TPTP Problem File: ITP125^2.p
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%------------------------------------------------------------------------------
% File : ITP125^2 : 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 : 0.33 v8.1.0, 0.50 v7.5.0
% Syntax : Number of formulae : 351 ( 66 unt; 58 typ; 0 def)
% Number of atoms : 917 ( 157 equ; 0 cnn)
% Maximal formula atoms : 13 ( 3 avg)
% Number of connectives : 3757 ( 85 ~; 22 |; 62 &;3057 @)
% ( 0 <=>; 531 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 9 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 130 ( 130 >; 0 *; 0 +; 0 <<)
% Number of symbols : 55 ( 54 usr; 13 con; 0-5 aty)
% Number of variables : 1061 ( 47 ^; 929 !; 47 ?;1061 :)
% ( 38 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 16:17:27.271
%------------------------------------------------------------------------------
% Could-be-implicit typings (9)
thf(ty_t_Interval_O_092_060I_062,type,
i: $tType ).
thf(ty_t_Product__Type_Oprod,type,
product_prod: $tType > $tType > $tType ).
thf(ty_t_MFOTL_Oformula,type,
formula: $tType > $tType ).
thf(ty_t_Trace_Oprefix,type,
prefix: $tType > $tType ).
thf(ty_t_Trace_Otrace,type,
trace: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_List_Olist,type,
list: $tType > $tType ).
thf(ty_t_Nat_Onat,type,
nat: $tType ).
thf(ty_tf_a,type,
a: $tType ).
% Explicit typings (49)
thf(sy_cl_HOL_Otype,type,
type:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__add,type,
semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni770627133id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict2144017051up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1598680935umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit1656338222tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit1037483654norder:
!>[A: $tType] : $o ).
thf(sy_c_Abstract__Monitor_Oprogress__axioms,type,
abstra417776764axioms:
!>[A: $tType,B: $tType] : ( ( ( trace @ A ) > ( list @ B ) > nat > $o ) > ( ( prefix @ A ) > nat ) > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_MFOTL_Oformula_OUntil,type,
until:
!>[A: $tType] : ( ( formula @ A ) > i > ( formula @ A ) > ( formula @ A ) ) ).
thf(sy_c_MFOTL_Osat,type,
sat:
!>[A: $tType] : ( ( trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) ) > ( list @ A ) > nat > ( formula @ A ) > $o ) ).
thf(sy_c_Monitor__Mirabelle__pzlrlsievl_Oprogress,type,
monito2027355080ogress:
!>[A: $tType] : ( ( trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) ) > ( formula @ A ) > nat > nat ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Trace_O_092_060tau_062,type,
tau:
!>[A: $tType] : ( ( trace @ A ) > nat > nat ) ).
thf(sy_c_Trace_Oplen,type,
plen:
!>[A: $tType] : ( ( prefix @ A ) > nat ) ).
thf(sy_c_Trace_Oprefix__of,type,
prefix_of:
!>[A: $tType] : ( ( prefix @ A ) > ( trace @ 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: prefix @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) ).
thf(sy_v__092_060sigma_062,type,
sigma: trace @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) ).
thf(sy_v__092_060sigma_062_H,type,
sigma2: trace @ ( product_prod @ ( list @ char ) @ ( list @ 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 (255)
thf(fact_0__092_060open_062k_A_060_AMonitor__Mirabelle__pzlrlsievl_Oprogress_A_092_060sigma_062_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_092_060close_062,axiom,
ord_less @ nat @ k @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>k < Monitor_Mirabelle_pzlrlsievl.progress \<sigma> \<phi>1 (plen \<pi>)\<close>
thf(fact_1_assms_I3_J,axiom,
ord_less @ nat @ i3 @ ( monito2027355080ogress @ a @ sigma @ phi @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% assms(3)
thf(fact_2_assms_I1_J,axiom,
prefix_of @ ( product_prod @ ( list @ char ) @ ( list @ 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_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X2: A] :
( ~ ( P @ X2 )
=> ? [Y2: A] :
( ( ord_less @ nat @ ( V @ Y2 ) @ ( V @ X2 ) )
& ~ ( P @ Y2 ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_12__092_060open_062j_____A_092_060le_062_AMonitor__Mirabelle__pzlrlsievl_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 @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>j__ \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma> \<phi>1 (plen \<pi>)\<close>
thf(fact_13_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X3: C] :
( ( ord_less @ C @ X3 @ Z )
=> ( F = F ) ) ) ).
% minf(11)
thf(fact_14_assms_I2_J,axiom,
prefix_of @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi @ sigma2 ).
% assms(2)
thf(fact_15__092_060open_062j_____A_092_060le_062_AMonitor__Mirabelle__pzlrlsievl_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 @ ( monito2027355080ogress @ a @ sigma @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>j__ \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma> \<phi>2 (plen \<pi>)\<close>
thf(fact_16_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_17_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_18_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_19_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_20_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_21_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y2: nat] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ Y2 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_22_progress__le,axiom,
! [A: $tType,Sigma: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Phi: formula @ A,J: nat] : ( ord_less_eq @ nat @ ( monito2027355080ogress @ A @ Sigma @ Phi @ J ) @ J ) ).
% progress_le
thf(fact_23_progress__mono,axiom,
! [A: $tType,J: nat,J2: nat,Sigma: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Phi: formula @ A] :
( ( ord_less_eq @ nat @ J @ J2 )
=> ( ord_less_eq @ nat @ ( monito2027355080ogress @ A @ Sigma @ Phi @ J ) @ ( monito2027355080ogress @ A @ Sigma @ Phi @ J2 ) ) ) ).
% progress_mono
thf(fact_24_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ T ) ) ) ).
% pinf(6)
thf(fact_25_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less_eq @ A @ T @ X3 ) ) ) ).
% pinf(8)
thf(fact_26_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less_eq @ A @ X3 @ T ) ) ) ).
% minf(6)
thf(fact_27_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less_eq @ A @ T @ X3 ) ) ) ).
% minf(8)
thf(fact_28_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J: nat] :
( ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ ( F2 @ I2 ) @ ( F2 @ J3 ) ) )
=> ( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_29_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_30_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_31_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_32_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_33_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N3: nat] :
( ( ord_less_eq @ nat @ M3 @ N3 )
& ( M3 != N3 ) ) ) ) ).
% nat_less_le
thf(fact_34_progress__prefix__conv,axiom,
! [A: $tType,Pi: prefix @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Sigma: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Sigma2: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Phi: formula @ A] :
( ( prefix_of @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Pi @ Sigma )
=> ( ( prefix_of @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Pi @ Sigma2 )
=> ( ( monito2027355080ogress @ A @ Sigma @ Phi @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Pi ) )
= ( monito2027355080ogress @ A @ Sigma2 @ Phi @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Pi ) ) ) ) ) ).
% progress_prefix_conv
thf(fact_35_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) )
=> ( P @ Y2 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_36_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y2: A] :
( ( ord_less @ B @ ( F2 @ Y2 ) @ ( F2 @ X2 ) )
=> ( P @ Y2 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% measure_induct
thf(fact_37_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_38_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ Z2 @ X2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_39_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T ) ) ) ).
% pinf(3)
thf(fact_40_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( X3 != T ) ) ) ).
% pinf(4)
thf(fact_41_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ~ ( ord_less @ A @ X3 @ T ) ) ) ).
% pinf(5)
thf(fact_42_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ Z @ X3 )
=> ( ord_less @ A @ T @ X3 ) ) ) ).
% pinf(7)
thf(fact_43_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F: D] :
? [Z: C] :
! [X3: C] :
( ( ord_less @ C @ Z @ X3 )
=> ( F = F ) ) ) ).
% pinf(11)
thf(fact_44_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
& ( Q @ X3 ) )
= ( ( P2 @ X3 )
& ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_45_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P2: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( P @ X2 )
= ( P2 @ X2 ) ) )
=> ( ? [Z2: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z2 )
=> ( ( Q @ X2 )
= ( Q2 @ X2 ) ) )
=> ? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ( ( P @ X3 )
| ( Q @ X3 ) )
= ( ( P2 @ X3 )
| ( Q2 @ X3 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_46_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T ) ) ) ).
% minf(3)
thf(fact_47_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( X3 != T ) ) ) ).
% minf(4)
thf(fact_48_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ( ord_less @ A @ X3 @ T ) ) ) ).
% minf(5)
thf(fact_49_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T: A] :
? [Z: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z )
=> ~ ( ord_less @ A @ T @ X3 ) ) ) ).
% minf(7)
thf(fact_50__092_060open_062j_A_092_060le_062_AMonitor__Mirabelle__pzlrlsievl_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 @ ( monito2027355080ogress @ a @ sigma2 @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma>' \<phi>1 (plen \<pi>)\<close>
thf(fact_51_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_52_Until_OIH_I1_J,axiom,
! [I: nat,V2: list @ a] :
( ( ord_less @ nat @ I @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) )
=> ( ( sat @ a @ sigma @ V2 @ I @ phi_1 )
= ( sat @ a @ sigma2 @ V2 @ I @ phi_1 ) ) ) ).
% Until.IH(1)
thf(fact_53_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_54_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_55_complete__interval,axiom,
! [A: $tType] :
( ( condit1037483654norder @ A )
=> ! [A2: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P @ A2 )
=> ( ~ ( P @ B2 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ A @ C2 @ B2 )
& ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ C2 ) )
=> ( P @ X3 ) )
& ! [D2: A] :
( ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ D2 ) )
=> ( P @ X2 ) )
=> ( ord_less_eq @ A @ D2 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_56_order_Onot__eq__order__implies__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order.not_eq_order_implies_strict
thf(fact_57_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_58_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( A3 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_59_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A3: A] :
( ( ord_less @ A @ B3 @ A3 )
| ( A3 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_60_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_61__092_060open_062j_A_092_060le_062_AMonitor__Mirabelle__pzlrlsievl_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 @ ( monito2027355080ogress @ a @ sigma2 @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma>' \<phi>2 (plen \<pi>)\<close>
thf(fact_62_Until_OIH_I2_J,axiom,
! [I: nat,V2: list @ a] :
( ( ord_less @ nat @ I @ ( monito2027355080ogress @ a @ sigma @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) )
=> ( ( sat @ a @ sigma @ V2 @ I @ phi_2 )
= ( sat @ a @ sigma2 @ V2 @ I @ phi_2 ) ) ) ).
% Until.IH(2)
thf(fact_63_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_64_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ B3 @ A3 )
& ( ord_less_eq @ A @ A3 @ B3 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_65_dual__order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_66_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_67_dual__order_Orefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_68_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less_eq @ A @ X @ Z4 ) ) ) ) ).
% order_trans
thf(fact_69_order__class_Oorder_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% order_class.order.antisym
thf(fact_70_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_71_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_72_order__class_Oorder_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( ord_less_eq @ A @ B3 @ A3 ) ) ) ) ) ).
% order_class.order.eq_iff
thf(fact_73_antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv
thf(fact_74_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z4 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z4 @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z4 )
=> ~ ( ord_less_eq @ A @ Z4 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z4 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_75_order_Otrans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_76_le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% le_cases
thf(fact_77_eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% eq_refl
thf(fact_78_linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linear
thf(fact_79_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% antisym
thf(fact_80_eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y4: A,Z3: A] : Y4 = Z3 )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% eq_iff
thf(fact_81_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_82_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_83_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_84_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_85_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F3: A > B,G: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_86_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B] :
( ! [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ ( G2 @ X2 ) )
=> ( ord_less_eq @ ( A > B ) @ F2 @ G2 ) ) ) ).
% le_funI
thf(fact_87_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funE
thf(fact_88_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F2: A > B,G2: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F2 @ G2 )
=> ( ord_less_eq @ B @ ( F2 @ X ) @ ( G2 @ X ) ) ) ) ).
% le_funD
thf(fact_89_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( A2
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_90_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F2: A > B,C3: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F2 @ B2 )
= C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_91_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_92_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_93_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_94_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_95_neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% neqE
thf(fact_96_neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% neq_iff
thf(fact_97_order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_98_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% dense
thf(fact_99_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_100_less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_asym
thf(fact_101_less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% less_asym'
thf(fact_102_less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_trans
thf(fact_103_less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% less_linear
thf(fact_104_less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% less_irrefl
thf(fact_105_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_106_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_107_dual__order_Oasym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_108_less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_not_eq
thf(fact_109_less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_not_sym
thf(fact_110_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A2: A] :
( ! [X2: A] :
( ! [Y2: A] :
( ( ord_less @ A @ Y2 @ X2 )
=> ( P @ Y2 ) )
=> ( P @ X2 ) )
=> ( P @ A2 ) ) ) ).
% less_induct
thf(fact_111_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_112_less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% less_imp_not_eq2
thf(fact_113_less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P ) ) ) ).
% less_imp_triv
thf(fact_114_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_115_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_116_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_117_less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% less_imp_not_less
thf(fact_118_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X5: A] : ( P3 @ X5 ) )
= ( ^ [P4: A > $o] :
? [N3: A] :
( ( P4 @ N3 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N3 )
=> ~ ( P4 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_119_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P @ B4 @ A4 )
=> ( P @ A4 @ B4 ) )
=> ( P @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_120_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_121_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_122_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_123_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_124_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit1656338222tinuum @ A )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_125_ex__has__least__nat,axiom,
! [A: $tType,P: A > $o,K: A,M: A > nat] :
( ( P @ K )
=> ? [X2: A] :
( ( P @ X2 )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( M @ X2 ) @ ( M @ Y2 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_126_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_127_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_128_le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less @ A @ X4 @ Y5 )
| ( X4 = Y5 ) ) ) ) ) ).
% le_less
thf(fact_129_less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ( X4 != Y5 ) ) ) ) ) ).
% less_le
thf(fact_130_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less @ B @ X2 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_131_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less_eq @ A @ X2 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_132_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F2: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X2: B,Y3: B] :
( ( ord_less_eq @ B @ X2 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F2 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_133_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F2: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C3 )
=> ( ! [X2: A,Y3: A] :
( ( ord_less @ A @ X2 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X2 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_134_not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% not_le
thf(fact_135_not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% not_less
thf(fact_136_le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% le_neq_trans
thf(fact_137_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_138_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_139_less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% less_imp_le
thf(fact_140_le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% le_less_trans
thf(fact_141_less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z4 )
=> ( ord_less @ A @ X @ Z4 ) ) ) ) ).
% less_le_trans
thf(fact_142_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z4: A,Y: A] :
( ! [X2: A] :
( ( ord_less @ A @ Z4 @ X2 )
=> ( ord_less_eq @ A @ Y @ X2 ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% dense_ge
thf(fact_143_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z4: A] :
( ! [X2: A] :
( ( ord_less @ A @ X2 @ Y )
=> ( ord_less_eq @ A @ X2 @ Z4 ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ).
% dense_le
thf(fact_144_le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% le_less_linear
thf(fact_145_le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% le_imp_less_or_eq
thf(fact_146_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y5: A] :
( ( ord_less_eq @ A @ X4 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_147_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_148_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_149_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_150_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less @ A @ A3 @ B3 )
| ( A3 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_151_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A,B3: A] :
( ( ord_less_eq @ A @ A3 @ B3 )
& ( A3 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_152_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_153_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_154_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z4: A,X: A,Y: A] :
( ( ord_less @ A @ Z4 @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z4 @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% dense_ge_bounded
thf(fact_155_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z4: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z4 ) ) )
=> ( ord_less_eq @ A @ Y @ Z4 ) ) ) ) ).
% dense_le_bounded
thf(fact_156_Until_Oprems,axiom,
ord_less @ nat @ ia @ ( monito2027355080ogress @ a @ sigma @ ( until @ a @ phi_1 @ i2 @ phi_2 ) @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% Until.prems
thf(fact_157_plen__mono,axiom,
! [A: $tType,Pi: prefix @ A,Pi2: prefix @ A] :
( ( ord_less_eq @ ( prefix @ A ) @ Pi @ Pi2 )
=> ( ord_less_eq @ nat @ ( plen @ A @ Pi ) @ ( plen @ A @ Pi2 ) ) ) ).
% plen_mono
thf(fact_158_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B5: B,A5: B] :
( ( ~ ( ord_less_eq @ B @ B5 @ A5 ) )
= ( ord_less @ B @ A5 @ B5 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_159__092_060open_062i_A_060_AMonitor__Mirabelle__pzlrlsievl_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 @ ( monito2027355080ogress @ a @ sigma2 @ ( until @ a @ phi_1 @ i2 @ phi_2 ) @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ).
% \<open>i < Monitor_Mirabelle_pzlrlsievl.progress \<sigma>' (formula.Until \<phi>1 I \<phi>2) (plen \<pi>)\<close>
thf(fact_160_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) )
& ! [Y2: A] :
( ( P @ Y2 )
=> ( ord_less_eq @ nat @ ( F2 @ Y2 ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_161_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,F2: A > nat,B2: nat] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_162_progress__axioms_Ointro,axiom,
! [B: $tType,A: $tType,Progress: ( prefix @ A ) > nat,Sat: ( trace @ A ) > ( list @ B ) > nat > $o] :
( ! [Pi3: prefix @ A,Pi4: prefix @ A] :
( ( ord_less_eq @ ( prefix @ A ) @ Pi3 @ Pi4 )
=> ( ord_less_eq @ nat @ ( Progress @ Pi3 ) @ ( Progress @ Pi4 ) ) )
=> ( ! [Sigma3: trace @ A,X2: nat] :
? [Pi5: prefix @ A] :
( ( prefix_of @ A @ Pi5 @ Sigma3 )
& ( ord_less_eq @ nat @ X2 @ ( Progress @ Pi5 ) ) )
=> ( ! [Pi3: prefix @ A,Sigma3: trace @ A,Sigma4: trace @ A,I2: nat,V3: list @ B] :
( ( prefix_of @ A @ Pi3 @ Sigma3 )
=> ( ( prefix_of @ A @ Pi3 @ Sigma4 )
=> ( ( ord_less @ nat @ I2 @ ( Progress @ Pi3 ) )
=> ( ( Sat @ Sigma3 @ V3 @ I2 )
= ( Sat @ Sigma4 @ V3 @ I2 ) ) ) ) )
=> ( abstra417776764axioms @ A @ B @ Sat @ Progress ) ) ) ) ).
% progress_axioms.intro
thf(fact_163_less__prefix__def,axiom,
! [A: $tType] :
( ( ord_less @ ( prefix @ A ) )
= ( ^ [X4: prefix @ A,Y5: prefix @ A] :
( ( ord_less_eq @ ( prefix @ A ) @ X4 @ Y5 )
& ~ ( ord_less_eq @ ( prefix @ A ) @ Y5 @ X4 ) ) ) ) ).
% less_prefix_def
thf(fact_164_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F3: A > B,G: A > B] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
& ~ ( ord_less_eq @ ( A > B ) @ G @ F3 ) ) ) ) ) ).
% less_fun_def
thf(fact_165_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ).
% arg_max_natI
thf(fact_166_arg__max__equality,axiom,
! [A: $tType,C: $tType] :
( ( order @ A )
=> ! [P: C > $o,K: C,F2: C > A] :
( ( P @ K )
=> ( ! [X2: C] :
( ( P @ X2 )
=> ( ord_less_eq @ A @ ( F2 @ X2 ) @ ( F2 @ K ) ) )
=> ( ( F2 @ ( lattices_ord_arg_max @ C @ A @ F2 @ P ) )
= ( F2 @ K ) ) ) ) ) ).
% arg_max_equality
thf(fact_167_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ~ ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ( ! [Y2: A] :
( ( P @ Y2 )
=> ~ ( ord_less @ B @ ( F2 @ X2 ) @ ( F2 @ Y2 ) ) )
=> ( Q @ X2 ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F2 @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_168_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_169_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_170_ex__prefix__of,axiom,
! [A: $tType,P5: prefix @ A] :
? [X_1: trace @ A] : ( prefix_of @ A @ P5 @ X_1 ) ).
% ex_prefix_of
thf(fact_171_prefix__of__antimono,axiom,
! [A: $tType,Pi: prefix @ A,Pi2: prefix @ A,S: trace @ A] :
( ( ord_less_eq @ ( prefix @ A ) @ Pi @ Pi2 )
=> ( ( prefix_of @ A @ Pi2 @ S )
=> ( prefix_of @ A @ Pi @ S ) ) ) ).
% prefix_of_antimono
thf(fact_172_prefix__of__imp__linear,axiom,
! [A: $tType,Pi: prefix @ A,Sigma: trace @ A,Pi2: prefix @ A] :
( ( prefix_of @ A @ Pi @ Sigma )
=> ( ( prefix_of @ A @ Pi2 @ Sigma )
=> ( ( ord_less_eq @ ( prefix @ A ) @ Pi @ Pi2 )
| ( ord_less_eq @ ( prefix @ A ) @ Pi2 @ Pi ) ) ) ) ).
% prefix_of_imp_linear
thf(fact_173_progress__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( abstra417776764axioms @ A @ B )
= ( ^ [Sat2: ( trace @ A ) > ( list @ B ) > nat > $o,Progress2: ( prefix @ A ) > nat] :
( ! [Pi6: prefix @ A,Pi7: prefix @ A] :
( ( ord_less_eq @ ( prefix @ A ) @ Pi6 @ Pi7 )
=> ( ord_less_eq @ nat @ ( Progress2 @ Pi6 ) @ ( Progress2 @ Pi7 ) ) )
& ! [Sigma5: trace @ A,X4: nat] :
? [Pi6: prefix @ A] :
( ( prefix_of @ A @ Pi6 @ Sigma5 )
& ( ord_less_eq @ nat @ X4 @ ( Progress2 @ Pi6 ) ) )
& ! [Pi6: prefix @ A,Sigma5: trace @ A,Sigma6: trace @ A,I4: nat,V4: list @ B] :
( ( prefix_of @ A @ Pi6 @ Sigma5 )
=> ( ( prefix_of @ A @ Pi6 @ Sigma6 )
=> ( ( ord_less @ nat @ I4 @ ( Progress2 @ Pi6 ) )
=> ( ( Sat2 @ Sigma5 @ V4 @ I4 )
= ( Sat2 @ Sigma6 @ V4 @ I4 ) ) ) ) ) ) ) ) ).
% progress_axioms_def
thf(fact_174_formula_Oinject_I9_J,axiom,
! [A: $tType,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_175__C21_C,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ K ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ia ) @ b ) )
=> ( ord_less @ nat @ K @ ( monito2027355080ogress @ a @ sigma @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ).
% "21"
thf(fact_176__C11_C,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ K ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ia ) @ b ) )
=> ( ord_less @ nat @ K @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ).
% "11"
thf(fact_177_that,axiom,
ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ k ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ia ) @ b ) ).
% that
thf(fact_178_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_179_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_180__C3_C,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ K ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ ia ) @ b ) )
=> ( ord_less @ nat @ K @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ).
% "3"
thf(fact_181__C1_C,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ K ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ ia ) @ b ) )
=> ( ord_less @ nat @ K @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ).
% "1"
thf(fact_182__C2_C,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ K ) @ ( plus_plus @ nat @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ ia ) @ b ) )
=> ( ord_less @ nat @ K @ ( monito2027355080ogress @ a @ sigma @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ).
% "2"
thf(fact_183__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 @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ ia ) @ b ) @ ( one_one @ nat ) ) @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ j ) ).
% \<open>\<tau> \<sigma> i + b + 1 \<le> \<tau> \<sigma> j__\<close>
thf(fact_184__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 @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ia ) @ b ) @ ( one_one @ nat ) ) @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ja ) ).
% \<open>\<tau> \<sigma>' i + b + 1 \<le> \<tau> \<sigma>' j\<close>
thf(fact_185_less___092_060tau_062D,axiom,
! [A: $tType,Sigma: trace @ A,I: nat,J: nat] :
( ( ord_less @ nat @ ( tau @ A @ Sigma @ I ) @ ( tau @ A @ Sigma @ J ) )
=> ( ord_less @ nat @ I @ J ) ) ).
% less_\<tau>D
thf(fact_186_ex__le___092_060tau_062,axiom,
! [A: $tType,I: nat,X: nat,S: trace @ A] :
? [J3: nat] :
( ( ord_less_eq @ nat @ I @ J3 )
& ( ord_less_eq @ nat @ X @ ( tau @ A @ S @ J3 ) ) ) ).
% ex_le_\<tau>
thf(fact_187__092_060tau_062__mono,axiom,
! [A: $tType,I: nat,J: nat,S: trace @ A] :
( ( ord_less_eq @ nat @ I @ J )
=> ( ord_less_eq @ nat @ ( tau @ A @ S @ I ) @ ( tau @ A @ S @ J ) ) ) ).
% \<tau>_mono
thf(fact_188_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_189_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_190_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_191_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_192_not__add__less2,axiom,
! [J: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I ) @ I ) ).
% not_add_less2
thf(fact_193_not__add__less1,axiom,
! [I: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J ) @ I ) ).
% not_add_less1
thf(fact_194_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_195_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_196_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_197_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_198_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_199_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_200_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_201_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_202_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_203_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_204_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_205_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_206_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_207_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X2: A] :
( ( P @ X2 )
=> ? [Y2: A] :
( ( P @ Y2 )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y2 ) @ ( F2 @ X2 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F2 @ Y3 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_208_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M: nat,K: nat] :
( ! [M4: nat,N2: nat] :
( ( ord_less @ nat @ M4 @ N2 )
=> ( ord_less @ nat @ ( F2 @ M4 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_209_progress__time__conv,axiom,
! [A: $tType,J: nat,Sigma: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Sigma2: trace @ ( product_prod @ ( list @ char ) @ ( list @ A ) ),Phi: formula @ A] :
( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( tau @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Sigma @ I2 )
= ( tau @ ( product_prod @ ( list @ char ) @ ( list @ A ) ) @ Sigma2 @ I2 ) ) )
=> ( ( monito2027355080ogress @ A @ Sigma @ Phi @ J )
= ( monito2027355080ogress @ A @ Sigma2 @ Phi @ J ) ) ) ).
% progress_time_conv
thf(fact_210_le___092_060tau_062__less,axiom,
! [A: $tType,Sigma: trace @ A,I: nat,J: nat] :
( ( ord_less_eq @ nat @ ( tau @ A @ Sigma @ I ) @ ( tau @ A @ Sigma @ J ) )
=> ( ( ord_less @ nat @ J @ I )
=> ( ( tau @ A @ Sigma @ I )
= ( tau @ A @ Sigma @ J ) ) ) ) ).
% le_\<tau>_less
thf(fact_211__092_060tau_062__prefix__conv,axiom,
! [A: $tType,P5: prefix @ A,S: trace @ A,S2: trace @ A,I: nat] :
( ( prefix_of @ A @ P5 @ S )
=> ( ( prefix_of @ A @ P5 @ S2 )
=> ( ( ord_less @ nat @ I @ ( plen @ A @ P5 ) )
=> ( ( tau @ A @ S @ I )
= ( tau @ A @ S2 @ I ) ) ) ) ) ).
% \<tau>_prefix_conv
thf(fact_212__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__pzlrlsievl_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__pzlrlsievl_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 @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ ia ) @ b ) @ ( one_one @ nat ) ) @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma2 @ J3 ) )
=> ( ( ord_less_eq @ nat @ J3 @ ( monito2027355080ogress @ a @ sigma2 @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) )
=> ~ ( ord_less_eq @ nat @ J3 @ ( monito2027355080ogress @ a @ sigma2 @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>j. \<lbrakk>\<tau> \<sigma>' i + b + 1 \<le> \<tau> \<sigma>' j; j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma>' \<phi>1 (plen \<pi>); j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma>' \<phi>2 (plen \<pi>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_213__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__pzlrlsievl_Oprogress_A_092_060sigma_062_A_092_060phi_0621_A_Iplen_A_092_060pi_062_J_059_Aj_A_092_060le_062_AMonitor__Mirabelle__pzlrlsievl_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 @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ ia ) @ b ) @ ( one_one @ nat ) ) @ ( tau @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ sigma @ J3 ) )
=> ( ( ord_less_eq @ nat @ J3 @ ( monito2027355080ogress @ a @ sigma @ phi_1 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) )
=> ~ ( ord_less_eq @ nat @ J3 @ ( monito2027355080ogress @ a @ sigma @ phi_2 @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ) ) ) ) ).
% \<open>\<And>thesis. (\<And>j. \<lbrakk>\<tau> \<sigma> i + b + 1 \<le> \<tau> \<sigma> j; j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma> \<phi>1 (plen \<pi>); j \<le> Monitor_Mirabelle_pzlrlsievl.progress \<sigma> \<phi>2 (plen \<pi>)\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_214_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_215_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_216_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_217_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_218_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_219_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_220_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_221_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_222_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_223_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A6: A,K: A,A2: A,B2: A] :
( ( A6
= ( plus_plus @ A @ K @ A2 ) )
=> ( ( plus_plus @ A @ A6 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_224_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B6 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_225_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_226_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_227_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_228_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A3: A,B3: A] : ( plus_plus @ A @ B3 @ A3 ) ) ) ) ).
% add.commute
thf(fact_229_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_230_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_231_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_232_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_233_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_234_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_235_add__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_236_add__left__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_237_less__eqE,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A2 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_238_add__right__mono,axiom,
! [A: $tType] :
( ( ordere779506340up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_239_le__iff__add,axiom,
! [A: $tType] :
( ( canoni770627133id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A3: A,B3: A] :
? [C4: A] :
( B3
= ( plus_plus @ A @ A3 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_240_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_241_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_242_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_243_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( I = J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_244_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_245_add__strict__mono,axiom,
! [A: $tType] :
( ( strict2144017051up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_246_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_247_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_248_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_249_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere236663937imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_250_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_251_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_252_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_253_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere223160158up_add @ A )
=> ! [I: A,J: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_254_discrete,axiom,
! [A: $tType] :
( ( unique1598680935umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A3: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
% Type constructors (37)
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A7: $tType,A8: $tType] :
( ( preorder @ A8 )
=> ( preorder @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A7: $tType,A8: $tType] :
( ( order @ A8 )
=> ( order @ ( A7 > A8 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A7: $tType,A8: $tType] :
( ( ord @ A8 )
=> ( ord @ ( A7 > A8 ) ) ) ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit1037483654norder @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1598680935umeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict2144017051up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le @ nat ).
thf(tcon_Nat_Onat___Groups_Ocanonically__ordered__monoid__add,axiom,
canoni770627133id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_1,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_2,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_3,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Oone,axiom,
one @ nat ).
thf(tcon_HOL_Obool___Orderings_Opreorder_4,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_5,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_6,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_7,axiom,
ord @ $o ).
thf(tcon_Trace_Oprefix___Orderings_Opreorder_8,axiom,
! [A7: $tType] : ( preorder @ ( prefix @ A7 ) ) ).
thf(tcon_Trace_Oprefix___Orderings_Oorder_9,axiom,
! [A7: $tType] : ( order @ ( prefix @ A7 ) ) ).
thf(tcon_Trace_Oprefix___Orderings_Oord_10,axiom,
! [A7: $tType] : ( ord @ ( prefix @ A7 ) ) ).
thf(tcon_Interval_O_092_060I_062___Orderings_Opreorder_11,axiom,
preorder @ i ).
thf(tcon_Interval_O_092_060I_062___Orderings_Olinorder_12,axiom,
linorder @ i ).
thf(tcon_Interval_O_092_060I_062___Orderings_Oorder_13,axiom,
order @ i ).
thf(tcon_Interval_O_092_060I_062___Orderings_Oord_14,axiom,
ord @ i ).
thf(tcon_Product__Type_Oprod___Orderings_Owellorder_15,axiom,
! [A7: $tType,A8: $tType] :
( ( ( wellorder @ A7 )
& ( wellorder @ A8 ) )
=> ( wellorder @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Opreorder_16,axiom,
! [A7: $tType,A8: $tType] :
( ( ( preorder @ A7 )
& ( preorder @ A8 ) )
=> ( preorder @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Olinorder_17,axiom,
! [A7: $tType,A8: $tType] :
( ( ( linorder @ A7 )
& ( linorder @ A8 ) )
=> ( linorder @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oorder_18,axiom,
! [A7: $tType,A8: $tType] :
( ( ( order @ A7 )
& ( order @ A8 ) )
=> ( order @ ( product_prod @ A7 @ A8 ) ) ) ).
thf(tcon_Product__Type_Oprod___Orderings_Oord_19,axiom,
! [A7: $tType,A8: $tType] :
( ( ( ord @ A7 )
& ( ord @ A8 ) )
=> ( ord @ ( product_prod @ A7 @ A8 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
ord_less @ nat @ k @ ( plen @ ( product_prod @ ( list @ char ) @ ( list @ a ) ) @ pi ) ).
%------------------------------------------------------------------------------