list_c1059388851t_unit,L2: list_c1059388851t_unit] :
( ( prefix1615116500t_unit @ L1 @ L2 )
=> ( ord_less_nat @ ( size_s1406904903t_unit @ L1 ) @ ( size_s1406904903t_unit @ L2 ) ) ) ).
% PrefixListMonotonicity
thf(fact_126_PrefixListMonotonicity,axiom,
! [L1: list_message_p_v,L2: list_message_p_v] :
( ( prefix47729710ge_p_v @ L1 @ L2 )
=> ( ord_less_nat @ ( size_s1168481041ge_p_v @ L1 ) @ ( size_s1168481041ge_p_v @ L2 ) ) ) ).
% PrefixListMonotonicity
thf(fact_127_nth__equalityI,axiom,
! [Xs2: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs2 )
= ( size_s1406904903t_unit @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1406904903t_unit @ Xs2 ) )
=> ( ( nth_co1649820636t_unit @ Xs2 @ I3 )
= ( nth_co1649820636t_unit @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_128_nth__equalityI,axiom,
! [Xs2: list_message_p_v,Ys: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs2 )
= ( size_s1168481041ge_p_v @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1168481041ge_p_v @ Xs2 ) )
=> ( ( nth_message_p_v @ Xs2 @ I3 )
= ( nth_message_p_v @ Ys @ I3 ) ) )
=> ( Xs2 = Ys ) ) ) ).
% nth_equalityI
thf(fact_129_Skolem__list__nth,axiom,
! [K: nat,P: nat > config256849571t_unit > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: config256849571t_unit] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_co1649820636t_unit @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_130_Skolem__list__nth,axiom,
! [K: nat,P: nat > message_p_v > $o] :
( ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ? [X5: message_p_v] : ( P @ I2 @ X5 ) ) )
= ( ? [Xs3: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K )
=> ( P @ I2 @ ( nth_message_p_v @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_131_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_c1059388851t_unit,Z2: list_c1059388851t_unit] : Y5 = Z2 )
= ( ^ [Xs3: list_c1059388851t_unit,Ys3: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs3 )
= ( size_s1406904903t_unit @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1406904903t_unit @ Xs3 ) )
=> ( ( nth_co1649820636t_unit @ Xs3 @ I2 )
= ( nth_co1649820636t_unit @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_132_list__eq__iff__nth__eq,axiom,
( ( ^ [Y5: list_message_p_v,Z2: list_message_p_v] : Y5 = Z2 )
= ( ^ [Xs3: list_message_p_v,Ys3: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs3 )
= ( size_s1168481041ge_p_v @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1168481041ge_p_v @ Xs3 ) )
=> ( ( nth_message_p_v @ Xs3 @ I2 )
= ( nth_message_p_v @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_133_dual__order_Oantisym,axiom,
! [B: nat,A: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_134_dual__order_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_135_dual__order_Otrans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_eq_nat @ C @ A ) ) ) ).
% dual_order.trans
thf(fact_136_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_eq_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_137_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_138_order__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_eq_nat @ X @ Z3 ) ) ) ).
% order_trans
thf(fact_139_order__class_Oorder_Oantisym,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_140_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_141_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_142_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_143_antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_144_le__cases3,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z3 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z3 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z3 )
=> ~ ( ord_less_eq_nat @ Z3 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z3 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_145_order_Otrans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_eq_nat @ A @ C ) ) ) ).
% order.trans
thf(fact_146_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_147_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_148_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_149_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_150_eq__iff,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% eq_iff
thf(fact_151_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_152_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_153_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_subst2
thf(fact_154_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_155_dual__order_Ostrict__implies__not__eq,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( A != B ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_156_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_157_not__less__iff__gr__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ( ord_less_nat @ Y @ X )
| ( X = Y ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_158_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_159_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A4: nat,B3: nat] :
( ( ord_less_nat @ A4 @ B3 )
=> ( P @ A4 @ B3 ) )
=> ( ! [A4: nat] : ( P @ A4 @ A4 )
=> ( ! [A4: nat,B3: nat] :
( ( P @ B3 @ A4 )
=> ( P @ A4 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_160_exists__least__iff,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [N3: nat] :
( ( P3 @ N3 )
& ! [M4: nat] :
( ( ord_less_nat @ M4 @ N3 )
=> ~ ( P3 @ M4 ) ) ) ) ) ).
% exists_least_iff
thf(fact_161_less__imp__not__less,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_imp_not_less
thf(fact_162_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans
thf(fact_163_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_164_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_165_less__imp__triv,axiom,
! [X: nat,Y: nat,P: $o] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ X )
=> P ) ) ).
% less_imp_triv
thf(fact_166_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_167_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_168_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X4: nat] :
( ! [Y4: nat] :
( ( ord_less_nat @ Y4 @ X4 )
=> ( P @ Y4 ) )
=> ( P @ X4 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_169_less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_not_sym
thf(fact_170_less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_171_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_172_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_less_eq_trans
thf(fact_173_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% ord_eq_less_trans
thf(fact_174_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_175_less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% less_linear
thf(fact_176_less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% less_trans
thf(fact_177_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_178_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_179_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_180_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_181_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_182_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_183_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_184_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_subst2
thf(fact_185_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_subst1
thf(fact_186_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% ord_less_eq_subst
thf(fact_187_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_188_MinPredicate,axiom,
! [P: nat > $o] :
( ? [X_12: nat] : ( P @ X_12 )
=> ? [N0: nat] :
( ( P @ N0 )
& ! [N5: nat] :
( ( P @ N5 )
=> ( ord_less_eq_nat @ N0 @ N5 ) ) ) ) ).
% MinPredicate
thf(fact_189_order_Onot__eq__order__implies__strict,axiom,
! [A: nat,B: nat] :
( ( A != B )
=> ( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_190_dual__order_Ostrict__implies__order,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ( ord_less_eq_nat @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_191_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_eq_nat @ B2 @ A3 )
& ( A3 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_192_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A3: nat] :
( ( ord_less_nat @ B2 @ A3 )
| ( A3 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_193_order_Ostrict__implies__order,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_194_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_195_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C @ B )
=> ( ord_less_nat @ C @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_196_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_eq_nat @ A3 @ B2 )
& ( A3 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_197_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A3: nat,B2: nat] :
( ( ord_less_nat @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_198_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans2
thf(fact_199_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C )
=> ( ord_less_nat @ A @ C ) ) ) ).
% order.strict_trans1
thf(fact_200_not__le__imp__less,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_eq_nat @ Y @ X )
=> ( ord_less_nat @ X @ Y ) ) ).
% not_le_imp_less
thf(fact_201_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ~ ( ord_less_eq_nat @ Y6 @ X3 ) ) ) ) ).
% less_le_not_le
thf(fact_202_le__imp__less__or__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_203_le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% le_less_linear
thf(fact_204_less__le__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% less_le_trans
thf(fact_205_le__less__trans,axiom,
! [X: nat,Y: nat,Z3: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z3 )
=> ( ord_less_nat @ X @ Z3 ) ) ) ).
% le_less_trans
thf(fact_206_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_207_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_208_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_209_le__neq__trans,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( A != B )
=> ( ord_less_nat @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_210_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_211_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_212_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_less_le_subst2
thf(fact_213_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_214_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
=> ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).
% order_le_less_subst2
thf(fact_215_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C )
=> ( ! [X4: nat,Y3: nat] :
( ( ord_less_nat @ X4 @ Y3 )
=> ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y3 ) ) )
=> ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_216_less__le,axiom,
( ord_less_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_eq_nat @ X3 @ Y6 )
& ( X3 != Y6 ) ) ) ) ).
% less_le
thf(fact_217_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X3: nat,Y6: nat] :
( ( ord_less_nat @ X3 @ Y6 )
| ( X3 = Y6 ) ) ) ) ).
% le_less
thf(fact_218_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_219_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_220_AssumpMinImplAllBigger,axiom,
minimalEnabled_p_v_s @ ( fe @ index ) @ ( ft @ index ) @ msg ).
% AssumpMinImplAllBigger
thf(fact_221__092_060open_062execution_OfirstOccurrence_A_Ife_Aindex_J_A_Ift_Aindex_J_Amsg_AOccM_092_060close_062,axiom,
firstO1414030372_p_v_s @ ( fe @ index ) @ ( ft @ index ) @ msga @ occM2 ).
% \<open>execution.firstOccurrence (fe index) (ft index) msg OccM\<close>
thf(fact_222_SameMsgOnLow,axiom,
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) )
=> ( ( nth_message_p_v @ ( ft @ index ) @ I4 )
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% SameMsgOnLow
thf(fact_223_OccM_H_I5_J,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ occM @ N5 )
=> ( ( ord_less_nat @ N5 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) )
=> ( msg
!= ( nth_message_p_v @ ( ft @ index ) @ N5 ) ) ) ) ).
% OccM'(5)
thf(fact_224_IPrefixList,axiom,
! [I4: nat] : ( prefix47729710ge_p_v @ ( ft @ I4 ) @ ( ft @ ( suc @ I4 ) ) ) ).
% IPrefixList
thf(fact_225_OccM_H_I1_J,axiom,
? [P4: p] : ( isReceiverOf_p_v @ P4 @ msg ) ).
% OccM'(1)
thf(fact_226_FirstOccMsg_H,axiom,
firstO1414030372_p_v_s @ ( fe @ index ) @ ( ft @ index ) @ msg @ occM ).
% FirstOccMsg'
thf(fact_227_AssumpOccMFirstOccurrence_I5_J,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ occM2 @ N5 )
=> ( ( ord_less_nat @ N5 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) )
=> ( msga
!= ( nth_message_p_v @ ( ft @ index ) @ N5 ) ) ) ) ).
% AssumpOccMFirstOccurrence(5)
thf(fact_228_OccM_H_I4_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ index ) @ occM ) @ msg ).
% OccM'(4)
thf(fact_229_AssumptionFairContr,axiom,
! [N5: nat] :
( ( ord_less_eq_nat @ n @ N5 )
=> ! [N02: nat] :
( ( ord_less_nat @ N02 @ ( size_s1168481041ge_p_v @ ( ft @ N5 ) ) )
=> ( ( ord_less_eq_nat @ ( size_s1168481041ge_p_v @ ( ft @ n ) ) @ N02 )
=> ( msg2
!= ( nth_message_p_v @ ( ft @ N5 ) @ N02 ) ) ) ) ) ).
% AssumptionFairContr
thf(fact_230_MessageStaysOrConsumed,axiom,
! [N1: nat,N22: nat,N: nat,Msg: message_p_v] :
( ( ( ord_less_eq_nat @ N1 @ N22 )
& ( ord_less_nat @ N22 @ ( size_s1406904903t_unit @ ( fe @ N ) ) )
& ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ N ) @ N1 ) @ Msg ) )
=> ( ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ N ) @ N22 ) @ Msg )
| ? [N03: nat] :
( ( ord_less_eq_nat @ N1 @ N03 )
& ( ord_less_nat @ N03 @ ( size_s1168481041ge_p_v @ ( ft @ N ) ) )
& ( ( nth_message_p_v @ ( ft @ N ) @ N03 )
= Msg ) ) ) ) ).
% MessageStaysOrConsumed
thf(fact_231__092_060open_062_092_060not_062_A_I_092_060exists_062i_060length_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_O_Alength_A_Ife_Aindex_J_A_N_A1_A_092_060le_062_Ai_A_092_060and_062_Amsg_____A_061_Aft_A_ISuc_Aindex_J_A_B_Ai_J_092_060close_062,axiom,
~ ? [I4: nat] :
( ( ord_less_nat @ I4 @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) @ one_one_nat ) )
& ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ one_one_nat ) @ I4 )
& ( msg2
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% \<open>\<not> (\<exists>i<length (fe (Suc index)) - 1. length (fe index) - 1 \<le> i \<and> msg__ = ft (Suc index) ! i)\<close>
thf(fact_232_AssumptionFair_I4_J,axiom,
isReceiverOf_p_v @ p2 @ msg2 ).
% AssumptionFair(4)
thf(fact_233_AssumpOccMFirstOccurrence_I1_J,axiom,
? [P4: p] : ( isReceiverOf_p_v @ P4 @ msga ) ).
% AssumpOccMFirstOccurrence(1)
thf(fact_234_AssumptionFair_I3_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ n ) @ n0 ) @ msg2 ).
% AssumptionFair(3)
% Conjectures (1)
thf(conj_0,conjecture,
( occM2
= ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ one_one_nat ) ) ).
%------------------------------------------------------------------------------