hcju/holps huggingface dataset:
REAL_MIN_LT
|- (!x. (!y. (!z. (((real_lt ((real_min x) y)) z) = (((real_lt x) z) \/ ((real_lt y) z))))))
NORM_CAUCHY_SCHWARZ_ABS
|- (!x. (!y. ((real_le (real_abs ((dot x) y))) ((real_mul (vector_norm x)) (vector_norm y)))))
BOREL_UNIONS
|- (!u. (((COUNTABLE u) /\ (!s. ((s IN u) ==> (borel s)))) ==> (borel (UNIONS u))))
HAS_COMPLEX_DERIVATIVE_DIFFERENTIABLE
|- (!f. (!x. ((((has_complex_derivative f) ((complex_derivative f) x)) (at x)) = ((complex_differentiable f) (at x)))))
HOMEOMORPHISM_IMP_CLOSED_MAP
|- (!f. (!g. (!s. (!t. (!u. ((((homeomorphism (s , t)) (f , g)) /\ ((closed_in ((subtopology euclidean) s)) u)) ==> ((closed_in ((subtopology euclidean) t)) ((IMAGE f) u))))))))
LIFT_SUM
|- (!k. (!x. ((lift ((sum k) x)) = ((vsum k) (lift o x)))))
CONTINUOUS_AT_IMP_CONTINUOUS_ON
|- (!f. (!s. ((!x. ((x IN s) ==> ((continuous f) (at x)))) ==> ((continuous_on f) s))))
IMAGE_o
|- (!f. (!g. (!s. (((IMAGE (f o g)) s) = ((IMAGE f) ((IMAGE g) s))))))
SIN_PI
|- ((sin pi) = (real_of_num (NUMERAL _0)))
VECTOR_MUL_LID
|- (!x. (((% (real_of_num (NUMERAL (BIT1 _0)))) x) = x))
RECTIFIABLE_PATH_JOIN
|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((rectifiable_path ((++ g1) g2)) = ((rectifiable_path g1) /\ (rectifiable_path g2))))))
HAS_MEASURE_MEASURABLE_MEASURE
|- (!s. (!m. (((has_measure s) m) = ((measurable s) /\ ((measure s) = m)))))
INTEGRAL_NORM_BOUND_INTEGRAL
|- (!f. (!g. (!s. ((((integrable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((real_le (vector_norm ((integral s) f))) (drop ((integral s) g)))))))
Tame_list.IMP_IMP_THM
|- (!a. (!b. ((a ==> (b ==> c)) = ((a /\ b) ==> c))))
DIAMETER_CLOSURE
|- (!s. ((diameter (closure s)) = (diameter s)))
REAL_LT_01
|- ((real_lt (real_of_num (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0))))
bool_elim
|- (!P. (((P T) /\ (P F)) ==> (!x. (P x))))
COLLINEAR_SUBSET
|- (!s. (!t. (((collinear t) /\ (s SUBSET t)) ==> (collinear s))))
PERMUTES_INVERSE_EQ
|- (!p. (!s. (((permutes p) s) ==> (!x. (!y. ((((inverse p) y) = x) = ((p x) = y)))))))
Sphere.packing
|- (!S. ((packing S) = (!u. (!v. (((S u) /\ ((S v) /\ (~ (u = v)))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (distance (u , v))))))))
FINE_UNIONS
|- (!d. (!ps. ((!p. ((p IN ps) ==> ((fine d) p))) ==> ((fine d) (UNIONS ps)))))
Trigonometry2.WHEN_K_POS_ARCV_STABLE
|- (((real_lt (real_of_num (NUMERAL _0))) k) ==> ((((arcV (vec (NUMERAL _0))) u) v) = (((arcV (vec (NUMERAL _0))) u) ((% k) v))))
HAS_REAL_MEASURE_DIFF_SUBSET
|- (!s1. (!s2. (!m1. (!m2. ((((has_real_measure s1) m1) /\ (((has_real_measure s2) m2) /\ (s2 SUBSET s1))) ==> ((has_real_measure (s1 DIFF s2)) ((real_sub m1) m2)))))))
HAS_REAL_DERIVATIVE_SUM
|- (!f. (!net. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> (((has_real_derivative (f a)) (f' a)) net)))) ==> (((has_real_derivative (\x. ((sum s) (\a. ((f a) x))))) ((sum s) f')) net)))))
NEGLIGIBLE_CONVEX_FRONTIER
|- (!s. ((convex s) ==> (negligible (frontier s))))
VECTOR_NEG_EQ_0
|- (((vector_neg x) = (vec (NUMERAL _0))) = (x = (vec (NUMERAL _0))))
SUM_INTEGRAL_LBOUND_INCREASING
|- (!f. (!g. (!m. (!n. (((m <= n) /\ ((!x. ((x IN (closed_segment ((CONS ((Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))) , (Cx (real_of_num n)))) NIL))) ==> (((has_complex_derivative g) (f x)) (at x)))) /\ (!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 (Re (f (Cx x)))) (Re (f (Cx y))))))))) ==> ((real_le (Re ((vector_sub (g (Cx (real_of_num n)))) (g (Cx ((real_sub (real_of_num m)) (real_of_num (NUMERAL (BIT1 _0))))))))) ((sum (m .. n)) (\k. (Re (f (Cx (real_of_num k))))))))))))
CX_ADD
|- (!x. (!y. ((Cx ((real_add x) y)) = ((vector_add (Cx x)) (Cx y)))))
VECTOR_EQ_NEG2
|- (!x. (!y. (((vector_neg x) = (vector_neg y)) = (x = y))))
STD_SIMPLEX
|- (((hull convex) ((vec (NUMERAL _0)) INSERT (GSPEC (\GEN%PVAR%1805. (?i. (((SETSPEC GEN%PVAR%1805) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))) = (GSPEC (\GEN%PVAR%1806. (?x. (((SETSPEC GEN%PVAR%1806) ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_le (real_of_num (NUMERAL _0))) (x $ i)))) /\ ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (x $ i)))) (real_of_num (NUMERAL (BIT1 _0)))))) x)))))
AFFINE_DEPENDENT_TRANSLATION_EQ
|- (!a. (!s. ((affine_dependent ((IMAGE (\x. ((vector_add a) x))) s)) = (affine_dependent s))))
REAL_LT_SUB_LADD
|- (!x. (!y. (!z. (((real_lt x) ((real_sub y) z)) = ((real_lt ((real_add x) z)) y)))))
fsigma
|- (!s. ((fsigma s) = (?g. ((COUNTABLE g) /\ ((!c. ((c IN g) ==> (closed c))) /\ ((UNIONS g) = s))))))
DROP_ADD
|- (!x. (!y. ((drop ((vector_add x) y)) = ((real_add (drop x)) (drop y)))))
absolutely_setcontinuous_on
|- (!s. (!f. (((absolutely_setcontinuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\ (!d. (!t. ((((division_of d) t) /\ ((t SUBSET s) /\ ((real_lt ((sum d) content)) r))) ==> ((real_lt ((sum d) (\k. (vector_norm (f k))))) e)))))))))))
NORM_0
|- ((vector_norm (vec (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
MULT_CLAUSES
|- ((!n. (((NUMERAL _0) * n) = (NUMERAL _0))) /\ ((!m. ((m * (NUMERAL _0)) = (NUMERAL _0))) /\ ((!n. (((NUMERAL (BIT1 _0)) * n) = n)) /\ ((!m. ((m * (NUMERAL (BIT1 _0))) = m)) /\ ((!m. (!n. (((SUC m) * n) = ((m * n) + n)))) /\ (!m. (!n. ((m * (SUC n)) = (m + (m * n))))))))))
IN_INTERVAL_1
|- (!a. (!b. (!x. (((x IN (closed_interval ((CONS (a , b)) NIL))) = (((real_le (drop a)) (drop x)) /\ ((real_le (drop x)) (drop b)))) /\ ((x IN (open_interval (a , b))) = (((real_lt (drop a)) (drop x)) /\ ((real_lt (drop x)) (drop b))))))))
FUNSPACE
|- (!s. (!m. (((mspace ((funspace s) m)) = (GSPEC (\GEN%PVAR%526. (?f. (((SETSPEC GEN%PVAR%526) ((!x. ((x IN s) ==> ((f x) IN (mspace m)))) /\ ((f IN (EXTENSIONAL s)) /\ ((mbounded m) ((IMAGE f) s))))) f))))) /\ (!f. (!g. (((mdist ((funspace s) m)) (f , g)) = (((COND (s = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\GEN%PVAR%527. (?x. (((SETSPEC GEN%PVAR%527) (x IN s)) ((mdist m) ((f x) , (g x)))))))))))))))
NORM_MUL
|- (!a. (!x. ((vector_norm ((% a) x)) = ((real_mul (real_abs a)) (vector_norm x)))))
VOLUME_FRUSTT_STRONG
|- (!v0. (!v1. (!h. (!a. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((bounded ((((frustt v0) v1) h) a)) /\ ((convex ((((frustt v0) v1) h) a)) /\ ((measurable ((((frustt v0) v1) h) a)) /\ ((measure ((((frustt v0) v1) h) a)) = (((COND ((v1 = v0) \/ (((real_le (real_of_num (NUMERAL (BIT1 _0)))) a) \/ ((real_lt h) (real_of_num (NUMERAL _0)))))) (real_of_num (NUMERAL _0))) ((real_mul pi) ((real_mul ((real_sub ((real_pow ((real_div h) a)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow h) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_div h) (real_of_num (NUMERAL (BIT1 (BIT1 _0)))))))))))))))))
MATRIX_ID
|- ((matrix (\x. x)) = (mat (NUMERAL (BIT1 _0))))
LOCALLY_PATH_CONNECTED_TRANSLATION_EQ
|- (!a. (!s. (((locally path_connected) ((IMAGE (\x. ((vector_add a) x))) s)) = ((locally path_connected) s))))
LIM_AT_ID
|- (!a. (((--> (\x. x)) a) (at a)))
CONVEX_CLOSURE_INTERIOR
|- (!s. (((convex s) /\ (~ ((interior s) = EMPTY))) ==> ((closure (interior s)) = (closure s))))
MATRIX_VECTOR_MUL_INV_EQ_0
|- (!A. ((((matrix_vector_mul (matrix_inv A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0)))))
CLOSURE_MINIMAL
|- (!s. (!t. (((s SUBSET t) /\ (closed t)) ==> ((closure s) SUBSET t))))
Hypermap_and_fan.RES_EMPTY
|- (!f. (((res f) EMPTY) = I))
SUM_ADD_NUMSEG
|- (!f. (!g. (!m. (!n. (((sum (m .. n)) (\i. ((real_add (f i)) (g i)))) = ((real_add ((sum (m .. n)) f)) ((sum (m .. n)) g)))))))
Seq.all_count
|- (!a. (!s. (((all a) s) = (((count a) s) = (sizel s)))))
pathstart
|- (!g. ((pathstart g) = (g (vec (NUMERAL _0)))))
Lp_ineqs_proofs.face_eq_power
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN ((darts_k k) (hypermap_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) d) = (GSPEC (\GEN%PVAR%8730. (?i. (((SETSPEC GEN%PVAR%8730) (i < k)) (((POWER (f_fan_pair_ext (V , E))) i) d))))))))))))
Tame_defs.split_at_rec
|- (((((split_at_rec c) bs) NIL) = (bs , NIL)) /\ ((((split_at_rec c) bs) ((CONS a) xs)) = (((COND (a = c)) (bs , xs)) (((split_at_rec c) ((APPEND bs) ((CONS a) NIL))) xs))))
INTERIOR_SEGMENT
|- ((!a. (!b. ((interior (closed_segment ((CONS (a , b)) NIL))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))) /\ (!a. (!b. ((interior (open_segment (a , b))) = (((COND ((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV))) EMPTY) (open_segment (a , b)))))))
Add_triangle.ext_contracted
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (d_fan ((vec (NUMERAL _0)) , (V , E)))) ==> (((ext_dart (V , E)) (contracted_dart d)) = d))))))
HAS_REAL_INTEGRAL_INTEGRAL
|- (!f. (!s. (((real_integrable_on f) s) = (((has_real_integral f) ((real_integral s) f)) s))))
INDEFINITE_INTEGRAL_UNIFORMLY_CONTINUOUS
|- (!f. (!a. (!b. (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((uniformly_continuous_on (\y. ((integral (closed_interval ((CONS ((fstcart y) , (sndcart y))) NIL))) f))) (closed_interval ((CONS (((pastecart a) a) , ((pastecart b) b))) NIL)))))))
ENDS_IN_SEGMENT
|- (!a. (!b. ((a IN (closed_segment ((CONS (a , b)) NIL))) /\ (b IN (closed_segment ((CONS (a , b)) NIL))))))
Sphere.rho
|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_sub const1) ((real_mul const1) (ly y))))))
integrable_on
|- (!f. (!i. (((integrable_on f) i) = (?y. (((has_integral f) y) i)))))
REAL_SERIES_SUB
|- (!x. (!x0. (!y. (!y0. (!s. (((((real_sums x) x0) s) /\ (((real_sums y) y0) s)) ==> (((real_sums (\n. ((real_sub (x n)) (y n)))) ((real_sub x0) y0)) s)))))))
NPRODUCT_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nproduct s) f) = ((nproduct s) g))))))
FSTCART_PASTECART
|- (!x. (!y. ((fstcart ((pastecart x) y)) = x)))
Ssrbool.iffLR
|- (!P. (!Q. ((P = Q) ==> (P ==> Q))))
Seq2.list_sum_cons
|- (!h. (!t. (!f. (((list_sum ((CONS h) t)) f) = ((real_add (f h)) ((list_sum t) f))))))
PATH_IMAGE_NONEMPTY
|- (!g. (~ ((path_image g) = EMPTY)))
Fan_defs.fully_surrounded
|- (!V. (!E. ((fully_surrounded (V , E)) = (!x. ((x IN (dart_of_fan (V , E))) ==> ((real_lt ((azim_dart (V , E)) x)) pi))))))
HAS_BOUNDED_VARIATION_COMPOSE_DECREASING
|- (!f. (!g. (!a. (!b. (((!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 y))) (drop (f x)))))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f b) , (f a))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))
Lp_ineqs_def.y4_fan
|- (!V. (!E. (!x. (((y4_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))
EXPOSED_FACE_OF_TRANSLATION_EQ
|- (!a. (!f. (!s. (((exposed_face_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((exposed_face_of f) s)))))
HAS_REAL_INTEGRAL_EQ
|- (!f. (!g. (!k. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ (((has_real_integral f) k) s)) ==> (((has_real_integral g) k) s))))))
TOPSPACE_EUCLIDEANREAL
|- ((topspace euclideanreal) = UNIV)
COVERING_SPACE_FINITE_SHEETS_EQ_PROPER_MAP
|- (!p. (!c. (!s. (((covering_space (c , p)) s) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2082. (?x. (((SETSPEC GEN%PVAR%2082) ((x IN c) /\ ((p x) = b))) x))))))) = (!k. (((k SUBSET s) /\ (compact k)) ==> (compact (GSPEC (\GEN%PVAR%2083. (?x. (((SETSPEC GEN%PVAR%2083) ((x IN c) /\ ((p x) IN k))) x))))))))))))
Tame_defs.d_tame
|- (!n. ((d_tame n) = (((COND (n = (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL _0))) (((COND (n = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((COND (n = (NUMERAL (BIT1 (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) (((COND (n = (NUMERAL (BIT0 (BIT1 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) tgt))))))
MEASURABLE_MIDPOINT_CONVEX_IMP_CONVEX_GEN
|- (!f. (!s. ((((measurable_on (lift o f)) s) /\ ((convex s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (f (midpoint (x , y)))) ((real_div ((real_add (f x)) (f y))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))) ==> (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((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)))) /\ (((closed_segment ((CONS (x , y)) NIL)) SUBSET (frontier s)) ==> (x = y))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))
Ssrbool.predI
|- (!p1. (!p2. (((predI p1) p2) = (\x. ((p1 x) /\ (p2 x))))))
COMPLEX_NORM_INV
|- (!z. ((vector_norm (complex_inv z)) = (real_inv (vector_norm z))))
SIMPLY_CONNECTED_TRANSLATION
|- (!a. (!s. ((simply_connected ((IMAGE (\x. ((vector_add a) x))) s)) = (simply_connected s))))
REAL_SECOND_MEAN_VALUE_THEOREM_GEN_FULL
|- (!f. (!g. (!a. (!b. (!u. (!v. (((~ ((closed_real_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((real_integrable_on f) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (open_real_interval (a , b))) ==> (((real_le u) (g x)) /\ ((real_le (g x)) v)))) /\ (!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 u) ((real_integral (closed_real_interval ((CONS (a , c)) NIL))) f))) ((real_mul v) ((real_integral (closed_real_interval ((CONS (c , b)) NIL))) f)))) (closed_real_interval ((CONS (a , b)) NIL))))))))))))
IMAGE_UNION
|- (!f. (!s. (!t. (((IMAGE f) (s UNION t)) = (((IMAGE f) s) UNION ((IMAGE f) t))))))
VECTOR_EQ_ADDR
|- ((((vector_add x) y) = x) = (y = (vec (NUMERAL _0))))
COMPACT_CLOSED_POINTIMAGES_IMP_CONTINUOUS_ON
|- (!f. (!s. (((!c. (((c SUBSET s) /\ (compact c)) ==> (compact ((IMAGE f) c)))) /\ (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1929. (?x. (((SETSPEC GEN%PVAR%1929) ((x IN s) /\ ((f x) = y))) x))))))) ==> ((continuous_on f) s))))
REAL_CNJ
|- (!z. ((real z) = ((cnj z) = z)))
REAL_LT_TRANS
|- (!x. (!y. (!z. ((((real_lt x) y) /\ ((real_lt y) z)) ==> ((real_lt x) z)))))
real_differentiable_on
|- (!f. (!s. (((real_differentiable_on f) s) = (!x. ((x IN s) ==> (?f'. (((has_real_derivative f) f') ((within (atreal x)) s))))))))
LINEAR_NEG
|- (!f. (!x. ((linear f) ==> ((f (vector_neg x)) = (vector_neg (f x))))))
HOMOTOPY_EQUIVALENT_CONTRACTIBILITY
|- (!s. (!t. (((homotopy_equivalent s) t) ==> ((contractible s) = (contractible t)))))
CONTINUOUS_AE_IMP_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET
|- (!f. (!s. (!m. ((((continuous_on f) (s DIFF m)) /\ ((lebesgue_measurable s) /\ (negligible m))) ==> ((measurable_on f) s)))))
FLOOR_UNIQUE
|- (!x. (!a. (((integer a) /\ (((real_le a) x) /\ ((real_lt x) ((real_add a) (real_of_num (NUMERAL (BIT1 _0))))))) = ((floor x) = a))))
INTEGRAL_CONST
|- (!a. (!b. (!c. (((integral (closed_interval ((CONS (a , b)) NIL))) (\x. c)) = ((% (content (closed_interval ((CONS (a , b)) NIL)))) c)))))
SUBSPACE_IMP_CONVEX
|- (!s. ((subspace s) ==> (convex s)))
AFF_DIM_SUBSET
|- (!s. (!t. ((s SUBSET t) ==> ((int_le (aff_dim s)) (aff_dim t)))))
INTEGRABLE_DECREASING_PRODUCT_UNIV
|- (!f. (!g. (!B. ((((integrable_on f) UNIV) /\ ((!x. (!y. (((real_le (drop x)) (drop y)) ==> ((real_le (g y)) (g x))))) /\ (!x. ((real_le (real_abs (g x))) B)))) ==> ((integrable_on (\x. ((% (g x)) (f x)))) UNIV)))))
ANALYTIC_AT_TWO
|- (!f. (!g. (!z. ((((analytic_on f) (z INSERT EMPTY)) /\ ((analytic_on g) (z INSERT EMPTY))) = (?s. ((open s) /\ ((z IN s) /\ (((holomorphic_on f) s) /\ ((holomorphic_on g) s)))))))))
EMPTY_GSPEC
|- ((GSPEC (\GEN%PVAR%14. (?x. (((SETSPEC GEN%PVAR%14) F) x)))) = EMPTY)
DIMINDEX_3
|- ((dimindex UNIV) = (NUMERAL (BIT1 (BIT1 _0))))
num_cases
|- (!P. (((P (NUMERAL _0)) /\ (!n. (P (SUC n)))) ==> (!m. (P m))))
List_hypermap.darts_k
|- (!H. (!k. (((darts_k k) H) = (GSPEC (\GEN%PVAR%8595. (?d. (((SETSPEC GEN%PVAR%8595) ((d IN (dart H)) /\ ((CARD ((face H) d)) = k))) d)))))))
Seq.cat_take_drop
|- (!n0. (!s. (((cat ((take n0) s)) ((dropl n0) s)) = s)))
MSPACE
|- (!s. (!d. ((is_metric_space (s , d)) ==> ((mspace (metric (s , d))) = s))))
Ssrnat.leq_exp2l
|- (!m. (!n1. (!n2. (((NUMERAL (BIT1 _0)) < m) ==> (((m EXP n1) <= (m EXP n2)) = (n1 <= n2))))))
Fan.th3
|- (!x. (!v. (!u. ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) ==> ((~ (x = v)) /\ ((~ (x = u)) /\ (((DISJOINT (x INSERT (v INSERT EMPTY))) (u INSERT EMPTY)) /\ (((DISJOINT (x INSERT (u INSERT EMPTY))) (v INSERT EMPTY)) /\ (((DISJOINT (x INSERT EMPTY)) (v INSERT (u INSERT EMPTY))) /\ (~ (u IN (aff (x INSERT (v INSERT EMPTY))))))))))))))
COUNTABLE_SUBSET
|- (!s. (!t. (((COUNTABLE t) /\ (s SUBSET t)) ==> (COUNTABLE s))))
pair_RECURSION
|- (!PAIR'. (?fn. (!a0. (!a1. ((fn (a0 , a1)) = ((PAIR' a0) a1))))))
RE_NEG
|- (!x. ((Re (vector_neg x)) = (real_neg (Re x))))
RIGHT_POLAR_DECOMPOSITION_INVERTIBLE
|- (!A. ((invertible A) = (?U. (?P. ((orthogonal_matrix U) /\ ((positive_definite P) /\ (((matrix_mul U) P) = A)))))))
CX_MUL
|- (!x. (!y. ((Cx ((real_mul x) y)) = ((complex_mul (Cx x)) (Cx y)))))
Topology.IMAGE_SEG
|- (!n. (!f. (((IMAGE f) (GSPEC (\GEN%PVAR%5485. (?i. (((SETSPEC GEN%PVAR%5485) (i < n)) i))))) = (GSPEC (\GEN%PVAR%5486. (?i. (((SETSPEC GEN%PVAR%5486) (i < n)) (f i))))))))
HAS_REAL_INTEGRAL
|- ((((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL))) = (((has_integral (lift o (f o drop))) (lift y)) (closed_interval ((CONS ((lift a) , (lift b))) NIL))))
Hypermap_and_fan.NODE_SET_AS_IMAGE
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (?f. ((!v. (!w. (((f v) = (f w)) ==> (v = w)))) /\ ((!v. (!x. ((x IN (f v)) ==> ((FST x) = v)))) /\ ((!v. ((FST (CHOICE (f v))) = v)) /\ ((node_set (hypermap_of_fan (V , E))) = ((IMAGE f) V)))))))))
REAL_OPEN_IN
|- (!s. ((real_open s) = ((open_in euclideanreal) s)))
REAL_NEGNEG
|- (!x. ((real_neg (real_neg x)) = x))
Localization.convex_local_fan
|- (!FF. (!V. (!E. ((convex_local_fan (V , (E , FF))) = ((local_fan (V , (E , FF))) /\ (!x. ((x IN FF) ==> (((real_le ((azim_in_fan x) E)) pi) /\ (V SUBSET ((wedge_in_fan_ge x) E))))))))))
CARD_ADD_FINITE
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((+_c s) t)))))
VECTOR_3
|- ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 _0))) = x) /\ ((((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT0 (BIT1 _0)))) = y) /\ (((vector ((CONS x) ((CONS y) ((CONS z) NIL)))) $ (NUMERAL (BIT1 (BIT1 _0)))) = z)))
SYMMETRIC_MATRIX_EQ_DIAGONALIZABLE_ALT
|- (!A. (((transp A) = A) = (?P. (?D. ((orthogonal_matrix P) /\ ((diagonal_matrix D) /\ (A = ((matrix_mul (transp P)) ((matrix_mul D) P)))))))))
LEBESGUE_MEASURABLE_OPEN_IN
|- (!s. (!t. ((((open_in ((subtopology euclidean) s)) t) /\ (lebesgue_measurable s)) ==> (lebesgue_measurable t))))
INTER_ASSOC
|- (!s. (!t. (!u. (((s INTER t) INTER u) = (s INTER (t INTER u))))))
CONVEX_ON_EQ
|- (!f. (!g. (!s. (((convex s) /\ ((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((convex_on f) s))) ==> ((convex_on g) s)))))
NUMSEG_EMPTY
|- (!m. (!n. (((m .. n) = EMPTY) = (n < m))))
MATRIX_VECTOR_MUL_ASSOC
|- (!A. (!B. (!x. (((matrix_vector_mul A) ((matrix_vector_mul B) x)) = ((matrix_vector_mul ((matrix_mul A) B)) x)))))
Tame_defs.tame_5a
|- (!H. ((tame_5a H) = (is_no_double_joins H)))
Nonlin_def.ldih6_x_div_sqrtdelta_posbranch
|- (ldih6_x_div_sqrtdelta_posbranch = (rotate6 ldih_x_div_sqrtdelta_posbranch))
AFFINE_HYPERPLANE
|- (!a. (!b. (affine (GSPEC (\GEN%PVAR%1401. (?x. (((SETSPEC GEN%PVAR%1401) (((dot a) x) = b)) x)))))))
CLOSED_HALFSPACE_IM_LE
|- (!b. (closed (GSPEC (\GEN%PVAR%4757. (?z. (((SETSPEC GEN%PVAR%4757) ((real_le (Im z)) b)) z))))))
Hypermap_and_fan.FACE_SUBSET_DART1_OF_FAN
|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart1_of_fan (V , E)))) ==> (((face (hypermap_of_fan (V , E))) x) SUBSET (dart1_of_fan (V , E)))))))
POSITIVE_SEMIDEFINITE_MUL_EQ
|- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> ((positive_semidefinite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))
real_compact
|- (!s. ((real_compact s) = (compact ((IMAGE lift) s))))
matrix_cmul
|- (!c. (!A. (((%% c) A) = (lambdai. (lambdaj. ((real_mul c) ((A $ i) $ j)))))))
Wrgcvdr_cizmrrh.X_IN_ITS_ORBIT
|- (x IN ((orbit_map f) x))
DIAGONAL_MATRIX_MUL
|- (!A. (!B. (((diagonal_matrix A) /\ (diagonal_matrix B)) ==> (diagonal_matrix ((matrix_mul A) B)))))
INSEG_WOSET
|- (!l. (!m. ((((inseg m) l) /\ (woset l)) ==> (woset m))))
INTEGRABLE_ON_EMPTY
|- (!f. ((integrable_on f) EMPTY))
Functional_equation.mk_126
|- (!f. ((mk_126 f) = (((((((compose6 f) proj_x1) proj_x2) two6) two6) two6) proj_x6)))
SUM_RESTRICT_SET
|- (!P. (!s. (!f. (((sum (GSPEC (\GEN%PVAR%199. (?x. (((SETSPEC GEN%PVAR%199) ((x IN s) /\ (P x))) x))))) f) = ((sum s) (\x. (((COND (P x)) (f x)) (real_of_num (NUMERAL _0)))))))))
IMAGE_LIFT_REAL_INTERVAL
|- ((((IMAGE lift) (closed_real_interval ((CONS (a , b)) NIL))) = (closed_interval ((CONS ((lift a) , (lift b))) NIL))) /\ (((IMAGE lift) (open_real_interval (a , b))) = (open_interval ((lift a) , (lift b)))))
RIGHT_IMP_EXISTS_THM
|- (!P. (!Q. ((P ==> (?x. (Q x))) = (?x. (P ==> (Q x))))))
COMPLEX_NORM_GE_RE_IM
|- (!z. (((real_le (real_abs (Re z))) (vector_norm z)) /\ ((real_le (real_abs (Im z))) (vector_norm z))))
MEASURABLE_NONNEGLIGIBLE_IMP_LARGE
|- (!s. (((measurable s) /\ ((real_lt (real_of_num (NUMERAL _0))) (measure s))) ==> (s =_c UNIV)))
SUP_EQ
|- (!s. (!t. ((!b. ((!x. ((x IN s) ==> ((real_le x) b))) = (!x. ((x IN t) ==> ((real_le x) b))))) ==> ((sup s) = (sup t)))))
UNION_EMPTY
|- ((!s. ((EMPTY UNION s) = s)) /\ (!s. ((s UNION EMPTY) = s)))
absolutely_continuous_on
|- (!f. (!s. (((absolutely_continuous_on f) s) = ((absolutely_setcontinuous_on (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))
CFUNSPACE
|- ((!top. (!m. ((mspace ((cfunspace top) m)) = (GSPEC (\GEN%PVAR%534. (?f. (((SETSPEC GEN%PVAR%534) ((!x. ((x IN (topspace top)) ==> ((f x) IN (mspace m)))) /\ ((f IN (EXTENSIONAL (topspace top))) /\ (((mbounded m) ((IMAGE f) (topspace top))) /\ (((topcontinuous top) (mtopology m)) f))))) f))))))) /\ (!f. (!g. (((mdist ((cfunspace top) m)) (f , g)) = (((COND ((topspace top) = EMPTY)) (real_of_num (NUMERAL _0))) (sup (GSPEC (\GEN%PVAR%535. (?x. (((SETSPEC GEN%PVAR%535) (x IN (topspace top))) ((mdist m) ((f x) , (g x)))))))))))))
ABSOLUTELY_INTEGRABLE_REFLECT_GEN
|- (!f. (!s. (((absolutely_integrable_on (\x. (f (vector_neg x)))) s) = ((absolutely_integrable_on f) ((IMAGE vector_neg) s)))))
CONTINUOUS_AT_ID
|- (!a. ((continuous (\x. x)) (at a)))
Real_ext.pow
|- ((((real_pow x) (NUMERAL _0)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (!n. (((real_pow x) (SUC n)) = ((real_mul x) ((real_pow x) n)))))
Functional_equation.proj_y2
|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((proj_y2 x1) x2) x3) x4) x5) x6) = (sqrt x2))))))))
SELECT_UNIQUE
|- (!P. (!x. ((!y. ((P y) = (y = x))) ==> ((@ P) = x))))
Sphere.voronoi_closed
|- (!S. (!v. (((voronoi_closed S) v) = (GSPEC (\GEN%PVAR%5242. (?x. (((SETSPEC GEN%PVAR%5242) (!w. ((S w) ==> ((real_le (distance (x , v))) (distance (x , w)))))) x)))))))
ANALYTIC_ON_INV
|- (!f. (!s. ((((analytic_on f) s) /\ (!z. ((z IN s) ==> (~ ((f z) = (Cx (real_of_num (NUMERAL _0)))))))) ==> ((analytic_on (\z. (complex_inv (f z)))) s))))
FRONTIER_INTER_SUBSET
|- (!s. (!t. ((frontier (s INTER t)) SUBSET ((frontier s) UNION (frontier t)))))
LIM_1_OVER_LOG
|- (((--> (\n. ((complex_div (Cx (real_of_num (NUMERAL (BIT1 _0))))) (clog (Cx (real_of_num n)))))) (Cx (real_of_num (NUMERAL _0)))) sequentially)
REAL_OF_INT_OF_REAL
|- (!x. ((integer x) ==> ((real_of_int (int_of_real x)) = x)))
SETDIST_EQ_0_COMPACT_CLOSED
|- (!s. (!t. (((compact s) /\ (closed t)) ==> (((setdist (s , t)) = (real_of_num (NUMERAL _0))) = ((s = EMPTY) \/ ((t = EMPTY) \/ (~ ((s INTER t) = EMPTY))))))))