hcju/isabelleps huggingface dataset:
has_integral_combine_division_topdown: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and \<D> :: "'b set set"   and K :: "'b set" assumes "f integrable_on S"   and "\<D> division_of K"   and "K \<subseteq> S" shows "(f has_integral (\<Sum>i\<in>\<D>. integral i f)) K"
l_span_cont_H: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set" assumes "R module M"   and "H \<subseteq> carrier M" shows "H \<subseteq> linear_span R M (carrier R) H"
matchT_MT_def: fixes indis :: "('a \<times> 'a) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "('b, 'c) com"   and d :: "('b, 'c) com" assumes "PL_Indis indis" shows "PL_Indis.matchT_MT tval aval indis c d \<equiv> \<forall>s t s'. (s, t) \<in> indis \<and> PL.transT tval aval (c, s) s' \<longrightarrow> (\<exists>t'. PL.MtransT tval aval (d, t) t' \<and> (s', t') \<in> indis)"
hdomain_not_mem: fixes r :: "hf"   and a :: "hf" shows "\<langle>hdomain r, a\<rangle> \<^bold>\<notin> r"
del_UnEdge_even': fixes v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "(v, w, v') \<in> G.E"   and "finite G.E" shows "(v' \<in> odd_nodes_set (del_unEdge v w v' G)) = even (degree v' G)"
NI_unrelated_def: shows "NI_unrelated \<equiv> \<forall>execs a n. realistic_executions execs \<longrightarrow> (let s_f = run n s0 execs in output_f s_f a = output_f (run n s0 (int.purge execs (current s_f))) a)"
fresh_psubst_I1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and zs :: "'varSort"   and z :: "'var" assumes "good X"   and "goodEnv rho"   and "fresh zs z X"   and "freshEnv zs z rho" shows "fresh zs z X #[rho]"
of_rat_add: fixes a :: "rat"   and b :: "rat" shows "of_rat (a + b) = of_rat a + of_rat b"
transformation_by_components_axioms_def: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow> 'c" shows "transformation_by_components_axioms A B F G t \<equiv> (\<forall>a. partial_magma.ide A a \<longrightarrow> partial_magma.in_hom B (t a) (F a) (G a)) \<and> (\<forall>f. partial_magma.arr A f \<longrightarrow> B (t (partial_magma.cod A f)) (F f) = B (G f) (t (partial_magma.dom A f)))"
inorder_del: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c" assumes "complete t"   and "sorted1 (inorder t)" shows "inorder (treeD (Tree23_Map.del x t)) = AList_Upd_Del.del_list x (inorder t)"
Sup_upper2: fixes u :: "'a"   and A :: "'a set"   and v :: "'a" assumes "u \<in> A"   and "v \<le> u" shows "v \<le> Sup A"
assms_1: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "0 \<le> f (g x)"
simps_1: fixes A :: "'a \<Rightarrow> 'b"   and uu :: "'a \<Rightarrow> 'b"   and uv :: "'a \<Rightarrow> 'b" shows "A_B.Dom (A_B.MkArr A uu uv) = A"
prv_imp_imp_exi: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<phi>" shows "prv (imp (exi x (imp \<phi> \<chi>)) (imp \<phi> (exi x \<chi>)))"
case_3: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Macron of Acute \<Rightarrow> f1 | Grave \<Rightarrow> f2 | Macron \<Rightarrow> f3) = f3"
poincare_between_poincare_line_uzv: fixes u :: "complex_homo"   and v :: "complex_homo"   and z :: "complex_homo" assumes "u \<noteq> v"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "z \<in> unit_disc"   and "poincare_between u z v" shows "z \<in> circline_set (poincare_line u v)"
simps: fixes \<delta> :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and \<delta>\<^sub>M :: "'b ltl_prop_equiv_quotient \<Rightarrow> 'b set \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0 :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and q\<^sub>0\<^sub>M :: "'b ltl \<Rightarrow> 'b ltl_prop_equiv_quotient"   and M_fin :: "('b ltl \<Rightarrow> nat option) \<Rightarrow> (('b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) \<times> 'b set \<times> 'b ltl_prop_equiv_quotient \<times> ('b ltl \<Rightarrow> ('b ltl_prop_equiv_quotient \<Rightarrow> nat option) option)) set"   and \<Sigma> :: "'b set set"   and \<phi> :: "'b ltl" shows "ltl_to_rabin_base_def.ltl_to_generalized_rabin \<delta> \<delta>\<^sub>M q\<^sub>0 q\<^sub>0\<^sub>M M_fin \<Sigma> \<phi> = (ltl_to_rabin_base_def.delta \<delta> \<delta>\<^sub>M q\<^sub>0\<^sub>M \<Sigma>, ltl_to_rabin_base_def.initial q\<^sub>0 q\<^sub>0\<^sub>M \<phi>, ltl_to_rabin_base_def.rabin_pairs \<delta>\<^sub>M q\<^sub>0\<^sub>M M_fin \<Sigma> \<phi>)"
simps_1: fixes n :: "nat"   and xs :: "Formula.formula list" shows "to_mregex_exec (regex.Skip n) xs = (MSkip n, xs)"
semiring_normalization_rules_29: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and x :: "'d" assumes "class.comm_semiring_1 times one plus zero" shows "times x x = power.power one times x 2"
prob_space_def: fixes M :: "'a measure" shows "prob_space M \<equiv> finite_measure M \<and> prob_space_axioms M"
of_nat_0_le_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
map_disc_iff: fixes f1 :: "'i \<Rightarrow> 'f"   and f2 :: "'j \<Rightarrow> 'g"   and a :: "('i, 'j, 'h) Resumption.resumption" shows "Resumption.resumption.is_Done (Resumption.resumption.map_resumption f1 f2 a) = Resumption.resumption.is_Done a"
coeff_xy_power: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "coeff (x_y ^ n) k = monom (of_nat (n choose (n - k)) * (- (1::'a)) ^ k) (n - k)"
ordering_axioms: shows "ordering (\<lambda>x y. y \<le> x) (\<lambda>x y. y < x)"
image_restrict_eq: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e set" shows "restrict f A ` A = f ` A"
wf_syscls_def: fixes P :: "'b prog" shows "wf_syscls P \<equiv> {Object} \<union> sys_xcpts \<subseteq> set (map fst P)"
CBF: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" shows "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) in v]"
take_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'b list" shows "xs \<up> m \<down> n = xs \<down> (n + m) \<up> m"
simps_1: fixes a :: "nat" shows "buffered_val [] a = None"
times_eq_iff: fixes A :: "'d set"   and B :: "'e set"   and C :: "'d set"   and D :: "'e set" shows "(A \<times> B = C \<times> D) = (A = C \<and> B = D \<or> (A = {} \<or> B = {}) \<and> (C = {} \<or> D = {}))"
inverse_zero: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and inverse :: "'e \<Rightarrow> 'e"   and one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse zero = zero"
AllN_1: fixes i :: "nat"   and init :: "real list" assumes "freeIn i F"   and "length init = i" shows "eval (lowerFm i 1 F) (init @ xs) = eval F (init @ [x] @ xs)"
A: shows "\<exists>u\<in>U. (u, D x) \<in> I"
Ctxt_validn_def: fixes n :: "nat"   and G :: "((Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Location \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) set" shows "( \<Turnstile>\<^sub>n G  ) = (\<forall>m\<le>n. \<forall>A. A \<in> G \<longrightarrow>  \<Turnstile>\<^sub>n Call : A )"
imageI: fixes x :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "x \<in> A" shows "f x \<in> f ` A"
spr_sim_r: shows "sim_r SPR.MC spr_simMC spr_sim"
eucl_eq_iff: fixes x :: "'b"   and y :: "'b" shows "(x = y) = (\<forall>i<DIM('b). x $\<^sub>e i = y $\<^sub>e i)"
obj_simps_3: fixes a :: "'a" assumes "S.obj a" shows "S.trg a = a"
A1'_def: shows "A1' \<equiv> \<lambda>view (m0, m1). coin_spmf \<bind> (\<lambda>S2. coin_spmf \<bind> (\<lambda>S3. coin_spmf \<bind> (\<lambda>S4. coin_spmf \<bind> (\<lambda>S5. R1_OT12 (S2, S3) c1 \<bind> (\<lambda>b. R1_OT12 (S4, S5) c1 \<bind> (\<lambda>c. Let (M, (m0, m1, S2, S3, S4, S5), view, b, c) D))))))"
degree_poly_lift2: fixes p :: "'b poly" shows "degree (poly_lift2 p) = degree p"
cases: fixes a1 :: "'d"   and a2 :: "'e"   and r :: "('e \<times> 'd) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'e" and b :: "'d" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
simps_4: fixes iinput :: "Proc \<Rightarrow> InputsOrNi"   and ioutput :: "Proc \<Rightarrow> InputsOrNi"   and ichosen :: "InputsOrNi"   and iallInput :: "InputsOrNi set"   and more :: "'a" shows "iallInput \<lparr>iinput = iinput, ioutput = ioutput, ichosen = ichosen, iallInput = iallInput, \<dots> = more\<rparr> = iallInput"
func_space_def: fixes S :: "'e set" shows "func_space S = \<lparr>carrier = S \<rightarrow>\<^sub>E carrier R, mult = \<lambda>f g. \<lambda>v\<in>S. \<zero>, one = \<lambda>v\<in>S. \<zero>, zero = \<lambda>v\<in>S. \<zero>, add = \<lambda>f g. \<lambda>v\<in>S. f v \<oplus> g v, smult = \<lambda>c f. \<lambda>v\<in>S. c \<otimes> f v\<rparr>"
f'g': shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f' g' \<eta>' \<epsilon>'"
prefix_prefix: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "prefix xs ys" shows "prefix xs (ys @ zs)"
antimono_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less" shows "order.antimono less_eq f = (\<forall>x y. less_eq x y \<longrightarrow> f y \<le> f x)"
Input: shows "a<x>.P \<longmapsto> \<alpha> \<prec> P'"
i_in_p: shows "to_F \<iota> \<in> fst (lnth Ns p)"
dis_is_lbl: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and u :: "nat"   and fo :: "nat \<times> nat \<Rightarrow> 'a"   and lo :: "nat \<Rightarrow> nat"   and no :: "nat \<Rightarrow> nat set"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set" assumes "discharge_invar c s t u fo lo no f l n" shows "Labeling c s t f l"
id_remove_1_mset_iff_notin: fixes M :: "'a multiset"   and a :: "'a" shows "(M = remove1_mset a M) = (a \<notin># M)"
in_set_takeD: fixes x :: "'g"   and n :: "nat"   and xs :: "'g list" assumes "x \<in> set (take n xs)" shows "x \<in> set xs"
unit_factor_snd_quot_of_fract: fixes x :: "'a fract" shows "unit_factor (snd (quot_of_fract x)) = (1::'a)"
containsDuplicateEdge_def: fixes g :: "graph"   and f :: "face"   and v :: "nat"   and "is" :: "nat list" shows "containsDuplicateEdge g f v is \<equiv> containsUnacceptableEdge (\<lambda>i j. duplicateEdge g f f\<^bsup>i\<^esup> \<bullet> v f\<^bsup>j\<^esup> \<bullet> v) is"
p_not_in_v2: shows "p \<^bold>\<notin> lan v2"
rel_F_pos_distr: fixes Co1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and Co1' :: "'q \<Rightarrow> 'r \<Rightarrow> bool"   and Co2 :: "'s \<Rightarrow> 't \<Rightarrow> bool"   and Co2' :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co3 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Co3' :: "'w \<Rightarrow> 'x \<Rightarrow> bool"   and Contra1 :: "'y \<Rightarrow> 'z \<Rightarrow> bool"   and Contra1' :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and Contra2 :: "'ab \<Rightarrow> 'ac \<Rightarrow> bool"   and Contra2' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and Contra3 :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and Contra3' :: "'af \<Rightarrow> 'ag \<Rightarrow> bool"   and uu :: "('ah \<times> 'ai \<times> 'aj \<times> 'ak \<times> 'al \<times> 'am \<times> 'an \<times> 'ao \<times> 'ap \<times> 'aq) itself"   and L1 :: "'ah \<Rightarrow> 'ai \<Rightarrow> bool"   and L2 :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and L3 :: "'an \<Rightarrow> 'ao \<Rightarrow> bool"   and L1' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and L2' :: "'al \<Rightarrow> 'am \<Rightarrow> bool"   and L3' :: "'ao \<Rightarrow> 'ap \<Rightarrow> bool" assumes "rel_F_pos_distr_cond Co1 Co1' Co2 Co2' Co3 Co3' Contra1 Contra1' Contra2 Contra2' Contra3 Contra3' uu" shows "rel_F L1 L2 L3 Co1 Co2 Co3 Contra1 Contra2 Contra3 OO rel_F L1' L2' L3' Co1' Co2' Co3' Contra1' Contra2' Contra3' \<le> rel_F (L1 OO L1') (L2 OO L2') (L3 OO L3') (Co1 OO Co1') (Co2 OO Co2') (Co3 OO Co3') (Contra1 OO Contra1') (Contra2 OO Contra2') (Contra3 OO Contra3')"
simps_6: fixes c :: "int"   and e :: "MIR.num" shows "isrlfm (Le (CN 0 c e)) = (0 < c \<and> numbound0 e)"
iso_kernel_image: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and f :: "'b \<Rightarrow> 'd" assumes "Group.group G"   and "Group.group H" shows "(f \<in> Group.iso G H) = (f \<in> hom G H \<and> kernel G H f = {\<one>\<^bsub>G\<^esub>} \<and> f ` carrier G = carrier H)"
comp_simp_1: fixes g :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" assumes "J.seq g f"   and "f = J.Zero" shows "g \<cdot>\<^sub>J f = g"
is_bisimulation_eqvt: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool"   and p :: "perm" assumes "is_bisimulation R" shows "is_bisimulation (p \<bullet> R)"
arr_dom_iff_arr: fixes f :: "'b" shows "J.arr (J.dom f) = J.arr f"
odlist_Abs_inject: fixes x :: "'c list"   and y :: "'c list" assumes "x \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)"   and "y \<in> rel_ext (\<lambda>x. sorted x \<and> distinct x)" shows "(odlist_Abs x = odlist_Abs y) = (x = y)"
sgn_mult_abs: fixes a :: "'a" shows "sgn a * \<bar>a\<bar> = a"
lens_prod_def: fixes X :: "'b \<Longrightarrow> 'd"   and Y :: "'c \<Longrightarrow> 'e" shows "X \<times>\<^sub>L Y = \<lparr>lens_get = map_prod get\<^bsub>X\<^esub> get\<^bsub>Y\<^esub>, lens_put = \<lambda>(u, v) (x, y). (put\<^bsub>X\<^esub> u x, put\<^bsub>Y\<^esub> v y)\<rparr>"
ideal_prod_is_ideal: fixes I :: "'a set"   and J :: "'a set" assumes "ideal I R"   and "ideal J R" shows "ideal (I \<cdot> J) R"
assms_2: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> n"
order_top_ow_axioms: shows "order_top_ow U\<^sub>2 (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>2) top\<^sub>2"
alw_iff_sdrop: fixes P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" shows "alw P \<omega> = (\<forall>m. P (sdrop m \<omega>))"
discr_Atm: fixes atm :: "'atom" shows "discr (Atm atm) = presAtm atm"
is_global_renaming_def: fixes \<alpha> :: "'d \<Rightarrow> 'd"   and \<beta> :: "'e \<Rightarrow> 'e"   and s :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option"   and s' :: "'d \<Rightarrow> (('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('e \<Rightarrow> ('d, 'e, 'f) val option) \<times> ('d, 'e, 'f) expr) option" shows "is_global_renaming \<alpha> \<beta> s s' \<equiv> \<forall>r. case s r of None \<Rightarrow> s' (\<alpha> r) = None | Some ls \<Rightarrow> s' (\<alpha> r) = Some (\<R>\<^sub>L \<alpha> \<beta> ls)"
lift_assnI: fixes s :: "'e"   and s' :: "'f"   and R :: "('e \<times> 'f) set"   and \<Phi> :: "'f \<Rightarrow> bool" assumes "(s, s') \<in> R"   and "\<Phi> s'" shows "lift_assn R \<Phi> s"
finite_set_in: fixes x :: "'x"   and FS :: "'x fset" shows "(x \<in> fset FS) = (x |\<in>| FS)"
map_ext: fixes xs :: "'b list"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<longrightarrow> f x = g x" shows "map f xs = map g xs"
transitive_star: fixes x :: "'a" assumes "transitive x" shows "x\<^sup>\<star> = (1::'a) \<squnion> x"
SCallInitDoneRed: fixes P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and M :: "char list"   and Ts :: "ty list"   and T :: "ty"   and pns :: "char list list"   and body :: "char list exp"   and D :: "char list"   and sh :: "sheap"   and sfs :: "sfields"   and vs :: "val list"   and h :: "heap"   and l :: "sfields" assumes "P \<turnstile> C sees M, Static :  Ts\<rightarrow>T = (pns, body) in D"   and "sh D = \<lfloor>(sfs, Done)\<rfloor> \<or> M = clinit \<and> sh D = \<lfloor>(sfs, Processing)\<rfloor>" shows "P \<turnstile> \<langle>C\<bullet>\<^sub>sM(map Val vs),(h, l, sh),False\<rangle> \<rightarrow> \<langle>C\<bullet>\<^sub>sM(map Val vs),(h, l, sh),True\<rangle>"
mu'_mu_small_i: fixes ii :: "nat"   and jj :: "nat" assumes "ii < i - 1" shows "gs2.\<mu> ii jj = fs.gs.\<mu> ii jj"
\<omega>_power_succ_gtr: fixes \<alpha> :: "V"   and n :: "nat" assumes "Ord \<alpha>" shows "\<omega> \<up> \<alpha> * ord_of_nat n < \<omega> \<up> ZFC_in_HOL.succ \<alpha>"
T_quasi_preserves_null2_2: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b"   and A :: "'b set"   and n :: "nat" assumes "qmpt M T"   and "A \<in> null_sets M" shows "qmpt.vimage_restr M (T ^^ n) A \<in> null_sets M"
rep_eq: fixes x :: "('c, 'd) lifted_boolean_algebra"   and xa :: "('c, 'd) lifted_boolean_algebra" shows "(x \<le> xa) = (\<forall>f. Rep_lifted_boolean_algebra x f \<le> Rep_lifted_boolean_algebra xa f)"
asset_iii_owner_iii: fixes s :: "(agent \<times> msg) set"   and m :: "nat"   and A :: "nat"   and n :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "(Asset m, \<lbrace>Num 4, PubKey A\<rbrace>) \<in> s"   and "(Owner n, \<lbrace>Num 3, PubKey A\<rbrace>) \<in> s" shows "False"
inverts_mat_sym: fixes A :: "complex mat"   and B :: "complex mat" assumes "inverts_mat A B"   and "dim_row B = dim_col A"   and "square_mat B" shows "inverts_mat B A"
inline_parametric': fixes S :: "'m \<Rightarrow> 'r \<Rightarrow> bool"   and C :: "'n \<Rightarrow> 's \<Rightarrow> bool"   and R :: "'o \<Rightarrow> 't \<Rightarrow> bool"   and C' :: "'p \<Rightarrow> 'u \<Rightarrow> bool"   and R' :: "'q \<Rightarrow> 'v \<Rightarrow> bool"   and A :: "'w \<Rightarrow> 'x \<Rightarrow> bool" shows "((S ===> C ===> rel_gpv'' (rel_prod R S) C' R') ===> rel_gpv'' A C R ===> S ===> rel_gpv'' (rel_prod A S) C' R') inline inline"
rtranclp_induct3: fixes r :: "'a \<times> 'b \<times> 'c \<Rightarrow> 'a \<times> 'b \<times> 'c \<Rightarrow> bool"   and ax :: "'a"   and ay :: "'b"   and az :: "'c"   and bx :: "'a"   and "by" :: "'b"   and bz :: "'c"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* (ax, ay, az) (bx, by, bz)"   and "P ax ay az"   and "\<And>a aa b ab ac ba. \<lbrakk>r\<^sup>*\<^sup>* (ax, ay, az) (a, aa, b); r (a, aa, b) (ab, ac, ba); P a aa b\<rbrakk> \<Longrightarrow> P ab ac ba" shows "P bx by bz"
eq_obs: shows "\<not> Policy.sp_spec_subj_obj u x READ"
eval_dioph_div_mult: fixes ks :: "int list"   and n :: "nat"   and xs :: "nat list" shows "eval_dioph ks (map (\<lambda>x. x div n) xs) * int n + eval_dioph ks (map (\<lambda>x. x mod n) xs) = eval_dioph ks xs"
execute_operator_sas_plus_def: fixes s :: "'c \<Rightarrow> 'd option"   and op :: "('c, 'd) sas_plus_operator" shows "s \<then>\<^sub>+ op \<equiv> s ++ map_of (effect_of op)"
Eq\<kappa>E: fixes v :: "i"   and x :: "\<kappa>"   and y :: "\<kappa>" assumes "[x \<^bold>=\<^sub>\<kappa> y in v]" shows "\<exists>o\<^sub>1 o\<^sub>2. Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> o\<^sub>1 = o\<^sub>2"
monom_list_degree_def: fixes xps :: "('c \<times> nat) list" shows "monom_list_degree xps \<equiv> sum_list (map snd xps)"
sim_conj: fixes F :: "'b formula"   and G :: "'b formula"   and \<Gamma> :: "'b formula set"   and H :: "'b formula" assumes "F \<triangleright> G \<triangleright> \<Gamma> \<union> AX10 \<turnstile>\<^sub>H H"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F"   and "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H G" shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H H"
lub_def: fixes A :: "'b set"   and x :: "'b" shows "lub A x \<equiv> (\<forall>b. (\<forall>a. a \<in> A \<longrightarrow> a \<le> b) \<longrightarrow> x \<le> b) \<and> (\<forall>a. a \<in> A \<longrightarrow> a \<le> x)"
lindep_span: fixes K :: "('a, 'b) ring_scheme"   and V :: "('a, 'c, 'd) module_scheme"   and U :: "'c set" assumes "vectorspace K V"   and "U \<subseteq> carrier V"   and "finite U" shows "module.lin_dep K V U = (\<exists>u\<in>U. u \<in> LinearCombinations.module.span K V (U - {u}))"
coboundedI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b \<le> c" shows "a \<cdot> b \<le> c"
beta_equiv_eq_1_2: fixes \<phi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and \<psi> :: "\<kappa> \<Rightarrow> \<kappa> \<Rightarrow> \<o>"   and v :: "i"   and x :: "\<nu>"   and y :: "\<nu>" assumes "IsProperInXY \<phi>"   and "IsProperInXY \<psi>"   and "\<And>x y. [\<phi> (x\<^sup>P) (y\<^sup>P) \<^bold>\<equiv> \<psi> (x\<^sup>P) (y\<^sup>P) in v]" shows "[\<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<phi> (x\<^sup>P) (y\<^sup>P)),x\<^sup>P,y\<^sup>P\<rparr> \<^bold>\<equiv> \<lparr>\<^bold>\<lambda>\<^sup>2 (\<lambda>x y. \<psi> (x\<^sup>P) (y\<^sup>P)),x\<^sup>P,y\<^sup>P\<rparr> in v]"
odd_oddpart: fixes n :: "nat" shows "odd (oddpart n) = (n \<noteq> 0)"
valuate_zero: fixes v :: "nat \<Rightarrow> 'a" shows "0 \<lbrace> v \<rbrace> = (0::'a)"
mat_to_polys_def: fixes ts :: "'c list"   and A :: "'d mat" shows "mat_to_polys ts A = map (row_to_poly ts) (rows A)"
vars_args_subseteq: fixes si :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "si \<in> set (args s)" shows "vars si \<subseteq> vars s"
set_of_appr_of_ivl_append_point: fixes xs :: "'b list"   and p :: "real list" shows "set_of_appr (xs @ appr_of_ivl ops p p) = (\<lambda>x. x @ p) ` set_of_appr xs"
remove_child_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and child :: "(_) node_ptr" assumes "h \<turnstile> ok remove_child ptr child" shows "ptr |\<in>| object_ptr_kinds h"
iso_char: fixes f :: "'s" shows "S.iso f = (S.arr f \<and> bij_betw (S.Fun f) (S.Dom f) (S.Cod f))"
le_imp_less_or_eq: fixes x :: "'b multiset"   and y :: "'b multiset" assumes "x \<subseteq># y" shows "x \<subset># y \<or> x = y"
scc_non_empty': fixes E :: "('a \<times> 'a) set"   and U :: "'a set" assumes "is_scc E U" shows "U \<noteq> {}"
cet_less_eq_int: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and e2 :: "cexpr" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : INTEG"   and "\<Gamma> \<turnstile>\<^sub>c e2 : INTEG" shows "\<Gamma> \<turnstile>\<^sub>c e1 \<le>\<^sub>c e2 : BOOL"
im_uminus: fixes T :: "'a \<Rightarrow> 'b"   and g :: "'a" assumes "UGroupHom T" shows "T (- g) = - T g"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M_ptr_not_in_heap: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "new_shadow_root_ptr |\<notin>| shadow_root_ptr_kinds h"
rel_spmf_mono: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b spmf"   and g :: "'c spmf"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "rel_spmf A f g"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_spmf B f g"
assms_1: shows "vars_term t \<subseteq> vars_term s"
assms: shows "round_robin n0 \<sigma> s = \<lfloor>(t, None, \<sigma>')\<rfloor>"
Inr_Inl_iff: fixes b :: "hf"   and a :: "hf" shows "(HF.Inr b = HF.Inl a) = False"
op_atLeastAtMost_ivl_def: shows "op_atLeastAtMost_ivl = atLeastAtMost"
HC_intros_8: fixes F :: "'b formula"   and G :: "'b formula" shows "AX10 \<turnstile>\<^sub>H (F \<^bold>\<and> G) \<^bold>\<rightarrow> F"