hcju/isabelleps huggingface dataset:
subst_Beta: fixes t :: "Lambda_Z.term"   and t' :: "Lambda_Z.term"   and x :: "name"   and s :: "Lambda_Z.term" assumes "t \<rightarrow>\<^sub>\<beta> t'" shows "subst x s t \<rightarrow>\<^sub>\<beta> subst x s t'"
comp_assoc: fixes h :: "'c"   and g :: "'c"   and f :: "'c" shows "C (C h g) f = C h (C g f)"
ToTC_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "'b \<times> 'b \<Rightarrow> 'b"   and f :: "'b \<times> 'b \<times> 'b" assumes "binary_endofunctor C T" shows "binary_endofunctor.ToTC C T f \<equiv> if partial_magma.arr (product_category.comp C (product_category.comp C C)) f then T (T (fst f, fst (snd f)), snd (snd f)) else partial_magma.null C"
fields_def2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and C :: "char list"   and FDTs :: "((char list \<times> char list) \<times> ty) list" assumes "P \<turnstile> C has_fields FDTs" shows "fields P C = FDTs"
lb_lnD: fixes prec :: "nat"   and x :: "float"   and y :: "float" assumes "lb_ln prec x = Some y" shows "real_of_float y \<le> ln (real_of_float x) \<and> 0 < real_of_float x"
pol_ct: fixes \<sigma>1 :: "'a"   and p :: "('b, 'c) epsym"   and \<sigma>2 :: "'d" shows "pol \<sigma>1 p = pol \<sigma>2 p"
Min_le: fixes A :: "'a DBMEntry set"   and x :: "'a DBMEntry" assumes "finite A"   and "x \<in> A" shows "linordered_monoid.Min A \<preceq> x"
nxt_geq_lNact: fixes n :: "nat"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf" assumes "\<exists>i\<ge>n. active c (t i)" shows "dynamic_component.lNAct active c t n \<le> dynamic_component.nxtAct active c t n"
min_set_idem: fixes X :: "'b set" shows "min_set (min_set X) = min_set X"
lin_indpt_list_length_eq_n_2: fixes xs :: "'a vec list" assumes "lin_indpt_list xs"   and "length xs = n" shows "basis (set xs)"
ide_compE: fixes g :: "'a \<times> 'a"   and f :: "'a \<times> 'a" assumes "VV.ide (VV.comp g f)" obtains "VV.seq g f" and "VV.seq f g" and "VV.comp g f = VV.dom f" and "VV.comp g f = VV.cod g"
mult_le_cancel_left1: 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 sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "less_eq c (times c b) = ((less zero c \<longrightarrow> less_eq one b) \<and> (less c zero \<longrightarrow> less_eq b one))"
elims_2: fixes c :: "nat \<times> nat \<Rightarrow> 'b"   and x :: "nat"   and xa :: "(nat \<times> nat) list"   and xb :: "nat" assumes "Graph.isPath c x xa xb" obtains u :: "nat" and v :: "nat" where "x = u__" and "xa = []" and "xb = v__" and "u__ = v__"   | u :: "nat" and xc :: "nat" and y :: "nat" and p :: "(nat \<times> nat) list" and v :: "nat" where "x = u" and "xa = (xc, y) # p" and "xb = v" and "u = xc \<and> (xc, y) \<in> Graph.E c \<and> Graph.isPath c y p v"
convol_def: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'c" shows "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
assms_2: shows "AE x in M. (\<lambda>xa. ereal (birkhoff_sum f xa x)) \<longlonglongrightarrow> - \<infinity>"
distr_bst_of_list_tree_sigma_count_space: shows "distr (random_perm xs) (tree_sigma (count_space A)) bst_of_list = distr (random_perm xs) (count_space (trees A)) bst_of_list"
simps_13: fixes x2 :: "'a" shows "Stuck \<noteq> Abrupt x2"
double_negation: fixes x :: "'b" shows "! (d (! (d (! (d x))))) = ! (d x)"
mkIde_as_mkArr: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set" assumes "set_category_given_img S img \<AA>"   and "A \<subseteq> Collect (category.terminal S)"   and "|A| <o \<AA>" shows "set_category_given_img.mkArr S img \<AA> A A (\<lambda>x. x) = set_category_given_img.mkIde S img \<AA> A"
intros_1: fixes b :: "(nat \<times> int) list"   and ds :: "nat set" shows "b \<in> grid b ds"
run_alt_state: fixes alt :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and m1 :: "('b, 'a) stateT"   and m2 :: "('b, 'a) stateT"   and s :: "'b" shows "run_state (alt_state alt m1 m2) s = alt (run_state m1 s) (run_state m2 s)"
image_respects_domain_permutation: fixes A :: "'b set"   and B :: "'c set" shows "(\<lambda>f. f ` A) respects domain_permutation A B"
jumpF_polyR_0_2: fixes q :: "real poly"   and a :: "real" shows "jumpF_polyR q 0 a = 0"
inject: fixes x2 :: "'l"   and y2 :: "'l" shows "(Some x2 = Some y2) = (x2 = y2)"
id_right: fixes f :: "'a" assumes "f \<in> Ar" shows "f \<bullet> Id (Dom f) = f"
F': shows "F \<in> opp_fold_A \<turnstile> ChamberComplex.chamber_system A"
Perp_def: 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" shows "Tarski_neutral_dimensionless.Perp Bet Cong A B C D \<equiv> \<exists>X. Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D"
balance_left_inv2_with_inv1_1: fixes lt :: "('a, 'b) rbt"   and rt :: "('a, 'b) rbt"   and k :: "'a"   and v :: "'b" assumes "inv2 lt"   and "inv2 rt"   and "bheight lt + 1 = bheight rt"   and "inv1 rt" shows "bheight (balance_left lt k v rt) = bheight lt + 1"
hyps: fixes p :: "'a list" assumes "length p < length pa__"   and "polynomial K p" shows "\<exists>q r. polynomial K q \<and> polynomial K r \<and> p = b \<otimes>\<^bsub>K [X]\<^esub> q \<oplus>\<^bsub>K [X]\<^esub> r \<and> (r = [] \<or> degree r < degree b)"
Nil_notin_lex: fixes ys :: "'a list"   and r :: "('a \<times> 'a) set" shows "([], ys) \<notin> lex r"
mapping_impl_choose2_code_3: shows "mapping_impl_choose2 mapping_Assoc_List mapping_Assoc_List = mapping_Assoc_List"
monoid_homomorphism_axioms: shows "monoid_homomorphism f A addA zeroA B addB zeroB"
splitit: fixes xs :: "'a list"   and q :: "'a"   and i :: "nat"   and thesis :: "bool"   and n :: "nat" assumes "index xs q < i \<and> i < length xs \<Longrightarrow> thesis"   and "index xs q = i \<Longrightarrow> thesis"   and "index xs q - n \<le> i \<and> i < index xs q \<Longrightarrow> thesis"   and "i < index xs q - n \<Longrightarrow> thesis"   and "i < length xs" shows "thesis"
RES_bind_choose: fixes x :: "'a"   and X :: "'a set"   and m :: "'b nres"   and f :: "'a \<Rightarrow> 'b nres" assumes "x \<in> X"   and "m \<le> f x" shows "m \<le> RES X \<bind> f"
s1_1_th: fixes n :: "nat"   and x :: "nat" shows "(\<lambda>y. nat_to_pr n (c_pair x y)) = nat_to_pr (s1_1 n x)"
comp_fun_commute_axioms: shows "comp_fun_commute (vadd \<circ> vprod_size)"
list_assn_prod_map: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> assn"   and B :: "'c \<Rightarrow> 'e \<Rightarrow> assn"   and xs :: "('b \<times> 'c) list"   and ys :: "('d \<times> 'e) list" shows "list_assn (A \<times>\<^sub>a B) xs ys = list_assn B (separators xs) (separators ys) * list_assn A (subtrees xs) (subtrees ys)"
congruent_triangle_asa_2: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "angle a1 b1 c1 = angle a2 b2 c2"   and "dist a1 b1 = dist a2 b2"   and "angle b1 a1 c1 = angle b2 a2 c2"   and "\<not> collinear {a1, b1, c1}" shows "dist a1 c1 = dist a2 c2"
mset_big_union_obtain: fixes x :: "'a"   and A :: "'a multiset multiset" assumes "x \<in># \<Sum>\<^sub># A" obtains a :: "'a multiset" where "a \<in># A" and "x \<in># a"
clauseFalseInPrefixToLastAssertedLiteral: fixes l :: "Literal"   and c :: "Literal list"   and M :: "(Literal \<times> bool) list" assumes "isLastAssertedLiteral l (oppositeLiteralList c) (elements M)"   and "clauseFalse c (elements M)"   and "uniq (elements M)" shows "clauseFalse c (elements (prefixToLevel (elementLevel l M) M))"
rbt_minus_def: fixes t1 :: "('a, 'c) rbt"   and t2 :: "('a, 'c) rbt" shows "rbt_minus t1 t2 = (let h1 = bheight t1; h2 = bheight t2 in if 2 * h1 \<le> h2 then rbtreeify (fold_rev (\<lambda>k v kvs. if rbt_lookup t2 k = None then (k, v) # kvs else kvs) t1 []) else RBT_Impl.fold (\<lambda>k v. rbt_delete k) t2 t1)"
select_convs_3: fixes variables_of :: "'c list"   and operators_of :: "'c strips_operator list"   and initial_of :: "'c \<Rightarrow> bool option"   and goal_of :: "'c \<Rightarrow> bool option"   and more :: "'d" shows "\<lparr>strips_problem.variables_of = variables_of, operators_of = operators_of, initial_of = initial_of, goal_of = goal_of, \<dots> = more\<rparr>\<^sub>I = initial_of"
carrier_reduced_homology_group: fixes p :: "int"   and X :: "'a topology" shows "carrier (reduced_homology_group p X) = kernel (homology_group p X) (homology_group p (discrete_topology {()})) (hom_induced p X {} (discrete_topology {()}) {} (\<lambda>x. ()))"
fixf_subset: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b"   and f :: "'b \<Rightarrow> 'b" assumes "A \<subseteq> B"   and "x \<in> fix (restrict f A) A" shows "x \<in> fix f B"
set1_G_bound: fixes x :: "('n, 't, 'o, 'p, 'q, 'r, 's) G" shows "|set1_G x| \<le>o bd_G"
to_nat_less_card: fixes a :: "'d" shows "Bij_Nat.to_nat a < CARD('d)"
stirling_integral_bigo: fixes \<alpha> :: "real"   and m :: "nat" assumes "\<alpha> \<in> {0<..<pi}"   and "0 < m" shows "stirling_integral m \<in> O[principal (complex_cone' \<alpha> - {0})](\<lambda>s. 1 / s ^ m)"
simple_restrict_none: fixes i :: "nat"   and X :: "'c option list"   and A :: "nat set" assumes "i < length X"   and "i \<notin> A" shows "restrict A X ! i = None"
unlabel_in: fixes l :: "'d"   and x :: "'e"   and A :: "('d \<times> 'e) list" assumes "(l, x) \<in> set A" shows "x \<in> set (unlabel A)"
min_elems_no: fixes x :: "'a list"   and A :: "'a list set"   and t :: "'a list" assumes "x \<in> min_elems A"   and "t \<in> A"   and "t \<le> x" shows "x = t"
map_sum_plus_oracle: fixes f :: "'b \<Rightarrow> 'f"   and g :: "'c \<Rightarrow> 'g"   and h :: "'h \<Rightarrow> 'd"   and k :: "'i \<Rightarrow> 'e"   and oracle1 :: "'a \<Rightarrow> 'f \<Rightarrow> ('h \<times> 'a) spmf"   and oracle2 :: "'a \<Rightarrow> 'g \<Rightarrow> ('i \<times> 'a) spmf" shows "map_fun id (map_fun (map_sum f g) (map_spmf (map_prod (map_sum h k) id))) (oracle1 \<oplus>\<^sub>O oracle2) = map_fun id (map_fun f (map_spmf (map_prod h id))) oracle1 \<oplus>\<^sub>O map_fun id (map_fun g (map_spmf (map_prod k id))) oracle2"
vars_term_subst: fixes t :: "('d, 'e) Term.term"   and \<sigma> :: "'e \<Rightarrow> ('d, 'c) Term.term" shows "fv (t \<cdot> \<sigma>) = \<Union> (fv ` \<sigma> ` fv t)"
interaction_bound_parametric': fixes R :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and C :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and A :: "'j \<Rightarrow> 'k \<Rightarrow> bool" assumes "bi_total R" shows "((C ===> (=)) ===> rel_gpv'' A C R ===> (=)) interaction_bound interaction_bound"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'c tree \<Rightarrow> 'c tree \<Rightarrow> nat \<Rightarrow> 'b"   and tree :: "'c tree" shows "P (case tree of ET \<Rightarrow> f1 | MKT x xa xb xc \<Rightarrow> f2 x xa xb xc) = ((tree = ET \<longrightarrow> P f1) \<and> (\<forall>x21 x22 x23 x24. tree = MKT x21 x22 x23 x24 \<longrightarrow> P (f2 x21 x22 x23 x24)))"
assms_2: shows "elapsed u u' \<le> (0::'a)"
get_owner_document_pure: fixes get_parent :: "(_) node_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr option) prog"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf :: "(_) heap \<Rightarrow> bool"   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"   and get_root_node :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr) prog"   and get_owner_document :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) document_ptr) prog"   and ptr :: "(_) object_ptr"   and h :: "(_) heap" assumes "l_get_owner_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_parent known_ptr type_wf get_disconnected_nodes get_disconnected_nodes_locs get_root_node get_owner_document" shows "pure (get_owner_document ptr) h"
elims: fixes x :: "'b list"   and xa :: "'b list"   and y :: "'b list" assumes "remove_all_cycles x xa = y" obtains xs :: "'b list" where "x = []" and "xa = xs__" and "y = xs__"   | xb :: "'b" and xs :: "'b list" and ys :: "'b list" where "x = xb # xs" and "xa = ys" and "y = remove_all_cycles xs (start_remove ys xb [])"
mem_ord_of_nat_iff: fixes x :: "V"   and n :: "nat" shows "(x \<in> elts (ord_of_nat n)) = (\<exists>m<n. x = ord_of_nat m)"
prems: shows "x \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (fflip (fflip f))"
degree_gcd1: fixes a :: "'d poly"   and b :: "'d poly" assumes "a \<noteq> 0" shows "degree (gcd a b) \<le> degree a"
comp_xs: shows "Result.those (map (compile_env_entry \<A> \<O>) xs) = Ok ys'"
zn_sg: shows "subgroup_generated (relative_homology_group (int n) (usphere n) (equator n)) {zn} = relative_homology_group (int n) (usphere n) (equator n)"
negligible_outer: fixes S :: "'a set" shows "negligible S = (\<forall>e>0. \<exists>T. S \<subseteq> T \<and> T \<in> lmeasurable \<and> Sigma_Algebra.measure lebesgue T < e)"
iterate_Suc: fixes n :: "nat"   and F :: "'a \<rightarrow> 'a"   and x :: "'a" shows "F\<^bsup>Suc n\<^esup>\<cdot>x = F\<cdot>(F\<^bsup>n\<^esup>\<cdot>x)"
unrest_var: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b" assumes "mwb_lens x"   and "x \<bowtie> y" shows "y \<sharp> x"
leg0_commutes: shows "assoc\<mu>\<nu>\<pi>.cod.leg0 \<cdot> assoc\<mu>\<nu>\<pi>.chine = assoc\<mu>\<nu>\<pi>.dom.leg0"
intersection_num_non_neg: fixes b1 :: "'b set"   and b2 :: "'b set" shows "0 \<le> b1 |\<inter>| b2"
mult_commute: fixes a :: "'a"   and b :: "'a" shows "a \<cdot> b = b \<cdot> a"
map_closed: fixes a :: "'a" assumes "a \<in> M" shows "Class a \<in> Partition"
root_int_ceiling_def: fixes p :: "nat"   and x :: "int" shows "root_int_ceiling p x = (if 0 \<le> x then root_int_ceiling_pos p x else - root_int_floor_pos p (- x))"
bind_spmf_cong: fixes p :: "'c spmf"   and q :: "'c spmf"   and f :: "'c \<Rightarrow> 'd spmf"   and g :: "'c \<Rightarrow> 'd spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q \<Longrightarrow> f x = g x" shows "p \<bind> f = q \<bind> g"
subdomain_iff: fixes R :: "('e, 'f) ring_scheme"   and H :: "'e set" assumes "ring R"   and "H \<subseteq> carrier R" shows "subdomain H R = domain (R\<lparr>carrier := H\<rparr>)"
similar_mat_witD_3: fixes n :: "nat"   and A :: "'b mat"   and B :: "'b mat"   and P :: "'b mat"   and Q :: "'b mat" assumes "n = dim_row A"   and "similar_mat_wit A B P Q" shows "A = P * B * Q"
RModHom_idhom: shows "RModuleHom UNIV (\<cdot>) V (\<cdot>) (id \<down> V)"
inject_3: fixes x5 :: "'b"   and y5 :: "'b" shows "(Addr x5 = Addr y5) = (x5 = y5)"
uw: shows "cbox u w = cbox u v \<union> cbox v w"
execn_nothrows_no_Abrupt: 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 "nothrows c"   and "\<forall>p\<in>dom \<Gamma>. nothrows (the (\<Gamma> p))"   and "\<not> isAbr s" shows "\<not> isAbr t"
transfer: shows "rel_fun pcr_real (rel_fun pcr_real pcr_real) (\<lambda>X Y n. X n + Y n) (+)"
defAss'_def: fixes g :: "'g"   and m :: "'node"   and v :: "'var" shows "defAss' g m v = (\<forall>ns. g \<turnstile> Entry g-ns\<rightarrow>m \<longrightarrow> (\<exists>n\<in>set ns. v \<in> defs g n))"
inorder_update: fixes t :: "('c \<times> 'd) tree23"   and x :: "'c"   and y :: "'d" assumes "sorted1 (inorder t)" shows "inorder (update x y t) = upd_list x y (inorder t)"
v_TS_def: shows "v_TS = \<lparr>init = v_init, trans = v_trans\<rparr>"
rep_eq: fixes x :: "ennreal"   and xa :: "ennreal" shows "(x \<le> xa) = (enn2ereal x \<le> enn2ereal xa)"
all_shared_share_acquired: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and a :: "nat" assumes "sharing_consistent \<S> \<O> sb"   and "a \<in> all_shared sb" shows "a \<in> dom (share sb \<S>) \<union> acquired True sb \<O>"
decode_state_at_valid_variable: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and k :: "nat"   and v :: "'b" assumes "(\<Phi>\<^sub>S\<inverse> \<Pi> \<A> k) v \<noteq> None" shows "v \<in> set (\<Pi>\<^sub>\<V>)"
Union_disjoint: fixes C :: "'a set set"   and g :: "'a \<Rightarrow> 'b" assumes "\<forall>A\<in>C. finite A"   and "\<forall>A\<in>C. \<forall>B\<in>C. A \<noteq> B \<longrightarrow> A \<inter> B = {}" shows "prod g (\<Union> C) = (prod \<circ> prod) g C"
a1: shows "low_equal s1 s2 \<and> t1 = snd (memory_read 10 address s1) \<and> t2 = snd (memory_read 10 address s2)"
ring_id_closed: shows "ring_id \<in> carrier (function_ring (carrier Zp) Zp)"
applyEdge_gState_progress_rel: fixes prog :: "program"   and g :: "gState\<^sub>I"   and p :: "pState"   and e :: "edge" assumes "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p"   and "cl_inv (g, p)"   and "e \<in> edgeSet (states prog !! pState.idx p)" shows "applyEdge prog e p g \<le> SPEC (\<lambda>g'. (g, g') \<in> gState_progress_rel prog)"
knows_subset_knows_Cons: fixes A :: "agent"   and evs :: "event list"   and e :: "event" shows "knows A evs \<subseteq> knows A (e # evs)"
intro: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a> :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i> :: "'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and r :: "'c"   and \<rho> :: "'c"   and f :: "'c"   and g :: "'c"   and u :: "'c"   and w :: "'c"   and \<theta> :: "'c" assumes "tabulation_data V H \<a> \<i> src trg r \<rho> f g"   and "uw\<theta>_axioms V H f u w \<theta>" shows "uw\<theta> V H \<a> \<i> src trg r \<rho> f g u w \<theta>"
tendsto_Re: fixes g :: "'a \<Rightarrow> quat"   and a :: "quat"   and F :: "'a filter" assumes "(g \<longlongrightarrow> a) F" shows "((\<lambda>x. quat.Re (g x)) \<longlongrightarrow> quat.Re a) F"
rassert_def: fixes c :: "'b upred" shows "{c}\<^sub>\<bottom> = II \<triangleleft> c \<triangleright>\<^sub>r true"
degree_def: fixes x :: "'c pdevs" shows "degree x = (LEAST i. \<forall>j\<ge>i. pdevs_apply x j = (0::'c))"
intro: fixes f\<^sub>a :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and f\<^sub>b :: "'d \<Rightarrow> 'd \<Rightarrow> 'd" assumes "Type_Semigroups.semigroup f\<^sub>a"   and "Type_Semigroups.semigroup f\<^sub>b" shows "semigroup_pair f\<^sub>a f\<^sub>b"
module_with_axioms: shows "module_with (+\<^sub>M\<^sub>_\<^sub>2) 0\<^sub>M\<^sub>_\<^sub>2 (-\<^sub>M\<^sub>_\<^sub>2) uminus\<^sub>M\<^sub>_\<^sub>2 (*s\<^sub>w\<^sub>i\<^sub>t\<^sub>h\<^sub>_\<^sub>2)"
measure_countable_negligible_Union_bounded: fixes S :: "nat \<Rightarrow> 'c set" assumes "\<And>n. S n \<in> lmeasurable"   and "pairwise (\<lambda>m n. negligible (S m \<inter> S n)) UNIV"   and "bounded (\<Union> (range S))" shows "(\<lambda>n. Sigma_Algebra.measure lebesgue (S n)) sums Sigma_Algebra.measure lebesgue (\<Union> (range S))"
isomorphic_to_identity_is_equivalence: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "natural_isomorphism (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R S.map \<phi>" shows "equivalence_functor (\<cdot>\<^sub>S) (\<cdot>\<^sub>S) R"
lmaps_to_axpr'_def: fixes b :: "bexp"   and bv :: "bool" shows "lmaps_to_axpr' b bv = lmaps_to_axpr b bv"
B_trusts_NS3_lemma: fixes A :: "agent"   and B :: "agent"   and evs :: "event list"   and NB :: "nat"   and NA :: "nat" assumes "A \<notin> bad"   and "B \<notin> bad"   and "evs \<in> ns_public"   and "Crypt (pubK B) (Nonce NB) \<in> parts (knows Spy evs)"   and "Says B A (Crypt (pubK A) \<lbrace>Nonce NA, Nonce NB\<rbrace>) \<in> set evs" shows "\<exists>C. Says A C (Crypt (pubK C) (Nonce NB)) \<in> set evs"
sub_inc_One_eq: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and n :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.sub minus plus one (Num.inc n) num.One = numeral.numeral one plus n"
lem: fixes n :: "nat"   and k :: "nat" assumes "n < Suc k"   and "n \<noteq> k" shows "n < k"
that: fixes y :: "'a"   and ys' :: "'a list" assumes "y \<in> x"   and "ys' \<in> listset xs"   and "ys = y # ys'" shows "thesis"
exhaust: fixes y :: "p_edge_kind" obtains x1 :: "(char list, val, char list \<times> label, char list) edge_kind" where "y = IEdge x1"   | x2 :: "char list \<times> expr list \<times> char list list" where "y = CEdge x2"
map_spmf_eq_map_spmf_iff: fixes f :: "'b \<Rightarrow> 'a"   and p :: "'b spmf"   and g :: "'c \<Rightarrow> 'a"   and q :: "'c spmf" shows "(map_spmf f p = map_spmf g q) = rel_spmf (\<lambda>x y. f x = g y) p q"
1: shows "rat_poly_inv (rat_poly_times (A - rat_poly_times (rat_poly_times B B) B) A) = rat_poly_times B B - rat_poly_times A A"
induct: fixes P :: "('b, 'c) RBT_Impl.rbt \<Rightarrow> bool"   and rbt :: "('b, 'c) RBT_Impl.rbt" assumes "P rbt.Empty"   and "\<And>x1 x2 x3 x4 x5. \<lbrakk>P x2; P x5\<rbrakk> \<Longrightarrow> P (Branch x1 x2 x3 x4 x5)" shows "P rbt"
boundedE: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "a \<le> inf b c" obtains "a \<le> b" and "a \<le> c"
not_numeral_less_one: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less (numeral.numeral one plus n) one"
two_composable_identity_arrows_of_spans_axioms: shows "two_composable_identity_arrows_of_spans (\<cdot>) prj0 prj1 f g"
le_imp_0_less: fixes z :: "int" assumes "0 \<le> z" shows "0 < 1 + z"
th001: fixes k :: "'b"   and f :: "'b \<Rightarrow> 'c" shows "(\<lambda>x. if x = k then f k else f x) = f"
bijections_in_range_permutation: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "{f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} \<in> {f \<in> A \<rightarrow>\<^sub>E B. bij_betw f A B} // range_permutation A B"
cop_F_CH_CD_on_disjoint: fixes ds :: "'b set"   and fp :: "'a set" assumes "cop_F_closed_inv ds fp"   and "cop_F_range_inv ds fp" shows "Xd ` CH fp \<inter> Xd ` (CD_on ds (- Rf CH fp) - fp) = {}"
INTERP_def: shows "INTERP = \<Union> (production ` N)"
prod_run: fixes r :: "nat \<Rightarrow> 'q \<times> 's" shows "prod.is_run r = (igba.is_run (fst \<circ> r) \<and> sa.is_run (snd \<circ> r) \<and> (\<forall>i. local.igba.L (fst (r i)) (local.sa.L (snd (r i)))))"
compact_Int_closed: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "topological_space.compact open S"   and "topological_space.closed open T" shows "topological_space.compact open (S \<inter> T)"
f1: shows "\<forall>a. a - \<pi> a \<in> orthogonal_complement M \<and> \<pi> a \<in> M"
affine_imp_polyhedron: fixes S :: "'b set" assumes "affine S" shows "polyhedron S"
rawpsubstT_atrm: fixes r :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r \<in> atrm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> atrm" shows "rawpsubstT r txs \<in> atrm"
bound: fixes s :: "'s" assumes "guard s"   and "I s" shows "f s \<le> bound"
is_extensional: fixes f :: "'a" assumes "\<not> S.arr f" shows "S.trg f = S.null"
mask_lt_2pn: fixes n :: "nat" assumes "n < LENGTH('b)" shows "mask n < 2 ^ n"
Fun_char: fixes f :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" assumes "arr f" shows "Fun f = (\<lambda>x\<^bold>\<in>Dom f. UP (happ (arr_to_hfun f) (DOWN x)))"