TPTP Problem File: ITP037^1.p
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%------------------------------------------------------------------------------
% File : ITP037^1 : TPTP v8.2.0. Released v7.5.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer Coincidence problem prob_129__7211024_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 : Coincidence/prob_129__7211024_1 [Des21]
% Status : Theorem
% Rating : 0.40 v8.2.0, 0.23 v8.1.0, 0.27 v7.5.0
% Syntax : Number of formulae : 550 ( 245 unt; 193 typ; 0 def)
% Number of atoms : 778 ( 464 equ; 0 cnn)
% Maximal formula atoms : 15 ( 2 avg)
% Number of connectives : 2327 ( 159 ~; 21 |; 61 &;1765 @)
% ( 0 <=>; 321 =>; 0 <=; 0 <~>)
% Maximal formula depth : 25 ( 6 avg)
% Number of types : 53 ( 52 usr)
% Number of type conns : 332 ( 332 >; 0 *; 0 +; 0 <<)
% Number of symbols : 144 ( 141 usr; 37 con; 0-10 aty)
% Number of variables : 1076 ( 37 ^; 978 !; 61 ?;1076 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Sledgehammer 2021-02-23 15:37:05.819
%------------------------------------------------------------------------------
% Could-be-implicit typings (52)
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plus_sf_sz: trm_sf_sz > trm_sf_sz > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__a_001tf__c,type,
times_a_c: trm_a_c > trm_a_c > trm_a_c ).
thf(sy_c_Syntax_Otrm_OTimes_001tf__sf_001tf__sz,type,
times_sf_sz: trm_sf_sz > trm_sf_sz > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_OVar_001tf__c_001tf__a,type,
var_c_a: c > trm_a_c ).
thf(sy_c_Syntax_Otrm_OVar_001tf__sz_001tf__sf,type,
var_sz_sf: sz > trm_sf_sz ).
thf(sy_c_Syntax_Otrm_Oset__trm_001tf__a_001tf__c,type,
set_trm_a_c2: trm_a_c > set_a ).
thf(sy_c_Syntax_Otrm_Oset__trm_001tf__sf_001tf__sz,type,
set_trm_sf_sz2: trm_sf_sz > set_sf ).
thf(sy_c_Transfer_Oleft__unique_001t__Syntax__Otrm_Itf__a_Mtf__c_J_001t__Frechet____Correctness__Oids__Ostrm_Itf__a_Mtf__c_J,type,
left_u968903956rm_a_c: ( trm_a_c > frechet_strm_a_c > $o ) > $o ).
thf(sy_c_Transfer_Oleft__unique_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J_001t__Frechet____Correctness__Oids__Ostrm_Itf__sf_Mtf__sz_J,type,
left_u638806404_sf_sz: ( trm_sf_sz > frechet_strm_sf_sz > $o ) > $o ).
thf(sy_c_Transfer_Oright__unique_001t__Syntax__Otrm_Itf__a_Mtf__c_J_001t__Frechet____Correctness__Oids__Ostrm_Itf__a_Mtf__c_J,type,
right_1933252621rm_a_c: ( trm_a_c > frechet_strm_a_c > $o ) > $o ).
thf(sy_c_Transfer_Oright__unique_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J_001t__Frechet____Correctness__Oids__Ostrm_Itf__sf_Mtf__sz_J,type,
right_963419005_sf_sz: ( trm_sf_sz > frechet_strm_sf_sz > $o ) > $o ).
thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__Syntax__Otrm_Itf__sf_Mtf__sz_J_Mtf__sz_J,type,
accp_P529224471_sz_sz: ( produc861709536_sz_sz > produc861709536_sz_sz > $o ) > produc861709536_sz_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_Mt__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_J,type,
member1057565379_sc_sz: produc1821101996_sc_sz > set_Pr1169339874_sc_sz > $o ).
thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_Mt__List__Olist_It__Syntax__Oformula_Itf__sf_Mtf__sc_Mtf__sz_J_J_J_Mt__Denotational____Semantics__Ointerp__Ointerp____ext_Itf__sf_Mtf__sc_Mtf__sz_Mt__Product____Type__Ounit_J_J,type,
member235934813t_unit: produc999471430t_unit > set_Pr2041213564t_unit > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__a_Mtf__c_J,type,
member_trm_a_c: trm_a_c > set_trm_a_c > $o ).
thf(sy_c_member_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J,type,
member_trm_sf_sz: trm_sf_sz > set_trm_sf_sz > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_c_member_001tf__sf,type,
member_sf: sf > set_sf > $o ).
thf(sy_v_I,type,
i: denota231621370t_unit ).
thf(sy_v__092_060nu_062,type,
nu: produc190496183real_c ).
thf(sy_v__092_060nu_062_H,type,
nu2: produc190496183real_c ).
thf(sy_v_args____,type,
args: c > trm_a_c ).
thf(sy_v_i,type,
i2: c ).
thf(sy_v_var____,type,
var: a ).
thf(sy_v_vid1,type,
vid1: sz ).
thf(sy_v_vid2,type,
vid2: sz ).
thf(sy_v_vid3,type,
vid3: sz ).
% Relevant facts (353)
thf(fact_0_free,axiom,
! [I: c] : ( dfree_a_c @ ( args @ I ) ) ).
% free
thf(fact_1_P__def,axiom,
p_sc_sf_sz = predic913887884_sf_sz ).
% P_def
thf(fact_2_strm_Odomain,axiom,
( ( domain512309999_sf_sz @ freche1244000341_sf_sz )
= dfree_sf_sz ) ).
% strm.domain
thf(fact_3_strm_Odomain,axiom,
( ( domain2047240575rm_a_c @ frechet_cr_strm_a_c )
= dfree_a_c ) ).
% strm.domain
thf(fact_4_dfree__DiffVar__simps,axiom,
! [X: sz] :
~ ( dfree_sf_sz @ ( diffVar_sz_sf @ X ) ) ).
% dfree_DiffVar_simps
thf(fact_5_dfree__DiffVar__simps,axiom,
! [X: c] :
~ ( dfree_a_c @ ( diffVar_c_a @ X ) ) ).
% dfree_DiffVar_simps
thf(fact_6_seq__sem_Ocases,axiom,
! [X: produc1821101996_sc_sz] :
~ ! [I2: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I2 @ S ) ) ).
% seq_sem.cases
thf(fact_7_singleton_Ocases,axiom,
! [X: produc861709536_sz_sz] :
~ ! [T: trm_sf_sz,I3: sz] :
( X
!= ( produc792909658_sz_sz @ T @ I3 ) ) ).
% singleton.cases
thf(fact_8_dfree__Differential__simps,axiom,
! [X: trm_sf_sz] :
~ ( dfree_sf_sz @ ( differential_sf_sz @ X ) ) ).
% dfree_Differential_simps
thf(fact_9_dfree__Differential__simps,axiom,
! [X: trm_a_c] :
~ ( dfree_a_c @ ( differential_a_c @ X ) ) ).
% dfree_Differential_simps
thf(fact_10_dfree__Var__simps,axiom,
! [X: c] : ( dfree_a_c @ ( var_c_a @ X ) ) ).
% dfree_Var_simps
thf(fact_11_dfree__Var__simps,axiom,
! [X: sz] : ( dfree_sf_sz @ ( var_sz_sf @ X ) ) ).
% dfree_Var_simps
thf(fact_12_dfree__Times__simps,axiom,
! [A: trm_a_c,B: trm_a_c] :
( ( dfree_a_c @ ( times_a_c @ A @ B ) )
= ( ( dfree_a_c @ A )
& ( dfree_a_c @ B ) ) ) ).
% dfree_Times_simps
thf(fact_13_dfree__Times__simps,axiom,
! [A: trm_sf_sz,B: trm_sf_sz] :
( ( dfree_sf_sz @ ( times_sf_sz @ A @ B ) )
= ( ( dfree_sf_sz @ A )
& ( dfree_sf_sz @ B ) ) ) ).
% dfree_Times_simps
thf(fact_14_dfree__Plus__simps,axiom,
! [A: trm_a_c,B: trm_a_c] :
( ( dfree_a_c @ ( plus_a_c @ A @ B ) )
= ( ( dfree_a_c @ A )
& ( dfree_a_c @ B ) ) ) ).
% dfree_Plus_simps
thf(fact_15_dfree__Plus__simps,axiom,
! [A: trm_sf_sz,B: trm_sf_sz] :
( ( dfree_sf_sz @ ( plus_sf_sz @ A @ B ) )
= ( ( dfree_sf_sz @ A )
& ( dfree_sf_sz @ B ) ) ) ).
% dfree_Plus_simps
thf(fact_16_dfree__Fun_Oprems,axiom,
denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ ( function_a_c @ var @ args ) ) ).
% dfree_Fun.prems
thf(fact_17_trm_Oinject_I3_J,axiom,
! [X31: a,X32: c > trm_a_c,Y31: a,Y32: c > trm_a_c] :
( ( ( function_a_c @ X31 @ X32 )
= ( function_a_c @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% trm.inject(3)
thf(fact_18_trm_Oinject_I3_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,Y31: sf,Y32: sz > trm_sf_sz] :
( ( ( function_sf_sz @ X31 @ X32 )
= ( function_sf_sz @ Y31 @ Y32 ) )
= ( ( X31 = Y31 )
& ( X32 = Y32 ) ) ) ).
% trm.inject(3)
thf(fact_19_trm_Oinject_I4_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,Y41: trm_sf_sz,Y42: trm_sf_sz] :
( ( ( plus_sf_sz @ X41 @ X42 )
= ( plus_sf_sz @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_20_trm_Oinject_I4_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,Y41: trm_a_c,Y42: trm_a_c] :
( ( ( plus_a_c @ X41 @ X42 )
= ( plus_a_c @ Y41 @ Y42 ) )
= ( ( X41 = Y41 )
& ( X42 = Y42 ) ) ) ).
% trm.inject(4)
thf(fact_21_trm_Oinject_I5_J,axiom,
! [X51: trm_sf_sz,X52: trm_sf_sz,Y51: trm_sf_sz,Y52: trm_sf_sz] :
( ( ( times_sf_sz @ X51 @ X52 )
= ( times_sf_sz @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_22_trm_Oinject_I5_J,axiom,
! [X51: trm_a_c,X52: trm_a_c,Y51: trm_a_c,Y52: trm_a_c] :
( ( ( times_a_c @ X51 @ X52 )
= ( times_a_c @ Y51 @ Y52 ) )
= ( ( X51 = Y51 )
& ( X52 = Y52 ) ) ) ).
% trm.inject(5)
thf(fact_23_trm_Oinject_I1_J,axiom,
! [X1: sz,Y1: sz] :
( ( ( var_sz_sf @ X1 )
= ( var_sz_sf @ Y1 ) )
= ( X1 = Y1 ) ) ).
% trm.inject(1)
thf(fact_24_trm_Oinject_I1_J,axiom,
! [X1: c,Y1: c] :
( ( ( var_c_a @ X1 )
= ( var_c_a @ Y1 ) )
= ( X1 = Y1 ) ) ).
% trm.inject(1)
thf(fact_25_trm_Oinject_I7_J,axiom,
! [X7: trm_sf_sz,Y7: trm_sf_sz] :
( ( ( differential_sf_sz @ X7 )
= ( differential_sf_sz @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_26_trm_Oinject_I7_J,axiom,
! [X7: trm_a_c,Y7: trm_a_c] :
( ( ( differential_a_c @ X7 )
= ( differential_a_c @ Y7 ) )
= ( X7 = Y7 ) ) ).
% trm.inject(7)
thf(fact_27_trm_Oinject_I6_J,axiom,
! [X6: sz,Y6: sz] :
( ( ( diffVar_sz_sf @ X6 )
= ( diffVar_sz_sf @ Y6 ) )
= ( X6 = Y6 ) ) ).
% trm.inject(6)
thf(fact_28_trm_Oinject_I6_J,axiom,
! [X6: c,Y6: c] :
( ( ( diffVar_c_a @ X6 )
= ( diffVar_c_a @ Y6 ) )
= ( X6 = Y6 ) ) ).
% trm.inject(6)
thf(fact_29_dfree__Fun__simps,axiom,
! [I: a,Args: c > trm_a_c] :
( ( dfree_a_c @ ( function_a_c @ I @ Args ) )
= ( ! [X2: c] : ( dfree_a_c @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_30_dfree__Fun__simps,axiom,
! [I: sf,Args: sz > trm_sf_sz] :
( ( dfree_sf_sz @ ( function_sf_sz @ I @ Args ) )
= ( ! [X2: sz] : ( dfree_sf_sz @ ( Args @ X2 ) ) ) ) ).
% dfree_Fun_simps
thf(fact_31_trm_Odistinct_I41_J,axiom,
! [X6: sz,X7: trm_sf_sz] :
( ( diffVar_sz_sf @ X6 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(41)
thf(fact_32_trm_Odistinct_I41_J,axiom,
! [X6: c,X7: trm_a_c] :
( ( diffVar_c_a @ X6 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(41)
thf(fact_33_trm_Odistinct_I39_J,axiom,
! [X51: trm_sf_sz,X52: trm_sf_sz,X7: trm_sf_sz] :
( ( times_sf_sz @ X51 @ X52 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(39)
thf(fact_34_trm_Odistinct_I39_J,axiom,
! [X51: trm_a_c,X52: trm_a_c,X7: trm_a_c] :
( ( times_a_c @ X51 @ X52 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(39)
thf(fact_35_trm_Odistinct_I37_J,axiom,
! [X51: trm_sf_sz,X52: trm_sf_sz,X6: sz] :
( ( times_sf_sz @ X51 @ X52 )
!= ( diffVar_sz_sf @ X6 ) ) ).
% trm.distinct(37)
thf(fact_36_trm_Odistinct_I37_J,axiom,
! [X51: trm_a_c,X52: trm_a_c,X6: c] :
( ( times_a_c @ X51 @ X52 )
!= ( diffVar_c_a @ X6 ) ) ).
% trm.distinct(37)
thf(fact_37_trm_Odistinct_I35_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,X7: trm_sf_sz] :
( ( plus_sf_sz @ X41 @ X42 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(35)
thf(fact_38_trm_Odistinct_I35_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,X7: trm_a_c] :
( ( plus_a_c @ X41 @ X42 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(35)
thf(fact_39_trm_Odistinct_I33_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,X6: sz] :
( ( plus_sf_sz @ X41 @ X42 )
!= ( diffVar_sz_sf @ X6 ) ) ).
% trm.distinct(33)
thf(fact_40_trm_Odistinct_I33_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,X6: c] :
( ( plus_a_c @ X41 @ X42 )
!= ( diffVar_c_a @ X6 ) ) ).
% trm.distinct(33)
thf(fact_41_trm_Odistinct_I31_J,axiom,
! [X41: trm_sf_sz,X42: trm_sf_sz,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( plus_sf_sz @ X41 @ X42 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_42_trm_Odistinct_I31_J,axiom,
! [X41: trm_a_c,X42: trm_a_c,X51: trm_a_c,X52: trm_a_c] :
( ( plus_a_c @ X41 @ X42 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(31)
thf(fact_43_trm_Odistinct_I29_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X7: trm_sf_sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(29)
thf(fact_44_trm_Odistinct_I29_J,axiom,
! [X31: a,X32: c > trm_a_c,X7: trm_a_c] :
( ( function_a_c @ X31 @ X32 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(29)
thf(fact_45_trm_Odistinct_I27_J,axiom,
! [X31: a,X32: c > trm_a_c,X6: c] :
( ( function_a_c @ X31 @ X32 )
!= ( diffVar_c_a @ X6 ) ) ).
% trm.distinct(27)
thf(fact_46_trm_Odistinct_I27_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X6: sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( diffVar_sz_sf @ X6 ) ) ).
% trm.distinct(27)
thf(fact_47_trm_Odistinct_I25_J,axiom,
! [X31: a,X32: c > trm_a_c,X51: trm_a_c,X52: trm_a_c] :
( ( function_a_c @ X31 @ X32 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_48_trm_Odistinct_I25_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(25)
thf(fact_49_trm_Odistinct_I23_J,axiom,
! [X31: a,X32: c > trm_a_c,X41: trm_a_c,X42: trm_a_c] :
( ( function_a_c @ X31 @ X32 )
!= ( plus_a_c @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_50_trm_Odistinct_I23_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz,X41: trm_sf_sz,X42: trm_sf_sz] :
( ( function_sf_sz @ X31 @ X32 )
!= ( plus_sf_sz @ X41 @ X42 ) ) ).
% trm.distinct(23)
thf(fact_51_trm_Odistinct_I11_J,axiom,
! [X1: sz,X7: trm_sf_sz] :
( ( var_sz_sf @ X1 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(11)
thf(fact_52_trm_Odistinct_I11_J,axiom,
! [X1: c,X7: trm_a_c] :
( ( var_c_a @ X1 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(11)
thf(fact_53_trm_Odistinct_I9_J,axiom,
! [X1: sz,X6: sz] :
( ( var_sz_sf @ X1 )
!= ( diffVar_sz_sf @ X6 ) ) ).
% trm.distinct(9)
thf(fact_54_trm_Odistinct_I9_J,axiom,
! [X1: c,X6: c] :
( ( var_c_a @ X1 )
!= ( diffVar_c_a @ X6 ) ) ).
% trm.distinct(9)
thf(fact_55_trm_Odistinct_I7_J,axiom,
! [X1: sz,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( var_sz_sf @ X1 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_56_trm_Odistinct_I7_J,axiom,
! [X1: c,X51: trm_a_c,X52: trm_a_c] :
( ( var_c_a @ X1 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(7)
thf(fact_57_trm_Odistinct_I5_J,axiom,
! [X1: sz,X41: trm_sf_sz,X42: trm_sf_sz] :
( ( var_sz_sf @ X1 )
!= ( plus_sf_sz @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_58_trm_Odistinct_I5_J,axiom,
! [X1: c,X41: trm_a_c,X42: trm_a_c] :
( ( var_c_a @ X1 )
!= ( plus_a_c @ X41 @ X42 ) ) ).
% trm.distinct(5)
thf(fact_59_trm_Odistinct_I3_J,axiom,
! [X1: c,X31: a,X32: c > trm_a_c] :
( ( var_c_a @ X1 )
!= ( function_a_c @ X31 @ X32 ) ) ).
% trm.distinct(3)
thf(fact_60_trm_Odistinct_I3_J,axiom,
! [X1: sz,X31: sf,X32: sz > trm_sf_sz] :
( ( var_sz_sf @ X1 )
!= ( function_sf_sz @ X31 @ X32 ) ) ).
% trm.distinct(3)
thf(fact_61_dfree_Odfree__Fun,axiom,
! [Args: c > trm_a_c,I: a] :
( ! [I3: c] : ( dfree_a_c @ ( Args @ I3 ) )
=> ( dfree_a_c @ ( function_a_c @ I @ Args ) ) ) ).
% dfree.dfree_Fun
thf(fact_62_dfree_Odfree__Fun,axiom,
! [Args: sz > trm_sf_sz,I: sf] :
( ! [I3: sz] : ( dfree_sf_sz @ ( Args @ I3 ) )
=> ( dfree_sf_sz @ ( function_sf_sz @ I @ Args ) ) ) ).
% dfree.dfree_Fun
thf(fact_63_dfree__Plus,axiom,
! [Theta_1: trm_a_c,Theta_2: trm_a_c] :
( ( dfree_a_c @ Theta_1 )
=> ( ( dfree_a_c @ Theta_2 )
=> ( dfree_a_c @ ( plus_a_c @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Plus
thf(fact_64_dfree__Plus,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_1 )
=> ( ( dfree_sf_sz @ Theta_2 )
=> ( dfree_sf_sz @ ( plus_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Plus
thf(fact_65_dfree__Times,axiom,
! [Theta_1: trm_a_c,Theta_2: trm_a_c] :
( ( dfree_a_c @ Theta_1 )
=> ( ( dfree_a_c @ Theta_2 )
=> ( dfree_a_c @ ( times_a_c @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Times
thf(fact_66_dfree__Times,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_1 )
=> ( ( dfree_sf_sz @ Theta_2 )
=> ( dfree_sf_sz @ ( times_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dfree_Times
thf(fact_67_dfree__Var,axiom,
! [I: sz] : ( dfree_sf_sz @ ( var_sz_sf @ I ) ) ).
% dfree_Var
thf(fact_68_dfree__Var,axiom,
! [I: c] : ( dfree_a_c @ ( var_c_a @ I ) ) ).
% dfree_Var
thf(fact_69_agree__times1,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( times_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T1 ) ) ) ).
% agree_times1
thf(fact_70_agree__times1,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( times_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T1 ) ) ) ).
% agree_times1
thf(fact_71_agree__times2,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( times_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T2 ) ) ) ).
% agree_times2
thf(fact_72_agree__times2,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( times_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T2 ) ) ) ).
% agree_times2
thf(fact_73_agree__plus1,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( plus_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T1 ) ) ) ).
% agree_plus1
thf(fact_74_agree__plus1,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( plus_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T1 ) ) ) ).
% agree_plus1
thf(fact_75_agree__plus2,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,T1: trm_sf_sz,T2: trm_sf_sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( plus_sf_sz @ T1 @ T2 ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ T2 ) ) ) ).
% agree_plus2
thf(fact_76_agree__plus2,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,T1: trm_a_c,T2: trm_a_c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( plus_a_c @ T1 @ T2 ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ T2 ) ) ) ).
% agree_plus2
thf(fact_77_mem__Collect__eq,axiom,
! [A: produc999471430t_unit,P: produc999471430t_unit > $o] :
( ( member235934813t_unit @ A @ ( collec377961627t_unit @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_78_mem__Collect__eq,axiom,
! [A: trm_sf_sz,P: trm_sf_sz > $o] :
( ( member_trm_sf_sz @ A @ ( collect_trm_sf_sz @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_79_mem__Collect__eq,axiom,
! [A: trm_a_c,P: trm_a_c > $o] :
( ( member_trm_a_c @ A @ ( collect_trm_a_c @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
! [A: sf,P: sf > $o] :
( ( member_sf @ A @ ( collect_sf @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_81_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_82_Collect__mem__eq,axiom,
! [A2: set_Pr2041213564t_unit] :
( ( collec377961627t_unit
@ ^ [X2: produc999471430t_unit] : ( member235934813t_unit @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
! [A2: set_trm_sf_sz] :
( ( collect_trm_sf_sz
@ ^ [X2: trm_sf_sz] : ( member_trm_sf_sz @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
! [A2: set_trm_a_c] :
( ( collect_trm_a_c
@ ^ [X2: trm_a_c] : ( member_trm_a_c @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
! [A2: set_sf] :
( ( collect_sf
@ ^ [X2: sf] : ( member_sf @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_86_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_87_agree__func,axiom,
! [Nu: produc1149990247eal_sz,Nu2: produc1149990247eal_sz,Var: sf,Args: sz > trm_sf_sz,I: sz] :
( ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( function_sf_sz @ Var @ Args ) ) )
=> ( denota102713844ree_sz @ Nu @ Nu2 @ ( static_FVDiff_sf_sz @ ( Args @ I ) ) ) ) ).
% agree_func
thf(fact_88_agree__func,axiom,
! [Nu: produc190496183real_c,Nu2: produc190496183real_c,Var: a,Args: c > trm_a_c,I: c] :
( ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( function_a_c @ Var @ Args ) ) )
=> ( denota1997846518gree_c @ Nu @ Nu2 @ ( static_FVDiff_a_c @ ( Args @ I ) ) ) ) ).
% agree_func
thf(fact_89_prod_Oinject,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz,Y1: list_f1238882004_sc_sz,Y2: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ X1 @ X22 )
= ( produc1822718231_sc_sz @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_90_prod_Oinject,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c,Y1: finite1398487019real_c,Y2: finite1398487019real_c] :
( ( ( produc394644079real_c @ X1 @ X22 )
= ( produc394644079real_c @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_91_prod_Oinject,axiom,
! [X1: trm_sf_sz,X22: sz,Y1: trm_sf_sz,Y2: sz] :
( ( ( produc792909658_sz_sz @ X1 @ X22 )
= ( produc792909658_sz_sz @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_92_prod_Oinject,axiom,
! [X1: produc866628903_sc_sz,X22: denota610675952t_unit,Y1: produc866628903_sc_sz,Y2: denota610675952t_unit] :
( ( ( produc502616768t_unit @ X1 @ X22 )
= ( produc502616768t_unit @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_93_prod_Oinject,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz,Y1: denota610675952t_unit,Y2: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ X1 @ X22 )
= ( produc789536734_sc_sz @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X22 = Y2 ) ) ) ).
% prod.inject
thf(fact_94_old_Oprod_Oinject,axiom,
! [A: list_f1238882004_sc_sz,B: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A @ B )
= ( produc1822718231_sc_sz @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_95_old_Oprod_Oinject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A3: finite1398487019real_c,B2: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_96_old_Oprod_Oinject,axiom,
! [A: trm_sf_sz,B: sz,A3: trm_sf_sz,B2: sz] :
( ( ( produc792909658_sz_sz @ A @ B )
= ( produc792909658_sz_sz @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_97_old_Oprod_Oinject,axiom,
! [A: produc866628903_sc_sz,B: denota610675952t_unit,A3: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( ( produc502616768t_unit @ A @ B )
= ( produc502616768t_unit @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_98_old_Oprod_Oinject,axiom,
! [A: denota610675952t_unit,B: produc866628903_sc_sz,A3: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A @ B )
= ( produc789536734_sc_sz @ A3 @ B2 ) )
= ( ( A = A3 )
& ( B = B2 ) ) ) ).
% old.prod.inject
thf(fact_99_trm_Oexhaust,axiom,
! [Y: trm_sf_sz] :
( ! [X12: sz] :
( Y
!= ( var_sz_sf @ X12 ) )
=> ( ! [X23: real] :
( Y
!= ( const_sf_sz @ X23 ) )
=> ( ! [X312: sf,X322: sz > trm_sf_sz] :
( Y
!= ( function_sf_sz @ X312 @ X322 ) )
=> ( ! [X412: trm_sf_sz,X422: trm_sf_sz] :
( Y
!= ( plus_sf_sz @ X412 @ X422 ) )
=> ( ! [X512: trm_sf_sz,X522: trm_sf_sz] :
( Y
!= ( times_sf_sz @ X512 @ X522 ) )
=> ( ! [X62: sz] :
( Y
!= ( diffVar_sz_sf @ X62 ) )
=> ~ ! [X72: trm_sf_sz] :
( Y
!= ( differential_sf_sz @ X72 ) ) ) ) ) ) ) ) ).
% trm.exhaust
thf(fact_100_trm_Oexhaust,axiom,
! [Y: trm_a_c] :
( ! [X12: c] :
( Y
!= ( var_c_a @ X12 ) )
=> ( ! [X23: real] :
( Y
!= ( const_a_c @ X23 ) )
=> ( ! [X312: a,X322: c > trm_a_c] :
( Y
!= ( function_a_c @ X312 @ X322 ) )
=> ( ! [X412: trm_a_c,X422: trm_a_c] :
( Y
!= ( plus_a_c @ X412 @ X422 ) )
=> ( ! [X512: trm_a_c,X522: trm_a_c] :
( Y
!= ( times_a_c @ X512 @ X522 ) )
=> ( ! [X62: c] :
( Y
!= ( diffVar_c_a @ X62 ) )
=> ~ ! [X72: trm_a_c] :
( Y
!= ( differential_a_c @ X72 ) ) ) ) ) ) ) ) ).
% trm.exhaust
thf(fact_101_prod__cases3,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B3: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B3 @ C ) ) ) ).
% prod_cases3
thf(fact_102_trm_Oinject_I2_J,axiom,
! [X22: real,Y2: real] :
( ( ( const_a_c @ X22 )
= ( const_a_c @ Y2 ) )
= ( X22 = Y2 ) ) ).
% trm.inject(2)
thf(fact_103_trm_Oinject_I2_J,axiom,
! [X22: real,Y2: real] :
( ( ( const_sf_sz @ X22 )
= ( const_sf_sz @ Y2 ) )
= ( X22 = Y2 ) ) ).
% trm.inject(2)
thf(fact_104_dfree__Const__simps,axiom,
! [R: real] : ( dfree_a_c @ ( const_a_c @ R ) ) ).
% dfree_Const_simps
thf(fact_105_dfree__Const__simps,axiom,
! [R: real] : ( dfree_sf_sz @ ( const_sf_sz @ R ) ) ).
% dfree_Const_simps
thf(fact_106_dfree__Const,axiom,
! [R: real] : ( dfree_a_c @ ( const_a_c @ R ) ) ).
% dfree_Const
thf(fact_107_dfree__Const,axiom,
! [R: real] : ( dfree_sf_sz @ ( const_sf_sz @ R ) ) ).
% dfree_Const
thf(fact_108_trm_Odistinct_I13_J,axiom,
! [X22: real,X31: a,X32: c > trm_a_c] :
( ( const_a_c @ X22 )
!= ( function_a_c @ X31 @ X32 ) ) ).
% trm.distinct(13)
thf(fact_109_trm_Odistinct_I13_J,axiom,
! [X22: real,X31: sf,X32: sz > trm_sf_sz] :
( ( const_sf_sz @ X22 )
!= ( function_sf_sz @ X31 @ X32 ) ) ).
% trm.distinct(13)
thf(fact_110_trm_Odistinct_I15_J,axiom,
! [X22: real,X41: trm_a_c,X42: trm_a_c] :
( ( const_a_c @ X22 )
!= ( plus_a_c @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_111_trm_Odistinct_I15_J,axiom,
! [X22: real,X41: trm_sf_sz,X42: trm_sf_sz] :
( ( const_sf_sz @ X22 )
!= ( plus_sf_sz @ X41 @ X42 ) ) ).
% trm.distinct(15)
thf(fact_112_trm_Odistinct_I17_J,axiom,
! [X22: real,X51: trm_sf_sz,X52: trm_sf_sz] :
( ( const_sf_sz @ X22 )
!= ( times_sf_sz @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_113_trm_Odistinct_I17_J,axiom,
! [X22: real,X51: trm_a_c,X52: trm_a_c] :
( ( const_a_c @ X22 )
!= ( times_a_c @ X51 @ X52 ) ) ).
% trm.distinct(17)
thf(fact_114_trm_Odistinct_I1_J,axiom,
! [X1: sz,X22: real] :
( ( var_sz_sf @ X1 )
!= ( const_sf_sz @ X22 ) ) ).
% trm.distinct(1)
thf(fact_115_trm_Odistinct_I1_J,axiom,
! [X1: c,X22: real] :
( ( var_c_a @ X1 )
!= ( const_a_c @ X22 ) ) ).
% trm.distinct(1)
thf(fact_116_trm_Odistinct_I21_J,axiom,
! [X22: real,X7: trm_sf_sz] :
( ( const_sf_sz @ X22 )
!= ( differential_sf_sz @ X7 ) ) ).
% trm.distinct(21)
thf(fact_117_trm_Odistinct_I21_J,axiom,
! [X22: real,X7: trm_a_c] :
( ( const_a_c @ X22 )
!= ( differential_a_c @ X7 ) ) ).
% trm.distinct(21)
thf(fact_118_trm_Odistinct_I19_J,axiom,
! [X22: real,X6: sz] :
( ( const_sf_sz @ X22 )
!= ( diffVar_sz_sf @ X6 ) ) ).
% trm.distinct(19)
thf(fact_119_trm_Odistinct_I19_J,axiom,
! [X22: real,X6: c] :
( ( const_a_c @ X22 )
!= ( diffVar_c_a @ X6 ) ) ).
% trm.distinct(19)
thf(fact_120_old_Oprod_Oinducts,axiom,
! [P: produc866628903_sc_sz > $o,Prod: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_121_old_Oprod_Oinducts,axiom,
! [P: produc190496183real_c > $o,Prod: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B3: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_122_old_Oprod_Oinducts,axiom,
! [P: produc861709536_sz_sz > $o,Prod: produc861709536_sz_sz] :
( ! [A4: trm_sf_sz,B3: sz] : ( P @ ( produc792909658_sz_sz @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_123_old_Oprod_Oinducts,axiom,
! [P: produc999471430t_unit > $o,Prod: produc999471430t_unit] :
( ! [A4: produc866628903_sc_sz,B3: denota610675952t_unit] : ( P @ ( produc502616768t_unit @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_124_old_Oprod_Oinducts,axiom,
! [P: produc1821101996_sc_sz > $o,Prod: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B3: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B3 ) )
=> ( P @ Prod ) ) ).
% old.prod.inducts
thf(fact_125_old_Oprod_Oexhaust,axiom,
! [Y: produc866628903_sc_sz] :
~ ! [A4: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] :
( Y
!= ( produc1822718231_sc_sz @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_126_old_Oprod_Oexhaust,axiom,
! [Y: produc190496183real_c] :
~ ! [A4: finite1398487019real_c,B3: finite1398487019real_c] :
( Y
!= ( produc394644079real_c @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_127_old_Oprod_Oexhaust,axiom,
! [Y: produc861709536_sz_sz] :
~ ! [A4: trm_sf_sz,B3: sz] :
( Y
!= ( produc792909658_sz_sz @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_128_old_Oprod_Oexhaust,axiom,
! [Y: produc999471430t_unit] :
~ ! [A4: produc866628903_sc_sz,B3: denota610675952t_unit] :
( Y
!= ( produc502616768t_unit @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_129_old_Oprod_Oexhaust,axiom,
! [Y: produc1821101996_sc_sz] :
~ ! [A4: denota610675952t_unit,B3: produc866628903_sc_sz] :
( Y
!= ( produc789536734_sc_sz @ A4 @ B3 ) ) ).
% old.prod.exhaust
thf(fact_130_Pair__inject,axiom,
! [A: list_f1238882004_sc_sz,B: list_f1238882004_sc_sz,A3: list_f1238882004_sc_sz,B2: list_f1238882004_sc_sz] :
( ( ( produc1822718231_sc_sz @ A @ B )
= ( produc1822718231_sc_sz @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_131_Pair__inject,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,A3: finite1398487019real_c,B2: finite1398487019real_c] :
( ( ( produc394644079real_c @ A @ B )
= ( produc394644079real_c @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_132_Pair__inject,axiom,
! [A: trm_sf_sz,B: sz,A3: trm_sf_sz,B2: sz] :
( ( ( produc792909658_sz_sz @ A @ B )
= ( produc792909658_sz_sz @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_133_Pair__inject,axiom,
! [A: produc866628903_sc_sz,B: denota610675952t_unit,A3: produc866628903_sc_sz,B2: denota610675952t_unit] :
( ( ( produc502616768t_unit @ A @ B )
= ( produc502616768t_unit @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_134_Pair__inject,axiom,
! [A: denota610675952t_unit,B: produc866628903_sc_sz,A3: denota610675952t_unit,B2: produc866628903_sc_sz] :
( ( ( produc789536734_sc_sz @ A @ B )
= ( produc789536734_sc_sz @ A3 @ B2 ) )
=> ~ ( ( A = A3 )
=> ( B != B2 ) ) ) ).
% Pair_inject
thf(fact_135_prod__cases,axiom,
! [P: produc866628903_sc_sz > $o,P2: produc866628903_sc_sz] :
( ! [A4: list_f1238882004_sc_sz,B3: list_f1238882004_sc_sz] : ( P @ ( produc1822718231_sc_sz @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_136_prod__cases,axiom,
! [P: produc190496183real_c > $o,P2: produc190496183real_c] :
( ! [A4: finite1398487019real_c,B3: finite1398487019real_c] : ( P @ ( produc394644079real_c @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_137_prod__cases,axiom,
! [P: produc861709536_sz_sz > $o,P2: produc861709536_sz_sz] :
( ! [A4: trm_sf_sz,B3: sz] : ( P @ ( produc792909658_sz_sz @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_138_prod__cases,axiom,
! [P: produc999471430t_unit > $o,P2: produc999471430t_unit] :
( ! [A4: produc866628903_sc_sz,B3: denota610675952t_unit] : ( P @ ( produc502616768t_unit @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_139_prod__cases,axiom,
! [P: produc1821101996_sc_sz > $o,P2: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B3: produc866628903_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ B3 ) )
=> ( P @ P2 ) ) ).
% prod_cases
thf(fact_140_surj__pair,axiom,
! [P2: produc866628903_sc_sz] :
? [X3: list_f1238882004_sc_sz,Y3: list_f1238882004_sc_sz] :
( P2
= ( produc1822718231_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_141_surj__pair,axiom,
! [P2: produc190496183real_c] :
? [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_142_surj__pair,axiom,
! [P2: produc861709536_sz_sz] :
? [X3: trm_sf_sz,Y3: sz] :
( P2
= ( produc792909658_sz_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_143_surj__pair,axiom,
! [P2: produc999471430t_unit] :
? [X3: produc866628903_sc_sz,Y3: denota610675952t_unit] :
( P2
= ( produc502616768t_unit @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_144_surj__pair,axiom,
! [P2: produc1821101996_sc_sz] :
? [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( P2
= ( produc789536734_sc_sz @ X3 @ Y3 ) ) ).
% surj_pair
thf(fact_145_dfree_Oinducts,axiom,
! [X: trm_a_c,P: trm_a_c > $o] :
( ( dfree_a_c @ X )
=> ( ! [I3: c] : ( P @ ( var_c_a @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const_a_c @ R2 ) )
=> ( ! [Args2: c > trm_a_c,I3: a] :
( ! [Ia: c] : ( dfree_a_c @ ( Args2 @ Ia ) )
=> ( ! [Ia: c] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function_a_c @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( dfree_a_c @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree_a_c @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus_a_c @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( dfree_a_c @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree_a_c @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times_a_c @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% dfree.inducts
thf(fact_146_dfree_Oinducts,axiom,
! [X: trm_sf_sz,P: trm_sf_sz > $o] :
( ( dfree_sf_sz @ X )
=> ( ! [I3: sz] : ( P @ ( var_sz_sf @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const_sf_sz @ R2 ) )
=> ( ! [Args2: sz > trm_sf_sz,I3: sf] :
( ! [Ia: sz] : ( dfree_sf_sz @ ( Args2 @ Ia ) )
=> ( ! [Ia: sz] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function_sf_sz @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree_sf_sz @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus_sf_sz @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( dfree_sf_sz @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dfree_sf_sz @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times_sf_sz @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( P @ X ) ) ) ) ) ) ) ).
% dfree.inducts
thf(fact_147_dfree_Osimps,axiom,
( dfree_a_c
= ( ^ [A5: trm_a_c] :
( ? [I4: c] :
( A5
= ( var_c_a @ I4 ) )
| ? [R3: real] :
( A5
= ( const_a_c @ R3 ) )
| ? [Args3: c > trm_a_c,I4: a] :
( ( A5
= ( function_a_c @ I4 @ Args3 ) )
& ! [X2: c] : ( dfree_a_c @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm_a_c,Theta_23: trm_a_c] :
( ( A5
= ( plus_a_c @ Theta_13 @ Theta_23 ) )
& ( dfree_a_c @ Theta_13 )
& ( dfree_a_c @ Theta_23 ) )
| ? [Theta_13: trm_a_c,Theta_23: trm_a_c] :
( ( A5
= ( times_a_c @ Theta_13 @ Theta_23 ) )
& ( dfree_a_c @ Theta_13 )
& ( dfree_a_c @ Theta_23 ) ) ) ) ) ).
% dfree.simps
thf(fact_148_dfree_Osimps,axiom,
( dfree_sf_sz
= ( ^ [A5: trm_sf_sz] :
( ? [I4: sz] :
( A5
= ( var_sz_sf @ I4 ) )
| ? [R3: real] :
( A5
= ( const_sf_sz @ R3 ) )
| ? [Args3: sz > trm_sf_sz,I4: sf] :
( ( A5
= ( function_sf_sz @ I4 @ Args3 ) )
& ! [X2: sz] : ( dfree_sf_sz @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm_sf_sz,Theta_23: trm_sf_sz] :
( ( A5
= ( plus_sf_sz @ Theta_13 @ Theta_23 ) )
& ( dfree_sf_sz @ Theta_13 )
& ( dfree_sf_sz @ Theta_23 ) )
| ? [Theta_13: trm_sf_sz,Theta_23: trm_sf_sz] :
( ( A5
= ( times_sf_sz @ Theta_13 @ Theta_23 ) )
& ( dfree_sf_sz @ Theta_13 )
& ( dfree_sf_sz @ Theta_23 ) ) ) ) ) ).
% dfree.simps
thf(fact_149_dfree_Ocases,axiom,
! [A: trm_a_c] :
( ( dfree_a_c @ A )
=> ( ! [I3: c] :
( A
!= ( var_c_a @ I3 ) )
=> ( ! [R2: real] :
( A
!= ( const_a_c @ R2 ) )
=> ( ! [Args2: c > trm_a_c] :
( ? [I3: a] :
( A
= ( function_a_c @ I3 @ Args2 ) )
=> ~ ! [I5: c] : ( dfree_a_c @ ( Args2 @ I5 ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( A
= ( plus_a_c @ Theta_12 @ Theta_22 ) )
=> ( ( dfree_a_c @ Theta_12 )
=> ~ ( dfree_a_c @ Theta_22 ) ) )
=> ~ ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( A
= ( times_a_c @ Theta_12 @ Theta_22 ) )
=> ( ( dfree_a_c @ Theta_12 )
=> ~ ( dfree_a_c @ Theta_22 ) ) ) ) ) ) ) ) ).
% dfree.cases
thf(fact_150_dfree_Ocases,axiom,
! [A: trm_sf_sz] :
( ( dfree_sf_sz @ A )
=> ( ! [I3: sz] :
( A
!= ( var_sz_sf @ I3 ) )
=> ( ! [R2: real] :
( A
!= ( const_sf_sz @ R2 ) )
=> ( ! [Args2: sz > trm_sf_sz] :
( ? [I3: sf] :
( A
= ( function_sf_sz @ I3 @ Args2 ) )
=> ~ ! [I5: sz] : ( dfree_sf_sz @ ( Args2 @ I5 ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( A
= ( plus_sf_sz @ Theta_12 @ Theta_22 ) )
=> ( ( dfree_sf_sz @ Theta_12 )
=> ~ ( dfree_sf_sz @ Theta_22 ) ) )
=> ~ ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( A
= ( times_sf_sz @ Theta_12 @ Theta_22 ) )
=> ( ( dfree_sf_sz @ Theta_12 )
=> ~ ( dfree_sf_sz @ Theta_22 ) ) ) ) ) ) ) ) ).
% dfree.cases
thf(fact_151_prod__induct3,axiom,
! [P: produc1821101996_sc_sz > $o,X: produc1821101996_sc_sz] :
( ! [A4: denota610675952t_unit,B3: list_f1238882004_sc_sz,C: list_f1238882004_sc_sz] : ( P @ ( produc789536734_sc_sz @ A4 @ ( produc1822718231_sc_sz @ B3 @ C ) ) )
=> ( P @ X ) ) ).
% prod_induct3
thf(fact_152_f0__def,axiom,
( f0_sf_sz
= ( ^ [F: sf] : ( function_sf_sz @ F @ empty_sz_sf ) ) ) ).
% f0_def
thf(fact_153_dsafe_Oinducts,axiom,
! [X: trm_sf_sz,P: trm_sf_sz > $o] :
( ( dsafe_sf_sz @ X )
=> ( ! [I3: sz] : ( P @ ( var_sz_sf @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const_sf_sz @ R2 ) )
=> ( ! [Args2: sz > trm_sf_sz,I3: sf] :
( ! [Ia: sz] : ( dsafe_sf_sz @ ( Args2 @ Ia ) )
=> ( ! [Ia: sz] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function_sf_sz @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( dsafe_sf_sz @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe_sf_sz @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus_sf_sz @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( dsafe_sf_sz @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe_sf_sz @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times_sf_sz @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta: trm_sf_sz] :
( ( dfree_sf_sz @ Theta )
=> ( P @ ( differential_sf_sz @ Theta ) ) )
=> ( ! [I3: sz] : ( P @ ( diffVar_sz_sf @ I3 ) )
=> ( P @ X ) ) ) ) ) ) ) ) ) ).
% dsafe.inducts
thf(fact_154_dsafe_Oinducts,axiom,
! [X: trm_a_c,P: trm_a_c > $o] :
( ( dsafe_a_c @ X )
=> ( ! [I3: c] : ( P @ ( var_c_a @ I3 ) )
=> ( ! [R2: real] : ( P @ ( const_a_c @ R2 ) )
=> ( ! [Args2: c > trm_a_c,I3: a] :
( ! [Ia: c] : ( dsafe_a_c @ ( Args2 @ Ia ) )
=> ( ! [Ia: c] : ( P @ ( Args2 @ Ia ) )
=> ( P @ ( function_a_c @ I3 @ Args2 ) ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( dsafe_a_c @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe_a_c @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( plus_a_c @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( dsafe_a_c @ Theta_12 )
=> ( ( P @ Theta_12 )
=> ( ( dsafe_a_c @ Theta_22 )
=> ( ( P @ Theta_22 )
=> ( P @ ( times_a_c @ Theta_12 @ Theta_22 ) ) ) ) ) )
=> ( ! [Theta: trm_a_c] :
( ( dfree_a_c @ Theta )
=> ( P @ ( differential_a_c @ Theta ) ) )
=> ( ! [I3: c] : ( P @ ( diffVar_c_a @ I3 ) )
=> ( P @ X ) ) ) ) ) ) ) ) ) ).
% dsafe.inducts
thf(fact_155_dsafe_Osimps,axiom,
( dsafe_sf_sz
= ( ^ [A5: trm_sf_sz] :
( ? [I4: sz] :
( A5
= ( var_sz_sf @ I4 ) )
| ? [R3: real] :
( A5
= ( const_sf_sz @ R3 ) )
| ? [Args3: sz > trm_sf_sz,I4: sf] :
( ( A5
= ( function_sf_sz @ I4 @ Args3 ) )
& ! [X2: sz] : ( dsafe_sf_sz @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm_sf_sz,Theta_23: trm_sf_sz] :
( ( A5
= ( plus_sf_sz @ Theta_13 @ Theta_23 ) )
& ( dsafe_sf_sz @ Theta_13 )
& ( dsafe_sf_sz @ Theta_23 ) )
| ? [Theta_13: trm_sf_sz,Theta_23: trm_sf_sz] :
( ( A5
= ( times_sf_sz @ Theta_13 @ Theta_23 ) )
& ( dsafe_sf_sz @ Theta_13 )
& ( dsafe_sf_sz @ Theta_23 ) )
| ? [Theta2: trm_sf_sz] :
( ( A5
= ( differential_sf_sz @ Theta2 ) )
& ( dfree_sf_sz @ Theta2 ) )
| ? [I4: sz] :
( A5
= ( diffVar_sz_sf @ I4 ) ) ) ) ) ).
% dsafe.simps
thf(fact_156_dsafe_Osimps,axiom,
( dsafe_a_c
= ( ^ [A5: trm_a_c] :
( ? [I4: c] :
( A5
= ( var_c_a @ I4 ) )
| ? [R3: real] :
( A5
= ( const_a_c @ R3 ) )
| ? [Args3: c > trm_a_c,I4: a] :
( ( A5
= ( function_a_c @ I4 @ Args3 ) )
& ! [X2: c] : ( dsafe_a_c @ ( Args3 @ X2 ) ) )
| ? [Theta_13: trm_a_c,Theta_23: trm_a_c] :
( ( A5
= ( plus_a_c @ Theta_13 @ Theta_23 ) )
& ( dsafe_a_c @ Theta_13 )
& ( dsafe_a_c @ Theta_23 ) )
| ? [Theta_13: trm_a_c,Theta_23: trm_a_c] :
( ( A5
= ( times_a_c @ Theta_13 @ Theta_23 ) )
& ( dsafe_a_c @ Theta_13 )
& ( dsafe_a_c @ Theta_23 ) )
| ? [Theta2: trm_a_c] :
( ( A5
= ( differential_a_c @ Theta2 ) )
& ( dfree_a_c @ Theta2 ) )
| ? [I4: c] :
( A5
= ( diffVar_c_a @ I4 ) ) ) ) ) ).
% dsafe.simps
thf(fact_157_dsafe_Ocases,axiom,
! [A: trm_sf_sz] :
( ( dsafe_sf_sz @ A )
=> ( ! [I3: sz] :
( A
!= ( var_sz_sf @ I3 ) )
=> ( ! [R2: real] :
( A
!= ( const_sf_sz @ R2 ) )
=> ( ! [Args2: sz > trm_sf_sz] :
( ? [I3: sf] :
( A
= ( function_sf_sz @ I3 @ Args2 ) )
=> ~ ! [I5: sz] : ( dsafe_sf_sz @ ( Args2 @ I5 ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( A
= ( plus_sf_sz @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe_sf_sz @ Theta_12 )
=> ~ ( dsafe_sf_sz @ Theta_22 ) ) )
=> ( ! [Theta_12: trm_sf_sz,Theta_22: trm_sf_sz] :
( ( A
= ( times_sf_sz @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe_sf_sz @ Theta_12 )
=> ~ ( dsafe_sf_sz @ Theta_22 ) ) )
=> ( ! [Theta: trm_sf_sz] :
( ( A
= ( differential_sf_sz @ Theta ) )
=> ~ ( dfree_sf_sz @ Theta ) )
=> ~ ! [I3: sz] :
( A
!= ( diffVar_sz_sf @ I3 ) ) ) ) ) ) ) ) ) ).
% dsafe.cases
thf(fact_158_dsafe_Ocases,axiom,
! [A: trm_a_c] :
( ( dsafe_a_c @ A )
=> ( ! [I3: c] :
( A
!= ( var_c_a @ I3 ) )
=> ( ! [R2: real] :
( A
!= ( const_a_c @ R2 ) )
=> ( ! [Args2: c > trm_a_c] :
( ? [I3: a] :
( A
= ( function_a_c @ I3 @ Args2 ) )
=> ~ ! [I5: c] : ( dsafe_a_c @ ( Args2 @ I5 ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( A
= ( plus_a_c @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe_a_c @ Theta_12 )
=> ~ ( dsafe_a_c @ Theta_22 ) ) )
=> ( ! [Theta_12: trm_a_c,Theta_22: trm_a_c] :
( ( A
= ( times_a_c @ Theta_12 @ Theta_22 ) )
=> ( ( dsafe_a_c @ Theta_12 )
=> ~ ( dsafe_a_c @ Theta_22 ) ) )
=> ( ! [Theta: trm_a_c] :
( ( A
= ( differential_a_c @ Theta ) )
=> ~ ( dfree_a_c @ Theta ) )
=> ~ ! [I3: c] :
( A
!= ( diffVar_c_a @ I3 ) ) ) ) ) ) ) ) ) ).
% dsafe.cases
thf(fact_159_strm_Oright__unique,axiom,
right_963419005_sf_sz @ freche1244000341_sf_sz ).
% strm.right_unique
thf(fact_160_strm_Oright__unique,axiom,
right_1933252621rm_a_c @ frechet_cr_strm_a_c ).
% strm.right_unique
thf(fact_161_dsafe__Const__simps,axiom,
! [R: real] : ( dsafe_a_c @ ( const_a_c @ R ) ) ).
% dsafe_Const_simps
thf(fact_162_dsafe__Const__simps,axiom,
! [R: real] : ( dsafe_sf_sz @ ( const_sf_sz @ R ) ) ).
% dsafe_Const_simps
thf(fact_163_dsafe__Fun__simps,axiom,
! [I: a,Args: c > trm_a_c] :
( ( dsafe_a_c @ ( function_a_c @ I @ Args ) )
= ( ! [X2: c] : ( dsafe_a_c @ ( Args @ X2 ) ) ) ) ).
% dsafe_Fun_simps
thf(fact_164_dsafe__Fun__simps,axiom,
! [I: sf,Args: sz > trm_sf_sz] :
( ( dsafe_sf_sz @ ( function_sf_sz @ I @ Args ) )
= ( ! [X2: sz] : ( dsafe_sf_sz @ ( Args @ X2 ) ) ) ) ).
% dsafe_Fun_simps
thf(fact_165_dsafe__Plus__simps,axiom,
! [A: trm_sf_sz,B: trm_sf_sz] :
( ( dsafe_sf_sz @ ( plus_sf_sz @ A @ B ) )
= ( ( dsafe_sf_sz @ A )
& ( dsafe_sf_sz @ B ) ) ) ).
% dsafe_Plus_simps
thf(fact_166_dsafe__Plus__simps,axiom,
! [A: trm_a_c,B: trm_a_c] :
( ( dsafe_a_c @ ( plus_a_c @ A @ B ) )
= ( ( dsafe_a_c @ A )
& ( dsafe_a_c @ B ) ) ) ).
% dsafe_Plus_simps
thf(fact_167_dsafe__Times__simps,axiom,
! [A: trm_sf_sz,B: trm_sf_sz] :
( ( dsafe_sf_sz @ ( times_sf_sz @ A @ B ) )
= ( ( dsafe_sf_sz @ A )
& ( dsafe_sf_sz @ B ) ) ) ).
% dsafe_Times_simps
thf(fact_168_dsafe__Times__simps,axiom,
! [A: trm_a_c,B: trm_a_c] :
( ( dsafe_a_c @ ( times_a_c @ A @ B ) )
= ( ( dsafe_a_c @ A )
& ( dsafe_a_c @ B ) ) ) ).
% dsafe_Times_simps
thf(fact_169_dsafe__Var__simps,axiom,
! [X: sz] : ( dsafe_sf_sz @ ( var_sz_sf @ X ) ) ).
% dsafe_Var_simps
thf(fact_170_dsafe__Var__simps,axiom,
! [X: c] : ( dsafe_a_c @ ( var_c_a @ X ) ) ).
% dsafe_Var_simps
thf(fact_171_dsafe__DiffVar__simps,axiom,
! [X: sz] : ( dsafe_sf_sz @ ( diffVar_sz_sf @ X ) ) ).
% dsafe_DiffVar_simps
thf(fact_172_dsafe__DiffVar__simps,axiom,
! [X: c] : ( dsafe_a_c @ ( diffVar_c_a @ X ) ) ).
% dsafe_DiffVar_simps
thf(fact_173_dsafe__Diff__simps,axiom,
! [A: trm_sf_sz] :
( ( dsafe_sf_sz @ ( differential_sf_sz @ A ) )
= ( dfree_sf_sz @ A ) ) ).
% dsafe_Diff_simps
thf(fact_174_dsafe__Diff__simps,axiom,
! [A: trm_a_c] :
( ( dsafe_a_c @ ( differential_a_c @ A ) )
= ( dfree_a_c @ A ) ) ).
% dsafe_Diff_simps
thf(fact_175_dfree__is__dsafe,axiom,
! [Theta3: trm_a_c] :
( ( dfree_a_c @ Theta3 )
=> ( dsafe_a_c @ Theta3 ) ) ).
% dfree_is_dsafe
thf(fact_176_dfree__is__dsafe,axiom,
! [Theta3: trm_sf_sz] :
( ( dfree_sf_sz @ Theta3 )
=> ( dsafe_sf_sz @ Theta3 ) ) ).
% dfree_is_dsafe
thf(fact_177_dsafe__Const,axiom,
! [R: real] : ( dsafe_a_c @ ( const_a_c @ R ) ) ).
% dsafe_Const
thf(fact_178_dsafe__Const,axiom,
! [R: real] : ( dsafe_sf_sz @ ( const_sf_sz @ R ) ) ).
% dsafe_Const
thf(fact_179_dsafe__Fun,axiom,
! [Args: c > trm_a_c,I: a] :
( ! [I3: c] : ( dsafe_a_c @ ( Args @ I3 ) )
=> ( dsafe_a_c @ ( function_a_c @ I @ Args ) ) ) ).
% dsafe_Fun
thf(fact_180_dsafe__Fun,axiom,
! [Args: sz > trm_sf_sz,I: sf] :
( ! [I3: sz] : ( dsafe_sf_sz @ ( Args @ I3 ) )
=> ( dsafe_sf_sz @ ( function_sf_sz @ I @ Args ) ) ) ).
% dsafe_Fun
thf(fact_181_dsafe__Plus,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dsafe_sf_sz @ Theta_1 )
=> ( ( dsafe_sf_sz @ Theta_2 )
=> ( dsafe_sf_sz @ ( plus_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Plus
thf(fact_182_dsafe__Plus,axiom,
! [Theta_1: trm_a_c,Theta_2: trm_a_c] :
( ( dsafe_a_c @ Theta_1 )
=> ( ( dsafe_a_c @ Theta_2 )
=> ( dsafe_a_c @ ( plus_a_c @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Plus
thf(fact_183_dsafe__Times,axiom,
! [Theta_1: trm_sf_sz,Theta_2: trm_sf_sz] :
( ( dsafe_sf_sz @ Theta_1 )
=> ( ( dsafe_sf_sz @ Theta_2 )
=> ( dsafe_sf_sz @ ( times_sf_sz @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Times
thf(fact_184_dsafe__Times,axiom,
! [Theta_1: trm_a_c,Theta_2: trm_a_c] :
( ( dsafe_a_c @ Theta_1 )
=> ( ( dsafe_a_c @ Theta_2 )
=> ( dsafe_a_c @ ( times_a_c @ Theta_1 @ Theta_2 ) ) ) ) ).
% dsafe_Times
thf(fact_185_dsafe__Var,axiom,
! [I: sz] : ( dsafe_sf_sz @ ( var_sz_sf @ I ) ) ).
% dsafe_Var
thf(fact_186_dsafe__Var,axiom,
! [I: c] : ( dsafe_a_c @ ( var_c_a @ I ) ) ).
% dsafe_Var
thf(fact_187_dsafe__DiffVar,axiom,
! [I: sz] : ( dsafe_sf_sz @ ( diffVar_sz_sf @ I ) ) ).
% dsafe_DiffVar
thf(fact_188_dsafe__DiffVar,axiom,
! [I: c] : ( dsafe_a_c @ ( diffVar_c_a @ I ) ) ).
% dsafe_DiffVar
thf(fact_189_dsafe__Diff,axiom,
! [Theta3: trm_sf_sz] :
( ( dfree_sf_sz @ Theta3 )
=> ( dsafe_sf_sz @ ( differential_sf_sz @ Theta3 ) ) ) ).
% dsafe_Diff
thf(fact_190_dsafe__Diff,axiom,
! [Theta3: trm_a_c] :
( ( dfree_a_c @ Theta3 )
=> ( dsafe_a_c @ ( differential_a_c @ Theta3 ) ) ) ).
% dsafe_Diff
thf(fact_191_local_Oempty__def,axiom,
( empty_c_a
= ( ^ [I4: c] : ( const_a_c @ zero_zero_real ) ) ) ).
% local.empty_def
thf(fact_192_local_Oempty__def,axiom,
( empty_sz_sf
= ( ^ [I4: sz] : ( const_sf_sz @ zero_zero_real ) ) ) ).
% local.empty_def
thf(fact_193_ids_Of0__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,F2: sf] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( f0_sf_sz @ F2 )
= ( function_sf_sz @ F2 @ empty_sz_sf ) ) ) ).
% ids.f0_def
thf(fact_194_dfree__Fun_OIH,axiom,
! [I: c] :
( ( denota1997846518gree_c @ nu @ nu2 @ ( static_FVDiff_a_c @ ( args @ I ) ) )
=> ( ( denota229585092_a_b_c @ i @ ( args @ I ) @ ( produc2010422875real_c @ nu ) @ ( produc314122909real_c @ nu ) )
= ( denota229585092_a_b_c @ i @ ( args @ I ) @ ( produc2010422875real_c @ nu2 ) @ ( produc314122909real_c @ nu2 ) ) ) ) ).
% dfree_Fun.IH
thf(fact_195_strm_Oleft__unique,axiom,
left_u638806404_sf_sz @ freche1244000341_sf_sz ).
% strm.left_unique
thf(fact_196_strm_Oleft__unique,axiom,
left_u968903956rm_a_c @ frechet_cr_strm_a_c ).
% strm.left_unique
thf(fact_197_trm_Oinduct,axiom,
! [P: trm_sf_sz > $o,Trm: trm_sf_sz] :
( ! [X3: sz] : ( P @ ( var_sz_sf @ X3 ) )
=> ( ! [X3: real] : ( P @ ( const_sf_sz @ X3 ) )
=> ( ! [X1a: sf,X2a: sz > trm_sf_sz] :
( ! [X2aa: trm_sf_sz] :
( ( member_trm_sf_sz @ X2aa @ ( image_sz_trm_sf_sz @ X2a @ top_top_set_sz ) )
=> ( P @ X2aa ) )
=> ( P @ ( function_sf_sz @ X1a @ X2a ) ) )
=> ( ! [X1a: trm_sf_sz,X2a: trm_sf_sz] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( plus_sf_sz @ X1a @ X2a ) ) ) )
=> ( ! [X1a: trm_sf_sz,X2a: trm_sf_sz] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( times_sf_sz @ X1a @ X2a ) ) ) )
=> ( ! [X3: sz] : ( P @ ( diffVar_sz_sf @ X3 ) )
=> ( ! [X3: trm_sf_sz] :
( ( P @ X3 )
=> ( P @ ( differential_sf_sz @ X3 ) ) )
=> ( P @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_198_trm_Oinduct,axiom,
! [P: trm_a_c > $o,Trm: trm_a_c] :
( ! [X3: c] : ( P @ ( var_c_a @ X3 ) )
=> ( ! [X3: real] : ( P @ ( const_a_c @ X3 ) )
=> ( ! [X1a: a,X2a: c > trm_a_c] :
( ! [X2aa: trm_a_c] :
( ( member_trm_a_c @ X2aa @ ( image_c_trm_a_c @ X2a @ top_top_set_c ) )
=> ( P @ X2aa ) )
=> ( P @ ( function_a_c @ X1a @ X2a ) ) )
=> ( ! [X1a: trm_a_c,X2a: trm_a_c] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( plus_a_c @ X1a @ X2a ) ) ) )
=> ( ! [X1a: trm_a_c,X2a: trm_a_c] :
( ( P @ X1a )
=> ( ( P @ X2a )
=> ( P @ ( times_a_c @ X1a @ X2a ) ) ) )
=> ( ! [X3: c] : ( P @ ( diffVar_c_a @ X3 ) )
=> ( ! [X3: trm_a_c] :
( ( P @ X3 )
=> ( P @ ( differential_a_c @ X3 ) ) )
=> ( P @ Trm ) ) ) ) ) ) ) ) ).
% trm.induct
thf(fact_199_agree__comm,axiom,
! [A2: produc1149990247eal_sz,B4: produc1149990247eal_sz,V: set_Sum_sum_sz_sz] :
( ( denota102713844ree_sz @ A2 @ B4 @ V )
=> ( denota102713844ree_sz @ B4 @ A2 @ V ) ) ).
% agree_comm
thf(fact_200_agree__comm,axiom,
! [A2: produc190496183real_c,B4: produc190496183real_c,V: set_Sum_sum_c_c] :
( ( denota1997846518gree_c @ A2 @ B4 @ V )
=> ( denota1997846518gree_c @ B4 @ A2 @ V ) ) ).
% agree_comm
thf(fact_201_prod_Ocollapse,axiom,
! [Prod: produc866628903_sc_sz] :
( ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ Prod ) @ ( produc4753477_sc_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_202_prod_Ocollapse,axiom,
! [Prod: produc861709536_sz_sz] :
( ( produc792909658_sz_sz @ ( produc2069561454_sz_sz @ Prod ) @ ( produc181739436_sz_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_203_prod_Ocollapse,axiom,
! [Prod: produc999471430t_unit] :
( ( produc502616768t_unit @ ( produc1216682964t_unit @ Prod ) @ ( produc687319826t_unit @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_204_prod_Ocollapse,axiom,
! [Prod: produc1821101996_sc_sz] :
( ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ Prod ) @ ( produc974239792_sc_sz @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_205_prod_Ocollapse,axiom,
! [Prod: produc190496183real_c] :
( ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_206_prod_Oexhaust__sel,axiom,
! [Prod: produc866628903_sc_sz] :
( Prod
= ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ Prod ) @ ( produc4753477_sc_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_207_prod_Oexhaust__sel,axiom,
! [Prod: produc861709536_sz_sz] :
( Prod
= ( produc792909658_sz_sz @ ( produc2069561454_sz_sz @ Prod ) @ ( produc181739436_sz_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_208_prod_Oexhaust__sel,axiom,
! [Prod: produc999471430t_unit] :
( Prod
= ( produc502616768t_unit @ ( produc1216682964t_unit @ Prod ) @ ( produc687319826t_unit @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_209_prod_Oexhaust__sel,axiom,
! [Prod: produc1821101996_sc_sz] :
( Prod
= ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ Prod ) @ ( produc974239792_sc_sz @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_210_prod_Oexhaust__sel,axiom,
! [Prod: produc190496183real_c] :
( Prod
= ( produc394644079real_c @ ( produc2010422875real_c @ Prod ) @ ( produc314122909real_c @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_211_surjective__pairing,axiom,
! [T3: produc866628903_sc_sz] :
( T3
= ( produc1822718231_sc_sz @ ( produc548504323_sc_sz @ T3 ) @ ( produc4753477_sc_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_212_surjective__pairing,axiom,
! [T3: produc861709536_sz_sz] :
( T3
= ( produc792909658_sz_sz @ ( produc2069561454_sz_sz @ T3 ) @ ( produc181739436_sz_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_213_surjective__pairing,axiom,
! [T3: produc999471430t_unit] :
( T3
= ( produc502616768t_unit @ ( produc1216682964t_unit @ T3 ) @ ( produc687319826t_unit @ T3 ) ) ) ).
% surjective_pairing
thf(fact_214_surjective__pairing,axiom,
! [T3: produc1821101996_sc_sz] :
( T3
= ( produc789536734_sc_sz @ ( produc1503602930_sc_sz @ T3 ) @ ( produc974239792_sc_sz @ T3 ) ) ) ).
% surjective_pairing
thf(fact_215_surjective__pairing,axiom,
! [T3: produc190496183real_c] :
( T3
= ( produc394644079real_c @ ( produc2010422875real_c @ T3 ) @ ( produc314122909real_c @ T3 ) ) ) ).
% surjective_pairing
thf(fact_216_agree__UNIV__eq,axiom,
! [Nu: produc1149990247eal_sz,Omega: produc1149990247eal_sz] :
( ( denota102713844ree_sz @ Nu @ Omega @ top_to1246370329_sz_sz )
=> ( Nu = Omega ) ) ).
% agree_UNIV_eq
thf(fact_217_agree__UNIV__eq,axiom,
! [Nu: produc190496183real_c,Omega: produc190496183real_c] :
( ( denota1997846518gree_c @ Nu @ Omega @ top_to1660572043um_c_c )
=> ( Nu = Omega ) ) ).
% agree_UNIV_eq
thf(fact_218_Vagree__univ,axiom,
! [A: finite824932053eal_sz,B: finite824932053eal_sz,C2: finite824932053eal_sz,D: finite824932053eal_sz] :
( ( denota102713844ree_sz @ ( produc1308130519eal_sz @ A @ B ) @ ( produc1308130519eal_sz @ C2 @ D ) @ top_to1246370329_sz_sz )
=> ( ( A = C2 )
& ( B = D ) ) ) ).
% Vagree_univ
thf(fact_219_Vagree__univ,axiom,
! [A: finite1398487019real_c,B: finite1398487019real_c,C2: finite1398487019real_c,D: finite1398487019real_c] :
( ( denota1997846518gree_c @ ( produc394644079real_c @ A @ B ) @ ( produc394644079real_c @ C2 @ D ) @ top_to1660572043um_c_c )
=> ( ( A = C2 )
& ( B = D ) ) ) ).
% Vagree_univ
thf(fact_220_ids_Oseq__sem_Ocases,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,X: produc1821101996_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [I2: denota610675952t_unit,S: produc866628903_sc_sz] :
( X
!= ( produc789536734_sc_sz @ I2 @ S ) ) ) ).
% ids.seq_sem.cases
thf(fact_221_ids_Oseq__sem_Oinduct,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P: denota610675952t_unit > produc866628903_sc_sz > $o,A0: denota610675952t_unit,A1: produc866628903_sc_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ! [I2: denota610675952t_unit,X_1: produc866628903_sc_sz] : ( P @ I2 @ X_1 )
=> ( P @ A0 @ A1 ) ) ) ).
% ids.seq_sem.induct
thf(fact_222_frechet_Osimps_I5_J,axiom,
! [I6: denota231621370t_unit,R: real,V2: finite1398487019real_c] :
( ( denota229585092_a_b_c @ I6 @ ( const_a_c @ R ) @ V2 )
= ( ^ [V3: finite1398487019real_c] : zero_zero_real ) ) ).
% frechet.simps(5)
thf(fact_223_prod__eqI,axiom,
! [P2: produc1821101996_sc_sz,Q: produc1821101996_sc_sz] :
( ( ( produc1503602930_sc_sz @ P2 )
= ( produc1503602930_sc_sz @ Q ) )
=> ( ( ( produc974239792_sc_sz @ P2 )
= ( produc974239792_sc_sz @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_224_prod__eqI,axiom,
! [P2: produc999471430t_unit,Q: produc999471430t_unit] :
( ( ( produc1216682964t_unit @ P2 )
= ( produc1216682964t_unit @ Q ) )
=> ( ( ( produc687319826t_unit @ P2 )
= ( produc687319826t_unit @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_225_prod__eqI,axiom,
! [P2: produc190496183real_c,Q: produc190496183real_c] :
( ( ( produc2010422875real_c @ P2 )
= ( produc2010422875real_c @ Q ) )
=> ( ( ( produc314122909real_c @ P2 )
= ( produc314122909real_c @ Q ) )
=> ( P2 = Q ) ) ) ).
% prod_eqI
thf(fact_226_prod_Oexpand,axiom,
! [Prod: produc1821101996_sc_sz,Prod2: produc1821101996_sc_sz] :
( ( ( ( produc1503602930_sc_sz @ Prod )
= ( produc1503602930_sc_sz @ Prod2 ) )
& ( ( produc974239792_sc_sz @ Prod )
= ( produc974239792_sc_sz @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_227_prod_Oexpand,axiom,
! [Prod: produc999471430t_unit,Prod2: produc999471430t_unit] :
( ( ( ( produc1216682964t_unit @ Prod )
= ( produc1216682964t_unit @ Prod2 ) )
& ( ( produc687319826t_unit @ Prod )
= ( produc687319826t_unit @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_228_prod_Oexpand,axiom,
! [Prod: produc190496183real_c,Prod2: produc190496183real_c] :
( ( ( ( produc2010422875real_c @ Prod )
= ( produc2010422875real_c @ Prod2 ) )
& ( ( produc314122909real_c @ Prod )
= ( produc314122909real_c @ Prod2 ) ) )
=> ( Prod = Prod2 ) ) ).
% prod.expand
thf(fact_229_prod__eq__iff,axiom,
( ( ^ [Y4: produc1821101996_sc_sz,Z: produc1821101996_sc_sz] : Y4 = Z )
= ( ^ [S2: produc1821101996_sc_sz,T4: produc1821101996_sc_sz] :
( ( ( produc1503602930_sc_sz @ S2 )
= ( produc1503602930_sc_sz @ T4 ) )
& ( ( produc974239792_sc_sz @ S2 )
= ( produc974239792_sc_sz @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_230_prod__eq__iff,axiom,
( ( ^ [Y4: produc999471430t_unit,Z: produc999471430t_unit] : Y4 = Z )
= ( ^ [S2: produc999471430t_unit,T4: produc999471430t_unit] :
( ( ( produc1216682964t_unit @ S2 )
= ( produc1216682964t_unit @ T4 ) )
& ( ( produc687319826t_unit @ S2 )
= ( produc687319826t_unit @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_231_prod__eq__iff,axiom,
( ( ^ [Y4: produc190496183real_c,Z: produc190496183real_c] : Y4 = Z )
= ( ^ [S2: produc190496183real_c,T4: produc190496183real_c] :
( ( ( produc2010422875real_c @ S2 )
= ( produc2010422875real_c @ T4 ) )
& ( ( produc314122909real_c @ S2 )
= ( produc314122909real_c @ T4 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_232_snd__conv,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz] :
( ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_233_snd__conv,axiom,
! [X1: trm_sf_sz,X22: sz] :
( ( produc181739436_sz_sz @ ( produc792909658_sz_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_234_snd__conv,axiom,
! [X1: produc866628903_sc_sz,X22: denota610675952t_unit] :
( ( produc687319826t_unit @ ( produc502616768t_unit @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_235_snd__conv,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz] :
( ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_236_snd__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc314122909real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X22 ) ).
% snd_conv
thf(fact_237_snd__eqD,axiom,
! [X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz,A: list_f1238882004_sc_sz] :
( ( ( produc4753477_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_238_snd__eqD,axiom,
! [X: trm_sf_sz,Y: sz,A: sz] :
( ( ( produc181739436_sz_sz @ ( produc792909658_sz_sz @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_239_snd__eqD,axiom,
! [X: produc866628903_sc_sz,Y: denota610675952t_unit,A: denota610675952t_unit] :
( ( ( produc687319826t_unit @ ( produc502616768t_unit @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_240_snd__eqD,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz,A: produc866628903_sc_sz] :
( ( ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_241_snd__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A: finite1398487019real_c] :
( ( ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) )
= A )
=> ( Y = A ) ) ).
% snd_eqD
thf(fact_242_fst__conv,axiom,
! [X1: list_f1238882004_sc_sz,X22: list_f1238882004_sc_sz] :
( ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_243_fst__conv,axiom,
! [X1: trm_sf_sz,X22: sz] :
( ( produc2069561454_sz_sz @ ( produc792909658_sz_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_244_fst__conv,axiom,
! [X1: produc866628903_sc_sz,X22: denota610675952t_unit] :
( ( produc1216682964t_unit @ ( produc502616768t_unit @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_245_fst__conv,axiom,
! [X1: denota610675952t_unit,X22: produc866628903_sc_sz] :
( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_246_fst__conv,axiom,
! [X1: finite1398487019real_c,X22: finite1398487019real_c] :
( ( produc2010422875real_c @ ( produc394644079real_c @ X1 @ X22 ) )
= X1 ) ).
% fst_conv
thf(fact_247_fst__eqD,axiom,
! [X: list_f1238882004_sc_sz,Y: list_f1238882004_sc_sz,A: list_f1238882004_sc_sz] :
( ( ( produc548504323_sc_sz @ ( produc1822718231_sc_sz @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_248_fst__eqD,axiom,
! [X: trm_sf_sz,Y: sz,A: trm_sf_sz] :
( ( ( produc2069561454_sz_sz @ ( produc792909658_sz_sz @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_249_fst__eqD,axiom,
! [X: produc866628903_sc_sz,Y: denota610675952t_unit,A: produc866628903_sc_sz] :
( ( ( produc1216682964t_unit @ ( produc502616768t_unit @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_250_fst__eqD,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz,A: denota610675952t_unit] :
( ( ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_251_fst__eqD,axiom,
! [X: finite1398487019real_c,Y: finite1398487019real_c,A: finite1398487019real_c] :
( ( ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) )
= A )
=> ( X = A ) ) ).
% fst_eqD
thf(fact_252_ids_Oempty__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( empty_c_a
= ( ^ [I4: c] : ( const_a_c @ zero_zero_real ) ) ) ) ).
% ids.empty_def
thf(fact_253_ids_Oempty__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( empty_sz_sf
= ( ^ [I4: sz] : ( const_sf_sz @ zero_zero_real ) ) ) ) ).
% ids.empty_def
thf(fact_254_ids_Osingleton_Ocases,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,X: produc861709536_sz_sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ~ ! [T: trm_sf_sz,I3: sz] :
( X
!= ( produc792909658_sz_sz @ T @ I3 ) ) ) ).
% ids.singleton.cases
thf(fact_255_ids_OP__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P2: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p_sc_sf_sz @ P2 )
= ( predic913887884_sf_sz @ P2 ) ) ) ).
% ids.P_def
thf(fact_256_agree__refl,axiom,
! [Nu: produc1149990247eal_sz,A2: set_Sum_sum_sz_sz] : ( denota102713844ree_sz @ Nu @ Nu @ A2 ) ).
% agree_refl
thf(fact_257_agree__refl,axiom,
! [Nu: produc190496183real_c,A2: set_Sum_sum_c_c] : ( denota1997846518gree_c @ Nu @ Nu @ A2 ) ).
% agree_refl
thf(fact_258_ids_Ostrm_Odomain,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( domain2047240575rm_a_c @ frechet_cr_strm_a_c )
= dfree_a_c ) ) ).
% ids.strm.domain
thf(fact_259_ids_Ostrm_Odomain,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( domain512309999_sf_sz @ freche1244000341_sf_sz )
= dfree_sf_sz ) ) ).
% ids.strm.domain
thf(fact_260_snd__zero,axiom,
( ( produc314122909real_c @ zero_z1506780526real_c )
= zero_z109254132real_c ) ).
% snd_zero
thf(fact_261_fst__zero,axiom,
( ( produc2010422875real_c @ zero_z1506780526real_c )
= zero_z109254132real_c ) ).
% fst_zero
thf(fact_262_zero__prod__def,axiom,
( zero_z659284464l_real
= ( produc705216881l_real @ zero_zero_real @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_263_zero__prod__def,axiom,
( zero_z72018569real_c
= ( produc2134005810real_c @ zero_zero_real @ zero_z1506780526real_c ) ) ).
% zero_prod_def
thf(fact_264_zero__prod__def,axiom,
( zero_z464587567real_c
= ( produc1327150128real_c @ zero_zero_real @ zero_z109254132real_c ) ) ).
% zero_prod_def
thf(fact_265_zero__prod__def,axiom,
( zero_z1830236699c_real
= ( produc171591948c_real @ zero_z1506780526real_c @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_266_zero__prod__def,axiom,
( zero_z993734942real_c
= ( produc1687028567real_c @ zero_z1506780526real_c @ zero_z1506780526real_c ) ) ).
% zero_prod_def
thf(fact_267_zero__prod__def,axiom,
( zero_z440955354real_c
= ( produc116546123real_c @ zero_z1506780526real_c @ zero_z109254132real_c ) ) ).
% zero_prod_def
thf(fact_268_zero__prod__def,axiom,
( zero_z1273814959c_real
= ( produc1715747632c_real @ zero_z109254132real_c @ zero_zero_real ) ) ).
% zero_prod_def
thf(fact_269_zero__prod__def,axiom,
( zero_z735295626real_c
= ( produc1563404851real_c @ zero_z109254132real_c @ zero_z1506780526real_c ) ) ).
% zero_prod_def
thf(fact_270_zero__prod__def,axiom,
( zero_z1506780526real_c
= ( produc394644079real_c @ zero_z109254132real_c @ zero_z109254132real_c ) ) ).
% zero_prod_def
thf(fact_271_top__prod__def,axiom,
( top_to543326711_set_c
= ( produc878845847_set_c @ top_top_set_c @ top_top_set_c ) ) ).
% top_prod_def
thf(fact_272_top__prod__def,axiom,
( top_to1303770207set_sz
= ( produc2039118377set_sz @ top_top_set_c @ top_top_set_sz ) ) ).
% top_prod_def
thf(fact_273_top__prod__def,axiom,
( top_to474099557_set_c
= ( produc760013927_set_c @ top_top_set_sz @ top_top_set_c ) ) ).
% top_prod_def
thf(fact_274_top__prod__def,axiom,
( top_to1999610481set_sz
= ( produc2156889set_sz @ top_top_set_sz @ top_top_set_sz ) ) ).
% top_prod_def
thf(fact_275_top__prod__def,axiom,
( top_to1128637210um_c_c
= ( produc1519884516um_c_c @ top_top_set_c @ top_to1660572043um_c_c ) ) ).
% top_prod_def
thf(fact_276_top__prod__def,axiom,
( top_to1783191648real_c
= ( produc280329770real_c @ top_top_set_c @ top_to1642137041real_c ) ) ).
% top_prod_def
thf(fact_277_top__prod__def,axiom,
( top_to1102260780um_c_c
= ( produc590684180um_c_c @ top_top_set_sz @ top_to1660572043um_c_c ) ) ).
% top_prod_def
thf(fact_278_top__prod__def,axiom,
( top_to1102487922real_c
= ( produc1915663706real_c @ top_top_set_sz @ top_to1642137041real_c ) ) ).
% top_prod_def
thf(fact_279_top__prod__def,axiom,
( top_to1309524330_set_c
= ( produc1767696108_set_c @ top_to1660572043um_c_c @ top_top_set_c ) ) ).
% top_prod_def
thf(fact_280_top__prod__def,axiom,
( top_to1246597548set_sz
= ( produc1251379220set_sz @ top_to1660572043um_c_c @ top_top_set_sz ) ) ).
% top_prod_def
thf(fact_281_ids_Ostrm_Oright__unique,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( right_963419005_sf_sz @ freche1244000341_sf_sz ) ) ).
% ids.strm.right_unique
thf(fact_282_directional__derivative__def,axiom,
( denota2112424896_a_b_c
= ( ^ [I7: denota231621370t_unit,T4: trm_a_c,V4: produc190496183real_c] : ( denota229585092_a_b_c @ I7 @ T4 @ ( produc2010422875real_c @ V4 ) @ ( produc314122909real_c @ V4 ) ) ) ) ).
% directional_derivative_def
thf(fact_283_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: denota610675952t_unit > produc866628903_sc_sz > $o,X: denota610675952t_unit,Y: produc866628903_sc_sz,A: produc1821101996_sc_sz] :
( ( P @ X @ Y )
=> ( ( A
= ( produc789536734_sc_sz @ X @ Y ) )
=> ( P @ ( produc1503602930_sc_sz @ A ) @ ( produc974239792_sc_sz @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_284_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,X: finite1398487019real_c,Y: finite1398487019real_c,A: produc190496183real_c] :
( ( P @ X @ Y )
=> ( ( A
= ( produc394644079real_c @ X @ Y ) )
=> ( P @ ( produc2010422875real_c @ A ) @ ( produc314122909real_c @ A ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_285_conjI__realizer,axiom,
! [P: denota610675952t_unit > $o,P2: denota610675952t_unit,Q2: produc866628903_sc_sz > $o,Q: produc866628903_sc_sz] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ P2 @ Q ) ) )
& ( Q2 @ ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_286_conjI__realizer,axiom,
! [P: finite1398487019real_c > $o,P2: finite1398487019real_c,Q2: finite1398487019real_c > $o,Q: finite1398487019real_c] :
( ( P @ P2 )
=> ( ( Q2 @ Q )
=> ( ( P @ ( produc2010422875real_c @ ( produc394644079real_c @ P2 @ Q ) ) )
& ( Q2 @ ( produc314122909real_c @ ( produc394644079real_c @ P2 @ Q ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_287_exI__realizer,axiom,
! [P: produc866628903_sc_sz > denota610675952t_unit > $o,Y: produc866628903_sc_sz,X: denota610675952t_unit] :
( ( P @ Y @ X )
=> ( P @ ( produc974239792_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) ) @ ( produc1503602930_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_288_exI__realizer,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,Y: finite1398487019real_c,X: finite1398487019real_c] :
( ( P @ Y @ X )
=> ( P @ ( produc314122909real_c @ ( produc394644079real_c @ X @ Y ) ) @ ( produc2010422875real_c @ ( produc394644079real_c @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_289_exE__realizer_H,axiom,
! [P: finite1398487019real_c > finite1398487019real_c > $o,P2: produc190496183real_c] :
( ( P @ ( produc314122909real_c @ P2 ) @ ( produc2010422875real_c @ P2 ) )
=> ~ ! [X3: finite1398487019real_c,Y3: finite1398487019real_c] :
~ ( P @ Y3 @ X3 ) ) ).
% exE_realizer'
thf(fact_290_sndI,axiom,
! [X: produc1821101996_sc_sz,Y: denota610675952t_unit,Z2: produc866628903_sc_sz] :
( ( X
= ( produc789536734_sc_sz @ Y @ Z2 ) )
=> ( ( produc974239792_sc_sz @ X )
= Z2 ) ) ).
% sndI
thf(fact_291_sndI,axiom,
! [X: produc190496183real_c,Y: finite1398487019real_c,Z2: finite1398487019real_c] :
( ( X
= ( produc394644079real_c @ Y @ Z2 ) )
=> ( ( produc314122909real_c @ X )
= Z2 ) ) ).
% sndI
thf(fact_292_eq__snd__iff,axiom,
! [B: produc866628903_sc_sz,P2: produc1821101996_sc_sz] :
( ( B
= ( produc974239792_sc_sz @ P2 ) )
= ( ? [A5: denota610675952t_unit] :
( P2
= ( produc789536734_sc_sz @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_293_eq__snd__iff,axiom,
! [B: finite1398487019real_c,P2: produc190496183real_c] :
( ( B
= ( produc314122909real_c @ P2 ) )
= ( ? [A5: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ A5 @ B ) ) ) ) ).
% eq_snd_iff
thf(fact_294_eq__fst__iff,axiom,
! [A: denota610675952t_unit,P2: produc1821101996_sc_sz] :
( ( A
= ( produc1503602930_sc_sz @ P2 ) )
= ( ? [B5: produc866628903_sc_sz] :
( P2
= ( produc789536734_sc_sz @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_295_eq__fst__iff,axiom,
! [A: finite1398487019real_c,P2: produc190496183real_c] :
( ( A
= ( produc2010422875real_c @ P2 ) )
= ( ? [B5: finite1398487019real_c] :
( P2
= ( produc394644079real_c @ A @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_296_fstI,axiom,
! [X: produc1821101996_sc_sz,Y: denota610675952t_unit,Z2: produc866628903_sc_sz] :
( ( X
= ( produc789536734_sc_sz @ Y @ Z2 ) )
=> ( ( produc1503602930_sc_sz @ X )
= Y ) ) ).
% fstI
thf(fact_297_fstI,axiom,
! [X: produc190496183real_c,Y: finite1398487019real_c,Z2: finite1398487019real_c] :
( ( X
= ( produc394644079real_c @ Y @ Z2 ) )
=> ( ( produc2010422875real_c @ X )
= Y ) ) ).
% fstI
thf(fact_298_range__snd,axiom,
( ( image_14429021real_c @ produc314122909real_c @ top_to226056039real_c )
= top_to1642137041real_c ) ).
% range_snd
thf(fact_299_range__fst,axiom,
( ( image_14429021real_c @ produc2010422875real_c @ top_to226056039real_c )
= top_to1642137041real_c ) ).
% range_fst
thf(fact_300_DomainpE,axiom,
! [R: trm_a_c > frechet_strm_a_c > $o,A: trm_a_c] :
( ( domain2047240575rm_a_c @ R @ A )
=> ~ ! [B3: frechet_strm_a_c] :
~ ( R @ A @ B3 ) ) ).
% DomainpE
thf(fact_301_Domainp__iff,axiom,
( domain2047240575rm_a_c
= ( ^ [T5: trm_a_c > frechet_strm_a_c > $o,X2: trm_a_c] :
? [X4: frechet_strm_a_c] : ( T5 @ X2 @ X4 ) ) ) ).
% Domainp_iff
thf(fact_302_Domainp__refl,axiom,
domain2047240575rm_a_c = domain2047240575rm_a_c ).
% Domainp_refl
thf(fact_303_Domainp_Ocases,axiom,
! [R: trm_a_c > frechet_strm_a_c > $o,A: trm_a_c] :
( ( domain2047240575rm_a_c @ R @ A )
=> ~ ! [B3: frechet_strm_a_c] :
~ ( R @ A @ B3 ) ) ).
% Domainp.cases
thf(fact_304_Domainp_Osimps,axiom,
( domain2047240575rm_a_c
= ( ^ [R3: trm_a_c > frechet_strm_a_c > $o,A5: trm_a_c] :
? [B5: trm_a_c,C3: frechet_strm_a_c] :
( ( A5 = B5 )
& ( R3 @ B5 @ C3 ) ) ) ) ).
% Domainp.simps
thf(fact_305_Domainp_ODomainI,axiom,
! [R: trm_a_c > frechet_strm_a_c > $o,A: trm_a_c,B: frechet_strm_a_c] :
( ( R @ A @ B )
=> ( domain2047240575rm_a_c @ R @ A ) ) ).
% Domainp.DomainI
thf(fact_306_Domainp_Oinducts,axiom,
! [R: trm_a_c > frechet_strm_a_c > $o,X: trm_a_c,P: trm_a_c > $o] :
( ( domain2047240575rm_a_c @ R @ X )
=> ( ! [A4: trm_a_c,B3: frechet_strm_a_c] :
( ( R @ A4 @ B3 )
=> ( P @ A4 ) )
=> ( P @ X ) ) ) ).
% Domainp.inducts
thf(fact_307_trm_Oset__cases,axiom,
! [E: sf,A: trm_sf_sz] :
( ( member_sf @ E @ ( set_trm_sf_sz2 @ A ) )
=> ( ! [Z22: sz > trm_sf_sz] :
( A
!= ( function_sf_sz @ E @ Z22 ) )
=> ( ! [Z1: sf,Z22: sz > trm_sf_sz] :
( ( A
= ( function_sf_sz @ Z1 @ Z22 ) )
=> ! [X3: trm_sf_sz] :
( ( member_trm_sf_sz @ X3 @ ( image_sz_trm_sf_sz @ Z22 @ top_top_set_sz ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ X3 ) ) ) )
=> ( ! [Z1: trm_sf_sz] :
( ? [Z22: trm_sf_sz] :
( A
= ( plus_sf_sz @ Z1 @ Z22 ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ Z1 ) ) )
=> ( ! [Z1: trm_sf_sz,Z22: trm_sf_sz] :
( ( A
= ( plus_sf_sz @ Z1 @ Z22 ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ Z22 ) ) )
=> ( ! [Z1: trm_sf_sz] :
( ? [Z22: trm_sf_sz] :
( A
= ( times_sf_sz @ Z1 @ Z22 ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ Z1 ) ) )
=> ( ! [Z1: trm_sf_sz,Z22: trm_sf_sz] :
( ( A
= ( times_sf_sz @ Z1 @ Z22 ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ Z22 ) ) )
=> ~ ! [Z3: trm_sf_sz] :
( ( A
= ( differential_sf_sz @ Z3 ) )
=> ~ ( member_sf @ E @ ( set_trm_sf_sz2 @ Z3 ) ) ) ) ) ) ) ) ) ) ).
% trm.set_cases
thf(fact_308_trm_Oset__cases,axiom,
! [E: a,A: trm_a_c] :
( ( member_a @ E @ ( set_trm_a_c2 @ A ) )
=> ( ! [Z22: c > trm_a_c] :
( A
!= ( function_a_c @ E @ Z22 ) )
=> ( ! [Z1: a,Z22: c > trm_a_c] :
( ( A
= ( function_a_c @ Z1 @ Z22 ) )
=> ! [X3: trm_a_c] :
( ( member_trm_a_c @ X3 @ ( image_c_trm_a_c @ Z22 @ top_top_set_c ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ X3 ) ) ) )
=> ( ! [Z1: trm_a_c] :
( ? [Z22: trm_a_c] :
( A
= ( plus_a_c @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ Z1 ) ) )
=> ( ! [Z1: trm_a_c,Z22: trm_a_c] :
( ( A
= ( plus_a_c @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ Z22 ) ) )
=> ( ! [Z1: trm_a_c] :
( ? [Z22: trm_a_c] :
( A
= ( times_a_c @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ Z1 ) ) )
=> ( ! [Z1: trm_a_c,Z22: trm_a_c] :
( ( A
= ( times_a_c @ Z1 @ Z22 ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ Z22 ) ) )
=> ~ ! [Z3: trm_a_c] :
( ( A
= ( differential_a_c @ Z3 ) )
=> ~ ( member_a @ E @ ( set_trm_a_c2 @ Z3 ) ) ) ) ) ) ) ) ) ) ).
% trm.set_cases
thf(fact_309_trm_Oset__intros_I1_J,axiom,
! [X31: a,X32: c > trm_a_c] : ( member_a @ X31 @ ( set_trm_a_c2 @ ( function_a_c @ X31 @ X32 ) ) ) ).
% trm.set_intros(1)
thf(fact_310_trm_Oset__intros_I1_J,axiom,
! [X31: sf,X32: sz > trm_sf_sz] : ( member_sf @ X31 @ ( set_trm_sf_sz2 @ ( function_sf_sz @ X31 @ X32 ) ) ) ).
% trm.set_intros(1)
thf(fact_311_trm_Osimps_I126_J,axiom,
! [X7: trm_a_c] :
( ( set_trm_a_c2 @ ( differential_a_c @ X7 ) )
= ( set_trm_a_c2 @ X7 ) ) ).
% trm.simps(126)
thf(fact_312_trm_Oset__intros_I7_J,axiom,
! [Yf: a,X7: trm_a_c] :
( ( member_a @ Yf @ ( set_trm_a_c2 @ X7 ) )
=> ( member_a @ Yf @ ( set_trm_a_c2 @ ( differential_a_c @ X7 ) ) ) ) ).
% trm.set_intros(7)
thf(fact_313_trm_Oset__intros_I2_J,axiom,
! [Y: trm_a_c,X32: c > trm_a_c,Ya: a,X31: a] :
( ( member_trm_a_c @ Y @ ( image_c_trm_a_c @ X32 @ top_top_set_c ) )
=> ( ( member_a @ Ya @ ( set_trm_a_c2 @ Y ) )
=> ( member_a @ Ya @ ( set_trm_a_c2 @ ( function_a_c @ X31 @ X32 ) ) ) ) ) ).
% trm.set_intros(2)
thf(fact_314_trm_Oset__intros_I2_J,axiom,
! [Y: trm_sf_sz,X32: sz > trm_sf_sz,Ya: sf,X31: sf] :
( ( member_trm_sf_sz @ Y @ ( image_sz_trm_sf_sz @ X32 @ top_top_set_sz ) )
=> ( ( member_sf @ Ya @ ( set_trm_sf_sz2 @ Y ) )
=> ( member_sf @ Ya @ ( set_trm_sf_sz2 @ ( function_sf_sz @ X31 @ X32 ) ) ) ) ) ).
% trm.set_intros(2)
thf(fact_315_ids_Osingleton_Ocong,axiom,
singleton_sz_sf = singleton_sz_sf ).
% ids.singleton.cong
thf(fact_316_ids_Of1_Ocong,axiom,
f1_sz_sf = f1_sz_sf ).
% ids.f1.cong
thf(fact_317_singleton_Opelims,axiom,
! [X: trm_sf_sz,Xa: sz,Y: trm_sf_sz] :
( ( ( singleton_sz_sf @ vid1 @ X @ Xa )
= Y )
=> ( ( accp_P529224471_sz_sz @ singleton_rel_sf_sz @ ( produc792909658_sz_sz @ X @ Xa ) )
=> ~ ( ( ( ( Xa = vid1 )
=> ( Y = X ) )
& ( ( Xa != vid1 )
=> ( Y
= ( const_sf_sz @ zero_zero_real ) ) ) )
=> ~ ( accp_P529224471_sz_sz @ singleton_rel_sf_sz @ ( produc792909658_sz_sz @ X @ Xa ) ) ) ) ) ).
% singleton.pelims
thf(fact_318_f1__def,axiom,
! [F2: sf,X: sz] :
( ( f1_sz_sf @ vid1 @ F2 @ X )
= ( function_sf_sz @ F2 @ ( singleton_sz_sf @ vid1 @ ( var_sz_sf @ X ) ) ) ) ).
% f1_def
thf(fact_319_proj__sing1,axiom,
! [Theta3: trm_sf_sz] :
( ( singleton_sz_sf @ vid1 @ Theta3 @ vid1 )
= Theta3 ) ).
% proj_sing1
thf(fact_320_expand__singleton,axiom,
! [T3: trm_sf_sz] :
( ( singleton_sz_sf @ vid1 @ T3 )
= ( ^ [I4: sz] : ( if_trm_sf_sz @ ( I4 = vid1 ) @ T3 @ ( const_sf_sz @ zero_zero_real ) ) ) ) ).
% expand_singleton
thf(fact_321_proj__sing2,axiom,
! [Y: sz,Theta3: trm_sf_sz] :
( ( vid1 != Y )
=> ( ( singleton_sz_sf @ vid1 @ Theta3 @ Y )
= ( const_sf_sz @ zero_zero_real ) ) ) ).
% proj_sing2
thf(fact_322_singleton_Oelims,axiom,
! [X: trm_sf_sz,Xa: sz,Y: trm_sf_sz] :
( ( ( singleton_sz_sf @ vid1 @ X @ Xa )
= Y )
=> ( ( ( Xa = vid1 )
=> ( Y = X ) )
& ( ( Xa != vid1 )
=> ( Y
= ( const_sf_sz @ zero_zero_real ) ) ) ) ) ).
% singleton.elims
thf(fact_323_singleton_Osimps,axiom,
! [I: sz,T3: trm_sf_sz] :
( ( ( I = vid1 )
=> ( ( singleton_sz_sf @ vid1 @ T3 @ I )
= T3 ) )
& ( ( I != vid1 )
=> ( ( singleton_sz_sf @ vid1 @ T3 @ I )
= ( const_sf_sz @ zero_zero_real ) ) ) ) ).
% singleton.simps
thf(fact_324_local_Oid__simps_I15_J,axiom,
vid3 != vid1 ).
% local.id_simps(15)
thf(fact_325_local_Oid__simps_I13_J,axiom,
vid2 != vid1 ).
% local.id_simps(13)
thf(fact_326_vne13,axiom,
vid1 != vid3 ).
% vne13
thf(fact_327_vne12,axiom,
vid1 != vid2 ).
% vne12
thf(fact_328_local_Oid__simps_I1_J,axiom,
vid1 != vid2 ).
% local.id_simps(1)
thf(fact_329_local_Oid__simps_I3_J,axiom,
vid1 != vid3 ).
% local.id_simps(3)
thf(fact_330_vne23,axiom,
vid2 != vid3 ).
% vne23
thf(fact_331_local_Oid__simps_I2_J,axiom,
vid2 != vid3 ).
% local.id_simps(2)
thf(fact_332_local_Oid__simps_I14_J,axiom,
vid3 != vid2 ).
% local.id_simps(14)
thf(fact_333_p1__def,axiom,
! [P2: sz,X: sz] :
( ( p1_sz_sf_sc @ vid1 @ P2 @ X )
= ( prop_sz_sf_sc @ P2 @ ( singleton_sz_sf @ vid1 @ ( var_sz_sf @ X ) ) ) ) ).
% p1_def
thf(fact_334_prod_Oswap__def,axiom,
( produc245128480t_unit
= ( ^ [P3: produc999471430t_unit] : ( produc789536734_sc_sz @ ( produc687319826t_unit @ P3 ) @ ( produc1216682964t_unit @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_335_prod_Oswap__def,axiom,
( produc704157711real_c
= ( ^ [P3: produc190496183real_c] : ( produc394644079real_c @ ( produc314122909real_c @ P3 ) @ ( produc2010422875real_c @ P3 ) ) ) ) ).
% prod.swap_def
thf(fact_336_formula_Oinject_I2_J,axiom,
! [X21: sz,X222: sz > trm_sf_sz,Y21: sz,Y22: sz > trm_sf_sz] :
( ( ( prop_sz_sf_sc @ X21 @ X222 )
= ( prop_sz_sf_sc @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% formula.inject(2)
thf(fact_337_pair__in__swap__image,axiom,
! [Y: produc866628903_sc_sz,X: denota610675952t_unit,A2: set_Pr1169339874_sc_sz] :
( ( member235934813t_unit @ ( produc502616768t_unit @ Y @ X ) @ ( image_904562809t_unit @ produc532048446_sc_sz @ A2 ) )
= ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_338_pair__in__swap__image,axiom,
! [Y: denota610675952t_unit,X: produc866628903_sc_sz,A2: set_Pr2041213564t_unit] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ Y @ X ) @ ( image_1426741113_sc_sz @ produc245128480t_unit @ A2 ) )
= ( member235934813t_unit @ ( produc502616768t_unit @ X @ Y ) @ A2 ) ) ).
% pair_in_swap_image
thf(fact_339_swap__simp,axiom,
! [X: produc866628903_sc_sz,Y: denota610675952t_unit] :
( ( produc245128480t_unit @ ( produc502616768t_unit @ X @ Y ) )
= ( produc789536734_sc_sz @ Y @ X ) ) ).
% swap_simp
thf(fact_340_swap__simp,axiom,
! [X: denota610675952t_unit,Y: produc866628903_sc_sz] :
( ( produc532048446_sc_sz @ ( produc789536734_sc_sz @ X @ Y ) )
= ( produc502616768t_unit @ Y @ X ) ) ).
% swap_simp
thf(fact_341_snd__swap,axiom,
! [X: produc190496183real_c] :
( ( produc314122909real_c @ ( produc704157711real_c @ X ) )
= ( produc2010422875real_c @ X ) ) ).
% snd_swap
thf(fact_342_fst__swap,axiom,
! [X: produc190496183real_c] :
( ( produc2010422875real_c @ ( produc704157711real_c @ X ) )
= ( produc314122909real_c @ X ) ) ).
% fst_swap
thf(fact_343_ids_Op1_Ocong,axiom,
p1_sz_sf_sc = p1_sz_sf_sc ).
% ids.p1.cong
thf(fact_344_ids_Op1__def,axiom,
! [Vid1: sz,Vid2: sz,Vid3: sz,Fid1: sf,Fid2: sf,Fid3: sf,Pid1: sc,Pid2: sc,Pid3: sc,Pid4: sc,P2: sz,X: sz] :
( ( ids_sz_sf_sc @ Vid1 @ Vid2 @ Vid3 @ Fid1 @ Fid2 @ Fid3 @ Pid1 @ Pid2 @ Pid3 @ Pid4 )
=> ( ( p1_sz_sf_sc @ Vid1 @ P2 @ X )
= ( prop_sz_sf_sc @ P2 @ ( singleton_sz_sf @ Vid1 @ ( var_sz_sf @ X ) ) ) ) ) ).
% ids.p1_def
thf(fact_345_formula_Oset__intros_I3_J,axiom,
! [Yb: trm_sf_sz,X222: sz > trm_sf_sz,Yc: sf,X21: sz] :
( ( member_trm_sf_sz @ Yb @ ( image_sz_trm_sf_sz @ X222 @ top_top_set_sz ) )
=> ( ( member_sf @ Yc @ ( set_trm_sf_sz2 @ Yb ) )
=> ( member_sf @ Yc @ ( set1_f989022726_sc_sz @ ( prop_sz_sf_sc @ X21 @ X222 ) ) ) ) ) ).
% formula.set_intros(3)
thf(fact_346_hpfree__ffree_Ointros_I10_J,axiom,
! [Args: sz > trm_sf_sz,P2: sz] :
( ! [Arg: trm_sf_sz] :
( ( member_trm_sf_sz @ Arg @ ( image_sz_trm_sf_sz @ Args @ top_top_set_sz ) )
=> ( dfree_sf_sz @ Arg ) )
=> ( ffree_sf_sc_sz @ ( prop_sz_sf_sc @ P2 @ Args ) ) ) ).
% hpfree_ffree.intros(10)
thf(fact_347_hpfree__ffree_Ointros_I15_J,axiom,
! [P: sc] : ( ffree_sf_sc_sz @ ( predic913887884_sf_sz @ P ) ) ).
% hpfree_ffree.intros(15)
thf(fact_348_map__prod__simp,axiom,
! [F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz,A: denota610675952t_unit,B: produc866628903_sc_sz] :
( ( produc403791768_sc_sz @ F2 @ G @ ( produc789536734_sc_sz @ A @ B ) )
= ( produc789536734_sc_sz @ ( F2 @ A ) @ ( G @ B ) ) ) ).
% map_prod_simp
thf(fact_349_fst__map__prod,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c,G: finite1398487019real_c > finite1398487019real_c,X: produc190496183real_c] :
( ( produc2010422875real_c @ ( produc368345510real_c @ F2 @ G @ X ) )
= ( F2 @ ( produc2010422875real_c @ X ) ) ) ).
% fst_map_prod
thf(fact_350_snd__map__prod,axiom,
! [F2: finite1398487019real_c > finite1398487019real_c,G: finite1398487019real_c > finite1398487019real_c,X: produc190496183real_c] :
( ( produc314122909real_c @ ( produc368345510real_c @ F2 @ G @ X ) )
= ( G @ ( produc314122909real_c @ X ) ) ) ).
% snd_map_prod
thf(fact_351_map__prod__imageI,axiom,
! [A: denota610675952t_unit,B: produc866628903_sc_sz,R4: set_Pr1169339874_sc_sz,F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz] :
( ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ A @ B ) @ R4 )
=> ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ ( F2 @ A ) @ ( G @ B ) ) @ ( image_1726193375_sc_sz @ ( produc403791768_sc_sz @ F2 @ G ) @ R4 ) ) ) ).
% map_prod_imageI
thf(fact_352_prod__fun__imageE,axiom,
! [C2: produc1821101996_sc_sz,F2: denota610675952t_unit > denota610675952t_unit,G: produc866628903_sc_sz > produc866628903_sc_sz,R4: set_Pr1169339874_sc_sz] :
( ( member1057565379_sc_sz @ C2 @ ( image_1726193375_sc_sz @ ( produc403791768_sc_sz @ F2 @ G ) @ R4 ) )
=> ~ ! [X3: denota610675952t_unit,Y3: produc866628903_sc_sz] :
( ( C2
= ( produc789536734_sc_sz @ ( F2 @ X3 ) @ ( G @ Y3 ) ) )
=> ~ ( member1057565379_sc_sz @ ( produc789536734_sc_sz @ X3 @ Y3 ) @ R4 ) ) ) ).
% prod_fun_imageE
% Helper facts (3)
thf(help_If_3_1_If_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J_T,axiom,
! [X: trm_sf_sz,Y: trm_sf_sz] :
( ( if_trm_sf_sz @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Syntax__Otrm_Itf__sf_Mtf__sz_J_T,axiom,
! [X: trm_sf_sz,Y: trm_sf_sz] :
( ( if_trm_sf_sz @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
dfree_a_c @ ( args @ i2 ) ).
%------------------------------------------------------------------------------