hcju/isabelleps huggingface dataset:
at_within_def: fixes "open" :: "'b set \<Rightarrow> bool"   and a :: "'b"   and s :: "'b set" assumes "class.topological_space open" shows "topological_space.at_within open a s = inf (topological_space.nhds open a) (principal (s - {a}))"
decides_subsets_def: fixes \<F> :: "nat set set"   and M :: "nat set" shows "decides_subsets \<F> M \<equiv> \<forall>T\<subseteq>M. finite T \<longrightarrow> decides \<F> T M"
user_seq_exe_def: fixes n :: "nat"   and s :: "'b sparc_state" shows "user_seq_exe n s \<equiv> \<forall>i t. i \<le> n \<and> SEQ i s = Some t \<longrightarrow> good_context t \<and> get_delayed_pool t = [] \<and> get_trap_set t = {}"
Inf_lower: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" shows "x \<le> \<Squnion> A"
HEndPhase2_HInv4a: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HEndPhase2 s s' p"   and "HInv4a s q" shows "HInv4a s' q"
variety_of_map_indets: fixes f :: "'e \<Rightarrow> 'c"   and F :: "(('e \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd) set" shows "\<V> (map_indets f ` F) = (\<lambda>a. a \<circ> f) -` \<V> F"
continuous_on_diff: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x - g x)"
sorted_list_of_multiset_mset: fixes xs :: "'k list" shows "sorted_list_of_multiset (mset xs) = sort xs"
f_def: shows "f = (\<lambda>r. Re (winding_number (g r) a))"
supp_list_append: fixes xs :: "'b list"   and ys :: "'b list" shows "supp (xs @ ys) = supp xs \<union> supp ys"
constant_transformation_axioms_def: fixes B :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and g :: "'ag" shows "constant_transformation_axioms B g \<equiv> partial_magma.arr B g"
update_convs_3: fixes current' :: "thread_id_t \<Rightarrow> thread_id_t"   and sp_impl_subj_subj :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and sp_impl_subj_obj :: "partition_id_t \<Rightarrow> obj_id_t \<Rightarrow> mode_t \<Rightarrow> bool"   and current :: "thread_id_t"   and obj :: "obj_id_t \<Rightarrow> obj_t"   and thread :: "thread_id_t \<Rightarrow> thread_t"   and more :: "'a" shows "current_update current' \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current, obj = obj, thread = thread, \<dots> = more\<rparr> = \<lparr>sp_impl_subj_subj = sp_impl_subj_subj, sp_impl_subj_obj = sp_impl_subj_obj, current = current' current, obj = obj, thread = thread, \<dots> = more\<rparr>"
CFG_wf_axioms: shows "CFG_wf sourcenode targetnode kind valid_edge (_Entry_) Def Use state_val"
append_\<tau>moves2: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and es :: "'addr expr1 list"   and pc :: "nat"   and xcp :: "'addr option"   and es' :: "'addr expr1 list" assumes "\<tau>moves2 P h stk es pc xcp" shows "\<tau>moves2 P h stk (es' @ es) (length (compEs2 es') + pc) xcp"
partition_on_mset_not_empty: fixes A :: "'b multiset"   and P :: "'b multiset multiset" assumes "A \<noteq> {#}"   and "partition_on_mset A P" shows "P \<noteq> {#}"
mset_subset_eqI: fixes A :: "'c multiset"   and B :: "'c multiset" assumes "\<And>a. count A a \<le> count B a" shows "A \<subseteq># B"
H_4: shows "x \<in> Pi\<^sub>E UNIV X"
cr_option_le_spmf_def: fixes x :: "'c option"   and p :: "'c spmf" shows "option_le_spmf.cr_option_le_spmf x p = ord_spmf (=) (return_pmf x) p"
PUpdate_def: fixes U :: "'b data \<Rightarrow> 'b pdata" shows "PUpdate U = (\<forall>d. Data.DataSpace d = PDataSpace (U d))"
HcompNml_assoc: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Nml v"   and "Src t = Trg u"   and "Src u = Trg v" shows "(t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v = t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> v"
subset_iff: fixes A :: "'e set"   and B :: "'e set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
nat_power_less_diff: fixes n :: "nat"   and q :: "nat"   and m :: "nat" assumes "2 ^ n * q < 2 ^ m" shows "q < 2 ^ (m - n)"
nodes_delete_edge: fixes v :: "'a"   and e :: "'b"   and v' :: "'a"   and G :: "('a, 'b, 'c) graph_scheme" shows "nodes (delete_edge v e v' G) = nodes G"
word_le_imp_diff_le: fixes k :: "'b word"   and n :: "'b word"   and m :: "'b word" assumes "k \<le> n"   and "n \<le> m" shows "n - k \<le> m"
hrealpow: fixes X :: "nat \<Rightarrow> real"   and m :: "nat" shows "star_n X ^ m = star_n (\<lambda>n. X n ^ m)"
suffix_simps_4: fixes \<sigma> :: "nat \<Rightarrow> 'b" shows "\<sigma> |\<^sub>s 0 = \<sigma>"
col_mat1: fixes i :: "nat"   and nr :: "nat"   and ze :: "'c"   and on :: "'c" assumes "i < nr" shows "col (mat1I ze on nr) i = vec1I ze on nr i"
weakSimulation_def: fixes \<Psi> :: "'b"   and 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" shows "\<Psi> \<rhd> P \<leadsto><Rel> Q \<equiv> (\<forall>\<Psi>'' \<alpha> Q'. \<Psi> \<rhd> Q \<longmapsto> \<alpha> \<prec> Q' \<longrightarrow> bn \<alpha> \<sharp>* \<Psi> \<longrightarrow> bn \<alpha> \<sharp>* P \<longrightarrow> \<alpha> \<noteq> \<tau> \<longrightarrow> (\<exists>P''. \<Psi> : Q \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'' \<and> (\<exists>P'. \<Psi> \<otimes> \<Psi>'' \<rhd> P'' \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi> \<otimes> \<Psi>'', P', Q') \<in> Rel))) \<and> (\<forall>Q'. \<Psi> \<rhd> Q \<longmapsto> \<tau> \<prec> Q' \<longrightarrow> (\<exists>P'. \<Psi> \<rhd> P \<Longrightarrow>\<^sup>^\<^sub>\<tau> P' \<and> (\<Psi>, P', Q') \<in> Rel))"
cong_scalar_left: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and division_segment :: "'c \<Rightarrow> 'c"   and b :: "'c"   and c :: "'c"   and a :: "'c"   and d :: "'c" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment"   and "unique_euclidean_semiring.cong modulo b c a" shows "unique_euclidean_semiring.cong modulo (times d b) (times d c) a"
mult_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b"   and "bot \<le> c" shows "a * c \<le> b * c"
timpl_closure'_timpls_trancl_supset': fixes c :: "('c set \<times> 'c set) set" shows "timpl_closure' c \<subseteq> timpl_closure' {(a, b). (a, b) \<in> c\<^sup>+ \<and> a \<noteq> b}"
t_in: shows "mix_pmf \<beta> v u \<in> local.\<P>"
normalize_monic: fixes p :: "'a poly" assumes "monic p" shows "normalize p = p"
nth_append_length: fixes xs :: "'b list"   and x :: "'b"   and ys :: "'b list" shows "(xs @ x # ys) ! length xs = x"
Suc: shows "map (lconjby (sum_list (altst k))) (lconjseq (if gcd_nat.greater_eq k 2 then altst j else altts j)) = lconjseq (altst j)"
finfun_snd_comp_conv: fixes f :: "'d \<Rightarrow> 'c \<times> 'b"   and g :: "('a, 'd) finfun" shows "finfun_snd (finfun_comp f g) = finfun_comp (snd \<circ> f) g"
times_canc_right: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and h :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "p * h = q * h"   and "h \<noteq> 0" shows "p = q"
swap: fixes y :: "'a"   and x :: "'a"   and l :: "'a list" shows "y # x # l <~~> x # y # l"
simps: fixes TI :: "('b \<times> 'b) list"   and a :: "'b"   and b :: "'b" shows "in_trancl TI a b = (if (a, b) \<in> set TI then True else list_ex (\<lambda>(c, d). c = a \<and> in_trancl (removeAll (c, d) TI) d b) TI)"
SDom_def: fixes \<sigma> :: "('e, 'f, 'g) subst" shows "SDom \<sigma> = {Inl x |x. x \<in> dom (SFunctions \<sigma>)} \<union> {Inr (Inl x) |x. x \<in> dom (SContexts \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPredicates \<sigma>)} \<union> {Inr (Inr x) |x. x \<in> dom (SPrograms \<sigma>)}"
q_def: fixes t :: "real" shows "q t = s0 + v * t"
g_fold_same: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" assumes "\<And>z. z \<in> set xs \<Longrightarrow> f z = f x" shows "g (fold merge xs x # ys) = g (x # xs @ ys)"
HInv5_def: fixes s :: "state" shows "HInv5 s = (\<forall>p. HInv5_inner s p)"
comp_inv_arr': fixes f :: "'a \<times> 'a" assumes "VV.iso f" shows "VV.comp (VV.inv f) f = VV.dom f"
mk_poly_def: fixes A :: "'b mat" shows "mk_poly A = mk_poly_sub (coeff_lift_hom.mat_hom A) (dim_col A - 1) (dim_col A - 1)"
map_update: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and k :: "nat"   and y :: "'c" shows "map f (xs[k := y]) = (map f xs)[k := f y]"
lappend_eq_LNil_iff: fixes xs :: "'b llist"   and ys :: "'b llist" shows "(xs @@ ys = LNil) = (xs = LNil \<and> ys = LNil)"
conditional_entropy_eq: fixes Y :: "'key \<Rightarrow> 'c"   and Py :: "'c \<Rightarrow> real"   and X :: "'key \<Rightarrow> 'd"   and Pxy :: "'d \<times> 'c \<Rightarrow> real" assumes "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) Y Py"   and "simple_distributed (point_measure keys (\<lambda>x. ennreal (K x))) (\<lambda>x. (X x, Y x)) Pxy" shows "K.conditional_entropy_Pow X Y = - (\<Sum>(x, y)\<in>(\<lambda>x. (X x, Y x)) ` space (point_measure keys (\<lambda>x. ennreal (K x))). Pxy (x, y) * log b (Pxy (x, y) / Py y))"
subst_map: fixes t :: "dB"   and ts :: "dB list"   and u :: "dB"   and i :: "nat" shows "(t \<degree>\<degree> ts)[u/i] = t[u/i] \<degree>\<degree> map (\<lambda>t. t[u/i]) ts"
Q_FreeS: shows "order.greater_eq (FreeGroup S) \<langle>\<Union>h\<in>FreeGroup S. lconjby h ` P'\<rangle>"
injI: fixes f :: "'g \<Rightarrow> 'h" assumes "\<And>x y. f x = f y \<Longrightarrow> x = y" shows "inj f"
sat_\<Phi>: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "Ik.wtE \<xi>" shows "Ik.satPB \<xi> \<Phi>"
str_eq_conv_Derivs: fixes A :: "'a list set" shows "\<approx>A = {(u, v). Derivs u A = Derivs v A}"
FunctorId2: fixes X :: "'a" assumes "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "F @@ X \<in> obj\<^bsub>CatCod F\<^esub> \<and> F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> (F @@ X)"
of_nat_fpxs_eq: fixes n :: "nat" shows "of_nat n = fpxs_const (of_nat n)"
fold_simps_1: fixes f :: "'f \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e" shows "fold f [] s = s"
ssubst_Pair_rhs: fixes r :: "'a"   and s :: "'b"   and R :: "('a \<times> 'b) set"   and s' :: "'b" assumes "(r, s) \<in> R"   and "s' = s" shows "(r, s') \<in> R"
vrestr_sym_diff: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" shows "f --` (A \<Delta> B) = f --` A \<Delta> f --` B"
abs_int_poly_inv: fixes p :: "int poly" shows "Polynomial.smult (sgn (lead_coeff p)) (abs_int_poly p) = p"
assms_1: shows "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>"
sub_AppE: fixes s :: "('a, 'b) tm"   and t :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "sub s (App t u)" obtains "App t u = s"   | "sub s u"   | "sub s t"
assms_1: shows "h' \<equiv> interpret_floatarith h vs"
trancl_imp_trans: fixes r :: "('a \<times> 'a) set" assumes "r\<^sup>+ \<subseteq> r" shows "trans r"
subst_not_fic1: fixes M :: "trm"   and a :: "coname"   and x :: "name"   and P :: "trm" shows "\<not> fic (M{a:=x.P}) a"
below_in_singleton_is_bottom: fixes P :: "'a set"   and x :: "'a" assumes "has_bottom P"   and "x \<in> P"   and "P .\<^bold>\<le> x = {x}" shows "x = bottom P"
pi_canonical: shows "\<downharpoonright>pi\<downharpoonleft> = pi"
AL_delete2: fixes b :: "'a"   and a :: "'a"   and l :: "('a \<times> 'b) list" assumes "b \<noteq> a" shows "delete l a\<down>b = l\<down>b"
viable_only_if_u: assumes "viable_market Mkt" shows "1 + r < u"
image_add_cball: fixes b :: "'a"   and a :: "'a"   and r :: "real" shows "(+) b ` cball a r = cball (a + b) r"
AE_cong_simp: fixes M :: "'b measure"   and N :: "'b measure"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "M = N"   and "\<And>x. x \<in> space N =simp=> P x = Q x" shows "almost_everywhere M P = almost_everywhere N Q"
vwalk_join_def: fixes p :: "'c list"   and q :: "'c list" shows "p \<oplus> q \<equiv> p @ tl q"
erH_rHom_coeff: fixes A :: "('e, 'f) Ring_scheme"   and B :: "('e, 'g) Ring_scheme"   and Y :: "'e"   and h :: "'a \<Rightarrow> 'e"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "Idomain S"   and "PolynRg A B Y"   and "h \<in> rHom S B"   and "pol_coeff S (n, f)" shows "pol_coeff B (n, cmp h f)"
toplevel_summands_in_lang: fixes w :: "'b list"   and s :: "'b rexp" shows "(w \<in> lang s) = (\<exists>r\<in>toplevel_summands s. w \<in> lang r)"
Var_Subset_sf: fixes i :: "name"   and j :: "name" shows "Sigma_fm (Var i SUBS Var j)"
shapiro_tauberian_def: fixes a :: "nat \<Rightarrow> real" shows "shapiro_tauberian a \<equiv> (\<forall>n>0. 0 \<le> a n) \<and> (\<lambda>x. dirichlet_prod' a (\<lambda>x. real_of_int \<lfloor>x\<rfloor>) x - x * ln x) \<in> O(\<lambda>x. x)"
in_var_plus: fixes x :: "'e \<Longrightarrow> 'g"   and y :: "'f \<Longrightarrow> 'g" shows "$(x +\<^sub>L y) = $x +\<^sub>L $y"
disj: shows "A \<inter> B \<inter> \<Union> S = {}"
conversepE: fixes r :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a1 :: "'b"   and a2 :: "'a" assumes "r\<inverse>\<inverse> a1 a2" obtains a :: "'a" and b :: "'b" where "a1 = b" and "a2 = a" and "r a b"
term_distinct_2: fixes name :: "name"   and u\<^sub>1 :: "'a"   and u\<^sub>2 :: "'a" shows "const name \<noteq> app u\<^sub>1 u\<^sub>2"
\<iota>_mprem: shows "D = main_prem_of \<iota>"
4: shows "bprv (imp (subst \<phi> \<langle>\<psi>\<rangle> xx) (softSubst \<chi> \<langle>\<chi>\<rangle> xx))"
set_plus_mono4: fixes a :: "'b"   and C :: "'b set"   and D :: "'b set" assumes "a \<in> C" shows "a +o D \<subseteq> D + C"
neg_equal_0_iff_equal: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.group_add minus plus zero uminus" shows "(uminus a = zero) = (a = zero)"
distinct_13: fixes x21 :: "nat"   and x22 :: "nat"   and x31 :: "nat set"   and x32 :: "nat set"   and x33 :: "'d" shows "x21:deliver(x22) \<noteq> x31\<not>x32:arrive(x33)"
inf2D2: fixes A :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and B :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and x :: "'g"   and y :: "'h" assumes "(A \<sqinter> B) x y" shows "B x y"
add_le_less_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<le> b"   and "c < d" shows "a + c < b + d"
distinct_sum_list_conv_Sum: fixes xs :: "'a list" assumes "distinct xs" shows "sum_list xs = \<Sum> (set xs)"
anonymous_profile_def: shows "anonymous_profile R2 = image_mset (weak_ranking \<circ> R2) (mset_set agents)"
simps_3: fixes n :: "num" shows "Num.sqr (num.Bit1 n) = num.Bit1 (num.Bit0 (Num.sqr n + n))"
rsquarefree_def': fixes p :: "'a poly" shows "rsquarefree p = (p \<noteq> 0 \<and> (\<forall>a. Polynomial.order a p \<le> 1))"
new_element_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_element_ptr :: "(_) element_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'e, 'g option) RElement_scheme \<Rightarrow> 'j" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_element \<rightarrow>\<^sub>r new_element_ptr"   and "ptr \<noteq> new_element_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
sup_ge2: fixes y :: "'a"   and x :: "'a" shows "y \<le> x + y"
distinct_1: fixes x1 :: "'b" shows "extT.Val x1 \<noteq> extT.Top"
max_coord_def: fixes M :: "nat"   and x :: "'b" shows "max_coord M x = snd (fold (\<lambda>a (b, c). let d = \<bar>x\<bar> \<bullet> a in if b \<le> d then (d, a) else (b, c)) (take M Basis_list) (0, 0::'b))"
intros_1: fixes u :: "'d \<Rightarrow> 'c"   and r :: "'d"   and d :: "'c" assumes "u r \<le> d" shows "dbm_entry_val u (Some r) None (Le d)"
assms_3: shows "(g \<longlongrightarrow> \<infinity>) F"
ns_mul_extE: fixes A :: "'a multiset"   and B :: "'a multiset"   and ns :: "('a \<times> 'a) set"   and s :: "('a \<times> 'a) set" assumes "(A, B) \<in> ns_mul_ext ns s" obtains A1 :: "'a multiset" and A2 :: "'a multiset" and B1 :: "'a multiset" and B2 :: "'a multiset" where "A = A1 + A2" and "B = B1 + B2" and "(A1, B1) \<in> multpw ns" and "\<And>b. b \<in># B2 \<Longrightarrow> \<exists>a. a \<in># A2 \<and> (a, b) \<in> s"
comp_inv_arr': fixes f :: "'a" assumes "B.iso f" shows "B.inv f \<cdot>\<^sub>B f = B.dom f"
terminatesI: fixes g :: "('a \<Rightarrow> bool) \<times> ('a \<Rightarrow> 'b \<times> 'a)" assumes "\<And>s. s \<in> terminates_on g" shows "terminates g"
samp_uni_spmf_mod_q: fixes x :: "nat" shows "spmf (sample_uniform q) (x mod q) = 1 / real q"
item_\<alpha>_of_inc_item: fixes x :: "item"   and s :: "symbol"   and k :: "nat" assumes "wellformed_item x"   and "next_symbol x = Some s" shows "item_\<alpha> (inc_item x k) = item_\<alpha> x @ [s]"
assms_2: fixes x :: "'a"   and y :: "'a" assumes "x \<in> S"   and "y \<in> S"   and "dist x y < e" shows "x = y"
augment_edge_rev_same: fixes v :: "nat"   and u :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and \<Delta> :: "'a" assumes "(v, u) \<in> E" shows "augment_edge f (u, v) \<Delta> (v, u) = f (v, u) - \<Delta>"
mono_comp: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less"   and "mono f"   and "order.mono less_eq g" shows "order.mono less_eq (f \<circ> g)"
dropD: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and m :: "'b match_expr"   and t :: "state" assumes "\<gamma>,p\<turnstile> \<langle>[Rule m Drop], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> t"   and "matches \<gamma> m Drop p" shows "t = Decision FinalDeny"
dres_nres_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>dres_nres_rel \<equiv> {(c, a). nres_of c \<le> \<Down> R a}"
same_upto_def: fixes n :: "nat"   and j :: "nat"   and A :: "'b mat"   and B :: "'b mat" shows "same_upto n j A B \<equiv> \<forall>i' j'. i' < n \<longrightarrow> j' < j \<longrightarrow> A $$ (i', j') = B $$ (i', j')"
f_NC_1: shows "f \<in> S \<rightarrow> carrier_vec nc"
Sup_UNIV: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup UNIV = top"
assms: shows "abs.low_mds_eq (mds\<^sub>A_of mds) (mem\<^sub>A_of mem\<^sub>1\<^sub>C) (mem\<^sub>A_of mem\<^sub>2\<^sub>C)"
qtable_mem_restr_UNIV: fixes n :: "nat"   and A :: "nat set"   and Q :: "'b option list \<Rightarrow> bool"   and X :: "'b option list set" shows "qtable n A (mem_restr UNIV) Q X = wf_table n A Q X"
PGC_weak: shows "\<forall>C. \<lfloor>PPA (Agent C) \<^bold>\<rightarrow> RightTo (Agent C) FWB\<rfloor>\<^sub>C"
const_term_not_zero: fixes p :: "'a list list" assumes "UP.const_term p \<noteq> \<zero>\<^bsub>poly_ring R\<^esub>" shows "p \<noteq> []"
sum_map_hom: fixes M1 :: "'c set"   and M2 :: "'c set" assumes "LinearCombinations.submodule K M1 V"   and "LinearCombinations.submodule K M2 V" shows "mod_hom K (direct_sum (vs M1) (vs M2)) V (\<lambda>v. fst v \<oplus>\<^bsub>V\<^esub> snd v)"
segment_segment: 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 "segment (Iod D (segment D b)) a = segment D a"
dvd_div_unit_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b"   and c :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times b one" shows "dvd.dvd times a (divide c b) = dvd.dvd times a c"
psubstInp_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option"   and inp :: "'a \<Rightarrow> ('b, 'c, 'varSort, 'var, 'd) qTerm set option" shows "inp %[rho] \<equiv> lift (psubst rho) inp"
limpt_closed_imp_exploding_subsequence: fixes T :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and l :: "'c" assumes "continuous_on T f"   and "closed T"   and "\<And>t. t \<in> T \<Longrightarrow> f t \<noteq> l"   and "l islimpt f ` T" obtains s :: "nat \<Rightarrow> 'b" where "(f \<circ> s) \<longlonglongrightarrow> l" and "\<And>i. s i \<in> T" and "\<And>C. \<lbrakk>compact C; C \<subseteq> T\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C"
iv: shows "remove1Prio (prio (getMinTree bq)) bq = remove1 (getMinTree bq) bq"
field_differentiable_within_Ln: fixes z :: "complex"   and S :: "complex set" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Ln field_differentiable at z within S"
coboundedI: fixes A :: "'b set"   and a :: "'b" assumes "finite A"   and "a \<in> A" shows "a \<le> \<Squnion>\<^sub>f\<^sub>i\<^sub>n A"
is_canon_rewriterD2: fixes rword :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> 't \<times> ('a \<Rightarrow>\<^sub>0 'b) \<Rightarrow> bool"   and A :: "('t \<Rightarrow>\<^sub>0 'b) set"   and u :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_canon_rewriter rword A u p" shows "p \<noteq> 0"
tseytin_tran_equisat: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula" shows "\<A> \<Turnstile> F = tseytin_update \<A> F \<Turnstile> tseytin_tran F"
assms_3: shows "(r |\<^bsub>rel\<^esub> s) = (r' |\<^bsub>rel\<^esub> s')"
erfc_has_field_derivative: fixes z :: "'a"   and A :: "'a set" shows "(erfc has_field_derivative - of_real (2 / sqrt pi) * exp (- z\<^sup>2)) (at z within A)"
monoE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a"   and f' :: "'a \<times> 'a" assumes "CC.mono g"   and "CC.seq g f"   and "CC.seq g f'"   and "CC.comp g f = CC.comp g f'" shows "f' = f"
supto_simp_2: fixes i :: "int"   and j :: "int" assumes "i \<le> j" shows "supto\<cdot>(MkI\<cdot>i)\<cdot>(MkI\<cdot>j) = MkI\<cdot>i :# supto\<cdot>(MkI\<cdot>i + 1)\<cdot>(MkI\<cdot>j)"
insort_key_conv_rec_list: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "insort_key f x xs = rec_list [x] (\<lambda>y ys zs. if f x \<le> f y then x # y # ys else y # zs) xs"
SigmaI: fixes a :: "'d"   and A :: "'d set"   and b :: "'e"   and B :: "'d \<Rightarrow> 'e set" assumes "a \<in> A"   and "b \<in> B a" shows "(a, b) \<in> Sigma A B"
class_defs_2: fixes sub :: "('a \<times> 'a) set"   and c1 :: "'a"   and c2 :: "'a" shows "class_les sub c1 c2 = (class_leq sub c1 c2 \<and> \<not> class_leq sub c2 c1)"
multiplicative_function_of_natD: fixes f :: "nat \<Rightarrow> nat" assumes "multiplicative_function (\<lambda>n. of_nat (f n))" shows "multiplicative_function f"
simps: fixes a1 :: "('s, 'v) tm"   and a2 :: "('s, 'v) tm" shows "gt_diff a1 a2 = (\<exists>t s. a1 = t \<and> a2 = s \<and> head t >\<^sub>h\<^sub>d head s \<and> is_Sym (head s) \<and> chkchop (>\<^sub>t) t s)"
similar_admI: fixes f :: "'b \<Rightarrow> Value"   and g :: "'b \<Rightarrow> CValue" assumes "cont f"   and "cont g" shows "adm (\<lambda>x. f x \<triangleleft>\<triangleright> g x)"
fdia_one: fixes x :: "('a \<times> 'a) set" shows "relfdia Id x = rel_ad (rel_ad x)"
order_eq_iff: fixes x :: "'c"   and y :: "'c" shows "(x = y) = (x \<le> y \<and> y \<le> x)"
trans: fixes b :: "'a"   and a :: "'a"   and c :: "'a" assumes "less_eq0 b a"   and "less_eq0 c b" shows "less_eq0 c a"
wp1n_RS3: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "wellformed_policy1_strong p" shows "wellformed_policy1_strong (rm_MT_rules C p)"