hcju/holps huggingface dataset:
LIFT_ADD
|- (!x. (!y. ((lift ((real_add x) y)) = ((vector_add (lift x)) (lift y)))))
Hypermap.power_inverse_element_lemma
|- (!s. (!p. (!n. (((FINITE s) /\ ((permutes p) s)) ==> (?j. (((POWER (inverse p)) n) = ((POWER p) j)))))))
BIT0
|- (!n. ((BIT0 n) = (n + n)))
RE_II
|- ((Re ii) = (real_of_num (NUMERAL _0)))
NSUM_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nsum s) f) = ((nsum s) g))))))
LINEAR_CMUL
|- (!f. (!c. (!x. ((linear f) ==> ((f ((% c) x)) = ((% c) (f x)))))))
LINEAR_SCALING
|- (!c. (linear (\x. ((% c) x))))
OPEN_INSIDE
|- (!s. ((closed s) ==> (open (inside s))))
REAL_POW_ZERO
|- (!n. (((real_pow (real_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))
CONVEX_HULL_REDUNDANT_SUBSET_GEN
|- (!s. (!t. (((compact s) /\ ((t SUBSET s) /\ ((DISJOINT (s DIFF t)) (GSPEC (\GEN%PVAR%2254. (?x. (((SETSPEC GEN%PVAR%2254) ((extreme_point_of x) ((hull convex) s))) x))))))) ==> (((hull convex) s) = ((hull convex) t)))))
HAS_SIZE_FINITE_IMAGE
|- (!s. (UNIV HAS_SIZE (dimindex s)))
MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_INTERVAL
|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!b. (lebesgue_measurable (GSPEC (\GEN%PVAR%3638. (?x. (((SETSPEC GEN%PVAR%3638) ((x IN s) /\ ((f x) IN (open_interval (a , b))))) x)))))))))))
EXISTS_IN_GSPEC
|- ((!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%25. (?x. (((SETSPEC GEN%PVAR%25) (P x)) (f x)))))) /\ (Q z))) = (?x. ((P x) /\ (Q (f x))))))) /\ ((!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%26. (?x. (?y. (((SETSPEC GEN%PVAR%26) ((P x) y)) ((f x) y))))))) /\ (Q z))) = (?x. (?y. (((P x) y) /\ (Q ((f x) y)))))))) /\ ((!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%27. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%27) (((P w) x) y)) (((f w) x) y)))))))) /\ (Q z))) = (?w. (?x. (?y. ((((P w) x) y) /\ (Q (((f w) x) y))))))))) /\ (!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%28. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%28) ((((P v) w) x) y)) ((((f v) w) x) y))))))))) /\ (Q z))) = (?v. (?w. (?x. (?y. (((((P v) w) x) y) /\ (Q ((((f v) w) x) y)))))))))))))
CONVEX_CONTAINS_SEGMENT
|- (!s. ((convex s) = (!a. (!b. (((a IN s) /\ (b IN s)) ==> ((closed_segment ((CONS (a , b)) NIL)) SUBSET s))))))
IN_INTER
|- (!s. (!t. (!x. ((x IN (s INTER t)) = ((x IN s) /\ (x IN t))))))
COMPLEX_MUL_ASSOC
|- (!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z)))))
AFF_DIM_EQ_0
|- (!s. (((aff_dim s) = (int_of_num (NUMERAL _0))) = (?a. (s = (a INSERT EMPTY)))))
List_hypermap_iso.e_fan_pair_ext_explicit
|- (!V. (!E. (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((e_fan_pair_ext (V , E)) (v , w)) = (w , v)))))))
INTEGRABLE_ON_SUBINTERVAL
|- (!f. (!s. (!a. (!b. ((((integrable_on f) s) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))))))
MATRIX_CMUL_LZERO
|- (!A. (((%% (real_of_num (NUMERAL _0))) A) = (mat (NUMERAL _0))))
PATH_JOIN
|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path ((++ g1) g2)) = ((path g1) /\ (path g2))))))
Lp_ineqs_proofs.dart4_y4'_hi
|- (!V. (!E. ((lp_fan (V , E)) ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le ((y4'_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))
Conforming.FAN80_FANADD
|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (!ds1. (!ds2. (!f10. (!f20. (!f30. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\ ((((((f1_fan x) V) E) f1) = f2) /\ ((((((f1_fan x) V) E) f2) = f3) /\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\ (((pr2 f1) = v) /\ (((pr2 f2) = u) /\ (((pr2 f3) = w) /\ (((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) /\ (((pr3 f1) = u) /\ (((pr3 f2) = w) /\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (v , (w , (((((sigma_fan x) V) E1) v) w))))) = ds1) /\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (w , (v , (((((sigma_fan x) V) E1) w) v))))) = ds2) /\ (((x , (w , (v , u))) = f10) /\ (((x , (v , (u , w))) = f20) /\ (((x , (u , (w , v))) = f30) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))))))))) ==> (fan80 (x , (V , E1))))))))))))))))))))
LIFT_DROP
|- ((!x. ((lift (drop x)) = x)) /\ (!x. ((drop (lift x)) = x)))
Functional_equation.proj_x2
|- (((((((proj_x2 x1) x2) x3) x4) x5) x6) = x2)
is_metric_space
|- (!s. (!d. ((is_metric_space (s , d)) = ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_of_num (NUMERAL _0))) (d (x , y)))))) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> (((d (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((d (x , y)) = (d (y , x)))))) /\ (!x. (!y. (!z. (((x IN s) /\ ((y IN s) /\ (z IN s))) ==> ((real_le (d (x , z))) ((real_add (d (x , y))) (d (y , z))))))))))))))
Seq.ncons
|- (!n. (!x. (((ncons n) x) = ((iter n) (CONS x)))))
Localization.h_dart
|- (!x. ((h_dart x) = ((real_div (vector_norm (FST x))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))
REAL_POW_EQ_1
|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL (BIT1 _0)))) = ((((real_abs x) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((real_lt x) (real_of_num (NUMERAL _0))) ==> (EVEN n))) \/ (n = (NUMERAL _0))))))
NSUM_OFFSET
|- (!p. (!f. (!m. (!n. (((nsum ((m + p) .. (n + p))) f) = ((nsum (m .. n)) (\i. (f (i + p)))))))))
REAL_EXP_LE_X
|- (!x. ((real_le ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x)) (exp x)))
BETWEEN_LINEAR_IMAGE_EQ
|- (!f. (!x. (!y. (!z. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((between (f x)) ((f y) , (f z))) = ((between x) (y , z))))))))
REAL_INV_1
|- ((real_inv (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))
POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC
|- (!A. ((positive_semidefinite A) ==> ((transp A) = A)))
Seq.filter_uniq
|- (!s. (!a. ((uniq s) ==> (uniq ((filter a) s)))))
DECREASING_LEFT_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 y))) (drop (f x)))))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (?l. (((--> f) l) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))
EXISTS_DOUBLE_ARC_EXPLICIT
|- (!g. (!a. (!b. (((simple_path g) /\ (((pathfinish g) = (pathstart g)) /\ ((a IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((b IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (((real_le (drop a)) (drop b)) /\ (~ ((g a) = (g b)))))))) ==> (?u. (?d. ((arc u) /\ ((arc d) /\ (((pathstart u) = (g a)) /\ (((pathfinish u) = (g b)) /\ (((pathstart d) = (g b)) /\ (((pathfinish d) = (g a)) /\ (((path_image u) = ((IMAGE g) (closed_interval ((CONS (a , b)) NIL)))) /\ (((path_image d) = ((IMAGE g) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF (open_interval (a , b))))) /\ ((((path_image u) INTER (path_image d)) = ((g a) INSERT ((g b) INSERT EMPTY))) /\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))))
COS_PI
|- ((cos pi) = (real_neg (real_of_num (NUMERAL (BIT1 _0)))))
BETWEEN_TRANS_2
|- (!a. (!b. (!c. (!d. ((((between a) (b , c)) /\ ((between d) (a , b))) ==> ((between a) (c , d)))))))
LIM_CMUL
|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((% c) (f x)))) ((% c) l)) net)))))
PATHSTART_LINEAR_IMAGE_EQ
|- (!f. (!g. ((linear f) ==> ((pathstart (f o g)) = (f (pathstart g))))))
Ssrnat.leq_pred
|- (!n. ((n - (NUMERAL (BIT1 _0))) <= n))
CLOSEST_POINT_LE
|- (!s. (!a. (!x. (((closed s) /\ (x IN s)) ==> ((real_le (distance (a , ((closest_point s) a)))) (distance (a , x)))))))
RADON_PARTITION
|- (!c. (((FINITE c) /\ (affine_dependent c)) ==> (?m. (?p. (((DISJOINT m) p) /\ (((m UNION p) = c) /\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p)))))))))
Tame_list.indexf
|- (!y. (!x. (!s. ((((indexf x) y) s) = (((COND (((MEM x) s) /\ ((MEM y) s))) ((((indexl y) s) + ((sizel s) - ((indexl x) s))) MOD (sizel s))) (sizel s))))))
IMAGE
|- (!s. (!f. (((IMAGE f) s) = (GSPEC (\GEN%PVAR%7. (?y. (((SETSPEC GEN%PVAR%7) (?x. ((x IN s) /\ (y = (f x))))) y)))))))
mdist
|- (!m. ((mdist m) = (SND (dest_metric m))))
Hypermap.lemma_in_orbit
|- (!f. (!n. (!x. ((((POWER f) n) x) IN ((orbit_map f) x)))))
CLOSED_IN_REFL
|- (!s. ((closed_in ((subtopology euclidean) s)) s))
REAL_LE_RDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le x) ((real_div y) z)) = ((real_le ((real_mul x) z)) y))))))
REAL_VECTOR_POLYNOMIAL_FUNCTION_o
|- (!f. (!g. (((vector_polynomial_function f) /\ (real_polynomial_function g)) ==> (real_polynomial_function (g o f)))))
Ssrbool.andbT
|- (!b. ((b /\ T) = b))
SPAN_OF_SUBSPACE
|- (!s. ((subspace s) ==> ((span s) = s)))
FINITE_SET_OF_LIST
|- (!l. (FINITE (set_of_list l)))
COMPACT_EQ_HEINE_BOREL
|- (!s. ((compact s) = (!f. (((!t. ((t IN f) ==> (open t))) /\ (s SUBSET (UNIONS f))) ==> (?f'. ((f' SUBSET f) /\ ((FINITE f') /\ (s SUBSET (UNIONS f')))))))))
Ssrnat.maxn_mulr
|- (!m. (!n1. (!n2. ((m * ((maxn n1) n2)) = ((maxn (m * n1)) (m * n2))))))
IN_SEGMENT_CX_GEN
|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS ((Cx a) , (Cx b))) NIL))) = (((Im x) = (real_of_num (NUMERAL _0))) /\ ((((real_le a) (Re x)) /\ ((real_le (Re x)) b)) \/ (((real_le b) (Re x)) /\ ((real_le (Re x)) a))))))))
REAL_LT_NEGTOTAL
|- (!x. ((x = (real_of_num (NUMERAL _0))) \/ (((real_lt (real_of_num (NUMERAL _0))) x) \/ ((real_lt (real_of_num (NUMERAL _0))) (real_neg x)))))
MATRIX_MUL_RID
|- (!A. (((matrix_mul A) (mat (NUMERAL (BIT1 _0)))) = A))
LOCALLY_COMPACT_PCROSS
|- (!s. (!t. ((((locally compact) s) /\ ((locally compact) t)) ==> ((locally compact) (s PCROSS t)))))
INVERTIBLE_DET_NZ
|- (!A. ((invertible A) = (~ ((det A) = (real_of_num (NUMERAL _0))))))
HAS_VECTOR_DERIVATIVE_REAL_COMPLEX
|- ((((has_complex_derivative f) f') (at (Cx (drop a)))) ==> (((has_vector_derivative (\x. (f (Cx (drop x))))) f') (at a)))
BOUNDED_INSIDE
|- (!s. ((bounded s) ==> (bounded (inside s))))
Nkezbfc_local.order
|- (!f. (!x. (!y. ((((order f) x) y) = (@n. (((((ITER n) f) x) = y) /\ (!i. ((((NUMERAL _0) < i) /\ (i < n)) ==> (~ ((((ITER i) f) x) = y))))))))))
VECTOR_MATRIX_MUL_TRANSP
|- (!A. (!x. (((vector_matrix_mul x) A) = ((matrix_vector_mul (transp A)) x))))
MULT_SUC
|- (!m. (!n. ((m * (SUC n)) = (m + (m * n)))))
Appendix.scs_J_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_J_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = J))))))))))
Fan.d20_fan
|- (!x. (!V. (!E. ((d20_fan (x , (V , E))) = (GSPEC (\GEN%PVAR%5471. (?x'. (?v. (((SETSPEC GEN%PVAR%5471) ((x' = x) /\ ((V v) /\ ((((set_of_edge v) V) E) = EMPTY)))) (x' , (v , (v , v))))))))))))
AFFINE_TRANSLATION
|- (!s. (!a. ((affine s) ==> (affine ((IMAGE (\x. ((vector_add a) x))) s)))))
CONNECTED_NEST
|- (!s. (((!n. ((compact (s n)) /\ (connected (s n)))) /\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m)))))) ==> (connected (INTERS (GSPEC (\GEN%PVAR%1154. (?n. (((SETSPEC GEN%PVAR%1154) (n IN UNIV)) (s n)))))))))
Seq.rot0
|- (!s. (((rot (NUMERAL _0)) s) = s))
CLOG_MUL
|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((clog ((complex_mul w) z)) = (((COND ((real_le (Im ((vector_add (clog w)) (clog z)))) (real_neg pi))) ((vector_add ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) (((COND ((real_gt (Im ((vector_add (clog w)) (clog z)))) pi)) ((vector_sub ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) ((vector_add (clog w)) (clog z))))))))
Ssrnat.fact0
|- ((FACT (NUMERAL _0)) = (NUMERAL (BIT1 _0)))
Pack2.NEGLIGIBLE_INTER_VORONOI_CLOSED
|- (!s. (!u. (!v. (((u IN s) /\ ((v IN s) /\ (~ (u = v)))) ==> (negligible (((voronoi_closed s) u) INTER ((voronoi_closed s) v)))))))
Tame_defs.tame_planar_hypermap
|- (!H. ((tame_planar_hypermap H) = ((tame_1 H) /\ ((tame_2 H) /\ ((tame_3 H) /\ ((tame_4 H) /\ ((tame_5a H) /\ ((tame_8 H) /\ ((tame_9a H) /\ ((tame_10 H) /\ ((tame_11a H) /\ ((tame_11b H) /\ ((tame_12o H) /\ (tame_13a H))))))))))))))
BOUNDED_UNIONS
|- (!f. (((FINITE f) /\ (!s. ((s IN f) ==> (bounded s)))) ==> (bounded (UNIONS f))))
simple_path
|- (!g. ((simple_path 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) \/ (((x = (vec (NUMERAL _0))) /\ (y = (vec (NUMERAL (BIT1 _0))))) \/ ((x = (vec (NUMERAL (BIT1 _0)))) /\ (y = (vec (NUMERAL _0))))))))))))
Lp_ineqs_proofs.lp_tau
|- (!V. (!E. ((lp_tau (V , E)) = (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) (tauVEF (V , (E , f)))))))))
SEPARATING_HYPERPLANE_SETS
|- (!s. (!t. (((convex s) /\ ((convex t) /\ ((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_ge ((dot a) x)) b))))))))))
WF_UREC
|- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (!f. (!g. (((!x. ((f x) = ((H f) x))) /\ (!x. ((g x) = ((H g) x)))) ==> (f = g)))))))
Functional_equation.mul6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((mul6 f) g) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
ORTHOGONAL_TRANSFORMATION_ORTHOGONAL_EIGENVECTORS
|- (!f. (!v. (!w. (!a. (!b. (((orthogonal_transformation f) /\ (((f v) = ((% a) v)) /\ (((f w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))
CONVEX_CONNECTED
|- (!s. ((convex s) ==> (connected s)))
Lp_ineqs_proofs.face5_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\w'. (LET_END ((LET (\u. (LET_END ((LET (\z. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , z)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (z , u)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (u , w')) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (v , w)) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w , z)) = (z , u)) /\ ((((f_fan_pair_ext (V , E)) (z , u)) = (u , w')) /\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w' , v)) = (v , w)) /\ (((inverse (f_fan_pair_ext (V , E))) (v , w)) = (w' , v)))))))))) /\ ((((v , w) IN (dart1_of_fan (V , E))) /\ (((w , z) IN (dart1_of_fan (V , E))) /\ (((z , u) IN (dart1_of_fan (V , E))) /\ (((u , w') IN (dart1_of_fan (V , E))) /\ ((w' , v) IN (dart1_of_fan (V , E))))))) /\ (((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , z) INSERT ((z , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY)))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) u) w'))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))
HOMOTOPIC_PATHS_RCANCEL
|- (!p. (!q. (!r. (!s. (((((homotopic_paths s) ((++ p) r)) ((++ q) r)) /\ (((pathfinish p) = (pathstart r)) /\ ((pathfinish q) = (pathstart r)))) ==> (((homotopic_paths s) p) q))))))
NSUM_SUBSET
|- (!u. (!v. (!f. (((FINITE u) /\ ((FINITE v) /\ (!x. ((x IN (u DIFF v)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum u) f) <= ((nsum v) f))))))
HAS_REAL_INTEGRAL_EMPTY
|- (!f. (((has_real_integral f) (real_of_num (NUMERAL _0))) EMPTY))
LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED
|- (!s. (((locally path_connected) s) ==> ((locally connected) s)))
DOT_LMUL_MATRIX
|- (!A. (!x. (!y. (((dot ((vector_matrix_mul x) A)) y) = ((dot x) ((matrix_vector_mul A) y))))))
RESTRICTION_FIXPOINT
|- (!s. (!f. ((((RESTRICTION s) f) = f) = (f IN (EXTENSIONAL s)))))
COMMUTING_WITH_DIAGONAL_MATRIX
|- (!A. (!D. ((diagonal_matrix D) ==> ((((matrix_mul A) D) = ((matrix_mul D) A)) = (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))) ==> ((((A $ i) $ j) = (real_of_num (NUMERAL _0))) \/ (((D $ i) $ i) = ((D $ j) $ j))))))))))
REAL_LE_RNEG
|- (!x. (!y. (((real_le x) (real_neg y)) = ((real_le ((real_add x) y)) (real_of_num (NUMERAL _0))))))
INT_POS
|- (!n. ((int_le (int_of_num (NUMERAL _0))) (int_of_num n)))
Fan.fan2
|- (!E. (!x. (!V. ((fan2 (x , (V , E))) = (~ (x IN V))))))
homeomorphic
|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((homeomorphism (s , t)) (f , g)))))))
BALL_MIN_INTER
|- (!a. (!r. (!s. ((ball (a , ((real_min r) s))) = ((ball (a , r)) INTER (ball (a , s)))))))
LE_CASES
|- (!m. (!n. ((m <= n) \/ (n <= m))))
Fan.e_fan
|- (!x. (!V. (!E. ((((e_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , (v , (((((sigma_fan x) V) E) w) v)))))))))))))))
Hypermap.edge
|- (!H. (!x. (((edge H) x) = ((orbit_map (edge_map H)) x))))
ARC_JOIN_EQ
|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\ ((arc g2) /\ (((path_image g1) INTER (path_image g2)) SUBSET ((pathstart g2) INSERT EMPTY))))))))
HAS_REAL_VECTOR_DERIVATIVE_AT
|- ((((has_real_derivative f) f') (atreal x)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) (at (lift x))))
IN_IMAGE_LIFT_DROP
|- ((!x. (!s. ((x IN ((IMAGE lift) s)) = ((drop x) IN s)))) /\ (!x. (!s. ((x IN ((IMAGE drop) s)) = ((lift x) IN s)))))
CONTINUOUS_ON_MUL
|- (!s. (!c. (!f. ((((continuous_on (lift o c)) s) /\ ((continuous_on f) s)) ==> ((continuous_on (\x. ((% (c x)) (f x)))) s)))))
HAS_BOUNDED_VARIATION_ON_ADD
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. ((vector_add (f x)) (g x)))) s)))))
Seq.nth_drop
|- (!x0. (!n0. (!s. (!i. ((((nth x0) ((dropl n0) s)) i) = (((nth x0) s) (n0 + i)))))))
PASTECART_CMUL
|- (!x1. (!y1. (!c. (((pastecart ((% c) x1)) ((% c) y1)) = ((% c) ((pastecart x1) y1))))))
TC_INC
|- (!R. (!x. (!y. (((R x) y) ==> (((TC R) x) y)))))
Packing3.HULL_INTER_SUBSET_INTER_HULL
|- (!P. (!s. (!t. (((hull P) (s INTER t)) SUBSET (((hull P) s) INTER ((hull P) t))))))
AFFINE_HULL_CONVEX_INTER_OPEN_IN
|- (!s. (!t. (((convex s) /\ (((open_in ((subtopology euclidean) ((hull affine) s))) t) /\ (~ ((s INTER t) = EMPTY)))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))
Nonlin_def.ups_126
|- (ups_126 = (((((((compose6 (promote3_to_6 ups_x)) proj_x1) proj_x2) proj_x6) dummy6) dummy6) dummy6))
Merge_ineq.delta_min_corner_4_784
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1) /\ (((real_le x1) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x2) /\ (((real_le x2) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x3) /\ (((real_le x3) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4) /\ (((real_le x4) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x5) /\ (((real_le x5) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6) /\ ((real_le x6) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) ==> ((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))
iterate
|- (!f. (!s. (!op. ((((iterate op) s) f) = (((COND (FINITE (((support op) f) s))) (((ITSET (\x. (\a. ((op (f x)) a)))) (((support op) f) s)) (neutral op))) (neutral op))))))
OPEN_IN_REFL
|- (!s. ((open_in ((subtopology euclidean) s)) s))
REAL_LT_ADDL
|- (!x. (!y. (((real_lt y) ((real_add x) y)) = ((real_lt (real_of_num (NUMERAL _0))) x))))