Datasets:

phanerozoic
/

HOL-Light

fact stringlengths 9 4.14k	type stringclasses 2 values	library stringclasses 29 values	imports listlengths 0 15	filename stringclasses 350 values	symbolic_name stringlengths 2 67	docstring null
ADD_0 : !m. m + 0 = m	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_0	null
ADD_SUC : !m n. m + (SUC n) = SUC(m + n)	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_SUC	null
ADD_CLAUSES : (!n. 0 + n = n) /\ (!m. m + 0 = m) /\ (!m n. (SUC m) + n = SUC(m + n)) /\ (!m n. m + (SUC n) = SUC(m + n))	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_CLAUSES	null
ADD_SYM : !m n. m + n = n + m	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_SYM	null
ADD_ASSOC : !m n p. m + (n + p) = (m + n) + p	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_ASSOC	null
ADD_AC : (m + n = n + m) /\ ((m + n) + p = m + (n + p)) /\ (m + (n + p) = n + (m + p))	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_AC	null
ADD_EQ_0 : !m n. (m + n = 0) <=> (m = 0) /\ (n = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	ADD_EQ_0	null
EQ_ADD_LCANCEL : !m n p. (m + n = m + p) <=> (n = p)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_ADD_LCANCEL	null
EQ_ADD_RCANCEL : !m n p. (m + p = n + p) <=> (m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_ADD_RCANCEL	null
EQ_ADD_LCANCEL_0 : !m n. (m + n = m) <=> (n = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_ADD_LCANCEL_0	null
EQ_ADD_RCANCEL_0 : !m n. (m + n = n) <=> (m = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_ADD_RCANCEL_0	null
BIT0 : !n. BIT0 n = n + n	theorem	core	[ "recursion.ml" ]	arith.ml	BIT0	null
BIT1 : !n. BIT1 n = SUC(n + n)	theorem	core	[ "recursion.ml" ]	arith.ml	BIT1	null
BIT0_THM : !n. NUMERAL (BIT0 n) = NUMERAL n + NUMERAL n	theorem	core	[ "recursion.ml" ]	arith.ml	BIT0_THM	null
BIT1_THM : !n. NUMERAL (BIT1 n) = SUC(NUMERAL n + NUMERAL n)	theorem	core	[ "recursion.ml" ]	arith.ml	BIT1_THM	null
ONE : 1 = SUC 0	theorem	core	[ "recursion.ml" ]	arith.ml	ONE	null
TWO : 2 = SUC 1	theorem	core	[ "recursion.ml" ]	arith.ml	TWO	null
ADD1 : !m. SUC m = m + 1	theorem	core	[ "recursion.ml" ]	arith.ml	ADD1	null
MULT_0 : !m. m * 0 = 0	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_0	null
MULT_SUC : !m n. m * (SUC n) = m + (m * n)	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_SUC	null
MULT_CLAUSES : (!n. 0 * n = 0) /\ (!m. m * 0 = 0) /\ (!n. 1 * n = n) /\ (!m. m * 1 = m) /\ (!m n. (SUC m) * n = (m * n) + n) /\ (!m n. m * (SUC n) = m + (m * n))	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_CLAUSES	null
MULT_SYM : !m n. m * n = n * m	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_SYM	null
LEFT_ADD_DISTRIB : !m n p. m * (n + p) = (m * n) + (m * p)	theorem	core	[ "recursion.ml" ]	arith.ml	LEFT_ADD_DISTRIB	null
RIGHT_ADD_DISTRIB : !m n p. (m + n) * p = (m * p) + (n * p)	theorem	core	[ "recursion.ml" ]	arith.ml	RIGHT_ADD_DISTRIB	null
MULT_ASSOC : !m n p. m * (n * p) = (m * n) * p	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_ASSOC	null
MULT_AC : (m * n = n * m) /\ ((m * n) * p = m * (n * p)) /\ (m * (n * p) = n * (m * p))	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_AC	null
MULT_EQ_0 : !m n. (m * n = 0) <=> (m = 0) \/ (n = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_EQ_0	null
EQ_MULT_LCANCEL : !m n p. (m * n = m * p) <=> (m = 0) \/ (n = p)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_MULT_LCANCEL	null
EQ_MULT_RCANCEL : !m n p. (m * p = n * p) <=> (m = n) \/ (p = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_MULT_RCANCEL	null
MULT_2 : !n. 2 * n = n + n	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_2	null
MULT_EQ_1 : !m n. (m * n = 1) <=> (m = 1) /\ (n = 1)	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_EQ_1	null
EXP_EQ_0 : !m n. (m EXP n = 0) <=> (m = 0) /\ ~(n = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_EQ_0	null
EXP_EQ_1 : !x n. x EXP n = 1 <=> x = 1 \/ n = 0	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_EQ_1	null
EXP_ZERO : !n. 0 EXP n = if n = 0 then 1 else 0	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_ZERO	null
EXP_ADD : !m n p. m EXP (n + p) = (m EXP n) * (m EXP p)	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_ADD	null
EXP_ONE : !n. 1 EXP n = 1	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_ONE	null
EXP_1 : !n. n EXP 1 = n	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_1	null
EXP_2 : !n. n EXP 2 = n * n	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_2	null
MULT_EXP : !p m n. (m * n) EXP p = m EXP p * n EXP p	theorem	core	[ "recursion.ml" ]	arith.ml	MULT_EXP	null
EXP_MULT : !m n p. m EXP (n * p) = (m EXP n) EXP p	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_MULT	null
EXP_EXP : !x m n. (x EXP m) EXP n = x EXP (m * n)	theorem	core	[ "recursion.ml" ]	arith.ml	EXP_EXP	null
LE_SUC_LT : !m n. (SUC m <= n) <=> (m < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_SUC_LT	null
LT_SUC_LE : !m n. (m < SUC n) <=> (m <= n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_SUC_LE	null
LE_SUC : !m n. (SUC m <= SUC n) <=> (m <= n)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_SUC	null
LT_SUC : !m n. (SUC m < SUC n) <=> (m < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_SUC	null
LE_0 : !n. 0 <= n	theorem	core	[ "recursion.ml" ]	arith.ml	LE_0	null
LT_0 : !n. 0 < SUC n	theorem	core	[ "recursion.ml" ]	arith.ml	LT_0	null
LE_REFL : !n. n <= n	theorem	core	[ "recursion.ml" ]	arith.ml	LE_REFL	null
LT_REFL : !n. ~(n < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_REFL	null
LT_IMP_NE : !m n:num. m < n ==> ~(m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_IMP_NE	null
LE_ANTISYM : !m n. (m <= n /\ n <= m) <=> (m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ANTISYM	null
LT_ANTISYM : !m n. ~(m < n /\ n < m)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ANTISYM	null
LET_ANTISYM : !m n. ~(m <= n /\ n < m)	theorem	core	[ "recursion.ml" ]	arith.ml	LET_ANTISYM	null
LTE_ANTISYM : !m n. ~(m < n /\ n <= m)	theorem	core	[ "recursion.ml" ]	arith.ml	LTE_ANTISYM	null
LE_TRANS : !m n p. m <= n /\ n <= p ==> m <= p	theorem	core	[ "recursion.ml" ]	arith.ml	LE_TRANS	null
LT_TRANS : !m n p. m < n /\ n < p ==> m < p	theorem	core	[ "recursion.ml" ]	arith.ml	LT_TRANS	null
LET_TRANS : !m n p. m <= n /\ n < p ==> m < p	theorem	core	[ "recursion.ml" ]	arith.ml	LET_TRANS	null
LTE_TRANS : !m n p. m < n /\ n <= p ==> m < p	theorem	core	[ "recursion.ml" ]	arith.ml	LTE_TRANS	null
LE_CASES : !m n. m <= n \/ n <= m	theorem	core	[ "recursion.ml" ]	arith.ml	LE_CASES	null
LT_CASES : !m n. (m < n) \/ (n < m) \/ (m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_CASES	null
LET_CASES : !m n. m <= n \/ n < m	theorem	core	[ "recursion.ml" ]	arith.ml	LET_CASES	null
LTE_CASES : !m n. m < n \/ n <= m	theorem	core	[ "recursion.ml" ]	arith.ml	LTE_CASES	null
LE_LT : !m n. (m <= n) <=> (m < n) \/ (m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_LT	null
LT_LE : !m n. (m < n) <=> (m <= n) /\ ~(m = n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_LE	null
NOT_LE : !m n. ~(m <= n) <=> (n < m)	theorem	core	[ "recursion.ml" ]	arith.ml	NOT_LE	null
NOT_LT : !m n. ~(m < n) <=> n <= m	theorem	core	[ "recursion.ml" ]	arith.ml	NOT_LT	null
LT_IMP_LE : !m n. m < n ==> m <= n	theorem	core	[ "recursion.ml" ]	arith.ml	LT_IMP_LE	null
EQ_IMP_LE : !m n. (m = n) ==> m <= n	theorem	core	[ "recursion.ml" ]	arith.ml	EQ_IMP_LE	null
LT_NZ : !n. 0 < n <=> ~(n = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_NZ	null
LE_1 : (!n. ~(n = 0) ==> 0 < n) /\ (!n. ~(n = 0) ==> 1 <= n) /\ (!n. 0 < n ==> ~(n = 0)) /\ (!n. 0 < n ==> 1 <= n) /\ (!n. 1 <= n ==> 0 < n) /\ (!n. 1 <= n ==> ~(n = 0))	theorem	core	[ "recursion.ml" ]	arith.ml	LE_1	null
LE_EXISTS : !m n. (m <= n) <=> (?d. n = m + d)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_EXISTS	null
LT_EXISTS : !m n. (m < n) <=> (?d. n = m + SUC d)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_EXISTS	null
LE_ADD : !m n. m <= m + n	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ADD	null
LE_ADDR : !m n. n <= m + n	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ADDR	null
LT_ADD : !m n. (m < m + n) <=> (0 < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ADD	null
LT_ADDR : !m n. (n < m + n) <=> (0 < m)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ADDR	null
LE_ADD_LCANCEL : !m n p. (m + n) <= (m + p) <=> n <= p	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ADD_LCANCEL	null
LE_ADD_RCANCEL : !m n p. (m + p) <= (n + p) <=> (m <= n)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ADD_RCANCEL	null
LT_ADD_LCANCEL : !m n p. (m + n) < (m + p) <=> n < p	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ADD_LCANCEL	null
LT_ADD_RCANCEL : !m n p. (m + p) < (n + p) <=> (m < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ADD_RCANCEL	null
LE_ADD2 : !m n p q. m <= p /\ n <= q ==> m + n <= p + q	theorem	core	[ "recursion.ml" ]	arith.ml	LE_ADD2	null
LET_ADD2 : !m n p q. m <= p /\ n < q ==> m + n < p + q	theorem	core	[ "recursion.ml" ]	arith.ml	LET_ADD2	null
LTE_ADD2 : !m n p q. m < p /\ n <= q ==> m + n < p + q	theorem	core	[ "recursion.ml" ]	arith.ml	LTE_ADD2	null
LT_ADD2 : !m n p q. m < p /\ n < q ==> m + n < p + q	theorem	core	[ "recursion.ml" ]	arith.ml	LT_ADD2	null
LT_MULT : !m n. (0 < m * n) <=> (0 < m) /\ (0 < n)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_MULT	null
LE_MULT2 : !m n p q. m <= n /\ p <= q ==> m * p <= n * q	theorem	core	[ "recursion.ml" ]	arith.ml	LE_MULT2	null
LT_LMULT : !m n p. ~(m = 0) /\ n < p ==> m * n < m * p	theorem	core	[ "recursion.ml" ]	arith.ml	LT_LMULT	null
LE_MULT_LCANCEL : !m n p. (m * n) <= (m * p) <=> (m = 0) \/ n <= p	theorem	core	[ "recursion.ml" ]	arith.ml	LE_MULT_LCANCEL	null
LE_MULT_RCANCEL : !m n p. (m * p) <= (n * p) <=> (m <= n) \/ (p = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	LE_MULT_RCANCEL	null
LT_MULT_LCANCEL : !m n p. (m * n) < (m * p) <=> ~(m = 0) /\ n < p	theorem	core	[ "recursion.ml" ]	arith.ml	LT_MULT_LCANCEL	null
LT_MULT_RCANCEL : !m n p. (m * p) < (n * p) <=> (m < n) /\ ~(p = 0)	theorem	core	[ "recursion.ml" ]	arith.ml	LT_MULT_RCANCEL	null
LT_MULT2 : !m n p q. m < n /\ p < q ==> m * p < n * q	theorem	core	[ "recursion.ml" ]	arith.ml	LT_MULT2	null
LE_SQUARE_REFL : !n. n <= n * n	theorem	core	[ "recursion.ml" ]	arith.ml	LE_SQUARE_REFL	null
LT_POW2_REFL : !n. n < 2 EXP n	theorem	core	[ "recursion.ml" ]	arith.ml	LT_POW2_REFL	null
WLOG_LE : (!m n. P m n <=> P n m) /\ (!m n. m <= n ==> P m n) ==> !m n. P m n	theorem	core	[ "recursion.ml" ]	arith.ml	WLOG_LE	null
WLOG_LT : (!m. P m m) /\ (!m n. P m n <=> P n m) /\ (!m n. m < n ==> P m n) ==> !m y. P m y	theorem	core	[ "recursion.ml" ]	arith.ml	WLOG_LT	null
WLOG_LE_3 : !P. (!x y z. P x y z ==> P y x z /\ P x z y) /\ (!x y z. x <= y /\ y <= z ==> P x y z) ==> !x y z. P x y z	theorem	core	[ "recursion.ml" ]	arith.ml	WLOG_LE_3	null
num_WF : !P. (!n. (!m. m < n ==> P m) ==> P n) ==> !n. P n	theorem	core	[ "recursion.ml" ]	arith.ml	num_WF	null
num_WOP : !P. (?n. P n) <=> (?n. P(n) /\ !m. m < n ==> ~P(m))	theorem	core	[ "recursion.ml" ]	arith.ml	num_WOP	null
num_MAX : !P. (?x. P x) /\ (?M. !x. P x ==> x <= M) <=> ?m. P m /\ (!x. P x ==> x <= m)	theorem	core	[ "recursion.ml" ]	arith.ml	num_MAX	null

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HOL-Light

A structured dataset of theorems and definitions from HOL Light, an interactive theorem prover for higher-order logic written in OCaml.

Source

Repository: https://github.com/jrh13/hol-light
License: BSD-2-Clause

Statistics

Property	Value
Total Entries	34,572
Theorems	32,614
Definitions	1,958
Source Files	556

Top Libraries

Library	Count
Multivariate	16,190
Library	7,326
core	2,584
100 (Flyspeck)	2,519
RichterHilbertAxiomGeometry	1,368

Schema

Column	Type	Description
`fact`	string	Theorem statement or definition body
`type`	string	"theorem" or "definition"
`library`	string	HOL Light library directory
`imports`	list[string]	OCaml needs statements
`filename`	string	Source .ml file
`symbolic_name`	string	Theorem/definition name