hcju/isabelleps huggingface dataset:
ithE: fixes x :: "'a"   and B :: "('b \<Rightarrow> 'a) set"   and i :: "'b" assumes "x \<in> ith B i" obtains f :: "'b \<Rightarrow> 'a" where "f \<in> B" and "f i = x"
sym: fixes b :: "'a"   and a :: "'a" shows "(cmp b a = Equiv) = (cmp a b = Equiv)"
alphaAbs_qAbs_iff_ex_equal_or_qFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGood X" shows "(qAbs xs x X $= qAbs xs' x' X') = (xs = xs' \<and> (\<exists>y. (y = x \<or> qFresh xs y X) \<and> (y = x' \<or> qFresh xs y X') \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs))"
finite_union_of_Int_eq: fixes P :: "'b set \<Rightarrow> bool" shows "(\<forall>S T. (finite union_of P) S \<and> (finite union_of P) T \<longrightarrow> (finite union_of P) (S \<inter> T)) = (\<forall>S T. P S \<and> P T \<longrightarrow> (finite union_of P) (S \<inter> T))"
pos_fun_distr: fixes R :: "'d \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'i \<Rightarrow> bool"   and R' :: "'h \<Rightarrow> 'f \<Rightarrow> bool"   and S' :: "'i \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_fun R S OO rel_fun R' S' \<le> rel_fun (R OO R') (S OO S')"
\<tau>exec_1_def: fixes P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and \<sigma> :: "'addr option \<times> 'heap \<times> 'addr frame list"   and \<sigma>' :: "'addr option \<times> 'heap \<times> 'addr frame list" shows "\<tau>exec_1 P t \<sigma> \<sigma>' = (P,t \<turnstile> \<sigma> -\<lbrace>\<rbrace>-jvm\<rightarrow> \<sigma>' \<and> \<tau>Move2 P \<sigma>)"
assms_3: shows "x = y \<otimes> a [^] h"
1: shows "y \<squnion> y\<^sup>T = x \<and> pd_kleene_allegory_class.acyclic y \<and> injective y"
discI_2: fixes tm :: "('c, 'd) tm"   and x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" assumes "tm = App x21 x22" shows "is_App tm"
nonempty_is_arg_max_ib: fixes ys :: "'a list" assumes "ys \<noteq> []" shows "is_arg_max length (\<lambda>xs. strict_border xs ys) (intrinsic_border ys)"
f: shows "f \<in> restrict_space M A \<rightarrow>\<^sub>M N"
ord_spmf_eqD_measure: fixes p :: "'c spmf"   and q :: "'c spmf"   and A :: "'c set" assumes "ord_spmf (=) p q" shows "Sigma_Algebra.measure (measure_spmf p) A \<le> Sigma_Algebra.measure (measure_spmf q) A"
\<delta>_5: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
merge: shows "h\<^sub>0 ++ h\<^sub>1 = h\<^sub>0' ++ h\<^sub>1'"
simps_9: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" shows "\<xi> \<Turnstile>\<^sub>n \<phi> R\<^sub>n \<psi> = (\<forall>i. suffix i \<xi> \<Turnstile>\<^sub>n \<psi> \<or> (\<exists>j<i. suffix j \<xi> \<Turnstile>\<^sub>n \<phi>))"
get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) shadow_root_ptr"   and getter :: "('c, 'a, 'b, 'j option) RShadowRoot_scheme \<Rightarrow> 'm" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| shadow_root_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t ptr getter"
that: assumes "vsv f"   and "n \<in>\<^sub>\<circ> \<omega>"   and "\<D>\<^sub>\<circ> f = A ^\<^sub>\<times> n"   and "\<R>\<^sub>\<circ> f = A" shows "thesis"
pfsubset_fcard_mono: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subset>| B" shows "fcard A < fcard B"
is_syz_sigE: fixes d :: "'a \<Rightarrow> nat"   and u :: "'t" assumes "is_syz_sig d u" obtains r :: "'t \<Rightarrow>\<^sub>0 'b" where "r \<noteq> 0" and "lt r = u" and "r \<in> dgrad_sig_set' (length fs) d" and "rep_list r = 0"
ccHeap_nonrec_eq: fixes x :: "Vars.var"   and e :: "exp"   and exp :: "exp"   and n :: "Arity" shows "ccHeap_nonrec x e exp\<cdot>n = CCfix_nonrec x e\<cdot>(Aexp_syn' n exp, ccExp_syn n exp)"
Nats_1: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a" assumes "class.semiring_1 one times plus zero" shows "one \<in> semiring_1.Nats one plus zero"
par_comm_def: shows "par_comm \<equiv> \<lambda>(guar, post). {c. (\<forall>i. Suc i < length c \<longrightarrow> c ! i -pc\<rightarrow> c ! Suc i \<longrightarrow> (snd (c ! i), snd (c ! Suc i)) \<in> guar) \<and> (All_None (fst (last c)) \<longrightarrow> snd (last c) \<in> post)}"
WF_def: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "WF(A)_v \<equiv> \<diamond>\<box>Enabled (<A>_v) \<longrightarrow> \<box>\<diamond><A>_v"
character_axioms: shows "character (G\<lparr>carrier := {\<one>}\<rparr>) (principal_char (G\<lparr>carrier := {\<one>}\<rparr>))"
as0_8: shows "map S CAs0 \<cdot>\<cdot>cl \<eta>s0 = map (S_M S M) CAs"
Ana_invar_subst_def: fixes \<M> :: "('fun, 'var) Term.term set" shows "Ana_invar_subst \<M> \<equiv> \<forall>f T K M \<delta>. Fun f T \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t \<M> \<longrightarrow> Ana (Fun f T) = (K, M) \<longrightarrow> Ana (Fun f T \<cdot> \<delta>) = (K \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>, M \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<delta>)"
sp_from_de_empty: fixes v1 :: "'d \<times> nat"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and g :: "('d \<times> nat, 'e) rgraph_scheme"   and es :: "('d \<times> nat) edge list"   and v2 :: "'d \<times> nat" assumes "v1 \<notin> subsumees subs"   and "out_edges g v1 = {}"   and "SubRel.subpath g v1 es v2 subs" shows "es = []"
eval_Cn: fixes n :: "nat"   and f :: "recf"   and gs :: "recf list"   and xs :: "nat list" assumes "recfn (length xs) (Cn n f gs)" shows "eval (Cn n f gs) xs = (if \<forall>g\<in>set gs. eval g xs \<down> then eval f (map (\<lambda>g. the (eval g xs)) gs) else None)"
gen_length_def: fixes n :: "nat"   and xs :: "'b list" shows "List.gen_length n xs = n + \<^bold>|xs\<^bold>|"
adjoint_pair_antipar_1: fixes f :: "'a"   and g :: "'a" assumes "C.adjoint_pair f g" shows "C.ide f"
Aeta_expand_fresh: fixes a :: "atom"   and n :: "Arity"   and e :: "exp" shows "a \<sharp> Aeta_expand n e = a \<sharp> e"
ennreal3_cases: fixes x :: "ennreal"   and xa :: "ennreal"   and xaa :: "ennreal" obtains r :: "real" and ra :: "real" and raa :: "real" where "0 \<le> r__" and "x = ennreal r__" and "0 \<le> ra__" and "xa = ennreal ra__" and "0 \<le> raa" and "xaa = ennreal raa"   | r :: "real" and ra :: "real" where "0 \<le> rb__" and "x = ennreal rb__" and "0 \<le> rab__" and "xa = ennreal rab__" and "xaa = top"   | r :: "real" and ra :: "real" where "0 \<le> rc__" and "x = ennreal rc__" and "xa = top" and "0 \<le> rac__" and "xaa = ennreal rac__"   | r :: "real" where "0 \<le> rd__" and "x = ennreal rd__" and "xa = top" and "xaa = top"   | r :: "real" and ra :: "real" where "x = top" and "0 \<le> re__" and "xa = ennreal re__" and "0 \<le> ra" and "xaa = ennreal ra"   | r :: "real" where "x = top" and "0 \<le> rf__" and "xa = ennreal rf__" and "xaa = top"   | r :: "real" where "x = top" and "xa = top" and "0 \<le> r" and "xaa = ennreal r"   | "x = top" and "xa = top" and "xaa = top"
iso_assoc: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "ide g"   and "ide h"   and "src f = trg g"   and "src g = trg h" shows "local.iso \<a>[f, g, h]"
FunctorCompDef: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme"   and f :: "'f"   and g :: "'f" assumes "Ftor F : A \<longrightarrow> B"   and "f \<approx>>\<^bsub>A\<^esub> g" shows "F ## f \<approx>>\<^bsub>B\<^esub> F ## g"
Lp_domination_1: fixes g :: "'b \<Rightarrow> real"   and M :: "'b measure"   and f :: "'b \<Rightarrow> real"   and p :: "ennreal" assumes "g \<in> borel_measurable M"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "AE x in M. \<bar>g x\<bar> \<le> \<bar>f x\<bar>" shows "g \<in> space\<^sub>N (\<LL> p M)"
plus_adds_pp_right: fixes s :: "'a"   and t :: "'a"   and v :: "'t" assumes "s + t adds\<^sub>p v" shows "t adds\<^sub>p v"
assms_2: shows "ct_suffix cts' (to_ct_list ts)"
get_info_def: fixes X :: "'f" shows "get_info X = SPEC (\<lambda>(h, Y). Y = X)"
tllist_all2_tappendI: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "('a, 'c) tllist"   and ys :: "('b, 'd) tllist"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xs' :: "'c \<Rightarrow> ('a, 'e) tllist"   and ys' :: "'d \<Rightarrow> ('b, 'f) tllist" assumes "tllist_all2 P Q xs ys"   and "\<lbrakk>tfinite xs; tfinite ys; Q (terminal xs) (terminal ys)\<rbrakk> \<Longrightarrow> tllist_all2 P R (xs' (terminal xs)) (ys' (terminal ys))" shows "tllist_all2 P R (tappend xs xs') (tappend ys ys')"
inv_char: fixes f :: "'s" assumes "local.iso f" shows "local.inv f = mkArr (Cod f) (Dom f) (inv_into (Dom f) (Fun f))"
conf_Null: 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 h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and T :: "ty" shows "(P,h \<turnstile> Null :\<le> T) = P \<turnstile> NT \<le> T"
not_UNIV_eq_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l' :: "'b"   and h' :: "'b" assumes "class.no_top less_eq less" shows "UNIV \<noteq> ord.atLeastAtMost less_eq l' h'"
not_le_real: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.semiring_real_line plus zero minus less_eq less times"   and "less a zero \<or> a = zero \<or> less zero a"   and "less b zero \<or> b = zero \<or> less zero b" shows "(\<not> less_eq b a) = less a b"
assms: fixes x :: "'i" assumes "x \<in> S" shows "\<bar>f x\<bar> \<noteq> \<infinity>"
uminus_assn_def: fixes P :: "assn" shows "\<not>\<^sub>A P \<equiv> Abs_assn (\<lambda>h. in_range h \<and> \<not> h \<Turnstile> P)"
s_TS_def: fixes init :: "nat list"   and initH :: "nat list"   and qs :: "nat list"   and n :: "nat" shows "s_TS init initH qs n = fst (TSdet init initH qs n)"
in_frontier_convex_hull: fixes S :: "'a set"   and x :: "'a" assumes "finite S"   and "card S \<le> Suc DIM('a)"   and "x \<in> S" shows "x \<in> frontier (convex hull S)"
intPF_def: shows "M.intPF \<equiv> local.intPF"
DAiniD: fixes p :: "('b, 'c) Combinators list" shows "DenyAll \<in> set (insertDeny p)"
vector_scalar_mult_def: fixes c :: "'b"   and x :: "('b, 'c) Finite_Cartesian_Product.vec" shows "c *s x = (\<chi>i. c * x $h i)"
that: fixes i :: "nat" assumes "i < length ts"   and "t = ts ! i" shows "thesis"
Direct: fixes new :: "qtname \<times> mdecl"   and old :: "qtname \<times> mdecl"   and G :: "prog" assumes "\<not> is_static new"   and "\<not> is_static old"   and "accmodi new \<noteq> Private"   and "msig new = msig old"   and "G\<turnstile>declclass new\<prec>\<^sub>C declclass old"   and "G\<turnstile>Method new declared_in declclass new"   and "G\<turnstile>Method old declared_in declclass old"   and "G \<turnstile>Method old inheritable_in pid (declclass new) "   and "G\<turnstile>resTy new\<preceq>resTy old" shows "G \<turnstile> new overrides old"
oprod_Preorder: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "Preorder r"   and "Preorder r'"   and "antisym r"   and "antisym r'" shows "Preorder (r *o r')"
disc_2: fixes x2 :: "'b" shows "is_Const (Const x2)"
resl3: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<leftarrow> y \<cdot> z = x \<leftarrow> z \<leftarrow> y"
is_selection: fixes K :: "'a set"   and A :: "'a set"   and \<phi> :: "'a" assumes "K = Cn A"   and "K .\<bottom>. \<phi> \<noteq> {}" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> \<subseteq> K .\<bottom>. \<phi>"
declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: fixes \<A> :: "('c strand_label \<times> ('a, 'd) stateful_strand_step) list"   and \<I> :: "'d \<Rightarrow> ('a, 'b) Term.term" shows "declassified\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> \<I> \<equiv> {t. \<langle>\<star>, receive\<langle>t\<rangle>\<rangle> \<in> set \<A>} \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I>"
pre_3: fixes \<sigma> :: "'\<sigma>" assumes "I {} \<sigma>" shows "P \<sigma>"
assms_2: fixes x :: "'a" shows "u 1 x \<le> 0"
Broadcast_nSend_empty1: fixes n :: "nat"   and nSend :: "nat \<Rightarrow> nat \<Rightarrow> 'a Frame list"   and recv :: "nat \<Rightarrow> 'a Frame list"   and t :: "nat" assumes "Broadcast n nSend recv"   and "\<forall>k<n. nSend k t = []" shows "recv t = []"
final_threadI: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x2 \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x2" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "final2 x"   and "wset s t = None" shows "r2.final_thread s t"
smooth_on_const: fixes k :: "enat"   and S :: "'a set"   and c :: "'b" shows "k-smooth_on S (\<lambda>x. c)"
partition_set: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list"   and yes :: "'b list"   and no :: "'b list" assumes "partition P xs = (yes, no)" shows "set yes \<union> set no = set xs"
s2_def: shows "s2 = (if conv_abscissa f = - \<infinity> then s \<bullet> (1::'a) - 1 else s \<bullet> (1::'a) * 2 / 3 + real_of_ereal (conv_abscissa f) * 1 / 3)"
signed_take_bit_nonnegative_iff: fixes n :: "nat"   and k :: "int" shows "(0 \<le> signed_take_bit n k) = (\<not> bit k n)"
I_r_closed: fixes I :: "'c set"   and R :: "('c, 'd) ring_scheme"   and a :: "'c"   and x :: "'c" assumes "ideal I R"   and "a \<in> I"   and "x \<in> carrier R" shows "a \<otimes>\<^bsub>R\<^esub> x \<in> I"
simps: fixes x :: "nat" shows "Prime x = (1 < x \<and> (\<forall>u<x. \<forall>v<x. u * v \<noteq> x))"
map_comp: fixes g :: "'c \<Rightarrow> 'b"   and f :: "'d \<Rightarrow> 'c"   and v :: "'d literal" shows "map_literal g (map_literal f v) = map_literal (g \<circ> f) v"
VectorSpaceHom_composite_left: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and T :: "'b \<Rightarrow> 'b"   and smult'' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'b \<Rightarrow> 'c" assumes "VectorSpaceEnd smult V T"   and "VectorSpaceHom smult V smult'' S" shows "VectorSpaceHom smult V smult'' (S \<circ> T)"
domain_subid: fixes x :: "'a" assumes "x \<le> (1::'a)" shows "x \<le> ad (ad x)"
conv_radius_norm: fixes f :: "nat \<Rightarrow> 'b" shows "conv_radius (\<lambda>x. norm (f x)) = conv_radius f"
raw_match_term_restriction: fixes t :: "Core.term"   and u :: "Core.term"   and subs :: "variable \<times> String.literal set \<Rightarrow> typ option"   and subs' :: "variable \<times> String.literal set \<Rightarrow> typ option"   and restriction :: "(variable \<times> String.literal set) set" assumes "raw_match_term t u subs = Some subs'"   and "tvs t \<subseteq> restriction" shows "raw_match_term t u (subs |` restriction) = Some (subs' |` restriction)"
assms_5: shows "\<forall>r\<in>range g \<inter> space N. \<exists>A\<in>sets N. range g \<inter> A = {r}"
assms_5: shows "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
upto_strict_2: fixes m :: "Integer" shows "supto\<cdot>m\<cdot>\<bottom> = \<bottom>"
borel_cantelli_limsup1: fixes A :: "nat \<Rightarrow> 'a set"   and M :: "'a measure" assumes "\<And>n. A n \<in> sets M"   and "\<And>n. emeasure M (A n) < \<infinity>"   and "summable (\<lambda>n. Sigma_Algebra.measure M (A n))" shows "limsup A \<in> null_sets M"
split_face_edges_f12_f21_vs: fixes f :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and f12 :: "face"   and f21 :: "face" assumes "pre_split_face f ram1 ram2 []"   and "(f12, f21) = split_face f ram1 ram2 []" shows "\<E> f12 \<union> \<E> f21 = \<E> f \<union> {(ram2, ram1), (ram1, ram2)}"
inner_image_box: assumes "1 \<in> Basis"   and "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i" shows "(\<lambda>x. x \<bullet> 1) ` box a b = {a \<bullet> 1<..<b \<bullet> 1}"
t3: shows "local.iso \<lbrace>t3\<rbrace>"
OF_eq: fixes \<gamma> :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "('a, 'c) flow_entry_match list"   and p :: "'b" assumes "no_overlaps \<gamma> f"   and "sorted_descending (map ofe_prio f)" shows "OF_match_linear \<gamma> f p = OF_priority_match \<gamma> f p"
continuous_openin_preimage_eq: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous_on S f = (\<forall>T. open T \<longrightarrow> openin (top_of_set S) (S \<inter> f -` T))"
weakSimClosedAux: fixes Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and p :: "(name \<times> name) list" assumes "eqvt Rel"   and "\<Psi> \<rhd> P \<leadsto><Rel> Q" shows "p \<bullet> \<Psi> \<rhd> p \<bullet> P \<leadsto><Rel> p \<bullet> Q"
assms: shows "\<tau>exec_1_d P t \<sigma> \<sigma>'"
Limsup_const: fixes F :: "'b filter"   and c :: "'d" assumes "F \<noteq> bot" shows "Limsup F (\<lambda>x. c) = c"
Gauss_Jordan_det_P_def: fixes A :: "(('d, 'e) vec, 'f) vec" shows "Gauss_Jordan_det_P A = Gauss_Jordan_upt_k_det_P A (ncols A - 1)"
find_base_vectors_1: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat" assumes "row_echelon_form A"   and "A \<in> carrier_mat nr nc" shows "set (find_base_vectors A) \<subseteq> mat_kernel A"
cSup_upper2: fixes x :: "'b multiset"   and X :: "'b multiset set"   and y :: "'b multiset" assumes "x \<in> X"   and "y \<subseteq># x"   and "subset_mset.bdd_above X" shows "y \<subseteq># Sup X"
VC_def: fixes ct :: "(char list \<times> nat \<times> unit list) list"   and P :: "'a" shows "VC ct P \<equiv> P"
eqpoll_finite_iff: fixes A :: "'b set"   and B :: "'c set" assumes "A \<approx> B" shows "finite A = finite B"
unintersect_coll_def: fixes X :: "'d set" shows "unintersect_coll X = SPEC ((\<subseteq>) X)"
m1_inv2i_servI: fixes x :: "'a m1r_state_scheme" assumes "\<And>A B Ra Kab Ts nla. \<lbrakk>A \<notin> bad; runs x Ra = Some (Init, [A, B], aKey Kab # aNum Ts # nla)\<rbrakk> \<Longrightarrow> \<exists>Rs. Kab = sesK (Rs $ sk) \<and> runs x Rs = Some (Serv, [A, B], [aNon (Ra $ na), aNum Ts])" shows "x \<in> m1_inv2i_serv"
ge_const_equiv: shows "(\<sim>\<^sub>C) \<le> (\<sim>\<^sub>L)"
subcocycle_AE_in_context: fixes u :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>m n. AE x in M. u (n + m) x \<le> u n x + u m ((T ^^ n) x)"   and "\<And>n. integrable M (u n)" shows "\<exists>w. subcocycle w \<and> (AE x in M. \<forall>n. w n x = u n x)"
word_nat_cases: fixes x :: "'b word" obtains n :: "nat" where "x = word_of_nat n" and "n < 2 ^ LENGTH('b)"
one_stalk_def: fixes V :: "'a set" shows "stk2.one_stalk V \<equiv> stk2.class_of V (one_str2 V)"
ta_seq_consist_imp_eq_upto_seq_inconsist_refl: fixes P :: "'a prog"   and vs :: "'b \<times> addr_loc \<Rightarrow> ('b val \<times> bool) option"   and obs :: "('b, 'c) obs_event action list" assumes "ta_seq_consist P vs (llist_of obs)" shows "eq_upto_seq_inconsist P obs obs vs"
nested_bind: fixes f :: "'b \<Rightarrow> ('e \<times> 'b) set \<times> bool"   and g :: "'e \<Rightarrow> 'd"   and h :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) set \<times> bool" shows "do y \<leftarrow> f; return (g y) od >>= h = do y \<leftarrow> f; h (g y) od"
inline1_inline_conv_inline2': fixes gpv :: "('b, 'c2, 'r2) gpv"   and s2 :: "'s2"   and s1 :: "'s1" shows "inline1 (\<lambda>(s2, s1) c2. map_gpv (\<lambda>((r, s2), s1). (r, s2, s1)) id (inline callee1 (callee2 s2 c2) s1)) gpv (s2, s1) = map_spmf (map_sum id (\<lambda>(x, rpv'', rpv', rpv). (x, \<lambda>r. rpv'' r \<bind> (\<lambda>(r1, s1). map_gpv (\<lambda>((r2, s2), s1). (r2, s2, s1)) id (inline callee1 (rpv' r1) s1)), rpv))) (local.inline2 gpv s2 s1)"
assms: shows "\<tau> \<equiv> gauss_sum 1 / complex_of_real (sqrt (real n))"
ss: shows "{b. \<exists>a. \<langle>a, b\<rangle> \<in>\<^sub>\<circ> r} \<subseteq> vsnd ` elts r"
select_convs_2: fixes CatDom :: "('g, 'h, 'i) Category_scheme"   and CatCod :: "('d, 'e, 'f) Category_scheme"   and MapM :: "'h \<Rightarrow> 'e"   and more :: "'j" shows "CatCod \<lparr>CatDom = CatDom, CatCod = CatCod, MapM = MapM, \<dots> = more\<rparr> = CatCod"
SUP_divide_ennreal: fixes f :: "'a \<Rightarrow> ennreal"   and I :: "'a set"   and c :: "ennreal" shows "\<Squnion> (f ` I) / c = (\<Squnion>i\<in>I. f i / c)"
finite_subset_induct': fixes F :: "'b set"   and A :: "'b set"   and P :: "'b set \<Rightarrow> bool" assumes "finite F"   and "F \<subseteq> A"   and "P {}"   and "\<And>a F. \<lbrakk>finite F; a \<in> A; F \<subseteq> A; a \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert a F)" shows "P F"
gaugeD_1: fixes \<gamma> :: "'b \<Rightarrow> 'b set"   and x :: "'b" assumes "gauge \<gamma>" shows "x \<in> \<gamma> x"
basis_exists: fixes V :: "'b set" obtains B :: "'b set" where "B \<subseteq> V" and "independent B" and "V \<subseteq> span B" and "card B = dim V"
prod_eq_prob_T: fixes P1 :: "'a stream \<Rightarrow> bool"   and P2 :: "'b stream \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'b" assumes "Measurable.pred (stream_space (count_space UNIV)) P1"   and "Measurable.pred (stream_space (count_space UNIV)) P2" shows "K1.T.prob x1 {\<omega> \<in> space (K1.T x1). P1 \<omega>} * K2.T.prob x2 {\<omega> \<in> space (K2.T x2). P2 \<omega>} = T.prob (x1, x2) {\<omega> \<in> space (T (x1, x2)). P1 (smap fst \<omega>) \<and> P2 (smap snd \<omega>)}"
sets_kernel: fixes K :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and a :: "'c" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "a \<in> space M" shows "sets (K a) = sets N"
coeffM_row_def: fixes A :: "(('c poly, 'd) vec, 'e) vec"   and n :: "nat"   and i :: "'e" shows "coeffM_row A n i = (\<chi>j. coeff (A $ i $ j) n)"
interval_3split: fixes i :: "nat"   and n :: "nat" assumes "i < n" shows "[0..<n] = [0..<i] @ [i] @ [i + 1..<n]"
card_fcs_gt_0: shows "0 < card G.face_cycle_sets"
wf_envs_progress_le_1: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and P' :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and db :: "(String.literal, event_data list set list) mapping"   and \<phi> :: "Formula.formula" assumes "wf_envs \<sigma> j P P' V db" shows "Monitor.progress \<sigma> P \<phi> j \<le> j"
b_meet_e: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<sqinter> e_aux a b c = b \<sqinter> (c \<squnion> a)"
l_comb_transpos: fixes A :: "'b set"   and H :: "'a set"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a"   and j :: "nat" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "s \<in> {l. l \<le> Suc n} \<rightarrow> A"   and "f \<in> {l. l \<le> Suc n} \<rightarrow> H"   and "j < Suc n" shows "\<Sigma>\<^sub>e M cmp (\<lambda>k. s k \<cdot>\<^sub>s f k) (transpos j (Suc n)) Suc n = \<Sigma>\<^sub>e M (\<lambda>k. cmp s (transpos j (Suc n)) k \<cdot>\<^sub>s cmp f (transpos j (Suc n)) k) Suc n"
sympI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
assms_1: shows "(cmp, cmp') \<in> R \<rightarrow> R \<rightarrow> comp_res_rel"
simps_2: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" shows "pred (P)\<^sub>\<surd> s = P s"
vareq_def: fixes x0 :: "'a"   and t :: "real" shows "Y.vareq x0 t = f' (Y.flow0 x0 t)"
Extend_SNOC_cut: fixes \<AA> :: "'i"   and x :: "'x"   and P :: "'v"   and k :: "'k" assumes "#\<^sub>V \<AA> = size x"   and "len P \<le> Length (SNOC x \<AA>)" shows "Extend k 0 (SNOC x \<AA>) P = SNOC (extend k (if eval P (Length \<AA>) then True else False) x) (Extend k 0 \<AA> (cut (Length \<AA>) P))"
indexed_var_def: fixes i :: "'c" shows "\<X>\<^bsub>i\<^esub> = indexed_const \<one> \<Otimes> i"