hcju/holps huggingface dataset:
CARD_EQ_CARD_IMP
|- (!s. (!t. (((FINITE t) /\ (s =_c t)) ==> ((CARD s) = (CARD t)))))
MDIST_REFL
|- (!m. (!x. ((x IN (mspace m)) ==> (((mdist m) (x , x)) = (real_of_num (NUMERAL _0))))))
tendsto
|- (!f. (!l. (!net. ((((--> f) l) net) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. ((real_lt (distance ((f x) , l))) e))) net)))))))
ADD_AC
|- (((m + n) = (n + m)) /\ ((((m + n) + p) = (m + (n + p))) /\ ((m + (n + p)) = (n + (m + p)))))
Conforming.conforming_fan
|- (!x. (!V. (!E. ((conforming_fan (x , (V , E))) = ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((conforming_bijection_fan (x , (V , E))) /\ ((conforming_half_space_fan (x , (V , E))) /\ ((conforming_solid_angle_fan (x , (V , E))) /\ (conforming_diagonal_fan (x , (V , E))))))))))))
IM_DEF
|- (!z. ((Im z) = (z $ (NUMERAL (BIT0 (BIT1 _0))))))
CLOSURE_HALFSPACE_LT
|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((closure (GSPEC (\GEN%PVAR%806. (?x. (((SETSPEC GEN%PVAR%806) ((real_lt ((dot a) x)) b)) x))))) = (GSPEC (\GEN%PVAR%807. (?x. (((SETSPEC GEN%PVAR%807) ((real_le ((dot a) x)) b)) x))))))))
INT_EQ_IMP_LE
|- (!x. (!y. ((x = y) ==> ((int_le x) y))))
LEBESGUE_MEASURABLE_LEBESGUE_MEASURABLE_PREIMAGE_BOREL
|- (!f. (!s. (!t. ((((measurable_on f) s) /\ ((lebesgue_measurable s) /\ (borel t))) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3660. (?x. (((SETSPEC GEN%PVAR%3660) ((x IN s) /\ ((f x) IN t))) x)))))))))
IN_UNION
|- (!s. (!t. (!x. ((x IN (s UNION t)) = ((x IN s) \/ (x IN t))))))
REALLIM_ATREAL
|- (!f. (!l. (!a. ((((---> f) l) (atreal a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_lt (real_abs ((real_sub x) a))) d)) ==> ((real_lt (real_abs ((real_sub (f x)) l))) e)))))))))))
INF
|- (!s. (((~ (s = EMPTY)) /\ (?b. (!x. ((x IN s) ==> ((real_le b) x))))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))
Collect_geom2.VBVYGGT
|- (!v1. (!v2. (!v3. (!v4. ((((CARD (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) /\ (~ (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))))) ==> (((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\ ((?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) , v)))))) /\ ((!p. (((p IN ((hull affine) (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))) /\ (?r. (!v. ((v IN (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) ==> (r = (distance (p , v))))))) ==> (p = (circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY)))))))) /\ ((LET (\x12. (LET_END ((LET (\x13. (LET_END ((LET (\x14. (LET_END ((LET (\x23. (LET_END ((LET (\x24. (LET_END ((LET (\x34. (LET_END ((LET (\chi11. (LET_END ((LET (\chi22. (LET_END ((LET (\chi33. (LET_END ((LET (\chi44. (LET_END ((circumcenter (v1 INSERT (v2 INSERT (v3 INSERT (v4 INSERT EMPTY))))) = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((delta x12) x13) x14) x23) x24) x34)))) ((vector_add ((% chi11) v1)) ((vector_add ((% chi22) v2)) ((vector_add ((% chi33) v3)) ((% chi44) v4))))))))) ((((((chi x34) x24) x14) x23) x13) x12))))) ((((((chi x34) x13) x23) x14) x24) x12))))) ((((((chi x12) x24) x23) x14) x13) x34))))) ((((((chi x12) x13) x14) x23) x24) x34))))) ((real_pow (distance (v3 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v2 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v4))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v3))) (NUMERAL (BIT0 (BIT1 _0)))))))) ((real_pow (distance (v1 , v2))) (NUMERAL (BIT0 (BIT1 _0)))))))))))))
Hypermap.lemma_in_face
|- (!H. (!x. (!n. ((((POWER (face_map H)) n) x) IN ((face H) x)))))
VSUM_SING_NUMSEG
|- (((vsum (n .. n)) f) = (f n))
MEASURABLE_ON_MEASURABLE_PREIMAGE_CLOSED
|- (!f. (!s. ((measurable s) ==> (((measurable_on f) s) = (!t. ((closed t) ==> (measurable (GSPEC (\GEN%PVAR%3646. (?x. (((SETSPEC GEN%PVAR%3646) ((x IN s) /\ ((f x) IN t))) x)))))))))))
SUBSET_ANTISYM
|- (!s. (!t. (((s SUBSET t) /\ (t SUBSET s)) ==> (s = t))))
Ssrnat.mul0n
|- (!n. (((NUMERAL _0) * n) = (NUMERAL _0)))
CAUCHY_CONTINUOUS_EXTENDS_TO_CLOSURE
|- (!f. (!s. ((!x. (((cauchy x) /\ (!n. ((x n) IN s))) ==> (cauchy (f o x)))) ==> (?g. (((continuous_on g) (closure s)) /\ (!x. ((x IN s) ==> ((g x) = (f x)))))))))
Appendix.BBprime2_v39
|- (!vv. (!s. (((BBprime2_v39 s) vv) = (((BBprime_v39 s) vv) /\ (((BBindex_v39 s) vv) = (BBindex_min_v39 s))))))
SERIES_DIRICHLET_COMPLEX_VERY_EXPLICIT
|- (!f. (!g. (!B. (!p. ((((real_lt (real_of_num (NUMERAL _0))) B) /\ (((NUMERAL (BIT1 _0)) <= p) /\ ((!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) f))) B)))) /\ ((!n. ((p <= n) ==> ((real (g n)) /\ ((real_le (real_of_num (NUMERAL _0))) (Re (g n)))))) /\ (!n. ((p <= n) ==> ((real_le (Re (g (n + (NUMERAL (BIT1 _0)))))) (Re (g n))))))))) ==> (!m. (!n. ((p <= m) ==> ((real_le (vector_norm ((vsum (m .. n)) (\k. ((complex_mul (f k)) (g k)))))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul B) (vector_norm (g m)))))))))))))
REAL_MAX_ACI
|- ((((real_max x) y) = ((real_max y) x)) /\ ((((real_max ((real_max x) y)) z) = ((real_max x) ((real_max y) z))) /\ ((((real_max x) ((real_max y) z)) = ((real_max y) ((real_max x) z))) /\ ((((real_max x) x) = x) /\ (((real_max x) ((real_max x) y)) = ((real_max x) y))))))
MEASURABLE_NEGLIGIBLE_SYMDIFF_EQ
|- (!s. (!t. ((negligible ((s DIFF t) UNION (t DIFF s))) ==> ((measurable s) = (measurable t)))))
Nonlin_def.dih_x_div_sqrtdelta_posbranch
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((dih_x_div_sqrtdelta_posbranch x1) x2) x3) x4) x5) x6) = ((LET (\d_x4. (LET_END ((LET (\d. (LET_END ((real_mul ((real_div (sqrt ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1))) d_x4)) (matan ((real_div ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ((real_mul x1) d))) ((real_pow d_x4) (NUMERAL (BIT0 (BIT1 _0)))))))))) ((((((delta_x x1) x2) x3) x4) x5) x6))))) ((((((delta_x4 x1) x2) x3) x4) x5) x6)))))))))
IM_COMPLEX_INV_LE_0
|- (!z. (((real_le (Im (complex_inv z))) (real_of_num (NUMERAL _0))) = ((real_le (real_of_num (NUMERAL _0))) (Im z))))
REAL_SUB_ADD2
|- (!x. (!y. (((real_add y) ((real_sub x) y)) = x)))
REAL_INTEGRAL
|- (((real_integrable_on f) s) ==> (((real_integral s) f) = (drop ((integral ((IMAGE lift) s)) (lift o (f o drop))))))
REAL_ADD_AC
|- ((((real_add m) n) = ((real_add n) m)) /\ ((((real_add ((real_add m) n)) p) = ((real_add m) ((real_add n) p))) /\ (((real_add m) ((real_add n) p)) = ((real_add n) ((real_add m) p)))))
absolutely_real_integrable_on
|- (!f. (!s. (((absolutely_real_integrable_on f) s) = (((real_integrable_on f) s) /\ ((real_integrable_on (\x. (real_abs (f x)))) s)))))
real_convex_on
|- (!s. (!f. (((real_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 (f ((real_add ((real_mul u) x)) ((real_mul v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))
CONTINUOUS_LIFT_PRODUCT
|- (!net. (!f. (!t. (((FINITE t) /\ (!i. ((i IN t) ==> ((continuous (\x. (lift ((f x) i)))) net)))) ==> ((continuous (\x. (lift ((product t) (f x))))) net)))))
MIN
|- (!m. (!n. (((MIN m) n) = (((COND (m <= n)) m) n))))
CONTINUOUS_AT_CEXP
|- (!z. ((continuous cexp) (at z)))
Collect_geom.DELTA_POS_4POINTS
|- (!x1. (!x2. (!x3. (!x4. ((real_le (real_of_num (NUMERAL _0))) ((((((delta ((real_pow (distance (x1 , x2))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x1 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x3))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x2 , x4))) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (distance (x3 , x4))) (NUMERAL (BIT0 (BIT1 _0))))))))))
CSQRT_PRINCIPAL
|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (Re (csqrt z))) \/ (((Re (csqrt z)) = (real_of_num (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) (Im (csqrt z))))))
Hypermap_iso.hyp_iso_card_components
|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) = (CARD ((face G) (f x)))) /\ (((CARD ((node H) x)) = (CARD ((node G) (f x)))) /\ ((CARD ((edge H) x)) = (CARD ((edge G) (f x))))))))))))
DIM_EQ_SPAN
|- (!s. (!t. (((s SUBSET t) /\ ((dim t) <= (dim s))) ==> ((span s) = (span t)))))
CLOSED_TRANSLATION
|- (!s. (!a. ((closed s) ==> (closed ((IMAGE (\x. ((vector_add a) x))) s)))))
Lp_ineqs_def.y7_fan
|- (!V. (!E. (!x. (((y7_fan (V , E)) x) = (vector_norm (FST (((POWER (inverse (f_fan_pair_ext (V , E)))) (NUMERAL (BIT0 (BIT1 _0)))) x)))))))
IN_ELIM_PAIR_THM
|- (!P. (!a. (!b. (((a , b) IN (GSPEC (\GEN%PVAR%18. (?x. (?y. (((SETSPEC GEN%PVAR%18) ((P x) y)) (x , y))))))) = ((P a) b)))))
Hypermap_and_fan.NODE_HYPERMAP_OF_FAN_ALT
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (((node (hypermap_of_fan (V , E))) (v , w)) = (GSPEC (\GEN%PVAR%5635. (?u. (((SETSPEC GEN%PVAR%5635) (u IN (((set_of_edge v) V) E))) (v , u)))))))))))
REAL_INTEGRABLE_EQ
|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_integrable_on f) s)) ==> ((real_integrable_on g) s)))))
PAIR
|- (!x. (((FST x) , (SND x)) = x))
COMPLEX_NORM_NZ
|- (!z. (((real_lt (real_of_num (NUMERAL _0))) (vector_norm z)) = (~ (z = (Cx (real_of_num (NUMERAL _0)))))))
SYMMETRIC_MATRIX
|- (!A. (((transp A) = A) = ((adjoint (\x. ((matrix_vector_mul A) x))) = (\x. ((matrix_vector_mul A) x)))))
AFFINE_BOUNDED_EQ_LOWDIM
|- (!s. ((affine s) ==> ((bounded s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL _0))))))
Float.FLOAT_ADD_PP
|- (!a. (!b. (!m. (!n. ((n <= m) ==> (((real_add ((float a) (int_of_num m))) ((float b) (int_of_num n))) = ((float ((int_add ((int_mul (int_of_num ((NUMERAL (BIT0 (BIT1 _0))) EXP (m - n)))) a)) b)) (int_of_num n))))))))
DIVISION_OF_NONTRIVIAL
|- (!s. (!a. (!b. ((((division_of s) (closed_interval ((CONS (a , b)) NIL))) /\ (~ ((content (closed_interval ((CONS (a , b)) NIL))) = (real_of_num (NUMERAL _0))))) ==> ((division_of (GSPEC (\GEN%PVAR%2756. (?k. (((SETSPEC GEN%PVAR%2756) ((k IN s) /\ (~ ((content k) = (real_of_num (NUMERAL _0)))))) k))))) (closed_interval ((CONS (a , b)) NIL)))))))
TRIVIAL_LIMIT_SEQUENTIALLY
|- (~ (trivial_limit sequentially))
TC_TRANS_R
|- (!R. (!x. (!y. (!z. ((((R x) y) /\ (((TC R) y) z)) ==> (((TC R) x) z))))))
COUNTABLE
|- (!t. ((COUNTABLE t) = (UNIV >=_c t)))
COMPLEX_POW_INV
|- (!x. (!n. (((complex_pow (complex_inv x)) n) = (complex_inv ((complex_pow x) n)))))
DOT_CAUCHY_SCHWARZ_EQUAL
|- (!x. (!y. ((((real_pow ((dot x) y)) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul ((dot x) x)) ((dot y) y))) = (collinear ((vec (NUMERAL _0)) INSERT (x INSERT (y INSERT EMPTY)))))))
CONTINUOUS_ATREAL
|- (!f. (!x. (((continuous f) (atreal x)) = (((--> f) (f x)) (atreal x)))))
HAS_REAL_INTEGRAL_RESTRICT_INTER
|- (!f. (!s. (!t. ((((has_real_integral (\x. (((COND (x IN s)) (f x)) (real_of_num (NUMERAL _0))))) i) t) = (((has_real_integral f) i) (s INTER t))))))
HAS_REAL_DERIVATIVE_ADD
|- (!f. (!f'. (!g. (!g'. (!net. (((((has_real_derivative f) f') net) /\ (((has_real_derivative g) g') net)) ==> (((has_real_derivative (\x. ((real_add (f x)) (g x)))) ((real_add f') g')) net)))))))
ROW_TRANSP
|- (!A. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((row i) (transp A)) = ((column i) A)))))
Trigonometry2.LEMMA_SUM_ALL_OVER_CYCLIC_SET
|- (!W. (!xicm. (((FINITE W) /\ (((CARD W) = n) /\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\ (((polar_cycle_on xicm) W) /\ (W p))))) ==> ((?p. (?q. ((W p) /\ ((W q) /\ (~ (p = q)))))) ==> (((sum ((NUMERAL _0) .. (n - (NUMERAL (BIT1 _0))))) (\i. ((arg_diff (((ITER i) xicm) p)) (((ITER (i + (NUMERAL (BIT1 _0)))) xicm) p)))) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))))
VECTOR_NEG_COMPONENT
|- (!x. (!i. (((vector_neg x) $ i) = (real_neg (x $ i)))))
Hypermap_iso.hyp_iso_inverse_comm
|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((((inverse (edge_map G)) (f x)) = (f ((inverse (edge_map H)) x))) /\ ((((inverse (node_map G)) (f x)) = (f ((inverse (node_map H)) x))) /\ (((inverse (face_map G)) (f x)) = (f ((inverse (face_map H)) x)))))))))))
AR_TRANSLATION
|- (!a. (!s. ((AR ((IMAGE (\x. ((vector_add a) x))) s)) = (AR s))))
AFFINE_INTERS
|- ((!s. ((s IN f) ==> (affine s))) ==> (affine (INTERS f)))
Pack2.KIUMVTC
|- (!p. (!r. (!S. ((packing S) ==> (FINITE (S INTER (ball (p , r))))))))
LIM_COMPLEX_RMUL
|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) net)))))
REAL_LE_MUL
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)))))
CLOSED_INTER
|- (!s. (!t. (((closed s) /\ (closed t)) ==> (closed (s INTER t)))))
Seq.subseq_uniq
|- (!s1. (!s2. (((subseq s1) s2) ==> ((uniq s2) ==> (uniq s1)))))
Seq.filter_subseq
|- (!a. (!s. ((subseq ((filter a) s)) s)))
SIMPLEX_TRANSLATION_EQ
|- (!a. (!s. (!n. (((simplex n) ((IMAGE (\x. ((vector_add a) x))) s)) = ((simplex n) s)))))
DIFFERENTIABLE_NORM_AT
|- (!a. ((~ (a = (vec (NUMERAL _0)))) ==> ((differentiable (\x. (lift (vector_norm x)))) (at a))))
CARD_EXP_CANTOR
|- (!s. (s <_c ((^_c UNIV) s)))
NOT_IMP
|- (!t1. (!t2. ((~ (t1 ==> t2)) = (t1 /\ (~ t2)))))
Ssrnat.muln0
|- (!n. ((n * (NUMERAL _0)) = (NUMERAL _0)))
Sphere.acs_sqrt_x1_d4
|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((acs_sqrt_x1_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x1)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))
LE_ADDR
|- (!m. (!n. (n <= (m + n))))
Audit_formal_proof.chk_ball_annulus_def
|- (!v. ((v IN ball_annulus) = (((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (vector_norm v)) /\ ((real_le (vector_norm v)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))))
COMPACT_INTER
|- (!s. (!t. (((compact s) /\ (compact t)) ==> (compact (s INTER t)))))
Add_triangle.f_split_fan_face
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((((LET (\w1. (\d2. (\f. (LET_END (((f (v , w)) = (w , w1)) /\ (((f (w , w1)) = (w1 , v)) /\ (((f (w1 , w)) = d2) /\ (((f ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1)) = (w1 , w)) /\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\ (~ ((d = (v , w)) \/ ((d = (w , w1)) \/ ((d = (w1 , w)) \/ (d = ((((((sigma_fan (vec (NUMERAL _0))) V) E) w1) v) , w1))))))) ==> ((f d) = ((f_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (f_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))
Lp_ineqs_proofs.f_fan_pair_ext_power_in_dart
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!k. (!d. ((d IN (dart_of_fan (V , E))) ==> ((((POWER (f_fan_pair_ext (V , E))) k) d) IN (dart_of_fan (V , E)))))))))
ORTHOGONAL_TRANSFORMATION_INJECTIVE
|- (!f. ((orthogonal_transformation f) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y))))))
CLOSURE_EQ_EMPTY
|- (!s. (((closure s) = EMPTY) = (s = EMPTY)))
Fan.MONO_SIGMA_FAN
|- (!x. (!V. (!E. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((v INSERT (w INSERT EMPTY)) IN E) /\ ((((((sigma_fan x) V) E) v) u) = (((((sigma_fan x) V) E) v) w))))) ==> (u = w))))))))
Hypermap.node
|- (!H. (!x. (((node H) x) = ((orbit_map (node_map H)) x))))
Trigonometry2.ups_x
|- (!x1. (!x2. (!x6. ((((ups_x x1) x2) x6) = ((real_add ((real_sub ((real_sub ((real_mul (real_neg x1)) x1)) ((real_mul x2) x2))) ((real_mul x6) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x1) x2))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul x2) x6)))))))))
Sphere.hplus
|- (hplus = ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0)))))))))))))))))
real_measurable
|- (!s. ((real_measurable s) = (?m. ((has_real_measure s) m))))
DIMINDEX_4
|- ((dimindex UNIV) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))
FORALL_DROP
|- ((!x. (P x)) = (!x. (P (drop x))))
MIDPOINT_IN_SEGMENT
|- ((!a. (!b. ((midpoint (a , b)) IN (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. (((midpoint (a , b)) IN (open_segment (a , b))) = (~ (a = b))))))
RELATIVE_ORIENTATION
|- (!A. (!f. (!s. (((rows A) = (GSPEC (\GEN%PVAR%2550. (?v. (((SETSPEC GEN%PVAR%2550) ((extreme_point_of v) s)) v))))) ==> (((relative_orientation f) s) = (real_sgn ((real_div (det ((maprows f) A))) (det A))))))))
REAL_POLY_NEG_CLAUSES
|- ((!x. ((real_neg x) = ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x))) /\ (!x. (!y. (((real_sub x) y) = ((real_add x) ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) y))))))
has_bounded_variation_on
|- (!f. (!s. (((has_bounded_variation_on f) s) = ((has_bounded_setvariation_on (\k. ((vector_sub (f (interval_upperbound k))) (f (interval_lowerbound k))))) s))))
REAL_SUB_RDISTRIB
|- (!x. (!y. (!z. (((real_mul ((real_sub x) y)) z) = ((real_sub ((real_mul x) z)) ((real_mul y) z))))))
Seq.nseq
|- (!n. (!x. (((nseq n) x) = (((ncons n) x) NIL))))
Hypermap_and_fan.FULLY_SURROUNDED
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) = (!v. ((v IN V) ==> ((surrounded_node (V , E)) v)))))))
Ssrnat.mul1n
|- (!n. (((NUMERAL (BIT1 _0)) * n) = n))
ORTHOGONAL_MATRIX_ORTHONORMAL_ROWS_PAIRWISE
|- (!A. ((orthogonal_matrix A) = (((CARD (rows A)) = (dimindex UNIV)) /\ ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((vector_norm ((row i) A)) = (real_of_num (NUMERAL (BIT1 _0)))))) /\ ((pairwise orthogonal) (rows A))))))
NUMSEG_OFFSET_IMAGE
|- (!m. (!n. (!p. (((m + p) .. (n + p)) = ((IMAGE (\i. (i + p))) (m .. n))))))
CONVERGENT_IMP_BOUNDED
|- (!s. (!l. ((((--> s) l) sequentially) ==> (bounded ((IMAGE s) UNIV)))))
HAS_DERIVATIVE_CONST
|- (!c. (!net. (((has_derivative (\x. c)) (\h. (vec (NUMERAL _0)))) net)))
Collect_geom.DELTA_COEFS
|- (!x12. (!x13. (!x14. (!x23. (!x24. (!x34. (((((((delta x12) x13) x14) x23) x24) x34) = ((real_add ((real_mul (real_neg x12)) ((real_pow x34) (NUMERAL (BIT0 (BIT1 _0)))))) ((real_add ((real_mul (((((b_coef x12) x13) x14) x23) x24)) x34)) (((((c_coef x12) x13) x14) x23) x24))))))))))
LIM_NULL_COMPLEX_SUB
|- (!net. (!f. (!g. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> (\x. ((vector_sub (f x)) (g x)))) (Cx (real_of_num (NUMERAL _0)))) net)))))
CLOSED_IN_SUBSET_TRANS
|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) u)) s) /\ ((s SUBSET t) /\ (t SUBSET u))) ==> ((closed_in ((subtopology euclidean) t)) s)))))
REAL_CONTINUOUS_AT_ID
|- (!x. ((real_continuous (\x. x)) (atreal x)))
AFFINE_HULL_LINEAR_IMAGE
|- (!f. (!s. ((linear f) ==> (((hull affine) ((IMAGE f) s)) = ((IMAGE f) ((hull affine) s))))))
int_sgn_th
|- (!x. ((real_of_int (int_sgn x)) = (real_sgn (real_of_int x))))
ABS_DROP
|- (!x. ((vector_norm x) = (real_abs (drop x))))
ARC_LINEAR_IMAGE_EQ
|- (!f. (!g. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((arc (f o g)) = (arc g)))))
higher_complex_derivative
|- ((((higher_complex_derivative (NUMERAL _0)) f) = f) /\ (!n. (((higher_complex_derivative (SUC n)) f) = (complex_derivative ((higher_complex_derivative n) f)))))
ANR_FROM_UNION_AND_INTER_LOCAL
|- (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\ ((ANR (s UNION t)) /\ (ANR (s INTER t))))) ==> ((ANR s) /\ (ANR t)))))
CONTINUOUS_ON_CLOSEST_POINT
|- (!s. (!t. (((convex s) /\ ((closed s) /\ (~ (s = EMPTY)))) ==> ((continuous_on (closest_point s)) t))))
REAL_MIN_LE
|- (!x. (!y. (!z. (((real_le ((real_min x) y)) z) = (((real_le x) z) \/ ((real_le y) z))))))
Functional_equation.functional_arclength_x_123'
|- (arclength_x_123' = (uni (acs , ((div6 ((add6 proj_x1) ((sub6 proj_x2) proj_x3))) (uni (sqrt , ((scalar6 ((mul6 proj_x1) proj_x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))
NEGLIGIBLE_IMP_LEBESGUE_MEASURABLE
|- (!s. ((negligible s) ==> (lebesgue_measurable s)))
CONVEX_HULL_LINEAR_IMAGE
|- (!f. (!s. ((linear f) ==> (((hull convex) ((IMAGE f) s)) = ((IMAGE f) ((hull convex) s))))))
Xwitccn.PERIODIC_PROPERTY
|- (((~ (k = (NUMERAL _0))) /\ ((periodic vv) k)) ==> (!i. ((vv (i MOD k)) = (vv i))))
DOT_POS_LT
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) ((dot x) x)) = (~ (x = (vec (NUMERAL _0))))))
HAS_VECTOR_DERIVATIVE_CONST
|- (!c. (!net. (((has_vector_derivative (\x. c)) (vec (NUMERAL _0))) net)))
REAL_SGN_POW
|- (!x. (!n. ((real_sgn ((real_pow x) n)) = ((real_pow (real_sgn x)) n))))
Misc_defs_and_lemmas.min_least
|- (!X. (!c. ((X c) ==> ((X (min_num X)) /\ ((min_num X) <= c)))))
SUM_CASES
|- (!s. (!P. (!f. (!g. ((FINITE s) ==> (((sum s) (\x. (((COND (P x)) (f x)) (g x)))) = ((real_add ((sum (GSPEC (\GEN%PVAR%213. (?x. (((SETSPEC GEN%PVAR%213) ((x IN s) /\ (P x))) x))))) f)) ((sum (GSPEC (\GEN%PVAR%214. (?x. (((SETSPEC GEN%PVAR%214) ((x IN s) /\ (~ (P x)))) x))))) g))))))))
Seq.foldr
|- (((((foldr f) z0) ((CONS x) s)) = ((f x) (((foldr f) z0) s))) /\ ((((foldr f) z0) NIL) = z0))
REAL_SUMMABLE
|- (((real_summable s) f) = ((summable s) (lift o f)))
DOT_RADD
|- (!x. (!y. (!z. (((dot x) ((vector_add y) z)) = ((real_add ((dot x) y)) ((dot x) z))))))
Real_ext.REAL_LE_LMUL_IMP
|- (!x. (!y. (!z. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le y) z)) ==> ((real_le ((real_mul x) y)) ((real_mul x) z))))))
CNJ_MUL
|- (!w. (!z. ((cnj ((complex_mul w) z)) = ((complex_mul (cnj w)) (cnj z)))))
Fatugpd.norm_mul_normalize
|- (!v. (((% (vector_norm v)) (normalize v)) = v))
Hypermap.lemma_transitive_permutation
|- (!L. (!x. (((in_dart_of_loop x) L) ==> ((dart_of_loop L) = ((orbit_map (loop_map L)) x)))))
COMPLEX_ADD_LID
|- (!x. (((vector_add (Cx (real_of_num (NUMERAL _0)))) x) = x))
Hypermap_and_fan.FINITE_ORBIT_MAP
|- (!s. (!f. (!x. (!n. (((FINITE s) /\ (((permutes f) s) /\ ((CARD ((orbit_map f) x)) = n))) ==> (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5631. (?k. (((SETSPEC GEN%PVAR%5631) (k < n)) (((POWER f) k) x)))))))))))
LEBESGUE_MEASURABLE_CLOSED_IN
|- (!s. (!t. ((((closed_in ((subtopology euclidean) s)) t) /\ (lebesgue_measurable s)) ==> (lebesgue_measurable t))))
COMPLEX_NORM_POW
|- (!z. (!n. ((vector_norm ((complex_pow z) n)) = ((real_pow (vector_norm z)) n))))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT3
|- (!x. (!V. (!E. (!E1. (!v. (!u. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ (((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) /\ ((fan80 (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))))))) ==> ((((((sigma_fan x) V) E1) w) v) = u)))))))))
SUBSET_UNION
|- ((!s. (!t. (s SUBSET (s UNION t)))) /\ (!s. (!t. (s SUBSET (t UNION s)))))
REVERSEPATH_LINEAR_IMAGE
|- (!f. (!g. ((linear f) ==> ((reversepath (f o g)) = (f o (reversepath g))))))
TRANSITIVE_STEPWISE_LE
|- (!R. (((!x. ((R x) x)) /\ ((!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))) /\ (!n. ((R n) (SUC n))))) ==> (!m. (!n. ((m <= n) ==> ((R m) n))))))
VECTOR_SUB_LZERO
|- (((vector_sub (vec (NUMERAL _0))) x) = (vector_neg x))
UNIONS_MONO_IMAGE
|- ((!x. ((x IN s) ==> ((f x) SUBSET (g x)))) ==> ((UNIONS ((IMAGE f) s)) SUBSET (UNIONS ((IMAGE g) s))))
OPEN_INTER
|- (!s. (!t. (((open s) /\ (open t)) ==> (open (s INTER t)))))
Functional_equation.functional_gchi4_x'
|- (gchi4_x' = ((mul6 (uni (gchi , proj_y4))) dih4_x'))
SETDIST_SYM
|- (!s. (!t. ((setdist (s , t)) = (setdist (t , s)))))
PI_NZ
|- (~ (pi = (real_of_num (NUMERAL _0))))
NSUM_SING_NUMSEG
|- (!f. (!n. (((nsum (n .. n)) f) = (f n))))