hcju/holps huggingface dataset:
Collect_geom.DELTA_X34
|- (!x12. (!x13. (!x14. (!x23. (!x24. (!x34. (((((((delta x12) x13) x14) x23) x24) x34) = ((real_add ((real_mul (real_neg x12)) ((real_pow x34) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul ((real_add ((real_mul (real_neg x13)) x14)) ((real_add ((real_mul (real_neg x23)) x24)) ((real_add ((real_mul x13) x24)) ((real_add ((real_mul x14) x23)) ((real_add ((real_mul (real_neg x12)) x12)) ((real_add ((real_mul x12) x14)) ((real_add ((real_mul x12) x24)) ((real_add ((real_mul x12) x13)) ((real_mul x12) x23)))))))))) x34)) ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24)))))))))))))))))))))
PATH_COMPONENT_IN
|- (!s. (!x. (!y. ((((path_component s) x) y) ==> ((x IN s) /\ (y IN s))))))
Seq.rotr
|- (!n. (!s. (((rotr n) s) = ((rot ((sizel s) - n)) s))))
CARD_LE_CONNECTED_COMPONENTS
|- (!f. (!s. (((continuous_on f) s) ==> ((GSPEC (\GEN%PVAR%1819. (?y. (((SETSPEC GEN%PVAR%1819) (y IN ((IMAGE f) s))) ((connected_component ((IMAGE f) s)) y))))) <=_c (GSPEC (\GEN%PVAR%1820. (?x. (((SETSPEC GEN%PVAR%1820) (x IN s)) ((connected_component s) x)))))))))
REAL_LE_MUL2
|- (!w. (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) w) /\ (((real_le w) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ ((real_le y) z)))) ==> ((real_le ((real_mul w) y)) ((real_mul x) z)))))))
HOMEOMORPHIC_TRANSLATION_LEFT_EQ
|- (!a. (!s. (!t. (((homeomorphic ((IMAGE (\x. ((vector_add a) x))) s)) t) = ((homeomorphic s) t)))))
rcone_gt
|- (rcone_gt = (rconesgn real_gt))
SPAN_LINEAR_IMAGE
|- (!f. (!s. ((linear f) ==> ((span ((IMAGE f) s)) = ((IMAGE f) (span s))))))
LIFT_NUM
|- (!n. ((lift (real_of_num n)) = (vec n)))
HAS_INTEGRAL_INTEGRAL
|- (!f. (!s. (((integrable_on f) s) = (((has_integral f) ((integral s) f)) s))))
MULT_EQ_0
|- (!m. (!n. (((m * n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) \/ (n = (NUMERAL _0))))))
HAS_REAL_DERIVATIVE_LMUL_ATREAL
|- (!f. (!f'. (!c. (!x. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative (\x. ((real_mul c) (f x)))) ((real_mul c) f')) (atreal x)))))))
REAL_COMPACT_CONTINUOUS_IMAGE
|- (!f. (!s. ((((real_continuous_on f) s) /\ (real_compact s)) ==> (real_compact ((IMAGE f) s)))))
INTER_COMM
|- (!s. (!t. ((s INTER t) = (t INTER s))))
Ssrnat.addn_eq0
|- (!m. (!n. (((m + n) = (NUMERAL _0)) = ((m = (NUMERAL _0)) /\ (n = (NUMERAL _0))))))
LOCALLY_PATH_CONNECTED_SPHERE
|- (!a. (!r. ((locally path_connected) (sphere (a , r)))))
Fnjlbxs.list_of_vector
|- (!v. (!d. (!n. ((((list_of_vector d) n) v) = ((TABLE (\k. (lambdai. (v $ ((k * d) + i))))) n)))))
COMPLEX_SUB_REFL
|- (!x. (((vector_sub x) x) = (Cx (real_of_num (NUMERAL _0)))))
LT_SUC_LE
|- (!m. (!n. ((m < (SUC n)) = (m <= n))))
FRONTIER_SUBSET_EQ
|- (!s. (((frontier s) SUBSET s) = (closed s)))
Hypermap_iso.iso_face_set
|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> ((face_set G) = ((IMAGE (IMAGE f)) (face_set H)))))))
MEASURABLE_SCALING
|- (!s. (!c. ((measurable s) ==> (measurable ((IMAGE (\x. ((% c) x))) s)))))
SPAN_MONO
|- (!s. (!t. ((s SUBSET t) ==> ((span s) SUBSET (span t)))))
More_list_hypermap.node_hypermap_of_list_explicit
|- (!L. (!n. (!d. (((good_list_nodes L) /\ (((MEM n) (list_of_nodes L)) /\ ((MEM d) n))) ==> (((node (hypermap_of_list L)) d) = (set_of_list n))))))
CLOSED_UNION
|- (!s. (!t. (((closed s) /\ (closed t)) ==> (closed (s UNION t)))))
IMAGE_CLAUSES
|- ((((IMAGE f) EMPTY) = EMPTY) /\ (((IMAGE f) (x INSERT s)) = ((f x) INSERT ((IMAGE f) s))))
HAS_REAL_DERIVATIVE_ID
|- (!net. (((has_real_derivative (\x. x)) (real_of_num (NUMERAL (BIT1 _0)))) net))
BOUNDED_UNION
|- (!s. (!t. ((bounded (s UNION t)) = ((bounded s) /\ (bounded t)))))
AFFINE_SCALING
|- (!s. (!c. ((affine s) ==> (affine ((IMAGE (\x. ((% c) x))) s)))))
CARD_LT_IMP_LE
|- (!s. (!t. ((s <_c t) ==> (s <=_c t))))
Ssrbool.contraLR
|- (!c. (!b. (((~ c) ==> (~ b)) ==> (b ==> c))))
JOINPATHS_LINEAR_IMAGE
|- (!f. (!g1. (!g2. ((linear f) ==> (((++ (f o g1)) (f o g2)) = (f o ((++ g1) g2)))))))
AFFINE_DEPENDENT_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((affine_dependent ((IMAGE f) s)) = (affine_dependent s)))))
MULT_AC
|- (((m * n) = (n * m)) /\ ((((m * n) * p) = (m * (n * p))) /\ ((m * (n * p)) = (n * (m * p)))))
ORTHOGONAL_TRANSFORMATION_NEG
|- (!f. ((orthogonal_transformation (\x. (vector_neg (f x)))) = (orthogonal_transformation f)))
RESTRICTION_IN_EXTENSIONAL
|- (!s. (!f. (((RESTRICTION s) f) IN (EXTENSIONAL s))))
NOT_EVEN
|- (!n. ((~ (EVEN n)) = (ODD n)))
between
|- (!a. (!x. (!b. (((between x) (a , b)) = ((distance (a , b)) = ((real_add (distance (a , x))) (distance (x , b))))))))
PATHFINISH_TRANSLATION
|- (!a. (!g. ((pathfinish ((\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathfinish g)))))
CARD_EXP_POWERSET
|- (!s. (((^_c UNIV) s) =_c (GSPEC (\GEN%PVAR%40. (?t. (((SETSPEC GEN%PVAR%40) (t SUBSET s)) t))))))
RPOW_POW
|- (!x. (!n. (((rpow x) (real_of_num n)) = ((real_pow x) n))))
Tame_general.CONTRAVENING_FAN
|- (!V. ((contravening V) ==> (FAN ((vec (NUMERAL _0)) , (V , (ESTD V))))))
VECTOR_MUL_COMPONENT
|- (!c. (!x. (!i. ((((% c) x) $ i) = ((real_mul c) (x $ i))))))
ABSOLUTELY_REAL_INTEGRABLE_INCREASING_PRODUCT
|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_real_interval ((CONS (a , b)) NIL))) /\ ((real_le x) y))) ==> ((real_le (f x)) (f y))))) /\ ((absolutely_real_integrable_on g) (closed_real_interval ((CONS (a , b)) NIL)))) ==> ((absolutely_real_integrable_on (\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))))))))
MEMBER_NOT_EMPTY
|- (!s. ((?x. (x IN s)) = (~ (s = EMPTY))))
LAMBDA_ETA
|- (!g. ((lambdai. (g $ i)) = g))
CARD_LE_INJ
|- (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ ((CARD s) <= (CARD t)))) ==> (?f. ((((IMAGE f) s) SUBSET t) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))
CARD_EQ_CONG
|- (!s. (!s'. (!t. (!t'. (((s =_c s') /\ (t =_c t')) ==> ((s =_c t) = (s' =_c t')))))))
midpoint
|- (!a. (!b. ((midpoint (a , b)) = ((% (real_inv (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((vector_add a) b)))))
CONTINUOUS_ON_COMPOSE
|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) ((IMAGE f) s))) ==> ((continuous_on (g o f)) s)))))
SYMMETRIC_MATRIX_INV
|- (!A. (((transp (matrix_inv A)) = (matrix_inv A)) = ((transp A) = A)))
AFFINE_PCROSS
|- (!s. (!t. (((affine s) /\ (affine t)) ==> (affine (s PCROSS t)))))
Nonlin_def.delta_x4
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x4 x1) x2) x3) x4) x5) x6) = ((real_add ((real_sub ((real_mul (real_neg x2)) x3)) ((real_mul x1) x4))) ((real_add ((real_mul x2) x5)) ((real_add ((real_sub ((real_mul x3) x6)) ((real_mul x5) x6))) ((real_mul x1) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))
HAS_BOUNDED_VARIATION_ON_COMPONENTWISE
|- (!f. (!s. (((has_bounded_variation_on f) s) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((has_bounded_variation_on (\x. (lift ((f x) $ i)))) s))))))
CONTINUOUS_ON_CX_LIFT
|- (!f. (!s. (((continuous_on (\x. (Cx (f x)))) s) = ((continuous_on (\x. (lift (f x)))) s))))
HOMOTOPIC_PATHS_IMP_PATH
|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> ((path p) /\ (path q))))))
CONVEX_HALFSPACE_LE
|- (!a. (!b. (convex (GSPEC (\GEN%PVAR%1428. (?x. (((SETSPEC GEN%PVAR%1428) ((real_le ((dot a) x)) b)) x)))))))
ORTHOGONAL_MUL
|- ((!a. (!x. (!y. (((orthogonal ((% a) x)) y) = ((a = (real_of_num (NUMERAL _0))) \/ ((orthogonal x) y)))))) /\ (!a. (!x. (!y. (((orthogonal x) ((% a) y)) = ((a = (real_of_num (NUMERAL _0))) \/ ((orthogonal x) y)))))))
Seq.rot_add_mod
|- (!m. (!n. (!s. ((n <= (sizel s)) ==> ((m <= (sizel s)) ==> (((rot m) ((rot n) s)) = ((rot (((COND ((m + n) <= (sizel s))) (m + n)) ((m + n) - (sizel s)))) s)))))))
REAL_MUL_RINV
|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul x) (real_inv x)) = (real_of_num (NUMERAL (BIT1 _0))))))
UNION_COMM
|- (!s. (!t. ((s UNION t) = (t UNION s))))
REAL_OF_NUM_EQ
|- (!m. (!n. (((real_of_num m) = (real_of_num n)) = (m = n))))
CONTINUOUS_ADD
|- (!f. (!g. (!net. ((((continuous f) net) /\ ((continuous g) net)) ==> ((continuous (\x. ((vector_add (f x)) (g x)))) net)))))
Ssrbool.predU
|- (!p1. (!p2. (((predU p1) p2) = (\x. ((p1 x) \/ (p2 x))))))
HOMOTOPIC_PATHS_JOIN
|- (!s. (!p. (!q. (!p'. (!q'. (((((homotopic_paths s) p) p') /\ ((((homotopic_paths s) q) q') /\ ((pathfinish p) = (pathstart q)))) ==> (((homotopic_paths s) ((++ p) q)) ((++ p') q'))))))))
REAL_SUM_INTEGRAL_LBOUND_INCREASING
|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL))) ==> (((has_real_derivative g) (f x)) ((within (atreal x)) (closed_real_interval ((CONS (((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0)))) , (real_of_num n))) NIL)))))) /\ (!x. (!y. ((((real_le ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) (real_of_num n)))) ==> ((real_le (f x)) (f y))))))) ==> ((real_le ((real_sub (g (real_of_num n))) (g ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))) ((sum (m .. n)) (\k. (f (real_of_num k))))))))))
DIMINDEX_NONZERO
|- (!s. (~ ((dimindex s) = (NUMERAL _0))))
Seq2.butlast
|- (!h. (!h2. (!t. (((butlast NIL) = NIL) /\ (((butlast ((CONS h) NIL)) = NIL) /\ ((butlast ((CONS h) ((CONS h2) t))) = ((CONS h) (butlast ((CONS h2) t)))))))))
Ssrnat.leq_eqVlt
|- (!m. (!n. ((m <= n) = ((m = n) \/ (m < n)))))
FINITE_NUMSEG
|- (!m. (!n. (FINITE (m .. n))))
Ssrnat.sqrn_sub
|- (!m. (!n. ((n <= m) ==> (((m - n) EXP (NUMERAL (BIT0 (BIT1 _0)))) = (((m EXP (NUMERAL (BIT0 (BIT1 _0)))) + (n EXP (NUMERAL (BIT0 (BIT1 _0))))) - ((NUMERAL (BIT0 (BIT1 _0))) * (m * n)))))))
FORALL_COMPLEX
|- ((!z. (P z)) = (!x. (!y. (P (complex (x , y))))))
Add_triangle.fan_split_fan_face
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> ((FAN ((vec (NUMERAL _0)) , ((split_fan_face (V , E)) (v , w)))) /\ (((dart_of_fan ((split_fan_face (V , E)) (v , w))) = ((dart1_of_fan (V , E)) UNION (((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , w) INSERT ((w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) INSERT EMPTY)))) /\ (fully_surrounded ((split_fan_face (V , E)) (v , w))))))))))))
AFFINE_HULL_EQ_EMPTY
|- (!s. ((((hull affine) s) = EMPTY) = (s = EMPTY)))
CEXP_CLOG
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((cexp (clog z)) = z)))
Collect_geom.DELTA_RRR_INTERPRETE
|- (((((((delta r) r) r) a) b) c) = ((real_add ((real_mul (real_neg a)) ((real_mul b) c))) ((real_mul r) (((ups_x a) b) c))))
MEASURE_TRANSLATION
|- (!a. (!s. ((measure ((IMAGE (\x. ((vector_add a) x))) s)) = (measure s))))
REAL_INV_0
|- ((real_inv (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
MBOUNDED
|- (!m. (!s. (((mbounded m) s) = ((s = EMPTY) \/ ((!x. ((x IN s) ==> (x IN (mspace m)))) /\ (?c. (?b. ((c IN (mspace m)) /\ (!x. ((x IN s) ==> ((real_le ((mdist m) (c , x))) b)))))))))))
LIM_TRANSFORM_WITHIN_OPEN
|- (!f. (!g. (!s. (!a. (!l. (((open s) /\ ((a IN s) /\ ((!x. (((x IN s) /\ (~ (x = a))) ==> ((f x) = (g x)))) /\ (((--> f) l) (at a))))) ==> (((--> g) l) (at a))))))))
ADD_0
|- (!m. ((m + (NUMERAL _0)) = m))
Hypermap.face_refl
|- (!H. (!x. (x IN ((face H) x))))
SUPPORTING_HYPERPLANE_FRONTIER
|- (!s. (!x. (((convex s) /\ (x IN (frontier s))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ (!y. ((y IN (closure s)) ==> ((real_le ((dot a) x)) ((dot a) y)))))))))
LIM_WITHIN
|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (at a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d))) ==> ((real_lt (distance ((f x) , l))) e))))))))))))
PASTECART_ADD
|- (!x1. (!y1. (!x2. (!y2. (((vector_add ((pastecart x1) y1)) ((pastecart x2) y2)) = ((pastecart ((vector_add x1) x2)) ((vector_add y1) y2)))))))
RIGHT_FORALL_OR_THM
|- (!P. (!Q. ((!x. (P \/ (Q x))) = (P \/ (!x. (Q x))))))
VECTOR_ADD_AC
|- ((((vector_add m) n) = ((vector_add n) m)) /\ ((((vector_add ((vector_add m) n)) p) = ((vector_add m) ((vector_add n) p))) /\ (((vector_add m) ((vector_add n) p)) = ((vector_add n) ((vector_add m) p)))))
SETDIST_POS_LE
|- (!s. (!t. ((real_le (real_of_num (NUMERAL _0))) (setdist (s , t)))))
SERIES_DROP_LE
|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\ ((((sums g) b) s) /\ (!x. ((x IN s) ==> ((real_le (drop (f x))) (drop (g x))))))) ==> ((real_le (drop a)) (drop b))))))))
IN_INTERVAL
|- ((!x. ((x IN (open_interval (a , b))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((real_lt (a $ i)) (x $ i)) /\ ((real_lt (x $ i)) (b $ i))))))) /\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((real_le (a $ i)) (x $ i)) /\ ((real_le (x $ i)) (b $ i))))))))
has_derivative
|- (!f. (!f'. (!net. ((((has_derivative f) f') net) = ((linear f') /\ (((--> (\y. ((% (real_inv (vector_norm ((vector_sub y) (netlimit net))))) ((vector_sub (f y)) ((vector_add (f (netlimit net))) (f' ((vector_sub y) (netlimit net)))))))) (vec (NUMERAL _0))) net))))))
INTEGER_SUB
|- (!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_sub x) y)))))
rconesgn
|- (!sgn. (!w. (!v. (!h. (((((rconesgn sgn) v) w) h) = (GSPEC (\GEN%PVAR%5166. (?x. (((SETSPEC GEN%PVAR%5166) ((sgn ((dot ((vector_sub x) v)) ((vector_sub w) v))) ((real_mul (distance (x , v))) ((real_mul (distance (w , v))) h)))) x)))))))))
OPERATIVE_FUNCTION_ENDPOINT_DIFF
|- (!f. ((operative vector_add) (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))))
Kepler_spec.atn_1_series
|- (!n. ((real_le (real_abs ((real_sub ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) ((sum ((NUMERAL _0) .. n)) (\i. ((real_div ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) i)) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 (BIT1 _0))))))))
REAL_INTEGRABLE_SPIKE
|- (!f. (!g. (!s. (!t. (((real_negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((real_integrable_on f) t) ==> ((real_integrable_on g) t)))))))
Seq2.mem_delete1_uniq
|- (!x. (!s. (((uniq s) /\ ((MEM x) s)) ==> (!y. (((MEM y) ((delete1 x) s)) = ((~ (y = x)) /\ ((MEM y) s)))))))
Ssrbool.andbA
|- (!b. (!c. (!p. ((b /\ (c /\ p)) = ((b /\ c) /\ p)))))
Ssrnat.subn_eq0
|- (!m. (!n. (((m - n) = (NUMERAL _0)) = (m <= n))))
REAL_LE_LSQRT
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ ((real_le x) ((real_pow y) (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((real_le (sqrt x)) y))))
REAL_NEG_ADD
|- (!x. (!y. ((real_neg ((real_add x) y)) = ((real_add (real_neg x)) (real_neg y)))))
INTEGRAL_COMBINE
|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ (((real_le (drop c)) (drop b)) /\ ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((vector_add ((integral (closed_interval ((CONS (a , c)) NIL))) f)) ((integral (closed_interval ((CONS (c , b)) NIL))) f)) = ((integral (closed_interval ((CONS (a , b)) NIL))) f)))))))
Pack2.PACKING
|- (!s. ((packing s) = (!u. (!v. (((u IN s) /\ ((v IN s) /\ ((real_lt (distance (u , v))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> (u = v))))))
COMPLEX_CMUL
|- (!c. (!x. (((% c) x) = ((complex_mul (Cx c)) x))))
Kepler_spec.abs_spec
|- (!x. ((((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_abs x) = x)) /\ (((real_lt x) (real_of_num (NUMERAL _0))) ==> ((real_abs x) = (real_neg x)))))
VECTOR_VARIATION_COMBINE
|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ (((real_le (drop c)) (drop b)) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> (((real_add ((vector_variation (closed_interval ((CONS (a , c)) NIL))) f)) ((vector_variation (closed_interval ((CONS (c , b)) NIL))) f)) = ((vector_variation (closed_interval ((CONS (a , b)) NIL))) f)))))))
OUTSIDE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((outside ((IMAGE f) s)) = ((IMAGE f) (outside s))))))
PERMUTATION_PERMUTES
|- (!p. ((permutation p) = (?s. ((FINITE s) /\ ((permutes p) s)))))
Appendix.scs_b_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_b_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = b))))))))))
Functional_equation.sub6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
PATH_CONNECTED_CONTINUOUS_IMAGE
|- (!f. (!s. ((((continuous_on f) s) /\ (path_connected s)) ==> (path_connected ((IMAGE f) s)))))
FSIGMA_PCROSS
|- (!s. (!t. (((fsigma s) /\ (fsigma t)) ==> (fsigma (s PCROSS t)))))
REAL_EXP_MONO_LE
|- (!x. (!y. (((real_le (exp x)) (exp y)) = ((real_le x) y))))
TAN_PERIODIC_PI
|- (!x. ((tan ((real_add x) pi)) = (tan x)))
HAS_BOUNDED_REAL_VARIATION_REFLECT_EQ
|- (!f. (!s. (((has_bounded_real_variation_on (\x. (f (real_neg x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE real_neg) s)))))
MDIST
|- (!s. (!d. ((is_metric_space (s , d)) ==> ((mdist (metric (s , d))) = d))))
CONTINUOUS_ON_LIFT_DOT
|- (!s. ((continuous_on (lift o (\y. ((dot a) y)))) s))
Fan.POWER_0
|- (!f. (((POWER f) (NUMERAL _0)) = I))
COMPLEX_DIFFERENTIABLE_CPOW_RIGHT
|- (!w. (!z. ((complex_differentiable (\z. ((cpow w) z))) (at z))))
ENDS_IN_REAL_INTERVAL
|- ((!a. (!b. ((a IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. ((b IN (closed_real_interval ((CONS (a , b)) NIL))) = (~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. (~ (a IN (open_real_interval (a , b)))))) /\ (!a. (!b. (~ (b IN (open_real_interval (a , b)))))))))
Add_triangle.fully_surrounded_dart_of_fan_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((dart_of_fan (V , E)) = (dart1_of_fan (V , E)))))))
LIMPT_APPROACHABLE
|- (!x. (!s. (((limit_point_of x) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?x'. ((x' IN s) /\ ((~ (x' = x)) /\ ((real_lt (distance (x' , x))) e)))))))))
URYSOHN_LOCAL
|- (!s. (!t. (!u. (!a. (!b. ((((closed_in ((subtopology euclidean) u)) s) /\ (((closed_in ((subtopology euclidean) u)) t) /\ ((s INTER t) = EMPTY))) ==> (?f. (((continuous_on f) u) /\ ((!x. ((x IN u) ==> ((f x) IN (closed_segment ((CONS (a , b)) NIL))))) /\ ((!x. ((x IN s) ==> ((f x) = a))) /\ (!x. ((x IN t) ==> ((f x) = b)))))))))))))
REAL_LT_SUB_RADD
|- (!x. (!y. (!z. (((real_lt ((real_sub x) y)) z) = ((real_lt x) ((real_add z) y))))))
inside
|- (!s. ((inside s) = (GSPEC (\GEN%PVAR%2006. (?x. (((SETSPEC GEN%PVAR%2006) ((~ (x IN s)) /\ (bounded ((connected_component (UNIV DIFF s)) x)))) x))))))
COMPONENT_LE_NORM
|- (!x. (!i. ((real_le (real_abs (x $ i))) (vector_norm x))))
OPEN_IN_PCROSS_EQ
|- (!s. (!s'. (!t. (!t'. (((open_in ((subtopology euclidean) (s PCROSS t))) (s' PCROSS t')) = ((s' = EMPTY) \/ ((t' = EMPTY) \/ (((open_in ((subtopology euclidean) s)) s') /\ ((open_in ((subtopology euclidean) t)) t')))))))))
ABSOLUTELY_SETCONTINUOUS_COMPARISON
|- (!f. (!g. (!s. ((((absolutely_setcontinuous_on f) s) /\ (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((real_le (vector_norm (g (closed_interval ((CONS (a , b)) NIL))))) (vector_norm (f (closed_interval ((CONS (a , b)) NIL))))))))) ==> ((absolutely_setcontinuous_on g) s)))))
Seq.belast
|- (!x. (!h. (!t. ((((belast x) NIL) = NIL) /\ (((belast x) ((CONS h) t)) = ((CONS x) ((belast h) t)))))))
SUM_0
|- (!s. (((sum s) (\n. (real_of_num (NUMERAL _0)))) = (real_of_num (NUMERAL _0))))
AGM_2
|- (!x. (!y. (!u. (!v. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le ((real_mul ((rpow x) u)) ((rpow y) v))) ((real_add ((real_mul u) x)) ((real_mul v) y))))))))
ALL2
|- (((((ALL2 P) NIL) NIL) = T) /\ (((((ALL2 P) ((CONS h1) t1)) NIL) = F) /\ (((((ALL2 P) NIL) ((CONS h2) t2)) = F) /\ ((((ALL2 P) ((CONS h1) t1)) ((CONS h2) t2)) = (((P h1) h2) /\ (((ALL2 P) t1) t2))))))
RPOW_LE2
|- (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) x) /\ (((real_le x) y) /\ ((real_le (real_of_num (NUMERAL _0))) z))) ==> ((real_le ((rpow x) z)) ((rpow y) z))))))
CONIC_HULL_EXPLICIT
|- (!s. (((hull conic) s) = (GSPEC (\GEN%PVAR%1457. (?c. (?x. (((SETSPEC GEN%PVAR%1457) (((real_le (real_of_num (NUMERAL _0))) c) /\ (x IN s))) ((% c) x))))))))
Ssrnat.add0n
|- (!n. (((NUMERAL _0) + n) = n))
BALL_SUBSET_CBALL
|- (!x. (!e. ((ball (x , e)) SUBSET (cball (x , e)))))
ACS_COS
|- (!x. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le x) pi)) ==> ((acs (cos x)) = x)))
LENGTH_LIST_OF_SET
|- (!s. ((FINITE s) ==> ((LENGTH (list_of_set s)) = (CARD s))))
CARD_EXP_ABSORB
|- (!s. (!t. (((INFINITE t) /\ ((UNIV <=_c s) /\ (s <=_c ((^_c UNIV) t)))) ==> (((^_c s) t) =_c ((^_c UNIV) t)))))
HAS_REAL_INTEGRAL_COMBINE
|- (!f. (!i. (!j. (!a. (!b. (!c. ((((real_le a) c) /\ (((real_le c) b) /\ ((((has_real_integral f) i) (closed_real_interval ((CONS (a , c)) NIL))) /\ (((has_real_integral f) j) (closed_real_interval ((CONS (c , b)) NIL)))))) ==> (((has_real_integral f) ((real_add i) j)) (closed_real_interval ((CONS (a , b)) NIL))))))))))
BOUNDED_INSERT
|- (!x. (!s. ((bounded (x INSERT s)) = (bounded s))))
CARD_LE_LT
|- (!s. (!t. ((s <=_c t) = ((s <_c t) \/ (s =_c t)))))
ABSOLUTELY_INTEGRABLE_CONVOLUTION_L1_LINF
|- (!bop. (!f. (!g. (!x. (((bilinear bop) /\ (((absolutely_integrable_on f) UNIV) /\ (((measurable_on g) UNIV) /\ (bounded ((IMAGE g) UNIV))))) ==> ((absolutely_integrable_on (\y. ((bop (f ((vector_sub x) y))) (g y)))) UNIV))))))
CONTINUOUS_COUNTABLE_RANGE_CONSTANT_EQ
|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\ (COUNTABLE ((IMAGE f) s))) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))
REAL_EQ_LDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> ((((real_div x) z) = y) = (x = ((real_mul y) z)))))))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT5
|- (!x. (!V. (!E. (!v. (!u. (!w. (!w1. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((fan80 (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ ((~ (w1 = (((((inverse1_sigma_fan x) V) E) w) u))) /\ (((w INSERT (w1 INSERT EMPTY)) IN E) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))))) ==> ((((((sigma_fan x) V) E1) w) w1) = (((((sigma_fan x) V) E) w) w1))))))))))
DIAMETER_EMPTY
|- ((diameter EMPTY) = (real_of_num (NUMERAL _0)))
vector_add
|- (!x. (!y. (((vector_add x) y) = (lambdai. ((real_add (x $ i)) (y $ i))))))
AFFINE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((affine ((IMAGE f) s)) = (affine s)))))
DOT_RZERO
|- (!x. (((dot x) (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0))))
Hypermap_iso.ext_dart
|- (!V. (!E. (!v. (!w. (((ext_dart (V , E)) (v , w)) = ((vec (NUMERAL _0)) , (v , (w , (((((extension_sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))
CLOSURE_UNIONS
|- (!f. ((FINITE f) ==> ((closure (UNIONS f)) = (UNIONS (GSPEC (\GEN%PVAR%573. (?s. (((SETSPEC GEN%PVAR%573) (s IN f)) (closure s)))))))))
REAL_CONTINUOUS_WITHIN
|- (!f. (!x. (!s. (((real_continuous f) ((within (at x)) s)) = (((---> f) (f x)) ((within (at x)) s))))))