hcju/isabelleps huggingface dataset:
sylvester_mat_sub_dim_1: fixes m :: "nat"   and n :: "nat"   and p :: "'c poly"   and q :: "'c poly" shows "dim_row (sylvester_mat_sub m n p q) = m + n"
card_payer_and_inversion: fixes xs :: "bool list"   and i :: "nat" assumes "xs \<in> inversion ` dc_crypto"   and "i < n" shows "card {dc \<in> dc_crypto. payer dc = Some i \<and> inversion dc = xs} = 2"
iUntil_disj_iUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1 \<or> Q t1. t1 \<U> t2 I. Q t2) = (P t1. t1 \<U> t2 I. Q t2)"
relop_f: fixes t :: "t"   and \<C> :: "t_context"   and uz :: "relop_f" assumes "is_float_t t" shows "\<C> \<turnstile> [Relop_f t uz] : [t, t] _> [T_i32]"
crename_AndL1: fixes R :: "trm"   and a :: "coname"   and b :: "coname"   and x :: "name"   and N :: "trm"   and y :: "name" assumes "R[a\<turnstile>c>b] = AndL1 x.N y"   and "x \<sharp> R" shows "\<exists>N'. R = AndL1 x.N' y \<and> N'[a\<turnstile>c>b] = N"
strip_eq_SKIP: fixes C :: "'a acom" shows "(strip C = SKIP) = (\<exists>P. C = SKIP {P})"
tuple_ext: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr0 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and pr1 :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b"   and g :: "'b" assumes "elementary_category_with_binary_products C pr0 pr1"   and "\<not> (partial_magma.arr C f \<and> partial_magma.arr C g \<and> partial_magma.dom C f = partial_magma.dom C g)" shows "elementary_category_with_binary_products.tuple C pr0 pr1 f g = partial_magma.null C"
all_unshared_takeWhile_subset: fixes P :: "'b memref \<Rightarrow> bool"   and sb :: "'b memref list" shows "all_unshared (takeWhile P sb) \<subseteq> all_unshared sb"
Sup2: fixes A :: "'c Assertion set"   and z :: "'c Assertion" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<le> z" shows "\<Squnion>A \<le> z"
lower_asymptotic_density_in_01_3: fixes A :: "nat set" shows "0 \<le> lower_asymptotic_density A"
assms_1: fixes e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau> :: "ty"   and \<pi>A :: "Syntax.term list"   and i :: "nat"   and \<pi>' :: "Syntax.term list"   and eV' :: "Syntax.term" assumes "{$$} \<turnstile> e, eP, eV : \<tau>"   and "\<lless>\<pi>A, eV\<ggreater> V\<rightarrow>i \<lless>\<pi>', eV'\<ggreater>" shows "\<exists>e' eP' \<pi> j \<pi>0 s s'. \<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater> \<and> \<lless>[], eP\<ggreater> P\<rightarrow>i \<lless>\<pi>, eP'\<ggreater> \<and> \<pi>A = \<pi> @ \<pi>' \<and> {$$} \<turnstile> e', eP', eV' : \<tau> \<or> j \<le> i \<and> \<lless>[], eP\<ggreater> P\<rightarrow>j \<lless>\<pi>0 @ [s], eP'\<ggreater> \<and> \<pi>A = \<pi>0 @ [s'] @ \<pi>' \<and> s \<noteq> s' \<and> hash s = hash s'"
endpoint_inV_def: fixes V :: "'b set"   and e :: "'b \<times> 'b" shows "endpoint_inV V e = (if fst e \<in> V then fst e else snd e)"
af_letter_subformulas\<^sub>\<mu>: fixes \<phi> :: "'b ltln"   and \<xi> :: "'b set" shows "subformulas\<^sub>\<mu> (af_letter \<phi> \<xi>) = subformulas\<^sub>\<mu> \<phi>"
finite_union_of_idempot: fixes P :: "'b set \<Rightarrow> bool" shows "finite union_of finite union_of P = finite union_of P"
induct: fixes P :: "'b rule list \<Rightarrow> bool"   and a0 :: "'b rule list" assumes "P []"   and "\<And>m a rs. P rs \<Longrightarrow> P (Rule m a # rs)" shows "P a0"
bindU_zeroU: fixes fmapU :: "(udom \<rightarrow> udom) \<rightarrow> udom\<cdot>'a \<rightarrow> udom\<cdot>'a"   and returnU :: "udom \<rightarrow> udom\<cdot>'a"   and bindU :: "udom\<cdot>'a \<rightarrow> (udom \<rightarrow> udom\<cdot>'a) \<rightarrow> udom\<cdot>'a"   and zeroU :: "udom\<cdot>'a"   and f :: "udom \<rightarrow> udom\<cdot>'a" assumes "class.monad_zero fmapU returnU bindU zeroU" shows "bindU\<cdot>zeroU\<cdot>f = zeroU"
c_int_def: shows "c = real_of_int (\<Prod>ia = 0..<m - 1. int m - 1 - int ia)"
that: fixes N :: "'a set" assumes "negligible N"   and "\<And>x e. \<lbrakk>x \<notin> N; 0 < e\<rbrakk> \<Longrightarrow> \<exists>d>0. \<forall>h. 0 < h \<and> h < d \<longrightarrow> norm (integral (cbox x (x + h *\<^sub>R One)) f /\<^sub>R h ^ DIM('a) - f x) < e" shows "thesis"
lexordp_eq_linear: fixes xs :: "'a list"   and ys :: "'a list" shows "lexordp_eq xs ys \<or> lexordp_eq ys xs"
power_less_zero_eq: 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 a :: "'a"   and n :: "nat" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less (power.power one times a n) zero = (odd n \<and> less a zero)"
leq_trans: fixes x :: "'a spmf"   and y :: "'a spmf"   and z :: "'a spmf" assumes "ord_spmf (=) x y"   and "ord_spmf (=) y z" shows "ord_spmf (=) x z"
exprel_def: shows "exprel \<equiv> {(xa, x). exprelp xa x}"
pcompose_uminus: fixes p :: "'b poly"   and r :: "'b poly" shows "- p \<circ>\<^sub>p r = - (p \<circ>\<^sub>p r)"
lm035: fixes a :: "('d \<times> 'c set) set"   and N :: "'d set"   and G :: "'c set" assumes "a\<inverse> \<in> injections (Range a) N"   and "Range a \<in> all_partitions G" shows "a \<in> allAllocations N G"
assms: fixes x :: "'a" assumes "x \<in> A" shows "y alg_dvd f x"
absolutely_integrable_spike: fixes T :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on T"   and "negligible S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x" shows "g absolutely_integrable_on T"
MakeCatMor: fixes C :: "('b, 'a, 'c) Category_scheme" shows "mor\<^bsub>MakeCat C\<^esub> = mor\<^bsub>C\<^esub>"
singleton: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and x :: "'b" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z {#x#} = x"
unit_factor_pCons: fixes a :: "'b"   and p :: "'b poly" shows "unit_factor (pCons a p) = (if p = 0 then [:unit_factor a:] else unit_factor p)"
fBex_fempty: fixes P :: "'a \<Rightarrow> bool" shows "fBex {||} P = False"
negligible_convex_interior: fixes S :: "'c set" assumes "convex S" shows "negligible S = (interior S = {})"
not_final_thread_deadlocked_final_thread: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and s :: "('d, 'b) locks \<times> (('b \<Rightarrow> ('a \<times> 'd \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'e wait_set_status option) \<times> 'b set"   and t :: "'b"   and xln :: "'a \<times> 'd \<Rightarrow>f nat" assumes "multithreaded final r"   and "thr s t = \<lfloor>xln\<rfloor>" shows "final_thread.not_final_thread final s t \<or> t \<in> multithreaded_base.deadlocked final r s \<or> final_thread.final_thread final s t"
D: shows "i \<in> offs_set_prev ns xs index key mi ma j"
case_ka: shows "\<forall>u<d * ka. \<forall>v<d * ka. u * v \<noteq> d * ka"
cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes ptr :: "(_) shadow_root_ptr" shows "cast ptr = document_ptr.Ext (Inl ptr)"
sigma_finite_countable: shows "\<exists>A. countable A \<and> A \<subseteq> sets (distr M N f) \<and> \<Union> A = space (distr M N f) \<and> (\<forall>a\<in>A. emeasure (distr M N f) a \<noteq> \<infinity>)"
reflexivity_rule: fixes G :: "('b, 'c) labeled_graph"   and l :: "'b" assumes "graph G"   and "maintained (reflexivity_rule l) G" shows "refl_on (vertices G) (getRel l G)"
idem: fixes a :: "'a" shows "a \<squnion> a = a"
shift_def: fixes r :: "real_int"   and x :: "real" shows " shift r x = Abs_real_int (left r + x, right r + x)"
fmran'_alt_def: fixes m :: "('d, 'c) fmap" shows "fmran' m = fset (fmran m)"
matchT_TMT_def2: fixes d :: "('test, 'atom) com"   and c :: "('test, 'atom) com" shows "matchT_TMT d c = (\<forall>s t t'. mustT c s \<and> mustT d t \<and> s \<approx> t \<and> (d, t) \<rightarrow>t t' \<longrightarrow> (\<exists>s'. (c, s) \<rightarrow>*t s' \<and> s' \<approx> t'))"
assms_1: shows "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'"
path_vector_deriv_line_integrals: shows "\<forall>x\<in>{0..1} - s. ((\<lambda>x. \<gamma> x \<bullet> i) has_vector_derivative vector_derivative (\<lambda>x. \<gamma> x \<bullet> i) (at x)) (at x)"
assms_1: shows "m1 \<in> smaps_over A B"
Group_FGModuleHomSet: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set" assumes "FGModule G smult V"   and "FGModule G smult' W" shows "Group (FGModuleHomSet G smult V smult' W)"
inv_equality: fixes y :: "cltn2"   and x :: "cltn2" assumes "y \<otimes>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = \<one>\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub>"   and "x \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>"   and "y \<in> carrier \<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>" shows "inv\<^bsub>\<lparr>carrier = UNIV, monoid.mult = cltn2_compose, one = cltn2_id\<rparr>\<^esub> x = y"
le1_1: shows "sum (wt (ParT cl) s) (BrnFT cl dl) < 1"
less_eq_finite_1_def: fixes x :: "finite_1"   and y :: "finite_1" shows "(x \<le> y) = True"
intra_path_get_procs: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'" shows "get_proc n = get_proc n'"
sublinear_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "sublinear t" shows "sublinear (wp (Embed t))"
linear_simps_5: fixes f :: "'c \<Rightarrow> 'd"   and s :: "real"   and v :: "'c" assumes "bounded_linear f" shows "f (s *\<^sub>R v) = s *\<^sub>R f v"
prob0_def: fixes n :: "nat"   and v :: "complex Matrix.mat"   and i :: "nat" shows "prob0 n v i \<equiv> \<Sum>j\<in>{k |k. k < 2 ^ n \<and> \<not> select_index n i k}. (cmod (v $$ (j, 0)))\<^sup>2"
treewidth_upper_bound_0: fixes G :: "('c, 'd) Graph_scheme" assumes "Graph G"   and "V\<^bsub>G\<^esub> = {}" shows "Graph.treewidth G = 0"
respects_hashes: fixes A :: "'a set"   and h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a" assumes "merge_on A h bo m"   and "a \<in> A" shows "(h a = h b) = (\<exists>ab. m a b = Some ab)"
subst_atm_list_tl: fixes As :: "'a list"   and \<sigma> :: "'s" shows "tl (As \<cdot>al \<sigma>) = tl As \<cdot>al \<sigma>"
assms_7: shows "\<guillemotleft>\<nu> : h \<Rightarrow> k\<guillemotright>"
B_trusts_YM4: fixes B :: "agent"   and A :: "agent"   and K :: "nat"   and NB :: "nat"   and X :: "msg"   and evs :: "event list" assumes "Gets B \<lbrace>Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, Nonce NB\<rbrace>, X\<rbrace> \<in> set evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> yahalom" shows "\<exists>NA. Says Server A \<lbrace>Nonce NB, Crypt (shrK A) \<lbrace>Agent B, Key K, Nonce NA\<rbrace>, Crypt (shrK B) \<lbrace>Agent A, Agent B, Key K, Nonce NB\<rbrace>\<rbrace> \<in> set evs"
map_dual_def_var2: fixes f :: "'b \<Rightarrow> 'c" shows "\<partial>\<^sup>- \<circ> map_dual f = f \<circ> \<partial>\<^sup>-"
adv: assumes "exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>g absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr)) \<surd>" shows "advantage (absorb \<A> (obsf_converter (w_adv_real |\<^sub>= w_usr))) (obsf_resource (sim |\<^sub>= 1\<^sub>C \<rhd> ideal_resource)) (obsf_resource real_resource) \<le> adv"
card_UNIV_finite_1_def: shows "card_UNIV_class.card_UNIV = Phantom(Enum.finite_1) 1"
cases: fixes x :: "tm \<times> tm \<times> tm" obtains i :: "name" and x :: "tm" and y :: "tm" and z :: "tm" where "atom i \<sharp> (x, y, z)" and "xa__ = (x, y, z)"
Red_I_without_red_F: fixes N :: "'f set" shows "Red_I_\<G> (N - Red_F_\<G> N) = Red_I_\<G> N"
lunit_in_hom: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and \<alpha> :: "'b \<times> 'b \<times> 'b \<Rightarrow> 'b"   and \<iota> :: "'b"   and a :: "'b" assumes "monoidal_category C T \<alpha> \<iota>"   and "partial_magma.ide C a" shows "partial_magma.in_hom C (monoidal_category.lunit C T \<alpha> \<iota> a) (T (monoidal_category.unity C \<iota>, a)) a"
rel_compp_Grp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_pmf R = (BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. set_pmf x \<subseteq> {(x, y). R x y}} (map_pmf snd)"
residual_rep_abs: fixes r :: "'a \<times> 'b" shows "alpha_residual (rep_residual (abs_residual r)) r"
a_ge_ak: shows "a ! k * xs ! k \<le> a \<bullet> xs"
add_neg_numeral_special_7: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and one :: "'d" assumes "class.neg_numeral minus plus zero uminus" shows "plus one (uminus one) = zero"
lsetp_into_lset: fixes xs :: "'c llist"   and x :: "'c" assumes "Coinductive_List.lsetp xs x" shows "x \<in> lset xs"
timpl_closure'_def: fixes TI :: "('g set \<times> 'g set) set" shows "timpl_closure' TI \<equiv> (timpl_closure'_step TI)\<^sup>*"
hom_mset_factors: fixes F :: "'a poly poly multiset"   and p :: "'a poly poly" assumes "mset_factors F p" shows "mset_factors (image_mset poly_y_x F) (poly_y_x p)"
sup2I1: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b"   and y :: "'c"   and B :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "A x y" shows "sup A B x y"
psorder_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "class.psorder less_eq less \<equiv> pseudo_ordering UNIV less_eq less"
split_domI': fixes args :: "('x \<Rightarrow>\<^sub>0 nat) \<times> 'x set \<times> ('x \<Rightarrow>\<^sub>0 nat) set" assumes "finite X"   and "fst (snd args) \<subseteq> X"   and "finite (snd (snd args))" shows "split_dom TYPE('b) args"
prj_char_2: shows "Maps.PRJ\<^sub>1 \<lbrakk>\<lbrakk>tab\<^sub>0 (f \<star> g)\<rbrakk>\<rbrakk> \<lbrakk>\<lbrakk>tab\<^sub>1 h\<rbrakk>\<rbrakk> = \<lbrakk>\<lbrakk>THfgh.p\<^sub>1\<rbrakk>\<rbrakk>"
elims: fixes x :: "real poly"   and xa :: "real poly"   and y :: "real poly list" assumes "smods_ext x xa = y" obtains p :: "real poly" and q :: "real poly" where "x = p" and "xa = q" and "y = (if p = 0 then [] else if p mod q \<noteq> 0 then p # smods_ext q (- (p mod q)) else p # smods_ext q (pderiv q))"
timpl_closure_set_mono_timpl_closure_set: fixes N :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and M :: "(('a, 'b, 'c) prot_fun, 'd) Term.term set"   and TI :: "('c set \<times> 'c set) set" assumes "N \<subseteq> timpl_closure_set M TI" shows "timpl_closure_set N TI \<subseteq> timpl_closure_set M TI"
rank_of_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set" assumes "\<E> \<subseteq> carrier"   and "\<E>' \<subseteq> \<E>" shows "indep_system.lower_rank_of (\<E>.indep_in \<E>) \<E>' = \<E>.rank_of \<E>'"
monad_commute_prob: shows "monad_commute return_pmf (\<bind>)"
lincomb_list_index: fixes i :: "nat"   and Xs :: "'a vec list"   and c :: "nat \<Rightarrow> 'a" assumes "i < n"   and "set Xs \<subseteq> carrier_vec n" shows "n.lincomb_list c Xs $ i = (\<Sum>j = 0..<length Xs. c j * Xs ! j $ i)"
Prf_def: fixes s :: "hf"   and k :: "hf"   and y :: "hf" shows "Prf s k y \<equiv> BuildSeq (\<lambda>x. x \<in> Axiom) (\<lambda>u v w. ModPon v w u \<or> Exists v u \<or> Subst v u) s k y"
simps_42: shows "subcomp sS4opt = {sA22, sA23, sA31, sA32, sA41}"
set_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "set (list.map f xs) = f ` set xs"
pure_hn_refineI: fixes Q :: "bool"   and c :: "'a"   and a :: "'b"   and R :: "('a \<times> 'b) set" assumes "Q \<longrightarrow> (c, a) \<in> R" shows "hn_refine (\<up> Q) (return c) (\<up> Q) (pure R) (RETURN a)"
executionally_equivalent_step: fixes e1 :: "((nat \<times> nat) \<times> transition) fset"   and s1 :: "nat"   and r1 :: "nat \<Rightarrow>f value option"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s2 :: "nat"   and r2 :: "nat \<Rightarrow>f value option"   and l :: "String.literal"   and i :: "value list"   and es :: "(String.literal \<times> value list) list" shows "executionally_equivalent e1 s1 r1 e2 s2 r2 ((l, i) # es) = ((\<forall>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. \<exists>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es) \<and> (\<forall>(s2', t2)|\<in>|possible_steps e2 s2 r2 l i. \<exists>(s1', t1)|\<in>|possible_steps e1 s1 r1 l i. evaluate_outputs t1 i r1 = evaluate_outputs t2 i r2 \<and> executionally_equivalent e1 s1' (evaluate_updates t1 i r1) e2 s2' (evaluate_updates t2 i r2) es))"
renaming_and_selected_part: fixes C :: "'at Literal set"   and Sel :: "'at set" shows "selected_part UNIV C = rename_clause Sel (validated_part Sel (rename_clause Sel C))"
assms_3: shows "fst a \<subseteq>\<^sub>f s"
coreflexive_symmetric: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.coreflexive x" shows "matrix_stone_relation_algebra.symmetric x"
ga_step_def: shows "ga_step \<equiv> stutter_extend_edges UNIV ga_gstep"
bl_to_bin_aux_cat: fixes bs :: "bool list"   and w :: "int" shows "bl_to_bin_aux bs (concat_bit 0 0 w) = concat_bit (0 + length bs) (bl_to_bin_aux bs 0) w"
set_child_nodes_locs_def: shows "set_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = (\<lambda>ptr. (if is_element_ptr_kind\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr then all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) RElement.child_nodes_update) else {}) \<union> (if is_document_ptr_kind ptr then all_args (put_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t (the (cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r ptr)) document_element_update) else {}))"
gfp_eqI: fixes F :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "mono F"   and "F x = x"   and "\<And>z. F z = z \<Longrightarrow> z \<le> x" shows "gfp F = x"
vmc_path_no_deadend_axioms_def: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c" shows "vmc_path_no_deadend_axioms G v0 \<equiv> \<not> \<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v0 \<rightarrow>\<^bsub>G\<^esub> w)"
determI: fixes t :: "('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" assumes "Transformers.additive t"   and "maximal t" shows "determ t"
singleton_in_stock_measure: fixes v :: "val"   and t :: "pdf_type" assumes "val_type v = t" shows "{v} \<in> sets (stock_measure t)"
simps_3: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and m :: "'c match_expr"   and a :: "action"   and rs :: "'c rule list" shows "approximating_bigstep_fun \<gamma> p (Rule m a # rs) Undecided = (if \<not> matches \<gamma> m a p then approximating_bigstep_fun \<gamma> p rs Undecided else case a of Accept \<Rightarrow> Decision FinalAllow | Drop \<Rightarrow> Decision FinalDeny | Log \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided | Reject \<Rightarrow> Decision FinalDeny | Empty \<Rightarrow> approximating_bigstep_fun \<gamma> p rs Undecided)"
inj_on_one_iff: fixes G :: "('e, 'f) monoid_scheme"   and H :: "('g, 'h) monoid_scheme"   and h :: "'e \<Rightarrow> 'g" assumes "group_hom G H h" shows "inj_on h (carrier G) = (\<forall>x. x \<in> carrier G \<longrightarrow> h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>\<^bsub>G\<^esub>)"
arccos_bounded: fixes y :: "real" assumes "- 1 \<le> y"   and "y \<le> 1" shows "0 \<le> arccos y \<and> arccos y \<le> pi"
properfactorI3: fixes p :: "'a"   and a :: "'a"   and b :: "'a" assumes "p = a \<otimes> b"   and "b \<notin> Units G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "properfactor G a p"
Proj_def: shows "Proj \<equiv> quot_type.abs proj_rel Abs_proj_space"
rtrancl_O_shift: fixes S :: "('b \<times> 'c) set"   and R :: "('c \<times> 'b) set" shows "(S O R)\<^sup>* O S = S O (R O S)\<^sup>*"
0: shows "\<forall>x\<in>ps. case x of (k, g) \<Rightarrow> real_of_int k * line_integral F basis g = 0"
dgrad_p_set_le_red_single: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c"   and f :: "'t \<Rightarrow>\<^sub>0 'c"   and t :: "'a" assumes "dickson_grading d"   and "red_single p q f t" shows "dgrad_p_set_le d {q} {f, p}"
orthogonal_matrix_card: fixes A :: "((real, 'b) vec, 'b) vec" assumes "orthogonal_matrix A" shows "card (columns A) = ncols A"
i_Exec_Comp_Stream_Acc_Output__eq_Msg_before_State_Idle_imp2: fixes k :: "nat"   and localState :: "'a \<Rightarrow> 'b"   and output_fun :: "'a \<Rightarrow> 'c message_af"   and trans_fun :: "'d message_af \<Rightarrow> 'a \<Rightarrow> 'a"   and input :: "nat \<Rightarrow> 'd message_af"   and c :: "'a"   and t :: "nat"   and m :: "'c message_af"   and t0 :: "nat"   and s :: "nat \<Rightarrow> 'a"   and t1 :: "nat" assumes "Suc 0 < k"   and "State_Idle localState output_fun trans_fun (i_Exec_Comp_Stream_Acc_LocalState k localState trans_fun input c t)"   and "m \<noteq> NoMsg"   and "t0 = t * k"   and "s = i_Exec_Comp_Stream trans_fun (input \<odot> k) c"   and "t1 \<in> [0\<dots>,k - Suc 0] \<oplus> t0"   and "output_fun (s t1) = m"   and "\<circle> t2 t1 [0\<dots>]. output_fun (s t3) = NoMsg. t3 \<U> t4 [0\<dots>] \<oplus> t2. output_fun (s t4) = NoMsg \<and> State_Idle localState output_fun trans_fun (localState (s t4))" shows "i_Exec_Comp_Stream_Acc_Output k output_fun trans_fun input c t = m"
neither: assumes "\<not> P"   and "\<not> Q" shows "R"
b_assn_def: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P :: "'a \<Rightarrow> bool" shows "b_assn A P \<equiv> \<lambda>x y. A x y * \<up> (P x)"
conj_disj_distr: fixes P :: "'b upred"   and Q :: "'b upred"   and R :: "'b upred" shows "(P \<and> (Q \<or> R)) = (P \<and> Q \<or> P \<and> R)"
Label_def: fixes T :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set" shows "Label T = label ` T"
list_all_rec_eq: fixes l :: "'c list"   and P :: "'c \<Rightarrow> bool" shows "(\<forall>x\<in>set l. P x) = list_all_rec P l"
bounded_clinear_def: fixes f :: "'b \<Rightarrow> 'c" shows "bounded_clinear f \<equiv> clinear f \<and> bounded_clinear_axioms f"
wf\<^sub>s\<^sub>t\<^sub>s'_update\<^sub>s\<^sub>t_rcv: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list" assumes "wf\<^sub>s\<^sub>t\<^sub>s' \<S> \<A>"   and "receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "wf\<^sub>s\<^sub>t\<^sub>s' (update\<^sub>s\<^sub>t \<S> (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S)) (\<A> @ [Step (send\<langle>t\<rangle>\<^sub>s\<^sub>t)])"
exec_elim_cases_4: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and s :: "('e, 'f) xstate"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>Skip,s\<rangle> \<Rightarrow> t" obtains s :: "'e" where "sb__ = Normal sc__" and "t = Normal sc__"   | f :: "'f" where "sb__ = Fault f" and "t = Fault f"   | "sb__ = Stuck" and "t = Stuck"   | s :: "'e" where "sb__ = Abrupt s" and "t = Abrupt s"