hcju/isabelleps huggingface dataset:
external_WT'_iff: fixes P :: "'a prog"   and h :: "'heap"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and U :: "ty" shows "P,h \<turnstile> a\<bullet>M(vs) : U = (\<exists>hT Ts Ts' D. typeof_addr h a = \<lfloor>hT\<rfloor> \<and> map typeof\<^bsub>h\<^esub> vs = map Some Ts \<and> P \<turnstile> class_type_of hT sees M: Ts'\<rightarrow>U = Native in D \<and> P \<turnstile> Ts [\<le>] Ts')"
simps_1: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and Lex :: "symbol \<Rightarrow> character list \<Rightarrow> nat \<Rightarrow> nat set"   and Sel :: "(symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set \<Rightarrow> (symbol \<times> character list) set"   and Doc :: "character list"   and k :: "nat" assumes "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel" shows "LocalLexing.\<Z> \<NN> \<TT> \<RR> \<SS> Lex Sel Doc k 0 = {}"
outstanding_vol_write_dropWhile_append: fixes sb :: "'b memref list"   and xs :: "'b memref list" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb \<noteq> {}" shows "dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) (sb @ xs) = dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb @ xs"
object_ptr_kinds_eq_h: shows "object_ptr_kinds h2 = object_ptr_kinds h |\<union>| {|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>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_character_data_ptr|}"
length_chop: fixes n :: "nat"   and xs :: "'a list" assumes "0 < n" shows "length (chop n xs) = nat \<lceil>real (length xs) / real n\<rceil>"
psi_l_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and l :: "nat" assumes "grover_state n f" shows "grover_state.psi_l n f l \<in> carrier_vec (grover_state.N n)"
ord_isom_Pre2: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and a :: "'c"   and f :: "'c \<Rightarrow> 'e" assumes "Worder D"   and "Worder E"   and "a \<in> carrier D"   and "ExPre D a"   and "ord_isom D E f" shows "f (Pre D a) = Pre E (f a)"
inject_3: fixes x3 :: "('a, 'b, 'c) formula"   and y3 :: "('a, 'b, 'c) formula" shows "(! x3 = ! y3) = (x3 = y3)"
norm_eq_sqrt_cinner: fixes x :: "'a" shows "norm x = sqrt (cmod \<langle>x, x\<rangle>)"
simps_1: fixes c :: "poly"   and n :: "nat"   and p :: "poly" shows "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
filter_is_subset: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "set (filter P xs) \<subseteq> set xs"
delete_aux_eq_Nil_conv: fixes k :: "'a"   and ts :: "('a \<times> 'b) list" shows "(AList.delete_aux k ts = []) = (ts = [] \<or> (\<exists>v. ts = [(k, v)]))"
prems_1: shows "valid_ownership_and_sharing \<S> (t # ts)"
open_existence_ivl0: fixes x :: "'a" assumes "x \<in> X" shows "\<exists>a b. a < 0 \<and> 0 < b \<and> {a..b} \<subseteq> existence_ivl0 x"
Abs_group_inverse: fixes y :: "'d monoid" assumes "y \<in> {G. Group.group G}" shows "Rep_group (Abs_group y) = y"
path_split_second: fixes prog :: "cmd"   and n :: "w_node"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and a :: "w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node"   and as' :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog n (as @ a # as') n'" shows "While_CFG.path prog (sourcenode a) (a # as') n'"
snd: fixes P :: "'a trm"   and x :: "'a"   and M :: "'a trm"   and P' :: "'a trm" assumes "substitutes P x M P'" shows "substitutes (Snd P) x M (Snd P')"
distincts_def: fixes xss :: "'b list list" shows "distincts xss \<equiv> distinct xss \<and> (\<forall>xs\<in>set xss. distinct xs \<and> xs \<noteq> []) \<and> (\<forall>xs\<in>set xss. \<forall>ys\<in>set xss. xs \<noteq> ys \<longrightarrow> set xs \<inter> set ys = {})"
take_bit_push_bit: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and m :: "nat"   and n :: "nat"   and a :: "'c" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "take_bit m (push_bit n a) = push_bit n (take_bit (m - n) a)"
eucl_abs: fixes x :: "'a" shows "\<bar>x\<bar> = (\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar> *\<^sub>R i)"
transitive: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs"   and Rel'' :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> Rel"   and "Q \<leadsto>\<^sup>^<Rel'> R"   and "Rel O Rel' \<subseteq> Rel''"   and "\<And>S T. (S, T) \<in> Rel \<Longrightarrow> S \<leadsto>\<^sup>^<Rel> T" shows "P \<leadsto>\<^sup>^<Rel''> R"
rank_insertList: fixes q :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_skew_invar q" shows "SkewBinomialHeapStruc.rank_skew_invar (SkewBinomialHeapStruc.insertList ts q)"
scaleC_right: fixes a :: "'a"   and r :: "complex"   and b :: "'b" shows "a ** r *\<^sub>C b = r *\<^sub>C (a ** b)"
mem_set_list_relE2: fixes xs :: "'c list"   and ys :: "'d list"   and R :: "('c \<times> 'd) set"   and y :: "'d" assumes "(xs, ys) \<in> \<langle>R\<rangle>list_rel"   and "y \<in> set ys" obtains x :: "'c" where "x \<in> set xs" and "(x, y) \<in> R"
mk_and_def: fixes x :: "'a ltln"   and y :: "'a ltln" shows "mk_and x y \<equiv> case x of true\<^sub>n \<Rightarrow> y | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> case y of true\<^sub>n \<Rightarrow> x | false\<^sub>n \<Rightarrow> false\<^sub>n | _ \<Rightarrow> x and\<^sub>n y"
vintersection_iff: fixes x :: "V"   and A :: "V"   and B :: "V" shows "(x \<in>\<^sub>\<circ> A \<inter>\<^sub>\<circ> B) = (x \<in>\<^sub>\<circ> A \<and> x \<in>\<^sub>\<circ> B)"
independent_backwards_2: fixes K :: "'a set"   and u :: "'a"   and Us :: "'a list" assumes "local.independent K (u # Us)" shows "local.independent K Us"
map_eq_append_conv: fixes f :: "'c \<Rightarrow> 'a"   and xs :: "'c list"   and ys :: "'a list"   and zs :: "'a list" shows "(map f xs = ys @ zs) = (\<exists>us vs. xs = us @ vs \<and> ys = map f us \<and> zs = map f vs)"
assms_5: shows "(d, t) \<rightarrow>c (d', t')"
index_last_take: fixes i :: "nat"   and xs :: "'e list" assumes "i < length xs" shows "xs ! i = last (take (Suc i) xs)"
intro: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool" assumes "M.Struct wtFsym wtPsym arOf resOf intT intF intP" shows "CM.Struct wtFsym wtPsym arOf resOf intT intF intP"
ptensor_mat_id: shows "tensor_P (1\<^sub>m ps2_P.d1) (1\<^sub>m ps2_P.d2) = 1\<^sub>m ps2_P.d0"
Ccompt: fixes C :: "('d, 'e, 'f) Category_scheme"   and f :: "'e"   and X :: "'d"   and Y :: "'d"   and g :: "'e"   and Z :: "'d" assumes "Category C"   and "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f ;;\<^bsub>C\<^esub> g maps\<^bsub>C\<^esub> X to Z"
shrK_in_initState_Server: fixes A :: "agent" shows "Key (shrK A) \<in> initState Kas"
not_gr_zero: fixes n :: "'c" shows "(\<not> (0::'c) < n) = (n = (0::'c))"
simps_5: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'f"   and x2 :: "'g" shows "(case \<lfloor>x2\<rfloor> of None \<Rightarrow> f1 | \<lfloor>x\<rfloor> \<Rightarrow> f2 x) = f2 x2"
untime_empty_imp_NoMsg: fixes s :: "'b message_af list"   and n :: "nat" assumes "untime s = []"   and "n < length s" shows "s ! n = NoMsg"
SPEC_rule_conj_leofI2: fixes m :: "'b nres"   and \<Phi> :: "'b \<Rightarrow> bool"   and \<Psi> :: "'b \<Rightarrow> bool" assumes "m \<le>\<^sub>n SPEC \<Phi>"   and "m \<le> SPEC \<Psi>" shows "m \<le> SPEC (\<lambda>s. \<Phi> s \<and> \<Psi> s)"
mem_vertices: fixes v :: "'form \<times> nat list" shows "(v |\<in>| vertices) = (fst v \<in> set conclusions \<and> (snd v = [] \<or> snd v \<in> (#) 0 ` it_paths (it' (fst v))))"
semiring_hom_def: fixes hom :: "'a \<Rightarrow> 'b" shows "semiring_hom hom \<equiv> comm_monoid_add_hom hom \<and> monoid_mult_hom hom"
expr_inj_term_simp: fixes e :: "expr" shows "\<langle>e\<rangle> = In1l e"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "('i, 'j, 'k, 'l) SLin_state_scheme" shows "P19 (s1, s2) = (abortVals s2 \<noteq> {} \<longrightarrow> abortVals s1 \<noteq> {})"
retraction_map_def: fixes X :: "'b topology"   and Y :: "'c topology"   and f :: "'b \<Rightarrow> 'c" shows "retraction_map X Y f \<equiv> \<exists>g. retraction_maps X Y f g"
Mul_Blacken_Roots: fixes n :: "nat" shows "\<turnstile> Mul_Blacken_Roots n  \<lbrace>\<acute>Mul_Proper n \<and> Roots \<subseteq> Blacks \<acute>M\<rbrace>"
sqrt_nat: fixes x :: "nat" shows "set (sqrt_nat x) = {y. y * y = x}"
mult_numeral_1_right: fixes a :: "'b" shows "a * Numeral1 = a"
FVT: shows "FVT \<theta> \<subseteq> semBV I ODE"
elims: fixes x :: "'a"   and xa :: "'a option"   and y :: "'a" assumes "the_default x xa = y" obtains d :: "'a" where "x = d" and "xa = None" and "y = d"   | uu_ :: "'a" and xb :: "'a" where "x = uu_" and "xa = Some xb" and "y = xb"
cr_pfun_def: shows "cr_pfun \<equiv> \<lambda>x y. x = Rep_pfun y"
split_face_distinct1: fixes f12 :: "face"   and f21 :: "face"   and oldF :: "face"   and ram1 :: "nat"   and ram2 :: "nat"   and newVertexList :: "nat list" assumes "(f12, f21) = split_face oldF ram1 ram2 newVertexList"   and "pre_split_face oldF ram1 ram2 newVertexList" shows "distinct (vertices f12)"
If: fixes b :: "bexp"   and A :: "char list set"   and c\<^sub>1 :: "com"   and A\<^sub>1 :: "char list set"   and c\<^sub>2 :: "com"   and A\<^sub>2 :: "char list set" assumes "vars b \<subseteq> A"   and "D A c\<^sub>1 A\<^sub>1"   and "D A c\<^sub>2 A\<^sub>2" shows "D A (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (A\<^sub>1 \<inter> A\<^sub>2)"
\<gamma>1_s: shows "set_mset (measure_conv r \<gamma>1) \<subseteq> r \<down>m {#fst \<beta>_step#}"
effect_def: fixes c :: "'d Heap"   and h :: "heap"   and h' :: "heap"   and r :: "'d" shows "effect c h h' r = (execute c h = Some (r, h'))"
inner_lv_rel_autoref: shows "(inner_lv_rel, (\<bullet>)) \<in> lv_rel \<rightarrow> lv_rel \<rightarrow> rnv_rel"
integrable_count_space_nat_iff: fixes f :: "nat \<Rightarrow> 'c" shows "integrable (count_space UNIV) f = summable (\<lambda>x. norm (f x))"
sel_1: fixes x1 :: "rat \<Rightarrow> rat \<Rightarrow> nat"   and x2 :: "rat \<Rightarrow> nat" shows "root_info.l_r (Root_Info x1 x2) = x1"
simps: fixes e :: "'a Point"   and pl :: "'a Plane" shows "inPlane e pl = coplanar e (pbasepoint pl) (pbasepoint pl \<leadsto> direction1 pl) (pbasepoint pl \<leadsto> direction2 pl)"
diff: fixes S :: "'a set"   and s :: "'a"   and t :: "'a" assumes "group_closurep S s"   and "group_closurep S t" shows "group_closurep S (s - t)"
congruence_def: shows "congruence \<equiv> {(P, Q) |P Q. P \<leadsto><weakBisim> Q \<and> Q \<leadsto><weakBisim> P}"
simps_5: fixes l1 :: "value"   and v :: "'b aexp"   and va :: "'b aexp" shows "AExp_Lexorder.less_aexp_aux (L l1) (Times v va) = True"
orthogonal_complement_orthoI: fixes x :: "'b"   and M :: "'b set"   and y :: "'b" assumes "x \<in> orthogonal_complement M"   and "y \<in> M" shows "is_orthogonal x y"
satisfies_eqI: fixes \<AA> :: "'i"   and \<phi> :: "('a, 'k) aformula"   and \<BB> :: "'i" assumes "local.wf (#\<^sub>V \<AA>) \<phi>"   and "#\<^sub>V \<AA> = #\<^sub>V \<BB>"   and "\<And>m k. LESS k m (#\<^sub>V \<AA>) \<Longrightarrow> m\<^bsup>\<AA>\<^esup>k = m\<^bsup>\<BB>\<^esup>k"   and "lformula \<phi>" shows "(\<AA> \<Turnstile> \<phi>) = (\<BB> \<Turnstile> \<phi>)"
rel_interior_translation_aux: fixes a :: "'b"   and S :: "'b set" shows "(+) a ` rel_interior S \<subseteq> rel_interior ((+) a ` S)"
my_taut_test_bool_def: shows "my_taut_test_bool = bool_taut_test \<circ> bool_expr_of_bexp"
ascii_of_idem: fixes c :: "char" assumes "\<not> digit7 c" shows "ascii_of c = c"
Upair_fsteq: fixes a :: "ZF"   and b :: "ZF"   and c :: "ZF" shows "(Upair a b = Upair a c) = (a = b \<and> a = c \<or> b = c)"
transfer: shows "pcr_real (\<lambda>n. 0) 0"
TauR_def: shows "\<tau> \<equiv> Abs_freeRes TauR_Rep"
classes_above_sub_classes_between_eq: fixes C :: "char list"   and D :: "char list"   and P :: "'b prog" assumes "P \<turnstile> C \<preceq>\<^sup>* D" shows "classes_above P C = classes_between P C D - {D} \<union> classes_above P D"
between_trivial: fixes A :: "'p"   and B :: "'p" shows "Bet A B B"
C: shows "(xs @ ipurge_tr_aux I D U (ys @ [y]) @ zs, Z) \<in> failures P"
assms: shows "\<exists>r o\<^sub>1 o\<^sub>2 o\<^sub>3. Some r = d\<^sub>3 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> Some o\<^sub>2 = d\<^sub>\<kappa> y \<and> Some o\<^sub>3 = d\<^sub>\<kappa> z \<and> (o\<^sub>1, o\<^sub>2, o\<^sub>3) \<in> ex3 r v"
intro: fixes r1 :: "('d, 'a, 'b, 'c, 'e, 'f) semantics"   and r2 :: "('d, 'a, 'g, 'h, 'e, 'f) semantics"   and bisim :: "'a \<Rightarrow> 'b \<times> 'c \<Rightarrow> 'g \<times> 'h \<Rightarrow> bool"   and \<tau>move1 :: "('d, 'a, 'b, 'c, 'e, 'f) \<tau>moves"   and \<tau>move2 :: "('d, 'a, 'g, 'h, 'e, 'f) \<tau>moves" assumes "\<And>t. delay_bisimulation_diverge (r1 t) (r2 t) (bisim t) (ta_bisim bisim) \<tau>move1 \<tau>move2" shows "FWdelay_bisimulation_diverge_axioms r1 r2 bisim \<tau>move1 \<tau>move2"
jkbpC_prefix_closed: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   and s :: "'e"   and envInit :: "'e list"   and envAction :: "'e \<Rightarrow> 'g list"   and envTrans :: "'g \<Rightarrow> ('b \<Rightarrow> 'd) \<Rightarrow> 'e \<Rightarrow> 'e"   and envVal :: "'e \<Rightarrow> 'c \<Rightarrow> bool" assumes "PreEnvironmentJView jkbp jview"   and "t \<leadsto> s \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview" shows "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"
pr1_simps_2: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "local.dom (pr1 a b) = local.prod a b"
tuple: shows "\<guillemotleft>tuple \<Lambda>_pr1 b_pr0 : local.prod a b \<rightarrow> local.prod (exp b c) b\<guillemotright>"
foldl_Nil: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'd"   and a :: "'d" shows "foldl f a [] = a"
N_def: shows "N = {a. a \<le> (0::'b)}"
order_tendsto_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and x :: "'b"   and F :: "'c filter" assumes "class.order_topology less_eq less open" shows "filterlim f (topological_space.nhds open x) F = ((\<forall>l. less l x \<longrightarrow> (\<forall>\<^sub>F x in F. less l (f x))) \<and> (\<forall>u. less x u \<longrightarrow> (\<forall>\<^sub>F x in F. less (f x) u)))"
supset_ld_is_ld: fixes A :: "'a vec set"   and B :: "'a vec set" assumes "lin_dep A"   and "A \<subseteq> B" shows "lin_dep B"
canonical_inj_is_hom: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set" assumes "Group.group G"   and "subgroup H G" shows "group_hom (G\<lparr>carrier := H\<rparr>) G id"
fne: fixes w :: "complex" assumes "w \<in> ball \<xi> r" shows "f w - f \<xi> = (w - \<xi>) ^ n * g w"
null: fixes x :: "'a" assumes "x \<in> X" shows "emeasure M {x} = 0"
that: fixes q :: "('a \<times> 'a set) set" assumes "q tagged_division_of cbox a b"   and "d fine q"   and "\<forall>(x, k)\<in>p. k \<subseteq> d x \<longrightarrow> (x, k) \<in> q" shows "thesis"
deduct_if_deduct_num: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and k :: "nat"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct_num arity public Ana M k t" shows "intruder_model.intruder_deduct arity public Ana M t"
rep_eq: fixes x :: "'b boolean_subset"   and xa :: "'b boolean_subset" shows "Rep_boolean_subset (x \<squnion> xa) = Rep_boolean_subset x \<squnion> Rep_boolean_subset xa"
lm3: fixes x :: "nat"   and key :: "nat" assumes "c_assoc_have_key (pr_gr x) key = 0" shows "c_assoc_have_key (pr_gr (Suc x)) key = 0"
pred_alt_def_1: fixes name :: "name"   and vs :: "value list" shows "vclosed (Vconstr name vs) = (True \<and> list_all vclosed vs)"
word_ao_nth: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" shows "(x OR y) !! n = (x !! n \<or> y !! n) \<and> (x AND y) !! n = (x !! n \<and> y !! n)"
prems: shows "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"
seq'_def: shows "seq' = (\<lambda>i. lookup (seq i))"
vec_nth_1_iff_cbox: fixes S :: "('c, 1) vec set"   and a :: "'c"   and b :: "'c" shows "((\<lambda>x. x $ 1) ` S = cbox a b) = (S = cbox (vec a) (vec b))"
sum_list_filter_nonzero: fixes xs :: "'b list" shows "sum_list (filter (\<lambda>x. x \<noteq> (0::'b)) xs) = sum_list xs"
Assign: fixes x :: "char list"   and a :: "aexp"   and s :: "char list \<Rightarrow> int" shows "(x ::= a, s) \<rightarrow> (SKIP, s(x := aval a s))"
my_U_is_defined_1: fixes p :: "'a pmf" assumes "p \<in> local.\<P>" shows "local.my_U p \<in> {0..1}"
00: fixes t :: "'trm"   and x :: "'var" shows "((t, x) \<in> set (zip (map Var us) (map snd txs))) = (\<exists>u\<in>var. t = Var u \<and> (Var u, x) \<in> set (zip (map Var us) (map snd txs)))"
i_set_subset_i_set0: shows "i_set \<subseteq> i_set0"
validity_is_local: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and dec :: "'g \<Rightarrow> 'h option"   and p :: "'f"   and w :: "'h"   and v :: "'h"   and n :: "nat" assumes "locally_similar \<sigma> \<tau>"   and "\<And>n. dec (\<sigma> n p) = Some w \<Longrightarrow> w = v"   and "dec (\<tau> n p) = Some w" shows "w = v"
inj_omega_sum: fixes n :: "nat" shows "inj_on omega_sum {l. length l = n}"
list_all_append: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" shows "list_all P (xs @ ys) = (list_all P xs \<and> list_all P ys)"
ide_compE: fixes g :: "'b"   and f :: "'b" assumes "B.ide (g \<cdot>\<^sub>B f)" obtains "B.seq g f" and "B.seq f g" and "g \<cdot>\<^sub>B f = B.dom f" and "g \<cdot>\<^sub>B f = B.cod g"
alpha_Tree_eqvt: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree"   and p :: "perm" assumes "t1 =\<^sub>\<alpha> t2" shows "p \<bullet> t1 =\<^sub>\<alpha> p \<bullet> t2"
abs_eq: fixes xc :: "('a, ('b, 'c) mapping) mapping"   and xb :: "'c"   and xa :: "'c \<Rightarrow> 'c"   and x :: "('a \<times> 'b) set" shows "upd_nested xc xb xa x = Mapping (\<lambda>a. case Mapping.lookup xc a of None \<Rightarrow> if a \<in> fst ` x then Some (upd_set' Mapping.empty xb xa {b. (a, b) \<in> x}) else None | Some m' \<Rightarrow> Some (upd_set' m' xb xa {b. (a, b) \<in> x}))"
lve_Entry_Exit_edge: fixes e :: "'f LDCFG_node \<times> 'g edge_kind \<times> 'f LDCFG_node"   and valid_edge :: "'h \<Rightarrow> bool"   and src :: "'h \<Rightarrow> 'f"   and trg :: "'h \<Rightarrow> 'f"   and knd :: "'h \<Rightarrow> 'g edge_kind"   and E :: "'f"   and X :: "'f" assumes "e = (NewEntry, (\<lambda>s. False)\<^sub>\<surd>, NewExit)" shows "lift_valid_edge valid_edge src trg knd E X e"
map_gpv'_comp: fixes f :: "'k \<Rightarrow> 'h"   and g :: "'l \<Rightarrow> 'i"   and h :: "'j \<Rightarrow> 'm"   and f' :: "'n \<Rightarrow> 'k"   and g' :: "'o \<Rightarrow> 'l"   and h' :: "'m \<Rightarrow> 'p"   and gpv :: "('n, 'o, 'p) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
basis_list_def: fixes n :: "nat"   and fs :: "'a vec list" shows "cof_vec_space.basis_list n fs = (set fs \<subseteq> carrier_vec n \<and> length fs = n \<and> carrier_vec n \<subseteq> LinearCombinations.module.span class_ring (module_vec TYPE('a) n) (set fs))"
frac_less: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and 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 x :: "'a"   and y :: "'a"   and w :: "'a"   and z :: "'a" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq zero x"   and "less x y"   and "less zero w"   and "less_eq w z" shows "less (divide x z) (divide y w)"
igSwapIPresIGWls_def: fixes MOD :: "('i, 'j, 'k, 'l, 'm, 'n, 'o, 'p, 'q) model_scheme" shows "igSwapIPresIGWls MOD \<equiv> \<forall>zs z1 z2 s X. igWls MOD s X \<longrightarrow> igWls MOD s (igSwap MOD zs z1 z2 X)"
bounded_clinear_eq_on: fixes A :: "'b \<Rightarrow> 'c"   and B :: "'b \<Rightarrow> 'c"   and G :: "'b set"   and t :: "'b" assumes "bounded_clinear A"   and "bounded_clinear B"   and "\<And>x. x \<in> G \<Longrightarrow> A x = B x"   and "t \<in> closure (cspan G)" shows "A t = B t"
scalar_product_upper_bound: fixes b :: "nat list"   and a :: "nat"   and "is" :: "int list" assumes "\<forall>x\<in>set b. len x \<le> a" shows "scalar_product b is \<le> (2 ^ a - 1) * sum_list (concat (map (\<lambda>i. if 0 < i then [i] else []) is))"
D_eq: shows "D = ((\<lambda>z. ball z (ee z / 3)) \<circ> p) ` k"
inducts: fixes P :: "ty \<Rightarrow> bool"   and ty :: "ty" assumes "\<And>list. P (TVar list)"   and "\<And>ty1 ty2. \<lbrakk>P ty1; P ty2\<rbrakk> \<Longrightarrow> P (ty1 \<rightarrow> ty2)" shows "P ty"
monom_valuate: fixes m :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" assumes "is_monom m" shows "m \<lbrace> v \<rbrace> = monom_coeff m *R v (monom_var m)"
returnU_llist_def: shows "returnU = (\<Lambda> x. LCons\<cdot>x\<cdot>LNil)"
uints_def: fixes n :: "nat" shows "uints n = range (take_bit n)"
divide_poly_main_field: fixes d :: "'b poly"   and q :: "'b poly"   and r :: "'b poly"   and dr :: "nat"   and n :: "nat" assumes "d \<noteq> 0" shows "divide_poly_main (lead_coeff d) q r d dr n = fst (pseudo_divmod_main (lead_coeff d) (smult (((1::'b) / lead_coeff d) ^ n) q) (smult (((1::'b) / lead_coeff d) ^ n) r) d dr n)"
simps_4: fixes x2 :: "'b"   and x1 :: "'a" shows "TargetTerm x2 \<noteq> SourceTerm x1"
fun_of_def: fixes S :: "('a \<times> 'b) set" shows "fun_of S \<equiv> \<lambda>x y. (x, y) \<in> S"
se_mono_for_sub: fixes c1 :: "('c, 'd) conf"   and l :: "('c, 'd) label"   and c1' :: "('c, 'd) conf"   and c2 :: "('c, 'd) conf"   and c2' :: "('c, 'd) conf" assumes "se c1 l c1'"   and "se c2 l c2'"   and "c2 \<sqsubseteq> c1" shows "c2' \<sqsubseteq> c1'"
hyp: shows "M.sumlist xs $ i = (\<Sum>j = 0..<length xs. xs ! j $ i)"
a: shows "e \<in> set_pmf (mix_pmf \<beta> p q)"
simps_2: fixes x21 :: "'a"   and x22 :: "'a"   and y21a :: "'a"   and y22a :: "'a" shows "(Edge x21 x22 = Edge y21a y22a) = (x21 = y21a \<and> x22 = y22a)"