hcju/holps huggingface dataset:
Lp_ineqs_proofs.y8_lo
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y8_fan (V , E)) d))))))))))
FACET_OF_TRANSLATION_EQ
|- (!a. (!f. (!s. (((facet_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((facet_of f) s)))))
COPLANAR_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((coplanar ((IMAGE f) s)) = (coplanar s)))))
REAL_EQ_RCANCEL_IMP
|- (!x. (!y. (!z. (((~ (z = (real_of_num (NUMERAL _0)))) /\ (((real_mul x) z) = ((real_mul y) z))) ==> (x = y)))))
joinpaths
|- (!g1. (!g2. (((++ g1) g2) = (\x. (((COND ((real_le (drop x)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (g1 ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x))) (g2 ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)) (vec (NUMERAL (BIT1 _0))))))))))
HAS_BOUNDED_VARIATION_ON_MUL
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (((has_bounded_variation_on g) s) /\ (is_interval s))) ==> ((has_bounded_variation_on (\x. ((% (drop (f x))) (g x)))) s)))))
REAL_POW_LT2
|- (!n. (!x. (!y. (((~ (n = (NUMERAL _0))) /\ (((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_lt x) y))) ==> ((real_lt ((real_pow x) n)) ((real_pow y) n))))))
SPAN_MUL
|- (!x. (!c. (!s. ((x IN (span s)) ==> (((% c) x) IN (span s))))))
COMPACT_IMP_CLOSED
|- (!s. ((compact s) ==> (closed s)))
FACE_OF_TRANS
|- (!s. (!t. (!u. ((((face_of s) t) /\ ((face_of t) u)) ==> ((face_of s) u)))))
ge_c
|- (!t. (!s. ((s >=_c t) = (t <=_c s))))
RSTC_SYM
|- (!R. (!x. (!y. ((((RSTC R) x) y) ==> (((RSTC R) y) x)))))
Seq.count
|- (!a. (!x. (!s'. ((((count a) NIL) = (NUMERAL _0)) /\ (((count a) ((CONS x) s')) = ((((COND (a x)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((count a) s')))))))
OPEN_NON_GENERAL_COMPONENT
|- (!c. (((!s. (!x. (!y. ((((c s) x) y) ==> ((x IN s) /\ (y IN s)))))) /\ ((!s. (!x. (!y. ((((c s) x) y) ==> (((c s) y) x))))) /\ ((!s. (!x. (!y. (!z. (((((c s) x) y) /\ (((c s) y) z)) ==> (((c s) x) z)))))) /\ ((!s. (!t. (!x. (!y. (((s SUBSET t) /\ (((c s) x) y)) ==> (((c t) x) y)))))) /\ (!s. (!x. (!y. (!e. (((y IN (ball (x , e))) /\ ((ball (x , e)) SUBSET s)) ==> (((c (ball (x , e))) x) y)))))))))) ==> (!s. (!x. ((open s) ==> (open (s DIFF ((c s) x))))))))
Sphere.lmfun
|- (!h. ((lmfun h) = (((COND ((real_le h) h0)) ((real_div ((real_sub h0) h)) ((real_sub h0) (real_of_num (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0)))))
num_WOP
|- (!P. ((?n. (P n)) = (?n. ((P n) /\ (!m. ((m < n) ==> (~ (P m))))))))
NPRODUCT_CLAUSES
|- ((!f. (((nproduct EMPTY) f) = (NUMERAL (BIT1 _0)))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((nproduct (x INSERT s)) f) = (((COND (x IN s)) ((nproduct s) f)) ((f x) * ((nproduct s) f)))))))))
INT_LT_DISCRETE
|- (!x. (!y. (((int_lt x) y) = ((int_le ((int_add x) (int_of_num (NUMERAL (BIT1 _0))))) y))))
SETDIST_LINEAR_IMAGE
|- (!f. (!s. (!t. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((setdist (((IMAGE f) s) , ((IMAGE f) t))) = (setdist (s , t)))))))
SEGMENT_CONVEX_HULL
|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((hull convex) (a INSERT (b INSERT EMPTY))))))
SUM_LE
|- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((real_le (f x)) (g x))))) ==> ((real_le ((sum s) f)) ((sum s) g))))))
AFF_GE_SUBSET_AFFINE_HULL
|- (!s. (!t. (((aff_ge s) t) SUBSET ((hull affine) (s UNION t)))))
Seq.has_pred1
|- (!x. (!s. (((has (pred1 x)) s) = ((MEM x) s))))
REAL_DIFFERENTIABLE_AT_RPOW_RIGHT
|- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_differentiable (\x. ((rpow a) x))) (atreal x)))))
Lp_main_estimate.packing_v_prime
|- (!V. (!f. ((packing V) ==> (packing ((v_prime V) f)))))
Functional_equation.proj_x5
|- (((((((proj_x5 x1) x2) x3) x4) x5) x6) = x5)
Nonlin_def.dummy6
|- (dummy6 = (constant6 (real_of_num (NUMERAL _0))))
Planarity.fan80
|- (!x. (!V. (!E. ((fan80 (x , (V , E))) = (!v. (!u. (((v INSERT (u INSERT EMPTY)) IN E) ==> (((real_lt (real_of_num (NUMERAL _0))) ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) /\ ((real_lt ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) pi)))))))))
OPEN_IN_INTER
|- (!top. (!s. (!t. ((((open_in top) s) /\ ((open_in top) t)) ==> ((open_in top) (s INTER t))))))
Ssrnat.leq_mul2l
|- (!m. (!n1. (!n2. (((m * n1) <= (m * n2)) = ((m = (NUMERAL _0)) \/ (n1 <= n2))))))
REAL_ABS_SIN_BOUND_LT
|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> ((real_lt (real_abs (sin x))) (real_abs x))))
VSUM_LMUL
|- (!f. (!c. (!s. (((vsum s) (\x. ((% c) (f x)))) = ((% c) ((vsum s) f))))))
Functional_equation.eta_all_atn
|- (!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)))))) ==> ((((((((eta2_126 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x2) x6)) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((((((eta2_135 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((((((eta2_456 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x4) x5) x6)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))
Hypermap.lemma_node_identity
|- (!H. (!x. (!y. ((y IN ((node H) x)) ==> (((node H) x) = ((node H) y))))))
UNIONS_SUBSET
|- (!f. (!t. (((UNIONS f) SUBSET t) = (!s. ((s IN f) ==> (s SUBSET t))))))
IN_COMPONENTS_SUBSET
|- (!s. (!c. ((c IN (components s)) ==> (c SUBSET s))))
Nonlinear_lemma.lmdih5_ldih5
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le (real_of_num (NUMERAL _0))) y5) /\ ((real_le y5) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((((((y_of_x lmdih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6) = (((((((y_of_x ldih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6)))))))))
Tame_inequalities.DELTA_X4_MONO_LT_4
|- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\ ((real_lt (real_of_num (NUMERAL _0))) x1)) ==> ((real_lt ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))
Tame_defs.tame_3
|- (!H. ((tame_3 H) = (is_edge_nondegenerate H)))
VECTOR_VARIATION_CONTINUOUS_LEFT
|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , c)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))
Rnsyjxm.in_face_imp_in_dart
|- (!x. (!H. (!d. (((x IN (dart H)) /\ (d IN ((face H) x))) ==> (d IN (dart H))))))
INT_LE_ANTISYM
|- (!x. (!y. ((((int_le x) y) /\ ((int_le y) x)) = (x = y))))
BEPPO_LEVI_INCREASING
|- (!f. (!s. (((!k. ((integrable_on (f k)) s)) /\ ((!k. (!x. ((x IN s) ==> ((real_le (drop ((f k) x))) (drop ((f (SUC k)) x)))))) /\ (bounded (GSPEC (\GEN%PVAR%3065. (?k. (((SETSPEC GEN%PVAR%3065) (k IN UNIV)) ((integral s) (f k))))))))) ==> (?g. (?k. ((negligible k) /\ (!x. ((x IN (s DIFF k)) ==> (((--> (\k. ((f k) x))) (g x)) sequentially)))))))))
SUM_EQ_SUPERSET
|- (!f. (!s. (!t. (((FINITE t) /\ ((t SUBSET s) /\ ((!x. ((x IN t) ==> ((f x) = (g x)))) /\ (!x. (((x IN s) /\ (~ (x IN t))) ==> ((f x) = (real_of_num (NUMERAL _0)))))))) ==> (((sum s) f) = ((sum t) g))))))
COLLINEAR_AFF_DIM
|- (!s. ((collinear s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL (BIT1 _0))))))
List_hypermap.uniq_list_of_elements
|- (!L. (uniq (list_of_elements L)))
MOD_MULT
|- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> (((m * n) MOD m) = (NUMERAL _0)))))
HAS_COMPLEX_DERIVATIVE_DIV_AT
|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\ ((((has_complex_derivative g) g') (at x)) /\ (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))) ==> (((has_complex_derivative (\x. ((complex_div (f x)) (g x)))) ((complex_div ((vector_sub ((complex_mul f') (g x))) ((complex_mul (f x)) g'))) ((complex_pow (g x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))))
COMPLETE_TRANSLATION_EQ
|- (!a. (!s. ((complete ((IMAGE (\x. ((vector_add a) x))) s)) = (complete s))))
Add_triangle.inE_eq_in_dart1
|- (!V. (!E. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) = ((v , w) IN (dart1_of_fan (V , E))))))))
REAL_POW_ONE
|- (!n. (((real_pow (real_of_num (NUMERAL (BIT1 _0)))) n) = (real_of_num (NUMERAL (BIT1 _0)))))
Trigonometry.XHVXJVB
|- (!x. ((vector_norm x) = (sqrt ((dot x) x))))
List_hypermap.e_list_ext
|- (!L. ((e_list_ext L) = ((res e_list) (darts_of_list L))))
Appendix.scs_d_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_d_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = d))))))))))
CLOG_CEXP
|- (!z. ((((real_lt (real_neg pi)) (Im z)) /\ ((real_le (Im z)) pi)) ==> ((clog (cexp z)) = z)))
COMPLEX_MUL_2
|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x) = ((vector_add x) x)))
Nonlin_def.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)))))))))))
HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ
|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homeomorphic ((IMAGE f) s)) t) = ((homeomorphic s) t))))))
DEPENDENT_BIGGERSET_GENERAL
|- (!s. (((FINITE s) ==> ((CARD s) > (dim s))) ==> (dependent s)))
REAL_SUMMABLE_RESTRICT
|- (!f. (!k. (((real_summable UNIV) (\n. (((COND (n IN k)) (f n)) (real_of_num (NUMERAL _0))))) = ((real_summable k) f))))
REAL_MEASURE_MEASURE
|- (!s. ((real_measure s) = (measure ((IMAGE lift) s))))
MATRIX_VECTOR_MUL_COMPONENT
|- (!A. (!x. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> ((((matrix_vector_mul A) x) $ k) = ((dot (A $ k)) x))))))
LEBESGUE_MEASURABLE_COUNTABLE_UNIONS
|- (!f. (((COUNTABLE f) /\ (!s. ((s IN f) ==> (lebesgue_measurable s)))) ==> (lebesgue_measurable (UNIONS f))))
REAL_NEG_GE0
|- (!x. (((real_le (real_of_num (NUMERAL _0))) (real_neg x)) = ((real_le x) (real_of_num (NUMERAL _0)))))
HAS_BOUNDED_VARIATION_ON_COMBINE
|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ ((real_le (drop c)) (drop b))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (((has_bounded_variation_on f) (closed_interval ((CONS (a , c)) NIL))) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (c , b)) NIL))))))))))
LINEAR_COMPOSE_CMUL
|- (!f. (!c. ((linear f) ==> (linear (\x. ((% c) (f x)))))))
REAL_EXP_INJ
|- (!x. (!y. (((exp x) = (exp y)) = (x = y))))
Lp_ineqs_proofs.dart4_tauVEF_eq_tauq
|- (!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)))) ==> ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d)))) = (((((((((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)))))))))))
Float.REAL_LT
|- (!m. (!n. (((real_lt (real_of_num m)) (real_of_num n)) = (m < n))))
ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG
|- (!v. (orthogonal_transformation (reflect_along v)))
Seq.last
|- (!x. (!h. (!t. ((((last x) NIL) = x) /\ (((last x) ((CONS h) t)) = ((last h) t))))))
CLOSED_IN
|- (!s. ((closed s) = ((closed_in euclidean) s)))
MATRIX_NEG
|- (!f. ((linear f) ==> ((matrix (\x. (vector_neg (f x)))) = (matrix_neg (matrix f)))))
FST
|- (!x. (!y. ((FST (x , y)) = x)))
Tame_lemmas.sum_tauVEF_upper_bound
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (scriptL V)) ==> ((real_le ((sum (face_set (hypermap_of_fan (V , E)))) (\f. (tauVEF (V , (E , f)))))) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))) sol0))))))))
NORMAL_MATRIX_KERNEL_TRANSP_EXPLICIT
|- (!A. (!x. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))))))
Ssrbool.XOR_DEF
|- (!p. (!q. (((XOR p) q) = (((COND p) (~ q)) q))))
Functional_equation.proj_y6
|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((proj_y6 x1) x2) x3) x4) x5) x6) = (sqrt x6))))))))
CONNECTED_CONTINUOUS_IMAGE
|- (!f. (!s. ((((continuous_on f) s) /\ (connected s)) ==> (connected ((IMAGE f) s)))))
Planarity.aff_ge_eq_aff_gt_union_aff_ge
|- (!x. (!v. (!w. ((~ (collinear (x INSERT (v INSERT (w INSERT EMPTY))))) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) = (((aff_gt (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) UNION (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) UNION ((aff_ge (x INSERT EMPTY)) (w INSERT EMPTY)))))))))
Misc_defs_and_lemmas.IN_ELIM_THM'
|- ((!P. (!x. ((x IN (GSPEC (\v. (P (SETSPEC v))))) = (P (\p. (\t. (p /\ (x = t)))))))) /\ ((!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))))))))
LIM_UNIQUE
|- (!net. (!f. (!l. (!l'. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ (((--> f) l') net))) ==> (l = l'))))))
tagged_division_of
|- (!s. (!i. (((tagged_division_of s) i) = (((tagged_partial_division_of s) i) /\ ((UNIONS (GSPEC (\GEN%PVAR%2759. (?k. (((SETSPEC GEN%PVAR%2759) (?x. ((x , k) IN s))) k))))) = i)))))
LIM_NULL_SUB
|- (!net. (!f. (!g. (((((--> f) (vec (NUMERAL _0))) net) /\ (((--> g) (vec (NUMERAL _0))) net)) ==> (((--> (\x. ((vector_sub (f x)) (g x)))) (vec (NUMERAL _0))) net)))))
Fan_defs.set_of_edge
|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\GEN%PVAR%5439. (?w. (((SETSPEC GEN%PVAR%5439) (((v INSERT (w INSERT EMPTY)) IN E) /\ (w IN V))) w))))))))
CX_SUB
|- (!x. (!y. ((Cx ((real_sub x) y)) = ((vector_sub (Cx x)) (Cx y)))))
UNIFORMLY_CONTINUOUS_ON_ADD
|- (!f. (!g. (!s. ((((uniformly_continuous_on f) s) /\ ((uniformly_continuous_on g) s)) ==> ((uniformly_continuous_on (\x. ((vector_add (f x)) (g x)))) s)))))
HAS_COMPLEX_DERIVATIVE_MUL_AT
|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\ (((has_complex_derivative g) g') (at x))) ==> (((has_complex_derivative (\x. ((complex_mul (f x)) (g x)))) ((vector_add ((complex_mul (f x)) g')) ((complex_mul f') (g x)))) (at x))))))))
Localization.AZIM_CYCLE_EQ_SIGMA_FAN_ALT
|- (!V. (!E. (!u. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (u IN (((set_of_edge v) V) E))) ==> (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) u) = (((((sigma_fan (vec (NUMERAL _0))) V) E) v) u)))))))
manhattan
|- (manhattan = (metric (UNIV , (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))))))
ABSOLUTE_RETRACT_PATH_IMAGE_ARC
|- (!g. (!s. (((arc g) /\ ((path_image g) SUBSET s)) ==> ((retract_of (path_image g)) s))))
INTEGRABLE_BOUNDED_VARIATION_PRODUCT
|- (!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. ((% (drop (g x))) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))
dist
|- (!x. (!y. ((distance (x , y)) = (vector_norm ((vector_sub x) y)))))
OPEN_BIJECTIVE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((open ((IMAGE f) s)) = (open s)))))
PERMUTES_INVERSES
|- (!p. (!s. (((permutes p) s) ==> ((!x. ((p ((inverse p) x)) = x)) /\ (!x. (((inverse p) (p x)) = x))))))
IN_COMPONENTS_NONEMPTY
|- (!s. (!c. ((c IN (components s)) ==> (~ (c = EMPTY)))))
REAL_ARCH_INV
|- (!e. (((real_lt (real_of_num (NUMERAL _0))) e) = (?n. ((~ (n = (NUMERAL _0))) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_inv (real_of_num n))) /\ ((real_lt (real_inv (real_of_num n))) e))))))
Add_triangle.azim_split_fan_face_add
|- (!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 (\w'. (LET_END ((LET (\d2. (LET_END ((LET (\F2. (LET_END ((((azim_dart (V , E)) d2) = ((real_add ((azim_dart F2) (w , w'))) ((azim_dart F2) d2))) /\ (((azim_dart (V , E)) (w' , v)) = ((real_add ((azim_dart F2) (w' , w))) ((azim_dart F2) (w' , v)))))))) ((split_fan_face (V , E)) (v , w)))))) ((f_fan_pair (V , E)) (v , w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))
positive_semidefinite
|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\ (!x. ((real_le (real_of_num (NUMERAL _0))) ((dot x) ((matrix_vector_mul A) x)))))))
SET_PROVE_CASES
|- (!P. (((P EMPTY) /\ (!a. (!s. ((~ (a IN s)) ==> (P (a INSERT s)))))) ==> (!s. (P s))))
Add_triangle.n_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. (\n. (LET_END (((n (w1 , w)) = ((n_fan_pair (V , E)) (w1 , v))) /\ (((n (w1 , v)) = (w1 , w)) /\ (((n d2) = (w , w1)) /\ (((n (w , w1)) = (w , v)) /\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\ (~ ((d = (w1 , w)) \/ ((d = (w1 , v)) \/ ((d = d2) \/ (d = (w , w1))))))) ==> ((n d) = ((n_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (n_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))
Dih2k_hypermap.F_SY
|- (!v. ((F_SY v) = (GSPEC (\GEN%PVAR%7490. (?i. (((SETSPEC GEN%PVAR%7490) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (((row i) v) , ((row (SUC (i MOD (dimindex UNIV)))) v))))))))
REAL_EQ_ADD_RCANCEL
|- (!x. (!y. (!z. ((((real_add x) z) = ((real_add y) z)) = (x = y)))))
complex_neg
|- ((vector_neg z) = (complex ((real_neg (Re z)) , (real_neg (Im z)))))
List_hypermap.elements_of_list
|- (!L. ((elements_of_list L) = (set_of_list (list_of_elements L))))
HAS_COMPLEX_DERIVATIVE_INV_AT
|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at x)) /\ (~ ((f x) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((has_complex_derivative (\x. (complex_inv (f x)))) ((complex_div (vector_neg f')) ((complex_pow (f x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))
Fnjlbxs.closed_ball_annulus
|- (closed ball_annulus)
HOMOTOPIC_PATHS_IMP_SUBSET
|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> (((path_image p) SUBSET s) /\ ((path_image q) SUBSET s))))))
OPEN_SURJECTIVE_LINEAR_IMAGE
|- (!f. (((linear f) /\ (!y. (?x. ((f x) = y)))) ==> (!s. ((open s) ==> (open ((IMAGE f) s))))))
CNJ_EQ_0
|- (!z. (((cnj z) = (Cx (real_of_num (NUMERAL _0)))) = (z = (Cx (real_of_num (NUMERAL _0))))))
Sphere.dih3_y
|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((dih3_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y3) y1) y2) y6) y4) y5))))))))
Fan_defs.dart_of_fan
|- (!V. (!E. ((dart_of_fan (V , E)) = ((GSPEC (\GEN%PVAR%5441. (?v. (((SETSPEC GEN%PVAR%5441) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))) UNION (GSPEC (\GEN%PVAR%5442. (?v. (?w. (((SETSPEC GEN%PVAR%5442) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w))))))))))
ANR_SPHERE
|- (!a. (!r. (ANR (sphere (a , r)))))
REAL_LE_REFL
|- (!x. ((real_le x) x))
MEASURE_UNIQUE
|- (!s. (!m. (((has_measure s) m) ==> ((measure s) = m))))
Hypermap_and_fan.SURROUNDED_IMP_CARD_SET_OF_EDGE_GE_3
|- (!V. (!E. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v IN V) /\ ((surrounded_node (V , E)) v))) ==> ((CARD (((set_of_edge v) V) E)) >= (NUMERAL (BIT1 (BIT1 _0))))))))
UNIFORMLY_CONTINUOUS_ON_VSUM
|- (!t. (!f. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> ((uniformly_continuous_on (f a)) t)))) ==> ((uniformly_continuous_on (\x. ((vsum s) (\a. ((f a) x))))) t)))))
REAL_INTEGRABLE_LMUL_EQ
|- (!f. (!s. (!c. (((real_integrable_on (\x. ((real_mul c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((real_integrable_on f) s))))))
CLOSEST_POINT_SELF
|- (!s. (!x. ((x IN s) ==> (((closest_point s) x) = x))))
Add_triangle.sigma_fan_inV
|- (!E. (!V. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) IN V)))))))
int_add_th
|- (!x. (!y. ((real_of_int ((int_add x) y)) = ((real_add (real_of_int x)) (real_of_int y)))))
Ssrbool.orbC
|- (!b. (!c. ((b \/ c) = (c \/ b))))
ARG_EQ_0_PI
|- (!z. ((((Arg z) = (real_of_num (NUMERAL _0))) \/ ((Arg z) = pi)) = (real z)))
INSIDE_NO_OVERLAP
|- (!s. (((inside s) INTER s) = EMPTY))
LIPSCHITZ_IMP_CONTINUOUS_ON
|- (!f. (!s. ((?B. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> ((continuous_on f) s))))
Collect_geom2.ETA_Y_POS_LE
|- ((real_le (real_of_num (NUMERAL _0))) (((eta_y ((dist3 v1) v2)) ((dist3 v1) v3)) ((dist3 v2) v3)))
Topology.AFF_GE_1_1
|- (!x. (!v. ((~ (x = v)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5522. (?y. (((SETSPEC GEN%PVAR%5522) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t2) /\ ((((real_add t1) t2) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (y = ((vector_add ((% t1) x)) ((% t2) v)))))))) y))))))))
REAL_CONTINUOUS_INV_WITHINREAL
|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\x. (real_inv (f x)))) ((within (atreal a)) s))))))
HAS_REAL_MEASURE_REAL_NEGLIGIBLE_SYMDIFF
|- (!s. (!t. (!m. ((((has_real_measure s) m) /\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> ((has_real_measure t) m)))))
CLOSED_TRANSLATION_EQ
|- (!a. (!s. ((closed ((IMAGE (\x. ((vector_add a) x))) s)) = (closed s))))
HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE
|- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1058. (?x. (((SETSPEC GEN%PVAR%1058) (((dot a) x) = b)) x))))) (GSPEC (\GEN%PVAR%1059. (?x. (((SETSPEC GEN%PVAR%1059) ((x $ k) = c)) x))))))))))
VECTOR_SUB
|- (((vector_sub x) y) = ((vector_add x) (vector_neg y)))
ARG
|- (!z. (((real_le (real_of_num (NUMERAL _0))) (Arg z)) /\ (((real_lt (Arg z)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (z = ((complex_mul (Cx (vector_norm z))) (cexp ((complex_mul ii) (Cx (Arg z)))))))))
Hypermap_iso.hyp_iso_ext
|- (!f. (!H. (!G. (((hyp_iso f) (H , G)) ==> (!g. ((!d. ((d IN (dart H)) ==> ((f d) = (g d)))) ==> ((hyp_iso g) (H , G))))))))
BETWEEN_DIST_LE
|- (!r. (!a. (!b. (!c. ((((real_le (distance (c , a))) r) /\ (((real_le (distance (c , b))) r) /\ ((between x) (a , b)))) ==> ((real_le (distance (c , x))) r))))))
SEGMENT_CLOSED_OPEN
|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((open_segment (a , b)) UNION (a INSERT (b INSERT EMPTY))))))
Seq.allP
|- (!a. (!s. ((!x. (((MEM x) s) ==> (a x))) = ((all a) s))))
Sphere.pathL
|- (!a. (!b. ((pathL (a , b)) = (a , ((real_div ((real_add a) b)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))
HAS_COMPLEX_DERIVATIVE_SUB
|- (!f. (!f'. (!g. (!g'. (!net. (((((has_complex_derivative f) f') net) /\ (((has_complex_derivative g) g') net)) ==> (((has_complex_derivative (\x. ((vector_sub (f x)) (g x)))) ((vector_sub f') g')) net)))))))
NORM_PASTECART_LE
|- (!x. (!y. ((real_le (vector_norm ((pastecart x) y))) ((real_add (vector_norm x)) (vector_norm y)))))
INT_NOT_LT
|- (!x. (!y. ((~ ((int_lt x) y)) = ((int_le y) x))))
Fan_defs.f_fan_pair_ext
|- (!V. (!E. (!x. (((f_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((f_fan_pair (V , E)) x)) x)))))
Hypermap_iso.IN_TRANS
|- (!x. (!s. (!t. (((t SUBSET s) /\ (x IN t)) ==> (x IN s)))))
Ssrnat.minnC
|- (!m. (!n. (((minn m) n) = ((minn n) m))))
REAL_EQ_ADD_LCANCEL_0
|- (!x. (!y. ((((real_add x) y) = x) = (y = (real_of_num (NUMERAL _0))))))
Sphere.periodic
|- (!n. (!f. (((periodic f) n) = (!i. ((f (i + n)) = (f i))))))
Seq.reshape
|- ((((reshape ((CONS n) sh)) s) = ((CONS ((take n) s)) ((reshape sh) ((dropl n) s)))) /\ (((reshape NIL) s) = NIL))