hcju/isabelleps huggingface dataset:
le_imp_diff_is_add: fixes a :: "'d"   and b :: "'d"   and c :: "'d" assumes "a \<le> b"   and "a \<le> b" shows "(b - a = c) = (b = c + a)"
assms_2: shows "is_offending_flows_min_set F G nP"
semiring_normalization_rules_18: fixes lx :: "'a"   and rx :: "'a"   and ry :: "'a" shows "lx * (rx * ry) = lx * rx * ry"
all_def: fixes P :: "'b \<Rightarrow> (nat \<Rightarrow> 'cmp) \<Rightarrow> nat \<Rightarrow> bool" shows "local.all P \<equiv> \<lambda>t n. \<forall>y. P y t n"
res_lim_Zp_cauchy: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and k :: "nat" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s" obtains N :: "nat" where "\<And>m. N < m \<longrightarrow> s m k = res_lim s k"
f1: shows "atom ` domA as \<sharp>* \<Gamma>"
real_of_float: fixes n :: "nat" shows "real_of_float (of_nat n) = real n"
cases: fixes a1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and a2 :: "'t \<times> ('l, 't, 'x, 'm, 'w, 'o) thread_action"   and a3 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "redT a1 a2 a3" obtains t :: "'t" and x :: "'x" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "a1 = s__" and "a2 = (tb__, ta)" and "a3 = s'__" and "tb__ \<turnstile> \<langle>x__, shr s__\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle>" and "thr s__ tb__ = \<lfloor>(x__, no_wait_locks)\<rfloor>" and "actions_ok s__ tb__ ta" and "redT_upd s__ tb__ ta x' m' s'__"   | s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and t :: "'t" and x :: "'x" and n :: "'l" and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" and ln :: "'l \<Rightarrow>f nat" where "a1 = s" and "a2 = (t, K$ [], [], [], [], [], convert_RA ln)" and "a3 = s'" and "thr s t = \<lfloor>(x, ln)\<rfloor>" and "\<not> waiting (wset s t)" and "may_acquire_all (locks s) t ln" and "0 < ln $ n" and "s' = (acquire_all (locks s) t ln, (thr s(t \<mapsto> (x, no_wait_locks)), shr s), wset s, interrupts s)"
comp_in_homE: fixes g :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and f :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and c :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "\<guillemotleft>g \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] c\<guillemotright>" obtains b :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" where "\<guillemotleft>f : a \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] b\<guillemotright>" and "\<guillemotleft>g : b \<rightarrow>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] c\<guillemotright>"
seq_if_composable: fixes \<nu> :: "'c"   and \<mu> :: "'c" assumes "\<nu> \<star>\<^sub>C \<mu> \<noteq> C.null" shows "src\<^sub>C \<nu> = trg\<^sub>C \<mu>"
simps_2: fixes x :: "('a, 'b) Combinators"   and xs :: "('a, 'b) Combinators list"   and l :: "'a set list" shows "all_in_list (x # xs) l = (in_list x l \<and> all_in_list xs l)"
em_def: fixes P' :: "(char list \<Rightarrow> int) \<Rightarrow> enat" shows "em P' = (\<lambda>(ps, n). P' (Partial_Evaluation.emb ps (\<lambda>_. 0)) \<le> enat n)"
simps_1: fixes var :: "nat"   and a :: "real mpoly"   and b :: "real mpoly"   and c :: "real mpoly"   and d :: "real mpoly"   and p :: "real mpoly" shows "quadratic_sub var a b c d (Eq p) = (let p1 = quadratic_part_1 var a b d (Eq p); p2 = quadratic_part_2 var c p1; A = isolate_variable_sparse p2 var 0; B = isolate_variable_sparse p2 var 1 in And (fm.Atom (Leq (A * B))) (fm.Atom (Eq (A\<^sup>2 - B\<^sup>2 * c))))"
edges_undef_simps_1: fixes w :: "'c \<times> 'd \<Rightarrow> enat" shows "edges_undef [] w"
card_eq_SucD: fixes A :: "'d set"   and k :: "nat" assumes "card A = Suc k" shows "\<exists>b B. A = insert b B \<and> b \<notin> B \<and> card B = k \<and> (k = 0 \<longrightarrow> B = {})"
intro: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and sep_size :: "nat" assumes "v0_v1_Digraph G v0 v1"   and "NoSmallSeparationsInduct_axioms G v0 v1 sep_size" shows "NoSmallSeparationsInduct G v0 v1 sep_size"
relcomp_unfold: fixes r :: "('b \<times> 'd) set"   and s :: "('d \<times> 'c) set" shows "r O s = {(x, z). \<exists>y. (x, y) \<in> r \<and> (y, z) \<in> s}"
inv_eq: fixes \<mu> :: "'a arrow_of_spans_data" assumes "local.iso \<mu>" shows "local.inv \<mu> = \<lparr>Chn = C.inv (Chn \<mu>), Dom = Cod \<mu>, Cod = Dom \<mu>\<rparr>"
bounded_iff: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "(a \<le> b \<cdot> c) = (a \<le> b \<and> a \<le> c)"
lines_through_intersect_S_twice_in_K2: fixes p :: "proj2" assumes "\<forall>l. proj2_incident p l \<longrightarrow> (\<exists>q r. q \<noteq> r \<and> q \<in> S \<and> r \<in> S \<and> proj2_incident q l \<and> proj2_incident r l)" shows "p \<in> hyp2"
init_fields_def: shows "init_fields \<equiv> map_of \<circ> map (\<lambda>(F, T). (F, default_val T))"
diff_disj: fixes x :: "'c"   and A :: "'c set" assumes "x \<notin> A" shows "A - {x} = A"
Un_set_offending_flows_bound_minus_subseteq: fixes m :: "'a SecurityInvariant_configured"   and V :: "'a set"   and E :: "('a \<times> 'a) set"   and X :: "('a \<times> 'a) set"   and E' :: "('a \<times> 'a) set" assumes "configured_SecurityInvariant m"   and "wf_graph \<lparr>nodes = V, edges = E\<rparr>"   and "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E\<rparr>) \<subseteq> X" shows "\<Union> (c_offending_flows m \<lparr>nodes = V, edges = E - E'\<rparr>) \<subseteq> X - E'"
IS: shows "\<nu> \<in> fml_sem I ([[$\<alpha> vid1**]]Pc pid1 \<rightarrow> [[$\<alpha> vid1]]Pc pid1)"
borel_measurable_iff_less: fixes f :: "'b \<Rightarrow> real"   and M :: "'b measure" shows "(f \<in> borel_measurable M) = (\<forall>a. {w \<in> space M. f w < a} \<in> sets M)"
lstep_def: fixes L :: "'c \<Rightarrow> ('a \<times> 'a) set"   and R :: "'c \<Rightarrow> ('a \<times> 'a) set"   and E :: "'c \<Rightarrow> ('a \<times> 'a) set"   and x :: "accent \<times> 'c" shows "lstep L R E x = (case x of (Acute, i) \<Rightarrow> (L i)\<inverse> | (Grave, i) \<Rightarrow> R i | (Macron, i) \<Rightarrow> E i)"
coeff_map_poly_hom: fixes p :: "'c poly poly"   and i :: "nat" shows "coeff (map_poly (pCons (0::'c)) p) i = pCons (0::'c) (coeff p i)"
object_ptr_kinds_eq3: shows "object_ptr_kinds h = object_ptr_kinds h'"
over_tagged_division_lemma: fixes p :: "('a \<times> 'a set) set"   and i :: "'a set"   and d :: "'a set \<Rightarrow> 'b" assumes "p tagged_division_of i"   and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = (0::'b)" shows "(\<Sum>(uu_, k)\<in>p. d k) = sum d (snd ` p)"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b multiset" shows "set_mset (image_mset f v) = f ` set_mset v"
Bind_def: fixes g :: "'a \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and ab :: "bool" shows "Bind g f ab \<equiv> \<lambda>P s. let a = g s in f a ab P s"
OclAnd_commute: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and Y) = (Y and X)"
evtEA: fixes n :: "nat"   and c :: "'kid"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'KS"   and \<gamma> :: "(nat \<Rightarrow> 'KS) \<Rightarrow> nat \<Rightarrow> bool" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>"   and "sb.eval c t t' n (\<diamond>\<^sub>b\<gamma>)" shows "\<exists>n'\<ge>\<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>. (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub> \<and> (\<forall>n''\<ge>\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n'\<^esub>. n'' \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n'\<^esub> \<longrightarrow> sb.eval c t t' n'' \<gamma>)) \<or> \<not> (\<exists>i\<ge>n'. \<parallel>c\<parallel>\<^bsub>t i\<^esub>) \<and> sb.eval c t t' n' \<gamma>"
sync_assoc: fixes P :: "'a process"   and C :: "'a set"   and Q :: "'a process"   and S :: "'a process" shows "(P \<lbrakk>C\<rbrakk> Q \<lbrakk>C\<rbrakk> S) = (P \<lbrakk>C\<rbrakk> (Q \<lbrakk>C\<rbrakk> S))"
fps_gcd: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "g \<noteq> 0" shows "gcd f g = fps_X ^ min (subdegree f) (subdegree g)"
fbox_def: fixes F :: "'d \<Rightarrow> 'e set"   and P :: "'e \<Rightarrow> bool" shows "|F] P = (\<lambda>s. \<forall>s'. s' \<in> F s \<longrightarrow> P s')"
rep_eq: fixes x :: "(nat, 'd) slist" shows "degree_slist x = degree_list (Rep_slist x)"
elims: fixes x :: "('c, 'd) tree"   and y :: "('c \<times> 'd) list" assumes "in_traverse_pairs x = y" obtains "x = Tip" and "y = []"   | l :: "('c, 'd) tree" and k :: "'c" and v :: "'d" and r :: "('c, 'd) tree" where "x = Node l k v r" and "y = in_traverse_pairs l @ (k, v) # in_traverse_pairs r"
circle_bot_edge_def: fixes i :: "real \<times> real"   and j :: "real \<times> real"   and d :: "real" assumes "circle i j d" shows "circle.circle_bot_edge d = (1, \<lambda>t. (x_coord t * d, - d * circle_y (x_coord t)))"
bounded_lattice: shows "class.bounded_lattice (\<sqinter>) (\<le>) (<) (\<squnion>) (0::'a) (1::'a)"
notin_fmap_minus: fixes x :: "'g \<times> 'h"   and F :: "'g \<Rightarrow> 'h option" shows "x \<notin> set_fmap (F -- x)"
ord_le_eq_trans: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat"   and c :: "'x \<Rightarrow>\<^sub>0 nat" assumes "lex_pm a b"   and "b = c" shows "lex_pm a c"
downset_set_def: fixes X :: "'a set" shows "\<Down> X = {y. \<exists>x\<in>X. y \<le> x}"
kleene_qfp: fixes bot :: "'b"   and A :: "'b set"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b"   and q :: "'b" assumes "bot \<in> A"   and "\<forall>q\<in>A. less_eq bot q"   and "omega_continuous A less_eq f"   and "extreme_bound A less_eq {f^n bot |. n} q" shows "sympartp less_eq (f q) q"
tab_def: shows "TTfgh.tab \<equiv> \<a>\<^sup>-\<^sup>1[f \<star> g, h, tab\<^sub>0 h \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0] \<cdot> ((f \<star> g) \<star> \<a>[h, tab\<^sub>0 h, fg\<^sub>0h\<^sub>1.p\<^sub>0]) \<cdot> ((f \<star> g) \<star> h.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>0) \<cdot> ((f \<star> g) \<star> fg\<^sub>0h\<^sub>1.\<phi>) \<cdot> \<a>[f \<star> g, tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0, fg\<^sub>0h\<^sub>1.p\<^sub>1] \<cdot> (Tfg.\<rho>\<sigma>.tab \<star> fg\<^sub>0h\<^sub>1.p\<^sub>1)"
i_expand_nth_interval_eq_nth_append_replicate_NoMsg: fixes k :: "nat"   and t :: "nat"   and t1 :: "nat"   and f :: "nat \<Rightarrow> 'a message_af" assumes "0 < k"   and "t * k \<le> t1"   and "t1 \<le> t * k + k - Suc 0" shows "f \<odot> k \<Down> Suc t1 \<up> (t * k) = f t # NoMsg\<^bsup>t1 - t * k\<^esup>"
inf_greatest: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and z :: "'a" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L"   and "z \<in> carrier L"   and "\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq>\<^bsub>L\<^esub> x" shows "z \<sqsubseteq>\<^bsub>L\<^esub> \<Sqinter>\<^bsub>L\<^esub>A"
is_TNil_ttl: fixes xs :: "('a, 'b) tllist" assumes "is_TNil xs" shows "is_TNil (ttl xs)"
infinite_imp_not_singleton: fixes A :: "'a set" assumes "infinite A" shows "\<nexists>a. A = {a}"
of_int_0_less_iff: fixes 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 times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and z :: "int" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less zero (ring_1.of_int one plus zero minus z) = (0 < z)"
Int_insert_right_if1: fixes a :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "a \<in> A" shows "A \<inter> insert a B = insert a (A \<inter> B)"
right_impl_ded: fixes b :: "'b"   and a :: "'b"   and c :: "'b" shows "b * a l\<rightarrow> c = b l\<rightarrow> a l\<rightarrow> c"
simps_2: fixes Q :: "hf set"   and x :: "'a"   and xs :: "'a list" shows "nextl Q (x # xs) = nextl (\<Union>q\<in>epsclo Q. nfa.nxt M q x) xs"
generate_check_def: fixes a :: "nat list"   and b :: "nat list" shows "generate_check a b = concat (map (\<lambda>ys. map (\<lambda>xs. (xs, ys)) (bounded_gen_check.gen_check (C\<^sub>1 b (fst ys)) a)) (c2.gen_check b))"
takeWhile_idem: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" shows "takeWhile P (takeWhile P xs) = takeWhile P xs"
arr_dom_iff_arr: fixes f :: "'a" shows "Left_a.arr (Left_a.dom f) = Left_a.arr f"
cSup_singleton: fixes x :: "'a multiset" shows "Sup {x} = x"
one_uint_def: shows "1 \<equiv> Abs_uint 1"
wf_tuple_Cons: fixes n :: "nat"   and A :: "nat set"   and x :: "'b option"   and xs :: "'b option list" shows "wf_tuple n A (x # xs) = ((if x = None then 0 \<notin> A else 0 \<in> A) \<and> (\<exists>m. n = Suc m \<and> wf_tuple m ((\<lambda>x. x - 1) ` (A - {0})) xs))"
less_eq_st_rep_iff_fun: fixes r1 :: "(char list \<times> 'a) list"   and r2 :: "(char list \<times> 'a) list" shows "local.less_eq_st_rep r1 r2 = (fun_rep r1 \<le> fun_rep r2)"
proper_itE: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" assumes "proper_it it it'" obtains l :: "'b list" where "it = foldli l" and "it' = foldli l"
eq1: shows "Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i)) = Sigma_Algebra.measure lebesgue ((\<lambda>x. \<chi>i. if i = m then x $ m + x $ n else x $ i) ` cbox a b) + Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i) \<inter> {x. (axis m 1 - axis n 1) \<bullet> x \<le> a $ m}) + Sigma_Algebra.measure lebesgue (cbox a (\<chi>i. if i = m then b $ m + b $ n else b $ i) \<inter> {x. b $ m \<le> (axis m 1 - axis n 1) \<bullet> x})"
hseqI: fixes \<mu> :: "'d"   and a :: "'d"   and b :: "'d"   and \<nu> :: "'d"   and c :: "'d" assumes "\<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow>\<^sub>D c\<guillemotright>" shows "D.hseq \<nu> \<mu>"
s2_2: shows "\<forall>x\<in>{0..1} - s2. g2 differentiable at x"
Ball_def: fixes A :: "'c set"   and P :: "'c \<Rightarrow> bool" shows "Ball A P = (\<forall>x. x \<in> A \<longrightarrow> P x)"
K2_centre_non_zero: shows "vector [0, 0, 1] \<noteq> 0"
smult_r_distr: fixes x :: "'a Matrix.vec"   and y :: "'a Matrix.vec"   and a :: "'a" assumes "True"   and "x \<in> carrier_vec n"   and "y \<in> carrier_vec n" shows "a \<cdot>\<^sub>v (x + y) = a \<cdot>\<^sub>v x + a \<cdot>\<^sub>v y"
multpw_cancel1: fixes ns :: "('a \<times> 'a) set"   and y :: "'a"   and x :: "'a"   and X :: "'a multiset"   and Y :: "'a multiset" assumes "trans ns"   and "(y, x) \<in> ns"   and "(add_mset x X, add_mset y Y) \<in> multpw ns" shows "(X, Y) \<in> multpw ns"
INF_lower2: fixes i :: "'a::type"   and A :: "'a::type set"   and f :: "'a::type \<Rightarrow> 'b::complete_lattice"   and u :: "'b::complete_lattice" assumes "(i::'a::type) \<in> (A::'a::type set)"   and "(f::'a::type \<Rightarrow> 'b::complete_lattice) (i::'a::type) \<le> (u::'b::complete_lattice)" shows "\<Sqinter> ((f::'a::type \<Rightarrow> 'b::complete_lattice) ` (A::'a::type set)) \<le> (u::'b::complete_lattice)"
map2_val_rneutr_def: fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and xs :: "'x"   and ys :: "'y" shows "map2_val_rneutr f xs ys = oalist_of_listx (map2_val_raw f id (map_val_raw (\<lambda>k. f k (0::'b))) (list_of_oalistx xs) (list_of_oalisty ys))"
inject_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Neg x2 = Neg y2) = (x2 = y2)"
sublens'_implies_sublens: fixes X :: "'d \<Longrightarrow> 'e"   and Y :: "'f \<Longrightarrow> 'e" assumes "vwb_lens X"   and "vwb_lens Y"   and "X \<subseteq>\<^sub>L' Y" shows "X \<subseteq>\<^sub>L Y"
homom3Tr1: fixes G :: "('e, 'f) Group_scheme"   and H :: "'e set"   and N :: "'e set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "H \<inter> N = gker\<^bsub>(\<natural>\<^bsub>G\<^esub>H),(\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N / N)\<^esub> (Pj (\<natural>\<^bsub>G\<^esub>H \<diamondop>\<^bsub>G\<^esub> N) N \<circ>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub> \<iota>\<^bsub>(\<natural>\<^bsub>G\<^esub>H)\<^esub>)"
c2_4: shows "D2 \<in> carrier_mat n2 nc2"
dis_log_def: fixes \<G> :: "'a cyclic_group" shows "dis_log \<G> \<equiv> 0 < order \<G>"
normalize: fixes A :: "'b set" shows "normalize (Gcd\<^sub>f\<^sub>i\<^sub>n A) = Gcd\<^sub>f\<^sub>i\<^sub>n A"
pac_ideal_mono: fixes A :: "int mpoly set"   and B :: "int mpoly set" assumes "A \<subseteq> B" shows "pac_ideal A \<subseteq> pac_ideal B"
poly_add_normalize_2: fixes R :: "('c, 'd) ring_scheme"   and p1 :: "'c list"   and p2 :: "'c list" assumes "ring R"   and "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "ring.poly_add R p1 p2 = ring.poly_add R p1 (ring.normalize R p2)"
m: shows "\<forall>z. norm z \<le> r \<longrightarrow> norm (poly cs z) \<le> m"
subspace_eq_real1: shows "module_on.subspace (*\<^sub>R) = subspace"
R28_bc_2: shows "pmf (sds R28) c = 0"
lunit_char_3: fixes f :: "'a" assumes "ide f" shows "trg f \<star> \<l>[f] = (\<i>[trg f] \<star> f) \<cdot> local.inv \<a>[trg f, trg f, f]"
tarjan_fin_def: fixes v :: "'v"   and s :: "'v tarjan_state" shows "tarjan_fin v s = (let ll = if stack s = [] then lowlink s else let u = hd (stack s) in lowlink s(u \<mapsto> min (\<zeta> s u) (\<zeta> s v)); s' = s\<lparr>lowlink := ll\<rparr> in ASSERT (v \<in> set (tj_stack s)) \<bind> (\<lambda>_. ASSERT (distinct (tj_stack s)) \<bind> (\<lambda>_. if \<zeta> s v = \<delta> s v then ASSERT (scc_root s v (scc_of E v)) \<bind> (\<lambda>_. tj_stack_pop (tj_stack s) v \<bind> (\<lambda>(tjs, scc). NOOP (s'\<lparr>tj_stack := tjs, sccs := insert scc (sccs s)\<rparr>))) else ASSERT (\<not> scc_root s v (scc_of E v)) \<bind> (\<lambda>_. NOOP s'))))"
ext_mbr_simp: fixes m :: "qtname \<times> memberdecl" shows "(declclass m, mbr m) = m"
case_2: fixes \<Gamma> :: "('k, 'l) web_scheme"   and f1 :: "'m"   and f2 :: "'m"   and f3 :: "'n \<Rightarrow> 'm" assumes "countable_bipartite_web \<Gamma>" shows "(case bi.SINK of bi.SOURCE \<Rightarrow> f1 | bi.SINK \<Rightarrow> f2 | bi.Inner x \<Rightarrow> f3 x) = f2"
poly_eq_0_iff_dvd: fixes p :: "'c poly"   and c :: "'c" shows "(poly p c = (0::'c)) = ([:- c, 1::'c:] dvd p)"
Uvariation_def: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and U :: "variable set" shows "Uvariation \<nu> \<nu>' U \<equiv> \<forall>i. i \<notin> U \<longrightarrow> \<nu> i = \<nu>' i"
cring_simprules_2: fixes R :: "('c, 'd) ring_scheme" assumes "cring R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
fields_def2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "fields P C = FDTs"
simps: fixes a1 :: "'d"   and a2 :: "'d" assumes "three TYPE('d)" shows "three.rel_12_21 a1 a2 = (a1 = three.three\<^sub>1 \<and> a2 = three.three\<^sub>2 \<or> a1 = three.three\<^sub>2 \<and> a2 = three.three\<^sub>1)"
antisym_conv: fixes y :: "'x \<Rightarrow>\<^sub>0 nat"   and x :: "'x \<Rightarrow>\<^sub>0 nat" assumes "y \<preceq> x" shows "(x \<preceq> y) = (x = y)"
step_z_norm: fixes A :: "('d \<times> ('e, real) cconstraint \<times> 'f \<times> 'e list \<times> 'd) set \<times> ('d \<Rightarrow> ('e, real) cconstraint)"   and l :: "'d"   and D :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and v :: "'e \<Rightarrow> nat"   and n :: "nat"   and l' :: "'d"   and D' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry"   and k :: "nat \<Rightarrow> nat" assumes "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>v,n\<^esub> \<langle>l', D'\<rangle>" shows "A \<turnstile> \<langle>l, D\<rangle> \<leadsto>\<^bsub>k,v,n\<^esub> \<langle>l', DBM_Normalization.norm (FW D' n) (\<lambda>x. real (k x)) n\<rangle>"
zero_closed: shows "\<zero>\<^bsub>S\<lparr>zero := h \<zero>\<rparr>\<^esub> \<in> carrier (S\<lparr>zero := h \<zero>\<rparr>)"
xs_x: shows "vfsequence (xs #\<^sub>\<circ> x)"
rep_eq: fixes x :: "'b"   and xa :: "'c \<Rightarrow> 'c"   and xb :: "('b, 'c) mapping" shows "rep (map_entry x xa xb) = (case rep xb x of None \<Rightarrow> rep xb | Some v \<Rightarrow> rep xb(x \<mapsto> xa v))"
unit_factor_mult_normalize: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "times (unit_factor a) (normalize a) = a"
assms_6: fixes j :: "'b" assumes "j \<in> B" shows "\<exists>x\<in>A. f x \<le>\<^sub>o\<^sub>w g j"
reverse_singleton: fixes x :: "'b" shows "reverse\<cdot>[x] = [x]"
Pb: shows "\<forall>u. (u \<^bold>\<in> Pb) = (u \<le> b)"
greedy_path_ltl: fixes p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a"   and \<sigma>' :: "'a \<Rightarrow> 'a"   and v0 :: "'a"   and v :: "'a"   and P :: "'a llist" assumes "greedy_conforming_path p \<sigma> \<sigma>' v0 = LCons v P" shows "P = LNil \<or> P = greedy_conforming_path p \<sigma> \<sigma>' (\<sigma> v0) \<or> P = greedy_conforming_path p \<sigma> \<sigma>' (\<sigma>' v0)"
fi: shows "f contour_integrable_on part_circlepath z r s t"
real_cond_exp_cong: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" assumes "sigma_finite_subalgebra M F"   and "AEeq M f g"   and "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "AEeq M (real_cond_exp M F f) (real_cond_exp M F g)"
simps_2: fixes v :: "'a"   and va :: "'a list" shows "local.coeff (v # va) = (\<lambda>i. if i = degree (v # va) then lead_coeff (v # va) else local.coeff (tl (v # va)) i)"
that: fixes P :: "bool" assumes "a = \<up> P" shows "thesis"