Lp_ineqs_proofs.y6_hi_std2

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> ((E = (ESTD V)) ==> (!d. ((d IN (dart_of_fan (V , E))) ==> ((real_le (y6_fan d)) ((DECIMAL (NUMERAL (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0)))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0)))))))))))))))))))


Hypermap.THREE

|- ((NUMERAL (BIT1 (BIT1 _0))) = (SUC (NUMERAL (BIT0 (BIT1 _0)))))


REAL_COMPACT_IMP_BOUNDED

|- (!s. ((real_compact s) ==> (real_bounded s)))


Real_ext.REAL_SV_NEG_0

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


PATH_CONNECTED_LINEAR_IMAGE_EQ

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


BASIS_ORTHOGONAL

|- (!b. ((FINITE b) ==> (?c. ((FINITE c) /\ (((CARD c) <= (CARD b)) /\ (((span c) = (span b)) /\ ((pairwise orthogonal) c)))))))


ABSOLUTELY_INTEGRABLE_CONTINUOUS

|- (!f. (!a. (!b. (((continuous_on f) (closed_interval ((CONS (a , b)) NIL))) ==> ((absolutely_integrable_on f) (closed_interval ((CONS (a , b)) NIL)))))))


REFLECT_ALONG_LINEAR_IMAGE

|- (!f. (!v. (!x. (((linear f) /\ (!x. ((vector_norm (f x)) = (vector_norm x)))) ==> (((reflect_along (f v)) (f x)) = (f ((reflect_along v) x)))))))


CEXP_ADD

|- (!w. (!z. ((cexp ((vector_add w) z)) = ((complex_mul (cexp w)) (cexp z)))))


PSUBSET

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


IN

|- (!P. (!x. ((x IN P) = (P x))))


STC_CLOSED

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


PRODUCT_EQ

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


MEASURABLE_CONVEX

|- (!s. (((convex s) /\ (bounded s)) ==> (measurable s)))


Add_triangle.sigma_in_dart1

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


Seq.rot

|- (!n. (!s. (((rot n) s) = ((cat ((dropl n) s)) ((take n) s)))))


ENR_ANR

|- (!s. ((ENR s) = ((ANR s) /\ ((locally compact) s))))


REAL_CONTINUOUS_ON

|- (!f. (!s. (((real_continuous_on f) s) = ((continuous_on (lift o (f o drop))) ((IMAGE lift) s)))))


Appendix.BBindex_min_v39

|- (!s. ((BBindex_min_v39 s) = (min_num ((IMAGE (BBindex_v39 s)) (BBprime_v39 s)))))


SUBSET

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


Tame_general.ly_EQ_lmfun

|- (!x. (((real_le (vector_norm (FST x))) ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) h0)) ==> ((lmfun (h_dart x)) = (ly (vector_norm (FST x))))))


LE_ADD

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


LEFT_OR_FORALL_THM

|- (!P. (!Q. (((!x. (P x)) \/ Q) = (!x. ((P x) \/ Q)))))


FINITE_IMAGE

|- (!f. (!s. ((FINITE s) ==> (FINITE ((IMAGE f) s)))))


ANR_IMP_ABSOLUTE_CLOSED_NEIGHBOURHOOD_RETRACT

|- (!s. (!u. (!s'. (((ANR s) /\ (((homeomorphic s) s') /\ ((closed_in ((subtopology euclidean) u)) s'))) ==> (?v. (?w. (((open_in ((subtopology euclidean) u)) v) /\ (((closed_in ((subtopology euclidean) u)) w) /\ ((s' SUBSET v) /\ ((v SUBSET w) /\ ((retract_of s') w)))))))))))


LIM_AT_INFINITY_COMPLEX_0

|- (!f. (!l. ((((--> f) l) at_infinity) = (((--> (f o complex_inv)) l) (at (Cx (real_of_num (NUMERAL _0))))))))


RIGHT_ADD_DISTRIB

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


IN_INSERT

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


SPAN_CONIC_HULL

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


VECTOR_POLYNOMIAL_FUNCTION_CONST

|- (!c. (vector_polynomial_function (\x. c)))


SERIES_COMPONENT

|- (!f. (!s. (!l. (!k. (((((sums f) l) s) /\ (((NUMERAL (BIT1 _0)) <= k) /\ (k <= (dimindex UNIV)))) ==> (((sums (\i. (lift ((f i) $ k)))) (lift (l $ k))) s))))))


COMPLEX_DERIVATIVE_SUB

|- (!f. (!g. (!z. ((((complex_differentiable f) (at z)) /\ ((complex_differentiable g) (at z))) ==> (((complex_derivative (\w. ((vector_sub (f w)) (g w)))) z) = ((vector_sub ((complex_derivative f) z)) ((complex_derivative g) z)))))))


Ssrnat.succnK

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


ISOMETRY_LINEAR

|- (!f. ((((f (vec (NUMERAL _0))) = (vec (NUMERAL _0))) /\ (!x. (!y. ((distance ((f x) , (f y))) = (distance (x , y)))))) ==> (linear f)))


INDEPENDENT_IMP_FINITE

|- (!s. ((independent s) ==> (FINITE s)))


Sphere.tame_table_d

|- (!r. (!s. (((tame_table_d r) s) = (((COND ((r + ((NUMERAL (BIT0 (BIT1 _0))) * s)) > (NUMERAL (BIT1 (BIT1 _0))))) ((real_add ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT1 (BIT1 (BIT1 (BIT1 (BIT1 _0))))))))))))) ((real_sub (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s)))) ((real_mul ((DECIMAL (NUMERAL (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT1 _0)))))))))))))) (NUMERAL (BIT0 (BIT0 (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT0 (BIT1 (BIT1 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))))))))) ((real_add (real_of_num r)) ((real_sub ((real_mul (real_of_num (NUMERAL (BIT0 (BIT1 _0))))) (real_of_num s))) (real_of_num (NUMERAL (BIT0 (BIT0 (BIT1 _0)))))))))) ((DECIMAL (NUMERAL _0)) (NUMERAL (BIT0 (BIT1 (BIT0 (BIT1 _0))))))))))


REAL_LE_HAUSDIST

|- (!s. (!t. (!a. (!b. (!c. (!z. (((~ (s = EMPTY)) /\ ((~ (t = EMPTY)) /\ ((!x. ((x IN s) ==> ((real_le (setdist ((x INSERT EMPTY) , t))) b))) /\ ((!y. ((y IN t) ==> ((real_le (setdist ((y INSERT EMPTY) , s))) c))) /\ (((z IN s) /\ ((real_le a) (setdist ((z INSERT EMPTY) , t)))) \/ ((z IN t) /\ ((real_le a) (setdist ((z INSERT EMPTY) , s))))))))) ==> ((real_le a) (hausdist (s , t))))))))))


Seq.map

|- ((((map f) ((CONS x) s)) = ((CONS (f x)) ((map f) s))) /\ (((map f) NIL) = NIL))


DIFF_CHAIN_AT

|- (!f. (!g. (!f'. (!g'. (!x. (((((has_derivative f) f') (at x)) /\ (((has_derivative g) g') (at (f x)))) ==> (((has_derivative (g o f)) (g' o f')) (at x))))))))


Hexagons.sqrt8_LE_CSTAB

|- ((real_le sqrt8) ((DECIMAL (NUMERAL (BIT1 (BIT0 (BIT1 (BIT1 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 _0))))))))))) (NUMERAL (BIT0 (BIT0 (BIT1 (BIT0 (BIT0 (BIT1 (BIT1 _0))))))))))


PASTECART_INJ

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


ORTHOGONAL_TRANSFORMATION_LINEAR

|- (!f. ((orthogonal_transformation f) ==> (linear f)))


LINEAR_CX_RE

|- (linear (Cx o Re))


ARITH_LT

|- ((!m. (!n. (((NUMERAL m) < (NUMERAL n)) = (m < n)))) /\ (((_0 < _0) = F) /\ ((!n. (((BIT0 n) < _0) = F)) /\ ((!n. (((BIT1 n) < _0) = F)) /\ ((!n. ((_0 < (BIT0 n)) = (_0 < n))) /\ ((!n. ((_0 < (BIT1 n)) = T)) /\ ((!m. (!n. (((BIT0 m) < (BIT0 n)) = (m < n)))) /\ ((!m. (!n. (((BIT0 m) < (BIT1 n)) = (m <= n)))) /\ ((!m. (!n. (((BIT1 m) < (BIT0 n)) = (m < n)))) /\ (!m. (!n. (((BIT1 m) < (BIT1 n)) = (m < n)))))))))))))


VECTOR_VARIATION_TRIANGLE

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


SUBSET_LE_DIM

|- (!s. (!t. ((s SUBSET (span t)) ==> ((dim s) <= (dim t)))))


IN_ELIM_PASTECART_THM

|- (!P. (!a. (!b. ((((pastecart a) b) IN (GSPEC (\GEN%PVAR%238. (?x. (?y. (((SETSPEC GEN%PVAR%238) ((P x) y)) ((pastecart x) y))))))) = ((P a) b)))))


CONTINUOUS_AT_COMPOSE

|- (!f. (!g. (!x. ((((continuous f) (at x)) /\ ((continuous g) (at (f x)))) ==> ((continuous (g o f)) (at x))))))


Lp_ineqs_proofs.tau_fan_eq_tauVEF

|- (!V. (!E. (!f. (((lp_tau (V , E)) /\ (f IN (face_set (hypermap_of_fan (V , E))))) ==> (((tau_fan (V , E)) f) = (tauVEF (V , (E , f))))))))


Hypermap_iso.hyp_iso_imp_iso

|- (!H. (!G. (!f. (((hyp_iso f) (H , G)) ==> ((iso H) G)))))


CONNECTED_COMPONENT_TRANSLATION

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


REAL_ADD_RDISTRIB

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


IM_ADD

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


List_hypermap.good_list_nodes

|- (!L. ((good_list_nodes L) = ((node_set (hypermap_of_list L)) = (set_of_list (nodes_of_list L)))))


LIM_POSINFINITY_SEQUENTIALLY

|- (!f. (!l. ((((--> f) l) at_posinfinity) ==> (((--> (\n. (f (real_of_num n)))) l) sequentially))))


Dih2k_hypermap.cstab

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


Nonlin_def.sqrt_x4

|- (((((((sqrt_x4 x1) x2) x3) x4) x5) x6) = (sqrt x4))


REAL_MUL_LINV

|- (!x. ((~ (x = (real_of_num (NUMERAL _0)))) ==> (((real_mul (real_inv x)) x) = (real_of_num (NUMERAL (BIT1 _0))))))


differentiable_on

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


Lp_ineqs_proofs.dart4_cross_eq_y4'

|- (!V. (!E. ((FAN ((vec (NUMERAL _0)) , (V , E))) ==> ((fully_surrounded (V , E)) ==> ((packing V) ==> ((V SUBSET ball_annulus) ==> (!d. ((d IN ((darts_k (NUMERAL (BIT0 (BIT0 (BIT1 _0))))) (hypermap_of_fan (V , E)))) ==> (((y4'_fan (V , E)) ((f_fan_pair_ext (V , E)) d)) = ((y4'_fan (V , E)) ((inverse (f_fan_pair_ext (V , E))) d)))))))))))


CONTINUOUS_AT_CACS

|- (!z. ((((Im z) = (real_of_num (NUMERAL _0))) ==> ((real_lt (real_abs (Re z))) (real_of_num (NUMERAL (BIT1 _0))))) ==> ((continuous cacs) (at z))))


PERMUTATION_SWAP

|- (!a. (!b. (permutation (swap (a , b)))))


REAL_SGN_MUL

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


COMPONENTS_TRANSLATION

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


Fan.POWER

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