hcju/isabelleps huggingface dataset:
Goedel_Form_def: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and num :: "'b set"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and eql :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and cnj :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and imp :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and all :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and exi :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and fls :: "'c"   and prv :: "'c \<Rightarrow> bool"   and bprv :: "'c \<Rightarrow> bool"   and enc :: "'c \<Rightarrow> 'b"   and S :: "'c"   and P :: "'c" shows "Goedel_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi fls prv bprv enc S P \<equiv> Deduct2_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv bprv \<and> Repr_SelfSubst var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc S \<and> HBL1 var trm fmla Var FvarsT substT Fvars subst num eql cnj imp all exi prv bprv enc P"
axioms_2: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set" assumes "RModule R smult M" shows "Group M"
s: shows "Partial_Cost_Model.config'_rand BIT (type1 init x y) [y, y] = type0 init y x"
big_def2: fixes \<sigma>' :: "'state"   and P :: "'prog"   and \<sigma> :: "'state" shows "(\<sigma>' \<in> big P \<sigma>) = (\<exists>n. \<sigma>' \<in> small_nstep P \<sigma> n \<and> \<sigma>' \<in> endset)"
idem: fixes a :: "'d DBMEntry" shows "linordered_monoid.min a a = a"
extend_set_strict_mono: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and A :: "'d set"   and B :: "'d set" assumes "Extend h" shows "(extend_set h A \<subseteq> extend_set h B) = (A \<subseteq> B)"
indep_sets_mono_sets: fixes F :: "'c \<Rightarrow> 'a set set"   and I :: "'c set"   and G :: "'c \<Rightarrow> 'a set set" assumes "indep_sets F I"   and "\<And>i. i \<in> I \<Longrightarrow> G i \<subseteq> F i" shows "indep_sets G I"
map_of_rev_zip_upto_length_eq_nth: fixes i :: "nat"   and B :: "'d list"   and d :: "nat" assumes "i < length B"   and "d = length B" shows "map_of (rev (zip [0..<d] B)) i = Some (B ! i)"
space: fixes \<Omega> :: "'b set"   and M :: "'b set set" assumes "Dynkin_system \<Omega> M" shows "\<Omega> \<in> M"
boolLessState_def: shows "boolLessState \<equiv> {(stateA, stateB). getM stateA = getM stateB \<and> (getConflictFlag stateA, getConflictFlag stateB) \<in> boolLess}"
round_diff_minimal: fixes z :: "'a"   and m :: "int" shows "\<bar>z - of_int (round z)\<bar> \<le> \<bar>z - of_int m\<bar>"
subg_sg_sg: fixes G :: "('e, 'f) Group_scheme"   and K :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<guillemotright> K "   and "\<natural>\<^bsub>G\<^esub>K \<guillemotright> H " shows "G \<guillemotright> H "
GrpI: fixes f :: "'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'a"   and A :: "'b set" assumes "f x = y"   and "x \<in> A" shows "BNF_Def.Grp A f x y"
Col_perm: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Col A B C" shows "Col A B C \<and> Col A C B \<and> Col B A C \<and> Col B C A \<and> Col C A B \<and> Col C B A"
int_word_uint: fixes x :: "int" shows "uint (word_of_int x) = x mod 2 ^ LENGTH('b)"
assms_2: shows "env_dom e1 \<inter> env_dom e2 = {}"
assms_2: fixes u :: "'t" assumes "v \<preceq>\<^sub>t u" shows "lookup p u = (0::'b)"
map_of_SomeD: fixes xs :: "('c \<times> 'b) list"   and k :: "'c"   and y :: "'b" assumes "map_of xs k = Some y" shows "(k, y) \<in> set xs"
residuated_conj1: fixes f :: "'a \<Rightarrow> 'a" assumes "residuated f" shows "conjugation_pair f (conjugate f)"
\<phi>L: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "\<phi>L \<phi> \<in> fmla"
filter_is_subset: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "set (filter P xs) \<subseteq> set xs"
restrict_restrict: fixes m :: "'a \<Rightarrow> 'b option"   and A :: "'a set"   and B :: "'a set" shows "m |` A |` B = m |` (A \<inter> B)"
fmrestrict_set_twice: fixes S :: "'c set"   and T :: "'c set"   and m :: "('c, 'd) fmap" shows "fmrestrict_set S (fmrestrict_set T m) = fmrestrict_set (S \<inter> T) m"
simps_2: fixes x :: "'d"   and xs :: "'d list"   and s :: "'d stream" shows "(x # xs) @- s = x ## xs @- s"
remove_key_lookup: fixes k0 :: "'c"   and f :: "'c \<Rightarrow>\<^sub>0 'b"   and k :: "'c" shows "lookup (remove_key k0 f) k = (lookup f k when k \<noteq> k0)"
fresh_eqvt: fixes pi :: "(name \<times> name) list"   and a :: "name"   and x :: "'a" shows "pi \<bullet> a \<sharp> x = pi \<bullet> a \<sharp> pi \<bullet> x"
a: fixes f :: "face"   and v :: "nat"   and "is" :: "nat list"   and a :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f"   and "\<forall>i\<in>set is. i < |vertices f|"   and "a < |vertices f|" shows "hideDupsRec ((f \<bullet> ^^ a) v) (map (\<lambda>k. (f \<bullet> ^^ k) v) is) = natToVertexListRec a v f is"
dom_1: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> y) \<le> d x"
refl_onD2: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and x :: "'b"   and y :: "'b" assumes "refl_on A r"   and "(x, y) \<in> r" shows "y \<in> A"
const_smallo_powr: fixes f :: "'b \<Rightarrow> real"   and F :: "'b filter"   and c :: "real"   and p :: "real" assumes "filterlim f at_top F"   and "F \<noteq> bot" shows "((\<lambda>_. c) \<in> o[F](\<lambda>x. f x powr p)) = (0 < p \<or> c = 0)"
change_tv_injection: fixes f :: "nat \<Rightarrow> nat" assumes "inj f" shows "inj (change_tv f)"
S_unique: fixes \<phi> :: "'fmla" shows "let SS = \<lambda>t1 t2. psubst S [(t1, xx), (t2, yy)] in \<phi> \<in> fmla \<longrightarrow> Fvars \<phi> = {xx} \<longrightarrow> bprv (all yy (all yy' (imp (cnj (SS \<langle>\<phi>\<rangle> (Var yy)) (SS \<langle>\<phi>\<rangle> (Var yy'))) (eql (Var yy) (Var yy')))))"
mset_eq_setD: fixes xs :: "'b list"   and ys :: "'b list" assumes "mset xs = mset ys" shows "set xs = set ys"
cSup_least: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and z :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq x z" shows "less_eq (Sup X) z"
gcd_mod_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.euclidean_semiring_gcd divide plus minus zero times one modulo euclidean_size unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero" shows "gcd a (modulo b a) = gcd a b"
append_eq_Cons_conv: fixes ys :: "'d list"   and zs :: "'d list"   and x :: "'d"   and xs :: "'d list" shows "(ys @ zs = x # xs) = (ys = [] \<and> zs = x # xs \<or> (\<exists>ys'. ys = x # ys' \<and> ys' @ zs = xs))"
slp_index_lookup_def: fixes vs :: "'a list"   and M :: "('b, nat) mapping"   and a :: "'b" shows "slp_index_lookup vs M a = slp_index vs (the (Mapping.lookup M a))"
measure_negligible_symdiff: fixes S :: "'c set"   and T :: "'c set" assumes "S \<in> lmeasurable"   and "negligible (S - T \<union> (T - S))" shows "Sigma_Algebra.measure lebesgue T = Sigma_Algebra.measure lebesgue S"
lir_fps_numerator_altdef: shows "lir_fps_numerator (length fs + 1 - length cs) cs f (eval_polyexp g) = lir_fps_numerator (length fs + 1 - length cs) cs ((!) fs) (eval_polyexp g)"
\<tau>exec_movesE: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "\<tau>exec_moves ci P t es h (stk, loc, pc, xcp) s'" obtains "exec_moves ci P t es h (stk, loc, pc, xcp) \<lbrace>\<rbrace> h s'" and "\<tau>moves2 P h stk es pc xcp"
whileAnno_def: fixes b :: "'d set"   and I :: "'d set"   and V :: "('d \<times> 'd) set"   and c :: "('d, 'e, 'f) com" shows "whileAnno b I V c = While b c"
map_ran_conv: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and al :: "('e \<times> 'd) list"   and k :: "'e" shows "map_of (map_ran f al) k = map_option (f k) (map_of al k)"
refine: fixes n :: "nat"   and ai :: "'b list list"   and a :: "'c set" assumes "DIM_precond TYPE('c) n"   and "(ai, a) \<in> clw_rel appr_rel" shows "(RETURN (ivl_rep_of_set_coll_impl n ai), ivl_rep_of_set_coll a) \<in> \<langle>lv_rel \<times>\<^sub>r lv_rel\<rangle>nres_rel"
le_funE: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'h"   and x :: "'g" assumes "f \<le> g" obtains "f x \<le> g x"
X_indep: shows "indep_vars (\<lambda>i. borel) X UNIV"
cont_def: fixes luba :: "'c set \<Rightarrow> 'c"   and orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd" shows "cont luba orda lubb ordb f = (\<forall>Y. Complete_Partial_Order.chain orda Y \<longrightarrow> Y \<noteq> {} \<longrightarrow> f (luba Y) = lubb (f ` Y))"
has_left_adjoint_functor: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" assumes "meta_adjunction C D F G \<phi> \<psi>" shows "left_adjoint_functor D C F"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a \<squnion> b = b)"
fun_relI: fixes A :: "('g \<times> 'h) set"   and f :: "'g \<Rightarrow> 'i"   and f' :: "'h \<Rightarrow> 'j"   and B :: "('i \<times> 'j) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
id_maps: fixes \<Gamma> :: "(name \<times> ty) list"   and a :: "name" shows "SN.id \<Gamma> maps a to Var a"
induct: fixes P :: "('b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b list"   and a2 :: "'b list" assumes "\<And>eq. P eq [] []"   and "\<And>eq a l a' l'. (eq a a' \<Longrightarrow> P eq l l') \<Longrightarrow> P eq (a # l) (a' # l')"   and "\<And>uu_ v va. P uu_ (v # va) []"   and "\<And>uu_ v va. P uu_ [] (v # va)" shows "P a0 a1 a2"
sublist_def: fixes xs :: "'a list"   and ys :: "'a list" shows "sublist xs ys = (\<exists>ps ss. ys = ps @ xs @ ss)"
that_1: shows "\<lfloor>x\<rfloor> \<le> \<lfloor>y\<rfloor>"
\<pi>: shows "product_cone J (\<cdot>) D a \<pi>"
proj_psi'_l_def: fixes l :: "nat" shows "proj_psi'_l l = outer_prod (psi'_l l) (psi'_l l)"
exhaust_sel: fixes tm :: "'a tm" obtains "tm = TM (tm.val tm) (un_TM2 tm)"
has_bochner_integral_nn_integral: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "0 \<le> x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "has_bochner_integral M f x"
signed_take_bit_eq_if_positive: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not"   and "\<not> bit a n" shows "ring_bit_operations.signed_take_bit zero times one bit take_bit or mask not n a = take_bit n a"
mem_set_list_relE1: fixes xs :: "'c list"   and ys :: "'d list"   and R :: "('c \<times> 'd) set"   and x :: "'c" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel"   and "x \<in> set xs" obtains y :: "'d" where "y \<in> set ys" and "(x, y) \<in> R"
exp_div_exp_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = times (zero_neq_one.of_bool one zero (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) m \<noteq> zero \<and> n \<le> m)) (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) (m - n))"
tab_simps_2: shows "src tab = src p\<^sub>0"
inj_on_image_mem_iff: fixes f :: "'b \<Rightarrow> 'c"   and B :: "'b set"   and a :: "'b"   and A :: "'b set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
equation_minus_iff: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.group_add minus plus zero uminus" shows "(a = uminus b) = (b = uminus a)"
sdiv_poly_def: fixes p :: "'a poly"   and a :: "'a" shows "sdiv_poly p a = map_poly (\<lambda>c. c div a) p"
in_homE: fixes f :: "('obj, 'obj) concrete_category.arr"   and a :: "('obj, 'obj) concrete_category.arr"   and b :: "('obj, 'obj) concrete_category.arr" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "arr f" and "local.dom f = a" and "cod f = b"
init_hta_def: fixes Qi :: "('a, unit) HashMap.hashmap"   and \<delta> :: "('a, 'b) ta_rule dlist" shows "init_hta Qi \<delta> \<equiv> \<lparr>hta_Qi = Qi, hta_\<delta> = \<delta>, hta_idx_f = None, hta_idx_s = None, hta_idx_sf = None\<rparr>"
exhaust: fixes y :: "msg" obtains x11 :: "nat" and x12 :: "nat" and x13 :: "nat" and x14 :: "k" and x15 :: "nat" and x16 :: "nat" and x17 :: "nat" and x18 :: "bool" where "y = Rreq x11 x12 x13 x14 x15 x16 x17 x18"   | x21 :: "nat" and x22 :: "nat" and x23 :: "nat" and x24 :: "nat" and x25 :: "nat" where "y = Rrep x21 x22 x23 x24 x25"   | x31 :: "nat \<Rightarrow> nat option" and x32 :: "nat" where "y = Rerr x31 x32"   | x41 :: "nat" and x42 :: "nat" where "y = Newpkt x41 x42"   | x51 :: "nat" and x52 :: "nat" and x53 :: "nat" where "y = Pkt x51 x52 x53"
normalize_gives_polynomial: fixes p :: "('c multiset \<Rightarrow> 'a) list"   and K :: "('c multiset \<Rightarrow> 'a) set" assumes "set p \<subseteq> K" shows "polynomial\<^bsub>L\<^esub> K (L.normalize p)"
qSkelInp_def: fixes qinp :: "'f \<Rightarrow> ('g, 'h, 'i, 'j, 'k) qTerm option" shows "qSkelInp qinp = lift qSkel qinp"
index_of_const_is_pr: shows "index_of_const \<in> PrimRec1"
in_renamed_in_unlabelled_local_1: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and r :: "'d"   and ls :: "('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr" assumes "bij \<alpha>"   and "bij \<beta>" shows "(\<alpha> r \<in> RID\<^sub>L (\<R>\<^sub>L \<alpha> \<beta> ls)) = (r \<in> RID\<^sub>L ls)"
unx_none_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "unapp (const name) = None"
Complex_eq_neg_1: fixes a :: "real"   and b :: "real" shows "(Complex a b = - 1) = (a = - 1 \<and> b = 0)"
pheap_pass1: fixes h :: "'b tree" assumes "pheap h" shows "pheap (pass\<^sub>1 h)"
compet_rand_def: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and c :: "real"   and S0 :: "'c list set" shows "compet_rand A c S0 = (\<forall>s\<in>S0. \<exists>b\<ge>0. \<forall>rs. static s rs \<longrightarrow> T_on_rand' A (fst A s \<bind> (\<lambda>is. return_pmf (s, is))) rs \<le> c * real (T_opt s rs) + b)"
assms: shows "\<forall>x\<in>cbox a b. \<exists>d>0. \<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow> f integrable_on cbox u v"
chamber_retraction1: fixes C :: "'a set" assumes "C \<in> g \<turnstile> folding_g.\<C>" shows "fixespointwise g C"
barityOf_lt_var_def: fixes uu :: "'a"   and Delta :: "('b, 'c, 'd, 'e, 'f, 'g) signature_scheme" shows "barityOf_lt_var uu Delta \<equiv> \<forall>delta. wlsOpSym Delta delta \<longrightarrow> |{i. barityOf Delta delta i \<noteq> None}| <o |UNIV|"
init_imp_InvGlobPointstampsEq: fixes c :: "('b, 't, 'loc) Combined.configuration" assumes "InitConfig c" shows "InvGlobPointstampsEq c"
Domainp_eq_Collect: fixes A :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and \<UU> :: "'e set" shows "(Domainp A = (\<lambda>x. x \<in> \<UU>)) = (\<UU> = Collect (Domainp A))"
sumResolvingBidEquivalence: fixes N :: "'c set"   and \<Omega> :: "'d list"   and a :: "('c \<times> 'd set) set"   and b :: "'c \<times> 'd set \<Rightarrow> 'e"   and r :: "integer" assumes "0 < card N"   and "distinct \<Omega>"   and "a \<subseteq> N \<times> (Pow (set \<Omega>) - {{}})" shows "sum (tiebids' (randomEl (takeAll (\<lambda>x. winningAllocationRel N (set \<Omega>) ((\<in>) x) b) (allAllocationsAlg N \<Omega>)) r) N (set \<Omega>)) a = sum (resolvingBidAlg N \<Omega> b r) a"
filterlim_at_infinity_conv_norm_at_top: fixes f :: "'a \<Rightarrow> 'b"   and G :: "'a filter" shows "filterlim f at_infinity G = (LIM x G. norm (f x) :> at_top)"
interior_cbox: fixes a :: "'c"   and b :: "'c" shows "interior (cbox a b) = box a b"
a_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier V"   and "y \<in> carrier V"   and "z \<in> carrier V" shows "x \<oplus>\<^bsub>V\<^esub> y \<oplus>\<^bsub>V\<^esub> z = x \<oplus>\<^bsub>V\<^esub> (y \<oplus>\<^bsub>V\<^esub> z)"
project_leadsETo_I: fixes F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "extend h F \<squnion> G \<in> extend_set h A leadsTo[givenBy f] extend_set h B" shows "F \<squnion> project h UNIV G \<in> A \<longmapsto> B"
connected_component_of_equiv: fixes X :: "'c topology"   and x :: "'c"   and y :: "'c" shows "connected_component_of X x y = (x \<in> topspace X \<and> y \<in> topspace X \<and> connected_component_of X x = connected_component_of X y)"
is_prefixpoint_def: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" shows "is_prefixpoint f x \<equiv> f x \<le> x"
iso_same_card: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme" assumes "G \<cong> H" shows "card (carrier G) = card (carrier H)"
eql2: shows "st1.ptensor_mat (1\<^sub>m 2) (1\<^sub>m st4.d2) = 1\<^sub>m st.d2"
extend_constrains: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and F :: "'c program"   and A :: "'c set"   and B :: "'c set" assumes "Extend h" shows "(extend h F \<in> extend_set h A co extend_set h B) = (F \<in> A co B)"
power_series_tendsto_at_left: fixes f :: "nat \<Rightarrow> real" assumes "\<And>i. 0 \<le> f i"   and "\<And>z. \<lbrakk>0 \<le> z; z < 1\<rbrakk> \<Longrightarrow> summable (\<lambda>n. f n * z ^ n)" shows "((\<lambda>z. ennreal (\<Sum>n. f n * z ^ n)) \<longlongrightarrow> (\<Sum>n. ennreal (f n))) (at_left 1)"
prems_1: shows "A \<triangleleft> x \<noteq> 0"
elims: fixes x :: "tcolor"   and xa :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and y :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "paint x xa = y" obtains c :: "tcolor" where "x = c__" and "xa = \<langle>\<rangle>" and "y = \<langle>\<rangle>"   | c :: "tcolor" and l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" and a :: "'c \<times> 'd" and uu_ :: "tcolor" and mkp :: "'c \<times> 'd" and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" where "x = c" and "xa = \<langle>l, (a, uu_, mkp), r\<rangle>" and "y = \<langle>l, (a, c, mkp), r\<rangle>"
set_sort: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "set (linorder.sort_key less_eq f xs) = set xs"
rtrancl_mono: assumes "Restr E V \<subseteq> E" shows "(Restr E V)\<^sup>* \<subseteq> E\<^sup>*"
nat_pow_consistent: fixes x :: "'c"   and n :: "nat"   and H :: "'c set" shows "x [^]\<^bsub>H\<^esub> n = x [^]\<^bsub>H\<lparr>carrier := H\<rparr>\<^esub> n"
monom_mult_monomial: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and s :: "'b"   and v :: "'a"   and d :: "'d" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair c s (monomial d v) = monomial (c * d) (term_powerprod.splus pair_of_term term_of_pair s v)"
prems_2: shows "i \<notin> block_nominals (ps, a)"
emeasure_PiM_emb': fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "J \<subseteq> I"   and "finite J"   and "X \<in> sets (Pi\<^sub>M J M)" shows "emeasure (Pi\<^sub>M I M) (emb I J X) = emeasure (Pi\<^sub>M J M) X"
RN_deriv_nn_integral: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "sigma_finite_measure M"   and "absolutely_continuous M N"   and "sets N = sets M"   and "f \<in> borel_measurable M" shows "integral\<^sup>N N f = \<integral>\<^sup>+ x. RN_deriv M N x * f x \<partial>M"
contract_conj_overlap_variant: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a"   and \<psi> :: "'a" assumes "K = Cn A" shows "(K \<div> \<phi>) \<inter> Cn {\<phi>} \<subseteq> K \<div> (\<phi> .\<and>. \<psi>)"
implication_frontier_eq_implied_frontier_alt: fixes c :: "('loc, 't) configuration"   and loc :: "'loc" assumes "inv_imps_work_sum c"   and "inv_implications_nonneg c"   and "\<And>loc. c_work c loc = {#}\<^sub>z" shows "frontier (c_imp c loc) = implied_frontier_alt c loc"
finite_measure_subadditive: fixes A :: "nat set set set"   and B :: "nat set set set" assumes "A \<in> ES.events"   and "B \<in> ES.events" shows "ES.prob (A \<union> B) \<le> ES.prob A + ES.prob B"
triangular_eq_diagonal_1x2: shows "Ball (carrier_mat 1 2) admits_triangular_reduction = Ball (carrier_mat 1 2) admits_diagonal_reduction"