TPTP Problem File: NUM923_5.p
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%------------------------------------------------------------------------------
% File : NUM923_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_23 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 179 ( 43 unt; 49 typ; 0 def)
% Number of atoms : 253 ( 96 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 144 ( 21 ~; 2 |; 2 &)
% ( 24 <=>; 95 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 6 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 28 ( 15 >; 13 *; 0 +; 0 <<)
% Number of predicates : 30 ( 29 usr; 0 prp; 1-3 aty)
% Number of functors : 18 ( 18 usr; 6 con; 0-6 aty)
% Number of variables : 496 ( 447 !; 5 ?; 496 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:22:37
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
tff(ty_tc_prod,type,
product_prod: ( $tType * $tType ) > $tType ).
%----Explicit typings (45)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ominus,type,
cl_Groups_Ominus:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere223160158up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
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_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B1: $tType] : ( ( A * B1 ) > product_prod(A,B1) ) ).
tff(sy_c_Product__Type_Ocurry,type,
product_curry:
!>[A: $tType,B1: $tType,C2: $tType] : ( ( fun(product_prod(A,B1),C2) * A * B1 ) > C2 ) ).
tff(sy_c_Product__Type_Ointernal__split,type,
produc1605651328_split:
!>[A: $tType,B1: $tType,C2: $tType] : ( ( fun(A,fun(B1,C2)) * product_prod(A,B1) ) > C2 ) ).
tff(sy_c_Product__Type_Oprod_Oprod__rec,type,
product_prod_rec:
!>[A: $tType,B1: $tType,T: $tType] : ( ( fun(A,fun(B1,T)) * product_prod(A,B1) ) > T ) ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Ois__sum2sq,type,
twoSqu1567020053sum2sq: int > $o ).
tff(sy_c_TwoSquares__Mirabelle__poiayhyqls_Osum2sq,type,
twoSqu196287499sum2sq: product_prod(int,int) > int ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
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_a,type,
a: int ).
tff(sy_v_b,type,
b: int ).
tff(sy_v_p,type,
p: int ).
tff(sy_v_q,type,
q: int ).
%----Relevant facts (99)
tff(fact_0_xzgcda__linear__aux1,axiom,
! [N: int,D1: int,C: int,M: int,B: int,R: int,A1: int] : plus_plus(int,times_times(int,minus_minus(int,A1,times_times(int,R,B)),M),times_times(int,minus_minus(int,C,times_times(int,R,D1)),N)) = minus_minus(int,plus_plus(int,times_times(int,A1,M),times_times(int,C,N)),times_times(int,R,plus_plus(int,times_times(int,B,M),times_times(int,D1,N)))) ).
tff(fact_1_square__diff__square__factored,axiom,
! [A: $tType] :
( comm_ring(A)
=> ! [Y: A,X: A] : minus_minus(A,times_times(A,X,X),times_times(A,Y,Y)) = times_times(A,plus_plus(A,X,Y),minus_minus(A,X,Y)) ) ).
tff(fact_2_mult__diff__mult,axiom,
! [A: $tType] :
( ring(A)
=> ! [B: A,A1: A,Y: A,X: A] : minus_minus(A,times_times(A,X,Y),times_times(A,A1,B)) = plus_plus(A,times_times(A,X,minus_minus(A,Y,B)),times_times(A,minus_minus(A,X,A1),B)) ) ).
tff(fact_3_eq__add__iff2,axiom,
! [A: $tType] :
( ring(A)
=> ! [D: A,Ba: A,C1: A,E: A,Aa: A] :
( ( plus_plus(A,times_times(A,Aa,E),C1) = plus_plus(A,times_times(A,Ba,E),D) )
<=> ( C1 = plus_plus(A,times_times(A,minus_minus(A,Ba,Aa),E),D) ) ) ) ).
tff(fact_4_eq__add__iff1,axiom,
! [A: $tType] :
( ring(A)
=> ! [D: A,Ba: A,C1: A,E: A,Aa: A] :
( ( plus_plus(A,times_times(A,Aa,E),C1) = plus_plus(A,times_times(A,Ba,E),D) )
<=> ( plus_plus(A,times_times(A,minus_minus(A,Aa,Ba),E),C1) = D ) ) ) ).
tff(fact_5_split__paired__All,axiom,
! [A: $tType,B1: $tType,P1: fun(product_prod(A,B1),bool)] :
( ! [X11: product_prod(A,B1)] : pp(aa(product_prod(A,B1),bool,P1,X11))
<=> ! [A3: A,B3: B1] : pp(aa(product_prod(A,B1),bool,P1,product_Pair(A,B1,A3,B3))) ) ).
tff(fact_6_Pair__eq,axiom,
! [A: $tType,B1: $tType,B6: B1,A6: A,Ba: B1,Aa: A] :
( ( product_Pair(A,B1,Aa,Ba) = product_Pair(A,B1,A6,B6) )
<=> ( ( Aa = A6 )
& ( Ba = B6 ) ) ) ).
tff(fact_7_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Ba: A,Aa: A] :
( ( plus_plus(A,Aa,Ba) = plus_plus(A,Aa,C1) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_8_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,Aa: A,Ba: A] :
( ( plus_plus(A,Ba,Aa) = plus_plus(A,C1,Aa) )
<=> ( Ba = C1 ) ) ) ).
tff(fact_9_mult__left__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [B: A,A1: A] : times_times(A,A1,times_times(A,A1,B)) = times_times(A,A1,B) ) ).
tff(fact_10_is__sum2sq__def,axiom,
! [X2: int] :
( twoSqu1567020053sum2sq(X2)
<=> ? [A3: int,B3: int] : twoSqu196287499sum2sq(product_Pair(int,int,A3,B3)) = X2 ) ).
tff(fact_11_Int2_Oaux1,axiom,
! [C: int,B: int,A1: int] :
( ( minus_minus(int,A1,B) = C )
=> ( A1 = plus_plus(int,C,B) ) ) ).
tff(fact_12_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B: A,A1: A] : times_times(A,times_times(A,A1,B),C) = times_times(A,A1,times_times(A,B,C)) ) ).
tff(fact_13_mult__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [X: A] : times_times(A,X,X) = X ) ).
tff(fact_14_times_Oidem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [A1: A] : times_times(A,A1,A1) = A1 ) ).
tff(fact_15_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
=> ( B = C ) ) ) ).
tff(fact_16_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_17_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_18_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_19_Pair__inject,axiom,
! [A: $tType,B1: $tType,B5: B1,A5: A,B: B1,A1: A] :
( ( product_Pair(A,B1,A1,B) = product_Pair(A,B1,A5,B5) )
=> ~ ( ( A1 = A5 )
=> ( B != B5 ) ) ) ).
tff(fact_20_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C1: A,Ba: A,Aa: A] :
( ( minus_minus(A,Aa,Ba) = minus_minus(A,C1,D) )
=> ( ( Aa = Ba )
<=> ( C1 = D ) ) ) ) ).
tff(fact_21_minus__apply,axiom,
! [A: $tType,B1: $tType] :
( cl_Groups_Ominus(A)
=> ! [X2: B1,B4: fun(B1,A),A4: fun(B1,A)] : aa(B1,A,minus_minus(fun(B1,A),A4,B4),X2) = minus_minus(A,aa(B1,A,A4,X2),aa(B1,A,B4,X2)) ) ).
tff(fact_22_fun__diff__def,axiom,
! [B1: $tType,A: $tType] :
( cl_Groups_Ominus(B1)
=> ! [B4: fun(A,B1),A4: fun(A,B1),X3: A] : aa(A,B1,minus_minus(fun(A,B1),A4,B4),X3) = minus_minus(B1,aa(A,B1,A4,X3),aa(A,B1,B4,X3)) ) ).
tff(fact_23_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [C: A,B: A,E3: A,A1: A] : plus_plus(A,times_times(A,A1,E3),plus_plus(A,times_times(A,B,E3),C)) = plus_plus(A,times_times(A,plus_plus(A,A1,B),E3),C) ) ).
tff(fact_24_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [C: A,B: A,A1: A] : times_times(A,plus_plus(A,A1,B),C) = plus_plus(A,times_times(A,A1,C),times_times(A,B,C)) ) ).
tff(fact_25_add__diff__add,axiom,
! [A: $tType] :
( ab_group_add(A)
=> ! [D1: A,B: A,C: A,A1: A] : minus_minus(A,plus_plus(A,A1,C),plus_plus(A,B,D1)) = plus_plus(A,minus_minus(A,A1,B),minus_minus(A,C,D1)) ) ).
tff(fact_26_add__diff__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] : minus_minus(A,plus_plus(A,A1,B),B) = A1 ) ).
tff(fact_27_diff__add__cancel,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B: A,A1: A] : plus_plus(A,minus_minus(A,A1,B),B) = A1 ) ).
tff(fact_28_split__paired__Ex,axiom,
! [A: $tType,B1: $tType,P1: fun(product_prod(A,B1),bool)] :
( ? [X11: product_prod(A,B1)] : pp(aa(product_prod(A,B1),bool,P1,X11))
<=> ? [A3: A,B3: B1] : pp(aa(product_prod(A,B1),bool,P1,product_Pair(A,B1,A3,B3))) ) ).
tff(fact_29_prod_Orecs,axiom,
! [B1: $tType,A: $tType,C2: $tType,Ba: C2,Aa: B1,F11: fun(B1,fun(C2,A))] : product_prod_rec(B1,C2,A,F11,product_Pair(B1,C2,Aa,Ba)) = aa(C2,A,aa(B1,fun(C2,A),F11,Aa),Ba) ).
tff(fact_30_int__distrib_I3_J,axiom,
! [W1: int,Z2: int,Z11: int] : times_times(int,minus_minus(int,Z11,Z2),W1) = minus_minus(int,times_times(int,Z11,W1),times_times(int,Z2,W1)) ).
tff(fact_31_int__distrib_I4_J,axiom,
! [Z2: int,Z11: int,W1: int] : times_times(int,W1,minus_minus(int,Z11,Z2)) = minus_minus(int,times_times(int,W1,Z11),times_times(int,W1,Z2)) ).
tff(fact_32_int__distrib_I1_J,axiom,
! [W1: int,Z2: int,Z11: int] : times_times(int,plus_plus(int,Z11,Z2),W1) = plus_plus(int,times_times(int,Z11,W1),times_times(int,Z2,W1)) ).
tff(fact_33_int__distrib_I2_J,axiom,
! [Z2: int,Z11: int,W1: int] : times_times(int,W1,plus_plus(int,Z11,Z2)) = plus_plus(int,times_times(int,W1,Z11),times_times(int,W1,Z2)) ).
tff(fact_34_crossproduct__eq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [Z: A,X2: A,Y2: A,W: A] :
( ( plus_plus(A,times_times(A,W,Y2),times_times(A,X2,Z)) = plus_plus(A,times_times(A,W,Z),times_times(A,X2,Y2)) )
<=> ( ( W = X2 )
| ( Y2 = Z ) ) ) ) ).
tff(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,M: A,A1: A] : plus_plus(A,times_times(A,A1,M),times_times(A,B,M)) = times_times(A,plus_plus(A,A1,B),M) ) ).
tff(fact_36_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : times_times(A,plus_plus(A,A1,B),C) = plus_plus(A,times_times(A,A1,C),times_times(A,B,C)) ) ).
tff(fact_37_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,A1: A] : times_times(A,A1,B) = times_times(A,B,A1) ) ).
tff(fact_38_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,Lx,Ry)) ) ).
tff(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),Ry) ) ).
tff(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,Lx,times_times(A,Ly,Rx)) ) ).
tff(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,times_times(A,Lx,Rx),Ly) ) ).
tff(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Lx,times_times(A,Ly,times_times(A,Rx,Ry))) ) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,times_times(A,Lx,Ly),Ry)) ) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),times_times(A,Ly,Ry)) ) ).
tff(fact_45_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_46_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D1)) = plus_plus(A,C,plus_plus(A,A1,D1)) ) ).
tff(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,A1: A] : plus_plus(A,A1,plus_plus(A,C,D1)) = plus_plus(A,plus_plus(A,A1,C),D1) ) ).
tff(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_49_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,plus_plus(A,A1,C),B) ) ).
tff(fact_50_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D1: A,C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),plus_plus(A,C,D1)) = plus_plus(A,plus_plus(A,A1,C),plus_plus(A,B,D1)) ) ).
tff(fact_51_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z1: A,Y: A,X: A] : times_times(A,X,plus_plus(A,Y,Z1)) = plus_plus(A,times_times(A,X,Y),times_times(A,X,Z1)) ) ).
tff(fact_52_crossproduct__noteq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [D: A,C1: A,Ba: A,Aa: A] :
( ( ( Aa != Ba )
& ( C1 != D ) )
<=> ( plus_plus(A,times_times(A,Aa,C1),times_times(A,Ba,D)) != plus_plus(A,times_times(A,Aa,D),times_times(A,Ba,C1)) ) ) ) ).
tff(fact_53_prod__induct6,axiom,
! [F1: $tType,E1: $tType,D2: $tType,C2: $tType,B1: $tType,A: $tType,X2: product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))))),P1: fun(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))))),bool)] :
( ! [A2: A,B2: B1,C3: C2,D3: D2,E2: E1,F2: F1] : pp(aa(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))))),bool,P1,product_Pair(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1)))),A2,product_Pair(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))),B2,product_Pair(C2,product_prod(D2,product_prod(E1,F1)),C3,product_Pair(D2,product_prod(E1,F1),D3,product_Pair(E1,F1,E2,F2)))))))
=> pp(aa(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))))),bool,P1,X2)) ) ).
tff(fact_54_prod__cases6,axiom,
! [A: $tType,B1: $tType,C2: $tType,D2: $tType,E1: $tType,F1: $tType,Y: product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1)))))] :
~ ! [A2: A,B2: B1,C3: C2,D3: D2,E2: E1,F2: F1] : Y != product_Pair(A,product_prod(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1)))),A2,product_Pair(B1,product_prod(C2,product_prod(D2,product_prod(E1,F1))),B2,product_Pair(C2,product_prod(D2,product_prod(E1,F1)),C3,product_Pair(D2,product_prod(E1,F1),D3,product_Pair(E1,F1,E2,F2))))) ).
tff(fact_55_prod__induct5,axiom,
! [E1: $tType,D2: $tType,C2: $tType,B1: $tType,A: $tType,X2: product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,E1)))),P1: fun(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,E1)))),bool)] :
( ! [A2: A,B2: B1,C3: C2,D3: D2,E2: E1] : pp(aa(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,E1)))),bool,P1,product_Pair(A,product_prod(B1,product_prod(C2,product_prod(D2,E1))),A2,product_Pair(B1,product_prod(C2,product_prod(D2,E1)),B2,product_Pair(C2,product_prod(D2,E1),C3,product_Pair(D2,E1,D3,E2))))))
=> pp(aa(product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,E1)))),bool,P1,X2)) ) ).
tff(fact_56_prod__cases5,axiom,
! [A: $tType,B1: $tType,C2: $tType,D2: $tType,E1: $tType,Y: product_prod(A,product_prod(B1,product_prod(C2,product_prod(D2,E1))))] :
~ ! [A2: A,B2: B1,C3: C2,D3: D2,E2: E1] : Y != product_Pair(A,product_prod(B1,product_prod(C2,product_prod(D2,E1))),A2,product_Pair(B1,product_prod(C2,product_prod(D2,E1)),B2,product_Pair(C2,product_prod(D2,E1),C3,product_Pair(D2,E1,D3,E2)))) ).
tff(fact_57_prod__induct4,axiom,
! [D2: $tType,C2: $tType,B1: $tType,A: $tType,X2: product_prod(A,product_prod(B1,product_prod(C2,D2))),P1: fun(product_prod(A,product_prod(B1,product_prod(C2,D2))),bool)] :
( ! [A2: A,B2: B1,C3: C2,D3: D2] : pp(aa(product_prod(A,product_prod(B1,product_prod(C2,D2))),bool,P1,product_Pair(A,product_prod(B1,product_prod(C2,D2)),A2,product_Pair(B1,product_prod(C2,D2),B2,product_Pair(C2,D2,C3,D3)))))
=> pp(aa(product_prod(A,product_prod(B1,product_prod(C2,D2))),bool,P1,X2)) ) ).
tff(fact_58_prod__cases4,axiom,
! [A: $tType,B1: $tType,C2: $tType,D2: $tType,Y: product_prod(A,product_prod(B1,product_prod(C2,D2)))] :
~ ! [A2: A,B2: B1,C3: C2,D3: D2] : Y != product_Pair(A,product_prod(B1,product_prod(C2,D2)),A2,product_Pair(B1,product_prod(C2,D2),B2,product_Pair(C2,D2,C3,D3))) ).
tff(fact_59_prod__cases3,axiom,
! [A: $tType,B1: $tType,C2: $tType,Y: product_prod(A,product_prod(B1,C2))] :
~ ! [A2: A,B2: B1,C3: C2] : Y != product_Pair(A,product_prod(B1,C2),A2,product_Pair(B1,C2,B2,C3)) ).
tff(fact_60_prod__induct3,axiom,
! [C2: $tType,B1: $tType,A: $tType,X2: product_prod(A,product_prod(B1,C2)),P1: fun(product_prod(A,product_prod(B1,C2)),bool)] :
( ! [A2: A,B2: B1,C3: C2] : pp(aa(product_prod(A,product_prod(B1,C2)),bool,P1,product_Pair(A,product_prod(B1,C2),A2,product_Pair(B1,C2,B2,C3))))
=> pp(aa(product_prod(A,product_prod(B1,C2)),bool,P1,X2)) ) ).
tff(fact_61_PairE,axiom,
! [A: $tType,B1: $tType,P: product_prod(A,B1)] :
~ ! [X1: A,Y1: B1] : P != product_Pair(A,B1,X1,Y1) ).
tff(fact_62_prod_Oexhaust,axiom,
! [A: $tType,B1: $tType,Y: product_prod(A,B1)] :
~ ! [A2: A,B2: B1] : Y != product_Pair(A,B1,A2,B2) ).
tff(fact_63_internal__split__conv,axiom,
! [B1: $tType,A: $tType,C2: $tType,Ba: C2,Aa: B1,C1: fun(B1,fun(C2,A))] : produc1605651328_split(B1,C2,A,C1,product_Pair(B1,C2,Aa,Ba)) = aa(C2,A,aa(B1,fun(C2,A),C1,Aa),Ba) ).
tff(fact_64_curry__conv,axiom,
! [A: $tType,B1: $tType,C2: $tType,Ba: C2,Aa: B1,F: fun(product_prod(B1,C2),A)] : product_curry(B1,C2,A,F,Aa,Ba) = aa(product_prod(B1,C2),A,F,product_Pair(B1,C2,Aa,Ba)) ).
tff(fact_65_curryI,axiom,
! [A: $tType,B1: $tType,Ba: B1,Aa: A,F: fun(product_prod(A,B1),bool)] :
( pp(aa(product_prod(A,B1),bool,F,product_Pair(A,B1,Aa,Ba)))
=> pp(product_curry(A,B1,bool,F,Aa,Ba)) ) ).
tff(fact_66_square__diff__one__factored,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [X: A] : minus_minus(A,times_times(A,X,X),one_one(A)) = times_times(A,plus_plus(A,X,one_one(A)),minus_minus(A,X,one_one(A))) ) ).
tff(fact_67_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_68_mult__1__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_69_mult__1,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_70_mult__1__left,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I11_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,one_one(A),A1) = A1 ) ).
tff(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,A1,one_one(A)) = A1 ) ).
tff(fact_73_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X2: A] :
( ( one_one(A) = X2 )
<=> ( X2 = one_one(A) ) ) ) ).
tff(fact_74_ext,axiom,
! [B1: $tType,A: $tType,G: fun(A,B1),F: fun(A,B1)] :
( ! [X1: A] : aa(A,B1,F,X1) = aa(A,B1,G,X1)
=> ( F = G ) ) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I4_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A] : plus_plus(A,M,M) = times_times(A,plus_plus(A,one_one(A),one_one(A)),M) ) ).
tff(fact_76_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A,M: A] : plus_plus(A,M,times_times(A,A1,M)) = times_times(A,plus_plus(A,A1,one_one(A)),M) ) ).
tff(fact_77_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [M: A,A1: A] : plus_plus(A,times_times(A,A1,M),M) = times_times(A,plus_plus(A,A1,one_one(A)),M) ) ).
tff(fact_78_curryD,axiom,
! [A: $tType,B1: $tType,Ba: B1,Aa: A,F: fun(product_prod(A,B1),bool)] :
( pp(product_curry(A,B1,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B1),bool,F,product_Pair(A,B1,Aa,Ba))) ) ).
tff(fact_79_curryE,axiom,
! [A: $tType,B1: $tType,Ba: B1,Aa: A,F: fun(product_prod(A,B1),bool)] :
( pp(product_curry(A,B1,bool,F,Aa,Ba))
=> pp(aa(product_prod(A,B1),bool,F,product_Pair(A,B1,Aa,Ba))) ) ).
tff(fact_80_le__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [D: A,Ba: A,C1: A,E: A,Aa: A] :
( ord_less_eq(A,plus_plus(A,times_times(A,Aa,E),C1),plus_plus(A,times_times(A,Ba,E),D))
<=> ord_less_eq(A,C1,plus_plus(A,times_times(A,minus_minus(A,Ba,Aa),E),D)) ) ) ).
tff(fact_81_le__add__iff1,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [D: A,Ba: A,C1: A,E: A,Aa: A] :
( ord_less_eq(A,plus_plus(A,times_times(A,Aa,E),C1),plus_plus(A,times_times(A,Ba,E),D))
<=> ord_less_eq(A,plus_plus(A,times_times(A,minus_minus(A,Aa,Ba),E),C1),D) ) ) ).
tff(fact_82_less__add__iff2,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [D: A,Ba: A,C1: A,E: A,Aa: A] :
( ord_less(A,plus_plus(A,times_times(A,Aa,E),C1),plus_plus(A,times_times(A,Ba,E),D))
<=> ord_less(A,C1,plus_plus(A,times_times(A,minus_minus(A,Ba,Aa),E),D)) ) ) ).
tff(fact_83_zle__add1__eq__le,axiom,
! [Z: int,W: int] :
( ord_less(int,W,plus_plus(int,Z,one_one(int)))
<=> ord_less_eq(int,W,Z) ) ).
tff(fact_84_zle__diff1__eq,axiom,
! [Z: int,W: int] :
( ord_less_eq(int,W,minus_minus(int,Z,one_one(int)))
<=> ord_less(int,W,Z) ) ).
tff(fact_85_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C1: A,Aa: A] :
( ord_less_eq(A,plus_plus(A,Aa,C1),plus_plus(A,Ba,C1))
<=> ord_less_eq(A,Aa,Ba) ) ) ).
tff(fact_86_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,Aa: A,C1: A] :
( ord_less_eq(A,plus_plus(A,C1,Aa),plus_plus(A,C1,Ba))
<=> ord_less_eq(A,Aa,Ba) ) ) ).
tff(fact_87_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,C1: A,Aa: A] :
( ord_less(A,plus_plus(A,Aa,C1),plus_plus(A,Ba,C1))
<=> ord_less(A,Aa,Ba) ) ) ).
tff(fact_88_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [Ba: A,Aa: A,C1: A] :
( ord_less(A,plus_plus(A,C1,Aa),plus_plus(A,C1,Ba))
<=> ord_less(A,Aa,Ba) ) ) ).
tff(fact_89_zless__imp__add1__zle,axiom,
! [Z1: int,W1: int] :
( ord_less(int,W1,Z1)
=> ord_less_eq(int,plus_plus(int,W1,one_one(int)),Z1) ) ).
tff(fact_90_add1__zle__eq,axiom,
! [Z: int,W: int] :
( ord_less_eq(int,plus_plus(int,W,one_one(int)),Z)
<=> ord_less(int,W,Z) ) ).
tff(fact_91_zless__add1__eq,axiom,
! [Z: int,W: int] :
( ord_less(int,W,plus_plus(int,Z,one_one(int)))
<=> ( ord_less(int,W,Z)
| ( W = Z ) ) ) ).
tff(fact_92_add__le__less__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D1: A,C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less(A,C,D1)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,D1)) ) ) ) ).
tff(fact_93_add__less__le__mono,axiom,
! [A: $tType] :
( ordere223160158up_add(A)
=> ! [D1: A,C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less_eq(A,C,D1)
=> ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,D1)) ) ) ) ).
tff(fact_94_order__le__neq__implies__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ( X != Y )
=> ord_less(A,X,Y) ) ) ) ).
tff(fact_95_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_96_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,Ba: A,Aa: A] :
( ( minus_minus(A,Aa,Ba) = minus_minus(A,C1,D) )
=> ( ord_less_eq(A,Aa,Ba)
<=> ord_less_eq(A,C1,D) ) ) ) ).
tff(fact_97_add__right__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,C)) ) ) ).
tff(fact_98_add__left__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B)) ) ) ).
%----Arities (28)
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Groups_Ominus,axiom,
! [T_1: $tType,T_2: $tType] :
( cl_Groups_Ominus(T_2)
=> cl_Groups_Ominus(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(int) ).
tff(arity_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere223160158up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(int) ).
tff(arity_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring(int) ).
tff(arity_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add(int) ).
tff(arity_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring(int) ).
tff(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(arity_Int_Oint___Groups_Ogroup__add,axiom,
group_add(int) ).
tff(arity_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Groups_Ominus,axiom,
cl_Groups_Ominus(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Groups_Oone,axiom,
one(int) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Groups_Ominus,axiom,
cl_Groups_Ominus(bool) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
times_times(int,twoSqu196287499sum2sq(product_Pair(int,int,a,b)),twoSqu196287499sum2sq(product_Pair(int,int,p,q))) = twoSqu196287499sum2sq(product_Pair(int,int,plus_plus(int,times_times(int,a,p),times_times(int,b,q)),minus_minus(int,times_times(int,a,q),times_times(int,b,p)))) ).
%------------------------------------------------------------------------------