hcju/isabelleps huggingface dataset:
h3: shows "\<turnstile>\<^sub>p {adjoint (exexH_k (n - 1)) * (adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)) * exexH_k (n - 1)} hadamard_n n {adjoint (tensor_P mat_Ph (1\<^sub>m K)) * (adjoint (exexH_k (n - 1)) * Q2 * exexH_k (n - 1)) * tensor_P mat_Ph (1\<^sub>m K)}"
transitive: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "P \<leadsto>[Rel] Q"   and "Q \<leadsto>[Rel'] R"   and "Rel O Rel' \<subseteq> Rel''" shows "P \<leadsto>[Rel''] R"
set_awalk_verts_co_path': fixes e :: "'a \<times> 'a"   and w :: "'a"   and V :: "'a set"   and u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "e \<in> parcs (subdivide G e w)"   and "w \<notin> pverts (subdivide G e w)"   and "V \<subseteq> pverts (subdivide G e w)"   and "pre_digraph.gen_iapath (with_proj (subdivide (subdivide G e w) e w)) V u p v" shows "set (pawalk_verts u (co_path e w p)) = set (pawalk_verts u p) - {w}"
is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) node_ptr" shows "is_element_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr = (case cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr of None \<Rightarrow> False | Some x \<Rightarrow> True)"
Call_lemma': fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and \<sigma> :: "'e"   and p :: "'d"   and Z :: "'e"   and c :: "('e, 'd, 'f) com" assumes "\<forall>q\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call q,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call q \<down> Normal s \<and> ((s, q), \<sigma>, p) \<in> termi_call_steps \<Gamma>} Call q {t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call q,Normal Z\<rangle> \<Rightarrow> Abrupt t}" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>the (\<Gamma> p) \<down> Normal \<sigma> \<and> (\<exists>cs css. \<Gamma>\<turnstile> ([the (\<Gamma> p)], [], Normal \<sigma>) \<rightarrow>\<^sup>* (c # cs, css, Normal s))} c {t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>c,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
interaction_bound_gpv_stop: fixes "consider" :: "'h \<Rightarrow> bool"   and gpv :: "('i, 'h, 'j) gpv" shows "interaction_bound consider (gpv_stop gpv) = interaction_bound consider gpv"
\<Phi>_sum_mset: fixes t :: "'a tree" shows "\<Phi> t = (\<Sum>s\<in>#subtrees_mset t. 6 * e * real (imbal s))"
partition_on_space: fixes A :: "'c set" assumes "A \<noteq> {}" shows "partition_on A {A}"
Keys_snd_sym_preproc_addnew: fixes gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and vs :: "'t list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" shows "Keys (set (snd (sym_preproc_addnew gs vs fs v))) \<union> insert v (set vs) = Keys (set fs) \<union> insert v (set (fst (sym_preproc_addnew gs vs fs v)))"
p: shows "(LEAST i. i < length as \<and> a \<in> as ! i) < length as \<and> a \<in> as ! (LEAST i. i < length as \<and> a \<in> as ! i)"
down_cast_same: assumes "is_down ucast" shows "ucast = scast"
\<L>\<^sub>P_is_word: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>"   and "a \<in> CFG.\<L>\<^sub>P \<NN> \<TT> \<RR> \<SS>" shows "CFG.is_word \<TT> a"
ring_simprules_8: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<oplus>\<^bsub>R\<^esub> x = x"
fmsubset_divides: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and b :: "'a" assumes "fmset G as \<subseteq># fmset G bs"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "a divides b"
somewhere_and_or_distr: fixes \<chi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<phi> :: "traffic \<Rightarrow> view \<Rightarrow> bool"   and \<psi> :: "traffic \<Rightarrow> view \<Rightarrow> bool" shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (\<chi> ts v \<and> (\<phi> ts v \<or> \<psi> ts v)) \<and> True)) \<and> True)) = ((\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (\<chi> ts v \<and> \<phi> ts v) \<and> True)) \<and> True)) \<or> (\<exists>va u. v=va\<parallel>u \<and> True \<and> (\<exists>v ua. u=v\<parallel>ua \<and> (\<exists>va u. v=va--u \<and> True \<and> (\<exists>v ua. u=v--ua \<and> (\<chi> ts v \<and> \<psi> ts v) \<and> True)) \<and> True)))"
fNext_geq_not_member: fixes S :: "'a fset"   and s :: "'a" assumes "fresh_fNext S \<le> s" shows "s |\<notin>| S"
decomps\<^sub>e\<^sub>s\<^sub>t_assignment_rhs_empty: fixes A' :: "('fun, 'var) extstrand_step list"   and M :: "('fun, 'var) Term.term set"   and N :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "A' \<in> decomps\<^sub>e\<^sub>s\<^sub>t M N \<I>" shows "assignment_rhs\<^sub>e\<^sub>s\<^sub>t A' = {}"
comparator_list: fixes comp\<^sub>'\<^sub>a :: "'b \<Rightarrow> 'b \<Rightarrow> order" assumes "comparator comp\<^sub>'\<^sub>a" shows "comparator (comparator_list comp\<^sub>'\<^sub>a)"
K_sets: fixes n :: "nat" shows "K n \<in> sets (Pi\<^sub>M (J n) (\<lambda>_. borel))"
noDA1C: fixes a :: "('b, 'c) Combinators"   and p :: "('b, 'c) Combinators list" assumes "noDenyAll1 (a # p)" shows "noDenyAll1 p"
deliver_is_valid: fixes m :: "'a \<times> 'b"   and i :: "nat" assumes "Deliver m \<in> set (history i)" shows "\<exists>j pre state. pre @ [Broadcast m] prefix of j \<and> apply_operations pre = Some state \<and> valid_msg state m"
h_is_min: shows "\<forall>z x. z < h x \<longrightarrow> f x z = 0"
take_map: fixes n :: "nat"   and f :: "'h \<Rightarrow> 'g"   and xs :: "'h list" shows "take n (map f xs) = map f (take n xs)"
A: shows "A \<in> carrier_mat (Suc l) (Suc l)"
elims: fixes x :: "'b heap"   and y :: "'b multiset" assumes "mset_heap x = y" obtains "x = Empty" and "y = {#}"   | xa :: "'b" and hs :: "'b heap list" where "x = Hp xa hs" and "y = {#xa#} + \<Sum>\<^sub># (mset (map mset_heap hs))"
infsetsum'_converges_def: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "infsetsum'_converges f A = (\<exists>x. (sum f \<longlongrightarrow> x) (finite_subsets_at_top A))"
full_chain_iff_chain: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist" shows "full_chain R xs = (chain R xs \<and> (lfinite xs \<longrightarrow> (\<forall>y. \<not> R (llast xs) y)))"
open_imp_locally_compact: fixes S :: "'a set" assumes "open S" shows "locally compact S"
adjoint_pair_antipar_4: fixes f :: "'a"   and g :: "'a" assumes "B.adjoint_pair f g" shows "src g = trg f"
mset_factorsD_1: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and F :: "'a multiset"   and p :: "'a"   and f :: "'a" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.mset_factors times one zero F p"   and "f \<in># F" shows "comm_semiring_1.irreducible times one zero f"
simps_8: fixes \<alpha> :: "('a, 'b, 'c) hp"   and \<sigma> :: "('a, 'b, 'c) subst" shows "Psubst \<alpha>** \<sigma> = Psubst \<alpha> \<sigma>**"
leg0_composite: shows "\<mu>\<nu>.leg0 = \<nu>.leg0 \<cdot> \<mu>\<nu>.prj\<^sub>0"
e_prim_elim: fixes v :: "val"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and e1 :: "exp"   and e2 :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E (EPrim f e1 e2) \<rho>" obtains n1 :: "nat" and n2 :: "nat" where "VNat n1 \<in> E e1 \<rho>" and "VNat n2 \<in> E e2 \<rho>" and "v = VNat (f n1 n2)"
has_bottom_def: fixes P :: "'b set" shows "codomain.has_bottom P \<equiv> \<exists>z\<in>P. \<forall>x\<in>P. codomain.greater_eq x z"
If: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and e :: "expr"   and c1 :: "stmt"   and c2 :: "stmt" assumes "E,dt\<Turnstile>e\<Colon>-PrimT Boolean"   and "E,dt\<Turnstile>c1\<Colon>\<surd>"   and "E,dt\<Turnstile>c2\<Colon>\<surd>" shows "E,dt\<Turnstile>If(e) c1 Else c2\<Colon>\<surd>"
map_iterator_genord_foldli_conv: fixes iti :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<times> 'g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and m :: "'f \<Rightarrow> 'g option"   and R :: "'f \<times> 'g \<Rightarrow> 'f \<times> 'g \<Rightarrow> bool" shows "map_iterator_genord iti m R = (\<exists>l0. distinct (map fst l0) \<and> m = map_of l0 \<and> sorted_wrt R l0 \<and> iti = foldli l0)"
acomplete_sym: fixes l :: "'a tree"   and x :: "'a"   and r :: "'a tree"   and y :: "'a" assumes "acomplete \<langle>l, x, r\<rangle>" shows "acomplete \<langle>r, y, l\<rangle>"
mydiv_cancel3: fixes k :: "nat"   and i :: "nat" assumes "0 < k" shows "i \<le> k * mydiv i k"
noDA1eq: fixes p :: "('a, 'b) Combinators list" assumes "noDenyAll p" shows "noDenyAll1 p"
mayDiverge_Seq_L: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c2 :: "('test, 'atom) com" assumes "mayDiverge c1 s" shows "mayDiverge (c1 ;; c2) s"
one_is_strong_fermat_liar: fixes n :: "nat" assumes "1 < n" shows "strong_fermat_liar 1 n"
inf_left_commute: fixes inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
ent_ex_preI: fixes P :: "'a \<Rightarrow> assn"   and Q :: "assn" assumes "\<And>x. P x \<Longrightarrow>\<^sub>A Q" shows "\<exists>\<^sub>Ax. P x \<Longrightarrow>\<^sub>A Q"
wt_wt': fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : T" shows "P,E,h,sh \<turnstile> e :' T"
get_root_node_si_root_not_shadow_root: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r root" shows "\<not> is_shadow_root_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r root"
strict_implies_not_eq: fixes b :: "'b multiset"   and a :: "'b multiset" assumes "b \<subset># a" shows "a \<noteq> b"
hyps: fixes c :: "nat \<Rightarrow> 'a" assumes "\<forall>w\<in>set ws. dim_vec w = n" shows "lincomb_list c ws = mat_of_cols n ws *\<^sub>v vec (length ws) c"
prm_inv_compose: fixes \<pi> :: "'b prm" shows "prm_inv \<pi> \<diamondop> \<pi> = \<epsilon>"
assms_10: fixes A :: "'a"   and B :: "'a"   and C :: "'a" assumes "rewrite A B"   and "rewrite A C" shows "\<exists>D. rewrite B D \<and> rewrite C D"
evaluate_list_total0: fixes es :: "exp0 list"   and s :: "'b state"   and env :: "v sem_env" assumes "\<And>e env s'. \<lbrakk>e \<in> set es; clock s' \<le> clock s\<rbrakk> \<Longrightarrow> \<exists>s'' r. evaluate True env s' e (s'', r)" shows "\<exists>s' r. evaluate_list True env s es (s', r)"
restricted_ideal_to_restricted_ideal_to\<^sub>ID: fixes \<V> :: "nat set"   and A :: "int mpoly multiset" shows "restricted_ideal_to \<V> (set_mset A) \<subseteq> restricted_ideal_to\<^sub>I \<V> A"
cond_precise: shows "\<forall>s. s \<in> wa_invar WAC \<and> \<alpha> s \<in> wa_cond WAA \<longrightarrow> s \<in> wa_cond WAC"
mk_poly2_sylvester_lower: fixes q :: "'b poly"   and i :: "nat"   and p :: "'b poly" assumes "degree q \<le> i"   and "i < degree p + degree q" shows "mk_poly2 (sylvester_mat p q) $ i = monom (1::'b) (degree p + degree q - Suc i) * q"
tl_drop_Suc: fixes i :: "nat"   and xs :: "'a list" shows "tl (drop i xs) = drop (Suc i) xs"
ctr: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and a :: "'c" shows "sscan f xs a = f (shd xs) a ## sscan f (stl xs) (f (shd xs) a)"
cases: fixes a :: "letter"   and a1 :: "letter list list"   and a2 :: "letter list list" assumes "R a a1 a2" obtains "a1 = []" and "a2 = []"   | vs :: "letter list list" and ws :: "letter list list" and w :: "letter list" where "a1 = w # vs" and "a2 = (a # w) # ws" and "R a vs ws"
IH: fixes J :: "'a multiset"   and K :: "'a multiset"   and I :: "'a multiset" assumes "n = size J"   and "J \<noteq> {#}"   and "\<forall>k\<in>#K. \<exists>j\<in>#J. (k, j) \<in> r" shows "(I + K, I + J) \<in> mult r"
assms_3: shows "ring_homomorphism \<phi> st.carrier_stalk st.add_stalk st.mult_stalk (st.zero_stalk V) (st.one_stalk V) pi.carrier_local_ring_at pi.add_local_ring_at pi.mult_local_ring_at pi.zero_local_ring_at pi.one_local_ring_at"
PiE_ext: fixes x :: "'c \<Rightarrow> 'd"   and k :: "'c set"   and s :: "'c \<Rightarrow> 'd set"   and y :: "'c \<Rightarrow> 'd" assumes "x \<in> Pi\<^sub>E k s"   and "y \<in> Pi\<^sub>E k s"   and "\<And>i. i \<in> k \<Longrightarrow> x i = y i" shows "x = y"
inject_4: fixes x4 :: "nat"   and y4 :: "nat" shows "(Key x4 = Key y4) = (x4 = y4)"
continuous_on_inv: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'b" assumes "continuous_on s f"   and "compact s"   and "\<forall>x\<in>s. g (f x) = x" shows "continuous_on (f ` s) g"
bounded_iff: fixes b :: "'a Word.word"   and c :: "'a Word.word"   and a :: "'a Word.word" shows "(signed.max b c \<le>s a) = (b \<le>s a \<and> c \<le>s a)"
integrable_cexpr_projection: fixes V :: "nat set"   and V' :: "nat set"   and v :: "nat"   and \<rho> :: "nat \<Rightarrow> val"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e :: "cexpr" assumes "finite V"   and "V \<inter> V' = {}"   and "v \<notin> V"   and "v \<notin> V'"   and "\<rho> \<in> space (state_measure V' \<Gamma>)"   and "\<Gamma> \<turnstile>\<^sub>c e : REAL"   and "free_vars e \<subseteq> insert v V \<union> V'"   and "integrable (state_measure (insert v V) \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge (insert v V) V' (\<sigma>, \<rho>)) e))" shows "AE x in stock_measure (\<Gamma> v). integrable (state_measure V \<Gamma>) (\<lambda>\<sigma>. extract_real (cexpr_sem (merge V (insert v V') (\<sigma>, \<rho>(v := x))) e))"
pm_of_poly_eq_zero_iff: fixes x :: "'a"   and p :: "'b poly" shows "(pm_of_poly x p = 0) = (p = 0)"
fdg: shows "(f12, f21, g') = splitFace g ram1 ram2 f vs"
path_conforms_VVpstar: fixes v :: "'a"   and p :: "Player"   and Ps :: "'a llist"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<notin> VV p"   and "path_conforms_with_strategy p Ps \<sigma>" shows "path_conforms_with_strategy p (LCons v Ps) \<sigma>"
assms: fixes i :: "nat" shows "coeff p i \<in> \<int>"
simps: fixes x :: "nat"   and y :: "nat" shows "lo x y = (if 1 < x \<and> 1 < y \<and> {u. loR [x, y, u]} \<noteq> {} then Max {u. loR [x, y, u]} else 0)"
homotopically_trivial_retraction_gen: fixes s :: "'b set"   and h :: "'b \<Rightarrow> 'c"   and t :: "'c set"   and k :: "'c \<Rightarrow> 'b"   and U :: "'d set"   and Q :: "('d \<Rightarrow> 'c) \<Rightarrow> bool"   and P :: "('d \<Rightarrow> 'b) \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "Retracts s h t k"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> t; Q f\<rbrakk> \<Longrightarrow> P (k \<circ> f)"   and "\<And>f. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f\<rbrakk> \<Longrightarrow> Q (h \<circ> f)"   and "\<And>h k. (\<And>x. x \<in> U \<Longrightarrow> h x = k x) \<Longrightarrow> Q h = Q k"   and "\<And>f g. \<lbrakk>continuous_on U f; f ` U \<subseteq> s; P f; continuous_on U g; g ` U \<subseteq> s; P g\<rbrakk> \<Longrightarrow> homotopic_with_canon P U s f g"   and "continuous_on U f"   and "f ` U \<subseteq> t"   and "Q f"   and "continuous_on U g"   and "g ` U \<subseteq> t"   and "Q g" shows "homotopic_with_canon Q U t f g"
top_def: shows "top = (THE x. \<forall>y. x = y \<squnion> - y)"
add_mset: fixes x :: "'k"   and N :: "'k multiset" shows "\<Sum>\<^sub># (add_mset x N) = x + \<Sum>\<^sub># N"
prime_dvd_power: fixes p :: "'a"   and x :: "'a"   and n :: "nat" assumes "prime p"   and "p dvd x ^ n" shows "p dvd x"
ccSUP_upper: fixes A :: "'d set"   and i :: "'d"   and f :: "'d \<Rightarrow> 'a" assumes "countable A"   and "i \<in> A" shows "f i \<le> Sup (f ` A)"
invf_inv: fixes x :: "'a" assumes "x \<in> carrier K - {\<zero>}" shows "x\<^bsup>\<hyphen> K\<^esup> \<in> carrier K \<and> x\<^bsup>\<hyphen> K\<^esup> \<cdot>\<^sub>r x = 1\<^sub>r"
max0: shows "0 \<le> max_list_non_empty (map (\<lambda>i. norm (coeff p i)) [0..<degree p])"
RP_saturated_if_fair_new_statement: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts"   and "FL.active_subset (lclss_of_state (lhd Sts)) = {}"   and "FL.passive_subset (Liminf_llist (lmap lclss_of_state Sts)) = {}" shows "FL.empty_ord.saturated (Liminf_llist (lmap lclss_of_state Sts))"
homotopy_eqv_contractible_sets: fixes S :: "'a set"   and T :: "'b set" assumes "contractible S"   and "contractible T"   and "(S = {}) = (T = {})" shows "S homotopy_eqv T"
Says_Server_imp_Gets: fixes B :: "agent"   and M :: "nat"   and A :: "agent"   and Na :: "nat"   and K :: "nat"   and Nb :: "nat"   and evs :: "event list" assumes "Says Server B \<lbrace>Nonce M, Crypt (shrK B) \<lbrace>Crypt (shrK A) \<lbrace>Nonce Na, Key K\<rbrace>, Nonce Nb, Key K\<rbrace>\<rbrace> \<in> set evs"   and "evs \<in> orb" shows "Gets Server \<lbrace>Nonce M, Agent A, Agent B, Crypt (shrK A) \<lbrace>Nonce Na, Nonce M, Agent A, Agent B\<rbrace>, Crypt (shrK B) \<lbrace>Nonce Nb, Nonce M, Nonce M, Agent A, Agent B\<rbrace>\<rbrace> \<in> set evs"
invar_NodeLeaf: fixes l :: "('b \<times> nat) tree"   and x :: "'b"   and lv :: "nat" shows "invar \<langle>l, (x, lv), \<langle>\<rangle>\<rangle> = (invar l \<and> lv = Suc (lvl l) \<and> lv = Suc 0)"
simps_3: fixes get_proc :: "'h \<Rightarrow> 'g"   and Main :: "'g" shows "lift_get_proc get_proc Main NewExit = Main"
Un_upper1: fixes A :: "'g set"   and B :: "'g set" shows "A \<subseteq> A \<union> B"
assms: shows "par_comp\<^sub>l\<^sub>s\<^sub>s\<^sub>t A Sec"
\<Phi>a_def: shows "\<Phi>a = (\<lambda>f. \<lambda>x'\<in>\<phi> ` S.set (S.dom f). \<phi> (S.Fun f (\<psi> x')))"
assms_4: shows "simple_ruleset (rs::'i common_primitive rule list)"
entails_trans_strong: fixes N1 :: "'f set"   and N2 :: "'f set"   and N3 :: "'f set" assumes "N1 \<Turnstile> N2"   and "N1 \<union> N2 \<Turnstile> N3" shows "N1 \<Turnstile> N3"
split_path_at_vertex_complete: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and pv1 :: "nat list"   and u :: "nat"   and pv2 :: "nat list" assumes "isPath s p t"   and "pathVertices_fwd s p = pv1 @ u # pv2" obtains p1 :: "(nat \<times> nat) list" and p2 :: "(nat \<times> nat) list" where "p = p1 @ p2" and "isPath s p1 u" and "pathVertices_fwd s p1 = pv1 @ [u]" and "isPath u p2 t" and "pathVertices_fwd u p2 = u # pv2"
map_of_delete: fixes k' :: "'c"   and k :: "'c"   and al :: "('c \<times> 'd) list" assumes "k' \<noteq> k" shows "map_of (delete k al) k' = map_of al k'"
transpI: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>r x y; r y z\<rbrakk> \<Longrightarrow> r x z" shows "transp r"
pfsubsetD: fixes A :: "'b fset"   and B :: "'b fset"   and c :: "'b" assumes "A |\<subset>| B"   and "c |\<in>| A" shows "c |\<in>| B"
assign_r_comp: fixes x :: "'d \<Longrightarrow> 'b"   and u :: "('d, 'b) uexpr"   and P :: "('b, 'c) urel" shows "x := u ;; P = P\<lbrakk>\<lceil>u\<rceil>\<^sub></$x\<rbrakk>"
Suc_mod_mult_self3: fixes k :: "nat"   and n :: "nat"   and m :: "nat" shows "Suc (k * n + m) mod n = Suc m mod n"
mat_max_ge: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b mat"   and B :: "'b mat" assumes "class.non_strict_order less_eq" shows "ord.mat_ge less_eq (ord.mat_max less_eq A B) A"
cosh_eq_iff: fixes x :: "real"   and y :: "real" assumes "0 \<le> x"   and "0 \<le> y" shows "(cosh x = cosh y) = (x = y)"
DN_simps_2: fixes \<mu> :: "('a E.term, 'a) arr" assumes "arr \<mu>" shows "src\<^sub>B (DN \<mu>) = DN (src \<mu>)"
pre_mangoldt_pos: fixes n :: "nat" shows "0 < pre_mangoldt n"
generate_in_carrier: fixes H :: "int set"   and h :: "int" assumes "H \<subseteq> carrier (Z 1)"   and "h \<in> generate (Z 1) H" shows "h \<in> carrier (Z 1)"
norm_eq_zero: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and dist :: "'e \<Rightarrow> 'e \<Rightarrow> real"   and norm :: "'e \<Rightarrow> real"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and scaleR :: "real \<Rightarrow> 'e \<Rightarrow> 'e"   and sgn :: "'e \<Rightarrow> 'e"   and uniformity :: "('e \<times> 'e) filter"   and "open" :: "'e set \<Rightarrow> bool"   and x :: "'e" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "(norm x = 0) = (x = zero)"
strategy_extends_backwards_VVp: fixes v0 :: "'a"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and w :: "'a" assumes "v0 \<in> VV p"   and "\<sigma> v0 = w"   and "v0 \<rightarrow> w"   and "strategy p \<sigma>"   and "winning_strategy p \<sigma> w" shows "winning_strategy p \<sigma> v0"
OverriderMethodDefs'_def: 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 R :: "char list \<times> char list list"   and M :: "char list"   and Cs :: "char list list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr" shows "OverriderMethodDefs' P R M Cs mthd \<equiv> (Cs, mthd) \<in> OverriderMethodDefs P R M"
eq: fixes ts' :: "'t list"   and fs' :: "('t \<Rightarrow>\<^sub>0 'b) list" shows "insert v (Keys (set (snd (sym_preproc_addnew gs ts' fs' v))) \<union> set ts') = insert v (Keys (set fs') \<union> set (fst (sym_preproc_addnew gs ts' fs' v)))"
ParPres: fixes \<Psi>' :: "'b"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi"   and U :: "('a, 'b, 'c) psi" assumes "(\<Psi>', S, T) \<in> Rel" shows "(\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel"
in_KeysE: fixes s :: "'c"   and F :: "('c \<Rightarrow>\<^sub>0 'd) set" assumes "s \<in> Keys F" obtains f :: "'c \<Rightarrow>\<^sub>0 'd" where "s \<in> keys f" and "f \<in> F"
ctwo_Cnotzero: shows "(ctwo, czero) \<notin> ordIso \<and> Card_order ctwo"
ba3_idempotent: fixes x :: "'a" shows "x \<squnion> x = x"
alg_closed_field_axiom: fixes p :: "'a poly" assumes "0 < degree p" shows "\<exists>z. poly p z = (0::'a)"
cases: fixes a1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a2 :: "'b \<Rightarrow> bool" assumes "invariant a1 a2" obtains rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool" and I :: "'b \<Rightarrow> bool" where "a1 = rel" and "a2 = I" and "\<And>x y. \<lbrakk>rel x y; I x\<rbrakk> \<Longrightarrow> I y"
powr_eventually_exp_ln': fixes x :: "real"   and p :: "real" assumes "0 < x" shows "\<forall>\<^sub>F x in nhds x. x powr p = exp (p * ln x)"
list_of_oalist_sorted: fixes xs :: "('c, 'd) oalist_ntm" shows "sorted_wrt (ko.lt (key_order_of_nat_term_order_inv (snd (list_of_oalist_ntm xs)))) (map fst (fst (list_of_oalist_ntm xs)))"
ordLeq_csum2: fixes p2 :: "('c \<times> 'c) set"   and p1 :: "('d \<times> 'd) set" assumes "Card_order p2" shows "p2 \<le>o p1 +c p2"