hcju/isabelleps huggingface dataset:
Sup_sup_closed: fixes X :: "'b set" assumes "Sup_closed_set X" shows "sup_closed_set X"
sees_method_mono: fixes P :: "'c prog"   and C' :: "String.literal"   and C :: "String.literal"   and wf_md :: "'c prog \<Rightarrow> String.literal \<Rightarrow> 'c mdecl \<Rightarrow> bool"   and M :: "String.literal"   and Ts :: "ty list"   and T :: "ty"   and m :: "'c option"   and D :: "String.literal" assumes "P \<turnstile> C' \<preceq>\<^sup>* C"   and "wf_prog wf_md P"   and "P \<turnstile> C sees M: Ts\<rightarrow>T = m in D" shows "\<exists>D' Ts' T' m'. P \<turnstile> C' sees M: Ts'\<rightarrow>T' = m' in D' \<and> P \<turnstile> Ts [\<le>] Ts' \<and> P \<turnstile> T' \<le> T"
times_disjoint: fixes ls :: "'c list"   and ms :: "'c list"   and ks :: "'c list"   and f :: "'c \<times> 'c \<Rightarrow> 'd"   and g :: "'c \<times> 'c \<Rightarrow> 'd"   and ns :: "'c list" assumes "disjoint ls ms" shows "ks \<langle>f\<rangle> ls \<odot> ms \<langle>g\<rangle> ns = mbot"
abs_eq: fixes x :: "'b \<times> 'b" assumes "eq_onp (\<lambda>x. x \<in> {(a, b). a \<le> b}) x x" shows "lower (Interval x) = fst x"
varFIX: fixes b :: "BExpr"   and \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> ((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list) \<times> (Location \<times> Location) set \<Rightarrow> bool"   and \<beta> :: "(Location \<times> Location) set"   and s :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "(Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list" assumes "(b, \<Phi>, \<beta>, s, t) \<in> var" shows "PhiWhile b \<Phi> (s, t, \<beta>)"
uE_def: fixes c :: "nat \<times> nat \<Rightarrow> 'a" shows "ek_analysis_defs.uE c \<equiv> Graph.E c \<union> (Graph.E c)\<inverse>"
nfoldli_nfoldli_prod_conv: fixes l2 :: "'b list"   and ctd :: "'a \<Rightarrow> bool"   and l1 :: "'c list"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'a \<Rightarrow> 'a nres"   and s :: "'a" shows "nfoldli l2 ctd (\<lambda>i. nfoldli l1 ctd (f i)) s = nfoldli (List.product l2 l1) ctd (\<lambda>(i, j). f i j) s"
num_mod_den_simps_2: shows "left num_mod_den = num"
simps_43: fixes x7 :: "jump"   and x2 :: "expr" shows "Jmp x7 \<noteq> Expr x2"
LENGTH_contr: assumes "Suc 0 < LENGTH('a)"   and "2 ^ LENGTH('a) \<le> Suc (Suc 0)" shows "False"
add_mset_remove_trivial_If: fixes a :: "'b"   and N :: "'b multiset" shows "add_mset a (remove1_mset a N) = (if a \<in># N then N else add_mset a N)"
primerec_2_3: shows "primerec rec_power (Suc (Suc 0))"
assms_2: shows "\<one>\<^bsub>S\<^esub> \<noteq> \<zero>\<^bsub>S\<^esub>"
windows_def: fixes as :: "'b list"   and ws :: "(nat \<times> nat) list" shows "windows as ws = ((\<forall>w\<in>set ws. window as w) \<and> sorted (map fst ws) \<and> sorted (map snd ws))"
sel_rews_1: shows "sthead\<cdot>\<bottom> = \<bottom>"
lemma7: fixes r :: "lam"   and r' :: "lam" assumes "r \<rightarrow>\<^sub>s r'"   and "NF r'" shows "r \<rightarrow>\<^sub>l r'"
aadd_poss_less: fixes b :: "ant"   and a :: "ant" assumes "b \<noteq> \<infinity>"   and "b \<noteq> - \<infinity>"   and "0 < a" shows "b < a + b"
path_app: fixes g :: "'g"   and ns :: "'node list"   and ms :: "'node list" assumes "local.path g ns"   and "local.path g ms"   and "last ns = hd ms" shows "local.path g (ns @ tl ms)"
ltake_eq_LNil_iff: fixes n :: "enat"   and xs :: "'c llist" shows "(ltake n xs = LNil) = (xs = LNil \<or> n = 0)"
completeE: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and s :: "'a set"   and f :: "nat \<Rightarrow> 'a" assumes "class.metric_space dist uniformity open"   and "uniform_space.complete open uniformity s"   and "\<forall>n. f n \<in> s"   and "uniform_space.Cauchy uniformity f" obtains l :: "'a" where "l \<in> s" and "filterlim f (topological_space.nhds open l) sequentially"
term_c: shows "\<Gamma>\<turnstile>c \<down> s"
nopaid: fixes xa :: "'a list \<times> 'b"   and r :: "'a" shows "\<forall>z\<in>set_pmf (snd A xa r). snd (fst z) = []"
lift_by0: fixes x :: "V" shows "lift x 0 = 0"
syzygy_module_list_def: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "syzygy_module_list bs = atomize_poly ` idx_pm_of_pm bs ` pmdl.syzygy_module (set bs)"
map_simp: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and g :: "'ah"   and f :: "'ag" assumes "constant_transformation A B g"   and "partial_magma.arr A f" shows "constant_transformation.map A B g f = g"
IS_PURE_def: fixes P :: "('e \<times> 'f) set \<Rightarrow> bool"   and R :: "'f \<Rightarrow> 'e \<Rightarrow> assn" shows "IS_PURE P R \<equiv> is_pure R \<and> P (the_pure R)"
iso_class_elems_isomorphic: fixes F :: "'a set"   and f :: "'a"   and f' :: "'a" assumes "B.is_iso_class F"   and "f \<in> F"   and "f' \<in> F" shows "f \<cong>\<^sub>B f'"
61393: fixes x :: "'a"   and y :: "'a" shows "- (- x \<sqinter> - - y) = - (- x \<sqinter> y)"
reflp_def: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "reflp r = (\<forall>x. r x x)"
discriminant_lemma: fixes a :: "real"   and b :: "real"   and r :: "real"   and c :: "real" assumes "a \<noteq> 0"   and "b = 2 * a * r"   and "a * r\<^sup>2 - 2 * a * r * r + c = 0" shows "\<forall>x. (a * x\<^sup>2 + b * x + c = 0) = (x = - r)"
case_2: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Package of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f2"
to_conjugate_space_inverse: fixes y :: "'d" assumes "y \<in> UNIV" shows "from_conjugate_space (to_conjugate_space y) = y"
lossless_response_AND: fixes R :: "'i \<times> 'c"   and W :: "'h \<times> 'b"   and s :: "'e" shows "lossless_spmf (response_AND R W s)"
simps: fixes S :: "nat rexp set" shows "L S = \<Union> (lang ` S)"
deriv_rec_0_idem: fixes f :: "floatarith" shows "deriv_rec f 0 = f"
subset_spec_ivl_coll_impl: fixes X :: "'c set"   and ivl :: "'c set"   and z :: "bool nres" assumes "SPEC (\<lambda>b. b \<longrightarrow> X \<subseteq> ivl) \<le> z" shows "subset_spec_coll X ivl \<le> z"
inducts: fixes x :: "'state"   and P :: "'state \<Rightarrow> bool" assumes "reach x"   and "P istate"   and "\<And>s a. \<lbrakk>reach s; P s\<rbrakk> \<Longrightarrow> P (snd (step s a))" shows "P x"
wp_under_wlp_DC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "wp_under_wlp a"   and "wp_under_wlp b" shows "wp_under_wlp (a \<Sqinter> b)"
BK3_msg_in_parts_knows_Spy: fixes A :: "agent"   and KA :: "nat"   and Timestamp :: "msg"   and B :: "msg"   and K :: "msg"   and X :: "msg"   and evs :: "event list" assumes "Gets A (Crypt KA \<lbrace>Timestamp, B, K, X\<rbrace>) \<in> set evs"   and "evs \<in> bankerb_gets" shows "X \<in> parts (knows Spy evs)"
to_list_correct_1: fixes m :: "('b, 'c) assoc_list" assumes "lm.invar m" shows "map_of (lm.to_list m) = lm.\<alpha> m"
constant_degree: fixes p :: "'a poly" shows "constant (poly p) = (degree p = 0)"
parts_image_Key: fixes N :: "nat set" shows "parts (Key ` N) = Key ` N"
Int_Un_distrib2: fixes B :: "'c set"   and C :: "'c set"   and A :: "'c set" shows "(B \<union> C) \<inter> A = B \<inter> A \<union> C \<inter> A"
eqv_def: fixes \<phi> :: "fm"   and \<chi> :: "fm" shows "eqv \<phi> \<chi> = (\<phi> IMP \<chi>) AND (\<chi> IMP \<phi>)"
get_def: shows "get \<equiv> \<lambda>s. ({(s, s)}, False)"
antipar_2: shows "src d\<^sub>0 = trg e\<^sub>0"
map_of_SomeD: fixes xs :: "('d \<times> 'c) list"   and k :: "'d"   and y :: "'c" assumes "Map.map_of xs k = Some y" shows "(k, y) \<in> set xs"
is_processT2: fixes P :: "'c process" shows "\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s"
subgroup_conjugation_is_hom: shows "(\<lambda>g. \<lambda>H\<in>{H. subgroup H G}. g <# H #> inv g) \<in> hom G (BijGroup {H. subgroup H G})"
h1: shows "all_list_constr_R (get_subsets_R (reduce_system_R p (qs, matrix, subsets, signs))) (length qs)"
closed_segment_subset_interval: fixes T :: "'b set"   and a :: "'b"   and b :: "'b" assumes "is_interval T"   and "a \<in> T"   and "b \<in> T" shows "{a--b} \<subseteq> T"
l2norm_lmult: fixes f :: "real \<Rightarrow> real"   and S :: "real set"   and c :: "real" assumes "f square_integrable S" shows "l2norm S (\<lambda>x. c * f x) = \<bar>c\<bar> * l2norm S f"
typeof_ClassCast: fixes h :: "heap" assumes "preallocated h" shows "typeof\<^bsub>h\<^esub> (Addr (addr_of_sys_xcpt ClassCast)) = \<lfloor>Class ClassCast\<rfloor>"
psubsetD: fixes A :: "'b set"   and B :: "'b set"   and c :: "'b" assumes "A \<subset> B"   and "c \<in> A" shows "c \<in> B"
leg_simps_3: shows "C.arr trg_trg.leg1"
semiring_normalization_rules_1: fixes a :: "'a"   and m :: "'a"   and b :: "'a" shows "a * m + b * m = (a + b) * m"
nontriv_solution_def: shows "nontriv_solution = (\<lambda>(a, b). (a, b) \<noteq> (1::'a, 0::'a) \<and> a\<^sup>2 = (1::'a) + of_nat D * b\<^sup>2)"
R10_ad_1: shows "pmf (sds R10) a = 1 / 2"
stable_no_blocking_onI: fixes X :: "'a set"   and ds :: "'b set" assumes "\<And>h X''. \<lbrakk>X'' = Ch h (X \<union> X''); X'' \<noteq> Ch h X; X'' \<subseteq> Y.CD_on ds (X \<union> X'')\<rbrakk> \<Longrightarrow> False" shows "Y.stable_no_blocking_on ds X"
range_ordering: fixes A :: "ordinal set" assumes "normal_set A" shows "range (OrdinalVeblen.ordering A) = A"
set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes document_ptr :: "(_) document_ptr"   and children :: "(_) node_ptr list" shows "set_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr children = (case children of [] \<Rightarrow> put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document_element_update None | [child] \<Rightarrow> case cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r child of None \<Rightarrow> error HierarchyRequestError | Some element_ptr \<Rightarrow> put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t document_ptr document_element_update (Some element_ptr) | child # aa # lista \<Rightarrow> error HierarchyRequestError)"
IH: fixes y :: "nat"   and a :: "real"   and b :: "real" assumes "y < i"   and "0 \<le> a \<and> a \<le> b \<or> odd y \<and> a \<le> b \<or> even y \<and> a \<le> 0 \<and> b \<le> a" shows "power_up prec a y \<le> power_up prec b y"
ineq_rec: fixes n :: "nat"   and r :: "nat"   and a :: "nat" assumes "N0 < n"   and "N0 < r" shows "u (a * n + r) \<le> real a * u n + u r"
assms: shows "At \<in> range aNon \<union> range aAgt \<union> range aNum"
trans: fixes r :: "'r"   and s :: "'r"   and t :: "'r" assumes "r = s"   and "s = t" shows "r = t"
perm_3: fixes pi :: "(name \<times> name) list" shows "pi \<bullet> \<tau> = \<tau>"
snd_sym_preproc_aux_pmdl: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and ks :: "'t list"   and ts :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pmdl (set gs \<union> set (snd (sym_preproc_aux gs ks (ts, fs)))) = pmdl (set gs \<union> set fs)"
fun_diff_def: fixes A :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd" shows "A - B = (\<lambda>x. A x - B x)"
simps: fixes as :: "'b iarray"   and n :: "integer" shows "sub' (as, n) = as !! nat_of_integer n"
runit_simps_4: fixes f :: "'a" assumes "ide f" shows "local.dom \<r>[f] = f \<star> src f"
Subring_tOp_ring_tOp: fixes S :: "('a, 'd) Ring_scheme"   and a :: "'a"   and b :: "'a" assumes "Subring R S"   and "a \<in> carrier S"   and "b \<in> carrier S" shows "a \<cdot>\<^sub>r\<^bsub>S\<^esub> b = a \<cdot>\<^sub>r b"
dest_fun_\<pi>_0: fixes i :: "nat"   and T :: "Unary_PCF.type" assumes "i < ar T" shows "dest_fun (\<pi> T i 0) = nth_drop i (dest_fun T)"
Span_smult_closed: fixes K :: "'a set"   and Us :: "'a list"   and k :: "'a"   and v :: "'a" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "k \<in> K"   and "v \<in> Span K Us" shows "k \<otimes> v \<in> Span K Us"
insert_dom_def: fixes p :: "'b \<Rightarrow> 'c"   and i :: "'b"   and j :: "'c" shows "insert_dom p i j \<equiv> \<lambda>i'. if i' < i then p i' else if i' = i then j else p (i' - (1::'b))"
count_add_mset: fixes b :: "'e"   and A :: "'e multiset"   and a :: "'e" shows "count (add_mset b A) a = (if b = a then Suc (count A a) else count A a)"
simps_3: fixes uu :: "typ"   and t :: "Core.term"   and k :: "nat" shows "loose_bvar1 (Abs uu t) k = loose_bvar1 t (k + 1)"
ih1: fixes t :: "trm" assumes "e\<^sub>1 \<Down> t" shows "Lam [x].e\<^sub>1' = t"
1: shows "make_mat n (Suc i) (snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i))))) $$ (0, Suc i) = snd (Smith_1x2 (Matrix.mat 1 2 (\<lambda>(a, b). if b = 0 then A $$ (0, 0) else A $$ (0, Suc i)))) $$ (0, 1)"
ex_unit_disc_fix_to_zero_positive_x_axis: fixes u :: "complex_homo"   and v :: "complex_homo" assumes "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<noteq> v" shows "\<exists>M. unit_disc_fix M \<and> moebius_pt M u = 0\<^sub>h \<and> moebius_pt M v \<in> positive_x_axis"
linear_homeomorphism_image: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "linear f"   and "inj f" obtains g :: "'d \<Rightarrow> 'c" where "homeomorphism (f ` S) S g f"
lt_max_keys: fixes v :: "'x \<Rightarrow>\<^sub>0 nat"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "v \<in> keys p" shows "v \<preceq> lpp p"
hyps_1: shows "P \<turnstile> T casts v to v' "
bounded_bilinear_blinfun_compose: shows "bounded_bilinear (o\<^sub>L)"
sep_conjE: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(P \<and>* Q) h" obtains x :: "'a" and y :: "'a" where "P x" and "Q y" and "x ## y" and "h = x + y"
vec_nth_inverse: fixes x :: "('d, 'e) Finite_Cartesian_Product.vec" shows "vec_lambda (($h) x) = x"
order_subst1: fixes a :: "'e"   and f :: "'f \<Rightarrow> 'e"   and b :: "'f"   and c :: "'f" assumes "a \<le> f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
abs_eq: fixes x :: "int" shows "double_of_integer (integer_of_int x) = double_of_int x"
mono: fixes a :: "'g"   and c :: "'g"   and b :: "'g"   and d :: "'g" assumes "a \<le> c"   and "b \<le> d" shows "min a b \<le> min c d"
assms: fixes x :: "'a" assumes "x \<in> A - B \<union> (B - A)" shows "f x = (0::'b)"
sup_bot_left: fixes x :: "'a" shows "(zero_class.zero::'a) + x = x"
r_append_zeros: fixes b :: "nat"   and z :: "nat" shows "eval r_append_zeros [b, z] \<down>= e_append_zeros b z"
abs_eq: fixes x :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" shows "col (of_fun x) = (\<lambda>j. \<chi>i. x i j)"
valuations_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and vv :: "nat \<Rightarrow> 'c \<Rightarrow> ant" shows "valuations K n vv = (\<forall>j\<le>n. valuation K (vv j))"
rational_boxes: fixes e :: "real"   and x :: "'b" assumes "0 < e" shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat>) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
C_eq_iD_aux2: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "noDenyAll1 p"   and "separated p"   and "p \<noteq> []"   and "x \<in> dom (Cp (list2FWpolicy p))" shows "Cp (list2FWpolicy (insertDenies p)) x = Cp (list2FWpolicy p) x"
hyp1: assumes "isTable E vs (deleteAround g a ps)" shows "ExcessNotAtRec (deleteAround g a ps) g = ListSum (ExcessNotAtRecList (deleteAround g a ps) g) E"
sums_iff_shift': fixes f :: "nat \<Rightarrow> 'b"   and n :: "nat"   and s :: "'b" shows "(\<lambda>i. f (i + n)) sums (s - sum f {..<n}) = f sums s"
Seq_def: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "P `;` Q \<equiv> action_of (action.relation_of P ;; action.relation_of Q)"
simps_3: fixes ao :: "'a option"   and uu :: "'a list" shows "proper_interval_Compl_set_aux ao uu [] = False"
lens_compat_def: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" shows "(X ##\<^sub>L Y) = (\<forall>s\<^sub>1 s\<^sub>2. s\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>2)"
borel_measurable_const: fixes c :: "'b"   and M :: "'a measure" shows "(\<lambda>x. c) \<in> borel_measurable M"
simps_5: fixes \<sigma> :: "(char list \<times> 'c list) trace"   and v :: "'c list"   and i :: "nat"   and \<phi> :: "'c MFOTL.formula" shows "MFOTL.sat \<sigma> v i (formula.Exists \<phi>) = (\<exists>z. MFOTL.sat \<sigma> (z # v) i \<phi>)"
size_subst: fixes f :: "form" shows "\<forall>m. size (subst m f) = size f"
del_point_blocks_def: fixes p :: "'a" shows "del_point_blocks p \<equiv> {#bl - {p}. bl \<in># \<B>#}"
WbisT_MtransT: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and s' :: "'state" assumes "c \<approx>wT d"   and "s \<approx> t"   and "(c, s) \<rightarrow>*t s'" obtains t' :: "'state" where "(d, t) \<rightarrow>*t t'" and "s' \<approx> t'"
less_funI: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<exists>k. f k < g k \<and> (\<forall>k'<k. f k' = g k')" shows "less_fun f g"
l_def: shows "l = int b ^ (nat \<lfloor>log (real b) (real a)\<rfloor> + 1)"
Tw_segment_segment: fixes E :: "('e, 'f) Order_scheme"   and a :: "'a" assumes "Worder E"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))"   and "a \<in> carrier D" shows "(Tw\<^bsub>D,E\<^esub>) ` segment D a = segment E ((Tw\<^bsub>D,E\<^esub>) a)"
mt: shows "((\<lambda>x. ((1 - x) ^ (n - 1) / fact (n - 1)) *\<^sub>R Df (X + x *\<^sub>R H) n H) has_integral f (X + H) - (\<Sum>i<n. (1 / fact i) *\<^sub>R Df X i H)) {0..1}"
result_evaluated: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and L :: "Vars.var list"   and \<Delta> :: "(Vars.var \<times> exp) list"   and z :: "exp" assumes "\<Gamma> : e \<Down>\<^bsub>L\<^esub> \<Delta> : z" shows "isVal z"
XX: shows "\<forall>X\<in>XX. well_founded X (\<sqsubset>) \<and> (\<forall>x\<in>X. \<forall>y\<in>\<Union> XX. y \<sqsubset> x \<longrightarrow> y \<in> X)"
Comp: fixes G :: "prog"   and s0 :: "st"   and c1 :: "stmt"   and n :: "nat"   and s1 :: "state"   and c2 :: "stmt"   and s2 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<midarrow>n\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c2\<midarrow>n\<rightarrow> s2" shows "G\<turnstile>Norm s0 \<midarrow>c1;; c2\<midarrow>n\<rightarrow> s2"
field_less_half_sum: fixes x :: "'a"   and y :: "'a" assumes "x < y" shows "x < (x + y) / (2::'a)"
bot_least: fixes a :: "'c \<times> 'c \<Rightarrow> 'd" shows "a \<preceq> mtop"
ground_chop: fixes t :: "('a, 'b) tm" assumes "is_App t"   and "ground t" shows "ground (chop t)"
assms_3: fixes s :: "'s" assumes "s \<in> S"   and "n s = 0" shows "x s = 0"
dmu_array_row: fixes mm :: "nat" assumes "mm \<le> length fs" shows "dmu_array_row (IArray fs) (IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) mm) mm = IArray.of_fun (\<lambda>i. IArray.of_fun (\<mu>' i) (Suc i)) (Suc mm)"
induct: fixes P :: "('b, 'c, 'd) sp\<^sub>\<mu> \<Rightarrow> bool"   and sp\<^sub>\<mu> :: "('b, 'c, 'd) sp\<^sub>\<mu>" assumes "\<And>x. (\<And>xa. xa \<in> range x \<Longrightarrow> P xa) \<Longrightarrow> P (Get x)"   and "\<And>x1a x2. P (Put x1a x2)" shows "P sp\<^sub>\<mu>"
lm12: fixes N :: "'c set"   and \<Omega> :: "'d set"   and a :: "('c \<times> 'd set) set" shows "summedBidVectorRel (pseudoAllocation a <| (N \<times> finestpart \<Omega>)) N \<Omega> = summedBidVectorRel (bidMaximizedBy a N \<Omega> Elsee 0) N \<Omega>"
intros: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t = head s"   and "chkchop_same (>\<^sub>t) t s"   and "\<forall>f\<in>local.ground_heads (head t). extf f (>\<^sub>t) (args t) (args s)" shows "gt_same t s"
IH_1: fixes \<Gamma>' :: "nat \<Rightarrow> pdf_type" assumes "\<And>x. x \<in> free_vars e1 \<Longrightarrow> \<Gamma> x = \<Gamma>' x" shows "\<Gamma>' \<turnstile> e1 : t1"
G'_def: shows "G' \<equiv> LG (on_triple g `` edges (snd R) \<union> E) {0..<max n (nextMax (Range g))}"
funcset_id: fixes A :: "'b set" shows "(\<lambda>x. x) \<in> A \<rightarrow> A"
number_partition_of_eq: shows "number_partition_of A B F = image_mset card (mset_set ((\<lambda>b. {x \<in> A. f x = b}) ` B - {{}}))"
Norm_Distr_2: fixes r :: "'c \<times> 'd \<Rightarrow> 'b"   and P1 :: "'f \<Rightarrow> 'c decision option"   and P2 :: "('g \<Rightarrow> 'd decision option) list"   and d :: "'e \<Rightarrow> 'f \<times> 'g"   and x :: "'e" shows "(r o\<^sub>f (P1 \<Otimes>\<^sub>2 list2policy P2 \<circ> d)) x = list2policy ((P1 \<Otimes>\<^sub>L P2) (\<Otimes>\<^sub>2) r d) x"
assms_5: shows "new_id \<notin> I ` insert_messages M"
eq: shows "{x \<in> U. f x \<in> V} = U \<inter> {x \<in> topspace X. f x \<in> V}"
cases: fixes a1 :: "('Var, 'AExp, 'BExp) Stmt"   and a2 :: "Mode \<Rightarrow> 'Var set"   and a3 :: "'Var \<Rightarrow> 'Val"   and a4 :: "('Var, 'AExp, 'BExp) Stmt"   and a5 :: "Mode \<Rightarrow> 'Var set"   and a6 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option"   and \<S>' :: "'Var set \<times> 'Var set"   and P' :: "'BExp set" assumes "\<langle>a1, a2, a3\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>',\<S>',P'\<^esub> \<langle>a4, a5, a6\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" and c :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" where "a1 = c" and "a2 = mds" and "a3 = mem\<^sub>1" and "a4 = c" and "a5 = mds" and "a6 = mem\<^sub>2" and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'" and "tyenv_wellformed mds \<Gamma> \<S> P" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" and "pred P mem\<^sub>1" and "pred P mem\<^sub>2" and "\<forall>x\<in>dom \<Gamma>. x \<notin> mds AsmNoReadOrWrite \<longrightarrow> type_max (the (\<Gamma> x)) mem\<^sub>1 \<le> dma mem\<^sub>1 x"
mat1: fixes nr :: "nat"   and ze :: "'a"   and on :: "'a" shows "mat nr nr (mat1I ze on nr)"
magQ_one: shows "\<lbrakk>1\<rbrakk>\<^sub>Q = (1::'a)"
extensional_restrict: fixes f :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "f \<in> extensional A" shows "restrict f A = f"
next_marker_free_state_def: fixes t :: "('a, 'b, 'c) event list"   and i :: "nat"   and cid :: "nat" shows "next_marker_free_state t i cid = (LEAST j. i \<le> j \<and> Marker \<notin> set (msgs (S t j) cid))"
Completeness: 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.ll \<NN> \<TT> \<RR> \<SS> Lex Sel Doc" shows "\<exists>\<alpha>. LocalLexing.pvalid_with \<NN> \<TT> \<RR> \<SS> Doc p (LocalLexing.finished_item \<SS> Doc \<alpha>) 0 [] \<and> LocalLexing.finished_item \<SS> Doc \<alpha> \<in> LocalLexing.\<II> \<TT> \<RR> \<SS> Lex Sel Doc"
Int_absorb2: fixes A :: "'b set"   and B :: "'b set" assumes "A \<subseteq> B" shows "A \<inter> B = A"
infsetsum_def: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set" shows "infsetsum f A = integral\<^sup>L (count_space A) f"