hcju/isabelleps huggingface dataset:
assms_5: fixes a :: "'a"   and b :: "'a" assumes "a \<in> set xs \<union> set ys"   and "b \<in> set ys"   and "rel a b" shows "a = b"
pderiv_sum: fixes f :: "'c \<Rightarrow> 'b poly"   and I :: "'c set" shows "pderiv (sum f I) = (\<Sum>i\<in>I. pderiv (f i))"
conforms_globsD: fixes x :: "abopt"   and s :: "st"   and G :: "prog"   and L :: "lname \<Rightarrow> ty option"   and r :: "oref"   and obj :: "obj" assumes "(x, s)\<Colon>\<preceq>(G, L)"   and "globs s r = Some obj" shows "G,s\<turnstile>obj\<Colon>\<preceq>\<surd>r"
vec_first_lincomb_list: fixes Xs :: "'b vec list"   and n :: "nat"   and m :: "nat"   and c :: "nat \<Rightarrow> 'b" assumes "set Xs \<subseteq> carrier_vec n"   and "m \<le> n" shows "vec_first (module.lincomb_list (module_vec TYPE('b) n) c Xs) m = module.lincomb_list (module_vec TYPE('b) m) c (map (\<lambda>v. vec_first v m) Xs)"
iff_simps_8: fixes p :: "fm"   and q :: "fm" assumes "p \<noteq> q"   and "p \<noteq> NOT q"   and "q \<noteq> NOT p"   and "p \<noteq> F"   and "q \<noteq> F"   and "p \<noteq> T"   and "q \<noteq> T" shows "Ferrack.iff p q = Iff p q"
AG_AX: fixes p :: "'b set" shows "\<^bold>A\<^bold>G p \<subseteq> \<^bold>A\<^bold>X p"
inject: fixes x2 :: "real"   and y2 :: "real" shows "(real_max_bot.R x2 = real_max_bot.R y2) = (x2 = y2)"
wf_M: shows "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (set M)"
arrE: fixes f :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))" and "Cod f \<in> Collect (functor (\<cdot>\<^sub>A) (\<cdot>\<^sub>B))" and "Map f \<in> Collect (natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (Dom f) (Cod f))"
alphaInp_refl: fixes qinp :: "'e \<Rightarrow> ('f, 'g, 'varSort, 'var, 'h) qTerm option" assumes "qGoodInp qinp" shows "(qinp, qinp) \<in> alphaInp"
def_X: shows "\<tau> \<Turnstile> \<delta> X"
inconsist_def: fixes i :: "nat"   and e :: "nat" shows "inconsist i e \<equiv> if \<exists>x<e_length e. \<phi> i x \<up> then None else if \<exists>x<e_length e. \<phi> i x \<down>\<noteq> e_nth e x then Some (LEAST x. x < e_length e \<and> \<phi> i x \<down>\<noteq> e_nth e x) else Some (e_length e)"
path_split2: fixes edges :: "(('a \<times> 'b) \<times> 'a \<times> 'c) set"   and v :: "'a"   and v' :: "'a"   and pth1 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list"   and e :: "('a \<times> 'b) \<times> 'a \<times> 'c"   and pth2 :: "(('a \<times> 'b) \<times> 'a \<times> 'c) list" shows "Pre_Port_Graph.path edges v v' (pth1 @ e # pth2) = (Pre_Port_Graph.path edges v (Pre_Port_Graph.edge_begin e) pth1 \<and> Pre_Port_Graph.path edges (Pre_Port_Graph.edge_begin e) v' (e # pth2))"
preprocess'_unsat_indices: fixes i :: "'a"   and s :: "('a \<times> QDelta ns_constraint) list"   and start :: "nat"   and v :: "nat \<Rightarrow> QDelta" assumes "i \<in> set (UnsatIndices (preprocess' s start))" shows "\<not> ({i}, v) \<Turnstile>\<^sub>i\<^sub>n\<^sub>s\<^sub>s  set s"
connected_by_dist: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and v :: "nat"   and v' :: "nat" shows "Graph.connected c v v' = (\<exists>d. Graph.dist c v d v')"
countable_Int1: fixes A :: "'a set"   and B :: "'a set" assumes "countable A" shows "countable (A \<inter> B)"
mk_pat_frees: fixes p :: "Term_Class.term" assumes "linear p" shows "patvars (mk_pat p) = frees p"
IH: fixes s :: "'s"   and as :: "'s list"   and v :: "ennreal" assumes "viterbi s t_end os = (as, v)" shows "length as = length os \<and> set as \<subseteq> \<S>"
LB_count: fixes vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'g) FSM"   and m :: "nat"   and FAIL :: "'g \<times> 'f"   and PM :: "('d, 'e, 'g \<times> 'f) FSM"   and V :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set"   and V'' :: "('d \<times> 'e) list set"   and T :: "'d list set"   and S :: "'g set" assumes "vs @ xs \<in> L M1"   and "OFSM M1"   and "OFSM M2"   and "asc_fault_domain M2 M1 m"   and "test_tools M2 M1 FAIL PM V \<Omega>"   and "V'' \<in> Perm V M1"   and "Prereq M2 M1 vs xs T S \<Omega> V''"   and "\<not> Rep_Pre M2 M1 vs xs"   and "\<not> Rep_Cov M2 M1 V'' vs xs" shows "LB M2 M1 vs xs T S \<Omega> V'' \<le> |M1|"
Kleene_iter_lpfp: fixes f :: "'b \<Rightarrow> 'b"   and p :: "'b"   and k :: "nat" assumes "mono f"   and "f p \<le> p" shows "(f ^^ k) bot \<le> p"
ceiling_add_le: fixes x :: "'a"   and y :: "'a" shows "\<lceil>x + y\<rceil> \<le> \<lceil>x\<rceil> + \<lceil>y\<rceil>"
succ_chain_swap_lr_exists: fixes l :: "('i, 'a) state list"   and i :: "nat"   and j :: "nat"   and x :: "nat" assumes "rel_chain l {(s, s'). \<triangle> (\<T> s) \<and> \<diamond> s \<and> \<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s \<and> \<nabla> s \<and> s \<succ>\<^sub>x s' \<and> \<B>\<^sub>i s' = \<B>\<^sub>i s \<and> \<U>\<^sub>c s' = \<U>\<^sub>c s}"   and "i < j"   and "j < length l"   and "x \<in> lvars (\<T> (l ! i))"   and "x \<in> rvars (\<T> (l ! j))" shows "\<exists>k\<ge>i. k < j \<and> k + 1 < length l \<and> x \<in> lvars (\<T> (l ! k)) \<and> x \<in> rvars (\<T> (l ! (k + 1)))"
Suc: fixes \<rho> :: "complex mat" shows "denote_while_n_iter M0 M1 DS (Suc n) \<rho> = DS (M1 * denote_while_n_iter M0 M1 DS n \<rho> * adjoint M1)"
path_sourcenode: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and n :: "'b"   and as :: "'a list"   and n' :: "'b" assumes "CFG sourcenode targetnode valid_edge Entry"   and "CFG.path sourcenode targetnode valid_edge n as n'"   and "as \<noteq> []" shows "hd (CFG.sourcenodes sourcenode as) = n"
add_eq_conv_ex: fixes a :: "'a"   and M :: "'a multiset"   and b :: "'a"   and N :: "'a multiset" shows "(a, M = b, N) = (M = N \<and> a = b \<or> (\<exists>K. M = b, K \<and> N = a, K))"
hb_broadcast_exists1: fixes m1 :: "'a \<times> ('a, 'b) operation"   and m2 :: "'a \<times> ('a, 'b) operation" assumes "hb m1 m2" shows "\<exists>i. Broadcast m1 \<in> set (history i)"
Diag_Diagonalize_2: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and t :: "'b local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Dom C (monoidal_language.Diagonalize C t) = monoidal_language.Diagonalize C (monoidal_language.Dom C t)"
cases_1: assumes "AlternativeSmallStep.inf \<Gamma> [c] [] s" shows "P"
Abs: fixes r :: "dB"   and r' :: "dB"   and ss :: "dB list"   and ss' :: "dB list" assumes "r \<rightarrow>\<^sub>s r'"   and "ss [\<rightarrow>\<^sub>s] ss'" shows "Abs r \<degree>\<degree> ss \<rightarrow>\<^sub>s Abs r' \<degree>\<degree> ss'"
open_Var_inj: fixes xT :: "char list \<times> Unary_PCF.type"   and t :: "expr"   and u :: "expr"   and i :: "nat" assumes "fresh xT t"   and "fresh xT u"   and "open_Var i xT t = open_Var i xT u" shows "t = u"
tp_closed: shows "(\<cdot>\<^sub>r) \<in> carrier R \<rightarrow> carrier R \<rightarrow> carrier R"
quotI: fixes a :: "'b" shows "{x. a \<sim> x} \<in> quot"
Sup_inter_less_eq: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and B :: "'b set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "less_eq (Sup (A \<inter> B)) (inf (Sup A) (Sup B))"
trelRS: shows "rel_weakly_reflects_barb_set TRel TWB {success}"
that: fixes b :: "nat \<Rightarrow> 'a" assumes "\<And>z. (\<Sum>i\<le>n. c i * z ^ i) = (z - a) * (\<Sum>i<n. b i * z ^ i)" shows "thesis"
msubset: shows "fmset G as \<subseteq># fmset G bs"
assignment_rhs\<^sub>e\<^sub>s\<^sub>t_append_subst_2: fixes \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and B :: "('fun, 'var) extstrand_step list" shows "assignment_rhs\<^sub>e\<^sub>s\<^sub>t (A @ B) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta> = (assignment_rhs\<^sub>e\<^sub>s\<^sub>t A \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<union> (assignment_rhs\<^sub>e\<^sub>s\<^sub>t B \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)"
short: shows "\<parallel>f\<parallel>\<^sup>2 ^ k * \<parallel>g\<parallel>\<^sup>2 ^ n < m\<^sup>2"
policy_refinement_def: fixes p :: "'c \<Rightarrow> 'd decision option"   and abs\<^sub>a :: "'e \<Rightarrow> 'c"   and abs\<^sub>b :: "'f \<Rightarrow> 'd"   and q :: "'e \<Rightarrow> 'f decision option" shows "p \<sqsubseteq>\<^bsub>abs\<^sub>a\<^esub>\<^sub>,\<^bsub>abs\<^sub>b\<^esub> q \<equiv> \<forall>a. case p a of \<bottom> \<Rightarrow> True | \<lfloor>allow y\<rfloor> \<Rightarrow> \<forall>a'\<in>{x. abs\<^sub>a x = a}. \<exists>b'. q a' = \<lfloor>allow b'\<rfloor> \<and> abs\<^sub>b b' = y | \<lfloor>deny y\<rfloor> \<Rightarrow> \<forall>a'\<in>{x. abs\<^sub>a x = a}. \<exists>b'. q a' = \<lfloor>deny b'\<rfloor> \<and> abs\<^sub>b b' = y"
row_polys_to_mat: fixes i :: "nat"   and ps :: "('c \<Rightarrow>\<^sub>0 'd) list"   and ts :: "'c list" assumes "i < length ps" shows "row (polys_to_mat ts ps) i = poly_to_row ts (ps ! i)"
lcm_fract_def: shows "lcm \<equiv> normalization_euclidean_semiring_class.lcm"
exhaust: fixes y :: "New" obtains x1 :: "CTypeId" where "y = new_instance x1"   | x21 :: "Arraytype" and x22 :: "nat" where "y = new_array x21 x22"
oarrivemsg_def: fixes P :: "(nat \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'e"   and a :: "'f node_action" shows "oarrivemsg P \<sigma> a \<equiv> case a of ii\<not>ni:arrive(m) \<Rightarrow> P \<sigma> m | _ \<Rightarrow> True"
Abs_process_inverse: fixes y :: "('b event list \<times> 'b event set) set \<times> 'b event list set" assumes "y \<in> {p. is_process p}" shows "Rep_process (Abs_process y) = y"
create_graph__incoming_name_exist: fixes \<phi> :: "'b ltlr" shows "create_graph \<phi> \<le> SPEC (\<lambda>nds. \<forall>nd\<in>nds. expand_init < name nd \<and> (\<forall>nm\<in>incoming nd. nm \<noteq> expand_init \<longrightarrow> nm \<in> name ` nds))"
H: shows "mset_set {0..<Suc (length N)} = add_mset 0 (mset_set {1..<Suc (length N)})"
bigoI: fixes f :: "'b \<Rightarrow> 'c"   and c :: "real"   and g :: "'b \<Rightarrow> 'c"   and F :: "'b filter" assumes "\<forall>\<^sub>F x in F. norm (f x) \<le> c * norm (g x)" shows "f \<in> O[F](g)"
a1: shows "s' = snd (fst (mul_instr_sub1 instr_name result s)) \<and> get_S (cpu_reg_val PSR s) = 0"
lift2_def: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b err"   and e\<^sub>1 :: "'c err"   and e\<^sub>2 :: "'d err" shows "lift2 f e\<^sub>1 e\<^sub>2 = (case e\<^sub>1 of Err \<Rightarrow> Err | OK x \<Rightarrow> case e\<^sub>2 of Err \<Rightarrow> Err | OK y \<Rightarrow> f x y)"
in_homE: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" obtains "C.arr f" and "C.dom f = a" and "C.cod f = b"
final: fixes s :: "'a" assumes "C s"   and "I (Some s)"   and "B s = None" shows "I None"
lm012: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R +< {x} \<times> R `` {x} \<subseteq> R"
trans: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "le a b"   and "le b c" shows "le a c"
SN_rel_alt_r_empty: fixes S :: "('a \<times> 'a) set" shows "SN_rel_alt {} S"
xy: shows "- (pprt a * pprt b - pprt a * nprt b - nprt a * pprt b + nprt a * nprt b) \<le> pprt a * pprt b + pprt a * nprt b + nprt a * pprt b + nprt a * nprt b"
d: fixes \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems k" shows "\<exists>c\<^sub>f' mem\<^sub>f'. (cms\<^sub>2 ! k, subst \<sigma> (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>f', snd (cms\<^sub>1' ! k), mem\<^sub>f'\<rangle> \<and> (cms\<^sub>1' ! k, subst \<sigma> mem\<^sub>1') \<approx> \<langle>c\<^sub>f', snd (cms\<^sub>1' ! k), mem\<^sub>f'\<rangle>"
SMP_def: shows "SMP \<equiv> \<lambda>M. {x. SMPp (\<lambda>x. x \<in> M) x}"
weakenBisimEqWeakBisim: shows "weakenBisim = weakBisim"
smooth_on_Taylor2E: fixes f :: "'c \<Rightarrow> real"   and X :: "'c" assumes "smooth_on UNIV f" obtains g :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> real" where "\<And>Y. f Y = f X + frechet_derivative f (at X) (Y - X) + (\<Sum>i\<in>Basis. \<Sum>j\<in>Basis. (Y - X) \<bullet> j * ((Y - X) \<bullet> i) * g i j Y)" and "\<And>i j. \<lbrakk>i \<in> Basis; j \<in> Basis\<rbrakk> \<Longrightarrow> smooth_on UNIV (g i j)"
cont_def: fixes f :: "'c \<Rightarrow> 'd" shows "cont f = (\<forall>Y. chain Y \<longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i))"
lift_hom: fixes G :: "'a set"   and binop :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and e :: "'a"   and T :: "'a \<Rightarrow> 'b" assumes "BinOpSetGroup G binop e"   and "\<forall>g\<in>G. \<forall>h\<in>G. T (binop g h) = T g + T h" shows "GroupHom (BinOpSetGroup.Abs_G_perm G binop ` G) (restrict0 (T \<circ> the_inv_into G (BinOpSetGroup.Abs_G_perm G binop)) (BinOpSetGroup.Abs_G_perm G binop ` G))"
invertible: fixes u :: "'a" assumes "u \<in> A" shows "f.source.additive.invertible u"
fin: fixes X :: "nat set"   and x :: "real" assumes "X \<subseteq> {n. real n \<le> x}" shows "finite X"
decomp_def: fixes t :: "('fun, 'var) Term.term" shows "decomp t \<equiv> case Ana t of (K, T) \<Rightarrow> send\<langle>t\<rangle>\<^sub>s\<^sub>t # map Send K @ map Receive T"
elim_p_order: shows "order x p - order x (pderiv p * q) = 1 - order x q"
intros_1: fixes lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" shows "\<langle>fst (fst lc), snd (fst lc), snd lc\<rangle> \<in> loc_reach lc"
distinct_29: fixes x31 :: "nat"   and x32 :: "'e"   and x5 :: "'e" shows "unicast x31 x32 \<noteq> send x5"
lcos_mult_one: fixes M :: "'a set" assumes "M \<subseteq> carrier G" shows "\<one> <# M = M"
nextl_snoc: fixes M :: "'b dfa"   and q :: "hf"   and xs :: "'b list"   and x :: "'b" assumes "dfa M" shows "dfa.nextl M q (xs @ [x]) = dfa.nxt M (dfa.nextl M q xs) x"
simps_1: fixes N :: "nat" shows "freediscrim (VAR N) = 0"
minus_add: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.group_add minus plus zero uminus" shows "uminus (plus a b) = plus (uminus b) (uminus a)"
sys_xcpts_defs_1: shows "NullPointer \<equiv> ''NullPointer''"
entails_\<G>_def: fixes N1 :: "'a literal multiset set"   and N2 :: "'a literal multiset set" shows "(N1 \<TTurnstile>\<G>e N2) = (\<forall>q\<in>UNIV. \<Union> (\<G>_F ` N1) \<TTurnstile>e \<Union> (\<G>_F ` N2))"
finite_measure_finite_Union: fixes S :: "'a set"   and A :: "'a \<Rightarrow> 's stream set"   and s :: "'s" assumes "finite S"   and "A ` S \<subseteq> T.events s"   and "disjoint_family_on A S" shows "T.prob s (\<Union> (A ` S)) = (\<Sum>i\<in>S. T.prob s (A i))"
sep_disj_prod_def: fixes m1 :: "'c \<times> 'd"   and m2 :: "'c \<times> 'd" shows "m1 ## m2 \<equiv> fst m1 ## fst m2 \<and> snd m1 ## snd m2"
older_seniors_cases_subset: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and y :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w" obtains "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n \<subset> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n"   | "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n = semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n"   | "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w y n \<subset> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n"
unowned_shared: shows "\<forall>a. (\<forall>i<length \<O>s. a \<notin> \<O>s ! i) \<longrightarrow> a \<in> dom \<S>"
notStuck_SeqD2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c1 :: "('e, 'd, 'f) com"   and c2 :: "('e, 'd, 'f) com"   and s :: "'e"   and s' :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Seq c1 c2,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "\<Gamma>\<turnstile> \<langle>c1,Normal s\<rangle> \<Rightarrow> s'" shows "\<Gamma>\<turnstile> \<langle>c2,s'\<rangle> \<Rightarrow>\<notin>{Stuck}"
filters_to_ideals: fixes X :: "'a set" shows "filterp X = idealp (\<partial> ` X)"
card_min_max: fixes i :: "nat_int" shows "i \<noteq> bot \<longrightarrow> |i| = maximum i - minimum i + 1"
gcd_mult_right: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and c :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (times a c) (times b c) = normalize (times (gcd b a) c)"
Undefined_def: fixes uu :: "unit" shows "Undefined uu = Collect HOL.undefined"
monoid_comm_monoidI: fixes G :: "('e, 'f) monoid_scheme" assumes "Group.monoid G"   and "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y = y \<otimes>\<^bsub>G\<^esub> x" shows "Group.comm_monoid G"
rangeI: fixes f :: "'l \<Rightarrow> 'k"   and x :: "'l" shows "f x \<in> range f"
False: shows "\<not> (c \<noteq> (0::'b) \<longrightarrow> f abs_summable_on A)"
oreachable_other': fixes \<sigma> :: "'d"   and p :: "'e"   and A :: "('d \<times> 'e, 'f) automaton"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'f \<Rightarrow> bool"   and U :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>' :: "'d" assumes "(\<sigma>, p) \<in> oreachable A S U"   and "U \<sigma> \<sigma>'" shows "(\<sigma>', p) \<in> oreachable A S U"
closure_involutive_aux': fixes X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and Z :: "('d \<Rightarrow> real) set" assumes "AlphaClosure X" shows "\<exists>U\<subseteq>{region X I r |I r. valid_region X k I r}. AlphaClosure.cla X k Z = \<Union> U"
snd_snd_Gauss_Jordan_column_k_PA_eq: fixes P :: "(('a, 'c) vec, 'c) vec"   and i :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and k :: "nat" shows "snd (snd (Gauss_Jordan_column_k_PA (P, i, A) k)) = snd (Gauss_Jordan_column_k (i, A) k)"
comm_monoid_add_isom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "comm_monoid_add_isom hom \<equiv> monoid_add_isom hom"
NoMessageLossStep: fixes cfg1 :: "('p, 'v, 's) configuration"   and m :: "('p, 'v) message"   and cfg2 :: "('p, 'v, 's) configuration"   and p :: "'p"   and m' :: "('p, 'v) message" assumes "cfg1 \<turnstile> m \<mapsto> cfg2"   and "isReceiverOf p m"   and "\<not> isReceiverOf p m'" shows "msgs cfg1 m' \<le> msgs cfg2 m'"
sum_list_distinct_conv_sum_set: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and xs :: "'c list"   and f :: "'c \<Rightarrow> 'd" assumes "class.monoid_add plus zero"   and "distinct xs" shows "sum_list (map f xs) = sum f (set xs)"
cont_case_prodI: fixes orda :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and ordb :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'e \<Rightarrow> 'a"   and lubb :: "'e set \<Rightarrow> 'e"   and luba :: "'d set \<Rightarrow> 'd" assumes "monotone (rel_prod orda ordb) (\<le>) (\<lambda>(x, y). f x y)"   and "\<And>x. cont lubb ordb Sup (\<le>) (f x)"   and "\<And>y. cont luba orda Sup (\<le>) (\<lambda>x. f x y)"   and "class.preorder orda (mk_less orda)"   and "class.preorder ordb (mk_less ordb)" shows "cont (prod_lub luba lubb) (rel_prod orda ordb) Sup (\<le>) (\<lambda>(x, y). f x y)"
setbydecision_append: fixes rs1 :: "'a common_primitive rule list"   and rs2 :: "'a common_primitive rule list"   and iface :: "iface" assumes "simple_ruleset (rs1 @ rs2)" shows "setbydecision TYPE('b) iface (rs1 @ rs2) FinalAllow = setbydecision TYPE('b) iface rs1 FinalAllow \<union> {ip. \<exists>p. approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs2 Undecided = Decision FinalAllow \<and> approximating_bigstep_fun (common_matcher, in_doubt_allow) (p\<lparr>p_iiface := iface_sel iface, p_src := ip\<rparr>) rs1 Undecided = Undecided}"
dimXAB: shows "\<forall>n. X n + Y n \<in> carrier_mat m m \<and> A + B \<in> carrier_mat m m"
has_integral_eq_rhs: fixes f :: "'a \<Rightarrow> 'b"   and j :: "'b"   and S :: "'a set"   and i :: "'b" assumes "(f has_integral j) S"   and "i = j" shows "(f has_integral i) S"
word_le_def: fixes a :: "'a Word.word"   and b :: "'a Word.word" shows "(a \<le> b) = (uint a \<le> uint b)"
filterlim_mono: fixes f :: "'b \<Rightarrow> 'c"   and F2 :: "'c filter"   and F1 :: "'b filter"   and F2' :: "'c filter"   and F1' :: "'b filter" assumes "filterlim f F2 F1"   and "F2 \<le> F2'"   and "F1' \<le> F1" shows "filterlim f F2' F1'"
G: fixes x :: "'a" assumes "x \<in> s" shows "norm (g x) \<le> G"
forall_spec: fixes P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "\<forall>b. P b \<longrightarrow> Q b"   and "P a" shows "Q a"
plus_V_succ_right: fixes x :: "V"   and y :: "V" shows "x + ZFC_in_HOL.succ y = ZFC_in_HOL.succ (x + y)"
test_comp_anti_iff: fixes p :: "'a"   and q :: "'a" assumes "test p"   and "test q" shows "(p \<le> q) = (!q \<le> !p)"
trans_tm_forget: fixes i :: "name"   and t :: "tm" assumes "atom i \<sharp> t" shows "trans_tm [i] t = trans_tm [] t"
locannot: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and bv :: "'b state \<times> (v, v) result"   and l :: "locn \<times> locn" assumes "evaluate ck env s e bv" shows "evaluate ck env s (Lannot e l) bv"
combine_common_factor: fixes a :: "'a"   and e :: "'a"   and b :: "'a"   and c :: "'a" shows "a \<sqinter> e \<squnion> (b \<sqinter> e \<squnion> c) = (a \<squnion> b) \<sqinter> e \<squnion> c"
assms_2: shows "filterlim g G (at x within A \<inter> {x. \<not> P x})"
chine_simps_2: shows "C.dom src.chine = src.apex"
diff_delta_ln_upper_15_def: shows "diff_delta_ln_upper_15 \<equiv> \<lambda>x. (x - 1) ^ 15 / ((1 + 56 * x + 2450 * x ^ 4 + 1960 * x ^ 3 + 588 * x\<^sup>2 + 8 * x ^ 7 + 196 * x ^ 6 + 1176 * x ^ 5)\<^sup>2 * x)"
sturm: shows "quasi_sturm_seq (p # q # r # ps'')"
bind_nonempty: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure" assumes "space M \<noteq> {}" shows "M \<bind> f = join (distr M (subprob_algebra (f (SOME x. x \<in> space M))) f)"
divp_2: fixes K :: "'a set" assumes "K \<in> p" shows "K \<subseteq> S"
simps_2: fixes h :: "'a"   and t :: "'a list" shows "uniq (h # t) = (h \<notin> set t \<and> uniq t)"
ACI_norm_wf: fixes n :: "nat"   and r :: "'b rexp" shows "local.wf n \<guillemotleft>r\<guillemotright> = local.wf n r"
unit_def: fixes x :: "'a" shows "unit x = (\<exists>y. x ** y = one \<and> y ** x = one)"
set_append: fixes xs :: "'e list"   and ys :: "'e list" shows "set (xs @ ys) = set xs \<union> set ys"
select_convs_3: fixes sprFst :: "('b, 'c, 'd) BEState"   and sprLst :: "('b, 'c, 'd) BEState"   and sprCRel :: "(('b, 'c, 'd) BEState \<times> ('b, 'c, 'd) BEState) set"   and more :: "'e" shows "sprCRel \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprCRel"
put\<^sub>O\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t_put_ptrs: fixes object_ptr :: "(_) object_ptr"   and object :: "(_) Object"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "put object_ptr object h = h'" shows "object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|object_ptr|}"
square_free_m_factor_2: fixes f :: "int poly"   and g :: "int poly" assumes "square_free_m (f * g)" shows "square_free_m g"
hd_append: fixes xs :: "'f list"   and ys :: "'f list" shows "hd (xs @ ys) = (if xs = [] then hd ys else hd xs)"
List_distinct: fixes h :: "'b \<Rightarrow> 'b ref"   and x :: "'b ref"   and as :: "'b list" assumes "List h x as" shows "distinct as"