hcju/isabelleps huggingface dataset:
cadjoint_exists: fixes G :: "'a \<Rightarrow> 'b" assumes "bounded_clinear G" shows "\<exists>F. is_cadjoint F G"
that: fixes a :: "'edge" assumes "valid_edge a"   and "n = sourcenode a"   and "intra_kind (kind a)"   and "distance (targetnode a) n' (x - 1)"   and "targetnode a = (SOME nx. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') n' (x - 1) \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = nx)" shows "thesis"
gamma_Step_subcomm: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and f1 :: "char list \<Rightarrow> aexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and f2 :: "char list \<Rightarrow> aexp \<Rightarrow> 'b st option \<Rightarrow> 'b st option"   and g1 :: "bexp \<Rightarrow> (char list \<Rightarrow> int) set \<Rightarrow> (char list \<Rightarrow> int) set"   and g2 :: "bexp \<Rightarrow> 'b st option \<Rightarrow> 'b st option"   and S :: "'b st option"   and C :: "'b st option acom" assumes "Gamma_semilattice num' plus' \<gamma>"   and "\<And>x e S. f1 x e (\<gamma>_option (\<gamma>_st \<gamma>) S) \<subseteq> \<gamma>_option (\<gamma>_st \<gamma>) (f2 x e S)"   and "\<And>b S. g1 b (\<gamma>_option (\<gamma>_st \<gamma>) S) \<subseteq> \<gamma>_option (\<gamma>_st \<gamma>) (g2 b S)" shows "Step f1 g1 (\<gamma>_option (\<gamma>_st \<gamma>) S) (ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) C) \<le> ACom.map_acom (\<gamma>_option (\<gamma>_st \<gamma>)) (Step f2 g2 S C)"
assms_2: fixes i :: "nat" shows "w i square_integrable S"
is_zero_row_upt_k_def: fixes i :: "'b"   and k :: "nat"   and A :: "(('c, 'd) vec, 'b) vec" shows "is_zero_row_upt_k i k A = (\<forall>j. mod_type_class.to_nat j < k \<longrightarrow> A $ i $ j = (0::'c))"
Span_min: fixes K :: "'a set"   and Us :: "'a list"   and E :: "'a set" assumes "subfield K R"   and "set Us \<subseteq> carrier R"   and "subgroup E (add_monoid R)"   and "K <#> set Us \<subseteq> E" shows "Span K Us \<subseteq> E"
i_preprocess_sat: fixes cs :: "('i \<times> 'a ns_constraint) list"   and t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and trans_v :: "(nat, 'a) mapping \<Rightarrow> (nat, 'a) mapping"   and U :: "'i list"   and I :: "'i set"   and v :: "(nat, 'a) mapping" assumes "preprocess cs = (t, as, trans_v, U)"   and "I \<inter> set U = {}"   and "(I, \<langle>v\<rangle>) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s set as"   and "\<langle>v\<rangle> \<Turnstile>\<^sub>t t" shows "(I, \<langle>trans_v v\<rangle>) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set cs"
get_curr_win_def: fixes uu :: "unit" shows "get_curr_win uu \<equiv> gets (\<lambda>s. ucast (get_CWP (cpu_reg_val PSR s))) >>= return"
Ord_Sup_succ_eq: fixes k :: "hf" assumes "Ord k" shows "\<Squnion>succ k = k"
indiff_set_Max_wrt_among: fixes carrier :: "'a set"   and A :: "'a set"   and S :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "finite carrier"   and "A \<subseteq> carrier"   and "A \<noteq> {}"   and "total_preorder_on carrier S" shows "indiff_set S (Max_wrt_among S A)"
StableD: fixes F :: "'b program"   and A :: "'b set" assumes "F \<in> Stable A" shows "F \<in> A Co A"
distinct_1: fixes x1 :: "nat"   and x21 :: "Lambda_Example.type"   and x22 :: "Lambda_Example.type" shows "Atom x1 \<noteq> x21 \<Rightarrow> x22"
right_impl_one: fixes a :: "'b" shows "a r\<rightarrow> a = (1::'b)"
pairwise_disjoint_path_components_of: fixes X :: "'b topology" shows "disjoint (path_components_of X)"
reduction_signs_are_distinct: fixes p :: "real poly"   and qs :: "real poly list"   and subsets :: "nat list list"   and signs :: "rat list list" assumes "satisfy_equation p qs subsets signs"   and "invertible_mat (M_mat signs subsets)"   and "distinct signs" shows "distinct (reduction_signs p qs signs subsets (M_mat signs subsets))"
simps_14: fixes e\<^sub>1 :: "'a exp"   and e\<^sub>2 :: "'a exp"   and A :: "'a set option" shows "\<D> (e\<^sub>1;; e\<^sub>2) A = (\<D> e\<^sub>1 A \<and> \<D> e\<^sub>2 (A \<squnion> \<A> e\<^sub>1))"
Cons: fixes x :: "'c"   and xs :: "'c list" shows "sum_list (x # xs) = x + sum_list xs"
unfold: shows "lrev_body3 = (\<Lambda> r v. Fixrec.run\<cdot>(match_lnil\<cdot>v\<cdot>(Fixrec.succeed\<cdot>ID) +++ match_lcons\<cdot>v\<cdot>(\<Lambda> x xs. Fixrec.succeed\<cdot>(r\<cdot>xs oo list2H\<cdot>(x :@ lnil)))))"
SN_rel_on_ideriv: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" shows "SN_rel_on R S T = (\<nexists>as. ideriv R S as \<and> as 0 \<in> T)"
simps_2: fixes uu :: "'b"   and uv :: "instr list"   and uw :: "nat \<Rightarrow> nat option"   and sb :: "'b memref list"   and ux :: "'c"   and uy :: "'d \<times> 'e"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat" shows "flush_all_until_volatile_write ((uu, uv, uw, sb, ux, uy) # ts) m = flush_all_until_volatile_write ts (flush (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) m)"
U_single_def: shows "U_single \<equiv> {\<lambda>x. if x = n then Some 1 else Some 0 |n. n \<in> UNIV}"
br'_step_abs: fixes \<Sigma> :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and \<delta> :: "('d, 'e) ta_rule set"   and \<Sigma>' :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)" assumes "\<Sigma> \<in> br'_invar \<delta>"   and "(\<Sigma>, \<Sigma>') \<in> br'_step \<delta>" shows "(br'_\<alpha> \<Sigma>, br'_\<alpha> \<Sigma>') \<in> br_step \<delta>"
ordinal_rec_def: fixes z :: "ordinal"   and s :: "ordinal \<Rightarrow> ordinal \<Rightarrow> ordinal" shows "ordinal_rec z s = ordinal_rec0 z s oLimit"
assms: shows "\<not> (\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>lnth t i\<^esub>)"
wf_mstate_minit_safe: fixes \<phi> :: "'a MFOTL.formula"   and R :: "'a list set" assumes "mmonitorable \<phi>" shows "wf_mstate \<phi> pnil R (minit_safe \<phi>)"
sees_methods_is_class: fixes P :: "'a prog"   and C :: "char list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "P \<turnstile> C sees_methods Mm" shows "is_class P C"
inverse_closed: fixes a :: "'ag" assumes "a \<in> U" shows "inverse a \<in> U"
v2_def: shows "w = \<lparr>ext = Abs_real_int (left (ext v) + x, left (ext v) + (x + y)), lan = lan v, own = own v\<rparr>"
IH: shows "higher_differentiable_on {0<..} f k \<and> (\<exists>p. real_polynomial_function p \<and> (\<forall>t>0. nth_derivative k f t 1 = p t / t ^ (2 * k) * exp (- inverse t)))"
closed_ca_lr: fixes d :: "ValD"   and P :: "db" assumes "d \<triangleleft> P" shows "closed P"
simps_1: fixes P :: "'b \<Rightarrow> bool" shows "dropWhile P [] = []"
f_smult: fixes r :: "'a"   and m :: "'c" assumes "r \<in> carrier K"   and "m \<in> carrier V" shows "T (r \<odot>\<^bsub>V\<^esub> m) = r \<odot>\<^bsub>W\<^esub> T m"
X4_ALL: fixes P :: "X4 set \<Rightarrow> bool" shows "(\<forall>X''. P X'') = (\<forall>X''\<in>set ` set (subseqs [X4.Xd1, X4.Xd1', X4.Xd2, Xd2']). P X'')"
dist_eqs_1: fixes cfun :: "Value \<rightarrow> Value"   and discr' :: "bool discr" shows "Fn\<cdot>cfun \<noteq> B\<cdot>discr'"
strictly_subsumes_def: fixes subst_atm :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and C :: "'b literal multiset"   and D :: "'b literal multiset" shows "substitution_ops.strictly_subsumes subst_atm C D = (substitution_ops.subsumes subst_atm C D \<and> \<not> substitution_ops.subsumes subst_atm D C)"
term_powerprod_def: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a" shows "term_powerprod pair_of_term term_of_pair \<equiv> (\<forall>v. term_of_pair (pair_of_term v) = v) \<and> (\<forall>p. pair_of_term (term_of_pair p) = p)"
fail_bind: fixes f :: "'e \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'c) set \<times> bool" shows "fail >>= f = fail"
midpoint_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "Cong A B B C" shows "B Midpoint A C"
cfg_ext_def: fixes init :: "'b"   and prod :: "'b \<Rightarrow> ('a + 'b) list fset"   and more :: "'c" shows "\<lparr>init = init, prod = prod, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons cfg_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple init prod) more"
minus_one_less: fixes x :: "real" shows "x - 1 < x"
p': fixes s :: "real" assumes "s \<in> {t..u}" shows "((\<lambda>t. f' (t, x t) $ (1, f (t, x t))) has_vector_derivative f'' (s, x s) $ (1, f (s, x s)) $ (1, f (s, x s)) + f' (s, x s) $ (0, f' (s, x s) $ (1, f (s, x s)))) (at s within {t..u})"
dvd_mult_cancel2: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "(n * m dvd m) = (n = 1)"
exhaust: fixes y :: "ipc_direction_t" obtains "y = SEND"   | "y = RECV"
components_are_iso: fixes a :: "'a" assumes "ide a" shows "local.iso (trg a)"
crename_AndR: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and c :: "coname"   and M :: "trm"   and d :: "coname"   and N :: "trm"   and e :: "coname" assumes "R[a\<turnstile>c>b] = AndR <c>.M <d>.N e"   and "c \<sharp> (a, b, d, e, N, R)"   and "d \<sharp> (a, b, c, e, M, R)"   and "e \<sharp> (a, b)" shows "\<exists>M' N'. R = AndR <c>.M' <d>.N' e \<and> M'[a\<turnstile>c>b] = M \<and> N'[a\<turnstile>c>b] = N \<and> c \<sharp> N' \<and> d \<sharp> M'"
zero_le: fixes x :: "'b multiset" shows "\<Empt> \<subseteq># x"
hcis_hrcis_eq: fixes a :: "real star" shows "hcis a = hrcis 1 a"
prod_norm: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "(\<Prod>x\<in>A. norm (f x)) = norm (prod f A)"
TryRedsVal: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and v :: "val"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>Val v,s',b'\<rangle>" shows "P \<turnstile> \<langle>try e catch(C V) e\<^sub>2,s,b\<rangle> \<rightarrow>* \<langle>Val v,s',b'\<rangle>"
config_rand_set: fixes A :: "('c list \<Rightarrow> 'd pmf) \<times> ('c list \<times> 'd \<Rightarrow> 'c \<Rightarrow> ((nat \<times> nat list) \<times> 'd) pmf)"   and init :: "'c list"   and qs :: "'c list" shows "\<forall>x\<in>set_pmf (config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs). set (fst x) = set init"
lemma6: fixes r :: "dB" assumes "WN r" shows "\<exists>r'. r \<rightarrow>\<^sub>l r'"
measurable_linear_image_interval: fixes f :: "'a \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a" assumes "linear f" shows "f ` cbox a b \<in> lmeasurable"
ereal_pos_sum_distrib_left: fixes r :: "ereal"   and f :: "'b \<Rightarrow> ereal"   and A :: "'b set" assumes "0 \<le> r"   and "r \<noteq> \<infinity>" shows "r * sum f A = (\<Sum>n\<in>A. r * f n)"
bdd_above_empty: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq {}"
nonconst_poly_irreducible_iff: fixes p :: "'b poly" assumes "degree p \<noteq> 0" shows "irreducible p = (irreducible (map_poly to_fract p) \<and> content p = (1::'b))"
join_compl_top: fixes x :: "'a" assumes "x \<in> carrier L" shows "\<^bold>- x \<squnion> x = \<top>"
tensor_mat_unitary: fixes m1 :: "'b mat"   and m2 :: "'b mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2"   and "unitary m1"   and "unitary m2" shows "unitary (tensor_mat m1 m2)"
mcalc2_iff: fixes \<sigma> :: "'tp"   and c :: "('fsym, 'psym) lit list" shows "(\<sigma> \<turnstile>2 c) = (infTp \<sigma> \<or> (\<forall>l x. l \<in> set c \<and> x \<in> nv2L l \<and> tpOfV x = \<sigma> \<longrightarrow> isGuard x (grdOf c l x)))"
1: assumes "i' \<noteq> i - 1"   and "j' = j" shows "A $$ (i', i - 1) = (0::'a)"
lips: fixes x :: "'M" assumes "x \<in> S" shows "\<exists>T. open T \<and> x \<in> T \<and> (\<forall>y\<in>S \<inter> T. norm (f y - f x) \<le> B * norm (y - x))"
Subobjs_Base: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list" assumes "is_class P C" shows "Subobjs P C [C]"
assms: shows "(s, s') \<in> br butlast ((\<noteq>) [])"
mult_less_cancel_right_disj: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and abs :: "'e \<Rightarrow> 'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and c :: "'e"   and b :: "'e" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less (times a c) (times b c) = (less zero c \<and> less a b \<or> less c zero \<and> less b a)"
self_codomain_iff_ide: fixes a :: "'a" shows "(a \<in> H.codomains a) = H.ide a"
adjoint_def: fixes A :: "'a mat" shows "adjoint A = mat (dim_col A) (dim_row A) (\<lambda>(i, j). conjugate (A $$ (j, i)))"
filter_rev_aux_alt: fixes a :: "'a list"   and P :: "'a \<Rightarrow> bool"   and l :: "'a list" shows "filter_rev_aux a P l = filter P (rev l) @ a"
mult_in_1: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" assumes "f \<in> L F (\<lambda>_. 1::'b)"   and "g \<in> L F (\<lambda>_. 1::'b)" shows "(\<lambda>x. f x * g x) \<in> L F (\<lambda>_. 1::'b)"
gprod_2_2_def: fixes R :: "('b \<times> 'b) set" shows "gprod_2_2 R \<equiv> {((a, b), c, d). a = c \<and> (b, d) \<in> R \<or> b = d \<and> (a, c) \<in> R}"
OAlist_tc_def: fixes xs :: "('a \<times> 'b) list" shows "OAlist_tc xs = Abs_oalist_tc (sort_oalist_tc xs)"
map_def: shows "\<psi>.map = transformation_by_components.map A B G (\<lambda>a. \<phi>.B.inv (\<phi> a))"
linear_injective_imp_surjective: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f"   and "inj f"   and "dim UNIV = dim UNIV" shows "surj f"
fin: shows "finite C\<^sub>1 \<and> finite C\<^sub>2"
FunctorCompM: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and f :: "'f"   and X :: "'d"   and Y :: "'d" assumes "FunctorM F"   and "f maps\<^bsub>CatDom F\<^esub> X to Y" shows "F ## f maps\<^bsub>CatCod F\<^esub> F @@ X to F @@ Y"
add_right: fixes a :: "'b \<Rightarrow>\<^sub>L 'a"   and b :: "'b"   and b' :: "'b" shows "blinfun_apply a (b + b') = blinfun_apply a b + blinfun_apply a b'"
distinct_mset_inter_mset_2: fixes D :: "'c multiset"   and C :: "'c multiset" assumes "distinct_mset D" shows "distinct_mset (C \<inter># D)"
comp_null_2: fixes f :: "'a" shows "f \<cdot> C.null = C.null"
taut_test_def: fixes ifex_of :: "'d \<Rightarrow> 'e ifex"   and b :: "'d" shows "taut_test ifex_of b = (ifex_of b = Trueif)"
basis_indep_card: fixes X :: "'a set"   and B :: "'a set" assumes "indep X"   and "basis B" shows "card X \<le> card B"
3: shows "reduced_GB (monomial (1::'b) ` S) \<subseteq> monomial (1::'b) ` S"
simps_3: fixes u0 :: "'a"   and pr :: "'a list"   and pl :: "'a list" shows "prep_wit_blue u0 (CIRC pr pl) = CIRC (u0 # pr) pl"
nnegD: fixes P :: "'a \<Rightarrow> 'b"   and x :: "'a" assumes "nneg P" shows "(0::'b) \<le> P x"
lunit_char_2: fixes a :: "'a" assumes "ide a" shows "\<I> \<otimes> \<l>[a] = (\<iota> \<otimes> a) \<cdot> \<a>\<^sup>-\<^sup>1[\<I>, \<I>, a]"
select_convs_2: fixes states :: "hf set"   and init :: "hf set"   and final :: "hf set"   and nxt :: "hf \<Rightarrow> 'b \<Rightarrow> hf set"   and eps :: "(hf \<times> hf) set"   and more :: "'c" shows "nfa.init \<lparr>nfa.states = states, init = init, final = final, nxt = nxt, eps = eps, \<dots> = more\<rparr> = init"
ring_record_simps_11: fixes carrier :: "'c set"   and mult :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and add :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and more :: "'d" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
P4: shows "C Out D D' \<and> Cong C D' A B"
undefined3_def: shows "undefined3 = case_sum3 (In1 \<circ> (\<lambda>x. undefined) (+) (\<lambda>x. Unit)) (\<lambda>x. In2 undefined) (\<lambda>x. In3 undefined)"
amalgamation_diverg: fixes i :: "nat"   and j :: "nat"   and x :: "nat" shows "(amalgamation i j x \<up>) = (\<phi> i x \<up> \<and> \<phi> j x \<up>)"
that: fixes S :: "'a set" assumes "H \<subseteq> S"   and "S \<in> subgroups_of_size (p ^ a)" shows "thesis"
card_Un_Int: fixes A :: "'c set"   and B :: "'c set" assumes "finite A"   and "finite B" shows "card A + card B = card (A \<union> B) + card (A \<inter> B)"
cases: fixes a :: "'d"   and s :: "'e + 'd" assumes "a \<in> Basic_BNFs.setr s" obtains x :: "'d" where "a = x" and "s = Inr x"
is_balanced_union: fixes A :: "nat set"   and B :: "nat set" assumes "A \<subseteq> {i. i < 2 ^ n}"   and "B \<subseteq> {i. i < 2 ^ n}"   and "card A = 2 ^ (n - 1)"   and "card B = 2 ^ (n - 1)"   and "A \<inter> B = {}" shows "A \<union> B = {i. i < 2 ^ n}"
insertion_fun_def: fixes f :: "nat \<Rightarrow> 'a"   and p :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow> 'a" shows "insertion_fun f p = (\<Sum>m. p m * (\<Prod>v. f v ^ poly_mapping.lookup m v))"
fg: fixes t :: "real" assumes "t \<le> dist x y" shows "f t = g t"
filters_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" assumes "class.pseudo_hoop_algebra times inf imp_l less_eq less one imp_r" shows "pseudo_hoop_algebra.filters times less_eq = {F. F \<noteq> {} \<and> (\<forall>a b. a \<in> F \<and> b \<in> F \<longrightarrow> times a b \<in> F) \<and> (\<forall>a b. a \<in> F \<and> less_eq a b \<longrightarrow> b \<in> F)}"
pfinite1: shows "pfinite_converter \<I> \<I>' conv1"
indicator_simps_2: fixes x :: "'d"   and S :: "'d set" assumes "x \<notin> S" shows "indicator S x = (0::'e)"
inf2D1: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h" assumes "(A \<sqinter> B) x y" shows "A x y"
rel_refl_strong: fixes x :: "'a option"   and Ra :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>z. z \<in> set_option x \<Longrightarrow> Ra z z" shows "rel_option Ra x x"
card_Diff_subset: fixes B :: "'c set"   and A :: "'c set" assumes "finite B"   and "B \<subseteq> A" shows "card (A - B) = card A - card B"
ibounded_byE: fixes \<A>1 :: "((bool list \<times> bool list) \<times> 'b, bool list, bool list \<times> bool list) gpv"   and \<A>2 :: "bool list \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, bool list, bool list \<times> bool list) gpv"   and q :: "enat" assumes "local.ind_cpa.ibounded_by (\<A>1, \<A>2) q" obtains q1 :: "enat" and q2 :: "enat" where "q1 + q2 \<le> q" and "interaction_any_bounded_by \<A>1 q1" and "\<And>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2"
gfp_least: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and f :: "'d \<Rightarrow> 'd"   and X :: "'d" 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"
good_item_simps_60: fixes rhoa :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and xa :: "'var"   and y :: "'var" assumes "goodEnv rhoa" shows "rhoa &[xa \<and> y]_zs &[xa \<and> y]_zs = rhoa"
rel_filter_neg_distr_cond'_def: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and C' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_filter_neg_distr_cond' C C' = (left_total C \<and> right_unique C \<or> right_total C' \<and> left_unique C')"
OPT2_C1: fixes x :: "'a"   and y :: "'a"   and qs :: "'a list" assumes "x \<noteq> y"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "real (T\<^sub>p [x, y] qs (OPT2 qs [x, y])) = real (length qs - 1) / 2"
pdom_res_twice: fixes A :: "'a set"   and B :: "'a set"   and f :: "('a, 'b) pfun" shows "A \<lhd>\<^sub>p (B \<lhd>\<^sub>p f) = (A \<inter> B) \<lhd>\<^sub>p f"
"case": fixes f :: "char list \<Rightarrow> hterm list \<Rightarrow> 'a"   and x1 :: "char list"   and x2 :: "hterm list" shows "(case HFun x1 x2 of HFun x xa \<Rightarrow> f x xa) = f x1 x2"
tendsto_componentwise_iff: fixes f :: "'a \<Rightarrow> 'b"   and l :: "'b"   and F :: "'a filter" shows "(f \<longlongrightarrow> l) F = (\<forall>i\<in>Basis. ((\<lambda>x. f x \<bullet> i) \<longlongrightarrow> l \<bullet> i) F)"
cong: fixes L :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b) set"   and L' :: "'c filter \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'b) set"   and Lr :: "'a filter \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) set"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter" assumes "landau_symbol L L' Lr"   and "\<forall>\<^sub>F x in F. f x = g x" shows "L F f = L F g"
path_cons: fixes xs :: "'a list"   and v :: "'a" assumes "H.path xs"   and "xs \<noteq> []"   and "v \<rightarrow>\<^bsub>H\<^esub> hd xs"   and "v \<notin> set xs" shows "H.path (v # xs)"
pol_expr_unique3: fixes c :: "nat \<times> (nat \<Rightarrow> 'a)"   and d :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "pol_coeff S c"   and "pol_coeff S d"   and "fst c < fst d" shows "(polyn_expr R X (fst c) c = polyn_expr R X (fst d) d) = ((\<forall>j\<le>fst c. snd c j = snd d j) \<and> (\<forall>j\<in>nset (Suc (fst c)) (fst d). snd d j = \<zero>\<^bsub>S\<^esub>))"
out_arcs_in_arcs: fixes x :: "'c"   and G :: "('d, 'c) pre_digraph"   and u :: "'d" assumes "x \<in> out_arcs G u" shows "x \<in> arcs G"
head_isnpolyh_Suc: fixes p :: "poly"   and n :: "nat" assumes "isnpolyh p (Suc n)" shows "head p = p"
assms: shows "w\<^sub>1 \<preceq>\<^sub>I w\<^sub>2"
assms_1: shows "fv\<^sub>s\<^sub>t S \<inter> bvars\<^sub>s\<^sub>t S = {}"
finite_borel_measure'_axioms_def: fixes M :: "'a measure" shows "finite_borel_measure'_axioms M \<equiv> sets M = sets borel"
consis: shows "sharing_consistent \<S> \<O> (x # sb)"
T_seq: fixes p :: "'a"   and x :: "'a"   and q :: "'a"   and y :: "'a"   and s :: "'a" shows "(p \<leadsto> x \<leadsto> q) \<cdot>  |x] (q \<leadsto> y \<leadsto> s) \<le> p \<leadsto> x \<cdot> y \<leadsto> s"
v11_app_if_vconverse_app: fixes y :: "V"   and x :: "V" assumes "y \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>"   and "(\<FF>\<lparr>ArrMap\<rparr>)\<inverse>\<^sub>\<circ>\<lparr>y\<rparr> = x" shows "\<FF>\<lparr>ArrMap\<rparr>\<lparr>x\<rparr> = y"
rel_U_pos_distr_imp: fixes Co1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'p \<Rightarrow> bool"   and Co2 :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2' :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Contra1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Contra1' :: "'u \<Rightarrow> 'v \<Rightarrow> bool"   and Contra2 :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra2' :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and tytok_G :: "(('z, 'n, 'q, 't, 'w, 'aa) U \<times> ('ab, 'o, 'r, 'u, 'x, 'aa) U \<times> ('ac, 'p, 's, 'v, 'y, 'aa) U \<times> 'z \<times> 'ab \<times> 'ac \<times> 'aa) itself"   and tytok_T :: "('z \<times> 'ab \<times> 'ac \<times> 'aa) itself" assumes "rel_G_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_G" shows "rel_U_pos_distr_cond Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok_T"