hcju/isabelleps huggingface dataset:
lookup_monom_mult: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and c :: "'d"   and t :: "'b"   and p :: "'a \<Rightarrow>\<^sub>0 'd"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.monom_mult pair_of_term term_of_pair c t p) v = (if term_powerprod.adds_pp pair_of_term t v then c * lookup p (term_powerprod.sminus pair_of_term term_of_pair v t) else (0::'d))"
pair_declclass_simp: fixes c :: "'a"   and x :: "'b" shows "declclass (c, x) = declclass c"
assms_4: shows "all_fulfill (in_all_resolvents_upon S E2) R"
simps_1: fixes x :: "'b common_primitive" shows "compress_extra (Match x) = Match x"
face_of_refl_eq: fixes S :: "'b set" shows "(S face_of S) = convex S"
distinct_1: fixes x1 :: "brick list"   and x21 :: "brick list"   and x22 :: "wall" shows "basic x1 \<noteq> x21 * x22"
is_non_volatile_Read\<^sub>s\<^sub>b_simps_2: fixes volatile :: "bool"   and a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set" shows "is_non_volatile_Read\<^sub>s\<^sub>b (Write\<^sub>s\<^sub>b volatile a sop v A L R W) = False"
remdupsListDiff: fixes a :: "'b list"   and b :: "'b list" shows "remdups (list_diff a b) = list_diff (remdups a) (remdups b)"
elims_3: fixes x :: "'b trm" assumes "\<not> is_compound x" obtains xa :: "'b" where "x = Var xa__"   | xa :: "'b" where "x = Const xa"
map_ide_simp: fixes a :: "'a" assumes "hom'.ide a" shows "\<psi>.map a = hom'.inv (d\<^sub>0e\<^sub>0.\<phi> a)"
cupcake_list_state_preserve0: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env :: "v sem_env"   and s :: "'a state"   and es :: "exp0 list"   and res :: "'a state \<times> (v list, v) result" assumes "cakeml_static_env static_cenv"   and "Big_Step_Unclocked_Single.evaluate_list (\<lambda>s e res. Big_Step_Unclocked_Single.evaluate env s e res \<and> (cakeml_static_env.is_cupcake_all_env static_cenv env \<longrightarrow> is_cupcake_exp e \<longrightarrow> s = fst res)) s es res"   and "list_all is_cupcake_exp es"   and "cakeml_static_env.is_cupcake_all_env static_cenv env" shows "s = fst res"
filter_map: fixes P :: "'e \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and xs :: "'f list" shows "filter P (map f xs) = map f (filter (P \<circ> f) xs)"
row_def: fixes i :: "'e"   and A :: "(('c, 'd) vec, 'e) vec" shows "row i A = (\<chi>j. A $ i $ j)"
sigma_sets_eqI: fixes A :: "'b set set"   and M :: "'b set"   and B :: "'b set set" assumes "\<And>a. a \<in> A \<Longrightarrow> a \<in> sigma_sets M B"   and "\<And>b. b \<in> B \<Longrightarrow> b \<in> sigma_sets M A" shows "sigma_sets M A = sigma_sets M B"
sin_cos_le1: fixes x :: "real"   and y :: "real" shows "\<bar>sin x * sin y + cos x * cos y\<bar> \<le> 1"
get_ancestors_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_ancestors ptr" shows "ptr |\<in>| object_ptr_kinds h"
fls_inverse_base2: fixes f :: "'a fls" shows "fls_nth (inverse f) (- fls_subdegree f) = inverse (fls_nth f (fls_subdegree f))"
rt_strictly_fresher_def': fixes i :: "nat"   and rt1 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt2 :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" shows "(rt1 \<sqsubset>\<^bsub>i\<^esub> rt2) = (nsqn\<^sub>r (the (rt1 i)) < nsqn\<^sub>r (the (rt2 i)) \<or> nsqn\<^sub>r (the (rt1 i)) = nsqn\<^sub>r (the (rt2 i)) \<and> \<pi>\<^sub>5 (the (rt2 i)) < \<pi>\<^sub>5 (the (rt1 i)))"
sa_impl_rel_eext_internal_def: fixes Rm :: "('i \<times> 'l) set"   and Rv :: "('g \<times> 'j) set"   and Rl :: "('h \<times> 'k) set" shows "sa_impl_rel_eext Rm Rv Rl \<equiv> \<langle>Rm, Rv \<rightarrow> Rl\<rangle>gen_sa_impl_rel_eext"
Action_def: fixes x2 :: "act"   and x1 :: "ccs" shows "x2.x1 \<equiv> Abs_ccs (Action_Rep x2 (Rep_ccs x1))"
iteration_double: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and star :: "'c \<Rightarrow> 'c"   and strong_iteration :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.dra plus times one zero less_eq less star strong_iteration" shows "strong_iteration (strong_iteration x) = strong_iteration one"
multrow_mat_def: fixes n :: "nat"   and k :: "nat"   and a :: "'a" shows "multrow_mat n k a = mat n n (\<lambda>(i, j). if k = i \<and> k = j then a else if i = j then 1::'a else (0::'a))"
lborel_affine_euclidean: fixes c :: "'b \<Rightarrow> real"   and t :: "'b" assumes "\<And>j. j \<in> Basis \<Longrightarrow> c j \<noteq> 0" shows "lborel = density (distr lborel borel (\<lambda>x. t + (\<Sum>j\<in>Basis. (c j * (x \<bullet> j)) *\<^sub>R j))) (\<lambda>x. ennreal (\<Prod>j\<in>Basis. \<bar>c j\<bar>))"
nonequiv_ex_Ostrowski_elem: fixes v :: "'a \<Rightarrow> ant"   and v' :: "'a \<Rightarrow> ant" assumes "valuation K v"   and "valuation K v'"   and "\<not> v_equiv K v v'" shows "\<exists>x\<in>carrier K. 0 \<le> v x \<and> v' x < 0"
assms_2: shows "theta\<inverse> \<subseteq> RetrT (theta\<inverse> \<union> BisT)"
map_def: fixes f :: "'a" shows "\<Omega>.map f = (if arr f then \<I> else null)"
pr_var_def: fixes x :: "'c \<Longrightarrow> 'd" shows "&x = x"
simps_1: fixes n :: "nat"   and fs :: "'a vec list"   and i :: "nat"   and j :: "nat" assumes "gram_schmidt_fs_lin_indpt n fs" shows "gram_schmidt_fs_lin_indpt.\<sigma> n fs 0 i j = (0::'a)"
sound_intros_8: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (P && Q)"
P2: shows "Bet B A B' \<and> Cong A B' B A"
N_fact_le: fixes m :: "nat" shows "N (fact m) \<le> 1 / (q - 1) * 1 / q ^ m"
Fields_i_i_o_def: shows "Fields_i_i_o \<equiv> \<lambda>x xa. pred.Pred (Fields x xa)"
assms: shows "cdens_ctxt_invar vs vs' \<Gamma> \<delta>"
listset_Cons_conv: fixes xs :: "'a list"   and A :: "'a set"   and As :: "'a set list" assumes "xs \<in> listset (A # As)" shows "\<exists>y ys. y \<in> A \<and> ys \<in> listset As \<and> xs = y # ys"
less_extended_def: fixes x :: "'a extended"   and y :: "'a extended" shows "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
conjugate_neg: fixes a :: "'a" shows "conjugate (- a) = - conjugate a"
simple_function_cong: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>t. t \<in> space M \<Longrightarrow> f t = g t" shows "simple_function M f = simple_function M g"
restrict_map_disj_left: fixes h\<^sub>0 :: "'c \<Rightarrow> 'd option"   and h\<^sub>1 :: "'c \<Rightarrow> 'd option"   and S :: "'c set" assumes "h\<^sub>0 \<bottom> h\<^sub>1" shows "h\<^sub>0 |` S \<bottom> h\<^sub>1"
independent_vnmD1: fixes C :: "'a pmf set"   and P :: "('a pmf \<times> 'a pmf) set" assumes "independent_vnm C P" shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
dense: fixes f :: "atom fm"   and l :: "real"   and x :: "real"   and xs :: "real list"   and u :: "real"   and y :: "real" assumes "nqfree f"   and "\<forall>y\<in>{l<..<x}. y \<notin> {(r - \<langle>cs,xs\<rangle>) / c |r c cs. Less r (c # cs) \<in> set (atoms\<^sub>0 f) \<and> 0 < c}"   and "\<forall>y\<in>{x<..<u}. y \<notin> UB f xs"   and "l < x"   and "x < u"   and "x \<notin> EQ f xs"   and "R.I f (x # xs)"   and "l < y"   and "y < u" shows "R.I f (y # xs)"
deriv_bodies: shows "\<forall>p\<in>Procs. \<forall>Z. \<Gamma>,\<Theta> \<union> (\<Union>p\<in>Procs. \<Union>Z. {(P p Z, p, Q p Z, A p Z)})\<turnstile>\<^bsub>/F \<^esub>(P p Z) the (\<Gamma> p) (Q p Z),(A p Z)"
AA25: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "'a \<Rightarrow> 'b"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and w :: "'a \<Rightarrow> 'c" assumes "|~ \<langle>P\<rangle>_v \<longrightarrow> \<langle>Q\<rangle>_w" shows "\<turnstile> \<diamond>\<langle>P\<rangle>_v \<longrightarrow> \<diamond>\<langle>Q\<rangle>_w"
sums_If_finite: fixes P :: "nat \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'c" assumes "finite {r. P r}" shows "(\<lambda>r. if P r then f r else (0::'c)) sums (\<Sum>r | P r. f r)"
2: shows "hom_induced p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0} (subtopology (nsphere n) {x. - 1 / 2 \<le> x k}) {x. - 1 / 2 \<le> x k \<and> x k \<le> 0} id \<in> Group.iso (relative_homology_group p (subtopology (nsphere n) {x. 0 \<le> x k}) {x. x k = 0}) (relative_homology_group p (subtopology (nsphere n) {x. - 1 / 2 \<le> x k}) {x. - 1 / 2 \<le> x k \<and> x k \<le> 0})"
fsubset_iff_pfsubset_eq: fixes A :: "'b fset"   and B :: "'b fset" shows "(A |\<subseteq>| B) = (A |\<subset>| B \<or> A = B)"
fls_lr_inverse_base_2: fixes f :: "'a fls"   and y :: "'a" shows "fls_nth (fls_right_inverse f y) (- fls_subdegree f) = y"
less_eq_gexp_def: fixes e1 :: "'b gexp"   and e2 :: "'b gexp" shows "e1 \<le> e2 \<equiv> e1 < e2 \<or> e1 = e2"
ereal_add_le_add_iff: fixes c :: "ereal"   and a :: "ereal"   and b :: "ereal" shows "(c + a \<le> c + b) = (a \<le> b \<or> c = \<infinity> \<or> c = - \<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
that: assumes "vsv a"   and "\<D>\<^sub>\<circ> a = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> a\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Obj\<rparr>" shows "thesis"
factor_bound: fixes f :: "int poly"   and g :: "int poly"   and n :: "nat"   and k :: "nat" assumes "f \<noteq> 0"   and "g dvd f"   and "degree g \<le> n" shows "\<bar>poly.coeff g k\<bar> \<le> factor_bound f n"
TID_nonneg: fixes x :: "'c Gromov_completion"   and y :: "'c Gromov_completion" shows "0 \<le> turn_into_distance extended_predist x y"
f1: shows "\<not> ((\<forall>(a, b, c)\<in>set a. a = 0 \<and> b = 0 \<and> c = 0) \<and> (\<forall>(a, b, c)\<in>set b. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c < 0) \<and> (\<forall>(a, b, c)\<in>set c. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<le> 0) \<and> (\<forall>(a, b, c)\<in>set d. \<exists>x. \<forall>y<x. a * y\<^sup>2 + b * y + c \<noteq> 0))"
distinct_replacefacesAt22: fixes i :: "nat"   and Fss :: "face list list"   and "is" :: "nat list"   and newFs :: "face list"   and olfF :: "face" assumes "i < |Fss|"   and "i \<notin> set is"   and "distinct is"   and "distinct (Fss ! i)"   and "distinct newFs"   and "set (Fss ! i) \<inter> set newFs \<subseteq> {olfF}" shows "distinct (replacefacesAt2 is olfF newFs Fss ! i)"
paths: shows "is_path_undir G v p' u \<and> is_path_undir G u' p'' v'"
abs_norm_cancel: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and a :: "'c" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "\<bar>norm a\<bar> = norm a"
null_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e_def: shows "null \<equiv> Abs_Pair\<^sub>b\<^sub>a\<^sub>s\<^sub>e \<lfloor>\<bottom>\<rfloor>"
wf: shows "wf_mstate \<phi> \<pi> R st"
inf_bot_right: fixes x :: "'c multiset" shows "x \<inter># {#} = {#}"
Card_cardinal_eq: fixes \<kappa> :: "V" assumes "Card \<kappa>" shows "vcard \<kappa> = \<kappa>"
distinct_1: fixes x1 :: "'d"   and x21 :: "'c"   and x22 :: "('c, 'd) Term.term list" shows "Var x1 \<noteq> Fun x21 x22"
cong_0_1_nat': fixes n :: "nat" shows "[0 = Suc 0] (mod n) = (n = Suc 0)"
3: fixes a :: "'a poly"   and ia :: "nat"   and ba :: "'a poly"   and j :: "nat" assumes "(a, ia) \<in> set bs"   and "(ba, j) \<in> set bs"   and "(a, ia) \<noteq> (ba, j)" shows "coprime a ba"
in_rel: fixes a :: "'e"   and b :: "'e" shows "(a = b) = (\<exists>z. z \<in> UNIV \<and> id z = a \<and> id z = b)"
items_simp: fixes H :: "msg set" assumes "H \<subseteq> range Agent \<union> range Number \<union> range Nonce \<union> range Key \<union> range Hash \<union> range Passwd \<union> range Pri_AgrK \<union> range Pub_AgrK" shows "items H = H"
Ex_first_occurrence: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n" shows "\<exists>m. P m \<and> (\<forall>i<m. \<not> P i)"
sympartpI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "r x y"   and "r y x" shows "sympartp r x y"
not_zero_less_neg_numeral: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and n :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "\<not> less zero (uminus (numeral.numeral one plus n))"
disc_1: fixes x1 :: "'b" shows "isl (Inl x1)"
Abs_dnelist_inverse: fixes y :: "'a list" assumes "y \<in> {xs. distinct xs \<and> xs \<noteq> []}" shows "list_of_dnelist (Abs_dnelist y) = y"
syzygy_module_list_closed_monom_mult: fixes s :: "'t \<Rightarrow>\<^sub>0 'c"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and c :: "'c"   and t :: "'a" assumes "s \<in> syzygy_module_list bs" shows "monom_mult c t s \<in> syzygy_module_list bs"
Eq\<^sub>0E: fixes v :: "i"   and F :: "\<o>"   and G :: "\<o>" assumes "[F \<^bold>=\<^sub>0 G in v]" shows "F = G"
lim: shows "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"
distinct_169: fixes x91 :: "expr"   and x92 :: "char list"   and x93 :: "char list list"   and x94 :: "expr"   and x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list" shows "x91\<bullet>x92{x93} := x94 \<noteq> Call x101 x102 x103 x104"
Min_le: fixes A :: "'a set"   and x :: "'a" assumes "finite A"   and "x \<in> A" shows "linorder.Min (\<sqsubseteq>) A \<sqsubseteq> x"
g'_def: fixes b :: "'c set" shows "g' b = (b \<subseteq> UNIV \<and> g b)"
pref_prod_short': fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v \<cdot> w"   and "\<^bold>|u\<^bold>| < \<^bold>|v\<^bold>|" shows "u <p v"
det_def: fixes A :: "'c mat" shows "det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'c))"
Hilbert_fun_cone_nonempty: fixes h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set"   and z :: "nat" assumes "h \<in> P[X]"   and "h \<noteq> 0"   and "U \<subseteq> X"   and "homogeneous h"   and "U \<noteq> {}" shows "Hilbert_fun (cone (h, U)) z = (if poly_deg h \<le> z then z - poly_deg h + (card U - 1) choose (card U - 1) else 0)"
assms_2: fixes t1 :: "('a, unit) RBT_Impl.rbt"   and t2 :: "('a, unit) RBT_Impl.rbt"   and color :: "color"   and a :: "'a"   and b :: "unit" assumes "P t1"   and "P t2"   and "t1 = rbt.Empty" shows "P (Branch color t1 a b t2)"
ideal_prod_assocTr1: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set"   and y :: "'a"   and n :: "nat" assumes "ideal R A"   and "ideal R B"   and "ideal R C"   and "y \<in> C" shows "\<forall>f\<in>{j. j \<le> n} \<rightarrow> set_mult R A B. \<Sigma>\<^sub>e R f n \<cdot>\<^sub>r y \<in> A \<diamondsuit>\<^sub>r (B \<diamondsuit>\<^sub>r C)"
dom_eqI': fixes a :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "a \<in> VV.domains f" shows "a = VV.dom f"
execn_Suc: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =Suc n\<Rightarrow> t"
sublmm_4: fixes a :: "('fun, 'var) stateful_strand_step"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "list_all tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p (a # A)"   and "fv\<^sub>s\<^sub>s\<^sub>t (a # A) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = {}"   and "\<forall>b\<in>set D. case b of (t, s) \<Rightarrow> (fv t \<union> fv s) \<inter> bvars\<^sub>s\<^sub>s\<^sub>t (a # A) = {}"   and "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (a # A) \<union> pair ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t (a # A) \<union> pair ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'" shows "\<forall>t\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` set D. \<forall>t'\<in>pair ` setops\<^sub>s\<^sub>s\<^sub>t A \<union> pair ` set D. (\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t'"
size_height_if_incomplete: fixes kdt :: "'a kdt" assumes "\<not> KD_Tree.complete kdt" shows "KD_Tree.size_kdt kdt < 2 ^ height kdt"
n_absorb2': fixes x :: "'a"   and y :: "'a" shows "!x \<cdot> (!x + !y) = !x"
M: assumes "(Spy, \<langle>n, AgrKey (PubK S)\<rangle>) \<in> u\<^sub>4"   and "Rev_PriK \<in> S"   and "\<forall>m. Sec_PriK m \<in> S \<longrightarrow> (n, m) \<notin> bad_id"   and "\<forall>S. Rev_PriK \<in> S \<longrightarrow> (Spy, \<langle>n, AgrKey (PubK S)\<rangle>) \<in> u\<^sub>4 \<longrightarrow> (\<exists>m. Sec_PriK m \<in> S \<and> (n, m) \<in> bad_id)" shows "False"
s_1: shows "pair (s, s') \<in> pair ` setops\<^sub>s\<^sub>s\<^sub>t S"
b: fixes Y :: "'a set"   and x :: "'a" assumes "Y \<noteq> {}" shows "\<Squnion>{x \<cdot> y |y. y \<in> Y} \<le> x \<cdot> \<Squnion>Y"
continuous_closedin_preimage_gen: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and T :: "'b set"   and U :: "'b set" assumes "continuous_on S f"   and "f ` S \<subseteq> T"   and "closedin (top_of_set T) U" shows "closedin (top_of_set S) (S \<inter> f -` U)"
state_ext: fixes m :: "('b, 'c) state"   and n :: "('b, 'c) state" assumes "\<And>s. run_state m s = run_state n s" shows "m = n"
open_Collect_less: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "continuous_on UNIV f"   and "continuous_on UNIV g" shows "open {x. f x < g x}"
m1: shows "all_security_requirements_fulfilled M \<lparr>nodes = V, edges = E\<rparr>"
elims_2: fixes x :: "'c list"   and xa :: "'c list" assumes "less_eq_list x xa" obtains ys :: "'c list" where "xa__ = []" and "xa = ys__"   | x :: "'c" and xs :: "'c list" and y :: "'c" and ys :: "'c list" where "xa__ = x # xs" and "xa = y # ys" and "x < y \<or> x = y \<and> less_eq_list xs ys"
items_le_paths_le: fixes k :: "nat"   and P :: "(symbol \<times> character list) list set" shows "items_le k (Gen P) = Gen (paths_le k P)"
ct_suffix_unfold_one: fixes xs :: "ct list"   and x :: "ct"   and ys :: "ct list"   and y :: "ct" shows "ct_suffix (xs @ [x]) (ys @ [y]) = (ct_eq x y \<and> ct_suffix xs ys)"
power_int_0_right: fixes x :: "'a" shows "x powi 0 = (1::'a)"
add_order_add_sym: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and t :: "'b"   and s :: "'b" assumes "class.group_add minus plus zero uminus" shows "monoid_add.add_order plus zero (plus t s) = monoid_add.add_order plus zero (plus s t)"
get_component_parent_inside: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and node_ptr :: "(_) node_ptr"   and parent :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 node_ptr \<in> set c"   and "h \<turnstile> get_parent node_ptr \<rightarrow>\<^sub>r Some parent" shows "parent \<in> set c"
assign_loc_reach: fixes c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and x :: "'Var"   and e :: "'AExp"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach \<langle>(x \<leftarrow> e) \<otimes> annos, mds, mem\<rangle>" shows "c' = Stop \<and> mds' = mds \<oplus> annos \<or> c' = (x \<leftarrow> e) \<otimes> annos \<and> mds' = mds"
lm097: fixes y :: "'c"   and f :: "('d \<times> 'c) set"   and x :: "'d" assumes "y \<in> f `` {x}"   and "runiq f" shows "f ,, x = y"
vproduct_vsingletonI: fixes f :: "V"   and c :: "V"   and A :: "V \<Rightarrow> V" assumes "f\<lparr>c\<rparr> \<in>\<^sub>\<circ> A c"   and "f = set {\<langle>c, f\<lparr>c\<rparr>\<rangle>}" shows "f \<in>\<^sub>\<circ> vproduct (set {c}) A"
assms_5: shows "\<forall>x. 0 < bv2int x"
poly_lift2_0_iff: fixes p :: "'c poly" shows "(poly_lift2 p = 0) = (p = 0)"
simps_1: fixes name :: "name" shows "unfree (Free name) = Some name"
proj_onto_def: fixes a :: "'d"   and S :: "'d set" shows "proj_onto a S = sum (proj a) S"
set_mset_mset: fixes xs :: "'a list" shows "set_mset (mset xs) = set xs"
exhaust: fixes y :: "('c, 'd) operation" obtains x11 :: "'c \<times> 'd \<times> bool" and x12 :: "'c option" where "y = Insert x11 x12"   | x2 :: "'c" where "y = Delete x2"
functions_of_partitions_of: fixes B :: "'c set"   and F :: "('d \<Rightarrow> 'c) set"   and A :: "'d set" assumes "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // range_permutation A B" shows "functions_of (partitions_of A B F) A B = F"
delta_hedging_cls_val_process: fixes N :: "bool stream measure"   and der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and n :: "nat"   and w :: "bool stream" shows "cls_val_process Mkt (delta_hedging N der matur) (Suc n) w = geom_proc (Suc n) w * delta_price N der matur n w + (1 + r) * (val_process Mkt (delta_hedging N der matur) n w - geom_proc n w * delta_price N der matur n w)"
i_cut_min_empty_2: fixes t :: "'a"   and I :: "'a set" assumes "t \<le> iMin I" shows "I \<down>< t = {}"
acute_distincts: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Acute A B C" shows "A \<noteq> B \<and> C \<noteq> B"
cfinitely_spanned: shows "\<exists>S. finite S \<and> cspan S = UNIV"
deriv: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Language.normalize c Q,A"
rule: fixes m :: "'v SecurityInvariant_configured"   and E :: "('v \<times> 'v) set" assumes "m \<in> set (get_ACS M)" shows "\<forall>F\<in>c_offending_flows m \<lparr>nodes = hosts \<T>, edges = flows_fix \<T> \<union> backflows (filternew_flows_state \<T>) - E\<rparr>. F \<subseteq> backflows (filternew_flows_state \<T>) - E"
zero_right: fixes x :: "'a set"   and y :: "'a set" assumes "x \<in> \<D>"   and "y \<in> \<D>"   and "x \<noteq> y"   and "x \<inter> {x. c \<le> x \<bullet> i} = y \<inter> {x. c \<le> x \<bullet> i}" shows "content (y \<inter> {x. c \<le> x \<bullet> i}) = 0"
fold_distr: fixes f :: "'b \<Rightarrow> 'c"   and list :: "'b list"   and e :: "'b" assumes "\<forall>x y. f (x + y) = f x + f y" shows "f (fold (+) list e) = fold (+) (map f list) (f e)"
simps_1: fixes x :: "'a" shows "cast_Ubx2 (OpUbx2 x) = Some x"
assms: shows "matrix_bounded_idempotent_semiring.covector f"
induct: fixes P :: "('b ivl \<times> 'b) tree \<Rightarrow> bool"   and a0 :: "('b ivl \<times> 'b) tree" assumes "\<And>l a m r. (l \<noteq> \<langle>\<rangle> \<Longrightarrow> P l) \<Longrightarrow> P \<langle>l, (a, m), r\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"