SIMPLEX_ORDERING_EXISTS

|- (!s. (((simplex ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))) s) ==> (?A. ((rows A) = (GSPEC (\GEN%PVAR%2546. (?v. (((SETSPEC GEN%PVAR%2546) ((extreme_point_of v) s)) v))))))))


IS_INTERVAL_IMP_ENR

|- (!s. ((is_interval s) ==> (ENR s)))


REAL_MUL_ASSOC

|- (!x. (!y. (!z. (((real_mul x) ((real_mul y) z)) = ((real_mul ((real_mul x) y)) z)))))


VECTOR_VARIATION_REFLECT2

|- (!f. (!s. (((vector_variation ((IMAGE vector_neg) s)) (\x. (f (vector_neg x)))) = ((vector_variation s) f))))


BALL_EQ_EMPTY

|- (!x. (!e. (((ball (x , e)) = EMPTY) = ((real_le e) (real_of_num (NUMERAL _0))))))


IN_SING

|- (!x. (!y. ((x IN (y INSERT EMPTY)) = (x = y))))


AFF_DIM_UNIV

|- ((aff_dim UNIV) = (int_of_num (dimindex UNIV)))


AFFINE_DEPENDENT_IMP_DEPENDENT

|- (!s. ((affine_dependent s) ==> (dependent s)))


BOUNDED_HALFSPACE_LT

|- (!a. (!b. ((bounded (GSPEC (\GEN%PVAR%844. (?x. (((SETSPEC GEN%PVAR%844) ((real_lt ((dot a) x)) b)) x))))) = ((a = (vec (NUMERAL _0))) /\ ((real_le b) (real_of_num (NUMERAL _0)))))))


ONE

|- ((NUMERAL (BIT1 _0)) = (SUC (NUMERAL _0)))


BARYCENTRE_LINEAR_IMAGE

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((barycentre ((IMAGE f) s)) = (f (barycentre s))))))


CLOSURE_SEGMENT

|- ((!a. (!b. ((closure (closed_segment ((CONS (a , b)) NIL))) = (closed_segment ((CONS (a , b)) NIL))))) /\ (!a. (!b. ((closure (open_segment (a , b))) = (((COND (a = b)) EMPTY) (closed_segment ((CONS (a , b)) NIL)))))))


SECOND_MEAN_VALUE_THEOREM_BONNET_FULL

|- (!f. (!g. (!a. (!b. (((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) /\ (((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_of_num (NUMERAL _0))) (g x)))) /\ (!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (g x)) (g y)))))))) ==> (?c. ((c IN (closed_interval ((CONS (a , b)) NIL))) /\ (((has_integral (\x. ((% (g x)) (f x)))) ((% (g b)) ((integral (closed_interval ((CONS (c , b)) NIL))) f))) (closed_interval ((CONS (a , b)) NIL))))))))))


CLOSURE_EMPTY

|- ((closure EMPTY) = EMPTY)


SERIES_LINEAR

|- (!f. (!h. (!l. (!s. (((((sums f) l) s) /\ (linear h)) ==> (((sums (\n. (h (f n)))) (h l)) s))))))


Tame_lemmas.fully_surrounded_sol_eq_sol_y

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) (v , w))) = (NUMERAL (BIT1 (BIT1 _0))))) ==> ((LET (\w'. (LET_END ((LET (\y1. (LET_END ((LET (\y2. (LET_END ((LET (\y3. (LET_END ((LET (\y4. (LET_END ((LET (\y5. (LET_END ((LET (\y6. (LET_END (((sol (vec (NUMERAL _0))) (((((dart_leads_into (vec (NUMERAL _0))) V) E) v) w)) = ((((((sol_y y1) y2) y3) y4) y5) y6))))) (distance (v , w)))))) (distance (v , w')))))) (distance (w , w')))))) (vector_norm w'))))) (vector_norm w))))) (vector_norm v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))


COMPACT_LINEAR_IMAGE_EQ

|- (!f. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> ((compact ((IMAGE f) s)) = (compact s)))))


NOT_IN_EMPTY

|- (!x. (~ (x IN EMPTY)))


Functional_equation.promote1_to_6

|- (!x2. (!x3. (!x4. (!x5. (!x6. (!f. (!x1. ((((((((promote1_to_6 f) x1) x2) x3) x4) x5) x6) = (f x1)))))))))


Hypermap_and_fan.FULLY_SURROUNDED_IMP_CARD_FACE_GE_3

|- (!V. (!E. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (fully_surrounded (V , E))) ==> (!x. ((x IN (dart_of_fan (V , E))) ==> ((CARD ((face (hypermap_of_fan (V , E))) x)) >= (NUMERAL (BIT1 (BIT1 _0)))))))))


real_sub

|- (!x. (!y. (((real_sub x) y) = ((real_add x) (real_neg y)))))


NOT_LT

|- (!m. (!n. ((~ (m < n)) = (n <= m))))


ARITH_ZERO

|- (((NUMERAL (NUMERAL _0)) = (NUMERAL _0)) /\ ((BIT0 _0) = _0))


COMPLEX_DIV_1

|- (!z. (((complex_div z) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = z))


Appendix.scs_str_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_str_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = str))))))))))


Yxionxl.PRO_EQU_ID1

|- ((((scs_k_v39 s) = k) /\ (is_scs_v39 s)) ==> (s = ((scs_prop_equ_v39 ((scs_prop_equ_v39 s) i)) (k - (i MOD k)))))


list_INDUCT

|- (!P. (((P NIL) /\ (!a0. (!a1. ((P a1) ==> (P ((CONS a0) a1)))))) ==> (!x. (P x))))


MEM_TRANSLATION

|- (!a. (!x. (!l. (((MEM ((vector_add a) x)) ((MAP (\x. ((vector_add a) x))) l)) = ((MEM x) l)))))


LINEAR_CONTINUOUS_AT

|- (!f. (!a. ((linear f) ==> ((continuous f) (at a)))))


HAS_REAL_MEASURE

|- (!s. (!m. (((has_real_measure s) m) = (((has_real_integral (\x. (((COND (x IN s)) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0))))) m) UNIV))))


MATRIX_NEG_MINUS1

|- (!A. ((matrix_neg A) = ((%% (real_neg (real_of_num (NUMERAL (BIT1 _0))))) A)))


Seq.rotr1_rcons

|- (!x. (!s. (((rotr (NUMERAL (BIT1 _0))) ((rcons s) x)) = ((CONS x) s))))


RELATIVE_INTERIOR_SUBSET

|- (!s. ((relative_interior s) SUBSET s))


Kepler_spec.BIJ_REAL3_TRIPLE

|- (((triple_of_real3 o real3_of_triple) = I) /\ ((real3_of_triple o triple_of_real3) = I))


SQRT_MONO_LT

|- (!x. (!y. ((((real_le (real_of_num (NUMERAL _0))) x) /\ ((real_lt x) y)) ==> ((real_lt (sqrt x)) (sqrt y)))))


FACE_OF_TRANSLATION_EQ

|- (!a. (!f. (!s. (((face_of ((IMAGE (\x. ((vector_add a) x))) f)) ((IMAGE (\x. ((vector_add a) x))) s)) = ((face_of f) s)))))


components

|- (!s. ((components s) = (GSPEC (\GEN%PVAR%904. (?x. (((SETSPEC GEN%PVAR%904) (x IN s)) ((connected_component s) x)))))))


Trigonometry2.EXISTS_PROJECTING_POINT2

|- (!p. (!u. (!v0. (?pp. ((pp IN (aff (u INSERT (v0 INSERT EMPTY)))) /\ (((dot ((vector_sub p) pp)) ((vector_sub u) v0)) = (real_of_num (NUMERAL _0))))))))


REAL_POW_ROOT

|- (!n. (!x. (((ODD n) \/ ((~ (n = (NUMERAL _0))) /\ ((real_le (real_of_num (NUMERAL _0))) x))) ==> (((real_pow ((root n) x)) n) = x))))


COND_ID

|- (!b. (!t. ((((COND b) t) t) = t)))


REAL_LTE_ANTISYM

|- (!x. (!y. (~ (((real_lt x) y) /\ ((real_le y) x)))))


EMPTY_AS_INTERVAL

|- (EMPTY = (closed_interval ((CONS ((vec (NUMERAL (BIT1 _0))) , (vec (NUMERAL _0)))) NIL)))


NEGLIGIBLE_ABSOLUTELY_CONTINUOUS_IMAGE

|- (!f. (!s. (!t. ((((absolutely_continuous_on f) s) /\ ((is_interval s) /\ ((negligible t) /\ (t SUBSET s)))) ==> (negligible ((IMAGE f) t))))))


Trigonometry.AVWKGNB

|- (!x3. (!x1. (!x2. (!x4. (!x5. (!x6. (((((((delta_x x1) x2) x3) x4) x5) x6) = ((real_add ((real_mul x1) ((real_mul x4) ((real_add (real_neg x1)) ((real_add x2) ((real_add ((real_sub x3) x4)) ((real_add x5) x6))))))) ((real_add ((real_mul x2) ((real_mul x5) ((real_add ((real_sub x1) x2)) ((real_add x3) ((real_add ((real_sub x4) x5)) x6)))))) ((real_sub ((real_sub ((real_sub ((real_sub ((real_mul x3) ((real_mul x6) ((real_add x1) ((real_add ((real_sub x2) x3)) ((real_add x4) ((real_sub x5) x6))))))) ((real_mul x2) ((real_mul x3) x4)))) ((real_mul x1) ((real_mul x3) x5)))) ((real_mul x1) ((real_mul x2) x6)))) ((real_mul x4) ((real_mul x5) x6))))))))))))


Lp_ineqs_def.y2_fan

|- (!x. ((y2_fan x) = (vector_norm (SND x))))


List_hypermap_iso.fan_list_nodes_iso

|- (!f. (!x. (!V. (!E. (((fan_list_nodes_iso (f , (V , E))) x) = (FST (f (x , (((COND ((((set_of_edge x) V) E) = EMPTY)) x) (CHOICE (((set_of_edge x) V) E)))))))))))


PATH_IMAGE_LINEAR_IMAGE

|- (!f. (!g. ((linear f) ==> ((path_image (f o g)) = ((IMAGE f) (path_image g))))))


REAL_LT_LDIV_EQ

|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt ((real_div x) z)) y) = ((real_lt x) ((real_mul y) z)))))))


MEASURABLE_ON_PREIMAGE_BOREL

|- (!f. (((measurable_on f) UNIV) = (!t. ((borel t) ==> (lebesgue_measurable (GSPEC (\GEN%PVAR%3549. (?x. (((SETSPEC GEN%PVAR%3549) ((f x) IN t)) x)))))))))


PATH_CONNECTED_IFF_PATH_COMPONENT

|- (!s. ((path_connected s) = (!x. (!y. (((x IN s) /\ (y IN s)) ==> (((path_component s) x) y))))))


Seq.filter

|- (!a. (!x. (!s'. ((((filter a) NIL) = NIL) /\ (((filter a) ((CONS x) s')) = (((COND (a x)) ((CONS x) ((filter a) s'))) ((filter a) s')))))))


HAS_COMPLEX_DERIVATIVE_CASN

|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> (((has_complex_derivative casn) (complex_inv (ccos (casn z)))) (at z))))


EVEN_MOD

|- (!n. ((EVEN n) = ((n MOD (NUMERAL (BIT0 (BIT1 _0)))) = (NUMERAL _0))))


AFF_DIM_HYPERPLANE

|- (!a. (!b. ((~ (a = (vec (NUMERAL _0)))) ==> ((aff_dim (GSPEC (\GEN%PVAR%1576. (?x. (((SETSPEC GEN%PVAR%1576) (((dot a) x) = b)) x))))) = ((int_sub (int_of_num (dimindex UNIV))) (int_of_num (NUMERAL (BIT1 _0))))))))


LIPSCHITZ_LIM

|- (!f. (!g. (!s. (!b. (((!n. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub ((f n) x)) ((f n) y)))) ((real_mul b) (vector_norm ((vector_sub x) y)))))))) /\ (!x. ((x IN s) ==> (((--> (\n. ((f n) x))) (g x)) sequentially)))) ==> (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (g x)) (g y)))) ((real_mul b) (vector_norm ((vector_sub x) y))))))))))))


Appendix.tgt

|- (tgt = ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))))))


CCOS_ADD

|- (!w. (!z. ((ccos ((vector_add w) z)) = ((vector_sub ((complex_mul (ccos w)) (ccos z))) ((complex_mul (csin w)) (csin z))))))


REAL_MUL_AC

|- ((((real_mul m) n) = ((real_mul n) m)) /\ ((((real_mul ((real_mul m) n)) p) = ((real_mul m) ((real_mul n) p))) /\ (((real_mul m) ((real_mul n) p)) = ((real_mul n) ((real_mul m) p)))))


Hypermap_iso.hyp_iso_inj

|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!x. (!y. (((x IN (dart H)) /\ ((y IN (dart H)) /\ ((f x) = (f y)))) ==> (x = y))))))))


OPEN_IN_IMP_SUBSET

|- (!top. (!s. (!t. (((open_in ((subtopology top) s)) t) ==> (t SUBSET s)))))


Hypermap_iso.hyp_iso_surj

|- (!G. (!H. (!f. (((hyp_iso f) (H , G)) ==> (!y. ((y IN (dart G)) ==> (?x. ((x IN (dart H)) /\ ((f x) = y)))))))))


Wrgcvdr_cizmrrh.wedge_in_fan_ge

|- (!E. (!v. (!w. (((wedge_in_fan_ge (v , w)) E) = (((COND ((CARD ((EE v) E)) > (NUMERAL (BIT1 _0)))) ((((wedge_ge (vec (NUMERAL _0))) v) w) ((((azim_cycle ((EE v) E)) (vec (NUMERAL _0))) v) w))) (GSPEC (\GEN%PVAR%6280. (?x. (((SETSPEC GEN%PVAR%6280) T) x)))))))))


Ssrnat.ltE

|- (!n. (!m. ((n < m) = ((SUC n) <= m))))


HOMOTOPIC_TRIVIALITY

|- (!s. (!t. ((!f. (!g. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) s) /\ (((IMAGE g) s) SUBSET t)))) ==> ((((homotopic_with (\x. T)) (s , t)) f) g)))) = (((s = EMPTY) \/ (path_connected t)) /\ (!f. ((((continuous_on f) s) /\ (((IMAGE f) s) SUBSET t)) ==> (?c. ((((homotopic_with (\x. T)) (s , t)) f) (\x. c)))))))))


complex_differentiable

|- (!f. (!net. (((complex_differentiable f) net) = (?f'. (((has_complex_derivative f) f') net)))))


MEASURABLE_BOUNDED_BY_INTEGRABLE_IMP_ABSOLUTELY_INTEGRABLE

|- (!f. (!g. (!s. ((((measurable_on f) s) /\ (((integrable_on g) s) /\ (!x. ((x IN s) ==> ((real_le (vector_norm (f x))) (drop (g x))))))) ==> ((absolutely_integrable_on f) s)))))


QUANTIFY_SURJECTION_THM

|- (!f. ((!y. (?x. ((f x) = y))) ==> (((!P. ((!x. (P x)) = (!x. (P (f x))))) /\ ((!P. ((?x. (P x)) = (?x. (P (f x))))) /\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE f) s))))) /\ (!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE f) s)))))))) /\ (!P. ((GSPEC (\GEN%PVAR%236. (?x. (((SETSPEC GEN%PVAR%236) (P x)) x)))) = ((IMAGE f) (GSPEC (\GEN%PVAR%237. (?x. (((SETSPEC GEN%PVAR%237) (P (f x))) x))))))))))


Lp_main_estimate.pent_diags_imp

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. (!a. (!b. ((LET (\f. (LET_END ((LET (\w'. (LET_END ((LET (\u. (LET_END ((LET (\z. (LET_END ((((v , w) IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) /\ ((~ (a = b)) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ (~ ((a INSERT (b INSERT EMPTY)) IN ((e_prime E) f))))))) ==> ((((a , b) = (w , w')) \/ ((a , b) = (w' , w))) \/ ((((a , b) = (z , v)) \/ ((a , b) = (v , z))) \/ ((((a , b) = (u , w)) \/ ((a , b) = (w , u))) \/ ((((a , b) = (w' , z)) \/ ((a , b) = (z , w'))) \/ (((a , b) = (v , u)) \/ ((a , b) = (u , v))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) u) w'))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w))))) ((face (hypermap_of_fan (V , E))) (v , w))))))))))


COUNTABLE_COMPACT_OPEN_IN

|- (!u. (COUNTABLE (GSPEC (\GEN%PVAR%1245. (?c. (((SETSPEC GEN%PVAR%1245) ((compact c) /\ ((open_in ((subtopology euclidean) u)) c))) c))))))


PCROSS_EQ_EMPTY

|- (!s. (!t. (((s PCROSS t) = EMPTY) = ((s = EMPTY) \/ (t = EMPTY)))))


int_neg_th

|- (!x. ((real_of_int (int_neg x)) = (real_neg (real_of_int x))))


NORM_FSTCART

|- (!x. ((real_le (vector_norm (fstcart x))) (vector_norm x)))


OPEN_IN

|- (!s. ((open s) = ((open_in euclidean) s)))


VECTOR_SUB_LDISTRIB

|- (((% c) ((vector_sub x) y)) = ((vector_sub ((% c) x)) ((% c) y)))


Ssrnat.fact_gt0

|- (!n. ((NUMERAL _0) < (FACT n)))


HULL_MINIMAL

|- (!P. (!s. (!t. (((s SUBSET t) /\ (P t)) ==> (((hull P) s) SUBSET t)))))


LEFT_ADD_DISTRIB

|- (!m. (!n. (!p. ((m * (n + p)) = ((m * n) + (m * p))))))


HOMEOMORPHIC_OPEN_INTERVALS_1

|- (!a. (!b. (!c. (!d. ((((real_lt (drop a)) (drop b)) /\ ((real_lt (drop c)) (drop d))) ==> ((homeomorphic (open_interval (a , b))) (open_interval (c , d))))))))


OPEN_IN_DELETE

|- (!u. (!s. (!a. (((open_in ((subtopology euclidean) u)) s) ==> ((open_in ((subtopology euclidean) u)) (s DELETE a))))))


Seq.unzip2

|- (unzip2 = (map SND))


has_derivative_at

|- (!f. (!f'. (!x. ((((has_derivative f) f') (at x)) = ((linear f') /\ (((--> (\y. ((% (real_inv (vector_norm ((vector_sub y) x)))) ((vector_sub (f y)) ((vector_add (f x)) (f' ((vector_sub y) x))))))) (vec (NUMERAL _0))) (at x)))))))


RELATIVE_INTERIOR_TRANSLATION

|- (!a. (!s. ((relative_interior ((IMAGE (\x. ((vector_add a) x))) s)) = ((IMAGE (\x. ((vector_add a) x))) (relative_interior s)))))


IN_SPHERE

|- (!x. (!y. (!e. ((y IN (sphere (x , e))) = ((distance (x , y)) = e)))))


Ssrnat.mulSnr

|- (!m. (!n. (((SUC m) * n) = ((m * n) + n))))


complex_div

|- (!w. (!z. (((complex_div w) z) = ((complex_mul w) (complex_inv z)))))


MIDPOINT_LINEAR_IMAGE

|- (!f. (!a. (!b. ((linear f) ==> ((midpoint ((f a) , (f b))) = (f (midpoint (a , b))))))))


BOUNDED_SUBSET_BALL

|- (!s. (!x. ((bounded s) ==> (?r. (((real_lt (real_of_num (NUMERAL _0))) r) /\ (s SUBSET (ball (x , r))))))))


PATH_APPROX_VECTOR_POLYNOMIAL_FUNCTION

|- (!g. (!e. (((path g) /\ ((real_lt (real_of_num (NUMERAL _0))) e)) ==> (?p. ((vector_polynomial_function p) /\ (((pathstart p) = (pathstart g)) /\ (((pathfinish p) = (pathfinish g)) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> ((real_lt (vector_norm ((vector_sub (p t)) (g t)))) e))))))))))


VEC_COMPONENT

|- (!k. (!i. (((vec k) $ i) = (real_of_num k))))


INT_LT_ANTISYM

|- (!x. (!y. (~ (((int_lt x) y) /\ ((int_lt y) x)))))


VECTOR_DERIVATIVE_UNIQUE_WITHIN_CLOSED_INTERVAL

|- (!f. (!a. (!b. (!x. (!f'. (!f''. ((((real_lt (drop a)) (drop b)) /\ ((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((((has_vector_derivative f) f') ((within (at x)) (closed_interval ((CONS (a , b)) NIL)))) /\ (((has_vector_derivative f) f'') ((within (at x)) (closed_interval ((CONS (a , b)) NIL))))))) ==> (f' = f''))))))))


REAL_CONTINUOUS_FRAC

|- (!x. ((~ (integer x)) ==> ((real_continuous frac) (atreal x))))


LOWDIM_EQ_INTER_HYPERPLANE

|- (!s. (!t. (((subspace s) /\ ((subspace t) /\ ((t SUBSET s) /\ (((dim t) + (NUMERAL (BIT1 _0))) = (dim s))))) ==> (?a. ((~ (a = (vec (NUMERAL _0)))) /\ (((GSPEC (\GEN%PVAR%191. (?x. (((SETSPEC GEN%PVAR%191) (((dot a) x) = (real_of_num (NUMERAL _0)))) x)))) INTER s) = t))))))


AFFINE_IMP_SUBSPACE

|- (!s. (((affine s) /\ ((vec (NUMERAL _0)) IN s)) ==> (subspace s)))


NORM_EQ_SQUARE

|- (!x. (((vector_norm x) = a) = (((real_le (real_of_num (NUMERAL _0))) a) /\ (((dot x) x) = ((real_pow a) (NUMERAL (BIT0 (BIT1 _0))))))))


columnvector

|- (!v. ((columnvector v) = (lambdai. (lambdaj. (v $ i)))))


HAS_REAL_DERIVATIVE_CARATHEODORY_WITHINREAL

|- (!f. (!f'. (!z. (!s. ((((has_real_derivative f) f') ((within (atreal z)) s)) = (?g. ((!w. (((real_sub (f w)) (f z)) = ((real_mul (g w)) ((real_sub w) z)))) /\ (((real_continuous g) ((within (atreal z)) s)) /\ ((g z) = f')))))))))


Ssrnat.leq_sub_add

|- (!m. (!n. (!p. (((m - n) <= p) = (m <= (n + p))))))


Ssrnat.muln_addr

|- (!m. (!n1. (!n2. ((m * (n1 + n2)) = ((m * n1) + (m * n2))))))


FACET_OF_LINEAR_IMAGE

|- (!f. (!c. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((facet_of ((IMAGE f) c)) ((IMAGE f) s)) = ((facet_of c) s))))))


Lp_ineqs_def.y5_fan

|- (!V. (!E. (!x. (((y5_fan (V , E)) x) = (distance ((inverse (f_fan_pair_ext (V , E))) x))))))


Ssrnat.minn_maxl

|- (!m1. (!m2. (!n. (((minn ((maxn m1) m2)) n) = ((maxn ((minn m1) n)) ((minn m2) n))))))


Conforming.properties_of_elements_in_node_fully_surroundedfan

|- (!x. (!V. (!E. (!ds. (!y. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((ds IN (node_set (hypermap1_of_fanx (x , (V , E))))) /\ (y IN ds)))) ==> (((pr2 y) INSERT ((pr3 y) INSERT EMPTY)) IN E)))))))


POSITIVE_SEMIDEFINITE_CMUL

|- (!c. (!A. (((positive_semidefinite A) /\ ((real_le (real_of_num (NUMERAL _0))) c)) ==> (positive_semidefinite ((%% c) A)))))


Tame_general.CONTRAVENING_IMP_FULLY_SURROUNDED

|- (!V. (((FAN ((vec (NUMERAL _0)) , (V , (ESTD V)))) /\ (contravening V)) ==> (fully_surrounded (V , (ESTD V)))))


RE_CMUL

|- (!a. (!z. ((Re ((% a) z)) = ((real_mul a) (Re z)))))


EDGE_OF_LINEAR_IMAGE

|- (!f. (!c. (!s. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((edge_of ((IMAGE f) c)) ((IMAGE f) s)) = ((edge_of c) s))))))


FORALL_VECTOR_3

|- ((!v. (P v)) = (!x. (!y. (!z. (P (vector ((CONS x) ((CONS y) ((CONS z) NIL)))))))))


Real_ext.REAL_PROP_LT_LMUL

|- (!x. (!y. (!z. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt y) z)) ==> ((real_lt ((real_mul x) y)) ((real_mul x) z))))))


FSIGMA_LOCALLY_COMPACT

|- (!s. (((locally compact) s) ==> (fsigma s)))


REAL_EXP_N

|- (!n. (!x. ((exp ((real_mul (real_of_num n)) x)) = ((real_pow (exp x)) n))))


Lp_ineqs_proofs.lp_cond_trans1

|- (!L. (!V. (!E. (!g. (!h. (!d. (((lp_cond (L , (g , h))) (V , E)) ==> ((lp_cond (((split_list_hyp L) d) , (g , h))) (V , ((add_diag (V , E)) (g d)))))))))))


BILINEAR_DROP_MUL

|- (bilinear (\x. (\y. ((% (drop x)) y))))


LOCALLY_CONNECTED_IM_KLEINEN

|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((x IN u) /\ ((u SUBSET v) /\ (!y. ((y IN u) ==> (?c. ((connected c) /\ ((c SUBSET v) /\ ((x IN c) /\ (y IN c))))))))))))))))


real_abs

|- (!x. ((real_abs x) = (((COND ((real_le (real_of_num (NUMERAL _0))) x)) x) (real_neg x))))


Hypermap_iso.inE

|- ((!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)))))))


RC_INC

|- (!R. (!x. (!y. (((R x) y) ==> (((RC R) x) y)))))


Functional_equation.functional_mud_234_x

|- (!y1. (!y5. (!y6. (((domain6 (\x1. (\x2. (\x3. (\x4. (\x5. (\x6. ((real_le (real_of_num (NUMERAL _0))) y1)))))))) (((mud_234_x_v1 y1) y5) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y1) y1))) proj_x2) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_234_x ((real_mul y1) y1)) ((real_mul y5) y5)) ((real_mul y6) y6)))))))))


REAL_LE_TRANS

|- (!x. (!y. (!z. ((((real_le x) y) /\ ((real_le y) z)) ==> ((real_le x) z)))))


Hypermap.lemma_face_identity

|- (!H. (!x. (!y. ((y IN ((face H) x)) ==> (((face H) x) = ((face H) y))))))


Tame_lemmas.reflection_at0

|- (!x. (((reflection (vec (NUMERAL _0))) x) = x))


HAS_REAL_MEASURE_DISJOINT_UNIONS_IMAGE_STRONG

|- (!f. (!s. (((FINITE (GSPEC (\GEN%PVAR%4933. (?x. (((SETSPEC GEN%PVAR%4933) ((x IN s) /\ (~ ((f x) = EMPTY)))) x))))) /\ ((!x. ((x IN s) ==> (real_measurable (f x)))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ (~ (x = y)))) ==> ((DISJOINT (f x)) (f y))))))) ==> ((has_real_measure (UNIONS ((IMAGE f) s))) ((sum s) (\x. (real_measure (f x))))))))


Trigonometry.KGLLRQT1

|- (!x. ((cos (real_neg x)) = (cos x)))


DIVISION_OF_TAGGED_DIVISION

|- (!s. (!i. (((tagged_division_of s) i) ==> ((division_of ((IMAGE SND) s)) i))))


LIM_AT_POSINFINITY

|- (!f. (!l. ((((--> f) l) at_posinfinity) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?b. (!x. (((real_ge x) b) ==> ((real_lt (distance ((f x) , l))) e)))))))))


ATN_ABS_LE_X

|- (!x. ((real_le (real_abs (atn x))) (real_abs x)))


Functional_equation.cos797

|- (cos797 = (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))))))))))))))


NORM_EQ_0

|- (!x. (((vector_norm x) = (real_of_num (NUMERAL _0))) = (x = (vec (NUMERAL _0)))))


REAL_SUMS_INFSUM

|- (!f. (!s. ((((real_sums f) ((real_infsum s) f)) s) = ((real_summable s) f))))


CX_ABS

|- (!x. ((Cx (real_abs x)) = (Cx (vector_norm (Cx x)))))


Seq2.card_set_of_list_uniq

|- (!l. ((uniq l) ==> ((CARD (set_of_list l)) = (sizel l))))


CONVEX_INTER

|- (!s. (!t. (((convex s) /\ (convex t)) ==> (convex (s INTER t)))))


Ssrnat.eqn_leq

|- (!m. (!n. ((m = n) = ((m <= n) /\ (n <= m)))))


MIDPOINT_IN_CONVEX

|- (!s. (!x. (!y. (((convex s) /\ ((x IN s) /\ (y IN s))) ==> ((midpoint (x , y)) IN s)))))


DIM_SUBSET_UNIV

|- (!s. ((dim s) <= (dimindex UNIV)))


REALLIM_WITHIN_SUBSET

|- (!f. (!l. (!a. (!s. (!t. (((((---> f) l) ((within (at a)) s)) /\ (t SUBSET s)) ==> (((---> f) l) ((within (at a)) t))))))))


BOUNDED_DIFFS

|- (!s. (!t. (((bounded s) /\ (bounded t)) ==> (bounded (GSPEC (\GEN%PVAR%586. (?x. (?y. (((SETSPEC GEN%PVAR%586) ((x IN s) /\ (y IN t))) ((vector_sub x) y))))))))))


LIMIT_POINT_FINITE

|- (!s. (!a. ((FINITE s) ==> (~ ((limit_point_of a) s)))))


RPOW_POS_LE

|- (!x. (!y. (((real_le (real_of_num (NUMERAL _0))) x) ==> ((real_le (real_of_num (NUMERAL _0))) ((rpow x) y)))))


sgn_le

|- (sgn_le = (\t. ((real_le t) (real_of_num (NUMERAL _0)))))


Fan.fan7

|- (!E. (!V. (!x. ((fan7 (x , (V , E))) = (!e1. (!e2. (((e1 IN (E UNION (GSPEC (\GEN%PVAR%5449. (?v. (((SETSPEC GEN%PVAR%5449) (v IN V)) (v INSERT EMPTY))))))) /\ (e2 IN (E UNION (GSPEC (\GEN%PVAR%5450. (?v. (((SETSPEC GEN%PVAR%5450) (v IN V)) (v INSERT EMPTY)))))))) ==> ((((aff_ge (x INSERT EMPTY)) e1) INTER ((aff_ge (x INSERT EMPTY)) e2)) = ((aff_ge (x INSERT EMPTY)) (e1 INTER e2))))))))))


SURJ

|- (!t. (!s. (!f. ((((SURJ f) s) t) = ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!x. ((x IN t) ==> (?y. ((y IN s) /\ ((f y) = x))))))))))


CX_DIV

|- (!x. (!y. ((Cx ((real_div x) y)) = ((complex_div (Cx x)) (Cx y)))))


MEASURE_INTERVAL_1

|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND ((real_le (drop a)) (drop b))) ((real_sub (drop b)) (drop a))) (real_of_num (NUMERAL _0)))))))


ROWS_TRANSP

|- (!A. ((rows (transp A)) = (columns A)))


REAL_SUB_LDISTRIB

|- (!x. (!y. (!z. (((real_mul x) ((real_sub y) z)) = ((real_sub ((real_mul x) y)) ((real_mul x) z))))))


DIST_1

|- (!x. (!y. ((distance (x , y)) = (real_abs ((real_sub (drop x)) (drop y))))))


SIMPLY_CONNECTED_FUNDAMENTAL_GROUP

|- (!s. ((simply_connected s) = ((path_connected s) /\ (!a. ((a IN s) ==> ((fundamental_group (s , a)) = (((homotopic_paths s) (linepath (a , a))) INSERT EMPTY)))))))


HAS_INTEGRAL_LIM_AT_POSINFINITY

|- (!f. (!l. ((((has_integral f) l) (GSPEC (\GEN%PVAR%2953. (?t. (((SETSPEC GEN%PVAR%2953) ((real_le (real_of_num (NUMERAL _0))) (drop t))) t))))) = ((!a. ((integrable_on f) (closed_interval ((CONS ((vec (NUMERAL _0)) , a)) NIL)))) /\ (((--> (\a. ((integral (closed_interval ((CONS ((vec (NUMERAL _0)) , (lift a))) NIL))) f))) l) at_posinfinity)))))


Local_lemmas1.localization

|- (!V. (!E. (!FF. (((localization (V , E)) FF) = (((v_prime V) FF) , ((e_prime E) FF))))))


Hypermap_and_fan.PAIR_IN_DART1_OF_FAN_IMP_NOT_EQ

|- (!V. (!E. (!v. (!w. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v , w) IN (dart1_of_fan (V , E)))) ==> (~ (v = w)))))))


INTEGRABLE_BOUNDED_VARIATION_COMPLEX_LMUL

|- (!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. ((complex_mul (g x)) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))


CONVEX_HULL_INSERT_REDUNDANT_POINT

|- (!s. (!a. (!b. (!c. (((a IN ((hull convex) (c INSERT s))) /\ ((b IN ((hull convex) (c INSERT s))) /\ (c IN (open_segment (a , b))))) ==> (((hull convex) (c INSERT s)) = ((hull convex) s)))))))


ORTHOGONAL_BASIS_BASIS

|- (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))) ==> (((orthogonal (basis i)) (basis j)) = (~ (i = j))))))


AFFINE_TRANSLATION_SUBSPACE

|- (!t. (((affine t) /\ (~ (t = EMPTY))) = (?a. (?s. ((subspace s) /\ (t = ((IMAGE (\x. ((vector_add a) x))) s)))))))


LOCALLY_CONNECTED

|- (!s. (((locally connected) s) = (!v. (!x. ((((open_in ((subtopology euclidean) s)) v) /\ (x IN v)) ==> (?u. (((open_in ((subtopology euclidean) s)) u) /\ ((connected u) /\ ((x IN u) /\ (u SUBSET v))))))))))


MATRIX_INV_INV

|- (!A. ((matrix_inv (matrix_inv A)) = A))


INVERSE_I

|- ((inverse I) = I)


INDEPENDENT_STDBASIS

|- (independent (GSPEC (\GEN%PVAR%152. (?i. (((SETSPEC GEN%PVAR%152) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (basis i))))))


mat

|- (!k. ((mat k) = (lambdai. (lambdaj. (((COND (i = j)) (real_of_num k)) (real_of_num (NUMERAL _0)))))))


Local_lemmas.PRESERABLE_AFF_GE_SUBSET

|- ((a IN ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))) ==> (((aff_ge (x INSERT EMPTY)) (a INSERT EMPTY)) SUBSET ((aff_ge (x INSERT EMPTY)) (b INSERT EMPTY))))


CONTINUOUS_ON

|- (!f. (!s. (((continuous_on f) s) = (!x. ((x IN s) ==> (((--> f) (f x)) ((within (at x)) s)))))))


SUBPATH_TRANSLATION

|- (!a. (!g. (!u. (!v. ((((subpath u) v) ((\x. ((vector_add a) x)) o g)) = ((\x. ((vector_add a) x)) o (((subpath u) v) g)))))))


Real_ext.REAL_PROP_LE_RMUL

|- (!x. (!y. (!z. ((((real_le x) y) /\ ((real_le (real_of_num (NUMERAL _0))) z)) ==> ((real_le ((real_mul x) z)) ((real_mul y) z))))))


FORALL_PAIR_THM

|- (!P. ((!p. (P p)) = (!p1. (!p2. (P (p1 , p2))))))


CNJ_DIV

|- (!w. (!z. ((cnj ((complex_div w) z)) = ((complex_div (cnj w)) (cnj z)))))


Ssrbool.andbN

|- (!b. ((b /\ (~ b)) = F))


ABSOLUTELY_CONTINUOUS_ISOMETRIC

|- (!f. (!g. (!s. ((!x. (!y. ((distance ((f x) , (f y))) = (distance ((g x) , (g y)))))) ==> (((absolutely_continuous_on f) s) = ((absolutely_continuous_on g) s))))))


CLOSURE_SPHERE

|- (!a. (!r. ((closure (sphere (a , r))) = (sphere (a , r)))))


COMPLEX_MUL_RID

|- (!x. (((complex_mul x) (Cx (real_of_num (NUMERAL (BIT1 _0))))) = x))


MATRIX_SUB_RZERO

|- (!A. (((matrix_sub A) (mat (NUMERAL _0))) = A))


vector_mul

|- (!c. (!x. (((% c) x) = (lambdai. ((real_mul c) (x $ i))))))


absolutely_integrable_on

|- (!f. (!s. (((absolutely_integrable_on f) s) = (((integrable_on f) s) /\ ((integrable_on (\x. (lift (vector_norm (f x))))) s)))))


REAL_MUL_SYM

|- (!x. (!y. (((real_mul x) y) = ((real_mul y) x))))


SUMMABLE_SUBSET

|- (!x. (!s. (!t. (((s SUBSET t) /\ ((summable t) (\i. (((COND (i IN s)) (x i)) (vec (NUMERAL _0)))))) ==> ((summable s) x)))))


Ssrnat.maxnC

|- (!m. (!n. (((maxn m) n) = ((maxn n) m))))


Ldurdpn.VECTOR_SCALE_CHANGE

|- ((~ (a = (real_of_num (NUMERAL _0)))) ==> ((((% a) x) = y) = (x = ((% ((real_div (real_of_num (NUMERAL (BIT1 _0)))) a)) y))))


PRODUCT_NEG_NUMSEG

|- (!f. (!m. (!n. (((product (m .. n)) (\i. (real_neg (f i)))) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((n + (NUMERAL (BIT1 _0))) - m))) ((product (m .. n)) f))))))


Hypermap_iso.darts_of_fan_disj1

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((DISJOINT (d1_fan ((vec (NUMERAL _0)) , (V , E)))) (d20_fan ((vec (NUMERAL _0)) , (V , E)))))))


REAL_EQ_ADD_RCANCEL_0

|- (!x. (!y. ((((real_add x) y) = y) = (x = (real_of_num (NUMERAL _0))))))


TRANSP_COMPONENT

|- (!A. (!i. (!j. ((((transp A) $ i) $ j) = ((A $ j) $ i)))))


Seq.rot_oversize

|- (!n. (!s. (((sizel s) <= n) ==> (((rot n) s) = s))))


DIV_UNIQ

|- (!m. (!n. (!q. (!r. (((m = ((q * n) + r)) /\ (r < n)) ==> ((m DIV n) = q))))))


CONTINUOUS_IMP_MEASURABLE_ON

|- (!f. (((continuous_on f) UNIV) ==> ((measurable_on f) UNIV)))


HAS_REAL_INTEGRAL_ON_SUPERSET

|- (!f. (!s. (!t. (((!x. ((~ (x IN s)) ==> ((f x) = (real_of_num (NUMERAL _0))))) /\ ((s SUBSET t) /\ (((has_real_integral f) i) s))) ==> (((has_real_integral f) i) t)))))


SERIES_COMPLEX_RMUL

|- (!f. (!l. (!c. (!s. ((((sums f) l) s) ==> (((sums (\x. ((complex_mul (f x)) c))) ((complex_mul l) c)) s))))))


LIFT_ADD

|- (!x. (!y. ((lift ((real_add x) y)) = ((vector_add (lift x)) (lift y)))))


Hypermap.power_inverse_element_lemma

|- (!s. (!p. (!n. (((FINITE s) /\ ((permutes p) s)) ==> (?j. (((POWER (inverse p)) n) = ((POWER p) j)))))))


BIT0

|- (!n. ((BIT0 n) = (n + n)))


RE_II

|- ((Re ii) = (real_of_num (NUMERAL _0)))


NSUM_EQ

|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((nsum s) f) = ((nsum s) g))))))


LINEAR_CMUL

|- (!f. (!c. (!x. ((linear f) ==> ((f ((% c) x)) = ((% c) (f x)))))))


LINEAR_SCALING

|- (!c. (linear (\x. ((% c) x))))


OPEN_INSIDE

|- (!s. ((closed s) ==> (open (inside s))))


REAL_POW_ZERO

|- (!n. (((real_pow (real_of_num (NUMERAL _0))) n) = (((COND (n = (NUMERAL _0))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))))


CONVEX_HULL_REDUNDANT_SUBSET_GEN

|- (!s. (!t. (((compact s) /\ ((t SUBSET s) /\ ((DISJOINT (s DIFF t)) (GSPEC (\GEN%PVAR%2254. (?x. (((SETSPEC GEN%PVAR%2254) ((extreme_point_of x) ((hull convex) s))) x))))))) ==> (((hull convex) s) = ((hull convex) t)))))


HAS_SIZE_FINITE_IMAGE

|- (!s. (UNIV HAS_SIZE (dimindex s)))


MEASURABLE_ON_LEBESGUE_MEASURABLE_PREIMAGE_OPEN_INTERVAL

|- (!f. (!s. ((lebesgue_measurable s) ==> (((measurable_on f) s) = (!a. (!b. (lebesgue_measurable (GSPEC (\GEN%PVAR%3638. (?x. (((SETSPEC GEN%PVAR%3638) ((x IN s) /\ ((f x) IN (open_interval (a , b))))) x)))))))))))


EXISTS_IN_GSPEC

|- ((!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%25. (?x. (((SETSPEC GEN%PVAR%25) (P x)) (f x)))))) /\ (Q z))) = (?x. ((P x) /\ (Q (f x))))))) /\ ((!P. (!f. ((?z. ((z IN (GSPEC (\GEN%PVAR%26. (?x. (?y. (((SETSPEC GEN%PVAR%26) ((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%27. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%27) (((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%28. (?v. (?w. (?x. (?y. (((SETSPEC GEN%PVAR%28) ((((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)))))))))))))


CONVEX_CONTAINS_SEGMENT

|- (!s. ((convex s) = (!a. (!b. (((a IN s) /\ (b IN s)) ==> ((closed_segment ((CONS (a , b)) NIL)) SUBSET s))))))


IN_INTER

|- (!s. (!t. (!x. ((x IN (s INTER t)) = ((x IN s) /\ (x IN t))))))


COMPLEX_MUL_ASSOC

|- (!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z)))))


AFF_DIM_EQ_0

|- (!s. (((aff_dim s) = (int_of_num (NUMERAL _0))) = (?a. (s = (a INSERT EMPTY)))))


List_hypermap_iso.e_fan_pair_ext_explicit

|- (!V. (!E. (!v. (!w. (((v , w) IN (dart_of_fan (V , E))) ==> (((e_fan_pair_ext (V , E)) (v , w)) = (w , v)))))))


INTEGRABLE_ON_SUBINTERVAL

|- (!f. (!s. (!a. (!b. ((((integrable_on f) s) /\ ((closed_interval ((CONS (a , b)) NIL)) SUBSET s)) ==> ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))))))


MATRIX_CMUL_LZERO

|- (!A. (((%% (real_of_num (NUMERAL _0))) A) = (mat (NUMERAL _0))))


PATH_JOIN

|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((path ((++ g1) g2)) = ((path g1) /\ (path g2))))))


Lp_ineqs_proofs.dart4_y4'_hi

|- (!V. (!E. ((lp_fan (V , E)) ==> ((!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((ineq ((CONS (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) , (y1 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y2 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y3 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0) , (y4 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y5 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) ((CONS (((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0)))))) , (y6 , ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)))) NIL))))))) ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((real_le ((y5_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((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)))))))))) /\ (((real_le ((y8_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ ((real_le ((y9_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))) ==> ((real_le ((y4'_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))


Conforming.FAN80_FANADD

|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (!ds1. (!ds2. (!f10. (!f20. (!f30. (((FAN (x , (V , E))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((fan80 (x , (V , E))) /\ ((ds IN (face_set (hypermap1_of_fanx (x , (V , E))))) /\ (((CARD ds) > (NUMERAL (BIT1 (BIT1 _0)))) /\ (((f1 INSERT (f2 INSERT (f3 INSERT EMPTY))) SUBSET ds) /\ ((((((f1_fan x) V) E) f1) = f2) /\ ((((((f1_fan x) V) E) f2) = f3) /\ ((~ (((((f1_fan x) V) E) f3) = f1)) /\ (((pr2 f1) = v) /\ (((pr2 f2) = u) /\ (((pr2 f3) = w) /\ (((v INSERT (u INSERT EMPTY)) IN E) /\ (((u INSERT (w INSERT EMPTY)) IN E) /\ ((~ ((w INSERT (v INSERT EMPTY)) IN E)) /\ (((((((sigma_fan x) V) E) u) w) = v) /\ (((pr3 f1) = u) /\ (((pr3 f2) = w) /\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (v , (w , (((((sigma_fan x) V) E1) v) w))))) = ds1) /\ ((((face (hypermap1_of_fanx (x , (V , E1)))) (x , (w , (v , (((((sigma_fan x) V) E1) w) v))))) = ds2) /\ (((x , (w , (v , u))) = f10) /\ (((x , (v , (u , w))) = f20) /\ (((x , (u , (w , v))) = f30) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))))))))) ==> (fan80 (x , (V , E1))))))))))))))))))))


LIFT_DROP

|- ((!x. ((lift (drop x)) = x)) /\ (!x. ((drop (lift x)) = x)))


Functional_equation.proj_x2

|- (((((((proj_x2 x1) x2) x3) x4) x5) x6) = x2)


is_metric_space

|- (!s. (!d. ((is_metric_space (s , d)) = ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (real_of_num (NUMERAL _0))) (d (x , y)))))) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> (((d (x , y)) = (real_of_num (NUMERAL _0))) = (x = y))))) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> ((d (x , y)) = (d (y , x)))))) /\ (!x. (!y. (!z. (((x IN s) /\ ((y IN s) /\ (z IN s))) ==> ((real_le (d (x , z))) ((real_add (d (x , y))) (d (y , z))))))))))))))


Seq.ncons

|- (!n. (!x. (((ncons n) x) = ((iter n) (CONS x)))))


Localization.h_dart

|- (!x. ((h_dart x) = ((real_div (vector_norm (FST x))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))


REAL_POW_EQ_1

|- (!x. (!n. ((((real_pow x) n) = (real_of_num (NUMERAL (BIT1 _0)))) = ((((real_abs x) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((real_lt x) (real_of_num (NUMERAL _0))) ==> (EVEN n))) \/ (n = (NUMERAL _0))))))


NSUM_OFFSET

|- (!p. (!f. (!m. (!n. (((nsum ((m + p) .. (n + p))) f) = ((nsum (m .. n)) (\i. (f (i + p)))))))))


REAL_EXP_LE_X

|- (!x. ((real_le ((real_add (real_of_num (NUMERAL (BIT1 _0)))) x)) (exp x)))


BETWEEN_LINEAR_IMAGE_EQ

|- (!f. (!x. (!y. (!z. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((between (f x)) ((f y) , (f z))) = ((between x) (y , z))))))))


REAL_INV_1

|- ((real_inv (real_of_num (NUMERAL (BIT1 _0)))) = (real_of_num (NUMERAL (BIT1 _0))))


POSITIVE_SEMIDEFINITE_IMP_SYMMETRIC

|- (!A. ((positive_semidefinite A) ==> ((transp A) = A)))


Seq.filter_uniq

|- (!s. (!a. ((uniq s) ==> (uniq ((filter a) s)))))


DECREASING_LEFT_LIMIT_1

|- (!f. (!a. (!b. (!c. (((!x. (!y. (((x IN (closed_interval ((CONS (a , b)) NIL))) /\ ((y IN (closed_interval ((CONS (a , b)) NIL))) /\ ((real_le (drop x)) (drop y)))) ==> ((real_le (drop (f y))) (drop (f x)))))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (?l. (((--> f) l) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))


EXISTS_DOUBLE_ARC_EXPLICIT

|- (!g. (!a. (!b. (((simple_path g) /\ (((pathfinish g) = (pathstart g)) /\ ((a IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((b IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ (((real_le (drop a)) (drop b)) /\ (~ ((g a) = (g b)))))))) ==> (?u. (?d. ((arc u) /\ ((arc d) /\ (((pathstart u) = (g a)) /\ (((pathfinish u) = (g b)) /\ (((pathstart d) = (g b)) /\ (((pathfinish d) = (g a)) /\ (((path_image u) = ((IMAGE g) (closed_interval ((CONS (a , b)) NIL)))) /\ (((path_image d) = ((IMAGE g) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) DIFF (open_interval (a , b))))) /\ ((((path_image u) INTER (path_image d)) = ((g a) INSERT ((g b) INSERT EMPTY))) /\ (((path_image u) UNION (path_image d)) = (path_image g)))))))))))))))))


COS_PI

|- ((cos pi) = (real_neg (real_of_num (NUMERAL (BIT1 _0)))))


BETWEEN_TRANS_2

|- (!a. (!b. (!c. (!d. ((((between a) (b , c)) /\ ((between d) (a , b))) ==> ((between a) (c , d)))))))


LIM_CMUL

|- (!f. (!l. (!c. ((((--> f) l) net) ==> (((--> (\x. ((% c) (f x)))) ((% c) l)) net)))))


PATHSTART_LINEAR_IMAGE_EQ

|- (!f. (!g. ((linear f) ==> ((pathstart (f o g)) = (f (pathstart g))))))


Ssrnat.leq_pred

|- (!n. ((n - (NUMERAL (BIT1 _0))) <= n))


CLOSEST_POINT_LE

|- (!s. (!a. (!x. (((closed s) /\ (x IN s)) ==> ((real_le (distance (a , ((closest_point s) a)))) (distance (a , x)))))))


RADON_PARTITION

|- (!c. (((FINITE c) /\ (affine_dependent c)) ==> (?m. (?p. (((DISJOINT m) p) /\ (((m UNION p) = c) /\ (~ ((DISJOINT ((hull convex) m)) ((hull convex) p)))))))))


Tame_list.indexf

|- (!y. (!x. (!s. ((((indexf x) y) s) = (((COND (((MEM x) s) /\ ((MEM y) s))) ((((indexl y) s) + ((sizel s) - ((indexl x) s))) MOD (sizel s))) (sizel s))))))


IMAGE

|- (!s. (!f. (((IMAGE f) s) = (GSPEC (\GEN%PVAR%7. (?y. (((SETSPEC GEN%PVAR%7) (?x. ((x IN s) /\ (y = (f x))))) y)))))))


mdist

|- (!m. ((mdist m) = (SND (dest_metric m))))


Hypermap.lemma_in_orbit

|- (!f. (!n. (!x. ((((POWER f) n) x) IN ((orbit_map f) x)))))


CLOSED_IN_REFL

|- (!s. ((closed_in ((subtopology euclidean) s)) s))


REAL_LE_RDIV_EQ

|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_le x) ((real_div y) z)) = ((real_le ((real_mul x) z)) y))))))


REAL_VECTOR_POLYNOMIAL_FUNCTION_o

|- (!f. (!g. (((vector_polynomial_function f) /\ (real_polynomial_function g)) ==> (real_polynomial_function (g o f)))))


Ssrbool.andbT

|- (!b. ((b /\ T) = b))


SPAN_OF_SUBSPACE

|- (!s. ((subspace s) ==> ((span s) = s)))


FINITE_SET_OF_LIST

|- (!l. (FINITE (set_of_list l)))


COMPACT_EQ_HEINE_BOREL

|- (!s. ((compact s) = (!f. (((!t. ((t IN f) ==> (open t))) /\ (s SUBSET (UNIONS f))) ==> (?f'. ((f' SUBSET f) /\ ((FINITE f') /\ (s SUBSET (UNIONS f')))))))))


Ssrnat.maxn_mulr

|- (!m. (!n1. (!n2. ((m * ((maxn n1) n2)) = ((maxn (m * n1)) (m * n2))))))


IN_SEGMENT_CX_GEN

|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS ((Cx a) , (Cx b))) NIL))) = (((Im x) = (real_of_num (NUMERAL _0))) /\ ((((real_le a) (Re x)) /\ ((real_le (Re x)) b)) \/ (((real_le b) (Re x)) /\ ((real_le (Re x)) a))))))))


REAL_LT_NEGTOTAL

|- (!x. ((x = (real_of_num (NUMERAL _0))) \/ (((real_lt (real_of_num (NUMERAL _0))) x) \/ ((real_lt (real_of_num (NUMERAL _0))) (real_neg x)))))


MATRIX_MUL_RID

|- (!A. (((matrix_mul A) (mat (NUMERAL (BIT1 _0)))) = A))


LOCALLY_COMPACT_PCROSS

|- (!s. (!t. ((((locally compact) s) /\ ((locally compact) t)) ==> ((locally compact) (s PCROSS t)))))


INVERTIBLE_DET_NZ

|- (!A. ((invertible A) = (~ ((det A) = (real_of_num (NUMERAL _0))))))


HAS_VECTOR_DERIVATIVE_REAL_COMPLEX

|- ((((has_complex_derivative f) f') (at (Cx (drop a)))) ==> (((has_vector_derivative (\x. (f (Cx (drop x))))) f') (at a)))


BOUNDED_INSIDE

|- (!s. ((bounded s) ==> (bounded (inside s))))


Nkezbfc_local.order

|- (!f. (!x. (!y. ((((order f) x) y) = (@n. (((((ITER n) f) x) = y) /\ (!i. ((((NUMERAL _0) < i) /\ (i < n)) ==> (~ ((((ITER i) f) x) = y))))))))))


VECTOR_MATRIX_MUL_TRANSP

|- (!A. (!x. (((vector_matrix_mul x) A) = ((matrix_vector_mul (transp A)) x))))


MULT_SUC

|- (!m. (!n. ((m * (SUC n)) = (m + (m * n)))))


Appendix.scs_J_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_J_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = J))))))))))


Fan.d20_fan

|- (!x. (!V. (!E. ((d20_fan (x , (V , E))) = (GSPEC (\GEN%PVAR%5471. (?x'. (?v. (((SETSPEC GEN%PVAR%5471) ((x' = x) /\ ((V v) /\ ((((set_of_edge v) V) E) = EMPTY)))) (x' , (v , (v , v))))))))))))


AFFINE_TRANSLATION

|- (!s. (!a. ((affine s) ==> (affine ((IMAGE (\x. ((vector_add a) x))) s)))))


CONNECTED_NEST

|- (!s. (((!n. ((compact (s n)) /\ (connected (s n)))) /\ (!m. (!n. ((m <= n) ==> ((s n) SUBSET (s m)))))) ==> (connected (INTERS (GSPEC (\GEN%PVAR%1154. (?n. (((SETSPEC GEN%PVAR%1154) (n IN UNIV)) (s n)))))))))


NOT_NEGLIGIBLE_UNIV

|- (~ (negligible UNIV))


Seq.rot0

|- (!s. (((rot (NUMERAL _0)) s) = s))


CLOG_MUL

|- (!w. (!z. (((~ (w = (Cx (real_of_num (NUMERAL _0))))) /\ (~ (z = (Cx (real_of_num (NUMERAL _0)))))) ==> ((clog ((complex_mul w) z)) = (((COND ((real_le (Im ((vector_add (clog w)) (clog z)))) (real_neg pi))) ((vector_add ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) (((COND ((real_gt (Im ((vector_add (clog w)) (clog z)))) pi)) ((vector_sub ((vector_add (clog w)) (clog z))) ((complex_mul ii) (Cx ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))))) ((vector_add (clog w)) (clog z))))))))


Ssrnat.fact0

|- ((FACT (NUMERAL _0)) = (NUMERAL (BIT1 _0)))


Pack2.NEGLIGIBLE_INTER_VORONOI_CLOSED

|- (!s. (!u. (!v. (((u IN s) /\ ((v IN s) /\ (~ (u = v)))) ==> (negligible (((voronoi_closed s) u) INTER ((voronoi_closed s) v)))))))


Tame_defs.tame_planar_hypermap

|- (!H. ((tame_planar_hypermap H) = ((tame_1 H) /\ ((tame_2 H) /\ ((tame_3 H) /\ ((tame_4 H) /\ ((tame_5a H) /\ ((tame_8 H) /\ ((tame_9a H) /\ ((tame_10 H) /\ ((tame_11a H) /\ ((tame_11b H) /\ ((tame_12o H) /\ (tame_13a H))))))))))))))


BOUNDED_UNIONS

|- (!f. (((FINITE f) /\ (!s. ((s IN f) ==> (bounded s)))) ==> (bounded (UNIONS f))))


simple_path

|- (!g. ((simple_path g) = ((path g) /\ (!x. (!y. (((x IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((y IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) /\ ((g x) = (g y)))) ==> ((x = y) \/ (((x = (vec (NUMERAL _0))) /\ (y = (vec (NUMERAL (BIT1 _0))))) \/ ((x = (vec (NUMERAL (BIT1 _0)))) /\ (y = (vec (NUMERAL _0))))))))))))


Lp_ineqs_proofs.lp_tau

|- (!V. (!E. ((lp_tau (V , E)) = (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((real_le (real_of_num (NUMERAL _0))) (tauVEF (V , (E , f)))))))))


SEPARATING_HYPERPLANE_SETS

|- (!s. (!t. (((convex s) /\ ((convex t) /\ ((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((DISJOINT s) t))))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ ((!x. ((x IN s) ==> ((real_le ((dot a) x)) b))) /\ (!x. ((x IN t) ==> ((real_ge ((dot a) x)) b))))))))))


WF_UREC

|- ((WF <<) ==> (!H. ((!f. (!g. (!x. ((!z. (((<< z) x) ==> ((f z) = (g z)))) ==> (((H f) x) = ((H g) x)))))) ==> (!f. (!g. (((!x. ((f x) = ((H f) x))) /\ (!x. ((g x) = ((H g) x)))) ==> (f = g)))))))


Functional_equation.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)))))))))))


ORTHOGONAL_TRANSFORMATION_ORTHOGONAL_EIGENVECTORS

|- (!f. (!v. (!w. (!a. (!b. (((orthogonal_transformation f) /\ (((f v) = ((% a) v)) /\ (((f w) = ((% b) w)) /\ (~ (a = b))))) ==> ((orthogonal v) w)))))))


CONVEX_CONNECTED

|- (!s. ((convex s) ==> (connected s)))


Lp_ineqs_proofs.face5_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!v. (!w. ((LET (\w'. (LET_END ((LET (\u. (LET_END ((LET (\z. (LET_END (((v , w) IN ((darts_k (NUMERAL (BIT1 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((((f_fan_pair_ext (V , E)) (v , w)) = (w , z)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT1 _0)))) (v , w)) = (z , u)) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT1 (BIT1 _0)))) (v , w)) = (u , w')) /\ (((((POWER (f_fan_pair_ext (V , E))) (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (v , w)) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w , z)) = (z , u)) /\ ((((f_fan_pair_ext (V , E)) (z , u)) = (u , w')) /\ ((((f_fan_pair_ext (V , E)) (u , w')) = (w' , v)) /\ ((((f_fan_pair_ext (V , E)) (w' , v)) = (v , w)) /\ (((inverse (f_fan_pair_ext (V , E))) (v , w)) = (w' , v)))))))))) /\ ((((v , w) IN (dart1_of_fan (V , E))) /\ (((w , z) IN (dart1_of_fan (V , E))) /\ (((z , 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 , z) INSERT ((z , u) INSERT ((u , w') INSERT ((w' , v) INSERT EMPTY)))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) u) w'))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) w') v))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))


HOMOTOPIC_PATHS_RCANCEL

|- (!p. (!q. (!r. (!s. (((((homotopic_paths s) ((++ p) r)) ((++ q) r)) /\ (((pathfinish p) = (pathstart r)) /\ ((pathfinish q) = (pathstart r)))) ==> (((homotopic_paths s) p) q))))))


SUBSET_HALFSPACES_IMP_COLLINEAR

|- ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%388. (?x. (((SETSPEC GEN%PVAR%388) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%389. (?x. (((SETSPEC GEN%PVAR%389) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%390. (?x. (((SETSPEC GEN%PVAR%390) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%391. (?x. (((SETSPEC GEN%PVAR%391) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%392. (?x. (((SETSPEC GEN%PVAR%392) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%393. (?x. (((SETSPEC GEN%PVAR%393) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%394. (?x. (((SETSPEC GEN%PVAR%394) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%395. (?x. (((SETSPEC GEN%PVAR%395) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%396. (?x. (((SETSPEC GEN%PVAR%396) ((real_lt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%397. (?x. (((SETSPEC GEN%PVAR%397) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%398. (?x. (((SETSPEC GEN%PVAR%398) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%399. (?x. (((SETSPEC GEN%PVAR%399) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%400. (?x. (((SETSPEC GEN%PVAR%400) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%401. (?x. (((SETSPEC GEN%PVAR%401) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%402. (?x. (((SETSPEC GEN%PVAR%402) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%403. (?x. (((SETSPEC GEN%PVAR%403) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%404. (?x. (((SETSPEC GEN%PVAR%404) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%405. (?x. (((SETSPEC GEN%PVAR%405) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%406. (?x. (((SETSPEC GEN%PVAR%406) ((real_le ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%407. (?x. (((SETSPEC GEN%PVAR%407) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%408. (?x. (((SETSPEC GEN%PVAR%408) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\GEN%PVAR%409. (?x. (((SETSPEC GEN%PVAR%409) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%410. (?x. (((SETSPEC GEN%PVAR%410) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\GEN%PVAR%411. (?x. (((SETSPEC GEN%PVAR%411) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%412. (?x. (((SETSPEC GEN%PVAR%412) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\GEN%PVAR%413. (?x. (((SETSPEC GEN%PVAR%413) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%414. (?x. (((SETSPEC GEN%PVAR%414) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\GEN%PVAR%415. (?x. (((SETSPEC GEN%PVAR%415) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%416. (?x. (((SETSPEC GEN%PVAR%416) (((dot a) x) = c)) x)))) SUBSET (GSPEC (\GEN%PVAR%417. (?x. (((SETSPEC GEN%PVAR%417) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%418. (?x. (((SETSPEC GEN%PVAR%418) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%419. (?x. (((SETSPEC GEN%PVAR%419) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%420. (?x. (((SETSPEC GEN%PVAR%420) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%421. (?x. (((SETSPEC GEN%PVAR%421) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%422. (?x. (((SETSPEC GEN%PVAR%422) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%423. (?x. (((SETSPEC GEN%PVAR%423) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%424. (?x. (((SETSPEC GEN%PVAR%424) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%425. (?x. (((SETSPEC GEN%PVAR%425) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%426. (?x. (((SETSPEC GEN%PVAR%426) ((real_ge ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%427. (?x. (((SETSPEC GEN%PVAR%427) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%428. (?x. (((SETSPEC GEN%PVAR%428) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%429. (?x. (((SETSPEC GEN%PVAR%429) ((real_lt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%430. (?x. (((SETSPEC GEN%PVAR%430) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%431. (?x. (((SETSPEC GEN%PVAR%431) ((real_le ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%432. (?x. (((SETSPEC GEN%PVAR%432) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%433. (?x. (((SETSPEC GEN%PVAR%433) (((dot b) x) = d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ ((!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%434. (?x. (((SETSPEC GEN%PVAR%434) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%435. (?x. (((SETSPEC GEN%PVAR%435) ((real_ge ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY))))))))) /\ (!a. (!b. (!c. (!d. (((GSPEC (\GEN%PVAR%436. (?x. (((SETSPEC GEN%PVAR%436) ((real_gt ((dot a) x)) c)) x)))) SUBSET (GSPEC (\GEN%PVAR%437. (?x. (((SETSPEC GEN%PVAR%437) ((real_gt ((dot b) x)) d)) x))))) ==> (collinear ((vec (NUMERAL _0)) INSERT (a INSERT (b INSERT EMPTY)))))))))))))))))))))))))))))))))


NSUM_SUBSET

|- (!u. (!v. (!f. (((FINITE u) /\ ((FINITE v) /\ (!x. ((x IN (u DIFF v)) ==> ((f x) = (NUMERAL _0)))))) ==> (((nsum u) f) <= ((nsum v) f))))))


HAS_REAL_INTEGRAL_EMPTY

|- (!f. (((has_real_integral f) (real_of_num (NUMERAL _0))) EMPTY))


LOCALLY_PATH_CONNECTED_IMP_LOCALLY_CONNECTED

|- (!s. (((locally path_connected) s) ==> ((locally connected) s)))


DOT_LMUL_MATRIX

|- (!A. (!x. (!y. (((dot ((vector_matrix_mul x) A)) y) = ((dot x) ((matrix_vector_mul A) y))))))


RESTRICTION_FIXPOINT

|- (!s. (!f. ((((RESTRICTION s) f) = f) = (f IN (EXTENSIONAL s)))))


COMMUTING_WITH_DIAGONAL_MATRIX

|- (!A. (!D. ((diagonal_matrix D) ==> ((((matrix_mul A) D) = ((matrix_mul D) A)) = (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ (j <= (dimindex UNIV))))) ==> ((((A $ i) $ j) = (real_of_num (NUMERAL _0))) \/ (((D $ i) $ i) = ((D $ j) $ j))))))))))


REAL_LE_RNEG

|- (!x. (!y. (((real_le x) (real_neg y)) = ((real_le ((real_add x) y)) (real_of_num (NUMERAL _0))))))


OPEN_INTERIOR

|- (!s. (open (interior s)))


INT_POS

|- (!n. ((int_le (int_of_num (NUMERAL _0))) (int_of_num n)))


Fan.fan2

|- (!E. (!x. (!V. ((fan2 (x , (V , E))) = (~ (x IN V))))))


homeomorphic

|- (!s. (!t. (((homeomorphic s) t) = (?f. (?g. ((homeomorphism (s , t)) (f , g)))))))


BALL_MIN_INTER

|- (!a. (!r. (!s. ((ball (a , ((real_min r) s))) = ((ball (a , r)) INTER (ball (a , s)))))))


LE_CASES

|- (!m. (!n. ((m <= n) \/ (n <= m))))


Fan.e_fan

|- (!x. (!V. (!E. ((((e_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , (v , (((((sigma_fan x) V) E) w) v)))))))))))))))


Hypermap.edge

|- (!H. (!x. (((edge H) x) = ((orbit_map (edge_map H)) x))))


ARC_JOIN_EQ

|- (!g1. (!g2. (((pathfinish g1) = (pathstart g2)) ==> ((arc ((++ g1) g2)) = ((arc g1) /\ ((arc g2) /\ (((path_image g1) INTER (path_image g2)) SUBSET ((pathstart g2) INSERT EMPTY))))))))


HAS_REAL_VECTOR_DERIVATIVE_AT

|- ((((has_real_derivative f) f') (atreal x)) = (((has_vector_derivative (lift o (f o drop))) (lift f')) (at (lift x))))


IN_IMAGE_LIFT_DROP

|- ((!x. (!s. ((x IN ((IMAGE lift) s)) = ((drop x) IN s)))) /\ (!x. (!s. ((x IN ((IMAGE drop) s)) = ((lift x) IN s)))))


CONTINUOUS_ON_MUL

|- (!s. (!c. (!f. ((((continuous_on (lift o c)) s) /\ ((continuous_on f) s)) ==> ((continuous_on (\x. ((% (c x)) (f x)))) s)))))


HAS_BOUNDED_VARIATION_ON_ADD

|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ ((has_bounded_variation_on g) s)) ==> ((has_bounded_variation_on (\x. ((vector_add (f x)) (g x)))) s)))))


Seq.nth_drop

|- (!x0. (!n0. (!s. (!i. ((((nth x0) ((dropl n0) s)) i) = (((nth x0) s) (n0 + i)))))))


PASTECART_CMUL

|- (!x1. (!y1. (!c. (((pastecart ((% c) x1)) ((% c) y1)) = ((% c) ((pastecart x1) y1))))))


TC_INC

|- (!R. (!x. (!y. (((R x) y) ==> (((TC R) x) y)))))


Packing3.HULL_INTER_SUBSET_INTER_HULL

|- (!P. (!s. (!t. (((hull P) (s INTER t)) SUBSET (((hull P) s) INTER ((hull P) t))))))


AFFINE_HULL_CONVEX_INTER_OPEN_IN

|- (!s. (!t. (((convex s) /\ (((open_in ((subtopology euclidean) ((hull affine) s))) t) /\ (~ ((s INTER t) = EMPTY)))) ==> (((hull affine) (s INTER t)) = ((hull affine) s)))))


Nonlin_def.ups_126

|- (ups_126 = (((((((compose6 (promote3_to_6 ups_x)) proj_x1) proj_x2) proj_x6) dummy6) dummy6) dummy6))


Merge_ineq.delta_min_corner_4_784

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x1) /\ (((real_le x1) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x2) /\ (((real_le x2) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x3) /\ (((real_le x3) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x4) /\ (((real_le x4) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x5) /\ (((real_le x5) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) /\ (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) x6) /\ ((real_le x6) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))))) ==> ((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)))))))))


iterate

|- (!f. (!s. (!op. ((((iterate op) s) f) = (((COND (FINITE (((support op) f) s))) (((ITSET (\x. (\a. ((op (f x)) a)))) (((support op) f) s)) (neutral op))) (neutral op))))))


OPEN_IN_REFL

|- (!s. ((open_in ((subtopology euclidean) s)) s))


REAL_LT_ADDL

|- (!x. (!y. (((real_lt y) ((real_add x) y)) = ((real_lt (real_of_num (NUMERAL _0))) x))))


Nonlinear_lemma.vol3f_x_lfun_alt

|- (!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))) x6))) ==> (((((((vol3f_x_lfun x1) x2) x3) x4) x5) x6) = ((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_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6))) ((real_add ((((((dih3_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((dih4_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_sub ((((((dih5_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((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 ((((((ldih_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((real_add ((((((ldih2_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)) ((((((ldih6_x x1) x2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x6)))))))))))))


Miqmcsn.scs_4M8_13

|- (scs_4M8_13 = ((((mk_unadorned_v39 (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 _0)))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) (((funlist_v39 ((CONS (((NUMERAL _0) , (NUMERAL (BIT1 _0))) , cstab)) ((CONS (((NUMERAL (BIT0 (BIT1 _0))) , (NUMERAL (BIT1 (BIT1 _0)))) , cstab)) ((CONS (((NUMERAL _0) , (NUMERAL (BIT0 (BIT1 _0)))) , (real_of_num (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))) ((CONS (((NUMERAL (BIT1 _0)) , (NUMERAL (BIT1 (BIT1 _0)))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))) NIL))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))


NORM_VSUM_PYTHAGOREAN

|- (!k. (!u. (((FINITE k) /\ ((pairwise (\i. (\j. ((orthogonal (u i)) (u j))))) k)) ==> (((real_pow (vector_norm ((vsum k) u))) (NUMERAL (BIT0 (BIT1 _0)))) = ((sum k) (\i. ((real_pow (vector_norm (u i))) (NUMERAL (BIT0 (BIT1 _0))))))))))


Merge_ineq.nonfunctional_mu6_x_y

|- (!x1. (!x2. (!x3. (!a. (!b. (!c. (((((((mu6_x x1) x2) x3) a) b) c) = (((mu_y (sqrt x1)) (sqrt x2)) (sqrt x3)))))))))


VSUM_CLAUSES_NUMSEG

|- ((!m. (((vsum (m .. (NUMERAL _0))) f) = (((COND (m = (NUMERAL _0))) (f (NUMERAL _0))) (vec (NUMERAL _0))))) /\ (!m. (!n. (((vsum (m .. (SUC n))) f) = (((COND (m <= (SUC n))) ((vector_add ((vsum (m .. n)) f)) (f (SUC n)))) ((vsum (m .. n)) f))))))


HOMOTOPIC_WITH_IMP_CONTINUOUS

|- (!P. (!X. (!Y. (!f. (!g. (((((homotopic_with P) (X , Y)) f) g) ==> (((continuous_on f) X) /\ ((continuous_on g) X))))))))


SIMPLE_PATH_TRANSLATION_EQ

|- (!a. (!g. ((simple_path ((\x. ((vector_add a) x)) o g)) = (simple_path g))))


Constants_approx.acs3_interval

|- ((interval_arith (acs ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))


COMPLEX_NORM_MUL

|- (!w. (!z. ((vector_norm ((complex_mul w) z)) = ((real_mul (vector_norm w)) (vector_norm z)))))


Ssrbool.andbAC

|- (!b. (!c. (!p. (((b /\ c) /\ p) = ((b /\ p) /\ c)))))


TAN_PI4

|- ((tan ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0))))))) = (real_of_num (NUMERAL (BIT1 _0))))


COUNTABLE_ANR_PATH_COMPONENTS

|- (!s. (!t. ((ANR s) ==> (COUNTABLE (GSPEC (\GEN%PVAR%2677. (?x. (((SETSPEC GEN%PVAR%2677) (x IN t)) ((path_component s) x)))))))))


CROSS_LMUL

|- (!c. (!x. (!y. (((cross ((% c) x)) y) = ((% c) ((cross x) y))))))


EXP_ONE

|- (!n. (((NUMERAL (BIT1 _0)) EXP n) = (NUMERAL (BIT1 _0))))


EMPTY_INTERIOR_FINITE

|- (!s. ((FINITE s) ==> ((interior s) = EMPTY)))


ENDS_IN_INTERVAL

|- ((!a. (!b. ((a IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. ((b IN (closed_interval ((CONS (a , b)) NIL))) = (~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY))))) /\ ((!a. (!b. (~ (a IN (open_interval (a , b)))))) /\ (!a. (!b. (~ (b IN (open_interval (a , b)))))))))


Trigonometry.KGLLRQT2

|- (!x. ((sin (real_neg x)) = (real_neg (sin x))))


sphere

|- (!x. (!e. ((sphere (x , e)) = (GSPEC (\GEN%PVAR%555. (?y. (((SETSPEC GEN%PVAR%555) ((distance (x , y)) = e)) y)))))))


CONNECTED_CONNECTED_IMP_CLOSED_POINTIMAGES

|- (!f. (!s. (((!c. (((c SUBSET s) /\ (connected c)) ==> (connected ((IMAGE f) c)))) /\ (!y. (connected (GSPEC (\GEN%PVAR%1938. (?x. (((SETSPEC GEN%PVAR%1938) ((x IN s) /\ ((f x) = y))) x))))))) ==> (!y. ((closed_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%1939. (?x. (((SETSPEC GEN%PVAR%1939) ((x IN s) /\ ((f x) = y))) x)))))))))


HAS_BOUNDED_VARIATION_ON_CONST

|- (!s. (!c. ((has_bounded_variation_on (\x. c)) s)))


OPEN_UNIV

|- (open UNIV)


COLLINEAR_TRANSLATION_EQ

|- (!a. (!s. ((collinear ((IMAGE (\x. ((vector_add a) x))) s)) = (collinear s))))


ORTHOGONAL_MATRIX_TRANSFORMATION

|- (!A. ((orthogonal_matrix A) = (orthogonal_transformation (\x. ((matrix_vector_mul A) x)))))


retract_of

|- (!s. (!t. (((retract_of t) s) = (?r. ((retraction (s , t)) r)))))


PRODUCT_CLAUSES

|- ((!f. (((product EMPTY) f) = (real_of_num (NUMERAL (BIT1 _0))))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((product (x INSERT s)) f) = (((COND (x IN s)) ((product s) f)) ((real_mul (f x)) ((product s) f)))))))))


Nonlin_def.sqrt_x6

|- (((((((sqrt_x6 x1) x2) x3) x4) x5) x6) = (sqrt x6))


Tame_defs2.mapAt

|- (((((mapAt NIL) f) xs) = xs) /\ ((((mapAt ((CONS n) ns)) f) xs) = (((COND (n < (LENGTH xs))) (((mapAt ns) f) (((list_update xs) n) (f ((EL n) xs))))) (((mapAt ns) f) xs))))


Packing3.EL_TABLE

|- (!f. (!n. (!i. ((i < n) ==> (((EL i) ((TABLE f) n)) = (f i))))))


COMPLEX_DIFFERENTIABLE_AT_CSIN

|- (!z. ((complex_differentiable csin) (at z)))


CONVEX_IMP_BOUNDED_ON_INTERVAL

|- (!f. (!a. (!b. (((convex_on f) (closed_interval ((CONS (a , b)) NIL))) ==> (?B. (((real_lt (real_of_num (NUMERAL _0))) B) /\ (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((real_le (real_abs (f x))) B)))))))))


SUM_CLAUSES

|- ((!f. (((sum EMPTY) f) = (real_of_num (NUMERAL _0)))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((sum (x INSERT s)) f) = (((COND (x IN s)) ((sum s) f)) ((real_add (f x)) ((sum s) f)))))))))


CONVEX_CONVEX_HULL

|- (!s. (convex ((hull convex) s)))


REAL_INV_NEG

|- (!x. ((real_inv (real_neg x)) = (real_neg (real_inv x))))


EVEN_MULT

|- (!m. (!n. ((EVEN (m * n)) = ((EVEN m) \/ (EVEN n)))))


COMPLEX_POW_2

|- (!x. (((complex_pow x) (NUMERAL (BIT0 (BIT1 _0)))) = ((complex_mul x) x)))


CLOSURE_IMAGE_CLOSURE

|- (!f. (!s. (((continuous_on f) (closure s)) ==> ((closure ((IMAGE f) (closure s))) = (closure ((IMAGE f) s))))))


Conforming.SIGMA_FAN_OF_FANADD1

|- (!x. (!V. (!E. (!E1. (!v. (!w. (((FAN (x , (V , E))) /\ ((FAN (x , (V , E1))) /\ ((!v. ((v IN V) ==> ((CARD (((set_of_edge v) V) E)) > (NUMERAL (BIT1 _0))))) /\ ((~ ((v INSERT (w INSERT EMPTY)) IN E)) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1))))) ==> (!v1. (!w1. ((((v1 INSERT (w1 INSERT EMPTY)) IN E) /\ (~ (v1 IN (v INSERT (w INSERT EMPTY))))) ==> ((((((sigma_fan x) V) E1) v1) w1) = (((((sigma_fan x) V) E) v1) w1))))))))))))


Lp_ineqs_def.ln_fan

|- (!v. ((ln_fan v) = (lmfun ((real_div (yn_fan v)) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))


Local_lemmas1.HAFL_CIRCLE_FORM_LOCAL_FAN

|- ((((local_fan (V , (E , FF))) /\ ((v IN V) /\ ((w IN V) /\ ((~ (v = w)) /\ ((!z. (!t. (((z IN (v INSERT (w INSERT EMPTY))) /\ (t IN (V DIFF (z INSERT EMPTY)))) ==> (~ (collinear ((vec (NUMERAL _0)) INSERT (z INSERT (t INSERT EMPTY)))))))) /\ (!x. ((x IN FF) ==> (((aff_gt ((vec (NUMERAL _0)) INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) SUBSET ((wedge_in_fan_gt x) E))))))))) /\ ((HS = (hypermap (HYP ((vec (NUMERAL _0)) , (V , (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))))))) /\ (fv = ((face HS) (v , ((rho_node1 FF) v)))))) ==> (local_fan (((v_prime V) fv) , (((e_prime (E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY))) fv) , fv))))


HAUSDIST_STILL_INSIDE_INTERIOR

|- (!s. (!t. (!x. (((bounded s) /\ ((bounded t) /\ ((convex s) /\ ((convex t) /\ ((~ (t = EMPTY)) /\ ((real_lt (hausdist (s , t))) (setdist ((x INSERT EMPTY) , (UNIV DIFF s))))))))) ==> (x IN (interior t))))))


CONNECTED_LINEAR_IMAGE

|- (!f. (!s. (((connected s) /\ (linear f)) ==> (connected ((IMAGE f) s)))))


HAS_REAL_DERIVATIVE_ATREAL_WITHIN

|- (!f. (!f'. (!x. (!s. ((((has_real_derivative f) f') (atreal x)) ==> (((has_real_derivative f) f') ((within (atreal x)) s)))))))


SIN_COS_INJ

|- (!x. (!y. ((((sin x) = (sin y)) /\ (((cos x) = (cos y)) /\ ((real_lt (real_abs ((real_sub x) y))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)))) ==> (x = y))))


Miqmcsn.SCS_3T7_OPP_IS_SCS

|- (is_scs_v39 (scs_opp_v39 scs_3T7))


MEASURABLE_ON_MAX

|- (!f. (!g. (!s. ((((measurable_on f) s) /\ ((measurable_on g) s)) ==> ((measurable_on (\x. (lambdai. ((real_max ((f x) $ i)) ((g x) $ i))))) s)))))


REAL_SUB_LNEG

|- (!x. (!y. (((real_sub (real_neg x)) y) = (real_neg ((real_add x) y)))))


HOMOTOPY_EQUIVALENT_INJECTIVE_LINEAR_IMAGE_LEFT_EQ

|- (!f. (!s. (!t. (((linear f) /\ (!x. (!y. (((f x) = (f y)) ==> (x = y))))) ==> (((homotopy_equivalent ((IMAGE f) s)) t) = ((homotopy_equivalent s) t))))))


Hypermap_iso.iso_components

|- (!G. (!f. (!H. (((hyp_iso f) (H , G)) ==> (!d. ((d IN (dart H)) ==> ((((node G) (f d)) = ((IMAGE f) ((node H) d))) /\ ((((face G) (f d)) = ((IMAGE f) ((face H) d))) /\ (((edge G) (f d)) = ((IMAGE f) ((edge H) d)))))))))))


INT_GE

|- (!x. (!y. (((int_ge x) y) = ((int_le y) x))))


CONTINUOUS_ON_SUB

|- (!f. (!g. (!s. ((((continuous_on f) s) /\ ((continuous_on g) s)) ==> ((continuous_on (\x. ((vector_sub (f x)) (g x)))) s)))))


LIMPT_OF_OPEN_IN

|- (!s. (!t. (!x. ((((open_in ((subtopology euclidean) s)) t) /\ (((limit_point_of x) s) /\ (x IN t))) ==> ((limit_point_of x) t)))))


BOUNDED_PATH_IMAGE

|- (!g. ((path g) ==> (bounded (path_image g))))


ABSOLUTELY_INTEGRABLE_SUB

|- (!f. (!g. (!s. ((((absolutely_integrable_on f) s) /\ ((absolutely_integrable_on g) s)) ==> ((absolutely_integrable_on (\x. ((vector_sub (f x)) (g x)))) s)))))


REAL_FLOOR_NEG

|- (!x. ((floor (real_neg x)) = (((COND (integer x)) (real_neg x)) (real_neg ((real_add (floor x)) (real_of_num (NUMERAL (BIT1 _0))))))))


CONVEX_ON_DIRECTIONAL_DERIVATIVE_FUNCTIONS

|- (!f. (!s. ((((convex_on f) s) /\ ((is_interval s) /\ (open s))) ==> (?l. (?r. ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (l x)) ((within (at x)) (GSPEC (\GEN%PVAR%3307. (?t. (((SETSPEC GEN%PVAR%3307) ((real_le (drop t)) (drop x))) t)))))))) /\ ((!x. ((x IN s) ==> (((has_vector_derivative (lift o f)) (r x)) ((within (at x)) (GSPEC (\GEN%PVAR%3308. (?t. (((SETSPEC GEN%PVAR%3308) ((real_le (drop x)) (drop t))) t)))))))) /\ ((!x. ((x IN s) ==> ((real_le (drop (l x))) (drop (r x))))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (drop (r x))) (drop (l y))))))))))))))


VSUM_IMAGE

|- (!f. (!g. (!s. (((FINITE s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y))))) ==> (((vsum ((IMAGE f) s)) g) = ((vsum s) (g o f)))))))


SWAP_IDEMPOTENT

|- (!a. (!b. (((swap (a , b)) o (swap (a , b))) = I)))


convex_on

|- (!s. (!f. (((convex_on f) s) = (!x. (!y. (!u. (!v. (((x IN s) /\ ((y IN s) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ (((real_le (real_of_num (NUMERAL _0))) v) /\ (((real_add u) v) = (real_of_num (NUMERAL (BIT1 _0)))))))) ==> ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_add ((real_mul u) (f x))) ((real_mul v) (f y))))))))))))


CONIC_LINEAR_IMAGE

|- (!f. (!s. (((conic s) /\ (linear f)) ==> (conic ((IMAGE f) s)))))


SEPARATING_HYPERPLANE_CLOSED_0

|- (!s. (((convex s) /\ ((closed s) /\ (~ ((vec (NUMERAL _0)) IN s)))) ==> (?a. (?b. ((~ (a = (vec (NUMERAL _0)))) /\ (((real_lt (real_of_num (NUMERAL _0))) b) /\ (!x. ((x IN s) ==> ((real_gt ((dot a) x)) b)))))))))


Lp_main_estimate.diag_not_in_dart

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!a. (!b. (!f. (((f IN (face_set (hypermap_of_fan (V , E)))) /\ ((a IN ((v_prime V) f)) /\ ((b IN ((v_prime V) f)) /\ ((~ ((a , b) IN f)) /\ (~ ((b , a) IN f)))))) ==> (~ ((a , b) IN (dart_of_fan (V , E))))))))))))


Tame_defs.tame_2

|- (!H. ((tame_2 H) = ((connected_hypermap H) /\ (simple_hypermap H))))


LENGTH

|- (((LENGTH NIL) = (NUMERAL _0)) /\ (!h. (!t. ((LENGTH ((CONS h) t)) = (SUC (LENGTH t))))))


VECTOR_4

|- ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 _0))) = w) /\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT1 _0)))) = x) /\ ((((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT1 (BIT1 _0)))) = y) /\ (((vector ((CONS w) ((CONS x) ((CONS y) ((CONS z) NIL))))) $ (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) = z))))


REAL_SERIES_RMUL

|- (!x. (!x0. (!c. (!s. ((((real_sums x) x0) s) ==> (((real_sums (\n. ((real_mul (x n)) c))) ((real_mul x0) c)) s))))))


CANTOR_BENDIXSON_GEN

|- (!s. (?t. (?u. (((closed_in ((subtopology euclidean) s)) t) /\ ((!x. ((x IN t) ==> ((condensation_point_of x) t))) /\ ((COUNTABLE u) /\ (s = (t UNION u))))))))


SIN_NPI

|- (!n. ((sin ((real_mul (real_of_num n)) pi)) = (real_of_num (NUMERAL _0))))


FLOOR_FRAC

|- (!x. ((integer (floor x)) /\ (((real_le (real_of_num (NUMERAL _0))) (frac x)) /\ (((real_lt (frac x)) (real_of_num (NUMERAL (BIT1 _0)))) /\ (x = ((real_add (floor x)) (frac x)))))))


covering_space

|- (!s. (!c. (!p. (((covering_space (c , p)) s) = (((continuous_on p) c) /\ ((((IMAGE p) c) = s) /\ (!x. ((x IN s) ==> (?t. ((x IN t) /\ (((open_in ((subtopology euclidean) s)) t) /\ (?v. (((UNIONS v) = (GSPEC (\GEN%PVAR%2050. (?x. (((SETSPEC GEN%PVAR%2050) ((x IN c) /\ ((p x) IN t))) x))))) /\ ((!u. ((u IN v) ==> ((open_in ((subtopology euclidean) c)) u))) /\ (((pairwise DISJOINT) v) /\ (!u. ((u IN v) ==> (?q. ((homeomorphism (u , t)) (p , q))))))))))))))))))))


TAN_ATN

|- (!x. ((((real_lt (real_neg ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) x) /\ ((real_lt x) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ==> ((atn (tan x)) = x)))


Hypermap_iso.hyp_iso_dart

|- (!H. (!f. (!G. (((hyp_iso f) (H , G)) ==> (!x. ((x IN (dart H)) ==> ((f x) IN (dart G))))))))


CARD_IMAGE_INJ

|- (!f. (!s. (((!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))) /\ (FINITE s)) ==> ((CARD ((IMAGE f) s)) = (CARD s)))))


integer

|- (!x. ((integer x) = (?n. ((real_abs x) = (real_of_num n)))))


Miqmcsn.SCS_4M8_13_BASIC

|- (scs_basic_v39 scs_4M8_13)


SEPARATING_HYPERPLANE_SET_POINT_INAFF

|- (!s. (!z. (((convex s) /\ ((~ (s = EMPTY)) /\ (~ (z IN s)))) ==> (?a. (?b. ((((vector_add z) a) IN ((hull affine) (z INSERT s))) /\ ((~ (a = (vec (NUMERAL _0)))) /\ (((real_le ((dot a) z)) b) /\ (!x. ((x IN s) ==> ((real_ge ((dot a) x)) b)))))))))))


Constants_approx.INTERVAL_SUB

|- ((((interval_arith x) (a , b)) /\ ((interval_arith y) (c , d))) ==> ((interval_arith ((real_sub x) y)) (((real_sub a) d) , ((real_sub b) c))))


Nonlin_def.vol3f_sqrt2_lmplus

|- (!y3. (!y4. (!y5. (!y1. (!y2. (!y6. (((((((vol3f_sqrt2_lmplus y1) y2) y3) y4) y5) y6) = ((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) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_add ((((((dih3_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_add ((((((dih4_y y1) y2) sqrt2) sqrt2) sqrt2) y6)) ((real_sub ((((((dih5_y y1) y2) sqrt2) sqrt2) sqrt2) 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 (lfun ((real_div y2) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih2_y y1) y2) sqrt2) sqrt2) sqrt2) y6))) ((real_mul (lfun ((real_div y6) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) ((((((dih6_y y1) y2) sqrt2) sqrt2) sqrt2) y6))))))))))))


SUM_EQ

|- (!f. (!g. (!s. ((!x. ((x IN s) ==> ((f x) = (g x)))) ==> (((sum s) f) = ((sum s) g))))))


Hypermap_iso.dart_of_fan_eq

|- (!V. (!E. ((dart_of_fan (V , E)) = ((dart1_of_fan (V , E)) UNION (GSPEC (\GEN%PVAR%7628. (?v. (((SETSPEC GEN%PVAR%7628) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v)))))))))


REAL_ABS_INV

|- (!x. ((real_abs (real_inv x)) = (real_inv (real_abs x))))


RC_CASES

|- (!R. (!a0. (!a1. ((((RC R) a0) a1) = (((R a0) a1) \/ (a1 = a0))))))


AFF_DIM_TRANSLATION_EQ

|- (!a. (!s. ((aff_dim ((IMAGE (\x. ((vector_add a) x))) s)) = (aff_dim s))))


SUM_RMUL

|- (!f. (!c. (!s. (((sum s) (\x. ((real_mul (f x)) c))) = ((real_mul ((sum s) f)) c)))))


Pack2.MEASURABLE_VORONOI_CLOSED

|- (!s. (!v. ((saturated s) ==> (measurable ((voronoi_closed s) v)))))


continuous_atreal

|- (((continuous f) (atreal x)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x'. (((real_lt (real_abs ((real_sub x') x))) d) ==> ((real_lt (distance ((f x') , (f x)))) e))))))))


CLOSEST_POINT_IN_SET

|- (!s. (!a. (((closed s) /\ (~ (s = EMPTY))) ==> (((closest_point s) a) IN s))))


FINITE_POWERSET

|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\GEN%PVAR%109. (?t. (((SETSPEC GEN%PVAR%109) (t SUBSET s)) t)))))))


Collect_geom2.COLLINEAR3_IMP_COPLANAR

|- (!v1. (!v2. (!v3. (!v. ((collinear (v1 INSERT (v2 INSERT (v3 INSERT EMPTY)))) ==> (coplanar_alt (v1 INSERT (v2 INSERT (v3 INSERT (v INSERT EMPTY))))))))))


CONTINUOUS_TRANSFORM_WITHIN_SET_IMP

|- (!f. (!a. (!s. (!t. ((((eventually (\x. ((x IN t) ==> (x IN s)))) (at a)) /\ ((continuous f) ((within (at a)) s))) ==> ((continuous f) ((within (at a)) t)))))))


REAL_LE_DIV

|- (!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_div x) y)))))


IM_CLOG_EQ_PI

|- (!z. ((~ (z = (Cx (real_of_num (NUMERAL _0))))) ==> (((Im (clog z)) = pi) = (((real_lt (Re z)) (real_of_num (NUMERAL _0))) /\ ((Im z) = (real_of_num (NUMERAL _0)))))))


Float.POW_MINUS1

|- (!n. (((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) ((NUMERAL (BIT0 (BIT1 _0))) * n)) = (real_of_num (NUMERAL (BIT1 _0)))))


DIST_POS_LE

|- (!x. (!y. ((real_le (real_of_num (NUMERAL _0))) (distance (x , y)))))


LEBESGUE_MEASURABLE_UNION

|- (!s. (!t. (((lebesgue_measurable s) /\ (lebesgue_measurable t)) ==> (lebesgue_measurable (s UNION t)))))


rpow

|- (!y. (!x. (((rpow x) y) = (((COND ((real_lt (real_of_num (NUMERAL _0))) x)) (exp ((real_mul y) (log x)))) (((COND (x = (real_of_num (NUMERAL _0)))) (((COND (y = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL _0)))) (((COND (?m. (?n. ((ODD m) /\ ((ODD n) /\ ((real_abs y) = ((real_div (real_of_num m)) (real_of_num n)))))))) (real_neg (exp ((real_mul y) (log (real_neg x)))))) (exp ((real_mul y) (log (real_neg x))))))))))


CONTINUOUS_ON_CNJ

|- (!s. ((continuous_on cnj) s))


REAL_ADD_LINV

|- (!x. (((real_add (real_neg x)) x) = (real_of_num (NUMERAL _0))))


int_lt

|- (!x. (!y. (((int_lt x) y) = ((real_lt (real_of_int x)) (real_of_int y)))))


VSUM_CLAUSES

|- ((!f. (((vsum EMPTY) f) = (vec (NUMERAL _0)))) /\ (!x. (!f. (!s. ((FINITE s) ==> (((vsum (x INSERT s)) f) = (((COND (x IN s)) ((vsum s) f)) ((vector_add (f x)) ((vsum s) f)))))))))


Ssrnat.add1n

|- (!n. (((NUMERAL (BIT1 _0)) + n) = (SUC n)))


Lp_ineqs_proofs.y5_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))))) ((y5_fan (V , E)) d))))))))))


INVERTIBLE_CMUL

|- (!A. (!c. ((invertible ((%% c) A)) = ((~ (c = (real_of_num (NUMERAL _0)))) /\ (invertible A)))))


REAL_SUB_0

|- (!x. (!y. ((((real_sub x) y) = (real_of_num (NUMERAL _0))) = (x = y))))


REAL_EQ_ADD_LCANCEL

|- (!x. (!y. (!z. ((((real_add x) y) = ((real_add x) z)) = (y = z)))))


REAL_POW_DIV

|- (!x. (!y. (!n. (((real_pow ((real_div x) y)) n) = ((real_div ((real_pow x) n)) ((real_pow y) n))))))


INTEGRABLE_RESTRICT_UNIV

|- (!f. (!s. (((integrable_on (\x. (((COND (x IN s)) (f x)) (vec (NUMERAL _0))))) UNIV) = ((integrable_on f) s))))


CSIN_EQ_1

|- (!z. (((csin z) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) = (?n. ((integer n) /\ (z = (Cx ((real_mul ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) n)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) pi)))))))


Ssrbool.andbb

|- (!b. ((b /\ b) = b))


DET_NEG

|- (!A. ((det (matrix_neg A)) = ((real_mul ((real_pow (real_neg (real_of_num (NUMERAL (BIT1 _0))))) (dimindex UNIV))) (det A))))


LINEAR_VSUM

|- (!f. (!g. (!s. (((linear f) /\ (FINITE s)) ==> ((f ((vsum s) g)) = ((vsum s) (f o g)))))))


BOUNDED_HAS_SUP

|- (!s. (((bounded ((IMAGE lift) s)) /\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le x) (sup s)))) /\ (!b. ((!x. ((x IN s) ==> ((real_le x) b))) ==> ((real_le (sup s)) b))))))


tagged_partial_division_of

|- (!i. (!s. (((tagged_partial_division_of s) i) = ((FINITE s) /\ ((!x. (!k. (((x , k) IN s) ==> ((x IN k) /\ ((k SUBSET i) /\ (?a. (?b. (k = (closed_interval ((CONS (a , b)) NIL)))))))))) /\ (!x1. (!k1. (!x2. (!k2. ((((x1 , k1) IN s) /\ (((x2 , k2) IN s) /\ (~ ((x1 , k1) = (x2 , k2))))) ==> (((interior k1) INTER (interior k2)) = EMPTY)))))))))))


Functional_equation.dih_all_atn

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) ((((((delta_x x1) x2) x3) x4) x5) x6)) /\ (((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6))))))) ==> ((((((((dih_x x1) x2) x3) x4) x5) x6) = ((((((dihatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih2_x x1) x2) x3) x4) x5) x6) = ((((((dih2atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih3_x x1) x2) x3) x4) x5) x6) = ((((((dih3atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih4_x x1) x2) x3) x4) x5) x6) = ((((((dih4atn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((dih5_x x1) x2) x3) x4) x5) x6) = ((((((dih5atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((dih6_x x1) x2) x3) x4) x5) x6) = ((((((dih6atn_x x1) x2) x3) x4) x5) x6))))))))))))))


CX_INJ

|- (!x. (!y. (((Cx x) = (Cx y)) = (x = y))))


AFFINE_IMP_CONVEX

|- (!s. ((affine s) ==> (convex s)))


REAL_LT_RDIV_EQ

|- (!x. (!y. (!z. (((real_lt (real_of_num (NUMERAL _0))) z) ==> (((real_lt x) ((real_div y) z)) = ((real_lt ((real_mul x) z)) y))))))


REAL_ABS_0

|- ((real_abs (real_of_num (NUMERAL _0))) = (real_of_num (NUMERAL _0)))


Ssrnat.leqNgt

|- (!m. (!n. ((m <= n) = (~ (n < m)))))


CLOSED_IN_CLOSED_INTER

|- (!u. (!s. ((closed s) ==> ((closed_in ((subtopology euclidean) u)) (u INTER s)))))


INVERTIBLE_I

|- (invertible (mat (NUMERAL (BIT1 _0))))


Constants_approx.pi_interval

|- ((interval_arith pi) (((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0)))))))))))))))))))))))))))))))) , ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 _0)))))))))))))))))))))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))))))))))))))))))))))


real_polynomial_function_INDUCT

|- (!real_polynomial_function'. (((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (real_polynomial_function' (\x. (x $ i))))) /\ ((!c. (real_polynomial_function' (\x. c))) /\ ((!f. (!g. (((real_polynomial_function' f) /\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\x. ((real_add (f x)) (g x))))))) /\ (!f. (!g. (((real_polynomial_function' f) /\ (real_polynomial_function' g)) ==> (real_polynomial_function' (\x. ((real_mul (f x)) (g x)))))))))) ==> (!a. ((real_polynomial_function a) ==> (real_polynomial_function' a)))))


has_complex_derivative

|- (!f. (!f'. (!net. ((((has_complex_derivative f) f') net) = (((has_derivative f) (\x. ((complex_mul f') x))) net)))))


REAL_MEASURE_UNIONS_LE_IMAGE

|- (!f. (!s. (((FINITE f) /\ (!a. ((a IN f) ==> (real_measurable (s a))))) ==> ((real_le (real_measure (UNIONS ((IMAGE s) f)))) ((sum f) (\a. (real_measure (s a))))))))


NSUM_UNION_RZERO

|- (!f. (!u. (!v. (((FINITE u) /\ (!x. (((x IN v) /\ (~ (x IN u))) ==> ((f x) = (NUMERAL _0))))) ==> (((nsum (u UNION v)) f) = ((nsum u) f))))))


HAS_REAL_INTEGRAL_NEG

|- (!f. (!k. (!s. ((((has_real_integral f) k) s) ==> (((has_real_integral (\x. (real_neg (f x)))) (real_neg k)) s)))))


SQRT_POS_LT

|- (!x. (((real_lt (real_of_num (NUMERAL _0))) x) ==> ((real_lt (real_of_num (NUMERAL _0))) (sqrt x))))


COUNTABLE_SING

|- (!x. (COUNTABLE (x INSERT EMPTY)))


Fan_defs.n_fan_pair

|- (!V. (!E. (!v. (!w. (((n_fan_pair (V , E)) (v , w)) = (v , (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))


ROTATION_LOWDIM_HORIZONTAL

|- (!s. (((dim s) < (dimindex UNIV)) ==> (?f. ((orthogonal_transformation f) /\ (((det (matrix f)) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((IMAGE f) s) SUBSET (GSPEC (\GEN%PVAR%456. (?z. (((SETSPEC GEN%PVAR%456) ((z $ (dimindex UNIV)) = (real_of_num (NUMERAL _0)))) z))))))))))


matrix_sub

|- (!A. (!B. (((matrix_sub A) B) = (lambdai. (lambdaj. ((real_sub ((A $ i) $ j)) ((B $ i) $ j)))))))


ITERATE_PAIR

|- (!op. ((monoidal op) ==> (!f. (!m. (!n. ((((iterate op) (((NUMERAL (BIT0 (BIT1 _0))) * m) .. (((NUMERAL (BIT0 (BIT1 _0))) * n) + (NUMERAL (BIT1 _0))))) f) = (((iterate op) (m .. n)) (\i. ((op (f ((NUMERAL (BIT0 (BIT1 _0))) * i))) (f (((NUMERAL (BIT0 (BIT1 _0))) * i) + (NUMERAL (BIT1 _0)))))))))))))


DISJOINT_EMPTY

|- (!s. (((DISJOINT EMPTY) s) /\ ((DISJOINT s) EMPTY)))


NEUTRAL_MUL

|- ((neutral *) = (NUMERAL (BIT1 _0)))


CONTINUOUS_LE_ON_CLOSURE

|- (!f. (!s. (!a. ((((continuous_on (lift o f)) (closure s)) /\ (!x. ((x IN s) ==> ((real_le (f x)) a)))) ==> (!x. ((x IN (closure s)) ==> ((real_le (f x)) a)))))))


HAS_COMPLEX_DERIVATIVE_CONST

|- (!c. (!net. (((has_complex_derivative (\x. c)) (Cx (real_of_num (NUMERAL _0)))) net)))


Conforming.NEGLIGIBLE_AFF_3

|- (!x. (!v. (!u. (negligible (aff (x INSERT (v INSERT (u INSERT EMPTY))))))))


le_c

|- (!t. (!s. ((s <=_c t) = (?f. ((!x. ((x IN s) ==> ((f x) IN t))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((f x) = (f y)))) ==> (x = y)))))))))


Seq.has_predC

|- (!a. (!s. (((has (predC a)) s) = (~ ((all a) s)))))


COVERING_SPACE_FINITE_SHEETS

|- (!p. (!c. (!s. (!b. ((((covering_space (c , p)) s) /\ (compact c)) ==> (FINITE (GSPEC (\GEN%PVAR%2084. (?x. (((SETSPEC GEN%PVAR%2084) ((x IN c) /\ ((p x) = b))) x))))))))))


SHIFTPATH_LINEAR_IMAGE

|- (!f. (!t. (!g. ((linear f) ==> (((shiftpath t) (f o g)) = (f o ((shiftpath t) g)))))))


FRONTIER_UNION_SUBSET

|- (!s. (!t. ((frontier (s UNION t)) SUBSET ((frontier s) UNION (frontier t)))))


real_INFINITE

|- (INFINITE UNIV)


BOUNDED_SUMS_IMAGE

|- (!f. (!g. (!t. (((bounded (GSPEC (\GEN%PVAR%588. (?x. (((SETSPEC GEN%PVAR%588) (x IN t)) (f x)))))) /\ (bounded (GSPEC (\GEN%PVAR%589. (?x. (((SETSPEC GEN%PVAR%589) (x IN t)) (g x))))))) ==> (bounded (GSPEC (\GEN%PVAR%590. (?x. (((SETSPEC GEN%PVAR%590) (x IN t)) ((vector_add (f x)) (g x)))))))))))


CONTINUOUS_RIGHT_INVERSE_IMP_QUOTIENT_MAP

|- (!f. (!g. (!s. (!t. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET t) /\ (((continuous_on g) t) /\ ((((IMAGE g) t) SUBSET s) /\ (!y. ((y IN t) ==> ((f (g y)) = y))))))) ==> (!u. ((u SUBSET t) ==> (((open_in ((subtopology euclidean) s)) (GSPEC (\GEN%PVAR%651. (?x. (((SETSPEC GEN%PVAR%651) ((x IN s) /\ ((f x) IN u))) x))))) = ((open_in ((subtopology euclidean) t)) u)))))))))


Dih2k_hypermap.EXISTS_POINT_DART_OF_HYP

|- (((FAN ((vec (NUMERAL _0)) , ((V_SY (vecmats l)) , (E_SY (vecmats l))))) /\ (((NUMERAL (BIT0 (BIT1 _0))) < (dimindex UNIV)) /\ (!i. (!j. ((((NUMERAL (BIT1 _0)) <= i) /\ ((i <= (dimindex UNIV)) /\ (((NUMERAL (BIT1 _0)) <= j) /\ ((j <= (dimindex UNIV)) /\ (((row i) (vecmats l)) = ((row j) (vecmats l))))))) ==> (i = j)))))) ==> (?x. (x IN ((darts_of_hyp (E_SY (vecmats l))) (V_SY (vecmats l))))))


FSIGMA_TRANSLATION

|- (!a. (!s. ((fsigma ((IMAGE (\x. ((vector_add a) x))) s)) = (fsigma s))))


COBOUNDED_UNIQUE_UNBOUNDED_COMPONENTS

|- (!s. (!c. (!c'. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ ((bounded (UNIV DIFF s)) /\ ((c IN (components s)) /\ ((~ (bounded c)) /\ ((c' IN (components s)) /\ (~ (bounded c'))))))) ==> (c' = c)))))


Ssrbool.orbF

|- (!b. ((b \/ F) = b))


real_variation

|- (!s. (!f. (((real_variation s) f) = ((vector_variation ((IMAGE lift) s)) (lift o (f o drop))))))


Fan.d1_fan

|- (!x. (!V. (!E. ((d1_fan (x , (V , E))) = (GSPEC (\GEN%PVAR%5465. (?x'. (?v. (?w. (?w1. (((SETSPEC GEN%PVAR%5465) ((x' = x) /\ (((v INSERT (w INSERT EMPTY)) IN E) /\ (w1 = (((((sigma_fan x) V) E) v) w))))) (x' , (v , (w , w1))))))))))))))


REAL_ABS_NZ

|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) = ((real_lt (real_of_num (NUMERAL _0))) (real_abs x))))


UNIONS_GSPEC

|- ((!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%31. (?x. (((SETSPEC GEN%PVAR%31) (P x)) (f x)))))) = (GSPEC (\GEN%PVAR%32. (?a. (((SETSPEC GEN%PVAR%32) (?x. ((P x) /\ (a IN (f x))))) a))))))) /\ ((!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%33. (?x. (?y. (((SETSPEC GEN%PVAR%33) ((P x) y)) ((f x) y))))))) = (GSPEC (\GEN%PVAR%34. (?a. (((SETSPEC GEN%PVAR%34) (?x. (?y. (((P x) y) /\ (a IN ((f x) y)))))) a))))))) /\ (!P. (!f. ((UNIONS (GSPEC (\GEN%PVAR%35. (?x. (?y. (?z. (((SETSPEC GEN%PVAR%35) (((P x) y) z)) (((f x) y) z)))))))) = (GSPEC (\GEN%PVAR%36. (?a. (((SETSPEC GEN%PVAR%36) (?x. (?y. (?z. ((((P x) y) z) /\ (a IN (((f x) y) z))))))) a)))))))))


COMPLEX_ADD_ASSOC

|- (!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z)))))


SNDCART_PASTECART

|- (!x. (!y. ((sndcart ((pastecart x) y)) = y)))


INTEGRABLE_INTEGRAL

|- (!f. (!i. (((integrable_on f) i) ==> (((has_integral f) ((integral i) f)) i))))


RETRACTION

|- (!s. (!t. (!r. (((retraction (s , t)) r) = ((t SUBSET s) /\ (((continuous_on r) s) /\ ((((IMAGE r) s) = t) /\ (!x. ((x IN t) ==> ((r x) = x))))))))))


HOMOTOPIC_WITH_COMPOSE_CONTINUOUS_LEFT

|- (!p. (!f. (!g. (!h. (!X. (!Y. (!Z. ((((((homotopic_with (\f. (p (h o f)))) (X , Y)) f) g) /\ (((continuous_on h) Y) /\ (((IMAGE h) Y) SUBSET Z))) ==> ((((homotopic_with p) (X , Z)) (h o f)) (h o g))))))))))


REAL_OF_NUM_SUB

|- (!m. (!n. ((m <= n) ==> (((real_sub (real_of_num n)) (real_of_num m)) = (real_of_num (n - m))))))


UNION_SUBSET

|- (!s. (!t. (!u. (((s UNION t) SUBSET u) = ((s SUBSET u) /\ (t SUBSET u))))))


TRANSP_MATRIX_INV

|- (!A. ((transp (matrix_inv A)) = (matrix_inv (transp A))))


REAL_EQ_SUB_RADD

|- (!x. (!y. (!z. ((((real_sub x) y) = z) = (x = ((real_add z) y))))))


BOUNDED_INTERVAL

|- ((!a. (!b. (bounded (closed_interval ((CONS (a , b)) NIL))))) /\ (!a. (!b. (bounded (open_interval (a , b))))))


COVERING_SPACE_LIFT_STRONG

|- (!p. (!c. (!s. (!f. (!u. (!a. (!z. ((((covering_space (c , p)) s) /\ ((a IN c) /\ ((z IN u) /\ ((simply_connected u) /\ (((locally path_connected) u) /\ (((continuous_on f) u) /\ ((((IMAGE f) u) SUBSET s) /\ ((f z) = (p a))))))))) ==> (?g. (((continuous_on g) u) /\ ((((IMAGE g) u) SUBSET c) /\ (((g z) = a) /\ (!y. ((y IN u) ==> ((p (g y)) = (f y))))))))))))))))


MEASURABLE_MEASURE_POS_LT

|- (!s. ((measurable s) ==> (((real_lt (real_of_num (NUMERAL _0))) (measure s)) = (~ (negligible s)))))


FSIGMA_CONTINUOUS_IMAGE

|- (!f. (!s. ((((continuous_on f) s) /\ (fsigma s)) ==> (fsigma ((IMAGE f) s)))))


MEASURABLE_MEASURABLE_INTER_LEBESGUE_MEASURABLE

|- (!s. (!t. (((measurable s) /\ (lebesgue_measurable t)) ==> (measurable (s INTER t)))))


Nonlin_def.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)))))))))))


STARLIKE_CONVEX_TWEAK_BOUNDARY_POINTS

|- (!s. (!t. (((convex s) /\ ((~ (s = EMPTY)) /\ (((relative_interior s) SUBSET t) /\ (t SUBSET (closure s))))) ==> (starlike t))))


Hypermap.lemma_face_subset

|- (!H. (!x. ((x IN (dart H)) ==> (((face H) x) SUBSET (dart H)))))


SURJECTIVE_IMAGE_EQ

|- (!s. (!t. (((!y. ((y IN t) ==> (?x. ((f x) = y)))) /\ (!x. (((f x) IN t) = (x IN s)))) ==> (((IMAGE f) s) = t))))


ARG_ROTATE2D_UNIQUE

|- (!t. (!a. (!z. (((~ (z = (Cx (real_of_num (NUMERAL _0))))) /\ ((Arg ((rotate2d t) z)) = a)) ==> (?n. ((integer n) /\ (t = ((real_add ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((real_mul n) pi))) ((real_sub a) (Arg z))))))))))


HYPERPLANE_EQ_UNIV

|- (!a. (!b. (((GSPEC (\GEN%PVAR%172. (?x. (((SETSPEC GEN%PVAR%172) (((dot a) x) = b)) x)))) = UNIV) = ((a = (vec (NUMERAL _0))) /\ (b = (real_of_num (NUMERAL _0)))))))


BOUNDED_HAS_INF

|- (!s. (((bounded ((IMAGE lift) s)) /\ (~ (s = EMPTY))) ==> ((!x. ((x IN s) ==> ((real_le (inf s)) x))) /\ (!b. ((!x. ((x IN s) ==> ((real_le b) x))) ==> ((real_le b) (inf s)))))))


Nonlinear_lemma.lmfun0

|- (!y. (((real_le ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) y) ==> ((lmfun ((real_div y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) = (real_of_num (NUMERAL _0)))))


SUBSET_REFL

|- (!s. (s SUBSET s))


Collect_geom.pow_g

|- (!x. ((real_le (real_of_num (NUMERAL _0))) ((real_pow x) (NUMERAL (BIT0 (BIT1 _0))))))


COMPLEX_NORM_DIV

|- (!w. (!z. ((vector_norm ((complex_div w) z)) = ((real_div (vector_norm w)) (vector_norm z)))))


SUM_DIFFS

|- (!m. (!n. (((sum (m .. n)) (\k. ((real_sub (f k)) (f (k + (NUMERAL (BIT1 _0))))))) = (((COND (m <= n)) ((real_sub (f m)) (f (n + (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0))))))


CARD_EQ_SIMPLE_PATH_IMAGE

|- (!g. ((simple_path g) ==> ((path_image g) =_c UNIV)))


INTEGRABLE_SUBINTERVALS_IMP_MEASURABLE

|- (!f. ((!a. (!b. ((integrable_on f) (closed_interval ((CONS (a , b)) NIL))))) ==> ((measurable_on f) UNIV)))


CLOSED_IN_SUBTOPOLOGY_EMPTY

|- (!top. (!s. (((closed_in ((subtopology top) EMPTY)) s) = (s = EMPTY))))


ITERATE_INJECTION

|- (!op. ((monoidal op) ==> (!f. (!p. (!s. (((FINITE s) /\ ((!x. ((x IN s) ==> ((p x) IN s))) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((p x) = (p y)))) ==> (x = y)))))) ==> ((((iterate op) s) (f o p)) = (((iterate op) s) f))))))))


PERMUTES_INSERT_LEMMA

|- (!p. (!a. (!s. (((permutes p) (a INSERT s)) ==> ((permutes ((swap (a , (p a))) o p)) s)))))


REAL_LTE_TOTAL

|- (!x. (!y. (((real_lt x) y) \/ ((real_le y) x))))


ORTHOGONAL_MATRIX_IMP_INVERTIBLE

|- (!A. ((orthogonal_matrix A) ==> (invertible A)))


Seq.all_filterP

|- (!a. (!s. ((((filter a) s) = s) = ((all a) s))))


INTEGRAL_0

|- (!s. (((integral s) (\x. (vec (NUMERAL _0)))) = (vec (NUMERAL _0))))


CONDENSATION_POINT_IMP_LIMPT

|- (!x. (!s. (((condensation_point_of x) s) ==> ((limit_point_of x) s))))


Real_ext.REAL_PROP_POS_ADD2

|- (!x. (!y. ((((real_lt (real_of_num (NUMERAL _0))) x) /\ ((real_lt (real_of_num (NUMERAL _0))) y)) ==> ((real_lt (real_of_num (NUMERAL _0))) ((real_add x) y)))))


VECTOR_VARIATION_AFFINITY2

|- (!m. (!c. (!f. (!s. (((vector_variation ((IMAGE (\x. ((vector_add ((% (real_inv m)) x)) (vector_neg ((% (real_inv m)) c))))) s)) (\x. (f ((vector_add ((% m) x)) c)))) = (((COND (m = (real_of_num (NUMERAL _0)))) (real_of_num (NUMERAL _0))) ((vector_variation s) f)))))))


Rogers.KHEJKCI_GEN

|- (!V. (!k. (!r. (!ul. (!vl. (((saturated V) /\ ((packing V) /\ ((((barV V) k) ul) /\ ((((barV V) r) vl) /\ ((initial_sublist ul) vl))))) ==> ((face_of ((voronoi_list V) vl)) ((voronoi_list V) ul))))))))


REAL_LT_ADDNEG

|- (!x. (!y. (!z. (((real_lt y) ((real_add x) (real_neg z))) = ((real_lt ((real_add y) z)) x)))))


LIM_ATREAL

|- (!f. (!l. (!a. ((((--> f) l) (atreal a)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. ((((real_lt (real_of_num (NUMERAL _0))) (real_abs ((real_sub x) a))) /\ ((real_lt (real_abs ((real_sub x) a))) d)) ==> ((real_lt (distance ((f x) , l))) e)))))))))))


FINITE_CARD_LT

|- (!s. ((FINITE s) = (s <_c UNIV)))


Ssrbool.negb_and

|- (!a. (!b. ((~ (a /\ b)) = ((~ a) \/ (~ b)))))


Fan_defs.e_fan_pair_ext

|- (!V. (!E. (!x. (((e_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((e_fan_pair (V , E)) x)) x)))))


FUN_IN_IMAGE

|- (!f. (!s. (!x. ((x IN s) ==> ((f x) IN ((IMAGE f) s))))))


Trigonometry2.ARG_DIFF_SUCCESSIBLE_IN_FIRST_PERIOD

|- (!W. (!xicm. (((FINITE W) /\ (((CARD W) = n) /\ ((!x. ((W x) ==> (~ (x = (vec (NUMERAL _0)))))) /\ ((polar_cycle_on xicm) W)))) ==> (!p. (!i. (!j. (((W p) /\ (((NUMERAL _0) <= i) /\ ((i <= j) /\ (j < n)))) ==> (((real_add ((arg_diff p) (((ITER i) xicm) p))) ((arg_diff (((ITER i) xicm) p)) (((ITER j) xicm) p))) = ((arg_diff p) (((ITER j) xicm) p))))))))))


Lp_ineqs_proofs.y3_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))))) ((y3_fan (V , E)) d))))))))))


REAL_LNEG_UNIQ

|- (!x. (!y. ((((real_add x) y) = (real_of_num (NUMERAL _0))) = (x = (real_neg y)))))


LIM_CONG_AT

|- ((!x. ((~ (x = a)) ==> ((f x) = (g x)))) ==> ((((--> (\x. (f x))) l) (at a)) = (((--> g) l) (at a))))