hcju/holps huggingface dataset:
MEASURABLE_ON_COMPOSE_CONTINUOUS_CLOSED_SET_0
|- (!f. (!g. (!s. (!t. (((closed s) /\ (((measurable_on f) t) /\ ((!x. ((f x) IN s)) /\ (((continuous_on g) s) /\ (((vec (NUMERAL _0)) IN s) /\ ((g (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))))) ==> ((measurable_on (g o f)) t))))))
REAL_INTERVAL_TRANSLATION
|- ((!c. (!a. (!b. ((closed_real_interval ((CONS (((real_add c) a) , ((real_add c) b))) NIL)) = ((IMAGE (\x. ((real_add c) x))) (closed_real_interval ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_real_interval (((real_add c) a) , ((real_add c) b))) = ((IMAGE (\x. ((real_add c) x))) (open_real_interval (a , b))))))))
BANACH_FIX
|- (!f. (!s. (!c. (((complete s) /\ ((~ (s = EMPTY)) /\ (((real_le (real_of_num (NUMERAL _0))) c) /\ (((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ ((((IMAGE f) s) SUBSET s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (distance ((f x) , (f y)))) ((real_mul c) (distance (x , y)))))))))))) ==> (?!x. ((x IN s) /\ ((f x) = x)))))))
DIFFERENTIABLE_IMP_CONTINUOUS_ON
|- (!f. (!s. (((differentiable_on f) s) ==> ((continuous_on f) s))))
CX_NEG
|- (!x. ((Cx (real_neg x)) = (vector_neg (Cx x))))
NPRODUCT_EQ_1
|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (NUMERAL (BIT1 _0))))) ==> (((nproduct s) f) = (NUMERAL (BIT1 _0))))))
COMPLEX_MUL_CSIN_CCOS
|- (!w. (!z. (((complex_mul (csin w)) (ccos z)) = ((complex_div ((vector_add (csin ((vector_add w) z))) (csin ((vector_sub w) z)))) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))
SERIES_DIRICHLET
|- (!f. (!g. (!N. (!k. (!m. (((bounded (GSPEC (\GEN%PVAR%1126. (?n. (((SETSPEC GEN%PVAR%1126) (n IN UNIV)) ((vsum (m .. n)) f)))))) /\ ((!n. ((N <= n) ==> ((real_le (g (n + (NUMERAL (BIT1 _0))))) (g n)))) /\ (((--> (lift o g)) (vec (NUMERAL _0))) sequentially))) ==> ((summable (from k)) (\n. ((% (g n)) (f n))))))))))
Nonlinear_lemma.vol3f_palt
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!r. (!f. (((y3 = r) /\ ((y4 = r) /\ (y5 = r))) ==> ((((((vol3f y1) y2) y6) r) f) = ((real_sub ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_div mm1) pi))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))) ((real_add ((((((dih3_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih4_y y1) y2) y3) y4) y5) y6)) ((real_sub ((((((dih5_y y1) y2) y3) y4) y5) y6)) ((real_mul (real_of_num (NUMERAL (BIT1 (BIT1 _0))))) pi))))))))) ((real_mul ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 _0))))))) ((real_div mm2) pi))) ((real_add ((real_mul (f ((real_div y1) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((real_add ((real_mul (f ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) y3) y4) y5) y6))) ((real_mul (f ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) y3) y4) y5) y6))))))))))))))))
IS_INTERVAL_INTERVAL
|- (!a. (!b. ((is_interval (open_interval (a , b))) /\ (is_interval (closed_interval ((CONS (a , b)) NIL))))))
CLOSURE_BALL
|- (!x. (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((closure (ball (x , e))) = (cball (x , e))))))
HULL_HULL
|- (!P. (!s. (((hull P) ((hull P) s)) = ((hull P) s))))
NADD_MUL_SYM
|- (!x. (!y. ((nadd_eq ((nadd_mul x) y)) ((nadd_mul y) x))))
EXP_ADD
|- (!m. (!n. (!p. ((m EXP (n + p)) = ((m EXP n) * (m EXP p))))))
IM_MUL_CX
|- (!x. (!z. (((Im ((complex_mul (Cx x)) z)) = ((real_mul x) (Im z))) /\ ((Im ((complex_mul z) (Cx x))) = ((real_mul (Im z)) x)))))
LIM_COMPLEX_MUL
|- (!net. (!f. (!g. (!l. (!m. (((((--> f) l) net) /\ (((--> g) m) net)) ==> (((--> (\x. ((complex_mul (f x)) (g x)))) ((complex_mul l) m)) net)))))))
rotation_matrix
|- (!Q. ((rotation_matrix Q) = ((orthogonal_matrix Q) /\ ((det Q) = (real_of_num (NUMERAL (BIT1 _0)))))))
Ssrnat.leq_addr
|- (!m. (!n. (n <= (n + m))))
Ssrnat.halfS
|- (!n. ((half (SUC n)) = (uphalf n)))
SIMPLEX_FURTHEST_LE
|- (!a. (!s. (((FINITE s) /\ (~ (s = EMPTY))) ==> (?y. ((y IN s) /\ (!x. ((x IN ((hull convex) s)) ==> ((real_le (vector_norm ((vector_sub x) a))) (vector_norm ((vector_sub y) a))))))))))
Packing3.VORONOI_SET_2
|- (!V. (!u. (!v. (((voronoi_set V) (u INSERT (v INSERT EMPTY))) = (((voronoi_closed V) v) INTER ((voronoi_closed V) u))))))
RESTRICTION_DEFINED
|- (!s. (!f. (!x. ((x IN s) ==> ((((RESTRICTION s) f) x) = (f x))))))
Trigonometry.JLWZFBH4
|- (!s. ((coplanar s) = (?u. (?v. (?w. (s SUBSET ((hull affine) (u INSERT (v INSERT (w INSERT EMPTY))))))))))
DIV_LE_EXCLUSION
|- (!a. (!b. (!c. (!d. (((~ (b = (NUMERAL _0))) /\ ((b * c) < ((a + (NUMERAL (BIT1 _0))) * d))) ==> ((c DIV d) <= (a DIV b)))))))
ARITH_SUC
|- ((!n. ((SUC (NUMERAL n)) = (NUMERAL (SUC n)))) /\ (((SUC _0) = (BIT1 _0)) /\ ((!n. ((SUC (BIT0 n)) = (BIT1 n))) /\ (!n. ((SUC (BIT1 n)) = (BIT0 (SUC n)))))))
REAL_NEG_RMUL
|- (!x. (!y. ((real_neg ((real_mul x) y)) = ((real_mul x) (real_neg y)))))
Conforming.SIGMA_FAN_OF_FANADD_AT_POINT2
|- (!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) v) u) = w)))))))))
Add_triangle.pair_expand
|- (!p. (p = ((FST p) , (SND p))))
REAL_LTE_ADD2
|- (!w. (!x. (!y. (!z. ((((real_lt w) x) /\ ((real_le y) z)) ==> ((real_lt ((real_add w) y)) ((real_add x) z)))))))
CARD_MUL_FINITE
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> (FINITE ((*_c s) t)))))
PATHFINISH_IN_PATH_IMAGE
|- (!g. ((pathfinish g) IN (path_image g)))
Pack2.MEASURE_VORONOI_CLOSED_OPEN
|- (!s. (!v. ((measure ((voronoi_closed s) v)) = (measure ((voronoi_open s) v)))))
csin
|- (!z. ((csin z) = ((complex_div ((vector_sub (cexp ((complex_mul ii) z))) (cexp ((complex_mul (vector_neg ii)) z)))) ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ii))))
SUBPATH_SCALING_LEMMA
|- (!u. (!v. (((IMAGE (\x. ((vector_add u) ((% (drop ((vector_sub v) u))) x)))) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) = (closed_segment ((CONS (u , v)) NIL)))))
AFF_DIM_OPEN_IN
|- (!s. (!t. (((~ (s = EMPTY)) /\ (((open_in ((subtopology euclidean) t)) s) /\ (affine t))) ==> ((aff_dim s) = (aff_dim t)))))
NSUM_ADD_NUMSEG
|- (!f. (!g. (!m. (!n. (((nsum (m .. n)) (\i. ((f i) + (g i)))) = (((nsum (m .. n)) f) + ((nsum (m .. n)) g)))))))
EQ_MULT_LCANCEL
|- (!m. (!n. (!p. (((m * n) = (m * p)) = ((m = (NUMERAL _0)) \/ (n = p))))))
CONTRACTIBLE_PCROSS_EQ
|- (!s. (!t. ((contractible (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((contractible s) /\ (contractible t)))))))
MDIST_SYM
|- (!m. (!x. (!y. (((x IN (mspace m)) /\ (y IN (mspace m))) ==> (((mdist m) (x , y)) = ((mdist m) (y , x)))))))
SUM_POS_LE_NUMSEG
|- (!m. (!n. (!f. ((!p. (((m <= p) /\ (p <= n)) ==> ((real_le (real_of_num (NUMERAL _0))) (f p)))) ==> ((real_le (real_of_num (NUMERAL _0))) ((sum (m .. n)) f))))))
FRONTIER_BIJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> ((frontier ((IMAGE f) s)) = ((IMAGE f) (frontier s))))))
DE_MORGAN_THM
|- (!t1. (!t2. (((~ (t1 /\ t2)) = ((~ t1) \/ (~ t2))) /\ ((~ (t1 \/ t2)) = ((~ t1) /\ (~ t2))))))
TRANSLATION_UNIV
|- (!a. (((IMAGE (\x. ((vector_add a) x))) UNIV) = UNIV))
Real_ext.REAL_PROP_ZERO_SQRT
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> (((sqrt x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0))))))
Collect_geom.conv
|- (!S. ((conv S) = (((affsign sgn_ge) EMPTY) S)))
LEFT_FORALL_OR_THM
|- (!P. (!Q. ((!x. ((P x) \/ Q)) = ((!x. (P x)) \/ Q))))
GDELTA_UNION
|- (!s. (!t. (((gdelta s) /\ (gdelta t)) ==> (gdelta (s UNION t)))))
INSIDE_TRANSLATION
|- (!a. (!s. ((inside ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (inside s)))))
INVERTIBLE_MATRIX_MUL
|- (!A. (!B. ((invertible ((matrix_mul A) B)) = ((invertible A) /\ (invertible B)))))
PAIR_EQ
|- (!x. (!y. (!a. (!b. (((x , y) = (a , b)) = ((x = a) /\ (y = b)))))))
FSIGMA_INTER
|- (!s. (!t. (((fsigma s) /\ (fsigma t)) ==> (fsigma (s INTER t)))))
COMPLEX_MUL_LID
|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT1 _0))))) x) = x))
HAS_REAL_INTEGRAL_REFLECT_GEN
|- (!f. (!i. (!s. ((((has_real_integral (\x. (f (real_neg x)))) i) s) = (((has_real_integral f) i) ((IMAGE real_neg) s))))))
ORDINAL_CHAINED
|- (!l. (!m. (((ordinal l) /\ (ordinal m)) ==> (((inseg m) l) \/ ((inseg l) m)))))
Ssrnat.minn
|- (!m. (!n. (((minn m) n) = (((COND (m < n)) m) n))))
FORALL_IN_INTERMEDIATE_CLOSURE
|- (!f. (!s. (!s'. (!t. (((closed t) /\ (((continuous_on f) s') /\ ((s SUBSET s') /\ ((s' SUBSET (closure s)) /\ (!x. ((x IN s) ==> ((f x) IN t))))))) ==> (!x. ((x IN s') ==> ((f x) IN t))))))))
HAS_BOUNDED_VARIATION_ABSOLUTELY_INTEGRABLE_DERIVATIVE
|- (!f. (!s. (!a. (!b. (((COUNTABLE s) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> ((differentiable f) (at x)))))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = ((absolutely_integrable_on (\x. ((vector_derivative f) (at x)))) (closed_interval ((CONS (a , b)) NIL)))))))))
Appendix.scs_prop_equ_v39
|- (!s. (!i. (((scs_prop_equ_v39 s) i) = ((LET (\shift1. (LET_END ((LET (\shift2. (LET_END ((LET (\shift2b. (LET_END (scs_v39 ((scs_k_v39 s) , ((scs_d_v39 s) , ((shift2 (scs_a_v39 s)) , ((shift2 (scs_am_v39 s)) , ((shift2 (scs_bm_v39 s)) , ((shift2 (scs_b_v39 s)) , ((shift2b (scs_J_v39 s)) , ((shift1 (scs_lo_v39 s)) , ((shift1 (scs_hi_v39 s)) , (shift1 (scs_str_v39 s))))))))))))))) (\f. (\j. (\j'. ((f (i + j)) (i + j'))))))))) (\f. (\j. (\j'. ((f (i + j)) (i + j'))))))))) (\f. (\j. (f (i + j))))))))
COS_PERIODIC
|- (!x. ((cos ((real_add x) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))) = (cos x)))
Hypermap_and_fan.TRIANGULAR_FACE
|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (((v , w) IN (dart1_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , w') INSERT ((w' , v) INSERT EMPTY)))) /\ (((((((sigma_fan (vec (NUMERAL _0))) V) E) w) w') = v) /\ ((((((sigma_fan (vec (NUMERAL _0))) V) E) w') v) = w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))
REAL_SERIES_LMUL
|- (!x. (!x0. (!c. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\n. ((real_mul c) (x n)))) ((real_mul c) x0)) s))))))
CONVEX_TRANSLATION
|- (!s. (!a. ((convex s) ==> (convex ((IMAGE (\x. ((vector_add a) x))) s)))))
COMPLEX_MUL_LZERO
|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL _0)))) x) = (Cx (real_of_num (NUMERAL _0)))))
FORALL_IN_PCROSS
|- ((!z. ((z IN (s PCROSS t)) ==> (P z))) = (!x. (!y. (((x IN s) /\ (y IN t)) ==> (P ((pastecart x) y))))))
NORM_LIFT
|- (!x. ((vector_norm (lift x)) = (real_abs x)))
MATRIX_MUL_RCANCEL
|- (!A. (!B. (!C. ((invertible C) ==> ((((matrix_mul A) C) = ((matrix_mul B) C)) = (A = B))))))
SERIES_RATIO
|- (!c. (!a. (!s. (!N. ((((real_lt c) (real_of_num (NUMERAL (BIT1 _0)))) /\ (!n. ((n >= N) ==> ((real_le (vector_norm (a (SUC n)))) ((real_mul c) (vector_norm (a n))))))) ==> (?l. (((sums a) l) s)))))))
LINEAR_CONTINUOUS_WITHIN
|- (!f. (!s. (!x. ((linear f) ==> ((continuous f) ((within (at x)) s))))))
IMAGE_STRETCH_INTERVAL
|- (!a. (!b. (!m. (((IMAGE (\x. (lambdak. ((real_mul (m k)) (x $ k))))) (closed_interval ((CONS (a , b)) NIL))) = (((COND ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) EMPTY) (closed_interval ((CONS ((lambdak. ((real_min ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))) , (lambdak. ((real_max ((real_mul (m k)) (a $ k))) ((real_mul (m k)) (b $ k)))))) NIL)))))))
Sphere.rotate3
|- (!f. (!x3. (!x1. (!x2. (!x6. (!x4. (!x5. ((((((((rotate3 f) x1) x2) x3) x4) x5) x6) = ((((((f x3) x1) x2) x6) x4) x5)))))))))
RECTIFIABLE_PATH_TRANSLATION_EQ
|- (!a. (!g. ((rectifiable_path ((\x. ((vector_add a) x)) o g)) = (rectifiable_path g))))
COVERING_SPACE_FINITE_SHEETS_EQ_CLOSED_MAP_STRONG
|- (!p. (!c. (!s. ((((covering_space (c , p)) s) /\ (!b. ((b IN s) ==> ((limit_point_of b) s)))) ==> ((!b. ((b IN s) ==> (FINITE (GSPEC (\GEN%PVAR%2080. (?x. (((SETSPEC GEN%PVAR%2080) ((x IN c) /\ ((p x) = b))) x))))))) = (!t. (((closed_in ((subtopology euclidean) c)) t) ==> ((closed_in ((subtopology euclidean) s)) ((IMAGE p) t)))))))))
SEGMENT_REFL
|- ((!a. ((closed_segment ((CONS (a , a)) NIL)) = (a INSERT EMPTY))) /\ (!a. ((open_segment (a , a)) = EMPTY)))
REAL_ABS_1
|- ((real_abs (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))
HAS_VECTOR_DERIVATIVE_AT_WITHIN
|- (!f. (!x. (!s. ((((has_vector_derivative f) f') (at x)) ==> (((has_vector_derivative f) f') ((within (at x)) s))))))
REAL_TENDSTO
|- (((--> s) l) = ((---> (drop o s)) (drop l)))
CLOSED_INTERVAL
|- (!a. (!b. (closed (closed_interval ((CONS (a , b)) NIL)))))
CARD_LE_REFL
|- (!s. (s <=_c s))
Lp_ineqs_def.tau_fan
|- (!V. (!E. (!f. (((tau_fan (V , E)) f) = (real_abs (tauVEF (V , (E , f))))))))
Appendix.scs_lo_v39_explicit
|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_lo_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = lo))))))))))
Ssrbool.predC
|- (!p. ((predC p) = (\x. (~ (p x)))))
REAL_POW_LE
|- (!x. (!n. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_pow x) n)))))
REAL_LE_BETWEEN
|- (!a. (!b. (((real_le a) b) = (?x. (((real_le a) x) /\ ((real_le x) b))))))
LOG_POW
|- (!n. (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((log ((real_pow x) n)) = ((real_mul (real_of_num n)) (log x))))))
Lvducxu.FACE_NODE_EDGE_ORBIT_INVERSE
|- ((((face H) x) = ((orbit_map (inverse (face_map H))) x)) /\ ((((node H) x) = ((orbit_map (inverse (node_map H))) x)) /\ (((edge H) x) = ((orbit_map (inverse (edge_map H))) x))))
STEINHAUS_SUMS
|- (!s. (!t. (((lebesgue_measurable s) /\ ((~ (negligible s)) /\ ((lebesgue_measurable t) /\ (~ (negligible t))))) ==> (~ ((interior (GSPEC (\GEN%PVAR%4605. (?x. (?y. (((SETSPEC GEN%PVAR%4605) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))) = EMPTY)))))
Ssrnat.subnn
|- (!n. ((n - n) = (NUMERAL _0)))
CONNECTED_FROM_OPEN_UNION_AND_INTER_LOCAL
|- (!u. (!s. (!t. ((((open_in ((subtopology euclidean) u)) s) /\ (((open_in ((subtopology euclidean) u)) t) /\ ((connected (s UNION t)) /\ (connected (s INTER t))))) ==> ((connected s) /\ (connected t))))))
RETRACT_OF_UNIV
|- (!s. (((retract_of s) UNIV) = ((AR s) /\ (closed s))))
EXPLICIT_SUBSET_RELATIVE_INTERIOR_CONVEX_HULL_MINIMAL
|- (!s. ((FINITE s) ==> ((GSPEC (\GEN%PVAR%1790. (?y. (((SETSPEC GEN%PVAR%1790) (?u. ((!x. ((x IN s) ==> ((real_lt (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y)))) SUBSET (relative_interior ((hull convex) s)))))
LEBESGUE_MEASURABLE_INTERVAL
|- ((!a. (!b. (lebesgue_measurable (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (lebesgue_measurable (open_interval (a , b))))))
HOMOTOPIC_PATHS_NEARBY_EXPLICIT
|- (!g. (!s. (!h. (((path g) /\ ((path h) /\ (((pathstart h) = (pathstart g)) /\ (((pathfinish h) = (pathfinish g)) /\ (!t. (!x. (((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (~ (x IN s))) ==> ((real_lt (vector_norm ((vector_sub (h t)) (g t)))) (vector_norm ((vector_sub (g t)) x)))))))))) ==> (((homotopic_paths s) g) h)))))
HOMOTOPIC_CONSTANT_MAPS
|- (!s. (!t. (!a. (!b. (((((homotopic_with (\x. T)) (s , t)) (\x. a)) (\x. b)) = ((s = EMPTY) \/ (((path_component t) a) b)))))))
NORM_LE_INFNORM
|- (!x. ((real_le (vector_norm x)) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (infnorm x))))
Tame_defs2.C_EL
|- (!n. (!l. (((#!# l) n) = ((EL n) l))))
Hypermap_and_fan.INVERSE_F_FAN_PAIR_EXT_EXPLICIT
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((inverse (f_fan_pair_ext (V , E))) = ((res (GABS (\f. (!v. (!w. ((GEQ (f (v , w))) ((((((sigma_fan (vec (NUMERAL _0))) V) E) v) w) , v))))))) (dart1_of_fan (V , E)))))))
Ssrbool.andb_id2l
|- (!a. (!b. (!c. ((a ==> (b = c)) ==> ((a /\ b) = (a /\ c))))))
Seq.uniq_perm_eq
|- (!s1. (!s2. ((uniq s1) ==> ((uniq s2) ==> ((!x. (((MEM x) s1) = ((MEM x) s2))) ==> ((perm_eq s1) s2))))))
FLOOR
|- (!x. ((integer (floor x)) /\ (((real_le (floor x)) x) /\ ((real_lt x) ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))
HAS_SIZE_UNION
|- (!s. (!t. (!m. (!n. (((s HAS_SIZE m) /\ ((t HAS_SIZE n) /\ ((DISJOINT s) t))) ==> ((s UNION t) HAS_SIZE (m + n)))))))
CARD_EQ_RESTRICTED_POWERSET
|- (!s. (!t. ((INFINITE s) ==> ((GSPEC (\GEN%PVAR%48. (?k. (((SETSPEC GEN%PVAR%48) ((k SUBSET s) /\ (k =_c t))) k)))) =_c (((COND (t <=_c s)) ((^_c s) t)) EMPTY)))))
toset
|- (!l. ((toset l) = ((poset l) /\ (!x. (!y. (((x IN (fl l)) /\ (y IN (fl l))) ==> ((l (x , y)) \/ (l (y , x)))))))))
ABSOLUTELY_CONTINUOUS_COMPARISON
|- (!f. (!g. (!s. ((((absolutely_continuous_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)))))))) ==> ((absolutely_continuous_on g) s)))))
EQ_ADD_RCANCEL
|- (!m. (!n. (!p. (((m + p) = (n + p)) = (m = n)))))
POLYHEDRON_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (polyhedron s)) ==> (polyhedron ((IMAGE f) s)))))
one_RECURSION
|- (!e. (?fn. ((fn one) = e)))
ITERATE_CLOSED
|- (!op. ((monoidal op) ==> (!P. (((P (neutral op)) /\ (!x. (!y. (((P x) /\ (P y)) ==> (P ((op x) y)))))) ==> (!f. (!s. ((!x. (((x IN s) /\ (~ ((f x) = (neutral op)))) ==> (P (f x)))) ==> (P (((iterate op) s) f)))))))))
HOMEOMORPHIC_MINIMAL
|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((!x. ((x IN s) ==> (((f x) IN t) /\ ((g (f x)) = x)))) /\ ((!y. ((y IN t) ==> (((g y) IN s) /\ ((f (g y)) = y)))) /\ (((continuous_on f) s) /\ ((continuous_on g) t)))))))))
Imjxphr.EXISTS_SMALL_LE_CONST
|- ((((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x6) /\ ((~ (collinear ((vec (NUMERAL _0)) INSERT (v1 INSERT (v2 INSERT EMPTY))))) /\ ((((real_pow (vector_norm v1)) (NUMERAL (BIT0 (BIT1 _0)))) = x1) /\ ((((real_pow (vector_norm v2)) (NUMERAL (BIT0 (BIT1 _0)))) = x2) /\ ((((real_pow (vector_norm ((vector_sub v1) v2))) (NUMERAL (BIT0 (BIT1 _0)))) = x6) /\ ((a = (real_neg (real_of_num (NUMERAL (BIT1 _0))))) /\ ((real_lt c) (distance (v2 , w))))))))))) ==> (?e. (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!t. ((((real_lt (real_of_num (NUMERAL _0))) t) /\ ((real_lt t) e)) ==> ((real_lt c) (distance (((((((((v3_defor_v1 a) v1) v2) x1) x2) x6) x6) ((real_sub x2) t)) , w))))))))
CLOSED_SEQUENTIAL_LIMITS
|- (!s. ((closed s) = (!x. (!l. (((!n. ((x n) IN s)) /\ (((--> x) l) sequentially)) ==> (l IN s))))))
Functional_equation.proj_x4
|- (((((((proj_x4 x1) x2) x3) x4) x5) x6) = x4)
BUTLAST
|- (((BUTLAST NIL) = NIL) /\ ((BUTLAST ((CONS h) t)) = (((COND (t = NIL)) NIL) ((CONS h) (BUTLAST t)))))
FINITE_NUMSEG_LT
|- (!n. (FINITE (GSPEC (\GEN%PVAR%119. (?m. (((SETSPEC GEN%PVAR%119) (m < n)) m))))))
CONTINUOUS_AT_DIST_CLOSEST_POINT
|- (!s. (!x. (((closed s) /\ (~ (s = EMPTY))) ==> ((continuous (\x. (lift (distance (x , ((closest_point s) x)))))) (at x)))))
NOT_ODD
|- (!n. ((~ (ODD n)) = (EVEN n)))
Real_ext.REAL_PROP_NN_SQRT
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) (sqrt x))))
REAL_EQ_MUL_LCANCEL
|- (!x. (!y. (!z. ((((real_mul x) y) = ((real_mul x) z)) = ((x = (real_of_num (NUMERAL _0))) \/ (y = z))))))
LAMBDA_UNPAIR_THM
|- (!f. ((GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((f x) y)))))) = (\p. ((f (FST p)) (SND p)))))
Sphere.asn797k
|- (!x2. (!x3. (!x4. (!x5. (!x6. (!k. (((((((asn797k k) x2) x3) x4) x5) x6) = ((real_mul k) (asn ((real_mul (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)))))))))))))) (sin ((real_div pi) k))))))))))))
TIETZE_CLOSED_INTERVAL
|- (!f. (!u. (!s. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((closed_in ((subtopology euclidean) u)) s) /\ (((continuous_on f) s) /\ (!x. ((x IN s) ==> ((f x) IN (closed_interval ((CONS (a , b)) NIL)))))))) ==> (?g. (((continuous_on g) u) /\ ((!x. ((x IN s) ==> ((g x) = (f x)))) /\ (!x. ((x IN u) ==> ((g x) IN (closed_interval ((CONS (a , b)) NIL))))))))))))))
FUNDAMENTAL_THEOREM_OF_CALCULUS_STRONG
|- (!f. (!f'. (!s. (!a. (!b. (((COUNTABLE s) /\ (((real_le (drop a)) (drop b)) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (f' x)) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (((has_integral f') ((vector_sub (f b)) (f a))) (closed_interval ((CONS (a , b)) NIL)))))))))
REAL_CONTINUOUS_WITHINREAL_SUBSET
|- (!f. (!s. (!t. ((((real_continuous f) ((within (atreal x)) s)) /\ (t SUBSET s)) ==> ((real_continuous f) ((within (atreal x)) t))))))
SERIES_CMUL
|- (!x. (!x0. (!c. (!s. ((((sums x) x0) s) ==> (((sums (\n. ((% c) (x n)))) ((% c) x0)) s))))))
VECTOR_1
|- (((vector ((CONS x) NIL)) $ (NUMERAL (BIT1 _0))) = x)
List_hypermap.e_list
|- (!d. ((e_list d) = ((SND d) , (FST d))))
Ssrnat.addnA
|- (!n. (!m. (!p. ((n + (m + p)) = ((n + m) + p)))))
UNION
|- (!s. (!t. ((s UNION t) = (GSPEC (\GEN%PVAR%0. (?x. (((SETSPEC GEN%PVAR%0) ((x IN s) \/ (x IN t))) x)))))))
DOT_SQUARE_NORM
|- (!x. (((dot x) x) = ((real_pow (vector_norm x)) (NUMERAL (BIT0 (BIT1 _0))))))
Seq.rcons
|- (!x. (!t. (!z. ((((rcons NIL) z) = ((CONS z) NIL)) /\ (((rcons ((CONS x) t)) z) = ((CONS x) ((rcons t) z)))))))
Functional_equation.dih4_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)))))) ==> (((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4_x' x1) x2) x3) x4) x5) x6)))))))))
LINEAR_0
|- (!f. ((linear f) ==> ((f (vec (NUMERAL _0))) = (vec (NUMERAL _0)))))
INTERIOR_CBALL
|- (!x. (!e. ((interior (cball (x , e))) = (ball (x , e)))))
DOT_BASIS
|- (!x. (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((((dot (basis i)) x) = (x $ i)) /\ (((dot x) (basis i)) = (x $ i))))))
NSUM_LE
|- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((f x) <= (g x))))) ==> (((nsum s) f) <= ((nsum s) g))))))
CONTINUOUS_GDELTA_PREIMAGE
|- (!f. (!s. (!t. ((((continuous_on f) s) /\ ((gdelta s) /\ (gdelta t))) ==> (gdelta (GSPEC (\GEN%PVAR%1321. (?x. (((SETSPEC GEN%PVAR%1321) ((x IN s) /\ ((f x) IN t))) x)))))))))
Seq.mem_undup
|- (!s. (!x. (((MEM x) (undup s)) = ((MEM x) s))))
REFLECT_UNIV
|- (((IMAGE vector_neg) UNIV) = UNIV)
SUBSET_INTERVAL_1
|- (!a. (!b. (!c. (!d. ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_lt (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_le (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))) /\ ((((closed_interval ((CONS (a , b)) NIL)) SUBSET (open_interval (c , d))) = (((real_lt (drop b)) (drop a)) \/ (((real_lt (drop c)) (drop a)) /\ (((real_le (drop a)) (drop b)) /\ ((real_lt (drop b)) (drop d)))))) /\ ((((open_interval (a , b)) SUBSET (closed_interval ((CONS (c , d)) NIL))) = (((real_le (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_lt (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))) /\ (((open_interval (a , b)) SUBSET (open_interval (c , d))) = (((real_le (drop b)) (drop a)) \/ (((real_le (drop c)) (drop a)) /\ (((real_lt (drop a)) (drop b)) /\ ((real_le (drop b)) (drop d)))))))))))))
CONTINUOUS_AT_WITHIN
|- (!f. (!x. (!s. (((continuous f) (at x)) ==> ((continuous f) ((within (at x)) s))))))
STC_CASES_L
|- (!R. (!x. (!z. ((((STC R) x) z) = (((R x) z) \/ ((((STC R) z) x) \/ (?y. ((((STC R) x) y) /\ ((R y) z)))))))))
RAT_LEMMA5
|- ((((real_lt (real_of_num (NUMERAL _0))) y1) /\ ((real_lt (real_of_num (NUMERAL _0))) y2)) ==> ((((real_div x1) y1) = ((real_div x2) y2)) = (((real_mul x1) y2) = ((real_mul x2) y1))))