hcju/isabelleps huggingface dataset:
base_prod: shows "f $$ fls_subdegree f * g $$ fls_subdegree g \<noteq> (0::'a)"
q: fixes t :: "real" assumes "t \<in> {0..1}" shows "q t = p (f t)"
subst_atm_mset_lists_nth: fixes AAs :: "'a multiset list"   and \<sigma>s :: "'s list"   and i :: "nat" assumes "length AAs = length \<sigma>s"   and "i < length AAs" shows "(AAs \<cdot>\<cdot>aml \<sigma>s) ! i = AAs ! i \<cdot>am \<sigma>s ! i"
nonempty_set_star_has_least_lemma: fixes S :: "nat set" assumes "S \<noteq> {}" shows "\<exists>n\<in>S. \<forall>m\<in>S. n \<le> m"
T_append2: fixes qs :: "'a list"   and as :: "'b list"   and step :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> nat"   and s :: "'c"   and qs' :: "'a list"   and as' :: "'b list" assumes "length qs = length as" shows "On_Off.T step t s (qs @ qs') (as @ as') = On_Off.T step t s qs as + On_Off.T step t (On_Off.steps step s qs as) qs' as'"
interchange: fixes f :: "('b \<Rightarrow> 'a) list"   and x :: "'b" shows "f \<diamondop> [x] = [\<lambda>f. f x] \<diamondop> f"
override_on_upd_lemma: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option"   and a :: "'a"   and b :: "'b"   and A :: "'a set" shows "(override_on f (g(a \<mapsto> b)) A)(a := g a) = override_on f g (insert a A)"
IH: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and n :: "nat" assumes "pre_subdivFace' g f u v n ovs"   and "minGraphProps g"   and "f \<in> \<F> g" shows "subdivFace' g f v n ovs = makeFaceFinal f g \<or> (\<forall>f'\<in>\<F> (subdivFace' g f v n ovs) - (\<F> g - {f}). \<exists>e\<in>\<E> f'. e \<notin> \<E> g)"
split_asm: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = (\<not> (bool = True \<and> \<not> P f1 \<or> bool = False \<and> \<not> P f2))"
listrelp_conj2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a list"   and y :: "'b list" assumes "listrelp (\<lambda>x y. R x y \<and> S x y) x y" shows "listrelp S x y"
cont: shows "continuous_on {0..1} (\<lambda>x. f (frac x))"
interpretation_subst_exists: shows "\<exists>\<I>. interpretation\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<I>"
outputTauTrans: fixes a :: "name"   and b :: "name"   and P :: "pi"   and P' :: "pi" assumes "a{b}.P \<longmapsto> \<tau> \<prec> P'" shows "False"
enum_CARD_1: fixes a :: "'c" shows "enum_class.enum = [a]"
dhops_update_changed: fixes rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and dip :: "nat"   and osn :: "nat"   and hops :: "nat"   and nhip :: "nat" assumes "rt \<noteq> update rt dip (osn, kno, val, hops, nhip, {})" shows "the (dhops (update rt dip (osn, kno, val, hops, nhip, {})) dip) = hops"
upper_iso: shows "isotone \<Y>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<X>\<^bsub>G \<circ>\<^sub>g F\<^esub> \<pi>\<^sub>*\<^bsub>G \<circ>\<^sub>g F\<^esub>"
fps_const_def: fixes c :: "'a" shows "fps_const c = Abs_fps (\<lambda>n. if n = 0 then c else (0::'a))"
nn_integral_distr: fixes T :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and M' :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" assumes "T \<in> M \<rightarrow>\<^sub>M M'"   and "f \<in> borel_measurable (distr M M' T)" shows "integral\<^sup>N (distr M M' T) f = \<integral>\<^sup>+ x. f (T x) \<partial>M"
of_nat_mult: fixes i :: "nat"   and j :: "nat" shows "\<guillemotleft>i * j\<guillemotright>\<^sub>\<nat> = \<guillemotleft>i\<guillemotright>\<^sub>\<nat> \<otimes> \<guillemotleft>j\<guillemotright>\<^sub>\<nat>"
vfsequence_vcons_vdistinct: fixes xs :: "V"   and x :: "V" assumes "vfsequence xs"   and "vdistinct (xs #\<^sub>\<circ> x)" shows "vdistinct xs"
vector_derivative_works: fixes f :: "real \<Rightarrow> 'c"   and net :: "real filter" shows "(f differentiable net) = (f has_vector_derivative vector_derivative f net) net"
assms: shows "subst_domain \<theta> \<inter> \<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r ` M) = {}"
i1_1: shows "flow0 x t1 = {a -- b}\<^bsub>i1\<^esub>"
eq_def: fixes S :: "'a set"   and s0 :: "'a"   and \<delta> :: "'a \<Rightarrow> 'a set"   and AP :: "'b set"   and L :: "'a \<Rightarrow> 'b set"   and AP' :: "'b set"   and \<phi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool"   and \<psi> :: "('a \<times> 'b set) stream list \<Rightarrow> bool" assumes "Shallow S s0 \<delta> AP L AP'" shows "Shallow.eq \<phi> \<psi> \<equiv> Shallow.con (Shallow.imp \<phi> \<psi>) (Shallow.imp \<psi> \<phi>)"
divide_inverse: fixes a :: "'c"   and b :: "'c" shows "a / b = a * inverse b"
useNodes_of_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and g :: "'g" shows "uninst.useNodes_of u g = fold (uninst.addN u g) (\<alpha>n g) Mapping.empty"
symmetric_bibd_axioms_def: fixes point_set :: "'a set"   and block_collection :: "'a set multiset" shows "symmetric_bibd_axioms point_set block_collection \<equiv> int (size block_collection) = int (card point_set)"
decompose_literal_vars: fixes L :: "'b literal"   and t :: "'b trm"   and s :: "'b trm"   and p :: "sign" assumes "decompose_literal L t s p" shows "vars_of t \<subseteq> vars_of_lit L"
Acc_SourcesNOT: fixes X :: "CSet"   and i :: "AbstrLevelsID"   and C :: "CSet" shows "(X \<notin> Acc i C) = (C \<notin> Sources i X)"
sequence_irpt_l': fixes P :: "'a \<Rightarrow> bool"   and M :: "'a \<Rightarrow> ('b \<times> 'a) option"   and M' :: "'a \<Rightarrow> ('c \<times> 'a) option" assumes "\<lbrace>P\<rbrace> M\<dagger>" shows "\<lbrace>P\<rbrace> M ;- M'\<dagger>"
scale_cancel_right: fixes a :: "'d"   and x :: "'d"   and b :: "'d" shows "(a * x = b * x) = (a = b \<or> x = (0::'d))"
M': shows "measure_pmf.random_variable (pair_pmf A B) (subprob_algebra N) M"
eventually_at_right: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "class.linorder_topology less_eq less open"   and "less x y" shows "eventually P (topological_space.at_within open x (ord.greaterThan less x)) = (\<exists>b. less x b \<and> (\<forall>y. less x y \<longrightarrow> less y b \<longrightarrow> P y))"
rewrite_def: fixes p :: "'a list"   and pc :: "nat"   and i :: "'a" shows "rewrite p pc i = p[pc := i]"
card_0_eq: fixes A :: "'g set" assumes "finite A" shows "(card A = 0) = (A = {})"
add_le_imp_le_right: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.ordered_ab_semigroup_add_imp_le plus minus less_eq less"   and "less_eq (plus a c) (plus b c)" shows "less_eq a b"
sum_of_naturals: fixes n :: "nat" shows "2 * \<Sum> {0..n} = n * Suc n"
compl_less_compl_iff: fixes x :: "'a"   and y :: "'a" shows "(- x < - y) = (y < x)"
continuous_on_If: fixes s :: "'a set"   and t :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and P :: "'a \<Rightarrow> bool" assumes "closed s"   and "closed t"   and "continuous_on s f"   and "continuous_on t g"   and "\<And>x. \<lbrakk>x \<in> s; \<not> P x\<rbrakk> \<Longrightarrow> f x = g x"   and "\<And>x. \<lbrakk>x \<in> t; P x\<rbrakk> \<Longrightarrow> f x = g x" shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
dvdm_pdivmod: fixes p :: "int"   and g :: "int poly"   and f :: "int poly" assumes "poly_mod_prime p" shows "poly_mod.dvdm p g f = (poly_mod.Mp p (snd (poly_mod.pdivmod2 p f g)) = 0)"
monom_mult_zero_right: fixes c :: "'c"   and t :: "'a" shows "local.punit.monom_mult c t 0 = 0"
open_segment_separate_right: fixes u :: "'a"   and a :: "'a"   and b :: "'a"   and v :: "'a"   and x :: "'a" assumes "u \<in> {a<--<b}"   and "v \<in> {a<--<u}"   and "dist x u < dist u v"   and "x \<in> {a<--<b}" shows "x \<in> {v<--<b}"
subst_open_Var: fixes xT :: "char list \<times> Unary_PCF.type"   and yU :: "char list \<times> Unary_PCF.type"   and u :: "expr"   and i :: "nat"   and v :: "expr" assumes "xT \<noteq> yU"   and "lc u" shows "subst xT u (open_Var i yU v) = open_Var i yU (subst xT u v)"
dispath_instr_privilege: fixes instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state"   and s' :: "'b sparc_state" assumes "snd (dispatch_instruction instr s) = False \<and> s' = snd (fst (dispatch_instruction instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
appendL_strict: fixes ys :: "'b LList" shows "appendL\<cdot>\<bottom>\<cdot>ys = \<bottom>"
enabled_def: fixes r :: "'rule"   and s :: "'state" shows "enabled r s \<equiv> \<exists>sl. eff r s sl"
directed_alt: fixes X :: "'a set" shows "directed X = (X \<noteq> {} \<and> (\<forall>x\<in>X. \<forall>y\<in>X. \<exists>z\<in>X. x \<le> z \<and> y \<le> z))"
distinct_7: fixes x2 :: "int"   and x3 :: "int" shows "Intv' x2 \<noteq> Greater' x3"
subject_reduction2: fixes 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"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l :: "char list \<Rightarrow> val option"   and e' :: "expr"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and T :: "ty" assumes "wf_C_prog P"   and "P,E \<turnstile> \<langle>e,(h, l)\<rangle> \<rightarrow> \<langle>e',(h', l')\<rangle>"   and "P,E \<turnstile> (h, l) \<surd>"   and "P,E,h \<turnstile> e : T" shows "P,E,h' \<turnstile> e' :\<^bsub>NT\<^esub> T"
Inl_in_eps_iff: fixes q :: "hf"   and Q :: "hf set" shows "(HF.Inl q \<in> ST.epsclo Q) = (HF.Inl q \<in> Q \<and> q \<in> dfa.states MS)"
degree_monic_char_poly: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "degree (char_poly A) = n \<and> coeff (char_poly A) n = (1::'a)"
alphaAbs_ex_qAFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_qAFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. qAFresh xs y X \<and> qAFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
stake_suffix: fixes i :: "nat"   and j :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and u :: "nat \<Rightarrow> 'b" assumes "i \<le> j" shows "stake j \<sigma> @- u |\<^sub>s i = \<sigma>(i \<rightarrow> j - i) @- u"
retract_of_UNIV: fixes S :: "'b set" shows "(S retract_of UNIV) = (AR S \<and> closed S)"
mult_right_ennreal_cancel: fixes a :: "ennreal"   and c :: "real"   and b :: "ennreal" shows "(a * ennreal c = b * ennreal c) = (a = b \<or> c \<le> 0)"
uint_bounded: fixes w :: "'b word" shows "uint w < 2 ^ LENGTH('b)"
index_valid_def: fixes as :: "('b \<times> 'c atom) set"   and s :: "('b, 'c) state" shows "index_valid as s = (\<forall>x b i. (Mapping.lookup (\<B>\<^sub>i\<^sub>l s) x = Some (i, b) \<longrightarrow> (i, Geq x b) \<in> as) \<and> (Mapping.lookup (\<B>\<^sub>i\<^sub>u s) x = Some (i, b) \<longrightarrow> (i, Leq x b) \<in> as))"
bounded_antilinear_axioms_def: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_antilinear_axioms f \<equiv> \<exists>K. \<forall>x. norm (f x) \<le> norm x * K"
injectivity: fixes x :: "'c poly"   and y :: "'c poly" assumes "poly_lift x = poly_lift y" shows "x = y"
H_while: fixes p :: "'a"   and r :: "'a"   and x :: "'a" assumes "H (\<tt>\<tt> p \<cdot> \<tt>\<tt> r) x p" shows "H p (while r do x) (\<tt>\<tt> p \<cdot> !r)"
on_ray_def: fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and betw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "TarskiAbsolute cong betw" shows "TarskiAbsolute.on_ray betw p a b = (betw a p b \<or> betw a b p)"
Tr: shows "Tr \<equiv> \<lambda>(s\<^sub>1, s\<^sub>2). Tr\<^sub>1 s\<^sub>1 \<union> Tr\<^sub>2 s\<^sub>2"
basic_monos_7: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A \<subseteq> B" shows "x \<in> A \<longrightarrow> x \<in> B"
allin: shows "xs all_in\<^sub>i range_tms e env"
isomorphic_trans: fixes G\<^sub>1 :: "nat set \<times> nat set set"   and G\<^sub>2 :: "nat set \<times> nat set set"   and G\<^sub>3 :: "nat set \<times> nat set set" assumes "G\<^sub>1 \<simeq> G\<^sub>2"   and "G\<^sub>2 \<simeq> G\<^sub>3" shows "G\<^sub>1 \<simeq> G\<^sub>3"
dim_lambda_inverse: fixes y :: "'a \<Rightarrow> 'b" assumes "y \<in> UNIV" shows "dim_nth (dim_lambda y) = y"
product_of_first_summand: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "\<not> O x z" shows "P (x \<otimes> (y \<oplus> z)) (x \<otimes> y)"
tab_succ_def: fixes l :: "('d \<times> 'e) list" shows "tab_succ l \<equiv> Mapping.lookup_default [] (fold (\<lambda>(u, v). Mapping.map_default u [] ((#) v)) l Mapping.empty)"
wellformed_induce_subgraph: fixes vs :: "'a set" shows "wf_digraph (G \<restriction> vs)"
single_addition_neg: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "int"   and m :: "int" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z"   and "n < 0"   and "m < 0" shows "n\<Odot>a\<^bsub>f,i,z\<^esub> \<^sub>f+ m\<Odot>a\<^bsub>f,i,z\<^esub> = (n + m)\<Odot>a\<^bsub>f,i,z\<^esub>"
Standard_hyperpow: fixes r :: "'b star"   and n :: "nat star" assumes "r \<in> Standard"   and "n \<in> Standard" shows "r pow n \<in> Standard"
whisker_right: fixes f :: "'d"   and \<nu> :: "'d"   and \<mu> :: "'d" assumes "D.ide f"   and "D.seq \<nu> \<mu>" shows "\<nu> \<cdot>\<^sub>D \<mu> \<star>\<^sub>D f = (\<nu> \<star>\<^sub>D f) \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D f)"
Suc_1: fixes u :: "'a list"   and v :: "'a list"   and w :: "'a list" assumes "length u = n"   and "length v = n"   and "length w = n" shows "foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u (map (\<lambda>xy. fst xy + snd xy) (zip v w))) (0::'a) = foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u v) (0::'a) + foldr (\<lambda>a. case a of (x, y) \<Rightarrow> (+) (x * y)) (zip u w) (0::'a)"
nprv_eql_reflI: fixes F :: "'fmla set"   and t :: "'trm" assumes "F \<subseteq> fmla"   and "finite F"   and "t \<in> trm" shows "nprv F (eql t t)"
vintersection_vdoubleton: fixes a :: "V"   and b :: "V" shows "a \<in>\<^sub>\<circ> set {a, b}"
norm_power2_cnj: fixes q :: "octo" shows "(norm q)\<^sup>2 = Ree (q * Octonions.cnj q)"
Comm2: fixes P :: "pi"   and a :: "name"   and b :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'"   and "Q \<longmapsto>\<^sub>e a<b> \<prec>\<^sub>e Q'" shows "P \<parallel> Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P' \<parallel> Q'"
jj_3: shows "ps (S t (j + 1)) p \<noteq> None"
hb_antisym: fixes history :: "nat \<Rightarrow> 'c event list"   and msg_id :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'c" assumes "causal_network history msg_id"   and "network.hb history x y"   and "network.hb history y x" shows "False"
eq_Nil_null: fixes xs :: "'d list" shows "(xs = []) = List.null xs"
nvalids_def: fixes n :: "nat"   and T :: "triple set" shows "|\<Turnstile>n: T \<equiv> \<forall>t\<in>T. \<Turnstile>n: t"
max_fun_diff_max: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "f \<noteq> g"   and "f \<in> rs.FINFUNC"   and "g \<in> rs.FINFUNC"   and "x \<in> {a \<in> Field s. f a \<noteq> g a}" shows "(x, s.max_fun_diff f g) \<in> s"
comp_assoc: fixes h :: "(bool, bool list) concrete_category.arr"   and g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" shows "(h \<cdot>\<^sub>J g) \<cdot>\<^sub>J f = h \<cdot>\<^sub>J g \<cdot>\<^sub>J f"
that: fixes ka :: "nat"   and kb :: "nat"   and xs :: "nat list"   and ys :: "nat list"   and zs :: "nat list" assumes "l = ka + kb - 1"   and "U = {xs, ys}"   and "Form_Body ka kb xs ys zs"   and "0 < kb"   and "kb \<le> ka"   and "ka \<le> Suc kb" shows "thesis"
simps_2: fixes x :: "'g"   and y :: "'g"   and xs :: "'g list" shows "remove1 x (y # xs) = (if x = y then xs else y # remove1 x xs)"
idmap_def: fixes A :: "'e set" shows "idmap A = (\<lambda>x\<in>A. x)"
rel_fun_cong: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'c"   and x :: "'a" assumes "(f, g) \<in> Id" shows "(f x, g x) \<in> Id"
p'_def: shows "p' = (atom i \<rightleftharpoons> atom i') + p"
eq_strict_2: fixes y :: "'r" shows "eq\<cdot>\<bottom>\<cdot>y = \<bottom>"
left_neutral: fixes a :: "'q" shows "(1::'q) * a = a"
VDM_Sound_emptyCtxt: fixes c :: "IMP"   and A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool" assumes "{} \<rhd> c : A" shows " \<Turnstile> c : A "
EntryPath_suffix: fixes g :: "'g"   and ns :: "'node list"   and ns' :: "'node list" assumes "old.EntryPath g ns"   and "suffix ns' ns"   and "ns' \<noteq> []" shows "old.EntryPath g ns'"
\<rr>_ide_simp: fixes f :: "'b" assumes "D.ide f" shows "D.\<rr> f = \<r>\<^sub>D[f]"
realizable_matched: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n' :: "'node SDG_node" assumes "matched n ns n'" shows "realizable n ns n'"
mem_ar_coset1: fixes I :: "'a set"   and a :: "'a"   and x :: "'a" assumes "ideal R I"   and "a \<in> carrier R"   and "x \<in> a \<uplus>\<^bsub>R\<^esub> I" shows "\<exists>h\<in>I. h \<plusminus> a = x"
canonical_fun: fixes f :: "(val \<times> val) fset"   and v :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "VFun f \<in> E v \<rho>"   and "isval v" shows "\<exists>x e. v = ELam x e"
\<psi>_in_hom: fixes x :: "'b"   and g :: "'c"   and y :: "'c" assumes "B.ide x"   and "\<guillemotleft>g : y \<rightarrow>\<^sub>C G' x\<guillemotright>" shows "\<guillemotleft>\<psi>' x g : F' y \<rightarrow>\<^sub>B x\<guillemotright>"
pathVertices_edge: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and e :: "nat \<times> nat" assumes "cf.isPath s p t"   and "e \<in> set p" shows "\<exists>vs1 vs2. cf.pathVertices_fwd s p = vs1 @ fst e # snd e # vs2"
okamoto_def: fixes \<G> :: "'a cyclic_group" shows "okamoto \<G> \<equiv> okamoto_base \<G> \<and> Cyclic_Group.cyclic_group \<G>"
iso_def: fixes f :: "'b" shows "B.iso f \<equiv> \<exists>g. B.inverse_arrows f g"
map_of_insertmap_SomeD': fixes fs :: "('f \<times> 'e) list"   and F :: "'f"   and y :: "'e"   and D :: "'g" assumes "map_of fs F = \<lfloor>y\<rfloor>" shows "map_of (map (\<lambda>(F, y). (F, D, y)) fs) F = \<lfloor>(D, y)\<rfloor>"
enat_coexhaust: fixes n :: "enat" obtains "n = 0"   | n' :: "enat" where "n = eSuc n'"
pp_isotone_sup: fixes x :: "'a"   and y :: "'a" shows "- - x \<le> - - (x \<squnion> y)"
and_mask_less_size: fixes n :: "nat"   and x :: "'b word" assumes "n < size x" shows "x AND mask n < 2 ^ n"
that: shows "(f \<longlongrightarrow> l) (at x within X - {x})"
ext_irrefl_before_trans_axioms: shows "ext_irrefl_before_trans lexext"
mult_botl: fixes x :: "'c \<times> 'd" shows "times_prod' bot_prod x = bot_prod"
norm_F: fixes x :: "'d" shows "norm (f' x) \<le> norm x * KF"
scott_continuous_def: fixes f :: "'a \<Rightarrow> 'a" shows "local.scott_continuous f \<equiv> f ` A \<subseteq> A \<and> (\<forall>X s. X \<subseteq> A \<longrightarrow> directed X (\<sqsubseteq>) \<longrightarrow> X \<noteq> {} \<longrightarrow> extreme_bound A (\<sqsubseteq>) X s \<longrightarrow> extreme_bound A (\<sqsubseteq>) (f ` X) (f s))"
transpos_ij_2: fixes i :: "nat"   and n :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "(\<tau>\<^bsub>i j\<^esub>) j = i"
Finite_Markov_Decision_Process_def: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set" shows "Finite_Markov_Decision_Process K S \<equiv> Markov_Decision_Process K \<and> Finite_Markov_Decision_Process_axioms K S"
apply_dist_to_dens_def: fixes dst :: "pdf_dist"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" shows "apply_dist_to_dens dst f = (\<lambda>\<rho> y. \<integral>\<^sup>+ x. f \<rho> x * dist_dens dst x y \<partial>stock_measure (dist_param_type dst))"
\<Sigma>0_def: shows "\<Sigma>0 = [patient1 \<mapsto> LR1]"
elims_3: fixes x :: "'a btree"   and xa :: "'a" assumes "\<not> isin x xa" obtains y :: "'a" where "x = Leaf" and "xa = y__"   | ts :: "('a btree \<times> 'a) list" and t :: "'a btree" and y :: "'a" where "x = Node ts t" and "xa = y" and "\<not> (case split ts y of (x, []) \<Rightarrow> isin t y | (x, (sub, sep) # rs) \<Rightarrow> if y = sep then True else isin sub y)"
binop: shows "binop \<rat>\<^sub>\<circ> vrat_plus"
h'_def: shows "h' \<equiv> \<lambda>u. h u / (u - z) ^ n"
open_prod_generated: shows "open = generate_topology {A \<times> B |A B. open A \<and> open B}"
arctan_eq_iff: fixes x :: "real"   and y :: "real" shows "(arctan x = arctan y) = (x = y)"
cong_gen_cong: fixes rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and eval :: "'b \<Rightarrow> 'a"   and retr :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "cong rel eval retr" shows "cong.cong rel eval (cong.gen_cong rel eval R)"
Nom_2: shows "Nom a at b in (ps, a) # branch"
enum_ex_bit1_def: fixes P :: "'a bit1 \<Rightarrow> bool" shows "enum_class.enum_ex P = (\<exists>b\<in>set enum_class.enum. P b)"
to_cs: fixes v :: "nat \<Rightarrow> rat"   and c :: "rat le_constraint" shows "v \<Turnstile>\<^sub>c (case lec_rel c of Leq_Rel \<Rightarrow> LEQ | Lt_Rel \<Rightarrow> LT) (lec_poly c) (lec_const c) = v \<Turnstile>\<^sub>l\<^sub>e c"
cosets_finite: fixes c :: "nat set"   and H :: "nat set" assumes "c \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G"   and "finite (carrier G)" shows "finite c"
Sard_lemma00: fixes a :: "real"   and i :: "'a"   and P :: "'a set"   and m :: "real"   and e :: "real" assumes "0 \<le> a"   and "a *\<^sub>R i \<noteq> (0::'a)"   and "i \<in> Basis"   and "P \<subseteq> {x. a *\<^sub>R i \<bullet> x = 0}"   and "0 \<le> m"   and "0 \<le> e" obtains S :: "'a set" where "S \<in> lmeasurable" and "{z. norm z \<le> m \<and> (\<exists>t\<in>P. norm (z - t) \<le> e)} \<subseteq> S" and "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (DIM('a) - 1)"
assms_2: shows "u \<in> [up M]\<^bsub>v,n\<^esub>"
alphaAbs_qFreshAbs_qSwapAbs_id: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" assumes "qGoodAbs A"   and "qFreshAbs zs z1 A"   and "qFreshAbs zs z2 A" shows "A $[[z1 \<and> z2]]_zs $= A"
transpose_def: shows "Square_Matrix.transpose \<equiv> map_fun to_fun of_fun (\<lambda>M i j. M j i)"
Exit_path: fixes sourcenode :: "'g \<Rightarrow> 'h"   and targetnode :: "'g \<Rightarrow> 'h"   and kind :: "'g \<Rightarrow> ('i, 'j, 'k, 'l) edge_kind"   and valid_edge :: "'g \<Rightarrow> bool"   and Entry :: "'h"   and get_proc :: "'h \<Rightarrow> 'l"   and get_return_edges :: "'g \<Rightarrow> 'g set"   and procs :: "('l \<times> 'i list \<times> 'i list) list"   and Main :: "'l"   and Exit :: "'h"   and n :: "'h" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "\<exists>as. CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges n as Exit"
eNorm_uminus: fixes N :: "'a quasinorm"   and x :: "'a" shows "eNorm N (- x) = eNorm N x"