TPTP Problem File: LCL752_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : LCL752_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 41
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_41 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 151 ( 44 unt; 37 typ; 0 def)
% Number of atoms : 234 ( 95 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 168 ( 48 ~; 12 |; 7 &)
% ( 25 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 5 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 24 ( 14 >; 10 *; 0 +; 0 <<)
% Number of predicates : 8 ( 7 usr; 0 prp; 1-3 aty)
% Number of functors : 27 ( 27 usr; 12 con; 0-5 aty)
% Number of variables : 284 ( 255 !; 4 ?; 284 :)
% ( 25 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:10
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Lambda_OdB,type,
dB: $tType ).
tff(ty_tc_List_Olist,type,
list: $tType > $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (32)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBI,type,
combi:
!>[A: $tType] : fun(A,A) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_InductTermi_OIT,type,
it: fun(dB,bool) ).
tff(sy_c_Lambda_OdB_OAbs,type,
abs: dB > dB ).
tff(sy_c_Lambda_OdB_OApp,type,
app: fun(dB,fun(dB,dB)) ).
tff(sy_c_Lambda_OdB_OVar,type,
var: fun(nat,dB) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: fun(dB,fun(nat,dB)) ).
tff(sy_c_Lambda_Osubst,type,
subst: fun(dB,fun(dB,fun(nat,dB))) ).
tff(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( fun(B,fun(A,B)) * B * list(A) ) > B ) ).
tff(sy_c_List_Olistsp,type,
listsp:
!>[A: $tType] : ( ( fun(A,bool) * list(A) ) > $o ) ).
tff(sy_c_List_Omap,type,
map:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * list(A) ) > list(B) ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fAll,type,
fAll:
!>[A: $tType] : fun(fun(A,bool),bool) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fconj,type,
fconj: fun(bool,fun(bool,bool)) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_ia,type,
ia: nat ).
tff(sy_v_ja,type,
ja: nat ).
tff(sy_v_n,type,
n: nat ).
tff(sy_v_rs,type,
rs: list(dB) ).
%----Relevant facts (98)
tff(fact_0_IT_OVar,axiom,
! [N1: nat,Rs: list(dB)] :
( listsp(dB,it,Rs)
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,var,N1),Rs))) ) ).
tff(fact_1_subst__map,axiom,
! [Ia: nat,U1: dB,Ts: list(dB),T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,foldl(dB,dB,app,T,Ts)),U1),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T),U1),Ia),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,U1),Ia),Ts)) ).
tff(fact_2_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),N1: nat,Rs: list(dB),M1: nat] :
( ( foldl(dB,dB,app,aa(nat,dB,var,M1),Rs) = foldl(dB,dB,app,aa(nat,dB,var,N1),Ss) )
<=> ( ( M1 = N1 )
& ( Rs = Ss ) ) ) ).
tff(fact_3_zero__less__diff,axiom,
! [M1: nat,N1: nat] :
( ord_less(nat,zero_zero(nat),minus_minus(nat,N1,M1))
<=> ord_less(nat,M1,N1) ) ).
tff(fact_4_zero__less__Suc,axiom,
! [N2: nat] : ord_less(nat,zero_zero(nat),suc(N2)) ).
tff(fact_5_less__Suc0,axiom,
! [N1: nat] :
( ord_less(nat,N1,suc(zero_zero(nat)))
<=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_6_subst__App,axiom,
! [K: nat,S1: dB,U: dB,T1: dB] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(dB,dB,aa(dB,fun(dB,dB),app,T1),U)),S1),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,T1),S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,U),S1),K)) ).
tff(fact_7_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),Ra: dB] :
( ( foldl(dB,dB,app,Ra,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( Ra = S ) ) ).
tff(fact_8_diff__Suc__Suc,axiom,
! [N2: nat,M2: nat] : minus_minus(nat,suc(M2),suc(N2)) = minus_minus(nat,M2,N2) ).
tff(fact_9_Suc__diff__diff,axiom,
! [K: nat,N2: nat,M2: nat] : minus_minus(nat,minus_minus(nat,suc(M2),N2),suc(K)) = minus_minus(nat,minus_minus(nat,M2,N2),K) ).
tff(fact_10_diff__0__eq__0,axiom,
! [N2: nat] : minus_minus(nat,zero_zero(nat),N2) = zero_zero(nat) ).
tff(fact_11_diff__self__eq__0,axiom,
! [M2: nat] : minus_minus(nat,M2,M2) = zero_zero(nat) ).
tff(fact_12_nat_Oinject,axiom,
! [Nat5: nat,Nat4: nat] :
( ( suc(Nat4) = suc(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_13_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB22: dB,DB12: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) = aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) )
<=> ( ( DB12 = DB13 )
& ( DB22 = DB23 ) ) ) ).
tff(fact_14_dB_Osimps_I1_J,axiom,
! [Nat5: nat,Nat4: nat] :
( ( aa(nat,dB,var,Nat4) = aa(nat,dB,var,Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_15_less__zeroE,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_16_less__nat__zero__code,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_17_neq0__conv,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_18_Suc__mono,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ord_less(nat,suc(M2),suc(N2)) ) ).
tff(fact_19_Suc__less__eq,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,suc(M1),suc(N1))
<=> ord_less(nat,M1,N1) ) ).
tff(fact_20_not__less__eq,axiom,
! [N1: nat,M1: nat] :
( ~ ord_less(nat,M1,N1)
<=> ord_less(nat,N1,suc(M1)) ) ).
tff(fact_21_lessI,axiom,
! [N2: nat] : ord_less(nat,N2,suc(N2)) ).
tff(fact_22_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( suc(X) = suc(Y) )
=> ( X = Y ) ) ).
tff(fact_23_Suc__n__not__n,axiom,
! [N2: nat] : suc(N2) != N2 ).
tff(fact_24_n__not__Suc__n,axiom,
! [N2: nat] : N2 != suc(N2) ).
tff(fact_25_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),N1: nat,M1: nat] :
( ( ord_less(nat,M1,N1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ( M1 = N1 )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> ( ( ord_less(nat,N1,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,N1),M1)) ) ) ) ).
tff(fact_26_less__not__refl3,axiom,
! [T1: nat,S1: nat] :
( ord_less(nat,S1,T1)
=> ( S1 != T1 ) ) ).
tff(fact_27_less__not__refl2,axiom,
! [M2: nat,N2: nat] :
( ord_less(nat,N2,M2)
=> ( M2 != N2 ) ) ).
tff(fact_28_less__irrefl__nat,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_29_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_30_nat__neq__iff,axiom,
! [N1: nat,M1: nat] :
( ( M1 != N1 )
<=> ( ord_less(nat,M1,N1)
| ord_less(nat,N1,M1) ) ) ).
tff(fact_31_less__not__refl,axiom,
! [N2: nat] : ~ ord_less(nat,N2,N2) ).
tff(fact_32_diff__commute,axiom,
! [K: nat,J1: nat,I: nat] : minus_minus(nat,minus_minus(nat,I,J1),K) = minus_minus(nat,minus_minus(nat,I,K),J1) ).
tff(fact_33_Suc__neq__Zero,axiom,
! [M2: nat] : suc(M2) != zero_zero(nat) ).
tff(fact_34_Zero__neq__Suc,axiom,
! [M2: nat] : zero_zero(nat) != suc(M2) ).
tff(fact_35_nat_Osimps_I3_J,axiom,
! [Nat3: nat] : suc(Nat3) != zero_zero(nat) ).
tff(fact_36_Suc__not__Zero,axiom,
! [M2: nat] : suc(M2) != zero_zero(nat) ).
tff(fact_37_nat_Osimps_I2_J,axiom,
! [Nat2: nat] : zero_zero(nat) != suc(Nat2) ).
tff(fact_38_Zero__not__Suc,axiom,
! [M2: nat] : zero_zero(nat) != suc(M2) ).
tff(fact_39_gr0I,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N2) ) ).
tff(fact_40_gr__implies__not0,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ( N2 != zero_zero(nat) ) ) ).
tff(fact_41_not__less0,axiom,
! [N2: nat] : ~ ord_less(nat,N2,zero_zero(nat)) ).
tff(fact_42_Suc__less__SucD,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,suc(M2),suc(N2))
=> ord_less(nat,M2,N2) ) ).
tff(fact_43_Suc__lessD,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,suc(M2),N2)
=> ord_less(nat,M2,N2) ) ).
tff(fact_44_less__SucE,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,suc(N2))
=> ( ~ ord_less(nat,M2,N2)
=> ( M2 = N2 ) ) ) ).
tff(fact_45_less__trans__Suc,axiom,
! [K: nat,J1: nat,I: nat] :
( ord_less(nat,I,J1)
=> ( ord_less(nat,J1,K)
=> ord_less(nat,suc(I),K) ) ) ).
tff(fact_46_Suc__lessI,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ( ( suc(M2) != N2 )
=> ord_less(nat,suc(M2),N2) ) ) ).
tff(fact_47_less__SucI,axiom,
! [N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ord_less(nat,M2,suc(N2)) ) ).
tff(fact_48_less__antisym,axiom,
! [M2: nat,N2: nat] :
( ~ ord_less(nat,N2,M2)
=> ( ord_less(nat,N2,suc(M2))
=> ( M2 = N2 ) ) ) ).
tff(fact_49_not__less__less__Suc__eq,axiom,
! [M1: nat,N1: nat] :
( ~ ord_less(nat,N1,M1)
=> ( ord_less(nat,N1,suc(M1))
<=> ( N1 = M1 ) ) ) ).
tff(fact_50_less__Suc__eq,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,suc(N1))
<=> ( ord_less(nat,M1,N1)
| ( M1 = N1 ) ) ) ).
tff(fact_51_diffs0__imp__equal,axiom,
! [N2: nat,M2: nat] :
( ( minus_minus(nat,M2,N2) = zero_zero(nat) )
=> ( ( minus_minus(nat,N2,M2) = zero_zero(nat) )
=> ( M2 = N2 ) ) ) ).
tff(fact_52_minus__nat_Odiff__0,axiom,
! [M2: nat] : minus_minus(nat,M2,zero_zero(nat)) = M2 ).
tff(fact_53_diff__less__mono2,axiom,
! [L: nat,N2: nat,M2: nat] :
( ord_less(nat,M2,N2)
=> ( ord_less(nat,M2,L)
=> ord_less(nat,minus_minus(nat,L,N2),minus_minus(nat,L,M2)) ) ) ).
tff(fact_54_less__imp__diff__less,axiom,
! [N2: nat,K: nat,J1: nat] :
( ord_less(nat,J1,K)
=> ord_less(nat,minus_minus(nat,J1,N2),K) ) ).
tff(fact_55_dB_Osimps_I5_J,axiom,
! [Nat: nat,DB21: dB,DB11: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) != aa(nat,dB,var,Nat) ).
tff(fact_56_dB_Osimps_I4_J,axiom,
! [DB21: dB,DB11: dB,Nat: nat] : aa(nat,dB,var,Nat) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) ).
tff(fact_57_subst__eq,axiom,
! [U: dB,K: nat] : aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,var,K)),U),K) = U ).
tff(fact_58_less__Suc__eq__0__disj,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,suc(N1))
<=> ( ( M1 = zero_zero(nat) )
| ? [J2: nat] :
( ( M1 = suc(J2) )
& ord_less(nat,J2,N1) ) ) ) ).
tff(fact_59_gr0__conv__Suc,axiom,
! [N1: nat] :
( ord_less(nat,zero_zero(nat),N1)
<=> ? [M3: nat] : N1 = suc(M3) ) ).
tff(fact_60_diff__less,axiom,
! [M2: nat,N2: nat] :
( ord_less(nat,zero_zero(nat),N2)
=> ( ord_less(nat,zero_zero(nat),M2)
=> ord_less(nat,minus_minus(nat,M2,N2),M2) ) ) ).
tff(fact_61_diff__less__Suc,axiom,
! [N2: nat,M2: nat] : ord_less(nat,minus_minus(nat,M2,N2),suc(M2)) ).
tff(fact_62_subst__lt,axiom,
! [U: dB,I: nat,J1: nat] :
( ord_less(nat,J1,I)
=> ( aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,aa(nat,dB,var,J1)),U),I) = aa(nat,dB,var,J1) ) ) ).
tff(fact_63_diff__Suc__less,axiom,
! [I: nat,N2: nat] :
( ord_less(nat,zero_zero(nat),N2)
=> ord_less(nat,minus_minus(nat,N2,suc(I)),N2) ) ).
tff(fact_64_Suc__pred,axiom,
! [N2: nat] :
( ord_less(nat,zero_zero(nat),N2)
=> ( suc(minus_minus(nat,N2,suc(zero_zero(nat)))) = N2 ) ) ).
tff(fact_65_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : minus_minus(A,A2,A2) = zero_zero(A) ) ).
tff(fact_66_listsp__conj__eq,axiom,
! [A: $tType,B2: fun(A,bool),A3: fun(A,bool),X3: list(A)] :
( listsp(A,combs(A,bool,bool,aa(fun(A,bool),fun(A,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(A,bool),fun(A,fun(bool,bool))),combb(bool,fun(bool,bool),A),fconj),A3),B2),X3)
<=> ( listsp(A,A3,X3)
& listsp(A,B2,X3) ) ) ).
tff(fact_67_map__ident,axiom,
! [A: $tType,X3: list(A)] : map(A,A,combi(A),X3) = X3 ).
tff(fact_68_gr0__implies__Suc,axiom,
! [N2: nat] :
( ord_less(nat,zero_zero(nat),N2)
=> ? [M: nat] : N2 = suc(M) ) ).
tff(fact_69_lift__Suc__mono__less,axiom,
! [A: $tType] :
( order(A)
=> ! [N3: nat,N1: nat,F: fun(nat,A)] :
( ! [N: nat] : ord_less(A,aa(nat,A,F,N),aa(nat,A,F,suc(N)))
=> ( ord_less(nat,N1,N3)
=> ord_less(A,aa(nat,A,F,N1),aa(nat,A,F,N3)) ) ) ) ).
tff(fact_70_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [M1: nat,N1: nat,F: fun(nat,A)] :
( ! [N: nat] : ord_less(A,aa(nat,A,F,N),aa(nat,A,F,suc(N)))
=> ( ord_less(A,aa(nat,A,F,N1),aa(nat,A,F,M1))
<=> ord_less(nat,N1,M1) ) ) ) ).
tff(fact_71_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,B1)
<=> ord_less(A,minus_minus(A,A1,B1),zero_zero(A)) ) ) ).
tff(fact_72_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,Xs: list(C),F: fun(C,B),A1: A,G: fun(A,fun(B,A))] : foldl(A,B,G,A1,map(C,B,F,Xs)) = foldl(A,C,combc(A,fun(C,B),fun(C,A),aa(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A))),aa(fun(fun(B,A),fun(fun(C,B),fun(C,A))),fun(fun(A,fun(B,A)),fun(A,fun(fun(C,B),fun(C,A)))),combb(fun(B,A),fun(fun(C,B),fun(C,A)),A),combb(B,A,C)),G),F),A1,Xs) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X2: A] : aa(A,B,F,X2) = aa(A,B,G,X2)
=> ( F = G ) ) ).
tff(fact_74_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_75_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C1: A,B1: A,A1: A] :
( ( minus_minus(A,A1,B1) = minus_minus(A,C1,D) )
=> ( ( A1 = B1 )
<=> ( C1 = D ) ) ) ) ).
tff(fact_76_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A2: A] : minus_minus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_77_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A1: A] :
( ( A1 = B1 )
<=> ( minus_minus(A,A1,B1) = zero_zero(A) ) ) ) ).
tff(fact_78_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B1: A,A1: A] :
( ( minus_minus(A,A1,B1) = zero_zero(A) )
<=> ( A1 = B1 ) ) ) ).
tff(fact_79_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,B1: A,A1: A] :
( ( minus_minus(A,A1,B1) = minus_minus(A,C1,D) )
=> ( ord_less(A,A1,B1)
<=> ord_less(A,C1,D) ) ) ) ).
tff(fact_80_zero__induct__lemma,axiom,
! [Ia: nat,K1: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,K1))
=> ( ! [N: nat] :
( pp(aa(nat,bool,P1,suc(N)))
=> pp(aa(nat,bool,P1,N)) )
=> pp(aa(nat,bool,P1,minus_minus(nat,K1,Ia))) ) ) ).
tff(fact_81_lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,I,K)
=> ( ( K != suc(I) )
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != suc(J) ) ) ) ) ).
tff(fact_82_Suc__lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,suc(I),K)
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != suc(J) ) ) ) ).
tff(fact_83_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat1: nat] : Y != suc(Nat1) ) ).
tff(fact_84_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero(nat) )
=> ? [M: nat] : N2 = suc(M) ) ).
tff(fact_85_nat__induct,axiom,
! [N1: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,zero_zero(nat)))
=> ( ! [N: nat] :
( pp(aa(nat,bool,P1,N))
=> pp(aa(nat,bool,P1,suc(N))) )
=> pp(aa(nat,bool,P1,N1)) ) ) ).
tff(fact_86_zero__induct,axiom,
! [K1: nat,P1: fun(nat,bool)] :
( pp(aa(nat,bool,P1,K1))
=> ( ! [N: nat] :
( pp(aa(nat,bool,P1,suc(N)))
=> pp(aa(nat,bool,P1,N)) )
=> pp(aa(nat,bool,P1,zero_zero(nat))) ) ) ).
tff(fact_87_dB_Osize_I1_J,axiom,
! [Nat: nat] : dB_size(aa(nat,dB,var,Nat)) = zero_zero(nat) ).
tff(fact_88_Beta,axiom,
! [Ss: list(dB),S: dB,Ra: dB] :
( pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),aa(dB,fun(dB,fun(nat,dB)),subst,Ra),S),zero_zero(nat)),Ss)))
=> ( pp(aa(dB,bool,it,S))
=> pp(aa(dB,bool,it,foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(Ra)),S),Ss))) ) ) ).
tff(fact_89_lifts__IT,axiom,
! [Ts: list(dB)] :
( listsp(dB,it,Ts)
=> listsp(dB,it,map(dB,dB,combc(dB,nat,dB,lift,zero_zero(nat)),Ts)) ) ).
tff(fact_90_dB_Osimps_I3_J,axiom,
! [DB4: dB,DB3: dB] :
( ( abs(DB3) = abs(DB4) )
<=> ( DB3 = DB4 ) ) ).
tff(fact_91_lift_Osimps_I2_J,axiom,
! [K: nat,T1: dB,S1: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1)),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,aa(nat,dB,aa(dB,fun(nat,dB),lift,S1),K)),aa(nat,dB,aa(dB,fun(nat,dB),lift,T1),K)) ).
tff(fact_92_Lambda,axiom,
! [R1: dB] :
( pp(aa(dB,bool,it,R1))
=> pp(aa(dB,bool,it,abs(R1))) ) ).
tff(fact_93_lift__IT,axiom,
! [I: nat,T1: dB] :
( pp(aa(dB,bool,it,T1))
=> pp(aa(dB,bool,it,aa(nat,dB,aa(dB,fun(nat,dB),lift,T1),I))) ) ).
tff(fact_94_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs: list(dB),Ra: dB] :
( ( foldl(dB,dB,app,abs(Ra),Rs) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( Ra = S )
& ( Rs = Ss ) ) ) ).
tff(fact_95_lift__map,axiom,
! [Ia: nat,Ts: list(dB),T: dB] : aa(nat,dB,aa(dB,fun(nat,dB),lift,foldl(dB,dB,app,T,Ts)),Ia) = foldl(dB,dB,app,aa(nat,dB,aa(dB,fun(nat,dB),lift,T),Ia),map(dB,dB,combc(dB,nat,dB,lift,Ia),Ts)) ).
tff(fact_96_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB: dB] : abs(DB) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_97_dB_Osimps_I8_J,axiom,
! [DB: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB) ).
%----Arities (4)
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
%----Helper facts (10)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_fAll_1_1_U,axiom,
! [A: $tType,X: A,P: fun(A,bool)] :
( ~ pp(aa(fun(A,bool),bool,fAll(A),P))
| pp(aa(A,bool,P,X)) ) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBI_1_1_U,axiom,
! [A: $tType,P: A] : aa(A,A,combi(A),P) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fconj_1_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(P)
| ~ pp(Q)
| pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q)) ) ).
tff(help_fconj_2_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(P) ) ).
tff(help_fconj_3_1_U,axiom,
! [Q: bool,P: bool] :
( ~ pp(aa(bool,bool,aa(bool,fun(bool,bool),fconj,P),Q))
| pp(Q) ) ).
%----Conjectures (2)
tff(conj_0,hypothesis,
listsp(dB,combs(dB,bool,bool,aa(fun(dB,bool),fun(dB,fun(bool,bool)),aa(fun(bool,fun(bool,bool)),fun(fun(dB,bool),fun(dB,fun(bool,bool))),combb(bool,fun(bool,bool),dB),fconj),it),aa(fun(dB,fun(nat,bool)),fun(dB,bool),aa(fun(fun(nat,bool),bool),fun(fun(dB,fun(nat,bool)),fun(dB,bool)),combb(fun(nat,bool),bool,dB),fAll(nat)),aa(fun(dB,fun(nat,fun(nat,bool))),fun(dB,fun(nat,bool)),aa(fun(fun(nat,fun(nat,bool)),fun(nat,bool)),fun(fun(dB,fun(nat,fun(nat,bool))),fun(dB,fun(nat,bool))),combb(fun(nat,fun(nat,bool)),fun(nat,bool),dB),aa(fun(fun(nat,bool),bool),fun(fun(nat,fun(nat,bool)),fun(nat,bool)),combb(fun(nat,bool),bool,nat),fAll(nat))),aa(fun(dB,fun(nat,fun(nat,dB))),fun(dB,fun(nat,fun(nat,bool))),aa(fun(fun(nat,fun(nat,dB)),fun(nat,fun(nat,bool))),fun(fun(dB,fun(nat,fun(nat,dB))),fun(dB,fun(nat,fun(nat,bool)))),combb(fun(nat,fun(nat,dB)),fun(nat,fun(nat,bool)),dB),aa(fun(fun(nat,dB),fun(nat,bool)),fun(fun(nat,fun(nat,dB)),fun(nat,fun(nat,bool))),combb(fun(nat,dB),fun(nat,bool),nat),aa(fun(dB,bool),fun(fun(nat,dB),fun(nat,bool)),combb(dB,bool,nat),it))),combc(dB,fun(nat,dB),fun(nat,fun(nat,dB)),aa(fun(dB,fun(dB,fun(nat,dB))),fun(dB,fun(fun(nat,dB),fun(nat,fun(nat,dB)))),aa(fun(fun(dB,fun(nat,dB)),fun(fun(nat,dB),fun(nat,fun(nat,dB)))),fun(fun(dB,fun(dB,fun(nat,dB))),fun(dB,fun(fun(nat,dB),fun(nat,fun(nat,dB))))),combb(fun(dB,fun(nat,dB)),fun(fun(nat,dB),fun(nat,fun(nat,dB))),dB),combb(dB,fun(nat,dB),nat)),subst),var))))),rs) ).
tff(conj_1,conjecture,
( ( ( n != ja )
| pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,var,ia),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,aa(nat,dB,var,ia)),ja),rs)))) )
& ( ( n = ja )
| ( ( ~ ord_less(nat,ja,n)
| pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,var,minus_minus(nat,n,suc(zero_zero(nat)))),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,aa(nat,dB,var,ia)),ja),rs)))) )
& ( ~ ord_less(nat,n,suc(ja))
| pp(aa(dB,bool,it,foldl(dB,dB,app,aa(nat,dB,var,n),map(dB,dB,combc(dB,nat,dB,combc(dB,dB,fun(nat,dB),subst,aa(nat,dB,var,ia)),ja),rs)))) ) ) ) ) ).
%------------------------------------------------------------------------------