TPTP Problem File: ITP125^1.p
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%------------------------------------------------------------------------------
% File : ITP125^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Monitor problem prob_453__6453916_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 : Monitor/prob_453__6453916_1 [Des21]
% Status : Theorem
% Rating : 0.30 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 315 ( 83 unt; 31 typ; 0 def)
% Number of atoms : 823 ( 183 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 2451 ( 87 ~; 25 |; 40 &;1887 @)
% ( 0 <=>; 412 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Number of types : 7 ( 6 usr)
% Number of type conns : 87 ( 87 >; 0 *; 0 +; 0 <<)
% Number of symbols : 26 ( 25 usr; 14 con; 0-4 aty)
% Number of variables : 866 ( 67 ^; 764 !; 35 ?; 866 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:32:11.449
%------------------------------------------------------------------------------
% Could-be-implicit typings (6)
thf(ty_n_t__Trace__Oprefix_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J_J,type,
prefix1027212443list_a: $tType ).
thf(ty_n_t__Trace__Otrace_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J_J,type,
trace_1367752404list_a: $tType ).
thf(ty_n_t__Interval__O__092__060I__062,type,
i: $tType ).
thf(ty_n_t__MFOTL__Oformula_Itf__a_J,type,
formula_a: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
% Explicit typings (25)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_MFOTL_Oformula_OUntil_001tf__a,type,
until_a: formula_a > i > formula_a > formula_a ).
thf(sy_c_MFOTL_Osat_001tf__a,type,
sat_a: trace_1367752404list_a > list_a > nat > formula_a > $o ).
thf(sy_c_Monitor__Mirabelle__prbptmgypa_Oprogress_001tf__a,type,
monito1457594016ress_a: trace_1367752404list_a > formula_a > nat > 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_001t__Trace__Oprefix_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J_J,type,
ord_le887097159list_a: prefix1027212443list_a > prefix1027212443list_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Trace__Oprefix_It__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J_J,type,
ord_le699472955list_a: prefix1027212443list_a > prefix1027212443list_a > $o ).
thf(sy_c_Trace_O_092_060tau_062_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J,type,
tau_Pr257024512list_a: trace_1367752404list_a > nat > nat ).
thf(sy_c_Trace_Oplen_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J,type,
plen_P694648887list_a: prefix1027212443list_a > nat ).
thf(sy_c_Trace_Oprefix__of_001t__Product____Type__Oprod_It__List__Olist_It__String__Ochar_J_Mt__List__Olist_Itf__a_J_J,type,
prefix1041802747list_a: prefix1027212443list_a > trace_1367752404list_a > $o ).
thf(sy_v_I____,type,
i2: i ).
thf(sy_v__092_060phi_062,type,
phi: formula_a ).
thf(sy_v__092_060phi_0621____,type,
phi_1: formula_a ).
thf(sy_v__092_060phi_0622____,type,
phi_2: formula_a ).
thf(sy_v__092_060pi_062,type,
pi: prefix1027212443list_a ).
thf(sy_v__092_060sigma_062,type,
sigma: trace_1367752404list_a ).
thf(sy_v__092_060sigma_062_H,type,
sigma2: trace_1367752404list_a ).
thf(sy_v_b____,type,
b: nat ).
thf(sy_v_i,type,
i3: nat ).
thf(sy_v_ia____,type,
ia: nat ).
thf(sy_v_j____,type,
j: nat ).
thf(sy_v_ja____,type,
ja: nat ).
thf(sy_v_k____,type,
k: nat ).
% Relevant facts (283)
thf(fact_0__092_060open_062k_A_060_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_nat @ k @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>k < Monitor_Mirabelle_prbptmgypa.progress \<sigma> \<phi>1 (plen \<pi>)\<close>
thf(fact_1_assms_I3_J,axiom,
ord_less_nat @ i3 @ ( monito1457594016ress_a @ sigma @ phi @ ( plen_P694648887list_a @ pi ) ) ).
% assms(3)
thf(fact_2_assms_I1_J,axiom,
prefix1041802747list_a @ pi @ sigma ).
% assms(1)
thf(fact_3_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_4_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_not_refl
thf(fact_5_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less_nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_6_less__not__refl3,axiom,
! [S: nat,T: nat] :
( ( ord_less_nat @ S @ T )
=> ( S != T ) ) ).
% less_not_refl3
thf(fact_7_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less_nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_8_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
=> ( P @ M2 ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_9_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M2: nat] :
( ( ord_less_nat @ M2 @ N2 )
& ~ ( P @ M2 ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_10_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_11__092_060open_062j_____A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_eq_nat @ j @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>j__ \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma> \<phi>1 (plen \<pi>)\<close>
thf(fact_12_assms_I2_J,axiom,
prefix1041802747list_a @ pi @ sigma2 ).
% assms(2)
thf(fact_13__092_060open_062j_____A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_A_092_060phi_0622_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_eq_nat @ j @ ( monito1457594016ress_a @ sigma @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>j__ \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma> \<phi>2 (plen \<pi>)\<close>
thf(fact_14_le__refl,axiom,
! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).
% le_refl
thf(fact_15_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_16_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_17_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_18_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_19_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B: nat] :
( ( P @ K )
=> ( ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq_nat @ Y2 @ B ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq_nat @ Y3 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_20_progress__le,axiom,
! [Sigma: trace_1367752404list_a,Phi: formula_a,J: nat] : ( ord_less_eq_nat @ ( monito1457594016ress_a @ Sigma @ Phi @ J ) @ J ) ).
% progress_le
thf(fact_21_progress__mono,axiom,
! [J: nat,J2: nat,Sigma: trace_1367752404list_a,Phi: formula_a] :
( ( ord_less_eq_nat @ J @ J2 )
=> ( ord_less_eq_nat @ ( monito1457594016ress_a @ Sigma @ Phi @ J ) @ ( monito1457594016ress_a @ Sigma @ Phi @ J2 ) ) ) ).
% progress_mono
thf(fact_22_pinf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_eq_nat @ X3 @ T ) ) ).
% pinf(6)
thf(fact_23_pinf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_eq_nat @ T @ X3 ) ) ).
% pinf(8)
thf(fact_24_minf_I6_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_eq_nat @ X3 @ T ) ) ).
% minf(6)
thf(fact_25_minf_I8_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_eq_nat @ T @ X3 ) ) ).
% minf(8)
thf(fact_26_less__mono__imp__le__mono,axiom,
! [F: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less_nat @ I2 @ J3 )
=> ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
=> ( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_27_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_28_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_29_le__eq__less__or__eq,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_nat @ M3 @ N3 )
| ( M3 = N3 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_30_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_31_nat__less__le,axiom,
( ord_less_nat
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq_nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_32_progress__prefix__conv,axiom,
! [Pi: prefix1027212443list_a,Sigma: trace_1367752404list_a,Sigma2: trace_1367752404list_a,Phi: formula_a] :
( ( prefix1041802747list_a @ Pi @ Sigma )
=> ( ( prefix1041802747list_a @ Pi @ Sigma2 )
=> ( ( monito1457594016ress_a @ Sigma @ Phi @ ( plen_P694648887list_a @ Pi ) )
= ( monito1457594016ress_a @ Sigma2 @ Phi @ ( plen_P694648887list_a @ Pi ) ) ) ) ) ).
% progress_prefix_conv
thf(fact_33_pinf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(1)
thf(fact_34_pinf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% pinf(2)
thf(fact_35_pinf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(3)
thf(fact_36_pinf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( X3 != T ) ) ).
% pinf(4)
thf(fact_37_pinf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ~ ( ord_less_nat @ X3 @ T ) ) ).
% pinf(5)
thf(fact_38_pinf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ Z @ X3 )
=> ( ord_less_nat @ T @ X3 ) ) ).
% pinf(7)
thf(fact_39_minf_I1_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(1)
thf(fact_40_minf_I2_J,axiom,
! [P: nat > $o,P2: nat > $o,Q: nat > $o,Q2: nat > $o] :
( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: nat] :
! [X2: nat] :
( ( ord_less_nat @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ).
% minf(2)
thf(fact_41_minf_I3_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(3)
thf(fact_42_minf_I4_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( X3 != T ) ) ).
% minf(4)
thf(fact_43_minf_I5_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ( ord_less_nat @ X3 @ T ) ) ).
% minf(5)
thf(fact_44_minf_I7_J,axiom,
! [T: nat] :
? [Z: nat] :
! [X3: nat] :
( ( ord_less_nat @ X3 @ Z )
=> ~ ( ord_less_nat @ T @ X3 ) ) ).
% minf(7)
thf(fact_45__092_060open_062j_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_H_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_eq_nat @ ja @ ( monito1457594016ress_a @ sigma2 @ phi_1 @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma>' \<phi>1 (plen \<pi>)\<close>
thf(fact_46_order__refl,axiom,
! [X: prefix1027212443list_a] : ( ord_le699472955list_a @ X @ X ) ).
% order_refl
thf(fact_47_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_48_Until_OIH_I1_J,axiom,
! [I: nat,V: list_a] :
( ( ord_less_nat @ I @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) )
=> ( ( sat_a @ sigma @ V @ I @ phi_1 )
= ( sat_a @ sigma2 @ V @ I @ phi_1 ) ) ) ).
% Until.IH(1)
thf(fact_49_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 )
=> ( ! [I3: nat] :
( ( ord_less_nat @ K2 @ I3 )
=> ( P @ I3 ) )
=> ( P @ K2 ) ) )
=> ( P @ M ) ) ) ).
% nat_descend_induct
thf(fact_50_complete__interval,axiom,
! [A: nat,B: nat,P: nat > $o] :
( ( ord_less_nat @ A @ B )
=> ( ( P @ A )
=> ( ~ ( P @ B )
=> ? [C: nat] :
( ( ord_less_eq_nat @ A @ C )
& ( ord_less_eq_nat @ C @ B )
& ! [X3: nat] :
( ( ( ord_less_eq_nat @ A @ X3 )
& ( ord_less_nat @ X3 @ C ) )
=> ( P @ X3 ) )
& ! [D: nat] :
( ! [X2: nat] :
( ( ( ord_less_eq_nat @ A @ X2 )
& ( ord_less_nat @ X2 @ D ) )
=> ( P @ X2 ) )
=> ( ord_less_eq_nat @ D @ C ) ) ) ) ) ) ).
% complete_interval
thf(fact_51_order_Onot__eq__order__implies__strict,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a] :
( ( A != B )
=> ( ( ord_le699472955list_a @ A @ B )
=> ( ord_le887097159list_a @ A @ B ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_52_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_53_dual__order_Ostrict__implies__order,axiom,
! [B: prefix1027212443list_a,A: prefix1027212443list_a] :
( ( ord_le887097159list_a @ B @ A )
=> ( ord_le699472955list_a @ B @ A ) ) ).
% dual_order.strict_implies_order
thf(fact_54_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_55_dual__order_Ostrict__iff__order,axiom,
( ord_le887097159list_a
= ( ^ [B2: prefix1027212443list_a,A2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_56_dual__order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( A2 != B2 ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_57_dual__order_Oorder__iff__strict,axiom,
( ord_le699472955list_a
= ( ^ [B2: prefix1027212443list_a,A2: prefix1027212443list_a] :
( ( ord_le887097159list_a @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_58_dual__order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [B2: nat,A2: nat] :
( ( ord_less_nat @ B2 @ A2 )
| ( A2 = B2 ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_59_order_Ostrict__implies__order,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a] :
( ( ord_le887097159list_a @ A @ B )
=> ( ord_le699472955list_a @ A @ B ) ) ).
% order.strict_implies_order
thf(fact_60_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_61__092_060open_062j_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_H_A_092_060phi_0622_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_eq_nat @ ja @ ( monito1457594016ress_a @ sigma2 @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma>' \<phi>2 (plen \<pi>)\<close>
thf(fact_62_Until_OIH_I2_J,axiom,
! [I: nat,V: list_a] :
( ( ord_less_nat @ I @ ( monito1457594016ress_a @ sigma @ phi_2 @ ( plen_P694648887list_a @ pi ) ) )
=> ( ( sat_a @ sigma @ V @ I @ phi_2 )
= ( sat_a @ sigma2 @ V @ I @ phi_2 ) ) ) ).
% Until.IH(2)
thf(fact_63_dual__order_Oantisym,axiom,
! [B: prefix1027212443list_a,A: prefix1027212443list_a] :
( ( ord_le699472955list_a @ B @ A )
=> ( ( ord_le699472955list_a @ A @ B )
=> ( A = B ) ) ) ).
% dual_order.antisym
thf(fact_64_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_65_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: prefix1027212443list_a,Z3: prefix1027212443list_a] : Y4 = Z3 )
= ( ^ [A2: prefix1027212443list_a,B2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ B2 @ A2 )
& ( ord_le699472955list_a @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_66_dual__order_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : Y4 = Z3 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ A2 @ B2 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_67_dual__order_Otrans,axiom,
! [B: prefix1027212443list_a,A: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ B @ A )
=> ( ( ord_le699472955list_a @ C2 @ B )
=> ( ord_le699472955list_a @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_68_dual__order_Otrans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_eq_nat @ C2 @ A ) ) ) ).
% dual_order.trans
thf(fact_69_linorder__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_eq_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ).
% linorder_wlog
thf(fact_70_dual__order_Orefl,axiom,
! [A: prefix1027212443list_a] : ( ord_le699472955list_a @ A @ A ) ).
% dual_order.refl
thf(fact_71_dual__order_Orefl,axiom,
! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).
% dual_order.refl
thf(fact_72_order__trans,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a,Z4: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X @ Y )
=> ( ( ord_le699472955list_a @ Y @ Z4 )
=> ( ord_le699472955list_a @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_73_order__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_eq_nat @ X @ Z4 ) ) ) ).
% order_trans
thf(fact_74_order__class_Oorder_Oantisym,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_le699472955list_a @ B @ A )
=> ( A = B ) ) ) ).
% order_class.order.antisym
thf(fact_75_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_76_ord__le__eq__trans,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( B = C2 )
=> ( ord_le699472955list_a @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_77_ord__le__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_le_eq_trans
thf(fact_78_ord__eq__le__trans,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( A = B )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ord_le699472955list_a @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_79_ord__eq__le__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% ord_eq_le_trans
thf(fact_80_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: prefix1027212443list_a,Z3: prefix1027212443list_a] : Y4 = Z3 )
= ( ^ [A2: prefix1027212443list_a,B2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A2 @ B2 )
& ( ord_le699472955list_a @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_81_order__class_Oorder_Oeq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : Y4 = Z3 )
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_82_antisym__conv,axiom,
! [Y: prefix1027212443list_a,X: prefix1027212443list_a] :
( ( ord_le699472955list_a @ Y @ X )
=> ( ( ord_le699472955list_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv
thf(fact_83_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_84_le__cases3,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z4 ) )
=> ( ( ( ord_less_eq_nat @ X @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z4 @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z4 )
=> ~ ( ord_less_eq_nat @ Z4 @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z4 @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_85_order_Otrans,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ord_le699472955list_a @ A @ C2 ) ) ) ).
% order.trans
thf(fact_86_order_Otrans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_eq_nat @ A @ C2 ) ) ) ).
% order.trans
thf(fact_87_le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% le_cases
thf(fact_88_eq__refl,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ( X = Y )
=> ( ord_le699472955list_a @ X @ Y ) ) ).
% eq_refl
thf(fact_89_eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% eq_refl
thf(fact_90_linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linear
thf(fact_91_antisym,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X @ Y )
=> ( ( ord_le699472955list_a @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_92_antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% antisym
thf(fact_93_eq__iff,axiom,
( ( ^ [Y4: prefix1027212443list_a,Z3: prefix1027212443list_a] : Y4 = Z3 )
= ( ^ [X4: prefix1027212443list_a,Y5: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X4 @ Y5 )
& ( ord_le699472955list_a @ Y5 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_94_eq__iff,axiom,
( ( ^ [Y4: nat,Z3: nat] : Y4 = Z3 )
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% eq_iff
thf(fact_95_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_96_ord__le__eq__subst,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > nat,C2: nat] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_97_ord__le__eq__subst,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_98_ord__le__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_le_eq_subst
thf(fact_99_ord__eq__le__subst,axiom,
! [A: prefix1027212443list_a,F: nat > prefix1027212443list_a,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_100_ord__eq__le__subst,axiom,
! [A: nat,F: prefix1027212443list_a > nat,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_101_ord__eq__le__subst,axiom,
! [A: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( A
= ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_102_ord__eq__le__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_103_order__subst2,axiom,
! [A: nat,B: nat,F: nat > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le699472955list_a @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_104_order__subst2,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > nat,C2: nat] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_105_order__subst2,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_le699472955list_a @ ( F @ B ) @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_106_order__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_subst2
thf(fact_107_order__subst1,axiom,
! [A: nat,F: prefix1027212443list_a > nat,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst1,axiom,
! [A: prefix1027212443list_a,F: nat > prefix1027212443list_a,B: nat,C2: nat] :
( ( ord_le699472955list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_109_order__subst1,axiom,
! [A: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le699472955list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_110_order__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_subst1
thf(fact_111_ord__eq__less__subst,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( A
= ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_112_ord__less__eq__subst,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ( F @ B )
= C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% ord_less_eq_subst
thf(fact_113_order__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_subst1
thf(fact_114_order__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_subst2
thf(fact_115_gt__ex,axiom,
! [X: nat] :
? [X_1: nat] : ( ord_less_nat @ X @ X_1 ) ).
% gt_ex
thf(fact_116_neqE,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% neqE
thf(fact_117_neq__iff,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
= ( ( ord_less_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ) ).
% neq_iff
thf(fact_118_order_Oasym,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% order.asym
thf(fact_119_less__imp__neq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_neq
thf(fact_120_less__asym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_asym
thf(fact_121_less__asym_H,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ~ ( ord_less_nat @ B @ A ) ) ).
% less_asym'
thf(fact_122_less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% less_trans
thf(fact_123_less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
| ( X = Y )
| ( ord_less_nat @ Y @ X ) ) ).
% less_linear
thf(fact_124_less__irrefl,axiom,
! [X: nat] :
~ ( ord_less_nat @ X @ X ) ).
% less_irrefl
thf(fact_125_ord__eq__less__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( A = B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_eq_less_trans
thf(fact_126_ord__less__eq__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( B = C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% ord_less_eq_trans
thf(fact_127_dual__order_Oasym,axiom,
! [B: nat,A: nat] :
( ( ord_less_nat @ B @ A )
=> ~ ( ord_less_nat @ A @ B ) ) ).
% dual_order.asym
thf(fact_128_less__imp__not__eq,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( X != Y ) ) ).
% less_imp_not_eq
thf(fact_129_less__not__sym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ~ ( ord_less_nat @ Y @ X ) ) ).
% less_not_sym
thf(fact_130_less__induct,axiom,
! [P: nat > $o,A: nat] :
( ! [X2: nat] :
( ! [Y3: nat] :
( ( ord_less_nat @ Y3 @ X2 )
=> ( P @ Y3 ) )
=> ( P @ X2 ) )
=> ( P @ A ) ) ).
% less_induct
thf(fact_131_antisym__conv3,axiom,
! [Y: nat,X: nat] :
( ~ ( ord_less_nat @ Y @ X )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv3
thf(fact_132_less__imp__not__eq2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( Y != X ) ) ).
% less_imp_not_eq2
thf(fact_133_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_134_linorder__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( X != Y )
=> ( ord_less_nat @ Y @ X ) ) ) ).
% linorder_cases
thf(fact_135_dual__order_Oirrefl,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% dual_order.irrefl
thf(fact_136_order_Ostrict__trans,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans
thf(fact_137_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_138_exists__least__iff,axiom,
( ( ^ [P3: nat > $o] :
? [X5: nat] : ( P3 @ X5 ) )
= ( ^ [P4: nat > $o] :
? [N3: nat] :
( ( P4 @ N3 )
& ! [M3: nat] :
( ( ord_less_nat @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ).
% exists_least_iff
thf(fact_139_linorder__less__wlog,axiom,
! [P: nat > nat > $o,A: nat,B: nat] :
( ! [A3: nat,B3: nat] :
( ( ord_less_nat @ A3 @ B3 )
=> ( P @ A3 @ B3 ) )
=> ( ! [A3: nat] : ( P @ A3 @ A3 )
=> ( ! [A3: nat,B3: nat] :
( ( P @ B3 @ A3 )
=> ( P @ A3 @ B3 ) )
=> ( P @ A @ B ) ) ) ) ).
% linorder_less_wlog
thf(fact_140_dual__order_Ostrict__trans,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans
thf(fact_141_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_142_order_Ostrict__implies__not__eq,axiom,
! [A: nat,B: nat] :
( ( ord_less_nat @ A @ B )
=> ( A != B ) ) ).
% order.strict_implies_not_eq
thf(fact_143_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_144_leD,axiom,
! [Y: prefix1027212443list_a,X: prefix1027212443list_a] :
( ( ord_le699472955list_a @ Y @ X )
=> ~ ( ord_le887097159list_a @ X @ Y ) ) ).
% leD
thf(fact_145_leD,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_nat @ X @ Y ) ) ).
% leD
thf(fact_146_leI,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% leI
thf(fact_147_le__less,axiom,
( ord_le699472955list_a
= ( ^ [X4: prefix1027212443list_a,Y5: prefix1027212443list_a] :
( ( ord_le887097159list_a @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_148_le__less,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_nat @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ).
% le_less
thf(fact_149_less__le,axiom,
( ord_le887097159list_a
= ( ^ [X4: prefix1027212443list_a,Y5: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_150_less__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ).
% less_le
thf(fact_151_order__le__less__subst1,axiom,
! [A: prefix1027212443list_a,F: nat > prefix1027212443list_a,B: nat,C2: nat] :
( ( ord_le699472955list_a @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le887097159list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_152_order__le__less__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ ( F @ B ) )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_153_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_le887097159list_a @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_154_order__le__less__subst2,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > nat,C2: nat] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_155_order__le__less__subst2,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_le887097159list_a @ ( F @ B ) @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_156_order__le__less__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_le_less_subst2
thf(fact_157_order__less__le__subst1,axiom,
! [A: prefix1027212443list_a,F: nat > prefix1027212443list_a,B: nat,C2: nat] :
( ( ord_le887097159list_a @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_158_order__less__le__subst1,axiom,
! [A: nat,F: prefix1027212443list_a > nat,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_159_order__less__le__subst1,axiom,
! [A: prefix1027212443list_a,F: prefix1027212443list_a > prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le887097159list_a @ A @ ( F @ B ) )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ! [X2: prefix1027212443list_a,Y2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X2 @ Y2 )
=> ( ord_le699472955list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_160_order__less__le__subst1,axiom,
! [A: nat,F: nat > nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ ( F @ B ) )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_eq_nat @ X2 @ Y2 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_161_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_le699472955list_a @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_le887097159list_a @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_le887097159list_a @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_162_order__less__le__subst2,axiom,
! [A: nat,B: nat,F: nat > nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( F @ B ) @ C2 )
=> ( ! [X2: nat,Y2: nat] :
( ( ord_less_nat @ X2 @ Y2 )
=> ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y2 ) ) )
=> ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).
% order_less_le_subst2
thf(fact_163_not__le,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_eq_nat @ X @ Y ) )
= ( ord_less_nat @ Y @ X ) ) ).
% not_le
thf(fact_164_not__less,axiom,
! [X: nat,Y: nat] :
( ( ~ ( ord_less_nat @ X @ Y ) )
= ( ord_less_eq_nat @ Y @ X ) ) ).
% not_less
thf(fact_165_le__neq__trans,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( A != B )
=> ( ord_le887097159list_a @ A @ B ) ) ) ).
% le_neq_trans
thf(fact_166_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_167_antisym__conv1,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ~ ( ord_le887097159list_a @ X @ Y )
=> ( ( ord_le699472955list_a @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_168_antisym__conv1,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% antisym_conv1
thf(fact_169_antisym__conv2,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X @ Y )
=> ( ( ~ ( ord_le887097159list_a @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_170_antisym__conv2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ~ ( ord_less_nat @ X @ Y ) )
= ( X = Y ) ) ) ).
% antisym_conv2
thf(fact_171_less__imp__le,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ( ord_le887097159list_a @ X @ Y )
=> ( ord_le699472955list_a @ X @ Y ) ) ).
% less_imp_le
thf(fact_172_less__imp__le,axiom,
! [X: nat,Y: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% less_imp_le
thf(fact_173_le__less__trans,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a,Z4: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X @ Y )
=> ( ( ord_le887097159list_a @ Y @ Z4 )
=> ( ord_le887097159list_a @ X @ Z4 ) ) ) ).
% le_less_trans
thf(fact_174_le__less__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% le_less_trans
thf(fact_175_less__le__trans,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a,Z4: prefix1027212443list_a] :
( ( ord_le887097159list_a @ X @ Y )
=> ( ( ord_le699472955list_a @ Y @ Z4 )
=> ( ord_le887097159list_a @ X @ Z4 ) ) ) ).
% less_le_trans
thf(fact_176_less__le__trans,axiom,
! [X: nat,Y: nat,Z4: nat] :
( ( ord_less_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z4 )
=> ( ord_less_nat @ X @ Z4 ) ) ) ).
% less_le_trans
thf(fact_177_le__less__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_nat @ Y @ X ) ) ).
% le_less_linear
thf(fact_178_le__imp__less__or__eq,axiom,
! [X: prefix1027212443list_a,Y: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X @ Y )
=> ( ( ord_le887097159list_a @ X @ Y )
| ( X = Y ) ) ) ).
% le_imp_less_or_eq
thf(fact_179_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_180_less__le__not__le,axiom,
( ord_le887097159list_a
= ( ^ [X4: prefix1027212443list_a,Y5: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X4 @ Y5 )
& ~ ( ord_le699472955list_a @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_181_less__le__not__le,axiom,
( ord_less_nat
= ( ^ [X4: nat,Y5: nat] :
( ( ord_less_eq_nat @ X4 @ Y5 )
& ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).
% less_le_not_le
thf(fact_182_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_183_order_Ostrict__trans1,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A @ B )
=> ( ( ord_le887097159list_a @ B @ C2 )
=> ( ord_le887097159list_a @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_184_order_Ostrict__trans1,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans1
thf(fact_185_order_Ostrict__trans2,axiom,
! [A: prefix1027212443list_a,B: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le887097159list_a @ A @ B )
=> ( ( ord_le699472955list_a @ B @ C2 )
=> ( ord_le887097159list_a @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_186_order_Ostrict__trans2,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ B @ C2 )
=> ( ord_less_nat @ A @ C2 ) ) ) ).
% order.strict_trans2
thf(fact_187_order_Oorder__iff__strict,axiom,
( ord_le699472955list_a
= ( ^ [A2: prefix1027212443list_a,B2: prefix1027212443list_a] :
( ( ord_le887097159list_a @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_188_order_Oorder__iff__strict,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_nat @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% order.order_iff_strict
thf(fact_189_order_Ostrict__iff__order,axiom,
( ord_le887097159list_a
= ( ^ [A2: prefix1027212443list_a,B2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_190_order_Ostrict__iff__order,axiom,
( ord_less_nat
= ( ^ [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
& ( A2 != B2 ) ) ) ) ).
% order.strict_iff_order
thf(fact_191_dual__order_Ostrict__trans1,axiom,
! [B: prefix1027212443list_a,A: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ B @ A )
=> ( ( ord_le887097159list_a @ C2 @ B )
=> ( ord_le887097159list_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_192_dual__order_Ostrict__trans1,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_eq_nat @ B @ A )
=> ( ( ord_less_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans1
thf(fact_193_dual__order_Ostrict__trans2,axiom,
! [B: prefix1027212443list_a,A: prefix1027212443list_a,C2: prefix1027212443list_a] :
( ( ord_le887097159list_a @ B @ A )
=> ( ( ord_le699472955list_a @ C2 @ B )
=> ( ord_le887097159list_a @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_194_dual__order_Ostrict__trans2,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ord_less_nat @ B @ A )
=> ( ( ord_less_eq_nat @ C2 @ B )
=> ( ord_less_nat @ C2 @ A ) ) ) ).
% dual_order.strict_trans2
thf(fact_195_Until_Oprems,axiom,
ord_less_nat @ ia @ ( monito1457594016ress_a @ sigma @ ( until_a @ phi_1 @ i2 @ phi_2 ) @ ( plen_P694648887list_a @ pi ) ) ).
% Until.prems
thf(fact_196_plen__mono,axiom,
! [Pi: prefix1027212443list_a,Pi2: prefix1027212443list_a] :
( ( ord_le699472955list_a @ Pi @ Pi2 )
=> ( ord_less_eq_nat @ ( plen_P694648887list_a @ Pi ) @ ( plen_P694648887list_a @ Pi2 ) ) ) ).
% plen_mono
thf(fact_197_verit__comp__simplify1_I3_J,axiom,
! [B4: nat,A4: nat] :
( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
= ( ord_less_nat @ A4 @ B4 ) ) ).
% verit_comp_simplify1(3)
thf(fact_198__092_060open_062i_A_060_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_H_A_Iformula_OUntil_A_092_060phi_0621_AI_A_092_060phi_0622_J_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less_nat @ ia @ ( monito1457594016ress_a @ sigma2 @ ( until_a @ phi_1 @ i2 @ phi_2 ) @ ( plen_P694648887list_a @ pi ) ) ).
% \<open>i < Monitor_Mirabelle_prbptmgypa.progress \<sigma>' (formula.Until \<phi>1 I \<phi>2) (plen \<pi>)\<close>
thf(fact_199_less__prefix__def,axiom,
( ord_le887097159list_a
= ( ^ [X4: prefix1027212443list_a,Y5: prefix1027212443list_a] :
( ( ord_le699472955list_a @ X4 @ Y5 )
& ~ ( ord_le699472955list_a @ Y5 @ X4 ) ) ) ) ).
% less_prefix_def
thf(fact_200_verit__la__disequality,axiom,
! [A: nat,B: nat] :
( ( A = B )
| ~ ( ord_less_eq_nat @ A @ B )
| ~ ( ord_less_eq_nat @ B @ A ) ) ).
% verit_la_disequality
thf(fact_201_verit__comp__simplify1_I1_J,axiom,
! [A: nat] :
~ ( ord_less_nat @ A @ A ) ).
% verit_comp_simplify1(1)
thf(fact_202_ex__prefix__of,axiom,
! [P5: prefix1027212443list_a] :
? [X_1: trace_1367752404list_a] : ( prefix1041802747list_a @ P5 @ X_1 ) ).
% ex_prefix_of
thf(fact_203_prefix__of__antimono,axiom,
! [Pi: prefix1027212443list_a,Pi2: prefix1027212443list_a,S: trace_1367752404list_a] :
( ( ord_le699472955list_a @ Pi @ Pi2 )
=> ( ( prefix1041802747list_a @ Pi2 @ S )
=> ( prefix1041802747list_a @ Pi @ S ) ) ) ).
% prefix_of_antimono
thf(fact_204_prefix__of__imp__linear,axiom,
! [Pi: prefix1027212443list_a,Sigma: trace_1367752404list_a,Pi2: prefix1027212443list_a] :
( ( prefix1041802747list_a @ Pi @ Sigma )
=> ( ( prefix1041802747list_a @ Pi2 @ Sigma )
=> ( ( ord_le699472955list_a @ Pi @ Pi2 )
| ( ord_le699472955list_a @ Pi2 @ Pi ) ) ) ) ).
% prefix_of_imp_linear
thf(fact_205_formula_Oinject_I9_J,axiom,
! [X91: formula_a,X92: i,X93: formula_a,Y91: formula_a,Y92: i,Y93: formula_a] :
( ( ( until_a @ X91 @ X92 @ X93 )
= ( until_a @ Y91 @ Y92 @ Y93 ) )
= ( ( X91 = Y91 )
& ( X92 = Y92 )
& ( X93 = Y93 ) ) ) ).
% formula.inject(9)
thf(fact_206__C21_C,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma2 @ K ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma2 @ ia ) @ b ) )
=> ( ord_less_nat @ K @ ( monito1457594016ress_a @ sigma @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ) ) ).
% "21"
thf(fact_207__C11_C,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma2 @ K ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma2 @ ia ) @ b ) )
=> ( ord_less_nat @ K @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) ) ) ).
% "11"
thf(fact_208_that,axiom,
ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma2 @ k ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma2 @ ia ) @ b ) ).
% that
thf(fact_209_nat__add__left__cancel__less,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_210_nat__add__left__cancel__le,axiom,
! [K: nat,M: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N ) )
= ( ord_less_eq_nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_211__C3_C,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma @ K ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma @ ia ) @ b ) )
=> ( ord_less_nat @ K @ ( plen_P694648887list_a @ pi ) ) ) ).
% "3"
thf(fact_212__C1_C,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma @ K ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma @ ia ) @ b ) )
=> ( ord_less_nat @ K @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) ) ) ).
% "1"
thf(fact_213__C2_C,axiom,
! [K: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ sigma @ K ) @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma @ ia ) @ b ) )
=> ( ord_less_nat @ K @ ( monito1457594016ress_a @ sigma @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ) ) ).
% "2"
thf(fact_214__092_060open_062_092_060tau_062_A_092_060sigma_062_Ai_A_L_Ab_A_L_A1_A_092_060le_062_A_092_060tau_062_A_092_060sigma_062_Aj_____092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma @ ia ) @ b ) @ one_one_nat ) @ ( tau_Pr257024512list_a @ sigma @ j ) ).
% \<open>\<tau> \<sigma> i + b + 1 \<le> \<tau> \<sigma> j__\<close>
thf(fact_215__092_060open_062_092_060tau_062_A_092_060sigma_062_H_Ai_A_L_Ab_A_L_A1_A_092_060le_062_A_092_060tau_062_A_092_060sigma_062_H_Aj_092_060close_062,axiom,
ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma2 @ ia ) @ b ) @ one_one_nat ) @ ( tau_Pr257024512list_a @ sigma2 @ ja ) ).
% \<open>\<tau> \<sigma>' i + b + 1 \<le> \<tau> \<sigma>' j\<close>
thf(fact_216_less___092_060tau_062D,axiom,
! [Sigma: trace_1367752404list_a,I: nat,J: nat] :
( ( ord_less_nat @ ( tau_Pr257024512list_a @ Sigma @ I ) @ ( tau_Pr257024512list_a @ Sigma @ J ) )
=> ( ord_less_nat @ I @ J ) ) ).
% less_\<tau>D
thf(fact_217_ex__le___092_060tau_062,axiom,
! [I: nat,X: nat,S: trace_1367752404list_a] :
? [J3: nat] :
( ( ord_less_eq_nat @ I @ J3 )
& ( ord_less_eq_nat @ X @ ( tau_Pr257024512list_a @ S @ J3 ) ) ) ).
% ex_le_\<tau>
thf(fact_218__092_060tau_062__mono,axiom,
! [I: nat,J: nat,S: trace_1367752404list_a] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ S @ I ) @ ( tau_Pr257024512list_a @ S @ J ) ) ) ).
% \<tau>_mono
thf(fact_219_less__add__eq__less,axiom,
! [K: nat,L: nat,M: nat,N: nat] :
( ( ord_less_nat @ K @ L )
=> ( ( ( plus_plus_nat @ M @ L )
= ( plus_plus_nat @ K @ N ) )
=> ( ord_less_nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_220_trans__less__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_less_add2
thf(fact_221_trans__less__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_less_add1
thf(fact_222_add__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_less_mono1
thf(fact_223_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_224_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_225_add__less__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ K @ L )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_226_add__lessD1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
=> ( ord_less_nat @ I @ K ) ) ).
% add_lessD1
thf(fact_227_add__leE,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ~ ( ( ord_less_eq_nat @ M @ N )
=> ~ ( ord_less_eq_nat @ K @ N ) ) ) ).
% add_leE
thf(fact_228_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M ) ) ).
% le_add1
thf(fact_229_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M @ N ) ) ).
% le_add2
thf(fact_230_add__leD1,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ M @ N ) ) ).
% add_leD1
thf(fact_231_add__leD2,axiom,
! [M: nat,K: nat,N: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N )
=> ( ord_less_eq_nat @ K @ N ) ) ).
% add_leD2
thf(fact_232_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus_nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_233_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_234_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_235_trans__le__add1,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).
% trans_le_add1
thf(fact_236_trans__le__add2,axiom,
! [I: nat,J: nat,M: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).
% trans_le_add2
thf(fact_237_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N3: nat] :
? [K3: nat] :
( N3
= ( plus_plus_nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_238_mono__nat__linear__lb,axiom,
! [F: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( F @ M4 ) @ ( F @ N2 ) ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_239_progress__time__conv,axiom,
! [J: nat,Sigma: trace_1367752404list_a,Sigma2: trace_1367752404list_a,Phi: formula_a] :
( ! [I2: nat] :
( ( ord_less_nat @ I2 @ J )
=> ( ( tau_Pr257024512list_a @ Sigma @ I2 )
= ( tau_Pr257024512list_a @ Sigma2 @ I2 ) ) )
=> ( ( monito1457594016ress_a @ Sigma @ Phi @ J )
= ( monito1457594016ress_a @ Sigma2 @ Phi @ J ) ) ) ).
% progress_time_conv
thf(fact_240_le___092_060tau_062__less,axiom,
! [Sigma: trace_1367752404list_a,I: nat,J: nat] :
( ( ord_less_eq_nat @ ( tau_Pr257024512list_a @ Sigma @ I ) @ ( tau_Pr257024512list_a @ Sigma @ J ) )
=> ( ( ord_less_nat @ J @ I )
=> ( ( tau_Pr257024512list_a @ Sigma @ I )
= ( tau_Pr257024512list_a @ Sigma @ J ) ) ) ) ).
% le_\<tau>_less
thf(fact_241__092_060tau_062__prefix__conv,axiom,
! [P5: prefix1027212443list_a,S: trace_1367752404list_a,S2: trace_1367752404list_a,I: nat] :
( ( prefix1041802747list_a @ P5 @ S )
=> ( ( prefix1041802747list_a @ P5 @ S2 )
=> ( ( ord_less_nat @ I @ ( plen_P694648887list_a @ P5 ) )
=> ( ( tau_Pr257024512list_a @ S @ I )
= ( tau_Pr257024512list_a @ S2 @ I ) ) ) ) ) ).
% \<tau>_prefix_conv
thf(fact_242__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062j_O_A_092_060lbrakk_062_092_060tau_062_A_092_060sigma_062_H_Ai_A_L_Ab_A_L_A1_A_092_060le_062_A_092_060tau_062_A_092_060sigma_062_H_Aj_059_Aj_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_H_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_059_Aj_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_H_A_092_060phi_0622_A_Iplen_A_092_060pi_062_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma2 @ ia ) @ b ) @ one_one_nat ) @ ( tau_Pr257024512list_a @ sigma2 @ J3 ) )
=> ( ( ord_less_eq_nat @ J3 @ ( monito1457594016ress_a @ sigma2 @ phi_1 @ ( plen_P694648887list_a @ pi ) ) )
=> ~ ( ord_less_eq_nat @ J3 @ ( monito1457594016ress_a @ sigma2 @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>j. \<lbrakk>\<tau> \<sigma>' i + b + 1 \<le> \<tau> \<sigma>' j; j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma>' \<phi>1 (plen \<pi>); j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma>' \<phi>2 (plen \<pi>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_243__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062j_O_A_092_060lbrakk_062_092_060tau_062_A_092_060sigma_062_Ai_A_L_Ab_A_L_A1_A_092_060le_062_A_092_060tau_062_A_092_060sigma_062_Aj_059_Aj_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_059_Aj_A_092_060le_062_AMonitor__Mirabelle__prbptmgypa_Oprogress_A_092_060sigma_062_A_092_060phi_0622_A_Iplen_A_092_060pi_062_J_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [J3: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ ( plus_plus_nat @ ( tau_Pr257024512list_a @ sigma @ ia ) @ b ) @ one_one_nat ) @ ( tau_Pr257024512list_a @ sigma @ J3 ) )
=> ( ( ord_less_eq_nat @ J3 @ ( monito1457594016ress_a @ sigma @ phi_1 @ ( plen_P694648887list_a @ pi ) ) )
=> ~ ( ord_less_eq_nat @ J3 @ ( monito1457594016ress_a @ sigma @ phi_2 @ ( plen_P694648887list_a @ pi ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>j. \<lbrakk>\<tau> \<sigma> i + b + 1 \<le> \<tau> \<sigma> j; j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma> \<phi>1 (plen \<pi>); j \<le> Monitor_Mirabelle_prbptmgypa.progress \<sigma> \<phi>2 (plen \<pi>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_244_add__less__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_right
thf(fact_245_add__left__cancel,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
= ( B = C2 ) ) ).
% add_left_cancel
thf(fact_246_add__right__cancel,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
= ( B = C2 ) ) ).
% add_right_cancel
thf(fact_247_add__le__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_left
thf(fact_248_add__le__cancel__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
= ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_cancel_right
thf(fact_249_add__less__cancel__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
= ( ord_less_nat @ A @ B ) ) ).
% add_less_cancel_left
thf(fact_250_one__reorient,axiom,
! [X: nat] :
( ( one_one_nat = X )
= ( X = one_one_nat ) ) ).
% one_reorient
thf(fact_251_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_252_add__mono__thms__linordered__semiring_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus_nat @ I @ K )
= ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_253_group__cancel_Oadd1,axiom,
! [A5: nat,K: nat,A: nat,B: nat] :
( ( A5
= ( plus_plus_nat @ K @ A ) )
=> ( ( plus_plus_nat @ A5 @ B )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add1
thf(fact_254_group__cancel_Oadd2,axiom,
! [B5: nat,K: nat,B: nat,A: nat] :
( ( B5
= ( plus_plus_nat @ K @ B ) )
=> ( ( plus_plus_nat @ A @ B5 )
= ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).
% group_cancel.add2
thf(fact_255_add_Oassoc,axiom,
! [A: nat,B: nat,C2: nat] :
( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C2 )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.assoc
thf(fact_256_add_Ocommute,axiom,
( plus_plus_nat
= ( ^ [A2: nat,B2: nat] : ( plus_plus_nat @ B2 @ A2 ) ) ) ).
% add.commute
thf(fact_257_add_Oleft__commute,axiom,
! [B: nat,A: nat,C2: nat] :
( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C2 ) )
= ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add.left_commute
thf(fact_258_add__left__imp__eq,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ( plus_plus_nat @ A @ B )
= ( plus_plus_nat @ A @ C2 ) )
=> ( B = C2 ) ) ).
% add_left_imp_eq
thf(fact_259_add__right__imp__eq,axiom,
! [B: nat,A: nat,C2: nat] :
( ( ( plus_plus_nat @ B @ A )
= ( plus_plus_nat @ C2 @ A ) )
=> ( B = C2 ) ) ).
% add_right_imp_eq
thf(fact_260_add__mono__thms__linordered__semiring_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_261_add__mono__thms__linordered__semiring_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_262_add__mono__thms__linordered__semiring_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_263_add__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_mono
thf(fact_264_add__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_left_mono
thf(fact_265_less__eqE,axiom,
! [A: nat,B: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ~ ! [C: nat] :
( B
!= ( plus_plus_nat @ A @ C ) ) ) ).
% less_eqE
thf(fact_266_add__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_right_mono
thf(fact_267_le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [A2: nat,B2: nat] :
? [C3: nat] :
( B2
= ( plus_plus_nat @ A2 @ C3 ) ) ) ) ).
% le_iff_add
thf(fact_268_add__le__imp__le__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_left
thf(fact_269_add__le__imp__le__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_eq_nat @ A @ B ) ) ).
% add_le_imp_le_right
thf(fact_270_add__mono__thms__linordered__field_I5_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_271_add__mono__thms__linordered__field_I2_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( I = J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_272_add__mono__thms__linordered__field_I1_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( K = L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_273_add__strict__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_strict_mono
thf(fact_274_add__strict__left__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).
% add_strict_left_mono
thf(fact_275_add__strict__right__mono,axiom,
! [A: nat,B: nat,C2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).
% add_strict_right_mono
thf(fact_276_add__less__imp__less__left,axiom,
! [C2: nat,A: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_left
thf(fact_277_add__less__imp__less__right,axiom,
! [A: nat,C2: nat,B: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
=> ( ord_less_nat @ A @ B ) ) ).
% add_less_imp_less_right
thf(fact_278_add__less__le__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_nat @ A @ B )
=> ( ( ord_less_eq_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_less_le_mono
thf(fact_279_add__le__less__mono,axiom,
! [A: nat,B: nat,C2: nat,D2: nat] :
( ( ord_less_eq_nat @ A @ B )
=> ( ( ord_less_nat @ C2 @ D2 )
=> ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).
% add_le_less_mono
thf(fact_280_add__mono__thms__linordered__field_I3_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_nat @ I @ J )
& ( ord_less_eq_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_281_add__mono__thms__linordered__field_I4_J,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ( ord_less_eq_nat @ I @ J )
& ( ord_less_nat @ K @ L ) )
=> ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_282_discrete,axiom,
( ord_less_nat
= ( ^ [A2: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A2 @ one_one_nat ) ) ) ) ).
% discrete
% Conjectures (1)
thf(conj_0,conjecture,
ord_less_nat @ k @ ( plen_P694648887list_a @ pi ) ).
%------------------------------------------------------------------------------