hcju/isabelleps huggingface dataset:
WTrt_elim_cases_5: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e\<^sub>1 :: "char list exp"   and C :: "char list"   and V :: "char list"   and e\<^sub>2 :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> try e\<^sub>1 catch(C V) e\<^sub>2 : T" obtains T\<^sub>1 :: "ty" where "P,E,h,sh \<turnstile> e\<^sub>1 : T\<^sub>1" and "P,E(V \<mapsto> Class C),h,sh \<turnstile> e\<^sub>2 : T" and "P \<turnstile> T\<^sub>1 \<le> T"
bin_to_bl_aux_cat_app: fixes n :: "nat"   and nw :: "nat"   and wa :: "int"   and w :: "int"   and bs :: "bool list" shows "bin_to_bl_aux (n + nw) (concat_bit nw wa w) bs = bin_to_bl n w @ bin_to_bl_aux nw wa bs"
valid_cfgI: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and s :: "'a"   and cfg :: "'a cfg" assumes "Finite_Markov_Decision_Process K S"   and "s \<in> S"   and "cfg \<in> Markov_Decision_Process.cfg_on K s" shows "cfg \<in> Finite_Markov_Decision_Process.valid_cfg K S"
dom_mkArr: fixes A :: "'s set"   and B :: "'s set"   and F :: "'s \<Rightarrow> 's" assumes "arr (mkArr A B F)" shows "local.dom (mkArr A B F) = mkIde A"
CM: fixes n_as :: "(nat \<times> 'a) list" shows "char_matrix (jordan_matrix n_as) ev \<in> carrier_mat (sum_list (map fst n_as)) (sum_list (map fst n_as))"
uints_unats: fixes n :: "nat" shows "uints n = int ` unats n"
ChiRel_HARoot: fixes x :: "'a"   and y :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "(x, y) \<in> ChiRel A" shows "y \<notin> States (HARoot A)"
fmran'I: fixes m :: "('b, 'a) fmap"   and x :: "'b"   and y :: "'a" assumes "fmlookup m x = Some y" shows "y \<in> fmran' m"
associated_eqI: 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 a :: "'a"   and b :: "'a" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times a b"   and "dvd.dvd times b a"   and "normalize a = a"   and "normalize b = b" shows "a = b"
rec_inseparable_0not1_1not0: shows "rec_inseparable {p. 0 \<in> nat_to_ce_set p \<and> 1 \<notin> nat_to_ce_set p} {p. 0 \<notin> nat_to_ce_set p \<and> 1 \<in> nat_to_ce_set p}"
indspaceD_add: fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set"   and x :: "('a, 'b) aezfun"   and y :: "('a, 'b) aezfun" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V"   and "x \<in> FH"   and "y \<in> FH" shows "f (x + y) = f x + f y"
axioms_2: fixes G :: "'c set"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup G M composition unit" shows "group G composition unit"
Smith_nx1_works: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A :: "'a mat"   and n :: "nat"   and P :: "'a mat"   and S :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op"   and "A \<in> carrier_mat n 1"   and "(P, S) = Smith_Impl.Smith_nx1 Smith_1x2 A" shows "is_SNF A (P, S, 1\<^sub>m 1)"
pr_def: fixes m :: "nat \<times> 'c \<Rightarrow> 'b"   and x :: "nat \<Rightarrow> 'b"   and i :: "'c" shows "local.pr m x i \<equiv> \<Sum>k = 0..<n. x k * m (k, i)"
nfa_accepting_replicate: fixes q :: "nat"   and N :: "bool list bdd list \<times> bool list" assumes "q < length (fst N)"   and "length (snd N) = length (fst N)" shows "nfa_accepting N ((replicate (length (fst N)) False)[q := True]) = snd N ! q"
iprev_greater_conv: fixes n :: "nat"   and I :: "nat set" shows "(n \<in> I \<and> iMin I < n) = (iprev n I < n)"
gamma2_differentiable_1: shows "finite (s' \<inter> {0..1})"
pell_valuation_uminus_uminus: fixes D :: "nat"   and x :: "int"   and y :: "int" assumes "pell D" shows "pell.pell_valuation D (- x, - y) = - pell.pell_valuation D (x, y)"
elims_2: fixes x :: "'b \<times> 'b" assumes "is_interval x" obtains l :: "'b" and u :: "'b" where "x = (l, u)" and "l < u"
assms: fixes b :: "'a set" assumes "b \<in># remove1_mset bl \<B>" shows "0 < bl |\<inter>| b"
h: shows "a * (\<Sum>i\<in>set_pmf (mix_pmf a p q). pmf p i * f i) + (1 - a) * (\<Sum>i\<in>set_pmf (mix_pmf a p q). pmf q i * f i) = a * (\<Sum>i\<in>set_pmf p. pmf p i * f i) + (1 - a) * (\<Sum>i\<in>set_pmf q. pmf q i * f i)"
attach_stateless_callee: fixes callee :: "'p \<Rightarrow> ('q, 'r, 's) gpv"   and "oracle" :: "'o \<Rightarrow> 'r \<Rightarrow> ('s \<times> 'o) spmf" shows "attach_callee (stateless_callee callee) oracle = \<dagger>(\<lambda>s q. exec_gpv oracle (callee q) s)"
r_E: shows "ltake (enat i) E = lappend (ltake (enat (ws i)) E) (LCons (lnth E (ws i)) (ldropn (Suc (ws i)) (ltake (enat i) E)))"
has_vector_derivative_eq_rhs: fixes f :: "real \<Rightarrow> 'a"   and X :: "'a"   and F :: "real filter"   and Y :: "'a" assumes "(f has_vector_derivative X) F"   and "X = Y" shows "(f has_vector_derivative Y) F"
C_eq_Cinf: fixes i :: "nat" shows "C\<^bsup>i\<^esup> \<noteq> C\<^sup>\<infinity>"
dvd_trans: fixes times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and a :: "'d"   and b :: "'d"   and c :: "'d" assumes "class.comm_monoid_mult times one"   and "dvd.dvd times a b"   and "dvd.dvd times b c" shows "dvd.dvd times a c"
aperiodic_eventually_recurrent: fixes C :: "'s option set" shows "K2.aperiodic C = (C \<in> UNIV // K2.communicating \<and> (\<forall>x\<in>C. \<forall>\<^sub>F m in sequentially. 0 < K2.p x x m))"
adjuster_lincomb: fixes w :: "'a vec"   and us :: "'a vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us = lincomb (\<lambda>u. - (w \<bullet>c u) / (u \<bullet>c u)) (set us)"
in_vimage_algebra: fixes A :: "'b set"   and M :: "'b measure"   and f :: "'c \<Rightarrow> 'b"   and X :: "'c set" assumes "A \<in> sets M" shows "f -` A \<inter> X \<in> sets (vimage_algebra X f M)"
cv_inv_version4: fixes S :: "(real, 'a) vec set"   and g :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and g' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec"   and h :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'b) vec"   and b :: "(real, 'b) vec" assumes "\<And>x. x \<in> S \<Longrightarrow> (g has_derivative g' x) (at x within S) \<and> invertible (matrix (g' x))"   and "\<And>x. x \<in> S \<Longrightarrow> continuous_on (g ` S) h \<and> h (g x) = x" shows "((\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) absolutely_integrable_on S \<and> integral S (\<lambda>x. \<bar>det (matrix (g' x))\<bar> *\<^sub>R f (g x)) = b) = (f absolutely_integrable_on g ` S \<and> integral (g ` S) f = b)"
take_replicate: fixes i :: "nat"   and k :: "nat"   and x :: "'a" shows "x\<^bsup>k\<^esup> \<down> i = x\<^bsup>min i k\<^esup>"
i_Exec_Stream_Init_output_strictly_causal: fixes n :: "nat"   and input1 :: "nat \<Rightarrow> 'd"   and input2 :: "nat \<Rightarrow> 'd"   and output1 :: "nat \<Rightarrow> 'e"   and output_fun :: "'f \<Rightarrow> 'e"   and trans_fun :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and c :: "'f"   and output2 :: "nat \<Rightarrow> 'e" assumes "input1 \<Down> n = input2 \<Down> n"   and "output1 = output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun input1 c"   and "output2 = output_fun \<circ> i_Exec_Comp_Stream_Init trans_fun input2 c" shows "output1 \<Down> Suc n = output2 \<Down> Suc n"
open_map_snd: fixes X :: "'d topology"   and Y :: "'e topology" shows "open_map (prod_topology X Y) Y snd"
hyps: fixes xs :: "'a list" assumes "length xs < length xsa__"   and "xs \<noteq> []"   and "hd xs \<noteq> (0::'a)"   and "last xs \<noteq> (0::'a)" shows "even (sign_changes xs) = (sgn (hd xs) = sgn (last xs))"
conj_iso: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "residuated f"   and "x \<le> y" shows "conjugate f x \<le> conjugate f y"
gcd_dvdI2: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize"   and "dvd.dvd times b c" shows "dvd.dvd times (gcd a b) c"
n_def: fixes m :: "nat"   and x :: "nat" assumes "n \<le> m" shows "token_succeeds x = (m < x \<or> (\<exists>j\<ge>i. rank x m = Some j) \<or> token_run x m \<in> F)"
flow_usolves_ode: fixes x0 :: "'a" assumes "t0 \<in> T"   and "x0 \<in> X" shows "(flow t0 x0 usolves_ode f from t0) (existence_ivl t0 x0) X"
omega_fix: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dual :: "'b \<Rightarrow> 'b"   and dual_star :: "'b \<Rightarrow> 'b"   and omega :: "'b \<Rightarrow> 'b"   and star :: "'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and x :: "'b" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "omega x = inf (times x (omega x)) one"
assms_1: shows "\<not> hb (i, Store e1 mo i) (r, Store e2 mo2 r)"
ide_eval_Ide: fixes t :: "'a local.term" assumes "Ide t" shows "ide \<lbrace>t\<rbrace>"
Contrapos1: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IMP B" shows "H \<turnstile> neg B IMP neg A"
2: shows "\<l>[\<I>\<^sub>1] \<cdot> (\<r>\<^sub>1[\<I>] \<otimes>\<^sub>1 \<l>[\<I>\<^sub>1]) = \<iota>\<^sub>1 \<cdot> (\<l>[\<I>\<^sub>1] \<otimes>\<^sub>1 \<l>[\<I>\<^sub>1])"
simps_2: fixes uu :: "'a"   and uv :: "real"   and l :: "'a kdt"   and r :: "'a kdt" shows "set_kdt (Node uu uv l r) = set_kdt l \<union> set_kdt r"
simps_3: fixes a :: "name"   and t :: "lam" shows "height (Lam [a].t) = height t + 1"
intro: fixes \<alpha> :: "'a \<Rightarrow> natural set"   and invar :: "'a \<Rightarrow> bool"   and empty :: "unit \<Rightarrow> 'a"   and memb :: "natural \<Rightarrow> 'a \<Rightarrow> bool"   and ins :: "natural \<Rightarrow> 'a \<Rightarrow> 'a"   and rem :: "natural \<Rightarrow> 'a \<Rightarrow> 'a"   and size :: "'a \<Rightarrow> nat"   and filter :: "(natural \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a" assumes "set_empty \<alpha> invar empty"   and "set_memb \<alpha> invar memb"   and "set_ins \<alpha> invar ins"   and "set_delete \<alpha> invar rem"   and "set_size \<alpha> invar size"   and "set_filter \<alpha> invar \<alpha> invar filter" shows "benchmark \<alpha> invar empty memb ins rem size filter"
convex: fixes S :: "'c set" shows "convex S = (\<forall>k u x. (\<forall>i. 1 \<le> i \<and> i \<le> k \<longrightarrow> 0 \<le> u i \<and> x i \<in> S) \<and> sum u {1..k} = 1 \<longrightarrow> (\<Sum>i = 1..k. u i *\<^sub>R x i) \<in> S)"
select_append_2: fixes ys :: "'a list"   and zs :: "'a list"   and k :: "nat" assumes "\<forall>y\<in>set ys. \<forall>z\<in>set zs. y \<le> z"   and "k \<in> {length ys..<length ys + length zs}" shows "select k (ys @ zs) = select (k - length ys) zs"
get_update_eq: fixes a :: "'b array"   and i :: "nat"   and v :: "'b"   and h :: "heap" shows "Array.get (Array.update a i v h) a = (Array.get h a)[i := v]"
indom_mI: fixes m :: "('d, 'c) fmap"   and x :: "'d"   and y :: "'c" assumes "fmlookup m x = Some y" shows "x \<in># dom_m m"
mult_right: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "f \<in> O[F](g)" shows "(\<lambda>x. f x * h x) \<in> O[F](\<lambda>x. g x * h x)"
"case": fixes f :: "int option \<Rightarrow> int option \<Rightarrow> 'a"   and x1 :: "int option"   and x2 :: "int option" shows "(case I x1 x2 of I x xa \<Rightarrow> f x xa) = f x1 x2"
filtermap_filtercomap: fixes f :: "'b \<Rightarrow> 'a"   and F :: "'a filter" shows "filtermap f (filtercomap f F) \<le> F"
length_0_conv: fixes xs :: "'a list" shows "(|xs| = 0) = (xs = [])"
prefix_def: fixes xs :: "'b list"   and ys :: "'b list" shows "prefix xs ys = (\<exists>zs. ys = xs @ zs)"
sigma_sets_empty_eq: fixes A :: "'c set" shows "sigma_sets A {} = {{}, A}"
span_alt: fixes B :: "'b set" shows "cspan B = {\<Sum>x | f x \<noteq> 0. f x *\<^sub>C x |f. {x. f x \<noteq> 0} \<subseteq> B \<and> finite {x. f x \<noteq> 0}}"
sound_modes: shows "\<forall>mem. sound_mode_use (add_initial_modes cmds, mem)"
pre_satisfied: shows "\<tau> \<Turnstile> PRE self a1"
aadd_pos_poss: fixes x :: "ant"   and y :: "ant" assumes "0 \<le> x"   and "0 < y" shows "0 < x + y"
prime_factors_product: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "y \<noteq> (0::'a)" shows "prime_factors (x * y) = prime_factors x \<union> prime_factors y"
simps_2: fixes t :: "('e, 'f) Term.term"   and \<theta> :: "'f \<Rightarrow> ('e, 'f) Term.term" shows "receive\<langle>t\<rangle> \<cdot>\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<theta> = receive\<langle>t \<cdot> \<theta>\<rangle>"
intro: fixes v\<^sub>e :: "real"   and v\<^sub>o :: "real"   and a\<^sub>e :: "real"   and a\<^sub>o :: "real"   and s\<^sub>e :: "real"   and s\<^sub>o :: "real" assumes "0 \<le> v\<^sub>e"   and "0 \<le> v\<^sub>o"   and "a\<^sub>e < 0"   and "a\<^sub>o < 0"   and "s\<^sub>e < s\<^sub>o" shows "safe_distance a\<^sub>e v\<^sub>e s\<^sub>e a\<^sub>o v\<^sub>o s\<^sub>o"
simps_2: fixes uu :: "action"   and rs :: "'b rule list" shows "rmMatchFalse (Rule (MatchNot MatchAny) uu # rs) = rmMatchFalse rs"
cinner_diff_right: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "cinner x (y - z) = cinner x y - cinner x z"
simps_2: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and I :: "('b, 'c, 'a) interp"   and uv :: "'b"   and r :: "real"   and \<nu> :: "(real, 'a) vec \<times> (real, 'a) vec"   and x :: "(real, 'a) vec" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "ids.extendf_deriv I uv (Const r) \<nu> x = (\<lambda>_. 0)"
hom_inv: fixes x :: "'a" assumes "x \<in> carrier A" shows "i (inv\<^bsub>A\<^esub> x) = inv\<^bsub>G\<^esub> i x"
seq2: fixes c1 :: "('a, 'b) MWLfCom"   and m :: "'b \<Rightarrow> 'c"   and c1' :: "('a, 'b) MWLfCom"   and V :: "('a, 'b) MWLfCom list"   and m' :: "'b \<Rightarrow> 'c"   and E :: "'a \<Rightarrow> ('b \<Rightarrow> 'c) \<Rightarrow> 'c"   and BMap :: "'c \<Rightarrow> bool"   and c2 :: "('a, 'b) MWLfCom" assumes "((c1, m), c1' # V, m') \<in> MWLf_semantics.MWLfSteps_det E BMap" shows "((c1;c2, m), (c1';c2) # V, m') \<in> MWLf_semantics.MWLfSteps_det E BMap"
associated_sym: fixes a :: "'a"   and b :: "'a" assumes "a \<sim> b" shows "b \<sim> a"
path_image_subpath: fixes u :: "real"   and v :: "real"   and g :: "real \<Rightarrow> 'a" shows "path_image (subpath u v g) = (if u \<le> v then g ` {u..v} else g ` {v..u})"
lock_ok_lasI: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions" assumes "\<And>l. lock_actions_ok (ls $ l) t (las $ l)" shows "lock_ok_las ls t las"
purgeIdle_simps_3: fixes st :: "'St"   and u :: "'U"   and c :: "'C"   and sl :: "('St, 'U, 'C) state list" shows "purgeIdle (State st u c # sl) = State st u c # purgeIdle sl"
iimg_def: fixes B :: "'b set"   and f :: "'b \<Rightarrow> 'b"   and a :: "'b" shows "iimg B f a \<equiv> if a \<in> f ` B then a else f a"
fresh_star_set_eq: fixes xs :: "'c list"   and c :: "'d" shows "set xs \<sharp>* c = xs \<sharp>* c"
simps: fixes S :: "'at Literal set set" shows "atoms_formula S = {A. \<exists>C. C \<in> S \<and> A \<in> atoms_clause C}"
Exists_in_extend: fixes S :: "(nat, 'a) form set"   and C :: "(nat, 'a) form set set"   and f :: "nat \<Rightarrow> (nat, 'a) form"   and n :: "nat"   and P :: "(nat, 'a) form" assumes "extend S C f n \<union> {f n} \<in> C"   and "Exists P = f n" shows "P[App (SOME k. k \<notin> \<Union> (params ` (extend S C f n \<union> {f n}))) []/0] \<in> extend S C f (Suc n)"
2: shows "invertible_mat P' \<and> invertible_mat Q' \<and> Smith_normal_form_mat S \<and> S = P' * D * Q' \<and> P' \<in> carrier_mat m m \<and> Q' \<in> carrier_mat n n \<and> S \<in> carrier_mat m n"
intuitionistic_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj" shows "sep_algebra.intuitionistic plus sep_disj P \<equiv> \<forall>h h'. P h \<and> sep_algebra.sep_substate plus sep_disj h h' \<longrightarrow> P h'"
hpair_nonzero: fixes x :: "hf"   and y :: "hf" shows "\<langle>x, y\<rangle> \<noteq> 0"
infinite_openin: fixes U :: "'b set"   and S :: "'b set"   and x :: "'b" assumes "openin (top_of_set U) S"   and "x \<in> S"   and "x islimpt U" shows "infinite S"
that: fixes g :: "real \<Rightarrow> real" assumes "continuous_on UNIV g"   and "g square_integrable S"   and "l2norm S (\<lambda>x. f x - g x) < e" shows "thesis"
rec_all2_less_def: fixes f :: "recf" shows "rec_all2_less f = (let cond1 = CN rec_eq [recf.Id 4 0, recf.Id 4 1]; cond2 = CN f [recf.Id 4 0, recf.Id 4 2, recf.Id 4 3] in CN (rec_all3 (CN rec_disj [cond1, cond2])) [recf.Id 3 0, recf.Id 3 0, recf.Id 3 1, recf.Id 3 2])"
not_measurable_UNIV: shows "UNIV \<notin> lmeasurable"
tail_del_arc: fixes G :: "('d, 'c) pre_digraph"   and a :: "'c" shows "tail (pre_digraph.del_arc G a) = tail G"
leofI: fixes m :: "'a nres"   and m' :: "'a nres" assumes "nofail m \<Longrightarrow> m \<le> m'" shows "m \<le>\<^sub>n m'"
assms_4: shows "pathfinish \<gamma> = pathstart \<gamma>"
cast\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>2\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_inv: fixes shadow_root :: "('m, 'n, 'o, 'p option) RShadowRoot_scheme" shows "cast (cast shadow_root) = Some shadow_root"
Disjoint_I: fixes i :: "name"   and t :: "tm"   and u :: "tm"   and H :: "fm set" assumes "insert (Var i IN t) (insert (Var i IN u) H) \<turnstile> Fls"   and "atom i \<sharp> (t, u)"   and "\<forall>B\<in>H. atom i \<sharp> B" shows "H \<turnstile> Disjoint t u"
assms_1: assumes "f = (\<lambda>u. False)" shows "P f"
fmlookup_dom'_iff: fixes x :: "'d"   and m :: "('d, 'e) fmap" shows "(x \<in> fmdom' m) = (\<exists>a. fmlookup m x = Some a)"
is_sup_uniq: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and s :: "'a"   and s' :: "'a" assumes "dlat le"   and "dpo.is_sup le x y s"   and "dpo.is_sup le x y s'" shows "s = s'"
G_is_functor: shows "functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G"
algebraic_representsI: fixes p :: "int poly"   and x :: "'b" assumes "p represents x" shows "algebraic x"
lambda_skolem: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "(\<forall>i. \<exists>x. P i x) = (\<exists>x. \<forall>i. P i (x $ i))"
set_empty: fixes xs :: "'d list" shows "(set xs = {}) = (xs = [])"
proots_count_leq_degree: fixes p :: "'a poly"   and s :: "'a set" assumes "p \<noteq> 0" shows "proots_count p s \<le> degree p"
lower_plus: fixes A :: "'a interval"   and B :: "'a interval" shows "lower (A + B) = lower A + lower B"
left_null_space_subset_span_basis: fixes A :: "(('a, 'c) vec, 'b) vec" shows "left_null_space A \<subseteq> vec.span (basis_left_null_space A)"
axioms_1: fixes \<eta> :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" assumes "NatTrans \<eta>" shows "NatTransP \<eta>"
PR_CONST_def: fixes x :: "'a" shows "PR_CONST x \<equiv> x"
integral_indicator_eq_restriction: fixes f :: "'b \<Rightarrow> 'c"   and R :: "'b set"   and S :: "'b set" assumes "f integrable_on R"   and "R \<subseteq> S" shows "integral S (\<lambda>x. indicat_real R x *\<^sub>R f x) = integral R f"
set_pmf_iff: fixes x :: "'e"   and M :: "'e pmf" shows "(x \<in> set_pmf M) = (pmf M x \<noteq> 0)"
div2_even_ext_nat: fixes x :: "nat"   and y :: "nat" assumes "x div 2 = y div 2"   and "even x = even y" shows "x = y"
isSPath_pathLE: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" assumes "cf.isPath s p t" shows "\<exists>p'. cf.isSimplePath s p' t"
ovalidNF_grab_asm: fixes G :: "bool"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd option"   and Q :: "'d \<Rightarrow> 'c \<Rightarrow> bool" assumes "G \<Longrightarrow> ovalidNF P f Q" shows "ovalidNF (\<lambda>s. G \<and> P s) f Q"
apsnd_conv: fixes f :: "'c \<Rightarrow> 'b"   and x :: "'a"   and y :: "'c" shows "apsnd f (x, y) = (x, f y)"
fminus_finsert_absorb: fixes x :: "'a"   and A :: "'a fset" assumes "x |\<notin>| A" shows "finsert x A |-| {|x|} = A"
that: fixes f :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "f \<in> F"   and "red_single p q f t" shows "thesis"
gauss_cnj_dvd_self_iff: fixes z :: "gauss_int" shows "(gauss_cnj z dvd z) = (ReZ z = 0 \<or> ImZ z = 0 \<or> \<bar>ReZ z\<bar> = \<bar>ImZ z\<bar>)"
var_inj_term_def: fixes v :: "Term.var" shows "\<langle>v\<rangle> = In2 v"
CBasis_prod_def: shows "CBasis = (\<lambda>u. (u, 0::'b)) ` CBasis \<union> Pair (0::'a) ` CBasis"
jkbpC_tLength_inv: fixes jkbp :: "'b \<Rightarrow> ('b, 'c, 'd) GC list"   and jview :: "'b \<Rightarrow> 'e Trace \<Rightarrow> 'f"   and t :: "'e Trace"   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"   and n :: "nat" assumes "PreEnvironmentJView jkbp jview"   and "t \<in> PreEnvironmentJView.jkbpC jkbp envInit envAction envTrans envVal jview"   and "tLength t = n" shows "t \<in> PreEnvironmentJView.jkbpCn jkbp envInit envAction envTrans envVal jview n"
rep_eq: fixes x :: "'a upred"   and xa :: "'a upred" shows "\<lbrakk>x \<Rightarrow> xa\<rbrakk>\<^sub>e = (\<lambda>A. \<lbrakk>x\<rbrakk>\<^sub>e A \<longrightarrow> \<lbrakk>xa\<rbrakk>\<^sub>e A)"
r1: shows "k-smooth_on (apply_chart d1 ` (domain d1 \<inter> domain c1)) (apply_chart c1 \<circ> inv_chart d1)"
unlabel_def: fixes ars :: "('c \<times> 'd \<times> 'c) set" shows "unlabel ars = {(a, c). \<exists>b. (a, b, c) \<in> ars}"
wf_mbuf2_def: fixes i :: "nat"   and ja :: "nat"   and jb :: "nat"   and P :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> event_data option list set \<Rightarrow> bool"   and buf :: "event_data option list set list \<times> event_data option list set list" shows "wf_mbuf2 i ja jb P Q buf = (i \<le> ja \<and> i \<le> jb \<and> (case buf of (xs, ys) \<Rightarrow> list_all2 P [i..<ja] xs \<and> list_all2 Q [i..<jb] ys))"
generalized_zipping_lemma1: assumes "N\<^bsub>\<V>1\<^esub> \<inter> E\<^bsub>ES2\<^esub> = {}"   and "N\<^bsub>\<V>2\<^esub> \<inter> E\<^bsub>ES1\<^esub> = {}" shows "\<forall>\<tau> lambda t1 t2. set \<tau> \<subseteq> E\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> set lambda \<subseteq> V\<^bsub>\<V>\<^esub> \<and> set t1 \<subseteq> E\<^bsub>ES1\<^esub> \<and> set t2 \<subseteq> E\<^bsub>ES2\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES1\<^esub> @ t1 \<in> Tr\<^bsub>ES1\<^esub> \<and> \<tau> \<upharpoonleft> E\<^bsub>ES2\<^esub> @ t2 \<in> Tr\<^bsub>ES2\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES1\<^esub> = t1 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> lambda \<upharpoonleft> E\<^bsub>ES2\<^esub> = t2 \<upharpoonleft> V\<^bsub>\<V>\<^esub> \<and> t1 \<upharpoonleft> C\<^bsub>\<V>1\<^esub> = [] \<and> t2 \<upharpoonleft> C\<^bsub>\<V>2\<^esub> = [] \<longrightarrow> (\<exists>t. \<tau> @ t \<in> Tr\<^bsub>ES1 \<parallel> ES2\<^esub> \<and> t \<upharpoonleft> V\<^bsub>\<V>\<^esub> = lambda \<and> t \<upharpoonleft> C\<^bsub>\<V>\<^esub> = [])"