hcju/isabelleps huggingface dataset:
mask_range_to_bl': fixes ptr :: "'b word"   and bits :: "nat" assumes "is_aligned ptr bits"   and "bits < LENGTH('b)" shows "mask_range ptr bits = {x. take (LENGTH('b) - bits) (to_bl x) = take (LENGTH('b) - bits) (to_bl ptr)}"
below_def: fixes L :: "(alt \<times> alt) set"   and a :: "alt"   and b :: "alt" shows "below L a b \<equiv> {(x, y). x <\<^bsub>L\<^esub> y \<and> x \<noteq> a \<and> y \<noteq> a} \<union> {(a, b)} \<union> {(x, a) |x. x <\<^bsub>L\<^esub> b \<and> x \<noteq> a} \<union> {(a, y) |y. b <\<^bsub>L\<^esub> y \<and> y \<noteq> a}"
uniformity_refl: fixes E :: "'a \<times> 'a \<Rightarrow> bool"   and x :: "'a" assumes "eventually E uniformity" shows "E (x, x)"
is_down: shows "is_down SCAST('a \<rightarrow> 'b)"
tendsto_ident_at: fixes "open" :: "'d set \<Rightarrow> bool"   and a :: "'d"   and s :: "'d set" assumes "class.topological_space open" shows "LIM x topological_space.at_within open a s. x :> topological_space.nhds open a"
restricted_deduct_if_deduct': fixes M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and m :: "('fun, 'var) Term.term" assumes "\<forall>m\<in>M. P m"   and "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'"   and "\<forall>t K T k. P t \<longrightarrow> Ana t = (K, T) \<longrightarrow> k \<in> set K \<longrightarrow> P k"   and "M \<turnstile> m"   and "P m" shows "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m"
word_of_int_eq_iff: fixes k :: "int"   and l :: "int" shows "(word_of_int k = word_of_int l) = (take_bit LENGTH('a) k = take_bit LENGTH('a) l)"
f: shows "f = (\<Prod>a\<in>gs. a ^ Suc (g a))"
r_inv_ex: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "\<exists>y\<in>carrier C. x \<otimes>\<^bsub>C\<^esub> y = \<one>\<^bsub>C\<^esub>"
upd_row_row: fixes M :: "('c, 'd) sq_matrix"   and i :: "'d" shows "upd_row M i (Square_Matrix.row M i) = M"
right_inverse: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and inverse :: "'e \<Rightarrow> 'e"   and a :: "'e" assumes "group f z inverse" shows "f a (inverse a) = z"
bernoulli_stream_component_probability: fixes N :: "bool stream measure"   and p :: "real" assumes "N = bernoulli_stream p"   and "0 \<le> p"   and "p \<le> 1" shows "\<forall>n. emeasure N {w \<in> space N. w !! n} = ennreal p"
mset_ran_combine_eq_diff: fixes a :: "int \<Rightarrow> 'b"   and r :: "int set"   and I :: "int set"   and a' :: "int \<Rightarrow> 'b" assumes "mset_ran a (r - I) = mset_ran a' (r - I)"   and "a = a' on I" shows "mset_ran a r = mset_ran a' r"
totient_power_Suc: fixes n :: "nat"   and m :: "nat" shows "totient (n ^ Suc m) = n ^ m * totient n"
addition2: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and h :: "nat \<Rightarrow> nat" assumes "f \<in> {j. j \<le> Suc n} \<rightarrow> carrier M"   and "h \<in> {j. j \<le> Suc n} \<rightarrow> {j. j \<le> Suc n}"   and "inj_on h {j. j \<le> Suc n}" shows "\<Sigma>\<^sub>e M cmp f h Suc n = \<Sigma>\<^sub>e M f Suc n"
in_num: fixes var :: "'a set"   and trm :: "'b set"   and fmla :: "'c set"   and Var :: "'a \<Rightarrow> 'b"   and FvarsT :: "'b \<Rightarrow> 'a set"   and substT :: "'b \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"   and Fvars :: "'c \<Rightarrow> 'a set"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c"   and num :: "'b set"   and n :: "'b" assumes "Syntax_with_Numerals var trm fmla Var FvarsT substT Fvars subst num"   and "n \<in> num" shows "n \<in> trm"
circular: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqsubseteq> y"   and "y \<sqsubseteq> z"   and "z \<sqsubseteq> x" shows "x = y \<and> y = z"
mask_eq_sum_exp: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "minus (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) one = comm_monoid_add.sum plus zero (power.power one times (numeral.numeral one plus (num.Bit0 num.One))) {q. q < n}"
simps_1: fixes N :: "nat"   and fas :: "floatarith list"   and xs :: "nat list"   and ds :: "nat list"   and vs :: "real list" shows "isnFDERIV N fas xs ds vs 0 = True"
fold_map_idx_count_1: fixes f :: "nat \<Rightarrow> bool \<Rightarrow> 'a \<Rightarrow> bool \<times> 'a"   and g :: "nat \<Rightarrow> 'a \<Rightarrow> nat"   and i :: "nat"   and c :: "bool"   and xs :: "'a list"   and y :: "nat" assumes "\<And>i c x y. fst (f i c x) = (c \<or> g y (snd (f i c x)) < g y x)"   and "\<And>i c x y. g y (snd (f i c x)) \<le> g y x"   and "\<And>x y y'. y < y' \<Longrightarrow> g y x < g y' x" shows "fst (fold_map_idx f i c xs) = (c \<or> foldl g y (snd (fold_map_idx f i c xs)) < foldl g y xs)"
lower_closed: fixes x :: "'a" assumes "x \<in> carrier \<X>\<^bsub>G\<^esub>" shows "\<pi>\<^sup>*\<^bsub>G\<^esub> x \<in> carrier \<Y>\<^bsub>G\<^esub>"
rep_eq: fixes x :: "\<nu> \<Rightarrow> \<o>" shows "eval\<Pi>\<^sub>1 (lambdabinder1 x) = (\<lambda>u s w. \<exists>xa. \<nu>\<upsilon> xa = u \<and> eval\<o> (x xa) s w)"
is_equivalence: shows "internal_equivalence (f \<star> f') (g' \<star> g) gg'.\<eta> gg'.\<epsilon>"
ground_aux_simps_2: fixes k :: "name"   and S :: "atom set" shows "ground_aux (Var k) S = (if atom k \<in> S then True else False)"
comp_right_conjugate: fixes x :: "'a" shows "conjugate (\<lambda>y. y * x) (\<lambda>y. y * x\<^sup>T)"
primitive_extractor_correct_2: fixes m :: "'c match_expr"   and disc :: "'c \<Rightarrow> bool"   and sel :: "'c \<Rightarrow> 'd"   and C :: "'d \<Rightarrow> 'c"   and as :: "'d negation_type list"   and ms :: "'c match_expr" assumes "normalized_nnf_match m"   and "wf_disc_sel (disc, sel) C"   and "primitive_extractor (disc, sel) m = (as, ms)" shows "normalized_nnf_match ms"
a'_insertion: shows "insertion (nth_default 0 (xs' @ x # xs)) (isolate_variable_sparse p var 2) = a'"
intros_2: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and root' :: "(att \<times> char list, att, nat) env"   and xs :: "operation list"   and root'' :: "(att \<times> char list, att, nat) env" assumes "root \<midarrow>x\<rightarrow> root'"   and "root' \<Midarrow>xs\<Rightarrow> root''" shows "root \<Midarrow>(x # xs)\<Rightarrow> root''"
last_snoc: fixes xs :: "'c list"   and x :: "'c" shows "last (xs @ [x]) = x"
distinct_13: fixes x11 :: "'a"   and x12 :: "'b" shows "x11 \<leftarrow> x12 \<noteq> Stop"
Abinds_reorder1: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and v :: "Vars.var"   and e :: "exp"   and Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "map_of \<Gamma> v = Some e" shows "ArityAnalysis.ABinds Aexp \<Gamma> = ArityAnalysis.ABind Aexp v e \<squnion> ArityAnalysis.ABinds Aexp (delete v \<Gamma>)"
nonneg: shows "nonneg_cexpr (V \<union> V') \<Gamma> e1"
that: fixes n :: "nat" assumes "n < length exec"   and "enabled (execConf n) msg"   and "\<forall>n'\<ge>n. n' < length trace \<longrightarrow> msg \<noteq> execMsg n'"   and "\<forall>n0. n0 < length exec \<and> enabled (execConf n0) msg \<and> (\<forall>n'\<ge>n0. n' < length trace \<longrightarrow> msg \<noteq> execMsg n') \<longrightarrow> n \<le> n0" shows "thesis"
has_contour_integral_sum: fixes s :: "'a set"   and f :: "'a \<Rightarrow> complex \<Rightarrow> complex"   and i :: "'a \<Rightarrow> complex"   and p :: "real \<Rightarrow> complex" assumes "finite s"   and "\<And>a. a \<in> s \<Longrightarrow> (f a has_contour_integral i a) p" shows "((\<lambda>x. \<Sum>a\<in>s. f a x) has_contour_integral sum i s) p"
simps_1: fixes xs :: "('b, 'c) Combinators list" shows "insertDeny (DenyAll # xs) = DenyAll # xs"
Units_l_cancel: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> Units \<G>"   and "y \<in> carrier \<G>"   and "z \<in> carrier \<G>" shows "(x \<otimes> y = x \<otimes> z) = (y = z)"
RG_proj_in_RG: fixes G :: "'a set"   and A :: "'c set"   and x :: "('b, 'c) aezfun" assumes "Group G" shows "aezfun_setspan_proj A x \<in> aezfun_setspan A"
rquot_accepts: fixes A :: "nat bdd list \<times> bool list"   and n :: "nat"   and bss :: "bool list list" assumes "wf_dfa A n"   and "list_all (is_alph n) bss" shows "dfa_accepts (rquot A n) bss = (\<exists>m. dfa_accepts A (bss @ zeros m n))"
is_rbt_rbt_unionwk: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and t1 :: "('c, 'd) rbt"   and t2 :: "('c, 'd) rbt"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd" assumes "class.linorder less_eq less"   and "ord.is_rbt less t1"   and "ord.is_rbt less t2" shows "ord.is_rbt less (ord.rbt_union_with_key less f t1 t2)"
apath_succ_decomp: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a"   and x :: "'a"   and y :: "'a" assumes "G.apath u p v"   and "(x, y) \<in> set p"   and "y \<noteq> v" shows "\<exists>p1 z p2. p = p1 @ (x, y) # (y, z) # p2 \<and> x \<noteq> z \<and> y \<noteq> z"
unlabel_def: fixes S :: "('d \<times> 'c) list" shows "unlabel S \<equiv> map snd S"
invertible_bezout_matrix: fixes bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and a :: "'c"   and b :: "'c"   and A :: "(('b, 'd) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec"   and j :: "'d" assumes "is_bezout_ext bezout"   and "a < b"   and "A $h a $h j \<noteq> (0::'b)" shows "invertible (bezout_matrix A a b j bezout)"
assms_8: shows "InvariantNoDecisionsWhenUnit (butlast (getF state)) (getM state) (currentLevel (getM state))"
assms: shows "\<langle>\<langle>r, A\<rangle>, s\<rangle> \<in>\<^sub>\<circ> CZH_Sets_BRelations.vimage D"
pred_cof_def: fixes L :: "char list set"   and t :: "sterm"   and t' :: "sterm" shows "pred_cof L t t' = (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> (\<exists>t''. t\<^bsup>[Fvar s,Fvar p]\<^esup> ->> t'' \<and> t' = \<sigma>[s,p] t''))"
else_: shows "eval (Cn 1 r_prod_encode [Cn 1 r_snoc [r_pdec1, Z], Cn 1 r_length [r_pdec1]]) [arg] \<down>= prod_encode (e_snoc b 0, e_length b)"
span_base: fixes a :: "'a"   and S :: "'a set" assumes "a \<in> S" shows "a \<in> ideal S"
PiE_def: fixes S :: "'a set"   and T :: "'a \<Rightarrow> 'b set" shows "Pi\<^sub>E S T = Pi S T \<inter> extensional S"
pw: shows "pairwise (\<lambda>S T. S \<inter> T \<subseteq> K) {X \<in> \<F>. \<forall>Y\<in>\<F>. \<not> X \<subset> Y}"
prems: shows "(A, B) \<in> set (pairs (A # As))"
that: fixes v :: "'c" assumes "f v = v'"   and "(u, v) \<in> (\<Union> (range (relations M)))\<^sup>*" shows "thesis"
allAllocationsInPowerset: fixes \<Omega> :: "'d set"   and N :: "'c set" shows "allAllocations N \<Omega> \<subseteq> Pow (N \<times> (Pow \<Omega> - {{}}))"
assms_2: shows "lookup q v \<noteq> (0::'b)"
fpairs_def: fixes r :: "V" shows "fpairs r = set {x. x \<in>\<^sub>\<circ> r \<and> (\<exists>a b. x = [a, b]\<^sub>\<circ>)}"
sum_to_tup_app: fixes f :: "('d + 'f) discr \<rightarrow> 'e" shows "sum_to_tup\<cdot>f = (\<Lambda> x. f\<cdot>(Discr (Inl (undiscr x))), \<Lambda> x. f\<cdot>(Discr (Inr (undiscr x))))"
first_pd_no_cd: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and \<pi> :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "IFC prob"   and "IFC_def.is_path prob \<pi>"   and "IFC_def.is_pd prob (\<pi> n) (\<pi> 0)"   and "\<forall>l<n. \<pi> l \<noteq> \<pi> n" shows "\<forall>l. \<not> IFC_def.is_cdi prob n \<pi> l"
merge_env_def: fixes merge :: "'i cset \<Rightarrow> ('i \<Rightarrow> 'h) \<Rightarrow> 'h"   and A :: "'i cset"   and f :: "'i \<Rightarrow> ('g, 'h) envT" shows "merge_env merge A f = EnvT (\<lambda>r. merge A (\<lambda>a. run_env (f a) r))"
has_derivative_def: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "(f has_derivative f') F = (bounded_linear f' \<and> ((\<lambda>y. (f y - f (Lim F (\<lambda>x. x)) - f' (y - Lim F (\<lambda>x. x))) /\<^sub>R norm (y - Lim F (\<lambda>x. x))) \<longlongrightarrow> (0::'d)) F)"
distrib_right: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and a :: "'k"   and b :: "'k"   and c :: "'k" assumes "class.semiring plus times" shows "times (plus a b) c = plus (times a c) (times b c)"
i_Streams_Connected_def: fixes outS :: "nat \<Rightarrow> 'd message_af"   and inS :: "nat \<Rightarrow> 'd message_af" shows "i_Streams_Connected outS inS \<equiv> inS = [NoMsg] \<frown> outS"
m_assoc: fixes x :: "'c"   and y :: "'c"   and z :: "'c" assumes "x \<in> carrier M"   and "y \<in> carrier M"   and "z \<in> carrier M" shows "x \<oplus>\<^bsub>M\<^esub> y \<oplus>\<^bsub>M\<^esub> z = x \<oplus>\<^bsub>M\<^esub> (y \<oplus>\<^bsub>M\<^esub> z)"
rtranclp_into_tranclp1: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>*\<^sup>* a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
module_prod_scale_eq_scaleR: shows "module_prod.scale (*\<^sub>R) (*\<^sub>R) = (*\<^sub>R)"
no_marker_left_in_final_state: fixes t :: "('a, 'b, 'c) event list"   and cid :: "nat" assumes "trace init t final" shows "Marker \<notin> set (msgs final cid)"
span_inv: fixes e :: "'a"   and v :: "'a"   and t :: "'a"   and g :: "'a"   and r :: "'a" assumes "e \<le> v * - v\<^sup>T"   and "vector v"   and "arc e"   and "t \<le> v * v\<^sup>T \<sqinter> g"   and "symmetric g"   and "v\<^sup>T = r\<^sup>T * t\<^sup>\<star>"   and "injective r"   and "r\<^sup>T \<le> v\<^sup>T"   and "r\<^sup>T * (v * v\<^sup>T \<sqinter> g)\<^sup>\<star> \<le> r\<^sup>T * t\<^sup>\<star>" shows "r\<^sup>T * ((v \<squnion> e\<^sup>T * top) * (v \<squnion> e\<^sup>T * top)\<^sup>T \<sqinter> g)\<^sup>\<star> \<le> r\<^sup>T * (t \<squnion> e)\<^sup>\<star>"
multiplicity_times_unit_left: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times c one" shows "factorial_semiring.multiplicity times one (times c p) x = factorial_semiring.multiplicity times one p x"
partn_def: fixes \<beta> :: "'b set"   and \<alpha> :: "nat"   and \<gamma> :: "nat"   and \<delta> :: "'c set" shows "partn \<beta> \<alpha> \<gamma> \<delta> \<equiv> \<forall>f\<in>[\<beta>]\<^bsup>\<gamma>\<^esup> \<rightarrow> \<delta>. \<exists>H\<in>[\<beta>]\<^bsup>\<alpha>\<^esup>. \<exists>\<xi>\<in>\<delta>. f ` [H]\<^bsup>\<gamma>\<^esup> \<subseteq> {\<xi>}"
in_spanE: fixes v :: "'b vec"   and n :: "nat"   and W :: "'b vec set" assumes "v \<in> LinearCombinations.module.span class_ring (module_vec TYPE('b) n) W" shows "\<exists>a A. v = module.lincomb (module_vec TYPE('b) n) a A \<and> finite A \<and> A \<subseteq> W"
xor_self: fixes x :: "'a" shows "x XOR x = (0::'a)"
entails_equiv_forward: fixes P :: "assn"   and Q :: "assn" assumes "P = Q" shows "P \<Longrightarrow>\<^sub>A Q"
single_list_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.linorder less_eq less" shows "linorder.single_list a = [a]"
assms_1: shows "p \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, U \<rightarrow>) P"
update_def: fixes k :: "'a"   and v :: "'b"   and m :: "(uint32, ('a, 'b) assoc_list) RBT.rbt" shows "HashMap_Impl.update k v m \<equiv> let hc = hashcode k in case rm.lookup hc m of None \<Rightarrow> rm.update hc (lm.update k v (lm.empty ())) m | Some bm \<Rightarrow> rm.update hc (lm.update k v bm) m"
S_M_grounding_of_clss: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and C :: "'a literal multiset"   and M :: "'a literal multiset set" assumes "selection S"   and "C \<in> grounding_of_clss M" obtains D :: "'a literal multiset" and \<sigma> :: "'s" where "D \<in> M \<and> C = D \<cdot> \<sigma> \<and> S_M S M C = S D \<cdot> \<sigma> \<and> is_ground_subst \<sigma>"
isTrue_iff_prv_neg_PPf: fixes n :: "'b"   and \<phi> :: "'c" assumes "n \<in> num"   and "\<phi> \<in> fmla"   and "Fvars \<phi> = {}" shows "isTrue (B.neg (PPf n \<langle>\<phi>\<rangle>)) = bprv (B.neg (PPf n \<langle>\<phi>\<rangle>))"
graph_symmetric: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow> v" shows "v \<rightarrow> u"
assms_1: shows "elem \<lbrakk>\<Psi> (l, u) i\<rbrakk> (l', u')"
string_of_word_base_ten_zeropad: fixes base :: "'a word"   and zero :: "nat"   and w :: "'a word" assumes "5 \<le> LENGTH('a)"   and "base = 10"   and "zero = 0" shows "string_of_word True base zero w = string_of_nat (unat w)"
mon_s_single: fixes fg :: "('b, 'c, 'd, 'a, 'e) flowgraph_rec_scheme"   and n :: "'b" shows "mon_s fg [n] = mon_n fg n"
closedin_product_topology: fixes X :: "'c \<Rightarrow> 'd topology"   and I :: "'c set"   and S :: "'c \<Rightarrow> 'd set" shows "closedin (product_topology X I) (Pi\<^sub>E I S) = (Pi\<^sub>E I S = {} \<or> (\<forall>i\<in>I. closedin (X i) (S i)))"
onormI: fixes f :: "'a \<Rightarrow> 'b"   and b :: "real"   and x :: "'a" assumes "\<And>x. norm (f x) \<le> b * norm x"   and "x \<noteq> (0::'a)"   and "norm (f x) = b * norm x" shows "onorm f = b"
Invariants_measurable_func: fixes f :: "'a \<Rightarrow> 'c"   and N :: "'c measure" assumes "f \<in> Invariants \<rightarrow>\<^sub>M N" shows "f \<in> M \<rightarrow>\<^sub>M N"
Lang_no_Nil: fixes w :: "'c list"   and G :: "('d \<times> ('d, 'c) RHS) list"   and S :: "'d" assumes "w \<in> Lang G S" shows "w \<noteq> []"
AE_alw_ev_same_loc_iff': fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and st :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> R_G.cfg_on (abss st)"   and "st \<in> S" shows "alternating cfg = alternating (repcs st cfg)"
log_less_cancel_iff: fixes a :: "real"   and x :: "real"   and y :: "real" assumes "1 < a"   and "0 < x"   and "0 < y" shows "(Transcendental.log a x < Transcendental.log a y) = (x < y)"
not_stable_on_cases: fixes ds :: "'b set"   and X :: "'a set" assumes "\<not> stable_on ds X" obtains "\<not> individually_rational_on ds X"   | "\<not> stable_no_blocking_on ds X"
ts't: shows "ts1' t = \<lfloor>(x', ln')\<rfloor>"
finite_set_mset: fixes M :: "'b multiset" shows "finite (set_mset M)"
assms_3: shows "b \<noteq> \<infinity>\<^sub>h"
eq_resource_on_refl: fixes A :: "'a set"   and res :: "('a, 'b) resource" shows "A \<turnstile>\<^sub>R res \<sim> res"
ereal_mult_right_mono: fixes a :: "ereal"   and b :: "ereal"   and c :: "ereal" assumes "a \<le> b"   and "0 \<le> c" shows "a * c \<le> b * c"
forest_separate: fixes x :: "'a" assumes "forest x" shows "coreflexive (x\<^sup>\<star> * x\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * x)"
fps_divide_nth_0': fixes g :: "'a fps"   and f :: "'a fps" assumes "subdegree g = 0" shows "(f / g) $ 0 = f $ 0 / g $ 0"
InvariantQCharacterizationAfterApplyDecide: fixes Vbl :: "nat set"   and state :: "State" assumes "\<not> Vbl \<subseteq> vars (elements (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 "getQ state = []" shows "let state' = applyDecide state Vbl in InvariantQCharacterization (getConflictFlag state') (getQ state') (getF state') (getM state')"
prime_dvd_power_int: fixes p :: "int"   and x :: "int"   and n :: "nat" assumes "prime p"   and "p dvd x ^ n" shows "p dvd x"
simps_2: fixes x :: "variable" shows "Existso x undeff = undeff"
substBinp_def: fixes xs :: "'varSort"   and X :: "('b, 'c, 'varSort, 'var, 'd) qTerm set"   and x :: "'var" shows "substBinp xs X x \<equiv> psubstBinp (idEnv [x \<leftarrow> X]_xs)"
simps_2: fixes v :: "'d"   and Q :: "'d \<Rightarrow> bool" shows "ospec (Some v) Q = Q v"
less_imp_not_less: fixes x :: "'b"   and y :: "'b" assumes "x < y" shows "(\<not> y < x) = True"
that: fixes f :: "'a \<times> real \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'a \<times> real"   and j :: "'b" assumes "linear f"   and "linear g"   and "\<And>z. g (f z) = z"   and "j \<in> Basis"   and "\<And>x. f (x, 0) \<bullet> j = 0" shows "thesis"
scale: fixes b :: "'a \<Rightarrow> real"   and r :: "real" assumes "\<forall>x. x \<in> src.diff_fun_space \<longrightarrow> X x \<in> UNIV"   and "b \<in> src.diff_fun_space" shows "X (r *\<^sub>R b) = r *\<^sub>R X b"
app_free_distinct: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and u\<^sub>1 :: "'c"   and u\<^sub>2 :: "'c"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "app u\<^sub>1 u\<^sub>2 \<noteq> free name"
length_drop: fixes n :: "nat"   and xs :: "'e list" shows "length (drop n xs) = length xs - n"
lbvjvm_def: fixes P :: "jvm_prog"   and mxs :: "nat"   and maxr :: "nat"   and T\<^sub>r :: "ty"   and et :: "ex_table"   and cert :: "(ty list \<times> ty err list) option err list"   and bs :: "bytecode" shows "lbvjvm P mxs maxr T\<^sub>r et cert bs \<equiv> wtl_inst_list bs cert (JVM_SemiType.sup P mxs maxr) (JVM_SemiType.le P mxs maxr) Err (OK None) (exec P mxs T\<^sub>r et bs) 0"
product_nn_integral_pair: fixes f :: "real \<Rightarrow> real \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a" assumes "(\<lambda>(x, y). f x y) \<in> borel_measurable (M \<Otimes>\<^sub>M M)"   and "x \<noteq> y" shows "\<integral>\<^sup>+ \<sigma>. f (\<sigma> x) (\<sigma> y) \<partial>Pi\<^sub>M {x, y} (\<lambda>_. M) = \<integral>\<^sup>+ z. f (fst z) (snd z) \<partial>M \<Otimes>\<^sub>M M"
finite_code_1: fixes dxs :: "'b set_dlist" shows "finite (DList_set dxs) = (case ID CEQ('b) of None \<Rightarrow> Code.abort STR ''finite DList_set: ceq = None'' (\<lambda>_. finite (DList_set dxs)) | Some x \<Rightarrow> True)"
prems: shows "\<nexists>n\<^sub>0. f n\<^sub>0 = Suc n"
restrict0_apply_in: fixes x :: "'c"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "x \<in> A" shows "restrict0 A f x = f x"
inject_4: fixes x5 :: "'a OCL_Types.type"   and y5 :: "'a OCL_Types.type" shows "(Set x5 = Set y5) = (x5 = y5)"
unfold_observe_none: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and d :: "nat \<Rightarrow>f value option"   and p :: "value option list"   and t :: "(String.literal \<times> value list) stream" shows "make_full_observation e None d p t = \<lparr>statename = None, datastate = d, action = shd t, output = p\<rparr> ## make_full_observation e None d [] (stl t)"
map_indets_inverseE_Polys: fixes f :: "'c \<Rightarrow> 'd"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e" assumes "inj_on f X"   and "p \<in> P[X]" shows "map_indets (the_inv_into X f) (map_indets f p) = p"
poly_def: fixes p :: "'a poly"   and a :: "'a" shows "poly p a = horner_sum id a (coeffs p)"
le_refl: fixes L :: "('c, 'd) gorder_scheme"   and x :: "'c" assumes "weak_partial_order L"   and "x \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> x"