hcju/holps huggingface dataset:
Conforming.DART_FANADD_EQ_DART_FAN_ADD_2DART
|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((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) /\ ((~ (v = w)) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))) ==> (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E1))))) = (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E))))) UNION ((v , w) INSERT ((w , v) INSERT EMPTY))))))))))))))))
vector_polynomial_function
|- (!f. ((vector_polynomial_function f) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (real_polynomial_function (\x. ((f x) $ i)))))))
INSERT_SUBSET
|- (!x. (!s. (!t. (((x INSERT s) SUBSET t) = ((x IN t) /\ (s SUBSET t))))))
SETDIST_UNIQUE
|- (!s. (!t. (!a. (!b. (!d. (((a IN s) /\ ((b IN t) /\ (((distance (a , b)) = d) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le (distance (a , b))) (distance (x , y))))))))) ==> ((setdist (s , t)) = d)))))))
Ssrnat.addnK
|- (!n. (!x. (((x + n) - n) = x)))
COMPLEMENT_CONNECTED_COMPONENT_UNIONS
|- (!s. (!x. ((s DIFF ((connected_component s) x)) = (UNIONS ((GSPEC (\GEN%PVAR%900. (?y. (((SETSPEC GEN%PVAR%900) (y IN s)) ((connected_component s) y))))) DELETE ((connected_component s) x))))))
CX_COS
|- (!x. ((Cx (cos x)) = (ccos (Cx x))))
Nonlin_def.sub6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
ZERO_AE_DERIVATIVE_IMP_CONSTANT_GEN
|- (!f. (!a. (!b. (!s. ((((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((negligible ((IMAGE f) s)) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (vec (NUMERAL _0))) (at x)))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = (f a)))))))))
CLOSED_SEGMENT_LINEAR_IMAGE
|- (!f. (!a. (!b. ((linear f) ==> ((closed_segment ((CONS ((f a) , (f b))) NIL)) = ((IMAGE f) (closed_segment ((CONS (a , b)) NIL))))))))
REAL_CONTINUOUS_DIV_WITHINREAL
|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\ (((real_continuous g) ((within (atreal a)) s)) /\ (~ ((g a) = (real_of_num (NUMERAL _0)))))) ==> ((real_continuous (\x. ((real_div (f x)) (g x)))) ((within (atreal a)) s))))))
Sphere.rotate4
|- (!f. (!x4. (!x2. (!x6. (!x1. (!x5. (!x3. ((((((((rotate4 f) x1) x2) x3) x4) x5) x6) = ((((((f x4) x2) x6) x1) x5) x3)))))))))
TAGGED_DIVISION_OF_ALT
|- (!p. (!s. (((tagged_division_of p) s) = (((tagged_partial_division_of p) s) /\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\ (x IN k))))))))))
SQRT_NEG
|- (!x. ((sqrt (real_neg x)) = (real_neg (sqrt x))))
COMPLEX_EQ_ADD_LCANCEL
|- (!x. (!y. (!z. ((((vector_add x) y) = ((vector_add x) z)) = (y = z)))))
CPRODUCT_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((cproduct s) f) = ((cproduct s) g))))))
UNBOUNDED_HALFSPACE_COMPONENT_GE
|- (!a. (!k. (~ (bounded (GSPEC (\GEN%PVAR%837. (?x. (((SETSPEC GEN%PVAR%837) ((real_ge (x $ k)) a)) x))))))))
SUMS_0
|- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (vec (NUMERAL _0))))) ==> (((sums f) (vec (NUMERAL _0))) s))))
HOLOMORPHIC_ON_CONST
|- (!c. (!s. ((holomorphic_on (\z. c)) s)))
HAS_REAL_DERIVATIVE_EXP
|- (!x. (((has_real_derivative exp) (exp x)) (atreal x)))
RE_SUB
|- (!x. (!y. ((Re ((vector_sub x) y)) = ((real_sub (Re x)) (Re y)))))
ROOT_MONO_LE
|- (!n. (!x. (!y. (((real_le x) y) ==> ((real_le ((root n) x)) ((root n) y))))))
VECTOR_VARIATION_COMPARISON
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (distance ((g x) , (g y)))) (distance ((f x) , (f y)))))))) ==> ((real_le ((vector_variation s) g)) ((vector_variation s) f))))))
HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS
|- (!f. (!g. (!s. (!r. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\ (((continuous_on g) s) /\ ((((IMAGE g) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\ (!x. ((x IN s) ==> (~ ((f x) = (vector_neg (g x)))))))))) ==> ((((homotopic_with (\x. T)) (s , (sphere ((vec (NUMERAL _0)) , r)))) f) g))))))
CONTINUOUS_ON_CASES_LOCAL
|- (!P. (!f. (!g. (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\ (((continuous_on f) s) /\ (((continuous_on g) t) /\ (!x. ((((x IN s) /\ (~ (P x))) \/ ((x IN t) /\ (P x))) ==> ((f x) = (g x)))))))) ==> ((continuous_on (\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))
SIMPLE_IMAGE_GEN
|- (!f. (!P. ((GSPEC (\GEN%PVAR%12. (?x. (((SETSPEC GEN%PVAR%12) (P x)) (f x))))) = ((IMAGE f) (GSPEC (\GEN%PVAR%13. (?x. (((SETSPEC GEN%PVAR%13) (P x)) x))))))))
SUM_PERMUTE
|- (!f. (!p. (!s. (((permutes p) s) ==> (((sum s) f) = ((sum s) (f o p)))))))
REAL_LE_MUL_EQ
|- ((!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) x) ==> (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_le (real_of_num (NUMERAL _0))) y))))) /\ (!x. (!y. (((real_lt (real_of_num (NUMERAL _0))) y) ==> (((real_le (real_of_num (NUMERAL _0))) ((real_mul x) y)) = ((real_le (real_of_num (NUMERAL _0))) x))))))
HULL_UNION_RIGHT
|- (!P. (!s. (!t. (((hull P) (s UNION t)) = ((hull P) (s UNION ((hull P) t)))))))
CARD_EQ_CLOSED_SETS
|- ((GSPEC (\GEN%PVAR%1254. (?s. (((SETSPEC GEN%PVAR%1254) (closed s)) s)))) =_c UNIV)
BOUNDED_SCALING
|- (!c. (!s. ((bounded s) ==> (bounded ((IMAGE (\x. ((% c) x))) s)))))
metric_tybij
|- ((!a. ((metric (dest_metric a)) = a)) /\ (!r. ((is_metric_space r) = ((dest_metric (metric r)) = r))))
CX_DEF
|- (!a. ((Cx a) = (complex (a , (real_of_num (NUMERAL _0))))))
NORM_LE_0
|- (!x. (((real_le (vector_norm x)) (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))
REAL_OF_NUM_MUL
|- (!m. (!n. (((real_mul (real_of_num m)) (real_of_num n)) = (real_of_num (m * n)))))
CONVEX_AFFSIGN
|- (!sgn. ((!x. (!y. (!u. (((sgn x) /\ ((sgn y) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0))))))) ==> (sgn ((real_add ((real_mul ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) x)) ((real_mul u) y))))))) ==> (!s. (!t. (convex (((affsign sgn) s) t))))))
SUM_POS_EQ_0_NUMSEG
|- (!f. (!m. (!n. (((!p. (((m <= p) /\ (p <= n)) ==> ((real_le (real_of_num (NUMERAL _0))) (f p)))) /\ (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))) ==> (!p. (((m <= p) /\ (p <= n)) ==> ((f p) = (real_of_num (NUMERAL _0)))))))))
DOMINATED_CONVERGENCE_AE
|- (!f. (!g. (!h. (!s. (!t. (((!k. ((integrable_on (f k)) s)) /\ (((integrable_on h) s) /\ ((negligible t) /\ ((!k. (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm ((f k) x))) (drop (h x)))))) /\ (!x. ((x IN (s DIFF t)) ==> (((--> (\k. ((f k) x))) (g x)) sequentially))))))) ==> (((integrable_on g) s) /\ (((--> (\k. ((integral s) (f k)))) ((integral s) g)) sequentially))))))))
REAL_SEQ_OFFSET
|- (!f. (!l. (!k. ((((---> f) l) sequentially) ==> (((---> (\i. (f (i + k)))) l) sequentially)))))
CONVEX_NEGATIONS
|- (!s. ((convex s) ==> (convex ((IMAGE vector_neg) s))))
HAS_REAL_INTEGRAL_SPIKE_FINITE
|- (!f. (!g. (!s. (!t. (!y. (((FINITE s) /\ ((!x. ((x IN (t DIFF s)) ==> ((g x) = (f x)))) /\ (((has_real_integral f) y) t))) ==> (((has_real_integral g) y) t)))))))
AFFINE_HULL_0_3_EXPLICIT
|- (!x. (!y. (!z. (((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))) = (?a. (?b. (?c. ((((vector_add ((% a) x)) ((vector_add ((% b) y)) ((% c) z))) = (vec (NUMERAL _0))) /\ (~ (((real_add a) ((real_add b) c)) = (real_of_num (NUMERAL _0))))))))))))
DIVMOD_ELIM_THM
|- (((P (m DIV n)) (m MOD n)) = (!q. (!r. ((((n = (NUMERAL _0)) /\ ((q = (NUMERAL _0)) /\ (r = m))) \/ ((m = ((q * n) + r)) /\ (r < n))) ==> ((P q) r)))))
Add_triangle.fan_hypermap_map1
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((face_map (hypermap_of_fan (V , E))) d) = ((f_fan_pair (V , E)) d)) /\ ((((node_map (hypermap_of_fan (V , E))) d) = ((n_fan_pair (V , E)) d)) /\ (((edge_map (hypermap_of_fan (V , E))) d) = ((e_fan_pair (V , E)) d)))))))))
SQRT_LE_0
|- (!x. (((real_le (real_of_num (NUMERAL _0))) (sqrt x)) = ((real_le (real_of_num (NUMERAL _0))) x)))
Functional_equation.functional_mud_135_x
|- (!y2. (!y4. (!y6. (((domain6 (\x1. (\x2. (\x3. (\x4. (\x5. (\x6. ((real_le (real_of_num (NUMERAL _0))) y2)))))))) (((mud_135_x_v1 y2) y4) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y2) y2))) proj_x1) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_135_x ((real_mul y2) y2)) ((real_mul y4) y4)) ((real_mul y6) y6)))))))))
Lp_ineqs_def.y4'_fan
|- (!V. (!E. (!x. (((y4'_fan (V , E)) x) = (distance ((SND x) , (FST ((inverse (f_fan_pair_ext (V , E))) x))))))))
Packing3.lemma1
|- (!f. (!P. (((FINITE f) /\ (P f)) ==> (?n. (?g. ((g SUBSET f) /\ ((g HAS_SIZE n) /\ (P g))))))))
MEASURABLE_ON_SPIKE
|- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((measurable_on f) t) ==> ((measurable_on g) t)))))))
CLOSURE_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((closure ((IMAGE f) s)) = ((IMAGE f) (closure s))))))
AFFINE_HULL_EMPTY
|- (((hull affine) EMPTY) = EMPTY)
UNIONS_2
|- ((UNIONS (s INSERT (t INSERT EMPTY))) = (s UNION t))
Hypermap_iso.fan_hypermaps_iso_explicit2
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso (ext_dart (V , E))) ((hypermap_of_fan (V , E)) , (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))))))))
HAS_BOUNDED_VARIATION_ON_SUBSET
|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\ (t SUBSET s)) ==> ((has_bounded_variation_on f) t)))))
CEXP_NZ
|- (!z. (~ ((cexp z) = (Cx (real_of_num (NUMERAL _0))))))
CLOSED_IN_DIFF
|- (!top. (!s. (!t. ((((closed_in top) s) /\ ((open_in top) t)) ==> ((closed_in top) (s DIFF t))))))
Ssrnat.addSn
|- (!m. (!n. (((SUC m) + n) = (SUC (m + n)))))
Hypermap.FACE_FINITE
|- (!H. (!x. (FINITE ((face H) x))))
Hypermap.lemma_face_representation
|- (!H. (!u. ((u IN (face_set H)) ==> (?x. ((x IN (dart H)) /\ (u = ((face H) x)))))))
INTERIOR_MAXIMAL_EQ
|- (!s. (!t. ((open s) ==> ((s SUBSET (interior t)) = (s SUBSET t)))))
INT_OF_NUM_MAX
|- (!m. (!n. (((int_max (int_of_num m)) (int_of_num n)) = (int_of_num ((MAX m) n)))))
REAL_INV_DIV
|- (!x. (!y. ((real_inv ((real_div x) y)) = ((real_div y) x))))
Seq.has
|- (!a. (!x. (!s'. ((((has a) NIL) = F) /\ (((has a) ((CONS x) s')) = ((a x) \/ ((has a) s')))))))
COMPLEX_NORM_II
|- ((vector_norm ii) = (real_of_num (NUMERAL (BIT1 _0))))
POW_2_SQRT
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((sqrt ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = x)))
Hypermap.lemma_atom_sub_node
|- (!H. (!L. (!x. ((((atom H) L) x) SUBSET ((node H) x)))))
OPEN_IN_SUBTOPOLOGY_REFL
|- (!top. (!u. (((open_in ((subtopology top) u)) u) = (u SUBSET (topspace top)))))
COMPLEX_DIFFERENTIABLE_DIV_WITHIN
|- (!f. (!g. (!z. (!s. ((((complex_differentiable f) ((within (at z)) s)) /\ (((complex_differentiable g) ((within (at z)) s)) /\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\z. ((complex_div (f z)) (g z)))) ((within (at z)) s)))))))
CTAN_ADD
|- (!w. (!z. (((~ ((ccos w) = (Cx (real_of_num (NUMERAL _0))))) /\ ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\ (~ ((ccos ((vector_add w) z)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((ctan ((vector_add w) z)) = ((complex_div ((vector_add (ctan w)) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_mul (ctan w)) (ctan z))))))))
HAS_REAL_MEASURE_TRANSLATION_EQ
|- (!s. (!m. (((has_real_measure ((IMAGE (\x. ((real_add a) x))) s)) m) = ((has_real_measure s) m))))
INT_OF_NUM_ADD
|- (!m. (!n. (((int_add (int_of_num m)) (int_of_num n)) = (int_of_num (m + n)))))
NOT_OUTSIDE_CONNECTED_COMPONENT_LE
|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> ((UNIV DIFF (outside s)) = (GSPEC (\GEN%PVAR%2014. (?x. (((SETSPEC GEN%PVAR%2014) (!B. (?y. (((real_le B) (vector_norm y)) /\ (~ (((connected_component (UNIV DIFF s)) x) y)))))) x)))))))
HAS_REAL_DERIVATIVE_MUL_ATREAL
|- (!f. (!f'. (!g. (!g'. (!x. (((((has_real_derivative f) f') (atreal x)) /\ (((has_real_derivative g) g') (atreal x))) ==> (((has_real_derivative (\x. ((real_mul (f x)) (g x)))) ((real_add ((real_mul (f x)) g')) ((real_mul f') (g x)))) (atreal x))))))))
CONVEX_IMP_PATH_CONNECTED
|- (!s. ((convex s) ==> (path_connected s)))
LT_EXISTS
|- (!m. (!n. ((m < n) = (?d. (n = (m + (SUC d)))))))
REAL_DIFFERENTIABLE_TRANSFORM
|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_differentiable_on f) s)) ==> ((real_differentiable_on g) s)))))
CONTINUOUS_CARD_LT_RANGE_CONSTANT_EQ
|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\ (((IMAGE f) s) <_c UNIV)) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))
subspace
|- (!s. ((subspace s) = (((vec (NUMERAL _0)) IN s) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> (((vector_add x) y) IN s)))) /\ (!c. (!x. ((x IN s) ==> (((% c) x) IN s))))))))
SUBSET_ANTISYM_EQ
|- (!s. (!t. (((s SUBSET t) /\ (t SUBSET s)) = (s = t))))
MEASURE_SCALING
|- (!s. (!c. ((measurable s) ==> ((measure ((IMAGE (\x. ((% c) x))) s)) = ((real_mul ((real_pow (real_abs c)) (dimindex UNIV))) (measure s))))))
COMPACT_TRANSLATION_EQ
|- (!a. (!s. ((compact ((IMAGE (\x. ((vector_add a) x))) s)) = (compact s))))
REAL_LE_MAX
|- (!x. (!y. (!z. (((real_le z) ((real_max x) y)) = (((real_le z) x) \/ ((real_le z) y))))))
Hypermap.card_orbit_le
|- (!f. (!n. (!x. (((~ (n = (NUMERAL _0))) /\ ((((POWER f) n) x) = x)) ==> ((CARD ((orbit_map f) x)) <= n)))))
INTEGRAL_SUB
|- (!f. (!g. (!k. (!l. (!s. ((((integrable_on f) s) /\ ((integrable_on g) s)) ==> (((integral s) (\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((integral s) f)) ((integral s) g)))))))))
conic_cap
|- (!r. (!v0. (!v1. (!a. (((((conic_cap v0) v1) r) a) = (((normball v0) r) INTER (((rcone_gt v0) v1) a)))))))
COMPACT_CLOSURE
|- (!s. ((compact (closure s)) = (bounded s)))
INF_INSERT_FINITE
|- (!x. (!s. ((FINITE s) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))
L1_LE_NORM
|- (!x. ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs (x $ i))))) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (vector_norm x))))
IN_SPHERE_0
|- (!x. (!e. ((x IN (sphere ((vec (NUMERAL _0)) , e))) = ((vector_norm x) = e))))
HAS_COMPLEX_DERIVATIVE_CARATHEODORY_WITHIN
|- (!f. (!f'. (!z. (!s. ((((has_complex_derivative f) f') ((within (at z)) s)) = (?g. ((!w. (((vector_sub (f w)) (f z)) = ((complex_mul (g w)) ((vector_sub w) z)))) /\ (((continuous g) ((within (at z)) s)) /\ ((g z) = f')))))))))
REAL_SEGMENT_INTERVAL
|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = (((COND ((real_le a) b)) (closed_real_interval ((CONS (a , b)) NIL))) (closed_real_interval ((CONS (b , a)) NIL)))))) /\ (!a. (!b. ((open_real_segment (a , b)) = (((COND ((real_le a) b)) (open_real_interval (a , b))) (open_real_interval (b , a)))))))
Lp_ineqs_proofs.y8_hi_std
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((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)))))))))))))))))))
LOCALLY_PATH_CONNECTED_PATH_IMAGE
|- (!p. ((path p) ==> ((locally path_connected) (path_image p))))
List_hypermap.mem_list_of_elements
|- (!x. (!L. (((MEM x) (list_of_elements L)) = (?y. ((MEM (x , y)) (list_of_darts L))))))
Appendix.scs_a_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_a_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = a))))))))))
ssreflect_eq_def
|- (!v. (!P. (((ssreflect_eq v) P) = P)))
BETWEEN_IMP_COLLINEAR
|- (!a. (!b. (!x. (((between x) (a , b)) ==> (collinear (a INSERT (x INSERT (b INSERT EMPTY))))))))
complex_pow
|- ((((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) /\ (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n)))))
SUM_EQ_NUMSEG
|- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (g i)))) ==> (((sum (m .. n)) f) = ((sum (m .. n)) g)))))))
INT_LE_RNEG
|- (!x. (!y. (((int_le x) (int_neg y)) = ((int_le ((int_add x) y)) (int_of_num (NUMERAL _0))))))
INTEGER_CASES
|- ((integer x) = ((?n. (x = (real_of_num n))) \/ (?n. (x = (real_neg (real_of_num n))))))
IMAGE_UNIONS
|- (!f. (!s. (((IMAGE f) (UNIONS s)) = (UNIONS ((IMAGE (IMAGE f)) s)))))
Seq.subseq_filter
|- (!s1. (!s2. (!a. (((subseq s1) ((filter a) s2)) = (((all a) s1) /\ ((subseq s1) s2))))))
HAS_BOUNDED_VARIATION_COMPOSE_INCREASING
|- (!f. (!g. (!a. (!b. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f x))) (drop (f y)))))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f a) , (f b))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))
rectifiable_path
|- (!g. ((rectifiable_path g) = ((path g) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))))))
Pack2.POLYHEDRON_VORONOI_CLOSED
|- (!s. (!v. (((packing s) /\ ((saturated s) /\ (v IN s))) ==> (polyhedron ((voronoi_closed s) v)))))
REAL_LE_LNEG
|- (!x. (!y. (((real_le (real_neg x)) y) = ((real_le (real_of_num (NUMERAL _0))) ((real_add x) y)))))
CEXP_NEG_RMUL
|- (!z. (((complex_mul (cexp z)) (cexp (vector_neg z))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))
vol_solid_triangle
|- (!r. (!v0. (!v1. (!v2. (!v3. ((((((vol_solid_triangle v0) v1) v2) v3) r) = ((LET (\a123. (LET_END ((LET (\a231. (LET_END ((LET (\a312. (LET_END ((real_mul ((real_add a123) ((real_add a231) ((real_sub a312) pi)))) ((real_div ((real_pow r) (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))) ((((dihV v0) v3) v1) v2))))) ((((dihV v0) v2) v3) v1))))) ((((dihV v0) v1) v2) v3))))))))
OPEN_IMP_GDELTA
|- (!s. ((open s) ==> (gdelta s)))
Functional_equation.rhazim_x'
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\ (((real_le (real_of_num (NUMERAL _0))) x2) /\ (((real_le (real_of_num (NUMERAL _0))) x3) /\ (((real_le (real_of_num (NUMERAL _0))) x4) /\ (((real_le (real_of_num (NUMERAL _0))) x5) /\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazim_x' x1) x2) x3) x4) x5) x6)))))))))
REAL_LE_SUB_LADD
|- (!x. (!y. (!z. (((real_le x) ((real_sub y) z)) = ((real_le ((real_add x) z)) y)))))
CLOSED_UNIONS
|- (!s. (((FINITE s) /\ (!t. ((t IN s) ==> (closed t)))) ==> (closed (UNIONS s))))
Ssrbool.contraR
|- (!c. (!b. (((~ c) ==> b) ==> ((~ b) ==> c))))
ACS_MONO_LE
|- (!x. (!y. ((((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_le (acs y)) (acs x)))))
Sphere.sol_y
|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((real_add ((((((dih_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih_y y2) y3) y1) y5) y6) y4)) ((real_sub ((((((dih_y y3) y1) y2) y6) y4) y5)) pi))))))))))
HAS_REAL_MEASURE_SCALING
|- (!s. (!m. (!y. (((has_real_measure s) y) ==> ((has_real_measure ((IMAGE (\x. ((real_mul m) x))) s)) ((real_mul (real_abs m)) y))))))
SERIES_0
|- (!s. (((sums (\n. (vec (NUMERAL _0)))) (vec (NUMERAL _0))) s))
REAL_LT_DIV
|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_div x) y)))))
NADD_LE_WELLDEF
|- (!x. (!x'. (!y. (!y'. ((((nadd_eq x) x') /\ ((nadd_eq y) y')) ==> (((nadd_le x) y) = ((nadd_le x') y')))))))
IM_SUB
|- (!x. (!y. ((Im ((vector_sub x) y)) = ((real_sub (Im x)) (Im y)))))
REAL_ADD_RID
|- (!x. (((real_add x) (real_of_num (NUMERAL _0))) = x))
GDELTA_PCROSS
|- (!s. (!t. (((gdelta s) /\ (gdelta t)) ==> (gdelta (s PCROSS t)))))
FCONS
|- ((!a. (!f. ((((FCONS a) f) (NUMERAL _0)) = a))) /\ (!a. (!f. (!n. ((((FCONS a) f) (SUC n)) = (f n))))))
MAPROWS_COMPOSE
|- (!f. (!g. ((maprows (f o g)) = ((maprows f) o (maprows g)))))
CBALL_EMPTY
|- (!x. (!e. (((real_lt e) (real_of_num (NUMERAL _0))) ==> ((cball (x , e)) = EMPTY))))
COMPLEX_MVT_LINE
|- (!f. (!f'. (!w. (!z. ((!u. ((u IN (closed_segment ((CONS (w , z)) NIL))) ==> (((has_complex_derivative f) (f' u)) (at u)))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\ (((real_sub (Re (f z))) (Re (f w))) = (Re ((complex_mul (f' u)) ((vector_sub z) w)))))))))))
CLOSED_CBALL
|- (!x. (!e. (closed (cball (x , e)))))
BARYCENTRE_IN_RELATIVE_INTERIOR
|- (!s. (((FINITE s) /\ (~ (s = EMPTY))) ==> ((barycentre s) IN (relative_interior ((hull convex) s)))))
LIFT_EQ
|- (!x. (!y. (((lift x) = (lift y)) = (x = y))))
Ssrnat.subKn
|- (!m. (!n. ((m <= n) ==> ((n - (n - m)) = m))))
root
|- (!x. (!n. (((root n) x) = ((real_mul (real_sgn x)) (exp ((real_div (log (real_abs x))) (real_of_num n)))))))
Functional_equation.rhazim_all_atn
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazimatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim2_x x1) x2) x3) x4) x5) x6) = ((((((rhazim2atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((rhazim3_x x1) x2) x3) x4) x5) x6) = ((((((rhazim3atn_x x1) x2) x3) x4) x5) x6)))))))))))
CONTINUOUS_NEG
|- (!f. (!net. (((continuous f) net) ==> ((continuous (\x. (vector_neg (f x)))) net))))
EXTREME_POINT_OF_CONVEX_HULL_INSERT
|- (!s. (!a. (((FINITE s) /\ (~ (a IN ((hull convex) s)))) ==> ((extreme_point_of a) ((hull convex) (a INSERT s))))))
AFF_GE_EQ_AFFINE_HULL
|- (!s. (!t. (((aff_ge s) EMPTY) = ((hull affine) s))))
SQRT_POW_2
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((real_pow (sqrt x)) (NUMERAL (BIT0 (BIT1 _0)))) = x)))
Lp_main_estimate.f_surr_in_e_prime
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v , w) IN f) \/ ((w , v) IN f)))))))))))
COMPLEX_TRAD
|- (!x. (!y. ((complex (x , y)) = ((vector_add (Cx x)) ((complex_mul ii) (Cx y))))))
Seq2.next_el_hd_cons2
|- (!h1. (!h2. (!t. (((next_el ((CONS h1) ((CONS h2) t))) h1) = h2))))
VSUM_ADD_NUMSEG
|- (!f. (!g. (!m. (!n. (((vsum (m .. n)) (\i. ((vector_add (f i)) (g i)))) = ((vector_add ((vsum (m .. n)) f)) ((vsum (m .. n)) g)))))))
Lp_ineqs_proofs.tauq_cross_split
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d)) = ((real_add ((((((taum (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((taum (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))
COMPACT_CBALL
|- (!x. (!e. (compact (cball (x , e)))))
Hypermap_and_fan.PAIR_IN_DART_OF_FAN
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart_of_fan (V , E)))) ==> ((v IN V) /\ (w IN V)))))))