hcju/isabelleps huggingface dataset:
acquired_by_instrs_mono: fixes A :: "nat set"   and B :: "nat set"   and "is" :: "instr list" assumes "A \<subseteq> B" shows "acquired_by_instrs is A \<subseteq> acquired_by_instrs is B"
proper_interval_set_Compl_aux_fusion_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and proper_interval :: "'b option \<Rightarrow> 'b option \<Rightarrow> bool"   and g1 :: "('b, 'c) generator"   and g2 :: "('b, 'd) generator"   and ao :: "'b option"   and n :: "nat"   and s1 :: "'c"   and s2 :: "'d" shows "proper_intrvl.proper_interval_set_Compl_aux_fusion less proper_interval g1 g2 ao n s1 s2 = proper_intrvl.proper_interval_set_Compl_aux less proper_interval ao n (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
simps: fixes \<alpha> :: "rat"   and n :: "nat"   and m :: "nat"   and upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "LLL_Impl.basis_reduction_main \<alpha> n m upw i state = (if i < m then case LLL_Impl.basis_reduction_step \<alpha> n m upw i state of (upw', i', state') \<Rightarrow> LLL_Impl.basis_reduction_main \<alpha> n m upw' i' state' else state)"
prefs_wf: fixes agents :: "'a set"   and alts :: "'b set"   and R :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and i :: "'a" assumes "pref_profile_wf agents alts R"   and "i \<in> agents" shows "finite_total_preorder_on alts (R i)"
\<sigma>_p: shows "\<forall>i<length (map vars_clause Cs). \<forall>x\<in>map vars_clause Cs ! i. \<sigma> x = (\<sigma>s ! i) x"
fdisjnt_swap: fixes M :: "'c fset"   and N :: "'c fset" assumes "fdisjnt M N" shows "fdisjnt N M"
iszlfm_gen: fixes p :: "fm"   and x :: "real"   and bs :: "real list" assumes "iszlfm p (x # bs)" shows "\<forall>y. iszlfm p (y # bs)"
intro: fixes F :: "'b set"   and \<delta> :: "'b \<Rightarrow> 'c \<Rightarrow> 'b" assumes "\<And>q \<nu>. q \<in> F \<Longrightarrow> \<delta> q \<nu> \<in> F" shows "mojmir_axioms \<delta> F"
Word_compareTr3: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme" assumes "Worder D"   and "Worder E"   and "\<exists>b\<in>carrier E. ord_equiv D (Iod E (segment E b))"   and "\<exists>a\<in>carrier D. ord_equiv (Iod D (segment D a)) E" shows "False"
r_inv: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "x \<otimes>\<^bsub>C\<^esub> inv\<^bsub>C\<^esub> x = \<one>\<^bsub>C\<^esub>"
u: shows "y = \<lbrakk>\<guillemotleft>u\<guillemotright>\<rbrakk>e"
dual_disjunctive: fixes x :: "'a" assumes "x \<in> disjunctive.disjunctive (\<squnion>) (\<squnion>) (*)" shows "x ^ o \<in> conjunctive.conjunctive (\<sqinter>) (\<sqinter>) (*)"
nn_integral_lborel_eq_integral: fixes f :: "'a \<Rightarrow> real" assumes "f \<in> borel_measurable borel"   and "\<And>x. 0 \<le> f x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel < \<infinity>" shows "\<integral>\<^sup>+ x. ennreal (f x) \<partial>lborel = ennreal (integral UNIV f)"
get_child_nodes_locs_impl: shows "get_child_nodes_locs = a_get_child_nodes_locs"
divides_fmsubset: fixes a :: "'a"   and b :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "a divides b"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "fmset G as \<subseteq># fmset G bs"
finite_dimensional_vector_space_axioms: shows "finite_dimensional_vector_space (module_prod.scale (*\<^sub>R) (*\<^sub>R)) Basis_pair"
eq_iff: fixes a :: "'b::order"   and b :: "'b::order" shows "((a::'b::order) = (b::'b::order)) = (b \<le> a \<and> a \<le> b)"
adjust_orthogonal: fixes us :: "'a vec list"   and w :: "'a vec" assumes "set us \<subseteq> carrier_vec n"   and "corthogonal us"   and "w \<in> carrier_vec n"   and "w \<notin> local.span (set us)" shows "corthogonal ((adjuster n w us + w) # us)"
S_compatible_le_scheme: fixes sch' :: "type_scheme"   and sch :: "type_scheme"   and S :: "nat \<Rightarrow> typ" assumes "sch' \<le> sch" shows "$ S sch' \<le> $ S sch"
z: shows "z = (p, Tree\<^sub>m (Unblinded (a, ts)))"
SD_pref_profileD_2: fixes i :: "'agent"   and q :: "'alt pmf"   and p :: "'alt pmf" assumes "i \<in> agents"   and "SD (R i) q p" shows "q \<in> lotteries_on alts"
pr0_simps_3: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b" shows "cod (hfsetcat.pr0 a b) = b"
assms: shows "\<forall>res. \<not> evaluate_whole_prog False env s prog res"
IH: fixes j :: "int" assumes "k < j"   and "j < i"   and "\<And>i. \<lbrakk>k < i; \<And>j. \<lbrakk>k < j; j < i\<rbrakk> \<Longrightarrow> P j\<rbrakk> \<Longrightarrow> P i" shows "P j"
suminf_le: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "\<And>n. f n \<le> g n"   and "summable f"   and "summable g" shows "suminf f \<le> suminf g"
induct: fixes P :: "fm \<Rightarrow> bool"   and a0 :: "fm" assumes "\<And>p q. \<lbrakk>P p; P q\<rbrakk> \<Longrightarrow> P (And p q)"   and "P T"   and "P F"   and "\<And>v. P (Le v)"   and "\<And>v. P (Lt v)"   and "\<And>v. P (Eq v)"   and "\<And>v. P (NEq v)"   and "\<And>v. P (NOT v)"   and "\<And>v va. P (Or v va)"   and "\<And>v va. P (Imp v va)"   and "\<And>v va. P (Iff v va)"   and "\<And>v. P (E v)"   and "\<And>v. P (A v)" shows "P a0"
successE: fixes f :: "'b Heap"   and h :: "heap" assumes "success f h" obtains r :: "'b" and h' :: "heap" where "execute f h = Some (r, h')"
C3: fixes xvec :: "name list"   and \<Psi>' :: "'b"   and U :: "('a, 'b, 'c) psi"   and S :: "('a, 'b, 'c) psi"   and T :: "('a, 'b, 'c) psi" assumes "xvec \<sharp>* \<Psi>'"   and "xvec \<sharp>* U" shows "(\<Psi>', (\<lparr>\<nu>*xvec\<rparr>S \<parallel> T) \<parallel> U, \<lparr>\<nu>*xvec\<rparr>S \<parallel> (T \<parallel> U)) \<in> Rel"
poly_shift_id: shows "poly_shift 0 = (\<lambda>x. x)"
retract_of_trans: fixes S :: "'a set"   and T :: "'a set"   and U :: "'a set" assumes "S retract_of T"   and "T retract_of U" shows "S retract_of U"
maximal_ltl: fixes P :: "'a llist" assumes "maximal_path P" shows "maximal_path (ltl P)"
Suc_in_imp_neq_iMOD: fixes n :: "nat"   and S :: "nat set"   and m :: "nat"   and r :: "nat" assumes "n \<in> S"   and "Suc n \<in> S"   and "m \<noteq> Suc 0" shows "S \<noteq> [ r, mod m ]"
ln_upper_13_def: fixes x :: "real" shows "ln_upper_13 x \<equiv> (353 + 8389 * x + 20149 * x ^ 4 + 50774 * x ^ 3 + 38524 * x\<^sup>2 + 1921 * x ^ 5 + 10 * x ^ 6) * (x - 1) / (70 * (1 + 42 * x + 525 * x ^ 4 + 700 * x ^ 3 + 315 * x\<^sup>2 + 126 * x ^ 5 + 7 * x ^ 6))"
fi_dvd_prod_hc: shows "fi dvd (\<Prod>c\<in>UNIV. h - [:c:])"
hom_mult: fixes x :: "'a poly"   and y :: "'a poly" shows "map_poly to_fract (x * y) = map_poly to_fract x * map_poly to_fract y"
ring_of_sets_axioms_def: fixes M :: "'c set set" shows "ring_of_sets_axioms M \<equiv> \<forall>a b. a \<in> M \<longrightarrow> b \<in> M \<longrightarrow> a \<union> b \<in> M"
ev_imp_shift: fixes \<phi> :: "'d stream \<Rightarrow> bool"   and xs :: "'d stream" assumes "ev \<phi> xs" shows "\<exists>xl xs2. xs = xl @- xs2 \<and> \<phi> xs2"
rank_nullity_theorem: fixes scaleC :: "'a \<Rightarrow> 'c \<Rightarrow> 'c"   and f :: "('a, 'b) vec \<Rightarrow> 'c" assumes "linear (*s) scaleC f" shows "finite_dimensional_vector_space.dimension cart_basis = vec.dim {x. f x = (0::'c)} + vector_space.dim scaleC (range f)"
simps_3: shows "stock_measure BOOL = count_space (range BoolVal)"
left_commute: fixes b :: "'a"   and a :: "'a"   and c :: "'a" shows "b \<sqinter> (a \<sqinter> c) = a \<sqinter> (b \<sqinter> c)"
order_one_triv_iff: shows "(order G = 1) = (carrier G = {\<one>})"
ccw_sorted_scaleR: fixes xs :: "(real \<times> real) list"   and r :: "real" assumes "ccw.sortedP 0 xs"   and "0 < r" shows "ccw.sortedP 0 (map ((*\<^sub>R) r) xs)"
self_finance_charact: fixes Mkt :: "('b, 'c) discrete_market"   and asset :: "'c"   and pf :: "'c \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> real"   and v :: "real" assumes "\<forall>n w. prices Mkt asset (Suc n) w \<noteq> 0"   and "portfolio pf" shows "self_financing Mkt (self_finance Mkt v pf asset)"
monotone_convergence_interval: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> real"   and a :: "'a"   and b :: "'a"   and g :: "'a \<Rightarrow> real" assumes "\<And>k. f k integrable_on cbox a b"   and "\<And>k x. x \<in> cbox a b \<Longrightarrow> f k x \<le> f (Suc k) x"   and "\<And>x. x \<in> cbox a b \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"   and "bounded (range (\<lambda>k. integral (cbox a b) (f k)))" shows "g integrable_on cbox a b \<and> (\<lambda>k. integral (cbox a b) (f k)) \<longlonglongrightarrow> integral (cbox a b) g"
fact_prod_Suc: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "nat" assumes "class.semiring_char_0 one times plus zero" shows "semiring_char_0.fact one plus zero n = semiring_1.of_nat one plus zero (prod Suc {0..<n})"
assms_6: shows "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"
transp_def: fixes r :: "'m \<Rightarrow> 'm \<Rightarrow> bool" shows "transp r = (\<forall>x y z. r x y \<longrightarrow> r y z \<longrightarrow> r x z)"
simps: fixes M :: "('e, 'f, 'd) FSM"   and q :: "'d"   and io :: "('e \<times> 'f) list" shows "io_targets M q io = {target (io || tr) q |tr. path M (io || tr) q \<and> length io = length tr}"
ith_typing_sub: fixes CT :: "nat \<Rightarrow> classDef option"   and \<Gamma> :: "nat \<Rightarrow> nat option"   and es :: "exp list"   and h :: "exp"   and t :: "exp list"   and Cs :: "nat list"   and h' :: "exp"   and Ci' :: "nat" assumes "CT;\<Gamma> \<turnstile>+ (es @ h # t) : Cs"   and "CT;\<Gamma> \<turnstile> h' : Ci'"   and "CT \<turnstile> Ci' <: Cs ! length es" shows "\<exists>Cs'. CT;\<Gamma> \<turnstile>+ (es @ h' # t) : Cs' \<and> CT \<turnstile>+ Cs' <: Cs"
l3_trans_def: shows "l3_trans \<equiv> \<Union>M N X Rb Ra A B K. (l3_step1 Ra A B \<union> l3_step2 Rb A B N X \<union> l3_step3 Ra A B N X \<union> l3_step4 Rb A B N X \<union> l3_dy M \<union> l3_lkr_others A \<union> l3_lkr_after A \<union> l3_skr Ra K)\<^sup>="
rb_aux_inv2_D2: 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 "j < length fs"   and "Inr j \<notin> set ps" shows "fs ! j \<in> ideal (rep_list ` set (filter (\<lambda>b. component_of_term (lt b) < Suc j) bs))"
skl1_inv5D: fixes Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and x :: "skl1_state"   and gny :: "msg"   and Nr :: "msg" assumes "guessed_runs Ra = \<lparr>role = Init, owner = A, partner = B\<rparr>"   and "in_progressS (progress x Ra) {xnx, xni, xnr, xgnx, xgny, xsk, xEnd}"   and "sklvl1.guessed_frame Ra xgny = Some gny"   and "sklvl1.guessed_frame Ra xnr = Some Nr"   and "can_signal x A B"   and "x \<in> skl1_inv5" shows "0 < signalsResp x (Running A B \<langle>NonceF (Ra $ ni), Nr, Exp gny (NonceF (Ra $ nx))\<rangle>)"
uminus_mono_iff: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(\<partial> \<circ> f = \<partial> \<circ> g) = (f = g)"
simps_2: shows "interpolation_poly Lagrange = lagrange_interpolation_poly"
dgrad_p_set_closed_monom_mult: fixes d :: "'a \<Rightarrow> nat"   and t :: "'a"   and m :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and c :: "'c" assumes "dickson_grading d"   and "d t \<le> m"   and "p \<in> punit.dgrad_p_set d m" shows "local.punit.monom_mult c t p \<in> punit.dgrad_p_set d m"
step2: shows "step p \<tau>' = map (\<lambda>pc. (pc, transf p \<tau>')) (rev (sorted_list_of_set (succs p)))"
comp_fun_idem_inf: shows "comp_fun_idem (\<sqinter>)"
fmap_ext: fixes m :: "('d, 'e) fmap"   and n :: "('d, 'e) fmap" assumes "\<And>x. fmlookup m x = fmlookup n x" shows "m = n"
Rep_uexpr: fixes x :: "('d, 'c) uexpr" shows "\<lbrakk>x\<rbrakk>\<^sub>e \<in> UNIV"
PAR: fixes xs :: "'varSort"   and s :: "'sort"   and us :: "'varSort" shows "(|vars xs| <o |UNIV| \<or> finite (vars xs)) \<and> (|terms s| <o |UNIV| \<or> finite (terms s)) \<and> (|abs (us, s)| <o |UNIV| \<or> finite (abs (us, s))) \<and> (\<forall>X\<in>terms s. wls s X) \<and> (\<forall>A\<in>abs (us, s). wlsAbs (us, s) A) \<and> (|envs| <o |UNIV| \<or> finite envs) \<and> Ball envs wlsEnv"
R_finrank_def: fixes M :: "'m set" shows "R_finrank M = (\<exists>n. \<forall>ms. set ms \<subseteq> M \<and> R_lin_independent ms \<longrightarrow> length ms \<le> n)"
find_Some_iff: fixes P :: "'d \<Rightarrow> bool"   and xs :: "'d list"   and x :: "'d" shows "(find P xs = Some x) = (\<exists>i<length xs. P (xs ! i) \<and> x = xs ! i \<and> (\<forall>j<i. \<not> P (xs ! j)))"
of_nat_less_hmset: fixes M :: "nat"   and N :: "nat" shows "(of_nat M < of_nat N) = (M < N)"
scc_ofI_apath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "S.apath u p v" shows "u \<in> S.scc_of v"
f3: shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> ((t |\<^bsub>rel\<^esub> u) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (v |\<^bsub>rel\<^esub> w)) = (r \<otimes> (t \<otimes> v) |\<^bsub>rel\<^esub> s \<otimes> (u \<otimes> w))"
perms_def: shows "perms = {\<pi>. \<pi> permutes Roots}"
n: shows "n \<le> length D \<and> E = take n D"
effect_to_assignments_construction_from_function_graph_2: fixes op :: "'a strips_operator" assumes "set (add_effects_of op) \<inter> set (delete_effects_of op) = {}" shows "effect_to_assignments op = map (\<lambda>v. (v, if ListMem v (delete_effects_of op) then False else True)) (add_effects_of op @ delete_effects_of op)"
from_nat_not_eq: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "nat"   and b :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "a \<noteq> mod_type.to_nat Rep b"   and "a < CARD('b)" shows "mod_type.from_nat Abs a \<noteq> b"
poly_weak_mono_def: fixes p :: "('b monom \<times> 'c) list"   and v :: "'b" shows "poly_weak_mono p v \<equiv> \<forall>\<alpha> \<beta>. (\<forall>x. v \<noteq> x \<longrightarrow> \<alpha> x = \<beta> x) \<longrightarrow> pos_assign \<beta> \<longrightarrow> \<beta> v \<le> \<alpha> v \<longrightarrow> eval_poly \<beta> p \<le> eval_poly \<alpha> p"
set_contI: fixes f :: "'c \<Rightarrow> 'd set" assumes "\<And>Y. chain Y \<Longrightarrow> f (\<Squnion>i. Y i) = \<Union> (f ` range Y)" shows "cont f"
map\<^sub>0_simps_1: fixes a :: "'a" assumes "B.obj a" shows "obj (UP.map\<^sub>0 a)"
trivial_RModuleHom: fixes smult' :: "('f, 'g) aezfun \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>r\<in>FG. smult' r (0::'b) = (0::'b)" shows "RModuleHom FG (\<cdot>) V smult' 0"
subsumes_in_all_contexts_directly_subsumes: fixes t2 :: "transition"   and t1 :: "transition"   and e1 :: "((nat \<times> nat) \<times> transition) fset"   and e2 :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and s' :: "nat" assumes "\<And>c. subsumes t2 c t1" shows "directly_subsumes e1 e2 s s' t2 t1"
healthy_wp_PC: fixes f :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and g :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and P :: "'b \<Rightarrow> real" assumes "healthy (wp f)"   and "healthy (wp g)"   and "unitary P" shows "healthy (wp (f \<^bsub>P\<^esub>\<oplus> g))"
lincomb_def: fixes a :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "lincomb a A = (\<Oplus>\<^bsub>V\<^esub>v\<in>A. a v \<odot>\<^bsub>V\<^esub> v)"
bounded: shows "\<exists>K. \<forall>a b. norm (a ** b) \<le> norm a * norm b * K"
valid_proj_action_valid_succ: fixes h :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and PROB :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set"   and vs :: "'c set"   and s :: "('c, 'd) fmap" assumes "h \<in> prob_proj PROB vs"   and "s \<in> valid_states PROB" shows "state_succ s h \<in> valid_states PROB"
case_distrib: fixes h :: "'b \<Rightarrow> 'a"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c list \<Rightarrow> 'b"   and list :: "'c list" shows "h (case list of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = (case list of [] \<Rightarrow> h f1 | x1 # x2 \<Rightarrow> h (f2 x1 x2))"
fold_types_def: fixes f :: "typ \<Rightarrow> 'a \<Rightarrow> 'a" shows "fold_types f = fold_term_types (\<lambda>x. f)"
add: fixes b1 :: "'b"   and b2 :: "'b" shows "g' (b1 + b2) = g' b1 + g' b2"
ss_hdw_inA: shows "ss ! hd w \<in> A"
lexn_n: fixes n :: "nat"   and x :: "'a"   and xs :: "'a list"   and y :: "'a"   and ys :: "'a list"   and r :: "('a \<times> 'a) set" assumes "0 < n" shows "((x # xs, y # ys) \<in> lexn r n) = ((length xs = n - 1 \<and> length ys = n - 1) \<and> ((x, y) \<in> r \<or> x = y \<and> (xs, ys) \<in> lexn r (n - 1)))"
unbounded: fixes A :: "ordinal set"   and x :: "ordinal" assumes "normal_set A" shows "\<exists>y\<in>A. x < y"
simps_1: fixes x1 :: "'b list"   and x2 :: "'b"   and y1 :: "'b list"   and y2 :: "'b" shows "(Infer x1 x2 = Infer y1 y2) = (x1 = y1 \<and> x2 = y2)"
finite_prefix: fixes n :: "nat" shows "finite {i. f i \<le> n}"
lemma_st_part_lub: fixes x :: "real star" assumes "x \<in> HFinite" shows "\<exists>t. isLub \<real> {s \<in> \<real>. s < x} t"
M: fixes R :: "real" assumes "0 \<le> R" shows "Ca * R ^ DIM('a) \<le> 1 + Cb * (lambda * R + C + norm (f (0::'a))) ^ DIM('b)"
component_prot_def: fixes n :: "'a"   and P :: "('a strand_label \<times> 'b) list set" shows "component_prot n P \<equiv> \<forall>l\<in>P. \<forall>s\<in>set l. is_LabelN n s \<or> is_LabelS s"
intro: fixes \<eta> :: "'c \<Rightarrow> 'd"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and G' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d"   and L :: "'c set" assumes "group_homomorphism \<eta> G composition unit G' composition' unit'"   and "normal_subgroup L G composition unit"   and "normal_subgroup_in_kernel_axioms \<eta> G unit' L" shows "normal_subgroup_in_kernel \<eta> G composition unit G' composition' unit' L"
Sup_less_comp: fixes P :: "'d \<Rightarrow> 'c \<Rightarrow> 'b"   and w :: "'d"   and S :: "'a \<Rightarrow> 'c" shows "Sup_less P w \<circ> S = Sup_less (\<lambda>w. P w \<circ> S) w"
simple_bochner_integral_partition: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'e"   and v :: "'e \<Rightarrow> 'd" assumes "simple_bochner_integrable M f"   and "simple_function M g"   and "\<And>x y. \<lbrakk>x \<in> space M; y \<in> space M; g x = g y\<rbrakk> \<Longrightarrow> f x = f y"   and "\<And>x. x \<in> space M \<Longrightarrow> f x = v (g x)" shows "simple_bochner_integral M f = (\<Sum>y\<in>g ` space M. Sigma_Algebra.measure M {x \<in> space M. g x = y} *\<^sub>R v y)"
awalk_verts_reachable_from: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and w :: "'a" assumes "G.awalk u p v"   and "w \<in> set (G.awalk_verts u p)" shows "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w"
is_rbt_fold_rbt_delete: fixes t :: "('a, 'c) rbt"   and xs :: "'a list" assumes "is_rbt t" shows "is_rbt (fold rbt_delete xs t)"
has_field_decl_above: fixes P :: "'a prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C has F,b:T in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
elims: fixes x :: "'d stream \<Rightarrow> 'c"   and xa :: "'d stream"   and y :: "'c" assumes "nxt x xa = y" obtains \<phi> :: "'d stream \<Rightarrow> 'c" and xs :: "'d stream" where "x = \<phi>" and "xa = xs" and "y = \<phi> (stl xs)"
reachable_props_2: fixes \<phi>' :: "'val"   and g :: "'g"   and \<phi> :: "'val" assumes "\<phi>' \<in> reachable g \<phi>" shows "unnecessaryPhi g \<phi>'"
assms_1: fixes X :: "'a quot"   and Y :: "'b quot" shows "f X Y \<equiv> g (pick X) (pick Y)"
intros_2: fixes X :: "msg"   and Y :: "msg"   and H :: "msg set" assumes "\<langle>X, Y\<rangle> \<in> analz H" shows "X \<in> analz H"
flipped_reflections_in_lconjseq: fixes ss :: "'w list" assumes "ss \<in> lists S" shows "order.greater_eq (set (lconjseq ss)) (flipped_reflections (sum_list ss))"
continuous_map_nsphere_projection: fixes n :: "nat"   and k :: "nat" shows "continuous_map (nsphere n) euclideanreal (\<lambda>x. x k)"
iMOD_0_icard: fixes r :: "nat" shows "icard [ r, mod 0 ] = eSuc 0"
tfn: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and x :: "'a"   and \<tau> :: "PreSimplyTyped.type"   and A :: "'a trm"   and \<sigma> :: "PreSimplyTyped.type" assumes "\<Gamma>(x \<mapsto> \<tau>) \<turnstile> A : \<sigma>" shows "\<Gamma> \<turnstile> Fn x \<tau> A : TArr \<tau> \<sigma>"
awalk_def: fixes G :: "('e, 'f) pre_digraph"   and u :: "'e"   and p :: "'f list"   and v :: "'e" shows "pre_digraph.awalk G u p v \<equiv> u \<in> verts G \<and> set p \<subseteq> arcs G \<and> pre_digraph.cas G u p v"
wp3Conc: fixes a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "wellformed_policy3Pr (a # p)" shows "wellformed_policy3Pr p"
\<epsilon>: shows "\<guillemotleft>\<epsilon> : f \<star> f\<^sub>a \<Rightarrow> trg f\<guillemotright> \<and> \<guillemotleft>\<epsilon> : trg f \<rightarrow> trg f\<guillemotright> \<and> arr \<epsilon> \<and> src \<epsilon> = trg f \<and> trg \<epsilon> = trg f \<and> local.dom \<epsilon> = f \<star> f\<^sub>a \<and> cod \<epsilon> = trg f"
P2_ideal_view_unfold: fixes x0 :: "'a"   and x1 :: "'a"   and \<sigma> :: "bool"   and z :: "'f"   and \<A> :: "(bool \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<times> 'a \<times> 'a \<times> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a \<times> 'a) \<times> nat) \<times> 'g) spmf) \<times> ('a \<times> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> 'e spmf)" shows "mal_def.ideal_view_2 (x0, x1) \<sigma> z (P2_S1, P2_S2) \<A> = P2_ideal_model (x0, x1) \<sigma> z \<A>"
simps_2: fixes dimen :: "nat"   and c :: "'a mat" shows "matr_option dimen (Some c) = c"
is_universal: fixes J :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and D :: "'a \<Rightarrow> 'b"   and a :: "'b"   and \<chi> :: "'a \<Rightarrow> 'b"   and a' :: "'b"   and \<chi>' :: "'a \<Rightarrow> 'b" assumes "limit_cone J C D a \<chi>"   and "Limit.cone J C D a' \<chi>'" shows "\<exists>!f. partial_magma.in_hom C f a' a \<and> (\<lambda>\<chi>\<in>{\<chi>. Limit.cone J C D (partial_magma.cod C f) \<chi>}. \<lambda>j. if partial_magma.arr J j then C (\<chi> j) f else partial_magma.null C) \<chi> = \<chi>'"
mono: fixes x :: "'b" shows "set.mono_body (\<lambda>f. U (F (C f)) x)"
bin_nth_of_bl_aux: fixes bl :: "bool list"   and w :: "int"   and n :: "nat" shows "bit (bl_to_bin_aux bl w) n = (n < length bl \<and> rev bl ! n \<or> length bl \<le> n \<and> bit w (n - length bl))"
awalk_verts_conv: fixes G :: "('b, 'c) pre_digraph"   and u :: "'b"   and p :: "'c list" shows "pre_digraph.awalk_verts G u p = (if p = [] then [u] else map (tail G) p @ [head G (last p)])"
iso_arcs_inv_iso: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism"   and a :: "'d" assumes "iG.digraph_isomorphism hom"   and "a \<in> arcs (app_iso hom G)" shows "iso_arcs (iG.inv_iso hom) (iso_arcs hom a) = a"
assert_image: fixes A :: "'c Assertion set" shows "assert ` A \<subseteq> assertion"
no_trailing_strip_while: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "no_trailing P (strip_while P xs)"
fpxs_of_fls_add: fixes f :: "'b fls"   and g :: "'b fls" shows "fpxs_of_fls (f + g) = fpxs_of_fls f + fpxs_of_fls g"
reachabilityInvariant_def: fixes cfile :: "CFile" shows "reachabilityInvariant cfile \<equiv> nextFreeBlockInvariant cfile \<and> unallocatedBlocksInvariant cfile \<and> lastBlockInvariant cfile"