├── .gitignore ├── README.md ├── hollight ├── LICENSE ├── test │ ├── aime-1983-p1.ml │ ├── aime-1983-p2.ml │ ├── aime-1983-p3.ml │ ├── aime-1984-p1.ml │ ├── aime-1984-p7.ml │ ├── aime-1994-p3.ml │ ├── aime-1995-p7.ml │ ├── algebra-2varlineareq-fp3zeq11-3tfm1m5zeqn68-feqn10-zeq7.ml │ ├── algebra-amgm-sum1toneqn-prod1tonleq1.ml │ ├── algebra-amgm-sumasqdivbgeqsuma.ml │ ├── algebra-ineq-nto1onlt2m1on.ml │ ├── algebra-others-exirrpowirrrat.ml │ ├── algebra-sqineq-at2malt1.ml │ ├── algebra-sqineq-unitcircatbpabsamblt1.ml │ ├── algebra-sqineq-unitcircatbpamblt1.ml │ ├── amc12-2000-p1.ml │ ├── amc12-2000-p12.ml │ ├── amc12-2000-p20.ml │ ├── amc12-2000-p6.ml │ ├── amc12-2001-p21.ml │ ├── amc12-2001-p5.ml │ ├── amc12a-2002-p13.ml │ ├── amc12a-2002-p6.ml │ ├── amc12a-2003-p23.ml │ ├── amc12a-2003-p5.ml │ ├── amc12a-2020-p10.ml │ ├── amc12a-2020-p15.ml │ ├── amc12a-2020-p4.ml │ ├── amc12a-2020-p7.ml │ ├── amc12a-2020-p9.ml │ ├── amc12a-2021-p12.ml │ ├── amc12a-2021-p14.ml │ ├── amc12a-2021-p18.ml │ ├── amc12a-2021-p19.ml │ ├── amc12a-2021-p22.ml │ ├── amc12a-2021-p25.ml │ ├── amc12a-2021-p3.ml │ ├── amc12a-2021-p8.ml │ ├── amc12a-2021-p9.ml │ ├── amc12b-2002-p19.ml │ ├── amc12b-2002-p2.ml │ ├── amc12b-2002-p4.ml │ ├── amc12b-2002-p7.ml │ ├── amc12b-2020-p13.ml │ ├── amc12b-2020-p2.ml │ ├── amc12b-2020-p21.ml │ ├── amc12b-2020-p22.ml │ ├── amc12b-2020-p6.ml │ ├── amc12b-2021-p1.ml │ ├── amc12b-2021-p13.ml │ ├── amc12b-2021-p18.ml │ ├── amc12b-2021-p3.ml │ ├── amc12b-2021-p4.ml │ ├── amc12b-2021-p9.ml │ ├── imo-1959-q1.ml │ ├── imo-2007-a6.ml │ ├── imo-2019-q1.ml │ ├── induction-11div10tonmn1ton.ml │ ├── induction-12dvd4expnp1p20.ml │ ├── induction-1pxpownlt1pnx.ml │ ├── induction-nfactltnexpnm1ngt3.ml │ ├── induction-sumkexp3eqsumksqsq.ml │ ├── mathb-algebra-44.ml │ ├── mathd-algebra-107.ml │ ├── mathd-algebra-113.ml │ ├── mathd-algebra-114.ml │ ├── mathd-algebra-125.ml │ ├── mathd-algebra-129.ml │ ├── mathd-algebra-137.ml │ ├── mathd-algebra-139.ml │ ├── mathd-algebra-141.ml │ ├── mathd-algebra-142.ml │ ├── mathd-algebra-143.ml │ ├── mathd-algebra-148.ml │ ├── mathd-algebra-153.ml │ ├── mathd-algebra-156.ml │ ├── mathd-algebra-158.ml │ ├── mathd-algebra-160.ml │ ├── mathd-algebra-17.ml │ ├── mathd-algebra-170.ml │ ├── mathd-algebra-171.ml │ ├── mathd-algebra-176.ml │ ├── mathd-algebra-184.ml │ ├── mathd-algebra-196.ml │ ├── mathd-algebra-208.ml │ ├── mathd-algebra-215.ml │ ├── mathd-algebra-24.ml │ ├── mathd-algebra-246.ml │ ├── mathd-algebra-263.ml │ ├── mathd-algebra-270.ml │ ├── mathd-algebra-275.ml │ ├── mathd-algebra-276.ml │ ├── mathd-algebra-288.ml │ ├── mathd-algebra-289.ml │ ├── mathd-algebra-293.ml │ ├── mathd-algebra-296.ml │ ├── mathd-algebra-302.ml │ ├── mathd-algebra-304.ml │ ├── mathd-algebra-313.ml │ ├── mathd-algebra-314.ml │ ├── mathd-algebra-320.ml │ ├── mathd-algebra-329.ml │ ├── mathd-algebra-33.ml │ ├── mathd-algebra-332.ml │ ├── mathd-algebra-338.ml │ ├── mathd-algebra-342.ml │ ├── mathd-algebra-346.ml │ ├── mathd-algebra-354.ml │ ├── mathd-algebra-362.ml │ ├── mathd-algebra-388.ml │ ├── mathd-algebra-392.ml │ ├── mathd-algebra-398.ml │ ├── mathd-algebra-400.ml │ ├── mathd-algebra-412.ml │ ├── mathd-algebra-419.ml │ ├── mathd-algebra-427.ml │ ├── mathd-algebra-432.ml │ ├── mathd-algebra-440.ml │ ├── mathd-algebra-478.ml │ ├── mathd-algebra-484.ml │ ├── mathd-algebra-487.ml │ ├── mathd-algebra-513.ml │ ├── mathd-algebra-76.ml │ ├── mathd-algebra-80.ml │ ├── mathd-numbertheory-100.ml │ ├── mathd-numbertheory-127.ml │ ├── mathd-numbertheory-135.ml │ ├── mathd-numbertheory-150.ml │ ├── mathd-numbertheory-175.ml │ ├── mathd-numbertheory-185.ml │ ├── mathd-numbertheory-207.ml │ ├── mathd-numbertheory-212.ml │ ├── mathd-numbertheory-222.ml │ ├── mathd-numbertheory-227.ml │ ├── mathd-numbertheory-229.ml │ ├── mathd-numbertheory-233.ml │ ├── mathd-numbertheory-234.ml │ ├── mathd-numbertheory-235.ml │ ├── mathd-numbertheory-237.ml │ ├── mathd-numbertheory-247.ml │ ├── mathd-numbertheory-254.ml │ ├── mathd-numbertheory-277.ml │ ├── mathd-numbertheory-299.ml │ ├── mathd-numbertheory-3.ml │ ├── mathd-numbertheory-321.ml │ ├── mathd-numbertheory-328.ml │ ├── mathd-numbertheory-34.ml │ ├── mathd-numbertheory-341.ml │ ├── mathd-numbertheory-342.ml │ ├── mathd-numbertheory-345.ml │ ├── mathd-numbertheory-427.ml │ ├── mathd-numbertheory-447.ml │ ├── mathd-numbertheory-451.ml │ ├── mathd-numbertheory-495.ml │ ├── mathd-numbertheory-551.ml │ ├── mathd-numbertheory-552.ml │ ├── mathd-numbertheory-618.ml │ ├── mathd-numbertheory-66.ml │ ├── mathd-numbertheory-85.ml │ ├── mathd-numbertheory-99.ml │ ├── numbertheory-aoddbdiv4asqpbsqmod8eq1.ml │ ├── numbertheory-notequiv2i2jasqbsqdiv8.ml │ └── others-chickenmcnuggets.ml └── valid │ ├── aime-1983-p9.ml │ ├── aime-1984-p15.ml │ ├── aime-1984-p5.ml │ ├── algebra-2complexrootspoly-xsqp49eqxp7itxpn7i.ml │ ├── algebra-2rootsintpoly-am10tap11eqasqpam110.ml │ ├── algebra-2rootspoly-apatapbeq2asqp2ab.ml │ ├── algebra-2varlineareq-xpeeq7-2xpeeq3-eeq11-xeqn4.ml │ ├── algebra-3rootspoly-amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta.ml │ ├── algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x.ml │ ├── algebra-amgm-prod1toneq1-sum1tongeqn.ml │ ├── algebra-amgm-sqrtxymulxmyeqxpy-xpygeq4.ml │ ├── algebra-amgm-sumasqdivbsqgeqsumbdiva.ml │ ├── algebra-binomnegdiscrineq-10alt28asqp1.ml │ ├── algebra-manipexpr-2erprsqpesqeqnrpnesq.ml │ ├── algebra-manipexpr-apbeq2cceqiacpbceqm2.ml │ ├── algebra-sqineq-2at2pclta2c2p41pc.ml │ ├── algebra-sqineq-2unitcircatblt1.ml │ ├── algebra-sqineq-36azm9asqle36zsq.ml │ ├── algebra-sqineq-4bap1lt4bsqpap1sq.ml │ ├── amc12-2000-p11.ml │ ├── amc12-2000-p15.ml │ ├── amc12-2000-p5.ml │ ├── amc12-2001-p2.ml │ ├── amc12-2001-p9.ml │ ├── amc12a-2002-p1.ml │ ├── amc12a-2002-p12.ml │ ├── amc12a-2002-p21.ml │ ├── amc12a-2003-p1.ml │ ├── amc12a-2003-p24.ml │ ├── amc12a-2003-p25.ml │ ├── amc12a-2020-p13.ml │ ├── amc12a-2020-p22.ml │ ├── amc12a-2021-p7.ml │ ├── amc12b-2002-p11.ml │ ├── amc12b-2002-p3.ml │ ├── amc12b-2002-p6.ml │ ├── amc12b-2003-p17.ml │ ├── amc12b-2003-p6.ml │ ├── amc12b-2003-p9.ml │ ├── amc12b-2004-p3.ml │ ├── amc12b-2020-p5.ml │ ├── amc12b-2021-p21.ml │ ├── imo-1990-p3.ml │ ├── imo-2006-a6.ml │ ├── induction-divisibility-3div2tooddnp1.ml │ ├── induction-divisibility-3divnto3m2n.ml │ ├── induction-divisibility-9div10tonm1.ml │ ├── induction-ineq-nsqlefactn.ml │ ├── induction-seq-mul2pnp1.ml │ ├── induction-sum-1oktkp1.ml │ ├── induction-sum-odd.ml │ ├── induction-sum2kp1npqsqm1.ml │ ├── mathd-algebra-10.ml │ ├── mathd-algebra-101.ml │ ├── mathd-algebra-104.ml │ ├── mathd-algebra-109.ml │ ├── mathd-algebra-11.ml │ ├── mathd-algebra-110.ml │ ├── mathd-algebra-116.ml │ ├── mathd-algebra-119.ml │ ├── mathd-algebra-123.ml │ ├── mathd-algebra-126.ml │ ├── mathd-algebra-13.ml │ ├── mathd-algebra-131.ml │ ├── mathd-algebra-132.ml │ ├── mathd-algebra-140.ml │ ├── mathd-algebra-144.ml │ ├── mathd-algebra-149.ml │ ├── mathd-algebra-15.ml │ ├── mathd-algebra-151.ml │ ├── mathd-algebra-159.ml │ ├── mathd-algebra-181.ml │ ├── mathd-algebra-182.ml │ ├── mathd-algebra-185.ml │ ├── mathd-algebra-190.ml │ ├── mathd-algebra-192.ml │ ├── mathd-algebra-206.ml │ ├── mathd-algebra-214.ml │ ├── mathd-algebra-22.ml │ ├── mathd-algebra-224.ml │ ├── mathd-algebra-234.ml │ ├── mathd-algebra-245.ml │ ├── mathd-algebra-247.ml │ ├── mathd-algebra-251.ml │ ├── mathd-algebra-267.ml │ ├── mathd-algebra-28.ml │ ├── mathd-algebra-282.ml │ ├── mathd-algebra-327.ml │ ├── mathd-algebra-35.ml │ ├── mathd-algebra-37.ml │ ├── mathd-algebra-421.ml │ ├── mathd-algebra-43.ml │ ├── mathd-algebra-455.ml │ ├── mathd-algebra-48.ml │ ├── mathd-algebra-51.ml │ ├── mathd-algebra-55.ml │ ├── mathd-algebra-59.ml │ ├── mathd-algebra-67.ml │ ├── mathd-algebra-69.ml │ ├── mathd-algebra-73.ml │ ├── mathd-algebra-77.ml │ ├── mathd-algebra-89.ml │ ├── mathd-algebra-96.ml │ ├── mathd-numbertheory-101.ml │ ├── mathd-numbertheory-102.ml │ ├── mathd-numbertheory-109.ml │ ├── mathd-numbertheory-110.ml │ ├── mathd-numbertheory-126.ml │ ├── mathd-numbertheory-13.ml │ ├── mathd-numbertheory-132.ml │ ├── mathd-numbertheory-133.ml │ ├── mathd-numbertheory-136.ml │ ├── mathd-numbertheory-149.ml │ ├── mathd-numbertheory-155.ml │ ├── mathd-numbertheory-156.ml │ ├── mathd-numbertheory-169.ml │ ├── mathd-numbertheory-198.ml │ ├── mathd-numbertheory-200.ml │ ├── mathd-numbertheory-202.ml │ ├── mathd-numbertheory-211.ml │ ├── mathd-numbertheory-22.ml │ ├── mathd-numbertheory-221.ml │ ├── mathd-numbertheory-232.ml │ ├── mathd-numbertheory-236.ml │ ├── mathd-numbertheory-24.ml │ ├── mathd-numbertheory-252.ml │ ├── mathd-numbertheory-257.ml │ ├── mathd-numbertheory-284.ml │ ├── mathd-numbertheory-30.ml │ ├── mathd-numbertheory-301.ml │ ├── mathd-numbertheory-303.ml │ ├── mathd-numbertheory-32.ml │ ├── mathd-numbertheory-326.ml │ ├── mathd-numbertheory-33.ml │ ├── mathd-numbertheory-335.ml │ ├── mathd-numbertheory-35.ml │ ├── mathd-numbertheory-37.ml │ ├── mathd-numbertheory-370.ml │ ├── mathd-numbertheory-403.ml │ ├── mathd-numbertheory-405.ml │ ├── mathd-numbertheory-412.ml │ ├── mathd-numbertheory-42.ml │ ├── mathd-numbertheory-43.ml │ ├── mathd-numbertheory-45.ml │ ├── mathd-numbertheory-458.ml │ ├── mathd-numbertheory-466.ml │ ├── mathd-numbertheory-48.ml │ ├── mathd-numbertheory-530.ml │ ├── mathd-numbertheory-629.ml │ ├── mathd-numbertheory-64.ml │ ├── mathd-numbertheory-640.ml │ ├── mathd-numbertheory-668.ml │ ├── mathd-numbertheory-690.ml │ ├── mathd-numbertheory-709.ml │ ├── mathd-numbertheory-739.ml │ ├── mathd-numbertheory-81.ml │ ├── mathd-numbertheory-84.ml │ ├── mathd-numbertheory-92.ml │ ├── numbertheory-2dvd4expn.ml │ ├── numbertheory-prmdvsneqnsqmodpeq0.ml │ ├── numbertheory-sqmod3in01d.ml │ ├── numbertheory-sqmod4in01d.ml │ └── numbertheory-xsqpysqintdenomeq.ml ├── isabelle ├── LICENSE ├── test │ ├── aime_1983_p1.thy │ ├── aime_1983_p2.thy │ ├── aime_1983_p3.thy │ ├── aime_1984_p1.thy │ ├── aime_1984_p7.thy │ ├── aime_1987_p5.thy │ ├── aime_1988_p8.thy │ ├── aime_1989_p8.thy │ ├── aime_1990_p15.thy │ ├── aime_1990_p4.thy │ ├── aime_1991_p9.thy │ ├── aime_1994_p3.thy │ ├── aime_1995_p7.thy │ ├── aime_1997_p9.thy │ ├── aime_1999_p11.thy │ ├── algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7.thy │ ├── algebra_9onxpypzleqsum2onxpy.thy │ ├── algebra_abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2.thy │ ├── algebra_absapbon1pabsapbleqsumabsaon1pabsa.thy │ ├── algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1.thy │ ├── algebra_amgm_sum1toneqn_prod1tonleq1.thy │ ├── algebra_amgm_sumasqdivbgeqsuma.thy │ ├── algebra_apbmpcneq0_aeq0anbeq0anceq0.thy │ ├── algebra_apbon2pownleqapownpbpowon2.thy │ ├── algebra_apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3.thy │ ├── algebra_bleqa_apbon2msqrtableqambsqon8b.thy │ ├── algebra_cubrtrp1oncubrtreq3_rcubp1onrcubeq5778.thy │ ├── algebra_ineq_nto1onlt2m1on.thy │ ├── algebra_others_exirrpowirrrat.thy │ ├── algebra_sqineq_at2malt1.thy │ ├── algebra_sqineq_unitcircatbpabsamblt1.thy │ ├── algebra_sqineq_unitcircatbpamblt1.thy │ ├── algebra_sum1onsqrt2to1onsqrt10000lt198.thy │ ├── amc12_2000_p1.thy │ ├── amc12_2000_p12.thy │ ├── amc12_2000_p20.thy │ ├── amc12_2000_p6.thy │ ├── amc12_2001_p21.thy │ ├── amc12_2001_p5.thy │ ├── amc12a_2002_p13.thy │ ├── amc12a_2002_p6.thy │ ├── amc12a_2003_p23.thy │ ├── amc12a_2003_p5.thy │ ├── amc12a_2008_p25.thy │ ├── amc12a_2009_p6.thy │ ├── amc12a_2009_p7.thy │ ├── amc12a_2013_p4.thy │ ├── amc12a_2019_p12.thy │ ├── amc12a_2020_p10.thy │ ├── amc12a_2020_p15.thy │ ├── amc12a_2020_p25.thy │ ├── amc12a_2020_p4.thy │ ├── amc12a_2020_p7.thy │ ├── amc12a_2020_p9.thy │ ├── amc12a_2021_p12.thy │ ├── amc12a_2021_p14.thy │ ├── amc12a_2021_p18.thy │ ├── amc12a_2021_p19.thy │ ├── amc12a_2021_p22.thy │ ├── amc12a_2021_p25.thy │ ├── amc12a_2021_p3.thy │ ├── amc12a_2021_p8.thy │ ├── amc12a_2021_p9.thy │ ├── amc12b_2002_p19.thy │ ├── amc12b_2002_p2.thy │ ├── amc12b_2002_p4.thy │ ├── amc12b_2002_p7.thy │ ├── amc12b_2020_p13.thy │ ├── amc12b_2020_p2.thy │ ├── amc12b_2020_p21.thy │ ├── amc12b_2020_p22.thy │ ├── amc12b_2020_p6.thy │ ├── amc12b_2021_p1.thy │ ├── amc12b_2021_p13.thy │ ├── amc12b_2021_p18.thy │ ├── amc12b_2021_p3.thy │ ├── amc12b_2021_p4.thy │ ├── amc12b_2021_p9.thy │ ├── imo_1959_p1.thy │ ├── imo_1960_p2.thy │ ├── imo_1962_p2.thy │ ├── imo_1963_p5.thy │ ├── imo_1964_p2.thy │ ├── imo_1965_p2.thy │ ├── imo_1968_p5_1.thy │ ├── imo_1969_p2.thy │ ├── imo_1974_p3.thy │ ├── imo_1977_p6.thy │ ├── imo_1981_p6.thy │ ├── imo_1982_p1.thy │ ├── imo_1983_p6.thy │ ├── imo_1984_p6.thy │ ├── imo_1985_p6.thy │ ├── imo_1992_p1.thy │ ├── imo_1997_p5.thy │ ├── imo_2001_p6.thy │ ├── imo_2007_p6.thy │ ├── imo_2019_p1.thy │ ├── induction_11div10tonmn1ton.thy │ ├── induction_12dvd4expnp1p20.thy │ ├── induction_1pxpownlt1pnx.thy │ ├── induction_nfactltnexpnm1ngt3.thy │ ├── induction_pord1p1on2powklt5on2.thy │ ├── induction_pprime_pdvdapowpma.thy │ ├── induction_prod1p1onk3le3m1onn.thy │ ├── induction_sumkexp3eqsumksq.thy │ ├── mathd_algebra_107.thy │ ├── mathd_algebra_113.thy │ ├── mathd_algebra_114.thy │ ├── mathd_algebra_125.thy │ ├── mathd_algebra_129.thy │ ├── mathd_algebra_137.thy │ ├── mathd_algebra_139.thy │ ├── mathd_algebra_141.thy │ ├── mathd_algebra_142.thy │ ├── mathd_algebra_143.thy │ ├── mathd_algebra_148.thy │ ├── mathd_algebra_153.thy │ ├── mathd_algebra_156.thy │ ├── mathd_algebra_158.thy │ ├── mathd_algebra_160.thy │ ├── mathd_algebra_17.thy │ ├── mathd_algebra_170.thy │ ├── mathd_algebra_171.thy │ ├── mathd_algebra_176.thy │ ├── mathd_algebra_184.thy │ ├── mathd_algebra_188.thy │ ├── mathd_algebra_196.thy │ ├── mathd_algebra_208.thy │ ├── mathd_algebra_209.thy │ ├── mathd_algebra_215.thy │ ├── mathd_algebra_24.thy │ ├── mathd_algebra_246.thy │ ├── mathd_algebra_263.thy │ ├── mathd_algebra_270.thy │ ├── mathd_algebra_275.thy │ ├── mathd_algebra_276.thy │ ├── mathd_algebra_288.thy │ ├── mathd_algebra_289.thy │ ├── mathd_algebra_293.thy │ ├── mathd_algebra_296.thy │ ├── mathd_algebra_302.thy │ ├── mathd_algebra_304.thy │ ├── mathd_algebra_313.thy │ ├── mathd_algebra_314.thy │ ├── mathd_algebra_320.thy │ ├── mathd_algebra_329.thy │ ├── mathd_algebra_33.thy │ ├── mathd_algebra_332.thy │ ├── mathd_algebra_338.thy │ ├── mathd_algebra_342.thy │ ├── mathd_algebra_346.thy │ ├── mathd_algebra_354.thy │ ├── mathd_algebra_359.thy │ ├── mathd_algebra_362.thy │ ├── mathd_algebra_388.thy │ ├── mathd_algebra_392.thy │ ├── mathd_algebra_398.thy │ ├── mathd_algebra_400.thy │ ├── mathd_algebra_412.thy │ ├── mathd_algebra_419.thy │ ├── mathd_algebra_427.thy │ ├── mathd_algebra_432.thy │ ├── mathd_algebra_44.thy │ ├── mathd_algebra_440.thy │ ├── mathd_algebra_441.thy │ ├── mathd_algebra_452.thy │ ├── mathd_algebra_459.thy │ ├── mathd_algebra_478.thy │ ├── mathd_algebra_484.thy │ ├── mathd_algebra_487.thy │ ├── mathd_algebra_513.thy │ ├── mathd_algebra_598.thy │ ├── mathd_algebra_756.thy │ ├── mathd_algebra_76.thy │ ├── mathd_algebra_80.thy │ ├── mathd_numbertheory_100.thy │ ├── mathd_numbertheory_1124.thy │ ├── mathd_numbertheory_12.thy │ ├── mathd_numbertheory_127.thy │ ├── mathd_numbertheory_135.thy │ ├── mathd_numbertheory_150.thy │ ├── mathd_numbertheory_175.thy │ ├── mathd_numbertheory_185.thy │ ├── mathd_numbertheory_207.thy │ ├── mathd_numbertheory_212.thy │ ├── mathd_numbertheory_222.thy │ ├── mathd_numbertheory_227.thy │ ├── mathd_numbertheory_229.thy │ ├── mathd_numbertheory_233.thy │ ├── mathd_numbertheory_234.thy │ ├── mathd_numbertheory_235.thy │ ├── mathd_numbertheory_237.thy │ ├── mathd_numbertheory_239.thy │ ├── mathd_numbertheory_247.thy │ ├── mathd_numbertheory_254.thy │ ├── mathd_numbertheory_277.thy │ ├── mathd_numbertheory_293.thy │ ├── mathd_numbertheory_296.thy │ ├── mathd_numbertheory_299.thy │ ├── mathd_numbertheory_3.thy │ ├── mathd_numbertheory_314.thy │ ├── mathd_numbertheory_320.thy │ ├── mathd_numbertheory_321.thy │ ├── mathd_numbertheory_328.thy │ ├── mathd_numbertheory_34.thy │ ├── mathd_numbertheory_341.thy │ ├── mathd_numbertheory_342.thy │ ├── mathd_numbertheory_343.thy │ ├── mathd_numbertheory_345.thy │ ├── mathd_numbertheory_353.thy │ ├── mathd_numbertheory_427.thy │ ├── mathd_numbertheory_430.thy │ ├── mathd_numbertheory_435.thy │ ├── mathd_numbertheory_447.thy │ ├── mathd_numbertheory_451.thy │ ├── mathd_numbertheory_457.thy │ ├── mathd_numbertheory_483.thy │ ├── mathd_numbertheory_495.thy │ ├── mathd_numbertheory_5.thy │ ├── mathd_numbertheory_517.thy │ ├── mathd_numbertheory_521.thy │ ├── mathd_numbertheory_541.thy │ ├── mathd_numbertheory_551.thy │ ├── mathd_numbertheory_552.thy │ ├── mathd_numbertheory_559.thy │ ├── mathd_numbertheory_582.thy │ ├── mathd_numbertheory_618.thy │ ├── mathd_numbertheory_66.thy │ ├── mathd_numbertheory_711.thy │ ├── mathd_numbertheory_728.thy │ ├── mathd_numbertheory_764.thy │ ├── mathd_numbertheory_765.thy │ ├── mathd_numbertheory_769.thy │ ├── mathd_numbertheory_85.thy │ ├── mathd_numbertheory_99.thy │ ├── numbertheory_2pownm1prime_nprime.thy │ ├── numbertheory_3pow2pownm1mod2pownp3eq2pownp2.thy │ ├── numbertheory_4x3m7y3neq2003.thy │ ├── numbertheory_aoddbdiv4asqpbsqmod8eq1.thy │ ├── numbertheory_exk2powkeqapb2mulbpa2_aeq1.thy │ ├── numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown.thy │ ├── numbertheory_notequiv2i2jasqbsqdiv8.thy │ └── numbertheory_x5neqy2p4.thy └── valid │ ├── aimeII_2020_p6.thy │ ├── aimeI_2000_p7.thy │ ├── aimeI_2001_p3.thy │ ├── aime_1983_p9.thy │ ├── aime_1984_p15.thy │ ├── aime_1984_p5.thy │ ├── aime_1987_p8.thy │ ├── aime_1988_p3.thy │ ├── aime_1988_p4.thy │ ├── aime_1990_p2.thy │ ├── aime_1991_p1.thy │ ├── aime_1991_p6.thy │ ├── aime_1994_p4.thy │ ├── aime_1996_p5.thy │ ├── aime_1997_p12.thy │ ├── algebra_2complexrootspoly_xsqp49eqxp7itxpn7i.thy │ ├── algebra_2rootsintpoly_am10tap11eqasqpam110.thy │ ├── algebra_2rootspoly_apatapbeq2asqp2ab.thy │ ├── algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4.thy │ ├── algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta.thy │ ├── algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x.thy │ ├── algebra_amgm_prod1toneq1_sum1tongeqn.thy │ ├── algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4.thy │ ├── algebra_amgm_sumasqdivbsqgeqsumbdiva.thy │ ├── algebra_apb4leq8ta4pb4.thy │ ├── algebra_binomnegdiscrineq_10alt28asqp1.thy │ ├── algebra_manipexpr_2erprsqpesqeqnrpnesq.thy │ ├── algebra_manipexpr_apbeq2cceqiacpbceqm2.thy │ ├── algebra_sqineq_2at2pclta2c2p41pc.thy │ ├── algebra_sqineq_2unitcircatblt1.thy │ ├── algebra_sqineq_36azm9asqle36zsq.thy │ ├── algebra_sqineq_4bap1lt4bsqpap1sq.thy │ ├── algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3.thy │ ├── amc12_2000_p11.thy │ ├── amc12_2000_p15.thy │ ├── amc12_2000_p5.thy │ ├── amc12_2001_p2.thy │ ├── amc12_2001_p9.thy │ ├── amc12a_2002_p1.thy │ ├── amc12a_2002_p12.thy │ ├── amc12a_2002_p21.thy │ ├── amc12a_2003_p1.thy │ ├── amc12a_2003_p24.thy │ ├── amc12a_2003_p25.thy │ ├── amc12a_2008_p15.thy │ ├── amc12a_2008_p2.thy │ ├── amc12a_2008_p4.thy │ ├── amc12a_2008_p8.thy │ ├── amc12a_2009_p15.thy │ ├── amc12a_2009_p2.thy │ ├── amc12a_2009_p25.thy │ ├── amc12a_2009_p5.thy │ ├── amc12a_2009_p9.thy │ ├── amc12a_2010_p10.thy │ ├── amc12a_2010_p11.thy │ ├── amc12a_2010_p22.thy │ ├── amc12a_2011_p18.thy │ ├── amc12a_2013_p7.thy │ ├── amc12a_2013_p8.thy │ ├── amc12a_2015_p10.thy │ ├── amc12a_2016_p2.thy │ ├── amc12a_2016_p3.thy │ ├── amc12a_2017_p2.thy │ ├── amc12a_2017_p7.thy │ ├── amc12a_2019_p21.thy │ ├── amc12a_2019_p9.thy │ ├── amc12a_2020_p13.thy │ ├── amc12a_2020_p22.thy │ ├── amc12a_2021_p7.thy │ ├── amc12b_2002_p11.thy │ ├── amc12b_2002_p3.thy │ ├── amc12b_2002_p6.thy │ ├── amc12b_2003_p17.thy │ ├── amc12b_2003_p6.thy │ ├── amc12b_2003_p9.thy │ ├── amc12b_2004_p3.thy │ ├── amc12b_2020_p5.thy │ ├── amc12b_2021_p21.thy │ ├── imo_1961_p1.thy │ ├── imo_1962_p4.thy │ ├── imo_1964_p1_1.thy │ ├── imo_1964_p1_2.thy │ ├── imo_1965_p1.thy │ ├── imo_1966_p4.thy │ ├── imo_1966_p5.thy │ ├── imo_1967_p3.thy │ ├── imo_1973_p3.thy │ ├── imo_1974_p5.thy │ ├── imo_1977_p5.thy │ ├── imo_1978_p5.thy │ ├── imo_1979_p1.thy │ ├── imo_1984_p2.thy │ ├── imo_1987_p4.thy │ ├── imo_1987_p6.thy │ ├── imo_1988_p6.thy │ ├── imo_1990_p3.thy │ ├── imo_1993_p5.thy │ ├── imo_2006_p6.thy │ ├── induction_divisibility_3div2tooddnp1.thy │ ├── induction_divisibility_3divnto3m2n.thy │ ├── induction_divisibility_9div10tonm1.thy │ ├── induction_ineq_nsqlefactn.thy │ ├── induction_seq_mul2pnp1.thy │ ├── induction_sum2kp1npqsqm1.thy │ ├── induction_sum_1oktkp1.thy │ ├── induction_sum_odd.thy │ ├── mathd_algebra_10.thy │ ├── mathd_algebra_101.thy │ ├── mathd_algebra_104.thy │ ├── mathd_algebra_109.thy │ ├── mathd_algebra_11.thy │ ├── mathd_algebra_110.thy │ ├── mathd_algebra_116.thy │ ├── mathd_algebra_119.thy │ ├── mathd_algebra_123.thy │ ├── mathd_algebra_126.thy │ ├── mathd_algebra_13.thy │ ├── mathd_algebra_131.thy │ ├── mathd_algebra_132.thy │ ├── mathd_algebra_140.thy │ ├── mathd_algebra_144.thy │ ├── mathd_algebra_149.thy │ ├── mathd_algebra_15.thy │ ├── mathd_algebra_151.thy │ ├── mathd_algebra_159.thy │ ├── mathd_algebra_181.thy │ ├── mathd_algebra_182.thy │ ├── mathd_algebra_185.thy │ ├── mathd_algebra_190.thy │ ├── mathd_algebra_192.thy │ ├── mathd_algebra_206.thy │ ├── mathd_algebra_214.thy │ ├── mathd_algebra_22.thy │ ├── mathd_algebra_224.thy │ ├── mathd_algebra_234.thy │ ├── mathd_algebra_245.thy │ ├── mathd_algebra_247.thy │ ├── mathd_algebra_251.thy │ ├── mathd_algebra_267.thy │ ├── mathd_algebra_28.thy │ ├── mathd_algebra_282.thy │ ├── mathd_algebra_31.thy │ ├── mathd_algebra_323.thy │ ├── mathd_algebra_327.thy │ ├── mathd_algebra_35.thy │ ├── mathd_algebra_37.thy │ ├── mathd_algebra_393.thy │ ├── mathd_algebra_405.thy │ ├── mathd_algebra_410.thy │ ├── mathd_algebra_421.thy │ ├── mathd_algebra_422.thy │ ├── mathd_algebra_43.thy │ ├── mathd_algebra_433.thy │ ├── mathd_algebra_437.thy │ ├── mathd_algebra_451.thy │ ├── mathd_algebra_455.thy │ ├── mathd_algebra_462.thy │ ├── mathd_algebra_48.thy │ ├── mathd_algebra_480.thy │ ├── mathd_algebra_482.thy │ ├── mathd_algebra_493.thy │ ├── mathd_algebra_509.thy │ ├── mathd_algebra_51.thy │ ├── mathd_algebra_510.thy │ ├── mathd_algebra_536.thy │ ├── mathd_algebra_547.thy │ ├── mathd_algebra_55.thy │ ├── mathd_algebra_568.thy │ ├── mathd_algebra_59.thy │ ├── mathd_algebra_616.thy │ ├── mathd_algebra_67.thy │ ├── mathd_algebra_69.thy │ ├── mathd_algebra_73.thy │ ├── mathd_algebra_77.thy │ ├── mathd_algebra_89.thy │ ├── mathd_algebra_96.thy │ ├── mathd_numbertheory_101.thy │ ├── mathd_numbertheory_102.thy │ ├── mathd_numbertheory_109.thy │ ├── mathd_numbertheory_110.thy │ ├── mathd_numbertheory_126.thy │ ├── mathd_numbertheory_13.thy │ ├── mathd_numbertheory_132.thy │ ├── mathd_numbertheory_136.thy │ ├── mathd_numbertheory_149.thy │ ├── mathd_numbertheory_155.thy │ ├── mathd_numbertheory_156.thy │ ├── mathd_numbertheory_169.thy │ ├── mathd_numbertheory_188.thy │ ├── mathd_numbertheory_198.thy │ ├── mathd_numbertheory_200.thy │ ├── mathd_numbertheory_202.thy │ ├── mathd_numbertheory_211.thy │ ├── mathd_numbertheory_22.thy │ ├── mathd_numbertheory_221.thy │ ├── mathd_numbertheory_232.thy │ ├── mathd_numbertheory_236.thy │ ├── mathd_numbertheory_24.thy │ ├── mathd_numbertheory_252.thy │ ├── mathd_numbertheory_257.thy │ ├── mathd_numbertheory_269.thy │ ├── mathd_numbertheory_284.thy │ ├── mathd_numbertheory_30.thy │ ├── mathd_numbertheory_301.thy │ ├── mathd_numbertheory_303.thy │ ├── mathd_numbertheory_32.thy │ ├── mathd_numbertheory_326.thy │ ├── mathd_numbertheory_33.thy │ ├── mathd_numbertheory_335.thy │ ├── mathd_numbertheory_35.thy │ ├── mathd_numbertheory_37.thy │ ├── mathd_numbertheory_370.thy │ ├── mathd_numbertheory_403.thy │ ├── mathd_numbertheory_405.thy │ ├── mathd_numbertheory_412.thy │ ├── mathd_numbertheory_42.thy │ ├── mathd_numbertheory_43.thy │ ├── mathd_numbertheory_45.thy │ ├── mathd_numbertheory_458.thy │ ├── mathd_numbertheory_461.thy │ ├── mathd_numbertheory_466.thy │ ├── mathd_numbertheory_48.thy │ ├── mathd_numbertheory_530.thy │ ├── mathd_numbertheory_543.thy │ ├── mathd_numbertheory_629.thy │ ├── mathd_numbertheory_64.thy │ ├── mathd_numbertheory_640.thy │ ├── mathd_numbertheory_668.thy │ ├── mathd_numbertheory_690.thy │ ├── mathd_numbertheory_709.thy │ ├── mathd_numbertheory_739.thy │ ├── mathd_numbertheory_780.thy │ ├── mathd_numbertheory_81.thy │ ├── mathd_numbertheory_84.thy │ ├── mathd_numbertheory_92.thy │ ├── mathd_numbertheory_961.thy │ ├── numbertheory_2dvd4expn.thy │ ├── numbertheory_aneqprodakp4_anmsqrtanp1eq2.thy │ ├── numbertheory_nckeqnm1ckpnm1ckm1.thy │ ├── numbertheory_prmdvsneqnsqmodpeq0.thy │ ├── numbertheory_sqmod3in01d.thy │ ├── numbertheory_sqmod4in01d.thy │ ├── numbertheory_sumkmulnckeqnmul2pownm1.thy │ └── numbertheory_xsqpysqintdenomeq.thy ├── lean ├── LICENSE ├── scripts │ ├── lint_style.py │ ├── mk_minif2f.sh │ └── simple_formatter.sh └── src │ ├── minif2f_import.lean │ ├── test.lean │ └── valid.lean ├── leanpkg.toml └── metamath ├── LICENSE ├── test ├── aime-1983-p1.mm ├── aime-1983-p2.mm ├── aime-1983-p3.mm ├── aime-1984-p1.mm ├── aime-1984-p7.mm ├── aime-1987-p5.mm ├── aime-1988-p8.mm ├── aime-1989-p8.mm ├── aime-1990-p15.mm ├── aime-1990-p4.mm ├── aime-1991-p9.mm ├── aime-1994-p3.mm ├── aime-1995-p7.mm ├── aime-1997-p9.mm ├── aime-1999-p11.mm ├── algebra-2varlineareq-fp3zeq11-3tfm1m5zeqn68-feqn10-zeq7.mm ├── algebra-9onxpypzleqsum2onxpy.mm ├── algebra-abpbcpcageq3-sumaonsqrtapbgeq3onsqrt2.mm ├── algebra-absapbon1pabsapbleqsumabsaon1pabsa.mm ├── algebra-absxm1pabsxpabsxp1eqxp2-0leqxleq1.mm ├── algebra-amgm-sum1toneqn-prod1tonleq1.mm ├── algebra-amgm-sumasqdivbgeqsuma.mm ├── algebra-apbmpcneq0-aeq0anbeq0anceq0.mm ├── algebra-apbon2pownleqapownpbpowon2.mm ├── algebra-apbpceq2-abpbcpcaeq1-aleq1on3anbleq1ancleq4on3.mm ├── algebra-bleqa-apbon2msqrtableqambsqon8b.mm ├── algebra-cubrtrp1oncubrtreq3-rcubp1onrcubeq5778.mm ├── algebra-ineq-nto1onlt2m1on.mm ├── algebra-others-exirrpowirrrat.mm ├── algebra-sqineq-at2malt1.mm ├── algebra-sqineq-unitcircatbpabsamblt1.mm ├── algebra-sqineq-unitcircatbpamblt1.mm ├── algebra-sum1onsqrt2to1onsqrt10000lt198.mm ├── amc12-2000-p1.mm ├── amc12-2000-p12.mm ├── amc12-2000-p20.mm ├── amc12-2000-p6.mm ├── amc12-2001-p21.mm ├── amc12-2001-p5.mm ├── amc12a-2002-p13.mm ├── amc12a-2002-p6.mm ├── amc12a-2003-p23.mm ├── amc12a-2003-p5.mm ├── amc12a-2008-p25.mm ├── amc12a-2009-p6.mm ├── amc12a-2009-p7.mm ├── amc12a-2013-p4.mm ├── amc12a-2019-p12.mm ├── amc12a-2020-p10.mm ├── amc12a-2020-p15.mm ├── amc12a-2020-p25.mm ├── amc12a-2020-p4.mm ├── amc12a-2020-p7.mm ├── amc12a-2020-p9.mm ├── amc12a-2021-p12.mm ├── amc12a-2021-p14.mm ├── amc12a-2021-p18.mm ├── amc12a-2021-p19.mm ├── amc12a-2021-p22.mm ├── amc12a-2021-p25.mm ├── amc12a-2021-p3.mm ├── amc12a-2021-p8.mm ├── amc12a-2021-p9.mm ├── amc12b-2002-p19.mm ├── amc12b-2002-p2.mm ├── amc12b-2002-p4.mm ├── amc12b-2002-p7.mm ├── amc12b-2020-p13.mm ├── amc12b-2020-p2.mm ├── amc12b-2020-p21.mm ├── amc12b-2020-p22.mm ├── amc12b-2020-p6.mm ├── amc12b-2021-p1.mm ├── amc12b-2021-p13.mm ├── amc12b-2021-p18.mm ├── amc12b-2021-p3.mm ├── amc12b-2021-p4.mm ├── amc12b-2021-p9.mm ├── imo-1959-p1.mm ├── imo-1960-p2.mm ├── imo-1962-p2.mm ├── imo-1963-p5.mm ├── imo-1964-p2.mm ├── imo-1965-p2.mm ├── imo-1968-p5-1.mm ├── imo-1969-p2.mm ├── imo-1974-p3.mm ├── imo-1977-p6.mm ├── imo-1981-p6.mm ├── imo-1982-p1.mm ├── imo-1983-p6.mm ├── imo-1984-p6.mm ├── imo-1985-p6.mm ├── imo-1992-p1.mm ├── imo-1997-p5.mm ├── imo-2001-p6.mm ├── imo-2007-p6.mm ├── imo-2019-p1.mm ├── induction-11div10tonmn1ton.mm ├── induction-12dvd4expnp1p20.mm ├── induction-1pxpownlt1pnx.mm ├── induction-nfactltnexpnm1ngt3.mm ├── induction-pord1p1on2powklt5on2.mm ├── induction-pprime-pdvdapowpma.mm ├── induction-prod1p1onk3le3m1onn.mm ├── induction-sumkexp3eqsumksqsq.mm ├── mathd-algebra-107.mm ├── mathd-algebra-113.mm ├── mathd-algebra-114.mm ├── mathd-algebra-125.mm ├── mathd-algebra-129.mm ├── mathd-algebra-137.mm ├── mathd-algebra-139.mm ├── mathd-algebra-141.mm ├── mathd-algebra-142.mm ├── mathd-algebra-143.mm ├── mathd-algebra-148.mm ├── mathd-algebra-153.mm ├── mathd-algebra-156.mm ├── mathd-algebra-158.mm ├── mathd-algebra-160.mm ├── mathd-algebra-17.mm ├── mathd-algebra-170.mm ├── mathd-algebra-171.mm ├── mathd-algebra-176.mm ├── mathd-algebra-184.mm ├── mathd-algebra-188.mm ├── mathd-algebra-196.mm ├── mathd-algebra-208.mm ├── mathd-algebra-209.mm ├── mathd-algebra-215.mm ├── mathd-algebra-24.mm ├── mathd-algebra-246.mm ├── mathd-algebra-263.mm ├── mathd-algebra-270.mm ├── mathd-algebra-275.mm ├── mathd-algebra-276.mm ├── mathd-algebra-288.mm ├── mathd-algebra-289.mm ├── mathd-algebra-293.mm ├── mathd-algebra-296.mm ├── mathd-algebra-302.mm ├── mathd-algebra-304.mm ├── mathd-algebra-313.mm ├── mathd-algebra-314.mm ├── mathd-algebra-320.mm ├── mathd-algebra-329.mm ├── mathd-algebra-33.mm ├── mathd-algebra-332.mm ├── mathd-algebra-338.mm ├── mathd-algebra-342.mm ├── mathd-algebra-346.mm ├── mathd-algebra-354.mm ├── mathd-algebra-359.mm ├── mathd-algebra-362.mm ├── mathd-algebra-388.mm ├── mathd-algebra-392.mm ├── mathd-algebra-398.mm ├── mathd-algebra-400.mm ├── mathd-algebra-412.mm ├── mathd-algebra-419.mm ├── mathd-algebra-427.mm ├── mathd-algebra-432.mm ├── mathd-algebra-44.mm ├── mathd-algebra-440.mm ├── mathd-algebra-441.mm ├── mathd-algebra-452.mm ├── mathd-algebra-459.mm ├── mathd-algebra-478.mm ├── mathd-algebra-484.mm ├── mathd-algebra-487.mm ├── mathd-algebra-513.mm ├── mathd-algebra-598.mm ├── mathd-algebra-756.mm ├── mathd-algebra-76.mm ├── mathd-algebra-80.mm ├── mathd-numbertheory-100.mm ├── mathd-numbertheory-1124.mm ├── mathd-numbertheory-12.mm ├── mathd-numbertheory-127.mm ├── mathd-numbertheory-135.mm ├── mathd-numbertheory-150.mm ├── mathd-numbertheory-175.mm ├── mathd-numbertheory-185.mm ├── mathd-numbertheory-207.mm ├── mathd-numbertheory-212.mm ├── mathd-numbertheory-222.mm ├── mathd-numbertheory-227.mm ├── mathd-numbertheory-229.mm ├── mathd-numbertheory-233.mm ├── mathd-numbertheory-234.mm ├── mathd-numbertheory-235.mm ├── mathd-numbertheory-237.mm ├── mathd-numbertheory-239.mm ├── mathd-numbertheory-247.mm ├── mathd-numbertheory-254.mm ├── mathd-numbertheory-277.mm ├── mathd-numbertheory-293.mm ├── mathd-numbertheory-296.mm ├── mathd-numbertheory-299.mm ├── mathd-numbertheory-3.mm ├── mathd-numbertheory-314.mm ├── mathd-numbertheory-320.mm ├── mathd-numbertheory-321.mm ├── mathd-numbertheory-328.mm ├── mathd-numbertheory-34.mm ├── mathd-numbertheory-341.mm ├── mathd-numbertheory-342.mm ├── mathd-numbertheory-343.mm ├── mathd-numbertheory-345.mm ├── mathd-numbertheory-353.mm ├── mathd-numbertheory-427.mm ├── mathd-numbertheory-430.mm ├── mathd-numbertheory-435.mm ├── mathd-numbertheory-447.mm ├── mathd-numbertheory-451.mm ├── mathd-numbertheory-457.mm ├── mathd-numbertheory-483.mm ├── mathd-numbertheory-495.mm ├── mathd-numbertheory-5.mm ├── mathd-numbertheory-517.mm ├── mathd-numbertheory-521.mm ├── mathd-numbertheory-541.mm ├── mathd-numbertheory-551.mm ├── mathd-numbertheory-552.mm ├── mathd-numbertheory-559.mm ├── mathd-numbertheory-582.mm ├── mathd-numbertheory-618.mm ├── mathd-numbertheory-66.mm ├── mathd-numbertheory-711.mm ├── mathd-numbertheory-728.mm ├── mathd-numbertheory-764.mm ├── mathd-numbertheory-765.mm ├── mathd-numbertheory-769.mm ├── mathd-numbertheory-85.mm ├── mathd-numbertheory-99.mm ├── numbertheory-2pownm1prime-nprime.mm ├── numbertheory-3pow2pownm1mod2pownp3eq2pownp2.mm ├── numbertheory-4x3m7y3neq2003.mm ├── numbertheory-aoddbdiv4asqpbsqmod8eq1.mm ├── numbertheory-exk2powkeqapb2mulbpa2-aeq1.mm ├── numbertheory-fxeq4powxp6powxp9powx-f2powmdvdf2pown.mm ├── numbertheory-notequiv2i2jasqbsqdiv8.mm └── numbertheory-x5neqy2p4.mm └── valid ├── aime-1983-p9.mm ├── aime-1984-p15.mm ├── aime-1984-p5.mm ├── aime-1987-p8.mm ├── aime-1988-p3.mm ├── aime-1988-p4.mm ├── aime-1990-p2.mm ├── aime-1991-p1.mm ├── aime-1991-p6.mm ├── aime-1994-p4.mm ├── aime-1996-p5.mm ├── aime-1997-p12.mm ├── aimeI-2000-p7.mm ├── aimeI-2001-p3.mm ├── aimeII-2020-p6.mm ├── algebra-2complexrootspoly-xsqp49eqxp7itxpn7i.mm ├── algebra-2rootsintpoly-am10tap11eqasqpam110.mm ├── algebra-2rootspoly-apatapbeq2asqp2ab.mm ├── algebra-2varlineareq-xpeeq7-2xpeeq3-eeq11-xeqn4.mm ├── algebra-3rootspoly-amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta.mm ├── algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x.mm ├── algebra-amgm-prod1toneq1-sum1tongeqn.mm ├── algebra-amgm-sqrtxymulxmyeqxpy-xpygeq4.mm ├── algebra-amgm-sumasqdivbsqgeqsumbdiva.mm ├── algebra-apb4leq8ta4pb4.mm ├── algebra-binomnegdiscrineq-10alt28asqp1.mm ├── algebra-manipexpr-2erprsqpesqeqnrpnesq.mm ├── algebra-manipexpr-apbeq2cceqiacpbceqm2.mm ├── algebra-sqineq-2at2pclta2c2p41pc.mm ├── algebra-sqineq-2unitcircatblt1.mm ├── algebra-sqineq-36azm9asqle36zsq.mm ├── algebra-sqineq-4bap1lt4bsqpap1sq.mm ├── algebra-xmysqpymzsqpzmxsqeqxyz-xpypzp6dvdx3y3z3.mm ├── amc12-2000-p11.mm ├── amc12-2000-p15.mm ├── amc12-2000-p5.mm ├── amc12-2001-p2.mm ├── amc12-2001-p9.mm ├── amc12a-2002-p1.mm ├── amc12a-2002-p12.mm ├── amc12a-2002-p21.mm ├── amc12a-2003-p1.mm ├── amc12a-2003-p24.mm ├── amc12a-2003-p25.mm ├── amc12a-2008-p15.mm ├── amc12a-2008-p2.mm ├── amc12a-2008-p4.mm ├── amc12a-2008-p8.mm ├── amc12a-2009-p15.mm ├── amc12a-2009-p2.mm ├── amc12a-2009-p25.mm ├── amc12a-2009-p5.mm ├── amc12a-2009-p9.mm ├── amc12a-2010-p10.mm ├── amc12a-2010-p11.mm ├── amc12a-2010-p22.mm ├── amc12a-2011-p18.mm ├── amc12a-2013-p7.mm ├── amc12a-2013-p8.mm ├── amc12a-2015-p10.mm ├── amc12a-2016-p2.mm ├── amc12a-2016-p3.mm ├── amc12a-2017-p2.mm ├── amc12a-2017-p7.mm ├── amc12a-2019-p21.mm ├── amc12a-2019-p9.mm ├── amc12a-2020-p13.mm ├── amc12a-2020-p22.mm ├── amc12a-2021-p7.mm ├── amc12b-2002-p11.mm ├── amc12b-2002-p3.mm ├── amc12b-2002-p6.mm ├── amc12b-2003-p17.mm ├── amc12b-2003-p6.mm ├── amc12b-2003-p9.mm ├── amc12b-2004-p3.mm ├── amc12b-2020-p5.mm ├── amc12b-2021-p21.mm ├── imo-1961-p1.mm ├── imo-1962-p4.mm ├── imo-1964-p1-1.mm ├── imo-1964-p1-2.mm ├── imo-1965-p1.mm ├── imo-1966-p4.mm ├── imo-1966-p5.mm ├── imo-1967-p3.mm ├── imo-1973-p3.mm ├── imo-1974-p5.mm ├── imo-1977-p5.mm ├── imo-1978-p5.mm ├── imo-1979-p1.mm ├── imo-1984-p2.mm ├── imo-1987-p4.mm ├── imo-1987-p6.mm ├── imo-1988-p6.mm ├── imo-1990-p3.mm ├── imo-1993-p5.mm ├── imo-2006-p6.mm ├── induction-divisibility-3div2tooddnp1.mm ├── induction-divisibility-3divnto3m2n.mm ├── induction-divisibility-9div10tonm1.mm ├── induction-ineq-nsqlefactn.mm ├── induction-seq-mul2pnp1.mm ├── induction-sum-1oktkp1.mm ├── induction-sum-odd.mm ├── induction-sum2kp1npqsqm1.mm ├── mathd-algebra-10.mm ├── mathd-algebra-101.mm ├── mathd-algebra-104.mm ├── mathd-algebra-109.mm ├── mathd-algebra-11.mm ├── mathd-algebra-110.mm ├── mathd-algebra-116.mm ├── mathd-algebra-119.mm ├── mathd-algebra-123.mm ├── mathd-algebra-126.mm ├── mathd-algebra-13.mm ├── mathd-algebra-131.mm ├── mathd-algebra-132.mm ├── mathd-algebra-140.mm ├── mathd-algebra-144.mm ├── mathd-algebra-149.mm ├── mathd-algebra-15.mm ├── mathd-algebra-151.mm ├── mathd-algebra-159.mm ├── mathd-algebra-181.mm ├── mathd-algebra-182.mm ├── mathd-algebra-185.mm ├── mathd-algebra-190.mm ├── mathd-algebra-192.mm ├── mathd-algebra-206.mm ├── mathd-algebra-214.mm ├── mathd-algebra-22.mm ├── mathd-algebra-224.mm ├── mathd-algebra-234.mm ├── mathd-algebra-245.mm ├── mathd-algebra-247.mm ├── mathd-algebra-251.mm ├── mathd-algebra-267.mm ├── mathd-algebra-28.mm ├── mathd-algebra-282.mm ├── mathd-algebra-31.mm ├── mathd-algebra-323.mm ├── mathd-algebra-327.mm ├── mathd-algebra-35.mm ├── mathd-algebra-37.mm ├── mathd-algebra-393.mm ├── mathd-algebra-405.mm ├── mathd-algebra-410.mm ├── mathd-algebra-421.mm ├── mathd-algebra-422.mm ├── mathd-algebra-43.mm ├── mathd-algebra-433.mm ├── mathd-algebra-437.mm ├── mathd-algebra-451.mm ├── mathd-algebra-455.mm ├── mathd-algebra-462.mm ├── mathd-algebra-48.mm ├── mathd-algebra-480.mm ├── mathd-algebra-482.mm ├── mathd-algebra-493.mm ├── mathd-algebra-509.mm ├── mathd-algebra-51.mm ├── mathd-algebra-510.mm ├── mathd-algebra-536.mm ├── mathd-algebra-547.mm ├── mathd-algebra-55.mm ├── mathd-algebra-568.mm ├── mathd-algebra-59.mm ├── mathd-algebra-616.mm ├── mathd-algebra-67.mm ├── mathd-algebra-69.mm ├── mathd-algebra-73.mm ├── mathd-algebra-77.mm ├── mathd-algebra-89.mm ├── mathd-algebra-96.mm ├── mathd-numbertheory-101.mm ├── mathd-numbertheory-102.mm ├── mathd-numbertheory-109.mm ├── mathd-numbertheory-110.mm ├── mathd-numbertheory-126.mm ├── mathd-numbertheory-13.mm ├── mathd-numbertheory-132.mm ├── mathd-numbertheory-136.mm ├── mathd-numbertheory-149.mm ├── mathd-numbertheory-155.mm ├── mathd-numbertheory-156.mm ├── mathd-numbertheory-169.mm ├── mathd-numbertheory-188.mm ├── mathd-numbertheory-198.mm ├── mathd-numbertheory-200.mm ├── mathd-numbertheory-202.mm ├── mathd-numbertheory-211.mm ├── mathd-numbertheory-22.mm ├── mathd-numbertheory-221.mm ├── mathd-numbertheory-232.mm ├── mathd-numbertheory-236.mm ├── mathd-numbertheory-24.mm ├── mathd-numbertheory-252.mm ├── mathd-numbertheory-257.mm ├── mathd-numbertheory-269.mm ├── mathd-numbertheory-284.mm ├── mathd-numbertheory-30.mm ├── mathd-numbertheory-301.mm ├── mathd-numbertheory-303.mm ├── mathd-numbertheory-32.mm ├── mathd-numbertheory-326.mm ├── mathd-numbertheory-33.mm ├── mathd-numbertheory-335.mm ├── mathd-numbertheory-35.mm ├── mathd-numbertheory-37.mm ├── mathd-numbertheory-370.mm ├── mathd-numbertheory-403.mm ├── mathd-numbertheory-405.mm ├── mathd-numbertheory-412.mm ├── mathd-numbertheory-42.mm ├── mathd-numbertheory-43.mm ├── mathd-numbertheory-45.mm ├── mathd-numbertheory-458.mm ├── mathd-numbertheory-461.mm ├── mathd-numbertheory-466.mm ├── mathd-numbertheory-48.mm ├── mathd-numbertheory-530.mm ├── mathd-numbertheory-543.mm ├── mathd-numbertheory-629.mm ├── mathd-numbertheory-64.mm ├── mathd-numbertheory-640.mm ├── mathd-numbertheory-668.mm ├── mathd-numbertheory-690.mm ├── mathd-numbertheory-709.mm ├── mathd-numbertheory-739.mm ├── mathd-numbertheory-780.mm ├── mathd-numbertheory-81.mm ├── mathd-numbertheory-84.mm ├── mathd-numbertheory-92.mm ├── mathd-numbertheory-961.mm ├── numbertheory-2dvd4expn.mm ├── numbertheory-aneqprodakp4-anmsqrtanp1eq2.mm ├── numbertheory-nckeqnm1ckpnm1ckm1.mm ├── numbertheory-prmdvsneqnsqmodpeq0.mm ├── numbertheory-sqmod3in01d.mm ├── numbertheory-sqmod4in01d.mm ├── numbertheory-sumkmulnckeqnmul2pownm1.mm └── numbertheory-xsqpysqintdenomeq.mm /.gitignore: -------------------------------------------------------------------------------- 1 | *.olean 2 | /_target 3 | /leanpkg.path 4 | .vscode 5 | minif2f.lean 6 | .vscode 7 | -------------------------------------------------------------------------------- /hollight/test/aime-1983-p1.ml: -------------------------------------------------------------------------------- 1 | let aime-1983-p1 = `!x:num y:num z:num w:num. 2 | ((1 < x) /\ (1 < y) /\ (1 < z)) /\ 3 | (0 <= w) /\ 4 | (ln (&w) / ln (&x) = &24) /\ 5 | (ln (&w) / ln (&y) = &40) /\ 6 | (ln (&w) / ln (&(x * y * z)) = &12) 7 | ==> 8 | (ln (&w) / ln (&z) = &60) 9 | `;; 10 | -------------------------------------------------------------------------------- /hollight/test/aime-1983-p2.ml: -------------------------------------------------------------------------------- 1 | let aime-1983-p2 = `!x p f. (&0 < p /\ p < &15) /\ (p <= x /\ x <= &15) /\ (f x = abs (x - p) + abs (x - &15) + abs (x - p - &15)) ==> &15 <= f x`;; 2 | -------------------------------------------------------------------------------- /hollight/test/aime-1983-p3.ml: -------------------------------------------------------------------------------- 1 | let aime-1983-p3 = `!f:real->real. 2 | (!x. f x = (x pow 2 + &18 * x + &30 - &2 * sqrt (x pow 2 + &18 * x + &45))) /\ 3 | (FINITE {x | f x = &0}) 4 | ==> 5 | product {x | f x = &0} (\x. x) = &20 6 | `;; 7 | 8 | -------------------------------------------------------------------------------- /hollight/test/aime-1984-p1.ml: -------------------------------------------------------------------------------- 1 | let aime-1984-p1 = `!u:num->real. 2 | (!n. rational (u n)) /\ 3 | (!n. u (n + 1) = u n + &1) /\ 4 | (sum (0..(98 - 1)) (\k. u (k + 1)) = &137) 5 | ==> 6 | (sum (0..(49 - 1)) (\k. u (2 * (k + 1))) = &93) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/aime-1984-p7.ml: -------------------------------------------------------------------------------- 1 | let aime-1984-p7 = `!f. (!n. 1000 <= n ==> f n = n - 3) /\ (!n. 0 < n /\ n < 1000 ==> f n = f (f (n + 5))) ==> f 84 = 997`;; 2 | -------------------------------------------------------------------------------- /hollight/test/aime-1994-p3.ml: -------------------------------------------------------------------------------- 1 | let aime-1994-p3 = `!x:int f:int->int. 2 | (f (x) + f (x - &1) = x pow 2) /\ 3 | (f (&19) = &94) 4 | ==> 5 | ((f (&94) == &561) (mod &1000)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/aime-1995-p7.ml: -------------------------------------------------------------------------------- 1 | let aime-1995-p7 = `!k:num m:num n:num t:real. 2 | (k > 0 /\ m > 0 /\ n > 0) /\ 3 | (gcd(m,n) = 1) /\ 4 | ((&1 + sin t) * (&1 + cos t) = &5 / &4) /\ 5 | ((&1 - sin t) * (&1 - cos t) = &m / &n - sqrt (&k)) 6 | ==> 7 | k + m + n = 27 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/algebra-2varlineareq-fp3zeq11-3tfm1m5zeqn68-feqn10-zeq7.ml: -------------------------------------------------------------------------------- 1 | let algebra-2varlineareq-fp3zeq11-3tfm1m5zeqn68-feqn10-zeq7 = `!f:complex z:complex. 2 | (f + Cx(&3) * z = Cx(&11)) /\ 3 | (Cx(&3) * (f - Cx(&1)) - Cx(&5) * z = Cx(-- &68)) 4 | ==> 5 | (f = Cx(-- &10)) /\ 6 | (z = Cx(&7)) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/algebra-amgm-sum1toneqn-prod1tonleq1.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-sum1toneqn-prod1tonleq1 = `!a:num->real n:num. (!k:num. a k >= &0) /\ ((sum (0..(n-1)) (\x. a x)) = &n) ==> (product (0..(n-1)) (\x. a x)) <= &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/algebra-amgm-sumasqdivbgeqsuma.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-sumasqdivbgeqsuma = `!a b c d. &0 < a /\ &0 < b /\ &0 < c /\ &0 < d ==> a pow 2 / b + b pow 2 / c + c pow 2 / d + d pow 2 / a >= a + b + c + d`;; 2 | -------------------------------------------------------------------------------- /hollight/test/algebra-ineq-nto1onlt2m1on.ml: -------------------------------------------------------------------------------- 1 | let algebra-ineq-nto1onlt2m1on = `!n:num. exp ((&1 / &n) * ln(&n)) < &2 - &1 / &n`;; 2 | -------------------------------------------------------------------------------- /hollight/test/algebra-others-exirrpowirrrat.ml: -------------------------------------------------------------------------------- 1 | let algebra-others-exirrpowirrrat = `?a:real b:real. ~rational a /\ ~rational b /\ rational (exp (b * ln a))`;; 2 | -------------------------------------------------------------------------------- /hollight/test/algebra-sqineq-at2malt1.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-at2malt1 = `!a:real. a * (&2 - a) <= &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/algebra-sqineq-unitcircatbpabsamblt1.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-unitcircatbpabsamblt1 = `!a:real b:real. 2 | (a pow 2 + b pow 2 = &1) 3 | ==> 4 | (a * b + abs (a - b) <= &1) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/algebra-sqineq-unitcircatbpamblt1.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-unitcircatbpamblt1 = `!a:real b:real. (a pow 2 + b pow 2 = &1) ==> a * b + (a - b) <= &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2000-p1.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p1 = let amc12-2000-p1 = `!i m n. ~(i = 0) /\ ~(m = 0) /\ ~(n = 0) /\ (i * m * n = 2001) ==> i + m + n <= 671`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2000-p12.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p12 = `!a m c. (a + m + c = 12) ==> a * m * c + a * m + m * c + a * c <= 112`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2000-p20.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p20 = `!x:real y z. (&0 < x /\ &0 < y /\ &0 < z) /\ (x + &1 / y = &4) /\ (y + &1 / z = &1) /\ (z + &1 / x = &7 / &3) ==> x * y * z = &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2000-p6.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p6 = let amc12-2000-p6 = `!p q. (prime p /\ prime q) /\ (4 <= p /\ p <= 18) /\ (4 <= q /\ q <= 18) ==> ~(p * q - (p + q) = 194)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2001-p21.ml: -------------------------------------------------------------------------------- 1 | let amc12-2001-p21 = `!a b c d. a * b * c * d = FACT 8 /\ a * b + a + b = 524 /\ b * c + b + c = 146 /\ c * d + c + d = 104 ==> a - d = 10`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12-2001-p5.ml: -------------------------------------------------------------------------------- 1 | let amc12-2001-p5 = `product {x | x < 10000 /\ ODD x} (\x. &x) = &(FACT 10000) / &((2 EXP 5000) * (FACT 5000))`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2002-p13.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2002-p13 = `!a b. (&0 < a /\ &0 < b) /\ ~(a = b) /\ (abs (a - &1 / a) = &1) /\ (abs (b - &1 / b) = &1) ==> a + b = sqrt(&5)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2002-p6.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2002-p6 = `!n. ~(n = 0) ==> (?m. m > n /\ ?p. m * p <= m + p )`;; 2 | 3 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2003-p23.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2003-p23 = `FINITE {k:num | k > 0 /\ (k * k) divides (nproduct (1..(10-1)) (\i. FACT i))} ==> CARD {k:num | k > 0 /\ (k * k) divides (nproduct (1..(10-1)) (\i. FACT i))} = 672`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2003-p5.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2003-p5 = `!a m c. (a <= 9 /\ m <= 9 /\ c <= 9) /\ (10 * (10 * (10 * (10 * a + m) + c) + 1) + 0 + (10 * (10 * (10 * (10 * a + m) + c) + 1) + 2) = 123422) ==> a + m + c = 14`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2020-p10.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2020-p10 = `!n:num. 2 | (n > 0) /\ 3 | (ln (ln (&n) / ln (&16)) / ln (&2) = ln (ln (&n) / ln (&4)) / ln (&4)) 4 | ==> 5 | n = 256 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2020-p15.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2020-p15 = `!a:complex b:complex. 2 | (a pow 3 - Cx(&8) = Cx(&0)) /\ 3 | (b pow 3 - Cx(&8) * b pow 2 - Cx(&8) * b + Cx(&64) = Cx(&0)) 4 | ==> 5 | norm (a - b) <= &2 * sqrt (&21) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2020-p9.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2020-p9 = `(FINITE {x:real | (&0 <= x) /\ (x <= &2 * pi) /\ (tan (&2 * x) = cos (x / &2))}) 2 | ==> 3 | (CARD {x:real | (&0 <= x) /\ (x <= &2 * pi) /\ (tan (&2 * x) = cos (x / &2))} = 5) 4 | `;; 5 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p12.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p12 = `!a:real b:real c:real d:real f:complex->complex. 2 | (!z. f z = z pow 6 - Cx(&10) * z pow 5 + Cx(a) * z pow 4 + Cx(b) * z pow 3 + Cx(c) * z pow 2 + Cx(d) * z + Cx(&16)) /\ 3 | (!z. f z = Cx(&0) ==> (Im z = &0 /\ &0 < Re z /\ (floor (Re z)) = Re z)) 4 | ==> 5 | b = &88 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p14.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p14 = `sum (1..(21-1)) (\k. ln (&(3 EXP (k EXP 2))) / ln (&(5 EXP k))) * sum (1..(101-1)) (\k. ln (&(25 EXP k)) / ln (&(9 EXP k))) = &21000`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p18.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p18 = `!f:real->real. 2 | (!x:real y:real. ((rational x) /\ (rational y) /\ (x > &0) /\ (y > &0)) ==> (f (x * y) = f (x) + f (y))) /\ 3 | (!p:num. prime p ==> f (&p) = &p) 4 | ==> 5 | f (&25 / &11) < &0 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p19.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p19 = `(FINITE {x:real | (&0 <= x) /\ (x <= pi) /\ (sin (pi / &2 * cos x) = cos (pi / &2 * sin x))}) 2 | ==> 3 | (CARD {x:real | (&0 <= x) /\ (x <= pi) /\ (sin (pi / &2 * cos x) = cos (pi / &2 * sin x))} = 2) 4 | `;; 5 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p22.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p22 = `!a:real b:real c:real f:real->real. 2 | (!x. f x = x pow 3 + a * x pow 2 + b * x + c) /\ 3 | ({x | f x = &0} = {cos (&2 * pi / &7), cos (&4 * pi / &7), cos (&6 * pi / &7)}) 4 | ==> 5 | a * b * c = &1 / &32 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p25.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p25 = `!n:num f:num->real. 2 | (n > 0 /\ !n. n > 0 ==> f n = sum {k | k divides n} (\k. &k) / (exp (&1 / &3 * ln (&n)))) /\ 3 | (!p. ~(p = n) ==> f p < f n) 4 | ==> 5 | n = 2520 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p3.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p3 = `!x:num y:num. 2 | (x + y = 17402) /\ 3 | (10 divides x) /\ 4 | (x DIV 10 = y) 5 | ==> 6 | (int_of_num (x) - &y = &14238) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p8.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p8 = `!d:num->num. 2 | (d 0 = 0) /\ 3 | (d 1 = 0) /\ 4 | (d 2 = 1) /\ 5 | (!n. n >=3 ==> d n = d (n - 1) + d (n - 3)) 6 | ==> 7 | EVEN (d 2021) /\ ODD (d 2022) /\ EVEN (d 2023) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/amc12a-2021-p9.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p9 = `&(nproduct (0..(7-1)) (\k. 2 EXP (2 EXP k) + 3 EXP (2 EXP k))) = int_of_num (3 EXP 128) - &(2 EXP 128)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2002-p19.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p19 = `!a:real b:real c:real. 2 | (&0 < a /\ &0 < b /\ &0 < c) /\ 3 | (a * (b + c) = &152) /\ 4 | (b * (c + a) = &162) /\ 5 | (c * (a + b) = &170) 6 | ==> a * b * c = &720`;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2002-p2.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p2 = `!x:int. (x = &4) ==> ( &3 * x - &2 ) * ( &4 * x + &1 ) - ( &3 * x - &2 ) * ( &4 * x ) + &1 = &11`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2002-p4.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p4 = `!n:num. 2 | (n > &0) /\ 3 | (?p. (&1 / &2 + &1 / &3 + &1 / &7 + &1 / &n) = &p) 4 | ==> 5 | (n = 42) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2002-p7.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p7 = `!a:num b:num c:num. 2 | ~(a = 0) /\ 3 | ~(b = 0) /\ 4 | ~(c = 0) /\ 5 | (b = a + 1) /\ 6 | (c = b + 1) /\ 7 | (a * b * c = 8 * (a + b + c) ) 8 | ==> a EXP 2 + ( b EXP 2 + c EXP 2 ) = 77`;; 9 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2020-p13.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p13 = `sqrt (ln (&6) / ln (&2) + ln (&6) / ln (&3)) = sqrt (ln (&3) / ln (&2)) + sqrt (ln (&2) / ln (&3))`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2020-p2.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p2 = `real_of_int (int_of_num (100 EXP 2) - &(7 EXP 2)) / real_of_int (int_of_num (70 EXP 2) - &(11 EXP 2)) * (&(70 - 11) * &(70 + 11) / &((100 - 7) * (100 + 7))) = &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2020-p21.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p21 = `FINITE {n | n > 0 /\ &(n + 1000) / &70 = floor (sqrt (&n))} ==> CARD {n | n > 0 /\ &(n + 1000) / &70 = floor (sqrt (&n))} = 6`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2020-p22.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p22 = `!t:real. ((exp (t * ln (&2)) - &3 * t) * t) / (exp(t * ln (&4))) <= &1 / &12`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2020-p6.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p6 = `!n:num. 9 <= n ==> ?x. &(x EXP 2) = real_of_int (int_of_num (FACT (n + 2)) - &(FACT (n + 1))) / &(FACT n)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p1.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p1 = `(FINITE {x:int | real_of_int (abs x) < &3 * pi}) ==> (CARD {x:int | real_of_int (abs x) < &3 * pi} = 19)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p13.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p13 = `FINITE {x | &0 < x /\ x <= &2 * pi /\ &1 - &3 * sin x + &5 * cos (&3 * x) = &0} ==> CARD {x | &0 < x /\ x <= &2 * pi /\ &1 - &3 * sin x + &5 * cos (&3 * x) = &0} = 6`;; 2 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p18.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p18 = `!z:complex. 2 | (&12 * (norm z) pow 2 = &2 * (norm (z + Cx (&2))) pow 2 + (norm (z pow 2 + Cx (&1))) pow 2 + &31) 3 | ==> 4 | z + Cx (&6) / z = -- Cx(&2)`;; 5 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p3.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p3 = `!x:real. 2 | (&2 + &1 / (&1 + &1 / (&2 + &2 / (&3 + x))) = &144 / &53) 3 | ==> 4 | (x = &3 / &4) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p4.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p4 = `!m:num a:num. 2 | (m > 0 /\ a > 0) /\ 3 | (&m / &a = &3 / &4) 4 | ==> 5 | (&84 * &m + &70 * &a) / (&m + &a) = &76 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/amc12b-2021-p9.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2021-p9 = `(ln (&80) / ln (&2)) / (ln (&2) / ln (&40)) - (ln (&160) / ln (&2)) / (ln (&2) / ln (&20)) = &2`;; 2 | -------------------------------------------------------------------------------- /hollight/test/imo-1959-q1.ml: -------------------------------------------------------------------------------- 1 | let imo-1959-q1 = `!n. n > 0 ==> gcd (21 * n + 4, 14 * n + 3) = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/imo-2007-a6.ml: -------------------------------------------------------------------------------- 1 | let imo-2007-a6 = `!a:num->real. 2 | (!k. a k >= &0) /\ 3 | sum (0..(100-1)) (\x. (a (x + 1)) pow 2) = &1 4 | ==> 5 | sum (0..(99-1)) (\x. ((a (x + 1)) pow 2 * a (x + 2))) + (a 100) pow 2 * a 1 < &12 / &25 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/imo-2019-q1.ml: -------------------------------------------------------------------------------- 1 | let imo-2019-q1 = `!f. (!a:int b:int. f (&2 * a) + &2 * f b = f (f (a + b))) <=> (!z:int. f z = &0 \/ ?c:int. !z. f z = &2 * z + c)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/induction-11div10tonmn1ton.ml: -------------------------------------------------------------------------------- 1 | let induction-11div10tonmn1ton = `!n. 11 divides num_of_int (&(10 EXP n) - (-- &1) pow n)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/induction-12dvd4expnp1p20.ml: -------------------------------------------------------------------------------- 1 | let induction-12dvd4expnp1p20 = `!n. 12 divides 4 EXP (n+1) + 20`;; 2 | -------------------------------------------------------------------------------- /hollight/test/induction-1pxpownlt1pnx.ml: -------------------------------------------------------------------------------- 1 | let induction-1pxpownlt1pnx = `!x n. n > 0 /\ -- &1 < x ==> (&1 + x) pow n <= &1 + &n * x`;; 2 | -------------------------------------------------------------------------------- /hollight/test/induction-nfactltnexpnm1ngt3.ml: -------------------------------------------------------------------------------- 1 | let induction-nfactltnexpnm1ngt3 = `!n. 3 <= n ==> FACT n < n EXP (n - 1)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/induction-sumkexp3eqsumksqsq.ml: -------------------------------------------------------------------------------- 1 | let induction-sumkexp3eqsumksqsq = `!n. nsum(0..(n-1)) (\k. k EXP 3) = (nsum(0..(n-1)) (\k. k EXP 2)) EXP 2`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathb-algebra-44.ml: -------------------------------------------------------------------------------- 1 | let mathb-algebra-44 = `!s t. s = &9 - &2 * t /\ t = &3 * s + &1 ==> s = &1 /\ t = &4`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-107.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-107 = `!x y. x pow 2 + &8 * x + y pow 2 - &6 * y = &0 ==> (x + &4) pow 2 + (y - &3) pow 2 = &5 pow 2`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-113.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-113 = `!x. x pow 2 - &14 * x + &3 >= &7 pow 2 - &14 * &7 + &3`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-114.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-114 = `!a:real. (a = &8) ==> exp (&1 / &3 * ln (exp (&1 / &3 * ln (&16 * (a pow 2))))) = &4`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-125.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-125 = `!x:int y:int. x > &0 /\ y > &0 /\ &5 * x = y /\ x - &3 + y - &3 = &30 ==> x = &6`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-129.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-129 = `!a:real. 2 | ~(a = &0) /\ 3 | ((&1 / &8) / (&1 / &4) - (&1 / a) = &1) 4 | ==> 5 | a = -- &2 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-137.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-137 = `!x:num. 2 | (&x + &4 / &100 * &x = &598) 3 | ==> 4 | (x = 575) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-139.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-139 = `!s:real->real->real. 2 | (!x y. 3 | ~(x = &0) /\ ~(y = &0) 4 | ==> 5 | (s x y = ((&1) / y - (&1) / x) / (x-y)) 6 | ) 7 | ==> 8 | (s (&3) (&11) = (&1) / (&33)) 9 | `;; 10 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-141.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-141 = `!a:real b:real. 2 | (a >= &0) /\ (b >= &0) /\ 3 | ((a * b) = &180) /\ 4 | (&2 * (a + b) = &54) 5 | ==> 6 | a pow 2 + b pow 2 = &369 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-142.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-142 = `!m:real b:real. 2 | (m * &7 + b = -- &1) /\ 3 | (m * (-- &1) + b = &7) 4 | ==> 5 | m + b = &5 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-143.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-143 = `!f:real->real g:real->real. 2 | (!x. f x = x + &1) /\ 3 | (!x. g x = x pow 2 + &3) 4 | ==> 5 | (f (g (&2)) = &8) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-148.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-148 = `!c:real f:real->real. 2 | (!x. f x = c * x pow 3 - &9 * x + &3) /\ 3 | (f (&2) = &9) 4 | ==> 5 | c = &3 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-153.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-153 = `!n:real. 2 | (n = &1 / &3) 3 | ==> 4 | floor (&10 * n) + floor (&100 * n) + floor (&1000 * n) + floor (&10000 * n) = &3702 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-156.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-156 = `!x y f g. (!t. f t = t pow 4) /\ (!t. g t = &5*t pow 2 - &6) /\ (f x = g x) /\ (f y = g y) /\ (x pow 2 < y pow 2) ==> y pow 2 - x pow 2 = &1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-158.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-158 = `!a:num. 2 | (EVEN a) /\ 3 | (&(nsum (0..(8-1)) (\k. 2 * k + 1)) - &(nsum (0..(5-1)) (\k. a + 2 * k)) = int_of_num 4) 4 | ==> 5 | a = 8 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-160.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-160 = `!n:real x:real. 2 | (n + x = &97) /\ 3 | (n + &5 * x = &265) 4 | ==> 5 | (n + &2 * x = &139) 6 | `;; 7 | 8 | 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-17.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-17 = `!a. sqrt (&4 + sqrt (&16 + &16 * a)) + sqrt (&1 + sqrt (&1 + a)) = &6 ==> a = &8`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-170.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-170 = `FINITE {n:int | abs (real_of_int n - &2) <= &5 + &6 / &10} ==> CARD {n:int | abs (real_of_int n - &2) <= &5 + &6 / &10} = 11`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-171.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-171 = `!f. (!x. f x = &5 * x + &4) ==> f (&1) = &9`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-176.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-176 = `!x:real. (x + &1) pow 2 * x = x pow 3 + &2 * x pow 2 + x`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-184.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-184 = `!a:real b:real. 2 | (&0 < a /\ &0 < b) /\ 3 | ((a pow 2) = &6 * b) /\ 4 | ((a pow 2) = &54 / b) 5 | ==> 6 | a = &3 * sqrt (&2) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-196.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-196 = `FINITE {x:real | abs (&2 - x) = &3} ==> sum {x:real | abs (&2 - x) = &3} (\k. k) = &4`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-208.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-208 = `sqrt (&1000000) - exp ((&1 / &3) * ln (&1000000)) = &900`;; 2 | 3 | 4 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-215.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-215 = `(FINITE {x:real | (x + &3) pow 2 = &121}) ==> (sum ({x:real | (x + &3) pow 2 = &121}) (\k. k) = -- &6)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-24.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-24 = `!x. (x / &50 = &40) ==> x = &2000`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-246.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-246 = `!a:real b:real f:real->real. 2 | (!x. f x = a * x pow 4 - b * x pow 2 + x + &5) /\ 3 | (f (-- &3) = &2) 4 | ==> 5 | f (&3) = &8 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-263.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-263 = `!y:real. 2 | (&0 <= &19 + &3 * y) /\ 3 | (sqrt (&19 + &3 * y) = &7) 4 | ==> 5 | (y = &10) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-270.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-270 = `!f:real->real. 2 | (!x. ~(x = -- &2) ==> f x = &1 / (x + &2)) 3 | ==> 4 | f (f (&1)) = &3 / &7 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-275.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-275 = `!x:real. 2 | (exp ((&3 * x - &3) * ln (exp ((&1 / &4) * ln(&11)))) = &1 / &5) 3 | ==> 4 | (exp ((&6 * x + &2) * ln (exp ((&1 / &4) * ln(&11)))) = &121 / &25) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-276.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-276 = `!a:int b:int. 2 | (!x:real. &10 * x pow 2 - x - &24 = (real_of_int a * x - &8) * (real_of_int b * x + &3)) 3 | ==> 4 | a + b = &12 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-288.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-288 = `!x:real y:real n:real. 2 | (n >= &0) /\ 3 | (x < &0 /\ y < &0) /\ 4 | (abs x = &6) /\ 5 | (sqrt ((x - &8) pow 2 + (y - &3) pow 2) = &15) /\ 6 | (sqrt (x pow 2 + y pow 2) = sqrt n) 7 | ==> 8 | (n = &52) 9 | `;; 10 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-289.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-289 = `!k:num t:num m:num n:num. 2 | (prime m /\ prime n) /\ 3 | (t < k) /\ 4 | (int_of_num (k EXP 2) - &(m * k) + &n = &0) /\ 5 | (int_of_num (t EXP 2) - &(m * t) + &n = &0) 6 | ==> 7 | m EXP n + n EXP m + k EXP t + t EXP k = 20 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-293.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-293 = `!x:real. (x >= &0) ==> sqrt (&60 * x) * sqrt (&12 * x) * sqrt (&63 * x) = &36 * x * sqrt (&35 * x)`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-296.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-296 = `abs ((int_of_num (3491) - &60) * &(3491 + 60) - &(3491 EXP 2)) = &3600`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-302.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-302 = `(ii / Cx(&2)) pow 2 = Cx(-- (&1 / &4))`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-304.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-304 = `91 EXP 2 = 8281`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-313.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-313 = `!v:complex i:complex z:complex. 2 | (v = i * z) /\ 3 | (v = Cx(&1) + ii) /\ 4 | (z = Cx(&2) - ii) 5 | ==> 6 | i = Cx(&1 / &5) + Cx(&3 / &5) * ii 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-314.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-314 = `!n:num. 2 | (n = 11) 3 | ==> 4 | (&1 / &4) pow (n + 1) * &(2 EXP (2 * n)) = &1 / &4 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-320.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-320 = `!x:real a:num b:num c:num. 2 | (x >= &0 /\ a > 0 /\ b > 0 /\ c > 0) /\ 3 | (&2 * x pow 2 = &4 * x + &9) /\ 4 | (x = (&a + sqrt (&b)) / &c) 5 | ==> 6 | a + b + c = 26 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-329.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-329 = `!x:real y:real. 2 | (&3 * y = x) /\ 3 | (&2 * x + &5 * y = &11) 4 | ==> 5 | (x + y = &4) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-33.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-33 = `!x y z. ~(x = &0) /\ &2 * x = &5 * y /\ &7 * y = &10 * z ==> z / x = &7 / &25`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-332.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-332 = `!x:real y:real. 2 | (x >= &0 /\ y >= &0) /\ 3 | ((x + y) / &2 = &7) /\ 4 | (sqrt (x * y) = sqrt (&19)) 5 | ==> 6 | x pow 2 * y pow 2 = &158 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-338.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-338 = `!a:real b:real c:real. 2 | (&3 * a + b + c = -- &3) /\ 3 | (a + &3 * b + c = &9) /\ 4 | (a + b + &3 * c = &19) 5 | ==> 6 | (a * b * c = -- &56) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-342.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-342 = `!a:real d:real. 2 | (sum (0..(5-1)) (\k. a + &k * d) = &70) /\ 3 | (sum (0..(10-1)) (\k. a + &k * d) = &210) 4 | ==> 5 | a = &42 / &5 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-346.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-346 = `!f:real->real g:real->real. 2 | (!x. f x = &2 * x - &3) /\ 3 | (!x. g x = x + &1) 4 | ==> 5 | (g (f (&5) - &1) = &7) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-354.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-354 = `!a:real d:real. 2 | (a + &6 * d = &30) /\ 3 | (a + &10 * d = &60) 4 | ==> 5 | a + &20 * d = &135 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-362.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-362 = `!a:real b:real. 2 | (a pow 2 * b pow 3 = &32 / &27) /\ 3 | (a / (b pow 3) = &27 / &4) 4 | ==> 5 | (a + b = &8 / &3) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-388.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-388 = `!x:real y:real z:real. 2 | (&3 * x + &4 * y - &12 * z = &10) /\ 3 | ((-- &2) * x - &3 * y + &9 * z = -- &4) 4 | ==> 5 | x = &14 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-392.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-392 = `!n:num. 2 | (EVEN n) /\ 3 | ((int_of_num (n) - &2) pow 2 + &n pow 2 + (&n + &2) pow 2 = &12296) 4 | ==> 5 | (int_of_real (real_of_int ((int_of_num (n) - &2) * &n * (&n + &2)) / &8) = &32736)`;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-398.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-398 = `!a:real b:real c:real. 2 | (&0 < a /\ &0 < b /\ &0 < c) /\ 3 | (&9 * b = &20 * c) /\ 4 | (&7 * a = &4 * b) 5 | ==> 6 | &63 * a = &80 * c 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-400.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-400 = `!x:real. 2 | (&5 + &500 / &100 * &10 = &110 / &100 * x) 3 | ==> 4 | (x = &50) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-412.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-412 = `!x:real y:real. 2 | (x + y = &25) /\ 3 | (x - y = &11) 4 | ==> 5 | x = &18 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-419.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-419 = `!a:real b:real. 2 | (a = -- &1) /\ 3 | (b = &5) 4 | ==> 5 | (--a - b pow 2 + &3 * (a * b) = -- &39) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-427.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-427 = `!x:real y:real z:real. 2 | (&3 * x + y = &17) /\ 3 | (&5 * y + z = &14) /\ 4 | (&3 * x + &5 * z = &41) 5 | ==> 6 | x + y + z = &12 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-432.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-432 = `!x:real. (x + &3) * (&2 * x - &6) = &2 * x pow 2 - &18`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-440.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-440 = `!x:real. (&3 / &2 / &3 = x / &10) ==> x = &5`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-478.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-478 = `!b:real h:real v:real. 2 | (&0 < b /\ &0 < h /\ &0 < v) /\ 3 | (v = &1 / &3 * (b * h)) /\ 4 | (b = &30) /\ 5 | (h = &13 / &2) 6 | ==> 7 | (v = &65) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-484.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-484 = `ln (&27) / ln (&3) = &3`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-487.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-487 = `!a:real b:real c:real d:real. 2 | (b = a pow 2) /\ 3 | (a + b = &1) /\ 4 | (d = c pow 2) /\ 5 | (c + d = &1) 6 | ==> 7 | (sqrt ((a - c) pow 2 + (b - d) pow 2) = sqrt (&10)) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-513.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-513 = `!a:real b:real. 2 | (&3 * a + &2 * b = &5) /\ 3 | (a + b = &2) 4 | ==> 5 | a = &1 /\ b = &1 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-76.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-76 = `!f:int->int. 2 | (!n. ODD (num_of_int (abs n)) ==> f n = n pow 2) /\ 3 | (!n. EVEN (num_of_int (abs n)) ==> f n = n pow 2 - &4 * n - &1) 4 | ==> 5 | f (&4) = -- &1 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-algebra-80.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-80 = `!x. ~(x = -- &1) /\ ((x - &9) / (x + &1) = &2) ==> x = -- &11`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-100.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-100 = `!n:num. 2 | (n > 0) /\ 3 | (gcd (n, 40) = 10) /\ 4 | (lcm (n, 40) = 280) 5 | ==> 6 | n = 70 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-127.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-127 = `(nsum (0..(101-1)) (\k. 2 EXP k)) MOD 7 = 3`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-150.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-150 = `!n:num. ~(prime (7 + 30 * n)) ==> 6 <= n`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-175.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-175 = `(2 EXP 2010) MOD 10 = 4`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-185.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-185 = `!n:num. (n MOD 5 = 3) ==> (2 * n) MOD 5 = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-207.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-207 = `8 * 9 EXP 2 + 5 * 9 + 2 = 695`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-212.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-212 = `(16 EXP 17 * 17 EXP 18 * 18 EXP 19) MOD 10 = 8`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-222.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-222 = `!b:num. 2 | (lcm (120, b) = 3720) /\ 3 | (gcd (120, b) = 8) 4 | ==> 5 | (b = 248) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-227.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-227 = `!x:num y:num n:num. (x > 0 /\ y > 0 /\ n > 0) /\ (&x / &4 + &y / &6 = &(x + y) / &n) ==> n = 5`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-229.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-229 = `(5 EXP 30) MOD 7 = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-233.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-233 = `!b:int. 2 | &0 <= b /\ b < &11 pow 2 /\ 3 | (b * &24 == &1) (mod (&11 pow 2)) 4 | ==> 5 | b = &116 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-234.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-234 = `!a:num b:num. 2 | (1 <= a /\ a <= 9 /\ b <= 9) /\ 3 | ((10 * a + b) EXP 3 = 912673) 4 | ==> 5 | (a + b = 16) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-235.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-235 = `(29 * 79 + 31 * 81) MOD 10 = 2`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-237.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-237 = `(nsum (0..(101-1)) (\k. k)) MOD 6 = 4`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-247.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-247 = `!n:num. ((3 * n) MOD 2 = 11) ==> n MOD 11 = 8`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-254.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-254 = `(239 + 174 + 83) MOD 10 = 6`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-277.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-277 = `!m:num n:num. 2 | (gcd(m, n) = 6) /\ 3 | (lcm(m, n) = 126) 4 | ==> 5 | 60 <= m + n 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-299.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-299 = `(1 * 3 * 5 * 7 * 9 * 11 * 13) MOD 10 = 5`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-3.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-3 = `(nsum (0..(10-1)) (\x. (x + 1) EXP 2)) MOD 10 = 5`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-321.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-321 = `!n:int. 2 | &0 <= n /\ n < &1399 /\ 3 | (n * &160 == &1) (mod &1399) 4 | ==> 5 | n = &1058 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-328.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-328 = `(5 EXP 999999) MOD 7 = 6`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-34.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-34 = `!x:num. 2 | (x < 100) /\ 3 | ((x * 9) MOD 100 = 1) 4 | ==> 5 | x = 89 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-341.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-341 = `!a:num b:num c:num. 2 | (a <= 9 /\ b <= 9 /\ c <= 9) /\ 3 | ((5 EXP 100) MOD 1000 = 10*(10*a + b) + c) 4 | ==> 5 | (a + b + c = 13) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-342.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-342 = `54 MOD 6 = 0`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-345.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-345 = `(2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) MOD 7 = 0`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-427.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-427 = `!a:num. 2 | (a = nsum ({d:num | d divides 500}) (\k. k)) 3 | ==> 4 | nsum ({x:num | (prime x /\ x divides a)}) (\k. k) = 25`;; 5 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-447.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-447 = `nsum {k:num | k >= 1 /\ k <= (50-1) /\ 3 divides k} (\k. k MOD 10) = 78`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-451.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-451 = `(FINITE {n:num | 2010 <= n /\ n <= 2019 /\ ?m. (CARD {p | p divides m} = 4 /\ nsum {p | p divides m} (\p. p) = n)}) 2 | ==> 3 | nsum {n:num | 2010 <= n /\ n <= 2019 /\ ?m. (CARD {p | p divides m} = 4 /\ nsum {p | p divides m} (\p. p) = n)} (\k. k) = 2016`;; 4 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-495.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-495 = `!a:num b:num. 2 | (0 < a /\ 0 < b) /\ 3 | (a MOD 10 = 2) /\ 4 | (b MOD 10 = 4) /\ 5 | (gcd (a, b) = 6) 6 | ==> 7 | (108 <= lcm (a, b)) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-551.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-551 = `1529 MOD 6 = 5`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-552.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-552 = `!f:num->num g:num->num h:num->num. 2 | (!x. (x > 0) ==> (f x = 12 * x + 7)) /\ 3 | (!x. (x > 0) ==> (g x = 5 * x + 2)) /\ 4 | (!x. (x > 0) ==> (h x = gcd (f x, g x))) /\ 5 | (FINITE {h x | x | x > 0}) 6 | ==> 7 | (nsum {h x | x | x > 0} (\k. k) = 12) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-618.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-618 = `!n:num p:num->num. 2 | (!x. &(p x) = int_of_num (x EXP 2) - &x + &41) /\ 3 | (1 < gcd (p (n), p (n+1))) 4 | ==> 5 | (41 <= n) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-66.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-66 = `194 MOD 11 = 7`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-85.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-85 = `1 * 3 EXP 3 + 2 * 3 EXP 2 + 2*3 + 2 = 53`;; 2 | -------------------------------------------------------------------------------- /hollight/test/mathd-numbertheory-99.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-99 = `!n:num. ((2 * n) MOD 47 = 15) ==> n MOD 47 = 31`;; 2 | -------------------------------------------------------------------------------- /hollight/test/numbertheory-aoddbdiv4asqpbsqmod8eq1.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-aoddbdiv4asqpbsqmod8eq1 = `!a:int b:num. 2 | (ODD (num_of_int (abs a))) /\ 3 | (4 divides b) 4 | ==> 5 | num_of_int (a pow 2 + &b pow 2) MOD 8 = 1 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/test/numbertheory-notequiv2i2jasqbsqdiv8.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-notequiv2i2jasqbsqdiv8 = `!a:int b:int. ~((?i j. a = &2 * i /\ b = &2 * j) <=> (?k. a pow 2 + b pow 2 = &8 * k))`;; 2 | -------------------------------------------------------------------------------- /hollight/test/others-chickenmcnuggets.ml: -------------------------------------------------------------------------------- 1 | let others-chickenmcnuggets = `!n:num. ?i j. n + 152 = 9 * i + 20 * j`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/aime-1983-p9.ml: -------------------------------------------------------------------------------- 1 | let aime-1983-p9 = `!x:real. 2 | (&0 < x /\ x < pi) ==> 3 | &12 <= ((&9 * (x pow 2 * (sin x) pow 2)) + &4) / (x * sin x)`;; 4 | -------------------------------------------------------------------------------- /hollight/valid/aime-1984-p5.ml: -------------------------------------------------------------------------------- 1 | let aime-1984-p5 = `!a:real b:real. 2 | (ln a / ln (&8) + ln (b pow 2) / ln (&4) = &5) /\ 3 | (ln b / ln (&8) + ln (a pow 2) / ln (&4) = &7) 4 | ==> 5 | a * b = &512 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/algebra-2complexrootspoly-xsqp49eqxp7itxpn7i.ml: -------------------------------------------------------------------------------- 1 | let algebra-2complexrootspoly-xsqp49eqxp7itxpn7i = `!x:complex. x pow 2 + Cx(&49) = (x + (Cx(&7) * ii)) * (x + (Cx(-- (&7)) * ii))`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-2rootsintpoly-am10tap11eqasqpam110.ml: -------------------------------------------------------------------------------- 1 | let algebra-2rootsintpoly-am10tap11eqasqpam110 = `!a:complex. (a - Cx(&10)) * (a + Cx(&11)) = a pow 2 + a - Cx(&110)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-2rootspoly-apatapbeq2asqp2ab.ml: -------------------------------------------------------------------------------- 1 | let algebra-2rootspoly-apatapbeq2asqp2ab = `!a:complex b:complex. (a + a) * (a + b) = Cx(&2) * a pow 2 + Cx(&2) * (a * b)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-2varlineareq-xpeeq7-2xpeeq3-eeq11-xeqn4.ml: -------------------------------------------------------------------------------- 1 | let algebra-2varlineareq-xpeeq7-2xpeeq3-eeq11-xeqn4 = `!x:complex e:complex. 2 | (x + e = Cx(&7)) /\ 3 | (Cx(&2) * x + e = Cx(&3)) 4 | ==> 5 | (e = Cx(&11) /\ x = Cx(-- (&4))) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/algebra-3rootspoly-amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta.ml: -------------------------------------------------------------------------------- 1 | let algebra-3rootspoly-amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta = `!b:complex c:complex d:complex a:complex. (a - d) * (a - c) * (a - b) = -- (((a pow 2 - (b + c) * a) + c * b) * d) + (a pow 2 - (b + c) * a + c * b) * a`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x = `!x:real. 2 | (x > &0) 3 | ==> 4 | &2 - sqrt(&2) >= &2 - x - &1 / (&2 * x) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/algebra-amgm-prod1toneq1-sum1tongeqn.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-prod1toneq1-sum1tongeqn = `!a:num->real n:num. 2 | (!k. a k >= &0) /\ 3 | (product (0..(n-1)) a = &1) 4 | ==> 5 | (sum (0..(n-1)) a >= &n) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/algebra-amgm-sqrtxymulxmyeqxpy-xpygeq4.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-sqrtxymulxmyeqxpy-xpygeq4 = `!x:real y:real. 2 | (&0 < x /\ &0 < y) /\ 3 | (y <= x) /\ 4 | (sqrt (x * y) * (x - y) = (x + y)) 5 | ==> 6 | (x + y >= &4) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/algebra-amgm-sumasqdivbsqgeqsumbdiva.ml: -------------------------------------------------------------------------------- 1 | let algebra-amgm-sumasqdivbsqgeqsumbdiva = `!a:real b:real c:real. 2 | (&0 < a /\ &0 < b /\ &0 < c) 3 | ==> 4 | (a pow 2 / b pow 2 + b pow 2 / c pow 2 + c pow 2 / a pow 2 >= b / a + c / b + a / c) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/algebra-binomnegdiscrineq-10alt28asqp1.ml: -------------------------------------------------------------------------------- 1 | let algebra-binomnegdiscrineq-10alt28asqp1 = `!a:real. &10 * a <= &28 * a pow 2 + &1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-manipexpr-2erprsqpesqeqnrpnesq.ml: -------------------------------------------------------------------------------- 1 | let algebra-manipexpr-2erprsqpesqeqnrpnesq = `!e:complex r:complex. Cx(&2) * (e * r) + (e pow 2 + r pow 2) = (-- r + (-- e)) pow 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-manipexpr-apbeq2cceqiacpbceqm2.ml: -------------------------------------------------------------------------------- 1 | let algebra-manipexpr-apbeq2cceqiacpbceqm2 = `!a:complex b:complex c:complex. 2 | (a + b = Cx(&2) * c) /\ 3 | (c = ii) 4 | ==> 5 | (a * c + b * c = Cx(-- &2)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/algebra-sqineq-2at2pclta2c2p41pc.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-2at2pclta2c2p41pc = `!a:real c:real. &2 * a * (&2 + c) <= a pow 2 + c pow 2 + &4 * (&1 + c)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-sqineq-2unitcircatblt1.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-2unitcircatblt1 = `!a:real b:real. a pow 2 + b pow 2 = &2 ==> a * b <= &1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-sqineq-36azm9asqle36zsq.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-36azm9asqle36zsq = `!z:real a:real. &36 * (a * z) - &9 * a pow 2 <= &36 * z pow 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/algebra-sqineq-4bap1lt4bsqpap1sq.ml: -------------------------------------------------------------------------------- 1 | let algebra-sqineq-4bap1lt4bsqpap1sq = `!a:real b:real. &4 * b * (a + &1) <= &4 * b pow 2 + (a + &1) pow 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/amc12-2000-p11.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p11 = `!a:real b:real. 2 | (~(a = &0) /\ ~(b = &0)) /\ 3 | (a * b = a - b) ==> 4 | a / b + b / a - a * b = &2`;; 5 | -------------------------------------------------------------------------------- /hollight/valid/amc12-2000-p15.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p15 = `!f:complex->complex. 2 | (!x. f (x / Cx(&3)) = x pow 2 + x + Cx(&1)) /\ 3 | (FINITE {y | f y = Cx(&7)}) 4 | ==> 5 | vsum {y | f y = Cx(&7)} (\y. y / Cx(&3)) = Cx(-- &1 / &9) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12-2000-p5.ml: -------------------------------------------------------------------------------- 1 | let amc12-2000-p5 = `!x:real p:real. 2 | (x < &2) /\ 3 | (abs (x - &2) = p) 4 | ==> 5 | (x - p = &2 - &2 * p) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12-2001-p2.ml: -------------------------------------------------------------------------------- 1 | let amc12-2001-p2 = `!a:num b:num n:num. 2 | (1 <= a /\ a <= 9) /\ 3 | (0 <= b /\ b <= 9) /\ 4 | (n = 10 * a + b) /\ 5 | (n = a * b + a + b) ==> 6 | b = 9`;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12-2001-p9.ml: -------------------------------------------------------------------------------- 1 | let amc12-2001-p9 = `!f:real->real. 2 | (!x y. (x > &0 /\ y > &0) ==> f (x * y) = f x / y) /\ 3 | (f (&500) = &3) 4 | ==> 5 | (f (&600) = &5 / &2) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2002-p1.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2002-p1 = `!f:complex->complex. 2 | (!x. f x = (Cx(&2) * x + Cx(&3)) * (x - Cx(&4)) + (Cx(&2) * x + Cx(&3)) * (x - Cx(&6))) /\ 3 | (FINITE {x | f x = Cx(&0)}) 4 | ==> 5 | vsum {x | f x = Cx(&0)} (\y. y) = Cx(&7 / &2) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2002-p12.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2002-p12 = `!f:real->real k:real. 2 | (!x. f x = x pow 2 - &63 * x + k) /\ 3 | ({x | f x = &0} SUBSET { x | ?n. &n = x /\ prime n}) 4 | ==> 5 | k = &122 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2002-p21.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2002-p21 = `!u:num->num. 2 | (u 0 = 4) /\ 3 | (u 1 = 7) /\ 4 | (!n. n >= 2 ==> u (n + 2) = (u n + u (n + 1)) MOD 10) 5 | ==> 6 | (!n. nsum (0..(n-1)) u > 10000 ==> 1999 <= n) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2003-p1.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2003-p1 = `!u:num->num v:num->num. 2 | (!n. u n = 2 * n + 2) /\ 3 | (!n. v n = 2 * n + 1) 4 | ==> 5 | (nsum (0..(2003 - 1)) u) - (nsum (0..(2003 - 1)) v) = 2003 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2003-p24.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2003-p24 = `!a:real b:real. 2 | (b <= a) /\ 3 | (&1 < b) ==> 4 | ln (a / b) / ln a + ln (b / a) / ln b <= &0`;; 5 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2003-p25.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2003-p25 = `!a:real b:real f:real->real. 2 | (&0 < b) /\ 3 | (!x. f x = sqrt (a * x pow 2 + b * x)) /\ 4 | ({x | &0 <= f x} = {f x | x | x IN {x | &0 <= f x}}) 5 | ==> 6 | (a = &0) \/ (a = -- &4) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2020-p13.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2020-p13 = `!a:num b:num c:num n:real. 2 | (n >= 0) /\ 3 | ~(n = 1) /\ 4 | (1 < a /\ 1 < b /\ 1 < c) /\ 5 | (exp ((&1) / (&a) * ln (&n * (exp ((&1) / (&b) * ln (&n * (exp ((&1) / (&c) * ln (&n)))))))) = exp ((&1) / (&36) * ln (&(n EXP 25)))) 6 | ==> 7 | b = 3 8 | `;; 9 | 10 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2020-p22.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2020-p22 = `(FINITE {n:num | (5 divides n) /\ (lcm (FACT 5, n) = 5 * gcd (FACT 10, n))}) ==> CARD {n:num | (5 divides n) /\ (lcm (FACT 5, n) = 5 * gcd (FACT 10, n))} = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/amc12a-2021-p7.ml: -------------------------------------------------------------------------------- 1 | let amc12a-2021-p7 = `!x:real y:real. &1 <= ((x * y) - &1) pow 2 + (x + y) pow 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2002-p11.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p11 = `!a:num b:num. 2 | (prime a) /\ 3 | (prime b) /\ 4 | (prime (a + b)) /\ 5 | (prime (a - b)) 6 | ==> prime (a + b + (a - b + (a + b)))`;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2002-p3.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p3 = `!n:num. 2 | (0 < n) /\ 3 | (prime (n EXP 2 + 2 - 3 * n)) 4 | ==> 5 | n = 3 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2002-p6.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2002-p6 = `!a:real b:real. 2 | (~(a = &0) /\ ~(b = &0)) /\ 3 | (!x. x pow 2 + a * x + b = (x - a) * (x - b)) 4 | ==> a = &1 /\ b = -- &2`;; 5 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2003-p17.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2003-p17 = `!x:real y:real. 2 | (&0 < x /\ &0 < y) /\ 3 | (ln (x * y pow 3) = &1) /\ 4 | (ln (x pow 2 * y) = &1) 5 | ==> 6 | ln (x * y) = &3 / &5 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2003-p6.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2003-p6 = `!a:real r:real u:num->real. 2 | (!k. u k = a * r pow k) /\ 3 | (u 1 = &2) /\ 4 | (u 3 = &6) 5 | ==> 6 | u 0 = &2 / (sqrt (&3)) \/ u 0 = -- (&2 / (sqrt (&3))) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2003-p9.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2003-p9 = `!a:real b:real f:real->real. 2 | (!x. f x = a * x + b) /\ 3 | (f (&6) - f (&2) = &12) 4 | ==> f (&12) - f (&2) = &30`;; 5 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2004-p3.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2004-p3 = `!x:num y:num. 2 EXP x * 3 EXP y = 1296 ==> x + y = 8`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/amc12b-2020-p5.ml: -------------------------------------------------------------------------------- 1 | let amc12b-2020-p5 = `!a:num b:num. 2 | (a > 0 /\ b > 0) /\ 3 | (&5 / &8 * &b - &2 / &3 * &a = &7) /\ 4 | (&b - &5 / &8 * &b - (&a - &2 / &3 * &a) = &7) 5 | ==> 6 | a = 42 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/imo-1990-p3.ml: -------------------------------------------------------------------------------- 1 | let imo-1990-p3 = `!n:num. 2 | (2 <= n) /\ 3 | ((n EXP 2) divides (2 EXP n + 1)) 4 | ==> n = 3`;; 5 | -------------------------------------------------------------------------------- /hollight/valid/imo-2006-a6.ml: -------------------------------------------------------------------------------- 1 | let imo-2006-a6 = `!a:real b:real c:real. 2 | (a * b * (a pow 2 - b pow 2)) + 3 | (b * c * (b pow 2 - c pow 2)) + 4 | (c * a * (c pow 2 - a pow 2)) <= 5 | (&9 * sqrt (&2)) / &32 * (a pow 2 + b pow 2 + c pow 2) pow 2`;; 6 | -------------------------------------------------------------------------------- /hollight/valid/induction-divisibility-3div2tooddnp1.ml: -------------------------------------------------------------------------------- 1 | let induction-divisibility-3div2tooddnp1 = `!n:num. 3 divides (2 EXP (2 * n + 1) + 1)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-divisibility-3divnto3m2n.ml: -------------------------------------------------------------------------------- 1 | let induction-divisibility-3divnto3m2n = `!n:num. 3 divides n EXP 3 + 2 * n`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-divisibility-9div10tonm1.ml: -------------------------------------------------------------------------------- 1 | let induction-divisibility-9div10tonm1 = `!n:num. (n > 0) ==> 9 divides (10 EXP n - 1)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-ineq-nsqlefactn.ml: -------------------------------------------------------------------------------- 1 | let induction-ineq-nsqlefactn = `!n:num. 4 <= n ==> n EXP 2 <= FACT n`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-seq-mul2pnp1.ml: -------------------------------------------------------------------------------- 1 | let induction-seq-mul2pnp1 = `!n:num u:num->num. 2 | (u 0 = 0) /\ 3 | (!n. u (n + 1) = 2 * u n + (n + 1)) 4 | ==> 5 | (u n = 2 EXP (n + 1) - (n + 2)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/induction-sum-1oktkp1.ml: -------------------------------------------------------------------------------- 1 | let induction-sum-1oktkp1 = `!n:num. sum(0..(n-1)) (\k. (&1) / (&((k + 1) * (k + 2)))) = (&n) / (&(n + 1))`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-sum-odd.ml: -------------------------------------------------------------------------------- 1 | let induction-sum-odd = `!n:num. nsum(0..(n-1)) (\k. 2 * k + 1) = n EXP 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/induction-sum2kp1npqsqm1.ml: -------------------------------------------------------------------------------- 1 | let induction-sum2kp1npqsqm1 = `!n:num. &(nsum (0..(n-1)) (\k. 2 * k + 3)) = int_of_num (n + 1) pow 2 - &1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-10.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-10 = `abs ((&120) / (&100) * (&30) - (&130) / (&100) * (&20)) = &10`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-101.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-101 = `!x:real. (x pow 2 - &5 * x - &4 <= &10) ==> x >= -- &2 /\ x <= &7`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-104.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-104 = `!x:real. 2 | ((&125) / (&8) = x / (&12)) 3 | ==> 4 | (x = (&375) / (&2)) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-109.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-109 = `!a:real b:real. 2 | (&3 * a + &2 * b = &12) /\ 3 | (a = &4) 4 | ==> 5 | b = &0 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-11.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-11 = `!a:real b:real. 2 | (~(a = b)) /\ 3 | (~(a = &2 * b)) /\ 4 | ((&4 * a + &3 * b) / (a - &2 * b) = &5) 5 | ==> 6 | ((a + &11 * b) / (a - b) = &2) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-110.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-110 = `!q:complex e:complex. 2 | (q = Cx(&2) - Cx(&2) * ii) /\ 3 | (e = Cx(&5) + Cx(&5) * ii) 4 | ==> 5 | q * e = Cx(&20) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-116.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-116 = `!k:real x:real. 2 | (x = (&13 - sqrt (&131)) / (&4)) /\ 3 | (&2 * x pow 2 - &13 * x + k = &0) 4 | ==> 5 | (k = (&19) / (&4)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-119.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-119 = `!d:real e:real. 2 | (&2 * d = &17 * e - &8) /\ 3 | (&2 * e = d - &9) 4 | ==> 5 | e = &2 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-123.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-123 = `!a:int b:int. 2 | (a > &0 /\ b > &0) /\ 3 | (a + b = &20) /\ 4 | (a = &3 * b) 5 | ==> 6 | a - b = &10 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-126.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-126 = `!x:real y:real. 2 | (&2 * &3 = x - &9) /\ 3 | (&2 * (-- &5) = y + &1) 4 | ==> 5 | (x = &15) /\ 6 | (y = -- &11) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-13.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-13 = `!a:real b:real. 2 | (!x. &4 * x / (x pow 2 - &8 * x + &15) = a / (x - &3) + b / (x - &5)) 3 | ==> 4 | a = -- &6 /\ b = &10 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-131.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-131 = `!a:real b:real f:real->real. 2 | (!x. f x = &2 * x pow 2 - &7 * x + &2) /\ 3 | (f a = &0) /\ 4 | (f b = &0) /\ 5 | ~(a = b) 6 | ==> 7 | &1 / (a - &1) + &1 / (b - &1) = -- &1 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-132.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-132 = `!x:real f:real->real g:real->real. 2 | (!x. f x = x + &2) /\ 3 | (!x. g x = x pow 2) /\ 4 | (f (g x) = g (f x)) 5 | ==> 6 | x = -- (&1) / (&2) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-140.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-140 = `!a:real b:real c:real. 2 | (&0 < a /\ &0 < b /\ &0 < c) /\ 3 | (!x. &24 * x pow 2 - &19 * x - &35 = (((a * x) - &5) * ((&2 * (b * x)) + c))) 4 | ==> 5 | (a * b - &3 * c = -- &9) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-144.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-144 = `!a:int b:int c:int d:int. 2 | (a > &0 /\ b > &0 /\ c > &0 /\ d > &0) /\ 3 | (c - b = d) /\ 4 | (b - a = d) /\ 5 | (a + b + c = &60) /\ 6 | (a + b > c) 7 | ==> 8 | d < &10 9 | `;; 10 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-149.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-149 = `!f:real->real. 2 | (!x. x < -- &5 ==> f x = x pow 2 + &5) /\ 3 | (!x. x >= -- &5 ==> f x = &3 * x - &8) /\ 4 | (FINITE {x | f x = &10}) 5 | ==> 6 | sum {x | f x = &10} (\k. k) = &6 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-15.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-15 = `!s:num->num->num. 2 | (!a:num b:num. (a > 0 /\ b > 0) ==> (s a b = a EXP b + b EXP a)) 3 | ==> 4 | (s 2 6 = 100) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-151.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-151 = `!c:int. 2 | (sqrt (&27) <= real_of_int c) /\ 3 | (!z:int. (sqrt (&27) <= real_of_int z) ==> (c <= z)) 4 | ==> 5 | real_of_int c - floor (sqrt (&26)) = &1 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-159.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-159 = `!b:real f:real->real. 2 | (!x. f x = &3 * x pow 4 - &7 * x pow 3 + &2 * x pow 2 - b * x + &1) /\ 3 | (f (&1) = &1) 4 | ==> 5 | (b = -- &2) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-181.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-181 = `!n:real. 2 | ~(n = &3) /\ 3 | ((n + &5) / (n - &3) = &2) 4 | ==> 5 | n = &11 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-182.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-182 = `!y:complex. Cx(&7) * (Cx(&3) * y + Cx(&2)) = Cx(&21) * y + Cx(&14)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-185.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-185 = `!f:int->int. 2 | (!x. f x = abs (x + &4)) /\ 3 | (FINITE {x | f x < &9}) 4 | ==> 5 | CARD {x | f x < &9 } = 17 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-190.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-190 = `(&3 / &8 + &7 / &8) / (&4 / &5) = &25 / &16`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-192.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-192 = `!q:complex e:complex d:complex. 2 | (q = Cx(&11) - (Cx(&5) * ii)) /\ 3 | (e = Cx(&11) + (Cx(&5) * ii)) /\ 4 | (d = Cx(&2) * ii) 5 | ==> 6 | q * e * d = Cx(&292) * ii 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-206.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-206 = `!a:real b:real f:real->real. 2 | (!x. f x = x pow 2 + a * x + b) /\ 3 | ~(&2 * a = b) /\ 4 | (f (&2 * a) = &0) /\ 5 | (f b = &0) 6 | ==> 7 | a + b = -- (&1) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-214.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-214 = `!a:real f:real->real. 2 | (!x. f x = a * (x - &2) pow 2 + &3) /\ 3 | (f (&4) = &4) 4 | ==> 5 | f (&6) = &7 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-22.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-22 = `ln (&(5 EXP 4)) / ln (&(5 EXP 2)) = &2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-224.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-224 = `(FINITE {n:num | (sqrt (&n) < &7 / &2) /\ (&2 < sqrt (&n))}) ==> (CARD {n:num | (sqrt (&n) < &7 / &2) /\ (&2 < sqrt (&n))} = 8)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-234.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-234 = `!d:real. 2 | (&27 / &125 * d = &9 / &25) 3 | ==> 4 | (&3 / &5 * d pow 3 = &25 / &9) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-245.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-245 = `!x:real. 2 | ~(x = &0) 3 | ==> 4 | inv (&4 / x) * ((&3 * x pow 3) / x) pow 2 * (inv (&1 / (&2 * x))) pow 3 = &18 * x pow 8 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-247.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-247 = `!t:real s:real n:int. 2 | (t = &2 * s - s pow 2) /\ 3 | (s = real_of_int (n pow 2) - exp (real_of_int (n) * ln (&2)) + &1) /\ 4 | (n = &3) 5 | ==> 6 | (t = &0) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-251.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-251 = `!x:real. 2 | ~(x = &0) /\ 3 | (&3 + &1 / x = &7 / x) 4 | ==> 5 | x = &2 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-267.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-267 = `!x:real. 2 | ~(x = &1) /\ 3 | ~(x = -- &2) /\ 4 | ((x + &1) / (x - &1) = (x - &2) / (x + &2)) 5 | ==> 6 | (x = &0) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-28.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-28 = `!c:real f:real->real. 2 | (!x. f x = &2 * x pow 2 + &5 * x + c) /\ 3 | (?x. f x <= &0) 4 | ==> 5 | c <= &25 / &8 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-327.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-327 = `!a:real. 2 | (&1 / &5 * abs (&9 + &2 * a) < &1) 3 | ==> 4 | -- &7 < a /\ a < -- &2 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-35.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-35 = `!p:real->real q:real->real. 2 | (!x. p x = &2 - x pow 2) /\ 3 | (!x. ~(x = &0) ==> q x = &6 / x) 4 | ==> 5 | p (q (&2)) = -- &7 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-37.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-37 = `!x:real y:real. 2 | (x + y = &7) /\ 3 | (&3 * x + y = &45) 4 | ==> 5 | (x pow 2 - y pow 2 = &217) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-421.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-421 = `!a:real b:real c:real d:real. 2 | (b = a pow 2 + &4 * a + &6) /\ 3 | (b = &1 / &2 * a pow 2 + a + &6) /\ 4 | (d = c pow 2 + &4 * c + &6) /\ 5 | (d = &1 / &2 * c pow 2 + c + &6) /\ 6 | (a <= c) 7 | ==> 8 | (c - a = &6) 9 | `;; 10 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-43.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-43 = `!a:real b:real f:real->real. 2 | (!x. f x = a * x + b) /\ 3 | (f (&7) = &4) /\ 4 | (f (&6) = &3) 5 | ==> 6 | f (&3) = &0 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-455.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-455 = `!x:real. 2 | (&2 * (&2 * (&2 * (&2 * x))) = &48) 3 | ==> 4 | x = &3 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-48.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-48 = `!q:complex e:complex. 2 | (q = Cx(&9) - Cx(&4) * ii) /\ 3 | (e = Cx(-- (&3)) - Cx(&4) * ii) 4 | ==> 5 | (q - e = Cx(&12)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-51.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-51 = `!a:real b:real. 2 | (&0 < a /\ &0 < b) /\ 3 | (a + b = &35) /\ 4 | (a = (&2 / &5) * b) 5 | ==> 6 | b - a = &15 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-55.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-55 = `!q:real p:real. 2 | (q = &2 - &4 + &6 - &8 + &10 - &12 + &14) /\ 3 | (p = &3 - &6 + &9 - &12 + &15 - &18 + &21) 4 | ==> 5 | (q / p = (&2) / (&3)) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-59.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-59 = `!b:real. exp (b * ln (&4)) + &(2 EXP 3) = &12 ==> b = &1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-67.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-67 = `!f:real->real g:real->real. 2 | (!x. f x = &5 * x + &3) /\ 3 | (!x. g x = x pow 2 - &2) 4 | ==> 5 | g (f (-- &1)) = &2`;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-69.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-69 = `!r:num s:num. 2 | (~(r = 0) /\ ~(s = 0)) /\ 3 | (r * s = 450) /\ 4 | ((&r + &5) * (int_of_num s - &3) = &450) 5 | ==> 6 | r = 25 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-73.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-73 = `!p:complex q:complex r:complex x:complex. 2 | ((x - p) * (x - q) = (r - p) * (r - q)) /\ 3 | ~(x = r) 4 | ==> 5 | (x = p + q - r) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-77.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-77 = `!a:real b:real f:real->real. 2 | (~(a = &0) /\ ~(b = &0)) /\ 3 | (!x. f x = x pow 2 + a * x + b) /\ 4 | (f a = &0) /\ 5 | (f b = &0) 6 | ==> 7 | a = &1 /\ b = -- &2 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-89.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-89 = `!b:real. 2 | ~(b = &0) 3 | ==> 4 | ((&7 * b pow 3) pow 2 * inv ((&4 * b pow 2) pow 3) = &49 / &64) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-algebra-96.ml: -------------------------------------------------------------------------------- 1 | let mathd-algebra-96 = `!x:real y:real z:real a:real. 2 | (&0 < x /\ &0 < y /\ &0 < z /\ &0 < a) /\ 3 | (ln x - ln y = a) /\ 4 | (ln y - ln z = &15) /\ 5 | (ln z - ln x = -- &7) 6 | ==> 7 | a = -- &8 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-101.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-101 = `(17 * 18) MOD 4 = 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-102.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-102 = `(2 EXP 8) MOD 5 = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-109.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-109 = `!v:num->num. (!n. v n = 2 * n - 1) ==> (nsum (1..(101-1)) v) MOD 7 = 4`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-110.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-110 = `!a:num b:num. 2 | (0 < a /\ 0 < b /\ b <= a) /\ 3 | ((a + b) MOD 10 = 2) /\ 4 | ((2 * a + b) MOD 10 = 1) 5 | ==> 6 | ((a - b) MOD 10 = 6) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-13.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-13 = `!u:num v:num. 2 | (~(u = 0) /\ ~(v = 0)) /\ 3 | ((14 * u) MOD 100 = 46) /\ 4 | ((14 * v) MOD 100 = 46) /\ 5 | (u < 50) /\ 6 | (v < 100) /\ 7 | (50 < v) 8 | ==> 9 | ((u + v) DIV 2 = 64) 10 | `;; 11 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-132.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-132 = `2004 MOD 12 = 0`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-133.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-133 = `gcd(180,168) = 12`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-136.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-136 = `!n:num. (123 * n + 17 = 39500) ==> n = 321`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-149.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-149 = `nsum {x:num | (x MOD 8 = 5) /\ (x MOD 6 = 3) /\ x < 50} (\k. k) = 66`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-155.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-155 = `CARD {x:num | x <= 1000-1 /\ x >= 100 /\ x MOD 19 = 7} = 52`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-156.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-156 = `!n:num. 2 | ~(n = 0) 3 | ==> 4 | gcd (n + 7, 2 * n + 1) <= 13 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-169.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-169 = `gcd ((FACT 20), 200000) = 40000`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-198.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-198 = `(5 EXP 2005) MOD 100 = 25`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-200.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-200 = `139 MOD 11 = 7`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-202.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-202 = `(19 EXP 19 + 99 EXP 99) MOD 10 = 8`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-211.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-211 = `CARD ({n:num | (n < 60) /\ (6 divides num_of_int (abs (&4 * int_of_num n - &2)))}) = 20`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-22.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-22 = `!b:num. 2 | (b < 10) /\ 3 | floor (sqrt (&(10 * b + 6))) * floor (sqrt (&(10 * b + 6))) = &(10 * b + 6) 4 | ==> 5 | (b = 3 \/ b = 1) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-221.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-221 = `(FINITE {x:num | (0 < x) /\ (x < 1000) /\ (CARD {d:num | d divides x} = 3)}) ==> (CARD {x:num | (0 < x) /\ (x < 1000) /\ (CARD {d:num | d divides x} = 3)} = 11)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-236.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-236 = `(1999 EXP 2000) MOD 5 = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-24.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-24 = `nsum (1..(10-1)) (\k. 11 EXP k) MOD 100 = 59`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-252.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-252 = `(FACT 7) MOD 23 = 3`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-257.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-257 = `!x:num. 2 | (1 <= x /\ x <= 100) /\ 3 | (77 divides (nsum (0..(101-1)) (\k. k - x))) 4 | ==> 5 | (x = 45) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-284.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-284 = `!a:num b:num. 2 | (1 <= a /\ a <= 9 /\ b <= 9) /\ 3 | (10 * a + b = 2 * (a + b)) 4 | ==> 5 | 10 * a + b = 18 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-30.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-30 = `(33818 EXP 2 + 33819 EXP 2 + 33820 EXP 2 + 33821 EXP 2 + 33822 EXP 2) MOD 17 = 0`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-301.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-301 = `!j:num. 2 | ~(j = 0) 3 | ==> 4 | ((3 * (7 * j + 3)) MOD 7 = 2) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-303.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-303 = `(FINITE {n:num | ((2 <= n) /\ ((171 == 80) (mod n)) /\ ((468 == 13) (mod n)))}) ==> (nsum {n:num | ((2 <= n) /\ ((171 == 80) (mod n)) /\ ((468 == 13) (mod n)))} (\k. k) = 111)`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-32.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-32 = `FINITE {n | n divides 36} ==> nsum {n | n divides 36} (\k. k) = 91`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-326.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-326 = `!n:num. 2 | ((int_of_num n - &1) * &n * (&n + &1) = &720) 3 | ==> 4 | (n + 1) = 10 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-33.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-33 = `!n:num. 2 | (n < 398) /\ 3 | ((n * 7) MOD 398 = 1) 4 | ==> 5 | (n = 57) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-335.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-335 = `!n:num. 2 | (n MOD 7 = 5) 3 | ==> 4 | ((5 * n) MOD 7 = 4) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-35.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-35 = ` 2 | (FINITE {n | n divides (num_of_int (int_of_real (sqrt (&196))))}) 3 | ==> 4 | (nsum {n | n divides (num_of_int (int_of_real (sqrt (&196))))} (\k. k) = 24) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-37.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-37 = `lcm(9999,100001) = 90900909`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-370.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-370 = `!n:num. 2 | (n MOD 7 = 3) 3 | ==> 4 | ((2 * n + 1) MOD 7 = 0) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-403.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-403 = `nsum {k | k divides 198 /\ ~(k = 198)} (\k. k) = 270`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-412.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-412 = `!x:num y:num. 2 | (x MOD 19 = 4) /\ 3 | (y MOD 19 = 7) 4 | ==> 5 | ((x + 1) EXP 2 * (y + 5) EXP 3) MOD 19 = 13 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-42.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-42 = `!u:num v:num. 2 | (~(u = 0) /\ ~(v = 0)) /\ 3 | ((27 * u) MOD 40 = 17) /\ 4 | ((27 * v) MOD 40 = 17) /\ 5 | (u < 40) /\ 6 | (v < 80) /\ 7 | (40 < v) 8 | ==> 9 | ((u + v) = 62) 10 | `;; 11 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-43.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-43 = `!n:num. 2 | (n > 0) /\ 3 | (15 EXP n divides FACT 942) /\ 4 | (!m. 15 EXP m divides FACT 942 ==> m <= n) 5 | ==> 6 | n = 233`;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-45.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-45 = `gcd(6432,132) + 11 = 23`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-458.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-458 = `!n:num. 2 | (n MOD 8 = 7) 3 | ==> 4 | (n MOD 4 = 3) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-466.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-466 = `(nsum (0..(11-1)) (\k. k)) MOD 9 = 1`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-48.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-48 = `!b:num. 2 | (0 < b) /\ 3 | (3 * b EXP 2 + 2 * b + 1 = 57) 4 | ==> 5 | (b = 4) 6 | `;; 7 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-530.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-530 = `!n:num k:num. 2 | (n > 0 /\ k > 0) /\ 3 | (&n / &k < &6) /\ 4 | (&5 < &n / &k) 5 | ==> 6 | &22 <= &(lcm(n, k)) / &(gcd(n, k)) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-629.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-629 = `(18 IN {t:num | (t > 0) /\ ((lcm (12, t)) EXP 3 = (12 * t) EXP 2)}) /\ (!t:num. (t IN {t:num | (t > 0) /\ ((lcm (12, t)) EXP 3 = (12 * t) EXP 2)}) ==> (t >= 18))`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-64.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-64 = ` 2 | 39 IN {x:num | ~(x = 0) /\ ((30 * x == 42) (mod 47))} /\ 3 | (!x:num. 4 | (x IN {x:num | ~(x = 0) /\ ((30 * x == 42) (mod 47))}) 5 | ==> 6 | (x >= 39) 7 | ) 8 | `;; 9 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-640.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-640 = `(91145 + 91146 + 91147 + 91148) MOD 4 = 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-709.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-709 = `!n:num. 2 | (n > 0) /\ 3 | (CARD {k | k divides (2 * n)} = 28) /\ 4 | (CARD {k | k divides (3 * n)} = 30) 5 | ==> 6 | CARD {k | k divides (6 * n)} = 35 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-739.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-739 = `(FACT 9) MOD 10 = 0`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-81.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-81 = `71 MOD 3 = 2`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-84.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-84 = `floor (&9 / &160 * &100) = &5`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/mathd-numbertheory-92.ml: -------------------------------------------------------------------------------- 1 | let mathd-numbertheory-92 = `!n:num. 2 | ((5 * n) MOD 17 = 8) 3 | ==> 4 | (n MOD 17 = 5) 5 | `;; 6 | -------------------------------------------------------------------------------- /hollight/valid/numbertheory-2dvd4expn.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-2dvd4expn = `!n:num. ~(n = 0) ==> 2 divides 4 EXP n`;; 2 | -------------------------------------------------------------------------------- /hollight/valid/numbertheory-prmdvsneqnsqmodpeq0.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-prmdvsneqnsqmodpeq0 = `!n:int p:num. 2 | (prime p) 3 | ==> 4 | (&p divides n) 5 | <=> 6 | (num_of_int(n pow 2) MOD p = 0) 7 | `;; 8 | -------------------------------------------------------------------------------- /hollight/valid/numbertheory-sqmod3in01d.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-sqmod3in01d = `!a:int. 2 | ((num_of_int(a pow 2) MOD 3) = 0) \/ 3 | ((num_of_int(a pow 2) MOD 3) = 1) 4 | `;; 5 | -------------------------------------------------------------------------------- /hollight/valid/numbertheory-sqmod4in01d.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-sqmod4in01d = `!a:int. 2 | ((num_of_int(a pow 2) MOD 4) = 0) \/ 3 | ((num_of_int(a pow 2) MOD 4) = 1) 4 | `;; 5 | -------------------------------------------------------------------------------- /hollight/valid/numbertheory-xsqpysqintdenomeq.ml: -------------------------------------------------------------------------------- 1 | let numbertheory-xsqpysqintdenomeq = `!x:real y:real 2 | (rational x /\ rational y) /\ 3 | integer(x pow 2 + y pow 2) 4 | ==> 5 | denominator x = denominator y 6 | `;; 7 | -------------------------------------------------------------------------------- /isabelle/test/algebra_ineq_nto1onlt2m1on.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory algebra_ineq_nto1onlt2m1on 6 | imports Complex_Main 7 | begin 8 | 9 | theorem algebra_ineq_nto1onlt2m1on: 10 | fixes n ::nat 11 | shows "(n::real) powr ((1::real) / n) < 2 - 1 / n" 12 | sorry 13 | 14 | end 15 | 16 | 17 | -------------------------------------------------------------------------------- /isabelle/test/algebra_others_exirrpowirrrat.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory algebra_others_exirrpowirrrat 6 | imports Complex_Main 7 | begin 8 | 9 | theorem algebra_others_exirrpowirrrat: 10 | "\ a b. a \ \ \ b \ \ \ a^b \ \" 11 | sorry 12 | 13 | end 14 | 15 | -------------------------------------------------------------------------------- /isabelle/test/amc12_2000_p12.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12_2000_p12 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12_2000_p12: 10 | fixes a m c :: nat 11 | assumes h0: "a + m + c = 12" 12 | shows "a*m*c + a*m + m*c + a*c \ 112" 13 | sorry 14 | 15 | end 16 | -------------------------------------------------------------------------------- /isabelle/test/amc12_2001_p5.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12_2001_p5 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12_2001_p5: 10 | shows "(\x\{x::nat. x<10000 \ odd x}. x) 11 | = fact 10000 / ((2^5000) * fact 5000)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2002_p6.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2002_p6 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12a_2002_p6: 10 | fixes n ::nat 11 | assumes "n>0" 12 | shows "\ m > n. \ p. m * p \ m + p" 13 | using assms by (metis lessI mult_zero_right zero_le) 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2013_p4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2013_p4 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | begin 8 | 9 | theorem amc12a_2013_p4 : 10 | "(2^2014 + 2^2012) / (2^2014 - 2^2012) = (5::real) / 3" 11 | sorry 12 | 13 | end 14 | -------------------------------------------------------------------------------- /isabelle/test/amc12a_2020_p9.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2020_p9 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12a_2020_p9: 10 | shows "card { x::real. 0 \ x \ x \ 2 * pi \ 11 | tan (2 * x) = cos (x / 2)} = 5" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2021_p14.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2021_p14 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12a_2021_p14 : 10 | shows "(\ k\{1..<21}. 11 | ln (3^(k^2)) / ln (5^k)) * (\ k \{1..<101}. ln (25^k) 12 | / ln (9^k))= 21000" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2021_p19.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2021_p19 6 | imports Complex_Main 7 | begin 8 | 9 | 10 | theorem amc12a_2021_p19: 11 | shows "card {x ::real. 0 \ x \ x \ pi \ sin 12 | (pi / 2 * cos x) = cos (pi / 2 * 13 | sin x)} = 2" 14 | sorry 15 | 16 | 17 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2021_p3.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2021_p3 6 | imports Complex_Main 7 | begin 8 | 9 | 10 | theorem amc12a_2021_p3: 11 | fixes x y :: nat 12 | assumes h0: "x + y = 17402" 13 | and h1: "10 dvd x" 14 | and h2: "x div 10 = y" 15 | shows "x - y = 14238" 16 | sorry 17 | 18 | end -------------------------------------------------------------------------------- /isabelle/test/amc12a_2021_p9.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2021_p9 6 | imports Complex_Main "HOL-Library.Code_Target_Numeral" 7 | begin 8 | 9 | theorem amc12a_2021_p9 : 10 | shows "(\ k<7. (2^(2^k) + 3^(2^k))) = (3::nat)^128 - 2^128" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2002_p2.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2002_p2 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2002_p2: 10 | fixes x ::int 11 | assumes h0: "x = 4" 12 | shows "(3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11" 13 | using assms by simp 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2020_p13.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2020_p13 6 | imports Complex_Main 7 | begin 8 | 9 | 10 | theorem amc12b_2020_p13 : 11 | shows "sqrt (ln 6 / ln 2 + ln 6 / ln 3) 12 | = sqrt (ln 3 / ln 2) + sqrt (ln 2 / ln 3)" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2020_p21.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2020_p21 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2020_p21: 10 | shows "card {n. (n + 1000) / 70 = floor (sqrt n)} = 6" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2020_p22.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2020_p22 6 | imports Complex_Main "HOL-Library.Code_Target_Numeral" 7 | begin 8 | 9 | theorem amc12b_2020_p22: 10 | fixes t :: real 11 | shows "((2 powr t - 3 * t) * t) / (4 powr t) \ 1 / 12" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2021_p1.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2021_p1 6 | imports Complex_Main 7 | begin 8 | 9 | 10 | theorem amc12b_2021_p1: 11 | shows "card {x::int. \real_of_int x\ < 3 * pi} = 19" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2021_p13.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2021_p13 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2021_p13: 10 | shows "card {x :: real. 0 < x \ x \ 2 * pi \ 1 - 3 * sin x 11 | + 5 * cos (3 * x) = 0} = 6" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2021_p18.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2021_p18 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2021_p18: 10 | fixes z :: complex 11 | assumes h0: "12 * norm z = 2 * 12 | norm (z + 2) + norm (z^2 + 1) + 31" 13 | shows "z + 6 / z = -2" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2021_p4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2021_p4 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2021_p4: 10 | fixes m a :: nat 11 | assumes h0: "m / a = 3 / (4::real)" 12 | shows "84 * m + 70 * a / (m + a) = (76::real)" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/amc12b_2021_p9.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12b_2021_p9 6 | imports Complex_Main 7 | begin 8 | 9 | theorem amc12b_2021_p9: 10 | shows "(ln 80 / ln 2) / (ln 2 / ln 40) - (ln 160 / ln 2) 11 | / (ln 2 / ln 20) = (2::real)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/imo_1959_p1.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory imo_1959_p1 6 | imports Complex_Main 7 | begin 8 | 9 | theorem imo_1959_p1: 10 | fixes n :: nat 11 | shows "gcd (21*n + 4) (14*n + 3) = 1" 12 | sorry 13 | 14 | end 15 | -------------------------------------------------------------------------------- /isabelle/test/imo_1963_p5.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory imo_1963_p5 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem imo_1963_p5 : 11 | "cos (pi / 7) - cos (2 * pi / 7) + cos (3 * pi / 7) = 1 / 2" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/imo_1977_p6.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory imo_1977_p6 imports Complex_Main 6 | begin 7 | 8 | theorem imo_1977_p6: 9 | fixes f :: "nat \ nat" 10 | assumes "\ n. f (f n) < f (n + 1)" 11 | and "\ n. f n >0" 12 | shows "\ n. f n = n" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/induction_11div10tonmn1ton.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory induction_11div10tonmn1ton 6 | imports Complex_Main 7 | begin 8 | 9 | theorem induction_11div10tonmn1ton: 10 | fixes n :: nat 11 | shows "(11::int) dvd (10^n - (-1)^n)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/induction_12dvd4expnp1p20.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory induction_12dvd4expnp1p20 6 | imports Complex_Main 7 | begin 8 | 9 | theorem induction_12dvd4expnp1p20: 10 | fixes n :: nat 11 | shows "(12::int) dvd 4^(n+1) + 20" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_141.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_141 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_algebra_141: 9 | fixes a b ::real 10 | assumes "(a * b)=180" 11 | and "2 * (a + b)=54" 12 | shows "a^2 + b^2 = 369" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_170.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_170 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | begin 8 | 9 | theorem mathd_algebra_170: 10 | "card { n::int. abs (n - 2) \ 5 + 6 / 10} = 11" 11 | sorry 12 | 13 | end 14 | -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_208.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_208 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_algebra_208 : 11 | "sqrt 1000000 - 1000000 powr (1/3) = 900" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_215.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_215 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_algebra_215: 9 | "(\ k \ {x::real. (x + 3)^2 = 121}. k) = -6" 10 | sorry 11 | 12 | end 13 | 14 | 15 | -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_24.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_24 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_algebra_24: 11 | fixes x :: real 12 | assumes "x / 50 = 40" 13 | shows "x = 2000" 14 | using assms by auto 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_296.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_296 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_algebra_296 : 9 | "abs (((3491 - 60) * (3491 + 60) - 3491^2)) = (3600::int)" 10 | by eval 11 | 12 | end 13 | -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_302.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_302 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_algebra_302: 11 | "(\ / 2)^2 = -(1 / 4)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_304.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_304 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_algebra_304: 11 | "91^2 = (8281::nat)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_419.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_419 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_algebra_419: 9 | fixes a b :: real 10 | assumes "a = -1" 11 | and "b = 5" 12 | shows "-a - b^2 + 3 * (a * b) = -39" 13 | using assms by auto 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_44.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_44 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_algebra_44: 9 | fixes s t :: real 10 | assumes "s = 9 - 2 * t" 11 | and "t = 3 * s + 1" 12 | shows "s = 1 \ t = 4" 13 | using assms by auto 14 | 15 | end 16 | 17 | 18 | -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_484.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_484 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_algebra_484 : 11 | "(ln 27) / (ln 3) = (3::real)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_algebra_80.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | theory mathd_algebra_80 imports Complex_Main 5 | begin 6 | 7 | theorem mathd_algebra_80: 8 | fixes x :: real 9 | assumes "x \ -1" 10 | and "(x - 9) / (x + 1) = 2" 11 | shows "x = -11" 12 | using assms by (auto simp:field_simps) 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_1124.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_1124 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_1124: 9 | fixes n :: nat 10 | assumes "n \ 9" 11 | and "18 dvd 374 * 10 + n" 12 | shows "n = 4" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_12.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_12 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_numbertheory_12 : 11 | "card {x::nat. 20 dvd x \ 15 \ x \ x < 86} =4" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_127.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_127 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_127 : 11 | "(\ k<101. 2^k) mod 7 = (3::nat)" sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_175.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_175 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_numbertheory_175 : 11 | "(2^2010) mod 10 = (4::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_207.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_207 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_207: 9 | "8 * 9^2 + 5 * 9 + 2 = (695::nat)" 10 | by eval 11 | 12 | end 13 | -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_229.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_229 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_229: 11 | "(5^30) mod 7 = (1::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_237.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_237 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_237 : 9 | "(\ k<101. k) mod 6 = (4::nat)" 10 | by eval 11 | 12 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_254.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_254 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_254: 11 | "(239 + 174 + 83) mod 10 = (6::nat)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_293.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_293 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_293: 9 | fixes n :: nat 10 | assumes "n \ 9" 11 | and "11 dvd 20 * 100 + 10 * n + 7" 12 | shows "n = 5" 13 | sorry 14 | 15 | end 16 | 17 | 18 | -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_299.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_299 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_299 : 9 | "(1 * 3 * 5 * 7 * 9 * 11 * 13) mod 10 = (5::nat)" by eval 10 | 11 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_3.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_3 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_3 : 9 | "(\ x < 10. ((x + 1)^2)) mod 10 = (5::nat)" 10 | by eval 11 | 12 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_328.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_328 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_numbertheory_328 : 11 | "(5^999999) mod 7 = (6::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_342.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_342 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_342: 9 | "54 mod 6 = (0::nat)" 10 | by eval 11 | 12 | end 13 | -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_343.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_343 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | begin 8 | 9 | theorem mathd_numbertheory_343 : 10 | "(\ k < 6. (2 * k + 1)) mod 10 = (5::nat)" 11 | by eval 12 | 13 | end 14 | -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_517.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_517 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_517 : 11 | "(121 * 122 * 123) mod 4 = (2::nat)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_551.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_551 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_551 : 11 | "1529 mod 6 = (5::nat)" 12 | by eval 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_66.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_66 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_66: 11 | "194 mod 11 = (7::nat)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_728.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_728 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem mathd_numbertheory_728: 11 | "(29^13 - 5^13) mod 7 = (0::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_769.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_769 imports Complex_Main 6 | begin 7 | 8 | theorem mathd_numbertheory_769 : 9 | "(129^34 + 96^38) mod 11 = (9::nat)" 10 | sorry 11 | 12 | end 13 | 14 | 15 | -------------------------------------------------------------------------------- /isabelle/test/mathd_numbertheory_85.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_85 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_numbertheory_85: 11 | "1 * 3^3 + 2 * 3^2 + 2*3 + 2 = (53::nat)" by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/numbertheory_4x3m7y3neq2003.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory numbertheory_4x3m7y3neq2003 imports Complex_Main 6 | begin 7 | 8 | theorem numbertheory_4x3m7y3neq2003: 9 | fixes x y :: int 10 | shows "4 * x^3 - 7 * y^3 \ 2003" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/test/numbertheory_x5neqy2p4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory numbertheory_x5neqy2p4 imports 6 | Complex_Main 7 | "HOL-Computational_Algebra.Computational_Algebra" 8 | begin 9 | 10 | theorem numbertheory_x5neqy2p4: 11 | fixes x y :: int 12 | shows "x^5 \ y^2 + 4" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/aime_1988_p3.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory aime_1988_p3 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem aime_1988_p3: 10 | fixes x :: real 11 | assumes h0 : "0 < x" 12 | and h1 : "log 2 (log 8 x) = log 8 (log 2 x)" 13 | shows "(log 2 x)^2 = 27" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/algebra_apb4leq8ta4pb4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory algebra_apb4leq8ta4pb4 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem algebra_apb4leq8ta4pb4: 10 | fixes a b :: real 11 | assumes h0 : "0 < a \ 0 < b" 12 | shows "(a+b)^4 \ 8 * (a^4 + b^4)" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/algebra_binomnegdiscrineq_10alt28asqp1.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory algebra_binomnegdiscrineq_10alt28asqp1 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem algebra_binomnegdiscrineq_10alt28asqp1: 10 | fixes a :: real 11 | shows "10 * a \ 28 * a^2 + 1" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/algebra_sqineq_2at2pclta2c2p41pc.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory algebra_sqineq_2at2pclta2c2p41pc imports 6 | Complex_Main 7 | begin 8 | 9 | theorem algebra_sqineq_2at2pclta2c2p41pc: 10 | fixes a c :: real 11 | shows "2 * a * (2+c) \ a^2 + c^2 + 4 * (1+c)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/algebra_sqineq_36azm9asqle36zsq.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory algebra_sqineq_36azm9asqle36zsq imports 6 | Complex_Main 7 | begin 8 | 9 | theorem algebra_sqineq_36azm9asqle36zsq: 10 | fixes z a :: real 11 | shows "36 * (a * z) - 9 * a^2 \ 36 * z^2" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/algebra_sqineq_4bap1lt4bsqpap1sq.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory algebra_sqineq_4bap1lt4bsqpap1sq imports 6 | Complex_Main 7 | begin 8 | 9 | theorem algebra_sqineq_4bap1lt4bsqpap1sq: 10 | fixes a b :: real 11 | shows "4 * b * (a+1) \ 4 * b^2 + (a+1)^2" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12_2000_p5.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | theory amc12_2000_p5 imports Complex_Main 5 | begin 6 | 7 | theorem amc12_2000_p5: 8 | fixes x p ::real 9 | assumes "x<2" 10 | and "\x -2\ = p" 11 | shows "x - p = 2 - 2 * p" 12 | using assms by auto 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2008_p15.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory amc12a_2008_p15 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem amc12a_2008_p15: 10 | fixes k :: nat 11 | assumes h0 : "k = 2008^2 + 2^2008" 12 | shows "(k^2 + 2^k) mod 10 = 6" 13 | sorry 14 | 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2008_p4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory amc12a_2008_p4 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem amc12a_2008_p4: 10 | "(\k::nat=1..501. ((4::real) * k + 4) / (4 * k)) = 502" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2009_p2.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory amc12a_2009_p2 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem amc12a_2009_p2: 10 | "(1 + (1 / (1 + (1 / (1 + 1))))) = (5::real) / 3" 11 | by fastforce 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2010_p22.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory amc12a_2010_p22 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | begin 8 | 9 | theorem amc12a_2010_p22: 10 | fixes x ::real 11 | shows "49 \ (\ k \ {1..<120}. abs (k * x - 1))" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2011_p18.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory amc12a_2011_p18 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem amc12a_2011_p18: 10 | fixes x y :: real 11 | assumes h0 : "abs (x+y) + abs (x-y) = 2" 12 | shows "x^2 - 6 * x + y^2 \ 9" 13 | sorry 14 | 15 | 16 | 17 | end -------------------------------------------------------------------------------- /isabelle/valid/amc12a_2015_p10.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | 6 | theory amc12a_2015_p10 imports 7 | Complex_Main 8 | begin 9 | 10 | theorem amc12a_2015_p10: 11 | fixes x y:: nat 12 | assumes h0: "0 ((7::nat) dvd (2^n + 1))" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/imo_1973_p3.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory imo_1973_p3 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem imo_1973_p3: 10 | fixes a b :: real 11 | assumes h0 : "\x. x^4 + a * x^3 + b * x^2 + a*x + 1 = 0" 12 | shows "4/5 \ a^2 + b^2" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/imo_1974_p5.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory imo_1974_p5 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem imo_1974_p5: 10 | 11 | fixes a b c d s :: real 12 | assumes h0 : "s=a/(a+b+d) + b/(a+b+c) + c/(b+c+d) + d/(a+c+d)" 13 | shows "1 s<2" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/imo_1987_p4.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory imo_1987_p4 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem imo_1987_p4: 10 | fixes f :: "nat \ nat" 11 | shows "\(n::nat). f (f n) \ n + 1987" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/imo_1990_p3.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory imo_1990_p3 imports 6 | Complex_Main 7 | begin 8 | theorem imo_1990_p3: 9 | fixes n :: nat 10 | assumes "2 \ n" 11 | and "n^2 dvd 2^n + 1" 12 | shows "n = 3" 13 | sorry 14 | 15 | end 16 | -------------------------------------------------------------------------------- /isabelle/valid/induction_divisibility_3divnto3m2n.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory induction_divisibility_3divnto3m2n 6 | imports 7 | Complex_Main 8 | begin 9 | 10 | theorem induction_divisibility_3divnto3m2n: 11 | fixes n::nat 12 | shows "3 dvd n^3 + 2 * n" 13 | sorry 14 | 15 | end 16 | -------------------------------------------------------------------------------- /isabelle/valid/induction_divisibility_9div10tonm1.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory induction_divisibility_9div10tonm1 6 | imports 7 | Complex_Main 8 | begin 9 | 10 | theorem induction_divisibility_9div10tonm1: 11 | fixes n::nat 12 | shows "(9::nat) dvd 10^n - 1" 13 | sorry 14 | 15 | end 16 | -------------------------------------------------------------------------------- /isabelle/valid/induction_sum_1oktkp1.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory induction_sum_1oktkp1 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem induction_sum_1oktkp1: 10 | fixes n :: nat 11 | shows "n=0 \ (\(k::nat) = 0..(n-1). (1::real)/((k+1)*(k+2))) = n / (n+1)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/induction_sum_odd.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory induction_sum_odd imports 6 | Complex_Main 7 | begin 8 | 9 | theorem induction_sum_odd: 10 | fixes n :: nat 11 | assumes "n > 0" 12 | shows "(\(k::nat) = 0..(n-1). 2 * k + 1) = n^2" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_10.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_10 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_10: 11 | "abs ((120::real) / 100 * 30 - 130 / 100 * 20) = 10" 12 | by eval 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_109.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_109 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_109: 10 | fixes a b :: real 11 | assumes h0 : "3*a+2*b=12" 12 | and h1 : "a=4" 13 | shows "b=0" 14 | using assms by linarith 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_116.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_116 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_116: 11 | fixes k x :: real 12 | assumes h0 : "x = (13 - sqrt 131) / 4" 13 | and h1 : "2 * x^2 - 13 * x + k = 0" 14 | shows "k = 19/4" 15 | sorry 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_182.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_182 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_182: 11 | fixes y:: complex 12 | shows "7*(3*y+2) = 21 * y + 14" 13 | by simp 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_190.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_190 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_190: 10 | "((3::real) / 8 + 7 / 8) / (4 / 5) = 25 / 16" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_22.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_22 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_22: 10 | "(log 2 (5^4)) / (log 2 (5^2)) = 2" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_251.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_251 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_251: 11 | fixes x :: real 12 | assumes h0: "x \ 0" 13 | and h1: "3 + 1/x = 7/x" 14 | shows "x = 2" 15 | sorry 16 | 17 | 18 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_327.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_327 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_327: 10 | fixes a :: real 11 | assumes h0 : "1 / 5 * abs(9 + 2 * a) < 1" 12 | shows "-7 < a \ a < -2" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_410.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_410 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_410: 10 | fixes x y :: real 11 | assumes h0 : "y = x^2 - 6 * x + 13" 12 | shows "4 \ y" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_462.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_462 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_462: 10 | "((1::real)/2 + 1/3) * (1/2 - 1/3) = 5/36" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_509.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_509 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_509: 11 | "sqrt ((5 / sqrt 80 + sqrt 845 / 9 + sqrt 45) / sqrt 5) = 13 / 6" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_51.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | theory mathd_algebra_51 imports Complex_Main 5 | begin 6 | 7 | theorem mathd_algebra_51: 8 | fixes a b ::real 9 | assumes "0 < a \ 0 < b" 10 | and "a + b = 35" 11 | and "a = (2/5) * b" 12 | shows "b - a = 15" 13 | using assms by auto 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_510.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_510 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_510: 10 | fixes x y :: real 11 | assumes h0 : "x+y=13" 12 | and h1 : "x*y=24" 13 | shows "sqrt (x^2 + y^2) = 11" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_536.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_algebra_536 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | "HOL-Number_Theory.Number_Theory" 8 | begin 9 | 10 | theorem mathd_algebra_536: 11 | "fact 3 * (2^3 + sqrt 9) / 2 = 33" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_568.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_568 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_algebra_568: 10 | fixes a :: real 11 | shows "(a-1) * (a+1) * (a+2) - (a-2) * (a+1) = a^3 + a^2" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_algebra_89.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_algebra_89 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_algebra_89: 11 | fixes b :: real 12 | assumes h0 : "b\0" 13 | shows "(7 * b^3)^2 * 1/((4 * b^2)^3) = 49 / 64" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_101.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_101 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_101: 10 | "(17 * 18) mod 4 = (2::nat)" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_102.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_102 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_102: 10 | "(2^8) mod 5 = (1::nat)" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_132.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_132 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_132: 10 | "2004 mod 12 = (0::nat)" 11 | by eval 12 | 13 | 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_136.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | theory mathd_numbertheory_136 imports Complex_Main 5 | begin 6 | 7 | theorem mathd_numbertheory_136: 8 | fixes n ::nat 9 | assumes "123 * n + 17 = 39500" 10 | shows "n = 321" 11 | using assms by auto 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_156.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_156 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_156: 10 | fixes n :: nat 11 | assumes h0: "n > 0" 12 | shows "gcd (n+7) (2*n+1) \ 13" 13 | sorry 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_169.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_169 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_169: 11 | "gcd (fact 20) 200000 = (40000::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_188.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_188 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_188: 10 | "gcd 180 168 = (12::nat)" 11 | by eval 12 | 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_198.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_198 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_198: 10 | "(5^2005) mod 100 = (25::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_200.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_200 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_200: 10 | "139 mod 11 = (7::nat)" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_202.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_202 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_202: 11 | "(19^19 + 99^99) mod 10 = (8::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_236.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_236 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_236: 10 | "(1999^2000) mod 5 = (1::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_24.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_24 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_24: 11 | "(\ k \{1..<10}. 11^k) mod 100 = (59::nat)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_252.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_252 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_252: 10 | "(fact 7) mod 23 = (3::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_269.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_269 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_269: 10 | "(2005^2 + 2005^0 + 2005^0 + 2005^5) mod 100 = (52::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_30.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_30 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_30: 11 | "(33818^2 + 33819^2 + 33820^2 + 33821^2 + 33822^2) mod 17 = (0::nat)" 12 | sorry 13 | 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_33.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_33 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_33: 10 | fixes n :: nat 11 | assumes h0 : "n < 398" 12 | and h1 : "(n * 7) mod 398 = 1" 13 | shows "n=57" 14 | sorry 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_37.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_37 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_37: 10 | "lcm 9999 100001 = (90900909::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_370.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_370 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_370: 11 | fixes n :: nat 12 | assumes h0 : "n mod 7 = (3::nat)" 13 | shows "(2*n+1) mod 7 = (0::nat)" 14 | sorry 15 | 16 | 17 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_45.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | theory mathd_numbertheory_45 imports Complex_Main 5 | begin 6 | 7 | theorem mathd_numbertheory_45 : 8 | "(gcd 6432 132) + 11 = (23::nat)" by eval 9 | 10 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_458.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_458 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_458: 10 | fixes n :: nat 11 | assumes h0 : "n mod 8 = (7::nat)" 12 | shows "n mod 4 = 3" 13 | sorry 14 | 15 | 16 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_48.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_48 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_48: 10 | fixes b :: nat 11 | assumes h0 : "0 k \ ({n::nat. n dvd (30^4)}). 1) - 2 = (123::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_629.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Wenda Li 3 | *) 4 | 5 | theory mathd_numbertheory_629 6 | imports Complex_Main "HOL-Computational_Algebra.Computational_Algebra" 7 | begin 8 | 9 | theorem mathd_numbertheory_629 : 10 | "(LEAST t::nat. (lcm 12 t)^3 = (12 * t)^2) = 18" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_640.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_640 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_640: 10 | "(91145+91146+91147+91148) mod 4 = (2::nat)" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_739.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_739 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_739: 10 | "(fact 9) mod 10 = (0::nat)" 11 | sorry 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_81.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_81 imports 6 | Complex_Main 7 | begin 8 | 9 | theorem mathd_numbertheory_81: 10 | "71 mod 3 = (2::nat)" 11 | by eval 12 | 13 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_84.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_84 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_84: 11 | "floor ((9::real) / 160 * 100) = (5::int)" 12 | by eval 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_92.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_92 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_92: 11 | fixes n :: nat 12 | assumes h0 : "(5 * n) mod 17 = 8" 13 | shows "n mod 17 = 5" 14 | sorry 15 | 16 | 17 | end -------------------------------------------------------------------------------- /isabelle/valid/mathd_numbertheory_961.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory mathd_numbertheory_961 imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem mathd_numbertheory_961: 11 | "2003 mod 11 = (1::nat)" 12 | by eval 13 | 14 | 15 | end -------------------------------------------------------------------------------- /isabelle/valid/numbertheory_2dvd4expn.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory numbertheory_2dvd4expn imports 6 | Complex_Main 7 | 8 | begin 9 | 10 | theorem numbertheory_2dvd4expn: 11 | fixes n :: nat 12 | assumes h0 : "n \ 0" 13 | shows "(2::nat) dvd 4^n" 14 | sorry 15 | 16 | 17 | end -------------------------------------------------------------------------------- /isabelle/valid/numbertheory_sqmod3in01d.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory numbertheory_sqmod3in01d imports 6 | Complex_Main 7 | begin 8 | 9 | theorem numbertheory_sqmod3in01d: 10 | fixes a :: int 11 | shows "a^2 mod 3 = 0 \ a^2 mod 3 = 1" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /isabelle/valid/numbertheory_sqmod4in01d.thy: -------------------------------------------------------------------------------- 1 | (* 2 | Authors: Albert Qiaochu Jiang 3 | *) 4 | 5 | theory numbertheory_sqmod4in01d imports 6 | Complex_Main 7 | begin 8 | 9 | theorem numbertheory_sqmod4in01d: 10 | fixes a :: int 11 | shows "(a^2 mod 4 = 0) \ (a^2 mod 4 = 1)" 12 | sorry 13 | 14 | end -------------------------------------------------------------------------------- /leanpkg.toml: -------------------------------------------------------------------------------- 1 | [package] 2 | name = "minif2f" 3 | version = "1.0" 4 | lean_version = "leanprover-community/lean:3.42.1" 5 | path = "lean/src" 6 | 7 | [dependencies] 8 | mathlib = {git = "https://github.com/leanprover-community/mathlib", rev = "cb2b02fff213ed6f65bebd64446baac64137dcda"} 9 | -------------------------------------------------------------------------------- /metamath/test/algebra-ineq-nto1onlt2m1on.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | algebra-ineq-nto1onlt2m1on.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | algebra-ineq-nto1onlt2m1on @p |- ( ph -> ( N ^ ( 1 / N ) ) < ( 2 - ( 1 / N ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/algebra-others-exirrpowirrrat.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | algebra-others-exirrpowirrrat @p |- E. a e. ( RR \ QQ ) E. b e. ( RR \ QQ ) ( a ^c b ) e. QQ @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/algebra-sqineq-at2malt1.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | algebra-sqineq-at2malt1.0 @e |- ( ph -> A e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | algebra-sqineq-at2malt1 @p |- ( ph -> ( A x. ( 2 - A ) ) <_ 1 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/algebra-sum1onsqrt2to1onsqrt10000lt198.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | algebra-sum1onsqrt2to1onsqrt10000lt198 @p |- ( ph -> sum_ k e. ( 2 ... ; ; ; ; 1 0 0 0 0 ) ( 1 / ( sqrt ` k ) ) < ; ; 1 9 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2002-p6.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2002-p6.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | amc12a-2002-p6 @p |- ( ph -> E. m e. NN ( m > N /\ E. n e. NN ( m x. n ) <_ ( m + n ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2003-p23.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | amc12a-2003-p23 @p |- ( # ` { k e. NN0 | ( k x. k ) || prod_ i e. ( 1 ... 9 ) ( ! ` i ) } ) = ; ; 6 7 2 @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2013-p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | amc12a-2013-p4 @p |- ( ph -> ( ( ( 2 ^ ; ; ; 2 0 1 4 ) + ( 2 ^ ; ; ; 2 0 1 2 ) ) / ( ( 2 ^ ; ; ; 2 0 1 4 ) - ( 2 ^ ; ; ; 2 0 1 2 ) ) ) = ( 5 / 3 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2020-p10.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2020-p10.0 @e |- ( ph -> N e. NN ) $@ 4 | amc12a-2020-p10.1 @e |- ( ph -> ( 2 logb ( ; 1 6 logb N ) ) = ( 4 logb ( 4 logb N ) ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 6 | amc12a-2020-p10 @p |- ( ph -> N = ; ; 2 5 6 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2020-p9.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2020-p9.0 @e |- D = { x e. ( 0 [,] ( 2 x. _pi ) ) | ( tan ` ( 2 x. x ) ) = ( cos ` ( x / 2 ) ) } $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12a-2020-p9 @p |- ( ph -> ( # ` D ) = 5 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2021-p14.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12a-2021-p14 @p |- ( ph -> ( sum_ k e. ( 1 ... ; 2 0 ) ( ( 5 ^ k ) logb ( 3 ^ ( k ^ 2 ) ) ) x. sum_ k e. ( 1 ... ; ; 1 0 0 ) ( ( 9 ^ k ) logb ( ; 2 5 ^ k ) ) ) = ; ; ; ; 2 1 0 0 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2021-p19.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2021-p19.0 @e |- D = { x e. ( 0 [,] _pi ) | ( sin ` ( ( _pi / 2 ) x. ( cos ` x ) ) ) = ( cos ` ( ( _pi / 2 ) x. ( sin ` x ) ) ) } $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12a-2021-p19 @p |- ( ph -> ( # ` D ) = 2 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12a-2021-p9.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12a-2021-p9 @p |- ( ph -> prod_ k e. ( 0 ... 6 ) ( ( 2 ^ ( 2 ^ k ) ) + ( 3 ^ ( 2 ^ k ) ) ) = ( ( 3 ^ ; ; 1 2 8 ) - ( 2 ^ ; ; 1 2 8 ) ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2002-p2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12b-2002-p2.0 @e |- ( ph -> X = 4 ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | amc12b-2002-p2 @p |- ( ph -> ( ( ( ( ( 3 x. X ) - 2 ) x. ( ( 4 x. X ) + 1 ) ) - ( ( ( 3 x. X ) - 2 ) x. ( 4 x. X ) ) ) + 1 ) = ; 1 1 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2002-p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12b-2002-p4.0 @e |- ( ph -> N e. NN ) $@ 4 | amc12b-2002-p4.1 @e |- ( ph -> ( ( ( 1 / 2 ) + ( 1 / 3 ) ) + ( ( 1 / 7 ) + ( 1 / N ) ) ) e. NN ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | amc12b-2002-p4 @p |- ( ph -> N = ; 4 2 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2020-p13.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2020-p13 @p |- ( ph -> ( sqrt ` ( ( 2 logb 6 ) + ( 3 logb 6 ) ) ) = ( ( sqrt ` ( 2 logb 3 ) ) + ( sqrt ` ( 3 logb 2 ) ) ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2020-p2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2020-p2 @p |- ( ph -> ( ( ( ( ; ; 1 0 0 ^ 2 ) - ( 7 ^ 2 ) ) / ( ( ; 7 0 ^ 2 ) - ( ; 1 1 ^ 2 ) ) ) x. ( ( ( ; 7 0 - ; 1 1 ) x. ( ; 7 0 + ; 1 1 ) ) / ( ( ; ; 1 0 0 - 7 ) x. ( ; ; 1 0 0 + 7 ) ) ) ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2020-p21.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2020-p21 @p |- ( ph -> ( # ` { n e. NN | ( ( n + ; ; ; 1 0 0 0 ) / ; 7 0 ) = ( |_ ` ( sqrt ` n ) ) } ) = 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2020-p22.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12b-2020-p22.0 @e |- ( ph -> T e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12b-2020-p22 @p |- ( ph -> ( ( ( ( 2 ^c T ) - ( 3 x. T ) ) x. T ) / ( 4 ^c T ) ) <_ ( 1 / ; 1 2 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2020-p6.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12b-2020-p6.0 @e |- ( ph -> N e. ( ZZ>= ` 9 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12b-2020-p6 @p |- ( ph -> ( sqrt ` ( ( ( ! ` ( N + 2 ) ) - ( ! ` ( N + 1 ) ) ) / ( ! ` N ) ) ) e. ZZ ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2021-p1.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2021-p1 @p |- ( ph -> ( # ` { x e. ZZ | ( abs ` x ) < ( 3 x. _pi ) } ) = ; 1 9 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2021-p13.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2021-p13 @p |- ( ph -> ( # ` { x e. ( 0 (,] ( 2 x. _pi ) ) | ( ( 1 - ( 3 x. ( sin ` x ) ) ) + ( 5 x. ( cos ` ( 3 x. x ) ) ) ) = 0 } ) = 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/amc12b-2021-p9.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 4 | amc12b-2021-p9 @p |- ( ph -> ( ( ( 2 logb ; 8 0 ) / ( ; 4 0 logb 2 ) ) - ( ( 2 logb ; ; 1 6 0 ) / ( ; 2 0 logb 2 ) ) ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/imo-1959-p1.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1959-p1.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | imo-1959-p1 @p |- ( ph -> ( ( ( ; 2 1 x. N ) + 4 ) gcd ( ( ; 1 4 x. N ) + 3 ) ) = 1 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/imo-1963-p5.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | imo-1963-p5 @p |- ( ph -> ( ( ( cos ` ( _pi / 7 ) ) - ( cos ` ( ( 2 x. _pi ) / 7 ) ) ) + ( cos ` ( ( 3 x. _pi ) / 7 ) ) ) = ( 1 / 2 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/imo-1974-p3.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1974-p3.0 @e |- ( ph -> N e. NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | imo-1974-p3 @p |- ( ph -> -. 5 || sum_ k e. ( 0 ... N ) ( ( ( ( 2 x. N ) + 1 ) _C ( ( 2 x. k ) + 1 ) ) x. ( 2 ^ ( 3 x. k ) ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-11div10tonmn1ton.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-11div10tonmn1ton.0 @e |- ( ph -> N e. NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | induction-11div10tonmn1ton @p |- ( ph -> ; 1 1 || ( ( ; 1 0 ^ N ) - ( -u 1 ^ N ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-12dvd4expnp1p20.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-12dvd4expnp1p20.0 @e |- ( ph -> N e. NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | induction-12dvd4expnp1p20 @p |- ( ph -> ; 1 2 || ( ( 4 ^ ( N + 1 ) ) + ; 2 0 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-nfactltnexpnm1ngt3.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-nfactltnexpnm1ngt3.0 @e |- ( ph -> N e. ( ZZ>= ` 3 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | induction-nfactltnexpnm1ngt3 @p |- ( ph -> ( ! ` N ) < ( N ^ ( N - 1 ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-pord1p1on2powklt5on2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-pord1p1on2powklt5on2.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | induction-pord1p1on2powklt5on2 @p |- ( ph -> prod_ k e. ( 1 ... N ) ( 1 + ( 1 / ( 2 ^ k ) ) ) < ( 5 / 2 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-prod1p1onk3le3m1onn.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-prod1p1onk3le3m1onn.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | induction-prod1p1onk3le3m1onn @p |- ( ph -> prod_ k e. ( 1 ... N ) ( 1 + ( 1 / ( k ^ 3 ) ) ) <_ ( 3 - ( 1 / N ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/induction-sumkexp3eqsumksqsq.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | induction-sumkexp3eqsumksqsq.0 @e |- ( ph -> N e. NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | induction-sumkexp3eqsumksqsq @p |- ( ph -> sum_ k e. ( 1 ... N ) ( k ^ 3 ) = ( sum_ k e. ( 1 ... N ) ( k ^ 2 ) ^ 2 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-113.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-113.0 @e |- ( ph -> X e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-113 @p |- ( ph -> ( ( ( X ^ 2 ) - ( ; 1 4 x. X ) ) + 3 ) >_ ( ( ( 7 ^ 2 ) - ( ; 1 4 x. 7 ) ) + 3 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-114.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-114.0 @e |- ( ph -> A = 8 ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-114 @p |- ( ph -> ( ( ; 1 6 x. ( ( A ^ 2 ) ^c ( 1 / 3 ) ) ) ^c ( 1 / 3 ) ) = 4 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-137.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-137.0 @e |- ( ph -> X e. NN0 ) $@ 4 | mathd-algebra-137.1 @e |- ( ph -> ( X + ( ( 4 / ; ; 1 0 0 ) x. X ) ) = ; ; 5 9 8 ) $@ 5 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 6 | mathd-algebra-137 @p |- ( ph -> X = ; ; 5 7 5 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-139.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-139.0 @e |- S = ( x e. RR* , y e. RR* |-> ( ( ( 1 / y ) - ( 1 / x ) ) / ( x - y ) ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-139 @p |- ( ph -> ( 3 S ; 1 1 ) = ( 1 / ; 3 3 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-170.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-170 @p |- ( ph -> ( card ` { n e. ZZ | ( abs ` ( n - 2 ) ) <_ ( 5 + ( 6 / ; 1 0 ) ) } ) = ; 1 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-171.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-171.0 @e |- F = ( x e. RR |-> ( ( 5 x. x ) + 4 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-171 @p |- ( ph -> ( F ` 1 ) = 9 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-176.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-176.0 @e |- ( ph -> X e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 5 | mathd-algebra-176 @p |- ( ph -> ( ( ( X + 1 ) ^ 2 ) x. X ) = ( ( ( X ^ 3 ) + ( 2 x. ( X ^ 2 ) ) ) + X ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-196.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-196.0 @e |- A = { x e. RR | ( abs ` ( 2 - x ) ) = 3 } $@ 4 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 5 | mathd-algebra-196 @p |- ( ph -> sum_ k e. A k = 4 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-208.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-208 @p |- ( ph -> ( ( sqrt ` ; ; ; ; ; ; 1 0 0 0 0 0 0 ) - ( ; ; ; ; ; ; 1 0 0 0 0 0 0 ^c ( 1 / 3 ) ) ) = ; ; 9 0 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-215.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-215.0 @e |- A = { x e. RR | ( ( x + 3 ) ^ 2 ) = ; ; 1 2 1 } $@ 4 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 5 | mathd-algebra-215 @p |- ( ph -> sum_ k e. A k = -u 6 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-24.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-24.0 @e |- ( ph -> X e. RR ) $@ 4 | mathd-algebra-24.1 @e |- ( ph -> ( X / ; 5 0 ) = ; 4 0 ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-24 @p |- ( ph -> X = ; ; ; 2 0 0 0 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-270.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-270.0 @e |- F = ( x e. ( RR \ { -u 2 } ) |-> ( 1 / ( x + 2 ) ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 5 | mathd-algebra-270 @p |- ( ph -> ( F ` ( F ` 1 ) ) = ( 3 / 7 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-296.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-296 @p |- ( ph -> ( abs ` ( ( ( ; ; ; 3 4 9 1 - ; 6 0 ) x. ( ; ; ; 3 4 9 1 + ; 6 0 ) ) - ( ; ; ; 3 4 9 1 ^ 2 ) ) ) = ; ; ; 3 6 0 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-302.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-302 @p |- ( ph -> ( ( _i / 2 ) ^ 2 ) = -u ( 1 / 4 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-304.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-304 @p |- ( ph -> ( ; 9 1 ^ 2 ) = ; ; ; 8 2 8 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-359.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-359.0 @e |- ( ph -> Y e. RR ) $@ 4 | mathd-algebra-359.1 @e |- ( ph -> ( ( Y + 6 ) + Y ) = ( 2 x. ; 1 2 ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-algebra-359 @p |- ( ph -> Y = 9 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-432.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-432.0 @e |- ( ph -> X e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 5 | mathd-algebra-432 @p |- ( ph -> ( ( X + 3 ) x. ( ( 2 x. X ) - 6 ) ) = ( ( 2 x. ( X ^ 2 ) ) - ; 1 8 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-440.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-440.0 @e |- ( ph -> X e. RR ) $@ 4 | mathd-algebra-440.1 @e |- ( ph -> ( ( 3 / 2 ) / 3 ) = ( X / ; 1 0 ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 6 | mathd-algebra-440 @p |- ( ph -> X = 5 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-algebra-484.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-484 @p |- ( ph -> ( 3 logb ; 2 7 ) = 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-1124.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-1124.0 @e |- ( ph -> N e. ( 0 ... 9 ) ) $@ 4 | mathd-numbertheory-1124.1 @e |- ( ph -> ; 1 8 || ; ; ; 3 7 4 N ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-numbertheory-1124 @p |- ( ph -> N = 4 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-12.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-12 @p |- ( ph -> ( # ` { n e. ( ; 1 5 ... ; 8 5 ) | ; 2 0 || n } ) = 4 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-127.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-127 @p |- ( ph -> ( sum_ k e. ( 0 ... ; ; 1 0 0 ) ( 2 ^ k ) mod 7 ) = 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-150.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-150.0 @e |- ( ph -> N e. NN ) $@ 4 | mathd-numbertheory-150.1 @e |- ( ph -> ( 7 + ( ; 3 0 x. N ) ) e/ Prime ) $@ 5 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 6 | mathd-numbertheory-150 @p |- ( ph -> 6 <_ N ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-175.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-175 @p |- ( ph -> ( ( 2 ^ ; ; ; 2 0 1 0 ) mod ; 1 0 ) = 4 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-185.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-185.0 @e |- ( ph -> N e. NN0 ) $@ 4 | mathd-numbertheory-185.1 @e |- ( ph -> ( N mod 5 ) = 3 ) $@ 5 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 6 | mathd-numbertheory-185 @p |- ( ph -> ( ( 2 x. N ) mod 5 ) = 1 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-207.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-207 @p |- ( ph -> ( ( ( 8 x. ( 9 ^ 2 ) ) + ( 5 x. 9 ) ) + 2 ) = ; ; 6 9 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-212.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-212 @p |- ( ph -> ( ( ( ( ; 1 6 ^ ; 1 7 ) x. ( ; 1 7 ^ ; 1 8 ) ) x. ( ; 1 8 ^ ; 1 9 ) ) mod ; 1 0 ) = 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-229.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-229 @p |- ( ph -> ( ( 5 ^ ; 3 0 ) mod 7 ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-235.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-235 @p |- ( ph -> ( ( ( ; 2 9 x. ; 7 9 ) + ( ; 3 1 x. ; 8 1 ) ) mod ; 1 0 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-237.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-237 @p |- ( ph -> ( sum_ k e. ( 1 ... ; ; 1 0 0 ) k mod 6 ) = 4 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-239.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-239 @p |- ( ph -> ( sum_ k e. ( 1 ... ; 1 2 ) k mod 4 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-254.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-254 @p |- ( ph -> ( ( ( ; ; 2 3 9 + ; ; 1 7 4 ) + ; 8 3 ) mod ; 1 0 ) = 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-293.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-293.0 @e |- ( ph -> N e. ( 0 ... 9 ) ) $@ 4 | mathd-numbertheory-293.1 @e |- ( ph -> ; 1 1 || ; ; ; 2 0 N 7 ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-numbertheory-293 @p |- ( ph -> N = 5 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-299.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-299 @p |- ( ph -> ( ( ( ( ( ( ( 1 x. 3 ) x. 5 ) x. 7 ) x. 9 ) x. ; 1 1 ) x. ; 1 3 ) mod ; 1 0 ) = 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-3.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-3 @p |- ( ph -> ( sum_ k e. ( 1 ... 9 ) ( k ^ 2 ) mod ; 1 0 ) = 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-328.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-328 @p |- ( ph -> ( ( 5 ^ ; ; ; ; ; 9 9 9 9 9 9 ) mod 7 ) = 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-342.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-342 @p |- ( ph -> ( ; 5 4 mod 6 ) = 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-343.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-343 @p |- ( ph -> ( prod_ k e. ( 0 ... 5 ) ( ( 2 x. k ) + 1 ) mod ; 1 0 ) = 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-345.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-345 @p |- ( ph -> ( ( ( ( ( ( ( ; ; ; 2 0 0 0 + ; ; ; 2 0 0 1 ) + ; ; ; 2 0 0 2 ) + ; ; ; 2 0 0 3 ) + ; ; ; 2 0 0 4 ) + ; ; ; 2 0 0 5 ) + ; ; ; 2 0 0 6 ) mod 7 ) = 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-353.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-353.0 @e |- ( ph -> S = sum_ k e. ( ; ; ; 2 0 1 0 ... ; ; ; 4 0 1 8 ) k ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | mathd-numbertheory-353 @p |- ( ph -> ( S mod ; ; ; 2 0 0 9 ) = 0 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-447.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Jun-2021.) @) 4 | mathd-numbertheory-447 @p |- ( ph -> sum_ k e. { n e. ( 1 ..^ ; 5 0 ) | 3 || n } ( k mod ; 1 0 ) = ; 7 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-451.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Jun-2021.) @) 4 | mathd-numbertheory-451 @p |- ( ph -> sum_ k e. { n e. ( ; ; ; 2 0 1 0 ... ; ; ; 2 0 1 9 ) | E. m e. NN ( ( # ` { p e. NN | p || m } ) = 4 /\ ( 1 sigma m ) = n ) } k = ; ; ; 2 0 1 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-457.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-457.0 @e |- ( ph -> N e. NN ) $@ 4 | mathd-numbertheory-457.1 @e |- ( ph -> ; ; ; ; 8 0 3 2 5 || ( ! ` N ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-numbertheory-457 @p |- ( ph -> ; 1 7 <_ N ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-517.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-517 @p |- ( ph -> ( ( ( ; ; 1 2 1 x. ; ; 1 2 2 ) x. ; ; 1 2 3 ) mod 4 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-551.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-551 @p |- ( ph -> ( ; ; ; 1 5 2 9 mod 6 ) = 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-66.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-66 @p |- ( ph -> ( ; ; 1 9 4 mod ; 1 1 ) = 7 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-728.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-728 @p |- ( ph -> ( ( ( ; 2 9 ^ ; 1 3 ) - ( 5 ^ ; 1 3 ) ) mod 7 ) = 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-769.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-769 @p |- ( ph -> ( ( ( ; ; 1 2 9 ^ ; 3 4 ) + ( ; 9 6 ^ ; 3 8 ) ) mod ; 1 1 ) = 9 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/mathd-numbertheory-85.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-85 @p |- ( ph -> ( ( ( ( 1 x. ( 3 ^ 3 ) ) + ( 2 x. ( 3 ^ 2 ) ) ) + ( 2 x. 3 ) ) + 2 ) = ; 5 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/numbertheory-notequiv2i2jasqbsqdiv8.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | numbertheory-notequiv2i2jasqbsqdiv8 @p |- -. ( ( E. i e. ZZ A = ( 2 x. i ) /\ E. j e. ZZ B = ( 2 x. j ) ) <-> E. k e. ZZ ( ( A ^ 2 ) + ( B ^ 2 ) ) = ( 8 x. k ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/test/numbertheory-x5neqy2p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | numbertheory-x5neqy2p4.0 @e |- ( ph -> X e. ZZ ) $@ 4 | numbertheory-x5neqy2p4.1 @e |- ( ph -> Y e. ZZ ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | numbertheory-x5neqy2p4 @p |- ( ph -> ( X ^ 5 ) =/= ( ( Y ^ 2 ) + 4 ) ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/aime-1988-p3.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | aime-1988-p3.0 @e |- ( ph -> X e. RR+ ) $@ 4 | aime-1988-p3.1 @e |- ( ph -> ( 2 logb ( 8 logb X ) ) = ( 8 logb ( 2 logb X ) ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | aime-1988-p3 @p |- ( ph -> ( ( 2 logb X ) ^ 2 ) = ; 2 7 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/aime-1990-p2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | aime-1990-p2 @p |- ( ph -> ( ( ( ; 5 2 + ( 6 x. ( sqrt ` ; 4 3 ) ) ) ^c ( 3 / 2 ) ) - ( ( ; 5 2 - ( 6 x. ( sqrt ` ; 4 3 ) ) ) ^c ( 3 / 2 ) ) ) = ; ; 8 2 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/aime-1994-p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | aime-1994-p4.0 @e |- ( ph -> N e. NN ) $@ 4 | aime-1994-p4.1 @e |- ( ph -> sum_ k e. ( 1 ... N ) ( |_ ` ( 2 logb k ) ) = ; ; ; 1 9 9 4 ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | aime-1994-p4 @p |- ( ph -> N = ; ; 3 1 2 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | algebra-amgm-faxinrrp2msqrt2geq2mxm1div2x @p |- ( ph -> A. x e. RR+ ( 2 - ( sqrt ` 2 ) ) >_ ( ( 2 - x ) - ( 1 / ( 2 x. x ) ) ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2008-p2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2008-p2.0 @e |- ( ph -> X e. RR ) $@ 4 | amc12a-2008-p2.1 @e |- ( ph -> ( X x. ( ( 1 / 2 ) + ( 2 / 3 ) ) ) = 1 ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | amc12a-2008-p2 @p |- ( ph -> X = ( 6 / 7 ) ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2008-p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | amc12a-2008-p4 @p |- ( ph -> prod_ k e. ( 1 ... ; ; 5 0 1 ) ( ( ( 4 x. k ) + 4 ) / ( 4 x. k ) ) = ; ; 5 0 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2009-p2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | amc12a-2009-p2 @p |- ( ph -> ( 1 + ( 1 / ( 1 + ( 1 / ( 1 + 1 ) ) ) ) ) = ( 5 / 3 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2010-p22.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2010-p22.0 @e |- ( ph -> X e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | amc12a-2010-p22 @p |- ( ph -> ; 4 9 <_ sum_ k e. ( 1 ... ; ; 1 1 9 ) ( abs ` ( ( k x. X ) - 1 ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2020-p22.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2020-p22.0 @e |- D = { n e. NN0 | ( 5 || n /\ ( ( ! ` 5 ) lcm n ) = ( 5 x. ( ( ! ` ; 1 0 ) gcd n ) ) ) } $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12a-2020-p22 @p |- ( ph -> ( # ` D ) = ; 4 8 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/amc12a-2021-p7.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12a-2021-p7.0 @e |- ( ph -> X e. RR ) $@ 4 | amc12a-2021-p7.1 @e |- ( ph -> Y e. RR ) $@ 5 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 6 | amc12a-2021-p7 @p |- ( ph -> 1 <_ ( ( ( ( X x. Y ) - 1 ) ^ 2 ) + ( ( X + Y ) ^ 2 ) ) ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/amc12b-2021-p21.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | amc12b-2021-p21.0 @e |- ( ph -> S = sum_ k e. { x e. RR+ | ( x ^c ( 2 ^c ( sqrt ` 2 ) ) ) = ( ( sqrt ` 2 ) ^c ( 2 ^c x ) ) } k ) $@ 4 | @( (Contributed by Kunhao Zheng, 3-Jun-2021.) @) 5 | amc12b-2021-p21 @p |- ( ph -> ( 2 <_ S /\ S < 6 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/imo-1964-p1-1.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1964-p1-1.0 @e |- ( ph -> N e. NN0 ) $@ 4 | imo-1964-p1-1.1 @e |- ( ph -> 7 || ( ( 2 ^ N ) - 1 ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | imo-1964-p1-1 @p |- ( ph -> 3 || N ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/imo-1964-p1-2.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1964-p1-2.0 @e |- ( ph -> N e. NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | imo-1964-p1-2 @p |- ( ph -> -. 7 || ( ( 2 ^ N ) + 1 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/imo-1987-p4.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1987-p4.0 @e |- ( ph -> F : NN0 --> NN0 ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | imo-1987-p4 @p |- ( ph -> E. n e. NN0 ( F ` ( F ` n ) ) =/= ( n + ; ; ; 1 9 8 7 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/imo-1990-p3.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | imo-1990-p3.0 @e |- ( ph -> N e. ( ZZ>= ` 2 ) ) $@ 4 | imo-1990-p3.1 @e |- ( ph -> ( ( ( 2 ^ N ) + 1 ) / ( N ^ 2 ) ) e. ZZ ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | imo-1990-p3 @p |- ( ph -> N = 3 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/imo-1993-p5.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | imo-1993-p5 @p |- ( ph -> E. f ( f : NN --> NN /\ ( ( ( A e. NN /\ B e. NN ) -> ( A < B <-> ( f ` A ) < ( f ` B ) ) ) /\ ( f ` 1 ) = 2 /\ A. n e. NN ( f ` ( f ` n ) ) = ( ( f ` n ) + n ) ) ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-10.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-algebra-10 @p |- ( ph -> ( abs ` ( ( ( ; ; 1 2 0 / ; ; 1 0 0 ) x. ; 3 0 ) - ( ( ; ; 1 3 0 / ; ; 1 0 0 ) x. ; 2 0 ) ) ) = ; 1 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-104.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-104.0 @e |- ( ph -> X e. RR ) $@ 4 | mathd-algebra-104.1 @e |- ( ph -> ( ; ; 1 2 5 / 8 ) = ( X / ; 1 2 ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-104 @p |- ( ph -> X = ( ; ; 3 7 5 / 2 ) ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-110.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-110.0 @e |- ( ph -> Q = ( 2 - ( 2 x. _i ) ) ) $@ 4 | mathd-algebra-110.1 @e |- ( ph -> E = ( 5 + ( 5 x. _i ) ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-110 @p |- ( ph -> ( Q x. E ) = ; 2 0 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-15.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-15.0 @e |- S = ( a e. NN , b e. NN |-> ( ( a ^ b ) + ( b ^ a ) ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-15 @p |- ( ph -> ( 2 S 6 ) = ; ; 1 0 0 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-182.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-182.0 @e |- ( ph -> Y e. CC ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-182 @p |- ( ph -> ( 7 x. ( ( 3 x. Y ) + 2 ) ) = ( ( ; 2 1 x. Y ) + ; 1 4 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-185.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-185.0 @e |- F = ( x e. RR |-> ( abs ` ( x + 4 ) ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-185 @p |- ( ph -> ( # ` { x e. ZZ | ( F ` x ) < 9 } ) = ; 1 7 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-190.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-190 @p |- ( ph -> ( ( ( 3 / 8 ) + ( 7 / 8 ) ) / ( 4 / 5 ) ) = ( ; 2 5 / ; 1 6 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-22.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-algebra-22 @p |- ( ph -> ( ( 5 ^ 2 ) logb ( 5 ^ 4 ) ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-224.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 31-May-2021.) @) 4 | mathd-algebra-224 @p |- ( ph -> ( card ` { x e. NN0 | ( ( sqrt ` x ) < ( 7 / 2 ) /\ 2 < ( sqrt ` x ) ) } ) = 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-35.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-35.0 @e |- P = ( x e. RR |-> ( 2 - ( x ^ 2 ) ) ) $@ 4 | mathd-algebra-35.1 @e |- Q = ( x e. RR* |-> ( 6 / x ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-35 @p |- ( ph -> ( P ` ( Q ` 2 ) ) = -u 7 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-405.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-405.0 @e |- ( ph -> X e. NN ) $@ 4 | mathd-algebra-405.1 @e |- ( ph -> ( ( ( X ^ 2 ) + ( 4 x. X ) ) + 4 ) < ; 2 0 ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-algebra-405 @p |- ( ph -> ( X = 1 \/ X = 2 ) ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-433.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-433.0 @e |- F = ( x e. RR |-> ( ( 3 x. ( sqrt ` ( ( 2 x. x ) - 7 ) ) ) - 8 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | mathd-algebra-433 @p |- ( ph -> ( F ` 8 ) = 1 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-462.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-algebra-462 @p |- ( ph -> ( ( ( 1 / 2 ) + ( 1 / 3 ) ) x. ( ( 1 / 2 ) - ( 1 / 3 ) ) ) = ( 5 / ; 3 6 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-48.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-48.0 @e |- ( ph -> Q = ( 9 - ( 4 x. _i ) ) ) $@ 4 | mathd-algebra-48.1 @e |- ( ph -> E = ( -u 3 - ( 4 x. _i ) ) ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-48 @p |- ( ph -> ( Q - E ) = ; 1 2 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-493.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-493.0 @e |- F = ( x e. RR |-> ( ( ( x ^ 2 ) - ( 4 x. ( sqrt ` x ) ) ) + 1 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | mathd-algebra-493 @p |- ( ph -> ( F ` ( F ` 4 ) ) = ; 7 0 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-509.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-algebra-509 @p |- ( ph -> ( sqrt ` ( ( ( ( 5 / ( sqrt ` ; 8 0 ) ) + ( ( sqrt ` ; ; 8 4 5 ) / 9 ) ) + ( sqrt ` ; 4 5 ) ) / ( sqrt ` 5 ) ) ) = ( ; 1 3 / 6 ) ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-536.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-algebra-536 @p |- ( ph -> ( ( ( ! ` 3 ) x. ( ( 2 ^ 3 ) + ( sqrt ` 9 ) ) ) / 2 ) = ; 3 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-547.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-547.0 @e |- ( ph -> X = 5 ) $@ 4 | mathd-algebra-547.1 @e |- ( ph -> Y = 2 ) $@ 5 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 6 | mathd-algebra-547 @p |- ( ph -> ( sqrt ` ( ( X ^ 3 ) - ( 2 ^ Y ) ) ) = ; 1 1 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-568.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-568.0 @e |- ( ph -> A e. RR ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | mathd-algebra-568 @p |- ( ph -> ( ( ( ( A - 1 ) x. ( A + 1 ) ) x. ( A + 2 ) ) - ( ( A - 2 ) x. ( A + 1 ) ) ) = ( ( A ^ 3 ) + ( A ^ 2 ) ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-59.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-59.0 @e |- ( ph -> B e. RR ) $@ 4 | mathd-algebra-59.1 @e |- ( ph -> ( ( 4 ^ B ) + ( 2 ^ 3 ) ) = ; 1 2 ) $@ 5 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 6 | mathd-algebra-59 @p |- ( ph -> B = 1 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-algebra-89.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-algebra-89.0 @e |- ( ph -> B e. RR* ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-algebra-89 @p |- ( ph -> ( ( ( 7 x. ( B ^ 3 ) ) ^ 2 ) x. ( ( 4 x. ( B ^ 2 ) ) ^ -u 3 ) ) = ( ; 4 9 / ; 6 4 ) ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-101.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-101 @p |- ( ph -> ( ( ; 1 7 x. ; 1 8 ) mod 4 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-102.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-102 @p |- ( ph -> ( ( 2 ^ 8 ) mod 5 ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-109.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-109.0 @e |- V = ( n e. NN |-> ( ( 2 x. n ) - 1 ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-numbertheory-109 @p |- ( ph -> ( sum_ k e. ( 1 ... ; ; 1 0 0 ) ( V ` k ) mod 7 ) = 4 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-132.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-132 @p |- ( ph -> ( ; ; ; 2 0 0 4 mod ; 1 2 ) = 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-149.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-149 @p |- ( ph -> sum_ k e. { n e. NN0 | ( n < ; 5 0 /\ ( n mod 8 ) = 5 /\ ( n mod 6 ) = 3 ) } k = ; 6 6 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-155.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-155 @p |- ( ph -> ( # ` { n e. ( ; ; 1 0 0 ... ; ; 9 9 9 ) | ( n mod ; 1 9 ) = 7 } ) = ; 5 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-156.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-156.0 @e |- ( ph -> N e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 5 | mathd-numbertheory-156 @p |- ( ph -> ( ( N + 7 ) gcd ( ( 2 x. N ) + 1 ) ) <_ ; 1 3 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-169.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-169 @p |- ( ph -> ( ( ! ` ; 2 0 ) gcd ; ; ; ; ; 2 0 0 0 0 0 ) = ; ; ; ; 4 0 0 0 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-198.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-198 @p |- ( ph -> ( ( 5 ^ ; ; ; 2 0 0 5 ) mod ; ; 1 0 0 ) = ; 2 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-200.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-200 @p |- ( ph -> ( ; ; 1 3 9 mod ; 1 1 ) = 7 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-202.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-202 @p |- ( ph -> ( ( ( ; 1 9 ^ ; 1 9 ) + ( ; 9 9 ^ ; 9 9 ) ) mod ; 1 0 ) = 8 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-211.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-211.0 @e |- D = { n e. NN | ( n < ; 6 0 /\ 6 || ( ( 4 x. n ) - 2 ) ) } $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | mathd-numbertheory-211 @p |- ( ph -> ( # ` D ) = ; 2 0 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-221.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Jun-2021.) @) 4 | mathd-numbertheory-221 @p |- ( ph -> ( # ` { x e. ( 1 ..^ ; ; ; 1 0 0 0 ) | ( # ` { p e. NN | p || x } ) = 3 } ) = ; 1 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-236.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-236 @p |- ( ph -> ( ( ; ; ; 1 9 9 9 ^ ; ; ; 2 0 0 0 ) mod 5 ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-24.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-24 @p |- ( ph -> ( sum_ k e. ( 1 ... 9 ) ( ; 1 1 ^ k ) mod ; ; 1 0 0 ) = ; 5 9 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-252.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-252 @p |- ( ph -> ( ( ! ` 7 ) mod ; 2 3 ) = 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-269.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-269 @p |- ( ph -> ( ( ( ( ( ; ; ; 2 0 0 5 ^ 2 ) + ( ; ; ; 2 0 0 5 ^ 0 ) ) + ( ; ; ; 2 0 0 5 ^ 0 ) ) + ( ; ; ; 2 0 0 5 ^ 5 ) ) mod ; ; 1 0 0 ) = ; 5 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-301.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-301.0 @e |- ( ph -> J e. NN ) $@ 4 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 5 | mathd-numbertheory-301 @p |- ( ph -> ( ( 3 x. ( ( 7 x. J ) + 3 ) ) mod 7 ) = 2 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-303.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Jun-2021.) @) 4 | mathd-numbertheory-303 @p |- ( ph -> sum_ k e. { n e. ( ZZ>= ` 2 ) | ( n || ( ; ; 1 7 1 - ; 8 0 ) /\ n || ( ; ; 4 6 8 - ; 1 3 ) ) } k = ; ; 1 1 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-32.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-32 @p |- ( ph -> sum_ k e. { n e. NN | n || ; 3 6 } k = ; 9 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-35.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-35 @p |- ( ph -> sum_ k e. { n e. NN | n || ( sqrt ` ; ; 1 9 6 ) } k = ; 2 4 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-37.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-37 @p |- ( ph -> ( ; ; ; 9 9 9 9 lcm ; ; ; ; ; 1 0 0 0 0 1 ) = ; ; ; ; ; ; ; 9 0 9 0 0 9 0 9 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-403.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Jun-2021.) @) 4 | mathd-numbertheory-403 @p |- ( ph -> sum_ k e. { n e. NN | ( n || ; ; 1 9 8 /\ n =/= ; ; 1 9 8 ) } k = ; ; 2 7 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-45.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-45 @p |- ( ph -> ( ( ; ; ; 6 4 3 2 gcd ; ; 1 3 2 ) + ; 1 1 ) = ; 2 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-458.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-458.0 @e |- ( ph -> N e. NN0 ) $@ 4 | mathd-numbertheory-458.1 @e |- ( ph -> ( N mod 8 ) = 7 ) $@ 5 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 6 | mathd-numbertheory-458 @p |- ( ph -> ( N mod 4 ) = 3 ) @= 7 | ? @. 8 | @} 9 | $) 10 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-461.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | mathd-numbertheory-461.0 @e |- ( ph -> N = ( # ` { n e. ( 1 ... 8 ) | ( n gcd 8 ) = 1 } ) ) $@ 4 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 5 | mathd-numbertheory-461 @p |- ( ph -> ( ( 3 ^ N ) mod 8 ) = 1 ) @= 6 | ? @. 7 | @} 8 | $) 9 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-466.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-466 @p |- ( ph -> ( sum_ k e. ( 1 ... ; 1 0 ) k mod 9 ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-543.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-543 @p |- ( ph -> ( ( 0 sigma ( ; 3 0 ^ 4 ) ) - 2 ) = ; ; 1 2 3 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-640.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-640 @p |- ( ph -> ( ( ( ( ; ; ; ; 9 1 1 4 5 + ; ; ; ; 9 1 1 4 6 ) + ; ; ; ; 9 1 1 4 7 ) + ; ; ; ; 9 1 1 4 8 ) mod 4 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-739.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 14-Jun-2021.) @) 4 | mathd-numbertheory-739 @p |- ( ph -> ( ( ! ` 9 ) mod ; 1 0 ) = 0 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-81.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-81 @p |- ( ph -> ( ; 7 1 mod 3 ) = 2 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-84.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 4 | mathd-numbertheory-84 @p |- ( ph -> ( |_ ` ( ( 9 / ; ; 1 6 0 ) x. ; ; 1 0 0 ) ) = 5 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/mathd-numbertheory-961.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | @( (Contributed by Kunhao Zheng, 9-Aug-2021.) @) 4 | mathd-numbertheory-961 @p |- ( ph -> ( ; ; ; 2 0 0 3 mod ; 1 1 ) = 1 ) @= 5 | ? @. 6 | @} 7 | $) 8 | -------------------------------------------------------------------------------- /metamath/valid/numbertheory-sqmod3in01d.mm: -------------------------------------------------------------------------------- 1 | $( 2 | @{ 3 | numbertheory-sqmod3in01d.0 @e |- ( ph -> A e. ZZ ) $@ 4 | @( (Contributed by Kunhao Zheng, 4-May-2021.) @) 5 | numbertheory-sqmod3in01d @p |- ( ph -> ( ( A ^ 2 ) mod 3 ) e. { 0 , 1 } ) @= 6 | ? @. 7 | @} 8 | $) 9 | --------------------------------------------------------------------------------