hcju/isabelleps huggingface dataset:
nat_filtration_borel_measurable_characterization: fixes p :: "real"   and M :: "bool stream measure"   and f :: "bool stream \<Rightarrow> 'b"   and n :: "nat" assumes "infinite_coin_toss_space p M"   and "f \<in> borel_measurable M" shows "(f \<in> borel_measurable (infinite_coin_toss_space.nat_filtration M n)) = (f \<circ> pseudo_proj_True n = f)"
less_TC_def: fixes x :: "V"   and y :: "V" shows "x \<sqsubset> y \<equiv> x \<in> elts (TC y)"
sel_2: fixes x1 :: "bool"   and x2 :: "'a \<Rightarrow> 'a language" shows "\<dd> (Lang x1 x2) = x2"
termination_aux: fixes \<Gamma> :: "(name \<times> ty) list"   and e :: "trm"   and T :: "ty"   and \<theta> :: "(name \<times> trm) list" assumes "\<Gamma> \<turnstile> e : T"   and "\<theta> Vcloses \<Gamma>" shows "\<exists>v. \<theta><e> \<Down> v \<and> v \<in> V T"
that: fixes a :: "'a"   and b :: "real" assumes "a \<noteq> (0::'a)"   and "S \<subseteq> {x. a \<bullet> x \<le> b}"   and "C = S \<inter> {x. a \<bullet> x = b}" shows "thesis"
adm_wo_def: fixes r :: "('d \<times> 'd) set"   and H :: "('d \<Rightarrow> 'e) \<Rightarrow> 'd \<Rightarrow> 'e" assumes "wo_rel r" shows "wo_rel.adm_wo r H \<equiv> \<forall>f g x. (\<forall>y\<in>underS r x. f y = g y) \<longrightarrow> H f x = H g x"
preserves_cod: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.cod \<a>\<^sub>B[fst f, fst (snd f), snd (snd f)] = B.HoVH (B.VVV.cod f)"
\<tau>_mono: fixes i :: "nat"   and j :: "nat"   and s :: "'a trace" assumes "i \<le> j" shows "\<tau> s i \<le> \<tau> s j"
simps_1: 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 abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.list_comb app f [] = f"
the_def: fixes A :: "'a Predicate.pred" shows "Predicate.the A = (THE x. pred.eval A x)"
nn_integral_0_iff: fixes u :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "u \<in> borel_measurable M" shows "(integral\<^sup>N M u = 0) = (emeasure M {x \<in> space M. u x \<noteq> 0} = 0)"
flow_fixed_point: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "flow0 x0 t = x0 + ivl_integral 0 t (\<lambda>t. f (flow0 x0 t))"
that: fixes a :: "'a"   and as :: "'a list" assumes "ys = xs @ a # as" shows "thesis"
insert_ops_appendD: fixes xs :: "('a \<times> 'a option) list"   and ys :: "('a \<times> 'a option) list" assumes "insert_ops (xs @ ys)" shows "insert_ops xs"
ms_of_greek_def: fixes as :: "(accent \<times> 'b) list" shows "ms_of_greek as = mset (map (\<lambda>x. case x of (xs, y, zs) \<Rightarrow> adj_msog xs zs (y, [])) (list_splits as))"
tranclp_DomainP: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "R\<^sup>+\<^sup>+ a b" shows "Domainp R a"
rel_gpv''_conversep: fixes A :: "'g \<Rightarrow> 'd \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'e \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_gpv'' A\<inverse>\<inverse> C\<inverse>\<inverse> R\<inverse>\<inverse> = (rel_gpv'' A C R)\<inverse>\<inverse>"
Subobjs_notSubobjsR: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and Cs :: "char list list" assumes "Subobjs P C Cs"   and "\<not> Subobjs\<^sub>R P C Cs" shows "\<exists>C' D. P \<turnstile> C \<preceq>\<^sup>* C' \<and> P \<turnstile> C' \<prec>\<^sub>S D \<and> Subobjs\<^sub>R P D Cs"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains l :: "name" and s :: "tm" and sl :: "name" and m :: "name" and n :: "name" and sm :: "name" and sn :: "name" and k :: "tm" and t :: "tm" where "atom l \<sharp> (s, sl, m, n, sm, sn)" and "atom sl \<sharp> (s, m, n, sm, sn)" and "atom m \<sharp> (s, n, sm, sn)" and "atom n \<sharp> (s, k, sm, sn)" and "atom sm \<sharp> (s, sn)" and "atom sn \<sharp> s" and "x = (s, k, t)"
cgenideal_ideal: fixes a :: "'a list" assumes "a \<in> carrier (A [X])" shows "ideal (PIdl\<^bsub>A [X]\<^esub> a) (A [X])"
goedel_second: assumes "consistent" shows "\<not> prv (neg (PP \<langle>fls\<rangle>))"
Gamma_real_pos_exp: fixes x :: "real" assumes "0 < x" shows "Gamma x = exp (ln_Gamma x)"
mem_adjoin: fixes H :: "'a set"   and x :: "'a"   and a :: "'a" assumes "subgroup H (G\<lparr>carrier := adjoin G H a\<rparr>)"   and "x \<in> H"   and "a \<in> carrier (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "x \<in> adjoin (G\<lparr>carrier := adjoin G H a\<rparr>) H a"
trapped_forward_def: fixes f :: "'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "'b"   and K :: "'b set" assumes "auto_ll_on_open f X" shows "auto_ll_on_open.trapped_forward f X x K = (auto_ll_on_open.flow0 f X x ` (auto_ll_on_open.existence_ivl0 f X x \<inter> {0..}) \<subseteq> K)"
space_lebesgue_on: fixes S :: "'a set" shows "space (lebesgue_on S) = S"
payload_inductive_cases_1: fixes X :: "tag"   and P :: "bool" assumes "Tag X \<in> payload" shows "P"
end_point_zero_point: fixes x :: "'a" assumes "path x" shows "end_points x = 0 \<or> is_point (end_points x)"
generalized_fw_joinD: fixes f1 :: "('b simple_match \<times> 'c) list"   and f2 :: "('b simple_match \<times> 'd) list"   and p :: "('b, 'e) simple_packet_scheme"   and u :: "'b simple_match"   and d1 :: "'c"   and d2 :: "'d" assumes "generalized_sfw (generalized_fw_join f1 f2) p = Some (u, d1, d2)" shows "\<exists>r1 r2. generalized_sfw f1 p = Some (r1, d1) \<and> generalized_sfw f2 p = Some (r2, d2) \<and> Some u = simple_match_and r1 r2"
bounded_rel_frontier: fixes S :: "'a set" assumes "bounded S" shows "bounded (rel_frontier S)"
prems_1: fixes ys' :: "'b list" assumes "ys' \<in> listset (x # xs)"   and "ys = map f ys'" shows "thesis"
typ_of_betapply: fixes Ts :: "typ list"   and uty :: "typ"   and t :: "Core.term"   and tty :: "typ"   and u :: "Core.term" assumes "typ_of1 Ts (Abs uty t) = Some (uty \<rightarrow> tty)"   and "typ_of1 Ts u = Some uty" shows "typ_of1 Ts (Abs uty t \<bullet> u) = Some tty"
is_monoid: fixes G :: "('a, 'b) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
distinct_7: fixes x21 :: "nat"   and x22 :: "v list"   and x3 :: "v list" shows "RSBreak x21 x22 \<noteq> RSReturn x3"
load_sub1_def: fixes instr :: "sparc_operation \<times> inst_operand list"   and rd :: "word5"   and s_val :: "word1" shows "load_sub1 instr rd s_val \<equiv> do curr_win \<leftarrow> get_curr_win (); address \<leftarrow> gets (get_addr (snd instr)); asi \<leftarrow> gets (\<lambda>s. ld_asi instr s_val); if (fst instr = load_store_type LDD \<or> fst instr = load_store_type LDDA) \<and> ucast address \<noteq> 0 \<or> fst instr \<in> {load_store_type LD, load_store_type LDA} \<and> ucast address \<noteq> 0 \<or> (fst instr = load_store_type LDUH \<or> fst instr = load_store_type LDUHA \<or> fst instr = load_store_type LDSH \<or> fst instr = load_store_type LDSHA) \<and> ucast address \<noteq> 0 then do raise_trap mem_address_not_aligned; return () od else load_sub3 instr curr_win rd asi address od"
ipd_1: shows "\<pi>' m' = ipd (\<pi> k)"
Int: fixes a :: "'a set"   and b :: "'a set" assumes "a \<in> generated_ring"   and "b \<in> generated_ring" shows "a \<inter> b \<in> generated_ring"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and 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" assumes "l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs"   and "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs" shows "l_set_disconnected_nodes_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs known_ptr get_child_nodes get_child_nodes_locs"
borel_Times: fixes A :: "'a set"   and B :: "'b set" assumes "A \<in> sets borel"   and "B \<in> sets borel" shows "A \<times> B \<in> sets borel"
hyps: fixes a :: "'c"   and b :: "'b" assumes "c < k"   and "to_nat a = to_nat b"   and "c \<le> to_nat a"   and "k < min (nrows A) (ncols A)" shows "diagonal_to_Smith_aux A [0..<k] bezout $ from_nat c $ from_nat c dvd diagonal_to_Smith_aux A [0..<k] bezout $ a $ b"
COMB_has_finite_config_set: fixes init :: "nat list"   and qs :: "nat list"   and h :: "nat list" assumes "distinct init"   and "set qs \<subseteq> set init" shows "finite (set_pmf (Partial_Cost_Model.config'_rand (COMB h) (fst (COMB h) init \<bind> (\<lambda>is. return_pmf (init, is))) qs))"
advantage_def: fixes \<A> :: "('a, 'b) distinguisher"   and res1 :: "('a, 'b) resource"   and res2 :: "('a, 'b) resource" shows "advantage \<A> res1 res2 = \<bar>spmf (connect \<A> res1) True - spmf (connect \<A> res2) True\<bar>"
setparts_1: shows "set (take i as') \<subseteq> set as'"
indRelTEQ_to_TRel_2: fixes P :: "('procS, 'procT) Proc"   and Q :: "('procS, 'procT) Proc"   and TRel :: "('procT \<times> 'procT) set" assumes "P \<sim>\<lbrakk>\<cdot>\<rbrakk>T<TRel> Q" shows "\<forall>SP TQ. SP \<in>S P \<and> TQ \<in>T Q \<longrightarrow> (\<lbrakk>SP\<rbrakk>, TQ) \<in> (TRel \<union> _Collect (T1, T2) (\<exists>S. T1 = \<lbrakk>S\<rbrakk> \<and> T2 = \<lbrakk>S\<rbrakk>))\<^sup>+"
add: fixes x :: "real"   and y :: "real"   and xa :: "'b" shows "(x + y) *\<^sub>R xa = x *\<^sub>R xa + y *\<^sub>R xa"
simps: fixes t :: "nat"   and a1 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)"   and a2 :: "nat \<times> nat \<times> (nat \<Rightarrow> thread)" shows "blstep t a1 a2 = (\<exists>c n ts c' n' s'. a1 = (c, n, ts) \<and> a2 = (c', n', ts(t := s')) \<and> t < N \<and> bstep_sng (c, n, ts t) (c', n', s'))"
LatticeDisjunctionElim1: fixes F :: "'a \<Rightarrow> bool"   and G :: "'a \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool" shows "\<turnstile> (F \<or> G \<leadsto> H) \<longrightarrow> (F \<leadsto> H)"
vimage2p_rel_fun: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'd"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "(vimage2p f g R ===> R) f g"
sympI: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>a b. r a b \<Longrightarrow> r b a" shows "symp r"
synth_simps_2: fixes K :: "nat"   and H :: "msg set" shows "(Key K \<in> synth H) = (Key K \<in> H)"
execute_instruction_def: fixes uu :: "unit" shows "execute_instruction uu \<equiv> do traps \<leftarrow> gets get_trap_set; if traps = {} then do exe_mode \<leftarrow> gets exe_mode_val; if exe_mode = True then do modify delayed_pool_write; fetch_result \<leftarrow> gets fetch_instruction; case fetch_result of Inl e1 \<Rightarrow> do raise_trap instruction_access_exception; return () od | Inr v1 \<Rightarrow> do dec \<leftarrow> gets (\<lambda>s. decode_instruction v1); case dec of Inl e2 \<Rightarrow> fail () | Inr v2 \<Rightarrow> do instr \<leftarrow> gets (\<lambda>s. v2); annul \<leftarrow> gets annul_val; if annul = False then do dispatch_instruction instr; execute_instr_sub1 instr; return () od else do set_annul False; npc_val \<leftarrow> gets (cpu_reg_val nPC); write_cpu npc_val PC; write_cpu (npc_val + 4) nPC; return () od od od od else return () od else do execute_trap (); return () od od"
iLast_iEx_iff_singleton: fixes t0 :: "nat"   and I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "iLast t0 I P = (\<diamond> t {iprev t0 I}. P t)"
poly_subst_def: fixes f :: "'d \<Rightarrow> ('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_subst f p = (\<Sum>t\<in>keys p. MPoly_Type_Class.punit.monom_mult (lookup p t) 0 (subst_pp f t))"
WHILEI_le_WHILEIT: fixes I :: "'c \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" shows "WHILE\<^bsup>I\<^esup> b f s \<le> WHILE\<^sub>T\<^bsup>I\<^esup> b f s"
foldr_Cons: fixes f :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list" shows "foldr f (x # xs) = f x \<circ> foldr f xs"
Fract_of_int_eq: fixes k :: "int" shows "Rat.Fract k 1 = rat_of_int k"
span_in_category_axioms: shows "span_in_category (\<cdot>) (Dom (trg f \<star> f))"
run_one_step_callcl_result: fixes d :: "nat"   and i :: "nat"   and s :: "s"   and vs :: "v list"   and ves :: "v list"   and cl :: "cl"   and s' :: "s"   and vs' :: "v list"   and res :: "res_step" assumes "run_one_step d i (s, vs, ves, Callcl cl) = (s', vs', res)" shows "(\<exists>r. res = RSNormal r) \<or> (\<exists>e. res = RSCrash e)"
fps_of_ratfps_euclidean_size: fixes x :: "'a ratfps" shows "euclidean_size x = euclidean_size (fps_of_ratfps x)"
galois_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b" shows "ord.galois less_eq l u \<equiv> \<forall>x y. less_eq (l x) y = less_eq x (u y)"
eq_Eq_conv: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a" assumes "comparator comp" shows "(comp x y = Eq) = (x = y)"
cong_mersenne_mod2: fixes k :: "int"   and n :: "nat" shows "[mersenne_mod2 k n = k] (mod 2 ^ n - 1)"
is_span: shows "Maps.span SPN_fgh.\<nu>\<pi>.leg0 SPN_fgh.\<nu>\<pi>.leg1"
iNext_True: fixes t0 :: "nat"   and I :: "nat set" shows "\<circle> t t0 I. True"
tLength_g0_conv: fixes t :: "'b Trace" shows "(0 < tLength t) = (\<exists>s t'. t = t' \<leadsto> s \<and> tLength t = Suc (tLength t'))"
inject: fixes a :: "'b"   and b :: "'c"   and a' :: "'b"   and b' :: "'c" shows "((a, b) = (a', b')) = (a = a' \<and> b = b')"
power_distrib: fixes h :: "'a"   and e :: "nat" assumes "h \<in> carrier \<G>" shows "\<^bold>g [^] e \<otimes> h [^] e = (\<^bold>g \<otimes> h) [^] e"
disc_iff_2: fixes xs :: "'c llist"   and ys :: "'d llist" shows "(\<not> lnull (lzip xs ys)) = (\<not> lnull xs \<and> \<not> lnull ys)"
non_empty_cycle_root_move_root: fixes r :: "'a"   and x :: "'a"   and q :: "'a" assumes "non_empty_cycle_root r x"   and "point q"   and "q \<le> x\<^sup>\<star> ; r" shows "non_empty_cycle_root q x"
ex_un_null: shows "\<exists>!n. \<forall>f. H.VV.comp n f = n \<and> H.VV.comp f n = n"
bij_lens_weak: fixes x :: "'a \<Longrightarrow> 'b" assumes "bij_lens x" shows "weak_lens x"
assms_2: shows "P \<subseteq> A\<^sup>\<omega>"
countable_Plus: fixes A :: "'a set"   and B :: "'b set" assumes "countable A"   and "countable B" shows "countable (A <+> B)"
simps_2: fixes a :: "'r"   and f :: "'r \<rightarrow> 's"   and l :: "'r tree"   and r :: "'r tree" assumes "a \<noteq> \<bottom>" shows "tree_map'\<cdot>f\<cdot>(Node\<cdot>a\<cdot>l\<cdot>r) = Node\<cdot>(f\<cdot>a)\<cdot>(tree_map'\<cdot>f\<cdot>l)\<cdot>(tree_map'\<cdot>f\<cdot>r)"
st_idempotent: fixes x :: "real star" assumes "x \<in> HFinite" shows "st (st x) = st x"
distinct_1: fixes x1 :: "'b"   and x2 :: "'c" shows "In1 x1 \<noteq> In2 x2"
ensures_extend_set_imp_project_ensures: fixes F :: "'a program"   and C :: "'c set"   and G :: "'c program"   and A :: "'c set"   and B :: "'c set"   and D :: "'a set" assumes "extend h F \<in> stable C"   and "G \<in> stable C"   and "extend h F \<squnion> G \<in> A ensures B"   and "A - B = C \<inter> extend_set h D" shows "F \<squnion> project h C G \<in> project_set h C \<inter> project_set h A ensures project_set h B"
basis_im_defines_hom: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and V :: "'b set"   and smult' :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and W :: "'c set"   and vs :: "'b list"   and ws :: "'c list" assumes "VectorSpace smult V"   and "VectorSpace smult' W"   and "set vs \<subseteq> V \<and> V = R_scalar_mult.RSpan UNIV smult vs \<and> R_scalar_mult.R_lin_independent UNIV smult vs"   and "set ws \<subseteq> W"   and "length ws = length vs" shows "\<exists>!T. VectorSpaceHom smult V smult' T \<and> map T vs = ws"
siftDown_heap_is_heap: fixes l :: "'c Tree"   and r :: "'c Tree"   and t :: "'c Tree"   and v :: "'c" assumes "is_heap l"   and "is_heap r"   and "t = T v l r" shows "is_heap (siftDown t)"
surj_fun_eq: fixes f :: "'d \<Rightarrow> 'c"   and X :: "'d set"   and g1 :: "'c \<Rightarrow> 'e"   and g2 :: "'c \<Rightarrow> 'e" assumes "f ` X = UNIV"   and "\<forall>x\<in>X. (g1 \<circ> f) x = (g2 \<circ> f) x" shows "g1 = g2"
real_cond_exp_sum: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> real"   and I :: "'b set" assumes "\<And>i. integrable M (f i)" shows "AE x in M. real_cond_exp M Invariants (\<lambda>x. \<Sum>i\<in>I. f i x) x = (\<Sum>i\<in>I. real_cond_exp M Invariants (f i) x)"
AE_measure_spmf_iff: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c spmf" shows "almost_everywhere (measure_spmf p) P = (\<forall>x\<in>set_spmf p. P x)"
external_to_internal_invariant: fixes S :: "('b, 'a) spec"   and Q :: "'a set"   and P :: "'b set" assumes "oreach S \<subseteq> Q"   and "Q \<subseteq> obs S ` P"   and "observable (obs S) P" shows "reach S \<subseteq> P"
Nil_prod_def: shows "Nil_prod \<equiv> Generator (\<lambda>_. Done)"
extends_to_prebicategory_with_units: shows "prebicategory_with_units (\<cdot>) (\<star>) \<a> \<i>'"
scale_eq_0_iff: fixes a :: "rat"   and x :: "'b" shows "(a *R x = (0::'b)) = (a = 0 \<or> x = (0::'b))"
qdsnd_setsum: fixes A :: "'a set"   and f :: "'a \<Rightarrow> QDelta" assumes "finite A" shows "qdsnd (sum f A) = (\<Sum>x\<in>A. qdsnd (f x))"
next_subset_dfs: fixes S :: "nat option bdd \<times> bool list list"   and xs :: "bool list list"   and x :: "bool list" assumes "subset_invariant A S"   and "list_all (nfa_is_node A) xs"   and "x \<in> set xs" shows "subset_memb x (dfs S xs)"
countableE_infinite: fixes S :: "'c set" assumes "countable S"   and "infinite S" obtains e :: "'c \<Rightarrow> nat" where "bij_betw e S UNIV"
assms: shows "SubstTerm v \<lbrakk>quot_dbtm u\<rbrakk>e x x'"
cs_hom_meet: fixes x :: "'a"   and y :: "'a" shows "d (x \<cdot> 1\<^sub>\<pi> \<sqinter> y \<cdot> 1\<^sub>\<pi>) = d (x \<cdot> 1\<^sub>\<pi>) \<sqinter> d (y \<cdot> 1\<^sub>\<pi>)"
subst_pp_trivial_not_zero: fixes t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "t \<noteq> 0" shows "subst_pp (\<lambda>_. 0) t = 0"
less_imp_not_less: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.preorder less_eq less"   and "less x y" shows "(\<not> less y x) = True"
CCfix_nonrec_eq: fixes x :: "Vars.var"   and e :: "exp"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CCfix_nonrec x e\<cdot>i = ccBind x e\<cdot>(Aheap_nonrec x e\<cdot>i, snd i) \<squnion> fv e G\<times> (ccNeighbors x (snd i) - (if isVal e then {} else {x})) \<squnion> snd i"
bst_splay: fixes t :: "'b tree"   and x :: "'b" assumes "bst t" shows "bst (splay x t)"
CP_T_eq_CP_O: fixes k :: "'key"   and obs :: "'observation list" assumes "k \<in> keys"   and "obs \<in> OB ` msgs" shows "prob ((\<lambda>x. (fst x, (t \<circ> OB) x)) -` {(k, t obs)} \<inter> msgs) / prob ((t \<circ> OB) -` snd ` {(k, t obs)} \<inter> msgs) = prob ((\<lambda>x. (fst x, OB x)) -` {(k, obs)} \<inter> msgs) / prob (OB -` snd ` {(k, obs)} \<inter> msgs)"
h4b: shows "\<exists>x. \<forall>y. insertion (nth_default 0 (L[var := y])) B = x"
k: shows "\<forall>y\<in>convex hull c. f y \<le> k"
mask_eq_exp_minus_1: 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"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "mask n = minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one"
Memop: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "nat \<Rightarrow> bool option"   and is' :: "instr list"   and \<theta>' :: "nat \<Rightarrow> nat option"   and sb' :: "'a memref list"   and m' :: "nat \<Rightarrow> nat"   and \<D>' :: "bool"   and \<O>' :: "nat set"   and \<R>' :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "(is, \<theta>, sb, m, \<D>, \<O>, \<R>, \<S>) \<rightarrow>\<^sub>s\<^sub>b\<^sub>h (is', \<theta>', sb', m', \<D>', \<O>', \<R>', \<S>')" shows "sbh_computation.concurrent_step (ts, m, \<S>) (ts[i := (p, is', \<theta>', sb', \<D>', \<O>', \<R>')], m', \<S>')"
Max_gr_iff: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'e set"   and x :: "'e" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "less x (linorder.Max less_eq A) = (\<exists>a\<in>A. less x a)"
S_transform_tau_transition_iff: fixes P :: "'state"   and P' :: "'state" shows "P \<Rightarrow>\<^sub>S P' = P \<Rightarrow> P'"
C: shows "\<exists>y'. y' \<in> ipurge_ref I D (D y) ys Y"
HPhase1or2ReadThen_HInv5_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv5_inner s p"   and "HInv4c s p"   and "Inv2c s" shows "HInv5_inner s' p"
singular_relboundary: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" shows "singular_relboundary p X S c = (\<exists>d e. singular_chain (Suc p) X d \<and> singular_chain p (subtopology X S) e \<and> chain_boundary (Suc p) d + e = c)"
vs_lt_n: shows "\<forall>i\<in>set vs. i < n"
rules2: shows "R = Ax \<union> R1 \<union> R2 \<union> R3"
Ord_linear_lt: fixes k :: "hf"   and l :: "hf" assumes "Ord k"   and "Ord l" obtains "k \<^bold>\<in> l"   | "k = l"   | "l \<^bold>\<in> k"
ir_If_False: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and b :: "bexp"   and c\<^sub>2 :: "com"   and c' :: "com"   and Q :: "(char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool"   and c\<^sub>1 :: "com" assumes "ir_hoare (\<lambda>s s'. P s s' \<and> \<not> bval b s) c\<^sub>2 c' Q" shows "ir_hoare P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) c' Q"