hcju/isabelleps huggingface dataset:
dlet5: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and err :: "v error_result"   and p :: "pat"   and mn :: "char list list"   and locs :: "locn \<times> locn" assumes "evaluate ck env s e (s', Rerr err) \<and> allDistinct (pat_bindings p [])" shows "evaluate_dec ck mn env s (Dlet locs p e) (s', Rerr err)"
run_asset_i_def: fixes n :: "nat"   and s :: "(agent \<times> msg) set" shows "run_asset_i n s \<equiv> SOME x. pred_asset_i n s x"
IH: fixes cs :: "'edge list" shows "upd_cs cs (as' @ [a]) = a # upd_cs cs as'"
cmp_simps_5: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.cod (\<Phi> (f, g)) = F (f \<star>\<^sub>C g)"
integrable_cong: fixes M :: "'b measure"   and N :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "M = N"   and "\<And>x. x \<in> space N \<Longrightarrow> f x = g x" shows "integrable M f = integrable N g"
image_diff_subset: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and B :: "'b set" shows "f ` A - f ` B \<subseteq> f ` (A - B)"
PTuple: fixes p :: "pat"   and T :: "ty"   and \<Delta>\<^sub>1 :: "(name \<times> ty) list"   and q :: "pat"   and U :: "ty"   and \<Delta>\<^sub>2 :: "(name \<times> ty) list" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>\<^sub>1"   and "\<turnstile> q : U \<Rightarrow> \<Delta>\<^sub>2" shows "\<turnstile> \<langle>\<langle>p, q\<rangle>\<rangle> : T \<otimes> U \<Rightarrow> \<Delta>\<^sub>2 @ \<Delta>\<^sub>1"
add_assoc: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> V"   and "y \<in> V"   and "z \<in> V" shows "x + y + z = x + (y + z)"
complete_finite_edges: fixes V :: "nat set" assumes "finite V" shows "finite (uedges (Ugraph_Lemmas.complete V))"
ext_inject: fixes edge :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more :: "'b"   and edge' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and more' :: "'b" shows "(\<lparr>edge = edge, \<dots> = more\<rparr> = \<lparr>edge = edge', \<dots> = more'\<rparr>) = (edge = edge' \<and> more = more')"
steps_SCons_iff: fixes E :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "Graph_Defs.steps E (x # y # xs) = (E x y \<and> Graph_Defs.steps E (y # xs))"
inf_sup_ord_4: fixes inf :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and less_eq :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and less :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> bool"   and sup :: "'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type"   and y :: "'b::type"   and x :: "'b::type" assumes "class.lattice (inf::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (less::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type)" shows "(less_eq::'b::type \<Rightarrow> 'b::type \<Rightarrow> bool) (y::'b::type) ((sup::'b::type \<Rightarrow> 'b::type \<Rightarrow> 'b::type) (x::'b::type) y)"
rawpsubst_exi: fixes \<phi> :: "'fmla"   and y :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "\<phi> \<in> fmla"   and "y \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "y \<notin> snd ` set txs"   and "y \<notin> \<Union> (FvarsT ` fst ` set txs)" shows "rawpsubst (exi y \<phi>) txs = exi y (rawpsubst \<phi> txs)"
truncate_up: fixes x :: "real"   and prec :: "nat" shows "x \<le> truncate_up prec x"
l8_15_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and X :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B X"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C X" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C X"
HCond_Idem: fixes P :: "'a hrel" shows "\<H> (\<H> P) = \<H> P"
x2_def: shows "x2 = Abs_real_int (s, right x)"
1: shows "{} \<turnstile> \<beta>(i::=\<guillemotleft>\<beta>\<guillemotright>) IFF (SyntaxN.Ex j (KRP \<guillemotleft>Var i\<guillemotright> (Var i) (Var j) AND \<alpha>(i::=Var j)))(i::=\<guillemotleft>\<beta>\<guillemotright>)"
lexext_compat_snoc_if_same_length: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" assumes "length ys = length xs"   and "lexext gt ys xs" shows "lexext gt (ys @ [x]) (xs @ [x])"
take_all: fixes xs :: "'b list"   and n :: "nat" assumes "length xs \<le> n" shows "xs \<down> n = xs"
iIN_iTILL_iTILL_conv: fixes n :: "nat"   and d :: "nat" assumes "0 < n" shows "[n\<dots>,d] = [\<dots>n + d] - [\<dots>n - Suc 0]"
insertI1: fixes a :: "'e"   and B :: "'e set" shows "a \<in> Set.insert a B"
cong: fixes f :: "'l \<Rightarrow> 'm"   and g :: "'l \<Rightarrow> 'm"   and x :: "'l"   and y :: "'l" assumes "f = g"   and "x = y" shows "f x = g y"
quasiconvexD: fixes C :: "real"   and X :: "'b set"   and x :: "'b"   and y :: "'b" assumes "quasiconvex C X"   and "x \<in> X"   and "y \<in> X" shows "\<exists>G. geodesic_segment_between G x y \<and> (\<forall>z\<in>G. infdist z X \<le> C)"
exhaustiveUnitPropagate_unfold: fixes state :: "State" shows "exhaustiveUnitPropagate state = (if getConflictFlag state \<or> getQ state = [] then state else exhaustiveUnitPropagate (applyUnitPropagate state))"
pm_of_idx_pm_monom_mult: fixes xs :: "'c list"   and c :: "'d"   and f :: "nat \<Rightarrow>\<^sub>0 'd" shows "pm_of_idx_pm xs (MPoly_Type_Class.punit.monom_mult c 0 f) = MPoly_Type_Class.punit.monom_mult c (0::'c) (pm_of_idx_pm xs f)"
get_component_subset: fixes heap_is_wellformed :: "(_) heap \<Rightarrow> bool"   and parent_child_rel :: "(_) heap \<Rightarrow> ((_) object_ptr \<times> (_) object_ptr) set"   and type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool"   and to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and get_parent_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   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"   and get_component :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and is_strongly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and is_weakly_dom_component_safe :: "(_) object_ptr set \<Rightarrow> (_) object_ptr set \<Rightarrow> (_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_root_node_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_ancestors :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and get_ancestors_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_element_by_id :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr option) prog"   and get_elements_by_tag_name :: "(_) object_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, (_) element_ptr list) prog"   and h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr" assumes "l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M heap_is_wellformed parent_child_rel type_wf known_ptr known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_disconnected_nodes get_element_by_id get_elements_by_tag_name"   and "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_component ptr \<rightarrow>\<^sub>r c"   and "ptr' \<in> set c" shows "h \<turnstile> get_component ptr' \<rightarrow>\<^sub>r c"
offs_next_prev: fixes ns :: "nat list"   and xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and j :: "nat" shows "(0 < offs_num (length ns) xs index key mi ma i \<and> offs_set_next ns xs index key mi ma i \<noteq> {} \<and> Min (offs_set_next ns xs index key mi ma i) = j) = (0 < offs_num (length ns) xs index key mi ma j \<and> offs_set_prev ns xs index key mi ma j \<noteq> {} \<and> Max (offs_set_prev ns xs index key mi ma j) = i)"
init_par_comp: fixes s :: "('c, 'e seq_action) automaton"   and t :: "('d, 'e seq_action) automaton" shows "init (s \<langle>\<langle> t) = init s \<times> init t"
span_add_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "('b \<Rightarrow>\<^sub>0 'c) set"   and y :: "'b \<Rightarrow>\<^sub>0 'c" assumes "x \<in> phull S" shows "(x + y \<in> phull S) = (y \<in> phull S)"
assms_1: shows "\<forall>y\<in>set ys \<union> set xs. \<forall>x\<in>set ys \<union> set xs. y >\<^sub>t x \<longrightarrow> h y >\<^sub>t h x"
LEX_imp_not_LEX: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "LEX P f g"   and "\<And>x y z. \<lbrakk>P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "\<And>x. \<not> P x x" shows "\<not> LEX P g f"
netglobal_def: fixes P :: "(nat \<Rightarrow> state) \<Rightarrow> bool" shows "opq.netglobal P \<equiv> \<lambda>s. P (fst (opq.initmissing (netgmap (\<lambda>(p, q). (fst (id p), snd (id p), q)) s)))"
sig_red_Un: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and B :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "sig_red sing_reg top_tail (A \<union> B) p q = (sig_red sing_reg top_tail A p q \<or> sig_red sing_reg top_tail B p q)"
mult_less_cancel_right1: 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 c :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less c (times b c) = ((less_eq zero c \<longrightarrow> less one b) \<and> (less_eq c zero \<longrightarrow> less b one))"
atm_of_subst_lit: fixes subst_atm :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and L :: "'a literal"   and \<sigma> :: "'b" shows "atm_of (substitution_ops.subst_lit subst_atm L \<sigma>) = subst_atm (atm_of L) \<sigma>"
norm_sum: fixes f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "norm (sum f A) \<le> (\<Sum>i\<in>A. norm (f i))"
g': assumes "x \<in> T \<union> closure S \<inter> closure T" shows "(g has_vector_derivative g' x) (at x within T \<union> closure S \<inter> closure T)"
NMPDA: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "DenyAll \<in> set p" shows "not_MT Cp p"
star_conv_concat: fixes A :: "'b list set" shows "star A = {concat ws |ws. set ws \<subseteq> A \<and> [] \<notin> set ws}"
correct': fixes \<D> :: "nat \<Rightarrow> ('c + 'e, 'd + 'f) distinguisher" assumes "\<And>\<eta>. \<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<D> \<eta> \<surd>"   and "\<And>\<eta>. interaction_any_bounded_by (\<D> \<eta>) (bound \<eta>)"   and "\<And>\<eta>. lossless \<Longrightarrow> plossless_gpv (\<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<D> \<eta>)" shows "(\<forall>\<eta>. wiring (\<I>_ideal \<eta>) (\<I>_real \<eta>) (cnv \<eta>) (w \<eta>)) \<and> Negligible.negligible (\<lambda>\<eta>. advantage (\<D> \<eta>) (ideal_resource \<eta>) (cnv \<eta> |\<^sub>= 1\<^sub>C \<rhd> real_resource \<eta>))"
push_down_state_rank_token_run: fixes q :: "'a ltl_prop_equiv_quotient"   and n :: "nat"   and i :: "nat" assumes "\<UU>.state_rank q n = Some i" shows "\<exists>x. \<UU>.token_run x n = q \<and> x \<le> n"
assms_3: shows "f : a \<mapsto>\<^bsub>\<DD>\<^esub> a"
interleave_append_tail_sym: fixes s :: "'a event list"   and t :: "'a event list"   and C :: "'a event set"   and u1 :: "'a event list"   and u2 :: "'a event list" assumes "s setinterleaves ((t, u1), C)"   and "set u2 \<inter> C = {}" shows "s @ u2 setinterleaves ((t, u1 @ u2), C)"
less_eq_fset_def: shows "(|\<subseteq>|) \<equiv> map_fun fset (map_fun fset id) (\<subseteq>)"
star_sum_var: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> + y\<^sup>\<star>)\<^sup>\<star> = (x + y)\<^sup>\<star>"
cases: fixes r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a1 :: "'c"   and a2 :: "'b" assumes "r\<inverse>\<inverse> a1 a2" obtains a :: "'b" and b :: "'c" where "a1 = b" and "a2 = a" and "r a b"
ennreal2_cases: fixes x :: "ennreal"   and xa :: "ennreal" obtains r :: "real" and ra :: "real" where "0 \<le> r__" and "x = ennreal r__" and "0 \<le> ra" and "xa = ennreal ra"   | r :: "real" where "0 \<le> rb__" and "x = ennreal rb__" and "xa = top"   | r :: "real" where "x = top" and "0 \<le> r" and "xa = ennreal r"   | "x = top" and "xa = top"
lsemimodule5: fixes next_behaviour :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and next_stimulus :: "'c \<times> 'd \<Rightarrow> 'c"   and a :: "'d" assumes "c2ka next_behaviour next_stimulus" shows "next_behaviour \<dd> a = (0::'d)"
relatedmain: shows "[c] \<approx>\<^bsub>d\<^esub> [c']"
update_CWP_def: fixes cwp_val :: "word5"   and psr_val :: "word32" shows "update_CWP cwp_val psr_val \<equiv> let tmp0 = psr_val AND 4294967264; s_val = ucast (get_S psr_val) in if s_val = 0 then (tmp0 OR ucast cwp_val) AND 4294967167 else (tmp0 OR ucast cwp_val) OR 128"
sin_inj: fixes \<alpha> :: "real"   and \<alpha>' :: "real" assumes "- pi / 2 \<le> \<alpha> \<and> \<alpha> \<le> pi / 2"   and "- pi / 2 \<le> \<alpha>' \<and> \<alpha>' \<le> pi / 2"   and "\<alpha> \<noteq> \<alpha>'" shows "sin \<alpha> \<noteq> sin \<alpha>'"
zx: shows "\<omega>_limit_set z \<subseteq> \<omega>_limit_set x"
Always_LeadsTo_weaken: fixes F :: "'b program"   and C :: "'b set"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set"   and B' :: "'b set" assumes "F \<in> Always C"   and "F \<in> A \<longmapsto>w A'"   and "C \<inter> B \<subseteq> A"   and "C \<inter> A' \<subseteq> B'" shows "F \<in> B \<longmapsto>w B'"
p7: shows "{OrdP (Var k)} \<turnstile> (SyntaxN.All lj (SeqQuoteP (Var i) (Var i') (Var si) ki IMP SeqQuoteP (Var j) (Var j') (Var sj) (Var lj) IMP HaddP ki (Var lj) (Var k) IMP (Var i IN Var j IMP PfP (Q_Mem (Var i') (Var j'))) AND (Var i SUBS Var j IMP PfP (Q_Subset (Var i') (Var j')))))(sj::=Var sj)"
assms: shows "minimal_sequence_to_failure_extending V M1 M2 vs xs"
simps_1: fixes hsx :: "'b tree"   and x :: "'b"   and hs :: "'b tree" shows "pass\<^sub>2 \<langle>hsx, x, hs\<rangle> = link \<langle>hsx, x, pass\<^sub>2 hs\<rangle>"
set_iterator_product_def: fixes it_a :: "('e \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e"   and it_b :: "'f \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('g \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "set_iterator_product it_a it_b \<equiv> \<lambda>c f. it_a c (\<lambda>a. it_b a c (\<lambda>b. f (a, b)))"
cases: fixes less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a1 :: "'a list"   and a2 :: "'a list" assumes "ord.lexordp less a1 a2" obtains y :: "'a" and ys :: "'a list" where "a1 = []" and "a2 = y__ # ys__"   | x :: "'a" and y :: "'a" and xs :: "'a list" and ys :: "'a list" where "a1 = x__ # xs__" and "a2 = ya__ # ysa__" and "less x__ ya__"   | x :: "'a" and y :: "'a" and xs :: "'a list" and ys :: "'a list" where "a1 = x # xs" and "a2 = y # ys" and "\<not> less x y" and "\<not> less y x" and "ord.lexordp less xs ys"
measurable_distr_eq1: fixes Mf :: "'d measure"   and Nf :: "'b measure"   and f :: "'d \<Rightarrow> 'b"   and Mf' :: "'c measure" shows "distr Mf Nf f \<rightarrow>\<^sub>M Mf' = Nf \<rightarrow>\<^sub>M Mf'"
single_pp_plus: fixes x :: "'b"   and a :: "'c"   and b :: "'c" shows "single_pp x a + single_pp x b = single_pp x (a + b)"
OclIncluding_valid_args_valid: fixes \<tau> :: "'d state \<times> 'd state"   and X :: "'d state \<times> 'd state \<Rightarrow> Set('e)"   and x :: "'d state \<times> 'd state \<Rightarrow> 'e" shows "(\<tau> \<Turnstile> \<upsilon> X->including\<^sub>S\<^sub>e\<^sub>t(x)) = (\<tau> \<Turnstile> \<delta> X \<and> \<tau> \<Turnstile> \<upsilon> x)"
j_eq: shows "to_nat (from_nat j) = to_nat (from_nat j)"
r0_2: shows "(\<lambda>n. eNorm (\<LL> p M) (f (r0 n))) \<longlonglongrightarrow> le"
closed_Collect_neg: fixes "open" :: "'a set \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool" assumes "class.topological_space open"   and "open {x. P x}" shows "topological_space.closed open {x. \<not> P x}"
inject: fixes x :: "C list"   and ya :: "C list" shows "(C x = C ya) = (x = ya)"
cases_scheme: fixes r :: "('b, 'c) simple_packet_scheme" obtains p_iiface :: "char list" and p_oiface :: "char list" and p_src :: "'b word" and p_dst :: "'b word" and p_proto :: "8 word" and p_sport :: "16 word" and p_dport :: "16 word" and p_tcp_flags :: "tcp_flag set" and p_payload :: "char list" and more :: "'c" where "r = \<lparr>p_iiface = p_iiface, p_oiface = p_oiface, p_src = p_src, p_dst = p_dst, p_proto = p_proto, p_sport = p_sport, p_dport = p_dport, p_tcp_flags = p_tcp_flags, p_payload = p_payload, \<dots> = more\<rparr>"
hmem_HF_iff: fixes x :: "hf"   and A :: "hf set" shows "(x \<^bold>\<in> HF A) = (x \<in> A \<and> finite A)"
plus_uint_def: shows "(+) \<equiv> map_fun Rep_uint (map_fun Rep_uint Abs_uint) (+)"
prems: shows "op_dghm \<GG> = op_dghm \<FF>"
interaction_bound_IO: fixes out :: "'out"   and c :: "('c, 'out, 'd) rpv"   and gpv :: "('c, 'out, 'd) gpv"   and input :: "'d" assumes "IO out c \<in> set_spmf (the_gpv gpv)" shows "(if consider out then eSuc (local.interaction_bound (c input)) else local.interaction_bound (c input)) \<le> local.interaction_bound gpv"
elims_2: fixes x :: "('e, 'f, 'g) FSM" assumes "well_formed x" obtains M :: "('e, 'f, 'g) FSM" where "x = M" and "finite_FSM M \<and> (\<forall>s1 x y. x \<notin> inputs M \<or> y \<notin> outputs M \<longrightarrow> succ M (x, y) s1 = {}) \<and> inputs M \<noteq> {} \<and> outputs M \<noteq> {}"
mult_le_cancel_right1: 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 c :: "'a"   and b :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times b c) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
lagrange_basis_poly_def: fixes xs :: "'c list"   and xj :: "'c" shows "lagrange_basis_poly xs xj \<equiv> let ys = filter (\<lambda>x. x \<noteq> xj) xs in \<Prod>xi\<leftarrow>ys. smult (inverse (xj - xi)) [:- xi, 1::'c:]"
discrete_cpo: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.discrete_cpo below" shows "below x y = (x = y)"
pairwise_imp_disjoint_family_on: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "pairwise R A"   and "\<And>m n. \<lbrakk>m \<in> A; n \<in> A; R m n\<rbrakk> \<Longrightarrow> f m \<inter> f n = {}" shows "disjoint_family_on f A"
call_source_Def_empty: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs" shows "Def (sourcenode a) = {}"
Append_preserves_ardenable: fixes rhs :: "'b trm set"   and r :: "'b rexp" assumes "ardenable rhs" shows "ardenable (Append_rexp_rhs rhs r)"
FEA: fixes \<Gamma> :: "'b formula list"   and \<AA> :: "'b formula"   and \<Theta> :: "'b formula list"   and \<BB> :: "'b formula"   and \<Delta> :: "'b formula list"   and \<Lambda> :: "'b formula list" assumes "\<Gamma> \<Rightarrow> \<AA> # \<Theta>"   and "\<BB> # \<Delta> \<Rightarrow> \<Lambda>" shows "\<AA> \<^bold>\<rightarrow> \<BB> # \<Gamma> @ \<Delta> \<Rightarrow> \<Theta> @ \<Lambda>"
unfolding_n_sound_complete: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and rs :: "'c rule list"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and n :: "nat"   and s :: "state"   and t :: "state" assumes "\<forall>rsg\<in>ran \<Gamma> \<union> {rs}. wf_chain \<Gamma> rsg" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>(process_call \<Gamma> ^^ n) rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
dvd_fact: fixes m :: "nat"   and n :: "nat" assumes "1 \<le> m"   and "m \<le> n" shows "m dvd fact n"
progress_induction_step: fixes T :: "'b set"   and X :: "'b set"   and C :: "'b set set"   and act :: "('b \<times> 'b) set"   and F :: "'b program"   and B :: "'b set" assumes "T \<inter> X \<in> C"   and "act \<in> Acts F"   and "X \<in> wens_set F B"   and "lattice C"   and "T \<in> C"   and "B \<in> C"   and "closed F T B C"   and "F \<in> stable T" shows "T \<inter> wens F act X \<in> C"
e1: shows "0 < e1 \<and> cball z e1 \<inter> affine hull S \<subseteq> S"
cinner_minus_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a"   and y :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (uminus y) = - cinner x y"
set_nebset_cases: fixes y :: "'d set" assumes "y \<in> {A. A \<noteq> {} \<and> |A| <o natLeq +c |UNIV|}" obtains x :: "'d set!['e]" where "y = set_nebset x"
f_pos: shows "\<forall>\<^sub>F x in at_top. 0 < f x"
ide_to_hf_hf_to_ide: fixes x :: "hf" assumes "x \<in> UNIV" shows "ide_to_hf (hf_to_ide x) = x"
KoenigLemma: fixes Tr :: "'b list set" assumes "infinite Tr"   and "\<forall>i. finite {t. \<exists>t'\<in>Tr. t = take i t'}" shows "\<exists>f. strict_mono f \<and> range f \<subseteq> {t. \<exists>t'\<in>Tr. t \<le> t'}"
self_codomain_iff_ide: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" shows "(a \<in> codomains a) = ide a"
Inv2c_def: fixes s :: "state" shows "Inv2c s = (\<forall>p. Inv2c_inner s p)"