hcju/isabelleps huggingface dataset:
cases: fixes a :: "'a set \<Rightarrow> 'a set" assumes "CK_nf a" obtains "a = id"   | "a = C"   | "a = K"   | "a = C \<circ> K"   | "a = K \<circ> C"   | "a = C \<circ> K \<circ> C"   | "a = K \<circ> C \<circ> K"   | "a = C \<circ> K \<circ> C \<circ> K"   | "a = K \<circ> C \<circ> K \<circ> C"   | "a = C \<circ> K \<circ> C \<circ> K \<circ> C"   | "a = K \<circ> C \<circ> K \<circ> C \<circ> K"   | "a = C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K"   | "a = K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C"   | "a = C \<circ> K \<circ> C \<circ> K \<circ> C \<circ> K \<circ> C"
sat_def: fixes H :: "'b pl set"   and p :: "'b pl" shows "(H |= p) = (\<forall>tt. (\<forall>q\<in>H. tt[[q]]) \<longrightarrow> tt[[p]])"
1: shows "val_Zp (x \<ominus> y) = val_Zp y"
one_fps_def: shows "1 = Abs_fps (\<lambda>n. if n = [] then 1::'d else (0::'d))"
singular_relcycle_diff: fixes p :: "nat"   and X :: "'c topology"   and S :: "'c set"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'c) \<Rightarrow>\<^sub>0 int" assumes "singular_relcycle p X S a"   and "singular_relcycle p X S b" shows "singular_relcycle p X S (a - b)"
assms_3: shows "\<forall>n. qty (Suc n) \<in> borel_measurable (F n)"
distinct_41: fixes x61 :: "('a, 'b, 'c) hp"   and x62 :: "('a, 'b, 'c) formula"   and x71 :: "'b"   and x72 :: "('a, 'b, 'c) formula" shows "(\<langle> x61 \<rangle> x62) \<noteq> InContext x71 x72"
execn_strip_to_execn: fixes F :: "'g set"   and \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and c :: "('f, 'e, 'g) com"   and s :: "('f, 'g) xstate"   and n :: "nat"   and t :: "('f, 'g) xstate" assumes "strip F \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<exists>t'. \<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t' \<and> (isFault t \<longrightarrow> isFault t') \<and> (t' \<in> Fault ` (- F) \<longrightarrow> t' = t) \<and> (\<not> isFault t' \<longrightarrow> t' = t)"
emeasure_distr: fixes f :: "'d \<Rightarrow> 'e"   and M :: "'d measure"   and N :: "'e measure"   and A :: "'e set" assumes "f \<in> M \<rightarrow>\<^sub>M N"   and "A \<in> sets N" shows "emeasure (distr M N f) A = emeasure M (f -` A \<inter> space M)"
pair_bidirected_digraph_axioms: shows "pair_bidirected_digraph H"
select_convs_1: fixes dpm :: "(nat \<times> nat \<times> nat) pointermap"   and dcl :: "nat \<times> nat \<times> nat \<Rightarrow> nat option"   and more :: "'a" shows "dpm \<lparr>dpm = dpm, dcl = dcl, \<dots> = more\<rparr> = dpm"
surjective_var: fixes x :: "'a" shows "surjective x = surjective_var x"
wiring_c1r22_c1r22_def: fixes cnv1 :: "('p, 'r, 't + 'v, 'x + 'z) converter"   and cnv2 :: "('q, 's, 'u + 'w, 'y + 'aa) converter" shows "wiring_c1r22_c1r22 cnv1 cnv2 \<equiv> (cnv1 |\<^sub>= cnv2) \<odot> parallel_wiring"
classes_above_classes_changed_sym: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "classes_above P' C \<inter> classes_changed P' P = {}"
SbisL_Sbis: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and tval :: "'d \<Rightarrow> 'b \<Rightarrow> bool"   and cl :: "('d, 'c, 'a) cmd list"   and dl :: "('d, 'c, 'a) cmd list"   and n :: "nat" assumes "PL_Indis cval indis"   and "PL_Indis.SbisL aval tval cval indis cl dl"   and "n < length cl \<or> n < length dl" shows "(cl ! n, dl ! n) \<in> PL_Indis.Sbis aval tval cval indis"
simps_2: fixes c :: "color"   and uu :: "('a, 'b) rbt"   and uv :: "'a"   and uw :: "'b"   and ux :: "('a, 'b) rbt" shows "color_of (Branch c uu uv uw ux) = c"
FOREACHoci_refine_genR: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and RR :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and RR' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "(('c set \<times> 'e) \<times> 'd set \<times> 'f) set"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres"   and R' :: "('e \<times> 'f) set" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; RR x y\<rbrakk> \<Longrightarrow> RR' (\<alpha> x) (\<alpha> y)"   and "((S, \<sigma>0), \<alpha> ` S, \<sigma>0') \<in> R"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi> it \<sigma>; \<forall>x\<in>S - it. \<forall>y\<in>it. RR x y; \<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<forall>x\<in>S - it. \<forall>y\<in>it. RR x y; \<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; \<forall>x\<in>S - it. \<forall>y\<in>it. RR x y; \<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y; x' = \<alpha> x; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; x \<in> it; \<forall>y\<in>it - {x}. RR x y; x' \<in> it'; \<forall>y'\<in>it' - {x'}. RR' x' y'; c \<sigma>; c' \<sigma>'\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). ((it - {x}, \<sigma>), it' - {x'}, \<sigma>') \<in> R} (f' x' \<sigma>')"   and "\<And>\<sigma> \<sigma>'. \<lbrakk>\<Phi> {} \<sigma>; \<Phi>' {} \<sigma>'; (({}, \<sigma>), {}, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; \<forall>x\<in>S - it. \<forall>y\<in>it. RR x y; \<forall>x\<in>S' - it'. \<forall>y\<in>it'. RR' x y; it' = \<alpha> ` it; ((it, \<sigma>), it', \<sigma>') \<in> R; it \<noteq> {}; it' \<noteq> {}; \<not> c \<sigma>; \<not> c' \<sigma>'\<rbrakk> \<Longrightarrow> (\<sigma>, \<sigma>') \<in> R'" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>RR,\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R' (FOREACH\<^sub>O\<^sub>C\<^bsup>RR',\<Phi>'\<^esup> S' c' f' \<sigma>0')"
subsumes_modulo_def: fixes C :: "('a, 'b) Term.term literal multiset"   and D :: "('a, 'b) Term.term literal multiset"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term option" shows "subsumes_modulo C D \<sigma> = (\<exists>\<tau>. dom \<tau> = vars_clause C \<union> dom \<sigma> \<and> extends_subst \<sigma> \<tau> \<and> subst_cls C (subst_of_map Var \<tau>) \<subseteq># D)"
valid_ref_def: fixes A :: "'b attree"   and A' :: "'b attree" shows "A \<sqsubseteq>\<^sub>V A' \<equiv> A \<sqsubseteq> A' \<and> \<turnstile>A'"
wpt_Seq: fixes c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "wp\<^sub>t (c\<^sub>1;; c\<^sub>2) Q = wp\<^sub>t c\<^sub>1 (wp\<^sub>t c\<^sub>2 Q)"
flow_has_vector_derivative: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "(flow0 x0 has_vector_derivative f (flow0 x0 t)) (at t)"
list_all2_Cons_append_singletonI: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and ys :: "'c list"   and y :: "'c" assumes "xs \<noteq> []"   and "list_all2 P (x # butlast xs) ys"   and "P (last xs) y" shows "list_all2 P (x # xs) (ys @ [y])"
map_zip_map: fixes f :: "'e \<times> 'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'e"   and xs :: "'g list"   and ys :: "'f list" shows "map f (zip (map g xs) ys) = map2 (\<lambda>x y. f (g x, y)) xs ys"
divide_inverse: fixes x :: "'b"   and y :: "'b" shows "x / y = x * inverse y"
s\<^sub>2: shows "s\<^sub>2 = (s\<^sub>1(r \<mapsto> (\<sigma>, \<tau>;;\<tau>', \<E> [VE (CV Unit)])))(r'' := None)"
symvar_var_imp_var: fixes sv :: "'c \<times> nat"   and e :: "('c \<Rightarrow> 'd) \<Rightarrow> 'd"   and s :: "'c \<Rightarrow> nat" assumes "sv \<in> Aexp.vars (adapt_aexp e s)" shows "fst sv \<in> Aexp.vars e"
a_carrier_vec: fixes l :: "nat"   and i :: "nat" assumes "l \<le> i"   and "i < length fs" shows "a i l \<in> carrier_vec n"
bres_bot_bot: shows "prod.bres bot_prod bot_prod = top_prod"
has_integral_eq: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and k :: "'b" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"   and "(f has_integral k) s" shows "(g has_integral k) s"
embed_bool_nneg: fixes P :: "'c \<Rightarrow> bool" shows "nneg \<guillemotleft> P \<guillemotright>"
convex_polychain_def: fixes xs :: "((real \<times> real) \<times> real \<times> real) list" shows "convex_polychain xs = (polychain xs \<and> (\<forall>i. Suc i < length xs \<longrightarrow> 0 < det3 (fst (xs ! i)) (snd (xs ! i)) (snd (xs ! Suc i))))"
finite_set_uprod: fixes x :: "'a uprod" shows "finite (set_uprod x)"
Eq\<^sub>EE: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "[x \<^bold>=\<^sub>E y in v]" shows "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = Semantics.d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = Semantics.d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
Rep_fin_string_inject: fixes x :: "fin_string"   and y :: "fin_string" shows "(Rep_fin_string x = Rep_fin_string y) = (x = y)"
induct: fixes x1 :: "nat \<times> nat + nat + nat \<times> nat"   and x2 :: "(symbol \<times> character list) list set + (symbol \<times> character list) set"   and P :: "nat \<times> nat + nat + nat \<times> nat \<Rightarrow> (symbol \<times> character list) list set + (symbol \<times> character list) set \<Rightarrow> bool" assumes "\<P>_\<Q>_\<Z>_graph x1 x2"   and "\<And>\<P>_\<Q>_\<Z>_sum. P (Inl (0, 0)) (Inl {[]})"   and "\<And>\<P>_\<Q>_\<Z>_sum k u. \<lbrakk>\<P>_\<Q>_\<Z>_graph (Inr (Inr (k, Suc u))) (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, Suc u)))); P (Inr (Inr (k, Suc u))) (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, Suc u)))); \<P>_\<Q>_\<Z>_graph (Inl (k, u)) (\<P>_\<Q>_\<Z>_sum (Inl (k, u))); P (Inl (k, u)) (\<P>_\<Q>_\<Z>_sum (Inl (k, u)))\<rbrakk> \<Longrightarrow> P (Inl (k, Suc u)) (Inl (limit (Append (projr (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, Suc u))))) k) (projl (\<P>_\<Q>_\<Z>_sum (Inl (k, u))))))"   and "\<And>\<P>_\<Q>_\<Z>_sum k. \<lbrakk>\<P>_\<Q>_\<Z>_graph (Inr (Inl k)) (\<P>_\<Q>_\<Z>_sum (Inr (Inl k))); P (Inr (Inl k)) (\<P>_\<Q>_\<Z>_sum (Inr (Inl k)))\<rbrakk> \<Longrightarrow> P (Inl (Suc k, 0)) (Inl (projl (\<P>_\<Q>_\<Z>_sum (Inr (Inl k)))))"   and "\<And>\<P>_\<Q>_\<Z>_sum k. P (Inr (Inr (k, 0))) (Inr {})"   and "\<And>\<P>_\<Q>_\<Z>_sum k u. \<lbrakk>\<P>_\<Q>_\<Z>_graph (Inr (Inr (k, u))) (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, u)))); P (Inr (Inr (k, u))) (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, u)))); \<P>_\<Q>_\<Z>_graph (Inl (k, u)) (\<P>_\<Q>_\<Z>_sum (Inl (k, u))); P (Inl (k, u)) (\<P>_\<Q>_\<Z>_sum (Inl (k, u)))\<rbrakk> \<Longrightarrow> P (Inr (Inr (k, Suc u))) (Inr (\<Y> (projr (\<P>_\<Q>_\<Z>_sum (Inr (Inr (k, u))))) (projl (\<P>_\<Q>_\<Z>_sum (Inl (k, u)))) k))"   and "\<And>\<P>_\<Q>_\<Z>_sum k. \<lbrakk>\<And>x. \<P>_\<Q>_\<Z>_graph (Inl (k, x)) (\<P>_\<Q>_\<Z>_sum (Inl (k, x))); \<And>x. P (Inl (k, x)) (\<P>_\<Q>_\<Z>_sum (Inl (k, x)))\<rbrakk> \<Longrightarrow> P (Inr (Inl k)) (Inl (natUnion (\<lambda>x1. projl (\<P>_\<Q>_\<Z>_sum (Inl (k, x1))))))" shows "P x1 x2"
f_bound_def: fixes outside :: "bool"   and ii :: "nat"   and fs :: "int vec list" shows "f_bound outside ii fs = (\<forall>i<m. \<parallel>fs ! i\<parallel>\<^sup>2 \<le> (if i \<noteq> ii \<or> outside then int (N * m) else int (4 ^ (m - 1) * N ^ m * m * m)))"
strict_implies_not_eq: fixes b :: "'a word"   and a :: "'a word" assumes "b <s a" shows "a \<noteq> b"
inter_aux_Par_def: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "'b \<Rightarrow> ('a, 'b, 'c) ann list option"   and F :: "'c set" shows "inter_aux_Par \<Gamma> \<Theta> F \<equiv> \<lambda>(cs, as, c, a). \<forall>i<length cs. interfree_aux \<Gamma> \<Theta> F (cs ! i, as ! i, c, a)"
os1: fixes M :: "lam" shows " M \<longrightarrow>\<^sub>1\<^sup>* M"
mult_Mp_2: fixes f :: "int poly"   and g :: "int poly" shows "q.eq_m (f * q.Mp g) (f * g)"
fps_exp_power_mult: fixes c :: "'a"   and n :: "nat" shows "fps_exp c ^ n = fps_exp (of_nat n * c)"
function_ring_car_memI: fixes R :: "('c, 'd) ring_scheme"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "\<And>a. a \<in> S \<Longrightarrow> f a \<in> carrier R"   and "\<And>a. a \<notin> S \<Longrightarrow> f a = undefined" shows "f \<in> carrier (function_ring S R)"
comp_assoc': fixes g :: "'a"   and f :: "'a"   and h :: "'a" assumes "seq g f"   and "seq h g" shows "(h \<cdot> g) \<cdot> f = h \<cdot> g \<cdot> f"
preserves_reflects_arr: fixes f :: "'a1" shows "A2_B.arr (F f) = A1.arr f"
hyps: assumes "k < ncols A"   and "j < ncols A" shows "span (cols_upt_k A k) = span (cols_upt_k (Gram_Schmidt_column_k A j) k)"
assms_2: shows "a \<equiv> \<lambda>k m. 1 / of_nat k * (\<Sum>d | d dvd gcd m k. g d * f (k div d) * of_nat d)"
l_new_character_data_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   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" shows "l_new_character_data_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_new_character_data type_wf \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_new_character_data_get_child_nodes_axioms get_child_nodes get_child_nodes_locs"
ebounds_def: fixes as :: "atom list" shows "ebounds as = concat (map (\<lambda>x. case x of Less real list \<Rightarrow> [] | Eq r [] \<Rightarrow> [] | Eq r (c # cs) \<Rightarrow> if c \<noteq> 0 then [(r / c, (- 1 / c) *\<^sub>s cs)] else []) as)"
single_valuedI: fixes r :: "('e \<times> 'f) set" assumes "\<And>x y z. \<lbrakk>(x, y) \<in> r; (x, z) \<in> r\<rbrakk> \<Longrightarrow> y = z" shows "single_valued r"
target: fixes T1 :: "'procT"   and T2 :: "'procT"   and TRel :: "('procT \<times> 'procT) set"   and SRel :: "('procS \<times> 'procS) set" assumes "(T1, T2) \<in> TRel" shows "TargetTerm T1 \<R>\<lbrakk>\<cdot>\<rbrakk>L<SRel,TRel> TargetTerm T2"
left_spec_None_iff: fixes xs :: "int list"   and i :: "nat" shows "(left_spec xs i = None) = (\<not> has_left xs i)"
abs_int_poly_represents: fixes p :: "int poly"   and x :: "'a" shows "abs_int_poly p represents x = p represents x"
abs_eq: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and x :: "int" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (Word.Word x) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('l) x))"
subs: shows "P \<turnstile> (S # Ss) [\<le>] Ts"
hyps: assumes "\<And>k. k < n + 1 \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d f (n + 1) = f 0 + matrix_sum d (\<lambda>k. f (k + 1)) n"
upclosed_set_iff: fixes dual :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "class.preorder_with_dual dual less_eq less" shows "ord.upclosed_set less_eq X = (\<forall>x\<in>X. \<forall>y. less_eq x y \<longrightarrow> y \<in> X)"
fun_below_iff: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" shows "(f \<sqsubseteq> g) = (\<forall>x. f x \<sqsubseteq> g x)"
associative: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "a + b + c = a + (b + c)"
wf_classes_iff_exe_wf_classes: fixes cs :: "('a \<times> 'a) list" shows "wf_subclass (set cs) = exe_wf_subclass cs"
elims_2: fixes x :: "term" assumes "linear x" obtains uu_ :: "name" where "x = Free uu_"   | uv_ :: "name" where "x = Const uv_"   | t\<^sub>1 :: "term" and t\<^sub>2 :: "term" where "x = t\<^sub>1 $ t\<^sub>2" and "linear t\<^sub>1 \<and> linear t\<^sub>2 \<and> \<not> is_free t\<^sub>1 \<and> fdisjnt (frees t\<^sub>1) (frees t\<^sub>2)"
eq_sym_conv: fixes a :: "'b"   and b :: "'b" shows "(a = b) = (b = a)"
IdInj: fixes C :: "('a, 'b, 'c) Category_scheme"   and X :: "'a"   and Y :: "'a" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "Y \<in> obj\<^bsub>C\<^esub>"   and "id\<^bsub>C\<^esub> X = id\<^bsub>C\<^esub> Y" shows "X = Y"
assms_4: shows "set V \<subseteq> \<V> g"
Let_def: fixes s :: "'n"   and f :: "'n \<Rightarrow> 'm" shows "Let s f \<equiv> f s"
mult_le_cancel_left_pos: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times"   and "less zero c" shows "less_eq (times c a) (times c b) = less_eq a b"
vsv_appI: fixes a :: "V"   and b :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> vreal_uminus" shows "-\<^sub>\<real> a = b"
inject_2: fixes x21 :: "'a"   and x22 :: "('a, 'b) Term.term list"   and y21 :: "'a"   and y22 :: "('a, 'b) Term.term list" shows "(Fun x21 x22 = Fun y21 y22) = (x21 = y21 \<and> x22 = y22)"
prod_mset_def: shows "\<Prod>\<^sub># \<equiv> comm_monoid_mset.F (*) (1::'a)"
sets_into_space: fixes x :: "'a set" assumes "x \<in> Dynkin \<Omega> M" shows "x \<subseteq> \<Omega>"
disjnt_insert1: fixes a :: "'c"   and X :: "'c set"   and Y :: "'c set" shows "disjnt (insert a X) Y = (a \<notin> Y \<and> disjnt X Y)"
subset_translation_eq: fixes a :: "'a"   and s :: "'a set"   and t :: "'a set" shows "((+) a ` s \<subseteq> (+) a ` t) = (s \<subseteq> t)"
blinding_of_hash_R1: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p h h (=)" shows "list_R1.rel_list_R1 bo \<le> vimage2p (list_R1.map_list_R1 h) (list_R1.map_list_R1 h) (=)"
ring_simprules_21: fixes R :: "('a, 'b) ring_scheme" assumes "ring R" shows "\<ominus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
NatTrans_def: fixes \<eta> :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans" shows "NatTrans \<eta> \<equiv> NatTransP \<eta> \<and> NatTransExt \<eta>"
borel_measurable_prod_ennreal: fixes S :: "'d set"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> ennreal"   and M :: "'e measure" assumes "\<And>i. i \<in> S \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. \<Prod>i\<in>S. f i x) \<in> borel_measurable M"
reachable_trans: fixes u :: "'a"   and v :: "'a"   and w :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> v"   and "v \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w" shows "u \<rightarrow>\<^sup>*\<^bsub>del_vert u\<^esub> w"
sub_consts_conjoin: fixes e :: "nat \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and g :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and cs :: "'b list"   and G :: "('b, 'c) form list" shows "eval e f g (sub_consts cs (conjoin G)) = eval e f g (conjoin (map (sub_consts cs) G))"
Not\<^sub>\<alpha>_def: shows "Not\<^sub>\<alpha> \<equiv> (rep_Tree\<^sub>\<alpha> ---> abs_Tree\<^sub>\<alpha>) tNot"
indep_system_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "indep_system \<E> (indep_in \<E>)"
secure_refinement_simple_def: fixes \<R>\<^sub>A :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) set"   and \<R> :: "((('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val)) \<times> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val)) set"   and abs_steps :: "('Com\<^sub>A \<times> (Mode \<Rightarrow> 'Var\<^sub>A set)) \<times> ('Var\<^sub>A \<Rightarrow> 'Val) \<Rightarrow> ('Com\<^sub>C \<times> (Mode \<Rightarrow> 'Var\<^sub>C set)) \<times> ('Var\<^sub>C \<Rightarrow> 'Val) \<Rightarrow> nat" shows "secure_refinement_simple \<R>\<^sub>A \<R> abs_steps \<equiv> closed_others \<R> \<and> preserves_modes_mem \<R> \<and> new_vars_private \<R> \<and> simple_refinement_safe \<R>\<^sub>A \<R> abs_steps \<and> bisim_simple \<R>\<^sub>A \<and> (\<forall>c\<^sub>1\<^sub>A mds\<^sub>A mem\<^sub>1\<^sub>A c\<^sub>1\<^sub>C mds\<^sub>C mem\<^sub>1\<^sub>C. (\<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<in> \<R> \<longrightarrow> (\<forall>c\<^sub>1\<^sub>C' mds\<^sub>C' mem\<^sub>1\<^sub>C'. \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C \<leadsto>\<^sub>C \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C \<longrightarrow> (\<exists>c\<^sub>1\<^sub>A' mds\<^sub>A' mem\<^sub>1\<^sub>A'. abs.neval \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A (abs_steps \<langle>c\<^sub>1\<^sub>A, mds\<^sub>A, mem\<^sub>1\<^sub>A\<rangle>\<^sub>A \<langle>c\<^sub>1\<^sub>C, mds\<^sub>C, mem\<^sub>1\<^sub>C\<rangle>\<^sub>C) \<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A \<and> (\<langle>c\<^sub>1\<^sub>A', mds\<^sub>A', mem\<^sub>1\<^sub>A'\<rangle>\<^sub>A, \<langle>c\<^sub>1\<^sub>C', mds\<^sub>C', mem\<^sub>1\<^sub>C'\<rangle>\<^sub>C) \<in> \<R>)))"
zmult_eq_1_iff: fixes m :: "int"   and n :: "int" shows "(m * n = 1) = (m = 1 \<and> n = 1 \<or> m = - 1 \<and> n = - 1)"
rep_eq: shows "Rep_stone_phi_pair top = stone_phi.pairs_top"
deriv: fixes x :: "real" assumes "a \<le> x" shows "(G has_real_derivative g x) (at x within {a..})"
simps_2: fixes rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and v :: "'b"   and va :: "'b list" shows "merge_wrt rel [] (v # va) = v # va"
distinct_37: fixes x4 :: "nat"   and x5 :: "nat" shows "Pan x4 \<noteq> Key x5"
that: fixes T :: "secLevel" assumes "\<Gamma> \<turnstile> e : T" shows "thesis"
more_temp_simps3_2: fixes F :: "'a \<Rightarrow> bool" shows "\<box>\<box>F \<equiv> \<box>F"
f_def: shows "f = monom (UP Zp) \<one> 2 \<oplus>\<^bsub>UP Zp\<^esub> to_polynomial Zp (\<ominus> (\<one> \<oplus> [p] \<cdot> \<one> [^] 3 \<otimes> a [^] 4))"
closed_segment_eq_open: fixes a :: "'a"   and b :: "'a" shows "closed_segment a b = open_segment a b \<union> {a, b}"
tendsto_sgn: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter" assumes "(f \<longlongrightarrow> l) F"   and "l \<noteq> (0::'c)" shows "((\<lambda>x. sgn (f x)) \<longlongrightarrow> sgn l) F"
Cauchy_def: fixes X :: "nat \<Rightarrow> 'a" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e)"
a_known_ptr_def: fixes ptr :: "(_) object_ptr" shows "NodeClass.known_ptr ptr = False"
tuple_pr: fixes a :: "'a"   and b :: "'a" assumes "ide a"   and "ide b" shows "\<langle>\<pp>\<^sub>1[a, b], \<pp>\<^sub>0[a, b]\<rangle> = a \<otimes> b"
hd_fail: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and t :: "pdf_type" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Fail t \<Rightarrow> (\<lambda>_ _. 0)"