File size: 129,301 Bytes
11999e4 |
1 |
[["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_binomnegdiscrineq_10alt28asqp1:\n fixes a :: real\n shows \"10 * a \\<le> 28 * a^2 + 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p15:\n fixes f :: \"complex \\<Rightarrow> complex\"\n assumes asm:\"\\<forall> x. f (x / 3) = x^2 + x + 1\"\n shows \"(\\<Sum>y\\<in>f -` {7}. y / 3) = - 1 / 9\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2008_p2:\n fixes x :: real\n assumes h0 : \"x * (1 / 2 + 2 / 3) = 1\"\n shows \"x = 6/7\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2003_p24:\n fixes a b::real\n assumes \"b\\<le>a\"\n and \"1<b\"\n shows \"ln (a/b) / ln a + ln (b/a) / ln b \\<le>0\" (is \"?L \\<le> _\")"], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_132:\n fixes x :: real\n and f g :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = x + 2\"\n and h1 : \"\\<And>x. g x = x^2\"\n and h2 : \"f (g x) = g (f x)\"\n shows \"x = -1/2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_64 :\n \"(LEAST x ::nat. [30 * x = 42] (mod 47)) = 39\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_155:\n \"card ({x::nat. x mod 19 = 7 \\<and> 100\\<le>x \\<and> x < 1000}) = 48\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_51:\n fixes a b ::real\n assumes \"0 < a \\<and> 0 < b\"\n and \"a + b = 35\"\n and \"a = (2/5) * b\"\n shows \"b - a = 15\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_149:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x < -5. f x = x^2 + 5\"\n and \"\\<forall> x \\<ge> -5. f x = 3 * x -8\"\n shows \"(\\<Sum> k \\<in> (f -` {10}). k) = 6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_709:\n fixes n :: nat\n assumes \"n>0\" \n and \"card ({k. k dvd (2*n)}) = 28\"\n and \"card ({k. k dvd (3*n)}) = 30\" \n shows \"card ({k. k dvd (6*n)}) = 35\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_sum_odd:\n fixes n :: nat\n assumes \"n > 0\"\n shows \"(\\<Sum>(k::nat) = 0..(n-1). 2 * k + 1) = n^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2019_p9:\n fixes a :: \"nat \\<Rightarrow> rat\"\n assumes \"a 1 = 1\"\n and \"a 2 = 3 / 7\"\n and \"\\<forall> n. a (n + 2) = (a n * a (n + 1)) / (2 * a n - a (n + 1))\" \n shows \"fst (quotient_of (a 2019)) + snd (quotient_of (a 2019)) = 8078\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_89:\n fixes b :: real\n assumes h0 : \"b\\<noteq>0\"\n shows \"(7 * b^3)^2 * 1/((4 * b^2)^3) = 49 / 64\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_140:\n fixes a b c :: real\n assumes h0 : \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and h1 : \"\\<forall>x. 24 * x^2 - 19 * x - 35 = ((a*x-5) * (2 * (b*x) + c))\"\n shows \"a * b - 3 * c = -9\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_405:\n fixes x :: nat\n assumes h0 : \"0 < x\"\n and h1 : \"x ^ 2 + 4 * x + 4 < 20\"\n shows \"x = 1 \\<or> x = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p3:\n fixes n ::nat\n assumes \"n>0\"\n and prime:\"prime (n^2+2-3*n)\"\n shows \"n=3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_326:\n fixes n :: nat\n assumes \"(n - 1) * n * (n + 1) = 720\" \n shows \"(n + 1) = 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_169:\n \"gcd (fact 20) 200000 = (40000::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_11:\n fixes a b :: real\n assumes h0 : \"a \\<noteq> b\"\n and h1 : \"a \\<noteq> 2 * b\"\n and h2 : \"(4*a+3*b) / (a-2*b) = 5\"\n shows \"(a+11*b) / (a-b) = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_109:\n fixes a b :: real\n assumes h0 : \"3*a+2*b=12\"\n and h1 : \"a=4\"\n shows \"b=0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_202:\n \"(19^19 + 99^99) mod 10 = (8::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_568:\n fixes a :: real\n shows \"(a-1) * (a+1) * (a+2) - (a-2) * (a+1) = a^3 + a^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_divisibility_9div10tonm1:\n fixes n::nat\n shows \"(9::nat) dvd 10^n - 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_437:\n fixes x y :: real\n and n :: int\n assumes \"x^3 = -45\"\n and \"y^3 = -101\"\n and \"x < n\"\n and \"n < y\" \n shows \"n = -4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_seq_mul2pnp1:\n fixes n :: nat\n and u :: \"nat \\<Rightarrow> nat\"\n assumes h0 : \"u 0 = 0\"\n and h1 : \"\\<And>(n::nat). u (n+1) = 2 * u n + (n+1)\"\n shows \"u n = 2 ^ (n+1) - (n+2)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_24:\n \"(\\<Sum> k \\<in>{1..<10}. 11^k) mod 100 = (59::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_493:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = x^2 - 4 * (sqrt x) + 1\"\n shows \"f (f 4) = 70\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p5:\n fixes x p ::real\n assumes \"x<2\"\n and \"\\<bar>x -2\\<bar> = p\"\n shows \"x - p = 2 - 2 * p\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_manipexpr_2erprsqpesqeqnrpnesq:\n fixes e r :: complex\n shows \"2 * (e * r) + (e^2 + r^2) = (-r + (-e))^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1964_p1_1:\n fixes n :: nat\n assumes \"7 dvd (2^n-1)\"\n shows \"3 dvd n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2019_p21:\n fixes z::complex\n assumes h0: \"z = (Complex (1/sqrt 2) (1/sqrt 2))\"\n shows \"(\\<Sum>k::nat=1..12. (z^(k^2))) * (\\<Sum> k::nat=1..12. 1/(z^(k^2))) =36\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_482:\n fixes m n :: nat\n and k :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"prime m\"\n and h1 : \"prime n\"\n and h2 : \"m \\<noteq> n\"\n and h3 : \"\\<And>x. f x = x^2 - 12*x + k\"\n and h4 : \"f m = 0\"\n and h5 : \"f n = 0\"\n shows \"k = 35\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_116:\n fixes k x :: real\n assumes h0 : \"x = (13 - sqrt 131) / 4\"\n and h1 : \"2 * x^2 - 13 * x + k = 0\"\n shows \"k = 19/4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_206:\n fixes a b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = x^2 + a*x + b\"\n and h1 : \"2 * a \\<noteq> b\"\n and h2 : \"f (2 * a) = 0\"\n and h3 : \"f b = 0\"\n shows \"a + b = -1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2003_p9:\n fixes a b ::real and f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. f x = a * x + b\"\n and \" f 6 - f 2 = 12\"\n shows \"f 12 - f 2 = 30\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_divisibility_3div2tooddnp1:\n fixes n ::nat\n shows \"(3::nat) dvd (2^(2 * n + 1) + 1)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1961_p1:\n fixes x y z a b :: real\n assumes h0 : \"0 < x \\<and> 0 < y \\<and> 0 < z\"\n and h1 : \"x \\<noteq> y\"\n and h2 : \"y \\<noteq> z\"\n and h3 : \"z \\<noteq> x\"\n and h4 : \"x + y + z = a\"\n and h5 : \"x^2 + y^2 + z^2 = b^2\"\n and h6 : \"x * y = z^2\"\n shows \"0<a \\<and> b^2 < a^2 \\<and> a^2 < 3*b^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_faxinrrp2msqrt2geq2mxm1div2x:\n \"\\<And>x. (x>0) \\<Longrightarrow> 2 - sqrt 2 \\<ge> 2 - x - 1/ (2 * x)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_sumasqdivbsqgeqsumbdiva:\n fixes a b c :: real\n assumes h0 : \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n shows \"a^2 / b^2 + b^2 / c^2 + c^2 / a^2 \\<ge> b / a + c / b + a / c\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_35:\n fixes k :: nat\n assumes \"k^2 = 196\"\n shows \"(\\<Sum> k \\<in> { n ::nat. n dvd k}. k) = (24::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1966_p4:\n fixes n :: nat\n and x :: real\n assumes h0 : \"\\<And>(k::nat). \\<And>(m::int). k\\<noteq>0 \\<Longrightarrow> x \\<noteq> m * pi / (2^k)\"\n and h1 : \"0 < n\"\n shows \"(\\<Sum>(k::nat) =1..n.(1 / sin ((2^k) * x))) = 1 / tan x - 1 / tan ((2^n) * x)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_780:\n fixes m x :: nat\n assumes h0 : \"10 \\<le> m\"\n and h1 : \"m \\<le> 99\"\n and h2 : \"(6 * x) mod m = 1\"\n and h3 : \"(x - 6^2) mod m = 0\"\n shows \"m = 43\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_32:\n \"(\\<Sum> k \\<in> { n ::nat. n dvd 36}. k) = 91\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_sum_1oktkp1:\n fixes n :: nat\n shows \"n=0 \\<or> (\\<Sum>(k::nat) = 0..(n-1). (1::real)/((k+1)*(k+2))) = n / (n+1)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_421:\n fixes a b c d :: real\n assumes h0 : \"b = a^2 + 4 * a + 6\"\n and h1 : \"b = 1 / 2 * a^2 + a + 6\"\n and h2 : \"d = c^2 + 4 * c + 6\"\n and h3 : \"d = 1 / 2 * c^2 + c + 6\"\n and h4 : \"a < c\"\n shows \"c-a=6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_151:\n shows \"ceiling (sqrt 27) - floor (sqrt 26) = 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p15:\n fixes n :: nat\n assumes \"0 < n\"\n and \"(\\<Sum> k \\<in> {1..<n+1}. (k * (\\<i>^k))) = 48 + 49 * \\<i>\" \n shows \"n = 97\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_35:\n fixes p q :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. p x = 2 - x^2\"\n and h1 : \"\\<And>x. (x\\<noteq>0) \\<Longrightarrow> q x = 6 / x\"\n shows \"p (q 2) = -7\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p7:\n fixes x y ::real\n shows \"1 \\<le> ((x * y) - 1)^2 + (x + y)^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_221:\n \"card {x ::nat. 0 < x \\<and> x < 1000 \\<and> card ({n. n dvd x}) = 3} = 11\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_640:\n \"(91145+91146+91147+91148) mod 4 = (2::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_136:\n fixes n ::nat\n assumes \"123 * n + 17 = 39500\"\n shows \"n = 321\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_xmysqpymzsqpzmxsqeqxyz_xpypzp6dvdx3y3z3:\n fixes x y z :: int\n assumes h0 : \"(x-y)^2 + (y-z)^2 + (z-x)^2 = x * y * z\"\n shows \"(x + y + z + 6) dvd (x^3 + y^3 + z^3)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1974_p5:\n fixes a b c d s :: real\n assumes \"a>0\" \"b>0\" \"c>0\" \"d>0\"\n assumes h0 : \"s=a/(a+b+d) + b/(a+b+c) + c/(b+c+d) + d/(a+c+d)\"\n shows \"1<s \\<and> s<2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_96:\n fixes x y z a ::real\n assumes \"x>0\" \"y>0\" \"z>0\" \n and \"ln x - ln y = a\"\n and \"ln y - ln z = 15\"\n and \"ln z - ln x=-7\"\n shows \"a = -8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_ineq_nsqlefactn:\n fixes n::nat\n assumes \" 4 \\<le> n\"\n shows \"n^2 \\<le> fact n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1987_p4:\n fixes f :: \"nat \\<Rightarrow> nat\"\n shows \"\\<exists>(n::nat). f (f n) \\<noteq> n + 1987\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_123:\n fixes a b :: nat\n assumes h0 : \"a + b = 20\"\n and h1 : \"a = 3 * b\"\n shows \"a - b = 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_234:\n fixes d :: real\n assumes h0 : \"27/125 * d = 9/25\"\n shows \"3/5 * d^3 = 25/9\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_370:\n fixes n :: nat\n assumes h0 : \"n mod 7 = (3::nat)\"\n shows \"(2*n+1) mod 7 = (0::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_2varlineareq_xpeeq7_2xpeeq3_eeq11_xeqn4:\n fixes x e :: complex\n assumes h0 : \"x + e = 7\"\n and h1 : \"2 * x + e = 3\"\n shows \"e=11 \\<and> x= (-4)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_81:\n \"71 mod 3 = (2::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_101:\n fixes x :: real\n assumes h0 : \"x^2 - 5 * x - 4 \\<le> 10\"\n shows \"x\\<ge> -2 \\<and> x \\<le> 7\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_458:\n fixes n :: nat\n assumes h0 : \"n mod 8 = (7::nat)\"\n shows \"n mod 4 = 3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2017_p2:\n fixes x y :: real\n assumes h0 : \"x \\<noteq> 0\"\n and h1 : \"y \\<noteq> 0\"\n and h2 : \"x + y = 4 * (x * y)\"\n shows \"1/x + 1/y = 4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_sqmod4in01d:\n fixes a :: int\n shows \"(a^2 mod 4 = 0) \\<or> (a^2 mod 4 = 1)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1993_p5:\n \"\\<exists> f :: nat \\<Rightarrow> nat. \n (\\<forall> a b. (a < b) \\<longleftrightarrow> f a < f b) \n \\<and> f 1 = 2 \\<and> (\\<forall> n. f (f n) = f n + n)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_22:\n fixes b :: nat\n assumes h0 : \"b < 10\"\n and h1 : \"\\<exists>a. (10*b+6) = a^2\"\n shows \"b=3 \\<or> b =1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aimeII_2020_p6:\n fixes t :: \"nat \\<Rightarrow> rat\"\n assumes \"t 1 = 20\"\n and \"t 2 = 21\"\n and \"\\<forall> n \\<ge> 3. t n = (5 * t (n - 1) + 1) / (25 * t (n - 2))\" \n shows \"let (a,b) = quotient_of (t 2020) in a +b = 626\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_10:\n \"abs ((120::real) / 100 * 30 - 130 / 100 * 20) = 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_190:\n \"((3::real) / 8 + 7 / 8) / (4 / 5) = 25 / 16\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_59:\n fixes b :: real\n assumes \"4 powr b + 2^3 = 12\"\n shows \"b=1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2003_p17:\n fixes x y ::real\n assumes \"x>0\" \"y>0\"\n and \"ln (x * y^3) =1\"\n and \"ln (x^2 * y) = 1\"\n shows \"ln (x*y) = 3/5\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_apb4leq8ta4pb4:\n fixes a b :: real\n assumes h0 : \"0 < a \\<and> 0 < b\"\n shows \"(a+b)^4 \\<le> 8 * (a^4 + b^4)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_236:\n \"(1999^2000) mod 5 = (1::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_126:\n fixes x :: nat\n assumes \"x>0\"\n shows \"(LEAST a. gcd a 40 = x + 3 \\<and> lcm a 40 = x * (x + 3)) = 8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_22:\n \"(log 2 (5^4)) / (log 2 (5^2)) = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2008_p15:\n fixes k :: nat\n assumes h0 : \"k = 2008^2 + 2^2008\"\n shows \"(k^2 + 2^k) mod 10 = 6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_36azm9asqle36zsq:\n fixes z a :: real\n shows \"36 * (a * z) - 9 * a^2 \\<le> 36 * z^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_prmdvsneqnsqmodpeq0:\n fixes n :: int\n and p :: nat\n assumes \"prime p\" \n shows \"p dvd n \\<longleftrightarrow> (n^2) mod p = 0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_251:\n fixes x :: real\n assumes h0: \"x \\<noteq> 0\"\n and h1: \"3 + 1/x = 7/x\"\n shows \"x = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_divisibility_3divnto3m2n:\n fixes n::nat\n shows \"3 dvd n^3 + 2 * n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_prod1toneq1_sum1tongeqn:\n fixes a :: \"nat \\<Rightarrow> real\"\n and n :: nat\n assumes \"\\<forall>i. a i \\<ge>0\"\n and \"prod a {..<n} = 1\" \n shows \"sum a {..<n} \\<ge> n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2008_p4:\n \"(\\<Prod>k::nat=1..501. ((4::real) * k + 4) / (4 * k)) = 502\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_2rootspoly_apatapbeq2asqp2ab:\n fixes a b :: complex\n shows \"(a+a) * (a+b) = 2 * a^2 + 2 * (a*b)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2002_p1:\n fixes f::\"complex \\<Rightarrow> complex\"\n assumes \"\\<forall> x. f x = (2 * x + 3) * (x - 4) + (2 * x + 3) * (x - 6)\"\n shows \"(\\<Sum> y \\<in> f -` {0}. y) = 7/2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2003_p25:\n fixes a b::real and f ::\"real \\<Rightarrow> real\"\n assumes \"b>0\" \n and \"\\<forall> x. f x = sqrt (a * x^2 + b * x)\"\n and \"{x. 0 \\<le> f x} = f ` {x. 0 \\<le> f x}\"\n shows \"a=0 \\<or> a = -4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_224:\n \"card { n :: nat. sqrt n < 7 / 2 \\<and> 2 < sqrt n} = 8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_284:\n fixes a b :: nat\n assumes h0 : \"1\\<le>a \\<and> a \\<le>9 \\<and> b \\<le>9\"\n and h1 : \"10 * a + b = 2 * (a+b)\"\n shows \"10 * a + b = 18\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_sqmod3in01d:\n fixes a :: int\n shows \"a^2 mod 3 = 0 \\<or> a^2 mod 3 = 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1962_p4:\n fixes x :: real\n assumes h0 : \"(cos x)^2 + (cos (2 * x))^2 + (cos (3 * x))^2 = 1\"\n shows \"(\\<exists>(m::int). x = pi/2 + m * pi) \\<or>\n (\\<exists>(m::int). x = pi/4 + m * pi/2) \\<or> \n (\\<exists>(m::int). x = pi/6 + m * pi/6) \\<or> \n (\\<exists>(m::int). x = 5*pi/6 + m * pi/6) \n \""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_252:\n \"(fact 7) mod 23 = (3::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_159:\n fixes b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = 3 * x^4 - 7 * x^3 + 2*x^2 - b*x +1\"\n and h1 : \"f 1 = 1\"\n shows \"b = -2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1965_p1:\n fixes x :: real\n assumes \"0 \\<le> x\"\n and \"x \\<le> 2 * pi\"\n and \"2 * cos x \\<le> abs (sqrt (1 + sin (2 * x)) \n - sqrt (1 - sin (2 * x)))\"\n and \"abs (sqrt (1 + sin (2 * x)) - sqrt (1 - sin (2 * x))) \\<le> sqrt 2\" \n shows \"pi / 4 \\<le> x \\<and> x \\<le> 7 * pi / 4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_690 :\n \"(LEAST a ::nat. [a = 2] (mod 3) \\<and> [a = 4] (mod 5) \n \\<and> [a = 6] (mod 7) \\<and> [a = 8] (mod 9)) = 314\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p9:\n fixes a b c::real\n and f::\"real \\<Rightarrow> real\"\n assumes h0:\"\\<forall> x. f (x+3) = 3 * x^2 + 7*x + 4\"\n and h1:\"\\<forall> x. f x = a * x^2 + b * x + c\"\n shows \"a+b+c=2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_536:\n \"fact 3 * (2^3 + sqrt 9) / 2 = 33\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_3rootspoly_amdtamctambeqnasqmbpctapcbtdpasqmbpctapcbta:\n fixes a b c d :: complex\n shows \"(a-d) * (a-c) * (a-b) = -(((a^2 - (b+c) * a) + c * b) * d) + (a^2 - (b+c) * a + c * b) * a\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1988_p3:\n fixes x :: real\n assumes h0 : \"0 < x\"\n and h1 : \"log 2 (log 8 x) = log 8 (log 2 x)\"\n shows \"(log 2 x)^2 = 27\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p11:\n fixes a b::nat\n assumes \"prime a\" and \"prime b\"\n and \"prime (a+b)\" and \"prime (a-b)\"\n shows \"prime (a + b + (a - b + (a + b)))\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_48:\n fixes q e :: complex\n assumes h0 : \"q = Complex 9 (-4)\"\n and h1 : \"e = Complex (-3) (-4)\"\n shows \"q - e = 12\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_181:\n fixes n :: real\n assumes h0 : \"n \\<noteq> 3\"\n and h1 : \"(n+5) / (n-3) = 2\"\n shows \"n=11\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_247:\n fixes t s :: real\n and n :: nat\n assumes h0 : \"t = 2 * s - s^2\"\n and h1 : \"s = n^2 - 2^n + 1\"\n and h2 : \"n=3\"\n shows \"t=0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1988_p4:\n fixes n :: nat\n and a :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"\\<And>n. abs (a n) < 1\"\n and h1 : \"(\\<Sum>(k::nat) = 0..(n-1). (abs (a k))) = 19 + abs(\\<Sum>(k::nat) = 0..(n-1). (a k))\"\n shows \"20 \\<le> n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_303:\n \"(\\<Sum> k \\<in> {n ::nat. 2 \\<le> n \\<and> [171 = 80] (mod n) \\<and> [468 = 13] (mod n)}. k) = 111\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2001_p9:\n fixes f:: \"real \\<Rightarrow> real\"\n assumes f_times:\"\\<forall> x > 0. \\<forall> y > 0. f (x * y) = f x / y\"\n and \"f 500 = 3\"\n shows \"f 600 = 5 / 2 \""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_33:\n fixes n :: nat\n assumes h0 : \"n < 398\"\n and h1 : \"(n * 7) mod 398 = 1\"\n shows \"n=57\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_119:\n fixes d e :: real\n assumes h0 : \"2 * d = 17 * e - 8\"\n and h1 : \"2 * e = d - 9\"\n shows \"e = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_sumkmulnckeqnmul2pownm1:\n fixes n k :: nat\n assumes h0 : \"0<n \\<and> 0<k\"\n and h1 : \"k\\<le>n\"\n shows \"n choose k = ((n-1) choose k) + ((n-1) choose (k-1))\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2003_p6:\n fixes a r::real and u::\"nat \\<Rightarrow> real\"\n assumes \"\\<forall> k. u k = a * r^k\"\n and \"u 1= 2\"\n and \"u 3=6\"\n shows \"u 0 = 2/ sqrt 3 \\<or> u 0 = - 2/sqrt 3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_102:\n \"(2^8) mod 5 = (1::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_48:\n fixes b :: nat\n assumes h0 : \"0<b\"\n and h1 : \"3 * b^2 + 2 * b + 1 = 57\"\n shows \"b=4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_269:\n \"(2005^2 + 2005^0 + 2005^0 + 2005^5) mod 100 = (52::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1966_p5:\n fixes x a :: \"nat \\<Rightarrow> real\"\n assumes \"a 1 > a 2\" and \"a 2 > a 3\" and \"a 3 > a 4\"\n assumes \n h6 : \"abs (a 1 - a 2) * x 2 + abs (a 1 - a 3) * x 3 + abs (a 1 - a 4) * x 4 = 1\"\n and h7 : \"abs (a 2 - a 1) * x 1 + abs (a 2 - a 3) * x 3 + abs (a 2 - a 4) * x 4 = 1\"\n and h8 : \"abs (a 3 - a 1) * x 1 + abs (a 3 - a 2) * x 2 + abs (a 3 - a 4) * x 4 = 1\"\n and h9 : \"abs (a 4 - a 1) * x 1 + abs (a 4 - a 2) * x 2 + abs (a 4 - a 3) * x 3 = 1\"\n shows \"x 2 = 0 \\<and> x 3 = 0 \\<and> x 1 = 1 / abs (a 1 - a 4) \\<and> x 4 = 1 / abs (a 1 - a 4)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_2complexrootspoly_xsqp49eqxp7itxpn7i:\n fixes x :: complex\n shows \"x^2 + 49 = (x + 7 * \\<i>) * (x - 7 * \\<i>)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2004_p3:\n fixes x y :: nat\n assumes \"2^x * 3^y = 1296\"\n shows \"x + y = 8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_110:\n fixes q e :: complex\n assumes h0 : \"q = Complex 2 (-2)\"\n and h1 : \"e = Complex 5 5\"\n shows \"q * e = 20\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_455:\n fixes x :: real\n assumes h0 : \"2 * (2 * (2 * (2 * x))) = 48\"\n shows \"x=3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_323:\n fixes \\<sigma>:: \"real \\<Rightarrow> real\"\n assumes \"bij \\<sigma>\"\n and \"\\<forall> x. \\<sigma> x = x^3 - 8\" \n shows \"Hilbert_Choice.inv \\<sigma> (\\<sigma> (Hilbert_Choice.inv \\<sigma> 19)) = 3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_73:\n fixes p q r x :: complex\n assumes h0 : \"(x-p) * (x-q) = (r-p) * (r-q)\"\n and h1 : \"x \\<noteq> r\"\n shows \"x = p + q -r\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_126:\n fixes x y :: real\n assumes h0 : \"2 * 3 = x - 9\"\n and h1 : \"2 * (-5) = y + 1\"\n shows \"x=15 \\<and> y = -11\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_547:\n fixes x y :: real\n assumes \"x=5\"\n and \"y=2\"\n shows \"sqrt (x^3 - y^2) = 11\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_403:\n \"(\\<Sum> k \\<in> ({n. n dvd 198 \\<and> n\\<noteq> 198}). k) = (270::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1987_p6:\n fixes p :: nat\n and f :: \"nat \\<Rightarrow> nat\"\n assumes h0 : \"\\<And>x. f x = x^2 + x + p\"\n and h1 : \"\\<And>(k::nat). (k\\<le>floor(sqrt (p/3))) \\<Longrightarrow> prime (f k)\"\n shows \"\\<And>i. (i \\<le> p - 2) \\<Longrightarrow> prime (f i)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2003_p1:\n fixes u v :: \"nat \\<Rightarrow> nat\"\n assumes u:\"\\<forall>n. u n = 2 *n\"\n and v:\"\\<forall>n. v n= 2* n -1\"\n shows \"(\\<Sum> k \\<in>{1..2003}. u k) - (\\<Sum> k \\<in>{1..2003}. v k) = 2003\"\n (is \"?L = ?R\")"], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_2006_p3:\n fixes a b c ::real\n shows \"(a * b * (a^2 - b^2)) + (b * c * (b^2 - c^2)) + \n (c * a * (c^2 - a^2)) \\<le> (9 * sqrt 2) / 32 * (a^2 + b^2 + c^2)^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_182:\n fixes y:: complex\n shows \"7*(3*y+2) = 21 * y + 14\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_37:\n fixes x y :: real\n assumes h0 : \"x+y=7\"\n and h1 : \"3 * x + y = 45\"\n shows \"x^2 - y^2 = 217\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aimeI_2000_p7:\n fixes x y z :: real\n and m :: rat\n assumes \"0 < x \\<and> 0 < y \\<and> 0 < z\"\n and \"x * y * z = 1\"\n and \"x + 1 / z = 5\"\n and \"y + 1 / x = 29\"\n and \"z + 1 / y = m\"\n and \"0 < m\" \n shows \"let (x,y) = quotient_of m in x + y = 5\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_sqrtxymulxmyeqxpy_xpygeq4:\n fixes x y :: real\n assumes h0 : \"0 < x \\<and> 0 < y\"\n and h1 : \"y \\<le> x\"\n and h2 : \"sqrt (x * y) * (x - y) = (x + y)\"\n shows \"x + y \\<ge> 4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_185:\n fixes f :: \"int \\<Rightarrow> int\"\n assumes h0 : \"\\<And>x. (f x = abs (x+4))\"\n shows \"card {(x::int). (f x < 9)} = 17\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2011_p18:\n fixes x y :: real\n assumes h0 : \"abs (x+y) + abs (x-y) = 2\"\n shows \"x^2 - 6 * x + y^2 \\<le> 8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_101:\n \"(17 * 18) mod 4 = (2::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p25:\n fixes a :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"a 1 = 1\"\n and h1 : \"a 2 = 1 / (sqrt 3)\"\n and h2 : \"\\<And>n. a (n+2) = (a n + a (n+1)) / (1 - (a n) * (a (n+1)))\"\n shows \"abs (a 2009) = 0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_37:\n \"lcm 9999 100001 = (90900909::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2016_p2:\n fixes x :: nat\n assumes h0 : \"10^x * 100^(2*x) = 1000^5\"\n shows \"x=3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_30:\n \"(33818^2 + 33819^2 + 33820^2 + 33821^2 + 33822^2) mod 17 = (0::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_335:\n fixes n :: nat\n assumes h0 : \"n mod 7 = 5\"\n shows \"(5 * n) mod 7 = 4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_211 :\n \"card {n::nat. n<60 \\<and> 6 dvd (4 * n - 2)} = 20\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_509:\n \"sqrt ((5 / sqrt 80 + sqrt 845 / 9 + sqrt 45) / sqrt 5) = 13 / 6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_451:\n fixes \\<sigma>:: \"real \\<Rightarrow> real\"\n assumes \"bij \\<sigma>\"\n and \"Hilbert_Choice.inv \\<sigma> (-15) = 0\"\n and \"Hilbert_Choice.inv \\<sigma> 0 = 3\"\n and \"Hilbert_Choice.inv \\<sigma> 3 = 9\"\n and \"Hilbert_Choice.inv \\<sigma> 9 = 20\" \n shows \"\\<sigma> (\\<sigma> 9) = 0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_480:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. x<0 \\<Longrightarrow> f x = -(x^2)-1\"\n and h1 : \"\\<And>x. (0 \\<le> x \\<and> x < 4) \\<Longrightarrow> f x = 2\"\n and h2 : \"\\<And>x. x\\<ge>4 \\<Longrightarrow> f x = sqrt x\"\n shows \"f pi = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_327:\n fixes a :: real\n assumes h0 : \"1 / 5 * abs(9 + 2 * a) < 1\"\n shows \"-7 < a \\<and> a < -2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_42:\n fixes u v :: nat\n assumes \"27 * u mod 40 = 17\"\n and \"27 * v mod 40 = 17\"\n and \"u < 40\"\n and \"v < 80\"\n and \"40 < v\" \n shows \"(u + v) = 62\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1984_p2:\n fixes a b :: nat\n assumes h0 : \"0 < a \\<and> 0 < b\"\n and h1 : \"\\<not> (7 dvd a)\"\n and h2 : \"\\<not> (7 dvd b)\"\n and h3 : \"\\<not> (7 dvd (a+b))\"\n and h4 : \"(7^7) dvd ((a+b)^7 - a^7 - b^7)\"\n shows \"19 \\<le> a + b\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_45 :\n \"(gcd 6432 132) + 11 = (23::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2002_p12:\n fixes f :: \"real => real\"\n and k :: real and a b::nat\n assumes \"\\<forall> x. f x = x^2 - 63 * x + k\"\n and \"f -` {0} = {of_nat a, of_nat b}\"\n and \"prime a\" and \"prime b\"\n shows \"k=122\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_2at2pclta2c2p41pc:\n fixes a c :: real\n shows \"2 * a * (2+c) \\<le> a^2 + c^2 + 4 * (1+c)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_77:\n fixes a b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"a \\<noteq> 0 \\<and> b \\<noteq> 0 \\<and> a \\<noteq> b\"\n and h1 : \"\\<And>x. f x = x^2 + a*x + b\"\n and h2 : \"f a = 0\"\n and h3 : \"f b = 0\"\n shows \"a=1 \\<and> b = -2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_xsqpysqintdenomeq:\n fixes x y :: rat\n assumes \"snd (quotient_of (x^2 + y^2)) = 1\"\n shows \"snd (quotient_of x) = snd (quotient_of y)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_668:\n fixes l r::int and a b::int\n assumes \"0\\<le>l\" \"l<7\" \"0\\<le>r\" \"r<7\"\n and \"[l * (2 + 3) = 1] (mod 7)\" \n and \"0\\<le>a \\<and> a<7 \\<and> [a*2=1] (mod 7)\"\n and \"0\\<le>b \\<and> b<7 \\<and> [b*3=1] (mod 7)\"\n and \"r = (a+b) mod 7\"\n shows \"l - r = 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1996_p5:\n fixes a b c r s t :: real\n and f g :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = x^3 + 3 * x^2 + 4*x -11\"\n and h1 : \"\\<And>x. g x = x^3 + r * x^2 + s*x + t\"\n and h2 : \"f a = 0\"\n and h3 : \"f b = 0\"\n and h4 : \"f c = 0\"\n and h5 : \"g (a+b) = 0\"\n and h6 : \"g (b+c) = 0\"\n and h7 : \"g (c+a) = 0\"\n and h8 : \"a \\<noteq> b\"\n and h9 : \"a \\<noteq> c\"\n and h10 : \"b \\<noteq> c\"\n shows \"t=23\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p5:\n fixes a b :: nat\n assumes \"(5::real) / 8 * b - 2 / 3 * a = 7\"\n and \"of_nat b - (5::real) / 8 * b - (a - 2 / 3 * a) = 7\"\n shows \"a = 42\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2010_p22:\n fixes x ::real \n shows \"49 \\<le> (\\<Sum> k \\<in> {1..<120}. abs (k * x - 1))\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1988_p6:\n fixes a b :: nat\n assumes h0 : \"0<a \\<and> 0<b\"\n and h1 : \"(a*b+1) dvd (a^2 + b^2)\"\n shows \"\\<exists>(x::nat). ((x^2) = (a^2+b^2)/(a*b+1))\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_739:\n \"(fact 9) mod 10 = (0::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_200:\n \"139 mod 11 = (7::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_13:\n fixes a b :: real\n assumes h0: \"\\<forall>(x::real). (x-3 \\<noteq> 0 \\<and> x - 5 \\<noteq> 0) \\<Longrightarrow> \n 4 * x / (x^2 - 8 * x + 15) = a / (x-3) + b / (x-5)\"\n shows \"a = -6 \\<and> b = 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p13:\n fixes a b c::nat\n assumes \"1 < a \\<and> 1 < b \\<and> 1 < c\"\n and \"\\<forall>n>1. (n * ((n * (n powr (1 / c))) powr (1 / b))) powr (1 / a) = (n^25) powr (1 / 36)\"\n shows \"b=3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_110:\n fixes a b :: nat\n assumes h0 : \"0 < a \\<and> 0 < b \\<and> b \\<le> a\"\n and h1 : \"(a+b) mod 10 = 2\"\n and h2 : \"(2*a + b) mod 10 = 1\"\n shows \"(a-b) mod 10 = 6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p6:\n fixes a b :: real\n assumes \"a \\<noteq> 0 \\<and> b \\<noteq> 0\"\n and \"\\<forall> x. x^2 + a * x + b = (x - a) * (x - b)\"\n shows \" a = 1 \\<and> b = -2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_267:\n fixes x :: real\n assumes h0 : \"x \\<noteq> 1\"\n and h1 : \"x \\<noteq> -2\"\n and h2 : \"(x + 1) / (x - 1) = (x - 2) / (x + 2)\"\n shows \"x=0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_232:\n fixes x y z::nat\n assumes \"x<31\" \"y<31\" \"z<31\"\n and \"[x *3 = 1] (mod 31)\"\n and \"[y * 5 = 1] (mod 31)\"\n and \"[z * (x + y) =1] (mod 31)\" \n shows \"z = 29\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_13:\n fixes u v :: nat\n assumes \"u>0 \\<and> v>0\"\n and \"(14 * u) mod 100 = 46\"\n and \"(14 * v) mod 100 = 46\"\n and \"u < 50\"\n and \"v < 100\"\n and \"50 < v\" \n shows \"(u + v) / 2 = 64\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_188:\n \"gcd 180 168 = (12::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_543 :\n \"(\\<Sum> k \\<in> ({n::nat. n dvd (30^4)}). 1) - 2 = (123::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p21:\n \"card {n :: nat. 5 dvd n \\<and> lcm (fact 5) n \n = 5 * gcd (fact 10) n} = 48\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2013_p7:\n fixes s :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"\\<And>n. s (n+2) = s (n+1) + s n\"\n and h1 : \"s 9 = 110\"\n and h2 : \"s 7 = 42\"\n shows \"s 4 = 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_28:\n fixes c :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<forall>x. f x = 2 * x^2 + 5 * x + c\"\n and h1 : \"\\<exists>x. f x \\<le> 0\"\n shows \"c \\<le> 25/8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_412:\n fixes x y :: nat\n assumes h0 : \"x mod 19 = (4:: nat)\"\n and h1 : \"y mod 19 = (7:: nat)\"\n shows \"(x+1)^2 * (y+5)^3 mod 19 = (13:: nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2010_p10:\n fixes p q :: real\n and a :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"\\<And>n. a (n+2) - a (n+1) = a (n+1) - a n\"\n and h1 : \"a 1 = p\"\n and h2 : \"a 2 = 9\"\n and h3 : \"a 3 = 3 * p - q\"\n and h4 : \"a 4 = 3 * p + q\"\n shows \"a 2010 = 8041\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_961:\n \"2003 mod 11 = (1::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2001_p2:\n fixes a b n::nat\n assumes \"1 \\<le> a \\<and> a \\<le> 9\"\n and \"0 \\<le> b \\<and> b \\<le> 9\"\n and \"n = 10 * a + b\"\n and \"n = a * b + a + b\"\n shows \"b=9\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_aneqprodakp4_anmsqrtanp1eq2:\n fixes a :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"a 0 = 1\"\n and h1 : \"\\<And>n. a (n+1) = (\\<Prod>(k::nat) =1..n. (a k))+4\"\n shows \"\\<And>n. (n\\<ge>1) \\<Longrightarrow> a n - sqrt (a (n+1)) = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1964_p1_2:\n fixes n :: nat\n shows \"\\<not> ((7::nat) dvd (2^n + 1))\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p21:\n \"2 \\<le> (\\<Sum> k \\<in> {x ::real. 0 < x \\<and> x powr (2 powr (sqrt 2))\n = (sqrt 2) powr (2 powr x)}. k) \\<and> \n (\\<Sum> k \\<in> {x :: real. 0 < x \\<and> x powr (2 powr (sqrt 2)) \n = (sqrt 2)powr (2 powr x)}. k) < 6\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_109:\n fixes v :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> n. v n = 2 * n - 1\" \n shows \"(\\<Sum> k \\<in>{1..<101}. v k) mod 7 = 4\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_92:\n fixes n :: nat\n assumes h0 : \"(5 * n) mod 17 = 8\"\n shows \"n mod 17 = 5\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_43:\n fixes n :: nat\n assumes h0 : \"15^n dvd (fact 942)\"\n and h1 : \"\\<And>(m::nat). ((15::nat)^m dvd (fact 942)) \\<Longrightarrow> m \\<le> n\"\n shows \"n=233\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1977_p5:\n fixes a b q r :: nat\n assumes h0 : \"r < a + b\"\n and h1 : \"a^2 + b^2 = (a+b) * q + r\"\n and h2 : \"q^2 + r = 1977\"\n shows \"(abs (int a - 22) = 15 \\<and> abs (int b - 22) = 28) \\<or> (abs (int a - 22) = 28 \\<and> abs (int b - 22) = 15)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_422:\n fixes x :: real and \\<sigma>::\"real \\<Rightarrow> real\"\n assumes \"bij \\<sigma>\"\n and \\<sigma>:\"\\<forall> x. \\<sigma> x = 5 * x - 12\"\n and \"\\<sigma> (x + 1) = (Hilbert_Choice.inv \\<sigma>) x\" \n shows \"x = 47 / 24\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_manipexpr_apbeq2cceqiacpbceqm2:\n fixes a b c :: complex\n assumes h0 : \"a+b = 2*c\"\n and h1 : \"c = \\<i>\"\n shows \"a*c+b*c=-2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_nckeqnm1ckpnm1ckm1:\n fixes n k ::nat\n assumes \"0 < n \\<and> 0 < k\"\n and \"k \\<le> n\" \n shows \"n choose k = (n - 1) choose k + (n - 1) choose (k - 1)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1990_p2:\n \"((52::real) + 6 * sqrt 43) powr (3/2) - ((52::real) - 6 * sqrt 43) powr (3/2) = 828\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_616:\n fixes f g :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = x^3 + 2 * x + 1\"\n and h1 : \"\\<And>x. g x = x - 1\"\n shows \"f (g 1) = 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2016_p3:\n fixes f :: \"real \\<Rightarrow> real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x y. f x y = x - y * floor (x/y)\"\n shows \"f ((3::real)/8) (- 2/5) = - 1/40\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_31:\n fixes x :: real\n and u :: \"nat \\<Rightarrow> real\"\n assumes \"\\<forall> n. u (n + 1) = sqrt (x + u n)\"\n and \"filterlim u at_top (nhds 9)\"\n shows \"9 = sqrt (x + 9)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_410:\n fixes x y :: real\n assumes h0 : \"y = x^2 - 6 * x + 13\"\n shows \"4 \\<le> y\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_4bap1lt4bsqpap1sq:\n fixes a b :: real\n shows \"4 * b * (a+1) \\<le> 4 * b^2 + (a+1)^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_198:\n \"(5^2005) mod 100 = (25::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_149:\n \"(\\<Sum> k\\<in> {x::nat. x<50 \\<and> x mod 8 =5 \\<and> x mod 6=3}. k) = 66\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_245:\n fixes x :: real\n assumes h0 : \"x \\<noteq> 0\"\n shows \"1/(4/x) * ((3*x^3)/x)^2 * (1/(1 / (2 * x)))^3 = 18 * x^8\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_301:\n fixes j :: nat \n assumes \"j>0\"\n shows \"(3 * (7 * j + 3)) mod 7 = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_132:\n \"2004 mod 12 = (0::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_257:\n fixes x :: nat\n assumes h0 : \"1 \\<le> x \\<and> x \\<le> 100\"\n and h1 : \"77 dvd ((\\<Sum>k::nat=0..100. k)-x)\"\n shows \"x=45\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2002_p21:\n fixes u:: \"nat\\<Rightarrow>nat\" and n::nat\n assumes \"u 0 =4\"\n and \"u 1=7\"\n and \"\\<forall> n \\<ge> 2. u (n + 2) = (u n + u (n + 1)) mod 10\"\n and \"(\\<Sum> k \\<in> {..n}. u k) > 10000\"\n shows \"1999 \\<le> n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1994_p4:\n fixes n :: nat\n assumes \"0 < n\"\n and \"(\\<Sum> k \\<in> {1..<n+1}. floor (ln k / ln 2)) = 1994\" \n shows \"n = 312\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2015_p10:\n fixes x y:: nat\n assumes h0: \"0<y\"\n and h1: \"y<x\"\n and h2: \"x+y + (x*y) = 80\"\n shows \"x=26\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p2:\n \"(1 + (1 / (1 + (1 / (1 + 1))))) = (5::real) / 3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_405:\n fixes a b c :: nat\n and t :: \"nat \\<Rightarrow> nat\"\n assumes h0 : \"t 0 = 0\"\n and h1 : \"t 1 = 1\"\n and h2 : \"\\<And>n. (n > 1) \\<Longrightarrow> t n = t (n-2) + t (n-1)\"\n and h3 : \"a mod 16 = 5\"\n and h4 : \"b mod 16 = 10\"\n and h5 : \"c mod 16 = 15\"\n shows \"(t a + t b + t c) mod 7 = 5\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1973_p3:\n fixes a b :: real\n assumes h0 : \"\\<exists>x. x^4 + a * x^3 + b * x^2 + a*x + 1 = 0\"\n shows \"4/5 \\<le> a^2 + b^2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p5:\n fixes x :: real\n assumes h0 : \"x^3 - (x+1) * (x-1) * x = 5\"\n shows \"x^3 = 125\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1979_p1:\n fixes p q :: nat \n assumes \"0 < q\"\n and \"(\\<Sum> k \\<in> {1..<1320}. ((-1) ^ (k + 1) * (1 / k)))\n = p / q\" \n shows \"1979 dvd p\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_43:\n fixes a b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = a * x + b\"\n and h1 : \"f 7 = 4\"\n and h2 : \"f 6 = 3\"\n shows \"f 3 = 0\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_462:\n \"((1::real)/2 + 1/3) * (1/2 - 1/3) = 5/36\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1984_p5:\n fixes a b ::real\n assumes \"(ln a) / (ln 8) + (ln (b^2)) / (ln 4) = 5\"\n \"(ln b) / (ln 8) + (ln (a^2)) / (ln 4) = 7\"\n shows \"a * b = 512\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2010_p11:\n fixes x b :: real\n assumes \"0 < b\"\n and \"7 powr (x + 7) = 8 powr x\"\n and \"x = ln (7^7) / ln b\" \n shows \"b = 8 / 7\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_2dvd4expn:\n fixes n :: nat\n assumes h0 : \"n \\<noteq> 0\"\n shows \"(2::nat) dvd 4^n\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aimeII_2001_p3:\n fixes x :: \"nat \\<Rightarrow> int\"\n assumes h0 : \"x 1 = 211\"\n and h1 : \"x 2 = 375\"\n and h2 : \"x 3 = 420\"\n and h3 : \"x 4 = 523\"\n and h4 : \"\\<And>(n::nat). ((n\\<ge>5) \\<Longrightarrow> (x n = x (n-1) - x (n-2) + x (n-3) - x (n-4)))\"\n shows \"x 531 + x 753 + x 975 = 898\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p11:\n fixes a b::real\n assumes \"a \\<noteq> 0\" \"b \\<noteq> 0\"\n and \"a * b = a - b\"\n shows \"a / b + b / a - a * b = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_156:\n fixes n :: nat\n assumes h0: \"n > 0\"\n shows \"gcd (n+7) (2*n+1) \\<le> 13\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_530:\n fixes n k :: nat\n assumes \"n / k < 6\"\n and \"5 < n / k\" \n shows \"22 \\<le> (lcm n k) / (gcd n k)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1978_p5:\n fixes n :: nat and f :: \"nat \\<Rightarrow> nat\"\n assumes \"inj f\" and \"f 0 = 0\"\n shows \"(\\<Sum> k \\<in>{1..<n+1}. 1 / k) \\<le> (\\<Sum> k \\<in>{1..<n+1}. (f k) / k^2)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_sum2kp1npqsqm1:\n fixes n :: nat \n shows \"(\\<Sum> k<n. 2 * k + 3) = (n + 1)^2 - 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_55:\n fixes q p :: real\n assumes h0 : \"q = 2 - 4 + 6 - 8 + 10 -12 + 14\"\n and h1 : \"p = 3 - 6 + 9 - 12 + 15 - 18 + 21\"\n shows \"q/p = 2/3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1983_p9:\n fixes x::real\n assumes \"0<x\" \"x<pi\"\n shows \"12 \\<le> ((9 * (x^2 * (sin x)^2)) + 4) / (x * sin x)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_131:\n fixes a b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = 2 * x^2 - 7 * x + 2\"\n and h1 : \"f a = 0\"\n and h2 : \"f b = 0\"\n and h3 : \"a \\<noteq> b\"\n shows \"1 / (a-1) + 1 / (b-1) = -1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2013_p8:\n fixes x y :: real\n assumes h0 : \"x\\<noteq>0\"\n and h1 : \"y\\<noteq>0\"\n and h2 : \"x\\<noteq>y\"\n and h3 : \"x + 2/x = y + 2/y\"\n shows \"x * y = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_466:\n \"(\\<Sum> k< 11. k) mod 9 = (1::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_192:\n fixes q e d :: complex\n assumes h0 : \"q = Complex 11 (-5)\"\n and h1 : \"e = Complex 11 5\"\n and h2 : \"d = Complex 0 2\"\n shows \"q * e * d = Complex 0 292\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_629 :\n \"(LEAST t::nat. (lcm 12 t)^3 = (12 * t)^2) = 18\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1984_p15:\n fixes x y z w::real\n assumes \"(x^2 / (2^2 - 1)) + (y^2 / (2^2 - 3^2)) \n + (z^2 / (2^2 - 5^2)) + (w^2 / (2^2 - 7^2)) = 1\"\n \"(x^2 / (4^2 - 1)) + (y^2 / (4^2 - 3^2)) \n + (z^2 / (4^2 - 5^2)) + (w^2 / (4^2 - 7^2)) = 1\"\n \"(x^2 / (6^2 - 1)) + (y^2 / (6^2 - 3^2)) \n + (z^2 / (6^2 - 5^2)) + (w^2 / (6^2 - 7^2)) = 1\"\n \"(x^2 / (8^2 - 1)) + (y^2 / (8^2 - 3^2)) \n + (z^2 / (8^2 - 5^2)) + (w^2 / (8^2 - 7^2)) = 1\"\n shows \"x^2 + y^2 + z^2 + w^2 = 36\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1990_p3:\n fixes n :: nat\n assumes \"2 \\<le> n\"\n and \"n^2 dvd 2^n + 1\"\n shows \"n = 3\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_2rootsintpoly_am10tap11eqasqpam110:\n fixes a :: complex\n shows \"(a-10) * (a+11) = a^2 + a -110\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_461:\n fixes n :: nat\n assumes \"n = card {k::nat. gcd k 8 = 1 \\<and> 1\\<le>k \\<and> k < 8}\" \n shows \"(3^n) mod 8 = (1::nat)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_144:\n fixes a b c d :: nat\n assumes h0:\"c - b = d\"\n and h1:\"b - a = d\"\n and h2: \"a+b+c = 60\"\n and h3: \"a + b > c\"\n shows \"d < 10\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_282:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. (x \\<in> \\<rat> ) \\<longrightarrow> f x = abs (floor x)\"\n and \"\\<forall> x. (x \\<notin> \\<rat>) \\<longrightarrow> f x = (ceiling x)^2\" \n shows \"f (8 powr (1/3)) + f (-pi) + f (sqrt 50) + f (9/2) = 79\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2008_p8:\n fixes x y::real\n assumes h0: \"0 < x \\<and> 0 < y\"\n and h1: \"y^3 = 1\"\n and h2: \"6 * x^2 = 2 * (6 * y^2)\"\n shows \"x^3 = 2 * sqrt 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_2unitcircatblt1:\n fixes a b :: real\n assumes \"a^2 + b^2 = 2\"\n shows \"a * b <= 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_393:\n fixes \\<sigma>::\"real \\<Rightarrow> real\" \n assumes \"bij \\<sigma>\"\n and \"\\<forall> x. \\<sigma> x = 4 * x^3 + 1\"\n shows \"Hilbert_Choice.inv \\<sigma> 33 = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_15:\n fixes s :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat\"\n assumes h0: \"\\<And>a b. s a b = a ^ b + b ^ a\"\n shows \"s 2 6 = 100\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_510:\n fixes x y :: real\n assumes h0 : \"x+y=13\"\n and h1 : \"x*y=24\"\n shows \"sqrt (x^2 + y^2) = 11\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1991_p1:\n fixes x y :: nat\n assumes h0 : \"0<x \\<and> 0<y\"\n and h1 : \"x*y + (x+y) = 71\"\n and h2 : \"x^2 * y + x * y^2=880\"\n shows \"x^2 + y^2=146\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_69:\n fixes r s :: nat\n assumes \"r * s = 450\"\n and \"(r + 5) * (s - 3) = 450\" \n shows \"r = 25\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1991_p6:\n fixes r :: real\n assumes \"(\\<Sum> k \\<in>{19::nat..<92}. (floor (r + k / 100))) = 546\" \n shows \"floor (100 * r) = 743\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1987_p8:\n fixes n :: nat\n assumes h0 : \"0 < n\"\n and h1 : \"\\<not> (\\<exists>!k. (8 / 15 < n / (n+k)) \\<and> n / (n+k) < 7/13)\"\n shows \"n \\<le> 112\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_433:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = 3 * sqrt (2 * x -7) - 8\"\n shows \"f 8 = 1\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1997_p11:\n fixes x :: real\n assumes h0 : \"x = (\\<Sum>(n::nat) =1..44. cos(n*pi/180)) / (\\<Sum>(n::nat) =1..44. sin(n*pi/180))\"\n shows \"floor (100*x) = 241\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2017_p7:\n fixes f :: \"nat \\<Rightarrow> real\"\n assumes h0 : \"f 1 = 2\"\n and h1 : \"\\<And>n. (1 < n \\<and> even n) \\<Longrightarrow> f n = f (n - 1) + 1\"\n and h2 : \"\\<And>n. (1 < n \\<and> odd n) \\<Longrightarrow> f n = f (n - 2) + 2\"\n shows \"f 2017 = 2018\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1967_p3:\n fixes k m n :: nat\n and c :: \"nat \\<Rightarrow> nat\"\n assumes h0 : \"0<k \\<and> 0<m \\<and> 0<n\"\n and h1 : \"\\<And>s. c s = s * (s+1)\"\n and h2 : \"prime (k+m+1)\"\n and h3 : \"n+1 < k + m + 1\"\n shows \"(\\<Prod>(i::nat) = 1..n.(c i)) dvd (\\<Prod>(i::nat) = 1..n.(c (m+i)) - c k)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_104:\n fixes x :: real\n assumes h0 : \"125/8 = x /12\"\n shows \"x = 375/2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_84:\n \"floor ((9::real) / 160 * 100) = (5::int)\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_214:\n fixes a :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = a * (x-2)^2 + 3\"\n and h1 : \"f 4 = 4\"\n shows \"f 6 = 7\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_67:\n fixes f g :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<And>x. f x = 5 * x + 3\"\n and h1 : \"\\<And>x. g x = x^2 - 2\"\n shows \"g (f (-1)) = 2\""], "miniF2F valid"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_11div10tonmn1ton:\n fixes n :: nat\n shows \"(11::int) dvd (10^n - (-1)^n)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_289:\n fixes k t m n :: nat\n assumes \"prime m \\<and> prime n\"\n and \"t < k\"\n and \"k^2 + n - m * k = 0\"\n and \"t^2 + n - m * t = 0\"\n and \"0 < t\"\n shows \"m^n + n^m + k^t + t^k = 20\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_521:\n fixes m n :: nat\n assumes \"even m\"\n and \"even n\"\n and \"m - n = 2\"\n and \"m * n = 288\"\n shows \"m = 18\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_769 :\n \"(129^34 + 96^38) mod 11 = (9::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1984_p1:\n fixes u :: \"nat \\<Rightarrow> rat\"\n assumes h0: \"\\<forall> n. u (n + 1) = u n + 1\"\n and h1: \"(\\<Sum> k < 98. u (k+1)) = 137\" \n shows \"(\\<Sum> k < 49. u (2 * (k+1))) = 93\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_148:\n fixes c :: real and f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. f x = c * x^3 - 9 * x + 3\"\n and \"f 2 = 9\" \n shows \"c = 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1989_p8:\n fixes a b c d e f g :: real\n assumes \"a + 4 * b + 9 * c + 16 * d + 25 * e + 36 * f + 49 * g = 1\"\n and \"4 * a + 9 * b + 16 * c + 25 * d + 36 * e + 49 * f + 64 * g = 12\"\n and \"9 * a + 16 * b + 25 * c + 36 * d + 49 * e + 64 * f + 81 * g = 123\" \n shows \"16 * a + 25 * b + 36 * c + 49 * d + 64 * e + 81 * f + 100 * g = 334\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_222:\n fixes b :: nat\n assumes \"lcm 120 b = 3720\"\n and \"gcd 120 b = 8\" \n shows \"b = 248\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_135:\n fixes n a b c::nat\n assumes \"n = 3^17 + 3^10\"\n and \"11 dvd (n + 1)\"\n and \"a\\<noteq>b\" \"b\\<noteq>c\" \"a\\<noteq>c\"\n and \"a\\<in>{0..9}\" \"b\\<in>{0..9}\" \"c\\<in>{0..9}\" \n and \"odd a \\<and> odd c\"\n and \"\\<not> 3 dvd b\"\n and \"digits n = [b,a,b,c,d,a,c,b,a]\" \n shows \"10*(10 * a + b) + c = 129\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1983_p6:\n fixes a b c ::real\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and \"c < a + b\"\n and \"b < a + c\"\n and \"a < b + c\"\n shows \"0 \\<le> a^2 * b * (a - b) + b^2 * c * (b - c) + c^2 * a * (c - a)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p13:\n shows \"card {x :: real. 0 < x \\<and> x \\<le> 2 * pi \\<and> 1 - 3 * sin x \n + 5 * cos (3 * x) = 0} = 6\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_478:\n fixes b h v ::real\n assumes \"0 < b \\<and> 0 < h \\<and> 0 < v\"\n and \"v = 1 / 3 * (b * h)\"\n and \"b = 30\"\n and \"h = 13 / 2\"\n shows \"v = 65\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1983_p3:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes h0 : \"\\<forall> x. f x = (x^2 + (18 * x + 30) - 2 \n * sqrt (x^2 + (18 * x + 45)))\"\n shows \"(\\<Prod> (f -` {0})) = 20\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_141:\n fixes a b ::real\n assumes \"(a * b)=180\"\n and \"2 * (a + b)=54\"\n shows \"a^2 + b^2 = 369\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_at2malt1:\n fixes a::real \n shows \"a * (2 - a) \\<le> 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_ineq_nto1onlt2m1on:\n fixes n ::nat \n shows \"(n::real) powr ((1::real) / n) < 2 - 1 / n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p1:\n shows \"card {x::int. \\<bar>real_of_int x\\<bar> < 3 * pi} = 19\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_582:\n fixes n :: nat\n assumes \"0 < n\"\n and \"3 dvd n\"\n shows \"((n + 4) + (n + 6) + (n + 8)) mod 9 = 0\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_apbpceq2_abpbcpcaeq1_aleq1on3anbleq1ancleq4on3:\n fixes a b c :: real\n assumes \"a \\<le> b \\<and> b \\<le> c\"\n and \"a + b + c = 2\"\n and \"a * b + b * c + c * a = 1\" \n shows \"0 \\<le> a \\<and> a \\<le> 1 / 3 \\<and> 1 / 3 \\<le> b \\<and> b \\<le> 1 \\<and> 1 \\<le> c \\<and> c \\<le> 4 / 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_1pxpownlt1pnx:\n fixes x :: real and n :: nat\n assumes \"-1 < x\"\n shows \"(1 + n*x) \\<le> (1 + x)^n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_263:\n fixes y :: real\n assumes \"0 \\<le> 19 + 3 * y\"\n and \"sqrt (19 + 3 * y) = 7\" \n shows \"y = 10\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p1:\n fixes i m k ::nat\n assumes h0 : \"i \\<noteq> 0 \\<and> m \\<noteq> 0 \\<and> k \\<noteq> 0\"\n and h1 : \"i*m*k = 2001\" \n and h2 : \"i \\<noteq> m \\<and> i \\<noteq> k \\<and> m \\<noteq> k\"\n shows \"i+m+k \\<le> 671\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_100:\n fixes n :: nat\n assumes \"gcd n 40 = 10\"\n and \"lcm n 40 = 280\" \n shows \"n = 70\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2002_p13:\n fixes a b :: real\n assumes h0: \"0 < a \\<and> 0 < b\"\n and h1: \"a \\<noteq> b\"\n and h2: \"abs (a - 1/a) = 1\"\n and h3: \"abs (b - 1/b) = 1\" \n shows \"a + b = sqrt 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p6:\n fixes p q ::nat\n assumes h0: \"prime p \\<and> prime q\"\n and h1: \"4 \\<le> p \\<and> p \\<le> 18\"\n and h2: \"4 \\<le> q \\<and> q \\<le> 18\" \n shows \"((p * q)::int) - (p + q) \\<noteq> 194\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_3pow2pownm1mod2pownp3eq2pownp2:\n fixes n :: nat\n assumes \"0 < n\" \n shows \"(3^(2^n) - 1) mod (2^(n + 3)) = (2::nat)^(n + 2)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_341:\n fixes a b c ::nat\n assumes \"a \\<le> 9 \\<and> b \\<le> 9 \\<and> c \\<le> 9\"\n and \"(5^100) mod 1000 = 10*(10*a + b) + c\"\n shows \"a + b + c = 13\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1987_p5:\n fixes x y ::int\n assumes \"y^2 + 3 * (x^2 * y^2) = 30 * x^2 + 517\"\n shows \"3 * (x^2 * y^2) = 588\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_1124:\n fixes n :: nat\n assumes \"n \\<le> 9\"\n and \"18 dvd 374 * 10 + n\"\n shows \"n = 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_332:\n fixes x y ::real\n assumes \"(x + y) / 2 = 7\"\n and \"sqrt (x * y) = sqrt 19\" \n shows \"x^2 + y^2 = 158\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p14 :\n shows \"(\\<Sum> k\\<in>{1..<21}.\n ln (3^(k^2)) / ln (5^k)) * (\\<Sum> k \\<in>{1..<101}. ln (25^k) \n / ln (9^k))= 21000\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_175 :\n \"(2^2010) mod 10 = (4::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_398:\n fixes a b c ::real\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and \"9 * b = 20 * c\"\n and \"7 * a = 4 * b\"\n shows \"63 * a = 80 * c\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_430:\n fixes a b c :: nat\n assumes \"1 \\<le> a \\<and> a \\<le> 9\"\n and \"1 \\<le> b \\<and> b \\<le> 9\"\n and \"1 \\<le> c \\<and> c \\<le> 9\"\n and \"a \\<noteq> b\"\n and \"a \\<noteq> c\"\n and \"b \\<noteq> c\"\n and \"a + b = c\"\n and \"10 * a + a - b = 2 * c\"\n and \"c * b = 10 * a + a + a\"\n shows \"a + b + c = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1963_p5 :\n \"cos (pi / 7) - cos (2 * pi / 7) + cos (3 * pi / 7) = 1 / 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2008_p25:\n fixes a b :: \"nat \\<Rightarrow> real\"\n assumes \"\\<forall> n. a (n + 1) = sqrt 3 * a n - b n\"\n and \"\\<forall> n. b (n + 1) = sqrt 3 * b n + a n\"\n and \"a 100 = 2\"\n and \"b 100 = 4\" \n shows \"a 1 + b 1 = 1 / (2^98)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_452:\n fixes a :: \"nat \\<Rightarrow> real\"\n assumes \"\\<forall> n. a (n + 2) - a (n + 1) = a (n + 1) - a n\"\n and \"a 1 = 2 / 3\"\n and \"a 9 = 4 / 5\" \n shows \"a 5 = 11 / 15\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p15:\n fixes a b :: complex\n assumes h0: \"a^3 - 8 = 0\"\n and h1: \"b^3 - 8 * b^2 - 8 * b + 64 = 0\" \n shows \"norm (a - b) \\<le> 2 * sqrt 21\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_notequiv2i2jasqbsqdiv8:\n fixes a b :: int\n shows \"\\<not> ((\\<exists> i j. a = 2*i \\<and> b=2*j) \\<longleftrightarrow> (\\<exists> k. a^2 + b^2 = 8*k))\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_76:\n fixes f :: \"int \\<Rightarrow> int\"\n assumes \"\\<forall>n. odd n \\<longrightarrow> f n = n^2\"\n and \"\\<forall> n. even n \\<longrightarrow> f n = n^2 - 4*n -1\" \n shows \"f 4 = -1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_484 :\n \"(ln 27) / (ln 3) = (3::real)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_85:\n \"1 * 3^3 + 2 * 3^2 + 2*3 + 2 = (53::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_209:\n fixes \\<sigma>::\"real \\<Rightarrow> real\"\n assumes \"bij \\<sigma>\"\n and \"(Hilbert_Choice.inv \\<sigma>) 2 = 10\"\n and \"(Hilbert_Choice.inv \\<sigma>) 10 = 1\"\n and \"(Hilbert_Choice.inv \\<sigma>) 1 = 2\" \n shows \"\\<sigma> (\\<sigma> 10) = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2001_p5:\n shows \"(\\<Prod>x\\<in>{x::nat. x<10000 \\<and> odd x}. x) \n = fact 10000 / ((2^5000) * fact 5000)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_427:\n fixes x y z :: real\n assumes \"3 * x + y = 17\"\n and \"5 * y + z = 14\"\n and \"3 * x + 5 * z = 41\" \n shows \"x + y + z = 12\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_275:\n fixes x :: real\n assumes \"(11 powr (1 / 4)) powr (3 * x - 3) = 1 / 5\" \n shows \"(11 powr (1 / 4)) powr (6 * x + 2) = 121 / 25\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1981_p6:\n fixes f :: \"nat \\<Rightarrow> nat \\<Rightarrow> nat\"\n assumes \"\\<forall> y. f 0 y = y + 1\"\n and \"\\<forall> x. f (x + 1) 0 = f x 1\"\n and \"\\<forall> x y. f (x + 1) (y + 1) = f x (f (x + 1) y)\" \n shows \"\\<forall> y. f 4 (y + 1) = 2^(f 4 y + 3) - 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_299 :\n \"(1 * 3 * 5 * 7 * 9 * 11 * 13) mod 10 = (5::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_296 :\n \"abs (((3491 - 60) * (3491 + 60) - 3491^2)) = (3600::int)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p2 :\n shows \"((100 ^ 2 - 7 ^ 2)) / (70 ^ 2 - 11 ^ 2) * \n ((70 - 11) * (70 + 11) / ((100 - 7) * (100 + 7))) = (1::real)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_absapbon1pabsapbleqsumabsaon1pabsa:\n fixes a b ::real\n shows \"abs (a + b) / (1 + abs (a + b)) \n \\<le> abs a / (1 + abs a) + abs b / (1 + abs b)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_234:\n fixes a b :: nat\n assumes \"1 \\<le> a \\<and> a \\<le> 9 \\<and> b \\<le> 9\"\n and \"(10 * a + b)^3 = 912673\" \n shows \"a + b = 16\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_fxeq4powxp6powxp9powx_f2powmdvdf2pown:\n fixes m n ::nat\n and f :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> x. f x = 4^x + 6^x + 9^x\"\n and \"0 < m \\<and> 0 < n\"\n and \"m \\<le> n\" \n shows \"f (2^m) dvd f (2^n)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_233:\n fixes b::int\n assumes \"0\\<le>b\"\n and \"b<11^2\"\n and \"[b * 24 = 1] (mod (11^2))\"\n shows \"b = 116\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_412:\n fixes x y :: real\n assumes \"x + y = 25\"\n and \"x - y = 11\" \n shows \"x = 18\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_125:\n fixes x y :: nat\n assumes \"5 * x = y\"\n and \"(x - 3) + (y - 3) = 30\" \n shows \"x = 6\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_188:\n fixes \\<sigma>:: \"real \\<Rightarrow> real\"\n assumes \"bij \\<sigma>\"\n and \"\\<sigma> 2 = (Hilbert_Choice.inv \\<sigma>) 2\" \n shows \"\\<sigma> (\\<sigma> 2) = 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_66:\n \"194 mod 11 = (7::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_44:\n fixes s t :: real\n assumes \"s = 9 - 2 * t\"\n and \"t = 3 * s + 1\" \n shows \"s = 1 \\<and> t = 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p12:\n fixes a b c d ::real\n and f :: \"complex \\<Rightarrow> complex\"\n assumes h0: \"\\<forall> z. f z = z^6 - 10 * z^5 + a * z^4 + b * z^3 \n + c * z^2 + d * z + 16\"\n and h1: \"\\<forall> z. f z = 0 \\<longrightarrow> (Im z = 0 \\<and> 0 < Re z \\<and> (floor (Re z)) = (Re z))\" \n shows \"b = 88\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p3:\n fixes x y :: nat\n assumes h0: \"x + y = 17402\"\n and h1: \"10 dvd x\"\n and h2: \"x div 10 = y\" \n shows \"x - y = 14238\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_107:\n fixes x y :: real\n assumes \"x^2 + 8 * x + y^2 - 6 * y = 0\" \n shows \"(x + 4)^2 + (y-3)^2 = 5^2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imosl_2007_algebra_p6:\n fixes a :: \"nat \\<Rightarrow> real\"\n assumes h0: \"(\\<Sum> x< 100. ((a (x + 1))^2)) = 1\" \n shows \"(\\<Sum>x<99. ((a (x + 1))^2 * a (x + 2))) + (a 100)^2 * a 1 < 12 / 25\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_2001_p6:\n fixes a b c d ::nat\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c \\<and> 0 < d\"\n and \"d < c\"\n and \"c < b\"\n and \"b < a\"\n and \"a * c + b * d = (b + d + a - c) * (b + d + c - a)\" \n shows \"\\<not> prime (a * b + c * d)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_495:\n fixes a b :: nat\n assumes \"0 < a \\<and> 0 < b\"\n and \"a mod 10 = 2\"\n and \"b mod 10 = 4\"\n and \"gcd a b = 6\"\n shows \"108 \\<le> lcm a b\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_513:\n fixes a b :: real\n assumes \"3 * a + 2 * b = 5\"\n and \"a + b = 2\" \n shows \"a = 1 \\<and> b = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1965_p2:\n fixes x y z :: real\n and a :: \"nat \\<Rightarrow> real\"\n assumes \"0 < a 0 \\<and> 0 < a 4 \\<and> 0 < a 8\"\n and \"a 1 < 0 \\<and> a 2 < 0\"\n and \"a 3 < 0 \\<and> a 5 < 0\"\n and \"a 7 < 0 \\<and> a 9 < 0\"\n and \"0 < a 0 + a 1 + a 2\"\n and \"0 < a 3 + a 4 + a 5\"\n and \"0 < a 6 + a 7 + a 8\"\n and \"a 0 * x + a 1 * y + a 2 * z = 0\"\n and \"a 3 * x + a 4 * y + a 5 * z = 0\"\n and \"a 6 * x + a 7 * y + a 8 * z = 0\" \n shows \"x = 0 \\<and> y = 0 \\<and> z = 0\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_728:\n \"(29^13 - 5^13) mod 7 = (3::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_346:\n fixes f g :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. f x = 2 * x - 3\"\n and \"\\<forall> x. g x = x + 1\"\n shows \"g (f 5 - 1) = 7\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1983_p2:\n fixes x p ::real and f :: \"real \\<Rightarrow> real\"\n assumes h0: \"0 < p \\<and> p < 15\"\n and h1: \"p \\<le> x \\<and> x \\<le> 15\"\n and h2: \"f x = abs (x - p) + abs (x - 15) + abs (x - p - 15)\" \n shows \"15 \\<le> f x\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_sum1toneqn_prod1tonleq1:\n fixes a :: \"nat \\<Rightarrow> real\" and n :: nat\n assumes h0: \"(\\<Sum> x< n. a x) = n\" and \"\\<forall>x. a x\\<ge>0\"\n shows \"(\\<Prod> x<n. a x) \\<le> 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_552:\n fixes f g h :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> x>0. f x = 12 * x + 7\"\n and \"\\<forall> x>0. g x = 5 * x + 2\"\n and \"\\<forall> x>0. h x = gcd (f x) (g x)\"\n shows \"(\\<Sum> k \\<in> range h. k) = 12\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1990_p15:\n fixes a b x y :: real\n assumes \"a * x + b * y = 3\"\n and \"a * x^2 + b * y^2 = 7\"\n and \"a * x^3 + b * y^3 = 16\"\n and \"a * x^4 + b * y^4 = 42\" \n shows \"a * x^5 + b * y^5 = 20\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_196:\n \"(\\<Sum> k \\<in> {x ::real. abs (2 - x) = 3}. k) = 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p9:\n shows \"(ln 80 / ln 2) / (ln 2 / ln 40) - (ln 160 / ln 2) \n / (ln 2 / ln 20) = (2::real)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_apbon2pownleqapownpbpowon2:\n fixes a b :: real and n :: nat\n assumes \"0 < a \\<and> 0 < b\"\n and \"0 < n\" \n shows \"((a + b) / 2)^n \\<le> (a^n + b^n) / 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_2varlineareq_fp3zeq11_3tfm1m5zeqn68_feqn10_zeq7:\n fixes f z::complex\n assumes h0: \"f + 3*z = 11\"\n and h1: \"3*(f - 1) - 5*z = -68\" \n shows \"f = -10 \\<and> z = 7\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1984_p7:\n fixes f :: \"nat \\<Rightarrow> nat\"\n assumes h0: \"\\<forall> n\\<ge>1000. f n = n - 3\"\n and h1: \"\\<forall> n < 1000. f n = f (f (n + 5))\"\n shows \"f 84 = 997\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_288:\n fixes x y :: real\n and n :: real\n assumes \"x < 0 \\<and> y < 0\"\n and \"abs y = 6\"\n and \"sqrt ((x - 8)^2 + (y - 3)^2) = 15\"\n and \"sqrt (x^2 + y^2) = sqrt n\"\n shows \"n = 52\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_296:\n fixes n :: nat\n assumes \"2 \\<le> n\"\n and \"\\<exists> x. x^3 = n\"\n and \"\\<exists> t. t^4 = n\" \n shows \"4096 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_150:\n fixes n :: nat\n assumes \" \\<not> prime (7 + 30 * n)\" \n shows \"6 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p7:\n fixes a b c :: nat\n assumes h0: \"b = a + 1\"\n and h1: \"c = b + 1\"\n and h2: \"a * b * c = 8 * (a + b + c)\" \n shows \"a^2 + b^2 + c^2 = 77\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_247:\n fixes n ::nat\n assumes \"(3 * n) mod 2 = 11\" \n shows \"n mod 11 = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1985_p6:\n fixes f :: \"nat \\<Rightarrow> real \\<Rightarrow> real\"\n assumes \"\\<forall> x>0. f 1 x = x\"\n and \"\\<forall> x>0. \\<forall> n. f (n + 1) x = f n x * (f n x + 1 / n)\" \n shows \"\\<exists>! a. \\<forall> n. 0 < f n a \\<and> f n a < f (n + 1) a \\<and> f (n + 1) a < 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_4x3m7y3neq2003:\n fixes x y :: int\n shows \"4 * x^3 - 7 * y^3 \\<noteq> 2003\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_unitcircatbpamblt1:\n fixes a b:: real\n assumes h0: \"a^2 + b^2 = 1\"\n shows \"a * b + (a - b) \\<le> 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_158:\n fixes a :: nat\n assumes \"even a\"\n and \"(\\<Sum> k < 8. (2 * k + 1)) - \n (\\<Sum> k<5. (a + 2 * k)) = 4\"\n shows \"a = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_12 :\n \"card {x::nat. 20 dvd x \\<and> 15 \\<le> x \\<and> x < 86} =4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_156:\n fixes x y :: real\n and f g :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall>t. f t = t^4\"\n and \"\\<forall>t. g t = 5 * t^2 - 6\"\n and \"f x = g x\"\n and \"f y = g y\"\n and \"x^2 < y^2\" \n shows \"y^2 - x^2 = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_246:\n fixes a b :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. f x = a * x^4 - b * x^2 + x + 5\"\n and \"f (-3) = 2\" \n shows \"f 3 = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_235 :\n \"(29 * 79 + 31 * 81) mod 10 = (2::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p13 :\n shows \"sqrt (ln 6 / ln 2 + ln 6 / ln 3) \n = sqrt (ln 3 / ln 2) + sqrt (ln 2 / ln 3)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_362:\n fixes a b :: real\n assumes \"a^2 * b^3 = 32 / 27\"\n and \"a / b^3 = 27 / 4\" \n shows \"a + b = 8 / 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_483:\n fixes a :: \"nat \\<Rightarrow> nat\"\n assumes \"a 1 = 1\"\n and \"a 2 = 1\"\n and \"\\<forall> n. a (n + 2) = a (n + 1) + a n\" \n and \"\\<forall> n. a n>0\"\n shows \"(a 100) mod 4 = 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_207:\n \"8 * 9^2 + 5 * 9 + 2 = (695::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p21:\n shows \"card {n. (n + 1000) / 70 = floor (sqrt n)} = 6\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_208 :\n \"sqrt 1000000 - 1000000 powr (1/3) = 900\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p9:\n shows \"card { x::real. 0 \\<le> x \\<and> x \\<le> 2 * pi \\<and> \n tan (2 * x) = cos (x / 2)} = 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_427:\n fixes a :: nat\n assumes \"a = (\\<Sum> k\\<in> {n. n dvd 500}. k)\" \n shows \"(\\<Sum> k \\<in> {n. prime n \\<and> n dvd a}. k) = 25\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p7:\n fixes a:: \"nat\\<Rightarrow>nat\"\nassumes h0: \"(a 0)^3 = 1\"\n and h1: \"(a 1)^3 = 8\"\n and h2: \"(a 2)^3 = 27\"\n and h3: \"(a 3)^3 = 64\"\n and h4: \"(a 4)^3 = 125\"\n and h5: \"(a 5)^3 = 216\"\n and h6: \"(a 6)^3 = 343\" \n shows \"(\\<Sum> k < 7. (6 * (a k)^2)) \n - 2 * (\\<Sum> k < 6. (a k)^2) = (658::int)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_359:\n fixes y :: real\n assumes \"y + 6 + y = 2 * 12\" \n shows \"y = 9\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_388:\n fixes x y z :: real\n assumes \"3 * x + 4 * y - 12 * z = 10\"\n and \"-2 * x - 3 * y + 9 * z = -4\"\n shows \"x = 14\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p25:\n fixes a :: rat\n assumes \n h1: \"(\\<Sum> k\\<in>{x::real. (floor x) * (x - (floor x)) \n = a * x^2}. k) = 420\" \n and h2: \"(a1,a2) = quotient_of a\"\n shows \"a1 + a2 = 929\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_185:\n fixes n ::nat\n assumes \"n mod 5 = 3\" \n shows \"(2 * n) mod 5 = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_absxm1pabsxpabsxp1eqxp2_0leqxleq1:\n fixes x ::real\n assumes \"abs (x - 1) + abs x + abs (x + 1) = x + 2\" \n shows \"0 \\<le> x \\<and> x \\<le> 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_314:\n fixes n :: nat\n assumes \"n = 11\" \n shows \"(1 / 4)^(n + 1) * 2^(2 * n) = 1 / (4::real)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_apbmpcneq0_aeq0anbeq0anceq0:\n fixes a b c :: rat\n and m n :: real\n assumes \"0 < m \\<and> 0 < n\"\n and \"m^3 = 2\"\n and \"n^3 = 4\"\n and \"real_of_rat a + real_of_rat b * m + real_of_rat c * n = 0\"\n shows \"a = 0 \\<and> b = 0 \\<and> c = 0\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p4:\n fixes n :: nat\n assumes h0 : \"fst (quotient_of (1 / 2 + 1 / 3 + 1 / 7 \n + 1 /(rat_of_nat n))) = 1\" \n shows \"n = 42\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_313:\n fixes v i z ::complex\n assumes \"v = i * z\"\n and \"v = 1 + \\<i>\"\n and \"z = 2 - \\<i>\" \n shows \"i = 1/5 + 3/5 * \\<i>\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_419:\n fixes a b :: real\n assumes \"a = -1\"\n and \"b = 5\"\n shows \"-a - b^2 + 3 * (a * b) = -39\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_prod1p1onk3le3m1onn:\n fixes n :: nat\n assumes \"0 < n\" \n shows \"(\\<Prod> k \\<in> {1..<n+1}. (1 + 1 / k^3)) \\<le> 3 - 1 / n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_129:\n fixes a :: real\n assumes \"a \\<noteq> 0\"\n and \"(inverse 8) / (inverse 4) - (inverse a) = 1\" \n shows \"a = -2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_184:\n fixes a b ::real\n assumes \"0 < a \\<and> 0 < b\"\n and \"(a^2) = 6*b\"\n and \"(a^2) = 54/b\" \n shows \"a = 3 * sqrt 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p20:\n fixes x y z :: real\n assumes h0: \"0 < x \\<and> 0 < y \\<and> 0 < z\"\n and h1: \"x + 1/y = 4\"\n and h2: \"y + 1/z = 1\"\n and h3: \"z + 1/x = 7/3\"\n shows \"x*y*z = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_293:\n fixes x :: real\n assumes \"0 \\<le> x\"\n shows \"sqrt (60 * x) * sqrt (12 * x) * sqrt (63 * x) \n = 36 * x * sqrt (35 * x)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_33:\n fixes x y z :: real \n assumes \"x \\<noteq> 0\"\n and \"2 * x = 5 * y\"\n and \"7 * y = 10 * z\"\n shows \"z / x = 7 / 25\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p22:\n fixes t :: real\n shows \"((2 powr t - 3 * t) * t) / (4 powr t) \\<le> 1 / 12\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_abpbcpcageq3_sumaonsqrtapbgeq3onsqrt2:\n fixes a b c :: real\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and \"3 \\<le> a * b + b * c + c * a\" \n shows \"3 / sqrt 2 \\<le> a / sqrt (a + b) \n + b / sqrt (b + c) + c / sqrt (c + a)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_160:\n fixes n x ::real\n assumes \"n + x = 97\"\n and \"n + 5 * x = 265\"\n shows \"n + 2 * x = 139\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1990_p4:\n fixes x ::real\n assumes \"0 < x\"\n and \"x^2 - 10 * x - 29 \\<noteq> 0\"\n and \"x^2 - 10 * x - 45 \\<noteq> 0\"\n and \"x^2 - 10 * x - 69 \\<noteq> 0\"\n and \"1 / (x^2 - 10 * x - 29) + 1 / (x^2 - 10 * x - 45)\n - 2 / (x^2 - 10 * x - 69) = 0\"\n shows \"x = 13\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_320:\n fixes n ::nat\n assumes \"n < 101\"\n and \"101 dvd (123456 - n)\" \n shows \"n = 34\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_459:\n fixes a b c d ::rat\n assumes \"3 * a = b + c + d\"\n and \"4 * b = a + c + d\"\n and \"2 * c = a + b + d\"\n and \"8 * a + 10 * b + 6 * c = 24\"\n shows \"fst (quotient_of d) + snd (quotient_of d) = 28\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2000_p12:\n fixes a m c :: nat\n assumes h0: \"a + m + c = 12\" \n shows \"a*m*c + a*m + m*c + a*c \\<le> 112\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_270:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x \\<noteq> -2. f x = 1 / (x + 2)\" \n shows \"f (f 1) = 3/7\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1964_p2:\n fixes a b c :: real\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and \"c < a + b\"\n and \"b < a + c\"\n and \"a < b + c\" \n shows \"a^2 * (b + c - a) + b^2 * (c + a - b) \n + c^2 * (a + b - c) \\<le> 3 * a * b * c\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_cubrtrp1oncubrtreq3_rcubp1onrcubeq5778:\n fixes r :: real\n assumes \"r powr (1 / 3) + 1 / r powr (1 / 3) = 3\" \n shows \"r^3 + 1 / r^3 = 5778\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_451:\n shows \"(\\<Sum> k \\<in> {n ::nat. 2010 \\<le> n \\<and> n \\<le> 2019 \n \\<and> (\\<exists> m. (card {i. i dvd m} = 4) \n \\<and> (\\<Sum> p \\<in> {i. i dvd m}. p) = n)}. k) = 2016\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_amgm_sumasqdivbgeqsuma:\n fixes a b c d ::real\n assumes h0: \"0 < a \\<and> 0 < b \\<and> 0 < c \\<and> 0 < d\" \n shows \"a^2 / b + b^2 / c + c^2 / d + d^2 / a \\<ge> a + b + c + d\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_354:\n fixes a d :: real\n assumes \"a + 6 * d = 30\"\n and \"a + 10 * d = 60\" \n shows \"a + 20 * d = 135\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p10:\n fixes n ::nat\n assumes \"n>0\"\n and h0: \"ln (ln n / ln 16) / ln 2 \n = ln (ln n / ln 4) / ln 4\" \n shows \"n = 256\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_nfactltnexpnm1ngt3:\n fixes n ::nat\n assumes \"3 \\<le> n\" \n shows \"fact n < n^(n - 1)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2020_p4:\n shows \"card {n :: nat. 1000\\<le>n \\<and> n\\<le>9999 \\<and> (\\<forall>d\\<in>set (digits n). even d) \\<and> 5 dvd n} = 100\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1969_p2:\n fixes m n ::real\n and k ::nat\n and a :: \"nat \\<Rightarrow>real\"\n and y :: \"real \\<Rightarrow> real\"\n assumes \"0 < k\"\n and \"\\<forall> x. y x = (\\<Sum> i < k. ((cos (a i + x)) / (2^i)))\"\n and \"y m = 0\"\n and \"y n = 0\"\n shows \"\\<exists> t::int. m - n = t * pi\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_343 :\n \"(\\<Prod> k < 6. (2 * k + 1)) mod 10 = (5::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_435:\n fixes k :: nat\n assumes \"0 < k\"\n and \"\\<forall> n. gcd (6 * n + k) (6 * n + 3) = 1\"\n and \"\\<forall> n. gcd (6 * n + k) (6 * n + 2) = 1\"\n and \"\\<forall> n. gcd (6 * n + k) (6 * n + 1) = 1\" \n shows \"5 \\<le> k\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_5:\n fixes n :: nat\n assumes \"10 \\<le> n\"\n and \"\\<exists> x. x^2 = n\"\n and \"\\<exists> t. t^3 = n\" \n shows \"64 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1991_p9:\n fixes x :: real\n and m :: rat\n assumes \"1 / cos x + tan x = 22 / 7\"\n and \"1 / sin x + 1 / tan x = m\" \n shows \"fst (quotient_of m) + snd (quotient_of m)= 44\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_9onxpypzleqsum2onxpy:\n fixes x y z :: real\n assumes \"0 < x \\<and> 0 < y \\<and> 0 < z\" \n shows \"9 / (x + y + z) \\<le> 2 / (x + y) + 2 / (y + z) + 2 / (z + x)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_441:\n fixes x :: real\n assumes \"x \\<noteq> 0\" \n shows \"12 / (x * x) * (x^4 / (14 * x)) * (35 / (3 * x)) = 10\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_756:\n fixes a b :: real\n assumes \"2 powr a = 32\"\n and \"a powr b = 125\"\n shows \"b powr a = 243\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_exk2powkeqapb2mulbpa2_aeq1:\n fixes a b :: nat\n assumes \"a>0 \\<and> b>0\"\n and \"\\<exists> k > 0. 2^k = (a + b^2) * (b + a^2)\" \n shows \"a = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_176:\n fixes x ::real\n shows \"(x + 1)^2 * x = x^3 + 2 * x^2 + x\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_aoddbdiv4asqpbsqmod8eq1:\n fixes a :: int and b :: nat\n assumes \"odd a\"\n and \"4 dvd b\" \n shows \"(a^2 + b^2) mod 8 = 1 \""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p6:\n fixes m n p q :: real\n assumes \"p = 2 powr m\"\n and \"q = 3 powr n\"\n shows \"p powr (2 * n) * (q powr m) = 12 powr (m * n)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_212 :\n \"(16^17 * 17^18 * 18^19) mod 10 = (8::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_342:\n fixes a d::real\n assumes \"(\\<Sum> k<5. (a + k * d)) = 70\"\n and \"(\\<Sum> k <10. (a + k * d)) = 210\" \n shows \"a = 42/5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1974_p3:\n fixes n ::nat \n shows \"\\<not> 5 dvd (\\<Sum> k \\<le> n. ( (2 * n + 1) choose\n (2 * k + 1)) * (2^(3 * k)))\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p9 :\n shows \"(\\<Prod> k<7. (2^(2^k) + 3^(2^k))) = (3::nat)^128 - 2^128\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_17:\n fixes a :: real\n assumes \"1 + a>0\"\n assumes \"sqrt (4 + sqrt (16 + 16 * a)) \n + sqrt (1 + sqrt (1 + a)) = 6\" \n shows \"a = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_447:\n \"(\\<Sum> k \\<in>{n. 3 dvd n \\<and> 0<n \\<and> n<50}. k mod 10) = (78::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p18:\n fixes f :: \"rat \\<Rightarrow> real\"\n assumes h0: \"\\<forall>x>0. \\<forall>y>0. f (x * y) = f x + f y\"\n and h1: \"\\<forall>p. prime p \\<longrightarrow> f (of_nat p) = p\"\n shows \"f (25 / 11) < 0\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_171:\n fixes f :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall>x. f x = 5 * x + 4\" \n shows \"f 1 = 9\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1992_p1:\n fixes p q r ::int\n assumes \"1 < p \\<and> p < q \\<and> q < r\"\n and \"(p - 1) * (q - 1) * (r - 1) dvd (p * q * r - 1)\" \n shows \"(p, q, r) = (2, 4, 8) \\<or> (p, q, r) = (3, 5, 15)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p4:\n fixes m a :: nat\n assumes h0: \"of_nat m / of_nat a = 3 / (4::real)\" \n shows \"(84 * m + 70 * a) / (m + a) = (76::real)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2019_p12:\n fixes x y :: real\n assumes \"x \\<noteq> 1 \\<and> y \\<noteq> 1\"\n and \"ln x / ln 2 = ln 16 / ln y\"\n and \"x * y = 64\" \n shows \"(ln (x / y) / ln 2)^2 = 20\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_765:\n fixes x :: int\n assumes \"x < 0\"\n and \"(24 * x) mod 1199 = 15\"\n shows \"x \\<le> -449\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1968_p5_1:\n fixes a :: real\n and f :: \"real \\<Rightarrow> real\"\n assumes \"0 < a\"\n and \"\\<forall> x. f (x + a) = 1 / 2 + sqrt (f x - (f x)^2)\" \n shows \"\\<exists> b > 0. \\<forall> x. f (x + b) = f x\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1983_p1:\n fixes x y z w :: nat\n assumes ht : \"1 < x \\<and> 1 < y \\<and> 1 < z\"\n and hw : \"0 \\<le> w\"\n and h0 : \"ln w / ln x = 24\"\n and h1 : \"ln w / ln y = 40\"\n and h2 : \"ln w / ln (x * y * z) = 12\"\n shows \"ln w / ln z = 60\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_2019_p1:\n fixes f :: \"int \\<Rightarrow> int\" \n shows \"(\\<forall> a b. f (2 * a) + (2 * f b) = f (f (a + b)))\n \\<longleftrightarrow> (\\<forall> z. f z = 0 \\<or> (\\<exists> c. \\<forall> z. f z = 2 * z + c))\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_143:\n fixes f g :: \"real \\<Rightarrow> real\"\n assumes \"\\<forall> x. f x = x + 1\"\n and \"\\<forall> x. g x = x^2 + 3\" \n shows \"f (g 2) = 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_x5neqy2p4:\n fixes x y :: int\n shows \"x^5 \\<noteq> y^2 + 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12_2001_p21:\n fixes a b c d ::nat\n assumes h0: \"a*b*c*d = fact 8\"\n and h1: \"a*b + a + b = 524\"\n and h2: \"b*c + b + c = 146\"\n and h3: \"c*d + c + d = 104\" \n shows \"int a - d = 10\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_227:\n fixes x y n ::nat\n assumes \"x / 4 + y / 6 = (x + y) / n\"\n and \"n\\<noteq>0\" \"x\\<noteq>0\" \"y\\<noteq>0\"\n shows \"n = 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p3:\n fixes x :: real\n assumes h0: \"2 + 1 / (1 + 1 / (2 + 2 / (3 + x))) = 144 / 53\" \n shows \"x = 3 / 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_551 :\n \"1529 mod 6 = (5::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_302:\n \"(\\<i> / 2)^2 = -(1 / 4)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2021_p18:\n fixes z :: complex\n assumes h0: \"12 * (norm z)^2 = 2 * \n (norm (z + 2))^2 + (norm (z^2 + 1))^2 + 31\"\n shows \"z + 6 / z = -2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_others_exirrpowirrrat:\n \"\\<exists> a b. a \\<notin> \\<rat> \\<and> b \\<notin> \\<rat> \\<and> a^b \\<in> \\<rat>\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_pord1p1on2powklt5on2:\n fixes n :: nat\n assumes \"0 < n\" \n shows \"(\\<Prod> k \\<in>{1..<n+1}. (1 + 1 / 2^k)) < 5 / 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_229:\n \"(5^30) mod 7 = (1::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_12dvd4expnp1p20:\n fixes n :: nat\n shows \"(12::int) dvd 4^(n+1) + 20\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1984_p6:\n fixes a b c d k m ::nat\n assumes \"0 < a \\<and> 0 < b \\<and> 0 < c \\<and> 0 < d\"\n and \"odd a \\<and> odd b \\<and> odd c \\<and> odd d\"\n and \"a < b \\<and> b < c \\<and> c < d\"\n and \"a * d = b * c\"\n and \"a + d = 2^k\"\n and \"b + c = 2^m\" \n shows \"a = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem numbertheory_2pownm1prime_nprime:\n fixes n ::nat\n assumes \"0 < n\"\n and \"prime (2^n - 1)\" \n shows \"prime n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_113:\n fixes x ::real\n shows \"x^2 - 14 * x + 3 \\<ge> 7^2 - 14 * 7 + 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sqineq_unitcircatbpabsamblt1:\n fixes a b :: real\n assumes h0: \"a^2 + b^2 = 1\" \n shows \"a * b + \\<bar>a - b\\<bar> \\<le> 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_353:\n fixes s :: nat\n assumes \"s = (\\<Sum> k\\<in> {2010..<4019}. k)\" \n shows \"s mod 2009 = 0\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_277:\n fixes m n ::nat\n assumes \"gcd m n = 6\"\n and \"lcm m n = 126\" \n shows \"60 \\<le> m + n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_320:\n fixes x :: real\n and a b c :: nat\n assumes \"2 * x^2 = 4 * x + 9\"\n and \"x = (a + sqrt b) / c\"\n and \"x\\<ge>0\" \"a>0\" \"b>0\" \"c>0\"\n and \"c = 2\"\n shows \"a + b + c = 26\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_487:\n fixes a b c d :: real\n assumes \"b = a^2\"\n and \"a + b = 1\"\n and \"d = c^2\"\n and \"c + d = 1\"\n and \"a \\<noteq> c\"\n shows \"sqrt ((a - c)^2 + (b - d)^2)= sqrt 10\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_sum1onsqrt2to1onsqrt10000lt198 :\n \"(\\<Sum> k \\<in> {2::nat..<10001}. (1 / sqrt k)) < 198\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_328 :\n \"(5^999999) mod 7 = (6::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_114:\n fixes a :: real\n assumes \"a = 8\" \n shows \"(16 * (a^2) powr (1 / 3)) powr (1 / 3) = 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1997_p5:\n fixes x y ::nat\n assumes \"0 < x \\<and> 0 < y\"\n and \"x^(y^2) = y^x\"\n shows \"(x, y) = (1, 1) \\<or> (x, y) = (16, 2) \\<or> (x, y) = (27, 3)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2002_p6:\n fixes m ::nat \n assumes \"m>0\"\n shows \"\\<exists> n. (n>0) \\<and> m * n \\<le> m + n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_329:\n fixes x y :: real\n assumes \"3 * y = x\"\n and \"2 * x + 5 * y = 11\" \n shows \"x + y = 4\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1997_p9:\n fixes a :: real\n assumes \"0 < a\"\n and \"1 / a - floor (1 / a) = a^2 - floor (a^2)\"\n and \"2 < a^2\"\n and \"a^2 < 3\" \n shows \"a^12 - 144 * (1 / a) = 233\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_pprime_pdvdapowpma:\n fixes p a :: nat\n assumes \"0 < a\"\n and \"prime p\" \n shows \"p dvd (a^p - a)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_618:\n fixes n :: nat\n and p :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> x. p x = x^2 - x + 41\"\n and \"1 < gcd (p n) (p (n+1))\"\n shows \"41 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2013_p4 :\n \"(2^2014 + 2^2012) / (2^2014 - 2^2012) = (5::real) / 3\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_276:\n fixes a b :: int\n assumes \"\\<forall> x :: real. 10 * x^2 - x - 24 = (a * x - 8) * (b * x + 3)\"\n shows \"a * b + b = 12\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_457:\n fixes n :: nat\n assumes \"0 < n\"\n and \"80325 dvd (fact n)\" \n shows \"17 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_321:\n fixes n::int\n assumes \"1\\<le>n \\<and> n\\<le> 1399 \\<and> [n*160 = 1] (mod 1399)\"\n shows \"n = 1058\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_127 :\n \"(\\<Sum> k<101. 2^k) mod 7 = (3::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_598:\n fixes a b c d :: real\n assumes \"(4 powr a) = 5\"\n and \"(5 powr b) = 6\"\n and \"(6 powr c) = 7\"\n and \"(7 powr d) = 8\" \n shows \"a * b * c * d = 3 / 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2003_p23:\n \"card {k::nat. (k * k) dvd (\\<Prod> i\\<in>{1..<10}. fact i)} = 672\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_314:\n fixes r n :: nat\n assumes \"r = 1342 mod 13\"\n and \"0 < n\"\n and \"1342 dvd n\"\n and \"n mod 13 < r\"\n shows \"6710 \\<le> n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2020_p6:\n fixes n :: nat\n assumes h0: \"9 \\<le> n\" \n shows \"\\<exists>x::nat. (real_of_nat x)^2 = (fact (n + 2) \n - fact (n + 1)) / fact n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_153:\n fixes n :: real\n assumes \"n = 1 / 3\" \n shows \"floor (10 * n) + floor (100 * n) \n + floor (1000 * n) + floor (10000 * n) = 3702\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1982_p1:\n fixes f :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> m n. f (m + n) - f m - f n = 0 \\<or> f (m + n) - f m - f n = 1\"\n and \"f 2 = 0\"\n and \"0 < f 3\"\n and \"f 9999 = 3333\" \n shows \"f 1982 = 660\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_541:\n fixes m n ::nat\n assumes \"1 < m\"\n and \"1 < n\"\n and \"m * n = 2005\" \n shows \"m + n = 406\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_237 :\n \"(\\<Sum> k<101. k) mod 6 = (4::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1959_p1:\n fixes n :: nat\n shows \"gcd (21*n + 4) (14*n + 3) = 1\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1999_p11:\n fixes m :: rat\n assumes \"(\\<Sum> k \\<in>{0::nat..<36}. \n sin (5 * k * pi / 180)) = tan (real_of_rat m * pi / 180)\"\n and \"(nn,dd) = quotient_of m\"\n and \"dd/nn < 90\" \n shows \"dd+nn = 177\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_239 :\n \"(\\<Sum> k \\<in>{1..<13}. k) mod 4 = (2::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1994_p3:\n fixes x :: int and f :: \"int \\<Rightarrow> int\"\n assumes h0: \"f x + f (x-1) = x^2\"\n and h1: \"f 19 = 94\"\n shows \"f 94 mod 1000 = 561\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1960_p2:\n fixes x :: real\n assumes \"0 \\<le> 1 + 2 * x\"\n and \"(1 - sqrt (1 + 2 * x))^2 \\<noteq> 0\"\n and \"(4 * x^2) / (1 - sqrt (1 + 2*x))^2 < 2*x + 9\"\n shows \"-(1 / 2) \\<le> x \\<and> x < 45 / 8\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1988_p8:\n fixes f :: \"nat \\<Rightarrow> nat \\<Rightarrow> real\"\n assumes \"\\<forall> x>0. f x x = x\"\n and \"\\<forall> x>0. \\<forall> y>0. f x y = f y x\"\n and \"\\<forall> x>0. \\<forall> y>0. (x + y) * f x y = y * (f x (x + y))\" \n shows \"f 14 52 = 364\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_293:\n fixes n :: nat\n assumes \"n \\<le> 9\"\n and \"11 dvd 20 * 100 + 10 * n + 7\" \n shows \"n = 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p2:\n fixes x ::int\n assumes h0: \"x = 4\" \n shows \"(3 * x - 2) * (4 * x + 1) - (3 * x - 2) * (4 * x) + 1 = 11\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_80:\n fixes x :: real\n assumes \"x \\<noteq> -1\"\n and \"(x - 9) / (x + 1) = 2\"\n shows \"x = -11\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_137:\n fixes x::nat\n assumes \" x + 4 / 100 * x = 598\" \n shows \"x = 575\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_559:\n fixes x y ::nat\n assumes \"x mod 3 = 2\"\n and \"y mod 5 = 4\"\n and \"x mod 10 = y mod 10\" \n shows \"14 \\<le> x\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem aime_1995_p7:\n fixes k m n :: nat and t :: real\n assumes h0: \"gcd m n = 1\"\n and h1: \"(1 + sin t) * (1 + cos t) = 5/4\"\n and h2: \"(1 - sin t) * (1- cos t) = m/n - sqrt k\"\n shows \"k + m + n = 27\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p22:\n fixes a b c ::real\n and f :: \"real \\<Rightarrow> real\"\n assumes h0: \"\\<forall> x. f x = x^3 + a * x^2 + b * x + c\"\n and h1: \"(f-`{0}) = {cos (2 * pi / 7), \n cos (4 * pi / 7), cos (6 * pi / 7)}\"\n shows \"a * b * c = 1 / 32\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_304:\n \"91^2 = (8281::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p25:\n fixes n :: nat and f :: \"nat \\<Rightarrow> real\"\n assumes\n h0: \"\\<forall> n. f n = (\\<Sum> k\\<in> {k::nat. k dvd n}. 1) / (n powr (1 / 3))\"\n and h1: \"\\<forall> p \\<noteq> n. f p < f n\" \n shows \"n = 2520\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem induction_sumkexp3eqsumksq:\n fixes n ::nat \n shows \"(\\<Sum> k<n. k^3) = (\\<Sum> k<n. k)^2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_254:\n \"(239 + 174 + 83) mod 10 = (6::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_711:\n fixes m n ::nat\n assumes \" 0 < m \\<and> 0 < n\"\n and \"gcd m n = 8\"\n and \"lcm m n = 112\"\n shows \"72 \\<le> m + n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_24:\n fixes x :: real\n assumes \"x / 50 = 40\"\n shows \"x = 2000\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_142:\n fixes m b ::real\n assumes \"m * 7 + b = -1\"\n and \"m * (-1) + b = 7\" \n shows \"m + b = 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem algebra_bleqa_apbon2msqrtableqambsqon8b:\n fixes a b :: real\n assumes \"0 < a \\<and> 0 < b\"\n and \"b \\<le> a\"\n shows \"(a + b) / 2 - sqrt (a * b) \\<le> (a - b)^2 / (8 * b)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_400:\n fixes x :: real\n assumes \"5 + 500 / 100 * 10 = 110 / 100 * x\" \n shows \"x = 50\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_139:\n fixes s :: \"real \\<Rightarrow> real \\<Rightarrow> real\"\n assumes \"\\<forall> x\\<noteq>0. \\<forall>y\\<noteq>0. s x y = (1/y - 1/x) / (x-y)\"\n shows \"s 3 11 = 1/33\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_764:\n fixes p :: nat\n assumes \"prime p\"\n and \"7 \\<le> p\" \nshows \"(\\<Sum> k \\<in> {1..<p-1}. (inv_mod k p)* (inv_mod (k+1) p)) = 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1977_p6:\n fixes f :: \"nat \\<Rightarrow> nat\"\n assumes \"\\<forall> n. f (f n) < f (n + 1)\" \n and \"\\<forall> n. f n >0\"\n shows \"\\<forall> n. f n = n\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12b_2002_p19:\n fixes a b c::real\n assumes h0: \"0 < a \\<and> 0 < b \\<and> 0 < c\"\n and h1: \"a * (b + c) = 152\"\n and h2: \"b * (c + a) = 162\"\n and h3: \"c * (a + b) = 170\"\n shows \"a * b * c = 720\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_392:\n fixes n :: nat\n assumes \"even n\"\n and \"(n - 2)^2 + n^2 + (n + 2)^2 = (12296::int)\" \n shows \"((n - 2) * n * (n + 2)) / 8 = (32736::int)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_34:\n fixes x::nat\n assumes \"x < 100\"\n and \"x*9 mod 100 = 1\" \n shows \"x = 89\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p8:\n fixes d :: \"nat \\<Rightarrow> nat\"\n assumes h0: \"d 0 = 0\"\n and h1: \"d 1 = 0\"\n and h2: \"d 2 = 1\"\n and h3: \"\\<forall> n\\<ge>3. d n = d (n - 1) + d (n - 3)\"\n shows \"even (d 2021) \\<and> odd (d 2022) \\<and> even (d 2023)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_99:\n fixes n :: nat\n assumes \"(2 * n) mod 47 = 15\" \n shows \"n mod 47 = 31\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2021_p19:\n shows \"card {x ::real. 0 \\<le> x \\<and> x \\<le> pi \\<and> sin \n (pi / 2 * cos x) = cos (pi / 2 * \n sin x)} = 2\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_170:\n \"card { n::int. abs (n - 2) \\<le> 5 + 6 / 10} = 11\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_338:\n fixes a b c :: real\n assumes \"3 * a + b + c = -3\"\n and \"a + 3 * b + c = 9\"\n and \"a + b + 3 * c = 19\" \n shows \"a * b * c = -56\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_432:\n fixes x ::real \n shows \"(x + 3) * (2 * x - 6) = 2 * x^2 - 18\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2003_p5:\n fixes a m c ::nat\n assumes h0: \"a \\<le> 9 \\<and> m \\<le> 9 \\<and> c \\<le> 9\"\n and h1: \"10*(10*(10*(10*a + m) + c) + 1) + 0 \n + (10*(10*(10*(10*a + m) + c) + 1) + 2) = 123422\"\n shows \"a + m + c = 14 \""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem amc12a_2009_p7:\n fixes x :: real\n and n :: nat \n and a :: \"nat \\<Rightarrow> real\"\n assumes \"\\<forall> n. a (n + 1) - a n = a (n + 2) - a (n + 1)\"\n and \"a 1 = 2 * x - 3\"\n and \"a 2 = 5 * x - 11\"\n and \"a 3 = 3 * x + 1\"\n and \"a n = 2009\" \n shows \"n = 502\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_440:\n fixes x ::real\n assumes \"3 / 2 / 3 = x / 10\"\n shows \"x = 5\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_517 :\n \"(121 * 122 * 123) mod 4 = (2::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_345 :\n \"(2000 + 2001 + 2002 + 2003 + 2004 + 2005 + 2006) mod 7 = (0::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_3 :\n \"(\\<Sum> x \\<le> 9. (x^2)) mod 10 = (5::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_numbertheory_342:\n \"54 mod 6 = (0::nat)\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem imo_1962_p2:\n fixes x :: real\n assumes \"0 \\<le> 3 - x\"\n and \"0 \\<le> x + 1\"\n and \"1 / 2 < sqrt (sqrt (3 - x) - sqrt (x + 1))\" \n shows \"-1 \\<le> x \\<and> x < 1 - sqrt 127 / 32\""], "miniF2F test"], ["/tiger/u/kefan/Software/Isabelle2022/src/HOL/Examples/Full.thy", "\n ", ["theorem mathd_algebra_215:\n \"(\\<Sum> k \\<in> {x::real. (x + 3)^2 = 121}. k) = -6\""], "miniF2F test"]] |