hcju/isabelleps huggingface dataset:
real_restrict_merge: fixes I :: "nat set"   and J :: "nat set"   and n :: "nat"   and tI :: "'a option list"   and tJ :: "'a option list" assumes "I \<inter> J = {}"   and "wf_tuple n I tI"   and "wf_tuple n J tJ" shows "restrict I (merge tI tJ) = restrict I tI \<and> restrict J (merge tI tJ) = restrict J tJ"
s: fixes p :: "'a"   and q :: "'a" shows "((p, q) \<in> s) = (\<exists>k u v. k \<parallel> p \<and> p \<parallel> u \<and> u \<parallel> v \<and> k \<parallel> q \<and> q \<parallel> v)"
unat_div_distrib: fixes v :: "'b word"   and w :: "'b word" shows "unat (v div w) = unat v div unat w"
split_asm: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'e \<Rightarrow> 'd"   and f2 :: "'f \<Rightarrow> 'd"   and sum :: "'e + 'f" shows "P (case sum of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = (\<not> ((\<exists>x1. sum = Inl x1 \<and> \<not> P (f1 x1)) \<or> (\<exists>x2. sum = Inr x2 \<and> \<not> P (f2 x2))))"
transitive_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "transitive B ((\<sqsubseteq>) \<restriction> A)"
assms: shows "p \<Turnstile>\<^sub>A (act TT, U \<rightarrow>) P"
last_ps_sgn_const: fixes ps :: "real poly list"   and x :: "real"   and y :: "real" assumes "quasi_sturm_seq ps" shows "sgn (poly (last ps) x) = sgn (poly (last ps) y)"
homeomorphic_simple_path_image_circle: fixes \<gamma> :: "real \<Rightarrow> 'a"   and r :: "real"   and a :: "complex" assumes "simple_path \<gamma>"   and "pathfinish \<gamma> = pathstart \<gamma>"   and "0 < r" shows "path_image \<gamma> homeomorphic sphere a r"
Int_lower2: fixes A :: "'c set"   and B :: "'c set" shows "A \<inter> B \<subseteq> B"
is_regular_spair_component_lt_cases: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "is_regular_spair p q" shows "component_of_term (lt (spair p q)) = component_of_term (lt p) \<or> component_of_term (lt (spair p q)) = component_of_term (lt q)"
IH: fixes x2a :: "('a, 'b) Term.term" assumes "x2a \<in> set T"   and "t \<sqsubset> x2a" shows "\<exists>f T. Fun f T \<sqsubseteq> x2a \<and> t \<in> set T"
Basis_le_norm: fixes b :: "'d"   and x :: "'d" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> norm x"
lins_def: fixes G :: "diagram" shows "lins G \<equiv> {\<pi>. distinct \<pi> \<and> set \<pi> = fset G^V <+> set G^E \<and> (\<forall>i j v e. i < length \<pi> \<and> j < length \<pi> \<and> \<pi> ! i = Inl v \<and> \<pi> ! j = Inr e \<and> v |\<in>| fst3 e \<longrightarrow> i < j) \<and> (\<forall>j k w e. j < length \<pi> \<and> k < length \<pi> \<and> \<pi> ! j = Inr e \<and> \<pi> ! k = Inl w \<and> w |\<in>| thd3 e \<longrightarrow> j < k)}"
partition_on_insert: fixes A :: "'c set"   and P :: "'c set set"   and X :: "'c set"   and A' :: "'c set" assumes "partition_on A P"   and "disjnt A X"   and "X \<noteq> {}"   and "A \<union> X = A'" shows "partition_on A' (insert X P)"
row_space_def: fixes A :: "'b mat" shows "row_space A = (\<lambda>v. v \<^sub>v* A) ` carrier_vec (dim_row A)"
wt_elim_cases_17: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and D :: "qtname"   and blk :: "stmt"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (Body D blk)\<Colon>T" obtains Ta :: "ty" where "T = Inl Ta" and "is_class (prg E) D" and "E,dt\<Turnstile>blk\<Colon>\<surd>" and "lcl E Result = Some Ta" and "is_type (prg E) Ta"
Par1F: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>\<^sup>^\<alpha> \<prec> P'" shows "P \<parallel> Q \<Longrightarrow>\<^sup>^\<alpha> \<prec> P' \<parallel> Q"
naturality: fixes f :: "'c" assumes "C.arr f" shows "src\<^sub>C (C.cod f) \<cdot>\<^sub>C src\<^sub>C f = src\<^sub>C f \<cdot>\<^sub>C src\<^sub>C (C.dom f)"
p'_pos_iff: fixes x :: "real" shows "(0 < ego2.p' x) = (if 0 < a\<^sub>e then - v\<^sub>e / a\<^sub>e < x else if a\<^sub>e < 0 then x < - v\<^sub>e / a\<^sub>e else 0 < v\<^sub>e)"
nu_seq_precongr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "\<nu> x \<le> \<nu> y" shows "\<nu> (x \<cdot> z) \<le> \<nu> (y \<cdot> z)"
left_hom_is_subcategory: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "partial_magma.arr V \<mu>" shows "subcategory V (weak_composition.left V H \<mu>)"
signed_take_bit_signed_take_bit: fixes m :: "nat"   and n :: "nat"   and a :: "'a" shows "signed_take_bit m (signed_take_bit n a) = signed_take_bit (min m n) a"
value_of_one: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "v 1\<^sub>r = 0"
"case": fixes f :: "(((common_metadata, common_metadata) blindable\<^sub>m \<times> (participant_metadata, participant_metadata) blindable\<^sub>m) \<times> view\<^sub>m list, (common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>m \<Rightarrow> 'a"   and x :: "(((common_metadata, common_metadata) blindable\<^sub>m \<times> (participant_metadata, participant_metadata) blindable\<^sub>m) \<times> view\<^sub>m list, (common_metadata blindable\<^sub>h \<times> participant_metadata blindable\<^sub>h) \<times> view\<^sub>h list) blindable\<^sub>m" shows "(case Transaction\<^sub>m x of Transaction\<^sub>m x \<Rightarrow> f x) = f x"
fInf_distl_var: fixes f :: "'b \<Rightarrow> 'c"   and G :: "('d \<Rightarrow> 'b) set" assumes "Inf_pres f" shows "f \<circ> \<Sqinter> G = \<Sqinter> ((\<circ>) f ` G)"
cs_sorted_list_of_cd: fixes \<pi> :: "nat \<Rightarrow> 'n"   and k :: "nat" shows "cs\<^bsup>\<pi>\<^esup> k = map \<pi> (sorted_list_of_set ({i. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> i} \<union> {k}))"
sup_ccINF: fixes B :: "'c set"   and a :: "'a"   and f :: "'c \<Rightarrow> 'a" assumes "countable B" shows "sup a (Inf (f ` B)) = (INF b\<in>B. sup a (f b))"
integral_cong: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "integral s f = integral s g"
frechet_derivative_eq_vector_derivative_1: fixes f :: "real \<Rightarrow> 'b"   and t :: "real" assumes "f differentiable at t" shows "frechet_derivative f (at t) 1 = vector_derivative f (at t)"
inc_never_one': fixes x :: "nat" shows "inc x \<noteq> Suc 0"
front_tickFree_charn: fixes s :: "'a event list" shows "front_tickFree s = (s = [] \<or> (\<exists>a t. s = t @ [a] \<and> tickFree t))"
sum_less_eqI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and y :: "'d" assumes "\<And>x. x \<in> A \<Longrightarrow> f x \<le> y" shows "sum f A \<le> y"
pdevs_mapping_eq: fixes xs :: "'b pdevs list" shows "pdevs_mapping xs = pdevs_applys xs"
\<psi>x: shows "\<guillemotleft>\<psi> (a', a) x : a' \<rightarrow> a\<guillemotright>"
fps_X_neq_one: shows "fps_X \<noteq> 1"
pair_in_Id_conv: fixes a :: "'a"   and b :: "'a" shows "((a, b) \<in> Id) = (a = b)"
add_cancel_left_right: fixes plus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and minus :: "'m \<Rightarrow> 'm \<Rightarrow> 'm"   and zero :: "'m"   and a :: "'m"   and b :: "'m" assumes "class.cancel_comm_monoid_add plus minus zero" shows "(plus a b = a) = (b = zero)"
comm_ring_hom_def: fixes hom :: "'c \<Rightarrow> 'd" shows "comm_ring_hom hom \<equiv> Ring_Hom.ring_hom hom"
basisD_1: fixes X :: "'a set" assumes "basis X" shows "indep X"
exhaust_sel: fixes idx_interval :: "'a idx_interval" obtains "idx_interval = IdxInterval (idx_interval.int idx_interval) (idx idx_interval)"
parts_crypt_mult: fixes s :: "(agent \<times> msg) set"   and SK :: "nat option \<times> nat set \<times> nat set"   and A :: "nat"   and B :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "Crypt (SesK SK) (A \<otimes> B) \<in> parts (used s)" shows "B \<in> fst (snd SK) \<and> (\<exists>n C. (Asset n, Token n (Auth_PriK n) B C SK) \<in> s) \<or> {A \<otimes> B, SesKey SK} \<subseteq> spied s"
less_eq_finite_5_def: fixes x :: "finite_5"   and y :: "finite_5" shows "(x \<le> y) = (case (x, y) of (a\<^sub>1, x) \<Rightarrow> True | (a\<^sub>2, a\<^sub>1) \<Rightarrow> False | (a\<^sub>2, a\<^sub>4) \<Rightarrow> False | (a\<^sub>2, _) \<Rightarrow> True | (a\<^sub>3, a\<^sub>3) \<Rightarrow> True | (a\<^sub>3, a\<^sub>5) \<Rightarrow> True | (a\<^sub>3, _) \<Rightarrow> False | (a\<^sub>4, a\<^sub>4) \<Rightarrow> True | (a\<^sub>4, a\<^sub>5) \<Rightarrow> True | (a\<^sub>4, _) \<Rightarrow> False | (a\<^sub>5, a\<^sub>5) \<Rightarrow> True | (a\<^sub>5, _) \<Rightarrow> False)"
remove_vertex_path_from_to_add: fixes x :: "'a"   and v :: "'a"   and xs :: "'a list"   and w :: "'a" assumes "v \<leadsto>xs\<leadsto>\<^bsub>H.remove_vertex x\<^esub> w" shows "v \<leadsto>xs\<leadsto>\<^bsub>H\<^esub> w"
inner_verts_def: fixes p :: "('a \<times> 'a) list" shows "inner_verts p \<equiv> tl (map fst p)"
ab_diff_conv_add_uminus: fixes a :: "'vs"   and b :: "'vs" assumes "a \<in> U\<^sub>V\<^sub>S"   and "b \<in> U\<^sub>V\<^sub>S" shows "a -\<^sub>V\<^sub>S b = a +\<^sub>V\<^sub>S -\<^sub>V\<^sub>S b"
not_neutral_contains_not_neutral: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" assumes "sup_monoid.sum g A \<noteq> bot" obtains a :: "'d" where "a \<in> A" and "g a \<noteq> bot"
ordinal_order_refl: fixes x :: "ordinal" shows "x \<le> x"
False: shows "\<not> subdegree f \<le> n - subdegree g"
missing_pattern_match_def: fixes m :: "String.literal"   and f :: "unit \<Rightarrow> 'b" shows "missing_pattern_match m f = f ()"
mult_right_leq: fixes y :: "'a vec"   and b :: "'a vec"   and A :: "'a mat"   and x :: "'a vec" assumes "dim_vec y = dim_vec b"   and "\<forall>i<dim_vec y. (0::'a) \<le> y $ i"   and "[A*\<^sub>vx]\<le>b" shows "(A *\<^sub>v x) \<bullet> y \<le> b \<bullet> y"
order_tendstoD_2: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and "open" :: "'e set \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'e"   and y :: "'e"   and F :: "'f filter"   and a :: "'e" assumes "class.order_topology less_eq less open"   and "filterlim f (topological_space.nhds open y) F"   and "less y a" shows "\<forall>\<^sub>F x in F. less (f x) a"
cases: fixes x :: "value" obtains uu_ :: "int" where "x = Num uu_"   | uv_ :: "String.literal" where "x = value.Str uv_"
map_uminus_psums: fixes xs :: "'a list" shows "map uminus (psums xs) = psums (map uminus xs)"
power_minus_odd: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "nat"   and a :: "'a" assumes "class.ring_1 one times plus zero minus uminus"   and "odd n" shows "power.power one times (uminus a) n = uminus (power.power one times a n)"
single_valued_def: fixes r :: "('e \<times> 'f) set" shows "single_valued r = (\<forall>x y. (x, y) \<in> r \<longrightarrow> (\<forall>z. (x, z) \<in> r \<longrightarrow> y = z))"
abs_msg_def: shows "abs_msg \<equiv> quot_type.abs (\<sim>) Abs_msg"
inject_1: fixes x1 :: "name"   and y1 :: "name" shows "(InputS_Rep x1 = InputS_Rep y1) = (x1 = y1)"
Cons_in_shuffles_leftI: fixes zs :: "'g list"   and xs :: "'g list"   and ys :: "'g list"   and z :: "'g" assumes "zs \<in> shuffles xs ys" shows "z # zs \<in> shuffles (z # xs) ys"
add_strict_left_mono: fixes a :: "complex"   and b :: "complex"   and c :: "complex" assumes "a <\<^sub>\<complex> b" shows "c + a <\<^sub>\<complex> c + b"
aboveS_notIn: fixes a :: "'a"   and r :: "('a \<times> 'a) set" shows "a \<notin> aboveS r a"
ev: shows "(cms ! k, mem) \<leadsto> (cms\<^sub>k', mem') \<and> k < length cms \<and> cms' = cms[k := cms\<^sub>k']"
composition: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and C :: "'e set" assumes "f \<in> A \<rightarrow> B"   and "g \<in> B \<rightarrow> C" shows "compose A g f \<in> A \<rightarrow> C"
simps_1: fixes x1 :: "float poly"   and x2 :: "float interval"   and y1 :: "float poly"   and y2 :: "float interval" shows "(TaylorModel x1 x2 = TaylorModel y1 y2) = (x1 = y1 \<and> x2 = y2)"
Basic: fixes P :: "'a set"   and f :: "'a \<Rightarrow> 'a"   and Q :: "'a set"   and \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and \<Theta> :: "('a set \<times> 'b \<times> 'a set \<times> 'a set) set"   and F :: "'c set"   and A :: "'a set" assumes "P \<subseteq> _Collect s (f s \<in> Q)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Basic f Q,A"
field_mute_fun_def: fixes x :: "'b"   and y :: "'b" shows "field_mute_fun x y = (x / y, 1::'b, y)"
in_set_tlD: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
lm062: fixes a :: "('a \<times> 'b set) set"   and G :: "'b set"   and N :: "'a set"   and bids :: "'a \<times> 'b set \<Rightarrow> 'c" assumes "a \<in> allAllocations N G" shows "sum (summedBidVector bids N G) a = sum (snd \<circ> summedBid bids) a"
valid_path_part_circlepath: fixes z :: "complex"   and r :: "real"   and s :: "real"   and t :: "real" shows "valid_path (part_circlepath z r s t)"
sl_def2: fixes P :: "'a prog"   and mxs :: "nat"   and mxl :: "nat" shows "JVM_SemiType.sl P mxs mxl \<equiv> (states P mxs mxl, JVM_SemiType.le P mxs mxl, JVM_SemiType.sup P mxs mxl)"
simps_2: fixes t :: "trm"   and k :: "nat"   and u :: "trm"   and us :: "trm list" shows "t[k \<mapsto>\<^sub>s u \<Colon> us] = t[k + \<parallel>us\<parallel> \<mapsto> u][k \<mapsto>\<^sub>s us]"
Pi_notEq: fixes i :: "nat"   and j :: "nat" assumes "i \<noteq> j" shows "Pi i \<noteq> Pi j"
get_dom_component_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_dom_component ptr" shows "ptr |\<in>| object_ptr_kinds h"
assms_1: shows "\<bar>\<Sum>i = M..N. a i / real i\<bar> < d"
append_same_eq: fixes ys :: "'l list"   and xs :: "'l list"   and zs :: "'l list" shows "(ys @ xs = zs @ xs) = (ys = zs)"
finite_abs_reach: fixes \<phi> :: "'b ltl" shows "finite (af_G_abs_opt.abs_reach (Abs \<phi>))"
pred_it_def: fixes g :: "'G"   and v :: "'V" shows "pred_it g v \<equiv> it_to_it (pred_list_it g v)"
notInKeys_notInVars: fixes x :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat"   and a :: "'a" assumes "x \<notin> keys m" shows "x \<notin> vars (MPoly_Type.monom m a)"
stone_phi_embed_strict_order_isomorphism: fixes x :: "'c stone_phi_pair"   and y :: "'c stone_phi_pair" shows "(x < y) = (stone_phi_embed x < stone_phi_embed y)"
intros: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and \<Gamma> :: "('a, 'b) web_scheme" assumes "\<And>x. d_OUT f x \<le> weight \<Gamma> x"   and "\<And>x. d_IN f x \<le> weight \<Gamma> x"   and "\<And>x. x \<notin> A \<Gamma> \<Longrightarrow> d_OUT f x \<le> d_IN f x"   and "\<And>a. a \<in> A \<Gamma> \<Longrightarrow> d_IN f a = 0"   and "\<And>b. b \<in> B \<Gamma> \<Longrightarrow> d_OUT f b = 0"   and "\<And>e. e \<notin> \<^bold>E\<^bsub>\<Gamma>\<^esub> \<Longrightarrow> f e = 0" shows "current \<Gamma> f"
natural_isomorphism_axioms: shows "natural_isomorphism C D F.G F.G F.G"
simps_2: fixes i :: "nat"   and f :: "'c \<Rightarrow> 'b"   and x :: "'c"   and h :: "'c" shows "nth_derivative (Suc i) f x h = nth_derivative i (\<lambda>x. frechet_derivative f (at x) h) x h"
Max_wrt_among_total_preorder: fixes A :: "'alt set" shows "Max_wrt_among (of_weak_ranking xs) A = {x \<in> \<Union> (set xs) \<inter> A. \<forall>y\<in>\<Union> (set xs) \<inter> A. of_weak_ranking xs y x}"
set_iterator_emp_foldli_conv: shows "set_iterator_emp = foldli []"
filterlim_filtermap: fixes f :: "'c \<Rightarrow> 'd"   and F1 :: "'d filter"   and g :: "'e \<Rightarrow> 'c"   and F2 :: "'e filter" shows "filterlim f F1 (filtermap g F2) = (LIM x F2. f (g x) :> F1)"
single_add: fixes k :: "'b"   and a :: "'c"   and b :: "'c" shows "Poly_Mapping.single k (a + b) = Poly_Mapping.single k a + Poly_Mapping.single k b"
wt_blocks: fixes Vs :: "String.literal list"   and Ts :: "ty list"   and vs :: "'addr val list"   and P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e :: "'addr expr"   and T :: "ty" assumes "length Vs = length Ts"   and "length vs = length Ts" shows "(P,E,h \<turnstile> blocks Vs Ts vs e : T) = (P,E(Vs [\<mapsto>] Ts),h \<turnstile> e : T \<and> (\<exists>Ts'. map typeof\<^bsub>h\<^esub> vs = map Some Ts' \<and> P \<turnstile> Ts' [\<le>] Ts))"
field_intro2: assumes "\<zero> \<noteq> \<one>"   and "\<And>x. x \<in> carrier R - {\<zero>} \<Longrightarrow> x \<in> Units R" shows "field R"
list_pred_cases: fixes P :: "'c \<Rightarrow> bool"   and xs :: "'c list" assumes "list_all P xs" obtains "xs = []"   | y :: "'c" and ys :: "'c list" where "xs = y # ys" and "P y" and "list_all P ys"
fMax_less_Length: fixes x :: "nat"   and m1 :: "nat"   and \<AA> :: "interp"   and k :: "WS1S_Prelim.order" assumes "x |\<in>| m1\<^bsup>\<AA>\<^esup>k" shows "fMax (m1\<^bsup>\<AA>\<^esup>k) < Length \<AA>"
dim_eq_card_independent: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'b set" assumes "vector_space scale"   and "\<not> module.dependent scale B" shows "vector_space.dim scale B = card B"
node\<^sub>i_bal_up\<^sub>i: fixes k :: "nat"   and ts :: "('a btree \<times> 'a) list"   and t :: "'a btree" assumes "bal_up\<^sub>i (node\<^sub>i k ts t)" shows "bal (Node ts t)"
closure_lessThan: fixes b :: "'a" shows "closure {..<b} = {..b}"
that: fixes v :: "'a list" assumes "w \<sim> v"   and "Lyndon v" shows "thesis"
sub_wrap_lt: fixes x :: "'b word"   and z :: "'b word" shows "(x < x - z) = (x < z)"
add_rel_well_defined: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and I :: "'c set set"   and x :: "'c set \<times> 'd"   and y :: "'c set \<times> 'd"   and z :: "'c set"   and z' :: "'c set" assumes "direct_lim S is_open \<FF> \<rho> b add_str mult_str zero_str one_str I"   and "x \<in> Sigma I \<FF>"   and "y \<in> Sigma I \<FF>"   and "z \<in> I"   and "z \<subseteq> fst x"   and "z \<subseteq> fst y"   and "z' \<in> I"   and "z' \<subseteq> fst x"   and "z' \<subseteq> fst y" shows "direct_lim.class_of \<FF> \<rho> I z (add_str z (\<rho> (fst x) z (snd x)) (\<rho> (fst y) z (snd y))) = direct_lim.class_of \<FF> \<rho> I z' (add_str z' (\<rho> (fst x) z' (snd x)) (\<rho> (fst y) z' (snd y)))"
pell_approximation_corollary: fixes x :: "real" assumes "x \<notin> \<rat>" shows "infinite {(t, u). 0 < u \<and> coprime (int u) t \<and> \<bar>real_of_int t - real u * x\<bar> < 1 / real u}"
c_def: shows "??.Stirling_Formula.c = integral {1..} (\<lambda>x. - ??.Stirling_Formula.P x) + ??.Stirling_Formula.g 1"
in_funswapid_image_iff: fixes x :: "'c"   and a :: "'c"   and b :: "'c"   and S :: "'c set" shows "(x \<in> (a \<rightleftharpoons>\<^sub>F b) ` S) = ((a \<rightleftharpoons>\<^sub>F b) x \<in> S)"
comm_group_def: fixes G :: "('a, 'b) monoid_scheme" shows "comm_group G \<equiv> Group.comm_monoid G \<and> Group.group G"
WT_elim_cases_7: fixes 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 e :: "expr"   and F :: "char list"   and Cs :: "char list list"   and v :: "expr"   and T :: "ty" assumes "P,E \<turnstile> e\<bullet>F{Cs} := v :: T" obtains C :: "char list" and T' :: "ty" where "P,E \<turnstile> e :: Class C" and "P \<turnstile> C has least F:T via Cs" and "P,E \<turnstile> v :: T'" and "P \<turnstile> T' \<le> T"
abP: shows "\<forall>i. a <\<^bsub>P i\<^esub> b"
finite_emeasure_space: shows "emeasure local.lim (space local.lim) \<noteq> \<top>"
int: fixes w :: "complex" assumes "w \<in> ball z r" shows "((\<lambda>u. f u / (u - w) ^ k) has_contour_integral g w) (circlepath z r)"
hom_0: fixes x :: "'a"   and d :: "nat" assumes "monom x d = 0" shows "x = (0::'a)"
X: fixes x :: "'a" assumes "x \<in> topspace X" shows "f x \<in> topspace Y \<and> g (f x) = x"
accs_exclusive_aux: fixes \<delta>n :: "('a, 'b) ta_rule set" assumes "accs \<delta>n n q"   and "\<delta>n = \<delta> \<union> \<delta>'"   and "\<delta>_states \<delta> \<inter> \<delta>_states \<delta>' = {}"   and "q \<in> \<delta>_states \<delta>" shows "accs \<delta> n q"
adopt_node_document_in_heap: fixes h :: "(_) heap"   and owner_document :: "(_) document_ptr"   and node :: "(_) node_ptr" assumes "Shadow_DOM.heap_is_wellformed h"   and "ShadowRootClass.known_ptrs h"   and "ShadowRootClass.type_wf h"   and "h \<turnstile> ok Shadow_DOM.adopt_node owner_document node" shows "owner_document |\<in>| document_ptr_kinds h"
assms_2: shows "\<forall>\<^sub>F x in at_top. f x < g x"
followed_by_summary: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and t :: "'c"   and s1 :: "'b"   and s2 :: "'b" assumes "dataflow_topology summary results_in" shows "results_in (results_in t s1) s2 = results_in t (s1 + s2)"
Inter_subset: fixes A :: "'b set set"   and B :: "'b set" assumes "\<And>X. X \<in> A \<Longrightarrow> X \<subseteq> B"   and "A \<noteq> {}" shows "\<Inter> A \<subseteq> B"
card_option_def: fixes n :: "nat" shows "card_option n = (if n = 0 then 0 else Suc n)"
intros_1: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'b mdecl list"   and FDTs :: "((char list \<times> char list) \<times> staticb \<times> ty) list"   and FDTs' :: "((char list \<times> char list) \<times> staticb \<times> ty) list" assumes "class P C = \<lfloor>(D, fs, ms)\<rfloor>"   and "C \<noteq> Object"   and "P \<turnstile> D has_fields FDTs"   and "FDTs' = map (\<lambda>(F, b, T). ((F, C), b, T)) fs @ FDTs" shows "P \<turnstile> C has_fields FDTs'"
finite_UNIV: assumes "class.finite TYPE('a)" shows "finite UNIV"
subset_le_dim: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "S \<subseteq> vec.span T" shows "vec.dim S \<le> vec.dim T"
prems_3: shows "A\<^sub>G \<sharp>* P"
wgraph_specs_6: fixes \<alpha>w :: "'a \<Rightarrow> 'b set \<Rightarrow> nat"   and \<alpha>g :: "'a \<Rightarrow> 'b ugraph"   and invar :: "'a \<Rightarrow> bool"   and adj :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list"   and empty :: "'a"   and add_edge :: "'b \<times> 'b \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"   and g :: "'a"   and u :: "'b"   and v :: "'b"   and d :: "nat" assumes "adt_wgraph \<alpha>w \<alpha>g invar adj empty add_edge"   and "invar g"   and "(u, v) \<notin> edges (\<alpha>g g)"   and "u \<noteq> v" shows "\<alpha>g (add_edge (u, v) d g) = ins_edge (u, v) (\<alpha>g g)"
ap_stream_homo: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e" shows "pure f \<diamondop> pure x = pure (f x)"
mset_bij: fixes f :: "nat \<Rightarrow> nat"   and xs :: "'b list"   and ys :: "'b list" assumes "bij_betw f {..<length xs} {..<length ys}"   and "\<And>i. i < length xs \<Longrightarrow> xs ! i = ys ! f i" shows "mset xs = mset ys"
inj_on_mk_uedge: shows "inj_on mk_uedge {(a, b). a < b}"
assms_3: shows "W B\<^sub>2 + w u \<le> real c"
diff: fixes x :: "'a"   and y :: "'a" assumes "vectorspace V"   and "x \<in> V"   and "y \<in> V" shows "f (x - y) = f x - f y"
last_appendR: fixes ys :: "'f list"   and xs :: "'f list" assumes "ys \<noteq> []" shows "last (xs @ ys) = last ys"
scaleR_left: fixes r :: "real"   and a :: "'a"   and b :: "'b" shows "r *\<^sub>R a ** b = r *\<^sub>R (a ** b)"
abs_finite_3_def: shows "abs = (\<lambda>x. case x of a\<^sub>3 \<Rightarrow> a\<^sub>2 | _ \<Rightarrow> x)"
hom_one: shows "(\<lambda>i\<in>I. (if i \<in> I then J i else (\<lambda>_. undefined)) (\<one>\<^bsub>DirProds Gs I\<^esub> i)) = \<one>\<^bsub>DirProds (f \<circ> Gs) I\<^esub>"
pre: shows "pre_subdivFace' g f r u n (ov # ovs)"
cong: fixes A :: "'d set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'a"   and h :: "'d \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> g x = h x" shows "F.F g A = F.F h B"