hcju/holps huggingface dataset:
INCREASING_RIGHT_LIMIT_1
|- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (?l. (((--> f) l) ((within (at c)) (closed_interval ((CONS (c , b)) NIL))))))))))
ARITH_EXP
|- ((!m. (!n. (((NUMERAL m) EXP (NUMERAL n)) = (NUMERAL (m EXP n))))) /\ (((_0 EXP _0) = (BIT1 _0)) /\ ((!m. (((BIT0 m) EXP _0) = (BIT1 _0))) /\ ((!m. (((BIT1 m) EXP _0) = (BIT1 _0))) /\ ((!n. ((_0 EXP (BIT0 n)) = ((_0 EXP n) * (_0 EXP n)))) /\ ((!m. (!n. (((BIT0 m) EXP (BIT0 n)) = (((BIT0 m) EXP n) * ((BIT0 m) EXP n))))) /\ ((!m. (!n. (((BIT1 m) EXP (BIT0 n)) = (((BIT1 m) EXP n) * ((BIT1 m) EXP n))))) /\ ((!n. ((_0 EXP (BIT1 n)) = _0)) /\ ((!m. (!n. (((BIT0 m) EXP (BIT1 n)) = ((BIT0 m) * (((BIT0 m) EXP n) * ((BIT0 m) EXP n)))))) /\ (!m. (!n. (((BIT1 m) EXP (BIT1 n)) = ((BIT1 m) * (((BIT1 m) EXP n) * ((BIT1 m) EXP n)))))))))))))))
ENR_IMP_ANR
|- (!s. ((ENR s) ==> (ANR s)))
Dih2k_hypermap.vecmat
|- (!f. (!j. (((vecmat j) f) = (lambdai. (f $ ((j * (dimindex UNIV)) + i))))))
Lp_ineqs_proofs.add_diag
|- (!V. (!v. (!w. (!E. (((add_diag (V , E)) (v , w)) = (((COND ((v , w) IN (dart1_of_fan (V , E)))) ((w INSERT ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) INSERT EMPTY)) INSERT E)) E))))))
Seq.perm_eq_mem
|- (!s1. (!s2. (((perm_eq s1) s2) ==> (!x. (((MEM x) s1) = ((MEM x) s2))))))
Hypermap.number_of_nodes
|- (!H. ((number_of_nodes H) = (CARD (node_set H))))
Pack2.BOUNDED_VORONOI_CLOSED
|- (!s. (!v. ((saturated s) ==> (bounded ((voronoi_closed s) v)))))
Hypermap.FINITE_HYPERMAP_ORBITS
|- (!H. ((FINITE (edge_set H)) /\ ((FINITE (node_set H)) /\ (FINITE (face_set H)))))
Hypermap.is_node_nondegenerate
|- (!H. ((is_node_nondegenerate H) = (!x. ((x IN (dart H)) ==> (~ (((node_map H) x) = x))))))
FRECHET_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL
|- (!f. (!f'. (!f''. (!x. (!a. (!b. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (a $ i)) (b $ i)))) /\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((((has_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\ (((has_derivative f) f'') ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (f' = f''))))))))
OPEN_COMPONENTS
|- (!u. (!s. (((open u) /\ (s IN (components u))) ==> (open s))))
QUOTIENT_MAP_FROM_COMPOSITION
|- (!f. (!g. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%667. (?x. (((SETSPEC GEN%PVAR%667) ((x IN s) /\ (((g o f) x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v)))))))) ==> (!v. ((v SUBSET u) ==> (((open_in ((subtopology euclidean) t)) (GSPEC (\GEN%PVAR%668. (?x. (((SETSPEC GEN%PVAR%668) ((x IN t) /\ ((g x) IN v))) x))))) = ((open_in ((subtopology euclidean) u)) v))))))))))
OUTSIDE_TRANSLATION
|- (!a. (!s. ((outside ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (outside s)))))
OPEN_IMP_LOCALLY_COMPACT
|- (!s. ((open s) ==> ((locally compact) s)))
TC_CASES
|- (!R. (!a0. (!a1. ((((TC R) a0) a1) = (((R a0) a1) \/ (?y. ((((TC R) a0) y) /\ (((TC R) y) a1))))))))
LIM_ZERO_NEGINFINITY
|- (!f. (!l. ((((--> (\x. (f ((real_div (real_of_num (NUMERAL (BIT1 _0)))) x)))) l) (atreal (real_of_num (NUMERAL _0)))) ==> (((--> f) l) at_neginfinity))))
NEGLIGIBLE_ON_INTERVALS
|- (!s. ((negligible s) = (!a. (!b. (negligible (s INTER (closed_interval ((CONS (a , b)) NIL))))))))
DIFFERENTIABLE_NEG
|- (!f. (!net. (((differentiable f) net) ==> ((differentiable (\z. (vector_neg (f z)))) net))))
INTERIOR_SUBSET_RELATIVE_INTERIOR
|- (!s. ((interior s) SUBSET (relative_interior s)))
REAL_CONVEX_ON
|- (!f. (!s. (((real_convex_on f) s) = ((convex_on (f o drop)) ((IMAGE lift) s)))))
Fan.pr2
|- (pr2 = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) v))))))))
SING_SUBSET
|- (!s. (!x. (((x INSERT EMPTY) SUBSET s) = (x IN s))))
REAL_MIN_MIN
|- (!x. (!y. (((real_le ((real_min x) y)) x) /\ ((real_le ((real_min x) y)) y))))
Nonlin_def.sol_euler_x_div_sqrtdelta
|- (!x4. (!x5. (!x3. (!x1. (!x2. (!x6. (((((((sol_euler_x_div_sqrtdelta x1) x2) x3) x4) x5) x6) = ((LET (\a. (LET_END ((real_div (matan ((real_div ((((((delta_x x1) x2) x3) x4) x5) x6)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_pow a) (NUMERAL (BIT0 (BIT1 _0)))))))) a)))) ((real_add (sqrt ((real_mul x1) ((real_mul x2) x3)))) ((real_add ((real_mul (sqrt x1)) ((real_div ((real_add x2) ((real_sub x3) x4))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_add ((real_mul (sqrt x2)) ((real_div ((real_add x1) ((real_sub x3) x5))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul (sqrt x3)) ((real_div ((real_add x1) ((real_sub x2) x6))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))))))))))
INJECTIVE_ON_PREIMAGE
|- (!f. (!s. (!u. ((!t. (!t'. (((t SUBSET u) /\ ((t' SUBSET u) /\ ((GSPEC (\GEN%PVAR%142. (?x. (((SETSPEC GEN%PVAR%142) ((x IN s) /\ ((f x) IN t))) x)))) = (GSPEC (\GEN%PVAR%143. (?x. (((SETSPEC GEN%PVAR%143) ((x IN s) /\ ((f x) IN t'))) x))))))) ==> (t = t')))) = (u SUBSET ((IMAGE f) s))))))
Functional_equation.proj_y3
|- (!x1. (!x2. (!x4. (!x5. (!x6. (!x3. (((((((proj_y3 x1) x2) x3) x4) x5) x6) = (sqrt x3))))))))
MEASURABLE_ON_TRANSLATION
|- (!f. (!s. (!a. (((measurable_on f) ((IMAGE (\x. ((vector_add a) x))) s)) ==> ((measurable_on (\x. (f ((vector_add a) x)))) s)))))
RELATIVE_INTERIOR_EQ_EMPTY
|- (!s. ((convex s) ==> (((relative_interior s) = EMPTY) = (s = EMPTY))))
CLOSED_HALFSPACE_IM_GE
|- (!b. (closed (GSPEC (\GEN%PVAR%4756. (?z. (((SETSPEC GEN%PVAR%4756) ((real_ge (Im z)) b)) z))))))
Add_triangle.f_fan_pair3
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((((v , w) IN (dart1_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0))))) ==> ((((f_fan_pair (V , E)) (v , w)) = (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) /\ ((((f_fan_pair (V , E)) (w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))) = ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v)) /\ ((((f_fan_pair (V , E)) ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v)) = (v , w)) /\ ((w , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) IN (dart1_of_fan (V , E))))))))))))
ABSOLUTELY_INTEGRABLE_SPIKE
|- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((absolutely_integrable_on f) t) ==> ((absolutely_integrable_on g) t)))))))
PERMUTES_EMPTY
|- (!p. (((permutes p) EMPTY) = (p = I)))
ARITH_EVEN
|- ((!n. ((EVEN (NUMERAL n)) = (EVEN n))) /\ (((EVEN _0) = T) /\ ((!n. ((EVEN (BIT0 n)) = T)) /\ (!n. ((EVEN (BIT1 n)) = F)))))
List_hypermap.mem_list_of_darts
|- (!d. (!L. (((MEM d) (list_of_darts L)) = (?l. (((MEM l) L) /\ ((MEM d) (list_pairs l)))))))
Planarity.PROPERTIES_TRIANGLE_FAN
|- (!x. (!V. (!E. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ (((w INSERT (v 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))))) /\ (fan80 (x , (V , E))))))))) ==> (((((((sigma_fan x) V) E) v) u) = w) /\ ((((((sigma_fan x) V) E) w) v) = u)))))))))
ARC_TRANSLATION_EQ
|- (!a. (!g. ((arc ((\x. ((vector_add a) x)) o g)) = (arc g))))
Ssrnat.sqrn_add
|- (!m. (!n. (((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))))))
STARLIKE_IMP_PATH_CONNECTED
|- (!s. ((starlike s) ==> (path_connected s)))
Hypermap.COM_POWER
|- (!n. (!f. (((POWER f) (SUC n)) = (f o ((POWER f) n)))))
REAL_SUB_REFL
|- (!x. (((real_sub x) x) = (real_of_num (NUMERAL _0))))
Seq.size_drop
|- (!n0. (!s. ((sizel ((dropl n0) s)) = ((sizel s) - n0))))
IMAGE_ID
|- (!s. (((IMAGE (\x. x)) s) = s))
REAL_ROOT_POW
|- (!n. (!x. (((ODD n) \/ ((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) x))) ==> (((root n) ((real_pow x) n)) = x))))
HOMOTOPIC_WITH_COMPOSE
|- (!P. (!Q. (!R. (!f. (!f'. (!g. (!g'. (!s. (!t. (!u. (((!f. (!g. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((P f) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET u) /\ (Q g)))))) ==> (R (g o f))))) /\ (((((homotopic_with P) (s , t)) f) f') /\ ((((homotopic_with Q) (t , u)) g) g'))) ==> ((((homotopic_with R) (s , u)) (g o f)) (g' o f')))))))))))))
COMPLEX_ADD_SYM
|- (!x. (!y. (((vector_add x) y) = ((vector_add y) x))))
ETA_AX
|- (!t. ((\x. (t x)) = t))
HOMEOMORPHIC_COMPACTNESS
|- (!s. (!t. (((homeomorphic s) t) ==> ((compact s) = (compact t)))))
ANR_OPEN_IN
|- (!s. (!t. ((((open_in ((subtopology euclidean) t)) s) /\ (ANR t)) ==> (ANR s))))
HOMOTOPY_EQUIVALENT_EMPTY
|- ((!s. (((homotopy_equivalent s) EMPTY) = (s = EMPTY))) /\ (!t. (((homotopy_equivalent EMPTY) t) = (t = EMPTY))))
REAL_SECOND_MEAN_VALUE_THEOREM_FULL
|- (!f. (!g. (!a. (!b. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ (!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 (g x)) (g y))))))) ==> (?c. ((c IN (closed_real_interval ((CONS (a , b)) NIL))) /\ (((has_real_integral (\x. ((real_mul (g x)) (f x)))) ((real_add ((real_mul (g a)) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul (g b)) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))
HAS_REAL_DERIVATIVE_WITHIN_SUBSET
|- (!f. (!s. (!t. (!x. (((((has_real_derivative f) f') ((within (atreal x)) s)) /\ (t SUBSET s)) ==> (((has_real_derivative f) f') ((within (atreal x)) t)))))))
RPOW_ZERO
|- (!y. (((rpow (real_of_num (NUMERAL _0))) y) = (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))
Lp_main_estimate.v_prime_subset
|- (!V. (!X. (!f. ((V SUBSET X) ==> (((v_prime V) f) SUBSET X)))))
Hypermap.lemma_in_subset
|- (!s. (!t. (!x. (((s SUBSET t) /\ (x IN s)) ==> (x IN t)))))
Hypermap.iterate_map_valuation
|- (!p. (!n. (!x. ((p (((POWER p) n) x)) = (((POWER p) (SUC n)) x)))))
AR_FROM_UNION_AND_INTER
|- (!s. (!t. (((closed s) /\ ((closed t) /\ ((AR (s UNION t)) /\ (AR (s INTER t))))) ==> ((AR s) /\ (AR t)))))
Hexagons.LE_sqrt8_2h0
|- ((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) sqrt8)
PROPER_LOCAL_HOMEOMORPHISM_IMP_COVERING_MAP
|- (!p. (!c. (!s. (((((IMAGE p) c) = s) /\ ((!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2100. (?x. (((SETSPEC GEN%PVAR%2100) ((x IN c) /\ ((p x) IN k))) x))))))) /\ (!x. ((x IN c) ==> (?t. (?u. (?q. ((x IN t) /\ (((open_in ((subtopology euclidean) c)) t) /\ (((open_in ((subtopology euclidean) s)) u) /\ ((homeomorphism (t , u)) (p , q)))))))))))) ==> ((covering_space (c , p)) s)))))
ARITH_EQ
|- ((!m. (!n. (((NUMERAL m) = (NUMERAL n)) = (m = n)))) /\ (((_0 = _0) = T) /\ ((!n. (((BIT0 n) = _0) = (n = _0))) /\ ((!n. (((BIT1 n) = _0) = F)) /\ ((!n. ((_0 = (BIT0 n)) = (_0 = n))) /\ ((!n. ((_0 = (BIT1 n)) = F)) /\ ((!m. (!n. (((BIT0 m) = (BIT0 n)) = (m = n)))) /\ ((!m. (!n. (((BIT0 m) = (BIT1 n)) = F))) /\ ((!m. (!n. (((BIT1 m) = (BIT0 n)) = F))) /\ (!m. (!n. (((BIT1 m) = (BIT1 n)) = (m = n)))))))))))))
SUBTOPOLOGY_UNIV
|- (!top. (((subtopology top) UNIV) = top))
ANALYTIC_ON_ADD
|- (!f. (!g. (!s. ((((analytic_on f) s) /\ ((analytic_on g) s)) ==> ((analytic_on (\z. ((vector_add (f z)) (g z)))) s)))))
Nonlin_def.edge2_126_x
|- (!d. (!x4. (!x5. ((((edge2_126_x d) x4) x5) = (((((((compose6 edge2_flatD_x1) (constant6 d)) proj_x1) proj_x2) proj_x6) (constant6 x4)) (constant6 x5))))))
GE
|- (!n. (!m. ((m >= n) = (n <= m))))
LOCALLY_NEGLIGIBLE_ALT
|- (!s. ((negligible s) = (!x. ((x IN s) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((x IN u) /\ (negligible u))))))))
LINEAR_INJECTIVE_IMP_SURJECTIVE
|- (!f. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (!y. (?x. ((f x) = y)))))
CX_TAN
|- (!x. ((Cx (tan x)) = (ctan (Cx x))))
FORALL_IN_INSERT
|- (!P. (!a. (!s. ((!x. ((x IN (a INSERT s)) ==> (P x))) = ((P a) /\ (!x. ((x IN s) ==> (P x))))))))
ABSOLUTELY_INTEGRABLE_BOUNDED_MEASURABLE_PRODUCT
|- (!op. (!f. (!g. (!s. (((bilinear op) /\ (((measurable_on f) s) /\ ((bounded ((IMAGE f) s)) /\ ((absolutely_integrable_on g) s)))) ==> ((absolutely_integrable_on (\x. ((op (f x)) (g x)))) s))))))
CONTRACTIBLE_TRANSLATION
|- (!a. (!s. ((contractible ((IMAGE (\x. ((vector_add a) x))) s)) = (contractible s))))
REAL_LT_NEG2
|- (!x. (!y. (((real_lt (real_neg x)) (real_neg y)) = ((real_lt y) x))))
Sphere.quadratic_root_plus
|- (!b. (!c. (!a. ((quadratic_root_plus (a , (b , c))) = ((real_div ((real_add (real_neg b)) (sqrt ((real_sub ((real_pow b) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul a) c)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) a))))))
NEUTRAL_REAL_ADD
|- ((neutral real_add) = (real_of_num (NUMERAL _0)))
COMPLEX_STONE_WEIERSTRASS_ALT
|- (!P. (!s. (((compact s) /\ ((!c. (P (\x. c))) /\ ((!f. ((P f) ==> (P (\x. (cnj (f x)))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((vector_add (f x)) (g x))))))) /\ ((!f. (!g. (((P f) /\ (P g)) ==> (P (\x. ((complex_mul (f x)) (g x))))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> (?f. ((P f) /\ (((continuous_on f) s) /\ (~ ((f x) = (f y)))))))))))))) ==> (!f. (!e. ((((continuous_on f) s) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?g. ((P g) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))
LOCALLY_PATH_CONNECTED_QUOTIENT_IMAGE
|- (!f. (!s. (((!t. ((t SUBSET ((IMAGE f) s)) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1904. (?x. (((SETSPEC GEN%PVAR%1904) ((x IN s) /\ ((f x) IN t))) x))))) = ((open_in ((subtopology euclidean) ((IMAGE f) s))) t)))) /\ ((locally path_connected) s)) ==> ((locally path_connected) ((IMAGE f) s)))))
RTC_INDUCT
|- (!R. (!P. (((!x. (!y. (((R x) y) ==> ((P x) y)))) /\ ((!x. ((P x) x)) /\ (!x. (!y. (!z. ((((P x) y) /\ ((P y) z)) ==> ((P x) z))))))) ==> (!x. (!y. ((((RTC R) x) y) ==> ((P x) y)))))))
Kepler_spec.sum_spec
|- (!f. (!n. ((((sum ((NUMERAL _0) .. (NUMERAL _0))) f) = (f (NUMERAL _0))) /\ (((sum ((NUMERAL _0) .. (SUC n))) f) = ((real_add ((sum ((NUMERAL _0) .. n)) f)) (f (SUC n)))))))
PCROSS
|- (!s. (!t. ((s PCROSS t) = (GSPEC (\GEN%PVAR%239. (?x. (?y. (((SETSPEC GEN%PVAR%239) ((x IN s) /\ (y IN t))) ((pastecart x) y)))))))))
Fan.extension_sigma_fan
|- (!x. (!V. (!E. (!v. (!u. ((((((extension_sigma_fan x) V) E) v) u) = (((COND (~ (u IN (((set_of_edge v) V) E)))) u) (((((sigma_fan x) V) E) v) u))))))))
IS_INTERVAL_PATH_CONNECTED_1
|- (!s. ((is_interval s) = (path_connected s)))
HAS_REAL_INTEGRAL_REFLECT
|- (!f. (!i. (!a. (!b. ((((has_real_integral (\x. (f (real_neg x)))) i) (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))) = (((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))))))))
UNIFORM_LIM_COMPLEX_INV
|- (!net. (!P. (!f. (!l. (!b. (((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\ (((real_lt (real_of_num (NUMERAL _0))) b) /\ ((eventually (\x. (!n. ((P n) ==> ((real_le b) (vector_norm (l n))))))) net))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub (complex_inv ((f n) x))) (complex_inv (l n))))) e))))) net)))))))))
DIVMOD_UNIQ
|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\ (r < n)) ==> (((m DIV n) = q) /\ ((m MOD n) = r)))))))
CONNECTED_BALL
|- (!x. (!e. (connected (ball (x , e)))))
BILINEAR_UNIFORMLY_CONTINUOUS_ON_COMPOSE
|- (!f. (!g. (!h. (!s. ((((uniformly_continuous_on f) s) /\ (((uniformly_continuous_on g) s) /\ ((bilinear h) /\ ((bounded ((IMAGE f) s)) /\ (bounded ((IMAGE g) s)))))) ==> ((uniformly_continuous_on (\x. ((h (f x)) (g x)))) s))))))
REAL_DIFFERENTIABLE_AT_SQRT
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_differentiable sqrt) (atreal x))))
Misc_defs_and_lemmas.NUM_COUNTABLE
|- (COUNTABLE UNIV)
Tame_defs.nextElem
|- (((((nextElem NIL) b) x) = b) /\ ((((nextElem ((CONS a) aas)) b) x) = (((COND (x = a)) (((COND (aas = NIL)) b) (HD aas))) (((nextElem aas) b) x))))
Hypermap.lemma_edge_subset
|- (!H. (!x. ((x IN (dart H)) ==> (((edge H) x) SUBSET (dart H)))))
HOMOTOPICALLY_TRIVIAL_RETRACTION_NULL_GEN
|- (!P. (!Q. (!s. (!t. (!u. (!h. (!k. (((((continuous_on h) s) /\ ((((IMAGE h) s) = t) /\ (((continuous_on k) t) /\ ((((IMAGE k) t) SUBSET s) /\ ((!y. ((y IN t) ==> ((h (k y)) = y))) /\ ((!f. ((((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET t) /\ (Q f))) ==> (P (k o f)))) /\ ((!f. ((((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET s) /\ (P f))) ==> (Q (h o f)))) /\ (!h. (!k. ((!x. ((x IN u) ==> ((h x) = (k x)))) ==> ((Q h) = (Q k)))))))))))) /\ (!f. ((((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET s) /\ (P f))) ==> (?c. ((((homotopic_with P) (u , s)) f) (\x. c)))))) ==> (!f. ((((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET t) /\ (Q f))) ==> (?c. ((((homotopic_with Q) (u , t)) f) (\x. c)))))))))))))
CONIC_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((conic ((IMAGE f) s)) = (conic s)))))
EXISTS_DROP
|- ((?x. (P x)) = (?x. (P (drop x))))
SUBSPACE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (subspace s)) ==> (subspace ((IMAGE f) s)))))
SPAN_FINITE
|- (!s. ((FINITE s) ==> ((span s) = (GSPEC (\GEN%PVAR%141. (?y. (((SETSPEC GEN%PVAR%141) (?u. (((vsum s) (\v. ((% (u v)) v))) = y))) y)))))))
FUNDAMENTAL_THEOREM_OF_CALCULUS
|- (!f. (!f'. (!a. (!b. ((((real_le (drop a)) (drop b)) /\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> (((has_vector_derivative f) (f' x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (((has_integral f') ((vector_sub (f b)) (f a))) (closed_interval ((CONS (a , b)) NIL))))))))
CARD_FRONTIER_INTERVAL_1
|- (!s. ((is_interval s) ==> ((FINITE (frontier s)) /\ ((CARD (frontier s)) <= (NUMERAL (BIT0 (BIT1 _0)))))))
JOINABLE_CONNECTED_COMPONENT_EQ
|- (!s. (!t. (!x. (!y. (((connected t) /\ ((t SUBSET s) /\ ((~ ((((connected_component s) x) INTER t) = EMPTY)) /\ (~ ((((connected_component s) y) INTER t) = EMPTY))))) ==> (((connected_component s) x) = ((connected_component s) y)))))))
REAL_DIFFERENTIABLE_AT_TAN
|- (!x. ((~ ((cos x) = (real_of_num (NUMERAL _0)))) ==> ((real_differentiable tan) (atreal x))))
HAS_VECTOR_DERIVATIVE_WITHIN_SUBSET
|- (!f. (!s. (!t. (!x. (((((has_vector_derivative f) f') ((within (at x)) s)) /\ (t SUBSET s)) ==> (((has_vector_derivative f) f') ((within (at x)) t)))))))
LOCALLY_CONNECTED_TRANSLATION_EQ
|- (!a. (!s. (((locally connected) ((IMAGE (\x. ((vector_add a) x))) s)) = ((locally connected) s))))
DIM_EQ_FULL
|- (!s. (((dim s) = (dimindex UNIV)) = ((span s) = UNIV)))
int_sub_th
|- (!x. (!y. ((real_of_int ((int_sub x) y)) = ((real_sub (real_of_int x)) (real_of_int y)))))
CONTINUOUS_ON_EMPTY
|- (!f. ((continuous_on f) EMPTY))
Lp_ineqs_proofs.pack_dist
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!x. (((packing V) /\ (x IN (dart_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance x))))))))
Add_triangle.face_split_fan_face_explicit
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((LET (\w'. (\H. (\H2. (LET_END ((((v , w) IN (dart_of_fan (V , E))) /\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face H) (v , w))))) ==> (((((face H2) (v , w)) = ((v , w) INSERT ((w , w') INSERT ((w' , v) INSERT EMPTY)))) /\ (((face H2) (w' , w)) = ((w' , w) INSERT (((face H) (v , w)) DIFF ((v , w) INSERT ((w' , v) INSERT EMPTY)))))) /\ ((((CARD ((face H2) (v , w))) = (NUMERAL (BIT1 (BIT1 _0)))) /\ ((CARD ((face H2) (w' , w))) = ((CARD ((face H) (v , w))) - (NUMERAL (BIT1 _0))))) /\ ((((v , w) IN ((face H) (v , w))) /\ (((w' , v) IN ((face H) (v , w))) /\ (~ ((w' , w) IN ((face H) (v , w)))))) /\ ((~ (w' = w)) /\ ((~ (v = w)) /\ (~ (v = w'))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) (hypermap_of_fan (V , E))) (hypermap_of_fan ((split_fan_face (V , E)) (v , w))))))))))
DOT_LMUL
|- (!c. (!x. (!y. (((dot ((% c) x)) y) = ((real_mul c) ((dot x) y))))))
CEXP_0
|- ((cexp (Cx (real_of_num (NUMERAL _0)))) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))
int_max_th
|- (!x. (!y. ((real_of_int ((int_max x) y)) = ((real_max (real_of_int x)) (real_of_int y)))))
Constants_approx.CONST_INTERVAL
|- (!x. ((interval_arith x) (x , x)))
HAS_DERIVATIVE_VSUM
|- (!f. (!net. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (((has_derivative (f a)) (f' a)) net)))) ==> (((has_derivative (\x. ((vsum s) (\a. ((f a) x))))) (\h. ((vsum s) (\a. ((f' a) h))))) net)))))
KREIN_MILMAN_MINKOWSKI
|- (!s. (((convex s) /\ (compact s)) ==> (s = ((hull convex) (GSPEC (\GEN%PVAR%2241. (?x. (((SETSPEC GEN%PVAR%2241) ((extreme_point_of x) s)) x))))))))
REAL_LOG_CONVEX_IMP_POS
|- (!f. (!s. (!x. ((((real_log_convex_on f) s) /\ (x IN s)) ==> ((real_le (real_of_num (NUMERAL _0))) (f x))))))
Lp_ineqs_def.rho_fan
|- (!v. ((rho_fan v) = (real_abs ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) (ln_fan v)))))))
MATRIX_CMUL
|- (!f. (!c. ((linear f) ==> ((matrix (\x. ((% c) (f x)))) = ((%% c) (matrix f))))))
IN_SEGMENT
|- (!a. (!b. (!x. (((x IN (closed_segment ((CONS (a , b)) NIL))) = (?u. (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le u) (real_of_num (NUMERAL (BIT1 _0)))) /\ (x = ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))) /\ ((x IN (open_segment (a , b))) = ((~ (a = b)) /\ (?u. (((real_lt (real_of_num (NUMERAL _0))) u) /\ (((real_lt u) (real_of_num (NUMERAL (BIT1 _0)))) /\ (x = ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))))))))
Sphere.rhazim
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((rhazim y1) y2) y3) y4) y5) y6) = ((real_mul (rho y1)) ((((((dih_y y1) y2) y3) y4) y5) y6)))))))))
HAS_REAL_INTEGRAL_RESTRICT
|- (!f. (!s. (!t. ((s SUBSET t) ==> ((((has_real_integral (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) t) = (((has_real_integral f) i) s))))))
BETWEEN_IN_SEGMENT
|- (!x. (!a. (!b. (((between x) (a , b)) = (x IN (closed_segment ((CONS (a , b)) NIL)))))))
HAS_MEASURE_ALMOST
|- (!s. (!s'. (!t. (!m. ((((has_measure s) m) /\ ((negligible t) /\ ((s UNION t) = (s' UNION t)))) ==> ((has_measure s') m))))))
VSUM_DELETE
|- (!f. (!s. (!a. (((FINITE s) /\ (a IN s)) ==> (((vsum (s DELETE a)) f) = ((vector_sub ((vsum s) f)) (f a)))))))
SUM_MULTICOUNT_GEN
|- (!R. (!s. (!t. (!k. (((FINITE s) /\ ((FINITE t) /\ (!j. ((j IN t) ==> ((CARD (GSPEC (\GEN%PVAR%203. (?i. (((SETSPEC GEN%PVAR%203) ((i IN s) /\ ((R i) j))) i))))) = (k j)))))) ==> (((sum s) (\i. (real_of_num (CARD (GSPEC (\GEN%PVAR%204. (?j. (((SETSPEC GEN%PVAR%204) ((j IN t) /\ ((R i) j))) j)))))))) = ((sum t) (\i. (real_of_num (k i))))))))))
REAL_LT_ADDNEG2
|- (!x. (!y. (!z. (((real_lt ((real_add x) (real_neg y))) z) = ((real_lt x) ((real_add z) y))))))
ABSORPTION
|- (!x. (!s. ((x IN s) = ((x INSERT s) = s))))
CONVEX_CBALL
|- (!x. (!e. (convex (cball (x , e)))))
CLOG_WORKS
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((cexp (clog z)) = z) /\ (((real_lt (real_neg pi)) (Im (clog z))) /\ ((real_le (Im (clog z))) pi)))))
REAL_SUB_LT
|- (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) ((real_sub x) y)) = ((real_lt y) x))))
PATH_IMAGE_SUBPATH_GEN
|- (!u. (!v. (!g. ((path_image (((subpath u) v) g)) = ((IMAGE g) (closed_segment ((CONS (u , v)) NIL)))))))
Hypermap_iso.image_set_inj_gen
|- (!u. (!f. ((!x. (!y. (((x IN u) /\ ((y IN u) /\ ((f x) = (f y)))) ==> (x = y)))) ==> (!s. (!t. (((s SUBSET u) /\ ((t SUBSET u) /\ (((IMAGE f) s) = ((IMAGE f) t)))) ==> (s = t)))))))
vector_derivative
|- (!f. (!net. (((vector_derivative f) net) = (@f'. (((has_vector_derivative f) f') net)))))
CONVEX_CONE_LINEAR_IMAGE
|- (!f. (!s. (((convex_cone s) /\ (linear f)) ==> (convex_cone ((IMAGE f) s)))))
Ssrnat.subn_add2l
|- (!p. (!m. (!n. (((p + m) - (p + n)) = (m - n)))))
EVENTUALLY_SEQUENTIALLY
|- (!p. (((eventually p) sequentially) = (?N. (!n. ((N <= n) ==> (p n))))))