Datasets:
Tasks:
Text Generation
Modalities:
Text
Sub-tasks:
language-modeling
Languages:
English
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100K - 1M
License:
| REAL_ARITH `!x y:real. (abs(x) - abs(y)) <= abs(x - y)`;; | |
| INT_ARITH | |
| `!a b a' b' D:int. | |
| (a pow 2 - D * b pow 2) * (a' pow 2 - D * b' pow 2) = | |
| (a * a' + D * b * b') pow 2 - D * (a * b' + a' * b) pow 2`;; | |
| REAL_ARITH | |
| `!x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11:real. | |
| x3 = abs(x2) - x1 /\ | |
| x4 = abs(x3) - x2 /\ | |
| x5 = abs(x4) - x3 /\ | |
| x6 = abs(x5) - x4 /\ | |
| x7 = abs(x6) - x5 /\ | |
| x8 = abs(x7) - x6 /\ | |
| x9 = abs(x8) - x7 /\ | |
| x10 = abs(x9) - x8 /\ | |
| x11 = abs(x10) - x9 | |
| ==> x1 = x10 /\ x2 = x11`;; | |
| REAL_ARITH `!x y:real. x < y ==> x < (x + y) / &2 /\ (x + y) / &2 < y`;; | |
| REAL_ARITH | |
| `((x1 pow 2 + x2 pow 2 + x3 pow 2 + x4 pow 2) pow 2) = | |
| ((&1 / &6) * ((x1 + x2) pow 4 + (x1 + x3) pow 4 + (x1 + x4) pow 4 + | |
| (x2 + x3) pow 4 + (x2 + x4) pow 4 + (x3 + x4) pow 4) + | |
| (&1 / &6) * ((x1 - x2) pow 4 + (x1 - x3) pow 4 + (x1 - x4) pow 4 + | |
| (x2 - x3) pow 4 + (x2 - x4) pow 4 + (x3 - x4) pow 4))`;; | |
| ARITH_RULE `x < 2 ==> 2 * x + 1 < 4`;; | |
| (**** Fails | |
| ARITH_RULE `~(2 * m + 1 = 2 * n)`;; | |
| ****) | |
| ARITH_RULE `x < 2 EXP 30 ==> (429496730 * x) DIV (2 EXP 32) = x DIV 10`;; | |
| (**** Fails | |
| ARITH_RULE `x <= 2 EXP 30 ==> (429496730 * x) DIV (2 EXP 32) = x DIV 10`;; | |
| ****) | |
| (**** Fails | |
| ARITH_RULE `1 <= x /\ 1 <= y ==> 1 <= x * y`;; | |
| ****) | |
| (**** Fails | |
| REAL_ARITH `!x y:real. x = y ==> x * y = y pow 2`;; | |
| ****) | |
| prioritize_real();; | |
| REAL_RING | |
| `s = (a + b + c) / &2 | |
| ==> s * (s - b) * (s - c) + s * (s - c) * (s - a) + | |
| s * (s - a) * (s - b) - (s - a) * (s - b) * (s - c) = | |
| a * b * c`;; | |
| REAL_RING `a pow 2 = &2 /\ x pow 2 + a * x + &1 = &0 ==> x pow 4 + &1 = &0`;; | |
| REAL_RING | |
| `(a * x pow 2 + b * x + c = &0) /\ | |
| (a * y pow 2 + b * y + c = &0) /\ | |
| ~(x = y) | |
| ==> (a * x * y = c) /\ (a * (x + y) + b = &0)`;; | |
| REAL_RING | |
| `p = (&3 * a1 - a2 pow 2) / &3 /\ | |
| q = (&9 * a1 * a2 - &27 * a0 - &2 * a2 pow 3) / &27 /\ | |
| x = z + a2 / &3 /\ | |
| x * w = w pow 2 - p / &3 | |
| ==> (z pow 3 + a2 * z pow 2 + a1 * z + a0 = &0 <=> | |
| if p = &0 then x pow 3 = q | |
| else (w pow 3) pow 2 - q * (w pow 3) - p pow 3 / &27 = &0)`;; | |
| REAL_FIELD `&0 < x ==> &1 / x - &1 / (&1 + x) = &1 / (x * (&1 + x))`;; | |
| REAL_FIELD | |
| `s pow 2 = b pow 2 - &4 * a * c | |
| ==> (a * x pow 2 + b * x + c = &0 <=> | |
| if a = &0 then | |
| if b = &0 then | |
| if c = &0 then T else F | |
| else x = --c / b | |
| else x = (--b + s) / (&2 * a) \/ x = (--b + --s) / (&2 * a))`;; | |
| (**** This needs an external SDP solver to assist with proof | |
| needs "Examples/sos.ml";; | |
| SOS_RULE `1 <= x /\ 1 <= y ==> 1 <= x * y`;; | |
| REAL_SOS | |
| `!a1 a2 a3 a4:real. | |
| &0 <= a1 /\ &0 <= a2 /\ &0 <= a3 /\ &0 <= a4 | |
| ==> a1 pow 2 + | |
| ((a1 + a2) / &2) pow 2 + | |
| ((a1 + a2 + a3) / &3) pow 2 + | |
| ((a1 + a2 + a3 + a4) / &4) pow 2 | |
| <= &4 * (a1 pow 2 + a2 pow 2 + a3 pow 2 + a4 pow 2)`;; | |
| REAL_SOS | |
| `!a b c:real. | |
| a >= &0 /\ b >= &0 /\ c >= &0 | |
| ==> &3 / &2 * (b + c) * (a + c) * (a + b) <= | |
| a * (a + c) * (a + b) + | |
| b * (b + c) * (a + b) + | |
| c * (b + c) * (a + c)`;; | |
| SOS_CONV `&2 * x pow 4 + &2 * x pow 3 * y - x pow 2 * y pow 2 + &5 * y pow 4`;; | |
| PURE_SOS | |
| `x pow 4 + &2 * x pow 2 * z + x pow 2 - &2 * x * y * z + | |
| &2 * y pow 2 * z pow 2 + &2 * y * z pow 2 + &2 * z pow 2 - &2 * x + | |
| &2 * y * z + &1 >= &0`;; | |
| ****) | |
| needs "Examples/cooper.ml";; | |
| COOPER_RULE `ODD n ==> 2 * n DIV 2 < n`;; | |
| COOPER_RULE `!n. n >= 8 ==> ?a b. n = 3 * a + 5 * b`;; | |
| needs "Rqe/make.ml";; | |
| REAL_QELIM_CONV `!x. &0 <= x ==> ?y. y pow 2 = x`;; | |