hcju/isabelleps huggingface dataset:
that: fixes \<pi> :: "'a ltl \<Rightarrow> nat option" assumes "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> the (\<pi> \<chi>) < max_rank_of \<Sigma> \<chi>"   and "accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (M_fin \<pi>, UNIV) w"   and "\<And>\<chi>. \<chi> \<in> dom \<pi> \<Longrightarrow> accepting_pair\<^sub>R (delta \<Sigma>) (initial \<phi>) (Acc \<Sigma> \<pi> \<chi>) w" shows "thesis"
in_zmset_conv_pos_neg_disj: fixes x :: "'a"   and M :: "'a zmultiset" shows "(x \<in>#\<^sub>z M) = (x \<in># mset_pos M \<or> x \<in># mset_neg M)"
almost_full_on_finite_subsetE: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set" assumes "reflp P"   and "almost_full_on P S" obtains T :: "'a set" where "finite T" and "T \<subseteq> S" and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. P t s"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) (\<lambda>f2. F (A1.dom f1, f2)) (\<lambda>f2. F (A1.cod f1, f2)) (\<lambda>f2. F (f1, f2))"
rel_chain_appendD_1: fixes l :: "'b list"   and l' :: "'b list"   and R :: "('b \<times> 'b) set" assumes "rel_chain (l @ l') R" shows "rel_chain l R"
SUP_spmf_neq_top': fixes x :: "'a"   and Y :: "'a spmf set" shows "(\<Squnion>p\<in>Y. ennreal (spmf p x)) \<noteq> \<top>"
the_ext_unique: fixes y' :: "'c"   and g :: "'d"   and f :: "'c" assumes "arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y y' g"   and "is_ext y' g f" shows "f = the_ext y' g"
distinct_of_odd_in_distinct_partitions: fixes p :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "p \<in> {p. p partitions n \<and> (\<forall>i. p i \<noteq> 0 \<longrightarrow> odd i)}" shows "distinct_of_odd p \<in> {p. p partitions n \<and> (\<forall>i. p i \<le> 1)}"
GmultD: fixes smult :: "('b, 'c) aezfun \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'c"   and v :: "'a" shows "aezfun_scalar_mult.Gmult smult g v = smult ((1::'b) \<delta>\<delta> g) v"
l_set_shadow_root_get_child_nodes_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_shadow_root_get_child_nodes type_wf set_shadow_root set_shadow_root_locs known_ptr get_child_nodes get_child_nodes_locs \<equiv> l_set_shadow_root type_wf set_shadow_root set_shadow_root_locs \<and> l_get_child_nodes type_wf known_ptr get_child_nodes get_child_nodes_locs \<and> l_set_shadow_root_get_child_nodes_axioms set_shadow_root_locs get_child_nodes_locs"
reflexive_inf_star: fixes y :: "'a"   and x :: "'a" assumes "reflexive y" shows "y \<sqinter> x\<^sup>\<star> = (1::'a) \<squnion> y \<sqinter> x\<^sup>+"
maxR_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "{x, y, z} \<subseteq> Field r" shows "maxR (maxR x y) z = maxR x (maxR y z)"
functions_of_def: fixes A :: "'b set"   and B :: "'c set" shows "functions_of A B = {f \<in> A \<rightarrow>\<^sub>E B. f ` A = B}"
subst_fm_Ex_with_renaming: fixes i' :: "name"   and A :: "fm"   and i :: "name"   and j :: "name"   and t :: "tm" assumes "atom i' \<sharp> (A, i, j, t)" shows "(SyntaxN.Ex i A)(j::=t) = SyntaxN.Ex i' (((i \<leftrightarrow> i') \<bullet> A)(j::=t))"
minimal_basis_red_rtrancl_lt: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and p :: "'t \<Rightarrow>\<^sub>0 'c"   and r :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_minimal_basis B"   and "p \<in> B"   and "(red (B - {p}))\<^sup>*\<^sup>* p r" shows "lt r = lt p"
conv_radius_def: fixes f :: "nat \<Rightarrow> 'a" shows "conv_radius f = inverse (limsup (\<lambda>n. ereal (root n (norm (f n)))))"
ExE: fixes v :: "i"   and \<phi> :: "'b \<Rightarrow> \<o>" assumes "[\<^bold>\<exists>x. \<phi> x in v]" shows "\<exists>y. [\<phi> y in v]"
CompFunCarAudioSystem_ran: shows "ran (CompFun CarAudioSystem) = {{}, {AudioPlayer_CTRL, CDPlayer_CTRL}, {CDMode_CTRL, TunerMode_CTRL}}"
g'_derivative: shows "\<forall>x\<in>{a..b}. (g has_vector_derivative g' x) (at x within {a..b})"
fgraph_def: fixes g :: "graph" shows "fgraph g = map vertices (faces g)"
enn2real_leI: fixes B :: "real"   and x :: "ennreal" assumes "0 \<le> B"   and "x \<le> ennreal B" shows "enn2real x \<le> B"
maxim_isMaxim: fixes A :: "'a set" assumes "finite A"   and "A \<noteq> {}"   and "A \<subseteq> Field r" shows "isMaxim A (maxim A)"
10: shows "minarc (v * - v\<^sup>T \<sqinter> g) * t\<^sup>T = bot"
nodes_transition: fixes a :: "'c"   and A :: "('c, 'd) nba"   and p :: "'d"   and q :: "'d" assumes "a \<in> alphabet A"   and "p \<in> nodes A"   and "q \<in> transition A a p" shows "q \<in> nodes A"
Dom_Diagonalize_rep: fixes f :: "'c local.term set" assumes "arr f" shows "Dom \<^bold>\<lfloor>rep f\<^bold>\<rfloor> = \<^bold>\<lfloor>rep (local.dom f)\<^bold>\<rfloor>"
runiq_singleton_rel: fixes x :: "'c"   and y :: "'d" shows "runiq {(x, y)}"
assms_5: shows "countable (support_flow (\<Squnion> Y))"
equivalence_of_bicategories_and_pseudonatural_equivalence_left_axioms: shows "equivalence_of_bicategories_and_pseudonatural_equivalence_left (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G \<eta>\<^sub>0 \<eta>\<^sub>1 \<epsilon>\<^sub>0 \<epsilon>\<^sub>1 F' \<Phi>\<^sub>F' \<tau>\<^sub>0 \<tau>\<^sub>1"
f_def: fixes x :: "nat" shows "f x = pr_index_enumerator x m"
qFreshAll_imp_ex_qAFreshAll: fixes V :: "'var set"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and fZs :: "'var \<Rightarrow> 'varSort set"   and A :: "('d, 'e, 'varSort, 'var, 'f) qAbs" assumes "finite V" shows "(qGood X \<longrightarrow> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFresh zs z X) \<longrightarrow> (\<exists>X'. X #= X' \<and> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qAFresh zs z X'))) \<and> (qGoodAbs A \<longrightarrow> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qFreshAbs zs z A) \<longrightarrow> (\<exists>A'. A $= A' \<and> (\<forall>z\<in>V. \<forall>zs\<in>fZs z. qAFreshAbs zs z A')))"
Subring_subset: fixes R :: "('d, 'e) Ring_scheme"   and S :: "('d, 'f) Ring_scheme" assumes "Subring R S" shows "carrier S \<subseteq> carrier R"
inf_notin_unit_disc: shows "\<infinity>\<^sub>h \<notin> unit_disc"
verts_set: shows "set (g_V G) = {0..<length (g_V G)}"
assms_1: shows "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<CC>"
Priority_Queue_axioms: shows "Priority_Queue local.empty is_empty local.insert value del_min heap mset_tree"
strict_trans2: fixes a :: "'b multiset"   and b :: "'b multiset"   and c :: "'b multiset" assumes "a \<subset># b"   and "b \<subseteq># c" shows "a \<subset># c"
d: shows "non_pivot_base A pp qj = vec nc (\<lambda>i. if i = qj then 1::'a else I i)"
sfilter_sfilter': fixes p :: "'r \<rightarrow> tr"   and q :: "'r \<rightarrow> tr"   and xs :: "[:'r:]" shows "sfilter\<cdot>p\<cdot>(sfilter\<cdot>q\<cdot>xs) = sfilter\<cdot>(\<Lambda> x. q\<cdot>x andalso p\<cdot>x)\<cdot>xs"
fmdom'_filter: fixes P :: "'d \<Rightarrow> bool"   and m :: "('d, 'e) fmap" shows "fmdom' (fmfilter P m) = Set.filter P (fmdom' m)"
invertible_iff_is_unit_JNF: fixes A :: "'b mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "invertible_mat A = (Determinant.det A dvd (1::'b))"
simps_1: fixes i :: "nat"   and k :: "nat" shows "take_nths i k [] = []"
finprod_singleton: fixes i :: "'d"   and A :: "'d set"   and f :: "'d \<Rightarrow> 'a" assumes "i \<in> A"   and "finite A"   and "f \<in> A \<rightarrow> carrier G" shows "(\<Otimes>j\<in>A. if i = j then f j else \<one>) = f i"
equivalence_functor_R: shows "equivalence_functor (\<cdot>) (\<cdot>) R"
x_in_csv: shows "x \<in> set (map (consistent_sign_vec_copr (qs1 @ qs2)) (characterize_root_list_p p))"
nat_minus_add_max: fixes n :: "nat"   and m :: "nat" shows "n - m + m = max n m"
exp_le_cancelI: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "exp x \<le> exp y"
K2_centre_in_K2: shows "K2_centre \<in> hyp2"
write_seen_committed_def: fixes ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('a, 'b) justifying_execution" shows "write_seen_committed ws J = (\<forall>n. \<forall>r'\<in>read_actions (justifying_exec (J n)) \<inter> committed (J n). let r = action_translation (J n) r'; r'' = inv_into (actions (justifying_exec (J (Suc n)))) (action_translation (J (Suc n))) r in action_translation (J (Suc n)) (justifying_ws (J (Suc n)) r'') = ws r)"
IH2': shows "quicksort_post R h (Suc p) hi xs'' xs'''"
atLeast_subset_iff: fixes x :: "'a"   and y :: "'a" shows "({x..} \<subseteq> {y..}) = (y \<le> x)"
comp_arr_dom: fixes f :: "'c local.term set"   and a :: "'c local.term set" assumes "\<F>C.arr f"   and "\<F>C.dom f = a" shows "f \<cdot> a = f"
unitary_mat_incr: fixes n :: "nat" shows "unitary (mat_incr n)"
t1: shows "\<Gamma> \<turnstile> e : PRODUCT t t"
SomeRecordExample_ext_def: fixes Field1 :: "nat message_af"   and Field2 :: "int message_af"   and Field3 :: "int message_af"   and more :: "'b" shows "\<lparr>Field1 = Field1, Field2 = Field2, Field3 = Field3, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons SomeRecordExample_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple Field1 (Record.iso_tuple_cons Record.tuple_iso_tuple Field2 Field3)) more"
Image_Collect_case_prod: fixes P :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f set" shows "{(x, y). P x y} `` A = {y. \<exists>x\<in>A. P x y}"
axioms_2: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'d \<Rightarrow> 'c"   and G :: "'c \<Rightarrow> 'd"   and \<phi> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and \<psi> :: "'c \<Rightarrow> 'd \<Rightarrow> 'c" assumes "meta_adjunction C D F G \<phi> \<psi>" shows "category D"
tendsto_const: fixes "open" :: "'d set \<Rightarrow> bool"   and k :: "'d"   and F :: "'e filter" assumes "class.topological_space open" shows "LIM x F. k :> topological_space.nhds open k"
span_def: fixes S :: "'b set" shows "span S = subspace hull S"
execute_serial_plan_split: fixes s :: "'a \<Rightarrow> bool option"   and \<pi>\<^sub>1 :: "'a strips_operator list"   and \<pi>\<^sub>2 :: "'a strips_operator list" assumes "are_all_operators_applicable s \<pi>\<^sub>1"   and "are_all_operators_non_interfering \<pi>\<^sub>1" shows "execute_serial_plan s (\<pi>\<^sub>1 @ \<pi>\<^sub>2) = execute_serial_plan (execute_serial_plan s \<pi>\<^sub>1) \<pi>\<^sub>2"
arr_dom: fixes f :: "'a \<times> 'a \<times> 'a" assumes "B.VVV.arr f" shows "B.VVV.arr (B.VVV.dom f)"
less_e: shows "lnorm (lebesgue_on S) 2 (\<lambda>x. f x - g x) < e"
lm103: fixes N :: "'c set"   and G :: "'d list" assumes "0 < card N"   and "distinct G" shows "{injections P N |P. P \<in> all_partitions (set G)} = set (map (\<lambda>l. set (injections_alg l N)) (all_partitions_list G))"
lprob_comp_stake: fixes pr :: "real"   and w :: "bool stream"   and matur :: "nat" shows "prod (prob_component pr w) {0..<matur} = lprob_comp pr (stake matur w)"
smult_sg_nsg: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set"   and N :: "'c set" assumes "Group G"   and "G \<guillemotright> H "   and "G \<triangleright> N" shows "G \<guillemotright> H \<diamondop>\<^bsub>G\<^esub> N "
simps_1: fixes uu :: "nat"   and uv :: "nat" shows "inl OpAdd [DNum uu, DNum uv] = Some OpAddNum"
finite_fv_Proof: fixes P :: "proofterm" shows "finite (fv_Proof P)"
r_null: fixes x :: "nat \<Rightarrow> 'a" assumes "x \<in> carrier P" shows "x \<otimes>\<^bsub>P\<^esub> \<zero>\<^bsub>P\<^esub> = \<zero>\<^bsub>P\<^esub>"
atLeast_Suc_atMost: fixes m :: "nat"   and n :: "nat"   and g :: "nat \<Rightarrow> 'g" assumes "m \<le> n" shows "prod g {m..n} = g m * prod g {Suc m..n}"
inject: fixes x21 :: "'c"   and x22 :: "'c Tree"   and x23 :: "'c Tree"   and y21 :: "'c"   and y22 :: "'c Tree"   and y23 :: "'c Tree" shows "(T x21 x22 x23 = T y21 y22 y23) = (x21 = y21 \<and> x22 = y22 \<and> x23 = y23)"
distinct_list_of_dlist: fixes dxs :: "'c set_dlist" assumes "ID CEQ('c) \<noteq> None" shows "distinct (list_of_dlist dxs)"
equivp_reflp: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c" assumes "equivp R" shows "R x x"
delete_edges_list_set: fixes G :: "'b graph"   and E :: "('b \<times> 'b) list" shows "delete_edges_list G E = FiniteGraph.delete_edges G (set E)"
ide_in_hom_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "horizontal_homs V src trg"   and "partial_magma.ide V f" shows "partial_magma.in_hom V f f f"
simps: fixes t :: "'b trm"   and s :: "'b trm" shows "subterms_of_eq (Eq t s) = subterms_of t \<union> subterms_of s"
prems: shows "isOK (check_allm_gen_index g f n (x # xs))"
\<phi>': fixes x :: "real" assumes "x \<in> {a..b}" shows "(\<phi> has_real_derivative \<phi>' x) (at x)"
assms: shows "\<lless>\<pi>\<^sub>1, eP\<ggreater> P\<rightarrow>i \<lless>\<pi>\<^sub>2, eP'\<ggreater>"
fcomp_obtain_middle: fixes a :: "V"   and c :: "V"   and f :: "V"   and g :: "V" assumes "[a, c]\<^sub>\<circ> \<in>\<^sub>\<circ> f \<circ>\<^sub>\<bullet> g" obtains b :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> g" and "[b, c]\<^sub>\<circ> \<in>\<^sub>\<circ> f"
prob0_fst_qubits_index: fixes n :: "nat" shows "{k |k. k < 2 ^ (n + 1) \<and> (\<forall>i\<in>{0..<n}. \<not> select_index (n + 1) i k)} = {0, 1}"
HomFtorFtor: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "Functor (Hom\<^bsub>C\<^esub>[X,\<midarrow>])"
poly_deg_times_le: fixes p :: "('d \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'e"   and q :: "('d \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'e" shows "poly_deg (p * q) \<le> poly_deg p + poly_deg q"
L: shows "(\<nexists>S. \<langle>n, PubKey S\<rangle> \<in> spied u\<^sub>4 \<and> Rev_PriK \<in> S \<and> (\<forall>m. Sec_PriK m \<in> S \<longrightarrow> (n, m) \<notin> bad_id)) \<and> \<langle>n, PubKey S\<rangle> \<in> spied v\<^sub>4 \<and> Rev_PriK \<in> S \<and> (\<forall>m. Sec_PriK m \<in> S \<longrightarrow> (n, m) \<notin> bad_id)"
max_0_1_6: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "ord.max less_eq (numeral.numeral one plus x) one = numeral.numeral one plus x"
Iff_E: fixes A :: "fm"   and B :: "fm"   and H :: "fm set"   and C :: "fm" assumes "insert A (insert B H) \<turnstile> C"   and "insert (Neg A) (insert (Neg B) H) \<turnstile> C" shows "insert (A IFF B) H \<turnstile> C"
less_imp_le: fixes x :: "'c list"   and y :: "'c list" assumes "strict_suffix x y" shows "suffix x y"
is_GB_cofactor_boundE_finite_Polys: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and b :: "nat"   and X :: "'x set" assumes "is_GB_cofactor_bound F b"   and "finite F"   and "F \<subseteq> P[X]" obtains G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" where "punit.is_Groebner_basis G" and "ideal G = ideal F" and "G \<subseteq> P[X]" and "\<And>g. g \<in> G \<Longrightarrow> \<exists>q. g = (\<Sum>f\<in>F. q f * f) \<and> (\<forall>f. q f \<in> P[X] \<and> poly_deg (q f * f) \<le> b)"
gfp_to_lfp_var: fixes f :: "'b \<Rightarrow> 'b" shows "gfp f = \<partial> (lfp (\<partial>\<^sub>F f))"
combine_is_bibd: shows "bibd \<V>\<^sup>+ \<B>\<^sup>+ \<k> (\<Lambda> + \<Lambda>')"
comp: fixes s0 :: "st"   and c1 :: "stmt"   and s1 :: "state"   and c2 :: "stmt"   and s2 :: "state"   and accC :: "qtname"   and L :: "lenv"   and C1 :: "assigned" assumes "G\<turnstile>Norm s0 \<midarrow>c1\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>c2\<rightarrow> s2"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c1\<Colon>\<surd>"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>c2\<Colon>\<surd>"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd (Norm s0))) \<guillemotright>In1r c1\<guillemotright> C1"   and "P L accC (Norm s0) (In1r c1) \<diamondsuit> s1"   and "\<And>Q. \<lbrakk>normal s1; \<And>C2. \<lbrakk>\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s1)) \<guillemotright>In1r c2\<guillemotright> C2; P L accC s1 (In1r c2) \<diamondsuit> s2\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q" shows "P L accC (Norm s0) (In1r (c1;; c2)) \<diamondsuit> s2"
happens_before_committed_weak_def: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "happens_before_committed_weak P E ws J = (\<forall>n. \<forall>r\<in>read_actions (justifying_exec (J n)) \<inter> committed (J n). let r' = action_translation (J n) r; w' = ws r'; w = inv_into (actions (justifying_exec (J n))) (action_translation (J n)) w' in (P,E \<turnstile> w' \<le>hb r') = (P,justifying_exec (J n) \<turnstile> w \<le>hb r) \<and> \<not> P,justifying_exec (J n) \<turnstile> r \<le>hb w)"
bound_isUb: fixes N :: "nat"   and x :: "real"   and y :: "real" assumes "\<forall>n\<ge>N. X n < x"   and "y \<in> S" shows "y \<le> x"
ppa_mem: fixes n :: "nat"   and P :: "nat \<Rightarrow> 'a set"   and A :: "'a set"   and i :: "nat"   and l :: "nat" assumes "\<forall>k\<le>Suc n. prime_ideal R (P k)"   and "ideal R A"   and "\<forall>l\<le>Suc n. \<not> A \<subseteq> P l"   and "\<forall>k\<le>Suc n. \<forall>l\<le>Suc n. k = l \<or> \<not> P k \<subseteq> P l"   and "i \<le> Suc n"   and "l \<le> n" shows "ppa R P A i l \<in> carrier R"
precedesReflexivity: fixes a :: "'b"   and l :: "'b list" assumes "a \<in> set l" shows "precedes a a l"
List_explode_goup_by_zeros: fixes xs :: "16 word list" shows "List_explode (goup_by_zeros xs) = map Some xs"
set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_def: fixes element_ptr :: "(_) element_ptr"   and children :: "(_) node_ptr list" shows "set_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r element_ptr children = put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr child_nodes_update children"
remaining_qty_predict': fixes asset :: "'b"   and pf :: "'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and v :: "real"   and n :: "nat" assumes "borel_adapt_stoch_proc F (prices Mkt asset)"   and "trading_strategy pf"   and "support_adapt Mkt pf" shows "remaining_qty Mkt v pf asset (Suc n) \<in> borel_measurable (F n)"
comp_in_homI: fixes f :: "'b"   and a :: "'b"   and b :: "'b"   and g :: "'b"   and c :: "'b" assumes "\<guillemotleft>f : a \<rightarrow>\<^sub>J b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow>\<^sub>J c\<guillemotright>" shows "\<guillemotleft>J g f : a \<rightarrow>\<^sub>J c\<guillemotright>"
permute_self: fixes p :: "perm" shows "p \<bullet> p = p"
compare_complex_def: fixes x :: "complex"   and y :: "complex" shows "compare x y = compare (Re x, Im x) (Re y, Im y)"
uniform_measure_def: fixes M :: "'a measure"   and A :: "'a set" shows "uniform_measure M A = density M (\<lambda>x. indicator A x / emeasure M A)"
borel_measurable_neq: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and g :: "'c \<Rightarrow> 'd" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "{w \<in> space M. f w \<noteq> g w} \<in> sets M"
fls_base_factor_to_fps_conv_fps_shift: fixes f :: "'a fls" assumes "0 \<le> fls_subdegree f" shows "fls_base_factor_to_fps f = fps_shift (nat (fls_subdegree f)) (fls_regpart f)"
cbigD: fixes \<sigma>' :: "'state"   and coll' :: "'coll"   and P :: "'prog"   and \<sigma> :: "'state" assumes "(\<sigma>', coll') \<in> cbig P \<sigma>" shows "\<exists>n. (\<sigma>', coll') \<in> csmall_nstep P \<sigma> n \<and> \<sigma>' \<in> endset"
induct: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "a \<in> A"   and "\<And>x. \<lbrakk>x \<in> A; \<And>y. \<lbrakk>y \<in> A; y \<sqsubset> x\<rbrakk> \<Longrightarrow> P y\<rbrakk> \<Longrightarrow> P x" shows "P a"
fst_uminus: fixes x :: "'d \<times> 'e" shows "fst (- x) = - fst x"
finite_Rep_perm: fixes p :: "perm" shows "finite {a. Rep_perm p a \<noteq> a}"
bind_spmf_cong_simp: fixes p :: "'l spmf"   and q :: "'l spmf"   and f :: "'l \<Rightarrow> 'm spmf"   and g :: "'l \<Rightarrow> 'm spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q =simp=> f x = g x" shows "p \<bind> f = q \<bind> g"
tab_def: shows "\<rho>\<sigma>.tab \<equiv> \<a>\<^sup>-\<^sup>1[r, s, s\<^sub>0 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0] \<cdot> (r \<star> \<a>[s, s\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>0]) \<cdot> (r \<star> \<sigma> \<star> r\<^sub>0s\<^sub>1.p\<^sub>0) \<cdot> (r \<star> r\<^sub>0s\<^sub>1.\<phi>) \<cdot> \<a>[r, r\<^sub>0, r\<^sub>0s\<^sub>1.p\<^sub>1] \<cdot> (\<rho> \<star> r\<^sub>0s\<^sub>1.p\<^sub>1)"
simps_1: fixes cs :: "('b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node) list" shows "CFGExit.valid_path_aux cs [] = True"
scale: fixes A :: "(('a, 'c) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and r :: "'a"   and b :: "('a, 'c) Finite_Cartesian_Product.vec" shows "A *v (r *s b) = r *s (A *v b)"
of_nat_1: shows "of_nat 1 = (1::'a)"
simps: fixes ires :: "cell list"   and rs :: "nat"   and l :: "cell list"   and r :: "cell list" shows "wcode_backto_standard_pos_2_O ires rs (l, r) = (\<exists>ml mr ln rn. l = Oc \<up> ml @ Bk # Bk # Bk \<up> ln @ Oc # ires \<and> r = Oc \<up> mr @ Bk \<up> rn \<and> ml + mr = Suc (Suc rs) \<and> 0 < mr)"
c_futures_failures: fixes yps :: "('e \<times> 'f) list"   and Y :: "('e \<times> 'f) set"   and step :: "'g \<Rightarrow> 'e \<Rightarrow> 'g"   and out :: "'g \<Rightarrow> 'e \<Rightarrow> 'f"   and s\<^sub>0 :: "'g"   and xps :: "('e \<times> 'f) list" shows "((yps, Y) \<in> futures (c_process step out s\<^sub>0) xps) = ((xps @ yps, Y) \<in> c_failures step out s\<^sub>0)"
Stream_snth: fixes x :: "'b"   and s :: "'b stream"   and n :: "nat" shows "(x ## s) !! n = (case n of 0 \<Rightarrow> x | Suc n \<Rightarrow> s !! n)"
FD_F: fixes P :: "'d process"   and Q :: "'d process" assumes "P \<sqsubseteq>\<^sub>F\<^sub>D Q" shows "P \<sqsubseteq>\<^sub>F Q"
dense_ge: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and z :: "'b"   and y :: "'b" assumes "class.dense_linorder less_eq less"   and "\<And>x. less z x \<Longrightarrow> less_eq y x" shows "less_eq y z"
simps_2: fixes M :: "'a"   and N :: "'a"   and P :: "('a, 'b, 'c) psi" shows "pOutput M N\<cdot>P = M\<langle>N\<rangle>.P"
f2: shows "\<exists>y. y \<in> (t |\<^bsub>rel\<^esub> u)"
right_idem: fixes a :: "'b"   and b :: "'b" shows "max (max a b) b = max a b"
ennreal_indicator: fixes A :: "'b set"   and x :: "'b" shows "ennreal (indicat_real A x) = indicator A x"
simps_2: 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 "sprLst \<lparr>sprFst = sprFst, sprLst = sprLst, sprCRel = sprCRel, \<dots> = more\<rparr> = sprLst"
Q'_def: shows "Q' \<equiv> insert Q_hit' (Q - {Q_hit})"
weakStatImpInputPres: fixes \<Psi> :: "'b"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<And>\<Psi>''. (\<Psi> \<otimes> \<Psi>'', M\<lparr>\<lambda>*xvec N\<rparr>.P, M\<lparr>\<lambda>*xvec N\<rparr>.Q) \<in> Rel" shows "\<Psi> \<rhd> M\<lparr>\<lambda>*xvec N\<rparr>.P \<lessapprox><Rel> M\<lparr>\<lambda>*xvec N\<rparr>.Q"