TPTP Problem File: NUM923^1.p
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%------------------------------------------------------------------------------
% File : NUM923^1 : TPTP v8.2.0. Released v5.3.0.
% Domain : Number Theory
% Problem : Sum of two squares line 23, 100 axioms selected
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla11] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla11]
% Names : s2s_100_thf_l23 [Bla11]
% Status : Theorem
% Rating : 1.00 v5.3.0
% Syntax : Number of formulae : 91 ( 46 unt; 16 typ; 0 def)
% Number of atoms : 113 ( 72 equ; 0 cnn)
% Maximal formula atoms : 3 ( 1 avg)
% Number of connectives : 665 ( 10 ~; 2 |; 2 &; 611 @)
% ( 17 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 26 ( 26 >; 0 *; 0 +; 0 <<)
% Number of symbols : 15 ( 14 usr; 4 con; 0-3 aty)
% Number of variables : 235 ( 0 ^; 231 !; 4 ?; 235 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-08-09 19:02:52
%------------------------------------------------------------------------------
%----Should-be-implicit typings (2)
thf(ty_ty_tc__Int__Oint,type,
int: $tType ).
thf(ty_ty_tc__prod_Itc__Int__Oint_Mtc__Int__Oint_J,type,
product_prod_int_int: $tType ).
%----Explicit typings (14)
thf(sy_c_All,type,
all: ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Ex,type,
ex: ( product_prod_int_int > $o ) > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_000tc__Int__Oint,type,
minus_minus_int: int > int > int ).
thf(sy_c_Groups_Oplus__class_Oplus_000tc__Int__Oint,type,
plus_plus_int: int > int > int ).
thf(sy_c_Groups_Otimes__class_Otimes_000tc__Int__Oint,type,
times_times_int: int > int > int ).
thf(sy_c_Orderings_Oord__class_Oless__eq_000tc__Int__Oint,type,
ord_less_eq_int: int > int > $o ).
thf(sy_c_Product__Type_OPair_000tc__Int__Oint_000tc__Int__Oint,type,
product_Pair_int_int: int > int > product_prod_int_int ).
thf(sy_c_Product__Type_Ocurry_000tc__Int__Oint_000tc__Int__Oint_000_Eo,type,
produc176579150_int_o: ( product_prod_int_int > $o ) > int > int > $o ).
thf(sy_c_TwoSquares__Mirabelle__fqdbopfjxb_Ois__sum2sq,type,
twoSqu1013291560sum2sq: int > $o ).
thf(sy_c_TwoSquares__Mirabelle__fqdbopfjxb_Osum2sq,type,
twoSqu1535104286sum2sq: product_prod_int_int > int ).
thf(sy_v_a,type,
a: int ).
thf(sy_v_b,type,
b: int ).
thf(sy_v_p,type,
p: int ).
thf(sy_v_q,type,
q: int ).
%----Relevant facts (74)
thf(fact_0_xzgcda__linear__aux1,axiom,
! [A_42: int,R: int,B_39: int,M_1: int,C_29: int,D_12: int,N: int] :
( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_42 @ ( times_times_int @ R @ B_39 ) ) @ M_1 ) @ ( times_times_int @ ( minus_minus_int @ C_29 @ ( times_times_int @ R @ D_12 ) ) @ N ) )
= ( minus_minus_int @ ( plus_plus_int @ ( times_times_int @ A_42 @ M_1 ) @ ( times_times_int @ C_29 @ N ) ) @ ( times_times_int @ R @ ( plus_plus_int @ ( times_times_int @ B_39 @ M_1 ) @ ( times_times_int @ D_12 @ N ) ) ) ) ) ).
thf(fact_1_mult__diff__mult,axiom,
! [X_6: int,Y_6: int,A_45: int,B_42: int] :
( ( minus_minus_int @ ( times_times_int @ X_6 @ Y_6 ) @ ( times_times_int @ A_45 @ B_42 ) )
= ( plus_plus_int @ ( times_times_int @ X_6 @ ( minus_minus_int @ Y_6 @ B_42 ) ) @ ( times_times_int @ ( minus_minus_int @ X_6 @ A_45 ) @ B_42 ) ) ) ).
thf(fact_2_eq__add__iff2,axiom,
! [A_44: int,E_4: int,C_31: int,B_41: int,D_11: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_44 @ E_4 ) @ C_31 )
= ( plus_plus_int @ ( times_times_int @ B_41 @ E_4 ) @ D_11 ) )
<=> ( C_31
= ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_41 @ A_44 ) @ E_4 ) @ D_11 ) ) ) ).
thf(fact_3_eq__add__iff1,axiom,
! [A_43: int,E_3: int,C_30: int,B_40: int,D_10: int] :
( ( ( plus_plus_int @ ( times_times_int @ A_43 @ E_3 ) @ C_30 )
= ( plus_plus_int @ ( times_times_int @ B_40 @ E_3 ) @ D_10 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_43 @ B_40 ) @ E_3 ) @ C_30 )
= D_10 ) ) ).
thf(fact_4_is__sum2sq__def,axiom,
! [X_5: int] :
( ( twoSqu1013291560sum2sq @ X_5 )
<=> ? [A_14: int,B_14: int] :
( ( twoSqu1535104286sum2sq @ ( product_Pair_int_int @ A_14 @ B_14 ) )
= X_5 ) ) ).
thf(fact_5_Int2_Oaux1,axiom,
! [A_42: int,B_39: int,C_29: int] :
( ( ( minus_minus_int @ A_42 @ B_39 )
= C_29 )
=> ( A_42
= ( plus_plus_int @ C_29 @ B_39 ) ) ) ).
thf(fact_6_zdiff__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( minus_minus_int @ Z1 @ Z2 ) )
= ( minus_minus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_7_zdiff__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( minus_minus_int @ Z1 @ Z2 ) @ W )
= ( minus_minus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_8_zadd__zmult__distrib2,axiom,
! [W: int,Z1: int,Z2: int] :
( ( times_times_int @ W @ ( plus_plus_int @ Z1 @ Z2 ) )
= ( plus_plus_int @ ( times_times_int @ W @ Z1 ) @ ( times_times_int @ W @ Z2 ) ) ) ).
thf(fact_9_zadd__zmult__distrib,axiom,
! [Z1: int,Z2: int,W: int] :
( ( times_times_int @ ( plus_plus_int @ Z1 @ Z2 ) @ W )
= ( plus_plus_int @ ( times_times_int @ Z1 @ W ) @ ( times_times_int @ Z2 @ W ) ) ) ).
thf(fact_10_diff__add__cancel,axiom,
! [A_41: int,B_38: int] :
( ( plus_plus_int @ ( minus_minus_int @ A_41 @ B_38 ) @ B_38 )
= A_41 ) ).
thf(fact_11_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A_40: int,B_37: int,C_28: int] :
( ( times_times_int @ ( times_times_int @ A_40 @ B_37 ) @ C_28 )
= ( times_times_int @ A_40 @ ( times_times_int @ B_37 @ C_28 ) ) ) ).
thf(fact_12_add__right__imp__eq,axiom,
! [B_36: int,A_39: int,C_27: int] :
( ( ( plus_plus_int @ B_36 @ A_39 )
= ( plus_plus_int @ C_27 @ A_39 ) )
=> ( B_36 = C_27 ) ) ).
thf(fact_13_add__imp__eq,axiom,
! [A_38: int,B_35: int,C_26: int] :
( ( ( plus_plus_int @ A_38 @ B_35 )
= ( plus_plus_int @ A_38 @ C_26 ) )
=> ( B_35 = C_26 ) ) ).
thf(fact_14_add__left__imp__eq,axiom,
! [A_37: int,B_34: int,C_25: int] :
( ( ( plus_plus_int @ A_37 @ B_34 )
= ( plus_plus_int @ A_37 @ C_25 ) )
=> ( B_34 = C_25 ) ) ).
thf(fact_15_add__right__cancel,axiom,
! [B_33: int,A_36: int,C_24: int] :
( ( ( plus_plus_int @ B_33 @ A_36 )
= ( plus_plus_int @ C_24 @ A_36 ) )
<=> ( B_33 = C_24 ) ) ).
thf(fact_16_add__left__cancel,axiom,
! [A_35: int,B_32: int,C_23: int] :
( ( ( plus_plus_int @ A_35 @ B_32 )
= ( plus_plus_int @ A_35 @ C_23 ) )
<=> ( B_32 = C_23 ) ) ).
thf(fact_17_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A_34: int,B_31: int,C_22: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_34 @ B_31 ) @ C_22 )
= ( plus_plus_int @ A_34 @ ( plus_plus_int @ B_31 @ C_22 ) ) ) ).
thf(fact_18_diff__eq__diff__eq,axiom,
! [A_33: int,B_30: int,C_21: int,D_9: int] :
( ( ( minus_minus_int @ A_33 @ B_30 )
= ( minus_minus_int @ C_21 @ D_9 ) )
=> ( ( A_33 = B_30 )
<=> ( C_21 = D_9 ) ) ) ).
thf(fact_19_zmult__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( times_times_int @ ( times_times_int @ Z1 @ Z2 ) @ Z3 )
= ( times_times_int @ Z1 @ ( times_times_int @ Z2 @ Z3 ) ) ) ).
thf(fact_20_zmult__commute,axiom,
! [Z: int,W: int] :
( ( times_times_int @ Z @ W )
= ( times_times_int @ W @ Z ) ) ).
thf(fact_21_zadd__assoc,axiom,
! [Z1: int,Z2: int,Z3: int] :
( ( plus_plus_int @ ( plus_plus_int @ Z1 @ Z2 ) @ Z3 )
= ( plus_plus_int @ Z1 @ ( plus_plus_int @ Z2 @ Z3 ) ) ) ).
thf(fact_22_zadd__left__commute,axiom,
! [X_5: int,Y_5: int,Z: int] :
( ( plus_plus_int @ X_5 @ ( plus_plus_int @ Y_5 @ Z ) )
= ( plus_plus_int @ Y_5 @ ( plus_plus_int @ X_5 @ Z ) ) ) ).
thf(fact_23_zadd__commute,axiom,
! [Z: int,W: int] :
( ( plus_plus_int @ Z @ W )
= ( plus_plus_int @ W @ Z ) ) ).
thf(fact_24_combine__common__factor,axiom,
! [A_32: int,E_2: int,B_29: int,C_20: int] :
( ( plus_plus_int @ ( times_times_int @ A_32 @ E_2 ) @ ( plus_plus_int @ ( times_times_int @ B_29 @ E_2 ) @ C_20 ) )
= ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A_32 @ B_29 ) @ E_2 ) @ C_20 ) ) ).
thf(fact_25_comm__semiring__class_Odistrib,axiom,
! [A_31: int,B_28: int,C_19: int] :
( ( times_times_int @ ( plus_plus_int @ A_31 @ B_28 ) @ C_19 )
= ( plus_plus_int @ ( times_times_int @ A_31 @ C_19 ) @ ( times_times_int @ B_28 @ C_19 ) ) ) ).
thf(fact_26_add__diff__add,axiom,
! [A_30: int,C_18: int,B_27: int,D_8: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_30 @ C_18 ) @ ( plus_plus_int @ B_27 @ D_8 ) )
= ( plus_plus_int @ ( minus_minus_int @ A_30 @ B_27 ) @ ( minus_minus_int @ C_18 @ D_8 ) ) ) ).
thf(fact_27_add__diff__cancel,axiom,
! [A_29: int,B_26: int] :
( ( minus_minus_int @ ( plus_plus_int @ A_29 @ B_26 ) @ B_26 )
= A_29 ) ).
thf(fact_28_crossproduct__eq,axiom,
! [W_1: int,Y_4: int,X_4: int,Z_2: int] :
( ( ( plus_plus_int @ ( times_times_int @ W_1 @ Y_4 ) @ ( times_times_int @ X_4 @ Z_2 ) )
= ( plus_plus_int @ ( times_times_int @ W_1 @ Z_2 ) @ ( times_times_int @ X_4 @ Y_4 ) ) )
<=> ( ( W_1 = X_4 )
| ( Y_4 = Z_2 ) ) ) ).
thf(fact_29_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A_28: int,M: int,B_25: int] :
( ( plus_plus_int @ ( times_times_int @ A_28 @ M ) @ ( times_times_int @ B_25 @ M ) )
= ( times_times_int @ ( plus_plus_int @ A_28 @ B_25 ) @ M ) ) ).
thf(fact_30_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A_27: int,B_24: int,C_17: int] :
( ( times_times_int @ ( plus_plus_int @ A_27 @ B_24 ) @ C_17 )
= ( plus_plus_int @ ( times_times_int @ A_27 @ C_17 ) @ ( times_times_int @ B_24 @ C_17 ) ) ) ).
thf(fact_31_crossproduct__noteq,axiom,
! [C_16: int,D_7: int,A_26: int,B_23: int] :
( ( ( A_26 != B_23 )
& ( C_16 != D_7 ) )
<=> ( ( plus_plus_int @ ( times_times_int @ A_26 @ C_16 ) @ ( times_times_int @ B_23 @ D_7 ) )
!= ( plus_plus_int @ ( times_times_int @ A_26 @ D_7 ) @ ( times_times_int @ B_23 @ C_16 ) ) ) ) ).
thf(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [X_3: int,Y_3: int,Z_1: int] :
( ( times_times_int @ X_3 @ ( plus_plus_int @ Y_3 @ Z_1 ) )
= ( plus_plus_int @ ( times_times_int @ X_3 @ Y_3 ) @ ( times_times_int @ X_3 @ Z_1 ) ) ) ).
thf(fact_33_Pair__inject,axiom,
! [A_25: int,B_22: int,A_24: int,B_21: int] :
( ( ( product_Pair_int_int @ A_25 @ B_22 )
= ( product_Pair_int_int @ A_24 @ B_21 ) )
=> ~ ( ( A_25 = A_24 )
=> ( B_22 != B_21 ) ) ) ).
thf(fact_34_Pair__eq,axiom,
! [A_23: int,B_20: int,A_22: int,B_19: int] :
( ( ( product_Pair_int_int @ A_23 @ B_20 )
= ( product_Pair_int_int @ A_22 @ B_19 ) )
<=> ( ( A_23 = A_22 )
& ( B_20 = B_19 ) ) ) ).
thf(fact_35_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A_21: int,B_18: int] :
( ( times_times_int @ A_21 @ B_18 )
= ( times_times_int @ B_18 @ A_21 ) ) ).
thf(fact_36_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [Lx_6: int,Rx_6: int,Ry_4: int] :
( ( times_times_int @ Lx_6 @ ( times_times_int @ Rx_6 @ Ry_4 ) )
= ( times_times_int @ Rx_6 @ ( times_times_int @ Lx_6 @ Ry_4 ) ) ) ).
thf(fact_37_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [Lx_5: int,Rx_5: int,Ry_3: int] :
( ( times_times_int @ Lx_5 @ ( times_times_int @ Rx_5 @ Ry_3 ) )
= ( times_times_int @ ( times_times_int @ Lx_5 @ Rx_5 ) @ Ry_3 ) ) ).
thf(fact_38_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [Lx_4: int,Ly_4: int,Rx_4: int] :
( ( times_times_int @ ( times_times_int @ Lx_4 @ Ly_4 ) @ Rx_4 )
= ( times_times_int @ Lx_4 @ ( times_times_int @ Ly_4 @ Rx_4 ) ) ) ).
thf(fact_39_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [Lx_3: int,Ly_3: int,Rx_3: int] :
( ( times_times_int @ ( times_times_int @ Lx_3 @ Ly_3 ) @ Rx_3 )
= ( times_times_int @ ( times_times_int @ Lx_3 @ Rx_3 ) @ Ly_3 ) ) ).
thf(fact_40_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [Lx_2: int,Ly_2: int,Rx_2: int,Ry_2: int] :
( ( times_times_int @ ( times_times_int @ Lx_2 @ Ly_2 ) @ ( times_times_int @ Rx_2 @ Ry_2 ) )
= ( times_times_int @ Lx_2 @ ( times_times_int @ Ly_2 @ ( times_times_int @ Rx_2 @ Ry_2 ) ) ) ) ).
thf(fact_41_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [Lx_1: int,Ly_1: int,Rx_1: int,Ry_1: int] :
( ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ ( times_times_int @ Rx_1 @ Ry_1 ) )
= ( times_times_int @ Rx_1 @ ( times_times_int @ ( times_times_int @ Lx_1 @ Ly_1 ) @ Ry_1 ) ) ) ).
thf(fact_42_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [Lx: int,Ly: int,Rx: int,Ry: int] :
( ( times_times_int @ ( times_times_int @ Lx @ Ly ) @ ( times_times_int @ Rx @ Ry ) )
= ( times_times_int @ ( times_times_int @ Lx @ Rx ) @ ( times_times_int @ Ly @ Ry ) ) ) ).
thf(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A_20: int,C_15: int] :
( ( plus_plus_int @ A_20 @ C_15 )
= ( plus_plus_int @ C_15 @ A_20 ) ) ).
thf(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A_19: int,C_14: int,D_6: int] :
( ( plus_plus_int @ A_19 @ ( plus_plus_int @ C_14 @ D_6 ) )
= ( plus_plus_int @ C_14 @ ( plus_plus_int @ A_19 @ D_6 ) ) ) ).
thf(fact_45_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A_18: int,C_13: int,D_5: int] :
( ( plus_plus_int @ A_18 @ ( plus_plus_int @ C_13 @ D_5 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_18 @ C_13 ) @ D_5 ) ) ).
thf(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A_17: int,B_17: int,C_12: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_17 @ B_17 ) @ C_12 )
= ( plus_plus_int @ A_17 @ ( plus_plus_int @ B_17 @ C_12 ) ) ) ).
thf(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A_16: int,B_16: int,C_11: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_16 @ B_16 ) @ C_11 )
= ( plus_plus_int @ ( plus_plus_int @ A_16 @ C_11 ) @ B_16 ) ) ).
thf(fact_48_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A_15: int,B_15: int,C_10: int,D_4: int] :
( ( plus_plus_int @ ( plus_plus_int @ A_15 @ B_15 ) @ ( plus_plus_int @ C_10 @ D_4 ) )
= ( plus_plus_int @ ( plus_plus_int @ A_15 @ C_10 ) @ ( plus_plus_int @ B_15 @ D_4 ) ) ) ).
thf(fact_49_split__paired__All,axiom,
! [P_2: product_prod_int_int > $o] :
( ( all @ P_2 )
<=> ! [A_14: int,B_14: int] : ( P_2 @ ( product_Pair_int_int @ A_14 @ B_14 ) ) ) ).
thf(fact_50_split__paired__Ex,axiom,
! [P_1: product_prod_int_int > $o] :
( ( ex @ P_1 )
<=> ? [A_14: int,B_14: int] : ( P_1 @ ( product_Pair_int_int @ A_14 @ B_14 ) ) ) ).
thf(fact_51_prod_Oexhaust,axiom,
! [Y_2: product_prod_int_int] :
~ ! [A_14: int,B_14: int] :
( Y_2
!= ( product_Pair_int_int @ A_14 @ B_14 ) ) ).
thf(fact_52_PairE,axiom,
! [P: product_prod_int_int] :
~ ! [X_2: int,Y_1: int] :
( P
!= ( product_Pair_int_int @ X_2 @ Y_1 ) ) ).
thf(fact_53_curryI,axiom,
! [F_3: product_prod_int_int > $o,A_13: int,B_13: int] :
( ( F_3 @ ( product_Pair_int_int @ A_13 @ B_13 ) )
=> ( produc176579150_int_o @ F_3 @ A_13 @ B_13 ) ) ).
thf(fact_54_curryD,axiom,
! [F_2: product_prod_int_int > $o,A_12: int,B_12: int] :
( ( produc176579150_int_o @ F_2 @ A_12 @ B_12 )
=> ( F_2 @ ( product_Pair_int_int @ A_12 @ B_12 ) ) ) ).
thf(fact_55_curryE,axiom,
! [F_1: product_prod_int_int > $o,A_11: int,B_11: int] :
( ( produc176579150_int_o @ F_1 @ A_11 @ B_11 )
=> ( F_1 @ ( product_Pair_int_int @ A_11 @ B_11 ) ) ) ).
thf(fact_56_curry__conv,axiom,
! [F: product_prod_int_int > $o,A_10: int,B_10: int] :
( ( produc176579150_int_o @ F @ A_10 @ B_10 )
<=> ( F @ ( product_Pair_int_int @ A_10 @ B_10 ) ) ) ).
thf(fact_57_le__add__iff1,axiom,
! [A_9: int,E_1: int,C_9: int,B_9: int,D_3: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_9 @ E_1 ) @ C_9 ) @ ( plus_plus_int @ ( times_times_int @ B_9 @ E_1 ) @ D_3 ) )
<=> ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A_9 @ B_9 ) @ E_1 ) @ C_9 ) @ D_3 ) ) ).
thf(fact_58_le__add__iff2,axiom,
! [A_8: int,E: int,C_8: int,B_8: int,D_2: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A_8 @ E ) @ C_8 ) @ ( plus_plus_int @ ( times_times_int @ B_8 @ E ) @ D_2 ) )
<=> ( ord_less_eq_int @ C_8 @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B_8 @ A_8 ) @ E ) @ D_2 ) ) ) ).
thf(fact_59_zadd__left__mono,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ord_less_eq_int @ ( plus_plus_int @ K @ I ) @ ( plus_plus_int @ K @ J ) ) ) ).
thf(fact_60_diff__eq__diff__less__eq,axiom,
! [A_7: int,B_7: int,C_7: int,D_1: int] :
( ( ( minus_minus_int @ A_7 @ B_7 )
= ( minus_minus_int @ C_7 @ D_1 ) )
=> ( ( ord_less_eq_int @ A_7 @ B_7 )
<=> ( ord_less_eq_int @ C_7 @ D_1 ) ) ) ).
thf(fact_61_add__le__imp__le__left,axiom,
! [C_6: int,A_6: int,B_6: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_6 @ A_6 ) @ ( plus_plus_int @ C_6 @ B_6 ) )
=> ( ord_less_eq_int @ A_6 @ B_6 ) ) ).
thf(fact_62_add__le__imp__le__right,axiom,
! [A_5: int,C_5: int,B_5: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A_5 @ C_5 ) @ ( plus_plus_int @ B_5 @ C_5 ) )
=> ( ord_less_eq_int @ A_5 @ B_5 ) ) ).
thf(fact_63_add__mono,axiom,
! [C_4: int,D: int,A_4: int,B_4: int] :
( ( ord_less_eq_int @ A_4 @ B_4 )
=> ( ( ord_less_eq_int @ C_4 @ D )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_4 @ C_4 ) @ ( plus_plus_int @ B_4 @ D ) ) ) ) ).
thf(fact_64_add__left__mono,axiom,
! [C_3: int,A_3: int,B_3: int] :
( ( ord_less_eq_int @ A_3 @ B_3 )
=> ( ord_less_eq_int @ ( plus_plus_int @ C_3 @ A_3 ) @ ( plus_plus_int @ C_3 @ B_3 ) ) ) ).
thf(fact_65_add__right__mono,axiom,
! [C_2: int,A_2: int,B_2: int] :
( ( ord_less_eq_int @ A_2 @ B_2 )
=> ( ord_less_eq_int @ ( plus_plus_int @ A_2 @ C_2 ) @ ( plus_plus_int @ B_2 @ C_2 ) ) ) ).
thf(fact_66_add__le__cancel__left,axiom,
! [C_1: int,A_1: int,B_1: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ C_1 @ A_1 ) @ ( plus_plus_int @ C_1 @ B_1 ) )
<=> ( ord_less_eq_int @ A_1 @ B_1 ) ) ).
thf(fact_67_add__le__cancel__right,axiom,
! [A: int,C: int,B: int] :
( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
<=> ( ord_less_eq_int @ A @ B ) ) ).
thf(fact_68_order__refl,axiom,
! [X_1: int] : ( ord_less_eq_int @ X_1 @ X_1 ) ).
thf(fact_69_linorder__le__cases,axiom,
! [X: int,Y: int] :
( ~ ( ord_less_eq_int @ X @ Y )
=> ( ord_less_eq_int @ Y @ X ) ) ).
thf(fact_70_zle__refl,axiom,
! [W: int] : ( ord_less_eq_int @ W @ W ) ).
thf(fact_71_zle__linear,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
| ( ord_less_eq_int @ W @ Z ) ) ).
thf(fact_72_zle__trans,axiom,
! [K: int,I: int,J: int] :
( ( ord_less_eq_int @ I @ J )
=> ( ( ord_less_eq_int @ J @ K )
=> ( ord_less_eq_int @ I @ K ) ) ) ).
thf(fact_73_zle__antisym,axiom,
! [Z: int,W: int] :
( ( ord_less_eq_int @ Z @ W )
=> ( ( ord_less_eq_int @ W @ Z )
=> ( Z = W ) ) ) ).
%----Conjectures (1)
thf(conj_0,conjecture,
( ( times_times_int @ ( twoSqu1535104286sum2sq @ ( product_Pair_int_int @ a @ b ) ) @ ( twoSqu1535104286sum2sq @ ( product_Pair_int_int @ p @ q ) ) )
= ( twoSqu1535104286sum2sq @ ( 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 ) ) ) ) ) ).
%------------------------------------------------------------------------------