hcju/isabelleps huggingface dataset:
src_hcomp: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr" assumes "S.arr \<mu>"   and "S.arr \<nu>"   and "S.src \<mu> = S.trg \<nu>" shows "S.src (\<mu> \<star>\<^sub>S \<nu>) = S.src \<nu>"
of_nat_push_bit: fixes 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 divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.of_nat one plus zero (push_bit m n) = push_bit m (semiring_1.of_nat one plus zero n)"
Times_eq_cancel2: fixes x :: "'a"   and C :: "'a set"   and A :: "'b set"   and B :: "'b set" assumes "x \<in> C" shows "(A \<times> C = B \<times> C) = (A = B)"
eval_def: shows "eval \<equiv> \<mu> eval. \<Lambda> v. Fixrec.run\<cdot>(match_Val\<cdot>v\<cdot>(\<Lambda> n. Fixrec.succeed\<cdot>(Just\<cdot>n)) +++ match_Add\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(mliftM2 (\<Lambda> a b. a + b)\<cdot>(eval\<cdot>x)\<cdot>(eval\<cdot>y))) +++ match_Throw\<cdot>v\<cdot>(Fixrec.succeed\<cdot>mfail) +++ match_Catch\<cdot>v\<cdot>(\<Lambda> x y. Fixrec.succeed\<cdot>(mcatch\<cdot>(eval\<cdot>x)\<cdot>(eval\<cdot>y))))"
per_union_commute: fixes R :: "('b \<times> 'b) set"   and i :: "'b"   and j :: "'b" shows "per_union R i j = per_union R j i"
\<phi>_in_terms_of_\<Phi>': fixes y :: "'d"   and f :: "'c"   and x :: "'c" assumes "D.ide y"   and "\<guillemotleft>f : F y \<rightarrow>\<^sub>C x\<guillemotright>" shows "\<phi> y f = (\<Phi>\<Psi>.\<psi>D (y, G x) \<circ> \<Phi>\<Psi>.\<Phi>.FUN (y, x) \<circ> \<phi>\<psi>.inC) f"
subst_rename: fixes y :: "name"   and t :: "trm"   and v :: "trm"   and x :: "name" assumes "y \<sharp> t" shows "([(y, x)] \<bullet> t)[y::=v] = t[x::=v]"
list_max_def: fixes l :: "'a list" shows "list_max l = fold max l"
Oops_range_spies2: fixes A :: "agent"   and authK :: "nat"   and servK :: "nat"   and B :: "agent"   and Ts :: "msg"   and servTicket :: "msg"   and evs :: "event list" assumes "Says Tgs A (Crypt authK \<lbrace>Key servK, Agent B, Ts, servTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "servK \<notin> range shrK \<and> servK \<in> symKeys"
tendstoD: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and l :: "'a"   and F :: "'b filter"   and e :: "real" assumes "class.metric_space dist uniformity open"   and "filterlim f (topological_space.nhds open l) F"   and "0 < e" shows "\<forall>\<^sub>F x in F. dist (f x) l < e"
lang_lang_trad: shows "subst \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, \<bb>, S, S, \<aa>]|}\<rparr> {|[Inr (init \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, \<bb>, S, S, \<aa>]|}\<rparr>)]|} = to_language (lang_trad \<lparr>init = (), prod = \<lambda>_. {|[], [\<aa>, \<bb>, S, S, \<aa>]|}\<rparr>)"
types_wellformed_def: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option" shows "A.types_wellformed \<Gamma> \<equiv> \<forall>x\<in>dom \<Gamma>. A.type_wellformed (the (\<Gamma> x))"
f_abs_def: shows "af_G_abs_opt.f_abs \<equiv> map_fun Rep (map_fun id Abs) af_G_letter_opt"
add_decreasing: fixes a :: "'a multiset"   and c :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># {#}"   and "c \<subseteq># b" shows "a + c \<subseteq># b"
subseq_stable: fixes r :: "nat \<Rightarrow> nat"   and n :: "nat"   and s :: "nat \<Rightarrow> nat" assumes "strict_mono r"   and "P n s" shows "P n (s \<circ> r)"
urel64_eq: fixes x :: "uint64"   and y :: "int"   and x' :: "uint64"   and y' :: "int" assumes "urel64 x y"   and "urel64 x' y'" shows "(x = x') = (y = y')"
assms_2: shows "t_wise_balance V B t \<Lambda>\<^sub>t"
permutes_less_3: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat"   and i :: "nat" assumes "p permutes {0..<n}" shows "p (inv p i) = i"
intros_3: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and \<Delta> :: "'b formula multiset"   and n :: "nat" assumes "\<Gamma> \<Rightarrow> F, \<Delta> \<down> n" shows "\<^bold>\<not> F, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
vprop: shows "open (SOME A. open A \<and> x \<in> A \<and> y \<notin> A) \<and> x \<in> (SOME A. open A \<and> x \<in> A \<and> y \<notin> A) \<and> y \<notin> (SOME A. open A \<and> x \<in> A \<and> y \<notin> A)"
intros_1: fixes A :: "fm" shows "A IMP A \<in> boolean_axioms"
basis_subspace_exists: fixes S :: "'b set" assumes "local.subspace S" obtains B :: "'b set" where "finite B" and "B \<subseteq> S" and "local.independent B" and "local.span B = S" and "card B = local.dim S"
convex_same_rel_interior_closure_straddle: fixes S :: "'b set"   and T :: "'b set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (rel_interior S \<subseteq> T \<and> T \<subseteq> closure S)"
extend_state_oracle_def: fixes callee :: "'s \<Rightarrow> 't \<Rightarrow> ('u \<times> 's) spmf" shows "\<dagger>callee = (\<lambda>(s', s) x. map_spmf (\<lambda>(y, s). (y, s', s)) (callee s x))"
in_angle_line: fixes P :: "'p"   and B :: "'p"   and A :: "'p"   and C :: "'p" assumes "P \<noteq> B"   and "A \<noteq> B"   and "C \<noteq> B"   and "Bet A B C" shows "P InAngle A B C"
host_shadow_root_rel_host: fixes get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_document :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and get_disconnected_document_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and h :: "(_) heap"   and shadow_root :: "(_) shadow_root_ptr"   and host :: "(_) element_ptr" assumes "l_get_parent_get_host_wf\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_tag_name known_ptr type_wf heap_is_wellformed parent_child_rel heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_disconnected_document get_disconnected_document_locs known_ptrs get_parent"   and "heap_is_wellformed h"   and "h \<turnstile> get_host shadow_root \<rightarrow>\<^sub>r host" shows "(cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r host, cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root) \<in> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_host_shadow_root_rel get_shadow_root h"
lincomb_Cons: fixes r :: "'f"   and rs :: "'f list"   and m :: "'v"   and ms :: "'v list" shows "lincomb' (r # rs) (m # ms) = r \<cdot> m + lincomb' rs ms"
normalized_nnf_match_normalize_match: fixes m :: "'c match_expr" shows "\<forall>m'\<in>set (normalize_match m). normalized_nnf_match m'"
image_iff: fixes z :: "'d"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
simps_6: fixes x31 :: "'c"   and x32 :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd" shows "Assign x31 x32 \<noteq> Skip"
neg_D: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> Robinson_Arithmetic.neg A"   and "H \<turnstile> A" shows "H \<turnstile> B"
pr_index_enumerator_is_pr: shows "pr_index_enumerator \<in> PrimRec2"
elims_3: fixes \<delta> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'e"   and \<gamma> :: "'e \<Rightarrow> 'f \<times> 'g \<Rightarrow> 'h"   and bot :: "'e"   and id1 :: "nat"   and id2 :: "nat"   and x :: "'i \<times> ('e, 'j, 'k, 'l) SLin_state_scheme" assumes "Idempotence \<delta> \<gamma> bot id1 id2"   and "\<not> Idempotence.P16 bot x" obtains s1 :: "'i" and s2 :: "('e, 'j, 'k, 'l) SLin_state_scheme" where "x = (s1, s2)" and "\<not> (dstate s2 \<noteq> bot \<longrightarrow> initialized s2)"
igSwapIGOp_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "igSwapIGOp MOD \<equiv> \<forall>zs z1 z2 delta inp binp. igWlsInp MOD delta inp \<and> igWlsBinp MOD delta binp \<longrightarrow> igSwap MOD zs z1 z2 (igOp MOD delta inp binp) = igOp MOD delta (igSwapInp MOD zs z1 z2 inp) (igSwapBinp MOD zs z1 z2 binp)"
locally_connected_quotient_image: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "locally connected S"   and "\<And>T. T \<subseteq> f ` S \<Longrightarrow> openin (top_of_set S) (S \<inter> f -` T) = openin (top_of_set (f ` S)) T" shows "locally connected (f ` S)"
midpoint_plus_self: fixes a :: "'a"   and b :: "'a" shows "midpoint a b + midpoint a b = a + b"
Rep_IF2: fixes x :: "'d IF2" shows "Rep_IF2 x \<in> car_init2 undefined"
Q: shows "Q = (\<lambda>((s, l, r), ss, x). dec_inv_1 ly x e (as, lm) (s, l, r) ires)"
div_mult_self4: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and b :: "'b"   and c :: "'b"   and a :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "b \<noteq> zero" shows "divide (plus (times b c) a) b = plus c (divide a b)"
norm_in_Hom: fixes t :: "'c local.term" assumes "Arr t" shows "\<^bold>\<parallel>t\<^bold>\<parallel> \<in> Hom (Dom t) (Cod t)"
add_increasing2: fixes c :: "'y multiset"   and b :: "'y multiset"   and a :: "'y multiset" assumes "{#} \<subseteq># c"   and "b \<subseteq># a" shows "b \<subseteq># a + c"
cet_or: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : BOOL"   and "\<Gamma> \<turnstile>\<^sub>c e2 : BOOL" shows "\<Gamma> \<turnstile>\<^sub>c e1 \<or>\<^sub>c e2 : BOOL"
exhaust: fixes y :: "'b up\<^sub>i" obtains x1 :: "'b tree234" where "y = T\<^sub>i x1"   | x21 :: "'b tree234" and x22 :: "'b" and x23 :: "'b tree234" where "y = Up\<^sub>i x21 x22 x23"
assms_1: shows "\<forall>t2<x. i t2 = []"
game_difference: fixes \<A> :: "((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 "interaction_bounded_by' (\<lambda>_. True) \<A> 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>"   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> \<surd>" shows "\<bar>spmf (connect \<A> (RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv_f) (Void, None, Map.empty))) True - spmf (connect \<A> (RES (local.lazy_channel_insec \<oplus>\<^sub>O lazy_channel_send \<oplus>\<^sub>O local.lazy_channel_recv) (Void, None, Map.empty))) True\<bar> \<le> real q / real (2 ^ \<eta>)"
adjunction_in_bicategory_axioms: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f f\<^sup>* \<eta> \<epsilon>"
progressing_Cons: fixes x :: "'b \<times> 'b"   and xs :: "('b \<times> 'b) list" shows "progressing (x # xs) = (xs = [] \<or> xs \<noteq> [] \<and> \<not> (fst x = snd (hd xs) \<and> snd x = fst (hd xs)) \<and> progressing xs)"
children: shows "h2 \<turnstile> get_child_nodes ptr \<rightarrow>\<^sub>r children"
hyps: fixes child_node :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child_node = child"   and "h \<turnstile> get_parent child_node \<rightarrow>\<^sub>r Some parent"   and "parent \<in> set c" shows "h \<turnstile> get_dom_component parent \<rightarrow>\<^sub>r c"
msgrel_imp_eqv_freeleft_aux: fixes U :: "freemsg" shows "freeleft U \<sim> freeleft U"
to_nat_from_nat_id: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and x :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "x < CARD('d)" shows "mod_type.to_nat Rep (mod_type.from_nat Abs x) = x"
l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_create_element_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr DocumentClass.known_ptrs DocumentClass.type_wf get_child_nodes get_child_nodes_locs get_disconnected_nodes get_disconnected_nodes_locs heap_is_wellformed parent_child_rel set_tag_name set_tag_name_locs set_disconnected_nodes set_disconnected_nodes_locs create_element"
nat_standard_induct: fixes P :: "nat \<Rightarrow> bool"   and x :: "nat" assumes "P 0"   and "\<And>n. P n \<Longrightarrow> P (Suc n)" shows "P x"
extends_ty_dom: fixes \<Delta> :: "binding list"   and \<Gamma> :: "binding list" assumes "\<Delta> extends \<Gamma>" shows "ty_dom \<Gamma> \<subseteq> ty_dom \<Delta>"
next_recvupd'_def: fixes c0 :: "('b, 'loc \<times> 't) Exchange.configuration"   and c1 :: "('b, 'loc \<times> 't) Exchange.configuration"   and p :: "'b"   and q :: "'b" shows "cri.next_recvupd' c0 c1 p q \<equiv> c_msg c0 p q \<noteq> [] \<and> c1 = c0\<lparr>c_msg := (c_msg c0)(p := (c_msg c0 p)(q := tl (c_msg c0 p q))), c_glob := (c_glob c0)(q := c_glob c0 q + hd (c_msg c0 p q))\<rparr>"
iTL_defs_10: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P \<equiv> P (iprev t0 I)"
dg_prod_ObjD_3: fixes a :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "a \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Obj\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "a\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>"
stake_nth: fixes m :: "nat"   and n :: "nat"   and s :: "'a stream" assumes "m < n" shows "stake n s ! m = s !! m"
multiplicity_prime_1_plus_i_aux: fixes c :: "nat" shows "multiplicity (1 + \<i>\<^sub>\<int>) (of_nat c) = 2 * multiplicity 2 c"
order_trans: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "order r"   and "x \<sqsubseteq>\<^bsub>r\<^esub> y"   and "y \<sqsubseteq>\<^bsub>r\<^esub> z" shows "x \<sqsubseteq>\<^bsub>r\<^esub> z"
simps_3: fixes w :: "nat"   and v :: "nat"   and va :: "'a tree"   and vb :: "'a tree"   and t\<^sub>2 :: "'a tree"   and a :: "'a" shows "sibling (Node w (Node v va vb) t\<^sub>2) a = (if a \<in> alphabet (Node v va vb) then sibling (Node v va vb) a else if a \<in> alphabet t\<^sub>2 then sibling t\<^sub>2 a else a)"
tabulation_axioms: shows "tabulation (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C r \<rho> f g"
assms_2: fixes x :: "'a" assumes "x \<in> X" shows "emeasure M {x} = 0"
rev_cases: fixes xs :: "'d list" obtains "xs = []"   | ys :: "'d list" and y :: "'d" where "xs = ys @ [y]"
dense_closed_sup: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top"   and "uminus x = bot"   and "uminus y = bot" shows "uminus (sup x y) = bot"
comb_sat_props: shows "1 < length qs \<longrightarrow> p \<noteq> 0 \<and> 0 < length qs \<and> (\<forall>q. List.member qs q \<longrightarrow> coprime p q) \<longrightarrow> satisfies_properties p qs (get_subsets (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs))))) (get_signs (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs))))) (get_matrix (snd (combine_systems p (take (length qs div 2) qs, calculate_data p (take (length qs div 2) qs)) (drop (length qs div 2) qs, calculate_data p (drop (length qs div 2) qs)))))"
CondT: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>1 :: "char list exp"   and e' :: "char list exp"   and s\<^sub>2 :: "heap \<times> sfields \<times> sheap"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>true,s\<^sub>1\<rangle>"   and "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>" shows "P \<turnstile> \<langle>if (e) e\<^sub>1 else e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>e',s\<^sub>2\<rangle>"
INF_top_conv_1: 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 B :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "(Inf (B ` A) = top) = (\<forall>x\<in>A. B x = top)"
that: fixes i :: "nat" assumes "i < num_params p" shows "xs ! i = ys ! i"
J_3: fixes j :: "nat" assumes "j \<in> J" shows "E j \<in> M.events"
UEA2: fixes \<BB> :: "'b formula"   and \<Gamma> :: "'b formula list"   and \<Theta> :: "'b formula list"   and \<AA> :: "'b formula" assumes "\<BB> # \<Gamma> \<Rightarrow> \<Theta>" shows "\<AA> \<^bold>\<and> \<BB> # \<Gamma> \<Rightarrow> \<Theta>"
cases: fixes base :: "nat"   and a :: "hmultiset" assumes "2 \<le> base"   and "well_base\<^sub>h base a" obtains M :: "hmultiset" where "a = M" and "\<forall>N\<in>#hmsetmset M. well_base\<^sub>h base N" and "well_base base (hmsetmset M)"
\<alpha>_limit_set_in_compact_subset: shows "\<alpha>_limit_set x \<subseteq> K"
trm_prm_simp_2: fixes \<pi> :: "'b prm"   and x :: "'b" shows "\<pi> \<cdot> Var x = Var (\<pi> $ x)"
cleanup_def: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal" shows "cleanup f = (\<lambda>(a, b). if f (b, a) < f (a, b) then f (a, b) - f (b, a) else 0)"
p_def: fixes z :: "nat"   and y :: "nat" shows "p z y = pA (c_pair (c_pair (c_fst z) (c_pair (c_snd z) (c_fst y))) 0) (c_snd y)"
oLimit_Suc: fixes f :: "nat \<Rightarrow> ordinal" assumes "f 0 < oLimit f" shows "oLimit (\<lambda>n. f (Suc n)) = oLimit f"
simps_1: fixes n :: "nat"   and uu :: "'a"   and i :: "nat" shows "unlift' n (Opaque uu) i = Var i"
card_seteq: fixes B :: "'e set"   and A :: "'e set" assumes "finite B"   and "A \<subseteq> B"   and "card B \<le> card A" shows "A = B"
len_takeWhile_comp2: fixes n1 :: "'b"   and xs :: "'b list"   and n2 :: "'b"   and x :: "'b" assumes "n1 \<in> set xs"   and "n2 \<notin> set xs" shows "length (takeWhile ((\<noteq>) n1) (rev (x # xs))) \<noteq> length (takeWhile ((\<noteq>) n2) (rev (x # xs)))"
additive_congruence: shows "normal_subgroup.Congruence I R (+) \<zero> = Ring_Congruence"
distinct_27: fixes x8 :: "'a" shows "OclVoid \<noteq> \<langle>x8\<rangle>\<^sub>\<T>"
wn_triangle3: fixes z :: "complex"   and a :: "complex"   and b :: "complex"   and c :: "complex" assumes "z \<in> interior (convex hull {a, b, c})"   and "0 < Im ((b - a) * cnj (b - z))"   and "0 < Im ((c - b) * cnj (c - z))"   and "0 < Im ((a - c) * cnj (a - z))" shows "winding_number (linepath a b +++ linepath b c +++ linepath c a) z = 1"
cases: fixes a1 :: "ccs"   and a2 :: "ccs"   and Rel :: "(ccs \<times> ccs) set" assumes "(a1, a2) \<in> bangRel Rel" obtains P :: "ccs" and Q :: "ccs" where "a1 = !Pa__" and "a2 = !Qa__" and "(Pa__, Qa__) \<in> Rel"   | R :: "ccs" and T :: "ccs" and P :: "ccs" and Q :: "ccs" where "a1 = R \<parallel> P" and "a2 = T \<parallel> Q" and "(R, T) \<in> Rel" and "(P, Q) \<in> bangRel Rel"
simps_18: fixes f1 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and f2 :: "(('b \<Rightarrow> 'c option) \<Rightarrow> bool) \<Rightarrow> 'a"   and f3 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> 'c option) list \<Rightarrow> 'a"   and f4 :: "(('b \<Rightarrow> 'c option) \<times> 'd \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> (('b \<Rightarrow> 'c option) \<Rightarrow> ('b \<Rightarrow> 'c option) \<Rightarrow> 'b \<Rightarrow> 'c option) \<Rightarrow> 'a"   and x2 :: "('b \<Rightarrow> 'c option) \<Rightarrow> bool" shows "(case (x2)\<^sub>\<surd> of \<Up>x \<Rightarrow> f1 x | (x)\<^sub>\<surd> \<Rightarrow> f2 x | x:xa\<hookrightarrow>\<^bsub>xb\<^esub>xc \<Rightarrow> f3 x xa xb xc | x\<hookleftarrow>\<^bsub>xa\<^esub>xb \<Rightarrow> f4 x xa xb) = f2 x2"
is_hashset_def: fixes s :: "'a set"   and ht :: "('a, unit) hashtable" shows "is_hashset s ht \<equiv> is_hashmap (map_of_set s) ht"
continuous_on_imp_set_integrable_cbox: fixes a :: "'a"   and b :: "'a"   and h :: "'a \<Rightarrow> 'b" assumes "continuous_on (cbox a b) h" shows "set_integrable lborel (cbox a b) h"
det_RETURN: fixes x :: "'a" shows "nres_of (dRETURN x) \<le> RETURN x"
div_free_divergence_refine_1: fixes P :: "'b process" shows "(D P = {}) = CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>D P"