Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated
• 3
Datasets:
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 |
End of preview. Expand
in Data Studio
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 |
About HOL Light
HOL Light is known for:
- Flyspeck project (formal proof of Kepler conjecture)
- Multivariate analysis library
- Floating-point verification
- Formalization of complex analysis
Creator
Charles Norton (phanerozoic)
- Downloads last month
- 6
Size of downloaded dataset files:
1.82 MB
Size of the auto-converted Parquet files:
1.82 MB
Number of rows:
34,572