hcju/isabelleps huggingface dataset:
assms: shows "\<forall>x. x \<rhd> (1::'a) \<rhd> (1::'a) = x"
Norm_nonneg: fixes N :: "'c quasinorm"   and x :: "'c" shows "0 \<le> Norm N x"
subsequent_half_segments_ccw: fixes i :: "nat"   and X :: "(real \<times> real) \<times> (real \<times> real) pdevs" assumes "Suc i < length (half_segments_of_aform X)" shows "ccw' (fst (half_segments_of_aform X ! i)) (snd (half_segments_of_aform X ! i)) (snd (half_segments_of_aform X ! Suc i))"
r_dettime_recfn: shows "recfn 2 r_dettime"
0: assumes "p = 0" shows "c = (0::'a) \<and> bs = []"
Pbefore_in_def: fixes x :: "'a"   and y :: "'a"   and A :: "('a list \<Rightarrow> 'b pmf) \<times> ('a list \<times> 'b \<Rightarrow> 'a \<Rightarrow> ((nat \<times> nat list) \<times> 'b) pmf)"   and qs :: "'a list"   and init :: "'a list" shows "Pbefore_in x y A qs init = map_pmf (\<lambda>p. x < y in fst p) (Partial_Cost_Model.config'_rand A (fst A init \<bind> (\<lambda>is. return_pmf (init, is))) qs)"
n2: fixes n' :: "nat" assumes "n' < n" shows "n' \<notin> length ` snd ` snd ` Q0"
le_union2: fixes v1 :: "val"   and v2 :: "val"   and v12 :: "val" assumes "v1 \<squnion> v2 = Some v12" shows "v2 \<sqsubseteq> v12"
has_mode\<^sub>A: fixes x :: "Eg1.var"   and mds\<^sub>C :: "Mode \<Rightarrow> var\<^sub>C set"   and m :: "Mode" assumes "Eg2_var\<^sub>C_of_Eg1 x \<in> mds\<^sub>C m" shows "x \<in> mds\<^sub>A_of mds\<^sub>C m"
bij_betw_id_iff: fixes A :: "'b set"   and B :: "'b set" shows "bij_betw id A B = (A = B)"
vfst_conv: fixes a :: "V"   and b :: "V" shows "vfst \<langle>a, b\<rangle> = a"
assms_3: shows "qf1 = (\<lambda>x y. x\<^sup>2 + y\<^sup>2 + b1 * x + b2 * y + c)"
scop_slice: fixes \<pi> :: "nat \<Rightarrow> 'n"   and i :: "nat" assumes "(\<pi>, i) \<in> scop" shows "\<pi> i \<in> slice \<inter> dom local.att"
inf_lower: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> carrier L"   and "x \<in> A" shows "\<Sqinter>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> x"
le_fininfE: fixes w\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action stream" assumes "w\<^sub>1 \<preceq>\<^sub>F\<^sub>I w\<^sub>2" obtains v\<^sub>2 :: "'action list" where "v\<^sub>2 \<le>\<^sub>F\<^sub>I w\<^sub>2" and "w\<^sub>1 \<preceq>\<^sub>F v\<^sub>2"
inf_disj_def: fixes n :: "nat"   and nS :: "nat \<Rightarrow> nat \<Rightarrow> 'a list" shows "inf_disj n nS \<equiv> \<forall>t i j. i < n \<and> j < n \<and> i \<noteq> j \<and> nS i t \<noteq> [] \<longrightarrow> nS j t = []"
pp_eqI: fixes s :: "('b, 'c) pp"   and t :: "('b, 'c) pp" assumes "\<And>u. lookup_pp s u = lookup_pp t u" shows "s = t"
arith_prog_mono: fixes c :: "nat"   and c' :: "nat"   and start :: "int"   and step :: "nat" assumes "c \<le> c'" shows "arith_prog start step c \<le> arith_prog start step c'"
le_less_linear: fixes x :: "'a word"   and y :: "'a word" shows "x \<le>s y \<or> y <s x"
IH_1: fixes lev :: "nat" assumes "\<And>x. lev < x \<Longrightarrow> \<not> loose_bvar1 t1 x"   and "\<not> occs (case v of (a, b) \<Rightarrow> Fv a b) t1" shows "bind_fv2 v lev (subst_bv1 t1 lev (case v of (a, b) \<Rightarrow> Fv a b)) = t1"
domains_comp: fixes g :: "'s"   and f :: "'s" assumes "seq g f" shows "domains (g \<cdot> f) = domains f"
eq: shows "fps_deriv (fps_exp a - 1) oo fps_inv (fps_exp a - 1) = fps_const a * (fps_X + 1)"
reachable_nodes_steps: fixes g :: "'b ugraph"   and r :: "'b" shows "(edges g)\<^sup>* `` reachable_nodes g r \<subseteq> reachable_nodes g r"
ipresSubstAll_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'h"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'varSort, 'l, 'm, 'var, 'h, 'i, 'n) model_scheme" shows "ipresSubstAll h hA MOD \<equiv> ipresSubst h MOD \<and> ipresSubstAbs h hA MOD"
split_sel: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'a"   and option :: "'b option" shows "P (case option of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = ((option = None \<longrightarrow> P f1) \<and> (option = \<lfloor>the option\<rfloor> \<longrightarrow> P (f2 (the option))))"
src_port_ipv4_def: fixes x :: "int \<times> ((int \<times> int \<times> int \<times> int) \<times> int) \<times> ((int \<times> int \<times> int \<times> int) \<times> int) \<times> 'a" shows "src_port x \<equiv> (snd \<circ> fst \<circ> snd) x"
existence_ivl_maximal_segment: fixes T :: "real set"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and X :: "'b set"   and x :: "real \<Rightarrow> 'b"   and t0 :: "real"   and t :: "real"   and x0 :: "'b" assumes "ll_on_open_it T f X"   and "(x solves_ode f) {t0--t} X"   and "x t0 = x0"   and "{t0--t} \<subseteq> T" shows "t \<in> ll_on_open.existence_ivl T f X t0 x0"
IH: fixes PROB :: "(('a, bool) fmap \<times> ('a, bool) fmap) set"   and s :: "('a, bool) fmap" assumes "as \<in> valid_plans PROB"   and "s \<in> valid_states PROB" shows "exec_plan s as \<in> valid_states PROB"
signed_funaction_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and s :: "'a"   and x :: "'a \<times> bool" shows "signed_funaction f s x \<equiv> map_prod (f s) (\<lambda>b. b \<noteq> (fst x = s)) x"
evalF_instance: fixes M :: "model"   and phi :: "vbl \<Rightarrow> object"   and u :: "vbl"   and A :: "formula" shows "evalF M phi (instanceF u A) = evalF M (vblcase (phi u) phi) A"
asm: fixes ts :: "'b triple set"   and A :: "'b triple set"   and G :: "prog" assumes "ts \<subseteq> A" shows "G,A|\<turnstile>ts"
TruePref_tax_Cd_not_x: fixes d :: "'b" assumes "d \<noteq> Xd x" shows "TruePref_tax.Cd d = Singleton_for_d.Cd d"
collapse: fixes optionT :: "('a, 'b) optionT" shows "OptionT (run_option optionT) = optionT"
simps_2: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y' :: "'a"   and y :: "'a" assumes "antiparallel_edges \<Delta>" shows "antiparallel_edges.split f (Edge x y', Vertex y) = (if y' = y then f (x, y) else 0)"
tensor_mat_le_one: fixes m1 :: "complex mat"   and m2 :: "complex mat" assumes "m1 \<in> carrier_mat st.d1 st.d1"   and "m2 \<in> carrier_mat st.d2 st.d2"   and "positive m1"   and "positive m2"   and "m1 \<le>\<^sub>L 1\<^sub>m st.d1"   and "m2 \<le>\<^sub>L 1\<^sub>m st.d2" shows "st.tensor_mat m1 m2 \<le>\<^sub>L 1\<^sub>m d"
id_remove_1_mset_iff_notin: fixes M :: "'c multiset"   and a :: "'c" shows "(M = remove1_mset a M) = (a \<notin># M)"
T_eq_rel_half: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and S :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a stream set"   and B :: "'b stream set"   and K :: "'a \<Rightarrow> 'a pmf"   and L :: "'b \<Rightarrow> 'b pmf"   and x :: "'a"   and y :: "'b" assumes "\<And>s t. R s t = (s \<in> S \<and> f s = t)"   and "A \<in> sets (stream_space (count_space UNIV))"   and "B \<in> sets (stream_space (count_space UNIV))"   and "PTA.rel_set_strong (stream_all2 R) A B"   and "rel_fun R (rel_pmf R) K L"   and "R x y" shows "emeasure (MC_syntax.T K x) A = emeasure (MC_syntax.T L y) B"
finfun_const_True_conv_top: shows "($) (K$ True) = top"
hyperpow_mult: fixes r :: "'b star"   and s :: "'b star"   and n :: "nat star" shows "(r * s) pow n = r pow n * s pow n"
boundedI: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "semilattice_order f less_eq less"   and "less_eq a b"   and "less_eq a c" shows "less_eq a (f b c)"
types_stable_def: fixes \<Gamma> :: "Eg1.var \<Rightarrow> bexp set option"   and \<S> :: "Eg1.var set \<times> Eg1.var set" shows "A.types_stable \<Gamma> \<S> \<equiv> \<forall>x\<in>dom \<Gamma>. A.type_stable \<S> (the (\<Gamma> x))"
wagner_1_gen: fixes x :: "'a"   and y :: "'a" shows "(x \<cdot> y\<^sup>\<star>)\<^sup>\<omega> \<le> (x + y)\<^sup>\<omega>"
strict_suffix_def: fixes xs :: "'a list"   and ys :: "'a list" shows "strict_suffix xs ys = (suffix xs ys \<and> xs \<noteq> ys)"
comp_in_homI: fixes f :: "'j"   and a :: "'j"   and b :: "'j"   and g :: "'j"   and c :: "'j" assumes "J.in_hom f a b"   and "J.in_hom g b c" shows "J.in_hom (g \<cdot>\<^sub>J f) a c"
ca_vanish: shows "(\<lambda>n. real_of_int (c (Suc n)) / real_of_int (a n)) \<longlonglongrightarrow> 0"
R1: fixes u :: "'a set"   and v :: "'a set" assumes "u \<in> nhds x"   and "v \<in> nhds y"   and "y \<notin> u" shows "x \<in> v"
bij_betw_compose_perm: fixes \<pi> :: "complex \<Rightarrow> complex"   and Y :: "complex multiset" assumes "\<pi> permutes Roots"   and "Y \<in> Roots_ms" shows "bij_betw (\<lambda>f. \<lambda>g\<in>X \<rightarrow>\<^sub>E Roots. f (restrict (\<pi> \<circ> g) X)) (tuples Y) (tuples Y)"
finite_QF: fixes k :: "nat"   and p :: "nat \<times> nat" shows "finite (QF k p)"
bs_insert_correct: fixes i :: "nat"   and s :: "integer" shows "bs_\<alpha> (bs_insert i s) = insert i (bs_\<alpha> s)"
step_3_main_inv: fixes A :: "'b mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n"   and "0 < k"   and "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.uppert A"   and "??.Jordan_Normal_Form_Existence.ev_block n A"   and "inv_upto n ??.Jordan_Normal_Form_Existence.jb A k" shows "??.Jordan_Normal_Form_Existence.inv_all n ??.Jordan_Normal_Form_Existence.jb (step_3_main n k A) \<and> ??.Jordan_Normal_Form_Existence.same_diag n A (step_3_main n k A)"
exhaust: fixes y :: "'b trm" obtains x1 :: "'b rexp" where "y = Lam x1"   | x21 :: "'b list set" and x22 :: "'b rexp" where "y = Trn x21 x22"
fmeasurableD2: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> fmeasurable M" shows "emeasure M A \<noteq> top"
foldli_transform: fixes \<alpha> :: "'e \<Rightarrow> 'f"   and invar :: "'e \<Rightarrow> bool"   and iti :: "('g \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'g \<Rightarrow> 'g) \<Rightarrow> 'g \<Rightarrow> 'g"   and S0 :: "'f set"   and R :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "set_iterator_abs_genord \<alpha> invar iti S0 R" shows "\<exists>lc. (\<forall>xc\<in>set lc. invar xc) \<and> distinct (map \<alpha> lc) \<and> S0 = set (map \<alpha> lc) \<and> sorted_wrt R (map \<alpha> lc) \<and> iti = foldli lc"
events_CHAOS: fixes A :: "'a set" shows "events_of (CHAOS A) = A"
Basis_pair_def: shows "Basis_pair = Basis \<times> {0::'b} \<union> {0::'a} \<times> Basis"
permute_list_map: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'c list"   and g :: "'c \<Rightarrow> 'd" assumes "f permutes {..<length xs}" shows "permute_list f (map g xs) = map g (permute_list f xs)"
igWlsBinp_def: fixes MOD :: "('c, 'd, 'varSort, 'sort, 'e, 'f, 'g, 'h, 'i) model_scheme"   and delta :: "'opSym"   and binp :: "'bindex \<Rightarrow> 'h option" shows "igWlsBinp MOD delta binp \<equiv> wlsOpS delta \<and> sameDom (barOf delta) binp \<and> liftAll2 (igWlsAbs MOD) (barOf delta) binp"
map_filter_neq_eq_filter_map: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "map f (filter (\<lambda>y. f x \<noteq> f y) xs) = filter ((\<noteq>) (f x)) (map f xs)"
function_smult_assoc1: fixes a :: "'a"   and b :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "b \<in> carrier R"   and "f \<in> carrier F" shows "b \<odot>\<^bsub>F\<^esub> (a \<odot>\<^bsub>F\<^esub> f) = b \<otimes> a \<odot>\<^bsub>F\<^esub> f"
FG_fdd0_closed: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and a :: "'b" assumes "FGModuleHom G smult V smult' T" shows "a \<delta>\<delta> (0::'a) \<in> aezfun_setspan G"
simps_2: fixes a :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) atm" shows "GE.wtL (Neg a) = GE.wtA a"
comp_null_2: fixes f :: "'a \<times> 'a" shows "VxV.comp f VxV.null = VxV.null"
extended_distance_set_middle'_1: fixes a :: "'c Bonk_Schramm_extension_unfolded"   and b :: "'c Bonk_Schramm_extension_unfolded" assumes "middle a b \<in> extended_distance_set" shows "a \<in> extended_distance_set \<inter> underS Bonk_Schramm_extension_unfolded_wo (middle a b)"
DiffI: fixes c :: "'k"   and A :: "'k set"   and B :: "'k set" assumes "c \<in> A"   and "c \<notin> B" shows "c \<in> A - B"
measurable_SUP2: fixes I :: "'b set"   and f :: "'c \<Rightarrow> 'd"   and N :: "'c measure"   and M :: "'b \<Rightarrow> 'd measure" assumes "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f \<in> N \<rightarrow>\<^sub>M M i"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> I\<rbrakk> \<Longrightarrow> space (M i) = space (M j)" shows "f \<in> N \<rightarrow>\<^sub>M \<Squnion> (M ` I)"
\<beta>_def: shows "\<beta> \<in> set noteq \<and> \<beta> < m"
n0_2: shows "s\<^bsub>01\<^esub> (f \<triangleright> n\<^sub>0) \<down>\<noteq> amalg01 f"
compl_fam_imp_triv_finprod: fixes S :: "'d \<Rightarrow> ('c \<Rightarrow> 'a) set"   and I :: "'d set"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'a" assumes "DP.compl_fam S I"   and "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I)"   and "finprod (DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I) f I = \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>"   and "f \<in> Pi I S" shows "\<forall>i\<in>I. f i = \<one>\<^bsub>DirProds (\<lambda>i. G\<lparr>carrier := S i\<rparr>) I\<^esub>"
mult_left_le_imp_le: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less_eq (times c a) (times c b)"   and "less zero c" shows "less_eq a b"
rec_predecessor_def: shows "rec_predecessor = CN (PR Z (recf.Id 3 0)) [recf.Id 1 0, recf.Id 1 0]"
fimage_iff: fixes z :: "'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b fset" shows "(z |\<in>| f |`| A) = fBex A (\<lambda>x. z = f x)"
tendsto_neg_powr_complex_of_real: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and s :: "complex" assumes "filterlim f at_top F"   and "Re s < 0" shows "((\<lambda>x. complex_of_real (f x) powr s) \<longlongrightarrow> 0) F"
isTable_subset: fixes qs :: "('c \<times> 'd) list"   and ps :: "('c \<times> 'd) list"   and E :: "'c \<Rightarrow> 'd"   and vs :: "'c list" assumes "set qs \<subseteq> set ps"   and "isTable E vs ps" shows "isTable E vs qs"
select_convs_1: fixes block :: "DiskBlock"   and proc :: "Proc"   and more :: "'a" shows "block \<lparr>block = block, proc = proc, \<dots> = more\<rparr> = block"
AE_distrD: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and M' :: "'d measure"   and P :: "'d \<Rightarrow> bool" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "almost_everywhere (distr M M' f) P" shows "AE x in M. P (f x)"
Interleaves_suffix_one_aux: fixes P :: "'b \<Rightarrow> 'b list \<Rightarrow> bool"   and x :: "'b"   and xs :: "'b list"   and zs :: "'b list" assumes "P x []" shows "\<not> xs @ [x] \<cong> {[], zs, P}"
fg: shows "adjunction_in_bicategory (\<cdot>) (\<star>) \<a> \<i> src trg f g \<eta> \<epsilon>"
simps_10: fixes f :: "'b \<Rightarrow> 'a"   and x21 :: "'b"   and x22 :: "'b FOL_Fitting.term list" shows "map_term f (App x21 x22) = App (f x21) (map (map_term f) x22)"
proj_O_def: shows "proj_O = mat N N (\<lambda>(i, j). if i = j then if f i then 1 else 0 else 0)"
simps_4: fixes Q :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and cfs :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" shows "transfer (Q\<hookleftarrow>\<^bsub>p\<^esub>f) (cf # cfs) = (case cfs of [] \<Rightarrow> [] | cf' # x \<Rightarrow> (f (fst cf) (fst cf'), snd cf') # x)"
assms: shows "DisjointPaths (remove_vertex v) v0 v1 paths"
bisim1Seq2: fixes P :: "'b prog"   and h :: "'heap"   and e2 :: "'addr expr1"   and exs :: "'addr expr1 \<times> 'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and e1 :: "'addr expr1" assumes "P,e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, pc, xcp)" shows "P,e1;; e2,h \<turnstile> exs \<leftrightarrow> (stk, loc, Suc (length (compE2 e1) + pc), xcp)"
RedBlock: fixes V :: "char list"   and T :: "ty"   and u :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list" shows "P \<turnstile> \<langle>{V:T; Val u},s\<rangle> \<rightarrow> \<langle>Val u,s\<rangle>"
Inf_seq_distrib: fixes C :: "'a::refinement_lattice set"   and d :: "'a::refinement_lattice" shows "\<Sqinter> (C::'a::refinement_lattice set) ; (d::'a::refinement_lattice) = (\<Sqinter>c::'a::refinement_lattice\<in>C. c ; d)"
subu_fin: shows "- \<infinity> < I.subcocycle_avg_ereal (\<lambda>n x. u n ((Tinv ^^ n) x))"
renamings_apart_var_disjoint: fixes Cs :: "'a literal multiset list" shows "var_disjoint (Cs \<cdot>\<cdot>cl renamings_apart Cs)"
not_eq_order_implies_strict: fixes a :: "'t"   and b :: "'t" assumes "a \<noteq> b"   and "a \<preceq>\<^sub>t b" shows "a \<prec>\<^sub>t b"
mirlfrqe_def: fixes p :: "fm" shows "mirlfrqe p = qelim (prep p) mirlfr"
pos_add_strict: fixes a :: "'c multiset"   and b :: "'c multiset"   and c :: "'c multiset" assumes "{#} \<subset># a"   and "b \<subset># c" shows "b \<subset># a + c"
negligible_Un: fixes S :: "'c set"   and T :: "'c set" assumes "negligible S"   and "negligible T" shows "negligible (S \<union> T)"
isin_abs: fixes t :: "'b trie_map"   and xs :: "'b list" shows "Trie_Map.isin t xs = Trie_Fun.isin (Trie_Map.abs t) xs"
SUP_mono': fixes Inf :: "'f set \<Rightarrow> 'f"   and Sup :: "'f set \<Rightarrow> 'f"   and inf :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and sup :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and bot :: "'f"   and top :: "'f"   and f :: "'g \<Rightarrow> 'f"   and g :: "'g \<Rightarrow> 'f"   and A :: "'g set" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>x. less_eq (f x) (g x)" shows "less_eq (Sup (f ` A)) (Sup (g ` A))"
unforwarded_empty_extend: fixes x :: "nat"   and sb :: "'b memref list"   and W :: "nat set" assumes "x \<in> unforwarded_non_volatile_reads sb {}"   and "x \<notin> W" shows "x \<in> unforwarded_non_volatile_reads sb W"
d_gchain_def: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and g :: "nat \<Rightarrow> 'e set" shows "d_gchain G n g = (if n = 0 then G \<guillemotright> g 0  else \<forall>l\<le>n. G \<guillemotright> g l  \<and> (\<forall>l\<le>n - Suc 0. g (Suc l) \<subseteq> g l))"
"1_3": fixes q :: "nat"   and u :: "char list \<Rightarrow> int" assumes "(c_, s1_) \<Rightarrow> q \<Down> u" shows "u = s2_ \<and> x_ = q"
no_outstanding_write_to_read_only_memory_axioms: shows "no_outstanding_write_to_read_only_memory \<S>\<^sub>s\<^sub>b ts\<^sub>s\<^sub>b"
l2_inv3I: fixes x :: "l2_state" assumes "\<And>Ra A B K. \<lbrakk>Auth B A (Aenc K (epubKF (Ra $ kE))) \<in> chan x; A \<notin> bad x \<and> B \<notin> bad x\<rbrakk> \<Longrightarrow> \<exists>Rb. guessed_runs Rb = \<lparr>role = Resp, owner = B, partner = A\<rparr> \<and> progress x Rb = Some {xpkE, xsk} \<and> guessed_frame Rb xpkE = Some (epubKF (Ra $ kE)) \<and> K = NonceF (Rb $ sk)" shows "x \<in> l2_inv3"
cdi_iff_no_strict_pd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob" shows "IFC_def.is_cdi prob i \<pi> k = (IFC_def.is_path prob \<pi> \<and> k < i \<and> \<pi> i \<noteq> IFC_def.return prob \<and> (\<forall>j\<in>{k..i}. (\<pi> k, \<pi> j) \<notin> IFC_def.pdt prob))"
nn_cond_exp_intg: fixes f :: "'a \<Rightarrow> ennreal"   and g :: "'a \<Rightarrow> ennreal" assumes "f \<in> borel_measurable F"   and "g \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. f x * nn_cond_exp M F g x \<partial>M = \<integral>\<^sup>+ x. f x * g x \<partial>M"
assms_3: shows "incid C (line A' B') \<or> incid B' (line A C)"
that: fixes cs' :: "'edge list"   and rs' :: "'edge list" assumes "valid_node m'"   and "valid_call_list cs' m'"   and "\<forall>i<length rs'. rs' ! i \<in> get_return_edges (cs' ! i)"   and "valid_return_list rs' m'"   and "length rs' = length cs'"   and "ms' = targetnodes rs'"   and "upd_cs cs as = cs'" shows "thesis"
map_tailrec_def: fixes f :: "'d \<Rightarrow> 'c"   and as :: "'d list" shows "map_tailrec f as = rev (map_tailrec_rev f as [])"
summable_sum: fixes I :: "'a set"   and f :: "'a \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<And>i. i \<in> I \<Longrightarrow> summable (f i)" shows "summable (\<lambda>n. \<Sum>i\<in>I. f i n)"
fst_conv: fixes x1 :: "'e"   and x2 :: "'f" shows "fst (x1, x2) = x1"
out_arcs_def: fixes G :: "('b, 'a) pre_digraph"   and v :: "'b" shows "out_arcs G v \<equiv> {e \<in> arcs G. tail G e = v}"
IH: assumes "Decomp t \<in> set A" shows "\<exists>f T. t = Fun f T"
verit_comp_simplify1_3: fixes b' :: "'a"   and a' :: "'a" shows "(\<not> b' \<le> a') = (a' < b')"
intros_3: fixes n\<^sub>0 :: "'node SDG_node"   and ns :: "'node SDG_node list"   and n\<^sub>1 :: "'node SDG_node"   and p :: "'pname"   and n\<^sub>2 :: "'node SDG_node"   and ns' :: "'node SDG_node list"   and n\<^sub>3 :: "'node SDG_node"   and n\<^sub>4 :: "'node SDG_node"   and V :: "'var"   and a :: "'edge"   and a' :: "'edge" assumes "matched n\<^sub>0 ns n\<^sub>1"   and "n\<^sub>1 -p\<rightarrow>\<^bsub>call\<^esub> n\<^sub>2"   and "matched n\<^sub>2 ns' n\<^sub>3"   and "n\<^sub>3 -p\<rightarrow>\<^bsub>ret\<^esub> n\<^sub>4 \<or> n\<^sub>3 -p:V\<rightarrow>\<^bsub>out\<^esub> n\<^sub>4"   and "valid_edge a"   and "a' \<in> get_return_edges a"   and "sourcenode a = parent_node n\<^sub>1"   and "targetnode a = parent_node n\<^sub>2"   and "sourcenode a' = parent_node n\<^sub>3"   and "targetnode a' = parent_node n\<^sub>4" shows "matched n\<^sub>0 (ns @ n\<^sub>1 # ns' @ [n\<^sub>3]) n\<^sub>4"
num_params_tm_lower_order_of_normed_le: fixes t :: "taylor_model"   and X :: "nat"   and prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list" assumes "num_params (tm_poly t) \<le> X" shows "num_params (tm_poly (tm_lower_order_of_normed prec ord I a t)) \<le> X"
star_slide_var: fixes x :: "'a list set" shows "times.opp_mult (\<cdot>) (x\<^sup>\<star>) x = times.opp_mult (\<cdot>) x (x\<^sup>\<star>)"
Seq_iff: fixes r :: "hf"   and d :: "hf" shows "Seq r d = (d \<le> hdomain r \<and> hfun_sigma r)"
simps_1: fixes SUC :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and wf0 :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and n :: "'b"   and uu :: "bool" shows "Formula_Operations.wf SUC wf0 n (FBool uu) = True"
cases: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and a1 :: "char list"   and a2 :: "char list list" assumes "Subobjs\<^sub>R P a1 a2" obtains C :: "char list" where "a1 = C__" and "a2 = [C__]" and "is_class P C__"   | C :: "char list" and D :: "char list" and Cs :: "char list list" where "a1 = C" and "a2 = C # Cs" and "P \<turnstile> C \<prec>\<^sub>R D" and "Subobjs\<^sub>R P D Cs"
is_tm_dghm_op: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>t\<^sub>m\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
cond_def: fixes p :: "'a"   and x :: "'a"   and y :: "'a" shows "if p then x else y fi = d p \<cdot> x + ad p \<cdot> y"