hcju/isabelleps huggingface dataset:
hom_one: shows "to_int_mod_ring 1 = 1"
int_simps_9: fixes P :: "'a \<Rightarrow> bool" shows "\<turnstile> P = #True = P"
abs_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and Abs :: "'c set \<Rightarrow> 'd"   and Rep :: "'d \<Rightarrow> 'c set"   and x :: "'c" assumes "quot_type R Abs Rep" shows "quot_type.abs R Abs x = Abs (Collect (R x))"
range_0_max_UNIV: shows "UNIV = {0..max_ip_addr}"
bounded_f'': shows "bounded ((\<lambda>xa. f'' (h * xa + t, x (h * xa + t))) ` {0..1})"
box_empty_imp: fixes a :: "'c"   and b :: "'c" assumes "box a b = {}" shows "content (cbox a b) = 0"
decoupled_def: fixes F :: "'b program"   and G :: "'b program" shows "decoupled F G \<equiv> \<forall>act\<in>Acts F. \<forall>B. G \<in> stable B \<longrightarrow> G \<in> stable (wp act B)"
coinduct: fixes X :: "'b stream \<Rightarrow> bool"   and x :: "'b stream" assumes "X x"   and "\<And>x. X x \<Longrightarrow> \<exists>a b w. x = a ## b ## w \<and> b \<le> a \<and> (X (b ## w) \<or> sdescending (b ## w))" shows "sdescending x"
hyps_2: shows "p\<^sub>2 = Point x\<^sub>2 y\<^sub>2"
enat_of_option_simps_3: fixes n :: "nat"   and x :: "nat option" shows "(enat n = enat_of_option x) = (x = Some n)"
ft: shows "farkas_coefficients_atoms_tableau (snd ` set (Atoms S)) (Tableau S) C"
d2_3: fixes K :: "'a set" assumes "K \<in> p2" shows "K \<noteq> {}"
length_last: fixes l :: "'a list"   and x :: "'a" shows "(l @ [x]) ! (length (l @ [x]) - 1) = x"
simps_2: fixes p :: "perm"   and \<phi> :: "'b" shows "p \<bullet> S_action.Pred \<phi> = S_action.Pred (p \<bullet> \<phi>)"
itypeI: fixes c :: "'g"   and I :: "interface" shows "c ::\<^sub>i I"
refpoint_eq: fixes c :: "'a list multiset"   and w :: "('b, 'c) label list"   and c' :: "'a list multiset" assumes "valid fg c" shows "((c, w, c') \<in> trcl (refpoint fg)) = ((c, w, c') \<in> trcl (tr fg))"
rbt_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invh l"   and "invc r"   and "invh r" shows "rbt (Set2_Join_RBT.join l x r)"
iMinI_ex: fixes I :: "'a set" assumes "\<exists>x. x \<in> I" shows "iMin I \<in> I"
snull_eq_snil: fixes xs :: "[:'a:]" shows "snull\<cdot>xs = eq\<cdot>xs\<cdot>[::]"
INT_extend_simps_10: fixes B :: "'c \<Rightarrow> 'a set"   and f :: "'b \<Rightarrow> 'c"   and A :: "'b set" shows "(\<Inter>a\<in>A. B (f a)) = \<Inter> (B ` f ` A)"
Gamma_series_minus_of_nat: fixes k :: "nat" shows "\<forall>\<^sub>F n in sequentially. Gamma_series (- of_nat k) n = (0::'a)"
lookup_tensor0: fixes "is" :: "nat list"   and ds :: "nat list" assumes "is \<lhd> ds" shows "lookup (tensor0 ds) is = (0::'m)"
eq_cbox_box: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b" shows "(cbox a b = box c d) = (cbox a b = {} \<and> box c d = {})"
R1: shows "\<forall>x y. R1 x y \<longrightarrow> (\<forall>a b. R1 a b \<longrightarrow> S x a = T y b)"
is_parserE: fixes p :: "char list \<Rightarrow> char list + 'a \<times> char list" assumes "is_parser p" obtains "\<And>s r x. p s = Inr (x, r) \<Longrightarrow> length r \<le> length s"
fmdrop_comm: fixes a :: "'a"   and b :: "'a"   and m :: "('a, 'b) fmap" shows "fmdrop a (fmdrop b m) = fmdrop b (fmdrop a m)"
qo_on_predecessor_subset_conv: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and x :: "'b"   and y :: "'b" assumes "qo_on P A"   and "x \<in> A"   and "y \<in> A" shows "({z \<in> A. P z x} \<subseteq> {z \<in> A. P z y}) = P x y"
assms: shows "to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract"
B: shows "trnl\<^sub>\<epsilon> u \<in> hom v (g \<star> u) \<rightarrow> hom (f \<star> v) u"
list_choose_hd: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "\<forall>x\<in>set xs. P x"   and "x \<in> set xs" shows "P (list.hd xs)"
inter_guards_Call: fixes p :: "'e"   and c2 :: "('d, 'e, 'f) com"   and c :: "('d, 'e, 'f) com" shows "((Call p \<inter>\<^sub>g c2) = Some c) = (c2 = Call p \<and> c = Call p)"
jviewInit: shows "\<forall>a s. clock_jviewInit a (envObs a s) = clock_jview a (tInit s)"
uint_take_bit_eq: fixes n :: "nat"   and w :: "'a word" shows "uint (take_bit n w) = take_bit n (uint w)"
reads_consistent_pending_write_antimono: fixes \<O> :: "nat set"   and m :: "nat \<Rightarrow> nat"   and sb :: "'b memref list" assumes "reads_consistent True \<O> m sb" shows "reads_consistent False \<O> m sb"
llength_ltl: fixes xs :: "'c llist" shows "llength (ltl xs) = epred (llength xs)"
cong_cong_prod_coprime_nat: fixes A :: "'b set"   and x :: "nat"   and y :: "nat"   and m :: "'b \<Rightarrow> nat" assumes "\<forall>i\<in>A. [x = y] (mod m i)"   and "\<forall>i\<in>A. \<forall>j\<in>A. i \<noteq> j \<longrightarrow> coprime (m i) (m j)" shows "[x = y] (mod prod m A)"
hyps: fixes z :: "'a" shows "pochhammer z (n + m) = pochhammer z n * pochhammer (z + of_nat n) m"
inf_set_def: fixes A :: "'k set"   and B :: "'k set" shows "A \<inter> B = Collect ((\<lambda>x. x \<in> A) \<sqinter> (\<lambda>x. x \<in> B))"
nested_submodules: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and W :: "'d set"   and X :: "'d set" assumes "module R M"   and "LinearCombinations.submodule R W M"   and "LinearCombinations.submodule R X M"   and "X \<subseteq> W" shows "LinearCombinations.submodule R X (M\<lparr>carrier := W\<rparr>)"
Abs_bit0'_def: fixes x :: "int" shows "Abs_bit0' x = Abs_bit0 (x mod int CARD('a bit0))"
of_nat_eq_1_iff: fixes n :: "nat" shows "(of_nat n = (1::'c)) = (n = 1)"
simps_2: fixes x :: "'b"   and r :: "'b tree" shows "del_min \<langle>\<langle>\<rangle>, x, r\<rangle> = \<langle>\<rangle>"
preorder_on_map: fixes f :: "'c \<Rightarrow> 'b" shows "Order_Predicates.preorder_on (f -` f -` carrier) (map_relation f (map_relation f le))"
cases: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c \<times> 'e"   and a2 :: "'d \<times> 'f" assumes "rel_prod R1 R2 a1 a2" obtains a :: "'c" and b :: "'d" and c :: "'e" and d :: "'f" where "a1 = (a, c)" and "a2 = (b, d)" and "R1 a b" and "R2 c d"
f1: shows "rk {A', B', P, Q} \<le> 4"
constrains_def: fixes A :: "'d set"   and B :: "'d set" shows "A co B \<equiv> {F. \<forall>act\<in>Acts F. act `` A \<subseteq> B}"
lexordp_induct: fixes xs :: "'a list"   and ys :: "'a list"   and P :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ord_class.lexordp xs ys"   and "\<And>y ys. P \<epsilon> (y # ys)"   and "\<And>x xs y ys. x < y \<Longrightarrow> P (x # xs) (y # ys)"   and "\<And>x xs ys. \<lbrakk>ord_class.lexordp xs ys; P xs ys\<rbrakk> \<Longrightarrow> P (x # xs) (x # ys)" shows "P xs ys"
PR_TAG_def: fixes x :: "'a" shows "PR_TAG x \<equiv> x"
complete_if_balanced_size_2powh: fixes kdt :: "'a kdt"   and h :: "nat" assumes "balanced kdt"   and "KD_Tree.size_kdt kdt = 2 ^ h" shows "KD_Tree.complete kdt"
Sigma_eq: fixes x :: "'a \<times> 'b" assumes "f x \<noteq> (0::'c)" shows "(x \<in> Sigma A B) = (x \<in> Sigma A' B')"
distr_const: fixes M :: "'b measure"   and c :: "'c"   and N :: "'c measure" assumes "prob_space M"   and "c \<in> space N" shows "distr M N (\<lambda>x. c) = return N c"
inject: fixes x2 :: "8 word"   and y2 :: "8 word" shows "(Proto x2 = Proto y2) = (x2 = y2)"
lsum_list_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and xs :: "'c llist" assumes "class.monoid_add plus zero" shows "monoid_add.lsum_list plus zero xs = (if lfinite xs then monoid_add.sum_list plus zero (list_of xs) else zero)"
assms_4: shows "Domainp (rel_gpv'' A C R) gpv"
fresh_distinct_list: fixes S :: "'b set"   and l :: "'b list" assumes "atom ` S \<sharp>* l" shows "S \<inter> set l = {}"
distinct_905: fixes x261 :: "t"   and x262 :: "binop_i"   and x291 :: "t"   and x292 :: "relop_i" shows "Binop_i x261 x262 \<noteq> Relop_i x291 x292"
assms_2: shows "\<parallel>c\<parallel>\<^bsub>t n\<^esub>"
hypsp: shows "r = Fract (fst p) (snd p) \<and> 0 < snd p \<and> coprime (fst p) (snd p)"
conv_abscissa_norm: fixes f :: "'a fds" shows "conv_abscissa (fds_norm f) = abs_conv_abscissa f"
lunit_naturality: fixes \<mu> :: "'a" assumes "Left.arr \<mu>" shows "Left.comp (Left.lunit (Left.cod \<mu>)) (Left.L \<mu>) = Left.comp \<mu> (Left.lunit (Left.dom \<mu>))"
fun_im: fixes f :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "{f a |a. a \<in> A} = {b. b \<in> f ` A}"
Blacken_Roots_def: shows "Blacken_Roots \<equiv> \<lbrace>\<acute>Proper\<rbrace> \<acute>ind := 0;; \<lbrace>\<acute>Proper \<and> \<acute>ind = 0\<rbrace>  WHILE \<acute>ind < length \<acute>M INV \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind \<le> length \<acute>M\<rbrace>  DO \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M\<rbrace>  IF \<acute>ind \<in> Roots THEN \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M \<and> \<acute>ind \<in> Roots\<rbrace> \<acute>M := \<acute>M[\<acute>ind := Black] FI;; \<lbrace>\<acute>Proper \<and> (\<forall>i<\<acute>ind + 1. i \<in> Roots \<longrightarrow> \<acute>M ! i = Black) \<and> \<acute>ind < length \<acute>M\<rbrace> \<acute>ind := \<acute>ind + 1 OD"
\<phi>_simps_5: shows "cod t\<^sub>0u\<^sub>1.\<phi> = u\<^sub>1 \<star> t\<^sub>0u\<^sub>1.p\<^sub>0"
R15_4: shows "pmf (sds R15) d = 1 / 2"
hyps_3: assumes "x \<sharp> P" shows "x \<sharp> P'"
D_eq_1: fixes r :: "('a \<times> 'a) set"   and \<tau> :: "'a list"   and \<sigma> :: "'a list"   and \<sigma>' :: "'a list"   and \<tau>' :: "'a list" assumes "trans r"   and "irrefl r"   and "Decreasing_Diagrams.D r \<tau> \<sigma> \<sigma>' \<tau>'" shows "(r|\<tau>'| -s r \<down>l \<sigma>, r|\<tau>|) \<in> mul_eq r"
append_child_def: shows "Shadow_DOM.append_child \<equiv> Shadow_DOM.i_append_child.a_append_child"
eval_poly_add: fixes p :: "'a poly"   and q :: "'a poly"   and x :: "'b" shows "eval_poly hom (p + q) x = eval_poly hom p x + eval_poly hom q x"
simps_1: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and a :: "'c"   and x :: "'b"   and xs :: "'b list" shows "applied_rule C a (x # xs) = (if a \<in> dom (C x) then \<lfloor>x\<rfloor> else applied_rule C a xs)"
Un_Int_eq_4: fixes T :: "'d set"   and S :: "'d set" shows "T \<inter> (S \<union> T) = T"
of_fun_inverse: fixes y :: "'b \<Rightarrow> 'b \<Rightarrow> 'c" assumes "y \<in> UNIV" shows "to_fun (of_fun y) = y"
polyneg0: fixes p :: "'b poly"   and n :: "nat" assumes "isnpolyh p n" shows "(~\<^sub>p p = 0\<^sub>p) = (p = 0\<^sub>p)"
LIST_SET_REV_TAG_def: shows "LIST_SET_REV_TAG \<equiv> \<lambda>x. x"
II1: shows "II1 \<in> local.shift cl n1 ` P n1"
iteration_star2: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<infinity> = x\<^sup>\<infinity>"
intro: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat" assumes "wf_dfa A n" shows "rquot_DFS A n"
mat_det_transpose: fixes M :: "complex \<times> complex \<times> complex \<times> complex" shows "mat_det (mat_transpose M) = mat_det M"
assert_Sup: fixes A :: "'c Assertion set" shows "{\<cdot> \<Squnion>A } = \<Squnion>(assert ` A)"
inorder_isempty: fixes t :: "'t" assumes "invar t" shows "is_empty t = (inorder t = [])"
simps_2: fixes f :: "'f \<Rightarrow> 'e option"   and x :: "'f"   and xs :: "'f list" shows "mapM f (x # xs) = f x \<bind> (\<lambda>y. mapM f xs \<bind> (\<lambda>ys. Some (y # ys)))"
find_indices_def: fixes x :: "'b"   and xs :: "'b list" shows "find_indices x xs \<equiv> filter (\<lambda>i. xs ! i = x) [0..<length xs]"
mismatchNilLeft: fixes Rel :: "(pi \<times> pi) set"   and a :: "name"   and P :: "pi" shows "\<zero> \<leadsto>[Rel] [a\<noteq>a]P"
conc_conc: fixes u :: "'a list"   and v :: "'a list"   and w :: "nat \<Rightarrow> 'a" shows "u \<frown> v \<frown> w = (u @ v) \<frown> w"
is_REL_state_simps_1: fixes t :: "nat"   and ts :: "nat \<Rightarrow> thread"   and tk :: "nat" assumes "t < N" shows "is_REL_state (ts(t := REL tk))"
star_left_unfold_equal: fixes x :: "'a" shows "top \<squnion> x \<sqinter> top = top"
fimage_fempty: fixes f :: "'e \<Rightarrow> 'd" shows "f |`| {||} = {||}"
safety_def: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" shows "safety P = (\<forall>\<sigma>. \<not> P \<sigma> \<longrightarrow> (\<exists>i. \<forall>\<beta>. \<not> P (stake i \<sigma> @- \<beta>)))"
CEMC_def: shows "CEMC \<equiv> CEM \<and> CMC \<and> CEMC_axioms"
\<pi>k: shows "(\<pi> @\<^bsup>n\<^esup> \<pi>') k = \<pi> k"
p_eq2: fixes p :: "nat"   and q :: "nat" assumes "QR p q" shows "int p = 2 * ((int p - 1) div 2) + 1"
less_eq_zmultiset_def: fixes M' :: "'a zmultiset"   and M :: "'a zmultiset" shows "(M' \<le> M) = (M' < M \<or> M' = M)"
evaluate_determ_3: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and r3a :: "'b state \<times> (v, v) result"   and r3b :: "'b state \<times> (v, v) result" assumes "BigStep.evaluate ck env s e r3a"   and "BigStep.evaluate ck env s e r3b" shows "r3a = r3b"
wf_cfg_def: fixes G :: "'a graph_rec" shows "wf_cfg G \<equiv> g_V0 G \<subseteq> set (g_V G)"
intros_2: fixes x :: "nat"   and z :: "nat"   and rel :: "(nat \<times> nat) set"   and n :: "nat"   and y :: "nat" assumes "(x, z) \<in> Clos rel n"   and "(z, y) \<in> Clos rel n"   and "z < n" shows "(x, y) \<in> Clos rel n"
dom_def: fixes f :: "'a1 \<times> 'a2" shows "local.dom f = (if domains f \<noteq> {} then SOME a. a \<in> domains f else null)"
preal_add_def: fixes r :: "preal"   and s :: "preal" shows "r + s \<equiv> Abs_preal (add_set (Rep_preal r) (Rep_preal s))"
raa_cor_2: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<not>\<phi> in v]"   and "[\<^bold>\<not>\<psi> in v] \<Longrightarrow> [\<phi> in v]"   and "[\<^bold>\<not>\<phi> in v]" shows "[\<psi> in v]"
star_inductl_var: fixes x :: "'a"   and y :: "'a" assumes "x \<cdot> y \<le> y" shows "qstar x \<cdot> y \<le> y"
sym_arcs: shows "symmetric (with_proj (list_digraph G_list))"
gp_def: shows "gp = (\<lambda>x. ennreal (g x))"
all_countable_subset_image: fixes f :: "'d \<Rightarrow> 'c"   and S :: "'d set"   and P :: "'c set \<Rightarrow> bool" shows "(\<forall>T. countable T \<and> T \<subseteq> f ` S \<longrightarrow> P T) = (\<forall>T. countable T \<and> T \<subseteq> S \<longrightarrow> P (f ` T))"
short_exact_sequenceD: fixes A :: "'b monoid"   and B :: "'b monoid"   and C :: "'b monoid"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "short_exact_sequence A B C f g" shows "exact_seq ([A, B, C], [f, g]) \<and> f \<in> epi B A \<and> g \<in> mon C B"
listrel1_eq_len: fixes xs :: "'g list"   and ys :: "'g list"   and r :: "('g \<times> 'g) set" assumes "(xs, ys) \<in> listrel1 r" shows "length xs = length ys"
p1: fixes y :: "xml" shows "monotone sum_bot.le_fun sum_bot_ord (p1 y)"
a1: shows "mmu_readable (get_acc_flag (snd p)) 8"
mem_hmsetmset_encodeD: fixes base :: "nat"   and M :: "hmultiset"   and e :: "nat"   and n :: "nat" assumes "2 \<le> base"   and "M \<in># hmsetmset (encode base e n)" shows "\<exists>e'\<ge>e. M = encode base 0 e'"
Glb_lower: fixes A :: "'a set"   and a :: "'a" assumes "A \<subseteq> L"   and "a \<in> A" shows "Glb A \<le> a"
Image_R': fixes n :: "nat"   and X :: "nat set" shows "R' n `` X = R `` (X \<inter> {..n}) \<inter> {..n} \<union> (if n + 1 \<in> X then R `` {n + 1..} \<inter> {..n} else {}) \<union> (if R `` (X \<inter> {..n}) \<inter> {n + 1..} = {} then {} else {n + 1}) \<union> (if n + 1 \<in> X \<and> R `` {n + 1..} \<inter> {n + 1..} \<noteq> {} then {n + 1} else {})"
winding_number_eq: fixes \<gamma> :: "real \<Rightarrow> complex"   and w :: "complex"   and S :: "complex set"   and z :: "complex" assumes "path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "w \<in> S"   and "z \<in> S"   and "connected S"   and "S \<inter> path_image \<gamma> = {}" shows "winding_number \<gamma> w = winding_number \<gamma> z"
adjoint_preds_skip: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and p :: "char"   and I :: "interp"   and \<omega> :: "variable \<Rightarrow> real" assumes "SPreds \<sigma> p = undeff" shows "Preds (USubst.adjoint \<sigma> I \<omega>) p = Preds I p"
assms_3: assumes "x \<in>\<^sub>\<circ> B" shows "P"
finite_list: fixes A :: "'g set" assumes "finite A" shows "\<exists>xs. set xs = A"
invertible_mat_four_block_mat_lower_right_id: fixes A :: "'b mat"   and m :: "nat"   and B :: "'b mat"   and n :: "nat"   and C :: "'b mat"   and D :: "'b mat" assumes "A \<in> carrier_mat m m"   and "B = 0\<^sub>m m (n - m)"   and "C = 0\<^sub>m (n - m) m"   and "D = 1\<^sub>m (n - m)"   and "m < n"   and "invertible_mat A" shows "invertible_mat (four_block_mat A B C D)"
subspace_is_vs: fixes W :: "'a vec set" assumes "subspace class_ring W V" shows "vectorspace class_ring (vs W)"
relation_of_Par: fixes A1 :: "('c, 'd) action"   and ns1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and cs :: "'c set"   and ns2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and A2 :: "('c, 'd) action" shows "action.relation_of A1 \<lbrakk> ns1 | cs | ns2 \<rbrakk> A2 = R (\<lambda>(S, S'). (\<nexists>tr1 tr2. (action.relation_of A1\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr1 = tr S)) (S, S') \<and> (spec False (wait S) (action.relation_of A2) ;; (\<lambda>(S, S'). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<and> (\<nexists>tr1 tr2. (spec False (wait S) (action.relation_of A1) ;; (\<lambda>(S, uu_). tr1 = tr S)) (S, S') \<and> (action.relation_of A2\<^sup>f\<^sub>f ;; (\<lambda>(S, S'). tr2 = tr S)) (S, S') \<and> tr_filter tr1 cs = tr_filter tr2 cs) \<turnstile> \<lambda>(S, S'). \<exists>s1 s2. ((\<lambda>(A, A'). action.relation_of A1\<^sup>t\<^sub>f (A, s1) \<and> action.relation_of A2\<^sup>t\<^sub>f (A, s2)) ;; M_par s1 ns1 s2 ns2 cs) (S, S'))"
final_thread_def: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b" shows "final_thread s t \<equiv> case thr s t of None \<Rightarrow> False | \<lfloor>(x, ln)\<rfloor> \<Rightarrow> final x \<and> ln = no_wait_locks \<and> wset s t = None"
upper_closed: fixes y :: "'b" assumes "y \<in> carrier \<Y>" shows "\<pi>\<^sub>* y \<in> carrier \<X>"
linorder_axioms_def: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool" shows "class.linorder_axioms less_eq \<equiv> \<forall>x y. less_eq x y \<or> less_eq y x"
chain_reparam_chain'_imp_finite_subdiv: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" assumes "finite one_chain1"   and "chain_reparam_chain' one_chain1 subdiv" shows "finite subdiv"
at_def: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> nat \<times> nat \<times> instr list \<times> (nat \<times> nat \<times> char list \<times> nat \<times> nat) list) list) list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and i :: "instr" shows "(P,C,M,pc \<triangleright> i) = (\<exists>is. drop pc (instrs_of P C M) = i # is)"
b_def_equiv: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "d_aux a b c \<le> e_aux a b c" shows "b_aux a b c = (b \<squnion> d_aux a b c) \<sqinter> e_aux a b c"
assert_disch3: fixes P :: "'d \<Rightarrow> bool"   and \<sigma> :: "'d" assumes "\<not> P \<sigma>" shows "\<not> (\<sigma> \<Turnstile> assert\<^sub>S\<^sub>E P)"
rep_eq: fixes x :: "'a st"   and xa :: "'a st" shows "(x \<le> xa) = less_eq_st_rep (rep_st x) (rep_st xa)"
Seq_MtransC: fixes c1 :: "('test, 'atom) com"   and s :: "'state"   and c1' :: "('test, 'atom) com"   and s' :: "'state"   and c2 :: "('test, 'atom) com" assumes "(c1, s) \<rightarrow>*c (c1', s')" shows "(c1 ;; c2, s) \<rightarrow>*c (c1' ;; c2, s')"
e2: shows "consistent_sign_vectors_R qs UNIV \<subseteq> set (remdups (map (signs_at qs) [0]))"
orbits_coverture: shows "\<Union> (orbits G E \<phi>) = E"
must_wait_mono: fixes A :: "'e set"   and B :: "'e set"   and s :: "('f, 'e) locks \<times> (('e \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'g) \<times> ('e \<Rightarrow> 'h wait_set_status option) \<times> 'e set"   and t :: "'e"   and lt :: "'f + 'e + 'e" assumes "\<And>x. x \<in> A \<longrightarrow> x \<in> B" shows "must_wait s t lt A \<longrightarrow> must_wait s t lt B"
tl_append2: fixes xs :: "'g list"   and ys :: "'g list" assumes "xs \<noteq> []" shows "tl (xs @ ys) = tl xs @ ys"
CompDefComp_1: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and g :: "'e"   and h :: "'e" assumes "Category C"   and "f \<approx>>\<^bsub>C\<^esub> g"   and "g \<approx>>\<^bsub>C\<^esub> h" shows "f \<approx>>\<^bsub>C\<^esub> g ;;\<^bsub>C\<^esub> h"
invertible_def: fixes u :: "'a" assumes "u \<in> G" shows "invertible u = (\<exists>v\<in>G. u \<cdot> v = \<one> \<and> v \<cdot> u = \<one>)"
subdivision_bidir: fixes G :: "('b, 'c) pre_digraph"   and rev_G :: "'c \<Rightarrow> 'c"   and H :: "('b, 'c) pre_digraph"   and rev_H :: "'c \<Rightarrow> 'c" assumes "subdivision (G, rev_G) (H, rev_H)" shows "bidirected_digraph H rev_H"
sturm_tarski_above: fixes p :: "real poly"   and a :: "real"   and q :: "real poly" assumes "poly p a \<noteq> 0" shows "taq {x. poly p x = 0 \<and> a < x} q = changes_gt_smods a p (pderiv p * q)"
order_trans_rules_23: fixes x :: "'b"   and y :: "'b"   and z :: "'b" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
dom_add_closed: fixes x :: "'a"   and y :: "'a" shows "ad (ad (ad (ad x) + ad (ad y))) = ad (ad x) + ad (ad y)"
gen_wf_uses'_phis'_def: shows "gen_wf_uses'_phis' \<equiv> CFG_Construct_linorder.uses'_phis' gen_wf_\<alpha>n gen_wf_inEdges' gen_wf_defs gen_wf_uses"
eval_abrupt: fixes G :: "prog"   and xc :: "abrupt"   and s :: "st"   and t :: "Term.term"   and w :: "vals"   and s' :: "state" shows "G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<rightarrow> (w, s') = (s' = (Some xc, s) \<and> w = undefined3 t \<and> G\<turnstile>(Some xc, s) \<midarrow>t\<succ>\<rightarrow> (undefined3 t, (Some xc, s)))"
UN_simps_8: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set set" shows "\<Union> (B ` \<Union> A) = (\<Union>y\<in>A. \<Union> (B ` y))"
us: shows "us \<equiv> getFrN (map snd txs) (map fst txs) [sdsj F] (length txs)"
tail_events_def: fixes A :: "'c \<Rightarrow> ('b \<Rightarrow> 'a) set set" shows "P.P.tail_events A = (\<Inter>n. sigma_sets (space (Pi\<^sub>M UNIV (\<lambda>_. M))) (\<Union> (A ` {n..})))"
exhaust: fixes y :: "'a literal" obtains x1 :: "'a" where "y = x1\<^sup>+"   | x2 :: "'a" where "y = x2\<inverse>"
unord_resolve_sound: fixes C :: "'c literal multiset"   and D :: "'c literal multiset"   and E :: "'c literal multiset"   and I :: "'c set" assumes "unord_resolve C D E"   and "I \<Turnstile> C"   and "I \<Turnstile> D" shows "I \<Turnstile> E"
not_Some_eq: fixes x :: "'b option" shows "(\<forall>y. x \<noteq> \<lfloor>y\<rfloor>) = (x = None)"
induct: fixes P :: "acc_modi \<Rightarrow> bool"   and acc_modi :: "acc_modi" assumes "P Private"   and "P Package"   and "P Protected"   and "P Public" shows "P acc_modi"
simps_3: fixes uw :: "nonce_t" shows "catomic (cNonce uw) = True"
uniform_limit_cong': fixes X :: "'c set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and h :: "'c \<Rightarrow> 'd"   and i :: "'c \<Rightarrow> 'd"   and F :: "'b filter" assumes "\<And>y x. x \<in> X \<Longrightarrow> f y x = g y x"   and "\<And>x. x \<in> X \<Longrightarrow> h x = i x" shows "uniform_limit X f h F = uniform_limit X g i F"
simps_2: fixes m1 :: "nat"   and m2 :: "nat" shows "FOV (FLess m1 m2) = {m1, m2}"
tagged_packet_untag_def: fixes p :: "('a, 'b) tagged_packet_scheme" shows "tagged_packet_untag p \<equiv> \<lparr>p_iiface = p_iiface p, p_oiface = p_oiface p, p_src = p_src p, p_dst = p_dst p, p_proto = p_proto p, p_sport = p_sport p, p_dport = p_dport p, p_tcp_flags = p_tcp_flags p, p_payload = p_payload p, \<dots> = tagged_packet.more p\<rparr>"