hcju/isabelleps huggingface dataset:
one_call_in_path_filter_conv: fixes x :: "'a"   and p :: "'a list" shows "one_call_in_path x p = (length (filter (\<lambda>x'. x' = x) p) \<le> 1)"
ab_semigroup_mult_hom_def: fixes hom :: "'b \<Rightarrow> 'c" shows "ab_semigroup_mult_hom hom \<equiv> semigroup_mult_hom hom"
asList_def: fixes A :: "'a set" shows "asList A \<equiv> SOME as. set as = A"
map_add_def: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" shows "m1 ++ m2 = (\<lambda>x. case m2 x of None \<Rightarrow> m1 x | Some y \<Rightarrow> Some y)"
m_comm: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier M"   and "y \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> y = y \<oplus>\<^bsub>M\<^esub> x"
val_Zp_dist_sym: fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp" shows "val_Zp_dist a b = val_Zp_dist b a"
Split: fixes c :: "('c, 'd) Old_Datatype.node set \<Rightarrow> ('c, 'd) Old_Datatype.node set \<Rightarrow> 'b"   and M :: "('c, 'd) Old_Datatype.node set"   and N :: "('c, 'd) Old_Datatype.node set" shows "Split c (Scons M N) = c M N"
eval_permutation_of_list_1: fixes x :: "'b" shows "permutation_of_list [] x = x"
I_hd_coeffs1: fixes \<phi> :: "atom fm"   and xs :: "int list" assumes "qfree \<phi>" shows "(\<exists>x. Z.I (hd_coeffs1 \<phi>) (x # xs)) = (\<exists>x. Z.I \<phi> (x # xs))"
Says_Kas_message_form: fixes A :: "agent"   and K :: "nat"   and authK :: "nat"   and Peer :: "agent"   and Ta :: "nat"   and authTicket :: "msg"   and evs :: "event list" assumes "Says Kas A (Crypt K \<lbrace>Key authK, Agent Peer, Number Ta, authTicket\<rbrace>) \<in> set evs"   and "evs \<in> kerbIV" shows "K = shrK A \<and> Peer = Tgs \<and> authK \<notin> range shrK \<and> authK \<in> authKeys evs \<and> authK \<in> symKeys \<and> authTicket = Crypt (shrK Tgs) \<lbrace>Agent A, Agent Tgs, Key authK, Number Ta\<rbrace> \<and> Key authK \<notin> used (before Says Kas A (Crypt K \<lbrace>Key authK, Agent Peer, Number Ta, authTicket\<rbrace>) on evs) \<and> Ta = CT (before Says Kas A (Crypt K \<lbrace>Key authK, Agent Peer, Number Ta, authTicket\<rbrace>) on evs)"
basis_def: fixes A :: "'c set" shows "ker.basis A = (ker.lin_indpt A \<and> ker.gen_set A \<and> A \<subseteq> carrier (V.vs kerT))"
fresh_append: fixes a :: "atom"   and xs :: "'b list"   and ys :: "'b list" shows "a \<sharp> xs @ ys = (a \<sharp> xs \<and> a \<sharp> ys)"
def_hashmap_size_ustate_wrapper_def: shows "def_hashmap_size = (\<lambda>_. def_hashmap_size TYPE('a) + def_hashmap_size TYPE('b))"
finite_lists_length_eqE: fixes A :: "'b set"   and n :: "nat" assumes "finite A" shows "finite {xs. set xs = A \<and> length xs = n}"
assms_1: shows "w \<in> remove_child_locs ptr owner_document"
holomorphic_uniform_limit: fixes z :: "complex"   and r :: "real"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and F :: "'a filter"   and g :: "complex \<Rightarrow> complex" assumes "\<forall>\<^sub>F n in F. continuous_on (cball z r) (f n) \<and> f n holomorphic_on ball z r"   and "uniform_limit (cball z r) f g F"   and "F \<noteq> bot" obtains "continuous_on (cball z r) g" and "g holomorphic_on ball z r"
W_cover: fixes p :: "'a" assumes "p \<in> carrier" shows "p \<in> \<Union> (range W)"
L: assumes "B = {#s#} + (B - {#s#})"   and "A = B - {#s#} + C" shows "P"
longest_Lyndon_sufI: fixes q :: "'a list"   and w :: "'a list" assumes "q \<le>s w"   and "Lyndon q"   and "\<forall>s. s \<le>s w \<and> Lyndon s \<longrightarrow> s \<le>s q" shows "LynSuf w = q"
x_in_dom: shows "x \<in> X \<union> vars_\<C> X"
listset_doubletonE: fixes ys :: "'a list"   and A :: "'a set"   and B :: "'a set" assumes "ys \<in> listset [A, B]" obtains a :: "'a" and b :: "'a" where "a \<in> A" and "b \<in> B" and "ys = [a, b]"
sset_shift: fixes xs :: "'c list"   and s :: "'c stream" shows "sset (xs @- s) = set xs \<union> sset s"
Times_insert_right: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'c multiset" shows "add_mset a A \<times># B = A \<times># B + image_mset (Pair a) B"
dghm_ObjMap_vsv: shows "vsv (\<GG>\<lparr>ObjMap\<rparr>)"
dres_foldli_ne_bot: fixes \<sigma> :: "'c dres"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c dres"   and l :: "'d list"   and c :: "'c dres \<Rightarrow> bool" assumes "\<sigma> \<noteq> dSUCCEED"   and "\<And>x \<sigma>. f x \<sigma> \<noteq> dSUCCEED" shows "foldli l c (\<lambda>x s. s \<bind> f x) \<sigma> \<noteq> dSUCCEED"
snd_foldl_ef_det_eq: fixes bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and n :: "'c"   and A :: "(('c, 'd) vec, 'e) vec"   and k :: "nat" shows "snd (foldl (echelon_form_of_column_k_det bezout) (n, A, 0) [0..<k]) = foldl (echelon_form_of_column_k bezout) (A, 0) [0..<k]"
ins_list_sorted: fixes xs :: "'b list"   and a :: "'b"   and x :: "'b"   and ys :: "'b list" assumes "sorted_less (xs @ [a])" shows "ins_list x (xs @ a # ys) = (if x < a then ins_list x xs @ a # ys else xs @ ins_list x (a # ys))"
abelian_monoidE_5: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_monoid R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y = y \<oplus>\<^bsub>R\<^esub> x"
bin_lcp_all_lcp: fixes ws :: "'a list list" assumes "ws \<in> lists {u\<^sub>0, u\<^sub>1}" shows "symcode.\<alpha> \<le>p concat ws \<cdot> symcode.\<alpha>"
zero_le_numeral: fixes n :: "num" shows "(0::'a) \<le> numeral n"
intro: fixes M :: "'b measure"   and F :: "nat \<Rightarrow> 'b measure" assumes "disc_filtr M F" shows "disc_filtr_prob_space_axioms M F"
jump_ok_ins_jump_ok: fixes ins :: "'a instr list"   and n :: "nat"   and n' :: "nat"   and pc :: "nat" assumes "jump_ok ins n n'"   and "pc < length ins" shows "ins_jump_ok (ins ! pc) (pc + n)"
oneof_length: fixes xs :: "char list list"   and s :: "char list"   and y :: "char list"   and r :: "char list" assumes "oneof xs s = Inr (y, r)" shows "length y + length r \<le> length s \<and> y \<in> set xs"
tconfI: fixes h :: "'heap"   and t :: "'thread_id"   and C :: "String.literal"   and P :: "'a prog" assumes "typeof_addr h (thread_id2addr t) = \<lfloor>Class_type C\<rfloor>"   and "P \<turnstile> C \<preceq>\<^sup>* Thread" shows "P,h \<turnstile> t \<surd>t"
abs_indicator: fixes A :: "'b set"   and x :: "'b" shows "\<bar>indicator A x\<bar> = indicator A x"
Obj_Src: fixes t :: "'a local.term" assumes "Arr t" shows "Obj (Src t)"
star_omega_1: fixes x :: "'a" shows "x\<^sup>\<star> \<cdot> x\<^sup>\<omega> = x\<^sup>\<omega>"
sum_prod: fixes f :: "nat \<Rightarrow> complex"   and b :: "nat"   and g :: "nat \<Rightarrow> complex"   and a :: "nat" shows "(\<Sum>i<a * b. f (i div b) * g (i mod b)) = sum f {..<a} * sum g {..<b}"
remove_child_types_preserved: fixes w :: "((_) heap, exception, unit) prog"   and ptr :: "(_) object_ptr"   and owner_document :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> remove_child_locs ptr owner_document"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
path_connected_sphere_gen: fixes S :: "'a set" assumes "convex S"   and "bounded S"   and "aff_dim S \<noteq> 1" shows "path_connected (rel_frontier S)"
assms_2: shows "a \<cdot> l \<le> l \<cdot> a"
Map_UP: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "Map (UP \<mu>) = \<mu>"
prime_power: fixes p :: "nat"   and k :: "nat" assumes "prime p"   and "0 < k" shows "totient (p ^ k) = p ^ k - p ^ (k - 1)"
minus_mult_commute: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = times a (uminus b)"
primerec_recs2_2: shows "primerec rec_nonstop (Suc (Suc (Suc 0)))"
scale_scale: fixes a :: "'b"   and b :: "'b"   and x :: "'b" shows "a * (b * x) = a * b * x"
covering_space_imp_continuous: fixes c :: "'c set"   and p :: "'c \<Rightarrow> 'd"   and S :: "'d set" assumes "covering_space c p S" shows "continuous_on c p"
assms_4: shows "cs\<^bsup>\<pi>\<^esup> (k + n) = cs\<^bsup>\<pi>'\<^esup> (k' + n')"
NEG_simp_2: fixes B :: "ty" shows "\<parallel>(B)\<parallel> = NEGn B (\<parallel><B>\<parallel>)"
1: fixes \<nu> :: "'a"   and \<mu> :: "'a" assumes "\<nu> \<star> \<mu> \<noteq> null" shows "src \<nu> = trg \<mu>"
red_inf_impl: fixes \<iota> :: "('f \<times> 'l) inference"   and NL :: "('f \<times> 'l) set" assumes "\<iota> \<in> std.Red_I NL" shows "to_F \<iota> \<in> no_labels.Red_I_\<G> (fst ` NL)"
assms_2: fixes i :: "nat" assumes "Suc i < length xs" shows "0 < det3 (fst (xs ! i)) (snd (xs ! i)) (snd (xs ! Suc i))"
fds_abs_summable_zeta_real_iff: fixes s :: "real" shows "fds_abs_converges fds_zeta s = (1 < s)"
distinct_47: fixes x51 :: "('d, 'f) ODE"   and x52 :: "('d, 'e, 'f) formula"   and x71 :: "('d, 'e, 'f) hp"   and x72 :: "('d, 'e, 'f) hp" shows "EvolveODE x51 x52 \<noteq> (x71 ;; x72)"
not_empty_pairE: fixes A :: "('a \<times> 'b) set" assumes "A \<noteq> {}" obtains a :: "'a" and b :: "'b" where "(a, b) \<in> A"
level_def: fixes p :: "(nat \<times> int) list" shows "level p = sum (lv p) {..<length p}"
Sup_set_fold: fixes xs :: "'a list" shows "\<Squnion> (set xs) = fold (\<squnion>) xs \<bottom>"
no_step_final': fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and cfg :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and cfg' :: "('b, 'a, 'c) com \<times> ('b, 'c) xstate"   and P :: "bool" assumes "\<Gamma>\<turnstile> cfg \<rightarrow> cfg'"   and "final cfg" shows "P"
oalist_inv_raw_def: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list" assumes "comparator comp" shows "comparator.oalist_inv_raw comp xs = ((0::'b) \<notin> snd ` set xs \<and> sorted_wrt (lt_of_comp comp) (map fst xs))"
cmax1: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set" assumes "Card_order r"   and "Card_order s"   and "s \<le>o r" shows "cmax r s =o r"
sets_PiM_single: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "sets (Pi\<^sub>M I M) = sigma_sets (\<Pi>\<^sub>E i\<in>I. space (M i)) {{f \<in> \<Pi>\<^sub>E i\<in>I. space (M i). f i \<in> A} |i A. i \<in> I \<and> A \<in> sets (M i)}"
vintersectionD1: fixes x :: "V"   and A :: "V"   and B :: "V" assumes "x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B" shows "x \<in>\<^sub>\<circ> A"
count_rem_components_alt: fixes bs :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list" shows "count_rem_components bs + card (const_lt_component ` (fst ` set bs - {0}) - {None}) = card (component_of_term ` Keys (fst ` set bs))"
complement: fixes a :: "'b set"   and A :: "'b set set" assumes "a \<in> sigma A" shows "- a \<in> sigma A"
irc_on_consistency_on: fixes A :: "'d set"   and f :: "'d set \<Rightarrow> 'd set" assumes "irc_on A f"   and "finite A" shows "consistency_on A f"
normalize_quot_aux_1: fixes b :: "'c"   and a :: "'c" assumes "b \<noteq> (0::'c)" shows "a = fst (normalize_quot (a, b)) * (gcd a b * unit_factor b)"
assms_2: fixes t :: "('b, 'a) Term.term" assumes "x \<in> vars_term t"   and "t \<in> subst_range \<sigma>" shows "P"
is_density_exprD_3: fixes vs :: "nat list"   and vs' :: "nat list"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "cexpr"   and t :: "pdf_type"   and e :: "cexpr" assumes "is_density_expr (vs, vs', \<Gamma>, \<delta>) t e" shows "nonneg_cexpr (shift_var_set (set vs')) (case_nat t \<Gamma>) e"
obj_sumE: fixes s :: "'d + 'e"   and P :: "bool" assumes "\<forall>x. s = Inl x \<longrightarrow> P"   and "\<forall>x. s = Inr x \<longrightarrow> P" shows "P"
simps_4: fixes indis :: "('a \<times> 'a) set"   and aval :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and tst :: "'c"   and c :: "('c, 'b) com" assumes "PL_Indis indis" shows "PL_Indis.SC_discr aval indis (while tst do c) = PL_Indis.SC_discr aval indis c"
terminates_rbt_keys_generator: shows "terminates (rbt_has_next, rbt_keys_next)"
intro: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.bounded_lattice inf less_eq less sup bot top"   and "class.distrib_lattice inf less_eq less sup"   and "class.boolean_algebra_axioms minus uminus inf sup bot top" shows "class.boolean_algebra minus uminus inf less_eq less sup bot top"
strict_mono_imp_strict_mono_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "strict_mono f" shows "SetInterval2.strict_mono_on f A"
intros_9: fixes H :: "fm set"   and A :: "fm"   and B :: "fm"   and i :: "name" assumes "H \<turnstile> A IMP B"   and "atom i \<sharp> B"   and "\<forall>C\<in>H. atom i \<sharp> C" shows "H \<turnstile> SyntaxN.Ex i A IMP B"
lowest_vertex_eq_center_iff: fixes x0 :: "real \<times> real"   and X :: "'a \<times> (real \<times> real) pdevs" shows "(lowest_vertex (x0, nlex_pdevs (snd X)) = x0) = (snd X = zero_pdevs)"
simps_4: fixes h1 :: "'b heap"   and h2 :: "'b heap" shows "exec Merge [h1, h2] = merge h1 h2"
bigtheta_mult_eq_set_mult: fixes F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" shows "\<Theta>[F](\<lambda>x. f x * g x) = set_mult \<Theta>[F](f) \<Theta>[F](g)"
the_def: fixes A :: "'b Predicate.pred" shows "Predicate.the A = (THE x. pred.eval A x)"
select_convs_1: fixes TDomain :: "'a"   and TExprSet :: "('a, 'b) Expression set"   and TCodomain :: "'a"   and more :: "'c" shows "TDomain \<lparr>TDomain = TDomain, TExprSet = TExprSet, TCodomain = TCodomain, \<dots> = more\<rparr> = TDomain"
coincide_on_term: fixes \<sigma> :: "('b \<times> 'b trm) list"   and \<eta> :: "('b \<times> 'b trm) list"   and t :: "'b trm" assumes "coincide_on \<sigma> \<eta> (vars_of t)" shows "t \<lhd> \<sigma> = t \<lhd> \<eta>"
simps_1: fixes c :: "('p, 's, 'm) configuration"   and p :: "'p"   and c' :: "('p, 's, 'm) configuration" shows "c \<turnstile> Snapshot p \<mapsto> c' = (can_occur (Snapshot p) c \<and> ps c' p = Some (states c p) \<and> (\<forall>p'. states c p' = states c' p') \<and> (\<forall>p'. p' \<noteq> p \<longrightarrow> ps c' p' = ps c p') \<and> (\<forall>i. (\<exists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i @ [Marker]) \<and> (\<forall>i. (\<exists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = (fst (cs c i), Recording)) \<and> (\<forall>i. (\<nexists>q. channel i = Some (p, q)) \<longrightarrow> msgs c' i = msgs c i) \<and> (\<forall>i. (\<nexists>q. channel i = Some (q, p)) \<longrightarrow> cs c' i = cs c i))"
sup_ge1: fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y \<le> x"
S\<^sub>e_def: shows "S\<^sub>e = {s \<in> S. p s = 0}"
hyps: assumes "distinct a'"   and "x \<in> set a'" shows "mset a' = {#x#} + mset (removeAll x a')"
z_2: shows "infinite {z' \<in> S. (case z' of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D)) = (case z of (t, u) \<Rightarrow> t\<^sup>2 - int (u\<^sup>2 * D))}"
abs_eq_iff: fixes x :: "'b \<times> 'b"   and y :: "'b \<times> 'b" shows "(abs_uprod x = abs_uprod y) = eq_upair x y"
invimage_vconverse: fixes r :: "V"   and A :: "V" shows "r\<inverse>\<^sub>\<circ> -`\<^sub>\<circ> A = r `\<^sub>\<circ> A"
ideI: fixes a :: "'a \<times> 'a" assumes "arr (fst a) \<and> arr (snd a) \<and> src (fst a) = trg (snd a)"   and "VxV.ide a" shows "VV.ide a"
assms_2: fixes tes :: "t list"   and tes' :: "t list" assumes "\<S>\<bullet>\<C> \<turnstile> [e] : tes _> tes'" shows "\<S>\<bullet>\<C> \<turnstile> [e'] : tes _> tes'"
img_dce_def: fixes M :: "'c measure"   and Y :: "'c \<Rightarrow> 'b"   and X :: "'c \<Rightarrow> real" shows "img_dce M Y X = (\<lambda>y. if Sigma_Algebra.measure M (Y -` {y} \<inter> space M) = 0 then 0 else (LINT w|M. X w * indicat_real (Y -` {y} \<inter> space M) w) / Sigma_Algebra.measure M (Y -` {y} \<inter> space M))"
StrongPostdomination_def: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Exit :: "'g" shows "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit \<equiv> Postdomination sourcenode targetnode kind valid_edge Entry Exit \<and> StrongPostdomination_axioms sourcenode targetnode valid_edge"
L_sorted: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and L_left :: "nat list"   and L_right :: "nat list" assumes "rtf_invar c s t f l n L_left L_right" shows "is_top_sorted (Network.adm_edges c f l) (L_left @ L_right)"
composite_simps_2: shows "trg (trg f \<star> f) = trg (trg f)"
gen_common_subdivision_imp_eq_line_integral_2: fixes one_chain1 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and one_chain2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and F :: "real \<times> real \<Rightarrow> real \<times> real"   and basis :: "(real \<times> real) set" assumes "common_sudiv_exists one_chain1 one_chain2"   and "boundary_chain one_chain1"   and "boundary_chain one_chain2"   and "\<forall>(k, \<gamma>)\<in>one_chain1. line_integral_exists F basis \<gamma>"   and "finite one_chain1"   and "finite one_chain2"   and "finite basis" shows "\<forall>(k, \<gamma>)\<in>one_chain2. line_integral_exists F basis \<gamma>"
simps_1: fixes \<nu> :: "'a"   and u :: "'a local.term" shows "\<^bold>\<langle>\<nu>\<^bold>\<rangle>\<^sub>0 \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u = u"
join_top: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "bounded_lattice L"   and "x \<in> carrier L" shows "x \<squnion>\<^bsub>L\<^esub> \<top>\<^bsub>L\<^esub> = \<top>\<^bsub>L\<^esub>"
indep_in_subI_subset: fixes \<E>' :: "'a set"   and \<E> :: "'a set"   and X :: "'a set" assumes "\<E>' \<subseteq> \<E>"   and "\<E>.indep_in \<E>' X" shows "\<E>.indep_in \<E> X"
is_rcoset_replistD_set: fixes H :: "'g set"   and gs :: "'g list" assumes "is_rcoset_replist H gs" shows "set gs \<subseteq> G"
rep_eq: fixes x :: "'a fset"   and xa :: "'a fset" shows "fset (x |\<inter>| xa) = fset x \<inter> fset xa"
eq: shows "(\<Prod>i = 0..<k. int n - int i) = 0"
simps_2: fixes A :: "nat \<Rightarrow> 'at"   and S :: "'at Literal set set"   and N :: "nat" shows "resolvents_sequence A S (Suc N) = simplify (resolvents_sequence A S N \<union> all_resolvents_upon (resolvents_sequence A S N) (A N))"
A_connected: fixes w :: "'a set \<Rightarrow> nat"   and g :: "'a ugraph"   and r :: "'a"   and A :: "('a \<times> 'a) set"   and u :: "'a"   and v :: "'a" assumes "Prim_Invar1_loc w g r A"   and "(u, v) \<in> A" shows "(v, r) \<in> A\<^sup>*"
g_2: shows "of_int_poly g = Polynomial.smult d G"
strict_coboundedI2: fixes c :: "'a"   and b :: "'a"   and a :: "'a" assumes "c \<sqsubset> b" shows "c \<sqsubset> ord.max (\<sqsubseteq>) a b"
eq_f: shows "\<forall>i. f (X $ i) = (\<Sum>j\<in>UNIV. A $ j $ i *s Y $ j)"
length_greater_0_conv: fixes xs :: "'d list" shows "(0 < length xs) = (xs \<noteq> [])"
monoid_homomorphism_axioms: shows "monoid_homomorphism contained.Class G (\<cdot>) \<one> contained.Partition ([\<cdot>]) (contained.Class \<one>)"
simps_2: fixes c :: "'a"   and s :: "'a FOL_Fitting.term"   and p :: "('a, 'b) form"   and z :: "('a, 'b) form list" shows "subcs c s (p # z) = subc c s p # subcs c s z"
simps_2: shows "not\<^sub>n false\<^sub>n = true\<^sub>n"
is_acc_def: fixes r :: "nat \<Rightarrow> 'q \<times> 's" shows "prod.is_acc r \<equiv> \<forall>n<prod.num_acc. \<exists>\<^sub>\<infinity>i. n \<in> prod.acc (r i)"
induct: fixes P :: "nat bdd list \<times> bool list \<Rightarrow> bool list list \<Rightarrow> bool"   and a0 :: "nat bdd list \<times> bool list"   and a1 :: "bool list list" assumes "\<And>M T. (\<And>x xa y. \<lbrakk>x = iter M T; (xa, y) = x; xa\<rbrakk> \<Longrightarrow> P M y) \<Longrightarrow> P M T" shows "P a0 a1"
iptables_bigstep_undecided_to_undecided_in_doubt_deny_approx: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided" shows "(\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided \<or> (\<beta>, in_doubt_deny),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalDeny"
bisim1AAss1: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr expr1"   and a' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and i :: "'addr expr1"   and e :: "'addr expr1" assumes "P,a,h \<turnstile> (a', xs) \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,a\<lfloor>i\<rceil> := e,h \<turnstile> (a'\<lfloor>i\<rceil> := e, xs) \<leftrightarrow> (stk, loc, pc, xcp)"
mn_unhalt_case': fixes f :: "recf"   and a :: "abc_inst list"   and b :: "nat"   and c :: "nat"   and args :: "nat list"   and B :: "abc_inst list"   and anything :: "nat list" assumes "rec_ci f = (a, b, c)"   and "\<forall>i. terminate f (args @ [i]) \<and> 0 < rec_exec f (args @ [i])"   and "B = [Dec (Suc (length args)) (length a + 5), Dec (Suc (length args)) (length a + 3), Goto (Suc (length a)), Inc (length args), Goto 0]" shows "{\<lambda>nl. nl = args @ 0 \<up> (max (Suc (length args)) c - length args) @ anything} a @ B \<up>"
reflects_iso: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b"   and f :: "'a"   and a' :: "'a"   and a :: "'a" assumes "fully_faithful_functor A B F"   and "partial_magma.in_hom A f a' a"   and "category.iso B (F f)" shows "category.iso A f"
match_result_sound_err: fixes env :: "v sem_env"   and s :: "'c state"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and err :: "v error_result"   and ck :: "bool" assumes "match_result env s v0 pes err_v = Rerr err" shows "evaluate_match ck env s v0 pes err_v (s, Rerr err)"
rotater_rev: fixes n :: "nat"   and ys :: "'b list" shows "rotater n ys = rev (rotate n (rev ys))"
in_intra_SDG_path_in_HRB_slice: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node"   and n'' :: "'b SDG_node"   and S :: "'b SDG_node set" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'"   and "n'' \<in> set ns"   and "n' \<in> S" shows "n'' \<in> SDG.HRB_slice sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses S"
less_def: fixes x :: "('b \<times> 'b) set"   and y :: "('b \<times> 'b) set" shows "(x \<subset> y) = (x \<subseteq> y \<and> x \<noteq> y)"
finite_extension_minimal: fixes K :: "'a set"   and xs :: "'a list" assumes "subring K R"   and "set xs \<subseteq> carrier R" shows "finite_extension K xs = \<Inter> {K'. subring K' R \<and> K \<subseteq> K' \<and> set xs \<subseteq> K'}"
fst_conv: fixes x1 :: "'l"   and x2 :: "'m" shows "fst (x1, x2) = x1"