hcju/isabelleps huggingface dataset:
horizontal_chop_split_add: fixes x :: "real"   and y :: "real"   and v :: "view" shows "0 \<le> x \<and> 0 \<le> y \<longrightarrow> \<parallel>ext v\<parallel> = x + y \<longrightarrow> (\<exists>u w. v=u\<parallel>w \<and> \<parallel>ext u\<parallel> = x \<and> \<parallel>ext w\<parallel> = y)"
abs_non_negative_sum_1: fixes x :: "int"   and y :: "int" assumes "Preliminaries.abs x + Preliminaries.abs y = 0" shows "Preliminaries.abs x = 0"
fps_deriv_eq_iff_ex: fixes f :: "'a fps"   and g :: "'a fps" shows "(fps_deriv f = fps_deriv g) = (\<exists>c. f = fps_const c + g)"
supI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "\<And>l. is_lub L l A \<Longrightarrow> P l"   and "A \<subseteq> carrier L" shows "P (\<Squnion>A)"
simps_2: fixes C :: "char list"   and F :: "char list"   and D :: "char list" shows "is_relevant_class (Getstatic C F D) = (\<lambda>P C. True)"
commutes_with_unit: shows "kernel.Class \<one> = kernel.Class \<one>"
valid_propagate_1: fixes \<sigma> :: "'b"   and P :: "bool" shows "(\<sigma> \<Turnstile> unit\<^sub>S\<^sub>E P) = P"
length_assoc_list_of_array: fixes a :: "'a array" shows "length (assoc_list_of_array a) = array_length a"
x_def: shows "x = gen_lucas_lehmer_sequence'' p 4 (p - 2)"
expansion_factor_sgn_invariant: fixes p :: "proj2"   and q :: "proj2"   and J :: "cltn2" assumes "p \<in> hyp2 \<union> S"   and "q \<in> hyp2 \<union> S"   and "is_K2_isometry J" shows "sgn (expansion_factor p J) = sgn (expansion_factor q J)"
that: fixes n :: "V" assumes "n \<in>\<^sub>\<circ> \<omega>"   and "n \<approx>\<^sub>\<circ> A" shows "thesis"
brickcount_zero_implies_cup: fixes x :: "brick" assumes "domain x = 0" shows "x = cup"
select_convs_2: fixes precondition_of :: "('c \<times> 'd) list"   and effect_of :: "('c \<times> 'd) list"   and more :: "'e" shows "effect_of \<lparr>sas_plus_operator.precondition_of = precondition_of, effect_of = effect_of, \<dots> = more\<rparr> = effect_of"
rep_eq: fixes x :: "'a list" shows "fset (fset_of_list x) = set x"
simps_2: fixes step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s :: "'g"   and xs :: "'e list"   and x :: "'e" shows "c_tr step out s (xs @ [x]) = c_tr step out s xs @ [(x, out (foldl step s xs) x)]"
prv_eqvI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (imp \<phi> \<chi>)"   and "bprv (imp \<chi> \<phi>)" shows "bprv (eqv \<phi> \<chi>)"
H_R2: fixes p :: "'a"   and x :: "'a"   and q :: "'a" assumes "H p x q" shows "x \<le> R p q"
fbox_one: fixes x :: "'a" shows " |(1::'a)] x = d x"
less_eq_bit0_def: fixes a :: "'a bit0"   and b :: "'a bit0" shows "(a \<le> b) = (Rep_bit0 a \<le> Rep_bit0 b)"
open_domain: fixes c :: "('b, 'c) chart" shows "open (domain c)"
monoid_axioms: shows "monoid (\<and>*) \<box>"
intro: fixes get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" assumes "l_get_host get_host get_host_locs"   and "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node_si get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"   and "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node_si get_host get_owner_document" shows "l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node_si get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_host get_host_locs get_owner_document"
card_of_well_order_on: fixes A :: "'b set" shows "well_order_on A |A|"
open_map_imp_subset: fixes X1 :: "'c topology"   and X2 :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and S :: "'c set" assumes "open_map X1 X2 f"   and "S \<subseteq> topspace X1" shows "f ` S \<subseteq> topspace X2"
vlist_of_vfsequence_def: shows "vlist_of_vfsequence = inv vfsequence_of_vlist"
octo_of_real_times_conv_scaleR: fixes x :: "real"   and y :: "octo" shows "octo_of_real x * y = x *\<^sub>R y"
Max_shift_ge: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat"   and m :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs"   and "exact_decomp (Suc m) qs" shows "Max (poly_deg ` fst ` set qs) \<le> Max (poly_deg ` fst ` set (local.shift k m qs))"
PDG_path_CFG_path: fixes n :: "'node"   and n' :: "'node" assumes "PDG.PDG_path sourcenode targetnode valid_edge Def Use weak_control_dependence n n'" obtains as :: "'edge list" where "n -as\<rightarrow>* n'"
primeE: fixes G :: "('c, 'd) monoid_scheme"   and p :: "'c" assumes "prime G p" obtains "p \<notin> Units G" and "\<forall>a\<in>carrier G. \<forall>b\<in>carrier G. p divides\<^bsub>G\<^esub> a \<otimes>\<^bsub>G\<^esub> b \<longrightarrow> p divides\<^bsub>G\<^esub> a \<or> p divides\<^bsub>G\<^esub> b"
homeomorphism_image1: fixes S :: "'c set"   and T :: "'d set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> 'c" assumes "homeomorphism S T f g" shows "f ` S = T"
on_line_def: fixes cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and betw :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "'a"   and a :: "'a"   and b :: "'a" assumes "TarskiAbsolute cong betw" shows "TarskiAbsolute.on_line betw p a b = (betw p a b \<or> betw a p b \<or> betw a b p)"
edge_succ_pred: fixes a :: "'b" shows "edge_succ M (edge_pred M a) = a"
simps_1: fixes sel_fun :: "(nat \<times> 'a) list \<Rightarrow> nat"   and mf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a"   and d :: "nat"   and rA :: "'a \<times> 'a mat" shows "sub3 sel_fun mf d 0 rA = rA"
fds_deriv_const: fixes c :: "'b" shows "fds_deriv (fds_const c) = 0"
coprime_id_is_unit: fixes a :: "'c" shows "comm_monoid_mult_class.coprime a a = is_unit a"
absolutely_integrable_componentwise_iff: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "(f absolutely_integrable_on A) = (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) absolutely_integrable_on A)"
lift_subst: fixes j :: "nat"   and i :: "nat"   and t :: "Core.term"   and s :: "Core.term" assumes "j < i + 1" shows "lift (subst_bv2 t j s) i = subst_bv2 (lift t (i + 1)) j (lift s i)"
\<theta>_simps_4: shows "local.dom \<theta>' = f \<star> w'"
has_domain_iff_arr: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" shows "(A_B.domains f \<noteq> {}) = A_B.arr f"
v_TS_defs_3: shows "v_trans = (\<Union>r v_f. \<Union> (range (v_round r v_f)))\<^sup>="
relpow_Suc_I2: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set"   and z :: "'a"   and n :: "nat" assumes "(x, y) \<in> R"   and "(y, z) \<in> R ^^ n" shows "(x, z) \<in> R ^^ Suc n"
rep_eq: fixes x :: "'c \<Rightarrow> 'b"   and xa :: "'a \<Rightarrow>\<^sub>0 'c" shows "lookup (Poly_Mapping.map x xa) = (\<lambda>k. x (lookup xa k) when lookup xa k \<noteq> (0::'c))"
finite_irrefl_trans_imp_wf: fixes r :: "('b \<times> 'b) set" assumes "finite r"   and "irrefl r"   and "trans r" shows "wf r"
symI: fixes r :: "('g \<times> 'g) set" assumes "\<And>a b. (a, b) \<in> r \<Longrightarrow> (b, a) \<in> r" shows "sym r"
stepfun_imp_chainp': fixes n :: "nat"   and f :: "nat \<Rightarrow> nat"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "nat \<Rightarrow> 'b" assumes "\<forall>i\<ge>n. i \<le> f i \<and> P (S i) (S (f i))" shows "\<forall>i. P (S ((f ^^ i) n)) (S ((f ^^ Suc i) n))"
Acts_totalize: fixes F :: "'b program" shows "Acts (totalize F) = totalize_act ` Acts F"
l_inv_ex: fixes x :: "nat" assumes "x \<in> carrier (units_of R)" shows "\<exists>y\<in>carrier (units_of R). y \<otimes>\<^bsub>units_of R\<^esub> x = \<one>\<^bsub>units_of R\<^esub>"
exec_Call_body: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and s :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "\<Gamma> p = Some bdy" shows "\<Gamma>\<turnstile> \<langle>Call p,s\<rangle> \<Rightarrow> t = \<Gamma>\<turnstile> \<langle>the (\<Gamma> p),s\<rangle> \<Rightarrow> t"
M_U_soundness: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<Phi> :: "('a, 'b) lit list set" assumes "M_MonotModel wtFsym wtPsym arOf resOf parOf intT intF intP \<Phi>" shows "CU.Model wtFsym wtPsym (length \<circ> arOf) (length \<circ> parOf) \<Phi> (InfModel.intTF any) (InfModel.intFF arOf resOf (MonotProblem.intTI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intFI wtFsym wtPsym arOf resOf parOf \<Phi>)) (InfModel.intPF parOf (MonotProblem.intTI wtFsym wtPsym arOf resOf parOf \<Phi>) (MonotProblem.intPI wtFsym wtPsym arOf resOf parOf \<Phi>))"
pdevs_val_project_coord_pdevs_inner_Basis: fixes b :: "'a"   and e :: "nat \<Rightarrow> real"   and sctn :: "'a sctn"   and X :: "'a pdevs" assumes "b \<in> Basis" shows "pdevs_val e (project_coord_pdevs sctn X) \<bullet> b = (if b = \<bar>normal sctn\<bar> then 0 else pdevs_val e X \<bullet> b)"
mantissa_exponent: fixes f :: "float" shows "real_of_float f = real_of_int (mantissa f) * 2 powr real_of_int (exponent f)"
preserves_comp: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" shows "E.map (g \<cdot> f) = E.map g \<cdot>\<^sub>D E.map f"
max_add_distrib_right: fixes x :: "'b multiset"   and y :: "'b multiset"   and z :: "'b multiset" shows "x + subset_mset.max y z = subset_mset.max (x + y) (x + z)"
lnull_lfilter: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c llist" shows "lnull (lfilter P xs) = (\<forall>x\<in>lset xs. \<not> P x)"
vsv_vimageI2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>)"
powr_0: fixes z :: "'b" shows "(0::'b) powr z = (0::'b)"
LeA_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" shows "A B C LeA D E F \<equiv> \<exists>P. P InAngle D E F \<and> A B C CongA D E P"
the_elem_set: fixes x :: "'a" shows "the_elem (set [x]) = x"
real_nonneg_vec_def: fixes v :: "complex Matrix.vec" shows "real_nonneg_vec v \<equiv> \<forall>a\<in>vec_elements v. a \<in> \<real> \<and> 0 \<le> Re a"
supp_\<beta>''': shows "{x. \<beta>''' x \<noteq> 0} \<subseteq> \<Sum>\<^sub># ` Roots_ms"
word_split_bl': fixes std :: "nat"   and c :: "'b word"   and b :: "'c word"   and a :: "'d word" assumes "std = size c - size b"   and "word_split c = (a, b)" shows "a = of_bl (take std (to_bl c)) \<and> b = of_bl (drop std (to_bl c))"
ltl_lfilter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b llist" shows "ltl (lfilter P xs) = lfilter P (ltl (ldropWhile (Not \<circ> P) xs))"
weak_action_modality_eq: fixes \<alpha>1 :: "'act"   and x1 :: "('idx, 'pred, 'act) formula"   and \<alpha>2 :: "'act"   and x2 :: "('idx, 'pred, 'act) formula" assumes "Act \<alpha>1 x1 = Act \<alpha>2 x2" shows "\<langle>\<langle>\<alpha>1\<rangle>\<rangle>x1 = \<langle>\<langle>\<alpha>2\<rangle>\<rangle>x2"
bv_set: fixes n :: "nat" shows "set_pmf (bv n) = {x. length x = n}"
nt: shows "\<not> dist v \<le> \<mu> c s v"
less_le_not_le: fixes y :: "'a"   and x :: "'a" shows "(y < x) = (y \<le> x \<and> \<not> x \<le> y)"
iso_is_arr: fixes f :: "'s" assumes "S.iso f" shows "S.arr f"
assms_1: shows "\<chi> \<noteq> \<chi>\<^sub>0"
hyps_3: fixes thesis :: "bool" assumes "\<And>a. \<lbrakk>a \<in> A; \<And>b. rel b a \<Longrightarrow> b \<notin> A\<rbrakk> \<Longrightarrow> thesis"   and "A \<noteq> {}" shows "thesis"
int_mod_pos_eq: fixes a :: "int"   and b :: "int"   and q :: "int"   and r :: "int" assumes "a = b * q + r"   and "0 \<le> r"   and "r < b" shows "a mod b = r"
HDomain_Incl_Mem_Ord: fixes H :: "fm set"   and r :: "tm"   and k :: "tm"   and k' :: "tm" assumes "H \<turnstile> HDomain_Incl r k"   and "H \<turnstile> k' IN k"   and "H \<turnstile> OrdP k" shows "H \<turnstile> HDomain_Incl r k'"
opp_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and p :: "'b point" assumes "class.ell_field divide inverse times one plus zero minus uminus" shows "ell_field.opp uminus p = (case p of Infinity \<Rightarrow> Infinity | Point x y \<Rightarrow> Point x (uminus y))"
assms_7: fixes A :: "'a"   and B :: "'a" assumes "equalish A B" shows "reflexive_rewrite A B"
sum_of_pr_is_pr2: fixes p :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "p \<in> PrimRec2" shows "(\<lambda>n m. \<Sum>x\<in>{i. i < n}. p x m) \<in> PrimRec2"
rem_effectless_works_5_i: fixes as :: "('a \<times> ('b, 'c) fmap) list" shows "subseq (rem_effectless_act as) as"
fmrestrict_set_dom: fixes m :: "'e \<rightharpoonup>\<^sub>f 'f" shows "fmrestrict_set (fmdom' m) m = m"
simps_2: fixes n :: "nat" shows "empty_boxes (Suc n) = empty_boxes n [+] [Dec n 2, Goto 0]"
VUnion_vempty_conv_left: fixes A :: "V" shows "(0 = \<Union>\<^sub>\<circ> A) = (\<forall>x\<in>elts A. x = 0)"
Reals_1: shows "(1::'b) \<in> \<real>"
is_path_undir_simps_1: fixes G :: "('a, 'b) graph"   and v :: "'a" assumes "valid_graph G" shows "is_path_undir G v [] v = (v \<in> nodes G)"
simps_2: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat option"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and i :: "(String.literal \<times> value list) stream" shows "stl (make_full_observation e s d p i) = make_full_observation e (fst (ltl_step e s d (shd i))) (snd (snd (ltl_step e s d (shd i)))) (fst (snd (ltl_step e s d (shd i)))) (stl i)"
UN_extend_simps_10: fixes B :: "'e \<Rightarrow> 'c set"   and f :: "'d \<Rightarrow> 'e"   and A :: "'d set" shows "(\<Union>a\<in>A. B (f a)) = \<Union> (B ` f ` A)"
prefixToLevelAppend_aux2: fixes i :: "nat"   and l :: "nat"   and a :: "('b \<times> bool) list"   and b :: "('b \<times> bool) list" assumes "i \<le> l"   and "currentLevel a + i \<le> l" shows "prefixToLevel_aux (a @ b) l i = a @ prefixToLevel_aux b l (i + currentLevel a)"
ball_empty: fixes P :: "'c \<Rightarrow> bool" shows "Ball {} P = True"
diff_diff_cancel: fixes i :: "nat"   and n :: "nat" assumes "order.greater_eq n i" shows "n - (n - i) = i"
measurable_extract_int_pair: shows "extract_int_pair \<in> stock_measure (PRODUCT INTEG INTEG) \<rightarrow>\<^sub>M count_space UNIV \<Otimes>\<^sub>M count_space UNIV"
nat_term_compare_Abs_nat_term_order_id: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "comparator cmp"   and "nat_term_comp cmp" shows "nat_term_compare (Abs_nat_term_order cmp) = cmp"
word_plus_and_or_coroll: fixes x :: "'a word"   and y :: "'a word" assumes "x AND y = 0" shows "x + y = x OR y"
monom_1_dvd_iff: fixes p :: "'b poly"   and n :: "nat" assumes "p \<noteq> 0" shows "(monom (1::'b) n dvd p) = (n \<le> order (0::'b) p)"
mult_eq_inf: fixes z :: "complex_homo"   and w :: "complex_homo" assumes "z *\<^sub>h w = \<infinity>\<^sub>h" shows "z = \<infinity>\<^sub>h \<or> w = \<infinity>\<^sub>h"
dom1v_unfold: shows "subst_domain (\<sigma>1(v := t)) = insert v (subst_domain \<sigma>1)"
real_is_int_def: fixes x :: "real" shows "is_int x = (x \<in> \<int>)"
intros: fixes W :: "'d list"   and \<sigma> :: "'d"   and Wtl :: "'d list"   and Wn :: "'d list"   and R :: "('d \<times> 'd) set"   and \<Sigma> :: "'d set"   and W' :: "'d list"   and \<Sigma>' :: "'d set" assumes "W = \<sigma> # Wtl"   and "distinct Wn"   and "set Wn = R `` {\<sigma>} - \<Sigma>"   and "W' = Wn @ Wtl"   and "\<Sigma>' = R `` {\<sigma>} \<union> \<Sigma>" shows "((\<Sigma>, W), \<Sigma>', W') \<in> dfs_step R"
group_commutes_pow: fixes x :: "'a mod_ring poly"   and y :: "'a mod_ring poly"   and n :: "nat" assumes "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x [^]\<^bsub>R\<^esub> n \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x [^]\<^bsub>R\<^esub> n"
assms: shows "y \<in>\<^sub>\<circ> vpairs r"
11: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
b_assn_subtyping_match_eqA_tL: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and x :: "'a"   and y :: "'b"   and P' :: "'a \<Rightarrow> bool" assumes "vassn_tag (hn_ctxt A x y) \<Longrightarrow> P' x" shows "hn_ctxt A x y \<Longrightarrow>\<^sub>t hn_ctxt (b_assn A P') x y"
eq_equiv_class: fixes r :: "('b \<times> 'b) set"   and a :: "'b"   and b :: "'b"   and A :: "'b set" assumes "r `` {a} = r `` {b}"   and "equiv A r"   and "b \<in> A" shows "(a, b) \<in> r"
distinct_idx: fixes f :: "'b \<Rightarrow> 'a"   and l :: "'b list"   and i :: "nat"   and j :: "nat" assumes "distinct (map f l)"   and "i < length l"   and "j < length l"   and "f (l ! i) = f (l ! j)" shows "i = j"
PolMon_def: shows "PolMon = policy2MON SE_LR_RBAC_ST_Policy"
pred_numeral_simps_1: shows "pred_numeral num.One = 0"
rebalance_last_tree_inorder: fixes split :: "('b btree \<times> 'b) list \<Rightarrow> 'b \<Rightarrow> ('b btree \<times> 'b) list \<times> ('b btree \<times> 'b) list"   and t :: "'b btree"   and sub :: "'b btree"   and ts :: "('b btree \<times> 'b) list"   and list :: "('b btree \<times> 'b) list"   and sep :: "'b"   and k :: "nat" assumes "split split"   and "height t = height sub"   and "ts = list @ [(sub, sep)]" shows "inorder (split.rebalance_last_tree k ts t) = inorder (Node ts t)"
lift_rcond_def: fixes b :: "('a, 'b) uexpr" shows "\<lceil>b\<rceil>\<^sub>\<leftarrow> = \<lceil>b\<rceil>\<^sub><"
S5Basic_1: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<diamond>\<^bold>\<box>\<phi> \<^bold>\<rightarrow> \<^bold>\<box>\<phi> in v]"
that: fixes l :: "char list \<Rightarrow> int \<Rightarrow> int"   and g :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "s = <l|g>" shows "thesis"
base: fixes G :: "'b pair_pre_digraph" assumes "pair_bidirected_digraph G" shows "pair_sd G G"
card_of_Sigma_cong2: fixes f :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and J :: "'b set"   and A :: "'b \<Rightarrow> 'c set" assumes "bij_betw f I J" shows "|SIGMA i:I. A (f i)| =o |Sigma J A|"
induct: fixes P :: "('f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option) \<Rightarrow> ('f, 'i, 'j) new_thread_action list \<Rightarrow> bool"   and a0 :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and a1 :: "('f, 'i, 'j) new_thread_action list" assumes "\<And>ts. P ts []"   and "\<And>ts ta tas. P (redT_updT' ts ta) tas \<Longrightarrow> P ts (ta # tas)" shows "P a0 a1"
p2: shows "Esigma n = Esigma g * Esigma N"
chine_simps_3: shows "C.cod \<nu>\<mu>.chine = \<nu>\<mu>.cod.apex"
share_all_shared_in: fixes a :: "nat"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "a \<in> dom (share sb \<S>)" shows "a \<in> dom \<S> \<or> a \<in> all_shared sb"
prm_set_apply_compose: fixes \<pi> :: "'b prm"   and \<sigma> :: "'b prm"   and S :: "'b set" shows "\<pi> {$} (\<sigma> {$} S) = \<pi> \<diamondop> \<sigma> {$} S"
abs_Gromov_completion_in_Gromov_boundary: fixes u :: "nat \<Rightarrow> 'b" assumes "Gromov_converging_at_boundary u" shows "abs_Gromov_completion u \<in> Gromov_boundary"
type_less_x_UnlimitedNatural: fixes \<tau> :: "'a basic_type" assumes "\<tau> < UnlimitedNatural" obtains "\<tau> = OclVoid"
Mapping_inject: fixes x :: "'a \<Rightarrow> 'b option"   and y :: "'a \<Rightarrow> 'b option" assumes "x \<in> UNIV"   and "y \<in> UNIV" shows "(Mapping x = Mapping y) = (x = y)"
rotate12: fixes A12 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and A4 :: "fmla"   and A5 :: "fmla"   and A6 :: "fmla"   and A7 :: "fmla"   and A8 :: "fmla"   and A9 :: "fmla"   and A10 :: "fmla"   and A11 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A12 (insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 H))))))))))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 (insert A5 (insert A6 (insert A7 (insert A8 (insert A9 (insert A10 (insert A11 (insert A12 H))))))))))) \<turnstile> B"
set_bset_inverse: fixes x :: "'a set['b]" shows "Abs_bset (set_bset x) = x"
k: shows "x - k *a a \<in> vs1.span (b - {a})"
kuhn_simplex_axioms: shows "kuhn_simplex p n b_t u_t t"
gt_trans: fixes u :: "('s, 'v) tm"   and t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "wary u"   and "wary t"   and "wary s"   and "local.kbo_std.gt u t"   and "local.kbo_std.gt t s" shows "local.kbo_std.gt u s"
second_vertices_in_V: shows "second_vertices \<subseteq> V"
finite_number_of_roots: fixes p :: "'a list" assumes "p \<in> carrier (poly_ring R)" shows "finite {x. is_root p x}"
ind_rule: fixes H :: "behavior \<Rightarrow> bool"   and sigma :: "behavior"   and P :: "state \<Rightarrow> bool"   and F :: "behavior \<Rightarrow> bool" assumes "sigma \<Turnstile> \<box>H"   and "sigma \<Turnstile> Init P"   and "\<turnstile> H \<longrightarrow> Init P \<and> \<not> \<box>F \<longrightarrow> Init P$ \<and> F" shows "sigma \<Turnstile> \<box>F"
eN\<tau>_def: shows "eN\<tau> \<equiv> instInpP (encF N) 0 \<tau>"
awalk_verts_arc2: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and e :: "'b" assumes "awalk u p v"   and "e \<in> set p" shows "head G e \<in> set (awalk_verts u p)"
MC_action_defs_5: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkNext send rcv cst p = (MClkFwd send rcv cst p \<or> MClkRetry send rcv cst p \<or> MClkReply send rcv cst p)"
sorted_wrt_upto: fixes i :: "int"   and j :: "int" shows "sorted_wrt (<) [i..j]"
assms: shows "prime P \<and> 0 < Q \<and> is_qfN (P * Q) N \<and> \<not> is_qfN P N"
st_not_Infinitesimal: fixes x :: "real star" assumes "st x \<noteq> 0" shows "x \<notin> Infinitesimal"
Reals_minus_iff: fixes a :: "'a" shows "(- a \<in> \<real>) = (a \<in> \<real>)"
insertion_sum: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a mpoly" shows "\<forall>n. insertion f (sum g {0..<n}) = (\<Sum>i = 0..<n. insertion f (g i))"
INTF_OF_REL_def: fixes R :: "('a \<times> 'b) set"   and I :: "'c itself" shows "INTF_OF_REL R I \<equiv> True"
rw: shows "mat_of_cols n (map ((*\<^sub>v) A) ss) = A * mat_of_cols nc ss"
ideal_of_UN: fixes A :: "'e \<Rightarrow> ('c \<Rightarrow> 'd) set"   and J :: "'e set" shows "\<I> (\<Union> (A ` J)) = (\<Inter>j\<in>J. \<I> (A j))"