TPTP Problem File: LCL748_5.p
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%------------------------------------------------------------------------------
% File : LCL748_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Logic Calculi
% Problem : Strong normalization of typed lambda calculus line 31
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : sn_31 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 152 ( 64 unt; 39 typ; 0 def)
% Number of atoms : 189 ( 127 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 101 ( 25 ~; 2 |; 8 &)
% ( 19 <=>; 47 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 37 ( 18 >; 19 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-2 aty)
% Number of functors : 22 ( 22 usr; 5 con; 0-5 aty)
% Number of variables : 289 ( 269 !; 0 ?; 289 :)
% ( 20 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:06
%------------------------------------------------------------------------------
%----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 (34)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[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: dB > $o ).
tff(sy_c_Lambda_Obeta,type,
beta: ( dB * dB ) > $o ).
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: nat > dB ).
tff(sy_c_Lambda_OdB_OdB__case,type,
dB_case:
!>[T2: $tType] : ( ( fun(nat,T2) * fun(dB,fun(dB,T2)) * fun(dB,T2) * dB ) > T2 ) ).
tff(sy_c_Lambda_OdB_OdB__rec,type,
dB_rec:
!>[T2: $tType] : ( ( fun(nat,T2) * fun(dB,fun(dB,fun(T2,fun(T2,T2)))) * fun(dB,fun(T2,T2)) * dB ) > T2 ) ).
tff(sy_c_Lambda_OdB_OdB__size,type,
dB_size: dB > nat ).
tff(sy_c_Lambda_Olift,type,
lift: ( dB * nat ) > dB ).
tff(sy_c_Lambda_Oliftn,type,
liftn: ( nat * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubst,type,
subst: ( dB * dB * nat ) > dB ).
tff(sy_c_Lambda_Osubstn,type,
substn: ( dB * dB * 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_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_ia,type,
ia: nat ).
tff(sy_v_r,type,
r: dB ).
%----Relevant facts (98)
tff(fact_0_Lambda,axiom,
! [R1: dB] :
( it(R1)
=> it(abs(R1)) ) ).
tff(fact_1_dB_Osimps_I3_J,axiom,
! [DB5: dB,DB4: dB] :
( ( abs(DB4) = abs(DB5) )
<=> ( DB4 = DB5 ) ) ).
tff(fact_2_dB_Osimps_I12_J,axiom,
! [A: $tType,DB4: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,abs(DB4)) = aa(dB,A,F3,DB4) ).
tff(fact_3_dB_Orecs_I3_J,axiom,
! [A: $tType,DB4: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,abs(DB4)) = aa(A,A,aa(dB,fun(A,A),F3,DB4),dB_rec(A,F1,F2,F3,DB4)) ).
tff(fact_4_lift_Osimps_I3_J,axiom,
! [K: nat,S1: dB] : lift(abs(S1),K) = abs(lift(S1,plus_plus(nat,K,one_one(nat)))) ).
tff(fact_5_abs,axiom,
! [T: dB,S1: dB] :
( beta(S1,T)
=> beta(abs(S1),abs(T)) ) ).
tff(fact_6_subst__lift,axiom,
! [S1: dB,K: nat,T: dB] : subst(lift(T,K),S1,K) = T ).
tff(fact_7_lift_Osimps_I2_J,axiom,
! [K: nat,T: dB,S1: dB] : lift(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,lift(S1,K)),lift(T,K)) ).
tff(fact_8_lift__preserves__beta,axiom,
! [I1: nat,S1: dB,R1: dB] :
( beta(R1,S1)
=> beta(lift(R1,I1),lift(S1,I1)) ) ).
tff(fact_9_dB_Osimps_I7_J,axiom,
! [Nat4: nat,DB3: dB] : abs(DB3) != var(Nat4) ).
tff(fact_10_dB_Osimps_I6_J,axiom,
! [DB3: dB,Nat4: nat] : var(Nat4) != abs(DB3) ).
tff(fact_11_beta__cases_I2_J,axiom,
! [S1: dB,R1: dB] :
( beta(abs(R1),S1)
=> ~ ! [T1: dB] :
( ( S1 = abs(T1) )
=> ~ beta(R1,T1) ) ) ).
tff(fact_12_dB_Osimps_I2_J,axiom,
! [DB23: dB,DB13: dB,DB21: dB,DB11: dB] :
( ( aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21) = aa(dB,dB,aa(dB,fun(dB,dB),app,DB13),DB23) )
<=> ( ( DB11 = DB13 )
& ( DB21 = DB23 ) ) ) ).
tff(fact_13_dB_Osimps_I1_J,axiom,
! [Nat3: nat,Nat2: nat] :
( ( var(Nat2) = var(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_14_appL,axiom,
! [U: dB,T: dB,S1: dB] :
( beta(S1,T)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),U),aa(dB,dB,aa(dB,fun(dB,dB),app,T),U)) ) ).
tff(fact_15_appR,axiom,
! [U: dB,T: dB,S1: dB] :
( beta(S1,T)
=> beta(aa(dB,dB,aa(dB,fun(dB,dB),app,U),S1),aa(dB,dB,aa(dB,fun(dB,dB),app,U),T)) ) ).
tff(fact_16_subst__App,axiom,
! [K: nat,S1: dB,U: dB,T: dB] : subst(aa(dB,dB,aa(dB,fun(dB,dB),app,T),U),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,subst(T,S1,K)),subst(U,S1,K)) ).
tff(fact_17_beta__cases_I1_J,axiom,
! [T: dB,I1: nat] : ~ beta(var(I1),T) ).
tff(fact_18_dB_Osimps_I4_J,axiom,
! [DB22: dB,DB12: dB,Nat4: nat] : var(Nat4) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) ).
tff(fact_19_dB_Osimps_I5_J,axiom,
! [Nat4: nat,DB22: dB,DB12: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB12),DB22) != var(Nat4) ).
tff(fact_20_subst__eq,axiom,
! [U: dB,K: nat] : subst(var(K),U,K) = U ).
tff(fact_21_dB_Orecs_I1_J,axiom,
! [A: $tType,Nat2: nat,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,var(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_22_dB_Osimps_I10_J,axiom,
! [A: $tType,Nat2: nat,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,var(Nat2)) = aa(nat,A,F1,Nat2) ).
tff(fact_23_dB_Osimps_I11_J,axiom,
! [A: $tType,DB21: dB,DB11: dB,F3: fun(dB,A),F2: fun(dB,fun(dB,A)),F1: fun(nat,A)] : dB_case(A,F1,F2,F3,aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21)) = aa(dB,A,aa(dB,fun(dB,A),F2,DB11),DB21) ).
tff(fact_24_dB_Orecs_I2_J,axiom,
! [A: $tType,DB21: dB,DB11: dB,F3: fun(dB,fun(A,A)),F2: fun(dB,fun(dB,fun(A,fun(A,A)))),F1: fun(nat,A)] : dB_rec(A,F1,F2,F3,aa(dB,dB,aa(dB,fun(dB,dB),app,DB11),DB21)) = aa(A,A,aa(A,fun(A,A),aa(dB,fun(A,fun(A,A)),aa(dB,fun(dB,fun(A,fun(A,A))),F2,DB11),DB21),dB_rec(A,F1,F2,F3,DB11)),dB_rec(A,F1,F2,F3,DB21)) ).
tff(fact_25_subst__preserves__beta,axiom,
! [I1: nat,T: dB,S1: dB,R1: dB] :
( beta(R1,S1)
=> beta(subst(R1,T,I1),subst(S1,T,I1)) ) ).
tff(fact_26_dB_Osimps_I8_J,axiom,
! [DB3: dB,DB2: dB,DB1: dB] : aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) != abs(DB3) ).
tff(fact_27_dB_Osimps_I9_J,axiom,
! [DB2: dB,DB1: dB,DB3: dB] : abs(DB3) != aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2) ).
tff(fact_28_substn_Osimps_I3_J,axiom,
! [K: nat,S1: dB,T: dB] : substn(abs(T),S1,K) = abs(substn(T,S1,plus_plus(nat,K,one_one(nat)))) ).
tff(fact_29_nat__add__left__cancel,axiom,
! [N: nat,M: nat,K1: nat] :
( ( plus_plus(nat,K1,M) = plus_plus(nat,K1,N) )
<=> ( M = N ) ) ).
tff(fact_30_nat__add__right__cancel,axiom,
! [N: nat,K1: nat,M: nat] :
( ( plus_plus(nat,M,K1) = plus_plus(nat,N,K1) )
<=> ( M = N ) ) ).
tff(fact_31_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,C1) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_32_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A2: A,B2: A] :
( ( plus_plus(A,B2,A2) = plus_plus(A,C1,A2) )
<=> ( B2 = C1 ) ) ) ).
tff(fact_33_liftn_Osimps_I3_J,axiom,
! [K: nat,S1: dB,N1: nat] : liftn(N1,abs(S1),K) = abs(liftn(N1,S1,plus_plus(nat,K,one_one(nat)))) ).
tff(fact_34_subst__Abs,axiom,
! [K: nat,S1: dB,T: dB] : subst(abs(T),S1,K) = abs(subst(T,lift(S1,zero_zero(nat)),plus_plus(nat,K,one_one(nat)))) ).
tff(fact_35_beta,axiom,
! [T: dB,S1: dB] : beta(aa(dB,dB,aa(dB,fun(dB,dB),app,abs(S1)),T),subst(S1,T,zero_zero(nat))) ).
tff(fact_36_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = plus_plus(A,A2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_37_add__is__0,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = zero_zero(nat) )
<=> ( ( M = zero_zero(nat) )
& ( N = zero_zero(nat) ) ) ) ).
tff(fact_38_liftn_Osimps_I2_J,axiom,
! [K: nat,T: dB,S1: dB,N1: nat] : liftn(N1,aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T),K) = aa(dB,dB,aa(dB,fun(dB,dB),app,liftn(N1,S1,K)),liftn(N1,T,K)) ).
tff(fact_39_substn_Osimps_I2_J,axiom,
! [K: nat,S1: dB,U: dB,T: dB] : substn(aa(dB,dB,aa(dB,fun(dB,dB),app,T),U),S1,K) = aa(dB,dB,aa(dB,fun(dB,dB),app,substn(T,S1,K)),substn(U,S1,K)) ).
tff(fact_40_substn__subst__n,axiom,
! [N1: nat,S1: dB,T: dB] : substn(T,S1,N1) = subst(T,liftn(N1,S1,zero_zero(nat)),N1) ).
tff(fact_41_liftn__0,axiom,
! [K: nat,T: dB] : liftn(zero_zero(nat),T,K) = T ).
tff(fact_42_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_43_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_44_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_45_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_46_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_47_add__eq__self__zero,axiom,
! [N1: nat,M1: nat] :
( ( plus_plus(nat,M1,N1) = M1 )
=> ( N1 = zero_zero(nat) ) ) ).
tff(fact_48_Nat_Oadd__0__right,axiom,
! [M1: nat] : plus_plus(nat,M1,zero_zero(nat)) = M1 ).
tff(fact_49_plus__nat_Oadd__0,axiom,
! [N1: nat] : plus_plus(nat,zero_zero(nat),N1) = N1 ).
tff(fact_50_substn__subst__0,axiom,
! [S1: dB,T: dB] : substn(T,S1,zero_zero(nat)) = subst(T,S1,zero_zero(nat)) ).
tff(fact_51_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B1: A] :
( ( plus_plus(A,B1,A1) = plus_plus(A,C,A1) )
=> ( B1 = C ) ) ) ).
tff(fact_52_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( plus_plus(A,A1,B1) = plus_plus(A,A1,C) )
=> ( B1 = C ) ) ) ).
tff(fact_53_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] :
( ( plus_plus(A,A1,B1) = plus_plus(A,A1,C) )
=> ( B1 = C ) ) ) ).
tff(fact_54_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B1: A,A1: A] : plus_plus(A,plus_plus(A,A1,B1),C) = plus_plus(A,A1,plus_plus(A,B1,C)) ) ).
tff(fact_55_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X2: A] :
( ( one_one(A) = X2 )
<=> ( X2 = one_one(A) ) ) ) ).
tff(fact_56_nat__add__assoc,axiom,
! [K: nat,N1: nat,M1: nat] : plus_plus(nat,plus_plus(nat,M1,N1),K) = plus_plus(nat,M1,plus_plus(nat,N1,K)) ).
tff(fact_57_nat__add__left__commute,axiom,
! [Z: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z)) = plus_plus(nat,Y,plus_plus(nat,X,Z)) ).
tff(fact_58_nat__add__commute,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,N1) = plus_plus(nat,N1,M1) ).
tff(fact_59_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( plus_plus(A,A2,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_60_beta__cases_I3_J,axiom,
! [U: dB,T: dB,S1: dB] :
( beta(aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T),U)
=> ( ! [S2: dB] :
( ( U = subst(S2,T,zero_zero(nat)) )
=> ( S1 != abs(S2) ) )
=> ( ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,T1),T) )
=> ~ beta(S1,T1) )
=> ~ ! [T1: dB] :
( ( U = aa(dB,dB,aa(dB,fun(dB,dB),app,S1),T1) )
=> ~ beta(T,T1) ) ) ) ) ).
tff(fact_61_dB_Osize_I1_J,axiom,
! [Nat4: nat] : dB_size(var(Nat4)) = zero_zero(nat) ).
tff(fact_62_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_63_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : plus_plus(A,zero_zero(A),A1) = A1 ) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : plus_plus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_66_add__0__iff,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [A2: A,B2: A] :
( ( B2 = plus_plus(A,B2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,B1: A,A1: A] : plus_plus(A,plus_plus(A,A1,B1),plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A1,C),plus_plus(A,B1,D)) ) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B1: A,A1: A] : plus_plus(A,plus_plus(A,A1,B1),C) = plus_plus(A,plus_plus(A,A1,C),B1) ) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B1: A,A1: A] : plus_plus(A,plus_plus(A,A1,B1),C) = plus_plus(A,A1,plus_plus(A,B1,C)) ) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A1,C),D) ) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D)) = plus_plus(A,C,plus_plus(A,A1,D)) ) ).
tff(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,A1: A] : plus_plus(A,A1,C) = plus_plus(A,C,A1) ) ).
tff(fact_73_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_74_Beta,axiom,
! [Ss: list(dB),S: dB,R: dB] :
( it(foldl(dB,dB,app,subst(R,S,zero_zero(nat)),Ss))
=> ( it(S)
=> it(foldl(dB,dB,app,aa(dB,dB,aa(dB,fun(dB,dB),app,abs(R)),S),Ss)) ) ) ).
tff(fact_75_dB_Osize_I3_J,axiom,
! [DB: dB] : dB_size(abs(DB)) = plus_plus(nat,dB_size(DB),suc(zero_zero(nat))) ).
tff(fact_76_dB_Osize_I2_J,axiom,
! [DB2: dB,DB1: dB] : dB_size(aa(dB,dB,aa(dB,fun(dB,dB),app,DB1),DB2)) = plus_plus(nat,plus_plus(nat,dB_size(DB1),dB_size(DB2)),suc(zero_zero(nat))) ).
tff(fact_77_nat_Oinject,axiom,
! [Nat3: nat,Nat2: nat] :
( ( suc(Nat2) = suc(Nat3) )
<=> ( Nat2 = Nat3 ) ) ).
tff(fact_78_add__Suc,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,suc(M1),N1) = suc(plus_plus(nat,M1,N1)) ).
tff(fact_79_add__Suc__right,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,M1,suc(N1)) = suc(plus_plus(nat,M1,N1)) ).
tff(fact_80_One__nat__def,axiom,
one_one(nat) = suc(zero_zero(nat)) ).
tff(fact_81_liftn__lift,axiom,
! [K: nat,T: dB,N1: nat] : liftn(suc(N1),T,K) = lift(liftn(N1,T,K),K) ).
tff(fact_82_one__is__add,axiom,
! [N: nat,M: nat] :
( ( suc(zero_zero(nat)) = plus_plus(nat,M,N) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_83_add__is__1,axiom,
! [N: nat,M: nat] :
( ( plus_plus(nat,M,N) = suc(zero_zero(nat)) )
<=> ( ( ( M = suc(zero_zero(nat)) )
& ( N = zero_zero(nat) ) )
| ( ( M = zero_zero(nat) )
& ( N = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_84_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( suc(X) = suc(Y) )
=> ( X = Y ) ) ).
tff(fact_85_Suc__n__not__n,axiom,
! [N1: nat] : suc(N1) != N1 ).
tff(fact_86_n__not__Suc__n,axiom,
! [N1: nat] : N1 != suc(N1) ).
tff(fact_87_add__Suc__shift,axiom,
! [N1: nat,M1: nat] : plus_plus(nat,suc(M1),N1) = plus_plus(nat,M1,suc(N1)) ).
tff(fact_88_Suc__neq__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_89_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_90_nat_Osimps_I3_J,axiom,
! [Nat1: nat] : suc(Nat1) != zero_zero(nat) ).
tff(fact_91_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_92_nat_Osimps_I2_J,axiom,
! [Nat: nat] : zero_zero(nat) != suc(Nat) ).
tff(fact_93_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_94_Var__apps__eq__Var__apps__conv,axiom,
! [Ss: list(dB),N: nat,Rs: list(dB),M: nat] :
( ( foldl(dB,dB,app,var(M),Rs) = foldl(dB,dB,app,var(N),Ss) )
<=> ( ( M = N )
& ( Rs = Ss ) ) ) ).
tff(fact_95_Abs__apps__eq__Abs__apps__conv,axiom,
! [Ss: list(dB),S: dB,Rs: list(dB),R: dB] :
( ( foldl(dB,dB,app,abs(R),Rs) = foldl(dB,dB,app,abs(S),Ss) )
<=> ( ( R = S )
& ( Rs = Ss ) ) ) ).
tff(fact_96_apps__eq__tail__conv,axiom,
! [S: dB,Ts: list(dB),R: dB] :
( ( foldl(dB,dB,app,R,Ts) = foldl(dB,dB,app,S,Ts) )
<=> ( R = S ) ) ).
tff(fact_97_Var__apps__neq__Abs__apps,axiom,
! [Ss: list(dB),R: dB,Ts: list(dB),N: nat] : foldl(dB,dB,app,var(N),Ts) != foldl(dB,dB,app,abs(R),Ss) ).
%----Arities (10)
tff(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (3)
tff(conj_0,hypothesis,
it(r) ).
tff(conj_1,hypothesis,
! [I: nat] : it(lift(r,I)) ).
tff(conj_2,conjecture,
it(lift(abs(r),ia)) ).
%------------------------------------------------------------------------------