hcju/isabelleps huggingface dataset:
simps_4: fixes x3 :: "'b" shows "SOURCE \<noteq> Inner x3"
f_aggregate_hd: fixes k :: "nat"   and xs :: "'b list"   and ag :: "'b list \<Rightarrow> 'b" assumes "0 < k"   and "k \<le> length xs" shows "hd (f_aggregate xs k ag) = ag (xs \<down> k)"
inner_add_left: 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 inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.real_inner minus dist norm plus zero uminus scaleR sgn uniformity open inner" shows "inner (plus x y) z = inner x z + inner y z"
span_span: fixes A :: "'a set" shows "cspan (cspan A) = cspan A"
intros: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T3 :: "ty" assumes "SemiType.sup P T1 T2 = OK T3" shows "is_lub_sup P T1 T2 T3"
distinct_62: fixes x7 :: "CTypeId"   and x6 :: "Arraytype" shows "CClassT x7 \<noteq> ArrT x6"
tao: shows "thread_oks ts \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>"
hd_toList: fixes xs :: "'b odlist"   and y :: "'b"   and ys :: "'b list" assumes "toList xs = y # ys" shows "ODList.hd xs = y"
the_elem_eq: fixes x :: "'b" shows "the_elem {x} = x"
le_2: shows "real_of_1 (p, l, r) \<le> real_of_rat r"
1: shows "\<guillemotleft>\<l>[f] : L f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> L \<l>[f] = (\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f]"
fmlookup_inject: fixes x :: "('c, 'd) fmap"   and y :: "('c, 'd) fmap" shows "(fmlookup x = fmlookup y) = (x = y)"
f_Exec_Stream_Acc_Output_Init_take: fixes n :: "nat"   and k :: "nat"   and output_fun :: "'e \<Rightarrow> 'd message_af"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and xs :: "'f message_af list"   and c :: "'e" shows "f_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun xs c \<down> Suc n = f_Exec_Comp_Stream_Acc_Output_Init k output_fun trans_fun (xs \<down> n) c"
un_dual_def: fixes dual :: "'b dual" shows "\<partial>\<^sup>- dual = (case dual of \<partial> x \<Rightarrow> x)"
finite_chain_with3_def: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and X :: "'b set" assumes "MinkowskiBetweenness \<E> \<P> betw" shows "MinkowskiBetweenness.finite_chain_with3 betw x y z X \<equiv> MinkowskiBetweenness.chain_with betw x y z X \<and> \<not> (\<exists>w\<in>X. betw w x y \<or> betw y z w)"
part_I_K_I: assumes "I \<circ> K \<circ> I = I" shows "I \<circ> K = K"
uneq_sets: shows "set (msgs c i) \<noteq> set (msgs c' i)"
TESL_interpretation_image: fixes \<Phi> :: "'b TESL_atomic list" shows "\<lbrakk>\<lbrakk> \<Phi> \<rbrakk>\<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L = \<Inter> (TESL_interpretation_atomic ` set \<Phi>)"
concurrent_history_steps_simulates_store_buffer_steps: fixes ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m :: "nat \<Rightarrow> nat"   and \<S> :: "'f"   and ts' :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> 'c \<times> 'd \<times> 'e) list"   and m' :: "nat \<Rightarrow> nat"   and \<S>' :: "'f"   and ts\<^sub>h :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and \<S>\<^sub>h :: "nat \<Rightarrow> bool option" assumes "ini.sb_concurrent_steps program_step (ts, m, \<S>) (ts', m', \<S>')"   and "ts \<sim>\<^sub>h ts\<^sub>h " shows "\<exists>ts\<^sub>h' \<S>\<^sub>h'. ini.sbh_concurrent_steps program_step (ts\<^sub>h, m, \<S>\<^sub>h) (ts\<^sub>h', m', \<S>\<^sub>h') \<and> ts' \<sim>\<^sub>h ts\<^sub>h' "
simps_1: fixes p :: "nat"   and a :: "floatarith"   and b :: "floatarith"   and vs :: "((real \<times> real pdevs) \<times> real) list" shows "approx_floatarith p (Add a b) vs = approx_floatarith p a vs \<bind> (\<lambda>a1. approx_floatarith p b vs \<bind> (\<lambda>a2. Some (add_aform' p a1 a2)))"
AR: shows "(a, a') \<in> \<langle>R\<rangle>array_rel"
mono: fixes c :: "'e"   and a :: "'e"   and d :: "'e"   and b :: "'e" assumes "c \<le> a"   and "d \<le> b" shows "sup c d \<le> sup a b"
Rep_cases: fixes Rep :: "'e \<Rightarrow> 'f"   and Abs :: "'f \<Rightarrow> 'e"   and A :: "'f set"   and y :: "'f" assumes "type_definition Rep Abs A"   and "y \<in> A" obtains x :: "'e" where "y = Rep x"
m_assoc: fixes x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a"   and z :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P"   and "y \<in> carrier P"   and "z \<in> carrier P" shows "x \<oplus>\<^bsub>P\<^esub> y \<oplus>\<^bsub>P\<^esub> z = x \<oplus>\<^bsub>P\<^esub> (y \<oplus>\<^bsub>P\<^esub> z)"
closed_def: fixes S :: "'b set" shows "closed S = open (- S)"
swp_apply_unequal: fixes x :: "'b"   and y :: "'b"   and s :: "'b \<times> 'b" assumes "x \<noteq> y" shows "swp_apply s x \<noteq> swp_apply s y"
llast_singleton: fixes x :: "'b" shows "llast <x> = x"
LSM_submodule: fixes M :: "('f, 'a, 'g) Module_scheme"   and T :: "'f set" assumes "R module M"   and "T \<subseteq> carrier M" shows "submodule R M (LSM\<^bsub>R\<^esub> M T)"
rep_eq: fixes x :: "Arity" shows "Aeta_expand x = eta_expand (Rep_Arity x)"
id: shows "map (\<lambda>j. gs.\<mu> i j \<cdot>\<^sub>v gs.gso j) [0..<Suc i] = map (\<lambda>j. gs.\<mu> i j \<cdot>\<^sub>v map gs.gso [0..<length fs] ! j) [0..<Suc i]"
finite_UNIV_word_def: shows "finite_UNIV = Phantom('a word) True"
of_char_eq_iff: fixes c :: "char"   and d :: "char" shows "(of_char c = of_char d) = (c = d)"
smap_shift: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list"   and s :: "'d stream" shows "smap f (xs @- s) = map f xs @- smap f s"
distinct_Basis_list: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and inner :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and Basis :: "'a set"   and abs :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Basis_list :: "'a list"   and eucl_down :: "int \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_down :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and eucl_truncate_up :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" assumes "class.executable_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less Basis_list eucl_down eucl_truncate_down eucl_truncate_up" shows "distinct Basis_list"
isomorphic_relative_homology_by_contractible: fixes X :: "'a topology"   and S :: "'a set"   and p :: "int" assumes "contractible_space (subtopology X S)"   and "topspace X \<inter> S \<noteq> {}" shows "relative_homology_group p X S \<cong> reduced_homology_group p X"
star_one: shows "(1::'a)\<^sup>\<star> = (1::'a)"
rel_spmf_pos_distr: fixes A :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_spmf A OO rel_spmf B \<le> rel_spmf (A OO B)"
not_arr_null: shows "\<not> S.VV.arr S.VV.null"
true_conj_zero_2: fixes P :: "'b upred" shows "(false \<and> P) = false"
disc_2: fixes xs :: "'b llist"   and ys :: "'b llist" assumes "\<not> lnull xs \<or> \<not> lnull ys" shows "\<not> lnull (lappend xs ys)"
cons_6: fixes x :: "real \<times> real" assumes "x \<in> set (x__ # xs)" shows "0 < snd x"
convexI: fixes s :: "'d set" assumes "\<And>x y u v. \<lbrakk>x \<in> s; y \<in> s; 0 \<le> u; 0 \<le> v; u + v = 1\<rbrakk> \<Longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s" shows "convex s"
snapshot_def: fixes PROB :: "(('d, 'e) fmap \<times> ('d, 'e) fmap) set"   and s :: "('d, 'e) fmap" shows "snapshot PROB s = {a |a. a \<in> PROB \<and> agree (fst a) s \<and> agree (snd a) s}"
eq: shows "p \<bind> (\<lambda>s. map_spmf fst (callee1 s a)) = q \<bind> (\<lambda>s. map_spmf fst (callee2 s a))"
compose_preserves_edge_preserving: fixes h1 :: "('e \<times> 'f) set"   and e1 :: "('g \<times> 'e \<times> 'e) set"   and e2 :: "('g \<times> 'f \<times> 'f) set"   and h2 :: "('f \<times> 'h) set"   and e3 :: "('g \<times> 'h \<times> 'h) set" assumes "edge_preserving h1 e1 e2"   and "edge_preserving h2 e2 e3" shows "edge_preserving (h1 O h2) e1 e3"
path2_last_in_ns: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m" shows "m \<in> set ns"
intro: fixes block_collection :: "'b set multiset"   and point_set :: "'b set" assumes "\<And>b. b \<in># block_collection \<Longrightarrow> b \<subseteq> point_set" shows "incidence_system point_set block_collection"
lfp_if_cont: fixes f :: "'a set \<Rightarrow> 'a set" assumes "cont f" shows "lfp f = (\<Union>n. (f ^^ n) {})"
loop_stepD: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and V :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 V"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> V \<inter> S"   and "real c < W B\<^sub>1 + w u"   and "W B\<^sub>2 + w u \<le> real c" shows "inv\<^sub>3 (P\<^sub>1 \<union> wrap B\<^sub>1) P\<^sub>2 {} (B\<^sub>2 \<union> {u}) (V - {u})"
rHyp_2: shows "ball 0 r \<subseteq> T"
infsetsum_Un_disjoint: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'c set" assumes "f abs_summable_on A"   and "f abs_summable_on B"   and "A \<inter> B = {}" shows "infsetsum f (A \<union> B) = infsetsum f A + infsetsum f B"
wtC_tC: fixes c :: "('fsym, 'psym) lit list" assumes "Ik.wtC c" shows "TE.wtC (tC c)"
intro: fixes trans :: "'a \<Rightarrow> exp \<Rightarrow> exp" assumes "\<And>a e. supp (trans a e) \<subseteq> supp e" shows "supp_bounded_transform trans"
fps_of_poly_0: shows "fps_of_poly 0 = 0"
restrict_extensional: fixes f :: "'ag \<Rightarrow> 'ah"   and A :: "'ag set" shows "restrict f A \<in> extensional A"
fvs_simp_2: fixes x :: "'b" shows "fvs (Var x) = {x}"
wf_newQuery: fixes J :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn0 :: "(nat set \<times> 'b option list set) set"   and QQ :: "(nat set \<times> 'b option list set) set"   and t :: "nat set \<times> 'b option list"   and QQn :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set" assumes "1 \<le> card J"   and "wf_query n J Q Qn0"   and "non_empty_query Q"   and "covering J Q"   and "\<forall>X\<in>Q. 1 \<le> card (fst X \<inter> J)"   and "QQ = New_max.newQuery J Q t"   and "QQn = New_max.newQuery J Qn t"   and "non_empty_query Qn"   and "Qn = New_max.filterQuery J Qn0" shows "rwf_query n J QQ QQn"
PDP: fixes x :: "'a"   and d :: "'a \<Rightarrow>\<^sub>L 'a" assumes "(x, d) \<in> CX"   and "x \<bullet> n = c" shows "(x, d - (blinfun_scaleR_left (f x) o\<^sub>L (blinfun_scaleR_left (inverse (f x \<bullet> n)) o\<^sub>L (blinfun_inner_left n o\<^sub>L d)))) \<in> PDP"
welltyped_Seq: fixes e1 :: "expr"   and e2 :: "expr" assumes "e1 ::: \<B>"   and "e2 ::: \<B>" shows "e1 ? e2 ::: \<B>"
select_convs_2: fixes mag :: "'f"   and dim :: "(int, 'e) dimvec"   and more :: "'g" shows "Quantity.dim \<lparr>mag = mag, dim = dim, \<dots> = more\<rparr> = dim"
simp_regular: fixes x :: "'c regular" shows "in_p_image (Rep_regular x)"
INV_def: fixes g' :: "gState" shows "INV g' = (g' \<in> (gState_progress_rel prog)\<^sup>* `` {g} \<and> gState_inv prog g')"
eq1: shows "sr *s (rowvector w *v y) = rowvector w *v (A *v y)"
Sum2: fixes Q :: "pi"   and V :: "Early_Semantics.residual"   and P :: "pi" assumes "Q \<longmapsto>\<^sub>e V" shows "P \<oplus> Q \<longmapsto>\<^sub>e V"
simps_159: fixes x101 :: "expr"   and x102 :: "char list option"   and x103 :: "char list"   and x104 :: "expr list"   and x71 :: "char list"   and x72 :: "expr" shows "Call x101 x102 x103 x104 \<noteq> x71:=x72"
sym_trans_gfp: fixes Retr :: "('a \<times> 'a) set \<Rightarrow> ('a \<times> 'a) set" assumes "mono Retr"   and "\<And>theta. sym theta \<and> trans theta \<Longrightarrow> sym (Retr theta) \<and> trans (Retr theta)" shows "sym (gfp Retr) \<and> trans (gfp Retr)"
ltln_of_dnf_prop_equiv: fixes \<phi> :: "'b ltln" shows "ltln_of_dnf (min_dnf \<phi>) \<sim>\<^sub>P \<phi>"
point_def: fixes x :: "'a" shows "point x \<equiv> is_vector x \<and> bijective x"
assms_2: shows "q \<le> x\<^sup>\<star> ; r"
that: fixes v\<^sub>1 :: "'a list"   and v\<^sub>2 :: "nat \<Rightarrow> 'a" assumes "v = v\<^sub>1 \<frown> v\<^sub>2"   and "length v\<^sub>1 = k" shows "thesis"
iMOD2_set_UN_set_eq: shows "iMOD2_set = iMOD2_UN_set"
simps_2: fixes x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "l_sort_inv_1 (x # xs, y # ys, []) = (l_sorted (y # ys) \<and> last (y # ys) \<le> x)"
B_def: shows "B = (\<lambda>(xa, x). ball xa (1 / real x)) ` (A \<times> UNIV)"
distinct_pds_def: fixes K :: "('c, 'd) Ring_scheme"   and n :: "nat"   and P :: "nat \<Rightarrow> ('c \<Rightarrow> ant) set" shows "distinct_pds K n P = ((\<forall>j\<le>n. P j \<in> Pds\<^bsub>K\<^esub>) \<and> (\<forall>l\<le>n. \<forall>m\<le>n. l \<noteq> m \<longrightarrow> P l \<noteq> P m))"
simps_3: fixes l :: "('b \<times> nat) option"   and t :: "'b" shows "lock_action_ok l t UnlockFail = (\<not> has_lock l t)"
retraction_def: fixes g :: "'a" shows "C.retraction g \<equiv> \<exists>f. C.ide (g \<cdot> f)"
cases: fixes x :: "'a poly \<times> 'a poly" obtains p :: "'a poly" and q :: "'a poly" where "x = (p, q)"
divide_less_eq_1_neg: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 a :: "'a"   and b :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less a zero" shows "less (divide b a) one = less a b"
rangeIt_def: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" shows "rangeIt x f = range (\<lambda>n. (f ^^ n) x)"
y: shows "y = real_alg_2 (abs_int_poly (reflect_poly p), inverse r, inverse l)"
mono: fixes x :: "'b" shows "mono_body (\<lambda>f. U (F (C f)) x)"
finfun_default_aux_def: fixes f :: "'b \<Rightarrow> 'a" shows "finfun_default_aux f = (if finite UNIV then undefined else THE b. finite {a. f a \<noteq> b})"
summable_add: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "summable f"   and "summable g" shows "summable (\<lambda>n. f n + g n)"
remove1_filter_not: fixes P :: "'k \<Rightarrow> bool"   and x :: "'k"   and xs :: "'k list" assumes "\<not> P x" shows "remove1 x (filter P xs) = filter P xs"
equiv_sym: fixes x :: "'a"   and y :: "'a" assumes "x \<sim> y" shows "y \<sim> x"
exact3_def: fixes R :: "('e, 'f) Ring_scheme"   and L0 :: "('g, 'e, 'h) Module_scheme"   and h0 :: "'g \<Rightarrow> 'i"   and L1 :: "('i, 'e, 'h) Module_scheme"   and h1 :: "'i \<Rightarrow> 'j"   and L2 :: "('j, 'e, 'h) Module_scheme" shows "exact3 R L0 h0 L1 h1 L2 \<equiv> h0 ` carrier L0 = ker\<^bsub>L1,L2\<^esub> h1"
circ_mult_sub_sup: shows "matrix_stone_algebra.comp_inf.coreflexive mtop"
permute_def: fixes permute :: "'b perm \<Rightarrow> 'c \<Rightarrow> 'c" shows "permute permute \<equiv> (\<forall>x. permute 0 x = x) \<and> (\<forall>p q x. permute (p + q) x = permute p (permute q x))"
ennreal_SUP_add: fixes f :: "nat \<Rightarrow> ennreal"   and g :: "nat \<Rightarrow> ennreal" assumes "incseq f"   and "incseq g" shows "(SUP i. f i + g i) = Sup (range f) + Sup (range g)"
finite_conv_nat_seg_image: fixes A :: "'a set" shows "finite A = (\<exists>n f. A = f ` {i. i < n})"
not_empty: fixes M :: "'b measure" assumes "prob_space M" shows "space M \<noteq> {}"
neg_equal_zero: fixes x :: "'a" assumes "x \<in> carrier R" shows "(\<ominus> x = x) = (x = \<zero>)"
filter_M_is_OK_I: fixes xs :: "'l list"   and h :: "'m"   and P :: "'l \<Rightarrow> ('m, 'n, bool) prog" assumes "\<And>x. x \<in> set xs \<Longrightarrow> h \<turnstile> ok P x"   and "\<And>x. x \<in> set xs \<Longrightarrow> pure (P x) h" shows "h \<turnstile> ok filter_M P xs"
measure_conv_def: fixes r :: "('a \<times> 'a) set"   and c :: "'b \<times> (bool \<times> 'a \<times> 'b) list" shows "measure_conv r c = r|labels_conv c|"
coerce_eq_emb: shows "COERCE('e, udom) = EMB('e)"
icard_the_enat: fixes A :: "'a set" assumes "finite A" shows "the_enat (icard A) = card A"
contractible_imp_path_connected: fixes S :: "'a set" assumes "contractible S" shows "path_connected S"
ivl_err_def: fixes ivl :: "real interval" shows "ivl_err ivl \<equiv> (((upper ivl + lower ivl) / 2, zero_pdevs), (upper ivl - lower ivl) / 2)"
stake_suffix_drop: fixes i :: "nat"   and \<sigma> :: "nat \<Rightarrow> 'b"   and j :: "nat" shows "\<sigma>(j \<rightarrow> i) = drop j (stake (i + j) \<sigma>)"
ess_def: fixes \<Phi> :: "\<mu> \<Rightarrow> i \<Rightarrow> bool"   and x :: "\<mu>" shows "\<Phi> ess x = (\<lambda>w. \<Phi> x w \<and> (\<forall>xa. (xa x m\<rightarrow> \<box> (\<lambda>v. \<forall>x. (\<Phi> x m\<rightarrow> xa x) v)) w))"
return_valid: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and p :: "'g" assumes "flowgraph fg" shows "proc_of fg (return fg p) = p"
matrix_equation_main_step_R: fixes p :: "real poly"   and signs :: "rat list list"   and qs :: "real poly list"   and I :: "nat list \<times> nat list" assumes "p \<noteq> 0"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "list_constr (fst I) (length qs)"   and "list_constr (snd I) (length qs)" shows "vec_of_list (mtx_row_R signs I) \<bullet> w_vec_R p qs signs = construct_NofI_R p (retrieve_polys qs (fst I)) (retrieve_polys qs (snd I))"
identify_blocks_main_iff: fixes k :: "nat"   and k' :: "nat"   and A :: "'c mat"   and list :: "(nat \<times> nat) list" assumes "k \<le> k'"   and "k' \<noteq> k \<longrightarrow> 0 < k \<longrightarrow> A $$ (k - 1, k) \<noteq> (1::'c)"   and "k' < n" shows "set (identify_blocks_main A k list) = set list \<union> {(i, j) |i j. i \<le> j \<and> j < k \<and> (\<forall>l\<ge>i. l < j \<longrightarrow> A $$ (l, Suc l) = (1::'c)) \<and> (Suc j \<noteq> k' \<longrightarrow> A $$ (j, Suc j) \<noteq> (1::'c)) \<and> (0 < i \<longrightarrow> A $$ (i - 1, i) \<noteq> (1::'c))}"
WT_gpv_absorb: fixes \<I>' :: "('h, 'i) \<I>"   and gpv :: "('h, 'i) distinguisher"   and \<I> :: "('j, 'k) \<I>"   and conv :: "('h, 'i, 'j, 'k) converter" assumes "\<I>' \<turnstile>g gpv \<surd>"   and "\<I>', \<I> \<turnstile>\<^sub>C conv \<surd>" shows "\<I> \<turnstile>g absorb gpv conv \<surd>"
assms_4: shows "h \<turnstile> get_shadow_root host \<rightarrow>\<^sub>r Some shadow_root_ptr"
eng: fixes h1 :: "'a"   and G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and h2 :: "'a" assumes "h1 \<in> generate G H"   and "h2 \<in> generate G H" shows "h1 \<otimes>\<^bsub>G\<^esub> h2 \<in> generate G H"
quot_of_fract_to_fract: fixes x :: "'a" shows "quot_of_fract (to_fract x) = (x, 1::'a)"
suminf_indicator: fixes A :: "nat \<Rightarrow> 'a set"   and x :: "'a" assumes "disjoint_family A" shows "(\<Sum>n. indicator (A n) x) = indicator (\<Union> (range A)) x"
index_smult_mat_3: fixes a :: "'d"   and A :: "'d mat" shows "dim_col (a \<cdot>\<^sub>m A) = dim_col A"
decompose: fixes ys :: "'a list"   and x :: "'a" assumes "ys \<noteq> []"   and "pairwise coprime (set ys)"   and "distinct ys"   and "\<And>y. y \<in> set ys \<Longrightarrow> is_unit (lift y)" shows "(\<Sum>i<length ys. lift (decompose x ys ! i) div lift (ys ! i)) = lift x div lift (prod_list ys)"
semilattice_inf_ow_axioms: shows "semilattice_inf_ow U\<^sub>2 inf\<^sub>2 le\<^sub>2 ls\<^sub>2"
G_def: shows "G \<equiv> \<lambda>t. C + K * integral {a..t} g"
cas_simp: fixes es :: "'b list"   and u :: "'a"   and v :: "'a" assumes "es \<noteq> []" shows "cas u es v = (tail G (hd es) = u \<and> cas (head G (hd es)) (tl es) v)"
compact_imp_Lindelof_space: fixes X :: "'a topology" assumes "compact_space X" shows "Lindelof_space X"
compact_space_subtopology: fixes X :: "'c topology"   and S :: "'c set" assumes "compactin X S" shows "compact_space (subtopology X S)"
fsize_def2: fixes t :: "(val \<times> val) fset" shows "fsize t = 1 + ffold (vadd \<circ> vprod_size) 0 t"
nonempty_alts: fixes agents :: "'c set"   and alts :: "'d set"   and R :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool" assumes "pref_profile_wf agents alts R" shows "alts \<noteq> {}"
prefix_prefix: fixes xs :: "'e list"   and ys :: "'e list"   and zs :: "'e list" assumes "prefix xs ys" shows "prefix xs (ys @ zs)"
simps: fixes a :: "Vertices list" shows "walk a = (a = [] \<or> (\<exists>v. a = [v] \<and> v \<in> V\<^bsub>T2\<^esub>) \<or> (\<exists>v w vs. a = v # w # vs \<and> v \<rightarrow>\<^bsub>T2\<^esub> w \<and> walk (w # vs)))"
False: shows "\<not> (\<exists>z\<in>s. g z = 0)"
relhoare_conseq_right: fixes pre :: "('b \<times> 'c) set"   and Ra :: "('b \<times> 'b) set"   and Rc :: "('c \<times> 'c) set"   and post' :: "('b \<times> 'c) set"   and post :: "('b \<times> 'c) set" assumes "{pre} Ra, Rc {> post'}"   and "post' \<subseteq> post" shows "{pre} Ra, Rc {> post}"
the_\<nu>_props_1: shows "\<guillemotleft>w.the_\<nu> : local.dom \<omega> \<Rightarrow> s\<^sub>1 \<star> w\<guillemotright>"
vimage2p_def: fixes f :: "'d \<Rightarrow> 'g"   and g :: "'e \<Rightarrow> 'h"   and R :: "'g \<Rightarrow> 'h \<Rightarrow> 'f" shows "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
Comp_Hom_Id: fixes A :: "'e set"   and f :: "'e \<Rightarrow> 'e"   and B :: "'e set" assumes "A \<in> {A. |A| <o \<AA>}"   and "f \<in> extensional A \<inter> (A \<rightarrow> B)" shows "compose A f (\<lambda>x\<in>A. x) = f"
v11_VLambda_iff: fixes f :: "V \<Rightarrow> V"   and A :: "V" shows "inj_on f (elts A) = v11 (VLambda A f)"
trancl_is_path: fixes u :: "'e"   and v :: "'e"   and E :: "('e \<times> 'e) set" assumes "(u, v) \<in> E\<^sup>+" obtains l :: "'e list" where "l \<noteq> []" and "path E u l v"
smult_one: fixes c :: "'c" shows "smult c 1 = [:c:]"
retraction_def: fixes g :: "'a \<times> 'a" shows "VV.retraction g \<equiv> \<exists>f. VV.ide (VV.comp g f)"
pat_vars_eqvt: fixes pi :: "(name \<times> name) list"   and p :: "pat" shows "pi \<bullet> pat_vars p = pat_vars (pi \<bullet> p)"
ugeq: fixes u :: "complex"   and z :: "complex" assumes "cmod u = 1"   and "z \<in> ball 0 r" shows "u * g z = g (u * z)"
tendsto_Pair: fixes f :: "'b \<Rightarrow> 'c"   and a :: "'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'd"   and b :: "'d" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. (f x, g x)) \<longlongrightarrow> (a, b)) F"
perm_swap_comp: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "perm_swap a b (f \<circ> g) x = perm_swap a b f (perm_swap a b g x)"
assms_2: fixes a :: "'a"   and b :: "'a" assumes "a \<in> P"   and "b \<in> P" shows "order.greater_eq a b = order.greater_eq (f b) (f a)"
proj_arcs_HG: fixes a :: "'b" assumes "a \<in> arcs G" shows "proj_arcs_H (proj_arcs_G a) = a"
dims_tensor0: fixes d :: "nat list" shows "dims (tensor0 d) = d"