hcju/holps huggingface dataset:
Miqmcsn.scs_3T4_prime2
|- (scs_3T4_prime2 = ((((mk_unadorned_v39 (NUMERAL (BIT1 (BIT1 _0)))) ((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))))))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) NIL))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) (NUMERAL (BIT1 (BIT1 _0))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) NIL)) cstab) (NUMERAL (BIT1 (BIT1 _0))))))
AFFINE_DEPENDENT_EXPLICIT
|- (!p. ((affine_dependent p) = (?s. (?u. ((FINITE s) /\ ((s SUBSET p) /\ ((((sum s) u) = (real_of_num (NUMERAL _0))) /\ ((?v. ((v IN s) /\ (~ ((u v) = (real_of_num (NUMERAL _0)))))) /\ (((vsum s) (\v. ((% (u v)) v))) = (vec (NUMERAL _0)))))))))))
PATH_VECTOR_POLYNOMIAL_FUNCTION
|- (!g. ((vector_polynomial_function g) ==> (path g)))
NORM_CEXP
|- (!z. ((vector_norm (cexp z)) = (exp (Re z))))
ARC_IMP_PATH
|- (!g. ((arc g) ==> (path g)))
Seq.can_inj
|- (!f. (!g. ((!x. ((g (f x)) = x)) ==> (!x. (!y. (((f x) = (f y)) ==> (x = y)))))))
REAL_POW_NZ
|- (!x. (!n. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (~ (((real_pow x) n) = (real_of_num (NUMERAL _0)))))))
Hypermap_iso.bij_ext
|- (!f. (!g. (!s. (!t. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> ((((BIJ f) s) t) ==> (((BIJ g) s) t)))))))
SUM_DIFF
|- (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((sum (s DIFF t)) f) = ((real_sub ((sum s) f)) ((sum t) f)))))))
REAL_SUB_RZERO
|- (!x. (((real_sub x) (real_of_num (NUMERAL _0))) = x))
TC_MONO
|- (!R. (!S. ((!x. (!y. (((R x) y) ==> ((S x) y)))) ==> (!x. (!y. ((((TC R) x) y) ==> (((TC S) x) y)))))))
Hypermap_iso.iso_number_of_nodes
|- (!f. (!G. (!H. (((hyp_iso f) (H , G)) ==> ((number_of_nodes G) = (number_of_nodes H))))))
HAS_SIZE
|- (!s. (!n. ((s HAS_SIZE n) = ((FINITE s) /\ ((CARD s) = n)))))
SPHERE_UNION_BALL
|- (!a. (!r. (((sphere (a , r)) UNION (ball (a , r))) = (cball (a , r)))))
BOHL
|- (!f. (!s. (!a. ((((continuous_on f) s) /\ ((convex s) /\ ((compact s) /\ (a IN (interior s))))) ==> ((?x. ((x IN s) /\ ((f x) = x))) \/ (?x. ((x IN (frontier s)) /\ (x IN (open_segment (a , (f x)))))))))))
REFLECT_REAL_INTERVAL
|- ((!a. (!b. (((IMAGE real_neg) (closed_real_interval ((CONS (a , b)) NIL))) = (closed_real_interval ((CONS ((real_neg b) , (real_neg a))) NIL))))) /\ (!a. (!b. (((IMAGE real_neg) (open_real_interval (a , b))) = (open_real_interval ((real_neg b) , (real_neg a)))))))
Seq.count_predT
|- (!s. (((count predT) s) = (sizel s)))
INTERIOR_SPHERE
|- (!a. (!r. ((interior (sphere (a , r))) = EMPTY)))
EXISTS_PAIR_THM
|- (!P. ((?p. (P p)) = (?p1. (?p2. (P (p1 , p2))))))
GEQ_DEF
|- (!a. (!b. (((GEQ a) b) = (a = b))))
CONIC_IMP_CONTRACTIBLE
|- (!s. ((conic s) ==> (contractible s)))
RE_CLOG_POS_LE
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((real_le (real_abs (Im (clog z)))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = ((real_le (real_of_num (NUMERAL _0))) (Re z)))))
ABSOLUTELY_SETCONTINUOUS_ON_NEG
|- (!f. (!s. (((absolutely_setcontinuous_on (\x. (vector_neg (f x)))) s) = ((absolutely_setcontinuous_on f) s))))
EXP_2
|- (!n. ((n EXP (NUMERAL (BIT0 (BIT1 _0)))) = (n * n)))
EXP_MONO_LE
|- (!x. (!y. (!n. (((x EXP n) <= (y EXP n)) = ((x <= y) \/ (n = (NUMERAL _0)))))))
FORALL_DOT_EQ_0
|- ((!y. ((!x. (((dot x) y) = (real_of_num (NUMERAL _0)))) = (y = (vec (NUMERAL _0))))) /\ (!x. ((!y. (((dot x) y) = (real_of_num (NUMERAL _0)))) = (x = (vec (NUMERAL _0))))))
PASTECART_SUB
|- (!x1. (!y1. (!x2. (!y2. (((vector_sub ((pastecart x1) y1)) ((pastecart x2) y2)) = ((pastecart ((vector_sub x1) x2)) ((vector_sub y1) y2)))))))
REAL_MUL_RID
|- (!x. (((real_mul x) (real_of_num (NUMERAL (BIT1 _0)))) = x))
epigraph
|- (!s. (!f. (((epigraph s) f) = (GSPEC (\GEN%PVAR%1703. (?xy. (((SETSPEC GEN%PVAR%1703) (((fstcart xy) IN s) /\ ((real_le (f (fstcart xy))) (drop (sndcart xy))))) xy)))))))
real_differentiable
|- (!f. (!net. (((real_differentiable f) net) = (?f'. (((has_real_derivative f) f') net)))))
INT_NOT_LE
|- (!x. (!y. ((~ ((int_le x) y)) = ((int_lt y) x))))
HAS_FRECHET_DERIVATIVE_UNIQUE_AT
|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) ==> (((frechet_derivative f) (at x)) = f')))))
LT_SUC
|- (!m. (!n. (((SUC m) < (SUC n)) = (m < n))))
ABSOLUTELY_INTEGRABLE_IMP_INTEGRABLE
|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((integrable_on f) s))))
PATHFINISH_LINEAR_IMAGE
|- (!f. (!g. ((linear f) ==> ((pathfinish (f o g)) = (f (pathfinish g))))))
Hypermap_iso.image_ext_eq
|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((IMAGE f) s) = ((IMAGE g) s))))))
HAS_COMPLEX_DERIVATIVE_CATN
|- (!z. ((((Re z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Im z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative catn) (complex_inv ((vector_add (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_pow z) (NUMERAL (BIT0 (BIT1 _0))))))) (at z))))
NEGLIGIBLE_LINEAR_IMAGE_EQ
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((negligible ((IMAGE f) s)) = (negligible s)))))
SIN_CIRCLE
|- (!x. (((real_add ((real_pow (sin x)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow (cos x)) (NUMERAL (BIT0 (BIT1 _0))))) = (real_of_num (NUMERAL (BIT1 _0)))))
BIT1
|- (!n. ((BIT1 n) = (SUC (n + n))))
Lp_ineqs_proofs.face4_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\w'. (LET_END ((LET (\u. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , u)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (u , w')) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w , u)) = (u , w')) /\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\ (((f_fan_pair_ext (V , E)) (w' , v)) = (v , w))))))) /\ ((((v , w) IN (dart1_of_fan (V , E))) /\ (((w , u) IN (dart1_of_fan (V , E))) /\ (((u , w') IN (dart1_of_fan (V , E))) /\ ((w' , v) IN (dart1_of_fan (V , E)))))) /\ (((face (hypermap_of_fan (V , E))) (v , w)) = ((v , w) INSERT ((w , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))
rational
|- (!x. ((rational x) = (?m. (?n. ((integer m) /\ ((integer n) /\ ((~ (n = (real_of_num (NUMERAL _0)))) /\ (x = ((real_div m) n)))))))))
complex_add
|- (((vector_add w) z) = (complex (((real_add (Re w)) (Re z)) , ((real_add (Im w)) (Im z)))))
CONNECTED_LIMIT_POINTS_EQ_CLOSURE
|- (!s. (((connected s) /\ (~ (?a. (s = (a INSERT EMPTY))))) ==> ((GSPEC (\GEN%PVAR%584. (?x. (((SETSPEC GEN%PVAR%584) ((limit_point_of x) s)) x)))) = (closure s))))
SERIES_ADD
|- (!x. (!x0. (!y. (!y0. (!s. (((((sums x) x0) s) /\ (((sums y) y0) s)) ==> (((sums (\n. ((vector_add (x n)) (y n)))) ((vector_add x0) y0)) s)))))))
Fan.power_map_points
|- (((((((((power_map_points f) x) V) E) v) w) (NUMERAL _0)) = w) /\ ((((((((power_map_points f) x) V) E) v) w) (SUC n)) = (((((f x) V) E) v) (((((((power_map_points f) x) V) E) v) w) n))))
TOPSPACE_EUCLIDEAN_SUBTOPOLOGY
|- (!s. ((topspace ((subtopology euclidean) s)) = s))
Nonlinear_lemma.sqrtxx
|- (!x. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((sqrt ((real_mul x) x)) = x)))
FINITE_IMP_COMPACT
|- (!s. ((FINITE s) ==> (compact s)))
REAL_POW_NEG
|- (!x. (!n. (((real_pow (real_neg x)) n) = (((COND (EVEN n)) ((real_pow x) n)) (real_neg ((real_pow x) n))))))
EQ_MULT_RCANCEL
|- (!m. (!n. (!p. (((m * p) = (n * p)) = ((m = n) \/ (p = (NUMERAL _0)))))))
FINITE_DIFF
|- (!s. (!t. ((FINITE s) ==> (FINITE (s DIFF t)))))
Fnjlbxs.vector_of_list
|- (!list. (!d. (((vector_of_list d) list) = (lambdai. (((COND (((i - (NUMERAL (BIT1 _0))) DIV d) < (LENGTH list))) (((EL ((i - (NUMERAL (BIT1 _0))) DIV d)) list) $ (((i - (NUMERAL (BIT1 _0))) MOD d) + (NUMERAL (BIT1 _0))))) (real_of_num (NUMERAL _0)))))))
Hypermap_and_fan.NOT_IN_DART1_OF_FAN
|- (!V. (!E. (!v. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (~ ((v , v) IN (dart1_of_fan (V , E))))))))
IN_REAL_INTERVAL
|- (!a. (!b. (!x. (((x IN (closed_real_interval ((CONS (a , b)) NIL))) = (((real_le a) x) /\ ((real_le x) b))) /\ ((x IN (open_real_interval (a , b))) = (((real_lt a) x) /\ ((real_lt x) b)))))))
SPAN_OPEN
|- (!s. (((open s) /\ (~ (s = EMPTY))) ==> ((span s) = UNIV)))
DIVISION
|- (!m. (!n. ((~ (n = (NUMERAL _0))) ==> ((m = (((m DIV n) * n) + (m MOD n))) /\ ((m MOD n) < n)))))
Seq.mask
|- ((((mask NIL) s') = NIL) /\ ((((mask m') NIL) = NIL) /\ (((mask ((CONS b) m')) ((CONS x) s')) = (((COND b) ((CONS x) ((mask m') s'))) ((mask m') s')))))
RELATIVE_FRONTIER_TRANSLATION
|- (!a. (!s. ((relative_frontier ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (relative_frontier s)))))
CONVEX_IMP_LOCALLY_CONNECTED
|- (!s. ((convex s) ==> ((locally connected) s)))
vector_sub
|- (!x. (!y. (((vector_sub x) y) = (lambdai. ((real_sub (x $ i)) (y $ i))))))
PAIRWISE_ORTHOGONAL_INDEPENDENT
|- (!s. ((((pairwise orthogonal) s) /\ (~ ((vec (NUMERAL _0)) IN s))) ==> (independent s)))
INTERIOR_CLOSURE_INTER_OPEN
|- (!s. (!t. (((open s) /\ (open t)) ==> ((interior (closure (s INTER t))) = ((interior (closure s)) INTER (interior (closure t)))))))
Lvducxu.FIRST_AAUHTVE2
|- ((FAN (x , (V , E))) ==> ((FINITE ((darts_of_hyp E) V)) /\ (((permutes (ee_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ (((permutes (nn_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ (((permutes (ff_of_hyp (x , (V , E)))) ((darts_of_hyp E) V)) /\ ((((ee_of_hyp (x , (V , E))) o ((nn_of_hyp (x , (V , E))) o (ff_of_hyp (x , (V , E))))) = I) /\ (((ee_of_hyp (x , (V , E))) o (ee_of_hyp (x , (V , E)))) = I)))))))
CONTINUOUS_COMPLEX_DIV
|- (!net. (!f. (!g. ((((continuous f) net) /\ (((continuous g) net) /\ (~ ((g (netlimit net)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((continuous (\x. ((complex_div (f x)) (g x)))) net)))))
SEPARATION_NORMAL
|- (!s. (!t. (((closed s) /\ ((closed t) /\ ((s INTER t) = EMPTY))) ==> (?u. (?v. ((open u) /\ ((open v) /\ ((s SUBSET u) /\ ((t SUBSET v) /\ ((u INTER v) = EMPTY))))))))))
FORALL_IN_GSPEC
|- ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%21. (?x. (((SETSPEC GEN%PVAR%21) (P x)) (f x)))))) ==> (Q z))) = (!x. ((P x) ==> (Q (f x))))))) /\ ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%22. (?x. (?y. (((SETSPEC GEN%PVAR%22) ((P x) y)) ((f x) y))))))) ==> (Q z))) = (!x. (!y. (((P x) y) ==> (Q ((f x) y)))))))) /\ ((!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%23. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%23) (((P w) x) y)) (((f w) x) y)))))))) ==> (Q z))) = (!w. (!x. (!y. ((((P w) x) y) ==> (Q (((f w) x) y))))))))) /\ (!P. (!f. ((!z. ((z IN (GSPEC (\GEN%PVAR%24. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%24) ((((P v) w) x) y)) ((((f v) w) x) y))))))))) ==> (Q z))) = (!v. (!w. (!x. (!y. (((((P v) w) x) y) ==> (Q ((((f v) w) x) y)))))))))))))
CX_INV
|- (!x. ((Cx (real_inv x)) = (complex_inv (Cx x))))
SIMPLY_CONNECTED_INJECTIVE_LINEAR_IMAGE
|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((simply_connected ((IMAGE f) s)) = (simply_connected s)))))
DIAMETER_LE
|- (!s. ((((~ (s = EMPTY)) \/ ((real_le (real_of_num (NUMERAL _0))) d)) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub x) y))) d))))) ==> ((real_le (diameter s)) d)))
SUM_LT
|- (!f. (!g. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((real_le (f x)) (g x)))) /\ (?x. ((x IN s) /\ ((real_lt (f x)) (g x)))))) ==> ((real_lt ((sum s) f)) ((sum s) g))))))
HELLY_CLOSED
|- (!f. ((((FINITE f) ==> ((CARD f) >= ((dimindex UNIV) + (NUMERAL (BIT1 _0))))) /\ ((!s. ((s IN f) ==> ((convex s) /\ (closed s)))) /\ ((?s. ((s IN f) /\ (bounded s))) /\ (!t. (((t SUBSET f) /\ ((FINITE t) /\ ((CARD t) = ((dimindex UNIV) + (NUMERAL (BIT1 _0)))))) ==> (~ ((INTERS t) = EMPTY))))))) ==> (~ ((INTERS f) = EMPTY))))
HAS_DERIVATIVE_ZERO_UNIQUE_STRONG_INTERVAL
|- (!f. (!a. (!b. (!k. (!y. (((COUNTABLE k) /\ (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (((f a) = y) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF k)) ==> (((has_derivative f) (\h. (vec (NUMERAL _0)))) ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = y)))))))))
List_hypermap.darts_of_list
|- (!L. ((darts_of_list L) = (set_of_list (list_of_darts L))))
Seq.size_nil
|- ((sizel NIL) = (NUMERAL _0))
INTERS_IMAGE
|- (!f. (!s. ((INTERS ((IMAGE f) s)) = (GSPEC (\GEN%PVAR%30. (?y. (((SETSPEC GEN%PVAR%30) (!x. ((x IN s) ==> (y IN (f x))))) y)))))))
mbounded
|- (!s. (!m. (((mbounded m) s) = (?c. (?b. (s SUBSET ((mcball m) (c , b))))))))
SEGMENT_TRANSLATION
|- ((!c. (!a. (!b. ((closed_segment ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\x. ((vector_add c) x))) (closed_segment ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_segment (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\x. ((vector_add c) x))) (open_segment (a , b))))))))
HAS_REAL_DERIVATIVE_FROM_COMPLEX_AT
|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at (Cx x))) /\ (!z. ((real z) ==> (real (f z))))) ==> (((has_real_derivative (Re o (f o Cx))) (Re f')) (atreal x))))))
CLOSURE_EQ
|- (!s. (((closure s) = s) = (closed s)))
TAGGED_DIVISION_UNIONS
|- (!iset. (!pfn. (((FINITE iset) /\ ((!i. ((i IN iset) ==> ((tagged_division_of (pfn i)) i))) /\ (!i1. (!i2. (((i1 IN iset) /\ ((i2 IN iset) /\ (~ (i1 = i2)))) ==> (((interior i1) INTER (interior i2)) = EMPTY)))))) ==> ((tagged_division_of (UNIONS ((IMAGE pfn) iset))) (UNIONS iset)))))
NSUM_UNION_NONZERO
|- (!f. (!s. (!t. (((FINITE s) /\ ((FINITE t) /\ (!x. ((x IN (s INTER t)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum (s UNION t)) f) = (((nsum s) f) + ((nsum t) f)))))))
IMAGE_SNDCART_PCROSS
|- (!s. (!t. (((IMAGE sndcart) (s PCROSS t)) = (((COND (s = EMPTY)) EMPTY) t))))
retraction
|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) SUBSET t) /\ (!x. ((x IN t) ==> ((r x) = x))))))))))
ABSOLUTELY_INTEGRABLE_NORM
|- (!f. (!s. (((absolutely_integrable_on f) s) ==> ((absolutely_integrable_on (\x. (lift (vector_norm (f x))))) s))))
HAS_MEASURE_LINEAR_IMAGE_SAME
|- (!f. (!s. (((linear f) /\ ((measurable s) /\ ((real_abs (det (matrix f))) = (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((has_measure ((IMAGE f) s)) (measure s)))))
ITERATE_DIFF
|- (!op. ((monoidal op) ==> (!f. (!s. (!t. (((FINITE s) /\ (t SUBSET s)) ==> (((op (((iterate op) (s DIFF t)) f)) (((iterate op) t) f)) = (((iterate op) s) f))))))))
REAL_DIV_1
|- (!x. (((real_div x) (real_of_num (NUMERAL (BIT1 _0)))) = x))
EVENTUALLY_AT
|- (!a. (!p. (((eventually p) (at a)) = (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (distance (x , a))) /\ ((real_lt (distance (x , a))) d)) ==> (p x))))))))
REAL_LET_TRANS
|- (!x. (!y. (!z. ((((real_le x) y) /\ ((real_lt y) z)) ==> ((real_lt x) z)))))
CARD_LE_ANTISYM
|- (!s. (!t. (((s <=_c t) /\ (t <=_c s)) = (s =_c t))))
VECTOR_SUB_COMPONENT
|- (!x. (!y. (!i. ((((vector_sub x) y) $ i) = ((real_sub (x $ i)) (y $ i))))))
Hypermap.EQ_SUC
|- (!m. (!n. (((SUC m) = (SUC n)) = (m = n))))
Packing3.VORONOI_LIST_SING
|- (!V. (!u. (((voronoi_list V) ((CONS u) NIL)) = ((voronoi_closed V) u))))
LIM_COMPONENTWISE
|- (!net. (!f. ((((--> f) l) net) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (((---> (\x. ((f x) $ i))) (l $ i)) net))))))
CONVEX_SUMS
|- (!s. (!t. (((convex s) /\ (convex t)) ==> (convex (GSPEC (\GEN%PVAR%1471. (?x. (?y. (((SETSPEC GEN%PVAR%1471) ((x IN s) /\ (y IN t))) ((vector_add x) y))))))))))
CONTENT_PASTECART
|- (!a. (!b. (!c. (!d. ((content (closed_interval ((CONS (((pastecart a) c) , ((pastecart b) d))) NIL))) = ((real_mul (content (closed_interval ((CONS (a , b)) NIL)))) (content (closed_interval ((CONS (c , d)) NIL)))))))))
NSUM_CLAUSES
|- ((!f. (((nsum EMPTY) f) = (NUMERAL _0))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((nsum (x INSERT s)) f) = (((COND (x IN s)) ((nsum s) f)) ((f x) + ((nsum s) f)))))))))
REAL_CONTINUOUS_AT_SQRT
|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_continuous sqrt) (atreal x))))
CROSS_LADD
|- (!x. (!y. (!z. (((cross ((vector_add x) y)) z) = ((vector_add ((cross x) z)) ((cross y) z))))))
CEXP_NEG
|- (!z. ((cexp (vector_neg z)) = (complex_inv (cexp z))))
Lp_ineqs_proofs.dart3_yf_eq
|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT1 (BIT1 _0)))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((f_fan_pair_ext (V , E)) d)) = (y2_fan d)) /\ (((y2_fan ((f_fan_pair_ext (V , E)) d)) = ((y3_fan (V , E)) d)) /\ ((((y3_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y1_fan d)) /\ ((((y4_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y5_fan (V , E)) d)) /\ ((((y5_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = (y6_fan d)) /\ ((y6_fan ((f_fan_pair_ext (V , E)) d)) = ((y4_fan (V , E)) d)))))))))))))))
LIM_TRANSFORM_WITHINREAL_SET_IMP
|- (!f. (!l. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (atreal a)) /\ (((--> f) l) ((within (atreal a)) s))) ==> (((--> f) l) ((within (atreal a)) t))))))))
INTEGER_CLOSED
|- ((!n. (integer (real_of_num n))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_add x) y))))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_sub x) y))))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_mul x) y))))) /\ ((!x. (!r. ((integer x) ==> (integer ((real_pow x) r))))) /\ ((!x. ((integer x) ==> (integer (real_neg x)))) /\ ((!x. ((integer x) ==> (integer (real_abs x)))) /\ ((!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_max x) y))))) /\ (!x. (!y. (((integer x) /\ (integer y)) ==> (integer ((real_min x) y)))))))))))))
Conforming.add_edge_imp_card_set_edge_ge1_fan
|- (!x. (!V. (!E. (!E1. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ (E1 = (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))) ==> (!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E1)) > (NUMERAL (BIT1 _0))))))))))
OPEN_IN_DIFF
|- (!top. (!s. (!t. ((((open_in top) s) /\ ((closed_in top) t)) ==> ((open_in top) (s DIFF t))))))
Sphere.bump
|- (!h. ((bump h) = ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) ((real_div ((real_pow ((real_sub h) h0)) (NUMERAL (BIT0 (BIT1 _0))))) ((real_pow ((real_sub hplus) h0)) (NUMERAL (BIT0 (BIT1 _0)))))))))
FACE_OF_CONIC
|- (!s. (!f. (((conic s) /\ ((face_of f) s)) ==> (conic f))))
KIRSZBRAUN
|- (!f. (!s. (!B. ((((real_le (real_of_num (NUMERAL _0))) 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)))))))) ==> (?g. ((!x. (!y. ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))) /\ (!x. ((x IN s) ==> ((g x) = (f x))))))))))
Hypermap.res
|- (!f. (!s. (!x. ((((res f) s) x) = (((COND (x IN s)) (f x)) x)))))
Lp_ineqs_def.y9_fan
|- (!V. (!E. (!x. (((y9_fan (V , E)) x) = (distance ((f_fan_pair_ext (V , E)) x))))))
REAL_EXP_NEG
|- (!x. ((exp (real_neg x)) = (real_inv (exp x))))
Sphere.rotate2
|- (!f. (!x2. (!x3. (!x1. (!x5. (!x6. (!x4. ((((((((rotate2 f) x1) x2) x3) x4) x5) x6) = ((((((f x2) x3) x1) x5) x6) x4)))))))))
Trigonometry2.AFF2
|- (!u. (!v. ((aff (u INSERT (v INSERT EMPTY))) = (GSPEC (\GEN%PVAR%5314. (?x. (((SETSPEC GEN%PVAR%5314) (?t. (x = ((vector_add ((% t) u)) ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) t)) v))))) x)))))))
AFFINE
|- (!V. ((affine V) = (!s. (!u. (((FINITE s) /\ ((~ (s = EMPTY)) /\ ((s SUBSET V) /\ (((sum s) u) = (real_of_num (NUMERAL (BIT1 _0))))))) ==> (((vsum s) (\x. ((% (u x)) x))) IN V))))))
PATHSTART_IN_PATH_IMAGE
|- (!g. ((pathstart g) IN (path_image g)))
MEM_EL
|- (!l. (!n. ((n < (LENGTH l)) ==> ((MEM ((EL n) l)) l))))
LAMBDA_UNIQUE
|- (!f. (!g. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((f $ i) = (g i)))) = ((lambda g) = f))))
INJ
|- (!t. (!s. (!f. ((((INJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))
Appendix.MMs_v39
|- (!vv. (!s. (((MMs_v39 s) vv) = (((BBprime2_v39 s) vv) /\ ((!i. (((scs_str_v39 s) i) ==> (((((azim (vec (NUMERAL _0))) (vv i)) (vv (SUC i))) (vv (i + ((scs_k_v39 s) - (NUMERAL (BIT1 _0)))))) = pi))) /\ ((!i. (((scs_lo_v39 s) i) ==> ((vector_norm (vv i)) = (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) /\ ((!i. (((scs_hi_v39 s) i) ==> ((vector_norm (vv i)) = ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) /\ ((!i. (!j. ((real_le (((scs_am_v39 s) i) j)) (distance ((vv i) , (vv j)))))) /\ (!i. (!j. ((real_le (distance ((vv i) , (vv j)))) (((scs_bm_v39 s) i) j))))))))))))
COLLINEAR_HYPERPLANE_2
|- (!a. (!b. ((((dimindex UNIV) <= (NUMERAL (BIT0 (BIT1 _0)))) /\ (~ (a = (vec (NUMERAL _0))))) ==> (collinear (GSPEC (\GEN%PVAR%1598. (?x. (((SETSPEC GEN%PVAR%1598) (((dot a) x) = b)) x))))))))
VECTOR_MUL_RZERO
|- (((% c) (vec (NUMERAL _0))) = (vec (NUMERAL _0)))
CONNECTED_COMPONENT_LINEAR_IMAGE
|- (!f. (!s. (!x. (((linear f) /\ ((!x. (!y. (((f x) = (f y)) ==> (x = y)))) /\ (!y. (?x. ((f x) = y))))) ==> (((connected_component ((IMAGE f) s)) (f x)) = ((IMAGE f) ((connected_component s) x)))))))
REAL_INFSUM_UNIQUE
|- (!f. (!l. (!s. ((((real_sums f) l) s) ==> (((real_infsum s) f) = l)))))
sgn_lt
|- (sgn_lt = (\t. ((real_lt t) (real_of_num (NUMERAL _0)))))
FORALL_4
|- (!P. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) ==> (P i))) = ((P (NUMERAL (BIT1 _0))) /\ ((P (NUMERAL (BIT0 (BIT1 _0)))) /\ ((P (NUMERAL (BIT1 (BIT1 _0)))) /\ (P (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))))))
MBOUNDED_MCBALL
|- (!m. (!c. (!b. ((mbounded m) ((mcball m) (c , b))))))
ORTHOGONAL_SYM
|- (!x. (!y. (((orthogonal x) y) = ((orthogonal y) x))))
LIM_NULL_COMPLEX_RMUL_BOUNDED
|- (!net. (!f. (!g. (!B. (((((--> f) (Cx (real_of_num (NUMERAL _0)))) net) /\ ((eventually (\a. (((f a) = (Cx (real_of_num (NUMERAL _0)))) \/ ((real_le (vector_norm (g a))) B)))) net)) ==> (((--> (\z. ((complex_mul (f z)) (g z)))) (Cx (real_of_num (NUMERAL _0)))) net))))))
CONTINUOUS_ON_SUBSET
|- (!f. (!s. (!t. ((((continuous_on f) s) /\ (t SUBSET s)) ==> ((continuous_on f) t)))))
Functional_equation.proj_y1
|- (!x2. (!x3. (!x4. (!x5. (!x6. (!x1. (((((((proj_y1 x1) x2) x3) x4) x5) x6) = (sqrt x1))))))))
inseg
|- (!m. (!l. (((inseg l) m) = (!x. (!y. ((l (x , y)) = ((m (x , y)) /\ ((fl l) y))))))))
CSQRT_CEXP_CLOG
|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> ((csqrt z) = (cexp ((complex_div (clog z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))
Lp_ineqs_def.rhazim2_fan
|- (!V. (!E. (!x. (((rhazim2_fan (V , E)) x) = ((rhazim_fan (V , E)) ((f_fan_pair_ext (V , E)) x))))))
CARD_EQ_CARD
|- (!s. (!t. (((FINITE s) /\ (FINITE t)) ==> ((s =_c t) = ((CARD s) = (CARD t))))))
Ssrnat.addn1
|- (!n. ((n + (NUMERAL (BIT1 _0))) = (SUC n)))
GDELTA_INTER
|- (!s. (!t. (((gdelta s) /\ (gdelta t)) ==> (gdelta (s INTER t)))))
Miqmcsn.SCS_4M8_13_IS_SCS
|- (is_scs_v39 scs_4M8_13)
LT_NZ
|- (!n. (((NUMERAL _0) < n) = (~ (n = (NUMERAL _0)))))
CLOSURE_APPROACHABLE
|- (!x. (!s. ((x IN (closure s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?y. ((y IN s) /\ ((real_lt (distance (y , x))) e))))))))
IN_MCBALL
|- (!m. (!x. (!r. (!y. ((y IN ((mcball m) (x , r))) = ((x IN (mspace m)) /\ ((y IN (mspace m)) /\ ((real_le ((mdist m) (x , y))) r))))))))
MATRIX_INV_0
|- ((matrix_inv (mat (NUMERAL _0))) = (mat (NUMERAL _0)))
POSITIVE_DEFINITE_2_DET
|- (!A. ((positive_definite A) = (((transp A) = A) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT1 _0))) $ (NUMERAL (BIT1 _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) ((A $ (NUMERAL (BIT0 (BIT1 _0)))) $ (NUMERAL (BIT0 (BIT1 _0))))) /\ ((real_lt (real_of_num (NUMERAL _0))) (det A)))))))
COMPLEX_NORM_CX
|- (!x. ((vector_norm (Cx x)) = (real_abs x)))
ABSOLUTELY_INTEGRABLE_CHANGE_OF_VARIABLES_LINEAR
|- (!f. (!g. (!s. ((linear g) ==> (((absolutely_integrable_on (\x. ((% (real_abs (det (matrix g)))) (f (g x))))) s) = ((absolutely_integrable_on f) ((IMAGE g) s)))))))
LIFT_SUB
|- (!x. (!y. ((lift ((real_sub x) y)) = ((vector_sub (lift x)) (lift y)))))