Lp_ineqs_proofs.dart4_y_invf_eq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y1_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y3_fan (V , E)) d)) /\ (((y2_fan ((inverse (f_fan_pair_ext (V , E))) d)) = (y1_fan d)) /\ ((((y3_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y7_fan (V , E)) d)) /\ ((((y5_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y8_fan (V , E)) d)) /\ (((y6_fan ((inverse (f_fan_pair_ext (V , E))) d)) = ((y5_fan (V , E)) d)) /\ ((((y7_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y2_fan d)) /\ ((((y8_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = ((y9_fan (V , E)) d)) /\ (((y9_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)) = (y6_fan d)))))))))))))))))


FINITE_INDUCT_STRONG

|- (!P. (((P EMPTY) /\ (!x. (!s. (((P s) /\ ((~ (x IN s)) /\ (FINITE s))) ==> (P (x INSERT s)))))) ==> (!s. ((FINITE s) ==> (P s)))))


Lp_details.quad_4240815464_a

|- (quad_nonlinear_v10 ==> (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. (!y7. (!y8. (!y9. ((ineq (((((((((dart_std4 y1) y2) y3) y4) y5) y6) y7) y8) y9)) (((real_lt (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((DECIMAL (NUMERAL (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) \/ (((real_lt ((real_add y2) y3)) y4) \/ (((real_gt ((real_add (((((((((tauq y1) y2) y3) y4) y5) y6) y7) y8) y9)) ((real_sub ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 _0))))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((((((dih_y y1) y2) y3) y4) y5) y6))) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 _0))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))) \/ ((real_lt ((((((delta_y y1) y2) y3) y4) y5) y6)) (real_of_num (NUMERAL _0)))))))))))))))))


LTE_CASES

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


POSITIVE_SEMIDEFINITE_SIMILAR

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


REAL_LE_NEGR

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


POLYTOPE_PCROSS

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


INTEGRABLE_SUBINTERVAL

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


Ssrbool.negb_or

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


Miqmcsn.K_SCS_OPP_3T7

|- ((scs_k_v39 (scs_opp_v39 scs_3T7)) = (NUMERAL (BIT1 (BIT1 _0))))


DROP_EQ

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


STONE_WEIERSTRASS_VECTOR_POLYNOMIAL_FUNCTION_SUBSPACE

|- (!f. (!s. (!e. (!t. (((compact s) /\ (((continuous_on f) s) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ ((subspace t) /\ (((IMAGE f) s) SUBSET t))))) ==> (?g. ((vector_polynomial_function g) /\ ((((IMAGE g) s) SUBSET t) /\ (!x. ((x IN s) ==> ((real_lt (vector_norm ((vector_sub (f x)) (g x)))) e)))))))))))


pathfinish

|- (!g. ((pathfinish g) = (g (vec (NUMERAL (BIT1 _0))))))


MAXIMAL_AFFINE_INDEPENDENT_SUBSET

|- (!s. (!b. (((b SUBSET s) /\ ((~ (affine_dependent b)) /\ (!b'. (((b SUBSET b') /\ ((b' SUBSET s) /\ (~ (affine_dependent b')))) ==> (b' = b))))) ==> (s SUBSET ((hull affine) b)))))


Yrtafyh.YRTAFYH

|- (!s. (!i. (!j. (((is_scs_v39 s) /\ ((scs_basic_v39 s) /\ (((NUMERAL (BIT1 (BIT1 _0))) < (scs_k_v39 s)) /\ ((((scs_diag (scs_k_v39 s)) i) j) /\ ((real_le (((scs_a_v39 s) i) j)) cstab))))) ==> ((is_scs_v39 (((scs_stab_diag_v39 s) i) j)) /\ (scs_basic_v39 (((scs_stab_diag_v39 s) i) j)))))))


REAL_CONTINUOUS_ON_EQ_CONTINUOUS_WITHIN

|- (!f. (!s. (((real_continuous_on f) s) = (!x. ((x IN s) ==> ((real_continuous f) ((within (atreal x)) s)))))))


REAL_CONVEX_BOUND_LT

|- (!x. (!y. (!a. (!u. (!v. ((((real_lt x) a) /\ (((real_lt y) a) /\ (((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_lt ((real_add ((real_mul u) x)) ((real_mul v) y))) a)))))))


Conforming.DART_FANADD_EQ_DART_FAN_ADD_2DART

|- (!x. (!V. (!E. (!E1. (!ds. (!f1. (!f2. (!f3. (!v. (!u. (!w. (((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) /\ ((~ (v = w)) /\ ((E UNION ((v INSERT (w INSERT EMPTY)) INSERT EMPTY)) = E1)))))))))))))))))) ==> (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E1))))) = (((IMAGE pr23) (dart (hypermap1_of_fanx (x , (V , E))))) UNION ((v , w) INSERT ((w , v) INSERT EMPTY))))))))))))))))


vector_polynomial_function

|- (!f. ((vector_polynomial_function f) = (!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> (real_polynomial_function (\x. ((f x) $ i)))))))


INSERT_SUBSET

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


SETDIST_UNIQUE

|- (!s. (!t. (!a. (!b. (!d. (((a IN s) /\ ((b IN t) /\ (((distance (a , b)) = d) /\ (!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le (distance (a , b))) (distance (x , y))))))))) ==> ((setdist (s , t)) = d)))))))


Ssrnat.addnK

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


COMPLEMENT_CONNECTED_COMPONENT_UNIONS

|- (!s. (!x. ((s DIFF ((connected_component s) x)) = (UNIONS ((GSPEC (\GEN%PVAR%900. (?y. (((SETSPEC GEN%PVAR%900) (y IN s)) ((connected_component s) y))))) DELETE ((connected_component s) x))))))


CX_COS

|- (!x. ((Cx (cos x)) = (ccos (Cx x))))


Nonlin_def.sub6

|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((sub6 f) g) x1) x2) x3) x4) x5) x6) = ((real_sub ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))


ZERO_AE_DERIVATIVE_IMP_CONSTANT_GEN

|- (!f. (!a. (!b. (!s. ((((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((negligible ((IMAGE f) s)) /\ (!x. ((x IN ((closed_interval ((CONS (a , b)) NIL)) DIFF s)) ==> (((has_vector_derivative f) (vec (NUMERAL _0))) (at x)))))) ==> (!x. ((x IN (closed_interval ((CONS (a , b)) NIL))) ==> ((f x) = (f a)))))))))


CLOSED_SEGMENT_LINEAR_IMAGE

|- (!f. (!a. (!b. ((linear f) ==> ((closed_segment ((CONS ((f a) , (f b))) NIL)) = ((IMAGE f) (closed_segment ((CONS (a , b)) NIL))))))))


REAL_CONTINUOUS_DIV_WITHINREAL

|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\ (((real_continuous g) ((within (atreal a)) s)) /\ (~ ((g a) = (real_of_num (NUMERAL _0)))))) ==> ((real_continuous (\x. ((real_div (f x)) (g x)))) ((within (atreal a)) s))))))


Sphere.rotate4

|- (!f. (!x4. (!x2. (!x6. (!x1. (!x5. (!x3. ((((((((rotate4 f) x1) x2) x3) x4) x5) x6) = ((((((f x4) x2) x6) x1) x5) x3)))))))))


TAGGED_DIVISION_OF_ALT

|- (!p. (!s. (((tagged_division_of p) s) = (((tagged_partial_division_of p) s) /\ (!x. ((x IN s) ==> (?t. (?k. (((t , k) IN p) /\ (x IN k))))))))))


SQRT_NEG

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


COMPLEX_EQ_ADD_LCANCEL

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


CPRODUCT_EQ

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


AFFINE_EMPTY

|- (affine EMPTY)


UNBOUNDED_HALFSPACE_COMPONENT_GE

|- (!a. (!k. (~ (bounded (GSPEC (\GEN%PVAR%837. (?x. (((SETSPEC GEN%PVAR%837) ((real_ge (x $ k)) a)) x))))))))


SUMS_0

|- (!f. (!s. ((!n. ((n IN s) ==> ((f n) = (vec (NUMERAL _0))))) ==> (((sums f) (vec (NUMERAL _0))) s))))


HOLOMORPHIC_ON_CONST

|- (!c. (!s. ((holomorphic_on (\z. c)) s)))


HAS_REAL_DERIVATIVE_EXP

|- (!x. (((has_real_derivative exp) (exp x)) (atreal x)))


RE_SUB

|- (!x. (!y. ((Re ((vector_sub x) y)) = ((real_sub (Re x)) (Re y)))))


ROOT_MONO_LE

|- (!n. (!x. (!y. (((real_le x) y) ==> ((real_le ((root n) x)) ((root n) y))))))


VECTOR_VARIATION_COMPARISON

|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (!x. (!y. (((x IN s) /\ ((y IN s) /\ ((real_lt (drop x)) (drop y)))) ==> ((real_le (distance ((g x) , (g y)))) (distance ((f x) , (f y)))))))) ==> ((real_le ((vector_variation s) g)) ((vector_variation s) f))))))


HOMOTOPIC_NON_ANTIPODAL_SPHEREMAPS

|- (!f. (!g. (!s. (!r. ((((continuous_on f) s) /\ ((((IMAGE f) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\ (((continuous_on g) s) /\ ((((IMAGE g) s) SUBSET (sphere ((vec (NUMERAL _0)) , r))) /\ (!x. ((x IN s) ==> (~ ((f x) = (vector_neg (g x)))))))))) ==> ((((homotopic_with (\x. T)) (s , (sphere ((vec (NUMERAL _0)) , r)))) f) g))))))


CONTINUOUS_ON_CASES_LOCAL

|- (!P. (!f. (!g. (!s. (!t. ((((closed_in ((subtopology euclidean) (s UNION t))) s) /\ (((closed_in ((subtopology euclidean) (s UNION t))) t) /\ (((continuous_on f) s) /\ (((continuous_on g) t) /\ (!x. ((((x IN s) /\ (~ (P x))) \/ ((x IN t) /\ (P x))) ==> ((f x) = (g x)))))))) ==> ((continuous_on (\x. (((COND (P x)) (f x)) (g x)))) (s UNION t))))))))


SIMPLE_IMAGE_GEN

|- (!f. (!P. ((GSPEC (\GEN%PVAR%12. (?x. (((SETSPEC GEN%PVAR%12) (P x)) (f x))))) = ((IMAGE f) (GSPEC (\GEN%PVAR%13. (?x. (((SETSPEC GEN%PVAR%13) (P x)) x))))))))


SUM_PERMUTE

|- (!f. (!p. (!s. (((permutes p) s) ==> (((sum s) f) = ((sum s) (f o p)))))))


REAL_LE_MUL_EQ

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


HULL_UNION_RIGHT

|- (!P. (!s. (!t. (((hull P) (s UNION t)) = ((hull P) (s UNION ((hull P) t)))))))


CARD_EQ_CLOSED_SETS

|- ((GSPEC (\GEN%PVAR%1254. (?s. (((SETSPEC GEN%PVAR%1254) (closed s)) s)))) =_c UNIV)


BOUNDED_SCALING

|- (!c. (!s. ((bounded s) ==> (bounded ((IMAGE (\x. ((% c) x))) s)))))


metric_tybij

|- ((!a. ((metric (dest_metric a)) = a)) /\ (!r. ((is_metric_space r) = ((dest_metric (metric r)) = r))))


CX_DEF

|- (!a. ((Cx a) = (complex (a , (real_of_num (NUMERAL _0))))))


NORM_LE_0

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


REAL_OF_NUM_MUL

|- (!m. (!n. (((real_mul (real_of_num m)) (real_of_num n)) = (real_of_num (m * n)))))


CONVEX_AFFSIGN

|- (!sgn. ((!x. (!y. (!u. (((sgn x) /\ ((sgn y) /\ (((real_le (real_of_num (NUMERAL _0))) u) /\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0))))))) ==> (sgn ((real_add ((real_mul ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) x)) ((real_mul u) y))))))) ==> (!s. (!t. (convex (((affsign sgn) s) t))))))


SUM_POS_EQ_0_NUMSEG

|- (!f. (!m. (!n. (((!p. (((m <= p) /\ (p <= n)) ==> ((real_le (real_of_num (NUMERAL _0))) (f p)))) /\ (((sum (m .. n)) f) = (real_of_num (NUMERAL _0)))) ==> (!p. (((m <= p) /\ (p <= n)) ==> ((f p) = (real_of_num (NUMERAL _0)))))))))


DOMINATED_CONVERGENCE_AE

|- (!f. (!g. (!h. (!s. (!t. (((!k. ((integrable_on (f k)) s)) /\ (((integrable_on h) s) /\ ((negligible t) /\ ((!k. (!x. ((x IN (s DIFF t)) ==> ((real_le (vector_norm ((f k) x))) (drop (h x)))))) /\ (!x. ((x IN (s DIFF t)) ==> (((--> (\k. ((f k) x))) (g x)) sequentially))))))) ==> (((integrable_on g) s) /\ (((--> (\k. ((integral s) (f k)))) ((integral s) g)) sequentially))))))))


REAL_SEQ_OFFSET

|- (!f. (!l. (!k. ((((---> f) l) sequentially) ==> (((---> (\i. (f (i + k)))) l) sequentially)))))


CONVEX_NEGATIONS

|- (!s. ((convex s) ==> (convex ((IMAGE vector_neg) s))))


HAS_REAL_INTEGRAL_SPIKE_FINITE

|- (!f. (!g. (!s. (!t. (!y. (((FINITE s) /\ ((!x. ((x IN (t DIFF s)) ==> ((g x) = (f x)))) /\ (((has_real_integral f) y) t))) ==> (((has_real_integral g) y) t)))))))


AFFINE_HULL_0_3_EXPLICIT

|- (!x. (!y. (!z. (((vec (NUMERAL _0)) IN ((hull affine) (x INSERT (y INSERT (z INSERT EMPTY))))) = (?a. (?b. (?c. ((((vector_add ((% a) x)) ((vector_add ((% b) y)) ((% c) z))) = (vec (NUMERAL _0))) /\ (~ (((real_add a) ((real_add b) c)) = (real_of_num (NUMERAL _0))))))))))))


DIVMOD_ELIM_THM

|- (((P (m DIV n)) (m MOD n)) = (!q. (!r. ((((n = (NUMERAL _0)) /\ ((q = (NUMERAL _0)) /\ (r = m))) \/ ((m = ((q * n) + r)) /\ (r < n))) ==> ((P q) r)))))


Add_triangle.fan_hypermap_map1

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart1_of_fan (V , E))) ==> ((((face_map (hypermap_of_fan (V , E))) d) = ((f_fan_pair (V , E)) d)) /\ ((((node_map (hypermap_of_fan (V , E))) d) = ((n_fan_pair (V , E)) d)) /\ (((edge_map (hypermap_of_fan (V , E))) d) = ((e_fan_pair (V , E)) d)))))))))


SQRT_LE_0

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


Functional_equation.functional_mud_135_x

|- (!y2. (!y4. (!y6. (((domain6 (\x1. (\x2. (\x3. (\x4. (\x5. (\x6. ((real_le (real_of_num (NUMERAL _0))) y2)))))))) (((mud_135_x_v1 y2) y4) y6)) ((mul6 (((((((compose6 mu6_x) (constant6 ((real_mul y2) y2))) proj_x1) proj_x3) dummy6) dummy6) dummy6)) (uni (sqrt , (((delta_135_x ((real_mul y2) y2)) ((real_mul y4) y4)) ((real_mul y6) y6)))))))))


Lp_ineqs_def.y4'_fan

|- (!V. (!E. (!x. (((y4'_fan (V , E)) x) = (distance ((SND x) , (FST ((inverse (f_fan_pair_ext (V , E))) x))))))))


Packing3.lemma1

|- (!f. (!P. (((FINITE f) /\ (P f)) ==> (?n. (?g. ((g SUBSET f) /\ ((g HAS_SIZE n) /\ (P g))))))))


MEASURABLE_ON_SPIKE

|- (!f. (!g. (!s. (!t. (((negligible s) /\ (!x. ((x IN (t DIFF s)) ==> ((g x) = (f x))))) ==> (((measurable_on f) t) ==> ((measurable_on g) t)))))))


CLOSURE_INJECTIVE_LINEAR_IMAGE

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


AFFINE_HULL_EMPTY

|- (((hull affine) EMPTY) = EMPTY)


UNIONS_2

|- ((UNIONS (s INSERT (t INSERT EMPTY))) = (s UNION t))


Hypermap_iso.fan_hypermaps_iso_explicit2

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((hyp_iso (ext_dart (V , E))) ((hypermap_of_fan (V , E)) , (hypermap1_of_fanx ((vec (NUMERAL _0)) , (V , E))))))))


HAS_BOUNDED_VARIATION_ON_SUBSET

|- (!f. (!s. (!t. ((((has_bounded_variation_on f) s) /\ (t SUBSET s)) ==> ((has_bounded_variation_on f) t)))))


CEXP_NZ

|- (!z. (~ ((cexp z) = (Cx (real_of_num (NUMERAL _0))))))


CLOSED_IN_DIFF

|- (!top. (!s. (!t. ((((closed_in top) s) /\ ((open_in top) t)) ==> ((closed_in top) (s DIFF t))))))


Ssrnat.addSn

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


Hypermap.FACE_FINITE

|- (!H. (!x. (FINITE ((face H) x))))


Hypermap.lemma_face_representation

|- (!H. (!u. ((u IN (face_set H)) ==> (?x. ((x IN (dart H)) /\ (u = ((face H) x)))))))


INTERIOR_MAXIMAL_EQ

|- (!s. (!t. ((open s) ==> ((s SUBSET (interior t)) = (s SUBSET t)))))


INT_OF_NUM_MAX

|- (!m. (!n. (((int_max (int_of_num m)) (int_of_num n)) = (int_of_num ((MAX m) n)))))


REAL_INV_DIV

|- (!x. (!y. ((real_inv ((real_div x) y)) = ((real_div y) x))))


Seq.has

|- (!a. (!x. (!s'. ((((has a) NIL) = F) /\ (((has a) ((CONS x) s')) = ((a x) \/ ((has a) s')))))))


COMPLEX_NORM_II

|- ((vector_norm ii) = (real_of_num (NUMERAL (BIT1 _0))))


POW_2_SQRT

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


Hypermap.lemma_atom_sub_node

|- (!H. (!L. (!x. ((((atom H) L) x) SUBSET ((node H) x)))))


OPEN_IN_SUBTOPOLOGY_REFL

|- (!top. (!u. (((open_in ((subtopology top) u)) u) = (u SUBSET (topspace top)))))


COMPLEX_DIFFERENTIABLE_DIV_WITHIN

|- (!f. (!g. (!z. (!s. ((((complex_differentiable f) ((within (at z)) s)) /\ (((complex_differentiable g) ((within (at z)) s)) /\ (~ ((g z) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((complex_differentiable (\z. ((complex_div (f z)) (g z)))) ((within (at z)) s)))))))


CTAN_ADD

|- (!w. (!z. (((~ ((ccos w) = (Cx (real_of_num (NUMERAL _0))))) /\ ((~ ((ccos z) = (Cx (real_of_num (NUMERAL _0))))) /\ (~ ((ccos ((vector_add w) z)) = (Cx (real_of_num (NUMERAL _0))))))) ==> ((ctan ((vector_add w) z)) = ((complex_div ((vector_add (ctan w)) (ctan z))) ((vector_sub (Cx (real_of_num (NUMERAL (BIT1 _0))))) ((complex_mul (ctan w)) (ctan z))))))))


HAS_REAL_MEASURE_TRANSLATION_EQ

|- (!s. (!m. (((has_real_measure ((IMAGE (\x. ((real_add a) x))) s)) m) = ((has_real_measure s) m))))


INT_OF_NUM_ADD

|- (!m. (!n. (((int_add (int_of_num m)) (int_of_num n)) = (int_of_num (m + n)))))


NOT_OUTSIDE_CONNECTED_COMPONENT_LE

|- (!s. ((((NUMERAL (BIT0 (BIT1 _0))) <= (dimindex UNIV)) /\ (bounded s)) ==> ((UNIV DIFF (outside s)) = (GSPEC (\GEN%PVAR%2014. (?x. (((SETSPEC GEN%PVAR%2014) (!B. (?y. (((real_le B) (vector_norm y)) /\ (~ (((connected_component (UNIV DIFF s)) x) y)))))) x)))))))


HAS_REAL_DERIVATIVE_MUL_ATREAL

|- (!f. (!f'. (!g. (!g'. (!x. (((((has_real_derivative f) f') (atreal x)) /\ (((has_real_derivative g) g') (atreal x))) ==> (((has_real_derivative (\x. ((real_mul (f x)) (g x)))) ((real_add ((real_mul (f x)) g')) ((real_mul f') (g x)))) (atreal x))))))))


CONVEX_IMP_PATH_CONNECTED

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


LT_EXISTS

|- (!m. (!n. ((m < n) = (?d. (n = (m + (SUC d)))))))


REAL_DIFFERENTIABLE_TRANSFORM

|- (!f. (!g. (!s. (((!x. ((x IN s) ==> ((f x) = (g x)))) /\ ((real_differentiable_on f) s)) ==> ((real_differentiable_on g) s)))))


CONTINUOUS_CARD_LT_RANGE_CONSTANT_EQ

|- (!s. ((connected s) = (!f. ((((continuous_on f) s) /\ (((IMAGE f) s) <_c UNIV)) ==> (?a. (!x. ((x IN s) ==> ((f x) = a))))))))


subspace

|- (!s. ((subspace s) = (((vec (NUMERAL _0)) IN s) /\ ((!x. (!y. (((x IN s) /\ (y IN s)) ==> (((vector_add x) y) IN s)))) /\ (!c. (!x. ((x IN s) ==> (((% c) x) IN s))))))))


INTERIOR_SUBSET

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


SUBSET_ANTISYM_EQ

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


MEASURE_SCALING

|- (!s. (!c. ((measurable s) ==> ((measure ((IMAGE (\x. ((% c) x))) s)) = ((real_mul ((real_pow (real_abs c)) (dimindex UNIV))) (measure s))))))


COMPACT_TRANSLATION_EQ

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


REAL_LE_MAX

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


Hypermap.card_orbit_le

|- (!f. (!n. (!x. (((~ (n = (NUMERAL _0))) /\ ((((POWER f) n) x) = x)) ==> ((CARD ((orbit_map f) x)) <= n)))))


INTEGRAL_SUB

|- (!f. (!g. (!k. (!l. (!s. ((((integrable_on f) s) /\ ((integrable_on g) s)) ==> (((integral s) (\x. ((vector_sub (f x)) (g x)))) = ((vector_sub ((integral s) f)) ((integral s) g)))))))))


conic_cap

|- (!r. (!v0. (!v1. (!a. (((((conic_cap v0) v1) r) a) = (((normball v0) r) INTER (((rcone_gt v0) v1) a)))))))


COMPACT_CLOSURE

|- (!s. ((compact (closure s)) = (bounded s)))


INF_INSERT_FINITE

|- (!x. (!s. ((FINITE s) ==> ((inf (x INSERT s)) = (((COND (s = EMPTY)) x) ((real_min x) (inf s)))))))


L1_LE_NORM

|- (!x. ((real_le ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs (x $ i))))) ((real_mul (sqrt (real_of_num (dimindex UNIV)))) (vector_norm x))))


IN_SPHERE_0

|- (!x. (!e. ((x IN (sphere ((vec (NUMERAL _0)) , e))) = ((vector_norm x) = e))))


HAS_COMPLEX_DERIVATIVE_CARATHEODORY_WITHIN

|- (!f. (!f'. (!z. (!s. ((((has_complex_derivative f) f') ((within (at z)) s)) = (?g. ((!w. (((vector_sub (f w)) (f z)) = ((complex_mul (g w)) ((vector_sub w) z)))) /\ (((continuous g) ((within (at z)) s)) /\ ((g z) = f')))))))))


REAL_SEGMENT_INTERVAL

|- ((!a. (!b. ((closed_real_segment ((CONS (a , b)) NIL)) = (((COND ((real_le a) b)) (closed_real_interval ((CONS (a , b)) NIL))) (closed_real_interval ((CONS (b , a)) NIL)))))) /\ (!a. (!b. ((open_real_segment (a , b)) = (((COND ((real_le a) b)) (open_real_interval (a , b))) (open_real_interval (b , a)))))))


Lp_ineqs_proofs.y8_hi_std

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le ((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)))))))))))))))))))


LOCALLY_PATH_CONNECTED_PATH_IMAGE

|- (!p. ((path p) ==> ((locally path_connected) (path_image p))))


List_hypermap.mem_list_of_elements

|- (!x. (!L. (((MEM x) (list_of_elements L)) = (?y. ((MEM (x , y)) (list_of_darts L))))))


Appendix.scs_a_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_a_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = a))))))))))


ssreflect_eq_def

|- (!v. (!P. (((ssreflect_eq v) P) = P)))


BETWEEN_IMP_COLLINEAR

|- (!a. (!b. (!x. (((between x) (a , b)) ==> (collinear (a INSERT (x INSERT (b INSERT EMPTY))))))))


complex_pow

|- ((((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0))))) /\ (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n)))))


SUM_EQ_NUMSEG

|- (!f. (!g. (!m. (!n. ((!i. (((m <= i) /\ (i <= n)) ==> ((f i) = (g i)))) ==> (((sum (m .. n)) f) = ((sum (m .. n)) g)))))))


INT_LE_RNEG

|- (!x. (!y. (((int_le x) (int_neg y)) = ((int_le ((int_add x) y)) (int_of_num (NUMERAL _0))))))


INTEGER_CASES

|- ((integer x) = ((?n. (x = (real_of_num n))) \/ (?n. (x = (real_neg (real_of_num n))))))


IMAGE_UNIONS

|- (!f. (!s. (((IMAGE f) (UNIONS s)) = (UNIONS ((IMAGE (IMAGE f)) s)))))


Seq.subseq_filter

|- (!s1. (!s2. (!a. (((subseq s1) ((filter a) s2)) = (((all a) s1) /\ ((subseq s1) s2))))))


HAS_BOUNDED_VARIATION_COMPOSE_INCREASING

|- (!f. (!g. (!a. (!b. (((!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 x))) (drop (f y)))))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((f a) , (f b))) NIL)))) ==> ((has_bounded_variation_on (g o f)) (closed_interval ((CONS (a , b)) NIL))))))))


RE

|- ((Re (complex (x , y))) = x)


rectifiable_path

|- (!g. ((rectifiable_path g) = ((path g) /\ ((has_bounded_variation_on g) (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))))))


Pack2.POLYHEDRON_VORONOI_CLOSED

|- (!s. (!v. (((packing s) /\ ((saturated s) /\ (v IN s))) ==> (polyhedron ((voronoi_closed s) v)))))


REAL_LE_LNEG

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


CEXP_NEG_RMUL

|- (!z. (((complex_mul (cexp z)) (cexp (vector_neg z))) = (Cx (real_of_num (NUMERAL (BIT1 _0))))))


vol_solid_triangle

|- (!r. (!v0. (!v1. (!v2. (!v3. ((((((vol_solid_triangle v0) v1) v2) v3) r) = ((LET (\a123. (LET_END ((LET (\a231. (LET_END ((LET (\a312. (LET_END ((real_mul ((real_add a123) ((real_add a231) ((real_sub a312) pi)))) ((real_div ((real_pow r) (NUMERAL (BIT1 (BIT1 _0))))) (real_of_num (NUMERAL (BIT1 (BIT1 _0))))))))) ((((dihV v0) v3) v1) v2))))) ((((dihV v0) v2) v3) v1))))) ((((dihV v0) v1) v2) v3))))))))


OPEN_IMP_GDELTA

|- (!s. ((open s) ==> (gdelta s)))


Functional_equation.rhazim_x'

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_le (real_of_num (NUMERAL _0))) x1) /\ (((real_le (real_of_num (NUMERAL _0))) x2) /\ (((real_le (real_of_num (NUMERAL _0))) x3) /\ (((real_le (real_of_num (NUMERAL _0))) x4) /\ (((real_le (real_of_num (NUMERAL _0))) x5) /\ ((real_le (real_of_num (NUMERAL _0))) x6)))))) ==> (((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazim_x' x1) x2) x3) x4) x5) x6)))))))))


REAL_LE_SUB_LADD

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


CLOSED_UNIONS

|- (!s. (((FINITE s) /\ (!t. ((t IN s) ==> (closed t)))) ==> (closed (UNIONS s))))


Ssrbool.contraR

|- (!c. (!b. (((~ c) ==> b) ==> ((~ b) ==> c))))


ACS_MONO_LE

|- (!x. (!y. ((((real_le (real_neg (real_of_num (NUMERAL (BIT1 _0))))) x) /\ (((real_le x) y) /\ ((real_le y) (real_of_num (NUMERAL (BIT1 _0)))))) ==> ((real_le (acs y)) (acs x)))))


Sphere.sol_y

|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((sol_y y1) y2) y3) y4) y5) y6) = ((real_add ((((((dih_y y1) y2) y3) y4) y5) y6)) ((real_add ((((((dih_y y2) y3) y1) y5) y6) y4)) ((real_sub ((((((dih_y y3) y1) y2) y6) y4) y5)) pi))))))))))


HAS_REAL_MEASURE_SCALING

|- (!s. (!m. (!y. (((has_real_measure s) y) ==> ((has_real_measure ((IMAGE (\x. ((real_mul m) x))) s)) ((real_mul (real_abs m)) y))))))


SERIES_0

|- (!s. (((sums (\n. (vec (NUMERAL _0)))) (vec (NUMERAL _0))) s))


REAL_LT_DIV

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


NADD_LE_WELLDEF

|- (!x. (!x'. (!y. (!y'. ((((nadd_eq x) x') /\ ((nadd_eq y) y')) ==> (((nadd_le x) y) = ((nadd_le x') y')))))))


IM_SUB

|- (!x. (!y. ((Im ((vector_sub x) y)) = ((real_sub (Im x)) (Im y)))))


REAL_ADD_RID

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


GDELTA_PCROSS

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


FCONS

|- ((!a. (!f. ((((FCONS a) f) (NUMERAL _0)) = a))) /\ (!a. (!f. (!n. ((((FCONS a) f) (SUC n)) = (f n))))))


MAPROWS_COMPOSE

|- (!f. (!g. ((maprows (f o g)) = ((maprows f) o (maprows g)))))


CBALL_EMPTY

|- (!x. (!e. (((real_lt e) (real_of_num (NUMERAL _0))) ==> ((cball (x , e)) = EMPTY))))


COMPLEX_MVT_LINE

|- (!f. (!f'. (!w. (!z. ((!u. ((u IN (closed_segment ((CONS (w , z)) NIL))) ==> (((has_complex_derivative f) (f' u)) (at u)))) ==> (?u. ((u IN (closed_segment ((CONS (w , z)) NIL))) /\ (((real_sub (Re (f z))) (Re (f w))) = (Re ((complex_mul (f' u)) ((vector_sub z) w)))))))))))


CLOSED_CBALL

|- (!x. (!e. (closed (cball (x , e)))))


BARYCENTRE_IN_RELATIVE_INTERIOR

|- (!s. (((FINITE s) /\ (~ (s = EMPTY))) ==> ((barycentre s) IN (relative_interior ((hull convex) s)))))


LIFT_EQ

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


Ssrnat.subKn

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


root

|- (!x. (!n. (((root n) x) = ((real_mul (real_sgn x)) (exp ((real_div (log (real_abs x))) (real_of_num n)))))))


Functional_equation.rhazim_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))))))) ==> ((((((((rhazim_x x1) x2) x3) x4) x5) x6) = ((((((rhazimatn_x x1) x2) x3) x4) x5) x6)) /\ ((((((((rhazim2_x x1) x2) x3) x4) x5) x6) = ((((((rhazim2atn_x x1) x2) x3) x4) x5) x6)) /\ (((((((rhazim3_x x1) x2) x3) x4) x5) x6) = ((((((rhazim3atn_x x1) x2) x3) x4) x5) x6)))))))))))


CONTINUOUS_NEG

|- (!f. (!net. (((continuous f) net) ==> ((continuous (\x. (vector_neg (f x)))) net))))


EXTREME_POINT_OF_CONVEX_HULL_INSERT

|- (!s. (!a. (((FINITE s) /\ (~ (a IN ((hull convex) s)))) ==> ((extreme_point_of a) ((hull convex) (a INSERT s))))))


AFF_GE_EQ_AFFINE_HULL

|- (!s. (!t. (((aff_ge s) EMPTY) = ((hull affine) s))))


SQRT_POW_2

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


Lp_main_estimate.f_surr_in_e_prime

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!f. (!v. (!w. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> (((v INSERT (w INSERT EMPTY)) IN ((e_prime E) f)) = (((v , w) IN f) \/ ((w , v) IN f)))))))))))


COMPLEX_TRAD

|- (!x. (!y. ((complex (x , y)) = ((vector_add (Cx x)) ((complex_mul ii) (Cx y))))))


Seq2.next_el_hd_cons2

|- (!h1. (!h2. (!t. (((next_el ((CONS h1) ((CONS h2) t))) h1) = h2))))


VSUM_ADD_NUMSEG

|- (!f. (!g. (!m. (!n. (((vsum (m .. n)) (\i. ((vector_add (f i)) (g i)))) = ((vector_add ((vsum (m .. n)) f)) ((vsum (m .. n)) g)))))))


Lp_ineqs_proofs.tauq_cross_split

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d)) = ((real_add ((((((taum (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((taum (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))


COMPACT_CBALL

|- (!x. (!e. (compact (cball (x , e)))))


Hypermap_and_fan.PAIR_IN_DART_OF_FAN

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


Lp_ineqs_proofs.y8_lo

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) ((y8_fan (V , E)) d))))))))))


FROM_0

|- ((from (NUMERAL _0)) = UNIV)


FACET_OF_TRANSLATION_EQ

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


COPLANAR_LINEAR_IMAGE_EQ

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


REAL_EQ_RCANCEL_IMP

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


joinpaths

|- (!g1. (!g2. (((++ g1) g2) = (\x. (((COND ((real_le (drop x)) ((real_div (real_of_num (NUMERAL (BIT1 _0)))) (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))) (g1 ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x))) (g2 ((vector_sub ((% (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) x)) (vec (NUMERAL (BIT1 _0))))))))))


HAS_BOUNDED_VARIATION_ON_MUL

|- (!f. (!g. (!s. ((((has_bounded_variation_on f) s) /\ (((has_bounded_variation_on g) s) /\ (is_interval s))) ==> ((has_bounded_variation_on (\x. ((% (drop (f x))) (g x)))) s)))))


REAL_POW_LT2

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


SPAN_MUL

|- (!x. (!c. (!s. ((x IN (span s)) ==> (((% c) x) IN (span s))))))


COMPACT_IMP_CLOSED

|- (!s. ((compact s) ==> (closed s)))


FACE_OF_TRANS

|- (!s. (!t. (!u. ((((face_of s) t) /\ ((face_of t) u)) ==> ((face_of s) u)))))


ge_c

|- (!t. (!s. ((s >=_c t) = (t <=_c s))))


RSTC_SYM

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


Seq.count

|- (!a. (!x. (!s'. ((((count a) NIL) = (NUMERAL _0)) /\ (((count a) ((CONS x) s')) = ((((COND (a x)) (NUMERAL (BIT1 _0))) (NUMERAL _0)) + ((count a) s')))))))


OPEN_NON_GENERAL_COMPONENT

|- (!c. (((!s. (!x. (!y. ((((c s) x) y) ==> ((x IN s) /\ (y IN s)))))) /\ ((!s. (!x. (!y. ((((c s) x) y) ==> (((c s) y) x))))) /\ ((!s. (!x. (!y. (!z. (((((c s) x) y) /\ (((c s) y) z)) ==> (((c s) x) z)))))) /\ ((!s. (!t. (!x. (!y. (((s SUBSET t) /\ (((c s) x) y)) ==> (((c t) x) y)))))) /\ (!s. (!x. (!y. (!e. (((y IN (ball (x , e))) /\ ((ball (x , e)) SUBSET s)) ==> (((c (ball (x , e))) x) y)))))))))) ==> (!s. (!x. ((open s) ==> (open (s DIFF ((c s) x))))))))


Sphere.lmfun

|- (!h. ((lmfun h) = (((COND ((real_le h) h0)) ((real_div ((real_sub h0) h)) ((real_sub h0) (real_of_num (NUMERAL (BIT1 _0)))))) (real_of_num (NUMERAL _0)))))


num_WOP

|- (!P. ((?n. (P n)) = (?n. ((P n) /\ (!m. ((m < n) ==> (~ (P m))))))))


NPRODUCT_CLAUSES

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


INT_LT_DISCRETE

|- (!x. (!y. (((int_lt x) y) = ((int_le ((int_add x) (int_of_num (NUMERAL (BIT1 _0))))) y))))


SETDIST_LINEAR_IMAGE

|- (!f. (!s. (!t. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> ((setdist (((IMAGE f) s) , ((IMAGE f) t))) = (setdist (s , t)))))))


SEGMENT_CONVEX_HULL

|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((hull convex) (a INSERT (b INSERT EMPTY))))))


SUM_LE

|- (!f. (!g. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((real_le (f x)) (g x))))) ==> ((real_le ((sum s) f)) ((sum s) g))))))


AFF_GE_SUBSET_AFFINE_HULL

|- (!s. (!t. (((aff_ge s) t) SUBSET ((hull affine) (s UNION t)))))


Seq.has_pred1

|- (!x. (!s. (((has (pred1 x)) s) = ((MEM x) s))))


REAL_DIFFERENTIABLE_AT_RPOW_RIGHT

|- (!a. (!x. (((real_lt (real_of_num (NUMERAL _0))) a) ==> ((real_differentiable (\x. ((rpow a) x))) (atreal x)))))


Lp_main_estimate.packing_v_prime

|- (!V. (!f. ((packing V) ==> (packing ((v_prime V) f)))))


Functional_equation.proj_x5

|- (((((((proj_x5 x1) x2) x3) x4) x5) x6) = x5)


Nonlin_def.dummy6

|- (dummy6 = (constant6 (real_of_num (NUMERAL _0))))


BOREL_UNIV

|- (borel UNIV)


Planarity.fan80

|- (!x. (!V. (!E. ((fan80 (x , (V , E))) = (!v. (!u. (((v INSERT (u INSERT EMPTY)) IN E) ==> (((real_lt (real_of_num (NUMERAL _0))) ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) /\ ((real_lt ((((azim x) v) u) (((((sigma_fan x) V) E) v) u))) pi)))))))))


OPEN_IN_INTER

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


Ssrnat.leq_mul2l

|- (!m. (!n1. (!n2. (((m * n1) <= (m * n2)) = ((m = (NUMERAL _0)) \/ (n1 <= n2))))))


REAL_ABS_SIN_BOUND_LT

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


VSUM_LMUL

|- (!f. (!c. (!s. (((vsum s) (\x. ((% c) (f x)))) = ((% c) ((vsum s) f))))))


Functional_equation.eta_all_atn

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((real_lt (real_of_num (NUMERAL _0))) x1) /\ (((real_lt (real_of_num (NUMERAL _0))) x2) /\ (((real_lt (real_of_num (NUMERAL _0))) x3) /\ (((real_lt (real_of_num (NUMERAL _0))) x4) /\ (((real_lt (real_of_num (NUMERAL _0))) x5) /\ ((real_lt (real_of_num (NUMERAL _0))) x6)))))) ==> ((((((((eta2_126 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x2) x6)) (NUMERAL (BIT0 (BIT1 _0))))) /\ ((((((((eta2_135 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x1) x3) x5)) (NUMERAL (BIT0 (BIT1 _0))))) /\ (((((((eta2_456 x1) x2) x3) x4) x5) x6) = ((real_pow (((eta_x x4) x5) x6)) (NUMERAL (BIT0 (BIT1 _0))))))))))))))


Hypermap.lemma_node_identity

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


UNIONS_SUBSET

|- (!f. (!t. (((UNIONS f) SUBSET t) = (!s. ((s IN f) ==> (s SUBSET t))))))


IN_COMPONENTS_SUBSET

|- (!s. (!c. ((c IN (components s)) ==> (c SUBSET s))))


Nonlinear_lemma.lmdih5_ldih5

|- (!y1. (!y2. (!y3. (!y4. (!y5. (!y6. ((((real_le (real_of_num (NUMERAL _0))) y5) /\ ((real_le y5) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0))) ==> ((((((((y_of_x lmdih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6) = (((((((y_of_x ldih5_x_div_sqrtdelta_posbranch) y1) y2) y3) y4) y5) y6)))))))))


Tame_inequalities.DELTA_X4_MONO_LT_4

|- (!x1. (!x2. (!x3. (!a. (!x5. (!x6. (!b. ((((real_lt a) b) /\ ((real_lt (real_of_num (NUMERAL _0))) x1)) ==> ((real_lt ((((((delta_x4 x1) x2) x3) b) x5) x6)) ((((((delta_x4 x1) x2) x3) a) x5) x6))))))))))


Tame_defs.tame_3

|- (!H. ((tame_3 H) = (is_edge_nondegenerate H)))


VECTOR_VARIATION_CONTINUOUS_LEFT

|- (!f. (!a. (!b. (!c. ((((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) /\ (c IN (closed_interval ((CONS (a , b)) NIL)))) ==> (((continuous (\x. (lift ((vector_variation (closed_interval ((CONS (a , x)) NIL))) f)))) ((within (at c)) (closed_interval ((CONS (a , c)) NIL)))) = ((continuous f) ((within (at c)) (closed_interval ((CONS (a , c)) NIL))))))))))


Rnsyjxm.in_face_imp_in_dart

|- (!x. (!H. (!d. (((x IN (dart H)) /\ (d IN ((face H) x))) ==> (d IN (dart H))))))


INT_LE_ANTISYM

|- (!x. (!y. ((((int_le x) y) /\ ((int_le y) x)) = (x = y))))


BEPPO_LEVI_INCREASING

|- (!f. (!s. (((!k. ((integrable_on (f k)) s)) /\ ((!k. (!x. ((x IN s) ==> ((real_le (drop ((f k) x))) (drop ((f (SUC k)) x)))))) /\ (bounded (GSPEC (\GEN%PVAR%3065. (?k. (((SETSPEC GEN%PVAR%3065) (k IN UNIV)) ((integral s) (f k))))))))) ==> (?g. (?k. ((negligible k) /\ (!x. ((x IN (s DIFF k)) ==> (((--> (\k. ((f k) x))) (g x)) sequentially)))))))))


SUM_EQ_SUPERSET

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


COLLINEAR_AFF_DIM

|- (!s. ((collinear s) = ((int_le (aff_dim s)) (int_of_num (NUMERAL (BIT1 _0))))))


List_hypermap.uniq_list_of_elements

|- (!L. (uniq (list_of_elements L)))


MOD_MULT

|- (!m. (!n. ((~ (m = (NUMERAL _0))) ==> (((m * n) MOD m) = (NUMERAL _0)))))


HAS_COMPLEX_DERIVATIVE_DIV_AT

|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\ ((((has_complex_derivative g) g') (at x)) /\ (~ ((g x) = (Cx (real_of_num (NUMERAL _0))))))) ==> (((has_complex_derivative (\x. ((complex_div (f x)) (g x)))) ((complex_div ((vector_sub ((complex_mul f') (g x))) ((complex_mul (f x)) g'))) ((complex_pow (g x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))))


COMPLETE_TRANSLATION_EQ

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


Add_triangle.inE_eq_in_dart1

|- (!V. (!E. (!v. (!w. (((v INSERT (w INSERT EMPTY)) IN E) = ((v , w) IN (dart1_of_fan (V , E))))))))


REAL_POW_ONE

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


Trigonometry.XHVXJVB

|- (!x. ((vector_norm x) = (sqrt ((dot x) x))))


List_hypermap.e_list_ext

|- (!L. ((e_list_ext L) = ((res e_list) (darts_of_list L))))


Appendix.scs_d_v39_explicit

|- (!k. (!d. (!a. (!b. (!alpha. (!beta. (!J. (!lo. (!str. ((scs_d_v39 (scs_v39 (k , (d , (a , (alpha , (beta , (b , (J , (lo , (hi , str))))))))))) = d))))))))))


CLOG_CEXP

|- (!z. ((((real_lt (real_neg pi)) (Im z)) /\ ((real_le (Im z)) pi)) ==> ((clog (cexp z)) = z)))


COMPLEX_MUL_2

|- (!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) x) = ((vector_add x) x)))


Nonlin_def.mul6

|- (!f. (!g. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((((mul6 f) g) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) ((((((g x1) x2) x3) x4) x5) x6)))))))))))


HOMEOMORPHIC_INJECTIVE_LINEAR_IMAGE_LEFT_EQ

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


DEPENDENT_BIGGERSET_GENERAL

|- (!s. (((FINITE s) ==> ((CARD s) > (dim s))) ==> (dependent s)))


REAL_SUMMABLE_RESTRICT

|- (!f. (!k. (((real_summable UNIV) (\n. (((COND (n IN k)) (f n)) (real_of_num (NUMERAL _0))))) = ((real_summable k) f))))


REAL_MEASURE_MEASURE

|- (!s. ((real_measure s) = (measure ((IMAGE lift) s))))


MATRIX_VECTOR_MUL_COMPONENT

|- (!A. (!x. (!k. ((((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV))) ==> ((((matrix_vector_mul A) x) $ k) = ((dot (A $ k)) x))))))


LEBESGUE_MEASURABLE_COUNTABLE_UNIONS

|- (!f. (((COUNTABLE f) /\ (!s. ((s IN f) ==> (lebesgue_measurable s)))) ==> (lebesgue_measurable (UNIONS f))))


REAL_NEG_GE0

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


HAS_BOUNDED_VARIATION_ON_COMBINE

|- (!f. (!a. (!b. (!c. ((((real_le (drop a)) (drop c)) /\ ((real_le (drop c)) (drop b))) ==> (((has_bounded_variation_on f) (closed_interval ((CONS (a , b)) NIL))) = (((has_bounded_variation_on f) (closed_interval ((CONS (a , c)) NIL))) /\ ((has_bounded_variation_on f) (closed_interval ((CONS (c , b)) NIL))))))))))


LINEAR_COMPOSE_CMUL

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


REAL_EXP_INJ

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


Lp_ineqs_proofs.dart4_tauVEF_eq_tauq

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> ((tauVEF (V , (E , ((face (hypermap_of_fan (V , E))) d)))) = (((((((((tauq (y1_fan d)) (y2_fan d)) ((y3_fan (V , E)) d)) ((y4'_fan (V , E)) d)) ((y5_fan (V , E)) d)) (y6_fan d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y9_fan (V , E)) d)))))))))))


Float.REAL_LT

|- (!m. (!n. (((real_lt (real_of_num m)) (real_of_num n)) = (m < n))))


ORTHOGONAL_TRANSFORMATION_REFLECT_ALONG

|- (!v. (orthogonal_transformation (reflect_along v)))


Seq.last

|- (!x. (!h. (!t. ((((last x) NIL) = x) /\ (((last x) ((CONS h) t)) = ((last h) t))))))


CLOSED_IN

|- (!s. ((closed s) = ((closed_in euclidean) s)))


MATRIX_NEG

|- (!f. ((linear f) ==> ((matrix (\x. (vector_neg (f x)))) = (matrix_neg (matrix f)))))


FST

|- (!x. (!y. ((FST (x , y)) = x)))


Tame_lemmas.sum_tauVEF_upper_bound

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (((real_le (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 _0))))))) (scriptL V)) ==> ((real_le ((sum (face_set (hypermap_of_fan (V , E)))) (\f. (tauVEF (V , (E , f)))))) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))) pi)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))) sol0))))))))


NORMAL_MATRIX_KERNEL_TRANSP_EXPLICIT

|- (!A. (!x. ((((matrix_mul (transp A)) A) = ((matrix_mul A) (transp A))) ==> ((((matrix_vector_mul (transp A)) x) = (vec (NUMERAL _0))) = (((matrix_vector_mul A) x) = (vec (NUMERAL _0)))))))


Ssrbool.XOR_DEF

|- (!p. (!q. (((XOR p) q) = (((COND p) (~ q)) q))))


Functional_equation.proj_y6

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (((((((proj_y6 x1) x2) x3) x4) x5) x6) = (sqrt x6))))))))


CONNECTED_CONTINUOUS_IMAGE

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


Planarity.aff_ge_eq_aff_gt_union_aff_ge

|- (!x. (!v. (!w. ((~ (collinear (x INSERT (v INSERT (w INSERT EMPTY))))) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) = (((aff_gt (x INSERT EMPTY)) (v INSERT (w INSERT EMPTY))) UNION (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) UNION ((aff_ge (x INSERT EMPTY)) (w INSERT EMPTY)))))))))


Misc_defs_and_lemmas.IN_ELIM_THM'

|- ((!P. (!x. ((x IN (GSPEC (\v. (P (SETSPEC v))))) = (P (\p. (\t. (p /\ (x = t)))))))) /\ ((!p. (!x. ((x IN (GSPEC (\v. (?y. (((SETSPEC v) (p y)) y))))) = (p x)))) /\ ((!P. (!x. (((GSPEC (\v. (P (SETSPEC v)))) x) = (P (\p. (\t. (p /\ (x = t)))))))) /\ ((!p. (!x. (((GSPEC (\v. (?y. (((SETSPEC v) (p y)) y)))) x) = (p x)))) /\ (!p. (!x. ((x IN (\y. (p y))) = (p x))))))))


LIM_UNIQUE

|- (!net. (!f. (!l. (!l'. (((~ (trivial_limit net)) /\ ((((--> f) l) net) /\ (((--> f) l') net))) ==> (l = l'))))))


tagged_division_of

|- (!s. (!i. (((tagged_division_of s) i) = (((tagged_partial_division_of s) i) /\ ((UNIONS (GSPEC (\GEN%PVAR%2759. (?k. (((SETSPEC GEN%PVAR%2759) (?x. ((x , k) IN s))) k))))) = i)))))


LIM_NULL_SUB

|- (!net. (!f. (!g. (((((--> f) (vec (NUMERAL _0))) net) /\ (((--> g) (vec (NUMERAL _0))) net)) ==> (((--> (\x. ((vector_sub (f x)) (g x)))) (vec (NUMERAL _0))) net)))))


Fan_defs.set_of_edge

|- (!v. (!E. (!V. ((((set_of_edge v) V) E) = (GSPEC (\GEN%PVAR%5439. (?w. (((SETSPEC GEN%PVAR%5439) (((v INSERT (w INSERT EMPTY)) IN E) /\ (w IN V))) w))))))))


CX_SUB

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


UNIFORMLY_CONTINUOUS_ON_ADD

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


HAS_COMPLEX_DERIVATIVE_MUL_AT

|- (!f. (!f'. (!g. (!g'. (!x. (((((has_complex_derivative f) f') (at x)) /\ (((has_complex_derivative g) g') (at x))) ==> (((has_complex_derivative (\x. ((complex_mul (f x)) (g x)))) ((vector_add ((complex_mul (f x)) g')) ((complex_mul f') (g x)))) (at x))))))))


Localization.AZIM_CYCLE_EQ_SIGMA_FAN_ALT

|- (!V. (!E. (!u. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ (u IN (((set_of_edge v) V) E))) ==> (((((azim_cycle (((set_of_edge v) V) E)) (vec (NUMERAL _0))) v) u) = (((((sigma_fan (vec (NUMERAL _0))) V) E) v) u)))))))


manhattan

|- (manhattan = (metric (UNIV , (GABS (\f. (!x. (!y. ((GEQ (f (x , y))) ((sum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. (real_abs ((real_sub (x $ i)) (y $ i)))))))))))))


ABSOLUTE_RETRACT_PATH_IMAGE_ARC

|- (!g. (!s. (((arc g) /\ ((path_image g) SUBSET s)) ==> ((retract_of (path_image g)) s))))


INTEGRABLE_BOUNDED_VARIATION_PRODUCT

|- (!f. (!g. (!a. (!b. ((((integrable_on f) (closed_interval ((CONS (a , b)) NIL))) /\ ((has_bounded_variation_on g) (closed_interval ((CONS (a , b)) NIL)))) ==> ((integrable_on (\x. ((% (drop (g x))) (f x)))) (closed_interval ((CONS (a , b)) NIL))))))))


dist

|- (!x. (!y. ((distance (x , y)) = (vector_norm ((vector_sub x) y)))))


OPEN_BIJECTIVE_LINEAR_IMAGE_EQ

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


PERMUTES_INVERSES

|- (!p. (!s. (((permutes p) s) ==> ((!x. ((p ((inverse p) x)) = x)) /\ (!x. (((inverse p) (p x)) = x))))))


IN_COMPONENTS_NONEMPTY

|- (!s. (!c. ((c IN (components s)) ==> (~ (c = EMPTY)))))


REAL_ARCH_INV

|- (!e. (((real_lt (real_of_num (NUMERAL _0))) e) = (?n. ((~ (n = (NUMERAL _0))) /\ (((real_lt (real_of_num (NUMERAL _0))) (real_inv (real_of_num n))) /\ ((real_lt (real_inv (real_of_num n))) e))))))


Add_triangle.azim_split_fan_face_add

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((LET (\w'. (LET_END ((LET (\d2. (LET_END ((LET (\F2. (LET_END ((((azim_dart (V , E)) d2) = ((real_add ((azim_dart F2) (w , w'))) ((azim_dart F2) d2))) /\ (((azim_dart (V , E)) (w' , v)) = ((real_add ((azim_dart F2) (w' , w))) ((azim_dart F2) (w' , v)))))))) ((split_fan_face (V , E)) (v , w)))))) ((f_fan_pair (V , E)) (v , w)))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)))))))))


positive_semidefinite

|- (!A. ((positive_semidefinite A) = (((transp A) = A) /\ (!x. ((real_le (real_of_num (NUMERAL _0))) ((dot x) ((matrix_vector_mul A) x)))))))


SET_PROVE_CASES

|- (!P. (((P EMPTY) /\ (!a. (!s. ((~ (a IN s)) ==> (P (a INSERT s)))))) ==> (!s. (P s))))


Add_triangle.n_split_fan_face

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> (!v. (!w. ((((v , w) IN (dart_of_fan (V , E))) /\ ((NUMERAL (BIT1 (BIT1 _0))) < (CARD ((face (hypermap_of_fan (V , E))) (v , w))))) ==> ((((LET (\w1. (\d2. (\n. (LET_END (((n (w1 , w)) = ((n_fan_pair (V , E)) (w1 , v))) /\ (((n (w1 , v)) = (w1 , w)) /\ (((n d2) = (w , w1)) /\ (((n (w , w1)) = (w , v)) /\ (!d. (((d IN (dart_of_fan ((split_fan_face (V , E)) (v , w)))) /\ (~ ((d = (w1 , w)) \/ ((d = (w1 , v)) \/ ((d = d2) \/ (d = (w , w1))))))) ==> ((n d) = ((n_fan_pair (V , E)) d))))))))))))) (((((sigma_fan (vec (NUMERAL _0))) V) E) v) w)) ((f_fan_pair (V , E)) (v , w))) (n_fan_pair ((split_fan_face (V , E)) (v , w)))))))))))


Dih2k_hypermap.F_SY

|- (!v. ((F_SY v) = (GSPEC (\GEN%PVAR%7490. (?i. (((SETSPEC GEN%PVAR%7490) (((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV)))) (((row i) v) , ((row (SUC (i MOD (dimindex UNIV)))) v))))))))


REAL_EQ_ADD_RCANCEL

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


complex_neg

|- ((vector_neg z) = (complex ((real_neg (Re z)) , (real_neg (Im z)))))


List_hypermap.elements_of_list

|- (!L. ((elements_of_list L) = (set_of_list (list_of_elements L))))


Sphere.aff

|- (aff = (hull affine))


Seq.flatten

|- (flatten = ((foldr cat) NIL))


HAS_COMPLEX_DERIVATIVE_INV_AT

|- (!f. (!f'. (!x. (((((has_complex_derivative f) f') (at x)) /\ (~ ((f x) = (Cx (real_of_num (NUMERAL _0)))))) ==> (((has_complex_derivative (\x. (complex_inv (f x)))) ((complex_div (vector_neg f')) ((complex_pow (f x)) (NUMERAL (BIT0 (BIT1 _0)))))) (at x))))))


Fnjlbxs.closed_ball_annulus

|- (closed ball_annulus)


HOMOTOPIC_PATHS_IMP_SUBSET

|- (!s. (!p. (!q. ((((homotopic_paths s) p) q) ==> (((path_image p) SUBSET s) /\ ((path_image q) SUBSET s))))))


OPEN_SURJECTIVE_LINEAR_IMAGE

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


CNJ_EQ_0

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


Sphere.dih3_y

|- (!y3. (!y1. (!y2. (!y6. (!y4. (!y5. (((((((dih3_y y1) y2) y3) y4) y5) y6) = ((((((dih_y y3) y1) y2) y6) y4) y5))))))))


Fan_defs.dart_of_fan

|- (!V. (!E. ((dart_of_fan (V , E)) = ((GSPEC (\GEN%PVAR%5441. (?v. (((SETSPEC GEN%PVAR%5441) ((v IN V) /\ ((((set_of_edge v) V) E) = EMPTY))) (v , v))))) UNION (GSPEC (\GEN%PVAR%5442. (?v. (?w. (((SETSPEC GEN%PVAR%5442) ((v INSERT (w INSERT EMPTY)) IN E)) (v , w))))))))))


ANR_SPHERE

|- (!a. (!r. (ANR (sphere (a , r)))))


REAL_LE_REFL

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


CONVEX_UNIV

|- (convex UNIV)


MEASURE_UNIQUE

|- (!s. (!m. (((has_measure s) m) ==> ((measure s) = m))))


Hypermap_and_fan.SURROUNDED_IMP_CARD_SET_OF_EDGE_GE_3

|- (!V. (!E. (!v. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((v IN V) /\ ((surrounded_node (V , E)) v))) ==> ((CARD (((set_of_edge v) V) E)) >= (NUMERAL (BIT1 (BIT1 _0))))))))


UNIFORMLY_CONTINUOUS_ON_VSUM

|- (!t. (!f. (!s. (((FINITE s) /\ (!a. ((a IN s) ==> ((uniformly_continuous_on (f a)) t)))) ==> ((uniformly_continuous_on (\x. ((vsum s) (\a. ((f a) x))))) t)))))


REAL_INTEGRABLE_LMUL_EQ

|- (!f. (!s. (!c. (((real_integrable_on (\x. ((real_mul c) (f x)))) s) = ((c = (real_of_num (NUMERAL _0))) \/ ((real_integrable_on f) s))))))


CLOSEST_POINT_SELF

|- (!s. (!x. ((x IN s) ==> (((closest_point s) x) = x))))


Add_triangle.sigma_fan_inV

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


int_add_th

|- (!x. (!y. ((real_of_int ((int_add x) y)) = ((real_add (real_of_int x)) (real_of_int y)))))


Ssrbool.orbC

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


ARG_EQ_0_PI

|- (!z. ((((Arg z) = (real_of_num (NUMERAL _0))) \/ ((Arg z) = pi)) = (real z)))


INSIDE_NO_OVERLAP

|- (!s. (((inside s) INTER s) = EMPTY))


LIPSCHITZ_IMP_CONTINUOUS_ON

|- (!f. (!s. ((?B. (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> ((continuous_on f) s))))


Collect_geom2.ETA_Y_POS_LE

|- ((real_le (real_of_num (NUMERAL _0))) (((eta_y ((dist3 v1) v2)) ((dist3 v1) v3)) ((dist3 v2) v3)))


Topology.AFF_GE_1_1

|- (!x. (!v. ((~ (x = v)) ==> (((aff_ge (x INSERT EMPTY)) (v INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5522. (?y. (((SETSPEC GEN%PVAR%5522) (?t1. (?t2. (((real_le (real_of_num (NUMERAL _0))) t2) /\ ((((real_add t1) t2) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (y = ((vector_add ((% t1) x)) ((% t2) v)))))))) y))))))))


REAL_CONTINUOUS_INV_WITHINREAL

|- (!f. (!s. (!a. ((((real_continuous f) ((within (atreal a)) s)) /\ (~ ((f a) = (real_of_num (NUMERAL _0))))) ==> ((real_continuous (\x. (real_inv (f x)))) ((within (atreal a)) s))))))


HAS_REAL_MEASURE_REAL_NEGLIGIBLE_SYMDIFF

|- (!s. (!t. (!m. ((((has_real_measure s) m) /\ (real_negligible ((s DIFF t) UNION (t DIFF s)))) ==> ((has_real_measure t) m)))))


CLOSED_TRANSLATION_EQ

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


HOMEOMORPHIC_HYPERPLANE_STANDARD_HYPERPLANE

|- (!a. (!b. (!k. (!c. (((~ (a = (vec (NUMERAL _0)))) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> ((homeomorphic (GSPEC (\GEN%PVAR%1058. (?x. (((SETSPEC GEN%PVAR%1058) (((dot a) x) = b)) x))))) (GSPEC (\GEN%PVAR%1059. (?x. (((SETSPEC GEN%PVAR%1059) ((x $ k) = c)) x))))))))))


VECTOR_SUB

|- (((vector_sub x) y) = ((vector_add x) (vector_neg y)))


ARG

|- (!z. (((real_le (real_of_num (NUMERAL _0))) (Arg z)) /\ (((real_lt (Arg z)) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (z = ((complex_mul (Cx (vector_norm z))) (cexp ((complex_mul ii) (Cx (Arg z)))))))))


Hypermap_iso.hyp_iso_ext

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


BETWEEN_DIST_LE

|- (!r. (!a. (!b. (!c. ((((real_le (distance (c , a))) r) /\ (((real_le (distance (c , b))) r) /\ ((between x) (a , b)))) ==> ((real_le (distance (c , x))) r))))))


SEGMENT_CLOSED_OPEN

|- (!a. (!b. ((closed_segment ((CONS (a , b)) NIL)) = ((open_segment (a , b)) UNION (a INSERT (b INSERT EMPTY))))))


Seq.allP

|- (!a. (!s. ((!x. (((MEM x) s) ==> (a x))) = ((all a) s))))


Sphere.pathL

|- (!a. (!b. ((pathL (a , b)) = (a , ((real_div ((real_add a) b)) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))))


HAS_COMPLEX_DERIVATIVE_SUB

|- (!f. (!f'. (!g. (!g'. (!net. (((((has_complex_derivative f) f') net) /\ (((has_complex_derivative g) g') net)) ==> (((has_complex_derivative (\x. ((vector_sub (f x)) (g x)))) ((vector_sub f') g')) net)))))))


NORM_PASTECART_LE

|- (!x. (!y. ((real_le (vector_norm ((pastecart x) y))) ((real_add (vector_norm x)) (vector_norm y)))))


INT_NOT_LT

|- (!x. (!y. ((~ ((int_lt x) y)) = ((int_le y) x))))


Fan_defs.f_fan_pair_ext

|- (!V. (!E. (!x. (((f_fan_pair_ext (V , E)) x) = (((COND (x IN (dart1_of_fan (V , E)))) ((f_fan_pair (V , E)) x)) x)))))


Hypermap_iso.IN_TRANS

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


Ssrnat.minnC

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


REAL_EQ_ADD_LCANCEL_0

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


Sphere.periodic

|- (!n. (!f. (((periodic f) n) = (!i. ((f (i + n)) = (f i))))))


Seq.reshape

|- ((((reshape ((CONS n) sh)) s) = ((CONS ((take n) s)) ((reshape sh) ((dropl n) s)))) /\ (((reshape NIL) s) = NIL))


FACET_OF_HALFSPACE_LE

|- (!f. (!a. (!b. (((facet_of f) (GSPEC (\GEN%PVAR%2216. (?x. (((SETSPEC GEN%PVAR%2216) ((real_le ((dot a) x)) b)) x))))) = ((~ (a = (vec (NUMERAL _0)))) /\ (f = (GSPEC (\GEN%PVAR%2217. (?x. (((SETSPEC GEN%PVAR%2217) (((dot a) x) = b)) x))))))))))


FINITE_RECURSION

|- (!f. (!b. ((!x. (!y. (!s. ((~ (x = y)) ==> (((f x) ((f y) s)) = ((f y) ((f x) s))))))) ==> (((((ITSET f) EMPTY) b) = b) /\ (!x. (!s. ((FINITE s) ==> ((((ITSET f) (x INSERT s)) b) = (((COND (x IN s)) (((ITSET f) s) b)) ((f x) (((ITSET f) s) b)))))))))))


REAL_CONVEX_ON_SECOND_DERIVATIVE

|- (!f. (!f'. (!f''. (!s. (((is_realinterval s) /\ ((~ (?a. (s = (a INSERT EMPTY)))) /\ ((!x. ((x IN s) ==> (((has_real_derivative f) (f' x)) ((within (atreal x)) s)))) /\ (!x. ((x IN s) ==> (((has_real_derivative f') (f'' x)) ((within (atreal x)) s))))))) ==> (((real_convex_on f) s) = (!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (f'' x))))))))))


Hypermap.lemma_dart_invariant

|- (!H. (!x. ((x IN (dart H)) ==> ((((edge_map H) x) IN (dart H)) /\ ((((node_map H) x) IN (dart H)) /\ (((face_map H) x) IN (dart H)))))))


COUNTABLE_RATIONAL

|- (COUNTABLE rational)


Seq.has_predU

|- (!a1. (!a2. (!s. (((has ((predU a1) a2)) s) = (((has a1) s) \/ ((has a2) s))))))


AFF_GT_INJECTIVE_LINEAR_IMAGE

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


REAL_LT_IMP_NZ

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


log_convex_on

|- (!s. (!f. (((log_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 (real_of_num (NUMERAL _0))) (f ((vector_add ((% u) x)) ((% v) y)))) /\ ((real_le (f ((vector_add ((% u) x)) ((% v) y)))) ((real_mul ((rpow (f x)) u)) ((rpow (f y)) v))))))))))))


Tame_defs.tame_9a

|- (!H. ((tame_9a H) = (!x. ((x IN (dart H)) ==> (((CARD ((face H) x)) >= (NUMERAL (BIT1 (BIT1 _0)))) /\ ((CARD ((face H) x)) <= (NUMERAL (BIT0 (BIT1 (BIT1 _0))))))))))


CLOSED_IN_CLOSED_TRANS

|- (!s. (!t. ((((closed_in ((subtopology euclidean) t)) s) /\ (closed t)) ==> (closed s))))


COMPLEX_MUL_RZERO

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


RE_CCOS

|- (!z. ((Re (ccos z)) = ((real_mul ((real_div ((real_add (exp (Im z))) (exp (real_neg (Im z))))) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) (cos (Re z)))))


HAS_SIZE_FACES_OF_SIMPLEX

|- (!n. (!s. (((simplex n) s) ==> ((GSPEC (\GEN%PVAR%2479. (?f. (((SETSPEC GEN%PVAR%2479) ((face_of f) s)) f)))) HAS_SIZE ((NUMERAL (BIT0 (BIT1 _0))) EXP (num_of_int ((int_add n) (int_of_num (NUMERAL (BIT1 _0))))))))))


segment

|- (((closed_segment ((CONS (a , b)) NIL)) = (GSPEC (\GEN%PVAR%559. (?u. (((SETSPEC GEN%PVAR%559) (((real_le (real_of_num (NUMERAL _0))) u) /\ ((real_le u) (real_of_num (NUMERAL (BIT1 _0)))))) ((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) u)) a)) ((% u) b))))))) /\ ((open_segment (a , b)) = ((closed_segment ((CONS (a , b)) NIL)) DIFF (a INSERT (b INSERT EMPTY)))))


REAL_CX

|- (!x. (real (Cx x)))


bilinear

|- (!f. ((bilinear f) = ((!x. (linear (\y. ((f x) y)))) /\ (!y. (linear (\x. ((f x) y)))))))


LIPSCHITZ_IMP_HAS_BOUNDED_VARIATION

|- (!f. (!s. (!B. (((bounded s) /\ (!x. (!y. (((x IN s) /\ (y IN s)) ==> ((real_le (vector_norm ((vector_sub (f x)) (f y)))) ((real_mul B) (vector_norm ((vector_sub x) y)))))))) ==> ((has_bounded_variation_on f) s)))))


WITHINREAL_UNIV

|- (!x. (((within (atreal x)) UNIV) = (atreal x)))


IN_COMPONENTS_CONNECTED

|- (!s. (!c. ((c IN (components s)) ==> (connected c))))


NUM_OF_INT

|- (!x. (((int_le (int_of_num (NUMERAL _0))) x) = ((int_of_num (num_of_int x)) = x)))


HOMOTOPIC_WITH_IMP_SUBSET

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


Hypermap_and_fan.N_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (n_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))


ORTHOGONAL_TRANSFORMATION_ID

|- (orthogonal_transformation (\x. x))


BASIS_EXISTS

|- (!v. (?b. ((b SUBSET v) /\ ((independent b) /\ ((v SUBSET (span b)) /\ (b HAS_SIZE (dim v)))))))


RE_DEF

|- (!z. ((Re z) = (z $ (NUMERAL (BIT1 _0)))))


ORTHOGONAL_SUBSPACE_DECOMP_UNIQUE

|- (!s. (!t. (!x. (!y. (!x'. (!y'. (((!a. (!b. (((a IN s) /\ (b IN t)) ==> ((orthogonal a) b)))) /\ ((x IN (span s)) /\ ((x' IN (span s)) /\ ((y IN (span t)) /\ ((y' IN (span t)) /\ (((vector_add x) y) = ((vector_add x') y'))))))) ==> ((x = x') /\ (y = y')))))))))


FACE_OF_HALFSPACE_LE

|- (!f. (!a. (!b. (((face_of f) (GSPEC (\GEN%PVAR%2153. (?x. (((SETSPEC GEN%PVAR%2153) ((real_le ((dot a) x)) b)) x))))) = ((f = EMPTY) \/ ((f = (GSPEC (\GEN%PVAR%2154. (?x. (((SETSPEC GEN%PVAR%2154) (((dot a) x) = b)) x))))) \/ (f = (GSPEC (\GEN%PVAR%2155. (?x. (((SETSPEC GEN%PVAR%2155) ((real_le ((dot a) x)) b)) x)))))))))))


DIAMETER_SUBSET

|- (!s. (!t. (((s SUBSET t) /\ (bounded t)) ==> ((real_le (diameter s)) (diameter t)))))


HAS_MEASURE_0

|- (!s. (((has_measure s) (real_of_num (NUMERAL _0))) = (negligible s)))


INTEGRAL_COMPONENT

|- (!f. (((integrable_on f) s) ==> ((((integral s) f) $ k) = (drop ((integral s) (\x. (lift ((f x) $ k))))))))


LIFT_NEG

|- (!x. ((lift (real_neg x)) = (vector_neg (lift x))))


SETDIST_EMPTY

|- ((!t. ((setdist (EMPTY , t)) = (real_of_num (NUMERAL _0)))) /\ (!s. ((setdist (s , EMPTY)) = (real_of_num (NUMERAL _0)))))


REAL_INV_POW

|- (!x. (!n. ((real_inv ((real_pow x) n)) = ((real_pow (real_inv x)) n))))


real_infsum

|- (!f. (!s. (((real_infsum s) f) = (@l. (((real_sums f) l) s)))))


CLOSED_IN_EMPTY

|- (!top. ((closed_in top) EMPTY))


CONTINUOUS_ON_ADD

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


SUM_SUB

|- (!f. (!g. (!s. ((FINITE s) ==> (((sum s) (\x. ((real_sub (f x)) (g x)))) = ((real_sub ((sum s) f)) ((sum s) g)))))))


NOT_LE

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


Planarity.exists_dartset_leads_into_fan

|- (!x. (!V. (!E. (!ds. (((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)))))))) ==> (?s. (!y. ((y IN ds) ==> (s = (((((dart_leads_into x) V) E) (pr2 y)) (pr3 y)))))))))))


real_uniformly_continuous_on

|- (!s. (!f. (((real_uniformly_continuous_on f) s) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (!x'. (((x IN s) /\ ((x' IN s) /\ ((real_lt (real_abs ((real_sub x') x))) d))) ==> ((real_lt (real_abs ((real_sub (f x')) (f x)))) e)))))))))))


SERIES_BOUND

|- (!f. (!g. (!s. (!a. (!b. (((((sums f) a) s) /\ ((((sums (lift o g)) (lift b)) s) /\ (!i. ((i IN s) ==> ((real_le (vector_norm (f i))) (g i)))))) ==> ((real_le (vector_norm a)) b)))))))


DIST_IN_CLOSED_SEGMENT

|- (!a. (!b. (!x. ((x IN (closed_segment ((CONS (a , b)) NIL))) ==> (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b))))))))


POLYHEDRON_NEGATIONS

|- (!s. ((polyhedron s) ==> (polyhedron ((IMAGE vector_neg) s))))


HAS_COMPLEX_DERIVATIVE_CSIN

|- (!z. (((has_complex_derivative csin) (ccos z)) (at z)))


SKOLEM_THM

|- (!P. ((!x. (?y. ((P x) y))) = (?y. (!x. ((P x) (y x))))))


COMPLEX_POLY_CLAUSES

|- ((!x. (!y. (!z. (((vector_add x) ((vector_add y) z)) = ((vector_add ((vector_add x) y)) z))))) /\ ((!x. (!y. (((vector_add x) y) = ((vector_add y) x)))) /\ ((!x. (((vector_add (Cx (real_of_num (NUMERAL _0)))) x) = x)) /\ ((!x. (!y. (!z. (((complex_mul x) ((complex_mul y) z)) = ((complex_mul ((complex_mul x) y)) z))))) /\ ((!x. (!y. (((complex_mul x) y) = ((complex_mul y) x)))) /\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL (BIT1 _0))))) x) = x)) /\ ((!x. (((complex_mul (Cx (real_of_num (NUMERAL _0)))) x) = (Cx (real_of_num (NUMERAL _0))))) /\ ((!x. (!y. (!z. (((complex_mul x) ((vector_add y) z)) = ((vector_add ((complex_mul x) y)) ((complex_mul x) z)))))) /\ ((!x. (((complex_pow x) (NUMERAL _0)) = (Cx (real_of_num (NUMERAL (BIT1 _0)))))) /\ (!x. (!n. (((complex_pow x) (SUC n)) = ((complex_mul x) ((complex_pow x) n))))))))))))))


CONVEX_ON_LEFT_SECANT

|- (!f. (!s. (((convex_on f) s) = (!a. (!b. (!x. (((a IN s) /\ ((b IN s) /\ (x IN (open_segment (a , b))))) ==> ((real_le ((real_div ((real_sub (f x)) (f a))) (vector_norm ((vector_sub x) a)))) ((real_div ((real_sub (f b)) (f a))) (vector_norm ((vector_sub b) a)))))))))))


Tame_defs2.list_update

|- (((((list_update NIL) i) v) = NIL) /\ ((((list_update ((CONS x) xs)) i) v) = (((COND (i = (NUMERAL _0))) ((CONS v) xs)) ((CONS x) (((list_update xs) (i - (NUMERAL (BIT1 _0)))) v)))))


Trigonometry2.TWO_PI_POS

|- ((real_lt (real_of_num (NUMERAL _0))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi))


HAS_COMPLEX_DERIVATIVE_AT_WITHIN

|- (!f. (!f'. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative f) f') ((within (at x)) s)))))))


RETRACT_OF_TRANSLATION_EQ

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


CONTRACTIBLE_INJECTIVE_LINEAR_IMAGE

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


BASIS_EXPANSION

|- (!x. (((vsum ((NUMERAL (BIT1 _0)) .. (dimindex UNIV))) (\i. ((% (x $ i)) (basis i)))) = x))


MEASURE_INTERVAL_2

|- ((!a. (!b. ((measure (closed_interval ((CONS (a , b)) NIL))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))) /\ (!a. (!b. ((measure (open_interval (a , b))) = (((COND (((real_le (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 _0)))) /\ ((real_le (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((real_mul ((real_sub (b $ (NUMERAL (BIT1 _0)))) (a $ (NUMERAL (BIT1 _0))))) ((real_sub (b $ (NUMERAL (BIT0 (BIT1 _0))))) (a $ (NUMERAL (BIT0 (BIT1 _0))))))) (real_of_num (NUMERAL _0)))))))


HOMOTOPIC_ORTHOGONAL_TRANSFORMATIONS_IMP

|- (!f. (!g. (((orthogonal_transformation f) /\ ((orthogonal_transformation g) /\ ((((homotopic_with (\x. T)) ((sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))) , (sphere ((vec (NUMERAL _0)) , (real_of_num (NUMERAL (BIT1 _0))))))) f) g))) ==> ((det (matrix f)) = (det (matrix g))))))


int_eq

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


connected_component

|- (!s. (!x. (!y. ((((connected_component s) x) y) = (?t. ((connected t) /\ ((t SUBSET s) /\ ((x IN t) /\ (y IN t)))))))))


HAS_REAL_INTEGRAL_SPIKE_INTERIOR

|- (!f. (!g. (!a. (!b. (!y. (((!x. ((x IN (open_real_interval (a , b))) ==> ((g x) = (f x)))) /\ (((has_real_integral f) y) (closed_real_interval ((CONS (a , b)) NIL)))) ==> (((has_real_integral g) y) (closed_real_interval ((CONS (a , b)) NIL)))))))))


TOPSPACE_EUCLIDEAN

|- ((topspace euclidean) = UNIV)


Hypermap.number_of_faces

|- (!H. ((number_of_faces H) = (CARD (face_set H))))


OPEN_UNIONS

|- ((!s. ((s IN f) ==> (open s))) ==> (open (UNIONS f)))


Lp_ineqs_proofs.add_diag_list

|- (!V. (!g. (!E. (!diags. (((add_diag_list (g , (V , E))) diags) = (((foldr (\d. (\E'. ((add_diag (V , E')) (g d))))) E) diags))))))


HULL_INC

|- (!P. (!s. (!x. ((x IN s) ==> (x IN ((hull P) s))))))


FACE_OF_CONVEX_HULLS

|- (!f. (!s. (((FINITE s) /\ ((f SUBSET s) /\ ((DISJOINT ((hull affine) f)) ((hull convex) (s DIFF f))))) ==> ((face_of ((hull convex) f)) ((hull convex) s)))))


REAL_LT_REFL

|- (!x. (~ ((real_lt x) x)))


REALLIM_SEQUENTIALLY

|- (!s. (!l. ((((---> s) l) sequentially) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (s n)) l))) e)))))))))


COUNTABLE_ALT

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


Lp_ineqs_proofs.y1_hi

|- (!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 (y1_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))


lt_c

|- (!t. (!s. ((s <_c t) = ((s <=_c t) /\ (~ (t <=_c s))))))


WLOG_LE

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


CBALL_INTERVAL

|- (!x. (!e. ((cball (x , e)) = (closed_interval ((CONS (((vector_sub x) (lift e)) , ((vector_add x) (lift e)))) NIL)))))


Real_ext.REAL_PROP_POS_MUL2

|- (!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_mul x) y)))))


Collect_geom.C_COEF_FORMULA

|- (!x12. (!x13. (!x14. (!x23. (!x24. ((((((c_coef x12) x13) x14) x23) x24) = ((real_add ((real_mul ((real_mul x14) x23)) x12)) ((real_add ((real_mul ((real_mul x14) x23)) x13)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x14)) ((real_add ((real_mul ((real_mul (real_neg x14)) x23)) x23)) ((real_add ((real_mul ((real_mul x14) x23)) x24)) ((real_add ((real_mul ((real_mul (real_neg x12)) x13)) x23)) ((real_add ((real_mul ((real_mul (real_neg x12)) x14)) x24)) ((real_add ((real_mul ((real_mul x13) x24)) x12)) ((real_add ((real_mul ((real_mul (real_neg x13)) x24)) x13)) ((real_add ((real_mul ((real_mul x13) x24)) x14)) ((real_add ((real_mul ((real_mul x13) x24)) x23)) ((real_mul ((real_mul (real_neg x13)) x24)) x24))))))))))))))))))


CASN_CSIN

|- (!z. ((((real_lt (real_abs (Re z))) ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) \/ (((real_abs (Re z)) = ((real_div pi) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) /\ ((Im z) = (real_of_num (NUMERAL _0))))) ==> ((casn (csin z)) = z)))


Dih2k_hypermap.SUC_NOT

|- (((NUMERAL (BIT1 _0)) < k) ==> (((NUMERAL (BIT1 _0)) <= (SUC (i MOD k))) /\ (((SUC (i MOD k)) <= k) /\ (~ (i = (SUC (i MOD k)))))))


LEBESGUE_DENSITY_THEOREM_INTEGRAL_CBALL

|- (!f. (!s. ((!a. (!b. ((absolutely_integrable_on f) (s INTER (closed_interval ((CONS (a , b)) NIL)))))) ==> (?k. ((negligible k) /\ (!x. ((~ (x IN k)) ==> (((--> (\e. ((% (real_inv (measure (cball (x , (drop e)))))) ((integral (s INTER (cball (x , (drop e))))) f)))) (((COND (x IN s)) (f x)) (vec (NUMERAL _0)))) ((within (at (vec (NUMERAL _0)))) (GSPEC (\GEN%PVAR%3724. (?t. (((SETSPEC GEN%PVAR%3724) ((real_lt (real_of_num (NUMERAL _0))) (drop t))) t)))))))))))))


Nonlinear_lemma.rho_alt

|- (!y. ((rho y) = ((real_add (real_of_num (NUMERAL (BIT1 _0)))) ((real_mul const1) ((real_div ((real_sub y) (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 _0)))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))


Ssrnat.subn_gt0

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


Fan.f1_fan

|- (!x. (!V. (!E. ((((f1_fan x) V) E) = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) (x , (w , ((((((inverse1_sigma_fan x) V) E) w) v) , v))))))))))))))


Lp_ineqs_def.y3_fan

|- (!V. (!E. (!x. (((y3_fan (V , E)) x) = (vector_norm (FST ((inverse (f_fan_pair_ext (V , E))) x)))))))


ANR_BALL

|- (!a. (!r. (ANR (ball (a , r)))))


CLOSED_IN_TRANS

|- (!s. (!t. (!u. ((((closed_in ((subtopology euclidean) t)) s) /\ ((closed_in ((subtopology euclidean) u)) t)) ==> ((closed_in ((subtopology euclidean) u)) s)))))


Nonlin_def.scalar6

|- (!f. (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. (!r. (((((((((scalar6 f) r) x1) x2) x3) x4) x5) x6) = ((real_mul ((((((f x1) x2) x3) x4) x5) x6)) r))))))))))


vertex_image

|- (!f. (!s. (((vertex_image f) s) = ((hull convex) ((IMAGE f) (GSPEC (\GEN%PVAR%2552. (?v. (((SETSPEC GEN%PVAR%2552) ((extreme_point_of v) s)) v)))))))))


WF_INSEG_WOSET

|- (WF (\x. (\y. ((woset x) /\ ((woset y) /\ (((inseg x) y) /\ (~ (x = y))))))))


List_hypermap.list_of_elements

|- (!L. ((list_of_elements L) = (undup (flatten L))))


CHOICE_DEF

|- (!s. ((~ (s = EMPTY)) ==> ((CHOICE s) IN s)))


SUM_SUPPORT

|- (!f. (!s. (((sum (((support real_add) f) s)) f) = ((sum s) f))))


Fnjlbxs.infnorm_lt

|- (!e. (!v. ((!i. ((((NUMERAL (BIT1 _0)) <= i) /\ (i <= (dimindex UNIV))) ==> ((real_lt (real_abs (v $ i))) e))) ==> ((real_lt (infnorm v)) e))))


I_O_ID

|- (!f. (((I o f) = f) /\ ((f o I) = f)))


Fan.collinear_imp_azim_is_rezo_fan

|- (!x. (!v. (!u. (!y. (!h1. (!h2. (((~ (v = x)) /\ ((~ (u = x)) /\ ((~ (collinear (x INSERT (v INSERT (u INSERT EMPTY))))) /\ (((real_le (real_of_num (NUMERAL _0))) y) /\ (((real_lt y) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) pi)) /\ (!e1. (!e2. (!e3. (((((orthonormal e1) e2) e3) /\ (((% (distance (v , x))) e3) = ((vector_sub v) x))) ==> (?psi. (?r1. (?r2. ((((vector_sub u) x) = ((vector_add ((% ((real_mul r1) (cos psi))) e1)) ((vector_add ((% ((real_mul r1) (sin psi))) e2)) ((% h1) ((vector_sub v) x))))) /\ ((((vector_sub u) x) = ((vector_add ((% ((real_mul r2) (cos ((real_add psi) y)))) e1)) ((vector_add ((% ((real_mul r2) (sin ((real_add psi) y)))) e2)) ((% h2) ((vector_sub v) x))))) /\ (((real_lt (real_of_num (NUMERAL _0))) r1) /\ ((real_lt (real_of_num (NUMERAL _0))) r2)))))))))))))))) ==> (y = (real_of_num (NUMERAL _0))))))))))


Collect_geom.ETA_Y_SYYM

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


rcone_ge

|- (rcone_ge = (rconesgn real_ge))


Hypermap.face_set

|- (!H. ((face_set H) = ((set_of_orbits (dart H)) (face_map H))))


STC

|- (!R. ((STC R) = (TC (SC R))))


IN_DELETE

|- (!s. (!x. (!y. ((x IN (s DELETE y)) = ((x IN s) /\ (~ (x = y)))))))


TRACE_COVARIANCE_CAUCHY_SCHWARZ

|- (!A. (!B. ((real_le (trace ((matrix_mul (transp A)) B))) ((real_mul (sqrt (trace ((matrix_mul (transp A)) A)))) (sqrt (trace ((matrix_mul (transp B)) B)))))))


Planarity.AFFINE_HULL_1

|- (!a. (((hull affine) (a INSERT EMPTY)) = (GSPEC (\GEN%PVAR%5568. (?u. (((SETSPEC GEN%PVAR%5568) (u = (real_of_num (NUMERAL (BIT1 _0))))) ((% u) a)))))))


Ssrnat.subn_sub

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


EMPTY_INTERIOR_LOWDIM

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


INTERVAL_TRANSLATION

|- ((!c. (!a. (!b. ((closed_interval ((CONS (((vector_add c) a) , ((vector_add c) b))) NIL)) = ((IMAGE (\x. ((vector_add c) x))) (closed_interval ((CONS (a , b)) NIL))))))) /\ (!c. (!a. (!b. ((open_interval (((vector_add c) a) , ((vector_add c) b))) = ((IMAGE (\x. ((vector_add c) x))) (open_interval (a , b))))))))


PATH_COMPONENT_EQ_EMPTY

|- (!s. (!x. ((((path_component s) x) = EMPTY) = (~ (x IN s)))))


NORM_POS_LE

|- (!x. ((real_le (real_of_num (NUMERAL _0))) (vector_norm x)))


SEPARATING_HYPERPLANE_AFFINE_AFFINE

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


AFFINE_HULL_SING

|- (!a. (((hull affine) (a INSERT EMPTY)) = (a INSERT EMPTY)))


LIM_COMPLEX_REAL

|- (!f. (!g. (!l. (!m. ((((eventually (\n. ((Re (g n)) = (f n)))) sequentially) /\ (((Re m) = l) /\ (((--> g) m) sequentially))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. ((N <= n) ==> ((real_lt (real_abs ((real_sub (f n)) l))) e)))))))))))


UNIONS_INSERT

|- ((UNIONS (s INSERT u)) = (s UNION (UNIONS u)))


CONNECTED_LINEAR_IMAGE_EQ

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


CLOSED_IN_COMPONENT

|- (!s. (!c. ((c IN (components s)) ==> ((closed_in ((subtopology euclidean) s)) c))))


REAL_ADD_LDISTRIB

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


RANK_DIM_IM

|- (!A. ((rank A) = (dim ((IMAGE (\x. ((matrix_vector_mul A) x))) UNIV))))


Trigonometry2.SIMPLIZE_COS_IF_OTHOR

|- (!v0. (!v1. (!w. (!p. (((~ (v0 = w)) /\ ((~ (v0 = v1)) /\ ((((vector_sub p) v0) = ((% k) ((vector_sub v1) v0))) /\ (((dot ((vector_sub w) p)) ((vector_sub v1) v0)) = (real_of_num (NUMERAL _0)))))) ==> ((cos (((arcV v0) v1) w)) = ((real_mul k) ((real_div (vector_norm ((vector_sub v1) v0))) (vector_norm ((vector_sub w) v0))))))))))


Hypermap.lemma_dart_invariant_power_face

|- (!H. (!x. (!n. ((x IN (dart H)) ==> ((((POWER (face_map H)) n) x) IN (dart H))))))


INTERVAL_UPPERBOUND_NONEMPTY

|- (!a. (!b. ((~ ((closed_interval ((CONS (a , b)) NIL)) = EMPTY)) ==> ((interval_upperbound (closed_interval ((CONS (a , b)) NIL))) = b))))


REAL_COMPACT_INTERVAL

|- (!a. (!b. (real_compact (closed_real_interval ((CONS (a , b)) NIL)))))


LIM_WITHINREAL

|- (!f. (!l. (!a. (!s. ((((--> f) l) ((within (atreal a)) s)) = (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?d. (((real_lt (real_of_num (NUMERAL _0))) d) /\ (!x. (((x IN s) /\ (((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))))))))))))


LINEAR_FROM_1

|- (!f. ((linear f) = (?c. (f = (\x. ((% (drop x)) c))))))


Sphere.eventually_radial

|- (!C. (!x. (((eventually_radial x) C) = (?r. (((real_gt r) (real_of_num (NUMERAL _0))) /\ (((radial r) x) (C INTER (ball (x , r)))))))))


ITER_POINTLESS

|- ((!f. (((ITER (NUMERAL _0)) f) = I)) /\ (!f. (!n. (((ITER (SUC n)) f) = (f o ((ITER n) f))))))


LINEAR_CONTINUOUS_ON

|- (!f. (!s. ((linear f) ==> ((continuous_on f) s))))


LIM_CONST

|- (!net. (!a. (((--> (\x. a)) a) net)))


REAL_MEASURABLE_IMP_REAL_LEBESGUE_MEASURABLE

|- (!s. ((real_measurable s) ==> (real_lebesgue_measurable s)))


INT_LE_REFL

|- (!x. ((int_le x) x))


RE_DIV_CX

|- (!z. (!x. ((Re ((complex_div z) (Cx x))) = ((real_div (Re z)) x))))


HAS_DERIVATIVE_ID

|- (!net. (((has_derivative (\x. x)) (\h. h)) net))


Hypermap_and_fan.F_FAN_PAIR_EXT_PERMUTES_DART_OF_FAN

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((permutes (f_fan_pair_ext (V , E))) (dart_of_fan (V , E))))))


REAL_POS

|- (!n. ((real_le (real_of_num (NUMERAL _0))) (real_of_num n)))


DROP_DIFFERENTIAL_NEG_AT_MAXIMUM

|- (!f. (!f'. (!x. (!s. (!e. (((x IN s) /\ ((convex s) /\ ((((has_derivative f) f') ((within (at x)) s)) /\ (((real_lt (real_of_num (NUMERAL _0))) e) /\ (!w. ((w IN (s INTER (ball (x , e)))) ==> ((real_le (drop (f w))) (drop (f x))))))))) ==> (!y. ((y IN s) ==> ((real_le (drop (f' ((vector_sub y) x)))) (real_of_num (NUMERAL _0)))))))))))


path_length

|- (!g. ((path_length g) = ((vector_variation (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) g)))


PERMUTES_IN_IMAGE

|- (!p. (!s. (!x. (((permutes p) s) ==> (((p x) IN s) = (x IN s))))))


Appendix.FZIOTEF_TRANS

|- (!S1. (!S2. (!S3. ((((scs_arrow_v39 S1) S2) /\ ((scs_arrow_v39 S2) S3)) ==> ((scs_arrow_v39 S1) S3)))))


NORM_CPOW_REAL

|- (!w. (!z. (((real w) /\ ((real_lt (real_of_num (NUMERAL _0))) (Re w))) ==> ((vector_norm ((cpow w) z)) = (exp ((real_mul (Re z)) (log (Re w))))))))


REAL_INTEGRABLE_ON

|- (((real_integrable_on f) s) = ((integrable_on (lift o (f o drop))) ((IMAGE lift) s)))


LIM_NULL_COMPARISON_COMPLEX

|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (vector_norm (f x))) (vector_norm (g x))))) net) /\ (((--> g) (Cx (real_of_num (NUMERAL _0)))) net)) ==> (((--> f) (Cx (real_of_num (NUMERAL _0)))) net)))))


EXTREME_POINT_OF_CONVEX_HULL

|- (!x. (!s. (((extreme_point_of x) ((hull convex) s)) ==> (x IN s))))


OPEN_MAP_CLOSED_SUPERSET_PREIMAGE

|- (!f. (!s. (!t. (!u. (!w. (((!k. (((open_in ((subtopology euclidean) s)) k) ==> ((open_in ((subtopology euclidean) t)) ((IMAGE f) k)))) /\ (((closed_in ((subtopology euclidean) s)) u) /\ ((w SUBSET t) /\ ((GSPEC (\GEN%PVAR%697. (?x. (((SETSPEC GEN%PVAR%697) ((x IN s) /\ ((f x) IN w))) x)))) SUBSET u)))) ==> (?v. (((closed_in ((subtopology euclidean) t)) v) /\ ((w SUBSET v) /\ ((GSPEC (\GEN%PVAR%698. (?x. (((SETSPEC GEN%PVAR%698) ((x IN s) /\ ((f x) IN v))) x)))) SUBSET u))))))))))


Tame_lemmas.reflection_eq

|- (!n. (!x. ((((dot x) n) = (real_of_num (NUMERAL _0))) ==> (((reflection n) x) = x))))


differentiable

|- (!f. (!net. (((differentiable f) net) = (?f'. (((has_derivative f) f') net)))))


HOLOMORPHIC_ON_MUL

|- (!f. (!g. (!s. ((((holomorphic_on f) s) /\ ((holomorphic_on g) s)) ==> ((holomorphic_on (\z. ((complex_mul (f z)) (g z)))) s)))))


REAL_ADD_ASSOC

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


HAS_COMPLEX_DERIVATIVE_CDIV_AT

|- (!f. (!f'. (!c. (!x. (!s. ((((has_complex_derivative f) f') (at x)) ==> (((has_complex_derivative (\x. ((complex_div (f x)) c))) ((complex_div f') c)) (at x))))))))


CONVEX_HULL_FINITE

|- (!s. (((hull convex) s) = (GSPEC (\GEN%PVAR%1491. (?y. (((SETSPEC GEN%PVAR%1491) (?u. ((!x. ((x IN s) ==> ((real_le (real_of_num (NUMERAL _0))) (u x)))) /\ ((((sum s) u) = (real_of_num (NUMERAL (BIT1 _0)))) /\ (((vsum s) (\x. ((% (u x)) x))) = y))))) y))))))


cross

|- (!a. (!b. (((cross a) b) = (vector ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT0 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0))))) ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT1 (BIT1 _0))))))) ((CONS ((real_sub ((real_mul (a $ (NUMERAL (BIT1 _0)))) (b $ (NUMERAL (BIT0 (BIT1 _0)))))) ((real_mul (a $ (NUMERAL (BIT0 (BIT1 _0))))) (b $ (NUMERAL (BIT1 _0)))))) NIL)))))))


REAL_MEASURABLE_ON_LEBESGUE_MEASURABLE_SUBSET

|- (!f. (!s. (!t. (((s SUBSET t) /\ (((real_measurable_on f) t) /\ (real_lebesgue_measurable s))) ==> ((real_measurable_on f) s)))))


ADMISSIBLE_COND

|- (!<<. (!p. (!P. (!s. (!h. (!k. ((((((admissible <<) p) s) P) /\ (((((admissible <<) (\f. (\x. (((p f) x) /\ ((P f) x))))) s) h) /\ ((((admissible <<) (\f. (\x. (((p f) x) /\ (~ ((P f) x)))))) s) k))) ==> ((((admissible <<) p) s) (\f. (\x. (((COND ((P f) x)) ((h f) x)) ((k f) x))))))))))))


Lp_ineqs_proofs.dart4_azim_cross_split

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((azim_dart (V , E)) d) = ((real_add ((((((dih_y (y1_fan d)) ((y3_fan (V , E)) d)) ((y7_fan (V , E)) d)) ((y8_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) ((y5_fan (V , E)) d))) ((((((dih_y (y1_fan d)) (y2_fan d)) ((y7_fan (V , E)) d)) ((y9_fan (V , E)) d)) ((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d))) (y6_fan d))))))))))))


REAL_LE_LCANCEL_IMP

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


INTEGRAL_UNIQUE

|- (!f. (!y. (!k. ((((has_integral f) y) k) ==> (((integral k) f) = y)))))


INTEGRAL_EQ_0

|- (!f. (!s. ((!x. ((x IN s) ==> ((f x) = (vec (NUMERAL _0))))) ==> (((integral s) f) = (vec (NUMERAL _0))))))


CONNECTED_COMPONENT_EQ

|- (!s. (!x. (!y. ((y IN ((connected_component s) x)) ==> (((connected_component s) y) = ((connected_component s) x))))))


REAL_LE_LADD

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


REAL_CONTINUOUS_COMPLEX_COMPONENTS_AT

|- (!z. (((real_continuous Re) (at z)) /\ ((real_continuous Im) (at z))))


REAL_MEASURABLE_SCALING

|- (!s. (!c. ((real_measurable s) ==> (real_measurable ((IMAGE (\x. ((real_mul c) x))) s)))))


OPEN_UNION

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


NUMSEG_LREC

|- (!m. (!n. ((m <= n) ==> ((m INSERT ((m + (NUMERAL (BIT1 _0))) .. n)) = (m .. n)))))


EQ_EXT

|- (!f. (!g. ((!x. ((f x) = (g x))) ==> (f = g))))


REAL_ABS_BETWEEN1

|- (!x. (!y. (!z. ((((real_lt x) z) /\ ((real_lt (real_abs ((real_sub y) x))) ((real_sub z) x))) ==> ((real_lt y) z)))))


ITERATE_BIJECTION

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


Ssrbool.andFb

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


REAL_CONTINUOUS_ON_RPOW

|- (!s. (!y. ((((real_of_num (NUMERAL _0)) IN s) ==> ((real_le (real_of_num (NUMERAL _0))) y)) ==> ((real_continuous_on (\x. ((rpow x) y))) s))))


NSUM_CLAUSES_LEFT

|- (!f. (!m. (!n. ((m <= n) ==> (((nsum (m .. n)) f) = ((f m) + ((nsum ((m + (NUMERAL (BIT1 _0))) .. n)) f)))))))


DIM_EQ_CARD

|- (!s. ((independent s) ==> ((dim s) = (CARD s))))


DIFFERENTIABLE_IMP_CONTINUOUS_AT

|- (!f. (!x. (((differentiable f) (at x)) ==> ((continuous f) (at x)))))


SUM_INJECTION

|- (!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)))))) ==> (((sum s) (f o p)) = ((sum s) f))))))


Hypermap_and_fan.CARD_FACE_GT_1

|- (!V. (!E. (!x. (((FAN ((vec (NUMERAL _0)) , (V , E))) /\ ((CARD ((face (hypermap_of_fan (V , E))) x)) > (NUMERAL (BIT1 _0)))) ==> (x IN (dart1_of_fan (V , E)))))))


COND_RAND

|- (!b. (!f. (!x. (!y. ((f (((COND b) x) y)) = (((COND b) (f x)) (f y)))))))


Nonlin_def.promote3_to_6

|- (!x4. (!x5. (!x6. (!f. (!x1. (!x2. (!x3. ((((((((promote3_to_6 f) x1) x2) x3) x4) x5) x6) = (((f x1) x2) x3)))))))))


HAS_DERIVATIVE_NEG_EQ

|- (!f. (!f'. (!net. ((((has_derivative (\x. (vector_neg (f x)))) (\h. (vector_neg (f' h)))) net) = (((has_derivative f) f') net)))))


Hypermap_iso.bij_disjoint_union

|- (!f. (!g. (!s. (!t. (!s'. (!t'. ((((DISJOINT s) t) /\ (((DISJOINT s') t') /\ ((((BIJ f) s) s') /\ (((BIJ g) t) t')))) ==> (((BIJ (\x. (((COND (x IN s)) (f x)) (g x)))) (s UNION t)) (s' UNION t')))))))))


ABSOLUTELY_INTEGRABLE_RESTRICT_UNIV

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


Fan_misc.INVERSE_SIGMA_FAN_EQ_INVERSE1_SIGMA_FAN

|- (!x. (!V. (!E. (!v. (!w. (((FAN (x , (V , E))) /\ ((v INSERT (w INSERT EMPTY)) IN E)) ==> ((((((inverse1_sigma_fan x) V) E) v) w) = (((((inverse_sigma_fan x) V) E) v) w))))))))


DROP_VEC

|- (!n. ((drop (vec n)) = (real_of_num n)))


Functional_equation.gchi_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))))))) ==> ((((((((gchi1_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x1))) ((((((dihatn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi2_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x2))) ((((((dih2atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi3_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x3))) ((((((dih3atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi4_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x4))) ((((((dih4atn_x x1) x2) x3) x4) x5) x6))) /\ ((((((((gchi5_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x5))) ((((((dih5atn_x x1) x2) x3) x4) x5) x6))) /\ (((((((gchi6_x x1) x2) x3) x4) x5) x6) = ((real_mul (gchi (sqrt x6))) ((((((dih6atn_x x1) x2) x3) x4) x5) x6)))))))))))))))


Tame_defs.contracted_dart

|- (!x. (!w1. (!v. (!w. ((contracted_dart (x , (v , (w , w1)))) = (v , w))))))


CONDENSATION_POINTS_EQ_EMPTY

|- (!s. (((GSPEC (\GEN%PVAR%1861. (?x. (((SETSPEC GEN%PVAR%1861) ((condensation_point_of x) s)) x)))) = EMPTY) = (COUNTABLE s)))


SPAN_SPAN

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


Planarity.properties_inside_collinear_fan

|- (!x. (!u. (!w. (!a. ((((real_lt (real_of_num (NUMERAL _0))) a) /\ (((real_lt a) (real_of_num (NUMERAL (BIT1 _0)))) /\ (~ (collinear (x INSERT (u INSERT (w INSERT EMPTY))))))) ==> ((~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (u INSERT EMPTY))))) /\ (~ (collinear (x INSERT (((vector_add ((% ((real_sub (real_of_num (NUMERAL (BIT1 _0)))) a)) u)) ((% a) w)) INSERT (w INSERT EMPTY)))))))))))


Fan.pr3

|- (pr3 = (GABS (\f. (!x. (!v. (!w. (!w1. ((GEQ (f (x , (v , (w , w1))))) w))))))))


RELATIVE_INTERIOR_EMPTY

|- ((relative_interior EMPTY) = EMPTY)


Miqmcsn.K_SCS_3T4_prime2

|- ((scs_k_v39 scs_3T4_prime2) = (NUMERAL (BIT1 (BIT1 _0))))


Lp_ineqs_proofs.y3_hi

|- (!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 ((y3_fan (V , E)) d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))))))))))


SIMPLICIAL_COMPLEX_IMP_TRIANGULATION

|- (!tr. ((simplicial_complex tr) ==> (triangulation tr)))


UNIFORM_LIM_COMPLEX_MUL

|- (!net. (!P. (!f. (!g. (!l. (!m. (!b1. (!b2. ((((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (l n))) b1))))) net) /\ (((eventually (\x. (!n. ((P n) ==> ((real_le (vector_norm (m n))) b2))))) net) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((f n) x)) (l n)))) e))))) net))) /\ (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((g n) x)) (m n)))) e))))) net)))))) ==> (!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> ((eventually (\x. (!n. ((P n) ==> ((real_lt (vector_norm ((vector_sub ((complex_mul ((f n) x)) ((g n) x))) ((complex_mul (l n)) (m n))))) e))))) net))))))))))))


ISO

|- (!g. (!f. (((ISO f) g) = ((!x. ((f (g x)) = x)) /\ (!y. ((g (f y)) = y))))))


MEM_EXISTS_EL

|- (!l. (!x. (((MEM x) l) = (?i. ((i < (LENGTH l)) /\ (x = ((EL i) l)))))))


CONVEX_SIMPLEX

|- (!n. (!s. (((simplex n) s) ==> (convex s))))


Hypermap.orbit_map

|- (!f. (!x. (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5390. (?n. (((SETSPEC GEN%PVAR%5390) (n >= (NUMERAL _0))) (((POWER f) n) x))))))))


VECTOR_MUL_RCANCEL

|- (!a. (!b. (!x. ((((% a) x) = ((% b) x)) = ((a = b) \/ (x = (vec (NUMERAL _0))))))))


Real_ext.REAL_PROP_POS_LMUL

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


COMPLEX_ADD_LINV

|- (!x. (((vector_add (vector_neg x)) x) = (Cx (real_of_num (NUMERAL _0)))))


TRIANGULATION_TRANSLATION

|- (!a. (!tr. ((triangulation ((IMAGE (IMAGE (\x. ((vector_add a) x)))) tr)) = (triangulation tr))))


COMPLEX_ADD_CSIN

|- (!w. (!z. (((vector_add (csin w)) (csin z)) = ((complex_mul (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))) ((complex_mul (csin ((complex_div ((vector_add w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0)))))))) (ccos ((complex_div ((vector_sub w) z)) (Cx (real_of_num (NUMERAL (BIT0 (BIT1 _0))))))))))))


REAL_LE_SETDIST_EQ

|- (!d. (!s. (!t. (((real_le d) (setdist (s , t))) = ((!x. (!y. (((x IN s) /\ (y IN t)) ==> ((real_le d) (distance (x , y)))))) /\ (((s = EMPTY) \/ (t = EMPTY)) ==> ((real_le d) (real_of_num (NUMERAL _0)))))))))


Ssrnat.leqSS

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


FINITE_SING

|- (!a. (FINITE (a INSERT EMPTY)))


Lp_ineqs_quads.invf_in_dart

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> (((inverse (f_fan_pair_ext (V , E))) d) IN (dart_of_fan (V , E))))))))


SUM_SUPERSET

|- (!f. (!u. (!v. (((u SUBSET v) /\ (!x. (((x IN v) /\ (~ (x IN u))) ==> ((f x) = (real_of_num (NUMERAL _0)))))) ==> (((sum v) f) = ((sum u) f))))))


homotopic_with

|- (!X. (!Y. (!p. (!q. (!P. (((((homotopic_with P) (X , Y)) p) q) = (?h. (((continuous_on h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) /\ ((((IMAGE h) ((closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL)) PCROSS X)) SUBSET Y) /\ ((!x. ((h ((pastecart (vec (NUMERAL _0))) x)) = (p x))) /\ ((!x. ((h ((pastecart (vec (NUMERAL (BIT1 _0)))) x)) = (q x))) /\ (!t. ((t IN (closed_interval ((CONS ((vec (NUMERAL _0)) , (vec (NUMERAL (BIT1 _0))))) NIL))) ==> (P (\x. (h ((pastecart t) x)))))))))))))))))


Seq.zip

|- ((((zip ((CONS x) s)) ((CONS y) t)) = ((CONS (x , y)) ((zip s) t))) /\ ((((zip NIL) t) = NIL) /\ (((zip s) NIL) = NIL)))


HAS_BOUNDED_VARIATION_ON_EMPTY

|- (!f. ((has_bounded_variation_on f) EMPTY))


COLLINEAR_DIST_BETWEEN

|- (!a. (!b. (!x. (((collinear (x INSERT (a INSERT (b INSERT EMPTY)))) /\ (((real_le (distance (x , a))) (distance (a , b))) /\ ((real_le (distance (x , b))) (distance (a , b))))) ==> ((between x) (a , b))))))


FORALL_BOOL_THM

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


Ssrbool.contraL

|- (!c. (!b. ((c ==> (~ b)) ==> (b ==> (~ c)))))


numseg

|- (!m. (!n. ((m .. n) = (GSPEC (\GEN%PVAR%150. (?x. (((SETSPEC GEN%PVAR%150) ((m <= x) /\ (x <= n))) x)))))))


BROUWER_REDUCTION_THEOREM

|- (!P. (!s. (((!f. (((!n. ((compact (f n)) /\ ((~ ((f n) = EMPTY)) /\ (P (f n))))) /\ (!n. ((f (SUC n)) SUBSET (f n)))) ==> (P (INTERS (GSPEC (\GEN%PVAR%1277. (?n. (((SETSPEC GEN%PVAR%1277) (n IN UNIV)) (f n))))))))) /\ ((compact s) /\ ((~ (s = EMPTY)) /\ (P s)))) ==> (?t. ((t SUBSET s) /\ ((compact t) /\ ((~ (t = EMPTY)) /\ ((P t) /\ (!u. (((u SUBSET s) /\ ((closed u) /\ ((~ (u = EMPTY)) /\ (P u)))) ==> (~ (u PSUBSET t))))))))))))


DIAMETER_TRANSLATION

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


Lp_main_estimate.f_surr_tau_fun_eq_tauVEF

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((V SUBSET ball_annulus) ==> (!f. ((f IN (face_set (hypermap_of_fan (V , E)))) ==> ((((tau_fun V) E) f) = (tauVEF (V , (E , f)))))))))))


REAL_EQ_RDIV_EQ

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


SUMMABLE_IMP_BOUNDED

|- (!f. (!k. (((summable k) f) ==> (bounded ((IMAGE f) k)))))


NEGLIGIBLE_CONVEX_HULL

|- (!s. (((FINITE s) /\ ((CARD s) <= (dimindex UNIV))) ==> (negligible ((hull convex) s))))


HAS_BOUNDED_REAL_VARIATION_TRANSLATION_EQ

|- (!a. (!f. (!s. (((has_bounded_real_variation_on (\x. (f ((real_add a) x)))) s) = ((has_bounded_real_variation_on f) ((IMAGE (\x. ((real_add a) x))) s))))))


ROWS_MAPROWS

|- (!f. (!A. ((rows ((maprows f) A)) = ((IMAGE f) (rows A)))))


HAS_DERIVATIVE_SEQUENCE

|- (!s. (!f. (!f'. (!g'. (((convex s) /\ ((!n. (!x. ((x IN s) ==> (((has_derivative (f n)) ((f' n) x)) ((within (at x)) s))))) /\ ((!e. (((real_lt (real_of_num (NUMERAL _0))) e) ==> (?N. (!n. (!x. (!h. (((n >= N) /\ (x IN s)) ==> ((real_le (vector_norm ((vector_sub (((f' n) x) h)) ((g' x) h)))) ((real_mul e) (vector_norm h)))))))))) /\ (?x. (?l. ((x IN s) /\ (((--> (\n. ((f n) x))) l) sequentially))))))) ==> (?g. (!x. ((x IN s) ==> ((((--> (\n. ((f n) x))) (g x)) sequentially) /\ (((has_derivative g) (g' x)) ((within (at x)) s)))))))))))


POSITIVE_SEMIDEFINITE_ADD

|- (!A. (!B. (((positive_semidefinite A) /\ (positive_semidefinite B)) ==> (positive_semidefinite ((matrix_add A) B)))))


REAL_CONTINUOUS_AT_ACS

|- (!x. (((real_lt (real_abs x)) (real_of_num (NUMERAL (BIT1 _0)))) ==> ((real_continuous acs) (atreal x))))


lim

|- (!f. (!net. (((lim net) f) = (@l. (((--> f) l) net)))))


MOD_LT

|- (!m. (!n. ((m < n) ==> ((m MOD n) = m))))


STC_TRANS

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


AFFINE_INDEPENDENT_IMP_FINITE

|- (!s. ((~ (affine_dependent s)) ==> (FINITE s)))


Miqmcsn.K_SCS_4M8_13

|- ((scs_k_v39 scs_4M8_13) = (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))


POLYTOPE_EMPTY

|- (polytope EMPTY)


SEGMENT_TO_FRONTIER_SIMPLE

|- (!s. (!x. (((bounded s) /\ (x IN s)) ==> (?a. (?b. ((a IN (frontier s)) /\ ((b IN (frontier s)) /\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))


RPOW_ONE

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


Hypermap_and_fan.PAIR_IN_DART1_OF_FAN

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


SEGMENT_TO_RELATIVE_FRONTIER_SIMPLE

|- (!s. (!x. (((bounded s) /\ ((x IN s) /\ (~ (s = (x INSERT EMPTY))))) ==> (?a. (?b. ((a IN (relative_frontier s)) /\ ((b IN (relative_frontier s)) /\ (x IN (closed_segment ((CONS (a , b)) NIL))))))))))


HAS_REAL_MEASURE_REAL_NEGLIGIBLE_UNIONS

|- (!m. (!f. (((FINITE f) /\ ((!s. ((s IN f) ==> ((has_real_measure s) (m s)))) /\ (!s. (!t. (((s IN f) /\ ((t IN f) /\ (~ (s = t)))) ==> (real_negligible (s INTER t))))))) ==> ((has_real_measure (UNIONS f)) ((sum f) m)))))


LE_MULT_RCANCEL

|- (!m. (!n. (!p. (((m * p) <= (n * p)) = ((m <= n) \/ (p = (NUMERAL _0)))))))


NPRODUCT_POS_LT

|- (!f. (!s. (((FINITE s) /\ (!x. ((x IN s) ==> ((NUMERAL _0) < (f x))))) ==> ((NUMERAL _0) < ((nproduct s) f)))))


CONVEX_HULL_SUBSET_AFFINE_HULL

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


Nonlin_def.sol_euler156_x_div_sqrtdelta

|- (sol_euler156_x_div_sqrtdelta = (rotate6 sol_euler_x_div_sqrtdelta))


IM_CNJ

|- (!z. ((Im (cnj z)) = (real_neg (Im z))))


Seq2.list_sum_eq0

|- (!s. (!f. ((!x. (((MEM x) s) ==> ((f x) = (real_of_num (NUMERAL _0))))) ==> (((list_sum s) f) = (real_of_num (NUMERAL _0))))))


IMAGE_FSTCART_PCROSS

|- (!s. (!t. (((IMAGE fstcart) (s PCROSS t)) = (((COND (t = EMPTY)) EMPTY) s))))


LTE_TRANS

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


has_bounded_real_variation_on

|- (!f. (!s. (((has_bounded_real_variation_on f) s) = ((has_bounded_variation_on (lift o (f o drop))) ((IMAGE lift) s)))))


INSIDE_OUTSIDE_INTERSECT_CONNECTED

|- (!s. (!t. (((connected t) /\ ((~ (((inside s) INTER t) = EMPTY)) /\ (~ (((outside s) INTER t) = EMPTY)))) ==> (~ ((s INTER t) = EMPTY)))))


OPEN_IN_INJECTIVE_LINEAR_IMAGE

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


LINEAR_SNDCART

|- (linear sndcart)


UNBOUNDED_COMPLEMENT_COMPONENT_CONVEX

|- (!s. (!c. (((convex s) /\ (c IN (components (UNIV DIFF s)))) ==> (~ (bounded c)))))


REAL_NEGLIGIBLE_COUNTABLE_UNIONS

|- (!s. ((!n. (real_negligible (s n))) ==> (real_negligible (UNIONS (GSPEC (\GEN%PVAR%4943. (?n. (((SETSPEC GEN%PVAR%4943) (n IN UNIV)) (s n)))))))))


sgn_ge

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


Merge_ineq.delta_y_sym

|- (!x1. (!x2. (!x3. (!x4. (!x5. (!x6. ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x2) x1) x3) x5) x4) x6)) /\ ((((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x1) x3) x2) x4) x6) x5)) /\ (((((((delta_y x1) x2) x3) x4) x5) x6) = ((((((delta_y x5) x4) x3) x2) x1) x6))))))))))


CLOSURE_HALFSPACE_COMPONENT_GT

|- (!a. (!k. ((closure (GSPEC (\GEN%PVAR%812. (?x. (((SETSPEC GEN%PVAR%812) ((real_gt (x $ k)) a)) x))))) = (GSPEC (\GEN%PVAR%813. (?x. (((SETSPEC GEN%PVAR%813) ((real_ge (x $ k)) a)) x)))))))


FINITE_PERMUTATIONS

|- (!s. ((FINITE s) ==> (FINITE (GSPEC (\GEN%PVAR%249. (?p. (((SETSPEC GEN%PVAR%249) ((permutes p) s)) p)))))))


MEASURE_SING

|- (!a. ((measure (a INSERT EMPTY)) = (real_of_num (NUMERAL _0))))


POLYTOPE_IMP_POLYHEDRON

|- (!p. ((polytope p) ==> (polyhedron p)))


Hypermap.orbit_cyclic

|- (!f. (!m. (!x. (((~ (m = (NUMERAL _0))) /\ ((((POWER f) m) x) = x)) ==> (((orbit_map f) x) = (GSPEC (\GEN%PVAR%5396. (?k. (((SETSPEC GEN%PVAR%5396) (k < m)) (((POWER f) k) x))))))))))


REALLIM_NULL_COMPARISON

|- (!net. (!f. (!g. ((((eventually (\x. ((real_le (real_abs (f x))) (g x)))) net) /\ (((---> g) (real_of_num (NUMERAL _0))) net)) ==> (((---> f) (real_of_num (NUMERAL _0))) net)))))


Real_ext.REAL_PROP_NN_INV

|- (!x. (((real_le (real_of_num (NUMERAL _0))) (real_inv x)) = ((real_le (real_of_num (NUMERAL _0))) x)))


QUANTIFY_SURJECTION_HIGHER_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))))) /\ ((!Q. ((!s. (Q s)) = (!s. (Q ((IMAGE (IMAGE f)) s))))) /\ ((!Q. ((?s. (Q s)) = (?s. (Q ((IMAGE (IMAGE f)) s))))) /\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\ ((!P. ((!g. (P g)) = (!g. (P (f o g))))) /\ ((!P. ((?g. (P g)) = (?g. (P (f o g))))) /\ ((!Q. ((!l. (Q l)) = (!l. (Q ((MAP f) l))))) /\ (!Q. ((?l. (Q l)) = (?l. (Q ((MAP f) l)))))))))))))))) /\ ((!P. ((GSPEC (\GEN%PVAR%238. (?x. (((SETSPEC GEN%PVAR%238) (P x)) x)))) = ((IMAGE f) (GSPEC (\GEN%PVAR%239. (?x. (((SETSPEC GEN%PVAR%239) (P (f x))) x))))))) /\ ((!Q. ((GSPEC (\GEN%PVAR%240. (?s. (((SETSPEC GEN%PVAR%240) (Q s)) s)))) = ((IMAGE (IMAGE f)) (GSPEC (\GEN%PVAR%241. (?s. (((SETSPEC GEN%PVAR%241) (Q ((IMAGE f) s))) s))))))) /\ (!R. ((GSPEC (\GEN%PVAR%242. (?l. (((SETSPEC GEN%PVAR%242) (R l)) l)))) = ((IMAGE (MAP f)) (GSPEC (\GEN%PVAR%243. (?l. (((SETSPEC GEN%PVAR%243) (R ((MAP f) l))) l))))))))))))


Ssrnat.maxn

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