hcju/holps huggingface dataset:
SETDIST_FRONTIER
|- ((!s. (!t. (((DISJOINT s) t) ==> ((setdist ((frontier s) , t)) = (setdist (s , t)))))) /\ (!s. (!t. (((DISJOINT s) t) ==> ((setdist (s , (frontier t))) = (setdist (s , t)))))))
OPEN_IMP_ANR
|- (!s. ((open s) ==> (ANR s)))
SPAN_VSUM
|- (!s. (!f. (!t. (((FINITE t) /\ (!x. ((x IN t) ==> ((f x) IN (span s))))) ==> (((vsum t) f) IN (span s))))))
o_THM
|- (!f. (!g. (!x. (((f o g) x) = (f (g x))))))
ANR_PCROSS_EQ
|- (!s. (!t. ((ANR (s PCROSS t)) = ((s = EMPTY) \/ ((t = EMPTY) \/ ((ANR s) /\ (ANR t)))))))
REAL_POW_MONO
|- (!m. (!n. (!x. ((((real_le (real_of_num (NUMERAL (BIT1 _0)))) x) /\ (m <= n)) ==> ((real_le ((real_pow x) m)) ((real_pow x) n))))))
HAS_BOUNDED_REAL_VARIATION_ON_LMUL
|- (!f. (!c. (!s. (((has_bounded_real_variation_on f) s) ==> ((has_bounded_real_variation_on (\x. ((real_mul c) (f x)))) s)))))
CLOSED_IN_UNIONS
|- (!top. (!s. (((FINITE s) /\ (!t. ((t IN s) ==> ((closed_in top) t)))) ==> ((closed_in top) (UNIONS s)))))
Lp_main_estimate.tau_fun_local
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun ((v_prime V) f)) ((e_prime E) f)) f) = (((tau_fun V) E) f)))))))
COMPLEX_DIFF_CHAIN_WITHIN
|- (!f. (!g. (!f'. (!g'. (!x. (!s. (((((has_complex_derivative f) f') ((within (at x)) s)) /\ (((has_complex_derivative g) g') ((within (at (f x))) ((IMAGE f) s)))) ==> (((has_complex_derivative (g o f)) ((complex_mul g') f')) ((within (at x)) s)))))))))
NEGLIGIBLE_AFFINE_HULL
|- (!s. (((FINITE s) /\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull affine) s))))
Functional_equation.proj_x3
|- (((((((proj_x3 x1) x2) x3) x4) x5) x6) = x3)
has_real_integral
|- (!f. (!y. (!s. ((((has_real_integral f) y) s) = (((has_integral (lift o (f o drop))) (lift y)) ((IMAGE lift) s))))))
ONORM
|- (!f. ((linear f) ==> ((!x. ((real_le (vector_norm (f x))) ((real_mul (onorm f)) (vector_norm x)))) /\ (!b. ((!x. ((real_le (vector_norm (f x))) ((real_mul b) (vector_norm x)))) ==> ((real_le (onorm f)) b))))))
Tame_general.CONTRAVENING_DART_DIST
|- (!V. (!x. (((contravening V) /\ (x IN (dart_of_fan (V , (ESTD V))))) ==> (((real_le ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (distance x)) /\ ((real_le (distance x)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))
REAL_NORM
|- (!z. ((real z) ==> ((vector_norm z) = (real_abs (Re z)))))
INTERIOR_TRANSLATION
|- (!a. (!s. ((interior ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (interior s)))))
INSIDE_OUTSIDE
|- (!s. ((inside s) = (UNIV DIFF (s UNION (outside s)))))
rowvector
|- (!v. ((rowvector v) = (lambdai. (lambdaj. (v $ j)))))
FRONTIER_COMPLEMENT
|- (!s. ((frontier (UNIV DIFF s)) = (frontier s)))
CONTINUOUS_ON_CONST
|- (!s. (!c. ((continuous_on (\x. c)) s)))
Topology.set_of_orbits_points_fan
|- (!x. (!V. (!E. (!v. (!u. ((((((set_of_orbits_points_fan x) V) E) v) u) = (GSPEC (\GEN%PVAR%5483. (?i. (((SETSPEC GEN%PVAR%5483) ((NUMERAL _0) <= i)) (((((((power_map_points sigma_fan) x) V) E) v) u) i)))))))))))
Tame_defs.split_at
|- (!c. (!xs. (((split_at c) xs) = (((split_at_rec c) NIL) xs))))
BOREL_IMP_LEBESGUE_MEASURABLE
|- (!s. ((borel s) ==> (lebesgue_measurable s)))
Seq.rev_cons
|- (!x. (!s. ((rev ((CONS x) s)) = ((rcons (rev s)) x))))
REAL_MUL
|- (!w. (!z. (((real w) /\ (real z)) ==> (real ((complex_mul w) z)))))
POSITIVE_DEFINITE_MUL_EQ
|- (!A. (!B. (((positive_definite A) /\ (positive_definite B)) ==> ((positive_definite ((matrix_mul A) B)) = (((matrix_mul A) B) = ((matrix_mul B) A))))))
REAL_NEG_MINUS1
|- (!x. ((real_neg x) = ((real_mul (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x)))
Add_triangle.f_fan_pair_power
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. (!i. ((d IN (dart1_of_fan (V , E))) ==> ((((POWER (f_fan_pair (V , E))) i) d) = (((POWER (f_fan_pair_ext (V , E))) i) d))))))))
AFFINE_LINEAR_IMAGE
|- (!f. (!s. (((affine s) /\ (linear f)) ==> (affine ((IMAGE f) s)))))
Sphere.acs_sqrt_x2_d4
|- (!x1. (!x3. (!x4. (!x5. (!x6. (!x2. (((((((acs_sqrt_x2_d4 x1) x2) x3) x4) x5) x6) = (acs ((real_div (sqrt x2)) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))))))
CNJ_EQ_CX
|- (!x. (!z. (((cnj z) = (Cx x)) = (z = (Cx x)))))
polytope
|- (!s. ((polytope s) = (?v. ((FINITE v) /\ (s = ((hull convex) v))))))
PATH_CONTINUOUS_IMAGE
|- (!f. (!g. (((path g) /\ ((continuous_on f) (path_image g))) ==> (path (f o g)))))
HAS_REAL_INTEGRAL_SUBSTITUTION_SIMPLE
|- (!f. (!g. (!g'. (!a. (!b. (!c. (!d. ((((real_continuous_on f) (closed_real_interval ((CONS (c , d)) NIL))) /\ ((!x. ((x IN (closed_real_interval ((CONS (a , b)) NIL))) ==> (((has_real_derivative g) (g' x)) ((within (atreal x)) (closed_real_interval ((CONS (a , b)) NIL)))))) /\ ((((IMAGE g) (closed_real_interval ((CONS (a , b)) NIL))) SUBSET (closed_real_interval ((CONS (c , d)) NIL))) /\ (((real_le a) b) /\ (((real_le c) d) /\ ((real_le (g a)) (g b))))))) ==> (((has_real_integral (\x. ((real_mul (f (g x))) (g' x)))) ((real_integral (closed_real_interval ((CONS ((g a) , (g b))) NIL))) f)) (closed_real_interval ((CONS (a , b)) NIL)))))))))))
REALLIM_CONG_AT
|- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((---> (\x. (f x))) l) (at a)) = (((---> g) l) (at a))))
relative_frontier
|- (!s. ((relative_frontier s) = ((closure s) DIFF (relative_interior s))))
REAL_INV_EQ_0
|- (!x. (((real_inv x) = (real_of_num (NUMERAL _0))) = (x = (real_of_num (NUMERAL _0)))))
FUN_EQ_THM
|- (!f. (!g. ((f = g) = (!x. ((f x) = (g x))))))
Ssrnat.distnEr
|- (!m. (!n. ((m <= n) ==> (((distn m) n) = (n - m)))))
Lp_ineqs_def.azim2_fan
|- (!V. (!E. (!x. (((azim2_fan (V , E)) x) = ((azim_dart (V , E)) ((f_fan_pair_ext (V , E)) x))))))
Lp_ineqs_proofs.y7_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))))) ((y7_fan (V , E)) d))))))))))
Nonlinear_lemma.dih_y_sym
|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (((((((dih_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y1) y3) y2) y4) y6) y5))))))))
SUM_OFFSET
|- (!p. (!f. (!m. (!n. (((sum ((m + p) .. (n + p))) f) = ((sum (m .. n)) (\i. (f (i + p)))))))))
Real_ext.REAL_PROP_POS_LADD
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))
Float.POW_2
|- (!x. (((real_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((real_mul x) x)))
LINEAR_ADD
|- (!f. (!x. (!y. ((linear f) ==> ((f ((vector_add x) y)) = ((vector_add (f x)) (f y)))))))
REAL_NEG_LMUL
|- (!x. (!y. ((real_neg ((real_mul x) y)) = ((real_mul (real_neg x)) y))))
COLLINEAR_3_EXPLICIT
|- (!x. (!y. (!z. ((collinear (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))) /\ (~ ((a = (real_of_num (NUMERAL _0))) /\ ((b = (real_of_num (NUMERAL _0))) /\ (c = (real_of_num (NUMERAL _0)))))))))))))))
Real_ext.REAL_PROP_NN_ADD2
|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_le (real_of_num (NUMERAL _0))) ((real_add x) y)))))
IM_CLOG_POS_LE
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((real_le (real_of_num (NUMERAL _0))) (Im (clog z))) = ((real_le (real_of_num (NUMERAL _0))) (Im z)))))
Hypermap.lemma_in_face_set
|- (!H. (!x. ((x IN (dart H)) = (((face H) x) IN (face_set H)))))
Lp_ineqs_proofs.y9_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> ((LET (\w'. (LET_END ((LET (\u. (LET_END (((y9_fan (V , E)) (v , w)) = (distance (w , u)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))))
CONTINUOUS_ON_COMPONENTWISE_LIFT
|- (!f. (!s. (((continuous_on f) s) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((continuous_on (\x. (lift ((f x) $ i)))) s))))))
Seq.size_rot
|- (!n0. (!s. ((sizel ((rot n0) s)) = (sizel s))))
HAS_COMPLEX_DERIVATIVE_ID
|- (!net. (((has_complex_derivative (\x. x)) (Cx (real_of_num (NUMERAL (BIT1 _0))))) net))
HOMEOMORPHIC_SYM
|- (!s. (!t. (((homeomorphic s) t) = ((homeomorphic t) s))))
conic
|- (!s. ((conic s) = (!x. (!c. (((x IN s) /\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (((% c) x) IN s))))))
FRAC_FLOOR
|- (!x. ((frac x) = ((real_sub x) (floor x))))
Hypermap_iso.fan_hypermaps_iso_explicit
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso contracted_dart) ((hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))) , (hypermap_of_fan (V , E)))))))
Sphere.INITIAL_SUBLIST
|- (!zl. (!xl. (((initial_sublist xl) zl) = (?yl. (zl = ((APPEND xl) yl))))))
INT_OF_NUM_OF_INT
|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) ==> ((int_of_num (num_of_int x)) = x)))
COMPLEX_POW_1
|- (!x. (((complex_pow x) (NUMERAL (BIT1 _0))) = x))
HULL_SUBSET
|- (!P. (!s. (s SUBSET ((hull P) s))))
CONNECTED_SEGMENT
|- ((!a. (!b. (connected (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. (connected (open_segment (a , b))))))
SUM_CLAUSES_NUMSEG
|- ((!m. (((sum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (real_of_num (NUMERAL _0))))) /\ (!m. (!n. (((sum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((real_add ((sum (m .. n)) f)) (f (SUC n)))) ((sum (m .. n)) f))))))
COMPACT_CONVEX_COLLINEAR_SEGMENT
|- (!s. (((~ (s = EMPTY)) /\ ((compact s) /\ ((convex s) /\ (collinear s)))) ==> (?a. (?b. (s = (closed_segment ((CONS (a , b)) NIL)))))))
DOT_EQ_0
|- (!x. ((((dot x) x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))
LEBESGUE_MEASURABLE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((lebesgue_measurable ((IMAGE f) s)) = (lebesgue_measurable s)))))
LT
|- ((!m. ((m < (NUMERAL _0)) = F)) /\ (!m. (!n. ((m < (SUC n)) = ((m = n) \/ (m < n))))))
IN_CLOSURE_OF
|- (!top. (!s. (!x. ((x IN ((closure_of top) s)) = ((x IN (topspace top)) /\ (!t. (((x IN t) /\ ((open_in top) t)) ==> (?y. ((y IN s) /\ (y IN t))))))))))
HAUSDIST_LINEAR_IMAGE
|- (!f. (!s. (!t. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((hausdist (((IMAGE f) s) , ((IMAGE f) t))) = (hausdist (s , t)))))))
FACE_OF_SING
|- (!x. (!s. (((face_of (x INSERT EMPTY)) s) = ((extreme_point_of x) s))))
Functional_equation.div6
|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((div6 f) g) x1) x2) x3) x4) x5) x6) = ((real_div ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))
RE_VSUM
|- (!f. (!s. ((FINITE s) ==> ((Re ((vsum s) f)) = ((sum s) (\x. (Re (f x))))))))
real_integrable_on
|- (!f. (!i. (((real_integrable_on f) i) = (?y. (((has_real_integral f) y) i)))))
FUBINI_INTEGRAL_INTERVAL
|- (!f. (!a. (!b. (!c. (!d. (((absolutely_integrable_on f) (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) ==> (((integral (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) f) = ((integral (closed_interval ((CONS (a , b)) NIL))) (\x. ((integral (closed_interval ((CONS (c , d)) NIL))) (\y. (f ((pastecart x) y)))))))))))))
Seq.head
|- (!x0. (!h. (!t. ((((headl x0) NIL) = x0) /\ (((headl x0) ((CONS h) t)) = h)))))
HAS_REAL_INTEGRAL_AFFINITY
|- (!f. (!i. (!a. (!b. (!m. (!c. (((((has_real_integral f) i) (closed_real_interval ((CONS (a , b)) NIL))) /\ (~ (m = (real_of_num (NUMERAL _0))))) ==> (((has_real_integral (\x. (f ((real_add ((real_mul m) x)) c)))) ((real_mul (real_inv (real_abs m))) i)) ((IMAGE (\x. ((real_mul (real_inv m)) ((real_sub x) c)))) (closed_real_interval ((CONS (a , b)) NIL)))))))))))
REAL_SGN_POW_2
|- (!x. ((real_sgn ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))) = (real_sgn (real_abs x))))
LINEPATH_LINEAR_IMAGE
|- (!f. ((linear f) ==> (!b. (!c. ((linepath ((f b) , (f c))) = (f o (linepath (b , c))))))))
VECTOR_SUB_RDISTRIB
|- (((% ((real_sub a) b)) x) = ((vector_sub ((% a) x)) ((% b) x)))
Packing3.SING_UNION_EQ_INSERT
|- (!s. (!x. (((x INSERT EMPTY) UNION s) = (x INSERT s))))
REAL_EXP_POS_LE
|- (!x. ((real_le (real_of_num (NUMERAL _0))) (exp x)))
HAS_BOUNDED_VARIATION_AFFINITY_EQ
|- (!m. (!c. (!f. (!s. (((has_bounded_variation_on (\x. (f ((vector_add ((% m) x)) c)))) s) = ((m = (real_of_num (NUMERAL _0))) \/ ((has_bounded_variation_on f) ((IMAGE (\x. ((vector_add ((% m) x)) c))) s))))))))
Fan.fan5
|- (!V. (!E. (!x. ((fan5 (x , (V , E))) = (!e. (!f. (((e IN E) /\ ((f IN E) /\ (~ (e = f)))) ==> ((((aff_gt (x INSERT EMPTY)) e) INTER ((aff_gt (x INSERT EMPTY)) f)) = EMPTY))))))))
CARD_EXP_ADD
|- (!s. (!t. (!u. (((^_c s) ((+_c t) u)) =_c ((*_c ((^_c s) t)) ((^_c s) u))))))
RELATIVE_INTERIOR_SEGMENT
|- ((!a. (!b. ((relative_interior (closed_segment ((CONS (a , b)) NIL))) = (((COND (a = b)) (a INSERT EMPTY)) (open_segment (a , b)))))) /\ (!a. (!b. ((relative_interior (open_segment (a , b))) = (open_segment (a , b))))))
Hypermap_iso.surj_res_inv_right
|- (!f. (!s. (!t. (!y. (((((SURJ f) s) t) /\ (y IN t)) ==> (((f (((res_inv f) s) y)) = y) /\ ((((res_inv f) s) y) IN s)))))))
REAL_DIFFERENTIABLE_ATREAL_WITHIN
|- (!f. (!s. (!z. (((real_differentiable f) (atreal z)) ==> ((real_differentiable f) ((within (atreal z)) s))))))
COMPLEX_INV_EQ_0
|- (!x. (((complex_inv x) = (Cx (real_of_num (NUMERAL _0)))) = (x = (Cx (real_of_num (NUMERAL _0))))))
HOMOTOPIC_WITH
|- ((!h. (!k. ((!x. ((x IN X) ==> ((h x) = (k x)))) ==> ((P h) = (P k))))) ==> (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x)))) /\ ((!x. ((x IN X) ==> ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x)))) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\x. (h ((pastecart t) x)))))))))))))
LINEAR_SURJECTIVE_IMP_INJECTIVE
|- (!f. (((linear f) /\ (!y. (?x. ((f x) = y)))) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y))))))
Fan_defs.e_fan_pair
|- (!V. (!E. (!w. (!v. (((e_fan_pair (V , E)) (v , w)) = (w , v))))))
CBALL_SING
|- (!x. (!e. ((e = (real_of_num (NUMERAL _0))) ==> ((cball (x , e)) = (x INSERT EMPTY)))))
DIST_EQ_0
|- (!x. (!y. (((distance (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))
Kepler_spec.dist_spec
|- (!x. (!y. (!z. (!x'. (!y'. (!z'. ((distance ((real3_of_triple (x , (y , z))) , (real3_of_triple (x' , (y' , z'))))) = (sqrt ((real_add ((real_pow ((real_sub x) x')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_add ((real_pow ((real_sub y) y')) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub z) z')) (NUMERAL (BIT0 (BIT1 _0))))))))))))))
Lp_ineqs_proofs.y6_hi_std2
|- (!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 (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))
Hypermap.THREE
|- ((NUMERAL (BIT1 (BIT1 _0))) = (SUC (NUMERAL (BIT0 (BIT1 _0)))))
REAL_COMPACT_IMP_BOUNDED
|- (!s. ((real_compact s) ==> (real_bounded s)))
Real_ext.REAL_SV_NEG_0
|- ((real_neg (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))
PATH_CONNECTED_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((path_connected ((IMAGE f) s)) = (path_connected s)))))
BASIS_ORTHOGONAL
|- (!b. ((FINITE b) ==> (?c. ((FINITE c) /\ (((CARD c) <= (CARD b)) /\ (((span c) = (span b)) /\ ((pairwise orthogonal) c)))))))
ABSOLUTELY_INTEGRABLE_CONTINUOUS
|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))
REFLECT_ALONG_LINEAR_IMAGE
|- (!f. (!v. (!x. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((reflect_along (f v)) (f x)) = (f ((reflect_along v) x)))))))
CEXP_ADD
|- (!w. (!z. ((cexp ((vector_add w) z)) = ((complex_mul (cexp w)) (cexp z)))))
PSUBSET
|- (!s. (!t. ((s PSUBSET t) = ((s SUBSET t) /\ (~ (s = t))))))
STC_CLOSED
|- (!R. (((STC R) = R) = ((!x. (!y. (((R x) y) ==> ((R y) x)))) /\ (!x. (!y. (!z. ((((R x) y) /\ ((R y) z)) ==> ((R x) z))))))))
PRODUCT_EQ
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((product s) f) = ((product s) g))))))
MEASURABLE_CONVEX
|- (!s. (((convex s) /\ (bounded s)) ==> (measurable s)))
Add_triangle.sigma_in_dart1
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (((v , w) IN (dart1_of_fan (V , E))) ==> ((v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) IN (dart1_of_fan (V , E)))))))))
Seq.rot
|- (!n. (!s. (((rot n) s) = ((cat ((dropl n) s)) ((take n) s)))))
ENR_ANR
|- (!s. ((ENR s) = ((ANR s) /\ ((locally compact) s))))
REAL_CONTINUOUS_ON
|- (!f. (!s. (((real_continuous_on f) s) = ((continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))
Appendix.BBindex_min_v39
|- (!s. ((BBindex_min_v39 s) = (min_num ((IMAGE (BBindex_v39 s)) (BBprime_v39 s)))))
SUBSET
|- (!s. (!t. ((s SUBSET t) = (!x. ((x IN s) ==> (x IN t))))))
Tame_general.ly_EQ_lmfun
|- (!x. (((real_le (vector_norm (FST x))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun (h_dart x)) = (ly (vector_norm (FST x))))))
LEFT_OR_FORALL_THM
|- (!P. (!Q. (((!x. (P x)) \/ Q) = (!x. ((P x) \/ Q)))))
FINITE_IMAGE
|- (!f. (!s. ((FINITE s) ==> (FINITE ((IMAGE f) s)))))
ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_RETRACT
|- (!s. (!u. (!s'. (((ANR s) /\ (((homeomorphic s) s') /\ ((closed_in ((subtopology euclidean) u)) s'))) ==> (?v. (?w. (((open_in ((subtopology euclidean) u)) v) /\ (((closed_in ((subtopology euclidean) u)) w) /\ ((s' SUBSET v) /\ ((v SUBSET w) /\ ((retract_of s') w)))))))))))
LIM_AT_INFINITY_COMPLEX_0
|- (!f. (!l. ((((--> f) l) at_infinity) = (((--> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))
RIGHT_ADD_DISTRIB
|- (!m. (!n. (!p. (((m + n) * p) = ((m * p) + (n * p))))))
IN_INSERT
|- (!x. (!y. (!s. ((x IN (y INSERT s)) = ((x = y) \/ (x IN s))))))
SPAN_CONIC_HULL
|- (!s. ((span ((hull conic) s)) = (span s)))
VECTOR_POLYNOMIAL_FUNCTION_CONST
|- (!c. (vector_polynomial_function (\x. c)))
SERIES_COMPONENT
|- (!f. (!s. (!l. (!k. (((((sums f) l) s) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> (((sums (\i. (lift ((f i) $ k)))) (lift (l $ k))) s))))))
COMPLEX_DERIVATIVE_SUB
|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))
Ssrnat.succnK
|- (!n. (((SUC n) - (NUMERAL (BIT1 _0))) = n))
ISOMETRY_LINEAR
|- (!f. ((((f (vec (NUMERAL _0))) = (vec (NUMERAL _0))) /\ (!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y)))))) ==> (linear f)))
INDEPENDENT_IMP_FINITE
|- (!s. ((independent s) ==> (FINITE s)))
Sphere.tame_table_d
|- (!r. (!s. (((tame_table_d r) s) = (((COND ((r + ((NUMERAL (BIT0 (BIT1 _0))) * s)) > (NUMERAL (BIT1 (BIT1 _0))))) ((real_add ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s)))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((real_add (real_of_num r)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))))))
REAL_LE_HAUSDIST
|- (!s. (!t. (!a. (!b. (!c. (!z. (((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((!x. ((x IN s) ==> ((real_le (setdist ((x INSERT EMPTY) , t))) b))) /\ ((!y. ((y IN t) ==> ((real_le (setdist ((y INSERT EMPTY) , s))) c))) /\ (((z IN s) /\ ((real_le a) (setdist ((z INSERT EMPTY) , t)))) \/ ((z IN t) /\ ((real_le a) (setdist ((z INSERT EMPTY) , s))))))))) ==> ((real_le a) (hausdist (s , t))))))))))
Seq.map
|- ((((map f) ((CONS x) s)) = ((CONS (f x)) ((map f) s))) /\ (((map f) NIL) = NIL))
DIFF_CHAIN_AT
|- (!f. (!g. (!f'. (!g'. (!x. (((((has_derivative f) f') (at x)) /\ (((has_derivative g) g') (at (f x)))) ==> (((has_derivative (g o f)) (g' o f')) (at x))))))))
Hexagons.sqrt8_LE_CSTAB
|- ((real_le sqrt8) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))
PASTECART_INJ
|- (!x. (!y. (!w. (!z. ((((pastecart x) y) = ((pastecart w) z)) = ((x = w) /\ (y = z)))))))
ORTHOGONAL_TRANSFORMATION_LINEAR
|- (!f. ((orthogonal_transformation f) ==> (linear f)))
ARITH_LT
|- ((!m. (!n. (((NUMERAL m) < (NUMERAL n)) = (m < n)))) /\ (((_0 < _0) = F) /\ ((!n. (((BIT0 n) < _0) = F)) /\ ((!n. (((BIT1 n) < _0) = F)) /\ ((!n. ((_0 < (BIT0 n)) = (_0 < n))) /\ ((!n. ((_0 < (BIT1 n)) = T)) /\ ((!m. (!n. (((BIT0 m) < (BIT0 n)) = (m < n)))) /\ ((!m. (!n. (((BIT0 m) < (BIT1 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT1 m) < (BIT0 n)) = (m < n)))) /\ (!m. (!n. (((BIT1 m) < (BIT1 n)) = (m < n)))))))))))))
VECTOR_VARIATION_TRIANGLE
|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((real_le ((vector_variation s) (\x. ((vector_add (f x)) (g x))))) ((real_add ((vector_variation s) f)) ((vector_variation s) g)))))))
SUBSET_LE_DIM
|- (!s. (!t. ((s SUBSET (span t)) ==> ((dim s) <= (dim t)))))
IN_ELIM_PASTECART_THM
|- (!P. (!a. (!b. ((((pastecart a) b) IN (GSPEC (\GEN%PVAR%238. (?x. (?y. (((SETSPEC GEN%PVAR%238) ((P x) y)) ((pastecart x) y))))))) = ((P a) b)))))
CONTINUOUS_AT_COMPOSE
|- (!f. (!g. (!x. ((((continuous f) (at x)) /\ ((continuous g) (at (f x)))) ==> ((continuous (g o f)) (at x))))))
Lp_ineqs_proofs.tau_fan_eq_tauVEF
|- (!V. (!E. (!f. (((lp_tau (V , E)) /\ (f IN (face_set (hypermap_of_fan (V , E))))) ==> (((tau_fan (V , E)) f) = (tauVEF (V , (E , f))))))))
Hypermap_iso.hyp_iso_imp_iso
|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> ((iso H) G)))))
CONNECTED_COMPONENT_TRANSLATION
|- (!a. (!s. (!x. (((connected_component ((IMAGE (\x. ((vector_add a) x))) s)) ((vector_add a) x)) = ((IMAGE (\x. ((vector_add a) x))) ((connected_component s) x))))))
REAL_ADD_RDISTRIB
|- (!x. (!y. (!z. (((real_mul ((real_add x) y)) z) = ((real_add ((real_mul x) z)) ((real_mul y) z))))))
IM_ADD
|- (!x. (!y. ((Im ((vector_add x) y)) = ((real_add (Im x)) (Im y)))))
List_hypermap.good_list_nodes
|- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))
LIM_POSINFINITY_SEQUENTIALLY
|- (!f. (!l. ((((--> f) l) at_posinfinity) ==> (((--> (\n. (f (real_of_num n)))) l) sequentially))))
Dih2k_hypermap.cstab
|- (cstab = ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))
Nonlin_def.sqrt_x4
|- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))
REAL_MUL_LINV
|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul (real_inv x)) x) = (real_of_num (NUMERAL (BIT1 _0))))))
differentiable_on
|- (!f. (!s. (((differentiable_on f) s) = (!x. ((x IN s) ==> ((differentiable f) ((within (at x)) s)))))))
Lp_ineqs_proofs.dart4_cross_eq_y4'
|- (!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)))) ==> (((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))
CONTINUOUS_AT_CACS
|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((continuous cacs) (at z))))
PERMUTATION_SWAP
|- (!a. (!b. (permutation (swap (a , b)))))
REAL_SGN_MUL
|- (!x. (!y. ((real_sgn ((real_mul x) y)) = ((real_mul (real_sgn x)) (real_sgn y)))))