hcju/isabelleps huggingface dataset:
acc_modi_le_Public: fixes a :: "acc_modi" assumes "a \<le> Public" shows "a = Private \<or> a = Package \<or> a = Protected \<or> a = Public"
dagger_unfoldr_eq: fixes x :: "'b \<times> 'c" shows "sup_prod one_prod (times_prod' (prod.qstar x) x) = prod.qstar x"
max_tstp_isl: fixes tstp :: "nat + nat"   and tstp' :: "nat + nat" assumes "isl tstp = isl tstp'" shows "isl (max_tstp tstp tstp') = isl tstp"
\<Sigma>_incr_upper: fixes f :: "'d \<Rightarrow> 'c"   and j :: "int"   and l :: "int" shows "\<Sigma> f j (l + 1) = \<Sigma> f j l + f (of_int l)"
get_shadow_root_new_shadow_root: fixes h :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> get_shadow_root_locs ptr'" shows "r h h'"
rl: shows "(reduced n \<circ> label) ` s = {..n}"
localRelS'_def: shows "localRelS' = (\<lambda>(u, v) (u', v') x y. u = u')"
basis_indep: fixes X :: "'a set" assumes "\<E>.basis X" shows "indep_in \<E> X"
card_eq: shows "card ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}) = card B"
fin_cut_same_Cons: fixes x :: "'c"   and y :: "'c"   and xs :: "'c list" shows "fin_cut_same x (y # xs) = (if fin_cut_same x xs = [] then if x = y then [] else [y] else y # fin_cut_same x xs)"
normalize_quot_idem: fixes x :: "'b \<times> 'b" shows "normalize_quot (normalize_quot x) = normalize_quot x"
funpow_mono2: fixes f :: "'d \<Rightarrow> 'd"   and i :: "nat"   and j :: "nat"   and x :: "'d"   and y :: "'d" assumes "mono f"   and "i \<le> j"   and "x \<le> y"   and "x \<le> f x" shows "(f ^^ i) x \<le> (f ^^ j) y"
recvmsg_simps_8: fixes P :: "'d \<Rightarrow> bool" shows "recvmsg P \<tau>\<^sub>s = True"
simps_4: shows "VARto stA6 = {data15, data16}"
psubstT_Var_Cons_aux: fixes y :: "'var"   and x :: "'var"   and t :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "y \<in> var"   and "x \<in> var"   and "t \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm"   and "x \<notin> snd ` set txs"   and "distinct (map snd txs)"   and "y \<noteq> x" shows "psubstT (Var y) ((t, x) # txs) = psubstT (Var y) txs"
pair_sigma_finite_axioms: shows "pair_sigma_finite (M x) (M y)"
nonPrincipalInvertLeft: fixes R1 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R2 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R3 :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and R :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and M1 :: "'b"   and M2 :: "'b"   and r :: "('a, 'b) sequent list \<times> ('a, 'b) sequent"   and ps :: "('a, 'b) sequent list"   and c :: "('a, 'b) sequent"   and R' :: "(('a, 'b) sequent list \<times> ('a, 'b) sequent) set"   and n :: "nat"   and M :: "'b"   and Ms :: "('a, 'b) form list"   and \<Gamma>' :: "('a, 'b) form multiset"   and \<Delta>' :: "('a, 'b) form multiset"   and \<Gamma> :: "('a, 'b) form multiset"   and \<Delta> :: "('a, 'b) form multiset"   and S :: "('a, 'b) sequent"   and Ps :: "('a, 'b) sequent list"   and n' :: "nat" assumes "R1 \<subseteq> upRules"   and "R2 \<subseteq> modRules2"   and "R3 \<subseteq> modRules2"   and "R = Ax \<union> R1 \<union> p_e R2 M1 M2 \<union> R3"   and "r \<in> R"   and "r = (ps, c)"   and "R' = Ax \<union> R1 \<union> R2 \<union> R3"   and "\<forall>m<n. \<forall>\<Gamma> \<Delta>. ( \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>, m) \<in> derivable (ext R R2 M1 M2) \<longrightarrow> (\<forall>r'\<in>R'. leftPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')) \<longrightarrow> (\<exists>m'\<le>m. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m') \<in> derivable (ext R R2 M1 M2))"   and "( \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>, n) \<in> derivable (ext R R2 M1 M2)"   and "\<forall>r'\<in>R'. leftPrincipal r' (Modal M Ms) R' \<longrightarrow> ( \<Gamma>' \<Rightarrow>* \<Delta>') \<in> set (fst r')"   and "\<not> leftPrincipal r (Modal M Ms) R'"   and "(r \<in> Ax \<or> r \<in> upRules \<or> r \<in> modRules2) \<and> extendRule S r = (Ps,  \<Gamma> \<oplus> Modal M Ms \<Rightarrow>* \<Delta>)"   and "n = n' + 1"   and "\<forall>p\<in>set Ps. \<exists>n\<le>n'. (p, n) \<in> derivable (ext R R2 M1 M2)"   and "Ps \<noteq> []" shows "\<exists>m\<le>n. ( \<Gamma> + \<Gamma>' \<Rightarrow>* \<Delta> + \<Delta>', m) \<in> derivable (ext R R2 M1 M2)"
map_list_R1_list_to_list_R1: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "map_list_R1 f (list_to_list_R1 xs) = list_to_list_R1 (map f xs)"
decrtm0: fixes t :: "tm"   and vs :: "'b list"   and x :: "'b"   and bs :: "'b list" assumes "tmbound0 t" shows "Itm vs (x # bs) t = Itm vs bs (decrtm0 t)"
mono_mndet_FD: fixes A :: "'d set"   and P :: "'d \<Rightarrow> 'd process"   and P' :: "'d \<Rightarrow> 'd process" assumes "\<forall>x\<in>A. P x \<le> P' x" shows "mndet A P \<le> mndet A P'"
assms: shows "\<forall>v. freedb e v \<longrightarrow> v < k"
stable_pair_on_CD_on: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set" assumes "stable_pair_on ds XD_XH" shows "match XD_XH = CD_on ds (fst XD_XH)"
map_IInf_map_us_inv: shows "map_IInf \<circ> map_us = id"
F: shows "set xs' \<subseteq> range p \<union> range q"
\<rho>_nonneg: fixes s :: "'s" shows "0 \<le> \<rho> s"
succ_rel_insert_alt: fixes a :: "'b"   and ref :: "'b"   and oid :: "'b"   and b :: "'b"   and list :: "'b list" assumes "a \<noteq> ref"   and "distinct (oid # a # b # list)" shows "insert_alt (succ_rel (a # b # list)) (oid, ref) = {(a, Some b)} \<union> insert_alt (succ_rel (b # list)) (oid, ref)"
deg_one: shows "deg R \<one>\<^bsub>P\<^esub> = 0"
cod_dom: fixes f :: "'s" assumes "S.arr f" shows "S.cod (S.dom f) = S.dom f"
zero_le: fixes x :: "'f multiset" shows "{#} \<subseteq># x"
divides_ff_def: fixes x :: "'b fract"   and y :: "'b fract" shows "divides_ff x y \<equiv> \<exists>r. y = x * to_fract r"
simple_extension_is_subfield: fixes K :: "'a set"   and x :: "'a" assumes "subfield K R"   and "x \<in> carrier R" shows "subfield (simple_extension K x) R = (algebraic over K) x"
sets_PiM_I_finite: fixes I :: "'b set"   and E :: "'b \<Rightarrow> 'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> E i \<in> sets (M i)" shows "Pi\<^sub>E I E \<in> sets (Pi\<^sub>M I M)"
IH: assumes "is_class P C" shows "\<exists>Cs. P \<turnstile> Path C to C' via Cs "
2: fixes v :: "i"   and x :: "\<nu>" shows "[(\<^bold>\<exists>!y. \<^bold>\<A>\<phi> y (x\<^sup>P)) \<^bold>\<equiv> \<lbrace>x\<^sup>P,H\<rbrace> in v]"
WHILE_refine_genR': fixes x :: "'c"   and x' :: "'d"   and R' :: "('c \<times> 'd) set"   and I' :: "'d \<Rightarrow> bool"   and b :: "'c \<Rightarrow> bool"   and b' :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and f' :: "'d \<Rightarrow> 'd nres"   and R :: "('c \<times> 'd) set" assumes "(x, x') \<in> R'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; I' x'\<rbrakk> \<Longrightarrow> b x = b' x'"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; b x; b' x'; I' x'\<rbrakk> \<Longrightarrow> f x \<le> \<Down> R' (f' x')"   and "\<And>x x'. \<lbrakk>(x, x') \<in> R'; \<not> b x; \<not> b' x'\<rbrakk> \<Longrightarrow> (x, x') \<in> R" shows "WHILE b f x \<le> \<Down> R (WHILE\<^bsup>I'\<^esup> b' f' x')"
dense_repr_set_fst: fixes p :: "'c list"   and K :: "'c set" assumes "set p \<subseteq> K" shows "fst ` set (A.dense_repr p) \<subseteq> K - {\<zero>\<^bsub>A\<^esub>}"
num_params_tm_add_le: fixes t1 :: "taylor_model"   and X :: "nat"   and t2 :: "taylor_model" assumes "num_params (tm_poly t1) \<le> X"   and "num_params (tm_poly t2) \<le> X" shows "num_params (tm_poly (tm_add t1 t2)) \<le> X"
le_funD: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "f \<le> g" shows "f x \<le> g x"
uniformly_continuous_on_extension_at_closure: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "uniformly_continuous_on X f"   and "x \<in> closure X" obtains l :: "'c" where "(f \<longlongrightarrow> l) (at x within X)"
gcd_mult_distrib': fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd_mult_normalize one zero times gcd lcm divide plus minus unit_factor normalize" shows "times (normalize c) (gcd a b) = gcd (times c a) (times c b)"
P_def: shows "P \<equiv> \<lambda>l. trms\<^sub>l\<^sub>s\<^sub>t (proj l A)"
in_ccProd: fixes x :: "Vars.var"   and y :: "Vars.var"   and S :: "Vars.var set"   and S' :: "Vars.var set" shows "x--y\<in>(S G\<times> S') = (x \<in> S \<and> y \<in> S' \<or> x \<in> S' \<and> y \<in> S)"
dickson_less_def: fixes d :: "'a \<Rightarrow> nat"   and m :: "nat"   and s :: "'a"   and t :: "'a" shows "dickson_less d m s t = (d s \<le> m \<and> d t \<le> m \<and> s \<prec> t)"
sel_2: fixes a :: "real" shows "Im (cis a) = sin a"
invoke_splits_1: fixes P :: "((_) heap, exception, 'n) prog \<Rightarrow> bool"   and Pred :: "(_) object_ptr \<Rightarrow> bool"   and f :: "(_) object_ptr \<Rightarrow> 'o \<Rightarrow> ((_) heap, exception, 'n) prog"   and xs :: "(((_) object_ptr \<Rightarrow> bool) \<times> ((_) object_ptr \<Rightarrow> 'o \<Rightarrow> ((_) heap, exception, 'n) prog)) list"   and ptr :: "(_) object_ptr"   and args :: "'o" shows "P (invoke ((Pred, f) # xs) ptr args) = ((\<not> Pred ptr \<longrightarrow> P (invoke xs ptr args)) \<and> (Pred ptr \<longrightarrow> P (Heap_Error_Monad.bind (check_in_heap ptr) (\<lambda>_. f ptr args))))"
ldropWhile_LNil: fixes P :: "'b \<Rightarrow> bool" shows "ldropWhile P LNil = LNil"
small_subsetI: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F g" shows "l F f \<subseteq> l F g"
AnalBinds_delete_to_fun_upd: fixes exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" shows "ExpAnalysis.AnalBinds exp (delete x \<Gamma>)\<cdot>ae = (ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae)(x := \<bottom>)"
i_l_m: fixes i :: "nat" assumes "i < l" shows "i < length fs"
times_set_def: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and imp_l :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and one :: "'b"   and imp_r :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "pseudo_hoop_algebra.times_set times A B = {a. \<exists>x\<in>A. \<exists>y\<in>B. a = times x y}"
extract_None_iff: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "(List.extract P xs = None) = (\<not> (\<exists>x\<in>set xs. P x))"
add_Pair: fixes a :: "'c"   and b :: "'d"   and c :: "'c"   and d :: "'d" shows "(a, b) + (c, d) = (a + c, b + d)"
exhaust_sel: fixes literal :: "'b literal" obtains "literal = Pos (atm_of literal)"   | "literal = Neg (atm_of literal)"
cases: fixes a1 :: "'var"   and a2 :: "'node SDG_node" assumes "a1 \<in> Use\<^bsub>SDG\<^esub> a2" obtains m :: "'node" and V :: "'var" and n :: "'node SDG_node" where "a1 = V__" and "a2 = nb__" and "valid_node m__" and "V__ \<in> Use m__" and "nb__ = CFG_node m__"   | n :: "'node SDG_node" and m :: "'node" and x :: "nat" and V :: "'var" where "a1 = Va__" and "a2 = nc__" and "valid_SDG_node nc__" and "nc__ = Actual_in (ma__, x__)" and "Va__ \<in> ParamUses ma__ ! x__"   | n :: "'node SDG_node" and m :: "'node" and x :: "nat" and p :: "'pname" and ins :: "'var list" and outs :: "'var list" and V :: "'var" where "a1 = V" and "a2 = n" and "valid_SDG_node n" and "n = Formal_out (m, x)" and "get_proc m = p" and "(p, ins, outs) \<in> set procs" and "V = outs ! x"
list_rel_compp: fixes A :: "('g \<times> 'i) set"   and B :: "('i \<times> 'h) set" shows "\<langle>A O B\<rangle>list_rel = \<langle>A\<rangle>list_rel O \<langle>B\<rangle>list_rel"
finite_Union: fixes A :: "'b set set" assumes "finite A"   and "\<And>M. M \<in> A \<Longrightarrow> finite M" shows "finite (\<Union> A)"
mat_mult: fixes nr :: "nat"   and n :: "nat"   and m1 :: "'c list list"   and nc :: "nat"   and m2 :: "'c list list"   and ze :: "'c"   and pl :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and ti :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" assumes "mat nr n m1"   and "mat n nc m2" shows "mat nr nc (mat_multI ze pl ti nr m1 m2)"
plane_rel_internal: fixes A :: "('c \<times> 'd) set" shows "plane_rel A = \<langle>A\<rangle>sctn_rel O br plane_of top"
vec_upd_def: fixes s :: "('c, 'd) vec"   and i :: "'d"   and a :: "'c" shows "HS_VC_Spartan.vec_upd s i a = (\<chi>j. ((($) s)(i := a)) j)"
select_convs_2: fixes c_work :: "'a \<Rightarrow> 'b zmultiset"   and c_pts :: "'a \<Rightarrow> 'b zmultiset"   and c_imp :: "'a \<Rightarrow> 'b zmultiset"   and more :: "'c" shows "c_pts \<lparr>c_work = c_work, c_pts = c_pts, c_imp = c_imp, \<dots> = more\<rparr> = c_pts"
in_degree_eq_card_arc_blocks: fixes v :: "'a" shows "in_degree G v = card (arc_to_block ` in_arcs G v)"
cases: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "Inf.inf r a" obtains x :: "'a" and y :: "'a" where "a = x" and "r x y" and "Inf.inf r y"
\<PP>_are_doc_tokens: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and p :: "(symbol \<times> character list) list"   and Doc :: "character list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "p \<in> LocalLexing.\<PP> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc" shows "LocalLexing.doc_tokens \<TT> Doc p"
subst_valid: shows "valid (Fsubst ($\<phi> p (singleton \<theta>) \<leftrightarrow> $\<phi> p (singleton \<theta>')) \<sigma>)"
path0: fixes subs :: "'b \<Rightarrow> 'b set"   and gamma :: "'b"   and P :: "(nat \<times> 'b) set \<Rightarrow> bool" shows "path subs gamma P 0 = gamma"
simps_1: shows "polypow 0 = (\<lambda>p. (1)\<^sub>p)"
distinct_41: shows "AtOp \<noteq> string_binop.GreaterOp"
intros: fixes A :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and gpv :: "('g, 'h, 'i) gpv"   and gpv' :: "('j, 'k, 'l) gpv" assumes "rel_spmf (rel_generat A C (R ===> rel_gpv'' A C R)) (the_gpv gpv) (the_gpv gpv')" shows "rel_gpv'' A C R gpv gpv'"
Y': shows "Y \<inter> {xs. \<not> lnull xs} \<noteq> {}"
prems_1: shows "local.sub1 q (Suc k) l (r, A) = (r'', A'')"
fmap_minus_direct_def: fixes F :: "'g \<Rightarrow> 'h option"   and x :: "'g \<times> 'h" shows "(F -- x) = (\<lambda>z. if fst x = z \<and> F (fst x) = Some (snd x) then None else F z)"
equivalence_functor_F: shows "equivalence_functor hom'.comp hom.comp (\<lambda>\<mu>. trg f \<star> \<mu> \<star> d)"
cfcomp2: fixes f :: "'e \<rightarrow> 'c"   and g :: "'d \<rightarrow> 'e"   and x :: "'d" shows "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
1: shows "|~ \<langle>Deq inp q1 mid\<rangle>_vars = Deq inp q1 mid"
abs_split_split: fixes n :: "nat"   and xs :: "('c \<times> 'd) list"   and p :: "'d" assumes "n < length xs"   and "\<forall>(uu_, s)\<in>set (take n xs). s < p"   and "case xs ! n of (x, s) \<Rightarrow> \<not> s < p" shows "abs_split xs p = (take n xs, drop n xs)"
aux3: fixes b :: "bool"   and x :: "'a" shows "the_default (0::'a) (if b then Some x else None) = (if b then x else (0::'a))"
gcdI: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times c a"   and "dvd.dvd times c b"   and "\<And>d. \<lbrakk>dvd.dvd times d a; dvd.dvd times d b\<rbrakk> \<Longrightarrow> dvd.dvd times d c"   and "normalize c = c" shows "c = gcd a b"
wf_iff_no_infinite_down_chain: fixes r :: "('d \<times> 'd) set" shows "wf r = (\<nexists>f. \<forall>i. (f (Suc i), f i) \<in> r)"
all_clauses_finite: fixes S :: "'b literal set set" assumes "finite S \<and> (\<forall>C\<in>S. finite C)" shows "finite (all_clauses S)"
nc_scomp_closed: fixes x :: "'a"   and y :: "'a" shows "(x \<sqinter> nc) \<cdot> (y \<sqinter> nc) \<le> nc"
connectedD_interval: fixes U :: "'a set"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "connected U"   and "x \<in> U"   and "y \<in> U"   and "x \<le> z"   and "z \<le> y" shows "z \<in> U"
writerT_induct: fixes P :: "'e\<cdot>('g, 'f) writerT \<Rightarrow> bool"   and y :: "'e\<cdot>('g, 'f) writerT" assumes "\<And>k. P (WriterT\<cdot>k)" shows "P y"
Sup_dual_def_var: 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 dual :: "'b \<Rightarrow> 'b"   and X :: "'b set" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "dual (Sup X) = Inf (dual ` X)"
less_numeral_extra_3: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less zero zero"
pr_key_secrecy: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and m :: "nat"   and n :: "nat"   and run :: "nat"   and K :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "User m \<noteq> Spy"   and "Says n run 5 (User m) (Card n) (Crypt (sesK K) (Passwd m)) \<in> set evs" shows "Key (sesK K) \<notin> analz (A \<union> spies evs)"
Re_winding_number: fixes \<gamma> :: "real \<Rightarrow> complex"   and z :: "complex" assumes "valid_path \<gamma>"   and "z \<notin> path_image \<gamma>" shows "Re (winding_number \<gamma> z) = Im (contour_integral \<gamma> (\<lambda>w. 1 / (w - z))) / (2 * pi)"
token_succeeds_def: fixes x :: "nat" shows "\<UU>.token_succeeds x = (\<exists>n. \<UU>.token_run x n \<in> {q. \<G> \<Turnstile>\<^sub>P Rep q})"
subcube_def: fixes a :: "real"   and b :: "real"   and cube :: "int \<times> (real \<Rightarrow> real \<times> real)" shows "subcube a b cube = (fst cube, subpath a b (snd cube))"
right_MPair: fixes X :: "msg"   and Y :: "msg" shows "right (MPair X Y) = Y"
fv_subset_subterms: fixes t :: "('c, 'd) Term.term"   and M :: "('c, 'd) Term.term set" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M" shows "fv t \<subseteq> fv\<^sub>s\<^sub>e\<^sub>t M"
lincomb_def: fixes rs :: "'r list"   and ms :: "'m list" shows "rs \<bullet>\<cdot> ms = sum_list (map2 (\<cdot>) rs ms)"
is_min2_def: fixes x :: "'c \<times> 'd"   and xs :: "('c \<times> 'd) set" shows "is_min2 x xs \<equiv> x \<in> xs \<and> (\<forall>y\<in>xs. snd x \<le> snd y)"
less: fixes fail :: "ennreal" assumes "fail \<le> 1"   and "\<not> 1 \<le> expectation_gpv fail \<I> (\<lambda>_. 1) (c input)" shows "expectation_gpv fail \<I> (\<lambda>_. 1) gpv < ennreal (weight_spmf (the_gpv gpv)) + fail * ennreal (pmf (the_gpv gpv) None)"
r_code_const1_aux_prim: shows "prim_recfn 3 r_code_const1_aux"
elims: fixes x :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and xa :: "('f, 'i, 'j) new_thread_action"   and y :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" assumes "redT_updT' x xa = y" obtains ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" and t' :: "'f" and xb :: "'i" and m :: "'j" where "x = tsa__" and "xa = NewThread t' xb m" and "y = tsa__(t' \<mapsto> (undefined, no_wait_locks))"   | ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option" and v :: "'f" and va :: "bool" where "x = ts" and "xa = ThreadExists v va" and "y = ts"
subset_insertI: fixes B :: "'k set"   and a :: "'k" shows "B \<subseteq> insert a B"
v: fixes n :: "nat" shows "dist (to_Gromov_completion (v n)) (u n) < 1 / real (n + 1)"
\<R>_divergent_divergent_aux2: fixes M :: "(nat \<Rightarrow> bool) set" assumes "\<forall>i. \<forall>P\<in>M. \<exists>j\<ge>i. P j"   and "M \<noteq> {}"   and "finite M" shows "\<forall>i. \<exists>j\<ge>i. \<exists>k>j. \<exists>P\<in>M. P j \<and> P k \<and> (\<forall>m<k. j < m \<longrightarrow> \<not> P m) \<and> (\<forall>Q\<in>M. \<exists>m\<le>k. j < m \<and> Q m)"
that: fixes r :: "'a \<Rightarrow> 'a" assumes "T = r ` S"   and "r ` S \<subseteq> S"   and "continuous_on S r"   and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x" shows "thesis"
real_hyp2_C_cross_ratio_endpoints_in_S: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2"   and d :: "hyp2" assumes "a \<noteq> b"   and "a b \<congruent>\<^sub>K c d" shows "cross_ratio (endpoint_in_S (Rep_hyp2 a) (Rep_hyp2 b)) (endpoint_in_S (Rep_hyp2 b) (Rep_hyp2 a)) (Rep_hyp2 a) (Rep_hyp2 b) = cross_ratio (endpoint_in_S (Rep_hyp2 c) (Rep_hyp2 d)) (endpoint_in_S (Rep_hyp2 d) (Rep_hyp2 c)) (Rep_hyp2 c) (Rep_hyp2 d)"
prod_pOp_mem_i: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and X :: "'c \<Rightarrow> 'd"   and Y :: "'c \<Rightarrow> 'd"   and i :: "'c" assumes "\<forall>k\<in>I. aGroup (A k)"   and "X \<in> carr_prodag I A"   and "Y \<in> carr_prodag I A"   and "i \<in> I" shows "prod_pOp I A X Y i = X i \<plusminus>\<^bsub>A i\<^esub> Y i"
intros_4: fixes f :: "'a set \<Rightarrow> 'a set"   and g :: "'a set \<Rightarrow> 'a set" assumes "IK f"   and "IK g" shows "IK (f \<circ> g)"
simps: fixes F :: "'c program"   and a1 :: "'c set"   and a2 :: "'c set" shows "leadsp F a1 a2 = ((\<exists>A B. a1 = A \<and> a2 = B \<and> F \<in> A ensures B) \<or> (\<exists>A B C. a1 = A \<and> a2 = C \<and> leadsp F A B \<and> leadsp F B C) \<or> (\<exists>S B. a1 = \<Union> S \<and> a2 = B \<and> (\<forall>A\<in>S. leadsp F A B)))"
bind_simps_1: fixes f :: "'c \<Rightarrow> 'b list" shows "List.bind \<epsilon> f = \<epsilon>"
prems_1: shows "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>2"
arr_inv: fixes f :: "'a" assumes "C.iso f" shows "C.arr (C.inv f)"
tab_simps_2: shows "src (\<rho> \<cdot> \<phi>) = src f"
simps_9: fixes S :: "'a ltl set"   and v :: "'a ltl" shows "eval\<^sub>G S (F v) = F v"
power_eq_if: fixes p :: "'b list"   and m :: "nat" shows "p \<^sup>@ m = (if m = 0 then \<epsilon> else p \<cdot> p \<^sup>@ (m - 1))"
pathstart_rec_join_2: fixes \<gamma> :: "real \<Rightarrow> real \<times> real"   and l :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" shows "pathstart (rec_join ((- 1, \<gamma>) # l)) = pathstart (reversepath \<gamma>)"