hcju/isabelleps huggingface dataset:
struct_specD1: fixes sel :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ap :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and ab :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and compl :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('a \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" assumes "punit.struct_spec sel ap ab compl" shows "sel_spec sel"
proj_def: fixes \<sigma> :: "'tp"   and a :: "univ" shows "proj \<sigma> a \<equiv> if intT \<sigma> a then a else pickT \<sigma>"
keys_MP_oalist: fixes xs :: "('b, 'c) oalist_ntm" shows "keys (MP_oalist xs) = set (map fst (fst (list_of_oalist_ntm xs)))"
gi_def: shows "gi = (\<lambda>x. (x - b) / c)"
vmc_path_no_deadend_axioms: shows "vmc_path_no_deadend G P' v0 p \<sigma>"
einterval_eq_2: fixes a :: "real" shows "einterval (ereal a) \<infinity> = {a<..}"
merge_adjacent_length: fixes ss :: "('b word \<times> 'b word) list"   and s :: "'b word"   and e :: "'b word" assumes "\<exists>(s', e')\<in>set ss. s \<le> e \<and> s' \<le> e' \<and> (word_next e = s' \<or> word_next e' = s)" shows "length (merge_adjacent (s, e) ss) = length ss"
lunit_in_hom_2: fixes f :: "'b" assumes "D.ide f" shows "\<guillemotleft>\<l>\<^sub>D[f] : trg\<^sub>D f \<star>\<^sub>D f \<Rightarrow>\<^sub>D f\<guillemotright>"
f1''f1: shows "1 \<le> (f1'' - f1) * c * real l0"
measurable_enabled: fixes K :: "'c \<Rightarrow> 'c pmf"   and s :: "'c" shows "Measurable.pred (stream_space (count_space UNIV)) (MC_syntax.enabled K s)"
map_def: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and f :: "'ah" assumes "diagonal_functor J C" shows "diagonal_functor.map J C f = (if partial_magma.arr C f then \<chi>.\<Phi>.Ya.Cop_S.MkArr (constant_functor.map J C (partial_magma.dom C f)) (constant_functor.map J C (partial_magma.cod C f)) (constant_transformation.map J C f) else partial_magma.null (concrete_category.COMP (Collect (functor J C)) (\<lambda>F G. Collect (natural_transformation J C F G)) (\<lambda>F G H \<tau> \<sigma>. vertical_composite.map J C \<sigma> \<tau>)))"
WT_converter_parallel_converter2: fixes \<I>1 :: "('a, 'b) \<I>"   and \<I>2 :: "('c, 'd) \<I>"   and conv1 :: "('a, 'b, 'c, 'd) converter"   and \<I>1' :: "('e, 'f) \<I>"   and \<I>2' :: "('g, 'h) \<I>"   and conv2 :: "('e, 'f, 'g, 'h) converter" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>1', \<I>2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1', \<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<surd>"
arr_map: fixes f :: "'c" assumes "C.arr f" shows "S.arr (Cones.map f)"
rel_fun_comp1: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S1 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'j"   and R2 :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and S2 :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and h :: "'k \<Rightarrow> 'l"   and R' :: "'g \<Rightarrow> 'k \<Rightarrow> bool" assumes "(R1 ===> S1) f g"   and "(R2 ===> S2) g h"   and "R' = R1 OO R2" shows "(R' ===> S1 OO S2) f h"
gen_build_def: fixes tol :: "'h \<Rightarrow> 'i list"   and upd :: "'i \<Rightarrow> 'j \<Rightarrow> 'g \<Rightarrow> 'g"   and emp :: "'g"   and f :: "'i \<Rightarrow> 'j"   and X :: "'h" shows "gen_build tol upd emp f X \<equiv> fold (\<lambda>x. upd x (f x)) (tol X) emp"
S_trms_instance_M: shows "has_all_wt_instances_of \<Gamma> (trms\<^sub>s\<^sub>t S) (set M)"
absorb2: fixes a :: "'a word"   and b :: "'a word" assumes "a \<le>s b" shows "signed.max a b = b"
elims: fixes x :: "('a, 'b) lit"   and y :: "Preliminaries.var set" assumes "varsL x = y" obtains at :: "('a, 'b) atm" where "x = Pos at__" and "y = varsA at__"   | at :: "('a, 'b) atm" where "x = Neg at" and "y = varsA at"
ipset_from_cidr_alt: fixes pre :: "'b word"   and len :: "nat" shows "ipset_from_cidr pre len = {pre && (mask len << LENGTH('b) - len)..pre || mask (LENGTH('b) - len)}"
Subcls: assumes "G\<turnstile>C\<prec>\<^sub>C D" shows "thesis"
terminal_tSup: fixes Y :: "('d, 'b) tllist set" assumes "\<forall>xs\<in>Y. is_TNil xs" shows "terminal (local.tSup Y) = flat_lub b (terminal ` Y)"
domain_eq: shows "Domainp (pcr_blinfun (=) (=)) = bounded_linear"
hyps: fixes ts :: "'t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option"   and queue :: "'t list" assumes "thread_oks ts ntas"   and "set queue = dom ts"   and "distinct queue" shows "distinct (enqueue_new_threads queue ntas)"
simps_4: fixes e :: "String.literal \<Rightarrow> 'a"   and f :: "String.literal \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "String.literal \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "fol fm"   and q :: "fol fm" shows "semantics e f g (Imp p q) = (semantics e f g p \<longrightarrow> semantics e f g q)"
gfp_least: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and f :: "'a \<Rightarrow> 'a"   and X :: "'a" 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"
inj_on_image_Int: fixes f :: "'b \<Rightarrow> 'c"   and C :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "inj_on f C"   and "order.greater_eq C A"   and "order.greater_eq C B" shows "f ` (A \<inter> B) = f ` A \<inter> f ` B"
iso_hcomp\<^sub>R\<^sub>W\<^sub>C_1: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "regular_weak_composition V H"   and "category.iso V \<mu>"   and "category.iso V \<nu>"   and "weak_composition.sources V H \<nu> \<inter> weak_composition.targets V H \<mu> \<noteq> {}" shows "category.iso V (H \<nu> \<mu>)"
cases: fixes a1 :: "'b val"   and a2 :: "'b rexp" assumes "\<turnstile> a1 : a2" obtains v1 :: "'b val" and r1 :: "'b rexp" and v2 :: "'b val" and r2 :: "'b rexp" where "a1 = Seq v1__ v2__" and "a2 = Times r1__ r2__" and "\<turnstile> v1__ : r1__" and "\<turnstile> v2__ : r2__"   | v1 :: "'b val" and r1 :: "'b rexp" and r2 :: "'b rexp" where "a1 = val.Left v1" and "a2 = Plus r1a__ r2a__" and "\<turnstile> v1 : r1a__"   | v2 :: "'b val" and r2 :: "'b rexp" and r1 :: "'b rexp" where "a1 = val.Right v2" and "a2 = Plus r1 r2" and "\<turnstile> v2 : r2"   | "a1 = Void" and "a2 = One"   | c :: "'b" where "a1 = Atm c" and "a2 = Atom c"   | r :: "'b rexp" where "a1 = Stars []" and "a2 = Star ra__"   | v :: "'b val" and r :: "'b rexp" and vs :: "'b val list" where "a1 = Stars (v # vs)" and "a2 = Star r" and "\<turnstile> v : r" and "\<turnstile> Stars vs : Star r"
lang_subset_lists: fixes r :: "'a rexp"   and S :: "'a set" assumes "atoms r \<subseteq> S" shows "lang S r \<subseteq> lists S"
divide_sgn: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "divide a (sgn b) = times a (sgn b)"
powr: fixes a :: "real"   and b :: "real" assumes "p powr a \<le> p powr b" shows "a \<le> b"
stoch_non_neg_vec_norm1: fixes v :: "(real, 'a) Finite_Cartesian_Product.vec" assumes "stoch_vec v"   and "non_neg_vec v" shows "norm1 v = 1"
assms_4: fixes x :: "'n" assumes "x \<in> S" shows "g (f x) = x"
simps_1: fixes x :: "'b"   and y :: "'b" shows "destNet (DenyAllFromTo x y) = y"
PO_l3_inv2: shows "reach l3 \<subseteq> l3_inv2"
nth_equal_first_eq: fixes x :: "'b"   and xs :: "'b list"   and n :: "nat" assumes "x \<notin> set xs"   and "n \<le> length xs" shows "((x # xs) ! n = x) = (n = 0)"
seqsim_stutstep: fixes s :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "stutstep s n" shows "s |\<^sub>s Suc n \<approx> s |\<^sub>s n"
poly_exp_eq_zero: fixes p :: "'a list"   and n :: "nat" shows "(poly (p %^ n) = poly []) = (poly p = poly [] \<and> n \<noteq> 0)"
merge_on_axioms_def: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option" shows "merge_on_axioms A h bo m \<equiv> (\<forall>a b. h a = h b \<longrightarrow> a \<in> A \<longrightarrow> (\<exists>ab. m a b = Some ab \<and> bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u))) \<and> (\<forall>a b. h a \<noteq> h b \<longrightarrow> a \<in> A \<longrightarrow> m a b = None)"
ennreal_mult'': fixes b :: "real"   and a :: "real" assumes "0 \<le> b" shows "ennreal (a * b) = ennreal a * ennreal b"
is_natural_transformation: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) G F \<tau>'.map"
add_le_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "i + k \<le> j + k"
simps_2: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and f :: "'b"   and ts :: "('b, 'c) Term.term list" shows "weight_fun.weight w w0 scf (Fun f ts) = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight_fun.weight w w0 scf ti * scff i) ts [0..<n]))"
return_time_pos: fixes P :: "'d set"   and x :: "'d" assumes "returns_to P x"   and "closed P" shows "0 < return_time P x"
suminf_split_initial_segment: fixes k :: "nat" assumes "summable f" shows "suminf f = (\<Sum>n. f (n + k)) + sum f {..<k}"
sds_wf: fixes R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "is_pref_profile R" shows "sds R \<in> lotteries"
prems_1: shows "insert_ops (pre @ [(oid, Some ref)] @ xs @ [x])"
not_frequently_False: fixes F :: "'a filter" shows "\<not> (\<exists>\<^sub>F x in F. False)"
sup_bot_left: fixes x :: "'d upred" shows "(false \<or> x) = x"
sum_sumlist: fixes c :: "nat \<Rightarrow> 'a" assumes "distinct Vsl" shows "(\<Sum>x\<in>set Vsl. sum_list (map c (find_indices x Vsl))) = sum c {0..<length Vsl}"
RedAAss: fixes h :: "'heap"   and a :: "'addr"   and T :: "ty"   and n :: "nat"   and i :: "32 word"   and w :: "'addr val"   and U :: "ty"   and P :: "'addr J_prog"   and h' :: "'heap"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>"   and "0 \<le>s i"   and "sint i < int n"   and "typeof\<^bsub>h\<^esub> w = \<lfloor>U\<rfloor>"   and "P \<turnstile> U \<le> T"   and "heap_write h a (ACell (nat (sint i))) w h'" shows "extTA,P,t \<turnstile> \<langle>addr a\<lfloor>Val (Intg i)\<rceil> := Val w,(h, l)\<rangle> -\<lbrace>WriteMem a (ACell (nat (sint i))) w\<rbrace>\<rightarrow> \<langle>unit,(h', l)\<rangle>"
emeasure_space_1: fixes M :: "'c measure" assumes "prob_space M" shows "emeasure M (space M) = 1"
"1_1": shows "w\<^sub>1 =\<^sub>F w\<^sub>2"
simps_1: fixes "is" :: "'b set"   and t :: "'b" shows "interrupt_action_ok' is (Interrupt t) = True"
r_null: fixes x :: "'a" assumes "x \<in> carrier R" shows "x \<otimes> \<zero> = \<zero>"
VA: shows "Vagree \<nu> \<omega> (\<Union>i\<in>SIGT (Differential \<theta>). case SFunctions \<sigma> i of None \<Rightarrow> {} | Some a \<Rightarrow> FVT a)"
csmall_to_csmall_nstep_prop2: fixes R :: "'prog \<Rightarrow> 'prog \<Rightarrow> 'coll \<Rightarrow> bool"   and Q :: "'state \<Rightarrow> bool"   and \<sigma>' :: "'state"   and coll :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state"   and n :: "nat"   and P' :: "'prog" assumes "\<And>P P' \<sigma> \<sigma>' coll. \<lbrakk>(\<sigma>', coll) \<in> csmall P \<sigma>; R P P' coll; Q \<sigma>\<rbrakk> \<Longrightarrow> (\<sigma>', coll) \<in> csmall P' \<sigma>"   and "\<And>P P' coll1 coll2. R P P' (combine coll1 coll2) = (R P P' coll1 \<and> R P P' coll2)"   and "\<And>P \<sigma> \<sigma>'. \<lbrakk>Q \<sigma>; \<sigma>' \<in> small P \<sigma>\<rbrakk> \<Longrightarrow> Q \<sigma>'"   and "(\<sigma>', coll) \<in> csmall_nstep P \<sigma> n"   and "R P P' coll"   and "Q \<sigma>" shows "(\<sigma>', coll) \<in> csmall_nstep P' \<sigma> n"
ide_dom: fixes f :: "'b" assumes "D.arr f" shows "D.ide (D.dom f)"
poly_tendsto_3: fixes p :: "real poly"   and x :: "real" shows "(poly p \<longlongrightarrow> poly p x) (at_right x)"
mediant_I_F_2: shows "mediant I = (1, 1)"
delay_bisimulation_diverge_final_flip: shows "delay_bisimulation_diverge_final trsys2 trsys1 (flip bisim) (flip tlsim) \<tau>move2 \<tau>move1 final2 final1"
zero_stalk_def: fixes V :: "'a set" shows "stk2.zero_stalk V \<equiv> stk2.class_of V (zero_str2 V)"
Units_eq: shows "Units (mult_of R) = carrier (mult_of R)"
mod_mod_trivial: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and 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 modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (modulo a b) b = modulo a b"
app_tl_hd_list: fixes f :: "nat \<Rightarrow> 'a"   and l :: "'b list"   and a :: "'a" assumes "tl (map f [0..<length l] @ [a]) \<noteq> []" shows "hd (map f [0..<length l] @ [a]) = map f [0..<length l] ! 0"
assms_1: shows "u \<turnstile>\<^bsub>v,n\<^esub> m"
sees_method_fun: fixes P :: "'b prog"   and C :: "String.literal"   and M :: "String.literal"   and TS :: "ty list"   and T :: "ty"   and m :: "'b option"   and D :: "String.literal"   and TS' :: "ty list"   and T' :: "ty"   and m' :: "'b option"   and D' :: "String.literal" assumes "P \<turnstile> C sees M: TS\<rightarrow>T = m in D"   and "P \<turnstile> C sees M: TS'\<rightarrow>T' = m' in D'" shows "TS' = TS \<and> T' = T \<and> m' = m \<and> D' = D"
dagger_unfoldl_distr: fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" shows "y \<union> (x O x\<^sup>*) O y = x\<^sup>* O y"
comp_ide_self: fixes a :: "'a" assumes "Left_a.ide a" shows "Left_a.comp a a = a"
GE_Guard_protCl: fixes f :: "'fsym"   and al :: "univ list" assumes "wtFsym f"   and "protCl (resOf f)"   and "list_all2 eintT (arOf f) al"   and "list_all2 (eintP \<circ> Guard) (arOf f) (map singl al)" shows "eintP (Guard (resOf f)) [intF f al]"
rotate_def: fixes x :: "wall"   and y :: "wall" shows "Tangle_Moves.rotate x y \<equiv> rotate_toppos x y \<or> rotate_topneg x y \<or> rotate_downpos x y \<or> rotate_downneg x y"
inner_simps_3: fixes x :: "'b"   and y :: "'b"   and z :: "'b" shows "x \<bullet> (y - z) = x \<bullet> y - x \<bullet> z"
Times_def: fixes fsm1 :: "('b, 'c) fsm_scheme"   and fsm2 :: "('b, 'd) fsm_scheme" shows "Times fsm1 fsm2 = \<lparr>states = states fsm1 * states fsm2, init = init fsm1 * init fsm2, final = final fsm1 * final fsm2, next = \<lambda>st x st'. \<exists>st1 st2 st1' st2'. st = \<langle>st1, st2\<rangle> \<and> st' = \<langle>st1', st2'\<rangle> \<and> next fsm1 st1 x st1' \<and> next fsm2 st2 x st2'\<rparr>"
adapted_init: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and x :: "nat \<Rightarrow> 'c \<Rightarrow> real" assumes "triv_init_disc_filtr_prob_space M F"   and "borel_adapt_stoch_proc F x" shows "\<exists>c. \<forall>w\<in>space M. x 0 w = c"
assms_3: shows "list.set \<alpha>' \<subseteq> ON"
glbI: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b set"   and x :: "'b" assumes "lb P B x"   and "\<And>y. lb P B y \<Longrightarrow> P\<^sup>=\<^sup>= y x" shows "glb P B x"
remove_vertex_walk: fixes xs :: "'a list"   and x :: "'a" assumes "walk xs"   and "x \<notin> set xs" shows "Digraph.walk (remove_vertex x) xs"
mem_restrI: fixes y :: "'c list"   and A :: "'c list set"   and n :: "nat"   and V :: "nat set"   and x :: "'c option list" assumes "y \<in> A"   and "length y = n"   and "wf_tuple n V x"   and "\<forall>i\<in>V. x ! i = Some (y ! i)" shows "mem_restr A x"
has_fields_fun: fixes P :: "'a prog"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
step: fixes Sup :: "'e set \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and f :: "'e \<Rightarrow> 'e" assumes "class.ccpo Sup less_eq less"   and "x \<in> ccpo.iterates Sup less_eq f" shows "f x \<in> ccpo.iterates Sup less_eq f"
H_OUT: fixes x :: "'v" assumes "x \<in> A \<Gamma>" shows "d_OUT (\<lambda>x. ennreal (H x)) x = weight \<Gamma> x"
natural_isomorphism_axioms: shows "natural_isomorphism (\<bullet>) (\<bullet>) L local.map \<ll>.map"
assms: fixes i :: "nat" assumes "pdevs_apply y i = (0::'a)" shows "pdevs_apply x i = (0::'b)"
measurable_bind2: fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and g :: "'d \<Rightarrow> 'e measure"   and R :: "'e measure" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "g \<in> N \<rightarrow>\<^sub>M subprob_algebra R" shows "(\<lambda>x. f x \<bind> g) \<in> M \<rightarrow>\<^sub>M subprob_algebra R"
assms_2: shows "Negligible.negligible (\<lambda>\<eta>. local.ddh.advantage \<eta> (R1_inter_adversary \<eta> D msgs))"
lookup_row_to_poly_not_zeroE: fixes ts :: "'b list"   and r :: "'a vec"   and t :: "'b" assumes "lookup (row_to_poly ts r) t \<noteq> (0::'a)" obtains i :: "nat" where "i < length ts" and "t = ts ! i"
Step1_4_6a: fixes p :: "PrIds"   and rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState" shows "\<turnstile> MClkReply memCh crCh cst p \<and> $S6 rmhist p \<and> S1 rmhist p$ \<and> unchanged (e p, r p, m p) \<longrightarrow> MemReturn memCh (resbar rmhist) p"
beta_eta_abs: fixes t :: "dB"   and t' :: "dB" assumes "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* t'" shows "Abs t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* Abs t'"
par_comp_prot_impl_par_comp_stateful: fixes \<P> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list set"   and Sec :: "('fun, 'var) Term.term set"   and \<A> :: "('lbl strand_label \<times> ('fun, 'var) stateful_strand_step) list" assumes "par_comp_prot_stateful \<P> Sec"   and "\<A> \<in> \<P>" shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<A> Sec"
map_default_def: fixes k :: "'c"   and v :: "'d"   and f :: "'d \<Rightarrow> 'd"   and m :: "('c, 'd) mapping" shows "Mapping.map_default k v f m = Mapping.map_entry k f (Mapping.default k v m)"
phi_wf: fixes g :: "'g"   and v :: "'val"   and vs :: "'val list" assumes "phi g v = Some vs" shows "length (predecessors g (defNode g v)) = length vs"
pfun_member_plus: fixes x :: "'a"   and y :: "'b"   and f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun" shows "((x, y) \<in>\<^sub>p f + g) = (x \<notin> pdom g \<and> (x, y) \<in>\<^sub>p f \<or> (x, y) \<in>\<^sub>p g)"
NB_Crypt_imp_Server_msg: fixes B :: "agent"   and evs :: "event list"   and NB :: "msg"   and K :: "nat"   and NA :: "msg"   and A :: "agent"   and X' :: "msg" assumes "B \<notin> bad"   and "evs \<in> otway"   and "Crypt (shrK B) \<lbrace>NB, Key K\<rbrace> \<in> parts (knows Spy evs)"   and "Says B Server \<lbrace>NA, Agent A, Agent B, X', Crypt (shrK B) \<lbrace>NA, NB, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs" shows "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"
ag_r_zero: fixes A :: "('c, 'd) aGroup_scheme"   and x :: "'c" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = x"
witness_2: fixes a :: "'c" assumes "a \<in> S" shows "j a \<in> T"
simps_10: fixes a :: "floatarith" shows "fold_const_fa (Cos a) = Cos (fold_const_fa a)"
assms_2: fixes i :: "nat" assumes "i < length ss"   and "i < length ts"   and "snd (r (ss ! i) (ts ! i))" shows "snd (r (map f ss ! i) (map f ts ! i))"
deg_minus_monom: fixes a :: "'a" assumes "a \<in> carrier R"   and "carrier R \<noteq> {\<zero>}" shows "deg R (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = 1"
distinct_55: fixes x5 :: "'a OCL_Types.type"   and x7 :: "'a OCL_Types.type" shows "Set x5 \<noteq> Bag x7"
rep_zero: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and equal :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and rep_nat :: "'a \<Rightarrow> nat"   and abs_nat :: "nat \<Rightarrow> 'a" assumes "class.nat minus plus zero equal less_eq less rep_nat abs_nat" shows "rep_nat zero = 0"
intro: fixes as :: "real list"   and k :: "nat"   and bs :: "real list" assumes "length as = k"   and "length bs = k"   and "k \<noteq> 0"   and "\<And>a. a \<in> set as \<Longrightarrow> 0 \<le> a"   and "\<And>b. b \<in> set bs \<Longrightarrow> b \<in> {0<..<1}" shows "akra_bazzi_params k as bs"
not_vend: shows "e \<noteq> (STR ''vend'', [])"
c1: shows "\<forall>x. insertion (nth_default 0 (xs @ x # \<Gamma>)) B = c 1"
DD_def: fixes ars :: "('a \<times> 'b \<times> 'a) set"   and r :: "('b \<times> 'b) set"   and d :: "('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> ('a \<times> ('b \<times> 'a) list) \<times> 'a \<times> ('b \<times> 'a) list" shows "DD ars r d = (diagram ars d \<and> D2 r d)"
a0i_trans_def: shows "a0i_trans \<equiv> \<Union>h d ds. (a0i_running h d \<union> a0i_commit h d \<union> a0i_corrupt ds)\<^sup>="