TPTP Problem File: ITP059^1.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP059^1 : TPTP v9.0.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer FLPTheorem problem prob_747__3300826_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_747__3300826_1 [Des21]
% Status : Theorem
% Rating : 0.25 v9.0.0, 0.30 v8.2.0, 0.15 v8.1.0, 0.18 v7.5.0
% Syntax : Number of formulae : 284 ( 127 unt; 39 typ; 0 def)
% Number of atoms : 528 ( 207 equ; 0 cnn)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 1888 ( 87 ~; 15 |; 42 &;1486 @)
% ( 0 <=>; 258 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 6 avg)
% Number of types : 8 ( 7 usr)
% Number of type conns : 148 ( 148 >; 0 *; 0 +; 0 <<)
% Number of symbols : 33 ( 32 usr; 12 con; 0-4 aty)
% Number of variables : 505 ( 13 ^; 456 !; 36 ?; 505 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:31:23.474
%------------------------------------------------------------------------------
% 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 (32)
thf(sy_c_AsynchronousSystem_Oconfiguration_Omsgs_001tf__p_001tf__v_001tf__s_001t__Product____Type__Ounit,type,
msgs_p1029620568t_unit: config256849571t_unit > message_p_v > nat ).
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_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_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_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_Olast_001t__AsynchronousSystem__Oconfiguration__Oconfiguration____ext_Itf__p_Mtf__v_Mtf__s_Mt__Product____Type__Ounit_J,type,
last_c571238084t_unit: list_c1059388851t_unit > config256849571t_unit ).
thf(sy_c_List_Olast_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
last_message_p_v: list_message_p_v > message_p_v ).
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_Olist_ONil_001t__AsynchronousSystem__Omessage_Itf__p_Mtf__v_J,type,
nil_message_p_v: list_message_p_v ).
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_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_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_msgInSet____,type,
msgInSet: message_p_v ).
thf(sy_v_msg____,type,
msg: message_p_v ).
thf(sy_v_n0____,type,
n0: nat ).
thf(sy_v_n1____,type,
n1: nat ).
thf(sy_v_nMsg____,type,
nMsg: nat ).
thf(sy_v_n____,type,
n: nat ).
thf(sy_v_p____,type,
p2: p ).
% Relevant facts (244)
thf(fact_0_AssumpOcc6_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ nMsg ).
% AssumpOcc6(1)
thf(fact_1_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I2_J,axiom,
! [P: config256849571t_unit > ( list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit ) > ( list_c1059388851t_unit > list_message_p_v > list_message_p_v ) > nat > $o,Q: config256849571t_unit > ( list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit ) > ( list_c1059388851t_unit > list_message_p_v > list_message_p_v ) > nat > $o,A4: config256849571t_unit,A5: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,A6: list_c1059388851t_unit > list_message_p_v > list_message_p_v,A7: nat] :
( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v] : ( P @ Cfg @ FStepCfg @ FStepMsg @ zero_zero_nat )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v,N: nat] :
( ( P @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( P @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v] : ( Q @ Cfg @ FStepCfg @ FStepMsg @ zero_zero_nat )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v,N: nat] :
( ( P @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( Q @ A4 @ A5 @ A6 @ A7 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(2)
thf(fact_2_infiniteExecutionCfg__infiniteExecutionMsg_Oinduct_I1_J,axiom,
! [P: config256849571t_unit > ( list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit ) > ( list_c1059388851t_unit > list_message_p_v > list_message_p_v ) > nat > $o,Q: config256849571t_unit > ( list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit ) > ( list_c1059388851t_unit > list_message_p_v > list_message_p_v ) > nat > $o,A0: config256849571t_unit,A1: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,A2: list_c1059388851t_unit > list_message_p_v > list_message_p_v,A3: nat] :
( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v] : ( P @ Cfg @ FStepCfg @ FStepMsg @ zero_zero_nat )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v,N: nat] :
( ( P @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( P @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v] : ( Q @ Cfg @ FStepCfg @ FStepMsg @ zero_zero_nat )
=> ( ! [Cfg: config256849571t_unit,FStepCfg: list_c1059388851t_unit > list_message_p_v > list_c1059388851t_unit,FStepMsg: list_c1059388851t_unit > list_message_p_v > list_message_p_v,N: nat] :
( ( P @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( ( Q @ Cfg @ FStepCfg @ FStepMsg @ N )
=> ( Q @ Cfg @ FStepCfg @ FStepMsg @ ( suc @ N ) ) ) )
=> ( P @ A0 @ A1 @ A2 @ A3 ) ) ) ) ) ).
% infiniteExecutionCfg_infiniteExecutionMsg.induct(1)
thf(fact_3_AssumptionSubset_I3_J,axiom,
firstO1414030372_p_v_s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msg @ nMsg ).
% AssumptionSubset(3)
thf(fact_4_SmallIndex,axiom,
! [NMsg: nat] :
( ( firstO1414030372_p_v_s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msg @ NMsg )
=> ( ord_less_nat @ NMsg @ ( size_s1406904903t_unit @ ( fe @ index ) ) ) ) ).
% SmallIndex
thf(fact_5_AssumptionSubset3_I3_J,axiom,
ord_less_nat @ nMsg @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) ).
% AssumptionSubset3(3)
thf(fact_6_AssumptionSubset2_I3_J,axiom,
ord_less_nat @ n1 @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) ).
% AssumptionSubset2(3)
thf(fact_7_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_8_zero__less__diff,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% zero_less_diff
thf(fact_9_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_10_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_11_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_12_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ M )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_13_diff__Suc__Suc,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_14_Suc__diff__diff,axiom,
! [M: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_15_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_16_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_17_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_18_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_19_not__gr__zero,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr_zero
thf(fact_20_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ A )
= zero_zero_nat ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_21_diff__zero,axiom,
! [A: nat] :
( ( minus_minus_nat @ A @ zero_zero_nat )
= A ) ).
% diff_zero
thf(fact_22_zero__diff,axiom,
! [A: nat] :
( ( minus_minus_nat @ zero_zero_nat @ A )
= zero_zero_nat ) ).
% zero_diff
thf(fact_23_Suc__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_eq
thf(fact_24_Suc__mono,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_25_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_26_bot__nat__0_Onot__eq__extremum,axiom,
! [A: nat] :
( ( A != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_27_AssumptionSubset_I2_J,axiom,
firstO1414030372_p_v_s @ ( fe @ ( suc @ index ) ) @ ( ft @ ( suc @ index ) ) @ msgInSet @ n1 ).
% AssumptionSubset(2)
thf(fact_28_AssumptionSubset_I1_J,axiom,
ord_less_eq_nat @ n1 @ nMsg ).
% AssumptionSubset(1)
thf(fact_29_zero__reorient,axiom,
! [X: nat] :
( ( zero_zero_nat = X )
= ( X = zero_zero_nat ) ) ).
% zero_reorient
thf(fact_30_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_31_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_32_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_33_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_34_infinite__descent,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ~ ( P @ N )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ~ ( P @ M2 ) ) )
=> ( P @ N2 ) ) ).
% infinite_descent
thf(fact_35_nat__less__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N )
=> ( P @ M2 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% nat_less_induct
thf(fact_36_less__irrefl__nat,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_irrefl_nat
thf(fact_37_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_38_less__not__refl2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( M != N2 ) ) ).
% less_not_refl2
thf(fact_39_less__not__refl,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ N2 ) ).
% less_not_refl
thf(fact_40_nat__neq__iff,axiom,
! [M: nat,N2: nat] :
( ( M != N2 )
= ( ( ord_less_nat @ M @ N2 )
| ( ord_less_nat @ N2 @ M ) ) ) ).
% nat_neq_iff
thf(fact_41_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_42_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_43_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_44_zero__less__iff__neq__zero,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( N2 != zero_zero_nat ) ) ).
% zero_less_iff_neq_zero
thf(fact_45_gr__implies__not__zero,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not_zero
thf(fact_46_not__less__zero,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less_zero
thf(fact_47_gr__zeroI,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr_zeroI
thf(fact_48_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_49_old_Onat_Oinducts,axiom,
! [P: nat > $o,Nat: nat] :
( ( P @ zero_zero_nat )
=> ( ! [Nat3: nat] :
( ( P @ Nat3 )
=> ( P @ ( suc @ Nat3 ) ) )
=> ( P @ Nat ) ) ) ).
% old.nat.inducts
thf(fact_50_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_51_Zero__not__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_52_Zero__neq__Suc,axiom,
! [M: nat] :
( zero_zero_nat
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_53_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_54_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_55_diff__induct,axiom,
! [P: nat > nat > $o,M: nat,N2: nat] :
( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
=> ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M @ N2 ) ) ) ) ).
% diff_induct
thf(fact_56_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( P @ N )
=> ( P @ ( suc @ N ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_57_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_58_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_59_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_60_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( zero_zero_nat
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_61_not__less__less__Suc__eq,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% not_less_less_Suc_eq
thf(fact_62_strict__inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] :
( ( J
= ( suc @ I2 ) )
=> ( P @ I2 ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( P @ ( suc @ I2 ) )
=> ( P @ I2 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_63_less__Suc__induct,axiom,
! [I: nat,J: nat,P: nat > nat > $o] :
( ( ord_less_nat @ I @ J )
=> ( ! [I2: nat] : ( P @ I2 @ ( suc @ I2 ) )
=> ( ! [I2: nat,J2: nat,K2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ( ord_less_nat @ J2 @ K2 )
=> ( ( P @ I2 @ J2 )
=> ( ( P @ J2 @ K2 )
=> ( P @ I2 @ K2 ) ) ) ) )
=> ( P @ I @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_64_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_65_Suc__less__SucD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_less_SucD
thf(fact_66_less__antisym,axiom,
! [N2: nat,M: nat] :
( ~ ( ord_less_nat @ N2 @ M )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
=> ( M = N2 ) ) ) ).
% less_antisym
thf(fact_67_Suc__less__eq2,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ ( suc @ N2 ) @ M )
= ( ? [M4: nat] :
( ( M
= ( suc @ M4 ) )
& ( ord_less_nat @ N2 @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_68_All__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ N2 )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ I3 ) ) ) ) ).
% All_less_Suc
thf(fact_69_not__less__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_nat @ M @ N2 ) )
= ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_70_less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) ) ) ).
% less_Suc_eq
thf(fact_71_Ex__less__Suc,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ N2 )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ I3 ) ) ) ) ).
% Ex_less_Suc
thf(fact_72_less__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% less_SucI
thf(fact_73_less__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_nat @ M @ N2 )
=> ( M = N2 ) ) ) ).
% less_SucE
thf(fact_74_Suc__lessI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ( suc @ M )
!= N2 )
=> ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).
% Suc_lessI
thf(fact_75_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_76_Suc__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_lessD
thf(fact_77_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_78_bot__nat__0_Oextremum__strict,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ zero_zero_nat ) ).
% bot_nat_0.extremum_strict
thf(fact_79_infinite__descent0,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ~ ( P @ N )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N )
& ~ ( P @ M2 ) ) ) )
=> ( P @ N2 ) ) ) ).
% infinite_descent0
thf(fact_80_gr__implies__not0,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( N2 != zero_zero_nat ) ) ).
% gr_implies_not0
thf(fact_81_less__zeroE,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_zeroE
thf(fact_82_not__less0,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% not_less0
thf(fact_83_not__gr0,axiom,
! [N2: nat] :
( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
= ( N2 = zero_zero_nat ) ) ).
% not_gr0
thf(fact_84_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% gr0I
thf(fact_85_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N: nat] :
( ( P @ ( suc @ N ) )
=> ( P @ N ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_86_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus_nat @ M @ zero_zero_nat )
= M ) ).
% minus_nat.diff_0
thf(fact_87_diffs0__imp__equal,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M )
= zero_zero_nat )
=> ( M = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_88_less__imp__diff__less,axiom,
! [J: nat,K: nat,N2: nat] :
( ( ord_less_nat @ J @ K )
=> ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).
% less_imp_diff_less
thf(fact_89_diff__less__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ( ord_less_nat @ M @ L )
=> ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).
% diff_less_mono2
thf(fact_90_lift__Suc__mono__less__iff,axiom,
! [F: nat > nat,N2: nat,M: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
= ( ord_less_nat @ N2 @ M ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_91_lift__Suc__mono__less,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_nat @ N2 @ N3 )
=> ( ord_less_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_92_less__Suc__eq__0__disj,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ( M = zero_zero_nat )
| ? [J3: nat] :
( ( M
= ( suc @ J3 ) )
& ( ord_less_nat @ J3 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_93_gr0__implies__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ? [M3: nat] :
( N2
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_94_All__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
=> ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
=> ( P @ ( suc @ I3 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_95_gr0__conv__Suc,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
= ( ? [M5: nat] :
( N2
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_96_Ex__less__Suc2,axiom,
! [N2: nat,P: nat > $o] :
( ( ? [I3: nat] :
( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
& ( P @ I3 ) ) )
= ( ( P @ zero_zero_nat )
| ? [I3: nat] :
( ( ord_less_nat @ I3 @ N2 )
& ( P @ ( suc @ I3 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_97_diff__less__Suc,axiom,
! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_98_Suc__diff__Suc,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ N2 @ M )
=> ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ).
% Suc_diff_Suc
thf(fact_99_diff__less,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( ord_less_nat @ zero_zero_nat @ M )
=> ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).
% diff_less
thf(fact_100_diff__Suc__less,axiom,
! [N2: nat,I: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I ) ) @ N2 ) ) ).
% diff_Suc_less
thf(fact_101_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_102__092_060open_062fe_Aindex_A_B_AnMsg_A_061_Afe_A_ISuc_Aindex_J_A_B_AnMsg_092_060close_062,axiom,
( ( nth_co1649820636t_unit @ ( fe @ index ) @ nMsg )
= ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ nMsg ) ) ).
% \<open>fe index ! nMsg = fe (Suc index) ! nMsg\<close>
thf(fact_103_ShorterTrace,axiom,
ord_less_nat @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) ).
% ShorterTrace
thf(fact_104_NotEmpty_I1_J,axiom,
( ( fe @ ( suc @ index ) )
!= nil_co1338500125t_unit ) ).
% NotEmpty(1)
thf(fact_105_IPrefixList,axiom,
! [I4: nat] : ( prefix47729710ge_p_v @ ( ft @ I4 ) @ ( ft @ ( suc @ I4 ) ) ) ).
% IPrefixList
thf(fact_106_IPrefixListEx,axiom,
! [I4: nat] : ( prefix1615116500t_unit @ ( fe @ I4 ) @ ( fe @ ( suc @ I4 ) ) ) ).
% IPrefixListEx
thf(fact_107_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_108_NotEmpty_I2_J,axiom,
( ( fe @ index )
!= nil_co1338500125t_unit ) ).
% NotEmpty(2)
thf(fact_109_AssumptionFair_I4_J,axiom,
isReceiverOf_p_v @ p2 @ msg ).
% AssumptionFair(4)
thf(fact_110_AssumpOcc6_I2_J,axiom,
( msg
!= ( nth_message_p_v @ ( ft @ index ) @ ( minus_minus_nat @ nMsg @ one_one_nat ) ) ) ).
% AssumpOcc6(2)
thf(fact_111_Occ1,axiom,
? [P2: p] : ( isReceiverOf_p_v @ P2 @ msg ) ).
% Occ1
thf(fact_112_AssumptionSubset2_I1_J,axiom,
? [P2: p] : ( isReceiverOf_p_v @ P2 @ msgInSet ) ).
% AssumptionSubset2(1)
thf(fact_113_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_114_le__zero__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_zero_eq
thf(fact_115_Suc__le__mono,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
= ( ord_less_eq_nat @ N2 @ M ) ) ).
% Suc_le_mono
thf(fact_116_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_117_bot__nat__0_Oextremum,axiom,
! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).
% bot_nat_0.extremum
thf(fact_118_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_119__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_120__092_060open_062_092_060not_062_A_I_092_060exists_062i_060length_A_Ift_A_ISuc_Aindex_J_J_O_Alength_A_Ift_Aindex_J_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 @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) )
& ( ord_less_eq_nat @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) @ I4 )
& ( msg
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ I4 ) ) ) ).
% \<open>\<not> (\<exists>i<length (ft (Suc index)). length (ft index) \<le> i \<and> msg = ft (Suc index) ! i)\<close>
thf(fact_121_NotConsumedIntermediate,axiom,
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) )
=> ( ( ord_less_eq_nat @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) @ I4 )
=> ( ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ I4 )
!= msg ) ) ) ).
% NotConsumedIntermediate
thf(fact_122_OccSameMsg,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ nMsg @ N4 )
=> ( ( ord_less_nat @ N4 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) )
=> ( ( nth_message_p_v @ ( ft @ index ) @ N4 )
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% OccSameMsg
thf(fact_123_Occ5,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ nMsg @ N4 )
=> ( ( ord_less_nat @ N4 @ ( size_s1168481041ge_p_v @ ( ft @ index ) ) )
=> ( msg
!= ( nth_message_p_v @ ( ft @ index ) @ N4 ) ) ) ) ).
% Occ5
thf(fact_124_diff__is__0__eq,axiom,
! [M: nat,N2: nat] :
( ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% diff_is_0_eq
thf(fact_125_diff__is__0__eq_H,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( minus_minus_nat @ M @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_126_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_127_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_128_AssumptionSubset3_I5_J,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ nMsg @ N4 )
=> ( ( ord_less_nat @ N4 @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) )
=> ( msg
!= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% AssumptionSubset3(5)
thf(fact_129_AssumptionSubset2_I5_J,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ n1 @ N4 )
=> ( ( ord_less_nat @ N4 @ ( size_s1168481041ge_p_v @ ( ft @ ( suc @ index ) ) ) )
=> ( msgInSet
!= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ N4 ) ) ) ) ).
% AssumptionSubset2(5)
thf(fact_130__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 )
& ( msg
= ( 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_131_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_132_AssumptionFairContr,axiom,
! [N4: nat] :
( ( ord_less_eq_nat @ n @ N4 )
=> ! [N0: nat] :
( ( ord_less_nat @ N0 @ ( size_s1168481041ge_p_v @ ( ft @ N4 ) ) )
=> ( ( ord_less_eq_nat @ ( size_s1168481041ge_p_v @ ( ft @ n ) ) @ N0 )
=> ( msg
!= ( nth_message_p_v @ ( ft @ N4 ) @ N0 ) ) ) ) ) ).
% AssumptionFairContr
thf(fact_133_AssumpOcc6_I3_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ index ) @ ( minus_minus_nat @ nMsg @ one_one_nat ) ) @ msg ).
% AssumpOcc6(3)
thf(fact_134_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_135_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_136_eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( M = N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% eq_imp_le
thf(fact_137_le__antisym,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M )
=> ( M = N2 ) ) ) ).
% le_antisym
thf(fact_138_nat__le__linear,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
| ( ord_less_eq_nat @ N2 @ M ) ) ).
% nat_le_linear
thf(fact_139_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_140_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 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_141_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_142_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_143_MinPredicate,axiom,
! [P: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ? [N02: nat] :
( ( P @ N02 )
& ! [N4: nat] :
( ( P @ N4 )
=> ( ord_less_eq_nat @ N02 @ N4 ) ) ) ) ).
% MinPredicate
thf(fact_144_lift__Suc__antimono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ N2 ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_145_lift__Suc__mono__le,axiom,
! [F: nat > nat,N2: nat,N3: nat] :
( ! [N: nat] : ( ord_less_eq_nat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
=> ( ( ord_less_eq_nat @ N2 @ N3 )
=> ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N3 ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_146_KeepProperty,axiom,
! [Low: nat,P: nat > $o,Q: nat > $o] :
( ! [I2: nat] :
( ( ord_less_eq_nat @ Low @ I2 )
=> ( ( P @ I2 )
=> ( ( P @ ( suc @ I2 ) )
& ( Q @ I2 ) ) ) )
=> ( ( P @ Low )
=> ! [I4: nat] :
( ( ord_less_eq_nat @ Low @ I4 )
=> ( Q @ I4 ) ) ) ) ).
% KeepProperty
thf(fact_147_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_148_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_149_zero__le,axiom,
! [X: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X ) ).
% zero_le
thf(fact_150_Suc__leD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% Suc_leD
thf(fact_151_le__SucE,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M @ N2 )
=> ( M
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_152_le__SucI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_153_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_154_le__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M @ N2 )
| ( M
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_155_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_156_not__less__eq__eq,axiom,
! [M: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).
% not_less_eq_eq
thf(fact_157_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N: nat] :
( ! [M2: nat] :
( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
=> ( P @ M2 ) )
=> ( P @ N ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_158_nat__induct__at__least,axiom,
! [M: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( P @ M )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ M @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_159_transitive__stepwise__le,axiom,
! [M: nat,N2: nat,R: nat > nat > $o] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ! [X3: nat] : ( R @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z: nat] :
( ( R @ X3 @ Y3 )
=> ( ( R @ Y3 @ Z )
=> ( R @ X3 @ Z ) ) )
=> ( ! [N: nat] : ( R @ N @ ( suc @ N ) )
=> ( R @ M @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_160_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_161_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_162_bot__nat__0_Oextremum__unique,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
= ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_163_bot__nat__0_Oextremum__uniqueI,axiom,
! [A: nat] :
( ( ord_less_eq_nat @ A @ zero_zero_nat )
=> ( A = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_164_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq_nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_165_less__imp__le__nat,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_imp_le_nat
thf(fact_166_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_167_less__or__eq__imp__le,axiom,
! [M: nat,N2: nat] :
( ( ( ord_less_nat @ M @ N2 )
| ( M = N2 ) )
=> ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_or_eq_imp_le
thf(fact_168_le__neq__implies__less,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( M != N2 )
=> ( ord_less_nat @ M @ N2 ) ) ) ).
% le_neq_implies_less
thf(fact_169_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J2: nat] :
( ( ord_less_nat @ I2 @ J2 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J2 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_170_eq__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_171_le__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_172_Nat_Odiff__diff__eq,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_173_diff__le__mono,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_174_diff__le__self,axiom,
! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).
% diff_le_self
thf(fact_175_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_176_diff__le__mono2,axiom,
! [M: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).
% diff_le_mono2
thf(fact_177_MinPredicate2,axiom,
! [P: nat > $o] :
( ? [X_1: nat] : ( P @ X_1 )
=> ? [N02: nat] :
( ( P @ N02 )
& ( ( N02 = zero_zero_nat )
| ~ ( P @ ( minus_minus_nat @ N02 @ one_one_nat ) ) ) ) ) ).
% MinPredicate2
thf(fact_178_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_179_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_180_Suc__leI,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ N2 )
=> ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).
% Suc_leI
thf(fact_181_Suc__le__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
= ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_eq
thf(fact_182_dec__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ I )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) ) )
=> ( P @ J ) ) ) ) ).
% dec_induct
thf(fact_183_inc__induct,axiom,
! [I: nat,J: nat,P: nat > $o] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( P @ J )
=> ( ! [N: nat] :
( ( ord_less_eq_nat @ I @ N )
=> ( ( ord_less_nat @ N @ J )
=> ( ( P @ ( suc @ N ) )
=> ( P @ N ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_184_Suc__le__lessD,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
=> ( ord_less_nat @ M @ N2 ) ) ).
% Suc_le_lessD
thf(fact_185_le__less__Suc__eq,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ( ord_less_nat @ N2 @ ( suc @ M ) )
= ( N2 = M ) ) ) ).
% le_less_Suc_eq
thf(fact_186_less__Suc__eq__le,axiom,
! [M: nat,N2: nat] :
( ( ord_less_nat @ M @ ( suc @ N2 ) )
= ( ord_less_eq_nat @ M @ N2 ) ) ).
% less_Suc_eq_le
thf(fact_187_less__eq__Suc__le,axiom,
( ord_less_nat
= ( ^ [N5: nat] : ( ord_less_eq_nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_188_le__imp__less__Suc,axiom,
! [M: nat,N2: nat] :
( ( ord_less_eq_nat @ M @ N2 )
=> ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).
% le_imp_less_Suc
thf(fact_189_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_eq_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_190_Suc__diff__le,axiom,
! [N2: nat,M: nat] :
( ( ord_less_eq_nat @ N2 @ M )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_191_less__diff__iff,axiom,
! [K: nat,M: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_nat @ M @ N2 ) ) ) ) ).
% less_diff_iff
thf(fact_192_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_193_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_194_diff__Suc__eq__diff__pred,axiom,
! [M: nat,N2: nat] :
( ( minus_minus_nat @ M @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_195_ex__least__nat__less,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K2: nat] :
( ( ord_less_nat @ K2 @ N2 )
& ! [I4: nat] :
( ( ord_less_eq_nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_196_nat__induct__non__zero,axiom,
! [N2: nat,P: nat > $o] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( P @ one_one_nat )
=> ( ! [N: nat] :
( ( ord_less_nat @ zero_zero_nat @ N )
=> ( ( P @ N )
=> ( P @ ( suc @ N ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_non_zero
thf(fact_197_Suc__pred_H,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( N2
= ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_pred'
thf(fact_198_Suc__diff__eq__diff__pred,axiom,
! [N2: nat,M: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
= ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_199_AssumptionSubset2_I6_J,axiom,
( ( n1 != zero_zero_nat )
=> ( ~ ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ ( minus_minus_nat @ n1 @ one_one_nat ) ) @ msgInSet )
| ( msgInSet
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ ( minus_minus_nat @ n1 @ one_one_nat ) ) ) ) ) ).
% AssumptionSubset2(6)
thf(fact_200_AssumptionSubset3_I6_J,axiom,
( ( nMsg != zero_zero_nat )
=> ( ~ ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ ( minus_minus_nat @ nMsg @ one_one_nat ) ) @ msg )
| ( msg
= ( nth_message_p_v @ ( ft @ ( suc @ index ) ) @ ( minus_minus_nat @ nMsg @ one_one_nat ) ) ) ) ) ).
% AssumptionSubset3(6)
thf(fact_201_EnabledIntermediate,axiom,
! [I4: nat] :
( ( ord_less_nat @ I4 @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ one_one_nat ) @ I4 )
=> ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ I4 ) @ msg ) ) ) ).
% EnabledIntermediate
thf(fact_202_length__greater__0__conv,axiom,
! [Xs: list_c1059388851t_unit] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1406904903t_unit @ Xs ) )
= ( Xs != nil_co1338500125t_unit ) ) ).
% length_greater_0_conv
thf(fact_203_length__greater__0__conv,axiom,
! [Xs: list_message_p_v] :
( ( ord_less_nat @ zero_zero_nat @ ( size_s1168481041ge_p_v @ Xs ) )
= ( Xs != nil_message_p_v ) ) ).
% length_greater_0_conv
thf(fact_204_AssumptionFair_I2_J,axiom,
ord_less_nat @ n0 @ ( size_s1406904903t_unit @ ( fe @ n ) ) ).
% AssumptionFair(2)
thf(fact_205_Occ4,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ index ) @ nMsg ) @ msg ).
% Occ4
thf(fact_206_AssumptionFair_I3_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ n ) @ n0 ) @ msg ).
% AssumptionFair(3)
thf(fact_207_AssumptionSubset3_I4_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ nMsg ) @ msg ).
% AssumptionSubset3(4)
thf(fact_208_AssumptionSubset2_I4_J,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ n1 ) @ msgInSet ).
% AssumptionSubset2(4)
thf(fact_209_MessageStaysOrConsumed,axiom,
! [N1: nat,N22: nat,N2: nat,Msg: message_p_v] :
( ( ( ord_less_eq_nat @ N1 @ N22 )
& ( ord_less_nat @ N22 @ ( size_s1406904903t_unit @ ( fe @ N2 ) ) )
& ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ N2 ) @ N1 ) @ Msg ) )
=> ( ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ N2 ) @ N22 ) @ Msg )
| ? [N03: nat] :
( ( ord_less_eq_nat @ N1 @ N03 )
& ( ord_less_nat @ N03 @ ( size_s1168481041ge_p_v @ ( ft @ N2 ) ) )
& ( ( nth_message_p_v @ ( ft @ N2 ) @ N03 )
= Msg ) ) ) ) ).
% MessageStaysOrConsumed
thf(fact_210__092_060open_062enabled_A_Ife_A_ISuc_Aindex_J_A_B_A_Ilength_A_Ife_A_ISuc_Aindex_J_J_A_N_A1_J_J_Amsg_092_060close_062,axiom,
enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ ( suc @ index ) ) ) @ one_one_nat ) ) @ msg ).
% \<open>enabled (fe (Suc index) ! (length (fe (Suc index)) - 1)) msg\<close>
thf(fact_211_length__0__conv,axiom,
! [Xs: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs )
= zero_zero_nat )
= ( Xs = nil_co1338500125t_unit ) ) ).
% length_0_conv
thf(fact_212_length__0__conv,axiom,
! [Xs: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs )
= zero_zero_nat )
= ( Xs = nil_message_p_v ) ) ).
% length_0_conv
thf(fact_213_Occ2,axiom,
enabled_p_v_s @ ( last_c571238084t_unit @ ( fe @ index ) ) @ msg ).
% Occ2
thf(fact_214_AssumptionCase1ImplThesis_H,axiom,
enabled_p_v_s @ ( last_c571238084t_unit @ ( fe @ n ) ) @ msg ).
% AssumptionCase1ImplThesis'
thf(fact_215_neq__if__length__neq,axiom,
! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs )
!= ( size_s1406904903t_unit @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_216_neq__if__length__neq,axiom,
! [Xs: list_message_p_v,Ys: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs )
!= ( size_s1168481041ge_p_v @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_217_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_c1059388851t_unit] :
( ( size_s1406904903t_unit @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_218_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_message_p_v] :
( ( size_s1168481041ge_p_v @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_219_length__induct,axiom,
! [P: list_c1059388851t_unit > $o,Xs: list_c1059388851t_unit] :
( ! [Xs2: list_c1059388851t_unit] :
( ! [Ys2: list_c1059388851t_unit] :
( ( ord_less_nat @ ( size_s1406904903t_unit @ Ys2 ) @ ( size_s1406904903t_unit @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_220_length__induct,axiom,
! [P: list_message_p_v > $o,Xs: list_message_p_v] :
( ! [Xs2: list_message_p_v] :
( ! [Ys2: list_message_p_v] :
( ( ord_less_nat @ ( size_s1168481041ge_p_v @ Ys2 ) @ ( size_s1168481041ge_p_v @ Xs2 ) )
=> ( P @ Ys2 ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_221_list_Osize_I3_J,axiom,
( ( size_s1406904903t_unit @ nil_co1338500125t_unit )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_222_list_Osize_I3_J,axiom,
( ( size_s1168481041ge_p_v @ nil_message_p_v )
= zero_zero_nat ) ).
% list.size(3)
thf(fact_223_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 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1406904903t_unit @ Xs3 ) )
=> ( ( nth_co1649820636t_unit @ Xs3 @ I3 )
= ( nth_co1649820636t_unit @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_224_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 ) )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ ( size_s1168481041ge_p_v @ Xs3 ) )
=> ( ( nth_message_p_v @ Xs3 @ I3 )
= ( nth_message_p_v @ Ys3 @ I3 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_225_Skolem__list__nth,axiom,
! [K: nat,P: nat > config256849571t_unit > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: config256849571t_unit] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_co1649820636t_unit @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_226_Skolem__list__nth,axiom,
! [K: nat,P: nat > message_p_v > $o] :
( ( ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ? [X4: message_p_v] : ( P @ I3 @ X4 ) ) )
= ( ? [Xs3: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs3 )
= K )
& ! [I3: nat] :
( ( ord_less_nat @ I3 @ K )
=> ( P @ I3 @ ( nth_message_p_v @ Xs3 @ I3 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_227_nth__equalityI,axiom,
! [Xs: list_c1059388851t_unit,Ys: list_c1059388851t_unit] :
( ( ( size_s1406904903t_unit @ Xs )
= ( size_s1406904903t_unit @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1406904903t_unit @ Xs ) )
=> ( ( nth_co1649820636t_unit @ Xs @ I2 )
= ( nth_co1649820636t_unit @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_228_nth__equalityI,axiom,
! [Xs: list_message_p_v,Ys: list_message_p_v] :
( ( ( size_s1168481041ge_p_v @ Xs )
= ( size_s1168481041ge_p_v @ Ys ) )
=> ( ! [I2: nat] :
( ( ord_less_nat @ I2 @ ( size_s1168481041ge_p_v @ Xs ) )
=> ( ( nth_message_p_v @ Xs @ I2 )
= ( nth_message_p_v @ Ys @ I2 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_229_EnabledOrConsumed,axiom,
( ( enabled_p_v_s @ ( nth_co1649820636t_unit @ ( fe @ n ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ n ) ) @ one_one_nat ) ) @ msg )
| ? [N03: nat] :
( ( ord_less_eq_nat @ n0 @ N03 )
& ( ord_less_nat @ N03 @ ( size_s1168481041ge_p_v @ ( ft @ n ) ) )
& ( ( nth_message_p_v @ ( ft @ n ) @ N03 )
= msg ) ) ) ).
% EnabledOrConsumed
thf(fact_230_Case2ImplThesis,axiom,
( ? [N0: nat] :
( ( ord_less_eq_nat @ n0 @ N0 )
& ( ord_less_nat @ N0 @ ( size_s1168481041ge_p_v @ ( ft @ n ) ) )
& ( ( nth_message_p_v @ ( ft @ n ) @ N0 )
= msg ) )
=> ? [N6: nat] :
( ( ord_less_eq_nat @ n @ N6 )
& ? [N03: nat] :
( ( ord_less_eq_nat @ n0 @ N03 )
& ( ord_less_nat @ N03 @ ( size_s1168481041ge_p_v @ ( ft @ N6 ) ) )
& ( msg
= ( nth_message_p_v @ ( ft @ N6 ) @ N03 ) ) ) ) ) ).
% Case2ImplThesis
thf(fact_231_EnabledInSuc,axiom,
enabled_p_v_s @ ( last_c571238084t_unit @ ( fe @ ( suc @ index ) ) ) @ msg ).
% EnabledInSuc
thf(fact_232_AssumptionSubset2_I2_J,axiom,
enabled_p_v_s @ ( last_c571238084t_unit @ ( fe @ ( suc @ index ) ) ) @ msgInSet ).
% AssumptionSubset2(2)
thf(fact_233_LastOfIndex,axiom,
( ( nth_co1649820636t_unit @ ( fe @ ( suc @ index ) ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ ( suc @ zero_zero_nat ) ) )
= ( last_c571238084t_unit @ ( fe @ index ) ) ) ).
% LastOfIndex
thf(fact_234_EnabledOrConsumedAtLast,axiom,
( ( enabled_p_v_s @ ( last_c571238084t_unit @ ( fe @ n ) ) @ msg )
| ? [N03: nat] :
( ( ord_less_eq_nat @ n0 @ N03 )
& ( ord_less_nat @ N03 @ ( size_s1168481041ge_p_v @ ( ft @ n ) ) )
& ( ( nth_message_p_v @ ( ft @ n ) @ N03 )
= msg ) ) ) ).
% EnabledOrConsumedAtLast
thf(fact_235_last__conv__nth,axiom,
! [Xs: list_c1059388851t_unit] :
( ( Xs != nil_co1338500125t_unit )
=> ( ( last_c571238084t_unit @ Xs )
= ( nth_co1649820636t_unit @ Xs @ ( minus_minus_nat @ ( size_s1406904903t_unit @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_236_last__conv__nth,axiom,
! [Xs: list_message_p_v] :
( ( Xs != nil_message_p_v )
=> ( ( last_message_p_v @ Xs )
= ( nth_message_p_v @ Xs @ ( minus_minus_nat @ ( size_s1168481041ge_p_v @ Xs ) @ one_one_nat ) ) ) ) ).
% last_conv_nth
thf(fact_237__092_060open_062_092_060And_062n0_Amsg_O_A_092_060forall_062n_O_Aexecution_OfirstOccurrence_A_Ife_An_J_A_Ift_An_J_Amsg_An0_A_092_060longrightarrow_062_Amsg_A_092_060in_062_D_Amsgs_A_Ilast_A_Ife_An_J_J_092_060close_062,axiom,
! [Msg: message_p_v,N04: nat,N7: nat] :
( ( firstO1414030372_p_v_s @ ( fe @ N7 ) @ ( ft @ N7 ) @ Msg @ N04 )
=> ( ord_less_nat @ zero_zero_nat @ ( msgs_p1029620568t_unit @ ( last_c571238084t_unit @ ( fe @ N7 ) ) @ Msg ) ) ) ).
% \<open>\<And>n0 msg. \<forall>n. execution.firstOccurrence (fe n) (ft n) msg n0 \<longrightarrow> msg \<in># msgs (last (fe n))\<close>
thf(fact_238_le__numeral__extra_I3_J,axiom,
ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).
% le_numeral_extra(3)
thf(fact_239_less__numeral__extra_I3_J,axiom,
~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).
% less_numeral_extra(3)
thf(fact_240_le__numeral__extra_I4_J,axiom,
ord_less_eq_nat @ one_one_nat @ one_one_nat ).
% le_numeral_extra(4)
thf(fact_241_less__numeral__extra_I4_J,axiom,
~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).
% less_numeral_extra(4)
thf(fact_242_less__numeral__extra_I1_J,axiom,
ord_less_nat @ zero_zero_nat @ one_one_nat ).
% less_numeral_extra(1)
thf(fact_243__092_060open_062_092_060forall_062n_O_A_092_060forall_062msg_H_092_060in_062firstOccSet_An_O_Amsg_H_A_092_060in_062_D_Amsgs_A_Ilast_A_Ife_An_J_J_092_060close_062,axiom,
! [N7: nat,X5: message_p_v] :
( ( member_message_p_v @ X5 @ ( firstOccSet @ N7 ) )
=> ( ord_less_nat @ zero_zero_nat @ ( msgs_p1029620568t_unit @ ( last_c571238084t_unit @ ( fe @ N7 ) ) @ X5 ) ) ) ).
% \<open>\<forall>n. \<forall>msg'\<in>firstOccSet n. msg' \<in># msgs (last (fe n))\<close>
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_nat @ ( minus_minus_nat @ nMsg @ ( suc @ zero_zero_nat ) ) @ ( minus_minus_nat @ ( size_s1406904903t_unit @ ( fe @ index ) ) @ ( suc @ zero_zero_nat ) ) ).
%------------------------------------------------------------------------------