hcju/isabelleps huggingface dataset:
uw\<theta>\<omega>\<nu>: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c"   and \<omega> :: "'c"   and \<nu> :: "'c" assumes "uw\<theta>\<omega>\<nu> V H \<a> \<i> src trg r \<rho> f g u w \<theta> \<omega> \<nu>" shows "partial_magma.ide V w \<and> partial_magma.in_hom V \<theta> (H f w) u \<and> partial_magma.in_hom V \<nu> (partial_magma.dom V \<omega>) (H g w) \<and> category.iso V \<nu> \<and> V (H r \<theta>) (V (\<a> r f w) (V (H \<rho> w) \<nu>)) = \<omega>"
dim_col_conjugate: fixes m :: "'b mat" shows "dim_col (conjugate m) = dim_col m"
fmrelD: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and m :: "'g \<rightharpoonup>\<^sub>f 'e"   and n :: "'g \<rightharpoonup>\<^sub>f 'f"   and x :: "'g" assumes "fmrel P m n" shows "rel_option P (fmlookup m x) (fmlookup n x)"
block_nominals: fixes p :: "('b, 'c) fm"   and block :: "('b, 'c) fm list \<times> 'c"   and i :: "'c" assumes "p on block"   and "i \<in> nominals p" shows "i \<in> block_nominals block"
erf_at_top: shows "(erf \<longlongrightarrow> 1) at_top"
UnE: fixes c :: "'d"   and A :: "'d set"   and B :: "'d set" assumes "c \<in> A \<union> B" obtains "c \<in> A"   | "c \<in> B"
value_scale_flow: fixes c :: "ennreal"   and f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "value_flow \<Delta> (\<lambda>e. c * f e) = c * value_flow \<Delta> f"
fmdom_drop_fset: fixes A :: "'a fset"   and m :: "('a, 'b) fmap" shows "fmdom (fmdrop_fset A m) = fmdom m |-| A"
tab_simps_3: shows "trg tab = trg (k\<^sup>* \<star> h)"
primroot_shorter_eq: fixes u :: "'b list" assumes "u \<noteq> \<epsilon>" shows "\<^bold>|\<rho> u\<^bold>| \<le> \<^bold>|u\<^bold>|"
double_negation: fixes P :: "'a upred" shows "(\<not> \<not> P) = P"
assms_2: shows "q \<cdot> b = (0::'a)"
simps: fixes a :: "(nat \<times> nat) convnet" shows "balanced_net a = ((\<exists>M. a = Input M) \<or> (\<exists>m A. a = Conv A m \<and> balanced_net m) \<or> (\<exists>m1 m2. a = Pool m1 m2 \<and> balanced_net m1 \<and> balanced_net m2 \<and> count_weights True m1 = count_weights True m2))"
is_process_Rep: fixes P :: "'a process" shows "is_process (Rep_process P)"
span_Diff_zero: fixes B :: "'c set" shows "ideal (B - {0::'c}) = ideal B"
filter_IFS_no_violations_def: fixes G :: "'b list_graph"   and M :: "'b SecurityInvariant list" shows "TopoS_Stateful_Policy_impl.filter_IFS_no_violations G M = TopoS_Stateful_Policy_impl.filter_IFS_no_violations_accu G M [] (edgesL G)"
not_gr_zero: fixes plus :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero :: "'f"   and less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and n :: "'f" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "(\<not> less zero n) = (n = zero)"
arr_Rel_ArrCod_in_Vset: fixes \<alpha> :: "V"   and T :: "V" assumes "arr_Rel \<alpha> T" shows "T\<lparr>ArrCod\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
simps_1: fixes cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool" shows "sorted_by cmp [] = True"
power_lookup_not_one_subset_keys: fixes f :: "'b \<Rightarrow> 'c"   and t :: "'b \<Rightarrow>\<^sub>0 nat" shows "{x. f x ^ lookup t x \<noteq> (1::'c)} \<subseteq> keys t"
empty: assumes "list_comb f xs = f"   and "xs = []" shows "thesis"
neq2: shows "p # \<Delta> \<noteq> \<Delta>"
chine_composites_5: shows "Chn \<lparr>Chn = \<mu>\<nu>_\<pi>_\<rho>.chine_assoc, Dom = Dom (((\<mu> \<star> \<nu>) \<star> \<pi>) \<star> \<rho>), Cod = Cod ((\<mu> \<star> \<nu>) \<star> \<pi> \<star> \<rho>)\<rparr> = \<mu>\<nu>_\<pi>_\<rho>.chine_assoc"
Taylor_exp: fixes z :: "complex"   and n :: "nat" shows "cmod (exp z - (\<Sum>k\<le>n. z ^ k / fact k)) \<le> exp \<bar>Re z\<bar> * cmod z ^ Suc n / fact n"
hd_rev: fixes xs :: "'g list" assumes "xs \<noteq> []" shows "hd (rev xs) = last xs"
Some_image_these_eq: fixes A :: "'b option set" shows "Some ` Option.these A = {x \<in> A. x \<noteq> None}"
sums_eval_fps: fixes z :: "'a"   and f :: "'a fps" assumes "ereal (norm z) < fps_conv_radius f" shows "(\<lambda>n. fps_nth f n * z ^ n) sums eval_fps f z"
thread_oks_redT_updT_conv: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and ta :: "('f, 'g, 'k) new_thread_action"   and tas :: "('f, 'i, 'j) new_thread_action list" shows "thread_oks (redT_updT' ts ta) tas = thread_oks (redT_updT ts ta) tas"
comp2: fixes g :: "'f \<Rightarrow> 'b" assumes "bounded_clinear g" shows "bounded_sesquilinear (\<lambda>x y. x ** g y)"
cod_closed: fixes f :: "'a" assumes "left a f" shows "left a (cod f)"
rel_simps_2: fixes m :: "num" shows "(num.Bit0 m \<le> num.One) = False"
monotone_preference_def: fixes B :: "'b set"   and P :: "('b \<times> 'b) set" shows "monotone_preference B P = (\<forall>x\<in>B. \<forall>y\<in>B. y < x \<longrightarrow> x \<succeq>[P] y \<and> (y, x) \<notin> P)"
domD: fixes a :: "'a"   and m :: "'a \<Rightarrow> 'b option" assumes "a \<in> dom m" shows "\<exists>b. m a = Some b"
measurable_stopping_time: fixes F :: "'b \<Rightarrow> 'c measure"   and T :: "'c \<Rightarrow> 'b"   and M :: "'c measure" assumes "stopping_time F T"   and "\<And>t. sets (F t) \<subseteq> sets M"   and "\<And>t. space (F t) = space M" shows "T \<in> borel_measurable M"
Ivl4: shows "{0..<4} = {0, 1, 2, 3}"
A: shows "rev xs \<cong> {ipurge_tr_rev_aux (I\<inverse>) D U (rev xs), rev (ipurge_tr_aux ((I\<inverse>)\<inverse>) D U (rev (rev xs))), \<lambda>y ys. \<exists>v\<in>sinks_aux ((I\<inverse>)\<inverse>) D U (rev ys). (D y, v) \<in> I\<inverse>}"
zdiv_int: fixes a :: "nat"   and b :: "nat" shows "int (a div b) = int a div int b"
assoc_simps_5: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "B.ide f"   and "B.ide g"   and "B.ide h"   and "src\<^sub>B f = trg\<^sub>B g"   and "src\<^sub>B g = trg\<^sub>B h" shows "B.cod \<a>\<^sub>B[f, g, h] = B.cod f \<star>\<^sub>B B.cod g \<star>\<^sub>B B.cod h"
strongly_accepts_1_19_plus: fixes \<F> :: "nat set set"   and M :: "nat set" assumes "thin_set \<F>"   and "infinite M"   and "decides_subsets \<F> M" obtains N :: "nat set" where "N \<subseteq> M" and "infinite N" and "\<And>S n. \<lbrakk>S \<subseteq> N; finite S; strongly_accepts \<F> S N; n \<in> N; S \<lless> {n}\<rbrakk> \<Longrightarrow> strongly_accepts \<F> (insert n S) N"
the_enat_minus_nat: fixes m :: "enat"   and n :: "nat" assumes "m \<noteq> \<infinity>" shows "the_enat (m - enat n) = the_enat m - n"
\<tau>instr_stk_drop_exec_moves: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and es :: "'addr expr1 list"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and fr' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and vs :: "'addr val list" assumes "exec_moves ci P t es h (stk, loc, pc, None) ta h' fr'" shows "\<tau>instr (compP2 P) h (stk @ vs) (compEs2 es ! pc) = \<tau>instr (compP2 P) h stk (compEs2 es ! pc)"
f_noteq_g_measurable: fixes f :: "'b \<Rightarrow> real"   and M :: "'b set set \<times> ('b set \<Rightarrow> real)"   and g :: "'b \<Rightarrow> real" assumes "f \<in> rv M"   and "g \<in> rv M" shows "{w. f w \<noteq> g w} \<in> measurable_sets M"
infdist_pos_not_in_closed: fixes S :: "'a set"   and x :: "'a" assumes "closed S"   and "S \<noteq> {}"   and "x \<notin> S" shows "0 < infdist x S"
poly_neg_number_rootat_def: fixes p :: "rat poly" shows "poly_neg_number_rootat p \<equiv> if even (degree p) then sgn (lead_coeff p) else - sgn (lead_coeff p)"
HNextPart_chosen: fixes s :: "state"   and s' :: "state" assumes "HNextPart s s'" shows "chosen s' = NotAnInput \<longrightarrow> (\<forall>p. outpt s' p = NotAnInput)"
reduce_eq_parreduce: fixes x :: "comb"   and y :: "comb" shows "(x \<rightarrow> y) = (x \<Rrightarrow> y)"
proots_uball_eq: fixes p :: "complex poly"   and r :: "real"   and z0 :: "complex" assumes "p \<noteq> 0"   and "0 < r" shows "proots_count p (ball z0 r) = proots_count (p \<circ>\<^sub>p [:z0, complex_of_real r:]) (ball 0 1)"
step_Normal_elim_cases_7: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "'e"   and u :: "('e, 'd, 'f) com list \<times> (('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list \<times> ('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (Skip # cs, css, Normal s) \<rightarrow> u" obtains "u = (cs, css, Normal s)"
DisN': fixes a :: "'c"   and ps :: "('b, 'c) fm list"   and branch :: "(('b, 'c) fm list \<times> 'c) list"   and p :: "('b, 'c) fm"   and q :: "('b, 'c) fm"   and A :: "'c set"   and n :: "nat" assumes "(\<^bold>\<not> (p \<^bold>\<or> q)) at a in' set ((ps, a) # branch)"   and "A, n \<turnstile> ((\<^bold>\<not> q) # (\<^bold>\<not> p) # ps, a) # branch" shows "A, n \<turnstile> (ps, a) # branch"
TBtheorem5a_notPQ: fixes PQ :: "specID"   and P :: "specID"   and Q :: "specID"   and ChSetP :: "chanID set"   and E :: "Expression"   and ChSetQ :: "chanID set" assumes "subcomponents PQ = {P, Q}"   and "correctCompositionOut PQ"   and "out_exprChannelSet P ChSetP E"   and "out_exprChannelSet Q ChSetQ E"   and "\<forall>x. x \<in> ChSetP \<longrightarrow> x \<in> loc PQ"   and "\<forall>x. x \<in> ChSetQ \<longrightarrow> x \<in> loc PQ" shows "\<not> eout PQ E"
BRParCases: fixes P :: "ccs"   and Q :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and F :: "ccs \<Rightarrow> bool" assumes "(P, Q \<parallel> !Q) \<in> bangRel Rel"   and "\<And>P R. \<lbrakk>(P, Q) \<in> Rel; (R, !Q) \<in> bangRel Rel\<rbrakk> \<Longrightarrow> F (P \<parallel> R)" shows "F P"
dg_Cod_app_in_Obj: fixes f :: "V" assumes "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" shows "\<CC>\<lparr>Cod\<rparr>\<lparr>f\<rparr> \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
wf_cdecl_supD: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "wf_cdecl G (C, c)"   and "C \<noteq> Object" shows "is_acc_class G (pid C) (super c) \<and> (super c, C) \<notin> (subcls1 G)\<^sup>+ \<and> (table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) entails \<lambda>new. \<forall>old sig. (G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old \<longrightarrow> G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le> accmodi new \<and> \<not> is_static old) \<and> (G,sig\<turnstile> new hides old \<longrightarrow> accmodi old \<le> accmodi new \<and> is_static old))"
closed': fixes x :: "'al" assumes "x \<in> \<UU>A" shows "f x \<in> \<UU>B"
prob1_deutsch_algo_const: assumes "is_const" shows "prob1 2 deutsch_algo 0 = 0"
obtains_1: fixes c :: "nat \<Rightarrow>f value option"   and t :: "(String.literal \<times> value list) list" assumes "obtains 1 c drinks2 0 <> t" shows "c $ 2 = Some (Num 0)"
alphaAbs_qAbs_iff_alphaAbs_all_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and X' :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and xs :: "'varSort"   and x :: "'var"   and xs' :: "'varSort"   and x' :: "'var" assumes "qGood X"   and "qGood X'" shows "(qAbs xs x X $= qAbs xs' x' X') = alphaAbs_all_qAFresh xs x X xs' x' X'"
decides_all_def: shows "decides_all \<equiv> \<lambda>S N. \<forall>T\<subseteq>S. decides \<F> T N"
extend_1_evals: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and es :: "expr list"   and t :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es'' :: "expr list"   and t'' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and es' :: "expr list"   and t' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "wwf_prog P"   and "P,E \<turnstile> \<langle>es,t\<rangle> [\<rightarrow>] \<langle>es'',t''\<rangle>"   and "P,E \<turnstile> \<langle>es'',t''\<rangle> [\<Rightarrow>] \<langle>es',t'\<rangle>" shows "P,E \<turnstile> \<langle>es,t\<rangle> [\<Rightarrow>] \<langle>es',t'\<rangle>"
eq_upto_lexmin_non_empty: fixes P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and C :: "(nat \<Rightarrow> 'b) set"   and i :: "nat" assumes "minimal_element P A"   and "C \<subseteq> SEQ A"   and "C \<noteq> {}" shows "eq_upto C (minimal_element.lexmin P A C) i \<noteq> {}"
strip_list_comb: fixes size :: "'b \<Rightarrow> nat"   and frees :: "'b \<Rightarrow> name fset"   and subst :: "'b \<Rightarrow> (name, 'b) fmap \<Rightarrow> 'b"   and "consts" :: "'b \<Rightarrow> name fset"   and app :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unapp :: "'b \<Rightarrow> ('b \<times> 'b) option"   and const :: "name \<Rightarrow> 'b"   and unconst :: "'b \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'b"   and unfree :: "'b \<Rightarrow> name option"   and abs_pred :: "('b \<Rightarrow> bool) \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b"   and ys :: "'b list" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.strip_comb unapp (term.list_comb app f ys) = (fst (term.strip_comb unapp f), snd (term.strip_comb unapp f) @ ys)"
of_int_uint: fixes w :: "'c word" shows "of_int (uint w) = unsigned w"
induct: fixes P :: "'b tree23 \<Rightarrow> bool"   and a0 :: "'b tree23" assumes "\<And>a. P \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>\<rangle>"   and "\<And>a b. P \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>, b, \<langle>\<rangle>\<rangle>"   and "\<And>v va vb a r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, a, r\<rangle>"   and "\<And>v va vb vc vd a r. P \<langle>v, va, vb, vc, vd\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd\<rangle>, a, r\<rangle>"   and "\<And>l a v va vb. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb\<rangle>\<rangle>"   and "\<And>l a v va vb vc vd. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd\<rangle>\<rangle>"   and "\<And>v va vb a m b r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, a, m, b, r\<rangle>"   and "\<And>v va vb vc vd a m b r. P \<langle>v, va, vb, vc, vd\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd\<rangle>, a, m, b, r\<rangle>"   and "\<And>l a v va vb b r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb\<rangle>, b, r\<rangle>"   and "\<And>l a v va vb vc vd b r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd\<rangle>, b, r\<rangle>"   and "\<And>l a m b v va vb. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb\<rangle>\<rangle>"   and "\<And>l a m b v va vb vc vd. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb, vc, vd\<rangle>\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"
reconstruction: fixes p :: "int"   and n :: "nat"   and sl_impl :: "(int poly, int \<times> int poly list, 'b) subseqs_foldr_impl"   and sli :: "int \<times> int poly list \<Rightarrow> int poly list \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> bool"   and m2 :: "int"   and state :: "'b"   and u :: "int poly"   and lu :: "int"   and d :: "nat"   and r :: "nat"   and vs :: "int poly list"   and res :: "int poly list"   and cands :: "(int \<times> int poly list) list"   and fs :: "int poly list"   and f :: "int poly"   and meas :: "nat \<times> (int \<times> int poly list) list" assumes "prime p"   and "p = p ^ n"   and "n \<noteq> 0"   and "correct_subseqs_foldr_impl (\<lambda>x. map_prod (p.mul_const x) ((#) x)) sl_impl sli"   and "p.reconstruction sl_impl m2 state u (Polynomial.smult lu u) lu d r vs res cands = fs"   and "f = u * prod_list res"   and "meas = (r - d, cands)"   and "d + d \<le> r"   and "r = length vs"   and "set cands \<subseteq> subseqs_foldr_impl.S (\<lambda>x. map_prod (p.mul_const x) ((#) x)) (lu, []) vs d"   and "d = 0 \<Longrightarrow> cands = []"   and "lu = lead_coeff u"   and "p.unique_factorization_m u (lu, mset vs)"   and "poly_mod.square_free_m p u"   and "comm_monoid_mult_class.coprime lu p"   and "\<And>v. v \<in> set vs \<Longrightarrow> p.Mp v = v"   and "\<And>ws. \<lbrakk>ws \<subseteq># mset vs; ws \<noteq> {#}; size ws < d \<or> size ws = d \<and> ws \<notin> (mset \<circ> snd) ` set cands\<rbrakk> \<Longrightarrow> p.test_dvd u ws"   and "\<And>f. f \<in> set res \<Longrightarrow> irreducible\<^sub>d f"   and "0 < degree u"   and "cands \<noteq> [] \<Longrightarrow> d < r"   and "\<forall>v n. v dvd Polynomial.smult lu u \<longrightarrow> degree v \<le> degree_bound vs \<longrightarrow> 2 * \<bar>coeff v n\<bar> < p"   and "f \<noteq> 0"   and "sli (lu, []) vs d state"   and "m2 = p div 2" shows "f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi)"
case_2: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b"   and x2 :: "nat" shows "(case Suc x2 of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = f2 x2"
\<G>\<F>_suffix: fixes \<phi> :: "'b ltln"   and i :: "nat"   and w :: "nat \<Rightarrow> 'b set" shows "\<G>\<F> \<phi> (suffix i w) = \<G>\<F> \<phi> w"
F'_1: shows "is_expansion_aux xs (\<lambda>x. f x - eval C x * b x powr 0) (b # basis)"
mset_append: fixes xs :: "'c list"   and ys :: "'c list" shows "mset (xs @ ys) = mset xs + mset ys"
rg_succ_ref2: fixes am :: "nat \<Rightarrow> nat list"   and u :: "nat" assumes "is_adj_map am"   and "u \<in> V" shows "rg_succ2 am cf u \<le> return (rg_succ am cf u)"
cts_wp_PC: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and p :: "'a \<Rightarrow> real" assumes "bd_cts (wp a)"   and "bd_cts (wp b)"   and "healthy (wp a)"   and "healthy (wp b)"   and "unitary p" shows "bd_cts (wp (a \<^bsub>p\<^esub>\<oplus> b))"
extremum_unique: fixes a :: "'c" shows "(a \<le> bot) = (a = bot)"
weak_snd_iso_thme: fixes H :: "'a set"   and N :: "'a set" assumes "subgroup H G"   and "N \<lhd> G" shows "G\<lparr>carrier := N <#> H\<rparr> Mod N \<cong> G\<lparr>carrier := H\<rparr> Mod N \<inter> H"
map_in_poly_ring_carrier: fixes p :: "'a list"   and f :: "'a \<Rightarrow> 'a list" assumes "p \<in> carrier (poly_ring R)"   and "\<And>a. a \<in> carrier R \<Longrightarrow> f a \<in> carrier (poly_ring R)"   and "\<And>a. a \<noteq> \<zero> \<Longrightarrow> f a \<noteq> []" shows "map f p \<in> carrier (poly_ring (poly_ring R))"
left_translations_of_monoid_axioms: shows "left_translations_of_monoid G (\<cdot>) \<one>"
Y_arr_ide_2: fixes b :: "'c"   and f :: "'c"   and a :: "'c"   and a' :: "'c" assumes "C.ide b"   and "\<guillemotleft>f : a \<rightarrow> a'\<guillemotright>" shows "Y f b = S.mkArr (Hom.set (b, a)) (Hom.set (b, a')) (\<phi> (b, a') \<circ> (\<cdot>) f \<circ> \<psi> (b, a))"
extensibleD: fixes R :: "('e, 'f) labeled_graph \<times> ('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "extensible R G f" obtains g :: "('f \<times> 'g) set" where "graph_homomorphism (snd R) G g" and "agree_on (fst R) f g"
endpoints_in_S_incident_unique: fixes a :: "proj2"   and b :: "proj2"   and p :: "proj2"   and l :: "proj2_line" assumes "a \<noteq> b"   and "a \<in> hyp2 \<union> S"   and "b \<in> hyp2 \<union> S"   and "p \<in> S"   and "proj2_incident a l"   and "proj2_incident b l"   and "proj2_incident p l" shows "p = endpoint_in_S a b \<or> p = endpoint_in_S b a"
numeral_plus_numeral: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and m :: "num"   and n :: "num" assumes "class.numeral plus" shows "plus (numeral.numeral one plus m) (numeral.numeral one plus n) = numeral.numeral one plus (m + n)"
of_nat_eq_star_of: shows "of_nat = hypnat_of_nat"
subfield_long_division_theorem_shell: fixes K :: "'a set"   and p :: "'a list"   and b :: "'a list" assumes "subfield K R"   and "p \<in> carrier (K [X])"   and "b \<in> carrier (K [X])"   and "b \<noteq> \<zero>\<^bsub>K [X]\<^esub>" shows "\<exists>q r. q \<in> carrier (K [X]) \<and> r \<in> carrier (K [X]) \<and> p = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = \<zero>\<^bsub>K [X]\<^esub> \<or> degree r < degree b)"
cases: fixes trsys :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and \<tau>move :: "'c \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "\<tau>trsys.\<tau>diverge trsys \<tau>move a" obtains s :: "'c" and s' :: "'c" where "a = s" and "\<tau>trsys.silent_move trsys \<tau>move s s'" and "\<tau>trsys.\<tau>diverge trsys \<tau>move s'"
add_node_def: fixes v :: "'b"   and G :: "'b graph" shows "FiniteGraph.add_node v G \<equiv> \<lparr>nodes = {v} \<union> nodes G, edges = edges G\<rparr>"
of_hypreal_inverse: fixes x :: "real star" shows "of_hypreal (inverse x) = inverse (of_hypreal x)"
enum_distinct: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" assumes "class.enum enum enum_all enum_ex" shows "distinct enum"
conjugation_is_hom: shows "(\<lambda>g. \<lambda>h\<in>carrier G. g \<otimes> h \<otimes> inv g) \<in> hom G (BijGroup (carrier G))"
induct: fixes P :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> 'c \<times> 'd \<Rightarrow> (('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree \<Rightarrow> bool"   and a0 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a1 :: "'c \<times> 'd"   and a2 :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree" assumes "\<And>uu_ t1 x t2 y t3. P (R uu_ t1 x t2) y t3"   and "\<And>x t1 y uv_ t2. P \<langle>\<rangle> x (B uv_ t1 y t2)"   and "\<And>v vc vf vb x t1 y uv_ t2. P (B vf v vc vb) x (B uv_ t1 y t2)"   and "\<And>x t1 y ux_ t2 z uw_ t3. P \<langle>\<rangle> x (R uw_ (B ux_ t1 y t2) z t3)"   and "\<And>v vc vf vb x t1 y ux_ t2 z uw_ t3. P (B vf v vc vb) x (R uw_ (B ux_ t1 y t2) z t3)"   and "\<And>x. P \<langle>\<rangle> x \<langle>\<rangle>"   and "\<And>x vc vf vb. P \<langle>\<rangle> x (R vf \<langle>\<rangle> vc vb)"   and "\<And>x va vg vj ve vc vf vb. P \<langle>\<rangle> x (R vf (R vj va vg ve) vc vb)"   and "\<And>v vc vf vb x. P (B vf v vc vb) x \<langle>\<rangle>"   and "\<And>v vc vf vb x vg vj ve. P (B vf v vc vb) x (R vj \<langle>\<rangle> vg ve)"   and "\<And>v vc vf vb x vd vk vn vi vg vj ve. P (B vf v vc vb) x (R vj (R vn vd vk vi) vg ve)" shows "P a0 a1 a2"
con_def: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<psi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool" shows "con \<phi> \<psi> \<equiv> neg (dis (neg \<phi>) (neg \<psi>))"
p: fixes a :: "int" shows "rat_of_int (f a) = poly p (rat_of_int a)"
elims_3: fixes x :: "Preliminaries.var \<Rightarrow> univ"   and xa :: "('fsym, 'psym) lit" assumes "\<not> F.satL x xa" obtains \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>'__" and "xa = Pos a__" and "\<not> F.satA \<xi>'__ a__"   | \<xi> :: "Preliminaries.var \<Rightarrow> univ" and a :: "('fsym, 'psym) atm" where "x = \<xi>" and "xa = Neg a" and "\<not> \<not> F.satA \<xi> a"
eventually_frequently_simps_11: fixes P :: "'a \<Rightarrow> bool"   and F :: "'a filter" shows "(\<not> frequently P F) = (\<forall>\<^sub>F x in F. \<not> P x)"
invertible_mat_def: fixes A :: "'c mat" shows "invertible_mat A \<equiv> square_mat A \<and> (\<exists>B. inverts_mat A B \<and> inverts_mat B A)"
lm11: fixes x :: "'c"   and P :: "('c \<times> 'd) set" shows "{x} \<times> UNIV \<inter> P = {x} \<times> P `` {x}"