hcju/isabelleps huggingface dataset:
argmaxLemma: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "argmax f A = {x \<in> A. f x = Max (f ` A)}"
sat_Since_rec: fixes \<sigma> :: "(char list \<times> 'b list) Trace.trace"   and v :: "'b list"   and i :: "nat"   and \<phi> :: "'b formula"   and I :: "\<I>"   and \<psi> :: "'b formula" shows "sat \<sigma> v i (Since \<phi> I \<psi>) = (mem 0 I \<and> sat \<sigma> v i \<psi> \<or> 0 < i \<and> enat (\<Delta> \<sigma> i) \<le> right I \<and> sat \<sigma> v i \<phi> \<and> sat \<sigma> v (i - 1) (Since \<phi> (subtract (\<Delta> \<sigma> i) I) \<psi>))"
thenOrdering_simps_4: fixes y :: "Ordering" shows "thenOrdering\<cdot>\<bottom>\<cdot>y = \<bottom>"
assms: shows "\<forall>p\<in>C. \<forall>q\<in>C. \<forall>r\<in>C. \<forall>\<alpha>\<in>{0<..1}. p \<succeq>[P] q = mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
cring_simprules_12: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "cring R"   and "x \<in> carrier R" shows "\<one>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = x"
YFtorFtor': fixes C :: "('d, 'e, 'f) LSCategory_scheme" assumes "LSCategory C" shows "FunctorM (YFtor' C)"
distinct_find_indices: fixes x :: "'b"   and xs :: "'b list" shows "distinct (find_indices x xs)"
chine_assoc'_props_2: shows "doms.Prj\<^sub>1\<^sub>1 \<cdot> doms.chine_assoc' = doms.Prj\<^sub>1"
ring_record_simps_11: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
sin_le_zero: fixes x :: "real" assumes "pi \<le> x"   and "x < 2 * pi" shows "sin x \<le> 0"
domintros_7: fixes v :: "exp0" shows "is_value_dom (Raise v)"
tllist_of_llist_inverse: fixes b :: "'g"   and xs :: "'f llist" shows "llist_of_tllist (tllist_of_llist b xs) = xs"
safe_combined2_def: fixes c :: "('b, 't, 'loc) Combined.configuration" shows "safe_combined2 c \<equiv> \<forall>loc1 loc2 t s p1 p2. 0 < zcount (c_caps (exchange_config c) p1) (loc1, t) \<and> s \<in>\<^sub>A path_summary loc1 loc2 \<and> init c p2 \<longrightarrow> (\<exists>t'. t' \<in>\<^sub>A frontier (c_imp (prop_config c p2) loc2) \<and> t' \<le> results_in t s)"
ANR_rel_frontier_convex: fixes S :: "'a set" assumes "bounded S"   and "convex S" shows "ANR (rel_frontier S)"
keyh: shows "\<exists>w. \<forall>y>w. a * y\<^sup>2 < a * (- b / a * y + - c / a)"
map_vector_inj: fixes v :: "('a, 'd) Finite_Cartesian_Product.vec"   and w :: "('a, 'd) Finite_Cartesian_Product.vec" shows "(map_vector hom v = map_vector hom w) = (v = w)"
abs_nat_pp_pp_def: fixes t :: "(nat, nat) pp" shows "abs_nat_pp t = pp_of_fun (\<lambda>n. abs_nat (lookup_pp t (rep_nat n)))"
inj_on_Inr: fixes A :: "V set" shows "inj_on ZFC_Cardinals.Inr A"
simps_3: fixes x :: "name"   and \<theta> :: "(name \<times> trm) list"   and s :: "trm" assumes "x \<sharp> \<theta>" shows "\<theta><\<Lambda> x . s> = \<Lambda> x . \<theta>
"
Assertion_wp: fixes x :: "'b"   and p :: "'b Assertion" shows "{\<cdot> wp x p } = x * {\<cdot> p } * \<top> \<sqinter> (1::'b)"
rwof_eq_DFS_invar: shows "rwof init cond step = DFS_invar G param"
SumAssert2: fixes \<Psi> :: "'b"   and Q :: "('a, 'b, 'c) psi"   and Rs :: "('a, 'b, 'c) residual"   and \<phi> :: "'c"   and P :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> Q \<longmapsto> Rs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded Q" shows "\<Psi> \<rhd> P \<oplus>\<^sub>\<phi> Q \<longmapsto> Rs"
TER_plus_web'': assumes "\<And>x. x \<notin> RF (TER g) \<Longrightarrow> d_OUT g x = 0" shows "\<E> (TER f \<union> TER g) \<subseteq> TER (f \<frown> g)"
equiv_type: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "equiv A r" shows "r \<subseteq> A \<times> A"
s: shows "\<sigma> = fst (fst \<Gamma> ! n)"
lsl_lax: fixes x :: "'b"   and y :: "'b" shows "\<nu> x \<cdot> \<nu> y \<le> \<nu> (x \<cdot> y)"
A3: fixes x :: "'a list" assumes "x \<in> set zs" shows "x \<noteq> []"
frontier_translation: fixes a :: "'a"   and s :: "'a set" shows "frontier ((+) a ` s) = (+) a ` frontier s"
beta_into_equiv: fixes s :: "dB"   and t :: "dB" assumes "s \<rightarrow>\<^sub>\<beta> t" shows "s \<leftrightarrow> t"
r_consistent: fixes r_psi :: "recf"   and \<psi> :: "nat \<Rightarrow> nat \<Rightarrow> nat option"   and i :: "nat"   and e :: "nat" assumes "recfn 2 r_psi"   and "\<forall>x y. eval r_psi [x, y] = \<psi> x y" shows "eval (r_consistent r_psi) [i, e] = consistent \<psi> i e"
None_eq_map_option_iff: fixes f :: "'d \<Rightarrow> 'b"   and x :: "'d option" shows "(None = map_option f x) = (x = None)"
emeasure_eq_0: fixes N :: "'c set"   and M :: "'c measure"   and K :: "'c set" assumes "N \<in> sets M"   and "emeasure M N = 0"   and "K \<subseteq> N" shows "emeasure M K = 0"
le_approx2: fixes P :: "'c process"   and Q :: "'c process"   and s :: "'c event list"   and X :: "'c event set" assumes "P \<sqsubseteq> Q"   and "s \<notin> D P" shows "((s, X) \<in> F Q) = ((s, X) \<in> F P)"
assms_2: fixes s :: "'a" assumes "I s"   and "b s" shows "f s \<le> SPEC (\<lambda>s'. (s', s) \<in> V)"
unsigned_of_int: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and k :: "int" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_int k) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('l) k))"
negligible: fixes S :: "'c set" shows "negligible S = (\<forall>T. (indicat_real S has_integral 0) T)"
approx_HFinite: fixes x :: "'a star"   and y :: "'a star" assumes "x \<in> HFinite"   and "x \<approx> y" shows "y \<in> HFinite"
measure_n_c: fixes X :: "char list set" assumes "finite X" shows "{(c, c \<triangle>\<^sub>c c') |c c'. strip c = strip c' \<and> c \<in> Com X \<and> c' \<in> Com X \<and> c' \<sqsubseteq> c \<and> \<not> c \<sqsubseteq> c \<triangle>\<^sub>c c'}\<inverse> \<subseteq> measure (m_c (n_o (n_st n_ivl X)))"
trap_arithmetic: fixes a :: "real"   and b :: "real" assumes "(a, b) \<in> trapG1" shows "p1d (a, b) (gx a b, gy a b) \<le> (- (27 / 25) - a\<^sup>2 + 2 * a * b) * p1 a b"
rename_gens_cancels_to: fixes f :: "bool \<Rightarrow> bool"   and l :: "(bool \<times> 'c) list"   and l' :: "(bool \<times> 'c) list"   and g :: "'c \<Rightarrow> 'd" assumes "inj f"   and "cancels_to l l'" shows "cancels_to (map (map_prod f g) l) (map (map_prod f g) l')"
update_cons_nil_env: fixes x :: "'h"   and opt :: "('f, 'g, 'h) env option"   and b :: "'g"   and es :: "'h \<Rightarrow> ('f, 'g, 'h) env option" shows "update [x] opt (Env b es) = Env b (es(x := opt))"
compat_low: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms\<^sub>1 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>1 :: "'a \<Rightarrow> 'b"   and cms\<^sub>2 :: "('c \<times> (Mode \<Rightarrow> 'a set)) list"   and mem\<^sub>2 :: "'a \<Rightarrow> 'b"   and mems :: "(('a \<Rightarrow> 'b) \<times> ('a \<Rightarrow> 'b)) list"   and i :: "nat"   and x :: "'a" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.makes_compatible dma \<C>_vars eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i < length cms\<^sub>1"   and "x \<in> sifum_security_init.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "dma mem\<^sub>1 x = Low"
rel_spmfI: fixes pq :: "('b \<times> 'd) spmf"   and P :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'b spmf"   and q :: "'d spmf" assumes "\<And>x y. (x, y) \<in> set_spmf pq \<Longrightarrow> P x y"   and "map_spmf fst pq = p"   and "map_spmf snd pq = q" shows "rel_spmf P p q"
insert_absorb: fixes a :: "'d"   and A :: "'d set" assumes "a \<in> A" shows "insert a A = A"
netmap_subnets_same: fixes s1 :: "'c net_state"   and i :: "nat"   and x :: "'c option"   and s2 :: "'c net_state" assumes "netmap s1 i = x"   and "netmap s2 i = x" shows "netmap (SubnetS s1 s2) i = x"
induct: fixes P :: "'b tree23 \<Rightarrow> 'b \<Rightarrow> 'b upD \<Rightarrow> bool"   and a0 :: "'b tree23"   and a1 :: "'b"   and a2 :: "'b upD" assumes "\<And>t1 a t2. P t1 a (TD t2)"   and "\<And>t1 b t2 a t3. P \<langle>t1, b, t2\<rangle> a (UF t3)"   and "\<And>t1 b t2 c t3 a t4. P \<langle>t1, b, t2, c, t3\<rangle> a (UF t4)"   and "\<And>b v. P \<langle>\<rangle> b (UF v)" shows "P a0 a1 a2"
condition1_part5: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and A :: "(('d, 'e) vec, 'f) vec"   and k :: "nat"   and a :: "'f"   and b :: "'f"   and m :: "'f" assumes "is_bezout_ext bezout"   and "echelon_form_upt_k A k"   and "is_zero_row_upt_k a (Suc k) (fst (echelon_form_of_column_k bezout (A, if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) k))"   and "a < b"   and "mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) < m"   and "A $ m $ mod_type_class.from_nat k \<noteq> (0::'d)"   and "(if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<noteq> nrows A"   and "k < ncols A" shows "is_zero_row_upt_k b (Suc k) (bezout_iterate (interchange_rows A (mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1)) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'d) \<and> mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1) \<le> n)) (nrows A - Suc 0) (mod_type_class.from_nat (if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1)) (mod_type_class.from_nat k) bezout)"
T_eq: shows "T = filter_a2_map I ts (tp - len) a2_map"
assms_1: shows "subset_dfs A (nfa_startnode A) = (bd, ls)"
simplicial_cone_def: fixes p :: "nat"   and v :: "nat \<Rightarrow> real" shows "simplicial_cone p v \<equiv> frag_extend (frag_of \<circ> simplex_cone p v)"
run_properties_1: fixes \<phi> :: "'a ltl"   and i :: "nat" shows "fst (run (\<delta>\<^sub>\<A> \<Sigma>) (\<iota>\<^sub>\<A> \<phi>) w i) = foldl \<up>af (Abs \<phi>) (prefix i w)"
mono: shows "\<forall>y z. y \<in> S \<and> y \<le> z \<longrightarrow> z \<in> S"
simps_5: fixes P :: "pol"   and i :: "nat"   and Q :: "pol"   and c :: "int" shows "PX P i Q \<langle>+\<rangle> Pc c = PX P i (Q \<langle>+\<rangle> Pc c)"
runiq_basic: fixes R :: "('c \<times> 'd) set" shows "runiq R = (\<forall>x y y'. (x, y) \<in> R \<and> (x, y') \<in> R \<longrightarrow> y = y')"
poly_degE: fixes p :: "('b \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'd" assumes "p \<noteq> 0" obtains t :: "'b \<Rightarrow>\<^sub>0 'c" where "t \<in> keys p" and "poly_deg p = deg_pm t"
measurable_differentiable_image: fixes S :: "(real, 'a) vec set"   and f :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec"   and f' :: "(real, 'a) vec \<Rightarrow> (real, 'a) vec \<Rightarrow> (real, 'a) vec" assumes "S \<in> sets lebesgue"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "(\<lambda>x. \<bar>det (matrix (f' x))\<bar>) integrable_on S" shows "f ` S \<in> lmeasurable"
suminf_mult: fixes f :: "nat \<Rightarrow> 'a"   and c :: "'a" assumes "summable f" shows "(\<Sum>n. c * f n) = c * suminf f"
eq: shows "f \<equiv> C (coset.fixp_fun (\<lambda>f. U (F (C f))))"
pdevs_apply_uminus_pdevs: fixes x :: "'b pdevs" shows "pdevs_apply (uminus_pdevs x) = - pdevs_apply x"
RootEx_Root: fixes F :: "('a, 'b, 'c) seqauto set"   and G :: "'a \<Rightarrow> ('a, 'b, 'c) seqauto set option" assumes "RootEx F G" shows "Root F G \<in> F"
inducts: fixes P :: "trm \<Rightarrow> bool"   and trm :: "trm" assumes "\<And>var. P (Var var)"   and "\<And>trm1 trm2. \<lbrakk>P trm1; P trm2\<rbrakk> \<Longrightarrow> P (App trm1 trm2)"   and "\<And>var trm. P trm \<Longrightarrow> P (Lam [var].trm)"   and "\<And>var trm1 trm2. \<lbrakk>P trm1; P trm2\<rbrakk> \<Longrightarrow> P (Let var be trm2 in trm1)" shows "P trm"
induct: fixes P :: "instr list \<Rightarrow> (int \<times> (char list \<Rightarrow> int) \<times> int list) list \<Rightarrow> bool"   and a0 :: "instr list"   and a1 :: "(int \<times> (char list \<Rightarrow> int) \<times> int list) list" assumes "\<And>P cf cf' cfs. P P (cf' # cfs) \<Longrightarrow> P P (cf # cf' # cfs)"   and "\<And>P. P P []"   and "\<And>P v. P P [v]" shows "P a0 a1"
intro: fixes s :: "nat \<Rightarrow> nat option" assumes "s \<in> \<R>"   and "learn_bc \<phi> U\<^sub>0 s" shows "r1_bc s"
FF_def: shows "G.FF \<equiv> \<lambda>\<mu>\<nu>. if C.VV.arr \<mu>\<nu> then (G (fst \<mu>\<nu>), G (snd \<mu>\<nu>)) else D.VV.null"
acyclic_class_hierarchy_def: fixes P :: "'b prog" shows "acyclic_class_hierarchy P = (\<forall>(C, D, fs, ml)\<in>set (classes P). C \<noteq> Object \<longrightarrow> \<not> P \<turnstile> D \<preceq>\<^sup>* C)"
simps: fixes V :: "nat set"   and Q :: "(nat set \<times> 'b option list set) set"   and st :: "nat set"   and t :: "'b option list" shows "New_max.newQuery V Q (st, t) = (\<lambda>tab. New_max.projectTable V (New_max.semiJoin tab (st, t))) ` Q"
that: fixes x :: "complex" assumes "f (f x) = x" shows "thesis"
elims_3: fixes x :: "'a infty"   and xa :: "'a infty" assumes "\<not> x < xa" obtains uu_ :: "'a infty" where "x = Infty" and "xa = uu_"   | a :: "'a" and b :: "'a" where "x = Num a" and "xa = Num b" and "\<not> a < b"
pcr_cr_eq: shows "pcr_list (=) = cr_list"
hd_prod: fixes u :: "'b list"   and v :: "'b list" assumes "u \<noteq> \<epsilon>" shows "(u \<cdot> v) ! 0 = u ! 0"
nth_list_of_vec: fixes i :: "nat"   and v :: "'b vec" assumes "i < dim_vec v" shows "list_of_vec v ! i = v $ i"
simps_5: fixes x51 :: "expr"   and x52 :: "expr"   and y51 :: "expr"   and y52 :: "expr" shows "(x51 \<cdot> x52 = y51 \<cdot> y52) = (x51 = y51 \<and> x52 = y52)"
reduce_root_def: fixes a :: "'a"   and p :: "'a poly" shows "reduce_root a p = p \<circ>\<^sub>p [:0::'a, a:]"
ordinal_times_1: fixes x :: "ordinal" shows "x * oSuc 0 = x"
assms_1: shows "zb \<noteq> \<infinity>\<^sub>h"
arccos_cos_eq_abs: fixes \<theta> :: "real" assumes "\<bar>\<theta>\<bar> \<le> pi" shows "arccos (cos \<theta>) = \<bar>\<theta>\<bar>"
gt_imp_ge: fixes gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and default :: "'c"   and x :: "'c"   and y :: "'c" assumes "order_pair gt default"   and "gt x y" shows "y \<le> x"
many_strongly_connected_iff_5: fixes x :: "'a" shows "many_strongly_connected x = (x\<^sup>\<star> ; x\<^sup>T \<le> x\<^sup>+)"
sas_plus_formalism_and_induced_strips_formalism_are_equally_expressive_i_b: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" assumes "is_valid_problem_sas_plus \<Psi>" shows "inj_on (map (sasp_op_to_strips \<Psi>)) (bounded_solution_set_sas_plus' \<Psi> k)"
i: fixes i :: "nat"   and x :: "nat" shows "eval (Cn 2 r_delay [recf.Id 2 0, recf.Id 2 1]) [i, x] = delay i x"
module_pair_axioms: shows "module_pair (*\<^sub>R) (*\<^sub>R)"
distinct_257: fixes x11a :: "floatarith"   and x151 :: "floatarith"   and x152 :: "nat" shows "Sqrt x11a \<noteq> Power x151 x152"
simps_9: fixes uu :: "'b ltln set" shows "false\<^sub>n[uu]\<^sub>\<nu> = false\<^sub>n"
P_Cons_5: shows "fps\<langle>l\<rangle>\<^sub>? = \<bottom>"
fps_hensel_aux_1: assumes "F' $ 0 = g * h"   and "\<forall>n>0. degree (F' $ n) < degree (F' $ 0)" shows "F' = hensel_fpxs1 * hensel_fpxs2"
cases: fixes a1 :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and a2 :: "'a expr list"   and a3 :: "'a expr list" assumes "a1 \<turnstile>\<^sub>L a2 \<Rrightarrow> a3" obtains \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" where "a1 = \<Gamma>__" and "a2 = []" and "a3 = []"   | \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type" and x :: "'a expr" and y :: "'a expr" and xs :: "'a expr list" and ys :: "'a expr list" where "a1 = \<Gamma>" and "a2 = x # xs" and "a3 = y # ys" and "\<Gamma> \<turnstile> x \<Rrightarrow> y" and "\<Gamma> \<turnstile>\<^sub>L xs \<Rrightarrow> ys"
reindex: fixes h :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and g :: "'c \<Rightarrow> 'a" assumes "inj_on h A" shows "sum g (h ` A) = sum (g \<circ> h) A"
MakeCatCod: fixes f :: "'g"   and C :: "('h, 'g, 'i) Category_scheme" assumes "f \<in> mor\<^bsub>C\<^esub>" shows "cod\<^bsub>C\<^esub> f = cod\<^bsub>MakeCat C\<^esub> f"
simps_2: fixes xs :: "'b list" shows "proper_interval None (Some xs) = (xs \<noteq> [])"
att_or_empty_back: fixes s' :: "'a set"   and s'' :: "'a set" assumes "s' \<subseteq> s''" shows "\<turnstile>[] \<oplus>\<^sub>\<or>\<^bsup>(s', s'')\<^esup>"
cycle_restrict: fixes p :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "permutation p"   and "b \<in> set (support p a)" shows "p b = cycle_of_list (support p a) b"
std_normal_moment_abs_odd: shows "has_bochner_integral lborel (\<lambda>x. std_normal_density x * \<bar>x\<bar> ^ (2 * k + 1)) (sqrt (2 / pi) * 2 ^ k * fact k)"
disj: shows "q = 0 \<or> b \<noteq> 0"
simps_5: fixes vb :: "('b, 'c) Combinators"   and vc :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators list" shows "noDenyAll1 (vb \<oplus> vc # va) = noDenyAll (vb \<oplus> vc # va)"
l_set_disconnected_nodes_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_disconnected_nodes_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs get_shadow_root get_shadow_root_locs get_host get_host_locs \<equiv> l_set_disconnected_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M set_disconnected_nodes set_disconnected_nodes_locs \<and> l_get_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_shadow_root get_shadow_root_locs \<and> l_get_host\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_shadow_root get_shadow_root_locs get_host get_host_locs type_wf"
uGraph_from_list_\<alpha>_weight_def: fixes L :: "('b \<times> 'a \<times> 'b) list"   and e :: "'b uprod" shows "uGraph_from_list_\<alpha>_weight L e = (THE w. \<exists>a' b'. Upair a' b' = e \<and> (a', w, b') \<in> set L)"
cr_vreal_def: fixes a :: "V"   and b :: "real" shows "cr_vreal a b = (a = b\<^sub>\<real>)"
u_Confidentiality_B: fixes B :: "agent"   and A :: "agent"   and servK :: "nat"   and Ts :: "nat"   and evs :: "event list" assumes "Crypt (shrK B) \<lbrace>Agent A, Agent B, Key servK, Number Ts\<rbrace> \<in> parts (knows Spy evs)"   and "\<not> expiredSK Ts evs"   and "A \<notin> bad"   and "B \<notin> bad"   and "B \<noteq> Tgs"   and "evs \<in> kerbV" shows "Key servK \<notin> analz (knows Spy evs)"
homotopy_eqv_simple_connectedness: fixes S :: "'a set"   and T :: "'b set" assumes "S homotopy_eqv T" shows "simply_connected S = simply_connected T"
segment_inc_if: fixes D :: "('c, 'd) Order_scheme"   and b :: "'c"   and a :: "'c" assumes "Order D"   and "b \<in> carrier D"   and "a \<in> segment D b" shows "a \<prec>\<^bsub>D\<^esub> b"
borel_adapt_stoch_proc_borel_measurable: fixes X :: "nat \<Rightarrow> 'a \<Rightarrow> 'c" assumes "borel_adapt_stoch_proc F X" shows "\<forall>n. random_variable borel (X n)"
radical_power: fixes n :: "nat"   and x :: "'b" assumes "0 < n" shows "radical (x ^ n) = radical x"
equivp_retr: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "equivp R" shows "equivp (retr R)"
simps_1: fixes \<sigma> :: "nat \<Rightarrow> exp option"   and x :: "nat" shows "substs \<sigma> (Var x) = (case \<sigma> x of None \<Rightarrow> Var x | Some p \<Rightarrow> p)"
sorted_descending_sort_descending_key: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "sorted_descending (map f (sort_descending_key f xs))"
that: fixes r :: "'a list"   and k :: "nat" assumes "r \<noteq> \<epsilon>"   and "1 < k"   and "r \<^sup>@ k = w"   and "w \<noteq> r" shows "thesis"
abs_eq: shows "\<top>\<^sub>S = Abs_scene (\<lambda>x y. y)"
analz_Key_image_insert_eq: fixes K :: "nat"   and evs :: "event list"   and KK :: "nat set" assumes "K \<in> symKeys"   and "Key K \<notin> used evs"   and "evs \<in> set_cr" shows "analz (Key ` insert K KK \<union> knows Spy evs) = insert (Key K) (analz (Key ` KK \<union> knows Spy evs))"
not_is_Eq_iff: fixes a :: "atom" shows "(\<forall>i j. a \<noteq> Eq i j) = (\<exists>i j. a = Less i j)"
det_state_cover_card: fixes M :: "('e, 'f, 'g) FSM"   and V :: "'e list set" assumes "is_det_state_cover M V"   and "finite (nodes M)" shows "card (d_reachable M (initial M)) = card V"
r: fixes b :: "lam" assumes "SN b" shows "P a b"
assms: shows "opnet np p \<Turnstile> (otherwith (=) (net_tree_ips p) inoclosed, U \<rightarrow>) P"
assms: shows "(A, B) \<in> s_mul_ext ns s"
mod_exp_nat_aux_def: shows "mod_exp_nat_aux = mod_exp_nat.efficient_power"
convex_on_ereal_alt: fixes C :: "'b set"   and f :: "'b \<Rightarrow> ereal" assumes "convex C" shows "Lower_Semicontinuous.convex_on C f = (\<forall>x\<in>C. \<forall>y\<in>C. \<forall>m. 0 \<le> m \<and> m \<le> 1 \<longrightarrow> f (m *\<^sub>R x + (1 - m) *\<^sub>R y) \<le> ereal m * f x + (1 - ereal m) * f y)"
lattice_def: fixes L :: "'b set set" shows "lattice L \<equiv> (\<forall>M\<subseteq>L. \<Inter> M \<in> L) \<and> (\<forall>M\<subseteq>L. \<Union> M \<in> L)"
simps_1: fixes y :: "real_max_bot" shows "MInfty + y = y"
vert_f': shows "\<V> f' = set vs1 \<union> set vs2 \<union> {a, v}"
1: shows "coreflexive (x * top * x\<^sup>T) \<and> coreflexive (x\<^sup>T * top * x)"
residue_add_comm: fixes k :: "nat"   and x :: "int"   and y :: "int" shows "x \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> y = y \<oplus>\<^bsub>residue_ring (p ^ k)\<^esub> x"
changes_itv_smods_def: fixes a :: "real"   and b :: "real"   and p :: "real poly"   and q :: "real poly" shows "changes_itv_smods a b p q = (let ps = smods p q in changes_poly_at ps a - changes_poly_at ps b)"
distinct_1: fixes x1 :: "b_e" shows "$x1 \<noteq> Trap"
elementLevelPrecedesLeq: fixes a :: "'b"   and b :: "'b"   and M :: "('b \<times> bool) list" assumes "precedes a b (elements M)" shows "elementLevel a M \<le> elementLevel b M"
combine_order: shows "card \<V> \<le> card \<V>\<^sup>+"
a_kernel_def: fixes G :: "('e, 'f) ring_scheme"   and H :: "('g, 'h) ring_scheme"   and h :: "'e \<Rightarrow> 'g" shows "a_kernel G H h = kernel (add_monoid G) (add_monoid H) h"
\<beta>: shows "{} \<turnstile> PfP \<guillemotleft>\<beta>\<guillemotright>"
simps_4: fixes f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'b" shows "(case Infinity of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = f1"
ipresVar_def: fixes h :: "('b, 'c, 'varSort, 'var, 'd) qTerm set \<Rightarrow> 'e"   and MOD :: "('f, 'g, 'varSort, 'h, 'i, 'var, 'e, 'j, 'k) model_scheme" shows "ipresVar h MOD \<equiv> \<forall>xs x. h (Var xs x) = igVar MOD xs x"