hcju/isabelleps huggingface dataset:
search_nonempty_neq0: fixes b :: "nat"   and i :: "nat"   and y\<^sub>0 :: "nat" assumes "s b \<down>= i"   and "\<phi> i y\<^sub>0 \<down>"   and "\<nexists>y. \<phi> i y \<down> \<and> e_length b \<le> y" shows "search b \<up>"
var_notin_notin_mult_monomsD: fixes y :: "char list"   and p :: "char list list"   and q :: "char list list" assumes "y \<in> set (mult_monoms p q)" shows "y \<in> set p \<or> y \<in> set q"
borel_measurable_vimage_open: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" shows "(f \<in> borel_measurable (lebesgue_on S)) = (\<forall>T. open T \<longrightarrow> {x \<in> S. f x \<in> T} \<in> sets (lebesgue_on S))"
emeasure_density: fixes f :: "'d \<Rightarrow> ennreal"   and M :: "'d measure"   and A :: "'d set" assumes "f \<in> borel_measurable M"   and "A \<in> sets M" shows "emeasure (density M f) A = set_nn_integral M A f"
add_edge_correct: fixes v :: "'c"   and v' :: "'c"   and G :: "'c list_graph" shows "FiniteGraph.add_edge v v' (list_graph_to_graph G) = list_graph_to_graph (FiniteListGraph.add_edge v v' G)"
interaction_bound_fixp_induct: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> enat) \<Rightarrow> bool"   and "consider" :: "'e \<Rightarrow> bool" assumes "lfp.admissible P"   and "P (\<lambda>_. 0)"   and "\<And>interaction_bound'. \<lbrakk>P interaction_bound'; \<And>gpv. interaction_bound' gpv \<le> interaction_bound consider gpv; \<And>gpv. interaction_bound' gpv \<le> (\<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))\<rbrakk> \<Longrightarrow> P (\<lambda>gpv. \<Squnion>generat\<in>set_spmf (the_gpv gpv). case generat of Pure x \<Rightarrow> 0 | IO out c \<Rightarrow> if consider out then eSuc (\<Squnion>input. interaction_bound' (c input)) else \<Squnion>input. interaction_bound' (c input))" shows "P (interaction_bound consider)"
ivl_subset_existence_ivl: fixes t :: "real"   and x0 :: "'c" assumes "t \<in> existence_ivl0 x0" shows "{0..t} \<subseteq> existence_ivl0 x0"
testN_post1: fixes k :: "nat" assumes "f k" shows "adjoint (testN k) * post * testN k = testN k"
inv_f_f: fixes x :: "'b" shows "inv (inv f) (inv f x) = x"
multi_inequalityTr0: fixes n :: "nat"   and x :: "nat \<Rightarrow> ant"   and m :: "nat" shows "(\<forall>j\<le>n. x j \<noteq> - \<infinity>) \<longrightarrow> (\<exists>L. \<forall>N>L. \<forall>l\<le>n. an m < x l + an N)"
A: fixes f :: "('a E.term, 'a) arr \<times> ('a E.term, 'a) arr" assumes "VV.arr f" shows "arr (fst f \<star> snd f)"
iffs: fixes ev_counter :: "nat"   and more :: "'a"   and ev_counter' :: "nat"   and more' :: "'a" shows "(\<lparr>ev_counter = ev_counter, \<dots> = more\<rparr> = \<lparr>ev_counter = ev_counter', \<dots> = more'\<rparr>) = (ev_counter = ev_counter' \<and> more = more')"
n_eq_1_imp: fixes s :: "'s"   and cfg :: "'s cfg" assumes "s \<in> S"   and "cfg \<in> cfg_on s"   and "n s = 1" shows "v cfg = 1"
All_mapping_tabulate: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "\<forall>x\<in>set xs. P x (f x)" shows "All_mapping (Mapping.tabulate xs f) P"
Jeroslow_Diagonalization_axioms: shows "Jeroslow_Diagonalization var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls dsj num prv enc F encF N ssap"
that: fixes c :: "real" assumes "\<And>s. s \<in> complex_cone' \<alpha> - {0} \<Longrightarrow> cmod (stirling_integral n s) \<le> c / cmod s ^ (n - 1)" shows "thesis"
finite_blocks: fixes b :: "'a set" assumes "b \<in># \<B>" shows "finite b"
union_set_fold: fixes xs :: "'b list"   and A :: "'b set" shows "set xs \<union> A = fold insert xs A"
Recv_given_channel: fixes ev :: "('a, 'b, 'c) event"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and c :: "('a, 'b, 'c) configuration" assumes "isRecv ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "can_occur ev c" shows "\<exists>s s' m. ev = Recv cid q p s s' m"
mem_mod_w32_low_equal: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and t1 :: "'b sparc_state"   and a :: "word36"   and bm :: "word4"   and data :: "word32"   and t2 :: "'b sparc_state" assumes "low_equal s1 s2 \<and> t1 = mem_mod_w32 10 a bm data s1 \<and> t2 = mem_mod_w32 10 a bm data s2" shows "low_equal t1 t2"
transfer: shows "IEEE.cr_float (0, 0, 0) 0"
nodes_connected_ind_sym: fixes F :: "(nat \<times> int \<times> nat) set"   and L :: "(nat \<times> int \<times> nat) list" assumes "F \<subseteq> set L" shows "sym {(x, y) |x y. nodes_connected \<lparr>nodes = fst ` set L \<union> (snd \<circ> snd) ` set L, edges = F\<rparr> x y}"
SystemDCSubst_def: shows "SystemDCSubst = \<lparr>SFunctions = Map.empty, SPredicates = Map.empty, SContexts = \<lambda>C. if C = pid1 then Some TT else if C = pid2 then Some (Geq (Differential (Var vid1)) (Differential (Const 0))) else if C = pid3 then Some (Geq ($f fid1 local.empty) (Const 0)) else None, SPrograms = Map.empty, SODEs = \<lambda>c. if c = vid1 then Some (OProd (OSing vid1 ($f fid1 local.empty)) (OSing vid2 (Var vid1))) else None\<rparr>"
subseq_fold_reduce: fixes n :: "nat"   and k :: "nat" shows "strict_mono (fold_reduce n k)"
UNIF1_subst_domain_Int: fixes \<sigma> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and E :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset"   and E' :: "(('d, 'c) Term.term \<times> ('d, 'c) Term.term) multiset" assumes "UNIF1 \<sigma> E E'" shows "subst_domain \<sigma> \<inter> vars_mset E' = {}"
Ll_def: shows "Ll = map Lli [0..<length (prems_of \<iota>F)]"
makes_compatible_invariant: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and cms\<^sub>1' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1' :: "'Var \<Rightarrow> 'Val" assumes "sound_mode_use (cms\<^sub>1, mem\<^sub>1)"   and "sound_mode_use (cms\<^sub>2, mem\<^sub>2)"   and "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "map snd cms\<^sub>1 = map snd cms\<^sub>2"   and "(cms\<^sub>1, mem\<^sub>1) \<rightarrow> (cms\<^sub>1', mem\<^sub>1')" obtains cms\<^sub>2'' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" and mem\<^sub>2'' :: "'Var \<Rightarrow> 'Val" and mems' :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list" where "map snd cms\<^sub>1' = map snd cms\<^sub>2'' \<and> (cms\<^sub>2, mem\<^sub>2) \<rightarrow> (cms\<^sub>2'', mem\<^sub>2'') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2'', mem\<^sub>2'') mems'"
permStarFresh: fixes xvec :: "name list"   and p :: "(name \<times> name) list"   and T :: "'d" assumes "xvec \<sharp>* p" shows "xvec \<sharp>* (p \<bullet> T) = xvec \<sharp>* T"
assms_3: shows "(cmod A)\<^sup>2 < (cmod B)\<^sup>2"
Aheap_def: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp" shows "Aheap \<Gamma> e = (\<Lambda> a. if nonrec \<Gamma> then (case hd \<Gamma> of (x, xa) \<Rightarrow> Aheap_nonrec x xa)\<cdot>(Aexp_syn' a e, ccExp_syn a e) else Afix \<Gamma>\<cdot>(Aexp_syn' a e \<squnion> (\<lambda>_. up\<cdot>0) f|` thunks \<Gamma>) f|` domA \<Gamma>)"
attractor_in_V: fixes W :: "'a set"   and p :: "Player" assumes "W \<subseteq> V" shows "attractor p W \<subseteq> V"
set_list_to_rbt: fixes xs :: "int list" shows "Tree2.set_tree (list_to_rbt xs) = set xs"
binrelchain_join: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and x :: "'b"   and y :: "'b"   and ys :: "'b list" assumes "binrelchain P (xs @ [x])"   and "binrelchain P (y # ys)"   and "P x y" shows "binrelchain P (xs @ x # y # ys)"
h3: shows "h2 \<turnstile> set_tag_name new_element_ptr tag \<rightarrow>\<^sub>h h3"
estep_Cn: fixes c :: "(recf \<times> nat list \<times> nat list) list \<times> nat option"   and n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list"   and ls :: "nat list"   and fs :: "(recf \<times> nat list \<times> nat list) list"   and rv :: "nat option" assumes "c = ((Cn n f gs, xs, ls) # fs, rv)" shows "estep_Cn (encode_config c) = encode_config (step c)"
sublist_imp_subseq: fixes xs :: "'a list"   and ys :: "'a list" assumes "sublist xs ys" shows "xs \<preceq> ys"
res: shows "res = filter (\<lambda>(a, i). a \<noteq> 1) r"
rep_simps_1: fixes r :: "'a" assumes "ide r" shows "arr (rep r)"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Cod (f \<star> src f))"
Id_onE: fixes c :: "'b \<times> 'b"   and A :: "'b set" assumes "c \<in> Id_on A" obtains x :: "'b" where "x \<in> A" and "c = (x, x)"
A_del_min: fixes h :: "'b tree" assumes "is_root h" shows "real (T_del_min h) + \<Phi> (del_min h) - \<Phi> h \<le> 2 * log 2 (real (size h + 1)) + 5"
Sem_deterministic: fixes c :: "'a com"   and s :: "'a"   and s1 :: "'a"   and s2 :: "'a" assumes "Sem c s s1"   and "Sem c s s2" shows "s1 = s2"
subst_def: fixes xs :: "'varSort"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and x :: "'var" shows "subst xs X x \<equiv> psubst (idEnv [x \<leftarrow> X]_xs)"
paval_extend: fixes ps1 :: "char list \<Rightarrow> int option"   and ps2 :: "char list \<Rightarrow> int option"   and a :: "aexp" assumes "ps1 ## ps2"   and "vars a \<subseteq> dom ps1" shows "paval a (ps1 + ps2) = paval a ps1"
zip_Cons1: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'c list" shows "zip (x # xs) ys = (case ys of [] \<Rightarrow> [] | y # ys \<Rightarrow> (x, y) # zip xs ys)"
betw_a_in_path: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and bc :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c"   and "bc \<in> \<P> \<and> b \<in> bc \<and> c \<in> bc \<and> b \<noteq> c" shows "a \<in> bc"
i_expand_nth_interval_eq_replicate_NoMsg: fixes k :: "nat"   and t :: "nat"   and t1 :: "nat"   and t2 :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" assumes "0 < k"   and "t * k < t1"   and "t1 \<le> t2"   and "t2 \<le> t * k + k" shows "f \<odot> k \<Down> t2 \<up> t1 = NoMsg\<^bsup>t2 - t1\<^esup>"
subst4Chain: fixes xvec :: "name list"   and Tvec :: "'a list"   and T :: "'a" assumes "length xvec = length Tvec"   and "distinct xvec"   and "xvec \<sharp>* Tvec" shows "xvec \<sharp>* T[xvec::=Tvec]"
simps_3: fixes x :: "'a"   and xs :: "'a list"   and e :: "'a"   and i :: "'a" shows "insert_rga (x # xs) (e, Some i) = (if x = i then x # insert_body xs e else x # insert_rga xs (e, Some i))"
news_compl: fixes n :: "'a"   and G :: "('a, 'b) form list" assumes "news n G" shows "news n (map compl G)"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and a1 :: "char list"   and a2 :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> a1 has_fields a2" obtains C :: "char list" and D :: "char list" and fs :: "(char list \<times> ty) list" and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list" and FDTs :: "((char list \<times> char list) \<times> ty) list" and FDTs' :: "((char list \<times> char list) \<times> ty) list" where "a1 = C" and "a2 = FDTs'" and "class P C = \<lfloor>(D__, fs__, ms__)\<rfloor>" and "C \<noteq> Object" and "P \<turnstile> D__ has_fields FDTs__" and "FDTs' = map (\<lambda>(F, T). ((F, C), T)) fs__ @ FDTs__"   | D :: "char list" and fs :: "(char list \<times> ty) list" and ms :: "(char list \<times> ty list \<times> ty \<times> 'b) list" and FDTs :: "((char list \<times> char list) \<times> ty) list" where "a1 = Object" and "a2 = FDTs" and "class P Object = \<lfloor>(D, fs, ms)\<rfloor>" and "FDTs = map (\<lambda>(F, T). ((F, Object), T)) fs"
tendsto_const: fixes "open" :: "'b set \<Rightarrow> bool"   and k :: "'b"   and F :: "'c filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
timpl_closure_FunI2: fixes T :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and c :: "('c set \<times> 'c set) set"   and S :: "(('a, 'b, 'c) prot_fun, 'd) Term.term list"   and f :: "('a, 'b, 'c) prot_fun"   and g :: "('a, 'b, 'c) prot_fun" assumes "\<And>i. i < length T \<Longrightarrow> \<exists>u. (T ! i, u) \<in> timpl_closure' c \<and> (S ! i, u) \<in> timpl_closure' c"   and "length T = length S"   and "f = g \<or> (\<exists>a b d. (a, d) \<in> c\<^sup>+ \<and> (b, d) \<in> c\<^sup>+ \<and> f = Abs a \<and> g = Abs b)" shows "\<exists>h U. (Fun f T, Fun h U) \<in> timpl_closure' c \<and> (Fun g S, Fun h U) \<in> timpl_closure' c"
extract_var_sum: fixes p :: "'b mpoly"   and p' :: "'b mpoly"   and v :: "nat" shows "extract_var (p + p') v = extract_var p v + extract_var p' v"
Schwartz_inequality_strong: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and g :: "real \<Rightarrow> real" assumes "f square_integrable S"   and "g square_integrable S" shows "l2product S (\<lambda>x. \<bar>f x\<bar>) (\<lambda>x. \<bar>g x\<bar>) \<le> l2norm S f * l2norm S g"
ortho: fixes l :: "'a vec"   and w :: "'a vec" assumes "l \<in> set Ls"   and "w \<in> set Ns" shows "w \<bullet> l = (0::'a)"
assms_3: shows "strict_border (take (i - 1) s) (take (j - 1) s)"
Reals_cases: fixes q :: "'a" assumes "q \<in> \<real>" obtains r :: "real" where "q = of_real r"
la: shows "list_all (\<lambda>ev. occurs_on ev \<noteq> q) (take (j - Suc i) (drop (Suc i) t))"
assms_2: shows "m \<le> SPEC (\<lambda>s. \<Psi> s \<longrightarrow> \<Phi> s)"
list_all_simps_1: fixes P :: "'e \<Rightarrow> bool"   and x :: "'e"   and xs :: "'e list" shows "list_all P (x # xs) = (P x \<and> list_all P xs)"
power_set_func_nat_trans: fixes f :: "'b \<Rightarrow> 'c" shows "\<eta> \<circ> id f = (`) f \<circ> \<eta>"
pred_prod_inject: fixes P1 :: "'d \<Rightarrow> bool"   and P2 :: "'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e" shows "pred_prod P1 P2 (a, b) = (P1 a \<and> P2 b)"
complex_homo_coords_eq_def: shows "(\<approx>) \<equiv> map_fun Rep_complex_homo_coords (map_fun Rep_complex_homo_coords id) (\<approx>\<^sub>v)"
conv_radius_mult_power_right: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" assumes "c \<noteq> (0::'b)" shows "conv_radius (\<lambda>n. f n * c ^ n) = conv_radius f / ereal (norm c)"
hom_functor_axioms: shows "hom_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>S) \<phi>C"
seq_offset_neg: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'a"   and k :: "nat" assumes "f \<longlonglongrightarrow> l" shows "(\<lambda>i. f (i - k)) \<longlonglongrightarrow> l"
row_2: fixes i :: "nat" assumes "\<not> i < nr1"   and "i < nr1 + nr2" shows "row (four_block_mat A1 B1 C1 D1) i = row C1 (i - nr1) @\<^sub>v row D1 (i - nr1)"
Not_Rex: fixes F :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "\<turnstile> (\<not> (\<exists>x. F x)) = (\<forall>x. \<not> F x)"
sp_ends_in_tgt_imp_mem: fixes e :: "('a \<times> nat) edge"   and v :: "'a \<times> nat"   and es :: "('a \<times> nat) edge list" assumes "tgt e \<notin> Graph.vertices g"   and "v \<noteq> tgt e"   and "SubRel.subpath (add_edge g e) v es (tgt e) subs" shows "e \<in> set es"
med01s_def: fixes t :: "l1_state" shows "med01s t \<equiv> \<lparr>ik = ik t, secret = secret t\<rparr>"
cases: fixes a :: "trm \<Rightarrow> (char list \<times> trm) list" assumes "rctxtp a" obtains E :: "trm \<Rightarrow> trm" and l :: "char list" and fs :: "(char list \<times> trm) list" where "a = (\<lambda>t. (l__, E__ t) \<Colon> fs)" and "ctxtp E__"   | v :: "trm" and E :: "trm \<Rightarrow> (char list \<times> trm) list" and l :: "char list" where "a = (\<lambda>t. (l, v) \<Colon> E t)" and "valuep v" and "rctxtp E"
ctor1_def: shows "ctor1 = Abs_IF1 \<circ> str_init1 undefined \<circ> F1map id Rep_IF1 Rep_IF2"
zero_freeword_def: shows "0 \<equiv> Abs_freeword []"
len_lexext_rev_hd_or_tl: fixes gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and ys :: "'b list"   and x :: "'b"   and xs :: "'b list" assumes "len_lexext_rev gt (y # ys) (x # xs)" shows "gt y x \<or> len_lexext_rev gt ys xs"
bind_eq_None_conv: fixes a :: "'b option"   and f :: "'b \<Rightarrow> 'a option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
lc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "R a b"   and "R a c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
split_asm: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "P (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (\<not> (option = None \<and> \<not> P f1 \<or> (\<exists>x2. option = Some x2 \<and> \<not> P (f2 x2))))"
ereal_le_epsilon2: fixes x :: "ereal"   and y :: "ereal" assumes "\<And>e. 0 < e \<Longrightarrow> x \<le> y + ereal e" shows "x \<le> y"
undirected_def: fixes G :: "'b graph" shows "FiniteGraph.undirected G = \<lparr>nodes = nodes G, edges = edges G \<union> {(b, a). (a, b) \<in> edges G}\<rparr>"
measurable_SUP: fixes I :: "'c set"   and F :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "countable I"   and "\<And>i. i \<in> I \<Longrightarrow> F i \<in> M \<rightarrow>\<^sub>M count_space UNIV" shows "(\<lambda>x. SUP i\<in>I. F i x) \<in> M \<rightarrow>\<^sub>M count_space UNIV"
angle_partition: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "B \<noteq> C" shows "Tarski_neutral_dimensionless.Acute Bet Cong A B C \<or> Tarski_neutral_dimensionless.Per Bet Cong A B C \<or> Tarski_neutral_dimensionless.Obtuse Bet Cong A B C"
assms_8: shows "abs_conv_abscissa f \<le> s0"
Join_project_LeadsTo: fixes C :: "'c set"   and F :: "'a program"   and G :: "'c program"   and A :: "'a set"   and B :: "'a set" assumes "C = reachable (extend h F \<squnion> G)"   and "F \<squnion> project h C G \<in> A \<longmapsto>w B" shows "extend h F \<squnion> G \<in> extend_set h A \<longmapsto>w extend_set h B"
simps_2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "CMP_BRANCH EQ x y z = y"
l_redu_subst1: fixes M :: "trm"   and M' :: "trm"   and y :: "name"   and c :: "coname"   and P :: "trm" assumes "M \<longrightarrow>\<^sub>l M'" shows "M{y:=<c>.P} \<longrightarrow>\<^sub>a* M'{y:=<c>.P}"
return_bind: fixes x :: "'d"   and f :: "'d \<Rightarrow> ['c]" shows "\<langle>x\<rangle> \<bind> f = f x"
generate_valid_topology_sound: fixes M :: "'b SecurityInvariant_configured list"   and V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "valid_reqs M"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>" shows "all_security_requirements_fulfilled M (generate_valid_topology M \<lparr>nodes = V, edges = E\<rparr>)"
simps: fixes B :: "ty"   and X :: "ctrm set" shows "NOTLEFT (NOT B) X = {x:NotL <a>.M x |a x M. fin (NotL <a>.M x) x \<and> <a>:M \<in> X}"
microstep_callI: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a" shows "(call(pn)) \<leadsto>\<^bsub>\<Gamma>\<^esub> \<Gamma> pn"
TBtheorem4a_P: fixes P :: "specID"   and E :: "Expression"   and PQ :: "specID"   and Q :: "specID" assumes "ine P E"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "\<exists>ch. ch \<in> ins P \<and> exprChannel ch E \<and> ch \<notin> loc PQ" shows "ine PQ E"
sym_iff_eq_refl: fixes x :: "'a"   and y :: "'a" assumes "x \<in> X"   and "y \<in> X" shows "(x \<sim> y) = (x = y \<and> y \<sqsubseteq> y)"
SourcesTrans: fixes C :: "CSet"   and i :: "AbstrLevelsID"   and S :: "CSet"   and Z :: "CSet" shows "C \<in> Sources i S \<and> S \<in> Sources i Z \<longrightarrow> C \<in> Sources i Z"
ennreal_Ex_less_of_nat: fixes x :: "ennreal" assumes "x < \<top>" shows "\<exists>n. x < of_nat n"
m_comm: fixes x :: "'b"   and y :: "'b" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "x * y = y * x"
fresh_finfun_const: fixes a :: "atom"   and b :: "'b" shows "a \<sharp> finfun_const b = a \<sharp> b"
concat_bit_assoc: fixes n :: "nat"   and k :: "int"   and m :: "nat"   and l :: "int"   and r :: "int" shows "concat_bit n k (concat_bit m l r) = concat_bit (m + n) (concat_bit n k l) r"
pos_poly_mult: fixes p :: "'b poly"   and q :: "'b poly" assumes "pos_poly p"   and "pos_poly q" shows "pos_poly (p * q)"
triangle_left': fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b"   and \<eta> :: "'b"   and \<epsilon> :: "'b" assumes "adjunction_in_bicategory V H \<a> \<i> src trg f g \<eta> \<epsilon>" shows "V (bicategory.lunit V H \<a> \<i> src trg f) (V (H \<epsilon> f) (V (bicategory.\<a>' V H \<a> src trg f g f) (V (H f \<eta>) (category.inv V (bicategory.runit V H \<a> \<i> src f))))) = f"
lift_streams_def: fixes xs :: "'b stream list" shows "lift_streams xs = foldr (smap2 (#)) xs (pure [])"
strict_implies_not_eq: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "less a b" shows "a \<noteq> b"
subgraphs_of_union_1: fixes G :: "('c, 'd) pre_digraph"   and G' :: "('c, 'd) pre_digraph" assumes "wf_digraph G"   and "wf_digraph G'"   and "compatible G G'" shows "subgraph G (Digraph_Component.union G G')"
scaleR_left: fixes r :: "real"   and a :: "'c \<Rightarrow>\<^sub>L 'a"   and b :: "'c" shows "blinfun_apply (r *\<^sub>R a) b = r *\<^sub>R blinfun_apply a b"
expand_init_def: shows "expand_init \<equiv> 0::'e"
load_tmps_write_tmps_distinct: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "load_tmps is \<inter> \<Union> (fst ` write_sops sb) = {}"
pullback_arr_cod_2: fixes f :: "'a" assumes "C.arr f" shows "C.inverse_arrows \<p>\<^sub>0[C.cod f, f] \<langle>f \<lbrakk>C.cod f, f\<rbrakk> C.dom f\<rangle>"
f0: fixes x :: "real" assumes "0 < x"   and "x \<le> a" shows "(f has_real_derivative f' x) (at x)"
points_of_path_iff_2: fixes x :: "'a" shows "(x + x\<^sup>T) ; 1 = x ; 1 + end_points x"
less_add_order_eq_0_contra: fixes k :: "nat"   and a :: "'a" assumes "order.greater k 0"   and "order.greater (add_order a) k" shows "a +^ k \<noteq> (0::'a)"
in_homE: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "'c"   and a :: "'c"   and b :: "'c" assumes "category C"   and "partial_magma.in_hom C f a b" obtains "partial_magma.arr C f" and "partial_magma.dom C f = a" and "partial_magma.cod C f = b"
a_consistent_if\<^sub>1: fixes ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and a :: "Arity"   and as :: "Arity list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and S :: "stack_elem list" assumes "a_consistent (ae, a, as) (\<Gamma>, scrut ? e1 : e2, S)" shows "a_consistent (ae, 0, a # as) (\<Gamma>, scrut, Alts e1 e2 # S)"
approx_mult_SComplex2: fixes a :: "complex star"   and x :: "complex star" assumes "a \<in> SComplex"   and "x \<approx> 0" shows "a * x \<approx> 0"
diff_preserves_multiset: fixes M :: "'k \<Rightarrow> nat"   and N :: "'k \<Rightarrow> nat" assumes "M \<in> multiset" shows "(\<lambda>a. M a - N a) \<in> multiset"
mergesort_remdups_correct: fixes l :: "'b list" shows "distinct (mergesort_remdups l) \<and> sorted (mergesort_remdups l) \<and> set (mergesort_remdups l) = set l"
B_def: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "QR.B p q = {x \<in> {0<..(int p * int q - 1) div 2}. x mod int p \<in> {(int p - 1) div 2<..int p - 1} \<and> x mod int q \<in> {0<..(int q - 1) div 2}}"