hcju/isabelleps huggingface dataset:
CastFail: fixes 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"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and h :: "nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option"   and l :: "char list \<Rightarrow> val option"   and D :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and C :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>addr a,(h, l)\<rangle>"   and "h a = \<lfloor>(D, fs)\<rfloor>"   and "(D, C) \<notin> (subcls1 P)\<^sup>*" shows "P \<turnstile> \<langle>Cast C e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>THROW ClassCast,(h, l)\<rangle>"
split_list_first_propE: fixes xs :: "'c list"   and P :: "'c \<Rightarrow> bool" assumes "\<exists>x\<in>set xs. P x" obtains ys :: "'c list" and x :: "'c" and zs :: "'c list" where "xs = ys @ x # zs" and "P x" and "\<forall>y\<in>set ys. \<not> P y"
content_prod_mset: fixes A :: "'a poly multiset" shows "content (\<Prod>\<^sub># A) = \<Prod>\<^sub># (image_mset content A)"
C1_def: shows "C1 = Max {B k / B 0 |k. k \<le> N + K + 1}"
entry_valid: fixes fg :: "('f, 'g, 'h, 'i, 'j) flowgraph_rec_scheme"   and p :: "'g" assumes "flowgraph fg" shows "proc_of fg (entry fg p) = p"
filter_empty_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "(filter P xs = []) = (\<forall>x\<in>set xs. \<not> P x)"
Hermite_intro: fixes associates :: "'a set"   and residues :: "'a \<Rightarrow> 'a set"   and A :: "(('a, 'b) vec, 'c) vec" assumes "Complete_set_non_associates associates"   and "Complete_set_residues residues"   and "echelon_form A"   and "\<forall>i. \<not> is_zero_row i A \<longrightarrow> A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> associates"   and "\<forall>i. \<not> is_zero_row i A \<longrightarrow> (\<forall>j<i. A $ j $ (LEAST n. A $ i $ n \<noteq> (0::'a)) \<in> residues (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))" shows "Hermite associates residues A"
vbrelation_vinE: fixes x :: "V" assumes "x \<in>\<^sub>\<circ> f" obtains a :: "V" and b :: "V" where "x = \<langle>a, b\<rangle>" and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> f" and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> f"
path_component_of_eq_overlap: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" shows "(path_component_of X x = path_component_of X y) = (x \<notin> topspace X \<and> y \<notin> topspace X \<or> path_component_of_set X x \<inter> path_component_of_set X y \<noteq> {})"
wpt_is_assertion: fixes x :: "'a" shows "wpt x \<in> assertion"
preserves_reflects_arr: fixes f :: "'a" shows "B.arr (B.R f) = B.arr f"
simps_4: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_letter (np(a)) \<nu> = (if a \<notin> \<nu> then true else false)"
rel_flip: fixes R :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b pmf"   and b :: "'c pmf" shows "rel_pmf R\<inverse>\<inverse> a b = rel_pmf R b a"
Pred\<^sub>\<alpha>: fixes f :: "'d"   and \<phi> :: "'b" shows "hereditarily_fs (Pred\<^sub>\<alpha> f \<phi>)"
SigmaI: fixes a :: "'a"   and A :: "'a set"   and b :: "'b"   and B :: "'a \<Rightarrow> 'b set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
sim_r_def: fixes M :: "('d, 'e, 'f) KripkeStructure"   and M' :: "('d, 'e, 'g) KripkeStructure"   and f :: "'f \<Rightarrow> 'g" shows "sim_r M M' f \<equiv> \<forall>a. \<forall>u\<in>worlds M. \<forall>v'. (f u, v') \<in> relations M' a \<longrightarrow> (\<exists>v. (u, v) \<in> relations M a \<and> f v = v')"
Max_wrt_nonempty: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le"   and "carrier \<noteq> {}"   and "finite carrier" shows "Max_wrt le \<noteq> {}"
log_exp2_le: fixes n :: "nat" assumes "0 < n" shows "2 ^ Discrete.log n \<le> n"
Inf_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "Inf A = (\<lambda>x. INF f\<in>A. f x)"
sum_multicount: fixes S :: "'a set"   and T :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and k :: "nat" assumes "finite S"   and "finite T"   and "\<forall>j\<in>T. card {i \<in> S. R i j} = k" shows "(\<Sum>i\<in>S. card {j \<in> T. R i j}) = k * card T"
WT_callee: fixes x :: "'e"   and s :: "'b" assumes "x \<in> outs_\<I> \<I>" shows "\<I>' \<turnstile>g callee s x \<surd>"
poly_of_vec_def: fixes v :: "'b vec" shows "poly_of_vec v \<equiv> let d = dim_vec v in \<Sum>i<d. monom (v $ (d - Suc i)) i"
sigma_semimultiplicative: fixes p :: "nat"   and m :: "nat"   and n :: "nat" assumes "prime p"   and "coprime p m" shows "sigma (p ^ n) * sigma m = sigma (p ^ n * m)"
initmissing_def: fixes \<sigma> :: "(nat \<Rightarrow> 'g option) \<times> 'a" shows "initmissing \<sigma> = (\<lambda>i. case fst \<sigma> i of None \<Rightarrow> someinit i | Some s \<Rightarrow> s, snd \<sigma>)"
max_def: fixes a :: "'a word"   and b :: "'a word" shows "signed.max a b = (if a \<le>s b then b else a)"
simps_1: fixes pf :: "'form" shows "nodeOf (pf, []) = Conclusion pf"
S4_successors: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "\<turnstile> $S4 rmhist p \<and> ImpNext p \<and> [HNext rmhist p]_(c p, r p, m p, rmhist!p) \<and> (\<forall>l. $MemInv mm l) \<longrightarrow> S4 rmhist p$ \<or> S5 rmhist p$"
at_set_avoiding_aux': fixes Xs :: "'a set"   and As :: "'a set"   and c :: "'b" assumes "at TYPE('a)"   and "finite Xs"   and "Xs \<subseteq> As"   and "finite As"   and "finite (supp c)" shows "\<exists>Ys pi. Ys \<sharp>* c \<and> Ys \<inter> As = {} \<and> pi \<bullet> Xs = Ys \<and> set pi \<subseteq> Xs \<times> Ys \<and> finite Ys \<and> distinctPerm pi"
Un_left_commute: fixes A :: "'g set"   and B :: "'g set"   and C :: "'g set" shows "A \<union> (B \<union> C) = B \<union> (A \<union> C)"
v: shows "v \<Turnstile>\<^sub>b \<B> (update\<B>\<I> (UBI_upd dir) i x c s)"
semiring_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semiring_ow U plus times \<equiv> ab_semigroup_add_ow U plus \<and> semigroup_mult_ow U times \<and> semiring_ow_axioms U plus times"
refines_def: fixes prog :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real"   and prog' :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "prog \<sqsubseteq> prog' \<equiv> \<forall>P. sound P \<longrightarrow> wp prog P \<tturnstile> wp prog' P"
monad_state_prob_envT: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and get :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b"   and put :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and sample :: "'d pmf \<Rightarrow> ('d \<Rightarrow> 'b) \<Rightarrow> 'b" assumes "monad_state_prob return bind get put sample" shows "monad_state_prob (return_env return) (bind_env bind) (get_env get) (put_env put) (sample_env sample)"
ideD_3: fixes a :: "'b" assumes "D.ide a" shows "D.cod a = a"
polymul_properties_2: fixes p :: "poly"   and n0 :: "nat"   and q :: "poly"   and n1 :: "nat"   and m :: "nat" assumes "isnpolyh p n0"   and "isnpolyh q n1"   and "m \<le> min n0 n1" shows "(p *\<^sub>p q = 0\<^sub>p) = (p = 0\<^sub>p \<or> q = 0\<^sub>p)"
takeWhile_append2: fixes xs :: "'b list"   and P :: "'b \<Rightarrow> bool"   and ys :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> P x" shows "takeWhile P (xs @ ys) = xs @ takeWhile P ys"
ANDLEFT1_eqvt_name: fixes pi :: "(name \<times> name) list"   and A :: "ty"   and B :: "ty"   and X :: "ntrm set" shows "pi \<bullet> ANDLEFT1 (A AND B) X = ANDLEFT1 (A AND B) (pi \<bullet> X)"
Inv2b_inner_def: fixes s :: "state"   and p :: "Proc"   and d :: "Disk" shows "Inv2b_inner s p d = ((d \<in> disksWritten s p \<longrightarrow> phase s p \<in> {1, 2} \<and> disk s d p = dblock s p) \<and> (phase s p \<in> {1, 2} \<longrightarrow> (blocksRead s p d \<noteq> {} \<longrightarrow> d \<in> disksWritten s p) \<and> \<not> hasRead s p d p))"
XC_cyl: fixes c :: "nat list" assumes "c \<in> C k" shows "XC c = cylinder S_edges (set (uwalk_edges c)) {}"
sets_in_extensional_aux: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" shows "{x \<in> space (Pi\<^sub>M I M). x \<in> extensional I} \<in> sets (Pi\<^sub>M I M)"
borel_measurable_sin: shows "sin \<in> borel_measurable borel"
induct: fixes x :: "point"   and P :: "point \<Rightarrow> bool" assumes "x \<in> constructible"   and "\<And>M. M \<in> points \<and> abscissa M \<in> \<rat> \<and> ordinate M \<in> \<rat> \<Longrightarrow> P M"   and "\<And>A B C D M. (A \<in> constructible \<and> P A) \<and> (B \<in> constructible \<and> P B) \<and> (C \<in> constructible \<and> P C) \<and> (D \<in> constructible \<and> P D) \<and> \<not> parallel A B C D \<and> is_intersection M A B C D \<Longrightarrow> P M"   and "\<And>A B C D E M. (A \<in> constructible \<and> P A) \<and> (B \<in> constructible \<and> P B) \<and> (C \<in> constructible \<and> P C) \<and> (D \<in> constructible \<and> P D) \<and> (E \<in> constructible \<and> P E) \<and> A \<noteq> B \<and> collinear A M B \<and> distance C M = distance D E \<Longrightarrow> P M"   and "\<And>A B C D E F M. (A \<in> constructible \<and> P A) \<and> (B \<in> constructible \<and> P B) \<and> (C \<in> constructible \<and> P C) \<and> (D \<in> constructible \<and> P D) \<and> (E \<in> constructible \<and> P E) \<and> (F \<in> constructible \<and> P F) \<and> \<not> (A = D \<and> distance B C = distance E F) \<and> distance A M = distance B C \<and> distance D M = distance E F \<Longrightarrow> P M" shows "P x"
rtranclp_converseI: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "r\<^sup>*\<^sup>* y x" shows "r\<inverse>\<inverse>\<^sup>*\<^sup>* x y"
sel: fixes x :: "'a" shows "of_phantom (Phantom('b) x) = x"
types_preserved_current_memory: fixes \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and c :: "i32" assumes "\<S>\<bullet>\<C> \<turnstile> [$Current_memory] : ts _> ts'" shows "\<S>\<bullet>\<C> \<turnstile> [$C ConstInt32 c] : ts _> ts'"
dom_char: fixes f :: "'a" shows "S.dom f = (if S.arr f then B.dom f else B.null)"
resultant_impl_generic: fixes div_exp :: "'b \<Rightarrow> 'b \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<And>x y n. to_fract x ^ n / to_fract y ^ (n - 1) \<in> range to_fract \<Longrightarrow> to_fract (div_exp x y n) = to_fract x ^ n / to_fract y ^ (n - 1)" shows "resultant_impl_generic div_exp = resultant"
new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_def: fixes h :: "(_) heap" shows "new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t h = (let new_shadow_root_ptr = shadow_root_ptr.Ref (Suc (fMax (shadow_root_ptr.the_ref |`| shadow_root_ptrs h))) in (new_shadow_root_ptr, put new_shadow_root_ptr (create_shadow_root_obj Open []) h))"
CallObjThrow: fixes 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"   and e :: "char list exp"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and M :: "char list"   and ps :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(ps),s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
index_map_mat_3: fixes f :: "'e \<Rightarrow> 'd"   and A :: "'e mat" shows "dim_col (map_mat f A) = dim_col A"
distinct_19: fixes x4 :: "'a \<times> nat"   and x5 :: "'a \<times> nat" shows "Actual_in x4 \<noteq> Actual_out x5"
matrix_vector_mul_diag_mat: fixes f :: "'e \<Rightarrow> 'd"   and s :: "('d, 'e) vec" shows "diag_mat f *v s = (\<chi>i. f i * s $ i)"
nat_pow_inv: fixes x :: "'a"   and i :: "nat" assumes "x \<in> carrier \<G>" shows "inv x [^] i = inv (x [^] i)"
assms_1: shows "normal sctn \<in> Basis \<or> - normal sctn \<in> Basis"
mr_cos_sprod_def: fixes M :: "('e, 'f, 'g) Module_scheme"   and H :: "'e set"   and a :: "'f"   and X :: "'e set" shows "mr_cos_sprod M H a X = {z. \<exists>x\<in>X. \<exists>h\<in>H. z = h \<plusminus>\<^bsub>M\<^esub> a \<cdot>\<^sub>s\<^bsub>M\<^esub> x}"
simps_1: fixes m :: "int"   and n :: "int" shows "plus_const (Const m) (Const n) = Const (m + n)"
sas_plus_equivalent_to_strips_i_a_VI: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and s :: "'c \<Rightarrow> 'd option"   and ops' :: "('c \<times> 'd) strips_operator list" assumes "is_valid_problem_sas_plus \<Psi>"   and "dom s \<subseteq> set (\<Psi>\<^sub>\<V>\<^sub>+)"   and "\<forall>v\<in>dom s. the (s v) \<in> \<R>\<^sub>+ \<Psi> v"   and "set ops' \<subseteq> set ((\<phi> \<Psi> )\<^sub>\<O>)"   and "are_all_operators_applicable_in s (map (strips_op_to_sasp \<Psi>) ops') \<and> SAS_Plus_Semantics.are_all_operator_effects_consistent (map (strips_op_to_sasp \<Psi>) ops')" shows "are_all_operators_applicable (\<phi>\<^sub>S \<Psi> s) ops'"
f2: fixes aa :: "'a"   and ab :: "'a" shows "aa \<notin> vs \<or> ab \<notin> vs' \<or> aa \<notin> fmdom' (snd a) \<or> ab \<notin> fmdom' (snd a)"
preserves_section_retraction: fixes e :: "'c"   and m :: "'c" assumes "C.ide (e \<cdot>\<^sub>C m)" shows "D.ide (G e \<cdot>\<^sub>D G m)"
basis_in_sub_cong: fixes \<E> :: "'a set"   and \<E>' :: "'a set"   and B :: "'a set" assumes "\<E> \<subseteq> \<E>'"   and "\<E>' \<subseteq> \<E>" shows "indep_system.basis_in (\<E>'\<^sub>1.indep_in \<E>) \<E>' B = \<E>'\<^sub>1.basis_in \<E>' B"
strict_implies_not_eq: fixes b :: "'a multiset"   and a :: "'a multiset" assumes "b \<subset># a" shows "a \<noteq> b"
arr_char: fixes f :: "'c local.term set \<times> 'c local.term set" shows "\<F>C.CC.arr f = (\<F>C.arr (fst f) \<and> \<F>C.arr (snd f))"
semigroup_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "semigroup f \<equiv> \<forall>a b c. f (f a b) c = f a (f b c)"
lookup_lift_keys_plus: fixes i :: "nat"   and xs :: "(('b, 'c) pp \<times> nat, 'a) oalist_ntm"   and t :: "('b, 'c) pp"   and k :: "nat" shows "lookup (MP_oalist (lift_keys_pprod i xs)) (t, i + k) = lookup (MP_oalist xs) (t, k)"
dotprod_gt0_iff: fixes x :: "nat list"   and y :: "nat list" assumes "length x = length y" shows "(0 < x \<bullet> y) = (\<exists>i<length y. 0 < x ! i \<and> 0 < y ! i)"
assms_1: shows "check_single \<C> e tn' = tm'"
find_md_m_match: fixes path :: "(ctx \<times> cld) list"   and m :: "char list"   and ctx :: "ctx"   and cl :: "cl"   and m' :: "char list"   and vds :: "vd list"   and mb :: "meth_body" shows "find_meth_def_in_path_f path m = Some (ctx, meth_def_def (meth_sig_def cl m' vds) mb) \<longrightarrow> m' = m"
extendFalsum: fixes S :: "'a sequent"   and \<Gamma> :: "'a form multiset"   and \<Delta> :: "'a form" assumes "extend S ( \<LM> ff  \<RM> \<Rightarrow>* Em) = ( \<Gamma> \<Rightarrow>* \<Delta>)" shows "ff \<in># \<Gamma>"
monoseq_arctan_series: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1" shows "monoseq (\<lambda>n. 1 / real (n * 2 + 1) * x ^ (n * 2 + 1))"
N1: shows "ennreal (1 - d2 / 2) < emeasure M {x \<in> space M. \<forall>B\<ge>N1. real (card ({n. \<exists>l\<in>{1..n}. u n x - u (n - l) x \<le> - c0 * real l} \<inter> {..<B})) < d2 / 2 * real B}"
RecvMarker_given_channel: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and ev :: "('a, 'b, 'c) event"   and cid :: "nat"   and p :: "'a"   and q :: "'a"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "isRecvMarker ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "distributed_system.can_occur channel trans send recv ev c" shows "ev = RecvMarker cid q p"
timesuu: shows "wtimes u\<^sub>1 u\<^sub>2 \<equiv>\<^sub>E ru1 * ru2"
pre_bfs_def: fixes src :: "nat"   and dst :: "nat" shows "pre_bfs src dst \<equiv> while\<^sub>T\<^bsup>outer_loop_invar src dst\<^esup> (\<lambda>(f, PRED, C, N, d). f = False \<and> C \<noteq> {}) (\<lambda>(f, PRED, C, N, d). (spec v. v \<in> C) \<bind> (\<lambda>v. let C = C - {v} in assert (v \<in> V) \<bind> (\<lambda>_. let succ = E `` {v} in assert (finite succ) \<bind> (\<lambda>_. add_succ_spec dst succ v PRED N \<bind> (\<lambda>(f, PRED, N). if f then return (f, PRED, C, N, d + 1) else assert (assn1 src dst (f, PRED, C, N, d)) \<bind> (\<lambda>_. if C = {} then let C = N; N = {}; d = d + 1 in return (f, PRED, C, N, d) else return (f, PRED, C, N, d))))))) (False, [src \<mapsto> src], {src}, {}, 0) \<bind> (\<lambda>(f, PRED, uu_, uu_, d). if f then return (Some (d, PRED)) else return None)"
rotate_drop_take: fixes n :: "nat"   and xs :: "'g list" shows "rotate n xs = drop (n mod length xs) xs @ take (n mod length xs) xs"
flip_def: fixes a :: "'e"   and b :: "'e" shows "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
tendsto_of_real_iff: fixes f :: "'b \<Rightarrow> real"   and c :: "real"   and F :: "'b filter" shows "((\<lambda>x. of_real (f x)) \<longlongrightarrow> of_real c) F = (f \<longlongrightarrow> c) F"
power_add: fixes a :: "'a"   and m :: "nat"   and n :: "nat" shows "power.power 1' (;) a (plus m n) = power.power 1' (;) a m ; power.power 1' (;) a n"
csqrt_exp_Ln: fixes z :: "complex" assumes "z \<noteq> 0" shows "csqrt z = exp (Ln z / 2)"
borel_measurable_affine: fixes f :: "'c \<Rightarrow> 'd"   and c :: "real"   and t :: "'c" assumes "f \<in> borel_measurable lebesgue"   and "c \<noteq> 0" shows "(\<lambda>x. f (t + c *\<^sub>R x)) \<in> borel_measurable lebesgue"
has_dom_def: fixes f :: "'e \<Rightarrow> 'e"   and S :: "'e set" shows "has_dom f S \<equiv> \<forall>s. s \<notin> S \<longrightarrow> f s = s"
AUC: fixes A :: "'a vec set" assumes "A \<subseteq> U" shows "A \<subseteq> carrier_vec n"
filter_insort_triv: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and cmp :: "'b comparator"   and xs :: "'b list" assumes "\<not> P x" shows "filter P (Sorting_Algorithms.insort cmp x xs) = filter P xs"
Fun_in_terms_of_comp: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "Fun f = restrict (img \<circ> local.comp f \<circ> mkPoint (local.dom f)) (Dom f)"
sort_oalist_id: fixes xs :: "('a \<times> 'b) list" assumes "tc.oalist_inv_raw xs" shows "sort_oalist_tc xs = xs"
nxt_geq_lNact: fixes n :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. \<parallel>c\<parallel>\<^bsub>t i\<^esub>" shows "\<langle>c \<Leftarrow> t\<rangle>\<^bsub>n\<^esub> \<le> \<langle>c \<rightarrow> t\<rangle>\<^bsub>n\<^esub>"
hash_map_blindable_simp: fixes f :: "'d \<Rightarrow> 'c"   and f' :: "'e \<Rightarrow> 'd"   and x :: "('e, 'c) blindable\<^sub>m" shows "hash_blindable f (map_blindable\<^sub>m f' id x) = hash_blindable (f \<circ> f') x"
transaction_strand_def: fixes T :: "('f, 'g, 'h, 'e) prot_transaction" shows "transaction_strand T \<equiv> transaction_receive T @ transaction_selects T @ transaction_checks T @ transaction_updates T @ transaction_send T"
cong_red_implies_ctx_red: fixes t :: "lam"   and t' :: "lam" assumes "t \<longrightarrow>o t'" shows "t \<longrightarrow>x t'"
lens_source_def: fixes X :: "'e \<Longrightarrow> 'd" shows "\<S>\<^bsub>X\<^esub> = {s. \<exists>v s'. s = put\<^bsub>X\<^esub> s' v}"
chine_composite: shows "f_src.chine = f.leg0 \<down>\<down> src.leg1"
semiconnex_iff_neg_antisymmetric: fixes A :: "'c set"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool" shows "semiconnex A less = antisymmetric A (\<lambda>x y. \<not> less x y)"
enn2real_nonneg: fixes x :: "ennreal" shows "0 \<le> enn2real x"
simps_3: fixes n1 :: "nat"   and n2 :: "nat" shows "(I n1 < I n2) = (n1 < n2)"
equal_deg_sum: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P"   and "g \<in> carrier P"   and "deg R f < n"   and "deg R g = n" shows "deg R (f \<oplus>\<^bsub>P\<^esub> g) = n"
eqg: fixes z :: "complex" assumes "z \<in> S" shows "z - b = (g__ z)\<^sup>2"
qr: shows "pseudo_divmod (Mp f) (Mp g) = (q, r)"
simps: fixes a1 :: "nat"   and a2 :: "hmultiset" shows "local.aligned\<^sub>d a1 a2 = (\<exists>M e. a1 = e \<and> a2 = M \<and> (\<forall>m\<in>#hmsetmset M. e \<le> local.decode 0 m))"
toArr_def: fixes f :: "('b, 'a list) concrete_category.arr" shows "toArr f \<equiv> hd (Map f)"
lcoset_subgroup_rel_equiv: fixes H :: "'g set" assumes "Group H \<and> order.greater_eq G H" shows "equiv G (LCoset_rel H)"
icard_Diff_singleton: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "icard (A - {x}) = icard A - 1"
space\<^sub>N_def: fixes N :: "'a quasinorm" shows "space\<^sub>N N = {f. eNorm N f < \<infinity>}"
Trg_Inv: fixes t :: "'c local.term" assumes "Can t" shows "Trg (Inv t) = Trg t"
fresh_trans_fm: fixes i :: "atom"   and e :: "name list"   and A :: "fm" shows "i \<sharp> trans_fm e A = (i \<sharp> A \<or> i \<in> atom ` set e)"
loc_f_def: fixes n :: "nat"   and sch1 :: "PrimScheme"   and sch2 :: "PrimScheme" shows "loc_f n sch1 sch2 = (if n = 0 then Base_zero else if n = 1 then Base_suc else if n = 2 then Base_fst else if n = 3 then Base_snd else if n = 4 then Comp_op sch1 sch2 else if n = 5 then Pair_op sch1 sch2 else if n = 6 then Rec_op sch1 sch2 else Base_zero)"
Int_UNIV_right: fixes A :: "'n set" shows "A \<inter> UNIV = A"
antisym_less_nmultiset: fixes X :: "'b nmultiset"   and Y :: "'b nmultiset" assumes "X < Y"   and "Y < X" shows "False"
extreme_unique: fixes X :: "'a set"   and x :: "'a"   and y :: "'a" assumes "X \<subseteq> A"   and "extreme X (\<lambda>x y. y \<sqsubseteq> x) x" shows "extreme X (\<lambda>x y. y \<sqsubseteq> x) y = (x = y)"
PLS_def: fixes s :: "('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "PLS s as \<equiv> length ` {as'. exec_plan s as' = exec_plan s as \<and> subseq as' as}"
fls_X_inv_times_comm: fixes f :: "'a fls" shows "fls_X_inv * f = f * fls_X_inv"
lookup_sum: fixes pp :: "'c \<Rightarrow> 'b \<Rightarrow>\<^sub>0 'a"   and X :: "'c set"   and i :: "'b" shows "lookup (sum pp X) i = (\<Sum>x\<in>X. lookup (pp x) i)"
uminus_Abs_freeword_singleton: fixes s :: "'b"   and b :: "bool" shows "- Abs_freeword [(s, b)] = Abs_freeword [(s, \<not> b)]"
security1: fixes \<A> :: "(bool, bool list \<times> bool list + bool list \<times> bool list \<times> 'hash, (bool list \<times> bool list \<times> 'hash) option + bool list option) gpv"   and q :: "nat"   and q' :: "nat"   and \<eta> :: "nat" assumes "lossless_gpv (\<I>_full \<oplus>\<^sub>\<I> \<I>_full) \<A>"   and "interaction_bounded_by' isl \<A> q"   and "interaction_bounded_by' (Not \<circ> isl) \<A> q'" shows "local.ind_cca.advantage \<eta> \<A> \<le> PRF.advantage \<eta> (reduction_prf \<eta> \<A>) + UPF.advantage1 \<eta> (guessing_many_one.reduction q' (\<lambda>_. reduction_upf \<eta> \<A>) ()) * real q' + real q * real q / real (card (prf_domain \<eta>))"
chooseNext_all: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g" assumes "CFG_SSA_Transformed \<alpha>e \<alpha>n invar inEdges' Entry oldDefs oldUses defs u p var"   and "uninst.redundant u p g" shows "uninst.chooseNext' u p g \<in> dom (p g) \<and> uninst.trivial u p g (uninst.chooseNext u p g)"
\<chi>\<sigma>_def: shows "\<chi>\<sigma> \<equiv> instInpP \<chi> (Suc 0) \<sigma>"
inverse_distrib: fixes a :: "'a"   and b :: "'a" shows "inverse (a \<cdot> b) = inverse a \<cdot> inverse b"