hcju/isabelleps huggingface dataset:
all_intersection_of: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool"   and R :: "'b set \<Rightarrow> bool" shows "(\<forall>S. (P intersection_of Q) S \<longrightarrow> R S) = (\<forall>T. P T \<and> T \<subseteq> Collect Q \<longrightarrow> R (\<Inter> T))"
a: fixes Y :: "'a set"   and x :: "'a" assumes "Y = {}" shows "\<Squnion>{x \<cdot> y |y. y \<in> Y} \<le> x \<cdot> \<Squnion>Y"
measurable_the_measure_pmf_Some: fixes p :: "'c spmf" shows "the \<in> restrict_space (measure_pmf p) (range Some) \<rightarrow>\<^sub>M count_space UNIV"
case_cong_weak: fixes option :: "'b option"   and option' :: "'b option"   and f1 :: "'c"   and f2 :: "'b \<Rightarrow> 'c" assumes "option = option'" shows "(case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option' of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x)"
parallel_converg_V01: fixes f :: "nat \<Rightarrow> nat option"   and x :: "nat" assumes "f \<in> V\<^sub>0\<^sub>1" shows "parallel (the (f 0)) (the (f 1)) x \<down>"
fmdrop_fmrestrict_set: fixes b :: "'a"   and S :: "'a set"   and m :: "('a, 'b) fmap" shows "fmdrop b (fmrestrict_set S m) = fmrestrict_set (S - {b}) m"
mrMTNone: fixes x :: "int \<times> (int \<times> int) \<times> (int \<times> int) \<times> DummyContent" shows "applied_rule_rev C x [] = \<bottom>"
uGraph_from_list_invar_def: fixes L :: "(nat \<times> int \<times> nat) list" shows "uGraph_from_list_invar L = (distinct (map \<alpha> L) \<and> (\<forall>p\<in>set L. case p of (u, w, v) \<Rightarrow> u \<noteq> v))"
noU: shows "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
lookup_union2: fixes k :: "'c"   and ys :: "'c \<rightharpoonup>\<^sub>f 'd"   and xs :: "'c \<rightharpoonup>\<^sub>f 'd" assumes "k |\<notin>| fmdom ys" shows "lookup (xs ++\<^sub>f ys) k = lookup xs k"
eval_implies_machine_star: fixes t :: "lam"   and t' :: "lam" assumes "t \<Down> t'" shows "<t,[]> \<mapsto>* <t',[]>"
moebius_translation_uminus: fixes v1 :: "complex" shows "- moebius_translation v1 = moebius_translation (- v1)"
split_min: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and i :: "'a"   and j :: "'a" assumes "class.linorder less_eq less" shows "P (ord.min less_eq i j) = ((less_eq i j \<longrightarrow> P i) \<and> (\<not> less_eq i j \<longrightarrow> P j))"
a3: shows "xa \<in> More_Modules.ideal (insert x (F \<union> B))"
setops\<^sub>s\<^sub>s\<^sub>t_cons_subset: fixes A :: "('c, 'd) stateful_strand_step list"   and a :: "('c, 'd) stateful_strand_step" shows "setops\<^sub>s\<^sub>s\<^sub>t A \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t (a # A)"
prv_cnj_com_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla" shows "bprv (imp (cnj \<phi> \<chi>) (cnj \<chi> \<phi>))"
transfer_rule_of_nat: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "R (0::'a) (0::'b)"   and "R (1::'a) (1::'b)"   and "(R ===> R ===> R) (+) (+)" shows "((=) ===> R) of_nat of_nat"
ringE_3: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "(x \<oplus>\<^bsub>R\<^esub> y) \<otimes>\<^bsub>R\<^esub> z = x \<otimes>\<^bsub>R\<^esub> z \<oplus>\<^bsub>R\<^esub> y \<otimes>\<^bsub>R\<^esub> z"
inj_def: fixes f :: "'a \<Rightarrow> 'b" shows "Fun.inj f = (\<forall>x y. f x = f y \<longrightarrow> x = y)"
power_half_series: shows "(\<lambda>n. (1 / 2) ^ Suc n) sums 1"
distribute_rely_parallel1: fixes i :: "'a::refinement_lattice"   and j :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "chaos \<sqsubseteq> (i::'a::refinement_lattice)"   and "chaos \<sqsubseteq> (j::'a::refinement_lattice)" shows "((c::'a::refinement_lattice) \<parallel> (d::'a::refinement_lattice)) // ((i::'a::refinement_lattice) \<parallel> (j::'a::refinement_lattice)) \<sqsubseteq> c // i \<parallel> d // j"
states_def: fixes c :: "('c, 'd) conf" shows "states c = {\<sigma>. \<exists>\<sigma>\<^sub>s\<^sub>y\<^sub>m. consistent \<sigma> \<sigma>\<^sub>s\<^sub>y\<^sub>m (store c) \<and> conjunct (pred c) \<sigma>\<^sub>s\<^sub>y\<^sub>m}"
exhaust_sel: fixes node :: "'a node" obtains "node = SOURCE"   | "node = Inner (node.inner node)"
bounds_compare_contradictory_3: fixes c :: "'a"   and b :: "'a option" assumes "c >\<^sub>u\<^sub>b b"   and "c \<le>\<^sub>u\<^sub>b b" shows "False"
has_derivative_within_If_eq: fixes P :: "'a \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and f' :: "'a \<Rightarrow> 'b"   and x :: "'a"   and S :: "'a set" shows "((\<lambda>x. if P x then f x else g x) has_derivative f') (at x within S) = (bounded_linear f' \<and> ((\<lambda>y. if P y then (f y - ((if P x then f x else g x) + f' (y - x))) /\<^sub>R norm (y - x) else (g y - ((if P x then f x else g x) + f' (y - x))) /\<^sub>R norm (y - x)) \<longlongrightarrow> (0::'b)) (at x within S))"
Un_upper2: fixes B :: "'f set"   and A :: "'f set" shows "B \<subseteq> A \<union> B"
cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r_inject: fixes x :: "(_) node_ptr"   and y :: "(_) node_ptr" shows "(cast x = cast y) = (x = y)"
deleteMin_correct: fixes q :: "('a, 'b) SkewBinomialHeap" assumes "q \<noteq> SkewBinomialHeap_loc.empty" shows "SkewBinomialHeap_loc.to_mset (SkewBinomialHeap_loc.deleteMin q) = SkewBinomialHeap_loc.to_mset q - {#SkewBinomialHeap_loc.findMin q#}"
simps_2: fixes m :: "n"   and n :: "n" shows "add2 (S m) n = S (add2 m n)"
that: fixes g :: "real \<Rightarrow> complex" assumes "valid_path g"   and "pathstart g = a"   and "pathfinish g = b"   and "path_image g \<subseteq> s - pts"   and "f contour_integrable_on g" shows "thesis"
of_int_vec_nth: fixes v :: "(int, 'b) vec"   and n :: "'b" shows "of_int_vec v $ n = of_int (v $ n)"
simps: fixes s :: "name"   and v :: "tm"   and i :: "tm"   and x :: "tm"   and x' :: "tm"   and k :: "name" assumes "atom s \<sharp> (v, i, x, x', k)"   and "atom k \<sharp> (v, i, x, x')" shows "AbstFormP v i x x' = VarP v AND OrdP i AND SyntaxN.Ex s (SyntaxN.Ex k (SeqAbstFormP v i x x' (Var s) (Var k)))"
encode_minsky_def: fixes M :: "(nat \<times> (nat, nat) Op) list" shows "encode_minsky M = list_to_nat (map (\<lambda>x. c_pair (fst x) (encode_Op (snd x))) M)"
circline_circle: fixes A :: "real"   and D :: "real"   and BC :: "complex"   and cl :: "complex set"   and \<mu> :: "complex"   and r2 :: "real"   and r :: "real" assumes "A \<noteq> 0"   and "A * D \<le> (cmod BC)\<^sup>2"   and "cl = circline A BC D"   and "\<mu> = - BC / cor A"   and "r2 = ((cmod BC)\<^sup>2 - A * D) / A\<^sup>2"   and "r = sqrt r2" shows "cl = circle \<mu> r"
ih_12: shows "InvariantConflictFlagCharacterization (getConflictFlag state') (getF state') (getM state')"
smallest_accepting_rank_def: fixes \<Sigma> :: "'b set"   and \<delta> :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and q\<^sub>0 :: "'c"   and w :: "nat \<Rightarrow> 'b"   and F :: "'c set" shows "mojmir_def.smallest_accepting_rank \<Sigma> \<delta> q\<^sub>0 w F \<equiv> if mojmir_def.accept \<delta> q\<^sub>0 w F then Some (LEAST i. finite (mojmir_def.fail \<Sigma> \<delta> q\<^sub>0 w F) \<and> finite (mojmir_def.merge \<Sigma> \<delta> q\<^sub>0 w F i) \<and> infinite (mojmir_def.succeed \<Sigma> \<delta> q\<^sub>0 w F i)) else None"
compl_conn_def: fixes compl :: "(('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> ((('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('t \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list \<Rightarrow> nat \<times> 'g \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'e) \<times> 'f) list \<times> 'g" shows "compl_conn compl = (\<forall>d m gs bs ps sps p q data. dickson_grading d \<longrightarrow> fst ` set gs \<subseteq> dgrad_p_set d m \<longrightarrow> is_Groebner_basis (fst ` set gs) \<longrightarrow> fst ` set bs \<subseteq> dgrad_p_set d m \<longrightarrow> set ps \<subseteq> set bs \<times> (set gs \<union> set bs) \<longrightarrow> sps \<noteq> [] \<longrightarrow> set sps \<subseteq> set ps \<longrightarrow> unique_idx (gs @ bs) data \<longrightarrow> (p, q) \<in> set sps \<longrightarrow> fst p \<noteq> 0 \<longrightarrow> fst q \<noteq> 0 \<longrightarrow> crit_pair_cbelow_on d m (fst ` (set gs \<union> set bs) \<union> fst ` set (fst (compl gs bs (ps -- sps) sps data))) (fst p) (fst q))"
simps_4: fixes uw :: "'b"   and t :: "'c tree" shows "U (Delete uw) [t] = 6 * \<phi> t + 3"
foldl_Cons': fixes x :: "'a"   and xs :: "'a list" shows "foldl (+) (0::'a) (x # xs) = x + foldl (+) (0::'a) xs"
raw_converter_invariant_def: fixes \<I> :: "('a, 'b) \<I>"   and \<I>' :: "('c, 'd) \<I>"   and callee :: "'e \<Rightarrow> 'a \<Rightarrow> ('b \<times> 'e, 'c, 'd) gpv"   and I :: "'e \<Rightarrow> bool" shows "raw_converter_invariant \<I> \<I>' callee I \<equiv> (\<forall>s x. x \<in> outs_\<I> \<I> \<longrightarrow> I s \<longrightarrow> results_gpv \<I>' (callee s x) \<subseteq> responses_\<I> \<I> x \<times> {s. I s}) \<and> (\<forall>x s. x \<in> outs_\<I> \<I> \<longrightarrow> I s \<longrightarrow> \<I>' \<turnstile>g callee s x \<surd>)"
GF_advice_inter_subformulas: fixes \<phi> :: "'b ltln"   and X :: "'b ltln set" shows "\<phi>[X \<inter> subformulas\<^sub>\<mu> \<phi>]\<^sub>\<Pi>\<^sub>1 = \<phi>[X]\<^sub>\<Pi>\<^sub>1"
indets_monomial_single_subset: fixes v :: "'a"   and k :: "nat"   and c :: "'b" shows "indets (monomial c (monomial k v)) \<subseteq> {v}"
of_int_prod: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> int"   and A :: "'b set" assumes "class.comm_ring_1 times one plus zero minus uminus" shows "ring_1.of_int one plus zero minus (prod f A) = comm_monoid_mult.prod times one (\<lambda>x. ring_1.of_int one plus zero minus (f x)) A"
cut_le_subset_mono: fixes A :: "'a set"   and B :: "'a set"   and t :: "'a" assumes "A \<subseteq> B" shows "A \<down>\<le> t \<subseteq> B \<down>\<le> t"
assms_4: shows "reachable x n ts (\<lambda>_. 0) (ts', ms, r)"
diff_fun_charts_euclD: fixes k :: "enat"   and g :: "'c \<Rightarrow> 'd" assumes "diff_fun k charts_eucl g" shows "k-smooth_on UNIV g"
map_vec_to_iarray_rw: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" shows "map (\<lambda>x. vec_to_iarray (A $ mod_type_class.from_nat x)) [0..<CARD('c)] ! mod_type_class.to_nat i = vec_to_iarray (A $ i)"
c: shows "(\<Sum>x\<in>A. real_of_int (f x) / real_of_int (g x)) = real_of_int c / real_of_int d"
delete2_def2: fixes x :: "'a"   and t :: "'a tree"   and n :: "nat"   and dl :: "nat" shows "local.delete2 x (t, n, dl) = (let t' = del x t in case t' of None \<Rightarrow> (t, n, dl) | Some t' \<Rightarrow> let n' = n - 1; dl' = dl + 1 in if bal_d n' dl' then (t', n', dl') else let t'' = Root_Balanced_Tree.bal_tree n' t' in (t'', n', 0))"
complex_cnj_mult: fixes x :: "complex"   and y :: "complex" shows "cnj (x * y) = cnj x * cnj y"
tendsto_ennrealI: fixes f :: "'a \<Rightarrow> real"   and x :: "real"   and F :: "'a filter" assumes "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. ennreal (f x)) \<longlongrightarrow> ennreal x) F"
face_cycle_set_succ: fixes a :: "'b" shows "H.face_cycle_set (H.face_cycle_succ a) = H.face_cycle_set a"
size_alist_def: fixes al :: "('a, 'b) alist" shows "size al = length (alist.impl_of al)"
bigtheta_trans2': fixes g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and f :: "'a \<Rightarrow> 'b"   and h :: "'a \<Rightarrow> 'b" assumes "g \<in> \<Theta>[F](f)"   and "g \<in> o[F](h)" shows "f \<in> o[F](h)"
rcis_zero_arg: fixes r :: "real" shows "rcis r 0 = complex_of_real r"
labeled_tiebreaker_lifting_def: fixes Bot_F :: "'a set"   and Inf_F :: "'a inference set"   and Bot_G :: "'b set"   and entails_G :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and Inf_G :: "'b inference set"   and Red_I_G :: "'b set \<Rightarrow> 'b inference set"   and Red_F_G :: "'b set \<Rightarrow> 'b set"   and \<G>_F :: "'a \<Rightarrow> 'b set"   and \<G>_I :: "'a inference \<Rightarrow> 'b inference set option"   and Prec_F :: "'b \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Inf_FL :: "('a \<times> 'c) inference set" shows "labeled_tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F Inf_FL \<equiv> tiebreaker_lifting Bot_F Inf_F Bot_G entails_G Inf_G Red_I_G Red_F_G \<G>_F \<G>_I Prec_F \<and> labeled_tiebreaker_lifting_axioms Inf_F Inf_FL"
euclidean_size_1: fixes divide :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and one :: "'d"   and modulo :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and euclidean_size :: "'d \<Rightarrow> nat"   and division_segment :: "'d \<Rightarrow> 'd" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "euclidean_size one = 1"
monadic_filter_rev_aux_def: fixes a :: "'b list"   and P :: "'b \<Rightarrow> bool nres"   and l :: "'b list" shows "monadic_filter_rev_aux a P l \<equiv> (rec\<^sub>T D. (\<lambda>(l, a). case l of [] \<Rightarrow> return a | v # l \<Rightarrow> P v \<bind> (\<lambda>c. let a = if c then v # a else a in D (l, a)))) (l, a)"
cong_mult_self_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo (times b a) zero a"
check_poly_weak_anti_mono_discrete: fixes p :: "('b monom \<times> 'a) list"   and v :: "'b" assumes "discrete"   and "check_poly_weak_anti_mono_discrete p v" shows "poly_weak_anti_mono p v"
is_alt_factorization_of_def: fixes fctrs :: "'b \<times> ('b \<times> nat) list"   and p :: "'b poly" shows "is_alt_factorization_of fctrs p = (interp_alt_factorization fctrs = p \<and> (0::'b) \<notin> set (map fst (snd fctrs)) \<and> distinct (map fst (snd fctrs)))"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_attribute :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> char list option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_attribute_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_attribute\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs"   and "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_attribute_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_attribute set_attribute_locs get_disconnected_nodes get_disconnected_nodes_locs"
prime_factors_of_prime: fixes p :: "nat" assumes "prime p" shows "prime_factors p = {p}"
prep: fixes bs :: "real list"   and p :: "fm" shows "Ifm bs (prep p) = Ifm bs p"
le_zero_eq: fixes n :: "'k multiset" shows "(n \<subseteq># {#}) = (n = {#})"
dilating_injects: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and A :: "nat set" assumes "dilating f sub r" shows "inj_on f A"
continuous: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'c filter"   and g :: "'c \<Rightarrow> 'a" assumes "bounded_linear f"   and "continuous F g" shows "continuous F (\<lambda>x. f (g x))"
a0i_running_def: fixes A :: "agent"   and B :: "agent"   and M :: "msg" shows "a0i_running A B M \<equiv> {(s, s'). s' = s\<lparr>a0n_state.signals := addSignal (a0n_state.signals s) (Running A B M)\<rparr>}"
minus_le_iff: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_group_add plus zero minus uminus less_eq less" shows "less_eq (uminus a) b = less_eq (uminus b) a"
unit_circle_fix_iff: fixes M :: "moebius" shows "unit_circle_fix M = (moebius_circline M unit_circle = unit_circle)"
nonzero_inverse_eq_divide: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "a \<noteq> zero" shows "inverse a = divide one a"
eventually_atLeastAtMostD_1: fixes f :: "real \<Rightarrow> real"   and l :: "real \<Rightarrow> real"   and u :: "real \<Rightarrow> real" assumes "\<forall>\<^sub>F x in at_top. f x \<in> {l x..u x}" shows "\<forall>\<^sub>F x in at_top. l x \<le> f x"
dagger_of_mat_on_ket: fixes A :: "complex mat"   and v :: "complex vec" assumes "dim_col A = dim_vec v" shows "A * |v\<rangle>\<^sup>\<dagger> = \<langle>|v\<rangle>| * A\<^sup>\<dagger>"
converse_Times: fixes A :: "'c set"   and B :: "'b set" shows "(A \<times> B)\<inverse> = B \<times> A"
add_leD2: fixes m :: "nat"   and k :: "nat"   and n :: "nat" assumes "m + k \<le> n" shows "k \<le> n"
JVM_le_Err_conv: fixes P :: "'a prog"   and m :: "nat"   and n :: "nat" shows "JVM_SemiType.le P m n = Err.le (sup_state_opt P)"
fstar_fiter_id: fixes f :: "'b \<Rightarrow> 'b" assumes "Inf_pres f" shows "fstar f = fiter_id f"
group_ow_def: fixes U :: "'a set"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and inverse :: "'a \<Rightarrow> 'a" shows "group_ow U f z inverse \<equiv> semigroup_ow U f \<and> group_ow_axioms U f z inverse"
semiring_normalization_rules_10: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "times a zero = zero"
map_poly_mult_2: fixes A :: "'b mat"   and nr :: "nat"   and n :: "nat"   and B :: "'b mat"   and nc :: "nat"   and p :: "'b poly" assumes "A \<in> carrier_mat nr n"   and "B \<in> carrier_mat n nc" shows "map_mat (\<lambda>a. [:a:] * p) (A * B) = map_mat (\<lambda>a. [:a:] * p) A * map_mat (\<lambda>a. [:a:]) B"
that: fixes ns :: "'node list" assumes "g \<turnstile> Entry g-ns\<rightarrow>n"   and "n \<notin> set (butlast ns)" shows "thesis"
has_bochner_integral_nn_integral: fixes f :: "'c \<Rightarrow> real"   and M :: "'c measure"   and x :: "real" assumes "f \<in> borel_measurable M"   and "AE x in M. 0 \<le> f x"   and "0 \<le> x"   and "\<integral>\<^sup>+ x. ennreal (f x) \<partial>M = ennreal x" shows "has_bochner_integral M f x"
create_append_commute: fixes i1 :: "'d"   and e1 :: "'c"   and i2 :: "'d"   and e2 :: "'c" shows "\<langle>Create i1 e1\<rangle> \<rhd> \<langle>Append i2 e2\<rangle> = \<langle>Append i2 e2\<rangle> \<rhd> \<langle>Create i1 e1\<rangle>"
assms_5: shows "ns2 O s2 \<subseteq> s2"
reflexive_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "reflexive B (\<sim>)"
singleton_simplex: fixes v :: "'a" assumes "v \<in> \<Union> X" shows "{v} \<in> X"
prj_simps_13: shows "C.dom \<mu>_\<nu>\<pi>_\<rho>.Prj\<^sub>1 = H\<mu>HH\<nu>\<pi>\<rho>.chine"
up_V: fixes Z :: "('c \<Rightarrow> real) set" assumes "Z \<subseteq> V" shows "Z\<^sup>\<up> \<subseteq> V"
atLeast_Suc_greaterThan: fixes k :: "nat" shows "{Suc (k::nat)..} = {k<..}"
matches: shows "\<forall>jj<j. t ! (i + jj) = s ! jj"
exhaust: fixes y :: "keymode" obtains "y = Signature"   | "y = Encryption"
deg_monom: fixes a :: "'a"   and n :: "nat" assumes "a \<noteq> \<zero>"   and "a \<in> carrier R" shows "deg R (monom P a n) = n"
NSdifferentiableI: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a"   and D :: "'a" assumes "NSDERIV f x :> D" shows "f NSdifferentiable x"
subsetAllocation: fixes a :: "('c \<times> 'd set) set"   and c :: "('c \<times> 'd set) set" assumes "a \<in> allocationsUniverse"   and "c \<subseteq> a" shows "c \<in> allocationsUniverse"
2: fixes g :: "'t \<Rightarrow>\<^sub>0 'b" shows "q g \<in> punit.dgrad_p_set d m"
is_root_def: fixes p :: "'a list"   and x :: "'a" shows "is_root p x = (x \<in> carrier R \<and> local.eval p x = \<zero> \<and> p \<noteq> [])"
unitary_regular: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary M" shows "mat_det M \<noteq> 0"
strict_runit: fixes f :: "'b" assumes "D.ide f" shows "\<r>\<^sub>D[f] = f"
sortedP_right_of_last: fixes zs :: "'a list"   and y :: "'a" assumes "sortedP zs"   and "y \<in> set zs"   and "y \<noteq> last zs" shows "le y (last zs)"
neighbors_finite: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and u :: "nat" assumes "neighbor_invar c s t f l n" shows "finite (n u)"
l5_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "Bet A B C"   and "Bet A B D" shows "Bet A C D \<or> Bet A D C"
pts_symbolic_to_pts_symbolic_c: fixes \<S> :: "('fun, 'var) strand_step list set"   and \<A>\<^sub>d :: "('fun, 'var) extstrand_step list"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) strand_step list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" assumes "(\<S>, to_st (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d)) \<Rightarrow>\<^sup>\<bullet>\<^sup>* (\<S>', \<A>')"   and "sem\<^sub>e\<^sub>s\<^sub>t_d {} \<I> (to_est \<A>')"   and "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> \<A>\<^sub>d"   and "wf\<^sub>s\<^sub>t\<^sub>s' \<S> (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d)"   and "wf\<^sub>e\<^sub>s\<^sub>t {} \<A>\<^sub>d"   and "Ana_invar_subst (\<Union> (ik\<^sub>s\<^sub>t ` dual\<^sub>s\<^sub>t ` \<S>) \<union> ik\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d \<union> (\<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>) \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d))"   and "well_analyzed \<A>\<^sub>d"   and "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)" shows "\<exists>\<A>\<^sub>d'. \<A>' = to_st (decomp_rm\<^sub>e\<^sub>s\<^sub>t \<A>\<^sub>d') \<and> (\<S>, \<A>\<^sub>d) \<Rightarrow>\<^sup>\<bullet>\<^sub>c\<^sup>* (\<S>', \<A>\<^sub>d') \<and> sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> \<A>\<^sub>d'"
pairwise_singleton: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a" shows "pairwise P {A}"
any_le_inf: fixes x :: "'b DBMEntry" shows "x \<le> \<infinity>"
UM: shows "UM_Model GE_wtFsym GE_wtPsym GE_arOf GE_resOf GE_parOf GE_\<Phi> D eintF eintP"
axis_def: fixes k :: "'d"   and x :: "'c" shows "axis k x = (\<chi>i. if i = k then x else (0::'c))"