hcju/isabelleps huggingface dataset:
mssnth_exp_series_stream: fixes n :: "nat" shows "mssnth exp_series_stream n = 1 / fact n"
empty_path_nodes: fixes n :: "'b LDCFG_node"   and n' :: "'b LDCFG_node" assumes "CFGExit.path n [] n'" shows "n = n'"
cInf_insert_If: fixes X :: "'a multiset set"   and a :: "'a multiset" assumes "subset_mset.bdd_below X" shows "Inf (insert a X) = (if X = {} then a else a \<inter># Inf X)"
ev_stl: fixes \<phi> :: "'b stream \<Rightarrow> bool"   and w :: "'b stream" shows "ev \<phi> (stl w) = (\<exists>u v. w = u @- v \<and> u \<noteq> [] \<and> \<phi> v)"
nrows_le: fixes A :: "'a matrix"   and n :: "nat" shows "(nrows A \<le> n) = (\<forall>j i. n \<le> j \<longrightarrow> Rep_matrix A j i = (0::'a))"
prime_factorization_subset_imp_dvd: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> (0::'a)"   and "prime_factorization x \<subseteq># prime_factorization y" shows "x dvd y"
oldUses_def: fixes n :: "'node"   and g :: "'g" assumes "n \<in> set (\<alpha>n g)" shows "oldUses g n = var g ` uses g n"
disc_12: fixes x21 :: "nat"   and x22 :: "'d"   and x23 :: "'d"   and x24 :: "'e"   and x25 :: "'e"   and x26 :: "'f" shows "\<not> isRecv (Send x21 x22 x23 x24 x25 x26)"
measure_spmf_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_fun (rel_pred A) (=)) (\<lambda>p. Sigma_Algebra.measure (measure_spmf p)) (\<lambda>q. Sigma_Algebra.measure (measure_spmf q))"
ordering_range: shows "OrdinalVeblen.ordering (range F) = F"
assms: shows "\<forall>n. trace (f n) \<le> x"
imp_nb: fixes p :: "fm"   and q :: "fm" assumes "bound0 p"   and "bound0 q" shows "bound0 (imp p q)"
D_def: shows "D = (\<lambda>n. 1 / 2 * complex_of_real (a + real n) powr - s)"
cmp_in_hom'_2: fixes \<mu> :: "'a"   and \<nu> :: "'a" assumes "C.arr \<mu>"   and "C.arr \<nu>"   and "src\<^sub>C \<mu> = trg\<^sub>C \<nu>" shows "\<guillemotleft>\<Phi> (\<mu>, \<nu>) : F (C.dom \<mu>) \<star>\<^sub>D F (C.dom \<nu>) \<Rightarrow>\<^sub>D F (C.cod \<mu> \<star>\<^sub>C C.cod \<nu>)\<guillemotright>"
null_eqI: fixes n :: "'a" assumes "\<And>f. n \<cdot> f = n \<and> f \<cdot> n = n" shows "n = B.null"
simps_5: fixes t :: "nat"   and e :: "expr"   and s\<^sub>1 :: "stmt"   and s\<^sub>2 :: "stmt" shows "valid_sops_stmt t (Cond e s\<^sub>1 s\<^sub>2) = (valid_sops_expr t e \<and> valid_sops_stmt t s\<^sub>1 \<and> valid_sops_stmt t s\<^sub>2)"
at_within_self_singleton: fixes i :: "'c" shows "at i within {i} = bot"
obj_simps_2: fixes a :: "'a" assumes "C.obj a" shows "src\<^sub>C a = a"
hb_completion_def: fixes s :: "('l, 'thread_id) locks \<times> (('thread_id \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('thread_id \<Rightarrow> 'w wait_set_status option) \<times> 'thread_id set"   and E :: "('thread_id \<times> ('addr, 'thread_id) obs_event action) list" shows "hb_completion s E = (\<forall>ttas s' t x ta x' m' i. s -\<triangleright>ttas\<rightarrow>* s' \<longrightarrow> non_speculative P (w_values P (\<lambda>_. {}) (map snd E)) (llist_of (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) \<longrightarrow> thr s' t = \<lfloor>(x, no_wait_locks)\<rfloor> \<longrightarrow> t \<turnstile> \<langle>x, shr s'\<rangle> -ta\<rightarrow> \<langle>x', m'\<rangle> \<longrightarrow> actions_ok s' t ta \<longrightarrow> non_speculative P (w_values P (w_values P (\<lambda>_. {}) (map snd E)) (concat (map (\<lambda>(t, ta). \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas))) (llist_of (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) \<longrightarrow> (\<exists>ta' x'' m''. t \<turnstile> \<langle>x, shr s'\<rangle> -ta'\<rightarrow> \<langle>x'', m''\<rangle> \<and> actions_ok s' t ta' \<and> take i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> = take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<and> ta_hb_consistent P (E @ concat (map (\<lambda>(t, ta). map (Pair t) \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>) ttas) @ map (Pair t) (take i \<lbrace>ta\<rbrace>\<^bsub>o\<^esub>)) (llist_of (map (Pair t) (drop i \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>))) \<and> (i < length \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> \<longrightarrow> i < length \<lbrace>ta'\<rbrace>\<^bsub>o\<^esub>) \<and> (if \<exists>ad al v. \<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i = NormalAction (ReadMem ad al v) then sim_action else (=)) (\<lbrace>ta\<rbrace>\<^bsub>o\<^esub> ! i) (\<lbrace>ta'\<rbrace>\<^bsub>o\<^esub> ! i)))"
inj_on_imageI2: fixes f' :: "'e \<Rightarrow> 'd"   and f :: "'c \<Rightarrow> 'e"   and A :: "'c set" assumes "inj_on (f' \<circ> f) A" shows "inj_on f A"
iso_def: fixes f :: "'a" shows "C.iso f \<equiv> \<exists>g. C.inverse_arrows f g"
of_nat_def: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and n :: "nat" assumes "class.semiring_1 one times plus zero" shows "semiring_1.of_nat one plus zero n = (plus one ^^ n) zero"
Rep: fixes Rep :: "'c \<Rightarrow> 'd"   and Abs :: "'d \<Rightarrow> 'c"   and A :: "'d set"   and x :: "'c" assumes "type_definition Rep Abs A" shows "Rep x \<in> A"
simps_1: fixes P :: "'d \<Rightarrow> bool"   and s :: "'d stream" shows "shd (sfilter P s) = shd (sdrop_while (Not \<circ> P) s)"
vassn_tagI: fixes \<Gamma> :: "assn"   and h :: "heap \<times> nat set" assumes "h \<Turnstile> \<Gamma>" shows "vassn_tag \<Gamma>"
simps_11: fixes f :: "'a"   and v :: "'a" shows "\<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<lfloor>\<^bold>\<otimes>\<^bold>\<rfloor> \<^bold>\<langle>v\<^bold>\<rangle> = \<^bold>\<langle>f\<^bold>\<rangle> \<^bold>\<otimes> \<^bold>\<langle>v\<^bold>\<rangle>"
1: shows "w \<sqinter> - (w \<sqinter> q) \<le> forest_components (w \<sqinter> - q \<squnion> (w \<sqinter> q)\<^sup>T)"
lookup_add: fixes f :: "'b \<Rightarrow>\<^sub>0 'a"   and g :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "lookup (f + g) k = lookup f k + lookup g k"
is_aligned_shiftr_shiftl: fixes w :: "'a word"   and n :: "nat" assumes "is_aligned w n" shows "w >> n << n = w"
decI: fixes u :: "'b list"   and G :: "'b list set" assumes "u \<in> \<langle>G\<rangle>" shows "concat (Dec G u) = u"
simps_1: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and prot :: "'c \<Rightarrow> bool"   and protFw :: "'c \<Rightarrow> bool"   and f :: "'a" assumes "ProblemIkTpart wtFsym wtPsym arOf resOf parOf \<Phi> infTp prot protFw" shows "ProblemIkTpart.TE_arOf arOf (Oldf f) = arOf f"
lemma_st_part_major: fixes x :: "real star"   and r :: "real star"   and t :: "real star" assumes "x \<in> HFinite"   and "r \<in> \<real>"   and "0 < r"   and "isLub \<real> {s \<in> \<real>. s < x} t" shows "\<bar>x - t\<bar> < r"
final_no_redT: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x :: "'x" assumes "s -t\<triangleright>ta\<rightarrow> s'"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>" shows "\<not> final x"
Goal: fixes \<Psi>' :: "'b"   and zvec :: "name list" assumes "A\<^sub>R \<sharp>* \<Psi>'"   and "A\<^sub>R \<sharp>* zvec"   and "A\<^sub>P \<sharp>* zvec"   and "zvec \<sharp>* R"   and "zvec \<sharp>* P"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<simeq> \<Psi>'" shows "\<exists>K'. \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> R \<longmapsto> K'\<lparr>N\<rparr> \<prec> R' \<and> \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>R \<turnstile> M \<leftrightarrow> K' \<and> zvec \<sharp>* K' \<and> A\<^sub>R \<sharp>* K'"
poly_roots_const: fixes c :: "'a" shows "poly_roots [:c:] = {#}"
F1set2_natural: fixes f1 :: "'i \<Rightarrow> 'l"   and f2 :: "'j \<Rightarrow> 'k" shows "F2set \<circ> F1map f1 f2 = (`) f2 \<circ> F2set"
StandardControlDependencePDG_axioms: shows "StandardControlDependencePDG sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val (_Exit_)"
not_coprime_iff_common_factor: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and p :: "'a"   and q :: "'a" assumes "class.comm_monoid_mult times one" shows "(\<not> comm_monoid_mult.coprime times one p q) = (\<exists>r. dvd.dvd times r p \<and> dvd.dvd times r q \<and> \<not> dvd.dvd times r one)"
Rep_dense_cases: fixes y :: "'c" assumes "y \<in> dense_elements" obtains x :: "'c dense" where "y = Rep_dense x"
ConjE: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and D :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" assumes "C \<tturnstile> D"   and "(P, c, Q) \<in> D" shows "C \<turnstile> {P} c {Q}"
PabsM: shows "absolutely_continuous (density M (\<lambda>x. 1 + h x)) P"
sublattice_def: shows "sublattice = (SIGMA cl:CompleteLattice. {S. S \<subseteq> pset cl \<and> \<lparr>pset = S, order = induced S (order cl)\<rparr> \<in> CompleteLattice})"
msllist_of_msstream_MSSCons: fixes x :: "'a"   and xs :: "'a msstream" shows "msllist_of_msstream (MSSCons x xs) = MSLCons x (msllist_of_msstream xs)"
simps_3: fixes A :: "('c, 'd) ioa"   and s :: "'c" shows "is_exec_frag_of A (s, []) = True"
every_path_distance: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'" obtains x :: "nat" where "distance n n' x" and "x \<le> length as"
simps_3: fixes t :: "'a E.term"   and u :: "'a E.term" shows "EVAL (t \<^bold>\<star> u) = EVAL t \<star>\<^sub>B EVAL u"
continuous_on_compose2: fixes t :: "'e set"   and g :: "'e \<Rightarrow> 'f"   and s :: "'g set"   and f :: "'g \<Rightarrow> 'e" assumes "continuous_on t g"   and "continuous_on s f"   and "f ` s \<subseteq> t" shows "continuous_on s (\<lambda>x. g (f x))"
divide_eq_eq_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "(divide b a = one) = (a \<noteq> zero \<and> a = b)"
diff_iff: fixes f :: "'a \<Rightarrow> 'c" shows "diff k charts1 charts2 f = (\<forall>x\<in>src.carrier. \<exists>c1\<in>src.atlas. \<exists>c2\<in>dest.atlas. x \<in> domain c1 \<and> f ` domain c1 \<subseteq> domain c2 \<and> k-smooth_on (codomain c1) (apply_chart c2 \<circ> f \<circ> inv_chart c1))"
pdevs_val_def: fixes e :: "nat \<Rightarrow> real"   and x :: "'a pdevs" shows "pdevs_val e x = (\<Sum>i. e i *\<^sub>R pdevs_apply x i)"
numbound0_I: fixes a :: "MIR.num"   and b :: "real"   and bs :: "real list"   and b' :: "real" assumes "numbound0 a" shows "Inum (b # bs) a = Inum (b' # bs) a"
hempty_iff: fixes z :: "hf" shows "(z = 0) = (\<forall>x. x \<^bold>\<notin> z)"
vimage_Int: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" shows "f -` (A \<inter> B) = f \<^sup>\<inverse> f -` B A"
restrict_map_def: fixes m :: "'d \<Rightarrow> 'e option"   and A :: "'d set" shows "m|\<^bsub>A\<^esub> = (\<lambda>x. if x \<in> A then m x else None)"
relabel: fixes f :: "nat \<times> nat \<Rightarrow> 'capacity"   and l :: "nat \<Rightarrow> nat"   and u :: "nat" assumes "Height_Bounded_Labeling c s t f l"   and "relabel_precond f l u" shows "((f, relabel_effect f l u), f, l) \<in> pr_algo_rel"
fst_monofun: fixes x :: "'b \<times> 'c"   and y :: "'b \<times> 'c" assumes "x \<sqsubseteq> y" shows "fst x \<sqsubseteq> fst y"
cnj_powr: fixes a :: "complex"   and b :: "complex" assumes "Im a = 0 \<Longrightarrow> 0 \<le> Re a" shows "cnj (a powr b) = cnj a powr cnj b"
right_adjoint_determines_left_up_to_iso: fixes f :: "'a"   and g :: "'a"   and f' :: "'a" assumes "B.adjoint_pair f g"   and "B.adjoint_pair f' g" shows "f \<cong>\<^sub>B f'"
R_def: shows "\<forall>j\<in>{0..<n}. R j \<in> {0..<n} - {e Num |e. e \<in> s \<and> e Col = j} \<and> inj_on R {0..<n}"
inject_1: fixes x1 :: "char list"   and y1 :: "char list" shows "(Pro x1 = Pro y1) = (x1 = y1)"
height_bound: shows "\<forall>u\<in>V. l u \<le> 2 * card V - 1"
gfp_star: fixes f :: "'b \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'b" assumes "mono f" shows "star_fun f \<circ> g = gfp (Omega_fun f g)"
wellformed'_2: fixes u :: "'a"   and v :: "'a" assumes "(u, v) \<in> parcs (list_digraph G)" shows "v \<in> pverts (list_digraph G)"
exhaust: fixes y :: "d" obtains x1 :: "int" where "y = DI x1"   | x2 :: "lambda \<times> (nat \<Rightarrow> contour option)" where "y = DC x2"   | x3 :: "prim" where "y = DP x3"   | "y = Stop"
that: shows "set xs \<subseteq> outs_\<I> \<I>"
DINING\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: fixes n :: "nat" shows "DINING\<^sub>n\<^sub>o\<^sub>r\<^sub>m n = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>D n,Up\<^sub>D\<rbrakk>"
space_empty_iff: fixes N :: "'b measure" shows "(space N = {}) = (sets N = {{}})"
static_field_type_is_static_def: fixes m :: "'a member_scheme" shows "is_static m \<equiv> static m"
collapse: fixes list :: "'e list" assumes "list \<noteq> []" shows "hd list # tl list = list"
power_closed: fixes x :: "'a"   and n :: "nat" assumes "x \<in> A" shows "x ^ n \<in> A"
fps_divide_cancel: fixes h :: "'a fps"   and f :: "'a fps"   and g :: "'a fps" assumes "h \<noteq> 0" shows "f * h / (g * h) = f / g"
simps_3: fixes v :: "val" shows "max_vars (Val v) = 0"
knd_def: fixes a :: "'g LDCFG_node \<times> 'f edge_kind \<times> 'g LDCFG_node" shows "knd a \<equiv> fst (snd a)"
prefixE: fixes xs :: "'d list"   and ys :: "'d list" assumes "prefix xs ys" obtains zs :: "'d list" where "ys = xs @ zs"
inv_exec: fixes ss :: "'s list"   and f :: "'op" assumes "\<forall>s\<in>set ss. inv s"   and "length ss = arity f" shows "inv (exec f ss)"
assms: shows "\<forall>I. I \<Turnstile> S1 \<longrightarrow> I \<Turnstile> S2"
nth_tl_length: fixes sl :: "'a list" assumes "tl sl \<noteq> []" shows "tl sl ! (length (tl sl) - Suc 0) = sl ! (length sl - Suc 0)"
prems_2: shows "\<forall>j\<in>{k..<l}. v \<notin> local.writes (path \<sigma> j)"
corec_disc_iff_2: fixes p :: "'d \<Rightarrow> bool"   and g21 :: "'d \<Rightarrow> 'c"   and q22 :: "'d \<Rightarrow> bool"   and g221 :: "'d \<Rightarrow> 'c llist"   and g222 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "(\<not> lnull (corec_llist p g21 q22 g221 g222 a)) = (\<not> p a)"
open_Int_closure_eq_empty: fixes S :: "'c set"   and T :: "'c set" assumes "open S" shows "(S \<inter> closure T = {}) = (S \<inter> T = {})"
lift_closed: fixes f :: "'e \<Rightarrow> 'a"   and gens :: "'e set"   and x :: "(bool \<times> 'e) list" assumes "f \<in> gens \<rightarrow> carrier G"   and "x \<in> lists (UNIV \<times> gens)" shows "G.lift f x \<in> carrier G"
TC_subcs: fixes G :: "('a, 'b) form list"   and c :: "'a"   and s :: "'a FOL_Fitting.term" assumes "infinite UNIV"   and "\<stileturn> G" shows "\<stileturn> subcs c s G"
anonymous_profile_auxdef: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> bool" shows "anonymous_profile R = image_mset (weak_ranking \<circ> R) (mset_set {i. R i \<noteq> (\<lambda>_ _. False)})"
map_simp: fixes f :: "'c \<times> 'c" assumes "C.VV.arr f" shows "C.VV.map f = f"
member_set_of_seq: fixes x :: "'a"   and xq :: "'a Predicate.seq" shows "(x \<in> set_of_seq xq) = Predicate.member xq x"
g_nz: shows "\<forall>\<^sub>F x in at_top. g x \<noteq> 0"
assms_3: shows "y \<in> cl_in \<E> (insert x X) - cl_in \<E> X"
simps_4: fixes f1 :: "'a"   and f2 :: "'a" shows "(case Nonfinal of Final \<Rightarrow> f1 | Nonfinal \<Rightarrow> f2) = f2"
fvi_And_Not: fixes b :: "nat"   and \<phi> :: "'c MFOTL.formula"   and \<psi> :: "'c MFOTL.formula" shows "MFOTL.fvi b (MFOTL.And_Not \<phi> \<psi>) = MFOTL.fvi b \<phi> \<union> MFOTL.fvi b \<psi>"
subprob_space_distr: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and M' :: "'d measure" assumes "subprob_space M"   and "f \<in> M \<rightarrow>\<^sub>M M'"   and "space M' \<noteq> {}" shows "subprob_space (distr M M' f)"
rr_val: shows "length (rev (tl (minus_poly_rev_list (map ((*) lc) (rev r)) (map ((*) (hd (rev r))) (rev d))))) = length r - 1"
2: shows "bprv (all yy' (imp (cnj (SS \<langle>\<chi>\<rangle> \<langle>\<psi>\<rangle>) (SS \<langle>\<chi>\<rangle> (Var yy'))) (eql \<langle>\<psi>\<rangle> (Var yy'))))"
H_PH: shows "Hermite_of_row_i ass res (Hermite_of_upt_row_i A k ass res) (mod_type_class.from_nat k) = P ** Hermite_of_upt_row_i A k ass res"
take_Nil: fixes n :: "nat" shows "[] \<down> n = []"
complex_roots_of_int_poly_1: fixes p :: "int poly" assumes "p \<noteq> 0" shows "set (complex_roots_of_int_poly p) = {x. poly (complex_of_int_poly p) x = 0}"
elemrcos_carrier: fixes a :: "'a"   and a' :: "'a" assumes "Group.group G"   and "a \<in> carrier G"   and "a' \<in> H #> a" shows "a' \<in> carrier G"
Drop_neq_Accept_unknown_match_tacs_1: shows "in_doubt_allow Drop \<noteq> in_doubt_allow Accept"
bigo_mult5: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<forall>x. f x \<noteq> (0::'d)" shows "O(f * g) \<subseteq> f *o O(g)"
ennreal_minus: fixes q :: "real"   and r :: "real" assumes "0 \<le> q" shows "ennreal r - ennreal q = ennreal (r - q)"
asm_3: shows "foldr (aux_2 g) (n # ns) (Mapping.empty, Mapping.empty) = (u', p')"
ordertype_infinite_ge_\<omega>: fixes A :: "V set" assumes "infinite A"   and "small A" shows "\<omega> \<le> ordertype A VWF"
elims: fixes x :: "taylor_model"   and xa :: "taylor_model"   and y :: "taylor_model" assumes "tm_add x xa = y" obtains p1 :: "float poly" and e1 :: "float interval" and p2 :: "float poly" and e2 :: "float interval" where "x = TaylorModel p1 e1" and "xa = TaylorModel p2 e2" and "y = TaylorModel (p1 +\<^sub>p p2) (e1 + e2)"
univ_commute': fixes A :: "'c set"   and r :: "('c \<times> 'c) set"   and f :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "equiv A r"   and "f respects r"   and "x \<in> A" shows "univ f (r `` {x}) = f x"
simps_2: fixes G :: "'b list_graph"   and v :: "'b"   and v' :: "'b"   and es :: "('b \<times> 'b) list" shows "FiniteListGraph.delete_edges G ((v, v') # es) = FiniteListGraph.delete_edges (FiniteListGraph.delete_edge v v' G) es"
vNM_utilityD_1: shows "relation \<subseteq> \<P> outcomes \<times> \<P> outcomes"
elims: fixes x :: "atom"   and y :: "atom fm" assumes "amin_inf x = y" obtains uu_ :: "nat" where "x = Less uu_ 0" and "y = FalseF"   | v :: "nat" where "x = Less 0 (Suc v__)" and "y = TrueF"   | i :: "nat" and j :: "nat" where "x = Less (Suc i__) (Suc j__)" and "y = Atom (Less i__ j__)"   | "x = Eq 0 0" and "y = TrueF"   | v :: "nat" where "x = Eq 0 (Suc va__)" and "y = FalseF"   | v :: "nat" where "x = Eq (Suc v) 0" and "y = FalseF"   | i :: "nat" and j :: "nat" where "x = Eq (Suc i) (Suc j)" and "y = Atom (Eq i j)"
subfieldE_3: fixes K :: "'e set"   and R :: "('e, 'f) ring_scheme" assumes "subfield K R" shows "K \<subseteq> carrier R"
asTerm_pick: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set" assumes "good X" shows "asTerm (pick X) = X"
multiset_prod_fact_remove: fixes x :: "'b"   and A :: "'b multiset" assumes "x \<in># A" shows "(\<Prod>y\<in>set_mset A. fact (count A y)) = count A x * (\<Prod>y\<in>set_mset (A - {#x#}). fact (count (A - {#x#}) y))"
translation_length_def: fixes f :: "'b \<Rightarrow> 'b" shows "translation_length f = Inf {dist x (f x) |x. True}"
mult_sm_eye_mm: fixes k :: "complex"   and A :: "complex \<times> complex \<times> complex \<times> complex" shows "k *\<^sub>s\<^sub>m eye *\<^sub>m\<^sub>m A = k *\<^sub>s\<^sub>m A"
sum_list_list_of_pdevs: fixes x :: "'b pdevs" shows "sum_list (map snd (list_of_pdevs x)) = sum_list (dense_list_of_pdevs x)"
card_SigmaI: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'd set" assumes "finite A"   and "\<forall>a\<in>A. finite (B a)" shows "card (Sigma A B) = (\<Sum>a\<in>A. card (B a))"
dim_vec_eq: fixes X :: "(real, 'a) vec set" shows "vec.dim X = dim X"
b: shows "steps0 (1, l, r) tp stpa = (s', l', r') \<and> stp \<le> stpa \<and> crsp ly (as, am) (s', l', r') ires"
elims: fixes x :: "'b tree"   and y :: "'b list" assumes "inorder x = y" obtains "x = \<langle>\<rangle>" and "y = []"   | l :: "'b tree" and xa :: "'b" and r :: "'b tree" where "x = \<langle>l, xa, r\<rangle>" and "y = inorder l @ [xa] @ inorder r"
v11_eq_iff: fixes a :: "V"   and b :: "V" assumes "a \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>"   and "b \<in>\<^sub>\<circ> \<AA>\<lparr>Obj\<rparr>" shows "(\<FF>\<lparr>ObjMap\<rparr>\<lparr>a\<rparr> = \<FF>\<lparr>ObjMap\<rparr>\<lparr>b\<rparr>) = (a = b)"
pos_oriented_circle_inf: fixes H :: "ocircline" assumes "\<infinity>\<^sub>h \<notin> ocircline_set H" shows "pos_oriented H = (\<infinity>\<^sub>h \<notin> disc H)"
iT_Plus_image_conv: fixes I :: "nat set"   and k :: "nat" shows "I \<oplus> k = (\<lambda>n. n + k) ` I"
distinct_15: fixes x21 :: "'c"   and x22 :: "'c \<Rightarrow> ('a, 'c) trm"   and x41 :: "('a, 'b, 'c) formula"   and x42 :: "('a, 'b, 'c) formula" shows "$\<phi> x21 x22 \<noteq> (x41 && x42)"
assms_2: fixes formula :: "('a, 'b, 'c) formula" assumes "P formula" shows "P (Formula.Not formula)"
less_irrefl: fixes x :: "('d, 'e) Term.term" shows "(x, x) \<notin> {\<rhd>}"
abs_eq: fixes xa :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a \<Rightarrow> 'b option" shows "Mapping.filter xa (Mapping x) = Mapping (\<lambda>k. case x k of None \<Rightarrow> None | Some v \<Rightarrow> if xa k v then Some v else None)"
assms_2: fixes s :: "char list"   and t :: "char list" shows "m s @ t = m (s @ t)"
r_right_minus_eq: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and b :: "'c" assumes "ring R"   and "a \<in> carrier R"   and "b \<in> carrier R" shows "(a \<ominus>\<^bsub>R\<^esub> b = \<zero>\<^bsub>R\<^esub>) = (a = b)"
elims_2: fixes x :: "object atom formula set"   and xa :: "ground_action list"   and xb :: "object atom formula set" assumes "ground_action_path x xa xb" obtains M :: "object atom formula set" and M' :: "object atom formula set" where "x = Ma__" and "xa = []" and "xb = M'__" and "Ma__ = M'__"   | M :: "object atom formula set" and \<alpha> :: "ground_action" and \<alpha>s :: "ground_action list" and M' :: "object atom formula set" where "x = M" and "xa = \<alpha> # \<alpha>s" and "xb = M'" and "M \<^sup>c\<TTurnstile>\<^sub>= ground_action.precondition \<alpha> \<and> ground_action_path (execute_ground_action \<alpha> M) \<alpha>s M'"
AndR_inv: fixes \<Gamma> :: "'b formula multiset"   and F :: "'b formula"   and G :: "'b formula"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> \<Rightarrow> F \<^bold>\<and> G, \<Delta>" shows "\<Gamma> \<Rightarrow> F, \<Delta> \<and> \<Gamma> \<Rightarrow> G, \<Delta>"
power_0: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and a :: "'g" shows "power.power one times a 0 = one"
rel_flip: fixes R1 :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and R2 :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "('a, 'c) tllist"   and b :: "('b, 'd) tllist" shows "tllist_all2 R1\<inverse>\<inverse> R2\<inverse>\<inverse> a b = tllist_all2 R1 R2 b a"
field_differentiable_ln_Gamma_complex: fixes z :: "complex"   and A :: "complex set" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "ln_Gamma field_differentiable at z within A"
c_nc: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<sqinter> nc = (0::'a)"
2: shows "uniform_limit (path_image \<gamma>) (\<lambda>n x. f' x * (inverse (x - n) ^ k - inverse (x - w) ^ k) / (n - w) / of_nat k) (\<lambda>x. f' x / (x - w) ^ Suc k) (at w)"