hcju/isabelleps huggingface dataset:
bind_returns_result_I: fixes h :: "'a"   and f :: "('a, 'b, 'c) prog"   and x :: "'c"   and h' :: "'a"   and g :: "'c \<Rightarrow> ('a, 'b, 'd) prog"   and y :: "'d" assumes "h \<turnstile> f \<rightarrow>\<^sub>r x"   and "h \<turnstile> f \<rightarrow>\<^sub>h h'"   and "h' \<turnstile> g x \<rightarrow>\<^sub>r y" shows "h \<turnstile> f \<bind> g \<rightarrow>\<^sub>r y"
fimg_def: fixes T :: "'b set set"   and f :: "'c \<Rightarrow> 'b"   and F :: "'c set set" shows "fimg T f F = {v. v \<subseteq> carrier\<^bsub>T\<^esub> \<and> (\<exists>u\<in>F. f ` u \<subseteq> v)}"
id_k: shows "id_upt_k (Gauss_Jordan A) k"
iFROM_div_Min: fixes n :: "nat"   and k :: "nat" shows "iMin ([n\<dots>] \<oslash> k) = n div k"
simps: fixes x :: "name"   and x' :: "name"   and y :: "name"   and z :: "name"   and f :: "tm"   and del :: "tm"   and k :: "tm"   and g :: "tm" assumes "atom x \<sharp> (x', y, z, f, del, k)"   and "atom x' \<sharp> (y, z, f, del, k)"   and "atom y \<sharp> (z, f, del, k)"   and "atom z \<sharp> (f, del, g, k)" shows "ShiftP f k del g = SyntaxN.All z (Var z IN g IFF SyntaxN.Ex x (SyntaxN.Ex x' (SyntaxN.Ex y (Var z EQ HPair (Var x') (Var y) AND HaddP del (Var x) (Var x') AND HPair (Var x) (Var y) IN f AND Var x IN k))))"
bits_1_div_2: 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" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "divide one (numeral.numeral one plus (num.Bit0 num.One)) = zero"
sub_commutative': fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> - y = - y \<squnion> - x"
Exit_no_SDG_edge_source: 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 Vopt :: "'c option"   and popt :: "('f \<times> bool) option"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses (CFG_node Exit) Vopt popt n'" shows "False"
x: fixes i :: "'a" assumes "i \<in> I" shows "x i \<noteq> 1"
uniformly_convergent_on_def: fixes X :: "'a set"   and f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b" shows "uniformly_convergent_on X f = (\<exists>l. uniform_limit X f l sequentially)"
nat_pow_closed: fixes x :: "'a mod_ring poly"   and n :: "nat" assumes "x \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> n \<in> carrier R"
measure_induct_rule: fixes f :: "'b \<Rightarrow> 'c"   and P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<And>x. (\<And>y. order.greater (f x) (f y) \<Longrightarrow> P y) \<Longrightarrow> P x" shows "P a"
map_FGco_def: fixes l1 :: "'o \<Rightarrow> 'ab"   and co1 :: "'p \<Rightarrow> 'ac"   and co2 :: "'r \<Rightarrow> 'ae"   and co3 :: "'s \<Rightarrow> 'af"   and co4 :: "'t \<Rightarrow> 'ag"   and co5 :: "'q \<Rightarrow> 'ad"   and co6 :: "'x \<Rightarrow> 'aj"   and contra1 :: "'ah \<Rightarrow> 'u"   and contra2 :: "'ai \<Rightarrow> 'v"   and contra3 :: "'ak \<Rightarrow> 'y"   and contra4 :: "'al \<Rightarrow> 'z" shows "map_FGco l1 co1 co2 co3 co4 co5 co6 contra1 contra2 contra3 contra4 = map_F l1 co1 co5 (map_G co1 co2 co3 co4 contra1 contra2) co3 co6 contra1 contra3 contra4"
return_pd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and x :: "'a" assumes "IFC prob"   and "x \<in> IFC_def.nodes prob" shows "IFC_def.is_pd prob (IFC_def.return prob) x"
ovalidNF_is_triple: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" shows "ovalidNF P f Q = triple_judgement P f (ovalidNF_property Q (\<lambda>s f. f s))"
eq_is_closed_def: fixes S :: "('a, 'b) eq_object_scheme"   and A :: "'a set" shows "is_closed\<^bsub>S\<^esub> A = (A \<subseteq> carrier S \<and> closure_of\<^bsub>S\<^esub> A = A)"
eval_rest: shows "P,E \<turnstile> \<langle>es'',s''\<rangle> [\<Rightarrow>] \<langle>es',s'\<rangle>"
assms_2: shows "compiler step2 step3 final2 final3 load2 load3 order2 match2 compile2"
generalized_sfw_mapsnd: fixes f :: "'d \<Rightarrow> 'b"   and fw :: "('a simple_match \<times> 'd) list"   and p :: "('a, 'c) simple_packet_scheme" shows "generalized_sfw (map (apsnd f) fw) p = map_option (apsnd f) (generalized_sfw fw p)"
shift_Nil_iff: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "(local.shift k m qs = []) = (qs = [])"
continuous_on_ennreal: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "continuous_on A f" shows "continuous_on A (\<lambda>x. ennreal (f x))"
higher_differentiable_on_open_subsetsI: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. x \<in> T \<and> open T \<and> higher_differentiable_on T f n" shows "higher_differentiable_on S f n"
infty_rel_internal_def: fixes R :: "('c \<times> 'd) set" shows "infty_rel R \<equiv> {(Num a, Num a') |a a'. (a, a') \<in> R} \<union> {(infty.Infty, infty.Infty)}"
subst_Var: fixes i :: "nat"   and s :: "dB"   and k :: "nat" shows "Var i[s/k] = (if k < i then Var (i - 1) else if i = k then s else Var i)"
Stone_Weierstrass_basic: fixes f :: "complex \<Rightarrow> real"   and e :: "real" assumes "continuous_on (sphere 0 1) f"   and "0 < e" shows "\<exists>g\<in>cx_poly. \<forall>x\<in>sphere 0 1. \<bar>f x - g x\<bar> < e"
simps_1: fixes sprFst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprLst :: "('a, 'b, 'c, 'd, 'e) BEState"   and sprCRel :: "(('a, 'b, 'c, 'd, 'e) BEState \<times> ('a, 'b, 'c, 'd, 'e) BEState) set"   and more :: "'f" shows "sprFst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprFst"
semiring_hom_axioms: shows "semiring_hom (inv (inv hom))"
length_reduct: fixes m :: "'a list list" assumes "m \<noteq> []" shows "length (reduct m) + 1 = length m"
proper_Rep_assn: fixes P :: "assn" shows "proper (Rep_assn P)"
refl: fixes a :: "'t" shows "a \<preceq>\<^sub>t a"
add_node_wf: fixes g :: "'b graph"   and v :: "'b" assumes "wf_graph g" shows "wf_graph (FiniteGraph.add_node v g)"
safe_cosafe: fixes r :: "'a Regex.regex"   and safe :: "safety \<Rightarrow> 'a \<Rightarrow> bool"   and fv :: "'a \<Rightarrow> 'b set"   and m :: "modality" assumes "\<And>x. \<lbrakk>x \<in> regex.atms r; safe Strict x\<rbrakk> \<Longrightarrow> safe Lax x"   and "Regex.safe_regex fv safe m Strict r" shows "Regex.safe_regex fv safe m Lax r"
open_seqp_action: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and \<sigma> :: "nat \<Rightarrow> 'b"   and i :: "nat"   and p :: "('b, 'c, 'a, 'd) seqp"   and a :: "'c seq_action"   and \<sigma>' :: "nat \<Rightarrow> 'b"   and p' :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>"   and "((\<sigma> i, p), a, \<sigma>' i, p') \<in> seqp_sos \<Gamma>" shows "((\<sigma>, p), a, \<sigma>', p') \<in> oseqp_sos \<Gamma> i"
strong_induct: fixes P :: "'a \<Rightarrow> stack \<Rightarrow> bool"   and z :: "'a"   and stack :: "stack" assumes "\<And>z. P z stack.Id"   and "\<And>name trm stack z. \<lbrakk>name \<sharp> z; name \<sharp> stack; \<And>z. P z stack\<rbrakk> \<Longrightarrow> P z [name]trm\<ggreater>stack" shows "P z stack"
while_nfoldli: fixes c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and l :: "'d list"   and \<sigma> :: "'c" shows "WHILE\<^sub>T (FOREACH_cond c) (FOREACH_body f) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>) \<le> nfoldli l c f \<sigma>"
ObjX: shows "X \<in> obj\<^bsub>C\<^esub>"
tcopy_correct1: fixes x :: "nat" assumes "0 < x" shows "{inv_begin1 x} tcopy {inv_end0 x}"
valid_reqs_IFS_D: fixes M :: "'b SecurityInvariant_configured list" assumes "valid_reqs M" shows "valid_reqs (get_IFS M)"
gr_zeroI: fixes n :: "'b multiset" assumes "n = {#} \<Longrightarrow> False" shows "{#} \<subset># n"
insertion_sum: fixes f :: "nat \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b mpoly"   and A :: "'c set" shows "insertion f (sum g A) = (\<Sum>x\<in>A. insertion f (g x))"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "'e \<Rightarrow> 'd"   and option :: "'e option" shows "P (case option of \<bottom> \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = ((option = \<bottom> \<longrightarrow> P f1) \<and> (\<forall>x2. option = \<lfloor>x2\<rfloor> \<longrightarrow> P (f2 x2)))"
bigI: fixes small :: "'a \<Rightarrow> 'b \<Rightarrow> 'b set"   and endset :: "'b set"   and \<sigma>' :: "'b"   and P :: "'a"   and \<sigma> :: "'b"   and n :: "nat" assumes "Semantics small endset"   and "\<sigma>' \<in> Semantics.small_nstep small P \<sigma> n"   and "\<sigma>' \<in> endset" shows "\<sigma>' \<in> Semantics.big small endset P \<sigma>"
Fun_def: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "Fun f = restrict (some_img \<circ> local.comp f \<circ> inv_into (hom unity (local.dom f)) some_img) (Dom f)"
S_Max_Le: shows "S_Max_Le = {u c' + d |c' d. 0 < v c' \<and> v c' \<le> n \<and> c \<noteq> c' \<and> M (v c) (v c') = Le d} \<union> {d |d. M (v c) 0 = Le d}"
dvd_times_right_cancel_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "a \<noteq> zero" shows "dvd.dvd times (times b a) (times c a) = dvd.dvd times b c"
onorm_eq_0: fixes f :: "'d \<Rightarrow> 'e" assumes "bounded_linear f" shows "(onorm f = 0) = (\<forall>x. f x = (0::'e))"
i_contract_prime: fixes R :: "('c, 'd) Ring_scheme"   and R' :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e"   and P :: "'e set" assumes "TwoRings R R'"   and "f \<in> rHom R R'"   and "prime_ideal R' P" shows "prime_ideal R (i_contract f R R' P)"
\<gamma>_altdef: shows "\<gamma> = big_circle +++ neg_line +++ small_circle +++ pos_line"
Abs_BetaD: fixes x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" assumes "Abs x s \<rightarrow>\<^sub>\<beta> t" shows "\<exists>u. t = Abs x u \<and> s \<rightarrow>\<^sub>\<beta> u"
next_p_union_frontier_change: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and loc' :: "'loc" assumes "next_propagate' c0 c1 loc t"   and "summary loc loc' \<noteq> {}\<^sub>A" shows "(\<Sum>loc'a\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c1 loc'a))))) (summary loc'a loc')) = (\<Sum>loc'a\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_imp c0 loc'a))))) (summary loc'a loc')) + after_summary (frontier_changes (c_imp c1 loc) (c_imp c0 loc)) (summary loc loc')"
bounded_clinear_scaleC_right: fixes c :: "complex" shows "bounded_clinear ((*\<^sub>C) c)"
coinduction: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<le> z + x \<cdot> y \<longrightarrow> y \<le> x\<^sup>\<infinity> \<cdot> z"
assms: fixes a :: "'a"   and b :: "'a" shows "f integrable_on cbox a b"
cases: fixes a1 :: "'a fset"   and a2 :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)"   and a3 :: "'a"   and a4 :: "'c option"   and a5 :: "'c"   and a6 :: "'a"   and a7 :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" assumes "closest_association_end a1 a2 a3 a4 a5 a6 a7" obtains classes :: "'a fset" and associations :: "'b \<rightharpoonup>\<^sub>f 'c \<rightharpoonup>\<^sub>f ('a \<times> nat \<times> enat \<times> bool \<times> bool)" and \<D> :: "'a" and "from" :: "'c option" and role :: "'c" and "end" :: "'a \<times> nat \<times> enat \<times> bool \<times> bool" and \<C> :: "'a" where "a1 = classes" and "a2 = associations" and "a3 = \<C>" and "a4 = from" and "a5 = role" and "a6 = \<D>" and "a7 = end" and "owned_association_end' classes associations \<D> from role end" and "\<C> \<le> \<D>" and "\<not> association_end_not_closest classes associations \<C> from role \<D> end"
Sup_least: fixes Inf :: "'g set \<Rightarrow> 'g"   and Sup :: "'g set \<Rightarrow> 'g"   and inf :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and bot :: "'g"   and top :: "'g"   and A :: "'g set"   and z :: "'g" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. x \<in> A \<Longrightarrow> less_eq x z" shows "less_eq (Sup A) z"
sup_least: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and y :: "'c"   and x :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x"   and "less_eq z x" shows "less_eq (sup y z) x"
simps_16: fixes x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "funs_term (Fun x21 x22) = insert x21 (\<Union> (funs_term ` set x22))"
index_elem: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set xs" shows "\<exists>i<length xs. x = xs ! i"
simps_244: shows "BinOr \<noteq> BinAnd"
implies_e2: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and K\<^sub>1 :: "clock"   and K\<^sub>2 :: "clock"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> (K\<^sub>1 implies K\<^sub>2) # \<Psi> \<triangleright> \<Phi> \<hookrightarrow>\<^sub>e K\<^sub>1 \<Up> n # K\<^sub>2 \<Up> n # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 implies K\<^sub>2) # \<Phi>"
e_aux_def: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "e_aux a b c = (a \<squnion> b) \<sqinter> (b \<squnion> c) \<sqinter> (c \<squnion> a)"
power_add_numeral2: fixes a :: "'b"   and m :: "num"   and n :: "num"   and b :: "'b" shows "a ^ numeral m * (a ^ numeral n * b) = a ^ numeral (m + n) * b"
StepInMsg: fixes cfg1 :: "('p, 'v, 's) configuration"   and p :: "'p"   and v :: "bool"   and cfg2 :: "('p, 'v, 's) configuration" shows "cfg1 \<turnstile> <p, inM v> \<mapsto> cfg2 = ((\<forall>s. (s = p \<longrightarrow> states cfg2 p = trans p (states cfg1 p) (Bool v)) \<and> (s \<noteq> p \<longrightarrow> states cfg2 s = states cfg1 s)) \<and> enabled cfg1 <p, inM v> \<and> msgs cfg2 = sends p (states cfg1 p) (Bool v) \<union># msgs cfg1 -# <p, inM v>)"
gram_schmidt0_result_3: fixes ws :: "complex vec list" assumes "set ws \<subseteq> carrier_vec d" shows "corthogonal (gram_schmidt0 d ws)"
insert_in_list_at_arbitrary_pos_thm: fixes xl :: "'a list"   and x :: "'a"   and l :: "'a list" shows "(xl \<in> insert_in_list_at_arbitrary_pos x l) = (\<exists>l1 l2. l = l1 @ l2 \<and> xl = l1 @ [x] @ l2)"
counit_simps_4: shows "src\<^sub>D (counit a) = trg\<^sub>D (\<tau>\<^sub>0 a)"
wt_elim_cases_28: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and c1 :: "stmt"   and tn :: "qtname"   and vn :: "vname"   and c2 :: "stmt"   and x :: "tys" assumes "E,dt\<Turnstile>In1r (Try c1 Catch(tn vn) c2)\<Colon>x" obtains "x = Inl (PrimT Void)" and "E,dt\<Turnstile>c1\<Colon>\<surd>" and "prg E\<turnstile>tn\<preceq>\<^sub>C SXcpt Throwable" and "lcl E (VName vn) = None" and "E\<lparr>lcl := lcl E(VName vn \<mapsto> Class tn)\<rparr>,dt\<Turnstile>c2\<Colon>\<surd>"
intro: fixes A1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "category A1"   and "category A2"   and "product_category A1 A2"   and "product_category A2 A1" shows "symmetry_functor A1 A2"
uniq_n: shows "\<forall>n1 n2. \<forall>j\<in>{0..<m}. P j n1 \<longrightarrow> P j n2 \<longrightarrow> n1 = n2"
select_convs_1: fixes NTDom :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NTCod :: "('d, 'e, 'f, 'g, 'h, 'i) Functor"   and NatTransMap :: "'d \<Rightarrow> 'g"   and more :: "'j" shows "NTDom \<lparr>NTDom = NTDom, NTCod = NTCod, NatTransMap = NatTransMap, \<dots> = more\<rparr> = NTDom"
distr_pair_fst: fixes M :: "'c measure"   and N :: "'d measure" assumes "prob_space M" shows "distr (N \<Otimes>\<^sub>M M) N fst = N"
ss424i: fixes x :: "'a" shows "is_total x = (\<forall>y. - (x ; y) \<le> x ; - y)"
n: shows "n = 0 \<or> n mod 30 = 11"
ring_seq_smult_eval: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int"   and k :: "nat" assumes "s \<in> carrier Zp\<^bsup>\<omega>\<^esup>"   and "a \<in> carrier Zp" shows "(a \<odot>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> s) k = a \<otimes> s k"
transitive: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "- x \<le> - y"   and "- y \<le> - z" shows "- x \<le> - z"
attractor_lowerbound: fixes W :: "'a set"   and S :: "'a set"   and p :: "Player" assumes "W \<union> S \<union> directly_attracted p S \<subseteq> S" shows "attractor p W \<subseteq> S"
ds_prop: fixes x :: "'a" shows "d x \<parallel> 1\<^sub>\<sigma> = d x"
cut_ge_empty: fixes t :: "'a" shows "{} \<down>\<ge> t = {}"
rel_sel: fixes K :: "'a set"   and \<phi> :: "'a" shows "\<gamma>\<^sub>T\<^sub>R K \<phi> \<equiv> if K .\<bottom>. \<phi> = {} then {K} else {B \<in> K .\<bottom>. \<phi>. \<forall>C\<in>K .\<bottom>. \<phi>. C \<preceq>\<^bsub>K\<^esub> B}"
mset_single_cases: fixes s :: "'a"   and c :: "'a multiset"   and r' :: "'a"   and c' :: "'a multiset" assumes "add_mset s c = add_mset r' c'" obtains "s = r'" and "c = c'"   | "c' = {#s#} + (c' - {#s#})" and "c = {#r'#} + (c - {#r'#})" and "c - {#r'#} = c' - {#s#}"
heterogeneous\<^sub>l\<^sub>s\<^sub>t_def: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and label_witness1 :: "'d"   and label_witness2 :: "'d"   and t :: "('a, 'b) Term.term"   and \<A> :: "('e strand_label \<times> ('a, 'b) strand_step) list"   and Secrets :: "('a, 'b) Term.term set" assumes "labeled_typed_model arity public Ana \<Gamma> label_witness1 label_witness2" shows "labeled_typed_model.heterogeneous\<^sub>l\<^sub>s\<^sub>t arity public Ana \<Gamma> t \<A> Secrets \<equiv> \<exists>l1 l2. \<exists>s1\<in>subterms t. \<exists>s2\<in>subterms t. l1 \<noteq> l2 \<and> labeled_typed_model.proj_specific arity public Ana \<Gamma> l1 s1 \<A> Secrets \<and> labeled_typed_model.proj_specific arity public Ana \<Gamma> l2 s2 \<A> Secrets"
n_o_mono: fixes S1 :: "ivl Abs_State.st option"   and X :: "char list set"   and S2 :: "ivl Abs_State.st option" assumes "domo S1 \<subseteq> X"   and "domo S2 \<subseteq> X"   and "S1 \<sqsubseteq> S2" shows "n_o (n_st n_ivl X) S1 \<le> n_o (n_st n_ivl X) S2"
norm_ell2_vec_of_basis_enum: fixes \<psi> :: "'b" shows "norm \<psi> = (let \<psi>' = vec_of_basis_enum \<psi> in sqrt (\<Sum>i = 0..<dim_vec \<psi>'. let z = \<psi>' $ i in (Re z)\<^sup>2 + (Im z)\<^sup>2))"
length_rev: fixes xs :: "'d list" shows "length (rev xs) = length xs"
moebius_pt_moebius_rotation_zero: fixes \<phi> :: "real" shows "moebius_pt (moebius_rotation \<phi>) 0\<^sub>h = 0\<^sub>h"
Rep_ccs_inject: fixes x :: "ccs"   and y :: "ccs" shows "(Rep_ccs x = Rep_ccs y) = (x = y)"
VxV_comp_eq_VV_comp: shows "VxV.comp = VV.comp"
subcls_is_class: fixes D :: "char list"   and C :: "char list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" assumes "P \<turnstile> D \<preceq>\<^sup>* C"   and "is_class P C" shows "is_class P D"
tendsto_ident_at: fixes "open" :: "'e set \<Rightarrow> bool"   and a :: "'e"   and s :: "'e set" assumes "class.topological_space open" shows "LIM x topological_space.at_within open a s. x :> topological_space.nhds open a"
sum_poly_mapping_eq_zeroI: fixes p :: "'e \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" assumes "p ` A \<subseteq> {0}" shows "sum p A = 0"
Red1Var: fixes s :: "'heap \<times> 'addr val list"   and V :: "nat"   and v :: "'addr val"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id" assumes "lcl s ! V = v"   and "V < length (lcl s)" shows "uf,P,t \<turnstile>1 \<langle>Var V,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Val v,s\<rangle>"
permute_list_nth: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list"   and i :: "nat" assumes "f permutes {..<length xs}"   and "i < length xs" shows "permute_list f xs ! i = xs ! f i"
extr_Inj: fixes At :: "atom"   and T :: "atom set"   and H :: "chmsg set" assumes "At \<in> T" shows "At \<in> extr T H"
profileI: fixes Is :: "'c set"   and A :: "'d set"   and P :: "'c \<Rightarrow> ('d \<times> 'd) set" assumes "\<And>i. i \<in> Is \<Longrightarrow> rpr A (P i)"   and "Is \<noteq> {}" shows "profile A Is P"
set_shadow_root_get_shadow_root_different_pointers: fixes ptr :: "(_) element_ptr"   and ptr' :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "ptr \<noteq> ptr'" shows "\<forall>w\<in>set_shadow_root_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
lderivs_Atom: fixes xs :: "'a list"   and as :: "'b" shows "lderivs xs (Atom as) \<in> {Zero, One, Atom as}"
ex_raw_match'_imp_tinstT: fixes T2 :: "typ"   and T1 :: "typ" assumes "\<exists>subs. raw_match' T2 T1 Map.empty = Some subs" shows "tinstT T1 T2"
union_with_inside: fixes S :: "'a set" shows "S \<union> inside S = - outside S"
RES_rule: fixes S :: "'a set"   and \<Phi> :: "'a \<Rightarrow> bool" assumes "\<And>x. x \<in> S \<Longrightarrow> \<Phi> x" shows "RES S \<le> SPEC \<Phi>"
simps_10: fixes v :: "nat"   and va :: "nat ifex"   and vb :: "nat ifex"   and uw :: "bdd" shows "Rmi_g 0 (IF v va vb) uw = False"
suminf_nonneg: fixes f :: "nat \<Rightarrow> 'b" assumes "summable f"   and "\<And>n. (0::'b) \<le> f n" shows "(0::'b) \<le> suminf f"
stable_on_range': fixes ds :: "'b set"   and X :: "'a set"   and x :: "'a" assumes "PdXXX.stable_on ds X"   and "x \<in> X" shows "x \<in> Field (Pd' (Xd x))"
step_C: fixes x :: "'a"   and n :: "nat" assumes "x \<in> space M" shows "Ionescu_Tulcea.C (K' x) (\<lambda>_. M) 0 1 (\<lambda>_. undefined) \<bind> Ionescu_Tulcea.C (K' x) (\<lambda>_. M) 1 n = K x \<bind> (\<lambda>y. Ionescu_Tulcea.C (K' x) (\<lambda>_. M) 1 n (case_nat y (\<lambda>_. undefined)))"
tl: shows "{x2 EQ x2', x3 EQ x3', x4 EQ x4'} \<turnstile> P x1' x2 x3 x4 IFF P x1' x2' x3' x4'"
NN_unique2: fixes \<phi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "bprv (all yy (imp (NN \<langle>\<phi>\<rangle> (Var yy)) (eql \<langle>neg \<phi>\<rangle> (Var yy))))"
subst_atms_empty_iff: fixes AA :: "'a set"   and \<eta> :: "'s" shows "(AA \<cdot>as \<eta> = {}) = (AA = {})"
pair_ord_spp_def: fixes x :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat"   and y :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) \<times> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) + nat" shows "pair_ord_spp x y = (sig_of_pair_spp x \<preceq>\<^sub>t sig_of_pair_spp y)"
pac_ideal_insert_already_in: fixes pq :: "int mpoly"   and A :: "int mpoly multiset" assumes "pq \<in> pac_ideal (set_mset A)" shows "pac_ideal (insert pq (set_mset A)) = pac_ideal (set_mset A)"