hcju/isabelleps huggingface dataset:
bisim1_Invoke_\<tau>Red: fixes P :: "'addr J1_prog"   and h :: "'heap"   and E :: "'addr expr1"   and e :: "'addr expr1"   and xs :: "'addr val list"   and vs :: "'addr val list"   and a :: "'addr"   and stk' :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and M :: "String.literal"   and n :: "nat"   and t :: "'thread_id" assumes "P,E,h \<turnstile> (e, xs) \<leftrightarrow> (rev vs @ Addr a # stk', loc, pc, None)"   and "pc < length (compE2 E)"   and "compE2 E ! pc = Invoke M (length vs)"   and "n + max_vars e \<le> length xs"   and "bsok E n" shows "\<exists>e' xs'. \<tau>red1r P t h (e, xs) (e', xs') \<and> P,E,h \<turnstile> (e', xs') \<leftrightarrow> (rev vs @ Addr a # stk', loc, pc, None) \<and> call1 e' = \<lfloor>(a, M, vs)\<rfloor>"
rotate4: fixes A4 :: "fmla"   and A1 :: "fmla"   and A2 :: "fmla"   and A3 :: "fmla"   and H :: "fmla set"   and B :: "fmla" assumes "insert A4 (insert A1 (insert A2 (insert A3 H))) \<turnstile> B" shows "insert A1 (insert A2 (insert A3 (insert A4 H))) \<turnstile> B"
Lyndon_mono_set: fixes ws :: "'a list list"   and u :: "'a list" assumes "Lyndon_mono ws"   and "u \<in> set ws" shows "Lyndon u"
minus_equality: fixes G :: "('c, 'd) ring_scheme"   and y :: "'c"   and x :: "'c" assumes "abelian_group G"   and "y \<oplus>\<^bsub>G\<^esub> x = \<zero>\<^bsub>G\<^esub>"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "\<ominus>\<^bsub>G\<^esub> x = y"
HEndPhase2_HInv3: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv3 s" shows "HInv3 s'"
Sigma_mset_def: fixes A :: "'b multiset"   and B :: "'b \<Rightarrow> 'c multiset" shows "Sigma_mset A B \<equiv> \<Sum>a\<in>#A. image_mset (Pair a) (B a)"
simps_2: fixes x :: "'b"   and y :: "'b"   and ys :: "'b list" shows "find_first x (y # ys) = (if x = y then Some 0 else map_option Suc (find_first x ys))"
supp_infinite_coinfinite: fixes S :: "atom set" assumes "infinite S"   and "infinite (UNIV - S)" shows "supp S = UNIV"
impE: fixes cid :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "sb.eval cid t t' n (\<gamma> \<longrightarrow>\<^sup>b \<gamma>')" shows "sb.eval cid t t' n \<gamma> \<longrightarrow> sb.eval cid t t' n \<gamma>'"
label_3: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis"   and "x \<in> cbox (0::'a) One \<and> x \<bullet> i = 1" shows "label x i = 1"
to_poly_closed: fixes a :: "'a" assumes "a \<in> carrier R" shows "to_polynomial R a \<in> carrier P"
f_Exec_Stream_Acc_Output_nth: fixes k :: "nat"   and n :: "nat"   and xs :: "'d message_af list"   and output_fun :: "'f \<Rightarrow> 'e message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f" assumes "0 < k"   and "n < length xs" shows "f_Exec_Comp_Stream_Acc_Output k output_fun trans_fun xs c ! n = last_message (map output_fun (f_Exec_Comp_Stream trans_fun (xs ! n # NoMsg\<^bsup>k - Suc 0\<^esup>) (f_Exec_Comp trans_fun (xs \<down> n \<odot> k) c)))"
vec_index_vCons: fixes a :: "'a"   and v :: "'a vec"   and n :: "nat" shows "vCons a v $ n = (if n = 0 then a else v $ (n - 1))"
prob_space_with_indicators_axioms: shows "prob_space_with_indicators M I A"
lprefix_llength_eq_imp_eq: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "llength xs = llength ys" shows "xs = ys"
mem_Epigraph: fixes x :: "'b"   and y :: "real"   and S :: "'b set"   and f :: "'b \<Rightarrow> ereal" shows "((x, y) \<in> Epigraph S f) = (x \<in> S \<and> f x \<le> ereal y)"
lhs: shows "\<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar> \<le> \<bar>spmf (R1_14 M (c0, c1) \<bind> D) True - spmf (R1_14_interm1 M (c0, c1) \<bind> D) True\<bar> + \<bar>spmf (R1_14_interm1 M (c0, c1) \<bind> D) True - spmf (R1_14_interm2 M (c0, c1) \<bind> D) True\<bar> + \<bar>spmf (R1_14_interm2 M (c0, c1) \<bind> D) True - spmf (S1_14 M out \<bind> D) True\<bar>"
reduced_GB_is_reduced_GB_finite: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite F" shows "punit.is_reduced_GB (punit.reduced_GB F)"
FSwSbImorph_iff: fixes h :: "'k \<Rightarrow> 'l"   and hA :: "'m \<Rightarrow> 'n"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'k, 'm, 'p) model_scheme"   and MOD' :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'o, 'l, 'n, 'q) model_scheme" shows "FSwSbImorph h hA MOD MOD' = (FSbImorph h hA MOD MOD' \<and> ipresIGSwapAll h hA MOD MOD')"
Sup_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "Sup A = (\<lambda>x. SUP f\<in>A. f x)"
CFG_axioms: shows "CFG \<NN> \<TT> \<RR> \<SS>"
simps_2: fixes v :: "'c"   and va :: "'c list"   and u :: "'c \<Rightarrow> 'd" shows "set_clocks (v # va) [] u = u"
filterlim_uminus_at_top: fixes f :: "'c \<Rightarrow> real"   and F :: "'c filter" shows "filterlim f at_top F = (LIM x F. - f x :> at_bot)"
ipurge_tr_rev_aux_append: fixes I :: "('b \<times> 'b) set"   and D :: "'a \<Rightarrow> 'b"   and U :: "'b set"   and xs :: "'a list"   and ys :: "'a list" shows "ipurge_tr_rev_aux I D U (xs @ ys) = ipurge_tr_rev_aux I D (sources_aux I D U ys) xs @ ipurge_tr_rev_aux I D U ys"
size_fset_overloaded_simps: fixes x :: "'b fset" shows "size x = (\<Sum>x\<in>fset x. Suc 0)"
left_edge_smooth: shows "(\<lambda>y. twoC (y, 0)) piecewise_C1_differentiable_on {0..1}"
emeasure_lborel_cbox_cart_eq: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "emeasure lborel (cbox a b) = ennreal (\<Prod>i\<in>UNIV. max 0 ((b - a) $ i))"
that: fixes n :: "nat" assumes "count M x = Suc n + count N x" shows "thesis"
T_def: shows "T \<equiv> {d. d dvd gcd a m * gcd k b}"
m_closed: fixes x :: "rat vec"   and y :: "rat vec" assumes "x \<in> Rn"   and "y \<in> Rn" shows "x + y \<in> Rn"
vinsert_vdoubleton: fixes a :: "V"   and b :: "V"   and c :: "V" shows "vinsert a (set {b, c}) = set {a, b, c}"
mset_merge_adj: fixes xss :: "'c list list" shows "mset_mset (merge_adj xss) = mset_mset xss"
vars_snd_rcv_strand_subset2_2: fixes S :: "('d, 'c) strand_step list" shows "\<Union> (set (map fv\<^sub>r\<^sub>c\<^sub>v S)) \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S"
simps_2: fixes x :: "'a"   and xs :: "'a list"   and y :: "'a" shows "count_list (x # xs) y = (if x = y then count_list xs y + 1 else count_list xs y)"
integer_of_nat_eq_of_nat: shows "integer_of_nat = of_nat"
valid_dbm_cases: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry" assumes "valid_dbm M" obtains "[M]\<^bsub>v,n\<^esub> \<subseteq> V" and "dbm_int M n"
P2_ideal_model'_def: fixes M :: "'grp \<times> 'grp"   and \<sigma> :: "bool"   and z :: "'e"   and \<A> :: "(bool \<Rightarrow> 'e \<Rightarrow> (('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<times> 'grp \<times> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> ((('grp \<times> 'grp \<times> 'grp) \<times> nat) \<times> 'f) spmf) \<times> ('grp \<times> 'grp \<Rightarrow> 'grp \<times> 'grp \<Rightarrow> 'f \<Rightarrow> 'd spmf)" shows "P2_ideal_model' M \<sigma> z \<A> = (let (x0, x1) = M; (\<A>1, \<A>2, \<A>3) = \<A> in \<A>1 \<sigma> z \<bind> (\<lambda>((h0, h1, a, b0, b1), s). assert_spmf (h0 \<in> carrier \<G> \<and> h1 \<in> carrier \<G> \<and> a \<in> carrier \<G> \<and> b0 \<in> carrier \<G> \<and> b1 \<in> carrier \<G>) \<bind> (\<lambda>_. \<A>2 (h0, h1, a, b0, b1) s \<bind> (\<lambda>(((in1, in2, in3), r), s'). let (h, a, b) = (h0 \<otimes> inv h1, a, b0 \<otimes> inv b1) in funct_DH_ZK (h, a, b) ((in1, in2, in3), r) \<bind> (\<lambda>(out_zk_funct, uu_). assert_spmf out_zk_funct \<bind> (\<lambda>_. let l = b0 \<otimes> inv (h0 [^] r) in P2_ideal_model_end (x0, x1) l ((h0, h1, a, b0, b1), s') \<A>3))))))"
commutative_square_comp_arr: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a"   and l :: "'a" assumes "commutative_square f g h k"   and "seq h l" shows "commutative_square f g (h \<cdot> l) (k \<cdot> l)"
whilefalse: fixes b :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and \<iota> :: "nat"   and c :: "('exp, 'id) MWLsCom" assumes "BMap (E b m) = False" shows "\<langle>while\<^bsub>\<iota>\<^esub> b do c od,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>None,m\<rangle>"
TS_def: fixes A :: "'p"   and B :: "'p"   and P :: "'p"   and Q :: "'p" shows "A B TS P Q \<equiv> \<not> Col P A B \<and> \<not> Col Q A B \<and> (\<exists>T. Col T A B \<and> Bet P T Q)"
MEM_SPLIT: fixes x :: "'b"   and l :: "'b list" assumes "\<not> ListMem x l" shows "\<forall>l1 l2. l \<noteq> l1 @ [x] @ l2"
cross_add_right: fixes x :: "(real, 3) vec"   and y :: "(real, 3) vec"   and z :: "(real, 3) vec" shows "x \<times> (y + z) = x \<times> y + x \<times> z"
map_prod_imageI: fixes a :: "'b"   and b :: "'c"   and R :: "('b \<times> 'c) set"   and f :: "'b \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e" assumes "(a, b) \<in> R" shows "(f a, g b) \<in> map_prod f g ` R"
disc_p: shows "(\<forall>a. \<not> disc2 (Prot a)) \<or> \<not> has_disc_negated disc False m"
reset_eq: fixes u :: "'c \<Rightarrow> real"   and r :: "'c list"   and r' :: "'c list" assumes "u \<in> V"   and "[[r\<rightarrow>0]u]\<^sub>\<R> = [[r'\<rightarrow>0]u]\<^sub>\<R>"   and "set r \<subseteq> \<X>"   and "set r' \<subseteq> \<X>" shows "[r\<rightarrow>0]u = [r'\<rightarrow>0]u"
iMODb_iIN_subset_conv: fixes r' :: "nat"   and m' :: "nat"   and c' :: "nat"   and n :: "nat"   and d :: "nat" shows "([ r', mod m', c' ] \<subseteq> [n\<dots>,d]) = (n \<le> r' \<and> r' + m' * c' \<le> n + d)"
big_by_small: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>"
int_pow_eq_id: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and i :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "(x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i = \<one>\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub>) = (int (GE.ord x) dvd i)"
axioms_1: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "attractive A less_eq" shows "semiattractive A less_eq"
seqr_inf_distl: fixes P :: "('b, 'd) urel"   and Q :: "('b, 'd) urel"   and R :: "('d, 'c) urel" shows "P \<sqinter> Q ;; R = (P ;; R) \<sqinter> (Q ;; R)"
N1_def: shows "N1 \<equiv> \<lambda>B. remdups (trms_list\<^sub>s\<^sub>s\<^sub>t B @ map (pair' Pair) (setops_list\<^sub>s\<^sub>s\<^sub>t B))"
empty_tokens_have_charslength_0: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and ts :: "(symbol \<times> character list) list" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"   and "\<And>t. t \<in> set ts \<Longrightarrow> chars_of_token t = []" shows "charslength ts = 0"
n_circ_8: fixes k :: "nat" shows "f contour_integrable_on n_circ k"
Extra_ax_imp_hfthm: fixes x :: "hf"   and e :: "(name, hf) finfun" assumes "x \<in> Extra_ax" obtains A :: "fm" where "x = \<lbrakk>\<guillemotleft>A\<guillemotright>\<rbrakk>e \<and> {} \<turnstile> A"
silent_moves2_into_silent_moves: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and tls :: "('t \<times> ('l, 't, 'x2, 'm2, 'w, 'o) thread_action) list"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "r2.mthr.silent_moves2 s tls s'" shows "\<tau>mRed2 s s'"
assms: shows "\<forall>a\<in>set A. \<not> is_Insert a \<and> \<not> is_Delete a"
f: shows "\<forall>j\<le>i. f (fst (\<pi> !! j)) (fst (\<pi>' !! j))"
pmdl_syzygy_module_list: fixes bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "pmdl (syzygy_module_list bs) = syzygy_module_list bs"
assms_2: shows "\<Inter> {rel_interior S |S. S \<in> I} \<noteq> {}"
drop_intsFrom_enumFrom: fixes n :: "int"   and i :: "int" assumes "0 \<le> n" shows "Data_List.drop\<cdot>(MkI\<cdot>n)\<cdot>[MkI\<cdot>i..] = [MkI\<cdot>(n + i)..]"
interpolation: fixes i :: "int" shows "poly (poly_of_list (map g [0..<k])) (unity_root k i) = (\<Sum>n<k. g n * unity_root k (i * int n))"
QAR_App: fixes t\<^sub>1 :: "trm"   and t\<^sub>1' :: "trm"   and t\<^sub>2 :: "trm" assumes "t\<^sub>1 \<leadsto> t\<^sub>1'" shows "App t\<^sub>1 t\<^sub>2 \<leadsto> App t\<^sub>1' t\<^sub>2"
map_ticket_addsimps_1: fixes f :: "nat \<Rightarrow> nat"   and t :: "thread" shows "(map_ticket f t = INIT) = (t = INIT)"
inv_mult_group: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier (G\<lparr>carrier := S\<rparr>)"   and "y \<in> carrier (G\<lparr>carrier := S\<rparr>)" shows "inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> (x \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> y) = inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> y \<otimes>\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> inv\<^bsub>G\<lparr>carrier := S\<rparr>\<^esub> x"
hoare_skip: fixes P :: "'a set"   and Q :: "'a set" assumes "P \<subseteq> Q" shows "{P} Id {> Q}"
rev_le: fixes xs :: "nat list"   and ys :: "nat list" shows "(rev xs \<le>\<^sub>v rev ys) = (xs \<le>\<^sub>v ys)"
AE_iff_measurable: fixes N :: "'a set"   and M :: "'a measure"   and P :: "'a \<Rightarrow> bool" assumes "N \<in> sets M"   and "{x \<in> space M. \<not> P x} = N" shows "almost_everywhere M P = (emeasure M N = 0)"
planning_by_cnf_dimacs_complete: fixes \<pi>s :: "char list list" assumes "valid_plan \<pi>s"   and "length \<pi>s \<le> h" shows "\<exists>M. M \<Turnstile> map_formula var_to_dimacs (\<Phi>\<^sub>\<forall> \<phi> prob_with_noop abs_prob  h)"
alpha'_1: fixes a :: "tyvrs"   and x :: "'a"   and b :: "tyvrs"   and y :: "'a" shows "([a].x = [b].y) = (a = b \<and> x = y \<or> a \<noteq> b \<and> [(b, a)] \<bullet> x = y \<and> b \<sharp> x)"
irs_SDG_path_matched: fixes n :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n'' :: "'node SDG_node"   and p :: "'pname"   and n' :: "'node SDG_node"   and V :: "'var" assumes "n irs-ns\<rightarrow>\<^sub>d* n''"   and "n'' s-p\<rightarrow>\<^bsub>ret\<^esub> n' \<or> n'' s-p:V\<rightarrow>\<^bsub>out\<^esub> n'" obtains nx :: "'node SDG_node" and nsx :: "'node SDG_node list" where "matched nx nsx n'" and "n \<in> set nsx" and "nx s-p\<rightarrow>\<^bsub>sum\<^esub> CFG_node (parent_node n')"
l2_inv8D: fixes x :: "l2_state" assumes "x \<in> l2_inv8" shows "secret x \<subseteq> {NonceF (R $ sk) |R. R = test \<or> R \<in> partners}"
rell_F_def: fixes L1 :: "'q \<Rightarrow> 'aa \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 'ab \<Rightarrow> bool"   and L3 :: "'s \<Rightarrow> 'ac \<Rightarrow> bool" shows "rell_F L1 L2 L3 = rel_F L1 L2 L3 (=) (=) (=) (=) (=) (=)"
gcd_proj1_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and m :: "'b"   and n :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "(gcd m n = normalize m) = dvd.dvd times m n"
lprefix_nitpick_simps: fixes xs :: "'c llist"   and ys :: "'c llist" shows "xs \<sqsubseteq> ys = (if lfinite xs then Coinductive_List.finite_lprefix xs ys else xs = ys)"
l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_root_node_si :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" shows "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 \<equiv> known_ptr = ShadowRootClass.known_ptr \<and> get_owner_document = l_get_owner_document\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_owner_document get_root_node_si get_disconnected_nodes get_host"
hyps: assumes "singleCombinators xs"   and "distinct xs"   and "allNetsDistinct xs" shows "wellformed_policy2 (removeShadowRules2 xs)"
less_sfirstD: fixes n :: "nat"   and P :: "'c stream \<Rightarrow> bool"   and \<omega> :: "'c stream" assumes "enat n < sfirst P \<omega>" shows "\<not> P (sdrop n \<omega>)"
subset_insertI2: fixes A :: "'e set"   and B :: "'e set"   and b :: "'e" assumes "A \<subseteq> B" shows "A \<subseteq> insert b B"
index_mat_2: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and g :: "nat \<Rightarrow> 'a Matrix.vec" assumes "i < nr"   and "j < nc" shows "mat\<^sub>r nr nc g $$ (i, j) = g i $v j"
hurwitz_zeta_shift: fixes a :: "real"   and s :: "complex"   and n :: "nat" assumes "0 < a"   and "s \<noteq> 1" shows "hurwitz_zeta (a + real n) s = hurwitz_zeta a s - (\<Sum>k<n. complex_of_real (a + real k) powr - s)"
that: fixes x :: "'a" assumes "x \<in> S" shows "\<exists>T. x \<in> T \<and> open T \<and> k-smooth_on T f"
rep_eq: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'b \<Rightarrow>\<^sub>0 'd" shows "lookup (Poly_Mapping.map x xa) = (\<lambda>k. x (lookup xa k) when lookup xa k \<noteq> (0::'d))"
finite_atms_of: fixes C :: "'a literal multiset" shows "finite (atms_of C)"
Mertens_bigo: shows "(\<lambda>x. (\<Sum>p | prime p \<and> real p \<le> x. ln (real p) / real p) - ln x) \<in> O(\<lambda>_. 1)"
norm_pn_in_cf: fixes x :: "'a itrm"   and f :: "dB" assumes "x \<in> CF" shows "norm_pn f x \<in> CF"
orderE: fixes b :: "'c"   and a :: "'c" assumes "b \<le> a" obtains "a = a \<squnion> b"
prodag_sameTr2: fixes I :: "'d set"   and A :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme"   and B :: "'d \<Rightarrow> ('e, 'f) aGroup_scheme" assumes "\<forall>k\<in>I. aGroup (A k)"   and "\<forall>k\<in>I. A k = B k" shows "prod_pOp I A = prod_pOp I B"
ResNil: fixes x :: "name" shows "<\<nu>x>\<zero> \<equiv>\<^sub>s \<zero>"
below_bool_def: fixes x :: "bool"   and y :: "bool" shows "(x \<sqsubseteq> y) = (x \<longrightarrow> y)"
setSpec: shows "setOf (binsert fst (i, a) t) = setOf t - eqs fst (i, a) \<union> {(i, a)}"
assms_5: shows "src\<^sub>C g = trg\<^sub>C h"
diff_eq_diff_less: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less"   and "minus a b = minus c d" shows "less a b = less c d"
subdg_cod_simp: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<BB>\<lparr>Arr\<rparr>" shows "\<BB>\<lparr>Cod\<rparr>\<lparr>f\<rparr> = \<AA>\<lparr>Cod\<rparr>\<lparr>f\<rparr>"
get_IFS_get_ACS_complies_1: fixes M :: "('a SecurityInvariant \<times> 'a SecurityInvariant_configured) list" assumes "\<forall>(m_impl, m_spec)\<in>set M. SecurityInvariant_complies_formal_def m_impl m_spec" shows "\<forall>(m_impl, m_spec)\<in>set (zip (TopoS_Composition_Theory_impl.get_IFS (get_impl M)) (TopoS_Composition_Theory.get_IFS (get_spec M))). SecurityInvariant_complies_formal_def m_impl m_spec"
fmrel_iff: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and m :: "('c, 'a) fmap"   and n :: "('c, 'b) fmap" shows "fmrel R m n = (\<forall>x. rel_option R (fmlookup m x) (fmlookup n x))"
binqueue_induct: fixes P :: "'c option list \<Rightarrow> bool"   and xs :: "'c option list" assumes "P []"   and "\<And>xs. P xs \<Longrightarrow> P (None # xs)"   and "\<And>x xs. P xs \<Longrightarrow> P (Some x # xs)" shows "P xs"
domain_export'': fixes x :: "'b"   and y :: "'b" shows "ad (ad (ad (ad x) \<cdot> y)) = ad (ad x) \<cdot> ad (ad y)"
bal_i_balance: fixes t :: "'a tree" shows "bal_i (size t) (height (balance_tree t))"
list_cases: fixes xs :: "'c list"   and P :: "'c list \<Rightarrow> bool" assumes "xs = [] \<Longrightarrow> P []"   and "\<And>x. xs = [x] \<Longrightarrow> P [x]"   and "\<And>x y ys. xs = x # y # ys \<Longrightarrow> P (x # y # ys)" shows "P xs"
wf_\<Gamma>_Call: fixes m :: "'a match_expr"   and chain :: "char list" assumes "wf_\<Gamma> [Rule m (Call chain)]" shows "wf_\<Gamma> (the (\<Gamma> chain)) \<and> (\<exists>rs. \<Gamma> chain = Some rs)"
assms_6: shows "prv (imp \<phi>2 \<chi>)"
coset_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier G" shows "M #>\<^bsub>add_monoid G\<^esub> \<zero> = M"
connectedin: fixes X :: "'b topology"   and S :: "'b set" shows "connectedin X S = (S \<subseteq> topspace X \<and> (\<nexists>E1 E2. openin X E1 \<and> openin X E2 \<and> S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 \<inter> S = {} \<and> E1 \<inter> S \<noteq> {} \<and> E2 \<inter> S \<noteq> {}))"
simps_1: fixes x :: "'b"   and y :: "'b" shows "(Some x \<sqsubseteq> Some y) = (x \<sqsubseteq> y)"
map_poly_hom_smult: fixes c :: "int"   and p :: "int poly" shows "Ring_Hom_Poly.of_int_poly (Polynomial.smult c p) = Polynomial.smult (of_int c) (Ring_Hom_Poly.of_int_poly p)"
awalkE: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.awalk u p v" obtains "set (pawalk_verts u p) \<subseteq> pverts (mk_symmetric G)" and "set p \<subseteq> parcs (mk_symmetric G)" and "pcas u p v" and "hd (pawalk_verts u p) = u" and "last (pawalk_verts u p) = v"
not_in_arcs_H: shows "{uv, rev_G uv} \<inter> arcs H = {}"
power_down_nonpos_iff: fixes prec :: "nat"   and b :: "real"   and n :: "nat" shows "(power_down prec b n \<le> 0) = (b < 0 \<and> odd n \<or> b = 0 \<and> n \<noteq> 0)"
nfoldli_transfer_plain: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and l :: "'c list"   and c :: "'d \<Rightarrow> bool"   and s :: "'d" assumes "\<And>x s. RETURN (f x s) \<le> f' x s" shows "RETURN (foldli l c f s) \<le> nfoldli l c f' s"
ntrp_stack_comp_s: fixes s :: "'n list"   and c :: "'n list multiset"   and ee :: "('p, 'ba) label list el_step"   and s' :: "'n list"   and c' :: "'n list multiset"   and r :: "'n list" assumes "((s, c), ee, s', c') \<in> ntrp fg"   and "mon_s fg r \<inter> mon_env fg [ee] = {}" shows "((s @ r, c), ee, s' @ r, c') \<in> ntrp fg"
WT_gpv_map_gpv': fixes g :: "'a \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'b"   and \<I> :: "('d, 'e) \<I>"   and gpv :: "('c, 'a, 'b) gpv"   and f :: "'c \<Rightarrow> 'f" assumes "map_\<I> g h \<I> \<turnstile>g gpv \<surd>" shows "\<I> \<turnstile>g map_gpv' f g h gpv \<surd>"
right_closed: fixes a :: "'a set \<times> 'b"   and b :: "'a set \<times> 'b" assumes "(a, b) \<in> {(x, y). stk1.rel x y}" shows "b \<in> Sigma I \<FF>1"