hcju/isabelleps huggingface dataset:
in_latticeI: fixes f :: "'a vec"   and n :: "nat"   and c :: "nat \<Rightarrow> int"   and fs :: "'a vec list" assumes "f = abelian_monoid.sumlist (module_vec TYPE('a) n) (map (\<lambda>i. of_int (c i) \<cdot>\<^sub>v fs ! i) [0..<length fs])" shows "f \<in> vec_module.lattice_of n fs"
fbind_fun_funion: fixes M :: "'b fset"   and f :: "'b \<Rightarrow> 'a fset"   and g :: "'b \<Rightarrow> 'a fset" shows "fbind M (\<lambda>x. f x |\<union>| g x) = fbind M f |\<union>| fbind M g"
Int_lower2: fixes A :: "'g set"   and B :: "'g set" shows "A \<inter> B \<subseteq> B"
a_kernel_def': fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" shows "a_kernel R S h = {x \<in> carrier R. h x = \<zero>\<^bsub>S\<^esub>}"
length_greater_0_conv: fixes xs :: "'b list" shows "(0 < length xs) = (xs \<noteq> [])"
mor_image2': fixes B1 :: "'d set"   and A :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('f, 'd, 'e) F2"   and B1' :: "'g set"   and B :: "'h set"   and s1' :: "'g \<Rightarrow> ('f, 'g, 'h) F1"   and s2' :: "'h \<Rightarrow> ('f, 'g, 'h) F2"   and fa :: "'d \<Rightarrow> 'g"   and f :: "'e \<Rightarrow> 'h"   and x :: "'e" assumes "mor B1 A s1 s2 B1' B s1' s2' fa f"   and "x \<in> A" shows "f x \<in> B"
lunit_in_vhom: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data"   and b :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.ide f"   and "Span.trg f = b" shows "Span.in_hom (Span.lunit f) (b \<circ> f) f"
iMODb_cut_greater1: fixes t :: "nat"   and r :: "nat"   and m :: "nat"   and c :: "nat" assumes "t \<in> [ r, mod m, c ]" shows "[ r, mod m, c ] \<down>> t = (if r + m * c \<le> t then {} else [ t + m, mod m, c - Suc ((t - r) div m) ])"
h5: assumes "0 \<le> Av"   and "Bv \<le> 0" shows "(Av + Bv * sqrt Cv < 0) = (Av < 0 \<and> Bv\<^sup>2 * Cv < Av\<^sup>2 \<or> Bv \<le> 0 \<and> (Av < 0 \<or> Av\<^sup>2 < Bv\<^sup>2 * Cv))"
measure_Diff_null_set: fixes A :: "'a set"   and M :: "'a measure"   and B :: "'a set" assumes "A \<in> sets M"   and "B \<in> null_sets M" shows "Sigma_Algebra.measure M (A - B) = Sigma_Algebra.measure M A"
of_nat_fun: fixes n :: "nat" shows "of_nat n = (\<lambda>x. of_nat n)"
Lower_cong: fixes A :: "'a set"   and A' :: "'a set" assumes "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=} A'" shows "Lower L A = Lower L A'"
in_codomains_char: fixes b :: "(bool, bool list) concrete_category.arr"   and f :: "(bool, bool list) concrete_category.arr" shows "(b \<in> codomains f) = ((f \<noteq> Null \<and> Dom f \<in> {False, True} \<and> Cod f \<in> {False, True} \<and> Map f \<in> Collect (G.path (Dom f) (Cod f))) \<and> b = MkIde (Cod f))"
efficient_funpow_code: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and y :: "'b"   and x :: "'b"   and n :: "nat" shows "efficient_funpow f y x n = (if n = 0 then y else if n = 1 then f x y else if even n then efficient_funpow f y (f x x) (n div 2) else efficient_funpow f (f x y) (f x x) (n div 2))"
le_sup_iff: fixes sup :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.semilattice_sup sup less_eq less" shows "less_eq (sup x y) z = (less_eq x z \<and> less_eq y z)"
empty: fixes g :: "'c \<Rightarrow> 'a" shows "sum_0 g {} = (0::'a)"
countable_infinite: fixes A :: "'a set"   and B :: "'b set" assumes "countable A"   and "infinite B" shows "|A| \<le>o |B|"
CatExp'_def: fixes A :: "('g, 'i, 'k) Category_scheme"   and B :: "('h, 'j, 'l) Category_scheme" shows "CatExp' A B \<equiv> \<lparr>Obj = {F. Ftor F : A \<longrightarrow> B}, Mor = {\<eta>. NatTrans \<eta> \<and> NTCatDom \<eta> = A \<and> NTCatCod \<eta> = B}, Dom = NTDom, Cod = NTCod, Id = IdNatTrans, Comp = (\<bullet>)\<rparr>"
that: fixes g :: "'a \<Rightarrow> 'c" assumes "continuous_map X Y g"   and "\<And>x i. \<lbrakk>i \<in> I; x \<in> topspace X \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x" shows "thesis"
liset_ltake: fixes A :: "'b set"   and k :: "nat"   and w :: "'b llist" shows "liset A (ltake (enat k) w) = liset A w \<inter> {..<k}"
fun_add_closed: fixes R :: "('a, 'b) ring_scheme"   and f :: "'c \<Rightarrow> 'a"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)" shows "f \<oplus>\<^bsub>function_ring S R\<^esub> g \<in> carrier (function_ring S R)"
wf_osig_translate_imp_exe_osig_conds: fixes oss :: "(String.literal \<times> String.literal) list \<times> (String.literal \<times> (String.literal \<times> String.literal set list) list) list" assumes "wf_osig (translate_osig oss)" shows "exe_osig_conds oss"
hyps: shows "poly_inv (monom_list_subst \<sigma> m)"
exhaustiveUnitPropagateTermination: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "InvariantUniq (getM state)"   and "InvariantConsistent (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl"   and "finite Vbl" shows "exhaustiveUnitPropagate_dom state"
simps: fixes M1 :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry"   and M2 :: "nat \<Rightarrow> nat \<Rightarrow> 'c DBMEntry" shows "And M1 M2 = (\<lambda>i j. min (M1 i j) (M2 i j))"
sfis_times: fixes a :: "real"   and f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and z :: "real" assumes "a \<in> sfis f M"   and "0 \<le> z" shows "z * a \<in> sfis (\<lambda>w. z * f w) M"
prems: shows "upd_rev_cs cs (as' @ [a']) = c' # cs'"
ind_is_open_def: fixes U :: "'a set" shows "ind_is_open U \<equiv> U \<subseteq> S \<and> (\<exists>V\<subseteq>X. is_open V \<and> U = S \<inter> V)"
cp\<^sub>P\<^sub>O\<^sub>S\<^sub>T: fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and res :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> 'res"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "POST self a1 res \<tau> = POST (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. res \<tau>) \<tau>"
eq_4: shows "H \<turnstile> x4 EQ x4'"
pdevs_apply_add_pdevs: fixes X :: "'a pdevs"   and Y :: "'a pdevs"   and n :: "nat" shows "pdevs_apply (add_pdevs X Y) n = pdevs_apply X n + pdevs_apply Y n"
isRecv_def: fixes event :: "('d, 'e, 'f) event" shows "isRecv event = (\<exists>x31 x32 x33 x34 x35 x36. event = Recv x31 x32 x33 x34 x35 x36)"
assms_4: fixes f :: "floatarith" assumes "f \<in> Mapping.keys M" shows "the (Mapping.lookup M f) < length slp"
weakPsiCongResChainPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list" assumes "\<Psi> \<rhd> P \<doteq> Q"   and "xvec \<sharp>* \<Psi>" shows "\<Psi> \<rhd> \<lparr>\<nu>*xvec\<rparr>P \<doteq> \<lparr>\<nu>*xvec\<rparr>Q"
sigma_sets_subseteq: fixes A :: "'c set set"   and B :: "'c set set"   and X :: "'c set" assumes "A \<subseteq> B" shows "sigma_sets X A \<subseteq> sigma_sets X B"
Sin_def: fixes x :: "floatarith" shows "Sin x = Cos (Pi * Num (Float 1 (- 1)) - x)"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_def: shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<equiv> map dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t\<^sub>p"
FOREACHoci_refine: fixes \<alpha> :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and S' :: "'d set"   and \<sigma>0 :: "'e"   and \<sigma>0' :: "'f"   and R :: "('e \<times> 'f) set"   and RR :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and RR' :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and \<Phi>'' :: "'c set \<Rightarrow> 'e \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi>' :: "'d set \<Rightarrow> 'f \<Rightarrow> bool"   and \<Phi> :: "'c set \<Rightarrow> 'e \<Rightarrow> bool"   and c :: "'e \<Rightarrow> bool"   and c' :: "'f \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e \<Rightarrow> 'e nres"   and f' :: "'d \<Rightarrow> 'f \<Rightarrow> 'f nres" assumes "inj_on \<alpha> S"   and "S' = \<alpha> ` S"   and "(\<sigma>0, \<sigma>0') \<in> R"   and "\<And>x y. \<lbrakk>x \<in> S; y \<in> S; RR x y\<rbrakk> \<Longrightarrow> RR' (\<alpha> x) (\<alpha> y)"   and "\<Phi>'' S \<sigma>0 (\<alpha> ` S) \<sigma>0'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; \<Phi> it \<sigma>; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> c \<sigma> = c' \<sigma>'"   and "\<And>it \<sigma> it' \<sigma>'. \<lbrakk>it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> \<Phi> it \<sigma>"   and "\<And>x it \<sigma> x' it' \<sigma>'. \<lbrakk>\<forall>y\<in>it - {x}. RR x y; x' = \<alpha> x; x \<in> it; x' \<in> it'; it' = \<alpha> ` it; it \<subseteq> S; it' \<subseteq> S'; \<Phi> it \<sigma>; \<Phi>' it' \<sigma>'; \<Phi>'' it \<sigma> it' \<sigma>'; c \<sigma>; c' \<sigma>'; (\<sigma>, \<sigma>') \<in> R\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> \<Down> {(\<sigma>, \<sigma>'). (\<sigma>, \<sigma>') \<in> R \<and> \<Phi>'' (it - {x}) \<sigma> (it' - {x'}) \<sigma>'} (f' x' \<sigma>')" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>RR,\<Phi>\<^esup> S c f \<sigma>0 \<le> \<Down> R (FOREACH\<^sub>O\<^sub>C\<^bsup>RR',\<Phi>'\<^esup> S' c' f' \<sigma>0')"
has_absolute_integral_change_of_variables_1': fixes S :: "real set"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> real"   and b :: "real" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (g has_real_derivative g' x) (at x within S)"   and "inj_on g S" shows "((\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>g' x\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
th3_5_1: fixes a :: "(real, 'b) vec"   and b :: "(real, 'b) vec"   and d :: "(real, 'b) vec"   and c :: "(real, 'b) vec" assumes "B\<^sub>\<real> a b d"   and "B\<^sub>\<real> b c d" shows "B\<^sub>\<real> a b c"
value_written_change_prefix: fixes n :: "enat"   and E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and P :: "'c prog" assumes "ltake n E = ltake n E'"   and "enat a < n" shows "value_written P E a = value_written P E' a"
snd_def: fixes prod :: "'e \<times> 'd" shows "snd prod = (case prod of (x1, x2) \<Rightarrow> x2)"
Path_update_new: fixes ps :: "ref list"   and alloc :: "ref list"   and p :: "ref"   and f :: "ref \<Rightarrow> ref"   and x :: "ref"   and q :: "ref" assumes "set ps \<subseteq> set alloc" shows "Path p (f(new (set alloc) := x)) q ps = Path p f q ps"
2: fixes s :: "real" shows "summable (\<lambda>x. cmod (indicat_real {0<..} s *\<^sub>R (complex_of_real s powr (z - 1) / complex_of_real (exp ((real x + a) * s)))))"
lookup_sum: fixes pp :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 'b"   and X :: "'d set"   and i :: "'c" shows "lookup (sum pp X) i = (\<Sum>x\<in>X. lookup (pp x) i)"
eq_iff: fixes a :: "'a list"   and b :: "'a list" shows "(a = b) = (prefix b a \<and> prefix a b)"
bounded_def: fixes d :: "'b \<Rightarrow> nat \<Rightarrow> nat"   and k :: "nat" shows "bounded d k = (\<forall>n<k. \<forall>x. d x n < k)"
catch_return: fixes x :: "'b"   and f :: "'c \<Rightarrow> 'a + 'b" shows "(try Inr x catch f) = Inr x"
mat_incr_mult_ket_k: fixes k :: "nat" assumes "k < K" shows "mat_incr K *\<^sub>v ket_k k = ket_k ((k + 1) mod K)"
Exit_noteq_Entry: assumes "(_Exit_) = (_Entry_)" shows "False"
poly_weak_mono_all_def: fixes p :: "('b monom \<times> 'c) list" shows "poly_weak_mono_all p \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. \<beta> x \<le> \<alpha> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> eval_poly \<beta> p \<le> eval_poly \<alpha> p"
FS: shows "(([u], {#}), LCall p # wsl, [v, u'], ch) \<in> ntrs fg"
valid_2: shows "Test__Lazy__Case_tree_map__tree = map_tree"
MaxI_ex2: fixes A :: "'a set" assumes "A \<noteq> {}"   and "finite A" shows "Max A \<in> A"
approximating_bigstep_wf: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and p :: "'d"   and rs :: "'c rule list" assumes "\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided" shows "wf_ruleset \<gamma> p rs"
powr_nat: fixes z :: "complex"   and n :: "nat" shows "z powr of_nat n = (if z = 0 then 0 else z ^ n)"
select_convs_6: fixes types_t :: "tf list"   and func_t :: "tf list"   and global :: "tg list"   and table :: "nat option"   and memory :: "nat option"   and local :: "t list"   and label :: "t list list"   and return :: "t list option"   and more :: "'a" shows "local \<lparr>types_t = types_t, func_t = func_t, global = global, table = table, memory = memory, local = local, label = label, return = return, \<dots> = more\<rparr> = local"
gen_bg_impl_rel_eext_def: fixes Rm :: "('f \<times> 'h) set"   and Rf :: "('e \<times> 'g set) set" shows "\<langle>Rm, Rf\<rangle>gen_bg_impl_rel_eext = {(\<lparr>bgi_F = Fi, \<dots> = mi\<rparr>, \<lparr>bg_F = F, \<dots> = m\<rparr>) |Fi mi F m. (Fi, F) \<in> Rf \<and> (mi, m) \<in> Rm}"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and 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 get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog" assumes "l_get_disconnected_nodes type_wf get_disconnected_nodes get_disconnected_nodes_locs"   and "l_get_root_node get_root_node get_parent"   and "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr get_disconnected_nodes get_root_node get_owner_document" shows "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document"
genby_set_lconjby_set_lconjby_closed: fixes g :: "'g"   and x :: "'g"   and A :: "'g set" assumes "g \<in> G"   and "x \<in> \<langle>\<Union>g\<in>G. lconjby g ` A\<rangle>" shows "lconjby g x \<in> \<langle>\<Union>g\<in>G. lconjby g ` A\<rangle>"
Left_equals_Right_coset: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and g :: "'c" assumes "normal_subgroup K G composition unit"   and "g \<in> G" shows "coset_notation.Left_Coset composition g K = coset_notation.Right_Coset composition K g"
poly_exp_bound: fixes b :: "'b"   and deg :: "nat" assumes "(0::'b) < b"   and "b < (1::'b)" shows "\<exists>p. \<forall>x. b ^ x * of_nat x ^ deg \<le> p"
has_vderiv_on_eq_rhs: fixes f :: "real \<Rightarrow> 'b"   and g' :: "real \<Rightarrow> 'b"   and T :: "real set"   and f' :: "real \<Rightarrow> 'b" assumes "(f has_vderiv_on g') T"   and "\<And>x. x \<in> T \<Longrightarrow> g' x = f' x" shows "(f has_vderiv_on f') T"
length_upto_enum_le_maxBound: fixes start :: "'b"   and "end" :: "'b" shows "length [start .e. end] \<le> Suc (fromEnum maxBound)"
split_beta: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and prod :: "'c \<times> 'd" shows "(case prod of (x, xa) \<Rightarrow> f x xa) = f (fst prod) (snd prod)"
the_eq: fixes A :: "'a Predicate.pred" shows "Predicate.the A = Predicate.singleton (\<lambda>x. Code.abort STR ''not_unique'' (\<lambda>_. Predicate.the A)) A"
emeasure_extend_measure_Pair: fixes M :: "'d measure"   and \<Omega> :: "'d set"   and I :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and G :: "'e \<Rightarrow> 'f \<Rightarrow> 'd set"   and \<mu> :: "'e \<Rightarrow> 'f \<Rightarrow> ennreal"   and \<mu>' :: "'d set \<Rightarrow> ennreal"   and i :: "'e"   and j :: "'f" assumes "M = extend_measure \<Omega> {(i, j). I i j} (\<lambda>(i, j). G i j) (\<lambda>(i, j). \<mu> i j)"   and "\<And>i j. I i j \<Longrightarrow> \<mu>' (G i j) = \<mu> i j"   and "\<And>i j. I i j \<Longrightarrow> G i j \<in> Pow \<Omega>"   and "positive (sets M) \<mu>'"   and "countably_additive (sets M) \<mu>'"   and "I i j" shows "emeasure M (G i j) = \<mu> i j"
crdt_ops_rem_last: fixes xs :: "('b \<times> 'c) list"   and x :: "'b \<times> 'c"   and deps :: "'c \<Rightarrow> 'b set" assumes "crdt_ops (xs @ [x]) deps" shows "crdt_ops xs deps"
summandTransition_2: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi" assumes "hnf P" shows "P \<longmapsto> a<x> \<prec> P' = (a<x>.P' \<in> summands P)"
del_list_sorted2: fixes xs :: "('c \<times> 'd) list"   and a :: "'c"   and b :: "'d"   and ys :: "('c \<times> 'd) list"   and x :: "'c" assumes "sorted1 (xs @ (a, b) # ys)"   and "x < a" shows "del_list x (xs @ (a, b) # ys) = del_list x xs @ (a, b) # ys"
It: fixes I :: "('a, 'b, 'c, 'd) Interpretation"   and A :: "'a" assumes "Interpretation I"   and "A \<in> Ty\<^bsub>iS\<^bsub>I\<^esub>\<^esub>" shows "Ty\<lbrakk>A\<rbrakk>\<^bsub>I\<^esub> \<in> obj\<^bsub>iC\<^bsub>I\<^esub>\<^esub>"
sum: fixes g :: "'c \<Rightarrow> real"   and A :: "'c set"   and x :: "'b" shows "sum g A *\<^sub>R x = (\<Sum>xa\<in>A. g xa *\<^sub>R x)"
connected_complement_absolute_retract: fixes S :: "'a set" assumes "compact S"   and "AR S"   and "2 \<le> DIM('a)" shows "connected (- S)"
zero_ik: shows "is_zero_row_upt_k a k A"
map\<^sub>0_def: fixes a :: "'a" shows "UP.map\<^sub>0 a \<equiv> S.src (S.UP a)"
entail_union: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N :: "'a set"   and N1 :: "'a set"   and N2 :: "'a set" assumes "consequence_relation Bot entails" shows "(entails N N1 \<and> entails N N2) = entails N (N1 \<union> N2)"
assms: fixes l :: "'a" assumes "l \<in> labels \<Gamma> p" shows "P (\<xi>, l)"
dom_llp1: fixes x :: "'a"   and y :: "'a" assumes "x \<le> ad (ad y) \<cdot> x" shows "ad (ad x) \<le> ad (ad y)"
tallyP': shows "card ({i \<in> Is. x \<^bsub>(P i)\<^esub>\<prec> y} - C) = card {i \<in> Is. y \<^bsub>(P i)\<^esub>\<prec> x}"
abs_dif_fiX_bound_C_aux1: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and x :: "real"   and y :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<bar>f l - g l\<bar> \<le> x"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>g l - g m\<bar> \<le> y"   and "p \<in> C"   and "q \<in> C"   and "r \<in> C" shows "\<bar>fiX f p r - fiX g q r\<bar> \<le> x + y"
hyps_1: fixes c2 :: "('a, 'b, 'c) com"   and c :: "('a, 'b, 'c) com"   and s :: "'a"   and n :: "nat" assumes "(a1 \<inter>\<^sub>g c2) = Some c"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> =n\<Rightarrow> Fault f" shows "\<Gamma>\<turnstile> \<langle>a1,Normal s\<rangle> =n\<Rightarrow> Fault f \<or> \<Gamma>\<turnstile> \<langle>c2,Normal s\<rangle> =n\<Rightarrow> Fault f"
wf_secondRecursiveCalls: fixes getIJ :: "(nat set \<times> 'b option list set) set \<Rightarrow> (nat set \<times> 'b option list set) set \<Rightarrow> nat set \<Rightarrow> nat set \<times> nat set"   and V :: "nat set"   and n :: "nat"   and Q :: "(nat set \<times> 'b option list set) set"   and Qn :: "(nat set \<times> 'b option list set) set"   and I :: "nat set"   and J :: "nat set"   and Qns :: "(nat set \<times> 'b option list set) set"   and Q_J_neg :: "(nat set \<times> 'b option list set) set"   and Q_J_pos :: "(nat set \<times> 'b option list set) set"   and t :: "nat set \<times> 'b option list" assumes "getIJ getIJ"   and "2 \<le> card V"   and "rwf_query n V Q Qn"   and "(I, J) = getIJ Q Qn V"   and "Qns \<subseteq> Qn"   and "Q_J_neg = New_max.filterQuery J Qns"   and "Q_J_pos = New_max.filterQuery J Q" shows "rwf_query n J (New_max.newQuery J Q_J_pos t) (New_max.newQuery J Q_J_neg t)"
dvd_power_iff_le: fixes k :: "nat"   and m :: "nat"   and n :: "nat" assumes "2 \<le> k" shows "(k ^ m dvd k ^ n) = (m \<le> n)"
sup_ge2: fixes y :: "'b"   and x :: "'b" shows "y \<le> sup x y"
combine_code_eq: fixes \<delta>\<^sub>L' :: "int"   and p\<^sub>0\<^sub>L :: "int \<times> int"   and p\<^sub>1\<^sub>L :: "int \<times> int"   and \<delta>\<^sub>R' :: "int"   and p\<^sub>0\<^sub>R :: "int \<times> int"   and p\<^sub>1\<^sub>R :: "int \<times> int"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int"   and l :: "int"   and ps :: "(int \<times> int) list"   and \<delta>' :: "int"   and c\<^sub>0' :: "int \<times> int"   and c\<^sub>1' :: "int \<times> int" assumes "\<delta>\<^sub>L' = dist_code p\<^sub>0\<^sub>L p\<^sub>1\<^sub>L"   and "\<delta>\<^sub>R' = dist_code p\<^sub>0\<^sub>R p\<^sub>1\<^sub>R"   and "(c\<^sub>0, c\<^sub>1) = combine (p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps"   and "(\<delta>', c\<^sub>0', c\<^sub>1') = combine_code (\<delta>\<^sub>L', p\<^sub>0\<^sub>L, p\<^sub>1\<^sub>L) (\<delta>\<^sub>R', p\<^sub>0\<^sub>R, p\<^sub>1\<^sub>R) l ps" shows "c\<^sub>0 = c\<^sub>0' \<and> c\<^sub>1 = c\<^sub>1'"
sethi_instr_privilege: fixes s' :: "'b sparc_state"   and instr :: "sparc_operation \<times> inst_operand list"   and s :: "'b sparc_state" assumes "s' = snd (fst (sethi_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
mono_generate: fixes K :: "'a set"   and H :: "'a set" assumes "K \<subseteq> H" shows "generate G K \<subseteq> generate G H"
new_keys_not_used: fixes K :: "nat"   and evs :: "event list" assumes "Key K \<notin> used evs"   and "K \<in> symKeys"   and "evs \<in> yahalom" shows "K \<notin> keysFor (parts (knows Spy evs))"
dim_poly_greater_ex_coeff: fixes d :: "nat"   and x :: "linear_poly" assumes "d < dim_poly x" shows "\<exists>i\<ge>d. Abstract_Linear_Poly.coeff x i \<noteq> 0"
a_one: fixes antidomain_op :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.antidomain_semiringl antidomain_op plus times one zero less_eq less" shows "antidomain_op one = zero"
zero_order_1: fixes x :: "'c multiset" shows "{#} \<subseteq># x"
NR\<^sub>n_NH\<^sub>N: fixes n :: "nat"   and u :: "nat" shows "NR\<^sub>N n u 0 = of_nat u * NH\<^sub>N n"