hcju/holps huggingface dataset:
Hypermap_and_fan.IN_DART_OF_FAN
|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (x IN (dart_of_fan (V , E)))) ==> (?v. (?w. ((x = (v , w)) /\ (((v , w) IN (dart_of_fan (V , E))) /\ ((v IN V) /\ (w IN V))))))))))
compact_in
|- (!top. (!s. (((compact_in s) top) = ((s SUBSET (topspace top)) /\ (!U. (((!u. ((u IN U) ==> ((open_in top) u))) /\ (s SUBSET (UNIONS U))) ==> (?V. ((FINITE V) /\ ((V SUBSET U) /\ (s SUBSET (UNIONS V)))))))))))
Collect_geom2.ETA_Y_POW2_EQ
|- (((real_div ((real_mul ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0))))))) (((ups_x ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_pow (((eta_y ((dist3 v2) v3)) ((dist3 v1) v3)) ((dist3 v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))))
REAL_INTEGRABLE_BY_PARTS
|- (!f. (!g. (!f'. (!g'. (!a. (!b. (!c. (((COUNTABLE c) /\ (((real_continuous_on (\x. ((real_mul (f x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN ((open_real_interval (a , b)) DIFF c)) ==> ((((has_real_derivative f) (f' x)) (atreal x)) /\ (((has_real_derivative g) (g' x)) (atreal x))))) /\ ((real_integrable_on (\x. ((real_mul (f x)) (g' x)))) (closed_real_interval ((CONS (a , b)) NIL)))))) ==> ((real_integrable_on (\x. ((real_mul (f' x)) (g x)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))
RELATIVE_BOUNDARY_OF_TRIANGLE
|- (!a. (!b. (!c. ((~ (collinear (a INSERT (b INSERT (c INSERT EMPTY))))) ==> ((((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))) DIFF (relative_interior ((hull convex) (a INSERT (b INSERT (c INSERT EMPTY)))))) = ((closed_segment ((CONS (a , b)) NIL)) UNION ((closed_segment ((CONS (b , c)) NIL)) UNION (closed_segment ((CONS (c , a)) NIL)))))))))
SC_RULES
|- (!R. ((!x. (!y. (((R x) y) ==> (((SC R) x) y)))) /\ (!x. (!y. ((((SC R) x) y) ==> (((SC R) y) x))))))
INTERVAL_REAL_INTERVAL
|- (((closed_interval ((CONS (a , b)) NIL)) = ((IMAGE lift) (closed_real_interval ((CONS ((drop a) , (drop b))) NIL)))) /\ ((open_interval (a , b)) = ((IMAGE lift) (open_real_interval ((drop a) , (drop b))))))
VECTOR_VARIATION_TRANSLATION2
|- (!a. (!f. (!s. (((vector_variation ((IMAGE (\x. ((vector_add (vector_neg a)) x))) s)) (\x. (f ((vector_add a) x)))) = ((vector_variation s) f)))))
Seq.in_cons
|- (!y. (!s. (!x. (((MEM x) ((CONS y) s)) = ((x = y) \/ ((MEM x) s))))))
TRACE_COVARIANCE_POS_LE
|- (!A. ((real_le (real_of_num (NUMERAL _0))) (trace ((matrix_mul (transp A)) A))))
MCBALL_SUBSET_MSPACE
|- (!m. (!x. (!r. (((mcball m) (x , r)) SUBSET (mspace m)))))
CONIC_RELATIVE_INTERIOR_INSERT
|- (!s. ((conic s) ==> (conic ((vec (NUMERAL _0)) INSERT (relative_interior s)))))
EXTREME_POINTS_OF_CONVEX_HULL
|- (!s. ((GSPEC (\GEN%PVAR%2195. (?x. (((SETSPEC GEN%PVAR%2195) ((extreme_point_of x) ((hull convex) s))) x)))) SUBSET s))
ADD_CLAUSES
|- ((!n. (((NUMERAL _0) + n) = n)) /\ ((!m. ((m + (NUMERAL _0)) = m)) /\ ((!m. (!n. (((SUC m) + n) = (SUC (m + n))))) /\ (!m. (!n. ((m + (SUC n)) = (SUC (m + n))))))))
ONORM_TRIANGLE
|- (!f. (!g. (((linear f) /\ (linear g)) ==> ((real_le (onorm (\x. ((vector_add (f x)) (g x))))) ((real_add (onorm f)) (onorm g))))))
SETDIST_LE_DIST
|- (!s. (!t. (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le (setdist (s , t))) (distance (x , y))))))))
ABSOLUTELY_INTEGRABLE_TRANSLATION
|- (!f. (!s. (!a. (((absolutely_integrable_on (\x. (f ((vector_add a) x)))) s) = ((absolutely_integrable_on f) ((IMAGE (\x. ((vector_add a) x))) s))))))
AFFINE_TRANSLATION_EQ
|- (!a. (!s. ((affine ((IMAGE (\x. ((vector_add a) x))) s)) = (affine s))))
LOCALLY_CONNECTED_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((locally connected) ((IMAGE f) s)) = ((locally connected) s)))))
Nonlin_def.delta_x1
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((delta_x1 x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul (real_neg x1)) x4)) ((real_add ((real_sub ((real_sub ((real_mul x2) x5)) ((real_mul x3) x5))) ((real_mul x2) x6))) ((real_add ((real_mul x3) x6)) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6)))))))))))))))
Packing3.BARV_CONS
|- (!V. (!k. (!ul. ((((barV V) k) ul) ==> (?h. (?t. ((ul = ((CONS h) t)) /\ (h = (HD ul)))))))))
AFFINE_HULL_FINITE_STEP_GEN
|- (!P. (((?u. ((!x. ((x IN EMPTY) ==> ((P x) (u x)))) /\ ((((sum EMPTY) u) = w) /\ (((vsum EMPTY) (\x. ((% (u x)) x))) = y)))) = ((w = (real_of_num (NUMERAL _0))) /\ (y = (vec (NUMERAL _0))))) /\ (((FINITE s) /\ ((!y. (((a IN s) /\ ((P a) y)) ==> ((P a) ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) /\ (!x. (!y. (((a IN s) /\ (((P a) x) /\ ((P a) y))) ==> ((P a) ((real_add x) y))))))) ==> ((?u. ((!x. ((x IN (a INSERT s)) ==> ((P x) (u x)))) /\ ((((sum (a INSERT s)) u) = w) /\ (((vsum (a INSERT s)) (\x. ((% (u x)) x))) = y)))) = (?v. (?u. (((P a) v) /\ ((!x. ((x IN s) ==> ((P x) (u x)))) /\ ((((sum s) u) = ((real_sub w) v)) /\ (((vsum s) (\x. ((% (u x)) x))) = ((vector_sub y) ((% v) a))))))))))))
MAP_REVERSE
|- (!f. (!l. ((REVERSE ((MAP f) l)) = ((MAP f) (REVERSE l)))))
ORDER_EXISTENCE_CARD
|- (!R. (!f. (!z. (!k. (((FINITE (GSPEC (\GEN%PVAR%74. (?n. (((SETSPEC GEN%PVAR%74) (n IN UNIV)) (R (((ITER n) f) z))))))) /\ (((CARD (GSPEC (\GEN%PVAR%75. (?n. (((SETSPEC GEN%PVAR%75) (n IN UNIV)) (R (((ITER n) f) z))))))) <= k) /\ (((R z) z) /\ ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!x. (!y. (((R (f x)) (f y)) = ((R x) y))))))))) ==> (?d. (((NUMERAL _0) < d) /\ ((d <= k) /\ (!n. (((R (((ITER n) f) z)) z) = ((num_divides d) n)))))))))))
SERIES_GOESTOZERO
|- (!s. (!x. (((summable s) x) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\n. ((n IN s) ==> ((real_lt (vector_norm (x n))) e)))) sequentially))))))
MAP_APPEND
|- (!f. (!l1. (!l2. (((MAP f) ((APPEND l1) l2)) = ((APPEND ((MAP f) l1)) ((MAP f) l2))))))
TC_CASES_L
|- (!R. (!x. (!z. ((((TC R) x) z) = (((R x) z) \/ (?y. ((((TC R) x) y) /\ ((R y) z))))))))
Sphere.dih5atn_x
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih5atn_x x1) x2) x3) x4) x5) x6) = ((LET (\e. (LET_END ((LET (\d. (LET_END ((real_add ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (atn ((real_div (real_neg e)) (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x5) d))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) (((((((rotate5 delta_x4) x1) x2) x3) x4) x5) x6)))))))))
Ssrnat.leq_pmul2r
|- (!m. (!n1. (!n2. (((NUMERAL _0) < m) ==> (((n1 * m) <= (n2 * m)) = (n1 <= n2))))))
IN_SET_OF_LIST
|- (!x. (!l. ((x IN (set_of_list l)) = ((MEM x) l))))
IN_FROM
|- (!m. (!n. ((m IN (from n)) = (n <= m))))
OPEN_IN_OPEN_INTER
|- (!u. (!s. ((open s) ==> ((open_in ((subtopology euclidean) u)) (u INTER s)))))
LT_MULT_LCANCEL
|- (!m. (!n. (!p. (((m * n) < (m * p)) = ((~ (m = (NUMERAL _0))) /\ (n < p))))))
CLOG_NEG_1
|- ((clog (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) = ((complex_mul ii) (Cx pi)))
ABSOLUTELY_CONTINUOUS_ON_IMP_CONTINUOUS
|- (!f. (!s. ((((absolutely_continuous_on f) s) /\ (is_interval s)) ==> ((continuous_on f) s))))
FRONTIER_INTERIORS
|- (!s. ((frontier s) = ((UNIV DIFF (interior s)) DIFF (interior (UNIV DIFF s)))))
MBALL_SUBSET_MCBALL
|- (!m. (!x. (!r. (((mball m) (x , r)) SUBSET ((mcball m) (x , r))))))
Fnjlbxs.vector_of_list_left_inverse
|- (!d. (!list. (((dimindex UNIV) = d) ==> (((d * (LENGTH list)) <= (dimindex UNIV)) ==> ((((list_of_vector d) (LENGTH list)) ((vector_of_list d) list)) = list)))))
real_log_convex_on
|- (!s. (!f. (((real_log_convex_on f) s) = (!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)))))))) ==> (((real_le (real_of_num (NUMERAL _0))) (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) /\ ((real_le (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))
COMPACT_PATH_IMAGE
|- (!g. ((path g) ==> (compact (path_image g))))
GDELTA_TRANSLATION
|- (!a. (!s. ((gdelta ((IMAGE (\x. ((vector_add a) x))) s)) = (gdelta s))))
Topology.complement_set
|- (!x. (!v. ((complement_set (x INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%5482. (?y. (((SETSPEC GEN%PVAR%5482) (~ (y IN (aff (x INSERT (v INSERT EMPTY)))))) y)))))))
Seq.nth_cat
|- (!x0. (!s1. (!s2. (!n. ((((nth x0) ((cat s1) s2)) n) = (((COND (n < (sizel s1))) (((nth x0) s1) n)) (((nth x0) s2) (n - (sizel s1)))))))))
CONVEX_INTERVAL
|- (!a. (!b. ((convex (closed_interval ((CONS (a , b)) NIL))) /\ (convex (open_interval (a , b))))))
TC_CASES_R
|- (!R. (!x. (!z. ((((TC R) x) z) = (((R x) z) \/ (?y. (((R x) y) /\ (((TC R) y) z))))))))
AFFINE_AFFINE_HULL
|- (!s. (affine ((hull affine) s)))
AFF_DIM_DIM
|- (!s. ((aff_dim s) = (((COND ((vec (NUMERAL _0)) IN ((hull affine) s))) (int_of_num (dim s))) ((int_sub (int_of_num (dim s))) (int_of_num (NUMERAL (BIT1 _0)))))))
CONTINUOUS_CLOSED_IN_PREIMAGE_GEN
|- (!f. (!s. (!t. (!u. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ ((closed_in ((subtopology euclidean) t)) u))) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%618. (?x. (((SETSPEC GEN%PVAR%618) ((x IN s) /\ ((f x) IN u))) x))))))))))
aff_ge_def
|- (aff_ge = (affsign sgn_ge))
METRIC_TOPCONTINUOUS
|- (!m. (!m'. (!f. ((((topcontinuous (mtopology m)) (mtopology m')) f) = ((!x. ((x IN (mspace m)) ==> ((f x) IN (mspace m')))) /\ (!a. (!e. ((((real_lt (real_of_num (NUMERAL _0))) e) /\ (a IN (mspace m))) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN (mspace m)) /\ ((real_lt ((mdist m) (a , x))) d)) ==> ((real_lt ((mdist m') ((f a) , (f x)))) e)))))))))))))
FACE_OF_SIMPLEX_SUBSET
|- (!n. (!s. (!f. ((((simplex n) s) /\ ((face_of f) s)) ==> (?c. ((c SUBSET (GSPEC (\GEN%PVAR%2470. (?x. (((SETSPEC GEN%PVAR%2470) ((extreme_point_of x) s)) x))))) /\ (f = ((hull convex) c))))))))
Nonlin_def.sqrt_x2
|- (((((((sqrt_x2 x1) x2) x3) x4) x5) x6) = (sqrt x2))
measurable_on
|- (!s. (!f. (((measurable_on f) s) = (?k. (?g. ((negligible k) /\ ((!n. ((continuous_on (g n)) UNIV)) /\ (!x. ((~ (x IN k)) ==> (((--> (\n. ((g n) x))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) sequentially))))))))))
VECTOR_VARIATION_CONTINUOUS_RIGHT
|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (c , b)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (c , b)) NIL))))))))))
SUBSPACE_IMP_AFFINE
|- (!s. ((subspace s) ==> (affine s)))
Nonlin_def.sqrt_x1
|- (((((((sqrt_x1 x1) x2) x3) x4) x5) x6) = (sqrt x1))
CONTINUOUS_LIFT_ABS_COMPONENT
|- (!net. (!f. (!k. (((continuous f) net) ==> ((continuous (\x. (lift (real_abs ((f x) $ k))))) net)))))
Nonlin_def.uni
|- (!f. (!x. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((uni (f , x)) x1) x2) x3) x4) x5) x6) = (f ((((((x x1) x2) x3) x4) x5) x6)))))))))))
NORM_BASIS
|- (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> ((vector_norm (basis k)) = (real_of_num (NUMERAL (BIT1 _0))))))
TRACE_COVARIANCE_POSITIVE_SEMIDEFINITE_LE
|- (!A. ((positive_semidefinite A) ==> ((real_le (trace ((matrix_mul (transp A)) A))) ((real_pow (trace A)) (NUMERAL (BIT0 (BIT1 _0)))))))
RC_REFL
|- (!R. (!x. (((RC R) x) x)))
DEPENDENT_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((dependent ((IMAGE f) s)) = (dependent s)))))
Seq.has_count
|- (!a. (!s. (((has a) s) = ((NUMERAL _0) < ((count a) s)))))
REAL_LE_ADDR
|- (!x. (!y. (((real_le x) ((real_add x) y)) = ((real_le (real_of_num (NUMERAL _0))) y))))
arc
|- (!g. ((arc g) = ((path g) /\ (!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((g x) = (g y)))) ==> (x = y)))))))
NEGLIGIBLE_IMP_MEASURABLE
|- (!s. ((negligible s) ==> (measurable s)))
POLYTOPE_EQ_BOUNDED_POLYHEDRON
|- (!s. ((polytope s) = ((polyhedron s) /\ (bounded s))))
INT_GT
|- (!x. (!y. (((int_gt x) y) = ((int_lt y) x))))
INTEGER_ADD
|- (!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_add x) y)))))
UNBOUNDED_OUTSIDE
|- (!s. ((bounded s) ==> (~ (bounded (outside s)))))
TAYLOR_CCOS
|- (!n. (!z. ((real_le (vector_norm ((vector_sub (ccos z)) ((vsum ((NUMERAL _0) .. n)) (\k. ((complex_mul ((complex_pow (vector_neg (Cx (real_of_num (NUMERAL (BIT1 _0)))))) k)) ((complex_div ((complex_pow z) ((NUMERAL (BIT0 (BIT1 _0))) * k))) (Cx (real_of_num (FACT ((NUMERAL (BIT0 (BIT1 _0))) * k))))))))))) ((real_mul (exp (real_abs (Im z)))) ((real_div ((real_pow (vector_norm z)) (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT0 (BIT1 _0)))))) (real_of_num (FACT (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))))))))
Ssrnat.distn
|- (!m. (!n. (((distn m) n) = ((m - n) + (n - m)))))
VECTOR_ADD_SUB
|- (((vector_sub ((vector_add x) y)) x) = y)
CEXP_CONVERGES
|- (!z. (((sums (\n. ((complex_div ((complex_pow z) n)) (Cx (real_of_num (FACT n)))))) (cexp z)) (from (NUMERAL _0))))
REAL
|- (!z. ((real z) = ((Cx (Re z)) = z)))
UNION_ACI
|- (((p UNION q) = (q UNION p)) /\ ((((p UNION q) UNION r) = (p UNION (q UNION r))) /\ (((p UNION (q UNION r)) = (q UNION (p UNION r))) /\ (((p UNION p) = p) /\ ((p UNION (p UNION q)) = (p UNION q))))))
CONNECTED_DISJOINT_UNIONS_OPEN_UNIQUE
|- (!f. (!f'. ((((pairwise DISJOINT) f) /\ (((pairwise DISJOINT) f') /\ ((!s. ((s IN f) ==> ((open s) /\ ((connected s) /\ (~ (s = EMPTY)))))) /\ ((!s. ((s IN f') ==> ((open s) /\ ((connected s) /\ (~ (s = EMPTY)))))) /\ ((UNIONS f) = (UNIONS f')))))) ==> (f = f'))))
EVENTUALLY_AT_INFINITY
|- (!p. (((eventually p) at_infinity) = (?b. (!x. (((real_ge (vector_norm x)) b) ==> (p x))))))
CONNECTED_INTERVAL
|- ((!a. (!b. (connected (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (connected (open_interval (a , b))))))
Miqmcsn.SCS_4M8_02_SLICE_02
|- ((scs_arrow_v39 (scs_4M8_02 INSERT EMPTY)) (((scs_prop_equ_v39 (scs_opp_v39 scs_3T7)) (NUMERAL (BIT0 (BIT1 _0)))) INSERT EMPTY))
CBALL_TRIVIAL
|- (!x. ((cball (x , (real_of_num (NUMERAL _0)))) = (x INSERT EMPTY)))
Lp_ineqs_proofs.f_fan_pair_ext_in_darts_k
|- (!V. (!E. (!k. (!d. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (d IN ((darts_k k) (hypermap_of_fan (V , E))))) ==> ((((f_fan_pair_ext (V , E)) d) IN ((darts_k k) (hypermap_of_fan (V , E)))) /\ (((inverse (f_fan_pair_ext (V , E))) d) IN ((darts_k k) (hypermap_of_fan (V , E))))))))))
MATRIX_MUL_LID
|- (!A. (((matrix_mul (mat (NUMERAL (BIT1 _0)))) A) = A))
HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_RIGHT_EQ
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homeomorphic s) ((IMAGE f) t)) = ((homeomorphic s) t))))))
BARYCENTRE_TRANSLATION
|- (!a. (!s. ((barycentre ((IMAGE (\x. ((vector_add a) x))) s)) = ((vector_add (((COND ((FINITE s) /\ (~ (s = EMPTY)))) a) (vec (NUMERAL _0)))) (barycentre s)))))
NORM_TRIANGLE_LE
|- (!x. (!y. (((real_le ((real_add (vector_norm x)) (vector_norm y))) e) ==> ((real_le (vector_norm ((vector_add x) y))) e))))
PATHSTART_TRANSLATION
|- (!a. (!g. ((pathstart ((\x. ((vector_add a) x)) o g)) = ((vector_add a) (pathstart g)))))
CLOG_II
|- ((clog ii) = ((complex_mul ii) (Cx ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))
NSUM_CONST
|- (!c. (!s. ((FINITE s) ==> (((nsum s) (\n. c)) = ((CARD s) * c)))))
CONTINUOUS_ON_PASTECART
|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) s)) ==> ((continuous_on (\x. ((pastecart (f x)) (g x)))) s)))))
Hypermap_iso.hyp_iso_comm
|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((((edge_map G) (f x)) = (f ((edge_map H) x))) /\ ((((node_map G) (f x)) = (f ((node_map H) x))) /\ (((face_map G) (f x)) = (f ((face_map H) x)))))))))))
HAS_COMPLEX_DERIVATIVE_CLOG
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_of_num (NUMERAL _0))) (Re z))) ==> (((has_complex_derivative clog) (complex_inv z)) (at z))))
JACOBIAN_WORKS
|- (!f. (!net. (((differentiable f) net) = (((has_derivative f) (\h. ((matrix_vector_mul ((jacobian f) net)) h))) net))))
HOMOTOPY_EQUIVALENT_TRANSLATION_RIGHT_EQ
|- (!a. (!s. (!t. (((homotopy_equivalent s) ((IMAGE (\x. ((vector_add a) x))) t)) = ((homotopy_equivalent s) t)))))
Add_triangle.f_fan_pair_eq_inv_n_o_e
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> (((f_fan_pair (V , E)) d) = ((inverse (n_fan_pair_ext (V , E))) ((e_fan_pair (V , E)) d))))))))
CONTINUOUS_ON_EQ_CONTINUOUS_AT
|- (!f. (!s. ((open s) ==> (((continuous_on f) s) = (!x. ((x IN s) ==> ((continuous f) (at x))))))))
Ssrnat.odd_add
|- (!m. (!n. ((odd (m + n)) = ((XOR (odd m)) (odd n)))))
REAL_MEASURABLE_ON_COMPOSE_CONTINUOUS_0
|- (!f. (!g. (!s. ((((real_measurable_on f) s) /\ (((real_continuous_on g) UNIV) /\ ((g (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0))))) ==> ((real_measurable_on (g o f)) s)))))
COMPACT_CONTINUOUS_IMAGE
|- (!f. (!s. ((((continuous_on f) s) /\ (compact s)) ==> (compact ((IMAGE f) s)))))
Seq2.list_sum_perm_eq
|- (!s1. (!s2. (!f. (((perm_eq s1) s2) ==> (((list_sum s1) f) = ((list_sum s2) f))))))
FLOOR_POS
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (?n. ((floor x) = (real_of_num n)))))
CONTINUOUS_CLOSED_IN_PREIMAGE_EQ
|- (!f. (!s. (((continuous_on f) s) = (!t. ((closed t) ==> ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%630. (?x. (((SETSPEC GEN%PVAR%630) ((x IN s) /\ ((f x) IN t))) x))))))))))
Seq.sumn
|- (sumn = ((foldr +) (NUMERAL _0)))
TRACE_POSITIVE_SEMIDEFINITE
|- (!A. ((positive_semidefinite A) ==> ((real_le (real_of_num (NUMERAL _0))) (trace A))))
Nonlin_def.unit6
|- (((((((unit6 x1) x2) x3) x4) x5) x6) = (real_of_num (NUMERAL (BIT1 _0))))
COLLINEAR_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((collinear ((IMAGE f) s)) = (collinear s)))))
Hypermap.lemma_power_loop_map_in_loop
|- (!L. (!x. (!k. (((in_dart_of_loop x) L) ==> ((in_dart_of_loop (((POWER (loop_map L)) k) x)) L)))))
Ssrnat.sub0n
|- (!n. (((NUMERAL _0) - n) = (NUMERAL _0)))
Nonlinear_lemma.halfbump_x4
|- (!x1. (!x2. (!x3. (!x5. (!x6. (!x4. (((((((halfbump_x4 x1) x2) x3) x4) x5) x6) = (halfbump_x x4))))))))
Seq.constant_nseq
|- (!n. (!x. (constant ((nseq n) x))))
REAL_OF_NUM_SUC
|- (!n. (((real_add (real_of_num n)) (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (SUC n))))
Lp_ineqs_def.list_defs_th
|- ((!fan. (!g. ((azim_list (g , fan)) = ((azim_dart fan) o g)))) /\ ((!fan. (!g. ((azim2_list (g , fan)) = ((azim2_fan fan) o g)))) /\ ((!fan. (!g. ((azim3_list (g , fan)) = ((azim3_fan fan) o g)))) /\ ((!h. ((yn_list h) = (yn_fan o h))) /\ ((!h. ((ln_list h) = (ln_fan o h))) /\ ((!h. ((rho_list h) = (rho_fan o h))) /\ ((!fan. (!g. ((rhazim_list (g , fan)) = ((rhazim_fan fan) o g)))) /\ ((!fan. (!g. ((rhazim2_list (g , fan)) = ((rhazim2_fan fan) o g)))) /\ ((!fan. (!g. ((rhazim3_list (g , fan)) = ((rhazim3_fan fan) o g)))) /\ ((!g. ((ye_list g) = (ye_fan o g))) /\ ((!fan. (!g. ((sol_list (g , fan)) = ((sol_fan fan) o (IMAGE g))))) /\ ((!fan. (!g. ((tau_list (g , fan)) = ((tau_fan fan) o (IMAGE g))))) /\ ((!g. ((y1_list g) = (y1_fan o g))) /\ ((!g. ((y2_list g) = (y2_fan o g))) /\ ((!fan. (!g. ((y3_list (g , fan)) = ((y3_fan fan) o g)))) /\ ((!fan. (!g. ((y4_list (g , fan)) = ((y4_fan fan) o g)))) /\ ((!fan. (!g. ((y5_list (g , fan)) = ((y5_fan fan) o g)))) /\ ((!g. ((y6_list g) = (y6_fan o g))) /\ ((!fan. (!g. ((y8_list (g , fan)) = ((y8_fan fan) o g)))) /\ ((!fan. (!g. ((y9_list (g , fan)) = ((y9_fan fan) o g)))) /\ (!fan. (!g. ((y4'_list (g , fan)) = ((y4'_fan fan) o g))))))))))))))))))))))))
Fan.n_fan
|- (!x. (!V. (!E. ((((n_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (v , ((((((sigma_fan x) V) E) v) w) , (((((sigma_fan x) V) E) v) (((((sigma_fan x) V) E) v) w))))))))))))))))
Sphere.rotate5
|- (!f. (!x5. (!x3. (!x4. (!x2. (!x6. (!x1. ((((((((rotate5 f) x1) x2) x3) x4) x5) x6) = ((((((f x5) x3) x4) x2) x6) x1)))))))))
INVERTIBLE_FIXPOINT_PROPERTY
|- (!s. (!t. (!i. (!r. ((((continuous_on i) t) /\ ((((IMAGE i) t) SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) SUBSET t) /\ (!y. ((y IN t) ==> ((r (i y)) = y))))))) ==> ((!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET s)) ==> (?x. ((x IN s) /\ ((f x) = x))))) ==> (!g. ((((continuous_on g) t) /\ (((IMAGE g) t) SUBSET t)) ==> (?y. ((y IN t) /\ ((g y) = y)))))))))))
RC_TC
|- (!R. ((RC (TC R)) = (TC (RC R))))
DIST_0
|- (!x. (((distance (x , (vec (NUMERAL _0)))) = (vector_norm x)) /\ ((distance ((vec (NUMERAL _0)) , x)) = (vector_norm x))))
VECTOR_ANGLE_RANGE
|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) ((vector_angle x) y)) /\ ((real_le ((vector_angle x) y)) pi))))
REAL_MUL_LID
|- (!x. (((real_mul (real_of_num (NUMERAL (BIT1 _0)))) x) = x))
Ssrnat.dist0n
|- (!n. (((distn (NUMERAL _0)) n) = n))
Hypermap.lemma_dart_inveriant_under_inverse_maps
|- (!H. (!x. ((x IN (dart H)) ==> ((((inverse (edge_map H)) x) IN (dart H)) /\ ((((inverse (node_map H)) x) IN (dart H)) /\ (((inverse (face_map H)) x) IN (dart H)))))))
ordinal
|- (!l. ((ordinal l) = ((woset l) /\ (!x. (((fl l) x) ==> (x = (@y. (~ ((less l) (y , x))))))))))
LOG_MUL
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((log ((real_mul x) y)) = ((real_add (log x)) (log y))))))
SUBSET_DIFF
|- (!s. (!t. ((s DIFF t) SUBSET s)))
CONNECTED_IMP_PERFECT
|- (!s. (!x. (((connected s) /\ ((~ (?a. (s = (a INSERT EMPTY)))) /\ (x IN s))) ==> ((limit_point_of x) s))))
Imjxphr.HYPER_MM_COLLINEAR
|- ((((scs_k_v39 s) = k) /\ ((is_scs_v39 s) /\ (((MMs_v39 s) w) /\ (((NUMERAL (BIT1 (BIT1 _0))) < k) /\ (((w l) = v2) /\ (((w (SUC l)) = v1) /\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\ (((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6))))))))) ==> (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x6) /\ (~ (collinear ((vec (NUMERAL _0)) INSERT (v1 INSERT (v2 INSERT EMPTY)))))))))
Fnjlbxs.lim_not_equal
|- (!a. (!b. (!f. (!g. (((~ (a = b)) /\ ((((--> f) a) sequentially) /\ (((--> g) b) sequentially))) ==> (?N. (!n. ((N <= n) ==> (~ ((f n) = (g n)))))))))))
Ssrnat.distnEl
|- (!m. (!n. ((n <= m) ==> (((distn m) n) = (m - n)))))
HOMEOMORPHISM_GRAPH_EXPLICIT
|- (!f. (!s. (((homeomorphism (s , (GSPEC (\GEN%PVAR%1033. (?x. (((SETSPEC GEN%PVAR%1033) (x IN s)) ((pastecart x) (f x)))))))) ((\x. ((pastecart x) (f x))) , fstcart)) = ((continuous_on f) s))))
HAS_COMPLEX_DERIVATIVE_CTAN
|- (!z. ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) ==> (((has_complex_derivative ctan) (complex_inv ((complex_pow (ccos z)) (NUMERAL (BIT0 (BIT1 _0)))))) (at z))))
outside
|- (!s. ((outside s) = (GSPEC (\GEN%PVAR%2007. (?x. (((SETSPEC GEN%PVAR%2007) ((~ (x IN s)) /\ (~ (bounded ((connected_component (UNIV DIFF s)) x))))) x))))))
INT_FORALL_ABS
|- (!P. ((!n. (P (int_of_num n))) = (!x. (P (int_abs x)))))
DIST_LIFT
|- (!x. (!y. ((distance ((lift x) , (lift y))) = (real_abs ((real_sub x) y)))))