hcju/holps huggingface dataset:
Yxionxl.PRO_EQU_ID1
|- ((((scs_k_v39 s) = k) /\ (is_scs_v39 s)) ==> (s = ((scs_prop_equ_v39 ((scs_prop_equ_v39 s) i)) (k - (i MOD k)))))
list_INDUCT
|- (!P. (((P NIL) /\ (!a0. (!a1. ((P a1) ==> (P ((CONS a0) a1)))))) ==> (!x. (P x))))
MEM_TRANSLATION
|- (!a. (!x. (!l. (((MEM ((vector_add a) x)) ((MAP (\x. ((vector_add a) x))) l)) = ((MEM x) l)))))
LINEAR_CONTINUOUS_AT
|- (!f. (!a. ((linear f) ==> ((continuous f) (at a)))))
HAS_REAL_MEASURE
|- (!s. (!m. (((has_real_measure s) m) = (((has_real_integral (\x. (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0))))) m) UNIV))))
MATRIX_NEG_MINUS1
|- (!A. ((matrix_neg A) = ((%% (real_neg (real_of_num (NUMERAL (BIT1 _0))))) A)))
Seq.rotr1_rcons
|- (!x. (!s. (((rotr (NUMERAL (BIT1 _0))) ((rcons s) x)) = ((CONS x) s))))
RELATIVE_INTERIOR_SUBSET
|- (!s. ((relative_interior s) SUBSET s))
Kepler_spec.BIJ_REAL3_TRIPLE
|- (((triple_of_real3 o real3_of_triple) = I) /\ ((real3_of_triple o triple_of_real3) = I))
SQRT_MONO_LT
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_lt x) y)) ==> ((real_lt (sqrt x)) (sqrt y)))))
FACE_OF_TRANSLATION_EQ
|- (!a. (!f. (!s. (((face_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((face_of f) s)))))
components
|- (!s. ((components s) = (GSPEC (\GEN%PVAR%904. (?x. (((SETSPEC GEN%PVAR%904) (x IN s)) ((connected_component s) x)))))))
Trigonometry2.EXISTS_PROJECTING_POINT2
|- (!p. (!u. (!v0. (?pp. ((pp IN (aff (u INSERT (v0 INSERT EMPTY)))) /\ (((dot ((vector_sub p) pp)) ((vector_sub u) v0)) = (real_of_num (NUMERAL _0))))))))
REAL_POW_ROOT
|- (!n. (!x. (((ODD n) \/ ((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) x))) ==> (((real_pow ((root n) x)) n) = x))))
COND_ID
|- (!b. (!t. ((((COND b) t) t) = t)))
REAL_LTE_ANTISYM
|- (!x. (!y. (~ (((real_lt x) y) /\ ((real_le y) x)))))
EMPTY_AS_INTERVAL
|- (EMPTY = (closed_interval ((CONS ((vec (NUMERAL (BIT1 _0))) , (vec (NUMERAL _0)))) NIL)))
NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE
|- (!f. (!s. (!t. ((((absolutely_continuous_on f) s) /\ ((is_interval s) /\ ((negligible t) /\ (t SUBSET s)))) ==> (negligible ((IMAGE f) t))))))
Trigonometry.AVWKGNB
|- (!x3. (!x1. (!x2. (!x4. (!x5. (!x6. (((((((delta_x x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul x1) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6))))))) ((real_add ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add x3) ((real_add ((real_sub x4) x5)) x6)))))) ((real_sub ((real_sub ((real_sub ((real_sub ((real_mul x3) ((real_mul x6) ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_add x4) ((real_sub x5) x6))))))) ((real_mul x2) ((real_mul x3) x4)))) ((real_mul x1) ((real_mul x3) x5)))) ((real_mul x1) ((real_mul x2) x6)))) ((real_mul x4) ((real_mul x5) x6))))))))))))
Lp_ineqs_def.y2_fan
|- (!x. ((y2_fan x) = (vector_norm (SND x))))
List_hypermap_iso.fan_list_nodes_iso
|- (!f. (!x. (!V. (!E. (((fan_list_nodes_iso (f , (V , E))) x) = (FST (f (x , (((COND ((((set_of_edge x) V) E) = EMPTY)) x) (CHOICE (((set_of_edge x) V) E)))))))))))
PATH_IMAGE_LINEAR_IMAGE
|- (!f. (!g. ((linear f) ==> ((path_image (f o g)) = ((IMAGE f) (path_image g))))))
REAL_LT_LDIV_EQ
|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt ((real_div x) z)) y) = ((real_lt x) ((real_mul y) z)))))))
MEASURABLE_ON_PREIMAGE_BOREL
|- (!f. (((measurable_on f) UNIV) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3549. (?x. (((SETSPEC GEN%PVAR%3549) ((f x) IN t)) x)))))))))
PATH_CONNECTED_IFF_PATH_COMPONENT
|- (!s. ((path_connected s) = (!x. (!y. (((x IN s) /\ (y IN s)) ==> (((path_component s) x) y))))))
Seq.filter
|- (!a. (!x. (!s'. ((((filter a) NIL) = NIL) /\ (((filter a) ((CONS x) s')) = (((COND (a x)) ((CONS x) ((filter a) s'))) ((filter a) s')))))))
HAS_COMPLEX_DERIVATIVE_CASN
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative casn) (complex_inv (ccos (casn z)))) (at z))))
EVEN_MOD
|- (!n. ((EVEN n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL _0))))
AFF_DIM_HYPERPLANE
|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((aff_dim (GSPEC (\GEN%PVAR%1576. (?x. (((SETSPEC GEN%PVAR%1576) (((dot a) x) = b)) x))))) = ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))))))
LIPSCHITZ_LIM
|- (!f. (!g. (!s. (!b. (((!n. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) ((real_mul b) (vector_norm ((vector_sub x) y)))))))) /\ (!x. ((x IN s) ==> (((--> (\n. ((f n) x))) (g x)) sequentially)))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))))))))
Appendix.tgt
|- (tgt = ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))
CCOS_ADD
|- (!w. (!z. ((ccos ((vector_add w) z)) = ((vector_sub ((complex_mul (ccos w)) (ccos z))) ((complex_mul (csin w)) (csin z))))))
REAL_MUL_AC
|- ((((real_mul m) n) = ((real_mul n) m)) /\ ((((real_mul ((real_mul m) n)) p) = ((real_mul m) ((real_mul n) p))) /\ (((real_mul m) ((real_mul n) p)) = ((real_mul n) ((real_mul m) p)))))
Hypermap_iso.hyp_iso_inj
|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!x. (!y. (((x IN (dart H)) /\ ((y IN (dart H)) /\ ((f x) = (f y)))) ==> (x = y))))))))
OPEN_IN_IMP_SUBSET
|- (!top. (!s. (!t. (((open_in ((subtopology top) s)) t) ==> (t SUBSET s)))))
Hypermap_iso.hyp_iso_surj
|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!y. ((y IN (dart G)) ==> (?x. ((x IN (dart H)) /\ ((f x) = y)))))))))
Wrgcvdr_cizmrrh.wedge_in_fan_ge
|- (!E. (!v. (!w. (((wedge_in_fan_ge (v , w)) E) = (((COND ((CARD ((EE v) E)) > (NUMERAL (BIT1 _0)))) ((((wedge_ge (vec (NUMERAL _0))) v) w) ((((azim_cycle ((EE v) E)) (vec (NUMERAL _0))) v) w))) (GSPEC (\GEN%PVAR%6280. (?x. (((SETSPEC GEN%PVAR%6280) T) x)))))))))
Ssrnat.ltE
|- (!n. (!m. ((n < m) = ((SUC n) <= m))))
HOMOTOPIC_TRIVIALITY
|- (!s. (!t. ((!f. (!g. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) s) /\ (((IMAGE g) s) SUBSET t)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g)))) = (((s = EMPTY) \/ (path_connected t)) /\ (!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET t)) ==> (?c. ((((homotopic_with (\x. T)) (s , t)) f) (\x. c)))))))))
complex_differentiable
|- (!f. (!net. (((complex_differentiable f) net) = (?f'. (((has_complex_derivative f) f') net)))))
MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE
|- (!f. (!g. (!s. ((((measurable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((absolutely_integrable_on f) s)))))
QUANTIFY_SURJECTION_THM
|- (!f. ((!y. (?x. ((f x) = y))) ==> (((!P. ((!x. (P x)) = (!x. (P (f x))))) /\ ((!P. ((?x. (P x)) = (?x. (P (f x))))) /\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE f) s))))) /\ (!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE f) s)))))))) /\ (!P. ((GSPEC (\GEN%PVAR%236. (?x. (((SETSPEC GEN%PVAR%236) (P x)) x)))) = ((IMAGE f) (GSPEC (\GEN%PVAR%237. (?x. (((SETSPEC GEN%PVAR%237) (P (f x))) x))))))))))
Lp_main_estimate.pent_diags_imp
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\f. (LET_END ((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)))) /\ ((~ (a = b)) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \/ ((a , b) = (w' , w))) \/ ((((a , b) = (z , v)) \/ ((a , b) = (v , z))) \/ ((((a , b) = (u , w)) \/ ((a , b) = (w , u))) \/ ((((a , b) = (w' , z)) \/ ((a , b) = (z , w'))) \/ (((a , b) = (v , u)) \/ ((a , b) = (u , v))))))))))) (((((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))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))
COUNTABLE_COMPACT_OPEN_IN
|- (!u. (COUNTABLE (GSPEC (\GEN%PVAR%1245. (?c. (((SETSPEC GEN%PVAR%1245) ((compact c) /\ ((open_in ((subtopology euclidean) u)) c))) c))))))
PCROSS_EQ_EMPTY
|- (!s. (!t. (((s PCROSS t) = EMPTY) = ((s = EMPTY) \/ (t = EMPTY)))))
int_neg_th
|- (!x. ((real_of_int (int_neg x)) = (real_neg (real_of_int x))))
NORM_FSTCART
|- (!x. ((real_le (vector_norm (fstcart x))) (vector_norm x)))
OPEN_IN
|- (!s. ((open s) = ((open_in euclidean) s)))
VECTOR_SUB_LDISTRIB
|- (((% c) ((vector_sub x) y)) = ((vector_sub ((% c) x)) ((% c) y)))
Ssrnat.fact_gt0
|- (!n. ((NUMERAL _0) < (FACT n)))
HULL_MINIMAL
|- (!P. (!s. (!t. (((s SUBSET t) /\ (P t)) ==> (((hull P) s) SUBSET t)))))
LEFT_ADD_DISTRIB
|- (!m. (!n. (!p. ((m * (n + p)) = ((m * n) + (m * p))))))
HOMEOMORPHIC_OPEN_INTERVALS_1
|- (!a. (!b. (!c. (!d. ((((real_lt (drop a)) (drop b)) /\ ((real_lt (drop c)) (drop d))) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))
OPEN_IN_DELETE
|- (!u. (!s. (!a. (((open_in ((subtopology euclidean) u)) s) ==> ((open_in ((subtopology euclidean) u)) (s DELETE a))))))
has_derivative_at
|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) = ((linear f') /\ (((--> (\y. ((% (real_inv (vector_norm ((vector_sub y) x)))) ((vector_sub (f y)) ((vector_add (f x)) (f' ((vector_sub y) x))))))) (vec (NUMERAL _0))) (at x)))))))
RELATIVE_INTERIOR_TRANSLATION
|- (!a. (!s. ((relative_interior ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (relative_interior s)))))
IN_SPHERE
|- (!x. (!y. (!e. ((y IN (sphere (x , e))) = ((distance (x , y)) = e)))))
Ssrnat.mulSnr
|- (!m. (!n. (((SUC m) * n) = ((m * n) + n))))
complex_div
|- (!w. (!z. (((complex_div w) z) = ((complex_mul w) (complex_inv z)))))
MIDPOINT_LINEAR_IMAGE
|- (!f. (!a. (!b. ((linear f) ==> ((midpoint ((f a) , (f b))) = (f (midpoint (a , b))))))))
BOUNDED_SUBSET_BALL
|- (!s. (!x. ((bounded s) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\ (s SUBSET (ball (x , r))))))))
PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION
|- (!g. (!e. (((path g) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?p. ((vector_polynomial_function p) /\ (((pathstart p) = (pathstart g)) /\ (((pathfinish p) = (pathfinish g)) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (p t)) (g t)))) e))))))))))
VEC_COMPONENT
|- (!k. (!i. (((vec k) $ i) = (real_of_num k))))
INT_LT_ANTISYM
|- (!x. (!y. (~ (((int_lt x) y) /\ ((int_lt y) x)))))
VECTOR_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL
|- (!f. (!a. (!b. (!x. (!f'. (!f''. ((((real_lt (drop a)) (drop b)) /\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((((has_vector_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\ (((has_vector_derivative f) f'') ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (f' = f''))))))))
REAL_CONTINUOUS_FRAC
|- (!x. ((~ (integer x)) ==> ((real_continuous frac) (atreal x))))
LOWDIM_EQ_INTER_HYPERPLANE
|- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((t SUBSET s) /\ (((dim t) + (NUMERAL (BIT1 _0))) = (dim s))))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ (((GSPEC (\GEN%PVAR%191. (?x. (((SETSPEC GEN%PVAR%191) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))) INTER s) = t))))))
AFFINE_IMP_SUBSPACE
|- (!s. (((affine s) /\ ((vec (NUMERAL _0)) IN s)) ==> (subspace s)))
NORM_EQ_SQUARE
|- (!x. (((vector_norm x) = a) = (((real_le (real_of_num (NUMERAL _0))) a) /\ (((dot x) x) = ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))
columnvector
|- (!v. ((columnvector v) = (lambdai. (lambdaj. (v $ i)))))
HAS_REAL_DERIVATIVE_CARATHEODORY_WITHINREAL
|- (!f. (!f'. (!z. (!s. ((((has_real_derivative f) f') ((within (atreal z)) s)) = (?g. ((!w. (((real_sub (f w)) (f z)) = ((real_mul (g w)) ((real_sub w) z)))) /\ (((real_continuous g) ((within (atreal z)) s)) /\ ((g z) = f')))))))))
Ssrnat.leq_sub_add
|- (!m. (!n. (!p. (((m - n) <= p) = (m <= (n + p))))))
Ssrnat.muln_addr
|- (!m. (!n1. (!n2. ((m * (n1 + n2)) = ((m * n1) + (m * n2))))))
FACET_OF_LINEAR_IMAGE
|- (!f. (!c. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((facet_of ((IMAGE f) c)) ((IMAGE f) s)) = ((facet_of c) s))))))
Lp_ineqs_def.y5_fan
|- (!V. (!E. (!x. (((y5_fan (V , E)) x) = (distance ((inverse (f_fan_pair_ext (V , E))) x))))))
Ssrnat.minn_maxl
|- (!m1. (!m2. (!n. (((minn ((maxn m1) m2)) n) = ((maxn ((minn m1) n)) ((minn m2) n))))))
Conforming.properties_of_elements_in_node_fully_surroundedfan
|- (!x. (!V. (!E. (!ds. (!y. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((ds IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\ (y IN ds)))) ==> (((pr2 y) INSERT ((pr3 y) INSERT EMPTY)) IN E)))))))
POSITIVE_SEMIDEFINITE_CMUL
|- (!c. (!A. (((positive_semidefinite A) /\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (positive_semidefinite ((%% c) A)))))
Tame_general.CONTRAVENING_IMP_FULLY_SURROUNDED
|- (!V. (((FAN ((vec (NUMERAL _0)) , (V , (ESTD V)))) /\ (contravening V)) ==> (fully_surrounded (V , (ESTD V)))))
RE_CMUL
|- (!a. (!z. ((Re ((% a) z)) = ((real_mul a) (Re z)))))
EDGE_OF_LINEAR_IMAGE
|- (!f. (!c. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((edge_of ((IMAGE f) c)) ((IMAGE f) s)) = ((edge_of c) s))))))
FORALL_VECTOR_3
|- ((!v. (P v)) = (!x. (!y. (!z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))
Real_ext.REAL_PROP_LT_LMUL
|- (!x. (!y. (!z. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt y) z)) ==> ((real_lt ((real_mul x) y)) ((real_mul x) z))))))
FSIGMA_LOCALLY_COMPACT
|- (!s. (((locally compact) s) ==> (fsigma s)))
REAL_EXP_N
|- (!n. (!x. ((exp ((real_mul (real_of_num n)) x)) = ((real_pow (exp x)) n))))
Lp_ineqs_proofs.lp_cond_trans1
|- (!L. (!V. (!E. (!g. (!h. (!d. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_hyp L) d) , (g , h))) (V , ((add_diag (V , E)) (g d)))))))))))
BILINEAR_DROP_MUL
|- (bilinear (\x. (\y. ((% (drop x)) y))))
LOCALLY_CONNECTED_IM_KLEINEN
|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((x IN u) /\ ((u SUBSET v) /\ (!y. ((y IN u) ==> (?c. ((connected c) /\ ((c SUBSET v) /\ ((x IN c) /\ (y IN c))))))))))))))))
real_abs
|- (!x. ((real_abs x) = (((COND ((real_le (real_of_num (NUMERAL _0))) x)) x) (real_neg x))))
Hypermap_iso.inE
|- ((!p. (!x. ((x IN (GSPEC (\v. (?y. (((SETSPEC v) (p y)) y))))) = (p x)))) /\ ((!P. (!x. (((GSPEC (\v. (P (SETSPEC v)))) x) = (P (\p. (\t. (p /\ (x = t)))))))) /\ ((!p. (!x. (((GSPEC (\v. (?y. (((SETSPEC v) (p y)) y)))) x) = (p x)))) /\ (!p. (!x. ((x IN (\y. (p y))) = (p x)))))))
RC_INC
|- (!R. (!x. (!y. (((R x) y) ==> (((RC R) x) y)))))
Functional_equation.functional_mud_234_x
|- (!y1. (!y5. (!y6. (((domain6 (\x1. (\x2. (\x3. (\x4. (\x5. (\x6. ((real_le (real_of_num (NUMERAL _0))) y1)))))))) (((mud_234_x_v1 y1) y5) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y1) y1))) proj_x2) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_234_x ((real_mul y1) y1)) ((real_mul y5) y5)) ((real_mul y6) y6)))))))))
REAL_LE_TRANS
|- (!x. (!y. (!z. ((((real_le x) y) /\ ((real_le y) z)) ==> ((real_le x) z)))))
Hypermap.lemma_face_identity
|- (!H. (!x. (!y. ((y IN ((face H) x)) ==> (((face H) x) = ((face H) y))))))
Tame_lemmas.reflection_at0
|- (!x. (((reflection (vec (NUMERAL _0))) x) = x))
HAS_REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG
|- (!f. (!s. (((FINITE (GSPEC (\GEN%PVAR%4933. (?x. (((SETSPEC GEN%PVAR%4933) ((x IN s) /\ (~ ((f x) = EMPTY)))) x))))) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((DISJOINT (f x)) (f y))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\x. (real_measure (f x))))))))
Trigonometry.KGLLRQT1
|- (!x. ((cos (real_neg x)) = (cos x)))
DIVISION_OF_TAGGED_DIVISION
|- (!s. (!i. (((tagged_division_of s) i) ==> ((division_of ((IMAGE SND) s)) i))))
LIM_AT_POSINFINITY
|- (!f. (!l. ((((--> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))
ATN_ABS_LE_X
|- (!x. ((real_le (real_abs (atn x))) (real_abs x)))
Functional_equation.cos797
|- (cos797 = (cos ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))
NORM_EQ_0
|- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))
REAL_SUMS_INFSUM
|- (!f. (!s. ((((real_sums f) ((real_infsum s) f)) s) = ((real_summable s) f))))
CX_ABS
|- (!x. ((Cx (real_abs x)) = (Cx (vector_norm (Cx x)))))
Seq2.card_set_of_list_uniq
|- (!l. ((uniq l) ==> ((CARD (set_of_list l)) = (sizel l))))
CONVEX_INTER
|- (!s. (!t. (((convex s) /\ (convex t)) ==> (convex (s INTER t)))))
Ssrnat.eqn_leq
|- (!m. (!n. ((m = n) = ((m <= n) /\ (n <= m)))))
MIDPOINT_IN_CONVEX
|- (!s. (!x. (!y. (((convex s) /\ ((x IN s) /\ (y IN s))) ==> ((midpoint (x , y)) IN s)))))
DIM_SUBSET_UNIV
|- (!s. ((dim s) <= (dimindex UNIV)))
REALLIM_WITHIN_SUBSET
|- (!f. (!l. (!a. (!s. (!t. (((((---> f) l) ((within (at a)) s)) /\ (t SUBSET s)) ==> (((---> f) l) ((within (at a)) t))))))))
BOUNDED_DIFFS
|- (!s. (!t. (((bounded s) /\ (bounded t)) ==> (bounded (GSPEC (\GEN%PVAR%586. (?x. (?y. (((SETSPEC GEN%PVAR%586) ((x IN s) /\ (y IN t))) ((vector_sub x) y))))))))))
LIMIT_POINT_FINITE
|- (!s. (!a. ((FINITE s) ==> (~ ((limit_point_of a) s)))))
RPOW_POS_LE
|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) ((rpow x) y)))))
sgn_le
|- (sgn_le = (\t. ((real_le t) (real_of_num (NUMERAL _0)))))
Fan.fan7
|- (!E. (!V. (!x. ((fan7 (x , (V , E))) = (!e1. (!e2. (((e1 IN (E UNION (GSPEC (\GEN%PVAR%5449. (?v. (((SETSPEC GEN%PVAR%5449) (v IN V)) (v INSERT EMPTY))))))) /\ (e2 IN (E UNION (GSPEC (\GEN%PVAR%5450. (?v. (((SETSPEC GEN%PVAR%5450) (v IN V)) (v INSERT EMPTY)))))))) ==> ((((aff_ge (x INSERT EMPTY)) e1) INTER ((aff_ge (x INSERT EMPTY)) e2)) = ((aff_ge (x INSERT EMPTY)) (e1 INTER e2))))))))))
SURJ
|- (!t. (!s. (!f. ((((SURJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!x. ((x IN t) ==> (?y. ((y IN s) /\ ((f y) = x))))))))))
CX_DIV
|- (!x. (!y. ((Cx ((real_div x) y)) = ((complex_div (Cx x)) (Cx y)))))
MEASURE_INTERVAL_1
|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))
ROWS_TRANSP
|- (!A. ((rows (transp A)) = (columns A)))
REAL_SUB_LDISTRIB
|- (!x. (!y. (!z. (((real_mul x) ((real_sub y) z)) = ((real_sub ((real_mul x) y)) ((real_mul x) z))))))
DIST_1
|- (!x. (!y. ((distance (x , y)) = (real_abs ((real_sub (drop x)) (drop y))))))
SIMPLY_CONNECTED_FUNDAMENTAL_GROUP
|- (!s. ((simply_connected s) = ((path_connected s) /\ (!a. ((a IN s) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))))
HAS_INTEGRAL_LIM_AT_POSINFINITY
|- (!f. (!l. ((((has_integral f) l) (GSPEC (\GEN%PVAR%2953. (?t. (((SETSPEC GEN%PVAR%2953) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))) = ((!a. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , a)) NIL)))) /\ (((--> (\a. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift a))) NIL))) f))) l) at_posinfinity)))))
Local_lemmas1.localization
|- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))
Hypermap_and_fan.PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (~ (v = w)))))))
INTEGRABLE_BOUNDED_VARIATION_COMPLEX_LMUL
|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on (\x. ((complex_mul (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))
CONVEX_HULL_INSERT_REDUNDANT_POINT
|- (!s. (!a. (!b. (!c. (((a IN ((hull convex) (c INSERT s))) /\ ((b IN ((hull convex) (c INSERT s))) /\ (c IN (open_segment (a , b))))) ==> (((hull convex) (c INSERT s)) = ((hull convex) s)))))))
ORTHOGONAL_BASIS_BASIS
|- (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))) ==> (((orthogonal (basis i)) (basis j)) = (~ (i = j))))))
AFFINE_TRANSLATION_SUBSPACE
|- (!t. (((affine t) /\ (~ (t = EMPTY))) = (?a. (?s. ((subspace s) /\ (t = ((IMAGE (\x. ((vector_add a) x))) s)))))))
LOCALLY_CONNECTED
|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((connected u) /\ ((x IN u) /\ (u SUBSET v))))))))))
MATRIX_INV_INV
|- (!A. ((matrix_inv (matrix_inv A)) = A))
INDEPENDENT_STDBASIS
|- (independent (GSPEC (\GEN%PVAR%152. (?i. (((SETSPEC GEN%PVAR%152) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))
mat
|- (!k. ((mat k) = (lambdai. (lambdaj. (((COND (i = j)) (real_of_num k)) (real_of_num (NUMERAL _0)))))))
Local_lemmas.PRESERABLE_AFF_GE_SUBSET
|- ((a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) SUBSET ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))
CONTINUOUS_ON
|- (!f. (!s. (((continuous_on f) s) = (!x. ((x IN s) ==> (((--> f) (f x)) ((within (at x)) s)))))))
SUBPATH_TRANSLATION
|- (!a. (!g. (!u. (!v. ((((subpath u) v) ((\x. ((vector_add a) x)) o g)) = ((\x. ((vector_add a) x)) o (((subpath u) v) g)))))))
Real_ext.REAL_PROP_LE_RMUL
|- (!x. (!y. (!z. ((((real_le x) y) /\ ((real_le (real_of_num (NUMERAL _0))) z)) ==> ((real_le ((real_mul x) z)) ((real_mul y) z))))))
FORALL_PAIR_THM
|- (!P. ((!p. (P p)) = (!p1. (!p2. (P (p1 , p2))))))
CNJ_DIV
|- (!w. (!z. ((cnj ((complex_div w) z)) = ((complex_div (cnj w)) (cnj z)))))
Ssrbool.andbN
|- (!b. ((b /\ (~ b)) = F))
ABSOLUTELY_CONTINUOUS_ISOMETRIC
|- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance ((g x) , (g y)))))) ==> (((absolutely_continuous_on f) s) = ((absolutely_continuous_on g) s))))))
CLOSURE_SPHERE
|- (!a. (!r. ((closure (sphere (a , r))) = (sphere (a , r)))))
COMPLEX_MUL_RID
|- (!x. (((complex_mul x) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = x))
MATRIX_SUB_RZERO
|- (!A. (((matrix_sub A) (mat (NUMERAL _0))) = A))
vector_mul
|- (!c. (!x. (((% c) x) = (lambdai. ((real_mul c) (x $ i))))))
absolutely_integrable_on
|- (!f. (!s. (((absolutely_integrable_on f) s) = (((integrable_on f) s) /\ ((integrable_on (\x. (lift (vector_norm (f x))))) s)))))
REAL_MUL_SYM
|- (!x. (!y. (((real_mul x) y) = ((real_mul y) x))))
SUMMABLE_SUBSET
|- (!x. (!s. (!t. (((s SUBSET t) /\ ((summable t) (\i. (((COND (i IN s)) (x i)) (vec (NUMERAL _0)))))) ==> ((summable s) x)))))
Ssrnat.maxnC
|- (!m. (!n. (((maxn m) n) = ((maxn n) m))))
Ldurdpn.VECTOR_SCALE_CHANGE
|- ((~ (a = (real_of_num (NUMERAL _0)))) ==> ((((% a) x) = y) = (x = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) a)) y))))
PRODUCT_NEG_NUMSEG
|- (!f. (!m. (!n. (((product (m .. n)) (\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((n + (NUMERAL (BIT1 _0))) - m))) ((product (m .. n)) f))))))
Hypermap_iso.darts_of_fan_disj1
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (d20_fan ((vec (NUMERAL _0)) , (V , E)))))))
REAL_EQ_ADD_RCANCEL_0
|- (!x. (!y. ((((real_add x) y) = y) = (x = (real_of_num (NUMERAL _0))))))
TRANSP_COMPONENT
|- (!A. (!i. (!j. ((((transp A) $ i) $ j) = ((A $ j) $ i)))))
Seq.rot_oversize
|- (!n. (!s. (((sizel s) <= n) ==> (((rot n) s) = s))))
DIV_UNIQ
|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\ (r < n)) ==> ((m DIV n) = q))))))
CONTINUOUS_IMP_MEASURABLE_ON
|- (!f. (((continuous_on f) UNIV) ==> ((measurable_on f) UNIV)))
HAS_REAL_INTEGRAL_ON_SUPERSET
|- (!f. (!s. (!t. (((!x. ((~ (x IN s)) ==> ((f x) = (real_of_num (NUMERAL _0))))) /\ ((s SUBSET t) /\ (((has_real_integral f) i) s))) ==> (((has_real_integral f) i) t)))))
SERIES_COMPLEX_RMUL
|- (!f. (!l. (!c. (!s. ((((sums f) l) s) ==> (((sums (\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) s))))))