hcju/isabelleps huggingface dataset:
inv_equality: fixes y :: "'b"   and x :: "'b" assumes "y \<otimes>\<^bsub>G i\<^esub> x = \<one>\<^bsub>G i\<^esub>"   and "x \<in> carrier (G i)"   and "y \<in> carrier (G i)" shows "inv\<^bsub>G i\<^esub> x = y"
Radon_s_lemma: fixes S :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "finite S"   and "sum f S = 0" shows "sum f {x \<in> S. 0 < f x} = - sum f {x \<in> S. f x < 0}"
a: shows "flip (Suc y) X ! Suc y = flip y zs ! y"
rotate_inv: fixes w1 :: "wall"   and w2 :: "wall" assumes "Tangle_Moves.rotate w1 w2" shows "kauff_mat w1 = kauff_mat w2"
Tau: fixes P :: "pi" shows "\<tau>.P \<Longrightarrow>\<^sub>l\<^sup>^\<tau> \<prec> P"
prefix_fininf_le_inf_transp: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item stream"   and w\<^sub>3 :: "'item stream" assumes "w\<^sub>1 \<le>\<^sub>F\<^sub>I w\<^sub>2"   and "w\<^sub>2 \<preceq>\<^sub>I w\<^sub>3" shows "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>3"
vreal_of_real_in_vrealE: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<real>\<^sub>\<circ>" obtains b :: "real" where "b\<^sub>\<real> = a"
simps_1: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> bool \<times> bool" shows "lex_ext_unbounded f [] [] = (False, True)"
exhaust_sel: fixes gauss_int :: "gauss_int" obtains "gauss_int = Gauss_Int (ReZ gauss_int) (ImZ gauss_int)"
list_def: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref" shows "list p h = (THE ps. List p h ps)"
thetaSeqZOD_ZObis: shows "thetaSeqZOD \<subseteq> ZObis"
indep_vars_def: fixes i :: "'i"   and M' :: "'d \<Rightarrow> 'e measure"   and X :: "'d \<Rightarrow> 'b \<Rightarrow> 'e"   and I :: "'d set" shows "D'.indep_vars i M' X I = ((\<forall>i\<in>I. D'.random_variable i (M' i) (X i)) \<and> D'.indep_sets i (\<lambda>i. sigma_sets (space (distr M (M' i) (X i))) {X i -` A \<inter> space (distr M (M' i) (X i)) |A. A \<in> sets (M' i)}) I)"
span_B: shows "{x. f x = (0::'d)} \<subseteq> local.span B"
assms: shows "A \<subseteq> {- 1<..<1}"
g_in_G: fixes i :: "nat" shows "(\<lambda>x. MAX n\<in>{..i}. gs n x) \<in> G"
choice_iff: fixes Q :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "(\<forall>x. \<exists>y. Q x y) = (\<exists>f. \<forall>x. Q x (f x))"
card_isolated_verts_da: shows "card H.isolated_verts = Suc (Suc (card G.isolated_verts))"
prefix_not_empty: fixes n :: "nat" shows "{i. f i \<le> n} \<noteq> {}"
of_int_of_nat_eq: fixes n :: "nat" shows "of_int (int n) = of_nat n"
elims_2: fixes x :: "Preliminaries.var \<Rightarrow> 'univ"   and xa :: "('fsym, 'psym) lit" assumes "satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> 'univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> satA \<xi> a"
at_least_2_labels: shows "label_witness1 \<noteq> label_witness2"
fun_upd_triv: fixes f :: "'e \<Rightarrow> 'f"   and x :: "'e" shows "f(x := f x) = f"
mult_unit_dvd_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times (times a b) c = dvd.dvd times a c"
function_one_closed: fixes R :: "('c, 'd) ring_scheme"   and S :: "'e set" assumes "ring_functions R" shows "\<one>\<^bsub>function_ring S R\<^esub> \<in> carrier (function_ring S R)"
distinct_3: fixes x31 :: "'d"   and x32 :: "'d ifex"   and x33 :: "'d ifex" shows "Trueif \<noteq> IF x31 x32 x33"
X_poly_plus_def: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c" shows "X_poly_plus R a = X_poly R \<oplus>\<^bsub>UP R\<^esub> to_polynomial R a"
isometry_vector_to_cblinfun: fixes x :: "'a" assumes "norm x = 1" shows "isometry (vector_to_cblinfun x)"
emeasure_count_space_prod_eq: fixes A :: "('c \<times> 'd) set" assumes "A \<in> sets (count_space UNIV \<Otimes>\<^sub>M count_space UNIV)" shows "emeasure (count_space UNIV \<Otimes>\<^sub>M count_space UNIV) A = emeasure (count_space UNIV) A"
induct: fixes P :: "nat set \<Rightarrow> (nat \<times> linear_poly) list \<Rightarrow> (nat \<times> linear_poly) list \<Rightarrow> bool"   and a0 :: "nat set"   and a1 :: "(nat \<times> linear_poly) list"   and a2 :: "(nat \<times> linear_poly) list" assumes "\<And>X t1. P X t1 []"   and "\<And>X t1 x p t2. \<lbrakk>x \<notin> X \<Longrightarrow> P X t1 t2; \<lbrakk>\<not> x \<notin> X; find (\<lambda>x. x \<notin> X) (Abstract_Linear_Poly.vars_list p) = None\<rbrakk> \<Longrightarrow> P X ((x, p) # t1) t2; \<And>x2 xa y xb ya xc yb. \<lbrakk>\<not> x \<notin> X; find (\<lambda>x. x \<notin> X) (Abstract_Linear_Poly.vars_list p) = Some x2; (xa, y) = pivot_tableau_eq t1 (x, p) t2 x2; (xb, ya) = y; (xc, yb) = xb\<rbrakk> \<Longrightarrow> P X xa ya\<rbrakk> \<Longrightarrow> P X t1 ((x, p) # t2)" shows "P a0 a1 a2"
uminus: fixes F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" shows "o[F](\<lambda>x. - g x) = o[F](g)"
SMP_append: fixes S :: "('fun, 'var) strand_step list"   and S' :: "('fun, 'var) strand_step list" shows "SMP (trms\<^sub>s\<^sub>t (S @ S')) = SMP (trms\<^sub>s\<^sub>t S) \<union> SMP (trms\<^sub>s\<^sub>t S')"
ext_funcset_to_sing_iff: fixes A :: "'b set"   and a :: "'c" shows "A \<rightarrow>\<^sub>E {a} = {\<lambda>x\<in>A. a}"
prob_space_T': fixes I :: "'s pmf" shows "prob_space (K.T' I)"
fold_pair_size_plus: fixes M :: "('d \<times> 'e) multiset"   and n :: "nat" shows "fold_mset ((+) \<circ> pair_size) 0 M + n = fold_mset ((+) \<circ> pair_size) n M"
NT_NF: fixes s :: "'c event list"   and P :: "'c process" assumes "s \<notin> T P" shows "(s, {}) \<notin> F P"
f_2: shows "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
env\<^sub>1'_def: shows "env\<^sub>1' = fmmap convert_term env\<^sub>1"
exhaust: fixes y :: "('a, 'b) lit" obtains x1 :: "('a, 'b) atm" where "y = Pos x1"   | x2 :: "('a, 'b) atm" where "y = Neg x2"
add_order_add_sym: fixes t :: "'a"   and s :: "'a" shows "add_order (t + s) = add_order (s + t)"
asymp_equiv_def: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<sim>[F] g) = ((\<lambda>x. if f x = (0::'b) \<and> g x = (0::'b) then 1::'b else f x / g x) \<longlongrightarrow> (1::'b)) F"
Mapping_inverse: fixes y :: "'b \<Rightarrow> 'c option" assumes "y \<in> UNIV" shows "mapping.rep (Mapping y) = y"
HLD_iff: fixes s :: "'a set"   and \<omega> :: "'a stream" shows "HLD s \<omega> = (shd \<omega> \<in> s)"
rimp_ppath_def: fixes X :: "'a ppath set"   and Z :: "'a ppath set" shows "rimp_ppath X Z = \<Union> {Y. pp_prod X Y \<subseteq> Z}"
nonzero_mult_div_cancel_left: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and a :: "'e"   and b :: "'e" assumes "class.semidom_divide divide plus minus zero times one"   and "a \<noteq> zero" shows "divide (times a b) a = b"
simps_4: shows "X5d X5.Xd2' = D2"
sec2: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common') \<turnstile>g \<A> \<surd>" shows "constructive_security_sim_obsf real_resource ideal_resource sim \<I>_real \<I>_common (absorb \<A> (obsf_converter (1\<^sub>C |\<^sub>= conv))) adv"
ordinal_times_distrib: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" shows "x * (y + z) = x * y + x * z"
codomains_comp: fixes g :: "'c \<times> 'c"   and f :: "'c \<times> 'c" assumes "CopxC.seq g f" shows "CopxC.codomains (g \<odot> f) = CopxC.codomains g"
callee_invariant': fixes y :: "'d"   and s' :: "'s"   and s :: "'s"   and x :: "'c" assumes "(y, s') \<in> set_spmf (right s x)"   and "I s"   and "x \<in> outs_\<I> \<I>r" shows "I s' \<and> y \<in> responses_\<I> \<I>r x"
strict_mono_compose: fixes r :: "'c \<Rightarrow> 'd"   and s :: "'e \<Rightarrow> 'c" assumes "strict_mono r"   and "strict_mono s" shows "strict_mono (\<lambda>x. r (s x))"
star_of_mult: fixes x :: "'a"   and y :: "'a" shows "star_of (x * y) = star_of x * star_of y"
gen_lossless_gpv_Pause: fixes co :: "bool"   and \<I> :: "('a, 'b) \<I>"   and out :: "'a"   and c :: "('c, 'a, 'b) rpv" shows "gen_lossless_gpv co \<I> (Generative_Probabilistic_Value.Pause out c) = (\<forall>input\<in>responses_\<I> \<I> out. gen_lossless_gpv co \<I> (c input))"
integrableD_3: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> real" assumes "integrable M f" shows "\<integral>\<^sup>+ x. ennreal (- f x) \<partial>M \<noteq> \<infinity>"
zero_less_measure_iff: fixes M :: "'a measure"   and A :: "'a set" shows "(0 < Sigma_Algebra.measure M A) = (Sigma_Algebra.measure M A \<noteq> 0)"
lattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "lattice L \<equiv> upper_semilattice L \<and> lower_semilattice L"
fds_divisor_sigma: fixes x :: "'b" shows "fds (divisor_sigma x) = fds_zeta * fds_shift x fds_zeta"
Reach_def: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" shows "Reach M l s t = (\<exists>k. (M, l, s, k, t) \<in> Reachable)"
Not\<^sub>\<alpha>_eq_iff: fixes t1\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>"   and t2\<^sub>\<alpha> :: "('e, 'f, 'g) Tree\<^sub>\<alpha>" shows "(Not\<^sub>\<alpha> t1\<^sub>\<alpha> = Not\<^sub>\<alpha> t2\<^sub>\<alpha>) = (t1\<^sub>\<alpha> = t2\<^sub>\<alpha>)"
directional_derivative_in_span: fixes a :: "'c"   and x :: "'c" shows "directional_derivative \<infinity> a x \<in> span (directional_derivative \<infinity> a ` Basis)"
cont_tf_thm_2: fixes v :: "i" shows "[ContingentlyFalse q\<^sub>0 \<^bold>\<or> ContingentlyFalse (q\<^sub>0\<^sup>-) in v]"
0: shows "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"
simps_1: fixes r :: "real" shows "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
eq_equivp: fixes eq :: "'b ltln \<Rightarrow> 'b ltln \<Rightarrow> bool" assumes "ltl_equivalence eq" shows "equivp eq"
finite_succeed_t'': fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and F :: "'d set"   and i :: "nat" assumes "mojmir \<Sigma> \<delta> q\<^sub>0 w F"   and "q\<^sub>0 \<in> F" shows "finite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i) = finite (mojmir_def.succeed_t \<Sigma> \<delta> q\<^sub>0 w F i)"
wf_rexp_of_alt: fixes n :: "nat"   and \<phi> :: "'a formula" assumes "wf_formula n \<phi>" shows "local.wf n (rexp_of_alt n \<phi>)"
homogeneous_monom_mult: fixes p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and c :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p" shows "homogeneous (MPoly_Type_Class.punit.monom_mult c t p)"
path_def: fixes Obj :: "'a set"   and Arr :: "'b set"   and Dom :: "'b \<Rightarrow> 'a"   and Cod :: "'b \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and p :: "'b list" assumes "graph Obj Arr Dom Cod" shows "graph.path Obj Arr Dom Cod x y p \<equiv> p = [] \<and> x = y \<and> x \<in> Obj \<or> p \<noteq> [] \<and> x = Dom (hd p) \<and> y = Cod (last p) \<and> (\<forall>n. 0 \<le> n \<and> n < length p \<longrightarrow> p ! n \<in> Arr) \<and> (\<forall>n. 0 \<le> n \<and> n < length p - 1 \<longrightarrow> Cod (p ! n) = Dom (p ! (n + 1)))"
simps_2: fixes n :: "nat"   and c :: "int"   and t :: "MIR.num" shows "ismaxcoeff (CN n c t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> ismaxcoeff t x)"
\<omega>_sum_less_self: fixes \<alpha> :: "V"   and \<alpha>s :: "V list" assumes "list.set (\<alpha> # \<alpha>s) \<subseteq> ON"   and "sorted_wrt (\<lambda>x y. y \<le> x) (\<alpha> # \<alpha>s)" shows "\<omega>_sum \<alpha>s < \<omega> \<up> \<alpha> + \<omega>_sum \<alpha>s"
arl_copy_def: shows "arl_copy \<equiv> \<lambda>(a, n). array_copy a \<bind> (\<lambda>a. return (a, n))"
to_nat_less_card: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and Rep :: "'d \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'd"   and j :: "'d" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep j < CARD('d)"
simps_15: fixes bs :: "real list"   and p :: "fm"   and q :: "fm" shows "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
get_element_by_id_result_in_tree_order: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and iden :: "char list"   and element_ptr :: "(_) element_ptr"   and to :: "(_) object_ptr list" assumes "h \<turnstile> Shadow_DOM.get_element_by_id ptr iden \<rightarrow>\<^sub>r Some element_ptr"   and "h \<turnstile> Shadow_DOM.to_tree_order ptr \<rightarrow>\<^sub>r to" shows "cast\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr \<in> set to"
list2policy_def: fixes l :: "('d \<Rightarrow> 'e decision option) list" shows "list2policy l = foldr (\<lambda>x y. x \<Oplus> y) l \<emptyset>"
symm: fixes x :: "'p"   and y :: "'p" shows "dist x y = dist y x"
scalar_right_one: fixes nn :: "nat"   and v :: "'a list"   and i :: "nat" assumes "vec nn v"   and "i < nn" shows "scalar_prod v (vec1 nn i) = v ! i"
assms: shows "desargues_config_3D A B C A' B' C' P \<alpha> \<beta> \<gamma>"
sup_least: fixes y :: "'a \<times> 'b"   and x :: "'a \<times> 'b"   and z :: "'a \<times> 'b" assumes "less_eq_prod y x"   and "less_eq_prod z x" shows "less_eq_prod (sup_prod y z) x"
simps_2: fixes Leg0 :: "'b"   and Leg1 :: "'b"   and more :: "'c" shows "Leg1 \<lparr>Leg0 = Leg0, Leg1 = Leg1, \<dots> = more\<rparr> = Leg1"
typeof_Boolean: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Boolean\<rfloor>" shows "\<exists>b. v = Bool b"
distinct_11: fixes x1 :: "'e"   and x7 :: "'e" shows "broadcast x1 \<noteq> receive x7"
start_points_in_root: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "start_points x \<le> r"
i_aggregate_i_append_mod: fixes xs :: "'d list"   and k :: "nat"   and f :: "nat \<Rightarrow> 'd"   and ag :: "'d list \<Rightarrow> 'd" assumes "length xs mod k = 0" shows "i_aggregate (xs \<frown> f) k ag = f_aggregate xs k ag \<frown> i_aggregate f k ag"
lprod_implies_mult: fixes as :: "'b list"   and bs :: "'b list"   and R :: "('b \<times> 'b) set" assumes "(as, bs) \<in> lprod R"   and "trans R" shows "(mset as, mset bs) \<in> mult R"
\<iota>_in_hom: shows "\<guillemotleft>\<iota>\<^sub>1 : \<I>\<^sub>1 \<otimes>\<^sub>1 \<I>\<^sub>1 \<rightarrow> \<I>\<^sub>1\<guillemotright>"
division_of_tagged_division: fixes s :: "('a \<times> 'a set) set"   and i :: "'a set" assumes "s tagged_division_of i" shows "snd ` s division_of i"
ts_cons: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and e0 :: "exp"   and C0 :: "nat"   and es :: "exp list"   and Cs :: "nat list" assumes "CT;\<Gamma> \<turnstile> e0 : C0"   and "CT;\<Gamma> \<turnstile>+ es : Cs" shows "CT;\<Gamma> \<turnstile>+ (e0 # es) : (C0 # Cs)"
sep_add_zero: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "class.pre_sep_algebra plus zero sep_disj" shows "plus x zero = x"
coprime_commute: fixes b :: "'a"   and a :: "'a" shows "algebraic_semidom_class.coprime b a = algebraic_semidom_class.coprime a b"
matrix_sum_trace_le: fixes n :: "nat"   and f :: "nat \<Rightarrow> complex mat"   and g :: "nat \<Rightarrow> complex mat" assumes "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> g k \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> trace (f k) \<le> trace (g k)" shows "trace (matrix_sum d f n) \<le> trace (matrix_sum d g n)"
simps: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" shows "mat_inv (a, b, c, d) = (1 / (a * d - b * c)) *\<^sub>s\<^sub>m (d, - b, - c, a)"
uminus_sum_list_map: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" assumes "class.ab_group_add plus zero minus uminus" shows "uminus (monoid_add.sum_list plus zero (map f xs)) = monoid_add.sum_list plus zero (map (uminus \<circ> f) xs)"
jumpF_poly_top_def: fixes q :: "real poly"   and p :: "real poly" shows "jumpF_poly_top q p = jumpF (\<lambda>x. poly q x / poly p x) at_top"
qSwapAll_sym: fixes zs :: "'f"   and x :: "'g"   and y :: "'g"   and X :: "('d, 'e, 'f, 'g, 'h) qTerm"   and A :: "('d, 'e, 'f, 'g, 'h) qAbs" shows "X #[[x \<and> y]]_zs = X #[[y \<and> x]]_zs \<and> A $[[x \<and> y]]_zs = A $[[y \<and> x]]_zs"
set_spmf_of_set: fixes A :: "'a set" shows "set_spmf (spmf_of_set A) = (if finite A then A else {})"
simps: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and a1 :: "'c rule list"   and a2 :: "state"   and a3 :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>a1, a2\<rangle> \<Rightarrow> a3 = ((\<exists>t. a1 = [] \<and> a2 = t \<and> a3 = t) \<or> (\<exists>m. a1 = [Rule m Accept] \<and> a2 = Undecided \<and> a3 = Decision FinalAllow \<and> matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Drop] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Reject] \<and> a2 = Undecided \<and> a3 = Decision FinalDeny \<and> matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Log] \<and> a2 = Undecided \<and> a3 = Undecided \<and> matches \<gamma> m p) \<or> (\<exists>m. a1 = [Rule m Empty] \<and> a2 = Undecided \<and> a3 = Undecided \<and> matches \<gamma> m p) \<or> (\<exists>m a. a1 = [Rule m a] \<and> a2 = Undecided \<and> a3 = Undecided \<and> \<not> matches \<gamma> m p) \<or> (\<exists>rs X. a1 = rs \<and> a2 = Decision X \<and> a3 = Decision X) \<or> (\<exists>rs\<^sub>1 t rs\<^sub>2 t'. a1 = rs\<^sub>1 @ rs\<^sub>2 \<and> a2 = Undecided \<and> a3 = t' \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> t \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>2, t\<rangle> \<Rightarrow> t') \<or> (\<exists>m chain rs\<^sub>1 m' rs\<^sub>2. a1 = [Rule m (Call chain)] \<and> a2 = Undecided \<and> a3 = Undecided \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some (rs\<^sub>1 @ [Rule m' Return] @ rs\<^sub>2) \<and> matches \<gamma> m' p \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs\<^sub>1, Undecided\<rangle> \<Rightarrow> Undecided) \<or> (\<exists>m chain rs t. a1 = [Rule m (Call chain)] \<and> a2 = Undecided \<and> a3 = t \<and> matches \<gamma> m p \<and> \<Gamma> chain = Some rs \<and> \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> t))"
assms: fixes x :: "'a"   and xs :: "'a list"   and a :: "'c \<times> 'd" shows "snd (h x xs a) = snd a"
whn_neq_hypnat_of_nat: fixes n :: "nat" shows "whn \<noteq> hypnat_of_nat n"
T5_def: shows "T5 \<equiv> unlabel (dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T))"
sigma_sets_subset': fixes a :: "'b set set"   and M :: "'b measure"   and \<Omega>' :: "'b set" assumes "a \<subseteq> sets M"   and "\<Omega>' \<in> sets M" shows "sigma_sets \<Omega>' a \<subseteq> sets M"
map: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a \<Rightarrow> 'c \<times> 'a" shows "map_state f (State x) = State (map_prod f id \<circ> x)"
islimpt_sequential: fixes x :: "'b"   and S :: "'b set" shows "x islimpt S = (\<exists>f. (\<forall>n. f n \<in> S - {x}) \<and> f \<longlonglongrightarrow> x)"
eq_fin_concat: fixes u :: "'action list"   and w\<^sub>1 :: "'action list"   and v :: "'action list"   and w\<^sub>2 :: "'action list" shows "(u @ w\<^sub>1 @ v =\<^sub>F u @ w\<^sub>2 @ v) = (w\<^sub>1 =\<^sub>F w\<^sub>2)"
vD_nsqn_sqn: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "ip \<in> vD rt" shows "nsqn rt ip = sqn rt ip"
trace_with_Tick_implies_tickFree_front: fixes P :: "('c event list \<times> 'c event set) set \<times> 'c event list set"   and t :: "'c event list" assumes "is_process P"   and "t @ [tick] \<in> TRACES P" shows "tickFree t"
attractive_def: fixes A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "attractive A less_eq \<equiv> semiattractive A less_eq \<and> attractive_axioms A less_eq"
low_mds_eq_def: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and mds :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.low_mds_eq dma mds mem\<^sub>1 mem\<^sub>2 \<equiv> \<forall>x. dma x = Low \<and> x \<notin> mds AsmNoRead \<longrightarrow> mem\<^sub>1 x = mem\<^sub>2 x"
healthy_nnegD: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and P :: "'c \<Rightarrow> real" assumes "healthy t"   and "sound P" shows "nneg (t P)"
writes_small_big: fixes SW :: "('x, 'y, 'z) prog set"   and setter :: "('x, 'y, 'aa) prog"   and h :: "'x"   and h' :: "'x"   and P :: "'x \<Rightarrow> 'x \<Rightarrow> bool" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> P h h'"   and "reflp P"   and "transp P" shows "P h h'"
hyps_4: assumes "\<nexists>w c. awalk w c w \<and> w \<in> set (awalk_verts u (q @ s)) \<and> awalk_cost f c < 0" shows "awalk_cost f (awalk_to_apath (q @ s)) \<le> awalk_cost f (q @ s)"
enat_of_option_simps_1: fixes x :: "nat option"   and n :: "nat" shows "(Dijkstra_Abstract.enat_of_option x = enat n) = (x = Some n)"
has_as_product_def: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and J :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'b \<Rightarrow> 'a"   and a :: "'a" assumes "category C" shows "category.has_as_product C J D a \<equiv> \<exists>\<pi>. product_cone J C D a \<pi>"
substitute: shows "integral {pathstart \<gamma> \<bullet> i..pathfinish \<gamma> \<bullet> i} (\<lambda>f_var. F (f_var *\<^sub>R i + g f_var) \<bullet> i) = integral {0..1} (\<lambda>x. F (\<gamma> x) \<bullet> i * vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x within {0..1}))"
nonzero_minus_divide_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "b \<noteq> zero" shows "uminus (divide a b) = divide a (uminus b)"
abs_eq_add_2_mult: fixes x :: "'a"   and i :: "int"   and y :: "'a" assumes "\<bar>x\<bar> = \<bar>x + (2::'a) * of_int i * y\<bar>"   and "- y < x"   and "x \<le> y"   and "i \<noteq> 0" shows "x = y \<and> i = - 1"