hcju/isabelleps huggingface dataset:
stake_cycle_eq: fixes u :: "'b list" assumes "u \<noteq> []" shows "stake (length u) (cycle u) = u"
schwefel_def: shows "schwefel = R\<^sub>e 58806 / R\<^sub>e (10 ^ 4) / R\<^sub>e (10 ^ 10) + Power (Var 0 - Power (Var 1) 2) 2 + Power (Var 1 - 1) 2 + Power (Var 0 - Power (Var 2) 2) 2 + Power (Var 2 - 1) 2"
slim_verts_def: shows "slim_verts \<equiv> verts3 (with_proj G) \<union> (\<Union>(u, p, uu_)\<in>slim_paths. set (pawalk_verts u p))"
subst_typ_no_tvsT_unchanged: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and T :: "typ" assumes "\<forall>(f, s)\<in>set insts. f \<notin> tvsT T" shows "subst_typ insts T = T"
SesKey_is_session_key: fixes Tgs_B :: "agent"   and A :: "agent"   and SesKey :: "nat"   and T :: "nat"   and evs :: "event list" assumes "Crypt (shrK Tgs_B) \<lbrace>Agent A, Agent Tgs_B, Key SesKey, Number T\<rbrace> \<in> parts (knows Spy evs)"   and "Tgs_B \<notin> bad"   and "evs \<in> kerbIV_gets" shows "SesKey \<notin> range shrK"
Domainp_mod_ring_rel32: shows "Domainp mod_ring_rel32 = (\<lambda>x. 0 \<le> x \<and> x < pp)"
add_Suc_right: fixes m :: "nat"   and n :: "nat" shows "plus m (Suc n) = Suc (plus m n)"
cond_subst_def: fixes \<sigma> :: "'e \<Rightarrow> 'e"   and b :: "(bool, 'e) uexpr"   and \<rho> :: "'e \<Rightarrow> 'e" shows "\<sigma> \<triangleleft> b \<triangleright>\<^sub>s \<rho> = (\<lambda>s. if \<lbrakk>b\<rbrakk>\<^sub>e s then \<sigma> s else \<rho> s)"
open_Int: fixes "open" :: "'b set \<Rightarrow> bool"   and S :: "'b set"   and T :: "'b set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<inter> T)"
rb_aux_inv2_ideal_subset: fixes bs :: "('t \<Rightarrow>\<^sub>0 'b) list"   and ss :: "'t list"   and ps :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list"   and j :: "nat" assumes "rb_aux_inv2 (bs, ss, ps)"   and "\<And>p0. p0 \<in> set ps \<Longrightarrow> j \<le> component_of_term (sig_of_pair p0)" shows "ideal (set (take j fs)) \<subseteq> ideal (rep_list ` set (filter (\<lambda>b. component_of_term (lt b) < j) bs))"
le_infI2: fixes b :: "'k multiset"   and x :: "'k multiset"   and a :: "'k multiset" assumes "b \<subseteq># x" shows "a \<inter># b \<subseteq># x"
set_sel: fixes a :: "'b option" assumes "a \<noteq> None" shows "the a \<in> set_option a"
is_dghmD_11: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<R>\<^sub>\<circ> (\<FF>\<lparr>ObjMap\<rparr>) \<subseteq>\<^sub>\<circ> \<BB>\<lparr>Obj\<rparr>"
I_def: shows "I \<equiv> \<lambda>f u. exp u * (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) 0) - (\<Sum>j\<le>degree f. poly ((pderiv ^^ j) f) u)"
abst_dbtm_fresh_ignore: fixes name :: "name"   and u :: "dbtm"   and j :: "nat" assumes "atom name \<sharp> u" shows "abst_dbtm name j u = u"
IH: fixes P :: "'a \<Rightarrow> real" assumes "unitary P" shows "unitary (t P)"
simps_1: fixes x :: "complex"   and y :: "complex" shows "Re (x * y) = Re x * Re y - Im x * Im y"
lem: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula" shows "(\<Gamma>::'b formula set) \<union> AX10 \<turnstile>\<^sub>H \<^bold>\<not> (F::'b formula) \<^bold>\<or> F"
candidate_values_renaming: fixes z :: "'a trm"   and CC :: "'a eclause"   and C' :: "'a literal set"   and C :: "'a literal set"   and s :: "'a trm"   and L :: "'a literal"   and L' :: "'a literal"   and \<sigma> :: "('a \<times> 'a trm) list"   and t' :: "'a trm"   and s' :: "'a trm"   and t :: "'a trm"   and S :: "'a eclause set"   and D :: "'a eclause" assumes "candidate_values z CC C' C s L L' \<sigma> t' s' t S"   and "finite C'"   and "finite (cl_ecl D)"   and "closed_under_renaming S"   and "Ball S well_constrained" shows "\<exists>CC_bis C'_bis L'_bis \<sigma>_bis t'_bis s'_bis t_bis. candidate_values z CC_bis C'_bis C s L L'_bis \<sigma>_bis t'_bis s'_bis t S \<and> vars_of_cl (cl_ecl D) \<inter> vars_of_cl (cl_ecl CC_bis) = {}"
sup_bot_right: fixes x :: "'b multiset" shows "x \<union># {#} = x"
border_positions: fixes xs :: "'a list"   and ys :: "'a list" assumes "border xs ys" shows "\<forall>i<length xs. ys ! i = ys ! (length ys - length xs + i)"
possible_both: fixes t :: "'b ttree"   and t' :: "'b ttree"   and x :: "'b" shows "possible (t \<otimes>\<otimes> t') x = (possible t x \<or> possible t' x)"
iterative_HSem': fixes x :: "'a"   and \<Gamma> :: "('a \<times> 'b) list"   and \<rho> :: "'a \<Rightarrow> 'c"   and ESem :: "'b \<Rightarrow> ('a \<Rightarrow> 'c) \<rightarrow> 'c"   and e :: "'b" assumes "x \<notin> domA \<Gamma>" shows "(\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> has_ESem.HSem ESem \<Gamma>\<cdot>xa)(x := ESem e\<cdot>xa)) = (\<mu> xa. (\<rho> ++\<^bsub>domA \<Gamma>\<^esub> has_ESem.HSem ESem \<Gamma>\<cdot>xa)(x := ESem e\<cdot>(has_ESem.HSem ESem \<Gamma>\<cdot>xa)))"
antisym: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "le y x" shows "x = y"
PNT4_imp_PNT1: assumes "\<theta> \<sim>[at_top] (\<lambda>x. x)" shows "\<pi> \<sim>[at_top] (\<lambda>x. x / ln x)"
factor_greater_one2: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n = m * k"   and "m < n"   and "k < n" shows "Suc 0 < k"
bound_on_all_plans_bounds_PLS_charles: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB s as. P PROB \<and> finite PROB \<and> as \<in> valid_plans PROB \<and> s \<in> valid_states PROB \<longrightarrow> (\<exists>x. x \<in> PLS_charles s as PROB \<and> x < f PROB)"
p_eq_p1_p2: fixes K1 :: "'a \<Rightarrow> 'a pmf"   and K2 :: "'b \<Rightarrow> 'b pmf"   and x1 :: "'a"   and x2 :: "'b"   and y1 :: "'a"   and y2 :: "'b"   and n :: "nat" shows "MC_syntax.p (MC_pair.Kp K1 K2) (x1, x2) (y1, y2) n = MC_syntax.p K1 x1 y1 n * MC_syntax.p K2 x2 y2 n"
continuous_orderD_1: fixes x :: "'c"   and S :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and c :: "'d" assumes "continuous (at x within S) g"   and "c < g x" shows "\<forall>\<^sub>F y in at x within S. c < g y"
weakCongSimParPres: fixes P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and \<Psi> :: "'b"   and R :: "('a, 'b, 'c) psi" assumes "\<And>\<Psi>'. \<Psi>' \<rhd> P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"   and "\<And>\<Psi>'. \<Psi>' \<rhd> P \<leadsto><Rel> Q"   and "\<And>\<Psi>'. \<Psi>' \<rhd> Q \<lessapprox><Rel> P"   and "eqvt Rel"   and "eqvt Rel'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>', T, S) \<in> Rel"   and "\<And>\<Psi>' S T \<Psi>''. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel"   and "\<And>\<Psi>' S T A\<^sub>U \<Psi>\<^sub>U U. \<lbrakk>(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel; extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>; A\<^sub>U \<sharp>* \<Psi>'; A\<^sub>U \<sharp>* S; A\<^sub>U \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel'"   and "\<And>\<Psi>' S T xvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; xvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel" shows "\<Psi> \<rhd> P \<parallel> R \<leadsto>\<guillemotleft>Rel'\<guillemotright> Q \<parallel> R"
dom_eqI: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "seq f a" shows "local.dom f = a"
simps_32: shows "cospan_shape.comp cospan_shape.Null cospan_shape.TT = cospan_shape.Null"
leg_simps_4: shows "C.dom HH\<nu>\<pi>\<rho>.leg1 = HH\<nu>\<pi>\<rho>.apex"
dim_image_le: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set" assumes "linear (*a) (*b) f" shows "vs2.dim (f ` S) \<le> vs1.dim S"
imethds_def: fixes G :: "prog"   and I :: "qtname" shows "Decl.imethds G I = iface_rec G I (\<lambda>I i ts. Un_tables ts \<oplus>\<oplus> (set_option \<circ> table_of (map (\<lambda>(s, m). (s, I, m)) (imethods i))))"
blocksFinal: fixes Vs :: "char list list"   and Ts :: "ty list"   and vs :: "val list"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "length Vs = length Ts"   and "length vs = length Ts"   and "final e" shows "P \<turnstile> \<langle>blocks (Vs, Ts, vs, e),(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e,(h, l, sh),b\<rangle>"
inImage_iimg: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a"   and B :: "'a set" assumes "a \<in> f ` B" shows "iimg B f a = a"
bezout_matrix_def: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and A :: "(('a, 'c) vec, 'b) vec"   and a :: "'b"   and b :: "'b"   and j :: "'c"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" assumes "class.bezout_ring times one plus zero minus uminus" shows "bezout_ring.bezout_matrix one zero A a b j bezout = (\<chi>x y. let (p, q, u, v, d) = bezout (A $ a $ j) (A $ b $ j) in if x = a \<and> y = a then p else if x = a \<and> y = b then q else if x = b \<and> y = a then u else if x = b \<and> y = b then v else if x = y then one else zero)"
effective_matrix_Tensor_elements2: fixes M1 :: "int poly fract list list"   and M2 :: "int poly fract list list" assumes "mat (rat_poly.row_length M1) (length M1) M1"   and "mat (rat_poly.row_length M2) (length M2) M2" shows "\<forall>i<rat_poly.row_length M1 *n rat_poly.row_length M2. \<forall>j<length M1 *n length M2. (M1 \<otimes> M2) ! j ! i = rat_poly_times (M1 ! (j div length M2) ! (i div rat_poly.row_length M2)) (M2 ! (j mod length M2) ! (i mod rat_poly.row_length M2))"
len_hchop_add: fixes v :: "view"   and v1 :: "view"   and v2 :: "view"   and ts :: "traffic"   and c :: "cars" shows "v=v1\<parallel>v2 \<longrightarrow> \<parallel>len v ts c\<parallel> = \<parallel>len v1 ts c\<parallel> + \<parallel>len v2 ts c\<parallel>"
prems_2: fixes x :: "'a" assumes "x \<in> X"   and "(a, x) \<in> r\<^sup>*"   and "(m, c) \<in> r\<^sup>*" shows "P"
fold_set_fold: fixes y :: "'c set"   and xs :: "'c list" shows "Finite_Set.fold insert y (set xs) = fold insert xs y"
inext_nth_image: fixes I :: "nat set"   and f :: "nat \<Rightarrow> nat"   and n :: "nat" assumes "I \<noteq> {}"   and "SetInterval2.strict_mono_on f I" shows "(f ` I) \<rightarrow> n = f (I \<rightarrow> n)"
left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "sup b (sup a c) = sup a (sup b c)"
\<psi>_minus_\<theta>_bound: fixes x :: "real" assumes "2 \<le> x" shows "\<psi> x - \<theta> x \<le> 2 * ln x * sqrt x"
c6: assumes "ip \<in> kD rt"   and "\<pi>\<^sub>2 r \<le> sqn rt ip"   and "\<pi>\<^sub>3 r = kno"   and "sqn rt ip = \<pi>\<^sub>2 r \<Longrightarrow> the (dhops rt ip) \<le> \<pi>\<^sub>5 r \<and> the (flag rt ip) = val" shows "P (rt(ip \<mapsto> the (rt ip)))"
wtE_pickE_pickE: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and xl :: "Preliminaries.var list"   and al :: "'h list" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP"   and "length xl = length al"   and "distinct xl"   and "\<And>i. i < length xl \<Longrightarrow> intT (Signature.tpOfV (xl ! i)) (al ! i)" shows "Struct.wtE intT (Struct.pickE intT xl al) \<and> (\<forall>i<length xl. Struct.pickE intT xl al (xl ! i) = al ! i)"
exhaust_sel: fixes tree :: "tree" obtains "tree = Leaf"   | "tree = Branching (ltree tree) (rtree tree)"
mult_Mp_2: fixes m :: "int"   and f :: "int poly"   and g :: "int poly" shows "poly_mod.Mp m (f * poly_mod.Mp m g) = poly_mod.Mp m (f * g)"
simps_3: fixes x :: "name"   and t :: "nterm" shows "all_frees (\<Lambda>\<^sub>n x. t) = finsert x (all_frees t)"
cring_simprules_15: fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" shows "x \<ominus> y = x \<oplus> \<ominus> y"
analz_insert_freshK3: fixes authK :: "nat"   and servK :: "nat"   and evs :: "event list"   and authK' :: "nat" assumes "AKcryptSK authK servK evs"   and "authK' \<noteq> authK"   and "authK' \<notin> range shrK"   and "evs \<in> kerbIV_gets" shows "(Key servK \<in> analz (insert (Key authK') (knows Spy evs))) = (servK = authK' \<or> Key servK \<in> analz (knows Spy evs))"
insert_if: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a"   and x :: "'c" assumes "finite A" shows "F g (insert x A) = (if x \<in> A then F g A else g x \<^bold>* F g A)"
gets_the_obind: fixes f :: "'b \<Rightarrow> 'd option"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> 'c option" shows "gets_the (f |>> g) = do x \<leftarrow> gets_the f; gets_the (g x) od"
s_Let2: fixes v :: "Syntax.term"   and x :: "Syntax.var"   and \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uw :: "mode" assumes "value v"   and "atom x \<sharp> (v, \<pi>)" shows "\<lless>\<pi>, Syntax.Let v x e\<ggreater> uw\<rightarrow> \<lless>\<pi>, e[v / x]\<ggreater>"
map_leI: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" assumes "\<And>x v. m1 x = Some v \<Longrightarrow> m2 x = Some v" shows "m1 \<subseteq>\<^sub>m m2"
card_equiv_class_restricted: fixes A :: "'a set"   and P :: "'a \<Rightarrow> bool"   and R :: "('a \<times> 'a) set" assumes "finite {x \<in> A. P x}"   and "equiv A R"   and "P respects R" shows "card {x \<in> A. P x} = sum card ({x \<in> A. P x} // R)"
ty: shows "\<Gamma> \<turnstile> (LET p = t IN u) : U"
widen_cls_dtype_att: fixes cf :: "CAttId" shows "CClassT (cls cf) \<le> dtype (att cf)"
spec_imp_iiws: fixes s :: "('loc, 't) configuration stream" assumes "spec s" shows "alw (holds inv_imps_work_sum) s"
case_sum_o_inj_2: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f" shows "case_sum f g \<circ> Inr = g"
pr_iso_test0_correct: fixes Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list"   and m :: "'a \<Rightarrow> 'b option" assumes "\<forall>F\<in>set Fs\<^sub>1. distinct F"   and "\<forall>F\<in>set Fs\<^sub>2. distinct F"   and "distinct Fs\<^sub>1"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>1)"   and "distinct Fs\<^sub>2"   and "inj_on (\<lambda>xs. {xs} // {\<cong>}) (set Fs\<^sub>2)"   and "inj_on m (dom m)" shows "pr_iso_test0 m Fs\<^sub>1 Fs\<^sub>2 = (\<exists>\<phi>. is_pr_iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<and> m \<subseteq>\<^sub>m Some \<circ> \<phi> \<and> inj_on \<phi> (dom m \<union> \<Union> (set ` set Fs\<^sub>1)))"
check_ipresIGCons: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" shows "ipresIGCons check check (errMOD MOD) MOD"
intro: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool" assumes "known_ptr = ShadowRootClass.known_ptr" shows "l_create_element\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms known_ptr"
ltrm_pow0: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "f \<in> carrier P" shows "monom P ((f [^]\<^bsub>P\<^esub> n) (deg R (f [^]\<^bsub>P\<^esub> n))) (deg R (f [^]\<^bsub>P\<^esub> n)) = monom P (f (deg R f)) (deg R f) [^]\<^bsub>P\<^esub> n"
sets_completionE: fixes A :: "'a set"   and M :: "'a measure" assumes "A \<in> sets (completion M)" obtains S :: "'a set" and N :: "'a set" and N' :: "'a set" where "A = S \<union> N" and "N \<subseteq> N'" and "N' \<in> null_sets M" and "S \<in> sets M"
assms_3: shows "\<not> FactoredSystem.inj last (state_set (state_list s as)) (valid_states PROB)"
AA2: fixes P :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool"   and v :: "'c \<Rightarrow> 'd"   and Q :: "(nat \<Rightarrow> 'c) \<Rightarrow> bool" shows "\<turnstile> \<box>[P]_v \<and> \<diamond>\<langle>Q\<rangle>_v \<longrightarrow> \<diamond>\<langle>P \<and> Q\<rangle>_v"
normal_map_fm: fixes f :: "atom \<Rightarrow> atom"   and \<phi> :: "atom fm" assumes "\<forall>a. (divisor (f a) \<noteq> 0) = (divisor a \<noteq> 0)" shows "Z.normal (map\<^sub>f\<^sub>m f \<phi>) = Z.normal \<phi>"
simps_4: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and tst :: "'c"   and c :: "('c, 'd, 'e) cmd" assumes "PL cval" shows "proper (While tst c) = proper c"
targets_cod: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "targets (cod \<mu>) = targets \<mu>"
eval_monom_smallo_eval_monom: fixes es1 :: "real list"   and es2 :: "real list"   and basis :: "(real \<Rightarrow> real) list" assumes "length es1 = length es2"   and "length es2 = length basis"   and "basis_wf basis"   and "compare_lists es1 es2 = LT" shows "eval_monom (1, es1) basis \<in> o(eval_monom (1, es2) basis)"
nba_g_V0: fixes A :: "('d, 'c) nba" shows "g_V0 (nba_g A) = initial A"
vD_addpreRT: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and npre :: "nat set" assumes "dip \<in> kD rt" shows "vD (the (addpreRT rt dip npre)) = vD rt"
pre_iso_test_def: fixes Fs :: "nat list list" shows "pre_iso_test Fs = ([] \<notin> set Fs \<and> (\<forall>F\<in>set Fs. distinct F) \<and> distinct (map rotate_min Fs))"
simps_3: fixes x :: "'d"   and \<theta> :: "('e, 'd) trm" shows "FVP (DiffAssign x \<theta>) = FVT \<theta>"
Ueq: shows "zs = concat [c, a, d, b] @ interact as bs"
iWeakUntil_disj_iWeakUntil_conv: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool"   and Q :: "nat \<Rightarrow> bool" shows "(P t1 \<or> Q t1. t1 \<W> t2 I. Q t2) = (P t1. t1 \<W> t2 I. Q t2)"
assm: shows "ts \<^bold>\<midarrow>r( c ) \<^bold>\<rightarrow> ts'"
asIMOD_igWlsAbsDisj: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" assumes "sWlsNE SEM"   and "sWlsDisj SEM" shows "igWlsAbsDisj (asIMOD SEM)"
suc_le_pow_2: fixes n :: "nat" assumes "1 < n" shows "Suc n < 2 ^ n"
normal_set_critical_set: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "normal_set (critical_set A x)"
GEN_OP_D: fixes c :: "'c"   and a :: "'d"   and R :: "('c \<times> 'd) set" assumes "GEN_OP c a R" shows "(c, a) \<in> R"
boundI: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and b :: "'c" assumes "\<And>x. x \<in> X \<Longrightarrow> r x b" shows "bound X r b"
cong_n: shows "[a ^ (n * d) = 1] (mod b)"
assms_1: shows "x \<in> env_dom (e\<lparr>a:X\<rparr>)"
rel_resumption_resumeD1: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and r1 :: "('f, 'h, 'j) Resumption.resumption"   and r2 :: "('g, 'i, 'j) Resumption.resumption"   and inp :: "'j" assumes "Resumption.resumption.rel_resumption A B r1 r2"   and "\<not> Resumption.resumption.is_Done r1" shows "Resumption.resumption.rel_resumption A B (Resumption.resumption.resume r1 inp) (Resumption.resumption.resume r2 inp)"
proj_stoch_proj_invariant: fixes p :: "real"   and M :: "bool stream measure"   and F :: "nat \<Rightarrow> bool stream measure"   and X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'b"   and n :: "nat"   and w :: "bool stream" assumes "infinite_cts_filtration p M F"   and "borel_adapt_stoch_proc F X" shows "proj_stoch_proc X n w = proj_stoch_proc X n (pseudo_proj_True n w)"
dg_length: shows "vcard (dg_Rel \<alpha>) = 4\<^sub>\<nat>"
lcoeff_monom: fixes a :: "'a" assumes "a \<in> carrier R"   and "carrier R \<noteq> {\<zero>}" shows "lcoeff (monom P \<one> 1 \<ominus>\<^bsub>P\<^esub> monom P a 0) = \<one>"
"1_3": shows "\<Gamma> \<union> AX10 \<turnstile>\<^sub>H F \<^bold>\<or> G"
twiddleVal_betaExtend: fixes \<beta> :: "(Location \<times> Location) set"   and v :: "Val"   and w :: "Val"   and \<gamma> :: "(Location \<times> Location) set" assumes "(\<beta>, v, w) \<in> twiddleVal"   and "Pbij_extends \<gamma> \<beta>" shows "(\<gamma>, v, w) \<in> twiddleVal"
Lcm_0_iff: fixes A :: "'a set" assumes "finite A" shows "(Lcm A = (0::'a)) = ((0::'a) \<in> A)"
exhaust: fixes y :: "key" obtains x1 :: "fresh_t" where "y = sesK x1"   | x2 :: "ltkey" where "y = ltK x2"
bij_betw_subset: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and A' :: "'d set"   and B :: "'c set"   and B' :: "'d set" assumes "bij_betw f A A'"   and "B \<subseteq> A"   and "f ` B = B'" shows "bij_betw f B B'"
l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_adopt_node_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.get_owner_document Shadow_DOM.get_parent Shadow_DOM.get_parent_locs Shadow_DOM.remove_child Shadow_DOM.remove_child_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs Shadow_DOM.adopt_node Shadow_DOM.adopt_node_locs ShadowRootClass.known_ptr ShadowRootClass.type_wf Shadow_DOM.get_child_nodes Shadow_DOM.get_child_nodes_locs ShadowRootClass.known_ptrs Shadow_DOM.set_child_nodes Shadow_DOM.set_child_nodes_locs Shadow_DOM.remove Shadow_DOM.heap_is_wellformed Shadow_DOM.parent_child_rel"
antichain_inverse: fixes y :: "'a set" assumes "y \<in> {A. finite A \<and> incomparable A}" shows "set_antichain (antichain y) = y"
intros_14: 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 v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" shows "P,E \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>Val v,s\<rangle>"
P1: shows "deg R ((p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> q0) < deg R ((p0 \<oplus>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) \<otimes>\<^bsub>P\<^esub> monom P (q (deg R q)) (deg R q))"
square_free_factorizationI': fixes p :: "'a poly"   and c :: "'a"   and bs :: "('a poly \<times> nat) list" assumes "p = smult c (\<Prod>(a, i)\<leftarrow>bs. a ^ Suc i)"   and "square_free (prod_list (map fst bs))"   and "\<And>b i. (b, i) \<in> set bs \<Longrightarrow> 0 < degree b"   and "p = 0 \<Longrightarrow> c = (0::'a) \<and> bs = []" shows "square_free_factorization p (c, bs)"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "infinite A" shows "F g A = \<^bold>1"
eval_wp_Embed: fixes t :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "wp (Embed t) = t"
inf_glb: fixes A :: "'a set" assumes "A \<subseteq> carrier L" shows "is_glb L (\<Sqinter>A) A"
coeff_closed: fixes p :: "nat \<Rightarrow> 'a"   and R :: "('a, 'b) ring_scheme"   and n :: "nat" assumes "p \<in> carrier (UP R)" shows "coeff (UP R) p n \<in> carrier R"
elims_2: fixes x :: "'b trm"   and xa :: "indices list"   and xb :: "'b trm" assumes "subterm x xa xb" obtains T :: "'b trm" and S :: "'b trm" where "x = T" and "xa = []" and "xb = S__" and "T = S__"   | xc :: "'b trm" and y :: "'b trm" and "next" :: "indices list" and S :: "'b trm" where "x = xc__ \<cdot> y__" and "xa = indices.Left # next__" and "xb = Sa__" and "subterm xc__ next__ Sa__"   | xc :: "'b trm" and y :: "'b trm" and "next" :: "indices list" and S :: "'b trm" where "x = xc \<cdot> y" and "xa = indices.Right # next" and "xb = S" and "subterm y next S"
bind_map_gpv: fixes f :: "'e \<Rightarrow> 'd"   and gpv :: "('e, 'b, 'c) gpv"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) gpv" shows "map_gpv f id gpv \<bind> g = gpv \<bind> g \<circ> f"
card_poly_roots_bound: fixes p :: "'b poly" assumes "p \<noteq> 0" shows "card {x. poly p x = (0::'b)} \<le> degree p"