hcju/isabelleps huggingface dataset:
gcd_add_dvd_left2: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times a c" shows "gcd (plus b c) a = gcd b a"
tau_U: shows "\<tau> U = 1\<^sub>\<pi>"
run_reachable: fixes G :: "('a, 'b) graph_rec_scheme"   and r :: "nat \<Rightarrow> 'a" assumes "graph_defs.is_run G r" shows "range r \<subseteq> (g_E G)\<^sup>* `` g_V0 G"
exhaust: fixes y :: "'a sequent" obtains x1 :: "'a form multiset" and x2 :: "'a form" where "y = ( x1 \<Rightarrow>* x2)"
prime_number_theorem: shows "primes_pi \<sim>[at_top] (\<lambda>x. x / ln x)"
mcp'_intro: fixes z :: "'d list"   and W :: "'d list set"   and p :: "'d list" assumes "mcp z W p" shows "mcp' z W = p"
llength_iterates: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "llength (iterates f x) = \<infinity>"
unit_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>C \<i>\<^sub>C[a] = a"
simps_2: fixes p :: "'b"   and v :: "'a" shows "unpackMessage <p, v> = Value v"
restrict_superlist: fixes ls :: "'c list"   and ks :: "'c list"   and ms :: "'c list"   and ns :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd" assumes "is_sublist ls ks"   and "is_sublist ms ns" shows "ks \<langle>ls \<langle>f\<rangle> ms\<rangle> ns = ls \<langle>f\<rangle> ms"
gfp_least: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<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 f :: "'b \<Rightarrow> 'b"   and X :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>u. less_eq u (f u) \<Longrightarrow> less_eq u X" shows "less_eq (complete_lattice.gfp Sup less_eq f) X"
preserves_arr: fixes f :: "'c" assumes "C.arr f" shows "arr (inclusion_of_generators f)"
TensorDiag_assoc: fixes t :: "'c local.term"   and u :: "'c local.term"   and v :: "'c local.term" assumes "Diag t"   and "Diag u"   and "Diag v" shows "(t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> v"
intro: fixes X :: "'f \<Longrightarrow> 'd"   and Y :: "'e \<Longrightarrow> 'd" assumes "\<And>\<sigma> v u. put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"   and "\<And>\<sigma> v. get\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) = get\<^bsub>X\<^esub> \<sigma>"   and "\<And>\<sigma> u. get\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) = get\<^bsub>Y\<^esub> \<sigma>" shows "X \<bowtie> Y"
pdevs_apply_pdevs_of_list_Nil: fixes i :: "nat" shows "pdevs_apply (pdevs_of_list []) i = (0::'b)"
nprv_zer_suc_contrE: fixes F :: "'fmla set"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "nprv F (eql zer (suc t))"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "t \<in> atrm" shows "nprv F \<phi>"
0: shows "{a. monom (remove_key v a) (monom (Poly_Mapping.single v (lookup a v)) (coeff p a)) \<noteq> 0} \<subseteq> S"
perp_foot_up_right_angle: fixes p :: "proj2"   and a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "p \<in> S"   and "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident p l"   and "proj2_incident b l" shows "right_angle p (perp_foot a l) (perp_up a l)"
bound_of_def: fixes P :: "'c \<Rightarrow> real" shows "bound_of P \<equiv> Sup (range P)"
Cons_sorted_less: fixes xs :: "'a list"   and p :: "'a" assumes "Sorted_Less.sorted (rev xs)"   and "\<forall>x\<in>set xs. x < p" shows "Sorted_Less.sorted (rev (p # xs))"
inject: fixes x1 :: "'a"   and x2 :: "'b"   and x3 :: "'c"   and x4 :: "'a"   and y1 :: "'a"   and y2 :: "'b"   and y3 :: "'c"   and y4 :: "'a" shows "(Trans x1 x2 x3 x4 = Trans y1 y2 y3 y4) = (x1 = y1 \<and> x2 = y2 \<and> x3 = y3 \<and> x4 = y4)"
fmember: fixes a :: "'b"   and as :: "'b list" shows "(a |\<in>| fset_of_list as) = List.member as a"
uniform_limit: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'e \<Rightarrow> 'b"   and X :: "'e set"   and l :: "'e \<Rightarrow> 'b"   and F :: "'d filter" assumes "bounded_linear f"   and "uniform_limit X g l F" shows "uniform_limit X (\<lambda>a b. f (g a b)) (\<lambda>a. f (l a)) F"
E: shows "adjoint_equivalence (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G F \<eta> \<epsilon>"
wf_no_path: fixes R :: "('a \<times> 'a) set" assumes "Domain R \<inter> Range R = {}" shows "wf R"
eqI: fixes \<tau>' :: "'c \<Rightarrow> 's" assumes "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (Y a) F \<tau>'"   and "\<tau>' a = \<tau> a" shows "\<tau>' = \<tau>"
bisim1'_imp_bisim1: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and n :: "nat"   and exs :: "'addr expr1 \<times> 'addr val list"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" assumes "P,e,n,h \<turnstile>' exs \<leftrightarrow> s" shows "P,e,h \<turnstile> exs \<leftrightarrow> s"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_2: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, receive\<langle>t\<rangle>) # A) = (l, send\<langle>t\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
path_connectedin_Euclidean_complements: fixes n :: "nat"   and S :: "(nat \<Rightarrow> real) set"   and T :: "(nat \<Rightarrow> real) set" assumes "closedin (Euclidean_space n) S"   and "closedin (Euclidean_space n) T"   and "subtopology (Euclidean_space n) S homeomorphic_space subtopology (Euclidean_space n) T" shows "path_connectedin (Euclidean_space n) (topspace (Euclidean_space n) - S) = path_connectedin (Euclidean_space n) (topspace (Euclidean_space n) - T)"
maximal_independent_subset: fixes V :: "'c set" obtains B :: "'c set" where "B \<subseteq> V" and "vs2.independent B" and "V \<subseteq> vs2.span B"
cases: fixes a :: "'b set"   and f :: "'b perm" assumes "a \<in> orbits f" obtains aa :: "'b" where "a = orbit f aa" and "aa \<in> affected f"
append_self_conv: fixes xs :: "'c list"   and ys :: "'c list" shows "(xs @ ys = xs) = (ys = [])"
h1: shows "maximal A (\<lambda>S. S \<subseteq> carrier V \<and> lin_indpt S)"
regions_partition: fixes v :: "'c \<Rightarrow> real" assumes "\<forall>x\<in>X. 0 \<le> v x" shows "\<exists>!R. R \<in> \<R> \<and> v \<in> R"
inter_sorted_correct_col: fixes xs :: "'a list"   and ys :: "'a list" assumes "Sorted_Less.sorted (rev xs)"   and "Sorted_Less.sorted (rev ys)" shows "inter_sorted_rev xs ys = rev (sorted_list_of_set (set xs \<inter> set ys))"
Arden_removes_cl: fixes Y :: "'b list set"   and yrhs :: "'b trm set" shows "rhss (Arden Y yrhs) = rhss yrhs - {Y}"
gauss_int_norm_not_3_mod_4: fixes z :: "gauss_int" shows "[gauss_int_norm z \<noteq> 3] (mod 4)"
commute: fixes a :: "'d"   and b :: "'d" shows "sup a b = sup b a"
supp_Cons: fixes x :: "'a"   and xs :: "'a list" shows "supp (x # xs) = supp x \<union> supp xs"
elims: fixes x :: "checker_type"   and y :: "checker_type" assumes "type_update_select x = y" obtains ts :: "t list" where "x = Type ts__" and "y = (if 3 \<le> length ts__ \<and> ts__ ! (length ts__ - 2) = ts__ ! (length ts__ - 3) then consume (Type ts__) [TAny, TSome T_i32] else Bot)"   | ts :: "ct list" where "x = TopType ts" and "y = (case length ts of 0 \<Rightarrow> TopType [TAny] | Suc 0 \<Rightarrow> type_update (TopType ts) [TSome T_i32] (TopType [TAny]) | Suc (Suc 0) \<Rightarrow> consume (TopType ts) [TSome T_i32] | Suc (Suc (Suc nat)) \<Rightarrow> type_update (TopType ts) [TAny, TAny, TSome T_i32] (select_return_top ts (ts ! (length ts - 2)) (ts ! (length ts - 3))))"   | "x = Bot" and "y = Bot"
elims_3: fixes x :: "t_context"   and xa :: "b_e list"   and xb :: "tf" assumes "\<not> b_e_type_checker x xa xb" obtains \<C> :: "t_context" and es :: "b_e list" and tn :: "t list" and tm :: "t list" where "x = \<C>" and "xa = es" and "xb = tn _> tm" and "\<not> c_types_agree (check \<C> es (Type tn)) tm"
Union_in_lattice: fixes M :: "'b set set"   and L :: "'b set set" assumes "M \<subseteq> L"   and "lattice L" shows "\<Union> M \<in> L"
joinable_connected_component_eq: fixes T :: "'a set"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "connected T"   and "T \<subseteq> S"   and "connected_component_set S x \<inter> T \<noteq> {}"   and "connected_component_set S y \<inter> T \<noteq> {}" shows "connected_component_set S x = connected_component_set S y"
hyps: fixes S' :: "nat set"   and S :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "\<forall>a\<in>all_shared sb. (a \<in> S') = (a \<in> S)" shows "release sb S' \<R> = release sb S \<R>"
viewnres_cases: fixes r :: "(('e, 'f) Node \<times> ('e, 'f) FingerTreeStruc) option" obtains "r = None"   | a :: "('e, 'f) Node" and t :: "('e, 'f) FingerTreeStruc" where "r = Some (a, t)"
keys_filter: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "('c, 'd) mapping" shows "Mapping.keys (Mapping.filter P m) \<subseteq> Mapping.keys m"
self_append_conv2: fixes ys :: "'c list"   and xs :: "'c list" shows "(ys = xs \<cdot> ys) = (xs = [])"
simps_1: fixes i :: "int"   and j :: "int" shows "plus_const (Const i) (Const j) = Const (i + j)"
psubst_fresh_Ctxt: fixes x :: "W.var"   and \<theta> :: "(W.tvar \<times> ty) list"   and \<Gamma> :: "(W.var \<times> tyS) list" shows "x \<sharp> \<theta><\<Gamma>> = x \<sharp> \<Gamma>"
game2_c_def: shows "game2_c = spmf_of_set (nlists UNIV len) \<bind> (\<lambda>r. exec_gpv rf_encrypt \<A>1 Map.empty \<bind> (\<lambda>(((p0, p1), \<sigma>), s1). if valid_plain p0 \<and> valid_plain p1 then coin_spmf \<bind> (\<lambda>b. let pb = if b then p0 else p1 in spmf_of_set (nlists UNIV len) \<bind> (\<lambda>pad. let cipher = (r, pb [\<oplus>] pad) in exec_gpv (rf_encrypt_bad r) (\<A>2 cipher \<sigma>) (s1(r \<mapsto> pad), False) \<bind> (\<lambda>(b', s2, bad). return_spmf (b' = b, bad)))) else coin_spmf \<bind> (\<lambda>b. return_spmf (b, False))))"
vconn_implies_hpar_4: fixes \<mu> :: "'a"   and f :: "'a"   and g :: "'a" assumes "\<guillemotleft>\<mu> : f \<Rightarrow>\<^sub>C g\<guillemotright>" shows "trg\<^sub>C g = trg\<^sub>C f"
numeral_eq_of_nat: fixes a :: "num" shows "numeral a = of_nat (numeral a)"
norm_eq: fixes x :: "'b"   and y :: "'c" shows "(norm x = norm y) = (x \<bullet> x = y \<bullet> y)"
simps_2: fixes x :: "'b"   and \<theta> :: "('a, 'b) trm"   and \<sigma> :: "('a, 'c, 'b) subst" shows "Osubst (OSing x \<theta>) \<sigma> = OSing x (Tsubst \<theta> \<sigma>)"
completion_density_eq: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "AE x in M. 0 < f x"   and "f \<in> borel_measurable M" shows "completion (density M f) = density (completion M) f"
code: fixes r :: "('a \<times> bool) language" shows "Times_Plus r = Lang (\<oo> r) (\<lambda>a. Times_Plus (Plus (\<dd> r (a, True)) (\<dd> r (a, False))))"
all_together: fixes \<eta> :: "nat"   and \<A> :: "nat \<Rightarrow> ((bool list \<times> bool list) insec_query + bool list + unit, (bool list \<times> bool list) option + unit + bool list option) distinguisher"   and q :: "nat" assumes "0 < \<eta>"   and "interaction_bounded_by' (\<lambda>_. True) (\<A> \<eta>) q"   and "plossless_gpv (\<I>_uniform (Set.Collect (valid_mac_query \<eta>)) (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>))) \<oplus>\<^sub>\<I> (\<I>_uniform (vld \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` vld \<eta>)))) (\<A> \<eta>)"   and "\<I>_uniform (Set.Collect (valid_mac_query \<eta>)) (insert None (Some ` (nlists UNIV \<eta> \<times> nlists UNIV \<eta>))) \<oplus>\<^sub>\<I> (\<I>_uniform (vld \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` vld \<eta>))) \<turnstile>g \<A> \<eta> \<surd>" shows "\<bar>spmf (connect (\<A> \<eta>) (CNV sim (Inl None) |\<^sub>= 1\<^sub>C \<rhd> A_CHAN.res)) True - spmf (connect (\<A> \<eta>) (MAC.res \<eta>)) True\<bar> \<le> real q / real (2 ^ \<eta>)"
orient_lit_semantics_neg: fixes I :: "'a trm \<Rightarrow> 'a trm \<Rightarrow> bool"   and l :: "'a literal"   and u :: "'a trm"   and v :: "'a trm"   and \<theta> :: "('a \<times> 'a trm) list"   and \<sigma> :: "('a \<times> 'a trm) list" assumes "fo_interpretation I"   and "orient_lit_inst l u v neg \<theta>"   and "validate_ground_lit I (subst_lit l \<sigma>)" shows "\<not> I (u \<lhd> \<sigma>) (v \<lhd> \<sigma>)"
subdegree_geI: fixes f :: "'a fps"   and n :: "nat" assumes "f \<noteq> 0"   and "\<And>i. i < n \<Longrightarrow> f $ i = (0::'a)" shows "n \<le> subdegree f"
surjective: fixes r :: "('a, 'b) network_scheme" shows "r = \<lparr>edge = edge r, capacity = capacity r, source = source r, sink = sink r, \<dots> = network.more r\<rparr>"
order_trans: fixes x :: "'c list"   and y :: "'c list"   and z :: "'c list" assumes "prefix x y"   and "prefix y z" shows "prefix x z"
height_list_split: fixes ls :: "('b btree \<times> 'b) list"   and a :: "'b btree"   and b :: "'b"   and rs :: "('b btree \<times> 'b) list"   and t :: "'b btree" shows "height_up\<^sub>i (Up\<^sub>i (Node ls a) b (Node rs t)) = height (Node (ls @ (a, b) # rs) t)"
inf_prefix_correct: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list" assumes "i < length (xs \<sqinter> ys)" shows "xs ! i = ys ! i"
refl: shows "\<forall>X x. derivation X \<longrightarrow> x \<in> X \<longrightarrow> f x \<sqsubseteq> f x"
apex_equivalence_lemma: fixes \<rho>' :: "'a"   and g' :: "'a"   and f' :: "'a"   and w :: "'a"   and \<theta> :: "'a"   and \<nu> :: "'a"   and w' :: "'a"   and \<theta>' :: "'a"   and \<nu>' :: "'a" assumes "\<guillemotleft>\<rho>' : g' \<Rightarrow> r \<star> f'\<guillemotright>"   and "ide w \<and> \<guillemotleft>\<theta> : f' \<star> w \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu> : g' \<Rightarrow> g' \<star> w\<guillemotright> \<and> local.iso \<nu> \<and> (r \<star> \<theta>) \<cdot> \<a>[r, f', w] \<cdot> (\<rho>' \<star> w) \<cdot> \<nu> = \<rho>'"   and "ide w' \<and> \<guillemotleft>\<theta>' : f' \<star> w' \<Rightarrow> f'\<guillemotright> \<and> \<guillemotleft>\<nu>' : g' \<Rightarrow> g' \<star> w'\<guillemotright> \<and> local.iso \<nu>' \<and> (r \<star> \<theta>') \<cdot> \<a>[r, f', w'] \<cdot> (\<rho>' \<star> w') \<cdot> \<nu>' = \<rho>'" shows "\<exists>\<phi>. \<guillemotleft>\<phi> : src f' \<Rightarrow> w' \<star> w\<guillemotright> \<and> local.iso \<phi>"
is_extensional: fixes f :: "'a" assumes "\<not> Left_a.arr f" shows "H\<^sub>L a f = Left_a.null"
trE: fixes p :: "tr" obtains "p = \<bottom>"   | "p = TT"   | "p = FF"
indRelLTPO_impl_TRel_is_weak_reduction_contrasimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "weak_reduction_contrasimulation (indRelLTPO TRel) (STCal Source Target)" shows "weak_reduction_contrasimulation (TRel\<^sup>+) Target"
prime_elem_imp_nonzero: fixes x :: "'a" assumes "ASSUMPTION (prime_elem x)" shows "x \<noteq> (0::'a)"
reachable_in_verts_2: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>Digraph_Component.union G H\<^esub> v" shows "v \<in> verts (Digraph_Component.union G H)"
simps: 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 a1 :: "'b SDG_node"   and a2 :: "'b SDG_node list"   and a3 :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses" shows "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses a1 a2 a3 = ((\<exists>n. a1 = n \<and> a2 = [] \<and> a3 = n \<and> SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n) \<or> (\<exists>n ns n'' n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' None None n') \<or> (\<exists>n ns n'' V n'. a1 = n \<and> a2 = ns @ [n''] \<and> a3 = n' \<and> SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'' \<and> SDG.SDG_edge sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n'' (Some V) None n' \<and> n'' \<noteq> n'))"
BRBang: fixes Rel :: "ccs \<Rightarrow> ccs \<Rightarrow> bool"   and P :: "ccs"   and Q :: "ccs" assumes "Rel P Q" shows "bangRelp Rel !P !Q"
elem_set_implies_set_eq_singleton: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and a :: "'b"   and b :: "'b" assumes "set_category S \<AA>"   and "a \<in> set_category_data.set S (set_category.some_img S \<AA>) b" shows "set_category_data.set S (set_category.some_img S \<AA>) a = {a}"
simps: fixes a :: "'c"   and R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set" shows "(a \<in> I \<cdot>\<^bsub>R\<^esub> J) = ((\<exists>i j. a = i \<otimes>\<^bsub>R\<^esub> j \<and> i \<in> I \<and> j \<in> J) \<or> (\<exists>s1 s2. a = s1 \<oplus>\<^bsub>R\<^esub> s2 \<and> s1 \<in> I \<cdot>\<^bsub>R\<^esub> J \<and> s2 \<in> I \<cdot>\<^bsub>R\<^esub> J))"
Bpow_def: fixes r :: "('d \<times> 'd) set"   and A :: "'c set" shows "Bpow r A \<equiv> {X. X \<subseteq> A \<and> |X| \<le>o r}"
foldr_refine: assumes "case (Qi, \<pi>i) of (Qi, \<pi>i) \<Rightarrow> Q_invar Qi \<and> M_invar \<pi>i"   and "\<And>s x. \<lbrakk>case s of (Qi, \<pi>i) \<Rightarrow> Q_invar Qi \<and> M_invar \<pi>i; x \<in> set (G_adj g u)\<rbrakk> \<Longrightarrow> (case foreach_impl_body u x s of (Qi, \<pi>i) \<Rightarrow> Q_invar Qi \<and> M_invar \<pi>i) \<and> map_prod Q_\<alpha> M_lookup (foreach_impl_body u x s) = foreach_body u x (map_prod Q_\<alpha> M_lookup s)" shows "(case foldr (foreach_impl_body u) (G_adj g u) (Qi, \<pi>i) of (Qi, \<pi>i) \<Rightarrow> Q_invar Qi \<and> M_invar \<pi>i) \<and> map_prod Q_\<alpha> M_lookup (foldr (foreach_impl_body u) (G_adj g u) (Qi, \<pi>i)) = foldr (foreach_body u) (G_adj g u) (map_prod Q_\<alpha> M_lookup (Qi, \<pi>i))"
HEndPhase1_HInv4c_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase1 s s' p"   and "HInv4c s q"   and "p \<noteq> q" shows "HInv4c s' q"
WT_calleeD: fixes \<I> :: "('a, 'b) \<I>"   and callee :: "'a \<Rightarrow> ('b \<times> 'c) spmf"   and ret :: "'b"   and s :: "'c"   and out :: "'a" assumes "\<I> \<turnstile>c callee \<surd>"   and "(ret, s) \<in> set_spmf (callee out)"   and "out \<in> outs_\<I> \<I>" shows "ret \<in> responses_\<I> \<I> out"
type_definition_phi: shows "type_definition Rep_phi Abs_phi {f. bounded_lattice_homomorphism f}"
in_measure_iff: fixes x :: "'a"   and y :: "'a"   and f :: "'a \<Rightarrow> nat" shows "((x, y) \<in> measure f) = (f x < f y)"
simps_2: fixes x2 :: "real" shows "MInfty \<noteq> real_max_bot.R x2"
simps_1: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'c btree" assumes "split split" shows "inorder_up\<^sub>i (T\<^sub>i t) = inorder t"
btree_abs_def: shows "btree_abs \<equiv> PRJ('a\<cdot>btree) oo EMB('a\<^sub>\<bottom> \<oplus> ('a\<cdot>btree)\<^sub>\<bottom> \<otimes> ('a\<cdot>btree)\<^sub>\<bottom>)"
lookup_tuple_since': shows "\<forall>as\<in>Mapping.keys tuple_since'. case Mapping.lookup tuple_since' as of Some t \<Rightarrow> t \<le> nt"
sub_inc_One_eq: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (Num.inc n) num.One = numeral.numeral one plus n"
sum_only_one_neq_0: fixes A :: "'a set"   and j :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "j \<in> A"   and "\<And>i. \<lbrakk>i \<in> A; i \<noteq> j\<rbrakk> \<Longrightarrow> g i = (0::'b)" shows "sum g A = g j"
orthogonal_transformation_id: shows "orthogonal_transformation (\<lambda>x. x)"
elims: fixes x :: "'a extB"   and xa :: "'a extB"   and y :: "'a extB" assumes "x + xa = y" obtains "xa__ = extB.Bot" and "xa = extB.Bot" and "ya__ = extB.Val top"   | x :: "'a" where "xa__ = extB.Bot" and "xa = extB.Val xb__" and "ya__ = extB.Val xb__"   | x :: "'a" where "xa__ = extB.Val xc__" and "xa = extB.Bot" and "ya__ = extB.Val xc__"   | x :: "'a" and y :: "'a" where "xa__ = extB.Val x" and "xa = extB.Val y" and "ya__ = extB.Val (x + y)"
label_wrtD: fixes B :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and C :: "'a set" assumes "label_wrt B f"   and "C \<in> \<C>" shows "bij_betw f C B"
FAssNull\<^sub>1: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat exp) list) list"   and e\<^sub>1 :: "nat exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and e\<^sub>2 :: "nat exp"   and v :: "val"   and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> val list"   and F :: "char list"   and D :: "char list" assumes "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>"   and "P \<turnstile>\<^sub>1 \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>2\<rangle>" shows "P \<turnstile>\<^sub>1 \<langle>e\<^sub>1\<bullet>F{D} := e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW NullPointer,s\<^sub>2\<rangle>"
diagonal_mat_def: fixes A :: "'c mat" shows "diagonal_mat A \<equiv> \<forall>i<dim_row A. \<forall>j<dim_col A. i \<noteq> j \<longrightarrow> A $$ (i, j) = (0::'c)"
INF_par_distrib: fixes X :: "'d::type set"   and d :: "'d::type \<Rightarrow> 'a::refinement_lattice"   and c :: "'a::refinement_lattice" assumes "(X::'d::type set) \<noteq> {}" shows "\<Sqinter> ((d::'d::type \<Rightarrow> 'a::refinement_lattice) ` (X::'d::type set)) \<parallel> (c::'a::refinement_lattice) = (\<Sqinter>x::'d::type\<in>X. d x \<parallel> c)"
arr_has_target: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "targets \<mu> \<noteq> {}"
elims: fixes x :: "'b literal"   and y :: "'b equation" assumes "atom x = y" obtains xa :: "'b equation" where "x = Pos xa__" and "y = xa__"   | xa :: "'b equation" where "x = Neg xa" and "y = xa"
all_zero: shows "\<forall>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k = (0::'a)"