hcju/isabelleps huggingface dataset:
step_elim_cases_5: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and c1 :: "('b, 'a, 'c) com"   and c2 :: "('b, 'a, 'c) com"   and s :: "('b, 'c) xstate"   and u :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate" assumes "\<Gamma>\<turnstile> (Seq c1 c2, s) \<rightarrow> u" obtains c\<^sub>1' :: "('b, 'a, 'c) com" and s' :: "('b, 'c) xstate" where "u = (Seq c\<^sub>1' c2, s')" and "\<Gamma>\<turnstile> (c1, s) \<rightarrow> (c\<^sub>1', s')"   | "c1 = Skip" and "u = (c2, s)"   | sa :: "'b" where "u = (Throw, Normal sa)" and "c1 = Throw" and "s = Normal sa"   | f :: "'c" where "s = Fault f__" and "u = (Skip, Fault f__)" and "redex c1 = Seq c1 c2"   | "s = Stuck" and "u = (Skip, Stuck)" and "redex c1 = Seq c1 c2"   | f :: "'b" where "s = Abrupt f" and "u = (Skip, Abrupt f)" and "redex c1 = Seq c1 c2"
il_sub_inf_left_zero: fixes x :: "'a" shows "bot \<sqinter> x = bot"
tops_def: shows "Sup_lattice.tops Inf (\<lambda>x y. y \<le> x) = Sup_lattice.Infs Inf (\<lambda>x y. y \<le> x) {}"
induct: fixes P :: "('f \<times> 'd) list \<Rightarrow> ('d \<times> 'e) list \<Rightarrow> bool"   and a0 :: "('f \<times> 'd) list"   and a1 :: "('d \<times> 'e) list" assumes "\<And>ys. P [] ys"   and "\<And>x xs ys. \<lbrakk>map_of ys (snd x) = None \<Longrightarrow> P (delete (fst x) xs) ys; \<And>x2. map_of ys (snd x) = Some x2 \<Longrightarrow> P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) ys" shows "P a0 a1"
bounded_subset_ballD: fixes S :: "'a set"   and x :: "'a" assumes "bounded S" shows "\<exists>r>0. S \<subseteq> ball x r"
evalO_n: fixes b :: "nat"   and n :: "nat" shows "evalO b ((S ^^ n) Z) = n"
I_or: fixes \<phi>\<^sub>1 :: "atom fm"   and \<phi>\<^sub>2 :: "atom fm"   and xs :: "real list" shows "R.I (or \<phi>\<^sub>1 \<phi>\<^sub>2) xs = R.I (Or \<phi>\<^sub>1 \<phi>\<^sub>2) xs"
invalidVertexList_shorten: fixes g :: "graph"   and f :: "face"   and vol :: "nat option list"   and v :: "nat option" assumes "invalidVertexList g f vol" shows "invalidVertexList g f (v # vol)"
assms_5: shows "\<nexists>l'. cs\<^bsup>\<pi>\<^esup> l = cs\<^bsup>\<pi>'\<^esup> l'"
PP_def: shows "PP = Pos ''P'' [Fun ''c'' []]"
Some: fixes x :: "real" shows "convert_atom_list var L (xs' @ x # \<Gamma>) = Some a"
induct: fixes P :: "'b list \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "'b list"   and a1 :: "'b list" assumes "\<And>x xs y ys. P xs ys \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>xs. P xs []"   and "\<And>y ys. P [] (y # ys)" shows "P a0 a1"
pole_lemma: fixes f :: "complex \<Rightarrow> complex"   and S :: "complex set"   and a :: "complex" assumes "f holomorphic_on S"   and "a \<in> interior S" shows "(\<lambda>z. if z = a then deriv f a else (f z - f a) / (z - a)) holomorphic_on S"
finite_Collect_subsets: fixes A :: "'f set" assumes "finite A" shows "finite {B. B \<subseteq> A}"
Im_Ln_eq_pi: fixes z :: "complex" assumes "z \<noteq> 0" shows "(Im (Ln z) = pi) = (Re z < 0 \<and> Im z = 0)"
unforwarded_non_volatile_reads_non_volatile_Read\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and W :: "nat set" shows "unforwarded_non_volatile_reads sb W \<subseteq> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b sb"
hash_oracle'_def: shows "hash_oracle' = (\<lambda>\<sigma> x. hash x \<bind> (\<lambda>h. Generative_Probabilistic_Value.Done (h, insert x \<sigma>)))"
mult_left_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "mone \<odot> x = x"
union'_language: fixes A :: "('c, 'd) dca"   and B :: "('c, 'e) dca" assumes "dca.alphabet A = dca.alphabet B" shows "DGCA.language (union' A B) = DCA.language A \<union> DCA.language B"
triangle_left: shows "(\<epsilon> \<star> f) \<cdot> \<a>\<^sup>-\<^sup>1[f, g, f] \<cdot> (f \<star> \<eta>) = f"
coinduct: fixes R :: "'c llist \<Rightarrow> 'c llist \<Rightarrow> bool"   and llist :: "'c llist"   and llist' :: "'c llist" assumes "R llist llist'"   and "\<And>llist llist'. R llist llist' \<Longrightarrow> lnull llist = lnull llist' \<and> (\<not> lnull llist \<longrightarrow> \<not> lnull llist' \<longrightarrow> lhd llist = lhd llist' \<and> R (ltl llist) (ltl llist'))" shows "llist = llist'"
assms_2: shows "f maps\<^bsub>C\<^esub> X to Y"
assms_3: shows "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"
assms_7: shows "(B, C) \<in> s_mul_ext ns s"
h_def: shows "h = (\<lambda>x. \<Sqinter>n\<in>{1..}. birkhoff_sum f n x)"
omega_top: fixes x :: "'a" shows "x\<^sup>\<omega> \<cdot> \<top> = x\<^sup>\<omega>"
freq_swapFourSyms: fixes t :: "'a tree"   and a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "consistent t"   and "a \<in> alphabet t"   and "b \<in> alphabet t"   and "c \<in> alphabet t"   and "d \<in> alphabet t" shows "freq (swapFourSyms t a b c d) = freq t"
fold_deriv_FAll: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "{\<langle>fold (deriv d0) xs (FAll k \<phi>)\<rangle> |xs. True} \<subseteq> {FAll k \<psi> |\<psi>. nf_ACI \<psi> \<and> conjuncts \<psi> \<subseteq> (\<Union>xs. conjuncts \<langle>fold (deriv d0) xs \<phi>\<rangle>)}"
Q5'_a_def: shows "Q5'_a \<equiv> [: \<lambda>(stk, mrk). {(stk', mrk'). stk = [] \<and> mrk' = mrk} :]"
Subset_Zero_D: fixes H :: "fm set"   and t :: "tm" assumes "H \<turnstile> t SUBS Zero" shows "H \<turnstile> t EQ Zero"
succs_is_node: fixes x :: "nat \<times> nat" assumes "prod_is_node A B x" shows "list_all (prod_is_node A B) (prod_succs A B x)"
dvd_add_triv_right_iff: fixes a :: "'a"   and b :: "'a" shows "(a dvd b + a) = (a dvd b)"
35: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and u :: "'a" shows "x \<sqinter> (y \<sqinter> z) \<squnion> u \<sqinter> z = (x \<sqinter> y \<squnion> u) \<sqinter> z"
class_les_irrefl: fixes cs :: "(String.literal \<times> String.literal) set"   and c :: "String.literal" assumes "wf_subclass_loc cs" shows "\<not> class_les cs c c"
cosine_sum_lemma: fixes x :: "real"   and n :: "nat" shows "(1 / 2 + (\<Sum>k = Suc 0..n. cos (real k * x))) * sin (x / 2) = sin ((real n + 1 / 2) * x) / 2"
OclValid_def: fixes \<tau> :: "'b state \<times> 'b state"   and P :: "'b state \<times> 'b state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "\<tau> \<Turnstile> P \<equiv> P \<tau> = true \<tau>"
llist_all2_True: fixes xs :: "'c llist"   and ys :: "'d llist" shows "llist_all2 (\<lambda>_ _. True) xs ys = (llength xs = llength ys)"
lang_verund: fixes r :: "nat list"   and rs :: "nat rexp list" shows "(r \<in> L (set rs)) = (r \<in> lang (verund rs))"
fold_congs_4: fixes r :: "('b, 'c) sparc_state_scheme"   and r' :: "('b, 'c) sparc_state_scheme"   and v' :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and f :: "(word8 \<Rightarrow> word36 \<Rightarrow> word8 option) \<Rightarrow> word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and f' :: "(word8 \<Rightarrow> word36 \<Rightarrow> word8 option) \<Rightarrow> word8 \<Rightarrow> word36 \<Rightarrow> word8 option" assumes "r = r'"   and "mem r' = v'"   and "\<And>v. v' = v \<Longrightarrow> f v = f' v" shows "mem_update f r = mem_update f' r'"
PDG_path_cdep: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and n :: "'b"   and n' :: "'b" assumes "PDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit control_dependence"   and "PDG.PDG_edge sourcenode targetnode valid_edge Def Use control_dependence n None n'" shows "PDG.PDG_path sourcenode targetnode valid_edge Def Use control_dependence n n'"
nj_set_def: shows "nj_set = {nj. \<exists>j\<in>{0..<m}. enat (Suc nj) < llength Ns \<and> prems_of \<iota> ! j \<notin> active_subset (snd (lnth Ns nj)) \<and> (\<forall>k>nj. enat k < llength Ns \<longrightarrow> prems_of \<iota> ! j \<in> active_subset (snd (lnth Ns k)))}"
sigma_eqI: fixes M :: "'c set set"   and \<Omega> :: "'c set"   and N :: "'c set set" assumes "M \<subseteq> Pow \<Omega>"   and "N \<subseteq> Pow \<Omega>"   and "sigma_sets \<Omega> M = sigma_sets \<Omega> N" shows "sigma \<Omega> M = sigma \<Omega> N"
measurable_epred: shows "epred \<in> count_space UNIV \<rightarrow>\<^sub>M count_space UNIV"
a_lcos_m_assoc: fixes M :: "'a set"   and g :: "'a"   and h :: "'a" assumes "M \<subseteq> carrier G"   and "g \<in> carrier G"   and "h \<in> carrier G" shows "g <+ (h <+ M) = g \<oplus> h <+ M"
a1: shows "low_equal s1 s2 \<and> Some t1 = memory_write 10 addr bm data s1"
bitlen_denormal_mantissa: fixes x :: "('a, 'b) IEEE.float" shows "bitlen \<bar>denormal_mantissa x\<bar> \<le> int LENGTH('b)"
smult_dvd_cancel: fixes a :: "'b"   and p :: "'b poly"   and q :: "'b poly" assumes "smult a p dvd q" shows "p dvd q"
prems_3: shows "set ps \<inter> set_kdt (Node k v l r) = {}"
less_preal_def: fixes x :: "preal"   and y :: "preal" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
assms_3: shows "KD_Tree.size_kdt l = KD_Tree.size_kdt r + 1"
mono_write0_FD: fixes P :: "'a process"   and Q :: "'a process"   and a :: "'a" assumes "P \<le> Q" shows "(a \<rightarrow> P) \<le> (a \<rightarrow> Q)"
Liminf_eq_Limsup: fixes F :: "'b filter"   and f :: "'b \<Rightarrow> 'd"   and f0 :: "'d" assumes "F \<noteq> bot"   and "Liminf F f = f0"   and "Limsup F f = f0" shows "(f \<longlongrightarrow> f0) F"
d'_at_d: shows "\<not> Interp D \<Turnstile> D'"
poly_N_xi_cond: fixes i :: "nat"   and j :: "nat" shows "poly (local.N i) (local.x j) = (if j < i then 0::'a else local.pprod j 0 i)"
inf_sup_absorb: fixes inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f"   and y :: "'f" assumes "class.lattice inf less_eq less sup" shows "inf x (sup x y) = x"
getMinTree_xlate: fixes q :: "('c, 'd) BsSkewBinomialTree list" assumes "q \<noteq> []" shows "bsmapt (getMinTree q) = SkewBinomialHeapStruc.getMinTree (map bsmapt q)"
sym_order_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sim> y"   and "z \<sqsubseteq> y"   and "x \<in> A"   and "y \<in> A"   and "z \<in> A" shows "z \<sqsubseteq> x"
convert_semantics: fixes fml :: "rat poly fml"   and x :: "real" shows "fml_sem fml x = lookup_sem (fst (convert fml)) (map (\<lambda>p. rpoly p x) (snd (convert fml)))"
WT_gpv_stop: fixes \<I> :: "('a, 'b) \<I>"   and gpv :: "('c, 'a, 'b) gpv" assumes "\<I> \<turnstile>g gpv \<surd>" shows "stop_\<I> \<I> \<turnstile>g gpv_stop gpv \<surd>"
dgrad_p_set_le_spoly_zero: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "dgrad_p_set_le d {spoly p 0} {p}"
assms_1: shows "strictly_maximal_literal P1 (Pos A)"
l2_inv2D: fixes A :: "agent"   and B :: "agent"   and KE :: "msg"   and x :: "l2_state" assumes "Auth A B \<langle>Number 0, KE\<rangle> \<in> chan x"   and "A \<notin> bad x \<and> B \<notin> bad x"   and "x \<in> l2_inv2" shows "\<exists>Ra. guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr> \<and> in_progress (progress x Ra) xpkE \<and> KE = epubKF (Ra $ kE)"
index_distinct: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and G1 :: "'b set"   and G2 :: "'b set"   and x :: "'b"   and y :: "'b" assumes "GDD point_set block_collection groups index"   and "G1 \<in> groups"   and "G2 \<in> groups"   and "G1 \<noteq> G2"   and "x \<in> G1"   and "y \<in> G2" shows "int (block_collection index {x, y}) = index"
a_2: shows "inverse (lead_coeff f) \<noteq> (0::'a)"
rbt_less_trans: fixes x :: "'a"   and t :: "('a, 'c) rbt"   and y :: "'a" assumes "t |\<guillemotleft> x"   and "x < y" shows "t |\<guillemotleft> y"
intros: fixes q :: "(char list multiset \<times> int) multiset"   and xs :: "char list multiset"   and n :: "int"   and p :: "(char list multiset \<times> int) multiset"   and r :: "(char list multiset \<times> int) multiset" shows "mult_poly_p q (add_mset (xs, n) p, r) (p, {#(remdups_mset (xs + ys), n * m). (ys, m) \<in># q#} + r)"
sum_a2: shows "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. n < m \<longrightarrow> sum a {n..m - 1} < e"
path_nret_inf_icd_seq: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "\<forall>i. \<pi> i \<noteq> IFC_def.return prob" obtains f :: "nat \<Rightarrow> nat" where "\<forall>i. IFC_def.is_icdi prob (f (Suc i)) \<pi> (f i)" and "range f = {i. \<forall>j>i. IFC_def.is_cdi prob j \<pi> i}" and "\<nexists>i. IFC_def.is_cdi prob (f 0) \<pi> i"
AbstrLevels_S4_A23: fixes i :: "AbstrLevelsID" assumes "sS4 \<in> AbstrLevel i" shows "sA23 \<notin> AbstrLevel i"
inverse_zero: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "inverse zero = zero"
ord_eq_le_trans: fixes a :: "'c word"   and b :: "'c word"   and c :: "'c word" assumes "a = b"   and "b \<le>s c" shows "a \<le>s c"
simps_2: fixes x :: "name"   and xs :: "name list"   and y :: "name"   and ys :: "name list" shows "composePerm (x # xs) (y # ys) = (x, y) # composePerm xs ys"
bigo: shows "fds_nth F' \<in> O(\<lambda>_. 1)"
lex_two_compat2: fixes ns1 :: "('b \<times> 'b) set"   and s1 :: "('b \<times> 'b) set"   and ns2 :: "('c \<times> 'c) set"   and s2 :: "('c \<times> 'c) set" assumes "ns1 O s1 \<subseteq> s1"   and "s1 O ns1 \<subseteq> s1"   and "s1 O s1 \<subseteq> s1"   and "ns1 O ns1 \<subseteq> ns1"   and "ns2 O s2 \<subseteq> s2" shows "lex_two s1 ns1 ns2 O lex_two s1 ns1 s2 \<subseteq> lex_two s1 ns1 s2"
cases: fixes a :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "a \<in> SAT \<Gamma> f" obtains x :: "'k" where "a = xb__" and "xb__ \<in> A \<Gamma>"   | x :: "'k" where "a = x" and "weight \<Gamma> x \<le> d_IN f x"
component_SKIP: fixes F :: "'c program" shows "\<bottom> \<le> F"
exhaust: fixes y :: "'a\<cdot>('b, 'c) stateT" obtains "y = \<bottom>"   | cfun :: "'c \<rightarrow> ('a \<times> 'c)\<cdot>'b" where "y = StateT\<cdot>cfun" and "cfun \<noteq> \<bottom>"
Macaulay_mat_def: fixes ps :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "Macaulay_mat ps = polys_to_mat (Keys_to_list ps) ps"
trace_in_def: shows "trace_in \<equiv> \<lambda>S t. lfp (\<lambda>p x1 x2. (\<exists>t' s s' \<omega>. x1 = s \<and> x2 = (t', s') ## \<omega> \<and> t < t' \<and> s \<in> S) \<or> (\<exists>t' s' \<omega> s. x1 = s \<and> x2 = (t', s') ## \<omega> \<and> t' \<le> t \<and> p s' \<omega>))"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') (\<Phi>'.Y a) \<Phi>'\<Psi>.map"
continuous_within_topological: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "continuous (at x within s) f = (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
list_rel_range: fixes R :: "('b \<times> 'a) set" assumes "Range R = UNIV" shows "Range (\<langle>R\<rangle>list_rel) = UNIV"
horizontal_homs_def: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b" shows "horizontal_homs V src trg \<equiv> (category V \<and> endofunctor V src) \<and> endofunctor V trg \<and> horizontal_homs_axioms V src trg"
setinv_closed: fixes K :: "'a set" assumes "K \<in> rcosets H" shows "set_inv K \<in> rcosets H"
wf_induct: fixes r :: "('e \<times> 'e) set"   and P :: "'e \<Rightarrow> bool"   and a :: "'e" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
gauss_sum_conv_gauss_sum_int: fixes k :: "nat" shows "gauss_sum k = gauss_sum_int (int k)"
borel_measurable_scaleR: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. f x *\<^sub>R g x) \<in> borel_measurable M"
\<epsilon>o_def: fixes a :: "'a" shows "F.\<epsilon>o a = (SOME e. terminal_arrow_from_functor (\<cdot>\<^sub>B) (\<cdot>\<^sub>A) F (F.Go a) a e)"
subspace_UNIV: fixes scale :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "module scale" shows "module.subspace scale UNIV"
insert_4: shows "\<forall>x\<in>insert x xs. \<forall>a\<in>ys. x < a"
transfer_deliver: fixes s :: "'s net_state"   and i :: "nat"   and d :: "nat"   and s' :: "'s net_state"   and n :: "net_tree"   and \<sigma> :: "nat \<Rightarrow> 'g"   and \<zeta> :: "'l net_state" assumes "(s, i:deliver(d), s') \<in> automaton.trans (pnet np n)"   and "s \<in> reachable (pnet np n) TT"   and "netgmap sr s = netmask (net_tree_ips n) (\<sigma>, \<zeta>)"   and "wf_net_tree n" obtains \<sigma>'' :: "nat \<Rightarrow> 'g" and \<zeta>'' :: "'l net_state" where "((\<sigma>, \<zeta>), i:deliver(d), \<sigma>'', \<zeta>'') \<in> automaton.trans (opnet onp n)" and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>'' j = \<sigma> j" and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>'', \<zeta>'')"
le_or_fail_def: fixes m :: "'b nres"   and m' :: "'b nres" shows "m \<le>\<^sub>n m' \<equiv> nofail m \<longrightarrow> m \<le> m'"
diameter_closure: fixes S :: "'b set" assumes "bounded S" shows "diameter (closure S) = diameter S"
sunflower_remove_element_lift: fixes S :: "'b set set"   and a :: "'b"   and F :: "'b set set" assumes "S \<subseteq> {A - {a} |A. A \<in> F \<and> a \<in> A}"   and "sunflower S" shows "\<exists>Sa. sunflower Sa \<and> Sa \<subseteq> F \<and> card Sa = card S \<and> Sa = insert a ` S"
head_in_verts: fixes e :: "'a \<times> 'a" assumes "e \<in> arcs (with_proj G)" shows "head (with_proj G) e \<in> verts (with_proj G)"
vfsequence_vdomain: shows "\<D>\<^sub>\<circ> \<BB> = vcard \<BB>"
simps_1: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d stream"   and a :: "'c" shows "shd (sscan f xs a) = f (shd xs) a"
less_3: fixes q :: "'a list" assumes "q \<in> carrier (poly_ring R)"   and "pirreducible (carrier R) q"   and "q pdivides p" shows "degree q \<le> 1"
phis_in_\<alpha>n: fixes g :: "('b, 'a, 'c) gen_cfg_wf"   and n :: "'b"   and v :: "'a \<times> 'b \<times> Def"   and vs :: "('a \<times> 'b \<times> Def) list" assumes "gen_wf.phis' g (n, v) = Some vs" shows "n \<in> set (gen_wf_\<alpha>n g)"
cardeq_lcoset: fixes M :: "'a set"   and g :: "'a" assumes "finite (carrier G)"   and "M \<subseteq> carrier G"   and "g \<in> carrier G" shows "card (g <# M) = card M"
gen_divisions_2: shows "gen_division s (cubeImage ` two_chain_typeII)"
order_subst1: fixes a :: "'b"   and f :: "'c \<Rightarrow> 'b"   and b :: "'c"   and c :: "'c" assumes "a \<le> f b"   and "b \<le> c"   and "\<And>x y. x \<le> y \<Longrightarrow> f x \<le> f y" shows "a \<le> f c"
has_dist_onD: fixes G :: "graph"   and m :: "nat"   and n :: "nat"   and V :: "nat set" assumes "has_dist_on G m n V" shows "\<exists>p. is_shortest_path_on G m n p V"
fusion_def: fixes S :: "('d \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> bool"   and S' :: "('d \<Rightarrow> bool) \<Rightarrow> 'e \<Rightarrow> bool"   and q :: "'d \<Rightarrow> bool"   and x :: "'e" shows "(S \<parallel> S') q x = (\<exists>p p'. p \<sqinter> p' \<le> q \<and> S p x \<and> S' p' x)"
inti: fixes i :: "'b" shows "LINT x|M. indicat_real A x * f i x = LINT x|M. indicat_real A x * real_cond_exp M F (f i) x"
simps_1: shows "Dijkstra_Abstract.enat_of_option None = \<infinity>"
left_tree_def: fixes s :: "'c"   and t :: "'c" shows "T.left_tree s t \<equiv> {u \<in> V\<^bsub>T\<^esub>. s \<in> set (u \<leadsto>\<^bsub>T\<^esub> t)}"
dghm_length: shows "vcard \<HH> = 4\<^sub>\<nat>"
col_space_def: fixes B :: "'a mat" shows "col_space B = local.span (set (cols B))"
intuitionistic_sep_conj_sep_true_P: fixes P :: "'a \<Rightarrow> bool"   and s :: "'a" assumes "(P \<and>* (\<lambda>s. True)) s"   and "intuitionistic P" shows "P s"
f1: shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r' |\<^bsub>rel\<^esub> s') = (r \<otimes> r' |\<^bsub>rel\<^esub> s \<otimes> s')"
Bseq_eventually_mono: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'b" assumes "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> norm (g n)"   and "Bseq g" shows "Bseq f"
disjoint_family_on_def: fixes A :: "'d \<Rightarrow> 'e set"   and S :: "'d set" shows "disjoint_family_on A S = (\<forall>m\<in>S. \<forall>n\<in>S. m \<noteq> n \<longrightarrow> A m \<inter> A n = {})"
oalist_inv_raw_foldr_update_by_pair: fixes ys :: "('a \<times> 'c) list"   and xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw ys" shows "oalist_inv_raw (foldr update_by_pair xs ys)"
extRules_def: shows "extRules \<equiv> \<lambda>R. {x. extRulesp (\<lambda>x. x \<in> R) x}"
inv_in_hom: fixes f :: "'c \<times> 'c"   and a :: "'c \<times> 'c"   and b :: "'c \<times> 'c" assumes "C.VV.iso f"   and "C.VV.in_hom f a b" shows "C.VV.in_hom (C.VV.inv f) b a"
exists_asSort_wls: fixes xs :: "'varSort" shows "\<exists>X. wls (asSort xs) X"
f_i_simps_2: fixes i :: "nat" shows "f_i (Suc i) = f_i i \<oplus> k_i i"
prefix_length_prefix: fixes ps :: "'c list"   and xs :: "'c list"   and qs :: "'c list" assumes "prefix ps xs"   and "prefix qs xs"   and "length ps \<le> length qs" shows "prefix ps qs"
le_supI2: fixes x :: "'a"   and b :: "'a"   and a :: "'a" assumes "x \<le> b" shows "x \<le> a \<squnion> b"
psi'_l_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "grover_state.psi'_l n f l \<in> carrier_vec (grover_state.N n)"
diff_add_eq_diff_diff_swap: fixes minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and uminus :: "'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.group_add minus plus zero uminus" shows "minus a (plus b c) = minus (minus a c) b"
levi_lemma: fixes t :: "'action list"   and u :: "'action list"   and v :: "'action list"   and w :: "'action list" assumes "t @ u =\<^sub>F v @ w" obtains p :: "'action list" and r :: "'action list" and s :: "'action list" and q :: "'action list" where "t =\<^sub>F p @ r" and "u =\<^sub>F s @ q" and "v =\<^sub>F p @ s" and "w =\<^sub>F r @ q" and "Ind (set r) (set s)"
cos_Arccos: fixes z :: "complex" shows "cos (Arccos z) = z"
inv_unique': fixes x :: "'b"   and y :: "'b" assumes "x \<in> carrier (G i)"   and "y \<in> carrier (G i)"   and "x \<otimes>\<^bsub>G i\<^esub> y = \<one>\<^bsub>G i\<^esub>"   and "y \<otimes>\<^bsub>G i\<^esub> x = \<one>\<^bsub>G i\<^esub>" shows "y = inv\<^bsub>G i\<^esub> x"
degree_add_eq_left: fixes q :: "'c poly"   and p :: "'c poly" assumes "degree q < degree p" shows "degree (p + q) = degree p"
multi_member_this: fixes x :: "'a"   and XS :: "'a multiset" shows "x \<in># \<LM> x  \<RM> + XS"
not_ks_2: fixes x :: "k" shows "(x \<noteq> unk) = (x = kno)"
euclidean_inner: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and Basis :: "'b set"   and x :: "'b"   and y :: "'b" assumes "class.euclidean_space minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis" shows "inner x y = (\<Sum>b\<in>Basis. inner x b * inner y b)"
J: shows "Crypt (SesK SK) (A \<otimes> B) \<in> parts (used v)"
comm_ring_hom_axioms: shows "comm_ring_hom poly_y_x"