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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