hcju/isabelleps huggingface dataset:
prim_invariant_def: fixes t :: "'a"   and v :: "'a"   and g :: "'a"   and r :: "'a" shows "prim_invariant t v g r \<equiv> prim_spanning_invariant t v g r \<and> (\<exists>w. minimum_spanning_tree w g r \<and> t \<le> w)"
prems_2: shows "G \<subseteq>\<^sub>m last (trace_parallel_plan_strips I (ops # \<pi>))"
product_functor_def: fixes A1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A2 :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and B1 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and B2 :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and F1 :: "'d \<Rightarrow> 'f"   and F2 :: "'e \<Rightarrow> 'g" shows "product_functor A1 A2 B1 B2 F1 F2 \<equiv> ((category A1 \<and> category A2) \<and> category B1 \<and> category B2) \<and> (functor A1 B1 F1 \<and> functor A2 B2 F2) \<and> product_category A1 A2 \<and> product_category B1 B2"
Gramian_beta: fixes i :: "nat" assumes "i < length fs" shows "\<beta> fs i = \<parallel>fs ! i\<parallel>\<^sup>2 - (\<Sum>j = 0..<i. (\<mu> i j)\<^sup>2 * \<beta> fs j)"
trans: shows "\<forall>z\<in>A. \<forall>y\<in>A. \<forall>x\<in>A. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
dvdm_power: fixes g :: "int poly" assumes "g dvd f" shows "p.dvdm u g = pl.dvdm u g"
cases: fixes x :: "arctic" obtains "x = MinInfty"   | n :: "int" where "x = Num_arc n"
valid_slide: fixes w :: "nat \<times> nat"   and as :: "'b list"   and t :: "'b tree" assumes "0 < fst w"   and "fst w \<le> snd w"   and "valid as t"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "snd w \<le> length as" shows "valid as (slide as t w) \<and> l (slide as t w) = fst w \<and> r (slide as t w) = snd w"
base: fixes data :: "nat \<times> nat \<times> 'd"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "P data bs [] (gs @ bs)"
gs2: fixes j :: "nat" assumes "j < m" shows "gs2.gso j \<in> Rn"
l'_def: shows "l' = nat ((int l - 1) mod int (length cs))"
simps_1: shows "braun \<langle>\<rangle> = True"
s'_def: fixes t :: "real" shows "delayed_safe_distance.other.s' t = (if t \<le> delayed_safe_distance.other.t_stop then delayed_safe_distance.other.p' t else 0)"
supply_def: fixes p :: "prog"   and us :: "nat list" shows "supply p us \<equiv> let vs = para p in if distinct vs \<and> length us = length vs then Some (map_of (zip vs us)) else None"
H3: fixes p :: "int mpoly"   and a :: "nat" shows "p - Var a = - Var a + p"
comp_assoc: fixes h :: "'b"   and g :: "'b"   and f :: "'b" shows "C' (C' h g) f = C' h (C' g f)"
ideal_generated_subset_generator: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and I :: "'b set" assumes "class.ring plus zero minus uminus times"   and "ring.ideal_generated plus zero uminus times A = I" shows "A \<subseteq> I"
mapOf_lookup3: fixes t :: "(int \<times> 'b) Tree"   and i :: "int" assumes "mapOf t i = None" shows "tlookup fst i t = None"
strong_induct: fixes P :: "'a \<Rightarrow> Syntax.term \<Rightarrow> bool"   and c :: "'a"   and "term" :: "Syntax.term" assumes "\<And>c. P c Unit"   and "\<And>x c. P c (Var x)"   and "\<And>x1 x2a c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (Lam x1 x2a)"   and "\<And>x1 x2a c. \<lbrakk>{atom x1} \<sharp>* c; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (Rec x1 x2a)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Inj1 x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Inj2 x)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (Syntax.Pair x1 x2a)"   and "\<And>x1 x2a x3 c. \<lbrakk>{atom x2a} \<sharp>* c; \<And>c. P c x1; \<And>c. P c x3\<rbrakk> \<Longrightarrow> P c (Syntax.Let x1 x2a x3)"   and "\<And>x1 x2a c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a\<rbrakk> \<Longrightarrow> P c (App x1 x2a)"   and "\<And>x1 x2a x3 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2a; \<And>c. P c x3\<rbrakk> \<Longrightarrow> P c (Case x1 x2a x3)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Prj1 x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Prj2 x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Roll x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Unroll x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Auth x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (Unauth x)"   and "\<And>x c. P c (Hash x)"   and "\<And>x1 x2a c. (\<And>c. P c x2a) \<Longrightarrow> P c (Hashed x1 x2a)" shows "P c term"
is_nan_minus: fixes x :: "('a, 'b) IEEE.float"   and y :: "('a, 'b) IEEE.float" assumes "IEEE.is_nan x \<or> IEEE.is_nan y" shows "IEEE.is_nan (x - y)"
inorder_update: fixes t :: "('c \<times> 'd) tree234"   and a :: "'c"   and b :: "'d" assumes "sorted1 (inorder t)" shows "inorder (update a b t) = upd_list a b (inorder t)"
Finite_completion_lemma: fixes I :: "'c set"   and F :: "'d program"   and A :: "'c \<Rightarrow> 'd set"   and A' :: "'c \<Rightarrow> 'd set"   and C :: "'d set" assumes "finite I" shows "(\<forall>i\<in>I. F \<in> A i \<longmapsto>w A' i \<union> C) \<longrightarrow> (\<forall>i\<in>I. F \<in> A' i Co A' i \<union> C) \<longrightarrow> F \<in> \<Inter> (A ` I) \<longmapsto>w \<Inter> (A' ` I) \<union> C"
distinct_insort_insert: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'a"   and x :: "'b" assumes "class.linorder less_eq less"   and "distinct xs" shows "distinct (linorder.insort_insert_key less_eq f x xs)"
erf_remainder_def: fixes n :: "nat"   and x :: "real" shows "erf_remainder n x = (- 1) ^ n * 2 * fact (2 * n) / (sqrt pi * 4 ^ n * fact n) * erf_remainder_integral n x"
edge_succ_cyclic: fixes G :: "('c, 'd) pre_digraph"   and M :: "'d pre_map"   and v :: "'c" assumes "digraph_map G M"   and "v \<in> verts G"   and "out_arcs G v \<noteq> {}" shows "cyclic_on (edge_succ M) (out_arcs G v)"
normed_vectorspace_with_fn_norm_axioms: shows "normed_vectorspace_with_fn_norm V norm"
simulation1: fixes s1 :: "'a"   and s2 :: "'c"   and tl1 :: "'b"   and s1' :: "'a" assumes "bisim s1 s2"   and "trsys1 s1 tl1 s1'"   and "\<not> False" shows "\<exists>s2' s2'' tl2. s2 -\<tau>2\<rightarrow>* s2' \<and> trsys2 s2' tl2 s2'' \<and> \<not> False \<and> bisim s1' s2'' \<and> tlsim tl1 tl2"
emeasure_single_in_space: fixes M :: "'b measure"   and x :: "'b" assumes "emeasure M {x} \<noteq> 0" shows "x \<in> space M"
sup_fun_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
ordLeq_cmax1: fixes r :: "('b \<times> 'b) set"   and s :: "('c \<times> 'c) set" assumes "Card_order r"   and "Card_order s" shows "r \<le>o cmax r s"
remove1_mset: fixes w :: "'b"   and vs :: "'b list" assumes "w \<in> set vs" shows "mset (remove1 w vs) + {#w#} = mset vs"
inter_scheme_simple: fixes l :: "nat"   and U :: "nat list set" assumes "Form l U"   and "0 < l" shows "inter_scheme l U \<in> WW \<and> 0 < length (inter_scheme l U)"
pmf_bind: fixes N :: "'c pmf"   and f :: "'c \<Rightarrow> 'b pmf"   and i :: "'b" shows "pmf (N \<bind> f) i = measure_pmf.expectation N (\<lambda>x. pmf (f x) i)"
dfn: fixes n :: "nat" shows "f n \<in> carrier_mat d d"
NewArray_\<tau>execI: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and T :: "ty" assumes "\<tau>exec_move ci P t e h s s'" shows "\<tau>exec_move ci P t (newA T\<lfloor>e\<rceil>) h s s'"
orthU: shows "pairwise orthogonal (S \<union> insert a' U)"
bounded_linear_sum: fixes I :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'f" assumes "\<And>i. i \<in> I \<Longrightarrow> bounded_linear (f i)" shows "bounded_linear (\<lambda>x. \<Sum>i\<in>I. f i x)"
fbrelationE1: fixes r :: "V"   and x :: "V" assumes "fbrelation r"   and "x \<in>\<^sub>\<circ> r" obtains a :: "V" and b :: "V" where "x = [a, b]\<^sub>\<circ>"
new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t: fixes h :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and h' :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "new\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_element_ptr, h')"   and "ptr \<noteq> cast (cast new_element_ptr)" shows "get ptr h = get ptr h'"
wit_swap_forall: fixes P :: "assn"   and c :: "'a Heap"   and A :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> 'a \<Rightarrow> assn" assumes "<P> c <\<lambda>_. true>"   and "\<forall>x. A x \<longrightarrow> <P> c <Q x>" shows "<P> c <\<lambda>r. \<not>\<^sub>A (\<exists>\<^sub>Ax. \<up> (A x) * \<not>\<^sub>A Q x r)>"
reaches_eq_imp: fixes x :: "'a"   and y :: "'a" shows "eq_imp (\<lambda>r' s. \<Union> (ran ` obj_fields ` set_option (sys_heap s r'))) (x reaches y)"
may_acquire_allE: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and ln :: "'c \<Rightarrow>f nat" assumes "may_acquire_all ls t ln" obtains "\<forall>l. 0 < ln $ l \<longrightarrow> may_lock (ls $ l) t"
abelian_group_axioms_def: fixes G :: "('e, 'f) ring_scheme" shows "abelian_group_axioms G \<equiv> comm_group (add_monoid G)"
v1: shows "viewL t = Some (a, s) \<and> ft_invar s \<and> toList t = a # toList s"
simps_16: fixes v :: "bexp"   and va :: "'a acom"   and vb :: "'a acom"   and vc :: "'a"   and vd :: "'a acom"   and ve :: "'a acom" shows "(IF v THEN va ELSE vb {vc} \<sqsubseteq> vd;; ve) = False"
F_def: fixes ds :: "'d set" shows "F ds = (\<lambda>(XD, XH). (F1 (undual XH), dual (F2 ds (F1 (undual XH)))))"
sqrt_rat: shows "\<bar>sqrt (real p)\<bar> = real m / real n"
hd_val: fixes v :: "val"   and V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal" assumes "countable_type (val_type v)" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Val v \<Rightarrow> (\<lambda>\<rho> x. branch_prob (V, V', \<Gamma>, \<delta>) \<rho> * indicator {v} x)"
generate_valid_topology_as_set: fixes M :: "'b SecurityInvariant_configured list"   and G :: "'b graph" shows "generate_valid_topology M G = delete_edges G (\<Union>m\<in>set M. \<Union> (c_offending_flows m G))"
completeness_def: shows "\<Sigma>_OR.completeness \<equiv> \<forall>h w e. (h, w) \<in> Rel_OR \<longrightarrow> e \<in> challenge_space \<longrightarrow> spmf (\<Sigma>_OR.completeness_game h w e) True = 1"
Cons_2: fixes i :: "nat"   and j :: "nat" assumes "i < j"   and "i < length (x # xs)"   and "j < length (x # xs)" shows "(x # xs) ! i \<noteq> (x # xs) ! j"
assms_3: shows "ord_Zp x \<noteq> ord_Zp y"
comm_ring_hom_axioms: shows "comm_ring_hom to_fract"
ope: fixes C :: "'a set" assumes "C \<in> \<B>" shows "openin (top_of_set S) C"
bigo_def: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "O[F](g) = {f. \<exists>c>0. \<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)}"
conjugate_zero_vec: fixes n :: "nat" shows "conjugate (0\<^sub>v n) = 0\<^sub>v n"
empty_no_car: shows "\<forall>ts v. \<parallel>ext v\<parallel> = 0 \<longrightarrow> (\<nexists>x. 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (clm ts) x = lan v \<and> |lan v| = 1 \<or> 0 < \<parallel>ext v\<parallel> \<and> len v ts x = ext v \<and> restrict v (res ts) x = lan v \<and> |lan v| = 1)"
roots_of_coprime_r_def: fixes qs :: "real poly list" shows "roots_of_coprime_r qs = {x. poly (coprime_r qs) x = 0}"
igAbsCongS_fromMOD: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "gAbsCongS MOD" shows "igAbsCongS (fromMOD MOD)"
presT: shows "rel_preserves_binary_pred (_Collect (P, Q) (\<exists>SP SQ. \<lbrakk>SP\<rbrakk> \<in>T P \<and> \<lbrakk>SQ\<rbrakk> \<in>T Q \<and> (\<lbrakk>SP\<rbrakk>, \<lbrakk>SQ\<rbrakk>) \<in> TRel)) Pred"
vs_conf_def: fixes P :: "'b prog"   and h :: "'heap"   and vs :: "'addr \<times> addr_loc \<Rightarrow> 'addr val set" shows "h.vs_conf P h vs = (\<forall>ad al v. v \<in> vs (ad, al) \<longrightarrow> (\<exists>T. h.addr_loc_type P h ad al T \<and> h.conf P h v T))"
distance0_2_def: fixes a\<^sub>e :: "real"   and v\<^sub>e :: "real"   and s\<^sub>e :: "real"   and a\<^sub>o :: "real"   and v\<^sub>o :: "real"   and s\<^sub>o :: "real"   and \<delta> :: "real" assumes "safe_distance_normal a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o \<delta>" shows "safe_distance_normal.distance0_2 v\<^sub>e a\<^sub>o v\<^sub>o \<delta> = v\<^sub>e * \<delta> + 1 / 2 * v\<^sub>o\<^sup>2 / a\<^sub>o"
get_root_node_locs_def: shows "get_root_node_locs \<equiv> l_get_root_node\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_root_node_locs get_parent_locs"
hurwitz_zeta_critical_strip: fixes s :: "complex"   and a :: "real" assumes "0 < Re s"   and "s \<noteq> 1"   and "0 < a" shows "(\<lambda>n. (\<Sum>i<n. complex_of_real (real i + a) powr - s) - of_nat n powr (1 - s) / (1 - s)) \<longlonglongrightarrow> hurwitz_zeta a s"
nth_normalize_ge_zero_iff: fixes p :: "(real, 2) vec"   and i :: "2" shows "(0 \<le> normalize p $ i) = (0 \<le> p $ i)"
set_insort_key: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d"   and xs :: "'d list" assumes "class.linorder less_eq less" shows "set (linorder.insort_key less_eq f x xs) = Set.insert x (set xs)"
inc: fixes A :: "nat \<Rightarrow> 'a set" assumes "range A \<subseteq> C \<inter> sigma_sets \<Omega> M"   and "A 0 = {}"   and "\<And>n. A n \<subseteq> A (Suc n)" shows "\<Union> (range A) \<in> C"
absorb2: fixes a :: "'b"   and b :: "'b" assumes "a \<le> b" shows "sup a b = b"
containsDuplicateEdge_eq1: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and "is" :: "nat list" shows "containsDuplicateEdge g f v is = containsDuplicateEdge' g f v is"
sqrt_irrat: fixes q :: "rat"   and b :: "nat"   and p :: "rat" assumes "q = 0 \<or> b \<in> sqrt_irrat"   and "real_of_rat p + real_of_rat q * sqrt (real b) = 0" shows "q = 0"
floatE_normed: fixes x :: "real" assumes "x \<in> float" obtains "x = 0"   | m :: "int" and e :: "int" where "x = real_of_int m * 2 powr real_of_int e" and "odd m" and "x \<noteq> 0"
compositionality_BSIA: fixes ES1 :: "'b ES_rec"   and ES2 :: "'b ES_rec"   and \<V> :: "'b V_rec"   and \<V>1 :: "'b V_rec"   and \<V>2 :: "'b V_rec"   and \<rho>1 :: "'b V_rec \<Rightarrow> 'b set"   and \<rho>2 :: "'b V_rec \<Rightarrow> 'b set"   and \<rho> :: "'b V_rec \<Rightarrow> 'b set" assumes "Compositionality ES1 ES2 \<V> \<V>1 \<V>2"   and "BSD \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSD \<V>2 Tr\<^bsub>ES2\<^esub>"   and "BSIA \<rho>1 \<V>1 Tr\<^bsub>ES1\<^esub>"   and "BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>"   and "\<rho>1 \<V>1 \<subseteq> \<rho> \<V> \<inter> E\<^bsub>ES1\<^esub>"   and "\<rho>2 \<V>2 \<subseteq> \<rho> \<V> \<inter> E\<^bsub>ES2\<^esub>" shows "BSIA \<rho> \<V> Tr\<^bsub>ES1 \<parallel> ES2\<^esub>"
stmt_var_inj_term: fixes t :: "stmt"   and w :: "Term.var" shows "\<langle>t\<rangle> \<noteq> \<langle>w\<rangle>"
integral_indicator: fixes M :: "'c measure"   and A :: "'c set" shows "integral\<^sup>L M (indicat_real A) = Sigma_Algebra.measure M (A \<inter> space M)"
hballE: fixes A :: "hf"   and P :: "hf \<Rightarrow> bool"   and x :: "hf" assumes "HBall A P" obtains "P x"   | "x \<^bold>\<notin> A"
retract_from_Un_Int: fixes S :: "'a set"   and T :: "'a set"   and U :: "'a set" assumes "closedin (top_of_set (S \<union> T)) S"   and "closedin (top_of_set (S \<union> T)) T"   and "S \<union> T retract_of U"   and "S \<inter> T retract_of T" shows "S retract_of U"
LPDi_Plus: fixes i :: "nat" shows "LPDi i (MPlus r s) \<subseteq> {MPlus r s} \<union> LPDi i r \<union> LPDi i s"
append_in_listsp_conv: fixes A :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'a list" shows "listsp A (xs @ ys) = (listsp A xs \<and> listsp A ys)"
order_trans: fixes x :: "'k multiset"   and y :: "'k multiset"   and z :: "'k multiset" assumes "x \<subseteq># y"   and "y \<subseteq># z" shows "x \<subseteq># z"
path_det: 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 n :: "'b"   and as :: "'a list"   and n' :: "'b"   and n'' :: "'b" assumes "CFG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "CFG.path sourcenode targetnode valid_edge n as n''" shows "n' = n''"
set_merge: fixes xs :: "'c list"   and ys :: "'c list" shows "set (merge xs ys) = set xs \<union> set ys"
fls_subdegree_of_int: fixes i :: "int" shows "fls_subdegree (of_int i) = 0"
try_spmf_lossless: fixes p :: "'a spmf"   and q :: "'a spmf" assumes "lossless_spmf p" shows "TRY p ELSE q = p"
sheaf_spec_morphisms_are_maps: fixes V :: "'a set set"   and U :: "'a set set" assumes "is_zariski_open V"   and "V \<subseteq> U" shows "Set_Theory.map (sheaf_spec_morphisms U V) (\<O> U) (\<O> V)"
range_map_indets: fixes f :: "'d \<Rightarrow> 'b" shows "range (map_indets f) = P[range f]"
cls': shows "class P Object = \<lfloor>(C', fs, ms)\<rfloor>"
approximating_bigstep_undecided_to_undecided_in_doubt_deny_approx: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "(\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided \<or> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Decision FinalAllow"
dual_undual: fixes x' :: "'b dual" shows "dual (undual x') = x'"
Cod_f: shows "\<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = b"
intro: fixes carrier :: "'b set"   and relation :: "('b \<times> 'b) set" assumes "preference carrier relation"   and "rational_preference_axioms carrier relation" shows "rational_preference carrier relation"
add_comm: fixes a :: "'a"   and b :: "'a"   and p\<^sub>1 :: "'a point"   and p\<^sub>2 :: "'a point" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "on_curve a b p\<^sub>1"   and "on_curve a b p\<^sub>2" shows "local.add a p\<^sub>1 p\<^sub>2 = local.add a p\<^sub>2 p\<^sub>1"
nn_integral_snd: fixes M1 :: "'d measure"   and M2 :: "'e measure"   and f :: "'d \<times> 'e \<Rightarrow> ennreal" assumes "pair_sigma_finite M1 M2"   and "f \<in> borel_measurable (M1 \<Otimes>\<^sub>M M2)" shows "\<integral>\<^sup>+ y. \<integral>\<^sup>+ x. f (x, y) \<partial>M1 \<partial>M2 = integral\<^sup>N (M1 \<Otimes>\<^sub>M M2) f"
vector_inf_closed: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv"   and "times x top = x"   and "times y top = y" shows "times (inf x y) top = inf x y"
prod_emb_trans: fixes J :: "'d set"   and K :: "'d set"   and L :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and X :: "('d \<Rightarrow> 'e) set" assumes "J \<subseteq> K"   and "K \<subseteq> L" shows "prod_emb L M K (prod_emb K M J X) = prod_emb L M J X"
simps_1: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" assumes "class.wellorder less_eq less" shows "wellorder.enumerate less_eq S 0 = ord.Least less_eq (\<lambda>n. n \<in> S)"
bounded_iff: fixes b :: "'b upred"   and c :: "'b upred"   and a :: "'b upred" shows "(a \<sqsubseteq> (b \<or> c)) = (a \<sqsubseteq> b \<and> a \<sqsubseteq> c)"
image_prod: fixes f :: "'g \<Rightarrow> 'e"   and g :: "'h \<Rightarrow> 'f"   and A :: "'g set"   and B :: "'h set" shows "(\<lambda>(x, y). (f x, g y)) ` (A \<times> B) = f ` A \<times> g ` B"
split_simple_path: fixes u :: "nat"   and p1 :: "(nat \<times> nat) list"   and p2 :: "(nat \<times> nat) list"   and v :: "nat" assumes "cf.isSimplePath u (p1 @ p2) v" shows "\<exists>w. cf.isSimplePath u p1 w \<and> cf.isSimplePath w p2 v"
h1: shows "abc_steps_l (0, lm) (A [+] B) nx = (length A, lma)"
os_ts__inangle: fixes B :: "'p"   and P :: "'p"   and A :: "'p"   and C :: "'p" assumes "B P TS A C"   and "B A OS C P" shows "P InAngle A B C"
char_empty: shows "\<chi> {} = (\<lambda>t. 0)"
density_unique2: fixes f :: "nat set set \<Rightarrow> ennreal"   and f' :: "nat set set \<Rightarrow> ennreal" assumes "random_variable borel f"   and "random_variable borel f'"   and "\<And>A. A \<in> events \<Longrightarrow> set_nn_integral P A f = set_nn_integral P A f'" shows "AE x in P. f x = f' x"
closed_closedin: fixes S :: "'c set" shows "closed S = closedin euclidean S"
assms: shows "(a, b) \<in> (B\<^sup>* O A O B\<^sup>*) ^^ n"
eqvt_apply: fixes p :: "perm"   and f :: "'d \<Rightarrow> 'c"   and x :: "'d" shows "p \<bullet> f x \<equiv> (p \<bullet> f) (p \<bullet> x)"
comp_ide_self: fixes a :: "'a" assumes "R.ide a" shows "R.comp a a = a"
iT_Mult_empty: fixes k :: "nat" shows "{} \<otimes> k = {}"
map_upd_Some_unfold: fixes m :: "'h \<Rightarrow> 'g option"   and a :: "'h"   and b :: "'g"   and x :: "'h"   and y :: "'g" shows "((m(a \<mapsto> b)) x = Some y) = (x = a \<and> b = y \<or> x \<noteq> a \<and> m x = Some y)"
surjective: fixes r :: "'a State_scheme" shows "r = \<lparr>getM = getM r, \<dots> = more r\<rparr>"
prems_2: shows "n \<in> set (\<alpha>n g)"
STAR_star_of_Int: fixes X :: "'a set" shows "*s* X \<inter> Standard = star_of ` X"
prime_factors_multiplicity: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and n :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize n) = {p. normalization_semidom.prime zero times one normalize p \<and> 0 < factorial_semiring.multiplicity times one p n}"
equalityI: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B"   and "B \<subseteq> A" shows "A = B"
rconf_In3: fixes G :: "prog"   and L :: "lenv"   and s :: "st"   and es :: "expr list"   and vs :: "val list"   and Ts :: "ty list" shows "G,L,s\<turnstile>In3 es\<succ>In3 vs\<Colon>\<preceq>Inr Ts = list_all2 (conf G s) vs Ts"
rotate_C_def: shows "rotate_C = (\<lambda>v. case v of A \<Rightarrow> B | B \<Rightarrow> D | C \<Rightarrow> C | D \<Rightarrow> A)"
simps_1: fixes x :: "'a \<Rightarrow> bool"   and ya :: "'a \<Rightarrow> bool" shows "(Pred x = Pred ya) = (x = ya)"
regular_pairs: fixes phi :: "'c \<Rightarrow> 'd Filters.filter" assumes "triple phi" shows "{(x, y). (x, y) \<in> triple.pairs phi \<and> triple.pairs_uminus phi (triple.pairs_uminus phi (x, y)) = (x, y)} = {(x, y). y = phi (- x)}"
mode: shows "mode = IntVir \<longrightarrow> a' \<noteq> Null"
rep_eq: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xa :: "('b, 'c) mapping"   and xb :: "('b, 'c) mapping" shows "rep (combine_with_key x xa xb) = (\<lambda>xc. combine_options (x xc) (rep xa xc) (rep xb xc))"