hcju/isabelleps huggingface dataset:
intro: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "ringed_space S is_open \<FF> \<rho> b add_str mult_str zero_str one_str"
integral_norm_bound_integral: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> real" assumes "integrable M f"   and "integrable M g"   and "\<And>x. x \<in> space M \<Longrightarrow> norm (f x) \<le> g x" shows "norm (integral\<^sup>L M f) \<le> integral\<^sup>L M g"
Uniq_I: fixes P :: "'c \<Rightarrow> bool" assumes "\<And>x y. \<lbrakk>P x; P y\<rbrakk> \<Longrightarrow> y = x" shows "Uniq P"
length_concat: fixes xss :: "'a list list" shows "length (concat xss) = sum_list (map length xss)"
f0: shows "length (mat_to_cols_list (A * B)) = length M"
instant_i: fixes \<Gamma> :: "'c constr list"   and n :: "nat"   and \<Phi> :: "'c TESL_atomic list" shows "\<Gamma>, n \<turnstile> [] \<triangleright> \<Phi> \<hookrightarrow>\<^sub>i \<Gamma>, Suc n \<turnstile> \<Phi> \<triangleright> []"
assms_2: assumes "r\<lparr>a\<rparr> = b" shows "P"
unfold: shows "split = (\<Lambda> pat xs. If prefix\<cdot>xs\<cdot>pat then (xs, sdrop\<cdot>(slength\<cdot>xs)\<cdot>pat) else split\<cdot>pat\<cdot>(stail\<cdot>xs))"
enat_of_option_simps_4: fixes x :: "nat option" shows "(\<infinity> = Common.enat_of_option x) = (x = None)"
pres_exec: fixes q :: "nat"   and \<tau> :: "nat list"   and p :: "nat"   and ss :: "nat list list" assumes "(q, \<tau>) \<in> set (step p (ss ! p))"   and "\<forall>n\<in>set (ss ! p). n < p"   and "ss ! p \<in> A"   and "p < n" shows "\<tau> \<in> A"
lexmin_minimal: fixes C :: "(nat \<Rightarrow> 'a) set"   and y :: "'a"   and i :: "nat" assumes "C \<subseteq> SEQ A"   and "C \<noteq> {}"   and "y \<in> A"   and "P y (lexmin C i)" shows "y \<notin> ith (eq_upto C (lexmin C) i) i"
preserves_reflects_arr: fixes A :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and B :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and F :: "'ag \<Rightarrow> 'ah"   and G :: "'ag \<Rightarrow> 'ah"   and \<tau> :: "'ag \<Rightarrow> 'ah"   and f :: "'ag" assumes "natural_transformation A B F G \<tau>" shows "partial_magma.arr B (\<tau> f) = partial_magma.arr A f"
Pbij_inverse_Pbij: fixes \<beta> :: "(Location \<times> Location) set" assumes "\<beta> \<in> Pbij" shows "Pbij_inverse \<beta> \<in> Pbij"
normalize_ex_bound': fixes cs :: "(String.literal \<times> String.literal) set"   and s :: "String.literal set"   and c :: "String.literal" assumes "wf_subclass_loc cs"   and "finite s"   and "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" shows "\<exists>c'\<in>normalize_sort cs s. class_les cs c' c"
select_result_E: fixes P :: "'x \<Rightarrow> bool"   and h :: "'z"   and f :: "('z, 'y, 'x) prog" assumes "P |h \<turnstile> f|\<^sub>r"   and "h \<turnstile> ok f" obtains x :: "'x" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "P x"
fpairs_iff: fixes x :: "V"   and r :: "V" shows "(x \<in>\<^sub>\<circ> fpairs r) = (x \<in>\<^sub>\<circ> r \<and> (\<exists>a b. x = [a, b]\<^sub>\<circ>))"
Rep_vector_cases: fixes y :: "'b" assumes "y \<in> vectors" obtains x :: "'b vector" where "y = Rep_vector x"
assms_4: shows "\<not> is_zero_row_upt_k j k A"
simps_4: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f" shows "(case None of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f1"
gcs_fun: fixes s :: "'a \<Rightarrow> 'b"   and t :: "'a \<Rightarrow> 'b" shows "gcs s t = (\<lambda>x. min (s x) (t x))"
simps: fixes A :: "'at" shows "canonic_int_fun_ordered A = (\<lambda>S. \<exists>C. C \<in> S \<and> strictly_maximal_literal C (Pos A) \<and> (\<forall>B. Pos B \<in> C \<longrightarrow> (B, A) \<in> atom_ordering \<longrightarrow> \<not> canonic_int_fun_ordered B S) \<and> (\<forall>B. Neg B \<in> C \<longrightarrow> (B, A) \<in> atom_ordering \<longrightarrow> canonic_int_fun_ordered B S))"
f: shows "f \<in> N \<rightarrow>\<^sub>M Pi\<^sub>M I M"
ENFsr_offending_imp_not_P: fixes F :: "('v \<times> 'v) set"   and s :: "'v"   and r :: "'v" assumes "F \<in> set_offending_flows G nP"   and "(s, r) \<in> F" shows "\<not> P (nP s) s (nP r) r"
monotone_Test: fixes X :: "(variable \<Rightarrow> real) set"   and Y :: "(variable \<Rightarrow> real) set"   and I :: "interp"   and \<phi> :: "fml" assumes "X \<subseteq> Y" shows "game_sem I (? \<phi>) X \<subseteq> game_sem I (? \<phi>) Y"
insert_remove: fixes A :: "'b set"   and x :: "'b" assumes "finite A" shows "card (insert x A) = Suc (card (A - {x}))"
switch_symm: fixes v :: "view"   and c :: "cars"   and u :: "view" shows "v = c > u \<longrightarrow> u = own v > v"
map_pred_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c \<Rightarrow> 'd option" shows "map_pred P m = (\<forall>x. case m x of None \<Rightarrow> True | Some y \<Rightarrow> P x y)"
tensor_assoc: fixes f0 :: "'d"   and f1 :: "'d"   and f2 :: "'d" assumes "D.arr f0"   and "D.arr f1"   and "D.arr f2" shows "(f0 \<otimes>\<^sub>D f1) \<otimes>\<^sub>D f2 = f0 \<otimes>\<^sub>D f1 \<otimes>\<^sub>D f2"
ring_of_sets_def: fixes \<Omega> :: "'c set"   and M :: "'c set set" shows "ring_of_sets \<Omega> M \<equiv> semiring_of_sets \<Omega> M \<and> ring_of_sets_axioms M"
is_constant_constructorsI_19: fixes x :: "'a ltln"   and n :: "nat" assumes "is_constant x" shows "is_constant (X\<^sub>n' n x)"
exhaust_sel: fixes list :: "'c list" obtains "list = []"   | "list = list.hd list # list.tl list"
box_borel: fixes a :: "'c"   and b :: "'c" shows "box a b \<in> sets borel"
abelian_groupE_1: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_group R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<in> carrier R"
assms_2: shows "mod_type_class.to_nat a = mod_type_class.to_nat b"
pc_l: shows "pc + length (i # ls) = length \<tau>s"
measurable_Un_null_set: fixes B :: "'b set"   and M :: "'b measure"   and A :: "'b set" assumes "B \<in> null_sets M" shows "(A \<union> B \<in> fmeasurable M \<and> A \<in> sets M) = (A \<in> fmeasurable M)"
hyps_3: assumes "f \<in> A \<rightarrow> carrier G"   and "g \<in> A \<rightarrow> carrier G" shows "(\<Otimes>x\<in>A. f x \<otimes> g x) = finprod G f A \<otimes> finprod G g A"
Pj_def: fixes G :: "('c, 'd) Group_scheme"   and H :: "'c set" shows "Pj G H = restrict ((\<bullet>\<^bsub>G\<^esub>) H) (carrier G)"
P_Gauss_Jordan_def: fixes A :: "(('d, 'f) vec, 'e) vec" shows "P_Gauss_Jordan A = fst (Gauss_Jordan_PA A)"
ltl_Release_StrongRelease_con_2: fixes \<xi> :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltlc"   and \<psi> :: "'b ltlc" shows "\<xi> \<Turnstile>\<^sub>c \<phi> M\<^sub>c \<psi> = \<xi> \<Turnstile>\<^sub>c \<phi> R\<^sub>c \<psi> and\<^sub>c F\<^sub>c \<phi>"
simps: fixes S :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list" shows "start_proof S = ([S], S)"
ExecRandomNoVar: fixes v :: "char list"   and \<sigma> :: "char list \<Rightarrow> nat option" assumes "v \<notin> dom \<sigma>" shows "Random v \<rhd> \<sigma> \<leadsto> None"
unit_def: fixes a :: "'b" shows "HoG.unit a \<equiv> THE \<psi>. \<guillemotleft>\<psi> : HoG.map\<^sub>0 a \<Rightarrow>\<^sub>D HoG.map a\<guillemotright> \<and> D.iso \<psi> \<and> \<psi> \<cdot>\<^sub>D \<i>\<^sub>D[HoG.map\<^sub>0 a] = (HoG.map \<i>\<^sub>B[a] \<cdot>\<^sub>D HoG.cmp (a, a)) \<cdot>\<^sub>D (\<psi> \<star>\<^sub>D \<psi>)"
a: shows "emeasure M A = emeasure M (A \<inter> B)"
src_trg: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "src (trg \<mu>) = trg \<mu>"
create_character_data_def: fixes get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val :: "(_) character_data_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_val_locs :: "(_) character_data_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and create_character_data :: "(_) document_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) character_data_ptr) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and document_ptr :: "(_) document_ptr"   and "text" :: "char list" assumes "l_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_val set_val_locs type_wf create_character_data known_ptr" shows "create_character_data document_ptr text = Heap_Error_Monad.bind new_character_data (\<lambda>new_character_data_ptr. Heap_Error_Monad.bind (set_val new_character_data_ptr text) (\<lambda>_. Heap_Error_Monad.bind (get_disconnected_nodes document_ptr) (\<lambda>disc_nodes. Heap_Error_Monad.bind (set_disconnected_nodes document_ptr (cast\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr # disc_nodes)) (\<lambda>_. return new_character_data_ptr))))"
Rep_one_or_two_inject: fixes x :: "'a one_or_two"   and y :: "'a one_or_two" shows "(Rep_one_or_two x = Rep_one_or_two y) = (x = y)"
borel_measurable_indicator: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets M" shows "indicator A \<in> borel_measurable M"
simps_2: fixes m :: "'a"   and l :: "('a, 'b) rose_tree\<^sub>m list"   and r :: "('a, 'b) rose_tree\<^sub>m list"   and z :: "('a \<times> ('a, 'b) rose_tree\<^sub>m list \<times> ('a, 'b) rose_tree\<^sub>m list) list"   and t :: "('a, 'b) rose_tree\<^sub>m" shows "tree_of_zipper\<^sub>m ((m, l, r) # z, t) = tree_of_zipper\<^sub>m (z, Tree\<^sub>m (Unblinded (m, l @ t # r)))"
WhileFalse: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com" assumes "s \<notin> b" shows "\<Gamma>\<turnstile>While b c \<down> Normal s"
Afix_def: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "ArityAnalysis.Afix Aexp \<Gamma> = (\<Lambda> ae. \<mu> x. ArityAnalysis.ABinds Aexp \<Gamma>\<cdot>x \<squnion> ae)"
tab_def: shows "TfTgh.tab \<equiv> \<a>\<^sup>-\<^sup>1[f, g \<star> h, (tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0) \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0] \<cdot> (f \<star> \<a>[g \<star> h, tab\<^sub>0 h \<star> g\<^sub>0h\<^sub>1.p\<^sub>0, f\<^sub>0gh\<^sub>1.p\<^sub>0]) \<cdot> (f \<star> Tgh.\<rho>\<sigma>.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>0) \<cdot> (f \<star> f\<^sub>0gh\<^sub>1.\<phi>) \<cdot> \<a>[f, tab\<^sub>0 f, f\<^sub>0gh\<^sub>1.p\<^sub>1] \<cdot> (f.tab \<star> f\<^sub>0gh\<^sub>1.p\<^sub>1)"
x: shows "x \<in> (\<Oplus>G\<leftarrow>Gs. G)"
compositionality: fixes f :: "'c \<Rightarrow> 'b"   and g :: "'d \<Rightarrow> 'c"   and multiset :: "'d multiset" shows "image_mset f (image_mset g multiset) = image_mset (f \<circ> g) multiset"
sub_def: fixes as :: "'a iarray"   and n :: "nat" shows "as !! n = list_of as ! n"
max_min_same_1: fixes x :: "'a"   and y :: "'a" shows "max x (min x y) = x"
vector_cases_append: fixes x :: "'a vec" shows "x = vNil \<or> (\<exists>v a. x = v @\<^sub>v [a]\<^sub>v)"
rreq_rrep_fresh_2: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and oip :: "nat"   and sip :: "nat" shows "rreq_rrep_fresh crt (Rrep hops dip dsn oip sip) = (sip \<noteq> dip \<longrightarrow> dip \<in> kD crt \<and> sqn crt dip = dsn \<and> the (dhops crt dip) = hops \<and> the (flag crt dip) = val)"
f1: shows "{j. select_index 3 1 j} = {2, 3, 6, 7}"
unit_in_vhom: shows "\<guillemotleft>\<eta> : src f \<Rightarrow> f\<^sup>* \<star> f\<guillemotright>"
keys_map_scale_subset: fixes k :: "'d"   and t :: "'c \<Rightarrow>\<^sub>0 'd" shows "keys (k \<cdot> t) \<subseteq> keys t"
simps: fixes M2 :: "('d, 'e, 'f) FSM"   and M1 :: "('d, 'e, 'g) FSM"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list"   and T :: "'d list set"   and S :: "'f set"   and \<Omega> :: "('d, 'e) ATC set"   and V'' :: "('d \<times> 'e) list set" shows "Prereq M2 M1 vs xs T S \<Omega> V'' = (finite T \<and> vs @ xs \<in> L M2 \<inter> L M1 \<and> S \<subseteq> nodes M2 \<and> (\<forall>s1\<in>S. \<forall>s2\<in>S. s1 \<noteq> s2 \<longrightarrow> (\<forall>io1\<in>RP M2 s1 vs xs V''. \<forall>io2\<in>RP M2 s2 vs xs V''. B M1 io1 \<Omega> \<noteq> B M1 io2 \<Omega>)))"
simps_3: fixes x :: "'b"   and y :: "'b"   and xs :: "'b list" shows "Missing_Lemmas.max_list (x # y # xs) = max x (Missing_Lemmas.max_list (y # xs))"
simps: fixes xs :: "nat list"   and ys :: "nat list"   and us :: "nat list"   and vs :: "nat list" shows "((xs, ys) <\<^sub>r\<^sub>l\<^sub>e\<^sub>x\<^sub>2 (us, vs)) = (xs @ ys <\<^sub>r\<^sub>l\<^sub>e\<^sub>x us @ vs)"
steps_appendI: fixes xs :: "'a list"   and x :: "'a"   and y :: "'a" assumes "steps (xs @ [x])"   and "E x y" shows "steps (xs @ [x, y])"
bounded_antilinear_eq_on: fixes A :: "'c \<Rightarrow> 'd"   and B :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and t :: "'c" assumes "bounded_antilinear A"   and "bounded_antilinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
simps: fixes a1 :: "'b"   and a2 :: "nat"   and a3 :: "'b fset" shows "(a1 |\<in>|\<^bsub>a2\<^esub> a3) = (\<exists>i s. a1 = list_of_fset s ! i \<and> a2 = i \<and> a3 = s \<and> i < size s)"
frechet_derivative_f_at_neg: fixes x :: "real" assumes "x \<in> {..<0}" shows "frechet_derivative f (at x) = (\<lambda>x. 0)"
delete_node_def: fixes v :: "'b"   and G :: "('b, 'c) graph_scheme" shows "FiniteGraph.delete_node v G \<equiv> \<lparr>nodes = nodes G - {v}, edges = {(e1, e2). (e1, e2) \<in> edges G \<and> e1 \<noteq> v \<and> e2 \<noteq> v}\<rparr>"
sub_map_finer_than_commute: fixes T :: "'c topology"   and f :: "'c \<Rightarrow> 'd"   and X :: "'d set" assumes "openin T (f -` X)" shows "subtopology (map_topology f T) X finer_than map_topology f (subtopology T (f -` X))"
trans: shows "\<forall>P Q R. (P, Q) \<in> Rel \<and> (Q, R) \<in> Rel \<and> (P \<in> ProcS \<and> Q \<in> ProcT \<or> P \<in> ProcT \<and> Q \<in> ProcS) \<longrightarrow> (P, R) \<in> Rel"
field_inverse: fixes a :: "'c" assumes "a \<noteq> (0::'c)" shows "inverse a * a = (1::'c)"
open_countable_Union_open_cbox: fixes S :: "'b set" assumes "open S" obtains \<D> :: "'b set set" where "countable \<D>" and "\<D> \<subseteq> Pow S" and "\<And>X. X \<in> \<D> \<Longrightarrow> \<exists>a b. X = cbox a b" and "\<Union> \<D> = S"
execute_parallel_operator_def: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list" shows "execute_parallel_operator s ops \<equiv> foldl (++) s (map (map_of \<circ> effect_to_assignments) ops)"
compose_const_1: fixes f :: "'e \<Rightarrow> 'd"   and a :: "'e" shows "f \<circ> (\<lambda>x. a) = (\<lambda>x. f a)"
inj_iff: fixes f :: "'c \<Rightarrow> 'd" shows "inj f = (inv f \<circ> f = id)"
unique_eigenvector_complex: fixes A :: "complex mat"   and n :: "nat"   and lam :: "complex"   and v :: "complex Matrix.vec"   and w :: "complex Matrix.vec" assumes "A \<in> carrier_mat n n"   and "order lam (char_poly A) = 1"   and "eigenvector A v lam"   and "eigenvector A w lam" shows "\<exists>a. v = a \<cdot>\<^sub>v w"
\<gamma>_def: shows "\<gamma> \<equiv> \<lambda>x. \<gamma>0 x \<inter> ball x (\<epsilon> / 8 / (norm (f x) + 1) * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / content (cbox a b))"
fcs_a_neq_a': shows "G.face_cycle_succ a' \<noteq> a"
multi_drop_mem_not_eq: fixes c :: "'b"   and B :: "'b multiset" assumes "c \<in># B" shows "B - {#c#} \<noteq> B"
sum_squares_le_zero_iff: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.linordered_ring_strict plus zero minus uminus less_eq less abs times" shows "less_eq (plus (times x x) (times y y)) zero = (x = zero \<and> y = zero)"
cterms_def: shows "cterms \<equiv> \<lambda>\<Gamma>. {x. ctermsp \<Gamma> x}"
call_zero_iff: fixes x :: "'a" shows "(\<forall>u. cinner x u = 0) = (x = (0::'a))"
enforce_return_pmf_None: fixes P :: "'c \<Rightarrow> bool" shows "enforce_spmf P (return_pmf None) = return_pmf None"
semantics_stateful_packet_tagging_intro_process_one: fixes \<Gamma> :: "char list \<Rightarrow> 'a rule list option"   and \<gamma> :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and packet_tagger :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and state_upate :: "'c \<Rightarrow> final_decision \<Rightarrow> 'd \<Rightarrow> 'c"   and \<sigma>\<^sub>0 :: "'c"   and built_in_chain :: "char list"   and default_policy :: "action"   and p :: "'d"   and ps :: "'d list"   and ps_processed_old :: "('d \<times> final_decision) list"   and \<sigma>_old :: "'c"   and X :: "final_decision"   and \<sigma>' :: "'c"   and ps_processed :: "('d \<times> final_decision) list" assumes "semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) (p # ps) ps_processed_old \<sigma>_old"   and "\<Gamma>,\<gamma>,packet_tagger \<sigma>_old p\<turnstile> \<langle>[Rule MatchAny (Call built_in_chain), Rule MatchAny default_policy], Undecided\<rangle> \<Rightarrow> Decision X"   and "\<sigma>' = state_upate \<sigma>_old X p"   and "ps_processed = ps_processed_old @ [(p, X)]" shows "semantics_stateful_packet_tagging \<Gamma> \<gamma> packet_tagger state_upate \<sigma>\<^sub>0 (built_in_chain, default_policy) ps ps_processed \<sigma>'"
map\<^sub>1_def: fixes f :: "'c" shows "map\<^sub>1 f = \<a>\<^sub>D\<^sup>-\<^sup>1[\<rho>\<^sub>0 (trg\<^sub>C f), \<sigma>\<^sub>0 (trg\<^sub>C f), F f] \<cdot>\<^sub>D (\<rho>\<^sub>0 (trg\<^sub>C f) \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[\<rho>\<^sub>0 (trg\<^sub>C f), G f, \<sigma>\<^sub>0 (src\<^sub>C f)] \<cdot>\<^sub>D (\<rho>\<^sub>1 f \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f)) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[H f, \<rho>\<^sub>0 (src\<^sub>C f), \<sigma>\<^sub>0 (src\<^sub>C f)]"
swap_id_idempotent: fixes a :: "'c"   and b :: "'c" shows "Fun.swap a b id \<circ> Fun.swap a b id = id"
antisym_conv2: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" assumes "ko.le ko x y" shows "(\<not> ko.lt ko x y) = (x = y)"
cmult': fixes c :: "'a"   and F :: "'b filter"   and f :: "'b \<Rightarrow> 'a" assumes "c \<noteq> (0::'a)" shows "O[F](\<lambda>x. f x * c) = O[F](f)"
xor_def2: fixes x :: "'c"   and y :: "'c" shows "x XOR y = (x OR y) AND (NOT x OR NOT y)"
sr_imp_eigen_vector: fixes u :: "(real, 'n) Finite_Cartesian_Product.vec" assumes "u \<in> X"   and "r u = sr" shows "eigen_vector A u sr"
ord_termI: fixes v :: "'t"   and w :: "'t" assumes "pp_of_term v \<preceq> pp_of_term w"   and "component_of_term v \<le> component_of_term w" shows "v \<preceq>\<^sub>t w"
Arbitrary_ExceptD: fixes v :: "('a \<times> bool list) list"   and n :: "nat"   and r :: "nat"   and b :: "bool" assumes "v \<in> lang n (Atom (Arbitrary_Except r b))"   and "r < n" shows "\<exists>x. v = [x] \<and> snd x ! r = b"
relation_subsumption_list_axioms: shows "relation_subsumption_list r subsumes"
UN_equiv_class_type: fixes A :: "'b set"   and r :: "('b \<times> 'b) set"   and f :: "'b \<Rightarrow> 'c set"   and X :: "'b set"   and B :: "'c set set" assumes "equiv A r"   and "f respects r"   and "X \<in> A // r"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> B" shows "\<Union> (f ` X) \<in> B"
lm010: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R = (R -- x) +< {x} \<times> R `` {x}"
misom2Tr: fixes H :: "'a set"   and N :: "'a set" assumes "submodule R M H"   and "submodule R M N"   and "H \<subseteq> N" shows "M /\<^sub>m H /\<^sub>m carrier (mdl M N /\<^sub>m H) \<cong>\<^bsub>R\<^esub> M /\<^sub>m N"
absorb2: fixes a :: "'t"   and b :: "'t" assumes "a \<preceq>\<^sub>t b" shows "ord_term_lin.max a b = b"
Vfrom_in_mono: fixes A :: "V"   and B :: "V"   and i :: "V"   and j :: "V" assumes "A \<subseteq>\<^sub>\<circ> B"   and "i \<in>\<^sub>\<circ> j" shows "Vfrom A i \<in>\<^sub>\<circ> Vfrom B j"
dtor_unfold: fixes s :: "'d \<Rightarrow> ('c, 'd coF) coG0"   and b :: "'d" shows "dtor_coG (dtor_unfold_coG s b) = map_pre_coG id (dtor_unfold_coG s) (s b)"
num_vars_geq_num_unique_vars: fixes t :: "('a, 'b) Term.term" shows "num_unique_vars t \<le> num_vars t"
ccw'_polychain_of_sorted_center_last: fixes c :: "real \<times> real"   and d :: "real \<times> real"   and x0 :: "real \<times> real"   and xs :: "(real \<times> real) list" assumes "(c, d) \<in> set (butlast (polychain_of x0 xs))"   and "ccw'.sortedP 0 xs"   and "xs \<noteq> []" shows "ccw' x0 d (snd (last (polychain_of x0 xs)))"
lnth_Suc_LCons: fixes x :: "'b"   and xs :: "'b llist"   and n :: "nat" shows "(x % xs) ?! Suc n = xs ?! n"
matches_dest_2: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and m' :: "'a match_expr"   and a :: "action"   and p :: "'b" assumes "matches \<gamma> (MatchAnd m m') a p"   and "\<not> matches \<gamma> m' a p" shows "False"
unif_sub: fixes \<sigma> :: "char list \<Rightarrow> fterm"   and L :: "fterm literal set" assumes "unifier\<^sub>l\<^sub>s \<sigma> L"   and "L \<noteq> {}" shows "\<exists>l. L \<cdot>\<^sub>l\<^sub>s \<sigma> = {l \<cdot>\<^sub>l \<sigma>}"
assms: shows "x |\<in>| A |\<times>| B"
core_argument: fixes p\<^sub>0 :: "int \<times> int"   and ps :: "(int \<times> int) list"   and \<delta> :: "real"   and ps\<^sub>L :: "(int \<times> int) set"   and ps\<^sub>R :: "(int \<times> int) set"   and l :: "int"   and p\<^sub>1 :: "int \<times> int" assumes "distinct (p\<^sub>0 # ps)"   and "sorted_snd (p\<^sub>0 # ps)"   and "0 \<le> \<delta>"   and "set (p\<^sub>0 # ps) = ps\<^sub>L \<union> ps\<^sub>R"   and "\<forall>p\<in>set (p\<^sub>0 # ps). real_of_int l - \<delta> \<le> real_of_int (fst p) \<and> real_of_int (fst p) \<le> real_of_int l + \<delta>"   and "\<forall>p\<in>ps\<^sub>L. fst p \<le> l"   and "\<forall>p\<in>ps\<^sub>R. l \<le> fst p"   and "sparse \<delta> ps\<^sub>L"   and "sparse \<delta> ps\<^sub>R"   and "p\<^sub>1 \<in> set ps"   and "dist (case p\<^sub>0 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) (case p\<^sub>1 of (x, y) \<Rightarrow> (real_of_int x, real_of_int y)) < \<delta>" shows "p\<^sub>1 \<in> set (take 7 ps)"
fst_sym_preproc_addnew_superset: fixes vs :: "'t list"   and gs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and fs :: "('t \<Rightarrow>\<^sub>0 'c) list"   and v :: "'t" shows "set vs \<subseteq> set (fst (sym_preproc_addnew gs vs fs v))"
select_convs_2: fixes set_dom :: "'b set"   and set_func :: "'b \<Rightarrow> 'b"   and set_cod :: "'b set"   and more :: "'c" shows "set_func \<lparr>set_dom = set_dom, set_func = set_func, set_cod = set_cod, \<dots> = more\<rparr> = set_func"
finite_juncts_1: fixes \<phi> :: "('a, 'b) aformula" shows "finite (disjuncts \<phi>)"
integral_std_normal_moment_even: fixes k :: "nat" shows "LBINT x. std_normal_density x * x ^ (2 * k) = fact (2 * k) / (2 ^ k * fact k)"
unitary_sound: fixes P :: "'a \<Rightarrow> real" assumes "unitary P" shows "sound P"