hcju/isabelleps huggingface dataset:
dom_def: fixes m :: "'b \<Rightarrow> 'c option" shows "dom m = {a. m a \<noteq> \<bottom>}"
kingman_theorem_nonergodic_1: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "subcocycle u"   and "- \<infinity> < subcocycle_avg_ereal u" shows "AE x in M. (\<lambda>n. u n x / real n) \<longlonglongrightarrow> subcocycle_lim u x"
assms_5: shows "ball a e \<inter> path_image p = {}"
that: fixes x :: "'a" assumes "P x"   and "\<And>y. P y \<Longrightarrow> weight x \<le> weight y" shows "thesis"
proj_stoch_proc_geom_disc_fct: fixes n :: "nat" shows "disc_fct (proj_stoch_proc geom_proc n)"
nth_sequence_of: fixes i :: "nat"   and A :: "'c set"   and enum :: "nat \<Rightarrow> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "i < card A" shows "sequence_of A enum f ! i = f (enum i)"
cases: fixes a1 :: "'b"   and a2 :: "('a, 'b, 'c) psi"   and a3 :: "('a, 'b, 'c) psi" assumes "a1 \<rhd> a2 \<approx> a3" obtains \<Psi> :: "'b" and P :: "('a, 'b, 'c) psi" and Q :: "('a, 'b, 'c) psi" where "a1 = \<Psi>" and "a2 = P" and "a3 = Q" and "\<Psi> \<rhd> P \<lessapprox><weakBisim> Q" and "\<Psi> \<rhd> P \<leadsto><weakBisim> Q" and "\<forall>\<Psi>'. \<Psi> \<otimes> \<Psi>' \<rhd> P \<approx> Q" and "\<Psi> \<rhd> Q \<approx> P"
inf_commute: fixes x :: "'a"   and y :: "'a" shows "x \<squnion> y = y \<squnion> x"
natural_transformation_axioms: shows "natural_transformation A2_B.comp A2_B.comp A2_B.map A2_B.map A2_B.map"
simps: fixes a1 :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list"   and a2 :: "'b constr list \<times> nat \<times> 'b TESL_atomic list \<times> 'b TESL_atomic list" shows "a1 \<hookrightarrow> a2 = ((\<exists>\<Gamma>\<^sub>1 n\<^sub>1 \<Psi>\<^sub>1 \<Phi>\<^sub>1 \<Gamma>\<^sub>2 n\<^sub>2 \<Psi>\<^sub>2 \<Phi>\<^sub>2. a1 = \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<and> a2 = \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2 \<and> \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>i \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2) \<or> (\<exists>\<Gamma>\<^sub>1 n\<^sub>1 \<Psi>\<^sub>1 \<Phi>\<^sub>1 \<Gamma>\<^sub>2 n\<^sub>2 \<Psi>\<^sub>2 \<Phi>\<^sub>2. a1 = \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<and> a2 = \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2 \<and> \<Gamma>\<^sub>1, n\<^sub>1 \<turnstile> \<Psi>\<^sub>1 \<triangleright> \<Phi>\<^sub>1 \<hookrightarrow>\<^sub>e \<Gamma>\<^sub>2, n\<^sub>2 \<turnstile> \<Psi>\<^sub>2 \<triangleright> \<Phi>\<^sub>2))"
assms_4: shows "a\<^sub>e < a\<^sub>o"
zeta_limit: shows "(\<lambda>k. \<Sum>i\<in>{0<..k}. 1 / of_nat i ^ n) \<longlonglongrightarrow> zeta"
induced_function_nn_integral_aux: fixes A :: "'a set"   and f :: "'a \<Rightarrow> ennreal"   and C :: "real" assumes "A \<in> sets M"   and "f \<in> borel_measurable M"   and "\<And>x. f x \<le> ennreal C"   and "0 \<le> C"   and "emeasure M {x \<in> space M. 0 < f x} < \<infinity>" shows "integral\<^sup>N M (induced_function A f) = set_nn_integral M (\<Union>n. (T ^^ n) --` A) f"
Example2_lemma2_aux: fixes j :: "nat"   and n :: "nat"   and b :: "nat \<Rightarrow> nat" assumes "j < n" shows "sum b {0..<n} = sum b {0..<j} + b j + (\<Sum>i = 0..<n - Suc j. b (Suc j + i))"
Rep: fixes x :: "'b word" shows "sint x \<in> sints LENGTH('b)"
hyps_2: assumes "chain C"   and "X \<subseteq> natUnion C" shows "\<exists>n. X \<subseteq> C n"
permute_boolI: fixes P :: "bool"   and p :: "perm" assumes "P" shows "p \<bullet> P"
prime_dvd_mult_nat: fixes p :: "nat"   and a :: "nat"   and b :: "nat" assumes "prime p" shows "(p dvd a * b) = (p dvd a \<or> p dvd b)"
dim_span: fixes S :: "'b set" shows "local.dim (local.span S) = local.dim S"
r_smn_aux: fixes n :: "nat"   and cs :: "nat list"   and m :: "nat"   and p :: "nat" assumes "0 < n"   and "length cs = m" shows "eval (r_smn_aux n m) (p # cs) \<down>= Partial_Recursive.list_encode (map (code_constn n) (p # cs) @ map (code_id n) [0..<n])"
permutes_inverses_1: fixes p :: "'c \<Rightarrow> 'c"   and S :: "'c set"   and x :: "'c" assumes "p permutes S" shows "p (inv p x) = x"
assoc_props_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and prj1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b arrow_of_spans_data"   and g :: "'b arrow_of_spans_data"   and h :: "'b arrow_of_spans_data" assumes "span_bicategory C prj0 prj1"   and "partial_magma.ide (span_vertical_category.vcomp C) f"   and "partial_magma.ide (span_vertical_category.vcomp C) g"   and "partial_magma.ide (span_vertical_category.vcomp C) h"   and "span_vertical_category.src C f = span_vertical_category.trg C g"   and "span_vertical_category.src C g = span_vertical_category.trg C h" shows "span_vertical_category.trg C \<lparr>Chn = three_composable_identity_arrows_of_spans.chine_assoc C prj0 prj1 f g h, Dom = Dom (span_bicategory.hcomp C prj0 prj1 (span_bicategory.hcomp C prj0 prj1 f g) h), Cod = Cod (span_bicategory.hcomp C prj0 prj1 f (span_bicategory.hcomp C prj0 prj1 g h))\<rparr> = span_vertical_category.trg C f"
axioms_1: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "statically_complete_calculus Bot Inf entails Red_I Red_F" shows "calculus Bot Inf entails Red_I Red_F"
WTrtFAss: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and F :: "char list"   and T :: "ty"   and D :: "char list"   and e\<^sub>2 :: "char list exp"   and T\<^sub>2 :: "ty" assumes "P,E,h,sh \<turnstile> e\<^sub>1 : Class C"   and "P \<turnstile> C has F,NonStatic:T in D"   and "P,E,h,sh \<turnstile> e\<^sub>2 : T\<^sub>2"   and "P \<turnstile> T\<^sub>2 \<le> T" shows "P,E,h,sh \<turnstile> e\<^sub>1\<bullet>F{D} := e\<^sub>2 : Void"
frrestriction_def: fixes r :: "V"   and A :: "V" shows "r \<restriction>\<^sup>r\<^sub>\<bullet> A = set {[a, b]\<^sub>\<circ> |a b. b \<in>\<^sub>\<circ> A \<and> [a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r}"
akiyama_tanigawa_def: shows "akiyama_tanigawa = gen_akiyama_tanigawa (\<lambda>n. 1 / real (Suc n))"
eval_tm_lower_bound: fixes \<AA> :: "interp"   and "is" :: "int list" shows "(2 ^ Length \<AA> - 1) * sum_list (concat (map (\<lambda>i. if i < 0 then [i] else []) is)) \<le> eval_tm \<AA> is"
UNIV_I: fixes x :: "'a" shows "x \<in> UNIV"
C1: fixes S :: "ccs"   and T :: "ccs"   and U :: "ccs" shows "(S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel"
no_loops: fixes v :: "'a" shows "(v, v) \<notin> E"
in_set_conv_nth: fixes x :: "'a"   and xs :: "'a list" shows "(x \<in> list.set xs) = (\<exists>i<length xs. xs ! i = x)"
bind_spmf_cong_simp: fixes p :: "'b spmf"   and q :: "'b spmf"   and f :: "'b \<Rightarrow> 'c spmf"   and g :: "'b \<Rightarrow> 'c spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q =simp=> f x = g x" shows "p \<bind> f = q \<bind> g"
cases: fixes a1 :: "bool"   and a2 :: "v sem_env"   and a3 :: "'b state"   and a4 :: "v"   and a5 :: "(pat \<times> exp0) list"   and a6 :: "v"   and a7 :: "'b state \<times> (v, v) result" assumes "evaluate_match a1 a2 a3 a4 a5 a6 a7" obtains ck :: "bool" and env :: "v sem_env" and v1 :: "v" and err_v :: "v" and s :: "'b state" where "a1 = ckb__" and "a2 = envb__" and "a3 = sb__" and "a4 = v1a__" and "a5 = []" and "a6 = err_va__" and "a7 = (sb__, Rerr (Rraise err_va__))"   | ck :: "bool" and env :: "v sem_env" and env' :: "(char list \<times> v) list" and v1 :: "v" and p :: "pat" and pes :: "(pat \<times> exp0) list" and e :: "exp0" and bv :: "'b state \<times> (v, v) result" and err_v :: "v" and s :: "'b state" where "a1 = ckc__" and "a2 = envc__" and "a3 = sc__" and "a4 = v1b__" and "a5 = (pa__, eb__) # pesb__" and "a6 = err_vb__" and "a7 = bv__" and "allDistinct (pat_bindings pa__ []) \<and> pmatch (c envc__) (refs sc__) pa__ v1b__ [] = Match env' \<and> evaluate ckc__ (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v envc__)) envc__) sc__ eb__ bv__"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and bv :: "'b state \<times> (v, v) result" and s :: "'b state" and err_v :: "v" where "a1 = ckd__" and "a2 = envd__" and "a3 = sd__" and "a4 = v1c__" and "a5 = (pb__, ec__) # pesc__" and "a6 = err_vc__" and "a7 = bv" and "allDistinct (pat_bindings pb__ []) \<and> pmatch (c envd__) (refs sd__) pb__ v1c__ [] = No_match \<and> evaluate_match ckd__ envd__ sd__ v1c__ pesc__ err_vc__ bv"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and s :: "'b state" and err_v :: "v" where "a1 = cke__" and "a2 = enve__" and "a3 = se__" and "a4 = v1d__" and "a5 = (pc__, ed__) # pesd__" and "a6 = err_vd__" and "a7 = (se__, Rerr (Rabort Rtype_error))" and "pmatch (c enve__) (refs se__) pc__ v1d__ [] = Match_type_error"   | ck :: "bool" and env :: "v sem_env" and v1 :: "v" and p :: "pat" and e :: "exp0" and pes :: "(pat \<times> exp0) list" and s :: "'b state" and err_v :: "v" where "a1 = ck" and "a2 = env" and "a3 = s" and "a4 = v1" and "a5 = (p, e) # pes" and "a6 = err_v" and "a7 = (s, Rerr (Rabort Rtype_error))" and "\<not> allDistinct (pat_bindings p [])"
not_may_lock_conv: fixes l :: "('c \<times> nat) option"   and t :: "'c" shows "(\<not> may_lock l t) = (\<exists>t'. t' \<noteq> t \<and> has_lock l t')"
lenlex_conv: fixes r :: "('a \<times> 'a) set" shows "lenlex r = {(xs, ys). length xs < length ys \<or> length xs = length ys \<and> (xs, ys) \<in> lex r}"
vec_simps_12: fixes ag :: "'a"   and bg :: "'a" shows "ag + bg = bg + ag"
sum_overlap: fixes w :: "i"   and x :: "i"   and y :: "i" assumes "CEM" shows "O w (x \<oplus> y) = (O w x \<or> O w y)"
rotate1_lr: fixes l :: "'b list" shows "rotate1 (rotater1 l) = l"
Shift_def: fixes Kl :: "'b list set"   and k :: "'b" shows "BNF_Greatest_Fixpoint.Shift Kl k = {kl. k # kl \<in> Kl}"
inf_conj_distrib: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "((c\<^sub>0::'a::refinement_lattice) \<sqinter> (c\<^sub>1::'a::refinement_lattice)) \<iinter> (d::'a::refinement_lattice) = c\<^sub>0 \<iinter> d \<sqinter> c\<^sub>1 \<iinter> d"
invert_side_of_triangle_2: fixes h :: "'d"   and f :: "'d"   and g :: "'d" assumes "D.arr h"   and "f \<cdot>\<^sub>D g = h"   and "D.iso g" shows "D.seq h (D.inv g) \<and> f = h \<cdot>\<^sub>D D.inv g"
productive_imp_produces_Max_atom: fixes S :: "'b literal multiset \<Rightarrow> 'b literal multiset"   and N :: "'b literal multiset set"   and C :: "'b literal multiset" assumes "ground_resolution_with_selection S"   and "ground_resolution_with_selection.production S N C \<noteq> {}" shows "ground_resolution_with_selection.production S N C = {Max (atms_of C)}"
intro: fixes alphabet1 :: "'b list"   and init1 :: "'c \<Rightarrow> 'd"   and delta1 :: "'b \<Rightarrow> 'd \<Rightarrow> 'd"   and accept1 :: "'d \<Rightarrow> bool"   and wellformed1 :: "'d \<Rightarrow> bool"   and Language1 :: "'d \<Rightarrow> 'b language"   and wf1 :: "'c \<Rightarrow> bool"   and Lang1 :: "'c \<Rightarrow> 'b language"   and alphabet2 :: "'e list"   and init2 :: "'f \<Rightarrow> 'g"   and delta2 :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and accept2 :: "'g \<Rightarrow> bool"   and wellformed2 :: "'g \<Rightarrow> bool"   and Language2 :: "'g \<Rightarrow> 'e language"   and wf2 :: "'f \<Rightarrow> bool"   and Lang2 :: "'f \<Rightarrow> 'e language"   and letter_eq :: "'b \<Rightarrow> 'e \<Rightarrow> bool" assumes "DA alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1"   and "DA alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2"   and "DAs_axioms alphabet1 alphabet2 letter_eq" shows "DAs alphabet1 init1 delta1 accept1 wellformed1 Language1 wf1 Lang1 alphabet2 init2 delta2 accept2 wellformed2 Language2 wf2 Lang2 letter_eq"
the_elem_inj: fixes X :: "'a set"   and Y :: "'a set" assumes "X \<in> carrier (R Quot a_kernel R S h)"   and "Y \<in> carrier (R Quot a_kernel R S h)"   and "the_elem (h ` X) = the_elem (h ` Y)" shows "X = Y"
intra_SDG_path_Append: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n'' :: "'b SDG_node"   and ns' :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' ns' n'"   and "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n''" shows "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n (ns @ ns') n'"
rell_FGf_def: fixes L1 :: "'p \<Rightarrow> 'z \<Rightarrow> bool"   and L2 :: "'q \<Rightarrow> 'aa \<Rightarrow> bool" shows "rell_FGf L1 L2 = rel_FGf L1 L2 (=) (=) (=) (=)"
termination_0_1_immediate: fixes guard :: "'a \<Rightarrow> bool"   and p :: "real"   and body :: "'a \<Rightarrow> 'a spmf"   and s :: "'a" assumes "\<And>s. guard s \<Longrightarrow> p \<le> spmf (map_spmf guard (body s)) False"   and "0 < p"   and "\<And>s. guard s \<Longrightarrow> lossless_spmf (body s)" shows "lossless_spmf (loop_spmf.while guard body s)"
cp: shows "char_poly A = [:- e, 1:] * (\<Prod>e\<leftarrow>es. [:- e, 1:])"
simps_1: shows "eq\<cdot>[]\<cdot>[] = TT"
real_alg_2_list_length_def: shows "real_alg_2_list_length \<equiv> (Rep_real_alg_2_list ---> id) length"
simps_9: fixes p :: "fm" shows "bound0 (NOT p) = bound0 p"
class_les_mgd: fixes oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and type :: "String.literal"   and mgd :: "String.literal \<Rightarrow> String.literal set list option"   and C' :: "String.literal"   and Ss' :: "String.literal set list"   and C :: "String.literal" assumes "wf_osig oss"   and "tcsigs oss type = Some mgd"   and "mgd C' = Some Ss'"   and "class_les (subclass oss) C' C" shows "mgd C \<noteq> None"
left_idem: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> (a \<cdot> b) = a \<cdot> b"
red_det: fixes c :: "com"   and s :: "char list \<Rightarrow> val option"   and c\<^sub>1 :: "com"   and s\<^sub>1 :: "char list \<Rightarrow> val option"   and c\<^sub>2 :: "com"   and s\<^sub>2 :: "char list \<Rightarrow> val option" assumes "\<langle>c,s\<rangle> \<rightarrow> \<langle>c\<^sub>1,s\<^sub>1\<rangle>"   and "\<langle>c,s\<rangle> \<rightarrow> \<langle>c\<^sub>2,s\<^sub>2\<rangle>" shows "c\<^sub>1 = c\<^sub>2 \<and> s\<^sub>1 = s\<^sub>2"
invocation_declclass_def: fixes G :: "prog"   and m :: "inv_mode"   and s :: "st"   and a' :: "val"   and statT :: "ref_ty"   and sig :: "sig" shows "invocation_declclass G m s a' statT sig = declclass (the (dynlookup G statT (invocation_class m s a' statT) sig))"
abs_eq: fixes xa :: "'a \<Longrightarrow> 'b"   and x :: "'b \<Rightarrow> 'c" shows "(xa \<sharp> mk\<^sub>e x) = (\<forall>b v. x (put\<^bsub>xa\<^esub> b v) = x b)"
4: shows "backward_terminating_path_root e (x\<^sup>T)"
game\<^sub>3_def: fixes f :: "bool \<Rightarrow> bool list \<Rightarrow> bool list \<Rightarrow> bool list spmf"   and x :: "nat"   and y :: "nat" shows "game\<^sub>3 f x y = coin_spmf \<bind> (\<lambda>b. exec_gpv oracle' (\<A>\<^sub>1 (\<^bold>g [^] x)) ({}, empty) \<bind> (\<lambda>(((m\<^sub>1, m\<^sub>2), \<sigma>), s, s_h). TRY assert_spmf (valid_plains m\<^sub>1 m\<^sub>2) \<bind> (\<lambda>_. f b m\<^sub>1 m\<^sub>2 \<bind> (\<lambda>h'. let cipher = (\<^bold>g [^] y, h') in exec_gpv oracle' (\<A>\<^sub>2 cipher \<sigma>) (s, s_h) \<bind> (\<lambda>(b', s', s_h'). return_spmf (b' = b, \<^bold>g [^] (x * y) \<in> s')))) ELSE (coin_spmf \<bind> (\<lambda>b. return_spmf (b, \<^bold>g [^] (x * y) \<in> s)))))"
DSourcesS5: shows "DSources level2 sS5 = {sS4}"
trans_inv_image: assumes "trans {(x, y). r x y}" shows "trans (inv_image {(x, y). r x y} f)"
dim_UNIV: shows "vs1.dim UNIV = DIM('a)"
subset_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (=)) (\<subseteq>) (\<subseteq>)"
select_convs_2: fixes om :: "'d \<Rightarrow> 'e"   and am :: "'b \<Rightarrow> 'c"   and more :: "'f" shows "\<lparr>om = om, am = am, \<dots> = more\<rparr> \<^bsub>\<a>\<^esub> = am"
hcomplex_hypreal_numeral: fixes w :: "num" shows "hcomplex_of_complex (numeral w) = hcomplex_of_hypreal (numeral w)"
emeasure_PiM_exponential_Ioi_countable: fixes J :: "'d set"   and I :: "'d set"   and R :: "'d \<Rightarrow> real"   and x :: "real" assumes "J \<subseteq> I"   and "countable J"   and "\<And>i. i \<in> I \<Longrightarrow> 0 < R i"   and "0 \<le> x"   and "integrable (count_space J) R" shows "emeasure (Pi\<^sub>M I (\<lambda>i. exponential (R i))) (prod_emb I (\<lambda>i. exponential (R i)) J (J \<rightarrow>\<^sub>E {x<..})) = ennreal (exp (- x * integral\<^sup>L (count_space J) R))"
span_base: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> S" shows "a \<in> cspan S"
length_takeWhile_le: fixes P :: "'g \<Rightarrow> bool"   and xs :: "'g list" shows "length (takeWhile P xs) \<le> length xs"
code: fixes xs :: "'c llist"   and ys :: "'c llist" shows "lappend xs ys = (case xs of LNil \<Rightarrow> ys | LCons x xs' \<Rightarrow> LCons x (lappend xs' ys))"
simps_2: fixes fj :: "'b"   and xj :: "'b"   and xis :: "'b list" shows "divided_differences_impl [] fj xj xis = [fj]"
converse_rtranclp_into_rtranclp: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>*\<^sup>* a c"
negInf_convert: fixes var :: "nat"   and L :: "atom list"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "length xs' = var" shows "(\<forall>f\<in>set L. eval (substNegInfinity var f) (xs' @ x # xs)) = (\<forall>f\<in>set L'. evalUni (substNegInfinityUni f) x)"
sets_restrict_stream_space: fixes S :: "'b set"   and M :: "'b measure" assumes "S \<in> sets M" shows "sets (restrict_space (stream_space M) (streams S)) = sets (stream_space (restrict_space M S))"
sets_1: shows "X \<in> sets (Pi\<^sub>M J M)"
cosh_ln_real: fixes x :: "real" assumes "0 < x" shows "cosh (ln x) = (x + inverse x) / 2"
type_less_x_Required: fixes \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a basic_type" assumes "\<tau> < \<sigma>[1]" obtains \<rho> :: "'a basic_type" where "\<tau> = \<rho>[1]" and "\<rho> < \<sigma>"
sum_distrib_right: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set"   and r :: "'a" shows "comm_monoid_add.sum (\<squnion>) bot f A * r = comm_monoid_add.sum (\<squnion>) bot (\<lambda>n. f n * r) A"
sup_ge1: fixes x :: "'b multiset"   and y :: "'b multiset" shows "x \<subseteq># x \<union># y"
abs_eq: fixes x :: "'b poly" shows "ratfps_of_poly x = Abs_ratfps (to_fract x)"
not_add_distrib: fixes a :: "'a"   and b :: "'a" shows "NOT (a + b) = NOT a - b"
wf_fdecl_def: fixes P :: "'a prog" shows "wf_fdecl P \<equiv> \<lambda>(F, T, fm). is_type P T"
i_set_cont_iprev: fixes I :: "nat set"   and n :: "nat" assumes "I \<in> i_set_cont"   and "n \<in> I"   and "iMin I < n" shows "iprev n I = n - Suc 0"
cong: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and h :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "set.F g A = set.F h B"
set_flatten: fixes xs :: "'a list list" shows "set (flatten xs) = \<Union> (set ` set xs)"
Ide_VcompNml: fixes t :: "'a local.term"   and u :: "'a local.term" assumes "Ide t"   and "Ide u"   and "Nml t"   and "Nml u"   and "Dom t = Cod u" shows "Ide (t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> u)"
part1_Un_part2: fixes V :: "'b set" assumes "infinite V" shows "part1 V \<union> part2 V = V"
prems_7: shows "P,E \<turnstile> s \<surd>"
frechet_derivative_snd: fixes f :: "'b \<Rightarrow> 'c \<times> 'd"   and x :: "'b" assumes "f differentiable at x" shows "frechet_derivative (\<lambda>x. snd (f x)) (at x) = (\<lambda>xa. snd (frechet_derivative f (at x) xa))"
AbstrLevel3: shows "AbstrLevel level3 = {sS1opt, sS3, sS4opt, sS7opt, sS9, sS10, sS11opt, sS12, sS13}"
scaleC_one: fixes x :: "'a" shows "1 *\<^sub>C x = x"
cases: fixes a1 :: "cld list"   and a2 :: "ctx"   and a3 :: "cl"   and a4 :: "nat" assumes "path_length a1 a2 a3 a4" obtains P :: "cld list" and ctx :: "ctx" where "a1 = P__" and "a2 = ctx__" and "a3 = cl_object" and "a4 = 0"   | P :: "cld list" and ctx :: "ctx" and fqn :: "fqn" and ctx' :: "ctx" and cld :: "cld" and cl :: "cl" and nn :: "nat" where "a1 = P" and "a2 = ctx" and "a3 = cl_fqn fqn" and "a4 = nn + 1" and "find_cld P ctx fqn (Some (ctx', cld))" and "superclass_name cld cl" and "path_length P ctx' cl nn"
fpxs_of_fls_const: fixes c :: "'b" shows "fpxs_of_fls (fls_const c) = fpxs_const c"
abs_Un_LtK: fixes K :: "msg set"   and S :: "msg set" assumes "K \<subseteq> range LtK" shows "local.abs (K \<union> S) = local.abs S"
Guard: fixes s :: "'d"   and g :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and f :: "'f" assumes "s \<in> g"   and "\<Gamma>\<turnstile>c \<down> Normal s" shows "\<Gamma>\<turnstile>Guard f g c \<down> Normal s"
nth_strict_mono_on: fixes f :: "nat \<Rightarrow> nat"   and i :: "nat" assumes "strict_mono_on f {0..<n}"   and "i < n" shows "f i = sorted_list_of_set (f ` {0..<n}) ! i"
fold_cong: fixes a :: "'c"   and b :: "'c"   and xs :: "'d list"   and ys :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "a = b"   and "xs = ys"   and "\<And>x. x \<in> set xs \<Longrightarrow> f x = g x" shows "List.fold f xs a = List.fold g ys b"
objects_def: fixes M :: "model" shows "objects M = fst (Rep_model M)"
diagonal_to_Smith_i_PQ: fixes P' :: "(('a, 'rows) vec, 'rows) vec"   and B' :: "(('a, 'cols) vec, 'rows) vec"   and Q' :: "(('a, 'cols) vec, 'cols) vec"   and xs :: "nat list"   and i :: "nat" assumes "(P', B', Q') = diagonal_to_Smith_i_PQ xs i bezout (P, B, Q)"   and "\<forall>x. x \<in> set xs \<longrightarrow> x < min (nrows A) (ncols A)"   and "isDiagonal B"   and "i \<notin> set xs"   and "i < min (nrows A) (ncols A)" shows "B' = P' ** A ** Q' \<and> invertible P' \<and> invertible Q' \<and> B' = diagonal_to_Smith_i xs B i bezout"
inputPres: fixes P :: "pi"   and x :: "name"   and Q :: "pi"   and a :: "name" assumes "\<forall>y. P[x::=y] \<sim> Q[x::=y]" shows "a<x>.P \<sim> a<x>.Q"
aezfun_cases: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> aezfun_set" obtains x :: "('b, 'a) aezfun" where "y = aezfun x"