hcju/isabelleps huggingface dataset:
StrongF_def: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b" shows "SF(F)_v \<equiv> \<box>\<diamond>Enabled \<langle>F\<rangle>_v \<longrightarrow> \<box>\<diamond>\<langle>F\<rangle>_v"
l_set_tag_name_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog 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" shows "l_set_tag_name_get_shadow_root type_wf set_tag_name set_tag_name_locs get_shadow_root get_shadow_root_locs \<equiv> l_set_tag_name type_wf set_tag_name set_tag_name_locs \<and> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_set_tag_name_get_shadow_root_axioms set_tag_name_locs get_shadow_root_locs"
content_cball_pos: fixes r :: "real"   and c :: "'a" assumes "0 < r" shows "0 < content (cball c r)"
simps_1: fixes xs :: "'d list" shows "shd (cycle xs) = hd xs"
assms_1: shows "constructive_security_aux_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common adv"
notin_fset: fixes x :: "'m"   and S :: "'m fset" shows "(x |\<notin>| S) = (x \<notin> fset S)"
pathfinish_in_path_image: fixes g :: "real \<Rightarrow> 'a" shows "pathfinish g \<in> path_image g"
i_Exec_Stream_Acc_Output_nth: fixes k :: "nat"   and output_fun :: "'b \<Rightarrow> 'a message_af"   and trans_fun :: "'c message_af \<Rightarrow> 'b \<Rightarrow> 'b"   and input :: "nat \<Rightarrow> 'c message_af"   and c :: "'b"   and n :: "nat" assumes "0 < k" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c n = last_message (map output_fun (f_Exec_Comp_Stream trans_fun (input n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (input \<Down> n \<odot> k) c)))"
regularAmicPair_def: fixes n :: "nat"   and m :: "nat" shows "regularAmicPair n m = (n Amic m \<and> (\<exists>M N g. g = gcd m n \<and> m = M * g \<and> n = N * g \<and> squarefree M \<and> squarefree N \<and> gcd g M = 1 \<and> gcd g N = 1))"
card_2_E: fixes A :: "'c set" assumes "card A = 2" obtains x :: "'c" and y :: "'c" where "x \<noteq> y" and "A = {x, y}"
sim: shows "\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q')"
assms: fixes p :: "'a poly" assumes "1 < degree p"   and "monic p"   and "coeff p (degree p - 1) = (0::'a)"   and "coeff p 0 \<noteq> (0::'a)" shows "\<not> irreducible p"
ap_spec_snd_subset: fixes ap :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and gs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ps :: "((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and data :: "nat \<times> 'g" assumes "ap_spec ap" shows "snd ` set (ap gs bs ps hs data) \<subseteq> snd ` set ps \<union> set gs \<union> set bs \<union> set hs"
additive_def: fixes M :: "'b set set"   and \<mu> :: "'b set \<Rightarrow> 'c" shows "Measure_Space.additive M \<mu> = (\<forall>x\<in>M. \<forall>y\<in>M. x \<inter> y = {} \<longrightarrow> \<mu> (x \<union> y) = \<mu> x + \<mu> y)"
height_node: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and r :: "('b \<times> nat) tree" shows "height (node l a r) = max (height l) (height r) + 1"
\<iota>_def: shows "\<F>C.\<iota>\<^sub>F\<^sub>M\<^sub>C \<equiv> \<l>[\<I>]"
Ge_def: fixes v :: "'a aexp"   and va :: "'a aexp" shows "Ge v va \<equiv> gNot (Lt v va)"
Lower_def: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set" shows "Lower L A = {l. \<forall>x. x \<in> A \<inter> carrier L \<longrightarrow> le L l x} \<inter> carrier L"
Conj_I: fixes H :: "fm set"   and A :: "fm"   and B :: "fm" assumes "H \<turnstile> A"   and "H \<turnstile> B" shows "H \<turnstile> A AND B"
in_basis_2: fixes v :: "'b" assumes "g v \<noteq> (0::'a)" shows "v \<in> basis"
naturally_isomorphic_reflexive: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd" assumes "functor A B F" shows "naturally_isomorphic A B F F"
intros: fixes \<sigma> :: "'b \<Rightarrow> bool list option" assumes "finite (dom \<sigma>)"   and "length ` ran \<sigma> \<subseteq> {len}" shows "invariant \<sigma>"
DEFL_eq_writer: shows "DEFL('d\<cdot>'e writer) = DEFL('e\<^sub>\<bottom> \<otimes> 'd\<^sub>\<bottom>)"
assms: shows "trace_core_eq core1 core2 E IA IU p q"
monoD: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "class.order less_eq less"   and "order.mono less_eq f"   and "less_eq x y" shows "f x \<le> f y"
Diff_disjoint: fixes A :: "'a set"   and B :: "'a set" shows "A \<inter> (B - A) = {}"
NA_Crypt_imp_Server_msg: fixes A :: "agent"   and evs :: "event list"   and B :: "agent"   and NA :: "msg"   and K :: "nat" assumes "A \<notin> bad"   and "evs \<in> otway"   and "Says A B \<lbrace>NA, Agent A, Agent B, Crypt (shrK A) \<lbrace>NA, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"   and "Crypt (shrK A) \<lbrace>NA, Key K\<rbrace> \<in> parts (knows Spy evs)" shows "\<exists>NB. Says Server B \<lbrace>NA, Crypt (shrK A) \<lbrace>NA, Key K\<rbrace>, Crypt (shrK B) \<lbrace>NB, Key K\<rbrace>\<rbrace> \<in> set evs"
last_take2: fixes n :: "nat"   and xs :: "'b list" assumes "0 < n"   and "n \<le> length xs" shows "last (xs \<down> n) = xs ! (n - Suc 0)"
IH: fixes s1 :: "('a, 'b) fmap"   and s2 :: "('a, 'b) fmap" assumes "s1 \<subseteq>\<^sub>f s2"   and "sat_precond_as s1 as" shows "exec_plan s1 as \<subseteq>\<^sub>f exec_plan s2 as"
qPsubst_preserves_alpha1: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm option"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm" assumes "qGoodEnv rho"   and "qGood X \<or> qGood Y"   and "X #= Y" shows "X #[[rho]] #= Y #[[rho]]"
l2_implements_l1: shows "implements med12s skl1 l2"
add_to_counter0_1: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and r1 :: "'b state \<times> (v, v) result" assumes "evaluate_match ck env s v pes err_v r1" shows "\<forall>s' r' extra. r1 = (s', r') \<and> r' \<noteq> Rerr (Rabort Rtimeout_error) \<and> ck = True \<longrightarrow> evaluate_match True env (update_clock (\<lambda>_. clock s + extra) s) v pes err_v (update_clock (\<lambda>_. clock s' + extra) s', r')"
neqE: fixes x :: "'t"   and y :: "'t" assumes "x \<noteq> y" obtains "lt_of_nat_term_order cmp_term x y"   | "lt_of_nat_term_order cmp_term y x"
cCons_i_def: fixes ops :: "'b arith_ops_record"   and x :: "'b"   and xs :: "'b list" shows "cCons_i ops x xs = (if xs = [] \<and> x = arith_ops_record.zero ops then [] else x # xs)"
disc_22: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "\<not> isRecvMarker (Send x21 x22 x23 x24 x25 x26)"
rel_conversep: fixes R :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "rel_pmf R\<inverse>\<inverse> = (rel_pmf R)\<inverse>\<inverse>"
pcr_cr_eq: shows "stream_from_llist_setup.pcr_stream (=) = stream_from_llist_setup.cr_stream"
g1_def: shows "g1 \<equiv> \<lambda>x. (- ln x) powr y * (x - ln x - 1) / (1 - x)"
is_element_ptr_implies_kind: fixes ptr :: "(_) node_ptr" assumes "is_element_ptr\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr" shows "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr"
maximalI_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'b"   and f :: "'b \<Rightarrow> 'a"   and S :: "'b set"   and y :: "'b" assumes "class.order less_eq less"   and "x \<in> order.maximal less_eq f S"   and "less_eq (f x) (f y)"   and "y \<in> S" shows "y \<in> order.maximal less_eq f S"
left: fixes M :: "trm"   and a :: "coname"   and N :: "trm"   and x :: "name" assumes "\<not> fic M a"   and "a \<sharp> N"   and "x \<sharp> M" shows "Cut <a>.M x.N \<longrightarrow>\<^sub>c M{a:=x.N}"
cases: fixes x :: "'b nonzero"   and a :: "'b nonzero" assumes "proj_rel x a" obtains c :: "real" where "a = c *\<^sub>R x" and "c \<noteq> 0"
s_App2: fixes v\<^sub>1 :: "Syntax.term"   and \<pi> :: "Syntax.term list"   and e\<^sub>2 :: "Syntax.term"   and m :: "mode"   and \<pi>' :: "Syntax.term list"   and e\<^sub>2' :: "Syntax.term" assumes "value v\<^sub>1"   and "\<lless>\<pi>, e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', e\<^sub>2'\<ggreater>" shows "\<lless>\<pi>, App v\<^sub>1 e\<^sub>2\<ggreater> m\<rightarrow> \<lless>\<pi>', App v\<^sub>1 e\<^sub>2'\<ggreater>"
tendsto_sandwich: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b"   and net :: "'c filter"   and h :: "'c \<Rightarrow> 'b"   and c :: "'b" assumes "class.order_topology less_eq less open"   and "\<forall>\<^sub>F n in net. less_eq (f n) (g n)"   and "\<forall>\<^sub>F n in net. less_eq (g n) (h n)"   and "filterlim f (topological_space.nhds open c) net"   and "filterlim h (topological_space.nhds open c) net" shows "filterlim g (topological_space.nhds open c) net"
ordLess_imp_ordLeq: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set" assumes "r <o r'" shows "r \<le>o r'"
simps_8: fixes x9 :: "Term.var"   and y9 :: "Term.var" shows "(Acc x9 = Acc y9) = (x9 = y9)"
app_inject: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c"   and u\<^sub>3 :: "'c"   and u\<^sub>4 :: "'c" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree"   and "app u\<^sub>1 u\<^sub>2 = app u\<^sub>3 u\<^sub>4" shows "u\<^sub>1 = u\<^sub>3 \<and> u\<^sub>2 = u\<^sub>4"
pf01: fixes t :: "'a" assumes "t \<in> S - U" shows "pf t ` S \<subseteq> {0..1}"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Sym x2 = Sym y2) = (x2 = y2)"
XY: shows "simple_function M (\<lambda>x. (X x, Y x))"
ChiRel_HAStates: fixes a :: "'d"   and b :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "(a, b) \<in> ChiRel A" shows "a \<in> HAStates A"
countable_Collect_finite_subset: fixes T :: "'c set" assumes "countable T" shows "countable {A. finite A \<and> A \<subseteq> T}"
rel_interior_convex_scaleR: fixes S :: "'a set"   and c :: "real" assumes "convex S" shows "(*\<^sub>R) c ` rel_interior S = rel_interior ((*\<^sub>R) c ` S)"
imbal_split_min: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "Root_Balanced_Tree.split_min t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "real (imbal t') - real (imbal t) \<le> 1"
sum_character: fixes x :: "i"   and y :: "i" shows "\<forall>v. O v (x \<oplus> y) = (O v x \<or> O v y)"
finish_simps_3: fixes u :: "'v"   and s :: "'v tarjan_state" shows "finished (finish u s) = finished s(u \<mapsto> counter s)"
prems: shows "crdt_ops (xs @ [(oid, oper)] @ ys @ [y]) deps"
distinct_94: fixes x8 :: "shift_type"   and x6 :: "arith_type" shows "shift_type x8 \<noteq> arith_type x6"
m1_inv1_ifresh_def: shows "m1_inv1_ifresh \<equiv> {s. \<forall>A A' B B' Ra Ra' Kab nl nl'. runs s Ra = Some (Init, [A, B], aKey Kab # nl) \<longrightarrow> runs s Ra' = Some (Init, [A', B'], aKey Kab # nl') \<longrightarrow> A \<notin> bad \<longrightarrow> B \<notin> bad \<longrightarrow> Kab \<notin> Domain (m1_state.leak s) \<longrightarrow> Ra = Ra'}"
cod_inv: fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.cod (VV.inv f) = VV.dom f"
HH2: shows "H2 = (A2, B2, C2, D2)"
edges: fixes u :: "'v"   and v :: "'v" assumes "\<pi> u = Some v" shows "(u, v) \<in> edges rg"
hoare1_sound: fixes P :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "\<turnstile>\<^sub>1 {P} c { e \<Down> Q}" shows "\<Turnstile>\<^sub>1 {P} c { e \<Down>Q}"
NFAND_def: shows "NFAND = nFAND \<circ> sorted_list_of_set"
assms_1: shows "\<parallel>c\<parallel>\<^bsub>t i\<^esub>"
inverse_eq_divide: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse a = divide one a"
ENR_def: fixes S :: "'b set" shows "ENR S \<equiv> \<exists>U. open U \<and> S retract_of U"
leftside: shows "V // per_union R x y = insert (per_union R x y `` {y}) (\<Union>x\<in>V - R `` {x} - R `` {y}. {R `` {x}})"
calc_intersect_x_axis_positive_lemma: fixes A :: "real"   and B :: "real" assumes "A\<^sup>2 < B\<^sup>2"   and "A \<noteq> 0" shows "(0 < (- B + sgn B * sqrt (B\<^sup>2 - A\<^sup>2)) / A) = (1 < - B / A)"
simps_2: fixes G :: "('a, 'b) graph"   and v :: "'a"   and v1 :: "'a"   and w :: "'b"   and v2 :: "'a"   and p :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_graph G" shows "valid_graph.is_path G v ((v1, w, v2) # p) v' = (v = v1 \<and> (v1, w, v2) \<in> edges G \<and> valid_graph.is_path G v2 p v')"
arcs_ends_def: fixes G :: "('b, 'c) pre_digraph" shows "arcs_ends G \<equiv> arc_to_ends G ` pre_digraph.arcs G"
tl_Nil: fixes xs :: "'b list" shows "(tl xs = \<epsilon>) = (xs = \<epsilon> \<or> (\<exists>x. xs = [x]))"
size_19: fixes x81 :: "'b ltln"   and x82 :: "'b ltln" shows "size (x81 U\<^sub>n x82) = size x81 + size x82 + Suc 0"
trace_eqI: fixes \<sigma> :: "'b trace"   and \<sigma>' :: "'b trace" assumes "\<And>i. \<Gamma> \<sigma> i = \<Gamma> \<sigma>' i"   and "\<And>i. \<tau> \<sigma> i = \<tau> \<sigma>' i" shows "\<sigma> = \<sigma>'"
cmp_img_def: fixes f :: "'d \<Rightarrow> 'e"   and cmp :: "'e \<Rightarrow> 'e \<Rightarrow> 'c"   and a :: "'d"   and b :: "'d" shows "cmp_img f cmp a b \<equiv> cmp (f a) (f b)"
l_cancel_one: fixes G :: "('e, 'f) monoid_scheme"   and x :: "'e"   and a :: "'e" assumes "Group.group G"   and "x \<in> carrier G"   and "a \<in> carrier G" shows "(x \<otimes>\<^bsub>G\<^esub> a = x) = (a = \<one>\<^bsub>G\<^esub>)"
diagonalize_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.arr g" shows "\<F>C.D (f \<otimes> g) = \<F>C.D (\<F>C.D f \<otimes> \<F>C.D g)"
WhileFalse: fixes s :: "'a"   and b :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> Normal s"
insert_remdups: fixes x :: "'d"   and xs :: "'d list" shows "List.insert x (remdups xs) = remdups (List.insert x xs)"
noq: fixes x :: "real" assumes "x \<in> {0..1}" shows "norm (g x - q x) < e / 4"
prod_set_simps_1: fixes x :: "'b"   and y :: "'c" shows "Basic_BNFs.fsts (x, y) = {x}"
assms_2: assumes "\<And>f. f \<in># F \<Longrightarrow> irreducible f"   and "F \<noteq> {#}"   and "\<Prod>\<^sub># F = p" shows "thesis"
closest_supchamberD_2: fixes F :: "'b set"   and D :: "'b set" assumes "F \<in> Y"   and "codomain.chamber D" shows "order.greater_eq (codomain.closest_supchamber F D) F"
that: fixes \<Gamma>' :: "(Vars.var \<times> exp) list"   and v' :: "exp" assumes "([], ccTransform 0 e, []) \<Rightarrow>\<^sup>* (\<Gamma>', v', [])"   and "isVal v'"   and "card (domA \<Gamma>') \<le> card (domA \<Gamma>)" shows "thesis"
univ_poly_consistent: fixes K :: "'c set" assumes "subring K S" shows "univ_poly (S\<lparr>carrier := K\<rparr>) = univ_poly S"
index_of_const_main: fixes n :: "nat" shows "nat_to_pr (index_of_const n) = (\<lambda>x. n)"
int_cases: shows "\<forall>i. i = 0 \<or> i < 0 \<or> 0 < i"
sim_f_tc: fixes M :: "('e, 'f, 'g) KripkeStructure"   and M' :: "('e, 'f, 'h) KripkeStructure"   and f :: "'g \<Rightarrow> 'h"   and u :: "'g"   and v :: "'g"   and as :: "'e set" assumes "sim M M' f"   and "(u, v) \<in> (\<Union> (relations M ` as))\<^sup>+" shows "(f u, f v) \<in> (\<Union> (relations M' ` as))\<^sup>+"
cases: fixes P :: "'a prog"   and a1 :: "String.literal"   and a2 :: "String.literal" assumes "P \<turnstile> a1 \<prec>\<^sup>1 a2" obtains C :: "String.literal" and D :: "String.literal" and rest :: "fdecl list \<times> 'a option mdecl list" where "a1 = C" and "a2 = D" and "class P C = \<lfloor>(D, rest)\<rfloor>" and "C \<noteq> Object"
vfsequence_vdomain_vlrestriction: fixes xs :: "V"   and k :: "V" assumes "vfsequence xs"   and "k \<in>\<^sub>\<circ> vcard xs" shows "\<D>\<^sub>\<circ> (xs \<restriction>\<^sup>l\<^sub>\<circ> k) = k"
R2P_def: fixes r :: "('c \<times> 'd) set" shows "R2P r \<equiv> \<lambda>x y. (x, y) \<in> r"
card_eq_dim: fixes B :: "('a, 'b) vec set"   and V :: "('a, 'b) vec set" assumes "B \<subseteq> V"   and "card B = vec.dim V"   and "finite B" shows "vec.independent B = (V \<subseteq> vec.span B)"
slg_rel_def: fixes R :: "('e \<times> 'f) set" shows "\<langle>R\<rangle>slg_rel \<equiv> {(succs, G). \<forall>v. (succs v, G `` {v}) \<in> \<langle>R\<rangle>list_set_rel}"
heap_WHILET_def: fixes b :: "'b \<Rightarrow> bool Heap"   and f :: "'b \<Rightarrow> 'b Heap"   and s :: "'b" shows "heap_WHILET b f s \<equiv> heap.fixp_fun (\<lambda>D s. b s \<bind> (\<lambda>bv. if bv then f s \<bind> D else return s)) s"
LIMSEQ_indicator_incseq: fixes A :: "nat \<Rightarrow> 'c set"   and x :: "'c" assumes "incseq A" shows "(\<lambda>i. indicator (A i) x) \<longlonglongrightarrow> indicator (\<Union> (range A)) x"
pwd_spy_id_sesk_rel: fixes n :: "nat" assumes "n \<in> bad_prikey \<inter> bad_id_shakey" shows "s\<^sub>0 \<Turnstile> fst (pwd_spy_id_sesk n)"
prime_gt_1_nat: fixes p :: "nat" assumes "prime p" shows "1 < p"
idx_find: fixes M :: "('a, 'b) message set"   and a :: "'a extended \<Rightarrow> 'a sort_key extended"   and s :: "('a, 'b option) insert_message list"   and x :: "nat"   and i :: "'a extended" assumes "integrate_insert_commute M a s"   and "x < length (ext_ids s)"   and "ext_ids s ! x = i" shows "idx s i = Inr x"
simps: fixes p :: "'a"   and x :: "'a"   and acc :: "nat" shows "divide_out_aux p (x, acc) = (if x = (0::'a) \<or> is_unit p \<or> \<not> p dvd x then (x, acc) else divide_out_aux p (x div p, acc + 1))"
step_preserves_le: fixes S :: "(char list \<Rightarrow> int) set"   and S' :: "'av Abs_State.st option"   and cs :: "(char list \<Rightarrow> int) set acom"   and ca :: "'av Abs_State.st option acom" assumes "S \<subseteq> local.\<gamma>\<^sub>o S'"   and "cs \<le> local.\<gamma>\<^sub>c ca" shows "step S cs \<le> local.\<gamma>\<^sub>c (local.step' S' ca)"
DAAux: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" shows "x \<in> dom (Cp DenyAll)"
LENGTH_state_set: fixes e :: "'a list"   and X :: "'a list" assumes "e \<in> state_set X" shows "length e \<le> length X"
bisimParComm: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> Q \<sim> Q \<parallel> P"
affected_times: fixes f :: "'b perm"   and g :: "'b perm" shows "affected (f * g) \<subseteq> affected f \<union> affected g"
FI_3: shows "Q_\<alpha> Qi' = Qinter (Q_\<alpha> Qi) (M_lookup \<pi>i) u"
trace_callee_closure_step: fixes callee1 :: "'d \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'd) spmf"   and callee2 :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g) spmf"   and A :: "'e set"   and p :: "'d spmf"   and q :: "'g spmf"   and p' :: "'d spmf"   and q' :: "'g spmf"   and a :: "'e" assumes "trace_callee_eq callee1 callee2 A p q"   and "trace_callee_closure callee1 callee2 A p q p' q'"   and "a \<in> A" shows "p' \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q' \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"
simps_2: fixes x :: "'b" shows "proper (Atm x) = True"
Int_empty_left: fixes B :: "'e set" shows "{} \<inter> B = {}"
mapping_eqI: fixes m :: "('a, 'b) mapping"   and m' :: "('a, 'b) mapping" assumes "\<And>x. Mapping.lookup m x = Mapping.lookup m' x" shows "m = m'"
i_Exec_Stream_Acc_Output_Init_eq_i_Exec_Stream_Acc_Output_Cons: fixes k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and input :: "nat \<Rightarrow> 'f message_af"   and c :: "'e" shows "i_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun input c = [output_fun c] \<frown> i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c"
imageE: fixes b :: "'b"   and f :: "'d \<Rightarrow> 'b"   and A :: "'d set" assumes "b \<in> f ` A" obtains x :: "'d" where "b = f x" and "x \<in> A"
lowest_tops_in: fixes k :: "'a ifex list"   and l :: "'a" assumes "lowest_tops k = Some l" shows "l \<in> set (concat (map ifex_vars k))"
assms: shows "has_bochner_integral (lebesgue_on {a..b}) f i"
expand_set: fixes g :: "'r \<Rightarrow> 'q" shows "Sum_any g = sum g {a. g a \<noteq> (0::'q)}"
lex_pair_def: shows "Abacus.lex_pair \<equiv> less_than <*lex*> less_than"
simps_13: fixes e :: "expr"   and e\<^sub>1 :: "expr"   and e\<^sub>2 :: "expr"   and A :: "char list set option" shows "\<D> (if (e) e\<^sub>1 else e\<^sub>2) A = (\<D> e A \<and> \<D> e\<^sub>1 (A \<squnion> \<A> e) \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e))"
B: fixes n :: "nat" shows "(\<Sum>k\<le>n. Sigma_Algebra.measure lebesgue (S k)) \<le> Sigma_Algebra.measure lebesgue (cbox a b)"
simps_9: fixes x31 :: "int poly"   and x32 :: "nat"   and x21 :: "rat"   and x22 :: "rat" shows "Real_Alg_Info x31 x32 \<noteq> Sqrt_Info x21 x22"
bits_one_mod_two_eq_one: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool" assumes "class.semiring_bits plus minus zero times one divide modulo bit" shows "modulo one (numeral.numeral one plus (num.Bit0 num.One)) = one"
that: fixes c :: "complex" assumes "\<And>z. f z = c" shows "thesis"
lemma_isUb_hypreal_of_real: fixes P :: "real star set"   and Y :: "real star" assumes "isUb \<real> P Y" shows "\<exists>Yo. isUb \<real> P (hypreal_of_real Yo)"
star_involutive: fixes x :: "'a" shows "x\<^sup>\<star> = x\<^sup>\<star>\<^sup>\<star>"
False: shows "\<not> (deg R p = 0 \<or> deg R q = 0)"
empty_append_eq_id: shows "(@) [] = (\<lambda>x. x)"
gt_ex: fixes x :: "'a" shows "\<exists>y. x < y"