hcju/isabelleps huggingface dataset:
simps_7: fixes f :: "'fun"   and \<Sigma> :: "'h option list" shows "lift_instr (Inca.instr.ICall f) \<Sigma> = Result.of_option () (get_arity f) \<bind> (\<lambda>ar. gen_pop_push (Ubx.instr.ICall f) (replicate ar None, None) \<Sigma>)"
Env: fixes P :: "'b com option"   and s :: "'b"   and t :: "'b" shows "(P, s) -e\<rightarrow> (P, t)"
setVar'_cl_inv: fixes g :: "'a gState_scheme"   and p :: "pState"   and gl :: "bool"   and v :: "String.literal"   and idx :: "integer option"   and val :: "integer" assumes "cl_inv (g, p)" shows "cl_inv (setVar' gl v idx val g p)"
a_in_principal: fixes a :: "'a" assumes "a \<in> carrier R" shows "a \<in> R \<diamondsuit>\<^sub>p a"
less_top_Value_Top: fixes x :: "'b" shows "top.Value x < top.Top"
compositionSym: fixes \<Psi> :: "'b"   and \<Psi>' :: "'b"   and \<Psi>'' :: "'b" assumes "\<Psi> \<simeq> \<Psi>'" shows "\<Psi>'' \<otimes> \<Psi> \<simeq> \<Psi>'' \<otimes> \<Psi>'"
Infinite_cong: fixes X :: "'d set"   and Y :: "'d set" assumes "X = Y" shows "infinite X = infinite Y"
lprefix_code_1: fixes ys :: "'c llist" shows "lprefix LNil ys = True"
nat_approx_posE: fixes e :: "'c" assumes "(0::'c) < e" obtains n :: "nat" where "(1::'c) / of_nat (Suc n) < e"
induct: fixes P :: "(real \<times> real) list \<Rightarrow> bool"   and a0 :: "(real \<times> real) list" assumes "P []"   and "\<And>x xs. (\<And>xa xb y xaa. \<lbrakk>xa = partition (coll 0 x) xs; (xb, y) = xa; xaa = x + sum_list xb\<rbrakk> \<Longrightarrow> P y) \<Longrightarrow> P (x # xs)" shows "P a0"
cases: fixes S :: "'a literal multiset \<Rightarrow> 'a literal multiset"   and a1 :: "'a literal multiset list"   and a2 :: "'a literal multiset"   and a3 :: "'a multiset list"   and a4 :: "'a list"   and a5 :: "'s"   and a6 :: "'a literal multiset" assumes "ord_resolve S a1 a2 a3 a4 a5 a6" obtains CAs :: "'a literal multiset list" and n :: "nat" and Cs :: "'a literal multiset list" and AAs :: "'a multiset list" and As :: "'a list" and \<sigma> :: "'s" and D :: "'a literal multiset" where "a1 = CAs" and "a2 = D + negs (mset As)" and "a3 = AAs" and "a4 = As" and "a5 = \<sigma>" and "a6 = (\<Sum>\<^sub># (mset Cs) + D) \<cdot> \<sigma>" and "length CAs = n" and "length Cs = n" and "length AAs = n" and "length As = n" and "n \<noteq> 0" and "\<forall>i<n. CAs ! i = Cs ! i + poss (AAs ! i)" and "\<forall>i<n. AAs ! i \<noteq> {#}" and "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))" and "eligible S \<sigma> As (D + negs (mset As))" and "\<forall>i<n. strictly_maximal_wrt (As ! i \<cdot>a \<sigma>) (Cs ! i \<cdot> \<sigma>)" and "\<forall>i<n. S (CAs ! i) = {#}"
that: fixes g :: "V"   and f :: "V"   and a :: "V"   and b :: "V"   and c :: "V"   and d :: "V" assumes "gf = [g, f]\<^sub>\<circ>"   and "ab = [a, b]\<^sub>\<circ>"   and "cd = [c, d]\<^sub>\<circ>"   and "g : a \<mapsto>\<^bsub>\<AA>\<^esub> c"   and "f : b \<mapsto>\<^bsub>\<BB>\<^esub> d" shows "thesis"
add_canc1: fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.cancellative_partial_semigroup times D"   and "D x y" shows "cancellative_partial_semigroup.rquot times D (times x y) x = y"
split_coeff_vec_index_sum: fixes x :: "rat vec"   and y :: "rat vec"   and v :: "linear_poly"   and l :: "nat"   and X :: "(nat, rat) mapping" assumes "(x, y) = split_i_j_x (dim_vec (lpoly_to_vec v)) l X" shows "(\<Sum>i = 0..<dim_vec x. Abstract_Linear_Poly.coeff v i * \<langle>X\<rangle> i) = (\<Sum>i = 0..<dim_vec x. lpoly_to_vec v $ i * x $ i)"
double_add_less_zero_iff_single_add_less_zero: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less (plus a a) zero = less a zero"
execn_noguards_no_Fault: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate"   and n :: "nat"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"   and "noguards c"   and "\<forall>p\<in>dom \<Gamma>. noguards (the (\<Gamma> p))"   and "\<not> isFault s" shows "\<not> isFault t"
discr0_transT: fixes c :: "('test, 'atom) com"   and s :: "'state"   and s' :: "'state" assumes "discr0 c"   and "mustT c s"   and "(c, s) \<rightarrow>t s'" shows "s \<approx> s'"
i1: shows "rk {C, D, M, P} + 2 \<le> 4"
\<psi>_def: fixes n :: "real"   and t :: "real" shows "\<psi> n t = char \<mu> (t / sqrt (\<sigma>\<^sup>2 * n))"
rm_vars_fv_obtain: fixes x :: "'a"   and t :: "('b, 'a) Term.term"   and X :: "'a set"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "x \<in> fv (t \<cdot> rm_vars X \<theta>) - X" shows "\<exists>y\<in>fv t - X. x \<in> fv (rm_vars X \<theta> y)"
mat_top_def: fixes i :: "'b"   and j :: "'c" shows "\<tau> i j \<equiv> True"
step_preserves_terminations: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and cs :: "('e, 'd, 'f) com list"   and css :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and s :: "('e, 'f) xstate"   and cs' :: "('e, 'd, 'f) com list"   and css' :: "(('e, 'd, 'f) com list \<times> ('e, 'd, 'f) com list) list"   and t :: "('e, 'f) xstate" assumes "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css', t)"   and "\<Gamma>\<turnstile>cs,css \<Down> s" shows "\<Gamma>\<turnstile>cs',css' \<Down> t"
homeomorphic_map_compose: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd"   and X'' :: "'e topology"   and g :: "'d \<Rightarrow> 'e" assumes "homeomorphic_map X Y f"   and "homeomorphic_map Y X'' g" shows "homeomorphic_map X X'' (g \<circ> f)"
that_1: fixes s :: "name" assumes "t = Free s" shows "thesis"
subalgebra_in_carrier: fixes K :: "'a set"   and V :: "'a set" assumes "subalgebra K V R" shows "V \<subseteq> carrier R"
supp_append: fixes xs :: "'a list"   and ys :: "'a list" shows "supp (xs @ ys) = supp xs \<union> supp ys"
InvCapsNonneg_def: fixes c :: "('b, 'loc \<times> 't) Exchange.configuration" shows "cri.InvCapsNonneg c = (\<forall>p t. 0 \<le> zcount (c_caps c p) t)"
Sup_empty: fixes Inf :: "'e set \<Rightarrow> 'e"   and Sup :: "'e set \<Rightarrow> 'e"   and inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and bot :: "'e"   and top :: "'e" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
sorted_list_subset_fusion_def: fixes eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and g1 :: "('a, 'd) generator"   and g2 :: "('a, 'e) generator"   and s1 :: "'d"   and s2 :: "'e" shows "sorted_list_subset_fusion eq g1 g2 s1 s2 = sorted_list_subset eq (list.unfoldr g1 s1) (list.unfoldr g2 s2)"
l_new_shadow_root_get_shadow_root\<^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 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" shows "l_new_shadow_root_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 \<equiv> 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"
ring_axioms: shows "ring carrier_direct_lim add_rel mult_rel \<lfloor>V, \<zero>\<^bsub>V\<^esub>\<rfloor> \<lfloor>V, \<one>\<^bsub>V\<^esub>\<rfloor>"
TS_c1'': fixes x :: "nat"   and y :: "nat"   and x0 :: "nat"   and y0 :: "nat"   and s :: "(nat list \<times> nat list) pmf"   and qs :: "nat list"   and h0 :: "nat list" assumes "x \<noteq> y"   and "{x, y} = {x0, y0}"   and "TS_inv s x [x0, y0]"   and "set qs \<subseteq> {x, y}"   and "qs \<in> lang (seq [Atom y, Atom x, Star (Times (Atom y) (Atom x)), Atom x])" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h0)) s qs) (last qs) [x0, y0] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h0)) s qs = real (length qs - 2)"
countable_basis: fixes "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.first_countable_topology open" obtains A :: "nat \<Rightarrow> 'b set" where "\<And>i. open (A i)" and "\<And>i. x \<in> A i" and "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> filterlim F (topological_space.nhds open x) sequentially"
PC_def: fixes a :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and P :: "'a \<Rightarrow> real"   and b :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "a \<^bsub>P\<^esub>\<oplus> b \<equiv> \<lambda>ab Q s. P s * a ab Q s + (1 - P s) * b ab Q s"
le: fixes \<omega> :: "'a" assumes "\<omega> \<in> \<Omega>" shows "T \<omega> \<le> S \<omega>"
normalize_ex_bound: fixes s :: "String.literal set"   and c :: "String.literal" assumes "finite s"   and "c \<in> s"   and "c \<notin> normalize_sort cs s"   and "sort_ex cs s" obtains c' :: "String.literal" where "c' \<in> normalize_sort cs s" and "class_les cs c' c"
gen_length_def: fixes n :: "nat"   and xs :: "'g list" shows "List.gen_length n xs = n + length xs"
conditional_entropy_eq_entropy_simple: fixes X :: "'a \<Rightarrow> 'd"   and Y :: "'a \<Rightarrow> 'e" assumes "simple_function M X"   and "simple_function M Y" shows "\<H>(X | Y) = entropy b (count_space (X ` space M) \<Otimes>\<^sub>M count_space (Y ` space M)) (\<lambda>x. (X x, Y x)) - \<H>(Y)"
sift_up_step_aux2: fixes heap :: "int \<Rightarrow> int"   and l :: "int"   and r :: "int"   and x :: "int"   and k :: "int" assumes "is_heap_except_up (\<le>) heap l r x"   and "l \<le> parent l x"   and "heap x \<le> heap (parent l x)"   and "bounded l r k"   and "parent l k = parent l x"   and "bounded l r (parent l (parent l k))" shows "swap heap (parent l x) x (parent l (parent l k)) \<le> swap heap (parent l x) x k"
ctr_2: fixes xs :: "'b llist" assumes "\<not> lnull xs" shows "lalternate xs = LCons (lhd xs) (case xs of LCons x xs \<Rightarrow> lalternate (ltl xs))"
QuoteP_imp_ConstP_F_hyps: fixes Us :: "name set"   and A :: "fm" assumes "Us \<subseteq> Vs"   and "{ConstP (F i) |i. i \<in> Us} \<turnstile> A" shows "quote_all p Us \<turnstile> A"
cases: fixes \<Gamma> :: "('a, 'b) web_scheme"   and \<epsilon> :: "ennreal" assumes "hindered_by \<Gamma> \<epsilon>" obtains f :: "'a \<times> 'a \<Rightarrow> ennreal" where "hindrance_by \<Gamma> f \<epsilon>" and "current \<Gamma> f" and "wave \<Gamma> f"
Ps: fixes i :: "'a" shows "\<exists>\<delta>>0. \<forall>x2. x2 \<noteq> x1 \<and> dist x2 x1 < \<delta> \<longrightarrow> dist (f i x2) (f i x1) < \<epsilon> / real CARD('a)"
finalRefE: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty" assumes "P,E,h,sh \<turnstile> e : T"   and "is_refT T"   and "final e" obtains "e = null"   | a :: "nat" and C :: "char list" where "e = addr a__" and "T = Class C"   | a :: "nat" where "e = Throw a"
eq_fin_length: fixes ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and w\<^sub>1 :: "'a list"   and w\<^sub>2 :: "'a list" assumes "traces ind"   and "(traces.eq_swap ind)\<^sup>*\<^sup>* w\<^sub>1 w\<^sub>2" shows "length w\<^sub>1 = length w\<^sub>2"
rel_pred_neg_distr_cond_def: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_pred_neg_distr_cond A B = (rel_pred (A OO B) \<le> rel_pred A OO rel_pred B)"
rec_ex1_def: fixes f :: "recf" shows "rec_ex1 f = CN rec_sign [rec_sigma1 f]"
vdomain_vrange_is_vempty: fixes a :: "V"   and b :: "V" shows "(\<D>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}) = 0) = (\<R>\<^sub>\<circ> (set {\<langle>a, b\<rangle>}) = 0)"
assms_1: shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return' c Q,A"
dualization_2: fixes F :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" shows "\<turnstile> \<not> \<diamond>F = \<box>\<not> F"
cases: fixes TA :: "('a, 'b) tree_automaton_rec"   and A :: "'b \<Rightarrow> nat option"   and a :: "('a set, 'b) ta_rule" assumes "ranked_tree_automaton TA A"   and "a \<in> ranked_tree_automaton.\<delta>ss TA A" obtains f :: "'b" and ss :: "'a set list" and s :: "'a set" where "a = s \<rightarrow> f ss" and "A f = Some (length ss)" and "listsp (\<lambda>s. s \<subseteq> ta_rstates TA) ss" and "s = ranked_tree_automaton.\<delta>ss_lhs TA f ss"
Rel_def: fixes r :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "Transfer.Rel r \<equiv> r"
modifies_spec: shows "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call p (Modif \<sigma>),{}"
PP_def: shows "PP \<equiv> (B ===> B ===> B) ===> B ===> (B ===> B ===> B) ===> (B ===> B) ===> ((=) ===> B ===> B) ===> (=)"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Cod (t \<^bold>\<cdot> u) = Cod t"
under_Field: fixes r :: "('a \<times> 'a) set"   and a :: "'a" shows "under r a \<subseteq> Field r"
chart_last_basis_defined: fixes xa :: "'b proj_space"   and b :: "'b" assumes "chart_last_domainP xa"   and "chart_basis_domainP b xa" shows "chart_last xa \<bullet> b \<noteq> 0"
set_iterator_filter_foldli_conv: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" shows "set_iterator_filter P (foldli xs) = foldli (filter P xs)"
less_QF1: fixes k :: "nat"   and j :: "nat" shows "QF k (j, m - Suc 0) \<lless> QF k (Suc j, 0)"
simps_3: fixes a :: "'b exp"   and b :: "'b exp" shows "ags (exp.Plus a b) = [a, b]"
restrict_Pi_cancel: fixes x :: "'a \<Rightarrow> 'b"   and I :: "'a set"   and A :: "'a \<Rightarrow> 'b set" shows "(restrict x I \<in> Pi I A) = (x \<in> Pi I A)"
FAssRedsThrow1: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and r :: "nat \<times> char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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 F :: "char list"   and Cs :: "char list list"   and e\<^sub>2 :: "expr" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>" shows "P,E \<turnstile> \<langle>e\<bullet>F{Cs} := e\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Throw r,s'\<rangle>"
AbGroup_inner_dirsum_pairwise_int0_double: fixes G :: "'a set"   and G' :: "'a set" assumes "AbGroup G"   and "AbGroup G'"   and "add_independentS [G, G']" shows "G \<inter> G' = 0"
assms_3: fixes P :: "'d"   and Q :: "'d" assumes "Prop P Q" shows "\<Psi> \<rhd> C P \<approx> C Q"
singleton_lst_proj_1: fixes l :: "'e"   and a :: "'d" shows "proj_unl l [(ln l, a)] = [a]"
holomorphic_Gamma: fixes A :: "complex set" assumes "A \<inter> \<int>\<^sub>\<le>\<^sub>0 = {}" shows "Gamma holomorphic_on A"
simps_1: fixes k :: "'a" shows "AList.delete_aux k [] = []"
Inf_lower: fixes x :: "'c \<times> 'd"   and A :: "('c \<times> 'd) set" assumes "x \<in> A" shows "less_eq_prod (Inf_prod A) x"
intro: fixes E' :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and E :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. E' a b \<Longrightarrow> E a b" shows "Subgraph E E'"
zero_less_mult_pos2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and b :: "'c"   and a :: "'c" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times b a)"   and "less zero a" shows "less zero b"
map_graph_selectors_2: fixes f :: "('g \<times> 'f) set"   and G :: "('e, 'g) labeled_graph" shows "edges (map_graph f G) = on_triple f `` edges G"
step: fixes f' :: "'d"   and g' :: "'f" assumes "\<And>x. orda (U1 f' x) (U1 f x)"   and "\<And>y. ordb (U2 g' y) (U2 g y)"   and "P (U1 f') (U2 g')" shows "P (U1 (F f')) (U2 (G g'))"
multiplicity_prime_fact: fixes p :: "nat"   and n :: "nat" assumes "prime p" shows "multiplicity p (fact n) = legendre_aux (real n) p"
subset_conjugation_is_hom: shows "(\<lambda>g. \<lambda>H\<in>{H. H \<subseteq> carrier G}. g <# H #> inv g) \<in> hom G (BijGroup {H. H \<subseteq> carrier G})"
destr_contstr_comp_id: fixes destr :: "'c \<Rightarrow> 'b"   and constr :: "'b \<Rightarrow> 'c" assumes "\<And>v. destr (constr v) = v" shows "destr \<circ> constr = id"
inf_le2: fixes x :: "'d multiset"   and y :: "'d multiset" shows "x \<inter># y \<subseteq># y"
iia_f_idem: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and B :: "'a set" assumes "f_range_on A f"   and "iia_on A f"   and "B \<subseteq> A" shows "f (f B) = f B"
A: shows "countable A \<and> A \<subseteq> sets (restr_to_subalg M F) \<and> \<Union> A = space (restr_to_subalg M F) \<and> (\<forall>a\<in>A. emeasure (restr_to_subalg M F) a \<noteq> \<infinity>)"
set_intros_1: fixes x21 :: "'b"   and x22 :: "'b list" shows "x21 \<in> ran\<^sub>l (x21 # x22)"
assms_1: shows "\<turnstile>\<^sub>3 {P} c { Q}"
A1: shows "c_pair x1 y1 = c_pair x2 y2"
ah_leq_il: fixes h1 :: "'b \<Rightarrow> 'b set"   and h2 :: "'b \<Rightarrow> 'b set"   and h1' :: "'b \<Rightarrow> 'b set"   and h2' :: "'b \<Rightarrow> 'b set" assumes "h1 [*] h2"   and "h1' \<le> h1"   and "h2' \<le> h2" shows "h1' [*] h2'"
ddh_0_def: fixes \<A> :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool spmf" shows "local.ddh.ddh_0 \<A> = sample_uniform (order \<G>) \<bind> (\<lambda>x. sample_uniform (order \<G>) \<bind> (\<lambda>y. \<A> (\<^bold>g [^] x) (\<^bold>g [^] y) (\<^bold>g [^] (x * y))))"
vsv_appI: fixes r :: "V"   and a :: "V"   and b :: "V" assumes "vsv r"   and "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "r\<lparr>a\<rparr> = b"
finite_measure_compl: fixes S :: "'a set" assumes "S \<in> events" shows "prob (space M - S) = prob (space M) - prob S"
lam_simps_1: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and c :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "ide a"   and "ide b"   and "ide c"   and "\<guillemotleft>f : hfsetcat.prod a b \<rightarrow> c\<guillemotright>" shows "arr (hfsetcat.\<Lambda> a b c f)"
z_def: shows "z \<equiv> getFr [out] [] [\<sigma>, \<phi>]"
lub_def: fixes S :: "'b set"   and po :: "'b potype" shows "lub S po = least (\<lambda>x. \<forall>y\<in>S. (y, x) \<in> order po) po"
add_data_cache_privilege: fixes s' :: "'b sparc_state"   and s :: "'b sparc_state"   and addr :: "word32"   and y :: "word32"   and m :: "word4" assumes "s' = add_data_cache s addr y m"   and "get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
HFinite_HInfinite_iff: fixes x :: "'a star" shows "(x \<in> HFinite) = (x \<notin> HInfinite)"
has_derivative_id: fixes a :: "'b" shows "(id has_derivative id) (at a)"
approx_def: fixes t :: "'a \<Rightarrow> 'b option"   and s :: "'a \<Rightarrow> 'b" shows "approx t s = (\<forall>x k. t x = Some k \<longrightarrow> s x = k)"
assms_3: fixes zs :: "'varSort"   and z :: "'var" shows "freshEnv zs z rho \<or> freshAbs zs z A"
num1_eqI: fixes a :: "1"   and b :: "1" shows "a = b"
t3: assumes "Can t3" shows "D.ide \<lbrace>t3\<rbrace>"
martingale_def: fixes M :: "'b measure"   and F :: "'c \<Rightarrow> 'b measure"   and X :: "'c \<Rightarrow> 'b \<Rightarrow> real" shows "martingale M F X = (Filtration.filtration M F \<and> (\<forall>t. integrable M (X t)) \<and> borel_adapt_stoch_proc F X \<and> (\<forall>t s. t \<le> s \<longrightarrow> (AE w in M. real_cond_exp M (F t) (X s) w = X t w)))"
abs_eq: fixes xa :: "'a"   and x :: "((real, 'a) vec, 'a) vec" shows "\<c>\<o>\<l> xa (to_mtx x) = column xa x"
distinct_1: fixes x11 :: "'f"   and x12 :: "'g"   and x21 :: "'h"   and x22 :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and x23 :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "qVar x11 x12 \<noteq> qOp x21 x22 x23"
assms_9: shows "Esigma (a * u) = Esigma a * (x + 1)"
rtrancl_listrel1_ConsI2: fixes x :: "'g"   and y :: "'g"   and r :: "('g \<times> 'g) set"   and xs :: "'g list"   and ys :: "'g list" assumes "(x, y) \<in> r\<^sup>*"   and "(xs, ys) \<in> (listrel1 r)\<^sup>*" shows "(x # xs, y # ys) \<in> (listrel1 r)\<^sup>*"
l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M ShadowRootClass.type_wf DocumentClass.type_wf set_tag_name set_tag_name_locs"
CallRedsThrowObj: fixes e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>0 :: "bool"   and a :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and b\<^sub>1 :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and M :: "char list"   and es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<bullet>M(es),s\<^sub>0,b\<^sub>0\<rangle> \<rightarrow>* \<langle>throw a,s\<^sub>1,b\<^sub>1\<rangle>"
Seq_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and P' :: "('c, 'd) action"   and Q' :: "('c, 'e) action" assumes "P \<preceq>S Q"   and "P' \<preceq>S Q'" shows "P `;` P' \<preceq>S Q `;` Q'"
assms: shows "\<forall>r\<in>R. Card_order r"
minus_eq: fixes x :: "'c"   and y :: "'c" shows "x - y = x + - y"
disc_18: fixes x31 :: "nat"   and x32 :: "'a"   and x33 :: "'a"   and x34 :: "'b"   and x35 :: "'b"   and x36 :: "'c" shows "\<not> isSnapshot (Recv x31 x32 x33 x34 x35 x36)"
lipschitz_on_closure: fixes C :: "real"   and U :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "C-lipschitz_on U f"   and "continuous_on (closure U) f" shows "C-lipschitz_on (closure U) f"
obtain_partition: fixes A :: "'c set"   and N :: "nat multiset" assumes "finite A"   and "number_partition (card A) N" shows "\<exists>P. partition_on A P \<and> image_mset card (mset_set P) = N"
assms: shows "((\<sigma>, \<zeta>), a, \<sigma>', \<zeta>') \<in> ocnet_sos (automaton.trans (opnet onp n))"
vimage_vrrestriction: fixes r :: "V"   and A :: "V"   and B :: "V" shows "(r \<restriction>\<^sup>r\<^sub>\<circ> A) `\<^sub>\<circ> B = A \<inter>\<^sub>\<circ> r `\<^sub>\<circ> B"
rep_eq: fixes x :: "'h set"   and xa :: "'i set" shows "Rep_\<I> (\<I>_uniform x xa) = (\<lambda>xb. if xb \<in> x then xa else {})"
reflect_poly_simps_3: fixes a :: "'a" shows "reflect_poly [:a:] = [:a:]"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> ManiPref'.CD_on ds X) = (x \<in> ManiPref'.Cd (Xd x) X \<and> Xd x \<in> ds)"
module_hom_uminus: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b" assumes "module scale" shows "module_hom scale scale uminus"
simps_4: fixes a :: "'b" shows "REV (Atom a) = Atom a"
simps_4: fixes G :: "('c, 'b) labeled_graph"   and l :: "'c" shows ":G:\<lbrakk>A_Lbl l\<rbrakk> = getRel l G"