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 |
|---|---|---|---|---|---|---|
LE_INDUCT : !P. (!m:num. P m m) /\ (!m n. m <= n /\ P m n ==> P m (SUC n)) ==> (!m n. m <= n ==> P m n) | theorem | core | [
"recursion.ml"
] | arith.ml | LE_INDUCT | null |
num_INDUCTION_DOWN : !(P:num->bool) m. (!n. m <= n ==> P n) /\ (!n. n < m /\ P(n + 1) ==> P n) ==> !n. P n | theorem | core | [
"recursion.ml"
] | arith.ml | num_INDUCTION_DOWN | null |
NOT_EVEN : !n. ~(EVEN n) <=> ODD n | theorem | core | [
"recursion.ml"
] | arith.ml | NOT_EVEN | null |
NOT_ODD : !n. ~(ODD n) <=> EVEN n | theorem | core | [
"recursion.ml"
] | arith.ml | NOT_ODD | null |
EVEN_OR_ODD : !n. EVEN n \/ ODD n | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_OR_ODD | null |
EVEN_AND_ODD : !n. ~(EVEN n /\ ODD n) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_AND_ODD | null |
EVEN_ADD : !m n. EVEN(m + n) <=> (EVEN m <=> EVEN n) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_ADD | null |
EVEN_MULT : !m n. EVEN(m * n) <=> EVEN(m) \/ EVEN(n) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_MULT | null |
EVEN_EXP : !m n. EVEN(m EXP n) <=> EVEN(m) /\ ~(n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_EXP | null |
ODD_ADD : !m n. ODD(m + n) <=> ~(ODD m <=> ODD n) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_ADD | null |
ODD_MULT : !m n. ODD(m * n) <=> ODD(m) /\ ODD(n) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_MULT | null |
ODD_EXP : !m n. ODD(m EXP n) <=> ODD(m) \/ (n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_EXP | null |
EVEN_DOUBLE : !n. EVEN(2 * n) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_DOUBLE | null |
ODD_DOUBLE : !n. ODD(SUC(2 * n)) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_DOUBLE | null |
EVEN_EXISTS_LEMMA : !n. (EVEN n ==> ?m. n = 2 * m) /\ (~EVEN n ==> ?m. n = SUC(2 * m)) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_EXISTS_LEMMA | null |
EVEN_EXISTS : !n. EVEN n <=> ?m. n = 2 * m | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_EXISTS | null |
ODD_EXISTS : !n. ODD n <=> ?m. n = SUC(2 * m) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_EXISTS | null |
EVEN_ODD_DECOMPOSITION : !n. (?k m. ODD m /\ (n = 2 EXP k * m)) <=> ~(n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_ODD_DECOMPOSITION | null |
SUB_0 : !m. (0 - m = 0) /\ (m - 0 = m) | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_0 | null |
SUB_PRESUC : !m n. PRE(SUC m - n) = m - n | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_PRESUC | null |
SUB_SUC : !m n. SUC m - SUC n = m - n | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_SUC | null |
SUB_REFL : !n. n - n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_REFL | null |
ADD_SUB : !m n. (m + n) - n = m | theorem | core | [
"recursion.ml"
] | arith.ml | ADD_SUB | null |
ADD_SUB2 : !m n. (m + n) - m = n | theorem | core | [
"recursion.ml"
] | arith.ml | ADD_SUB2 | null |
SUB_EQ_0 : !m n. (m - n = 0) <=> m <= n | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_EQ_0 | null |
ADD_SUBR2 : !m n. m - (m + n) = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | ADD_SUBR2 | null |
ADD_SUBR : !m n. n - (m + n) = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | ADD_SUBR | null |
SUB_ADD : !m n. n <= m ==> ((m - n) + n = m) | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_ADD | null |
SUB_ADD_LCANCEL : !m n p. (m + n) - (m + p) = n - p | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_ADD_LCANCEL | null |
SUB_ADD_RCANCEL : !m n p. (m + p) - (n + p) = m - n | theorem | core | [
"recursion.ml"
] | arith.ml | SUB_ADD_RCANCEL | null |
LEFT_SUB_DISTRIB : !m n p. m * (n - p) = m * n - m * p | theorem | core | [
"recursion.ml"
] | arith.ml | LEFT_SUB_DISTRIB | null |
RIGHT_SUB_DISTRIB : !m n p. (m - n) * p = m * p - n * p | theorem | core | [
"recursion.ml"
] | arith.ml | RIGHT_SUB_DISTRIB | null |
SUC_SUB1 : !n. SUC n - 1 = n | theorem | core | [
"recursion.ml"
] | arith.ml | SUC_SUB1 | null |
EVEN_SUB : !m n. EVEN(m - n) <=> m <= n \/ (EVEN(m) <=> EVEN(n)) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_SUB | null |
ODD_SUB : !m n. ODD(m - n) <=> n < m /\ ~(ODD m <=> ODD n) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_SUB | null |
FACT_LT : !n. 0 < FACT n | theorem | core | [
"recursion.ml"
] | arith.ml | FACT_LT | null |
FACT_LE : !n. 1 <= FACT n | theorem | core | [
"recursion.ml"
] | arith.ml | FACT_LE | null |
FACT_NZ : !n. ~(FACT n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | FACT_NZ | null |
FACT_MONO : !m n. m <= n ==> FACT m <= FACT n | theorem | core | [
"recursion.ml"
] | arith.ml | FACT_MONO | null |
EXP_LT_0 : !n x. 0 < x EXP n <=> ~(x = 0) \/ (n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_LT_0 | null |
LT_EXP : !x m n. x EXP m < x EXP n <=> 2 <= x /\ m < n \/ (x = 0) /\ ~(m = 0) /\ (n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | LT_EXP | null |
LE_EXP : !x m n. x EXP m <= x EXP n <=> if x = 0 then (m = 0) ==> (n = 0) else (x = 1) \/ m <= n | theorem | core | [
"recursion.ml"
] | arith.ml | LE_EXP | null |
EQ_EXP : !x m n. x EXP m = x EXP n <=> if x = 0 then (m = 0 <=> n = 0) else (x = 1) \/ m = n | theorem | core | [
"recursion.ml"
] | arith.ml | EQ_EXP | null |
EXP_MONO_LE_IMP : !x y n. x <= y ==> x EXP n <= y EXP n | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_MONO_LE_IMP | null |
EXP_MONO_LT_IMP : !x y n. x < y /\ ~(n = 0) ==> x EXP n < y EXP n | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_MONO_LT_IMP | null |
EXP_MONO_LE : !x y n. x EXP n <= y EXP n <=> x <= y \/ n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_MONO_LE | null |
EXP_MONO_LT : !x y n. x EXP n < y EXP n <=> x < y /\ ~(n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_MONO_LT | null |
EXP_MONO_EQ : !x y n. x EXP n = y EXP n <=> x = y \/ n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | EXP_MONO_EQ | null |
DIVMOD_EXIST : !m n. ~(n = 0) ==> ?q r. (m = q * n + r) /\ r < n | theorem | core | [
"recursion.ml"
] | arith.ml | DIVMOD_EXIST | null |
DIVMOD_EXIST_0 : !m n. ?q r. if n = 0 then q = 0 /\ r = m else m = q * n + r /\ r < n | theorem | core | [
"recursion.ml"
] | arith.ml | DIVMOD_EXIST_0 | null |
DIVISION : !m n. ~(n = 0) ==> (m = m DIV n * n + m MOD n) /\ m MOD n < n | theorem | core | [
"recursion.ml"
] | arith.ml | DIVISION | null |
DIV_ZERO : !n. n DIV 0 = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_ZERO | null |
MOD_ZERO : !n. n MOD 0 = n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_ZERO | null |
DIVISION_SIMP : (!m n. m DIV n * n + m MOD n = m) /\ (!m n. n * m DIV n + m MOD n = m) | theorem | core | [
"recursion.ml"
] | arith.ml | DIVISION_SIMP | null |
EQ_DIVMOD : !p m n. m DIV p = n DIV p /\ m MOD p = n MOD p <=> m = n | theorem | core | [
"recursion.ml"
] | arith.ml | EQ_DIVMOD | null |
MOD_LT_EQ : !m n. m MOD n < n <=> ~(n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_LT_EQ | null |
MOD_LT_EQ_LT : !m n. m MOD n < n <=> 0 < n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_LT_EQ_LT | null |
DIVMOD_UNIQ_LEMMA : !m n q1 r1 q2 r2. ((m = q1 * n + r1) /\ r1 < n) /\ ((m = q2 * n + r2) /\ r2 < n) ==> (q1 = q2) /\ (r1 = r2) | theorem | core | [
"recursion.ml"
] | arith.ml | DIVMOD_UNIQ_LEMMA | null |
DIVMOD_UNIQ : !m n q r. (m = q * n + r) /\ r < n ==> (m DIV n = q) /\ (m MOD n = r) | theorem | core | [
"recursion.ml"
] | arith.ml | DIVMOD_UNIQ | null |
MOD_UNIQ : !m n q r. (m = q * n + r) /\ r < n ==> (m MOD n = r) | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_UNIQ | null |
DIV_UNIQ : !m n q r. (m = q * n + r) /\ r < n ==> (m DIV n = q) | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_UNIQ | null |
MOD_LT : !m n. m < n ==> m MOD n = m | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_LT | null |
MOD_EQ_SELF : !m n. m MOD n = m <=> n = 0 \/ m < n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EQ_SELF | null |
MOD_CASES : !n p. n < 2 * p ==> n MOD p = if n < p then n else n - p | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_CASES | null |
MOD_ADD_CASES : !m n p. m < p /\ n < p ==> (m + n) MOD p = if m + n < p then m + n else (m + n) - p | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_ADD_CASES | null |
MOD_EQ : !m n p q. m = n + q * p ==> m MOD p = n MOD p | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EQ | null |
DIV_LE : !m n. m DIV n <= m | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_LE | null |
DIV_MUL_LE : !m n. n * (m DIV n) <= m | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_MUL_LE | null |
MOD_LE_TWICE : !m n. 0 < m /\ m <= n ==> 2 * n MOD m <= n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_LE_TWICE | null |
DIV_LT : !m n. m < n ==> m DIV n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_LT | null |
MOD_MOD : !m n p. (m MOD (n * p)) MOD n = m MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MOD | null |
MOD_MOD_REFL : !m n. (m MOD n) MOD n = m MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MOD_REFL | null |
MOD_MOD_LE : !m n p. ~(n = 0) /\ n <= p ==> (m MOD n) MOD p = m MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MOD_LE | null |
MOD_EVEN_2 : !m n. EVEN n ==> m MOD n MOD 2 = m MOD 2 | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EVEN_2 | null |
DIV_MULT2 : !m n p. ~(m = 0) ==> ((m * n) DIV (m * p) = n DIV p) | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_MULT2 | null |
MOD_MULT2 : !m n p. (m * n) MOD (m * p) = m * n MOD p | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MULT2 | null |
MOD_EXISTS : !m n. (?q. m = n * q) <=> if n = 0 then (m = 0) else (m MOD n = 0) | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EXISTS | null |
LE_RDIV_EQ : !a b n. ~(a = 0) ==> (n <= b DIV a <=> a * n <= b) | theorem | core | [
"recursion.ml"
] | arith.ml | LE_RDIV_EQ | null |
RDIV_LT_EQ : !a b n. ~(a = 0) ==> (b DIV a < n <=> b < a * n) | theorem | core | [
"recursion.ml"
] | arith.ml | RDIV_LT_EQ | null |
LE_LDIV_EQ : !a b n. ~(a = 0) ==> (b DIV a <= n <=> b < a * (n + 1)) | theorem | core | [
"recursion.ml"
] | arith.ml | LE_LDIV_EQ | null |
LDIV_LT_EQ : !a b n. ~(a = 0) ==> (n < b DIV a <=> a * (n + 1) <= b) | theorem | core | [
"recursion.ml"
] | arith.ml | LDIV_LT_EQ | null |
LE_LDIV : !a b n. ~(a = 0) /\ b <= a * n ==> b DIV a <= n | theorem | core | [
"recursion.ml"
] | arith.ml | LE_LDIV | null |
DIV_MONO : !m n p. m <= n ==> m DIV p <= n DIV p | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_MONO | null |
DIV_MONO_LT : !m n p. ~(p = 0) /\ m + p <= n ==> m DIV p < n DIV p | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_MONO_LT | null |
DIV_EQ_0 : !m n. ~(n = 0) ==> ((m DIV n = 0) <=> m < n) | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_EQ_0 | null |
MOD_DIV_EQ_0 : !m n. ~(n = 0) ==> (m MOD n) DIV n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_DIV_EQ_0 | null |
MOD_EQ_0 : !m n. (m MOD n = 0) <=> ?q. m = q * n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EQ_0 | null |
DIV_EQ_SELF : !m n. m DIV n = m <=> m = 0 \/ n = 1 | theorem | core | [
"recursion.ml"
] | arith.ml | DIV_EQ_SELF | null |
MOD_REFL : !n. n MOD n = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_REFL | null |
EVEN_MOD : !n. EVEN(n) <=> n MOD 2 = 0 | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_MOD | null |
ODD_MOD : !n. ODD(n) <=> n MOD 2 = 1 | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_MOD | null |
MOD_2_CASES : !n. n MOD 2 = if EVEN n then 0 else 1 | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_2_CASES | null |
EVEN_MOD_EVEN : !m n. EVEN n ==> (EVEN(m MOD n) <=> EVEN m) | theorem | core | [
"recursion.ml"
] | arith.ml | EVEN_MOD_EVEN | null |
ODD_MOD_EVEN : !m n. EVEN n ==> (ODD(m MOD n) <=> ODD m) | theorem | core | [
"recursion.ml"
] | arith.ml | ODD_MOD_EVEN | null |
HALF_DOUBLE : (!n. (2 * n) DIV 2 = n) /\ (!n. (n * 2) DIV 2 = n) | theorem | core | [
"recursion.ml"
] | arith.ml | HALF_DOUBLE | null |
DOUBLE_HALF : (!n. EVEN n ==> 2 * n DIV 2 = n) /\ (!n. EVEN n ==> n DIV 2 * 2 = n) | theorem | core | [
"recursion.ml"
] | arith.ml | DOUBLE_HALF | null |
MOD_MULT_RMOD : !m n p. (m * (p MOD n)) MOD n = (m * p) MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MULT_RMOD | null |
MOD_MULT_LMOD : !m n p. ((m MOD n) * p) MOD n = (m * p) MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MULT_LMOD | null |
MOD_MULT_MOD2 : !m n p. ((m MOD n) * (p MOD n)) MOD n = (m * p) MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_MULT_MOD2 | null |
MOD_EXP_MOD : !m n p. ((m MOD n) EXP p) MOD n = (m EXP p) MOD n | theorem | core | [
"recursion.ml"
] | arith.ml | MOD_EXP_MOD | null |
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