hcju/isabelleps huggingface dataset:
elims: fixes x :: "'a word \<times> 'a word"   and y :: "'a wordinterval" assumes "iprange_interval x = y" obtains ip_start :: "'a word" and ip_end :: "'a word" where "x = (ip_start, ip_end)" and "y = WordInterval ip_start ip_end"
abs_summable_on_cong_neutral: fixes A :: "'b set"   and B :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>x. x \<in> A - B \<Longrightarrow> f x = (0::'c)"   and "\<And>x. x \<in> B - A \<Longrightarrow> g x = (0::'c)"   and "\<And>x. x \<in> A \<inter> B \<Longrightarrow> f x = g x" shows "(f abs_summable_on A) = (g abs_summable_on B)"
assm: shows "|lan v| = 1 \<and> 0 < \<parallel>ext v\<parallel> \<and> \<not> (0 < \<parallel>ext v\<parallel> \<and> |lan v| = 1 \<and> (\<forall>c. \<parallel>len v ts c\<parallel> = 0 \<or> restrict v (clm ts) c = bot \<and> restrict v (res ts) c = bot))"
allE: fixes P :: "'f \<Rightarrow> bool"   and x :: "'f" assumes "\<forall>x. P x" obtains "P x"
ccw'_sortedP_polychain_of2: fixes u :: "real \<times> real"   and v :: "real \<times> real"   and x0 :: "real \<times> real" assumes "ccw'.sortedP 0 [u, v]" shows "ccw'.sortedP x0 (map snd (polychain_of x0 [u, v]))"
Gromov_product_le_infdist: fixes G :: "'b set"   and x :: "'b"   and y :: "'b"   and e :: "'b" assumes "geodesic_segment_between G x y" shows "Gromov_product_at e x y \<le> infdist e G"
axioms_1: fixes \<alpha> :: "V"   and \<CC> :: "V" assumes "finite_digraph \<alpha> \<CC>" shows "digraph \<alpha> \<CC>"
hm_lookup_rule: fixes m :: "'b \<Rightarrow> 'a option"   and ht :: "('b, 'a) hashtable"   and k :: "'b" shows "<is_hashmap m ht> hm_lookup k ht <\<lambda>r. is_hashmap m ht * \<up> (r = m k)>"
rrts: shows "set rrts = {x. poly (map_poly complex_of_real q) x = 0 \<and> x \<in> \<real>}"
less_le: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'e"   and y :: "'e" assumes "class.order less_eq less" shows "less x y = (less_eq x y \<and> x \<noteq> y)"
convol_def: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g" shows "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
pcr_cr_eq: shows "pcr_mapping (=) (=) = cr_mapping"
hyps_4: shows "pmdl (set gs \<union> set (snd (sym_preproc_aux gs (ks @ [v]) (sym_preproc_addnew gs vs' fs v)))) = pmdl (set gs \<union> set (snd (sym_preproc_addnew gs vs' fs v)))"
\<psi>D: fixes T :: "'v \<Rightarrow> 'w" assumes "T \<in> GRepHomSet (\<star>) W" shows "\<psi>_condition T (\<psi> T)"
vdomain_dg_prod_Dom: shows "\<D>\<^sub>\<circ> (dg_prod I \<AA>\<lparr>Dom\<rparr>) = dg_prod I \<AA>\<lparr>Arr\<rparr>"
mult_sound: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "sound P"   and "sound Q" shows "sound (\<lambda>s. P s * Q s)"
lesubstep_union: fixes A\<^sub>1 :: "(nat \<times> 'a) set"   and r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and B\<^sub>1 :: "(nat \<times> 'a) set"   and A\<^sub>2 :: "(nat \<times> 'a) set"   and B\<^sub>2 :: "(nat \<times> 'a) set" assumes "A\<^sub>1 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>1"   and "A\<^sub>2 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>2" shows "A\<^sub>1 \<union> A\<^sub>2 {\<sqsubseteq>\<^bsub>r\<^esub>} B\<^sub>1 \<union> B\<^sub>2"
distr_pair_PiM_eq_PiM: fixes I :: "'a set"   and M :: "'a \<Rightarrow> 'b measure"   and i' :: "'a" assumes "\<And>i. i \<in> I \<Longrightarrow> prob_space (M i)"   and "prob_space (M i')" shows "distr (M i' \<Otimes>\<^sub>M Pi\<^sub>M I M) (Pi\<^sub>M (insert i' I) M) (\<lambda>(x, X). X(i' := x)) = Pi\<^sub>M (insert i' I) M"
aI_aHom: fixes A :: "('c, 'd) aGroup_scheme" assumes "aGroup A" shows "aI\<^bsub>A\<^esub> \<in> aHom A A"
IH: shows "eval_poly \<alpha> p = eval_poly \<beta> p"
m_div_def: fixes G :: "('a, 'b) ring_scheme"   and x :: "'a"   and y :: "'a" shows "x \<oslash>\<^bsub>G\<^esub> y = (if y = \<zero>\<^bsub>G\<^esub> then \<zero>\<^bsub>G\<^esub> else x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y)"
vec_index_of_poly_n: fixes i :: "nat"   and n :: "nat"   and p :: "'a poly" assumes "i < n" shows "vec_of_poly_n p n $ i = (if i < n - Suc (degree p) then 0::'a else coeff p (n - i - 1))"
homeomorphic_interiors_same_dimension: fixes S :: "'a set"   and T :: "'b set" assumes "S homeomorphic T"   and "DIM('a) = DIM('b)" shows "interior S homeomorphic interior T"
\<Phi>_preserves_incl: fixes m :: "'t" assumes "S'.incl m" shows "S.incl (\<Psi> m)"
init_notempty: shows "\<forall>j. init (np j) \<noteq> {}"
intros: fixes A :: "'a set"   and B :: "'a set"   and G :: "('a, 'b) graph_scheme"   and S :: "'a set" assumes "\<And>x y p. \<lbrakk>x \<in> A; y \<in> B; path G x p y\<rbrakk> \<Longrightarrow> (\<exists>z\<in>set p. z \<in> S) \<or> x \<in> S" shows "separating_gen G A B S"
coeffs: shows "coeff (pCons b c2) (degree q) mod lead_coeff q = 0"
extremum: fixes a :: "'b upred" shows "a \<sqsubseteq> false"
a1: shows "Esigma (a * r * c) = Esigma a * Esigma (r * c)"
cd_trans: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and j :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and i :: "nat"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob j \<pi> i"   and "IFC_def.is_cdi prob k \<pi> j" shows "IFC_def.is_cdi prob k \<pi> i"
pmf_mix: fixes a :: "real"   and p :: "'b pmf"   and q :: "'b pmf"   and x :: "'b" assumes "a \<in> {0..1}" shows "pmf (mix_pmf a p q) x = a * pmf p x + (1 - a) * pmf q x"
rev_eq_Cons_iff: fixes xs :: "'g list"   and y :: "'g"   and ys :: "'g list" shows "(rev xs = y # ys) = (xs = rev ys @ [y])"
simps_6: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat" shows "(case Newpkt x21 x22 of Pkt x xa \<Rightarrow> f1 x xa | Newpkt x xa \<Rightarrow> f2 x xa) = f2 x21 x22"
set_pmf_eq': fixes p :: "'a pmf" shows "set_pmf p = {x. 0 < pmf p x}"
n2_type2: fixes t1 :: "'b bro"   and h :: "nat"   and t2 :: "'b bro"   and a :: "'b" assumes "t1 \<in> B h"   and "t2 \<in> Um h" shows "n2 t1 a t2 \<in> T (Suc h)"
irrefl: fixes a :: "'k multiset" shows "\<not> a \<subset># a"
s: fixes i :: "nat" shows "simple_bochner_integrable M (s i)"
noContr_def: shows "noContr \<equiv> \<forall>\<phi>\<in>fmla. Fvars \<phi> = {} \<longrightarrow> prv (neg (cnj (PP \<langle>\<phi>\<rangle>) (PP \<langle>neg \<phi>\<rangle>)))"
projection_is_projection_on: fixes M :: "'b set" assumes "convex M"   and "closed M"   and "M \<noteq> {}" shows "is_projection_on (projection M) M"
plossless_converter_5: shows "a \<in> outs_\<I> \<I>"
arc_reversepath: fixes g :: "real \<Rightarrow> 'a" assumes "arc g" shows "arc (reversepath g)"
rep_eq: fixes x :: "'a set"   and xa :: "('a, 'b) pfun" shows "Rep_pfun (x \<lhd>\<^sub>p xa) = Rep_pfun xa |` x"
assms_6: shows "v \<notin> circline_set x_axis"
cod_simp: fixes f :: "'c local.term set" assumes "\<F>\<^sub>SC.arr f" shows "\<F>\<^sub>SC.cod f = cod f"
h2: shows "h \<turnstile> (case parent_opt of None \<Rightarrow> return () | Some parent \<Rightarrow> remove_child parent child) \<rightarrow>\<^sub>h h2"
height_del: fixes t :: "('c \<times> 'd) tree234"   and x :: "'c" assumes "bal t" shows "height (Tree234_Map.del x t) = height t"
intros_5: fixes \<E> :: "('a, String.literal) phantom" shows "OclVoid \<sqsubset>\<^sub>B Enum \<E>"
DERIV_arctan_series: fixes x :: "real" assumes "\<bar>x\<bar> < 1" shows "((\<lambda>x'. \<Sum>k. (- 1) ^ k * (1 / real (k * 2 + 1) * x' ^ (k * 2 + 1))) has_real_derivative (\<Sum>k. (- 1) ^ k * x ^ (k * 2))) (at x)"
closure_linear_image_subset: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "linear f" shows "f ` closure S \<subseteq> closure (f ` S)"
E_conv_def: fixes b_of_a :: "'d \<Rightarrow> 'c"   and E :: "char list \<Rightarrow> 'd" shows "E_conv b_of_a E \<equiv> \<lambda>x. b_of_a (E x)"
relcomp_distrib: fixes R :: "('b \<times> 'd) set"   and S :: "('d \<times> 'c) set"   and T :: "('d \<times> 'c) set" shows "R O (S \<union> T) = R O S \<union> R O T"
Lim_ident_at: fixes x :: "'a"   and s :: "'a set" assumes "at x within s \<noteq> bot" shows "Lim (at x within s) (\<lambda>x. x) = x"
zero_lens_indep: fixes X :: "'c \<Longrightarrow> 'b" shows "0\<^sub>L \<bowtie> X"
has_fields_fun: fixes P :: "'c prog"   and C :: "String.literal"   and FDTs :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list"   and FDTs' :: "((String.literal \<times> String.literal) \<times> ty \<times> fmod) list" assumes "P \<turnstile> C has_fields FDTs"   and "P \<turnstile> C has_fields FDTs'" shows "FDTs' = FDTs"
distinct_lists_no_parent: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and node_ptr :: "(_) node_ptr" assumes "local.a_distinct_lists h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "node_ptr \<in> set disc_nodes" shows "\<nexists>parent_ptr. parent_ptr |\<in>| object_ptr_kinds h \<and> node_ptr \<in> set |h \<turnstile> get_child_nodes parent_ptr|\<^sub>r"
powser_inside: fixes f :: "nat \<Rightarrow> 'b"   and x :: "'b"   and z :: "'b" assumes "summable (\<lambda>n. f n * x ^ n)"   and "norm z < norm x" shows "summable (\<lambda>n. f n * z ^ n)"
cSUP_lessD: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and y :: "'a"   and i :: "'b" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "less (Sup (f ` A)) y"   and "i \<in> A" shows "less (f i) y"
sraux2: fixes c :: "nat"   and n :: "nat"   and ts :: "nat \<Rightarrow> thread"   and ci :: "nat"   and ni :: "nat"   and tsi :: "nat \<Rightarrow> thread"   and t :: "nat" assumes "sim_rel1 (c, n, ts) (ci, ni, tsi)"   and "t < N" shows "tsi t = map_ticket (\<lambda>x. x mod N) (ts t)"
A: fixes i :: "'agent"   and A :: "'alt set" assumes "i \<in> agents" shows "Max_wrt_among (R i) A \<subseteq> A"
pp'_def: shows "pp' = endpoint_in_S (Rep_hyp2 a') (Rep_hyp2 b')"
l_comb_sc: fixes A :: "'b set"   and H :: "'a set"   and r :: "'b"   and s :: "nat \<Rightarrow> 'b"   and n :: "nat"   and g :: "nat \<Rightarrow> 'a" assumes "ideal R A"   and "H \<subseteq> carrier M"   and "r \<in> carrier R"   and "s \<in> {j. j \<le> n} \<rightarrow> A"   and "g \<in> {j. j \<le> n} \<rightarrow> H" shows "r \<cdot>\<^sub>s \<Sigma>\<^sub>e M (\<lambda>k. s k \<cdot>\<^sub>s g k) n = \<Sigma>\<^sub>e M (\<lambda>k. r \<cdot>\<^sub>s (s k \<cdot>\<^sub>s g k)) n"
cSup_eq_Max: fixes X :: "'d set" assumes "finite X"   and "X \<noteq> {}" shows "Sup X = Max X"
SDG_edge_sum_SDG_edge: fixes n :: "'node SDG_node"   and Vopt :: "'var option"   and popt :: "('pname \<times> bool) option"   and n' :: "'node SDG_node" assumes "SDG_edge n Vopt popt n'" shows "sum_SDG_edge n Vopt popt False n'"
paths_singles: fixes xs :: "'b list"   and S :: "'b set" shows "(xs \<in> paths (singles S)) = (\<forall>x\<in>S. one_call_in_path x xs)"
transfer_Ifun: fixes f :: "('a \<Rightarrow> 'b) star"   and F :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and x :: "'a star"   and X :: "nat \<Rightarrow> 'a" assumes "f \<equiv> star_n F"   and "x \<equiv> star_n X" shows "f \<star> x \<equiv> star_n (\<lambda>n. F n (X n))"
det_subPA_rw: fixes I' :: "nat set" assumes "I' \<subseteq> {0..<n}"   and "card I' = k" shows "Determinant.det (submatrix (P * A) I I') = (\<Sum>J' | J' \<subseteq> {0..<m} \<and> card J' = k. Determinant.det (submatrix P I J') * Determinant.det (submatrix A J' I'))"
scalar_prod_smult_distrib: fixes v\<^sub>1 :: "'b vec"   and n :: "nat"   and v\<^sub>2 :: "'b vec"   and a :: "'b" assumes "v\<^sub>1 \<in> carrier_vec n"   and "v\<^sub>2 \<in> carrier_vec n" shows "v\<^sub>1 \<bullet> (a \<cdot>\<^sub>v v\<^sub>2) = a * (v\<^sub>1 \<bullet> v\<^sub>2)"
elims: fixes x :: "opubx"   and y :: "opinl" assumes "deubx x = y" obtains "x = OpAddNumUbx" and "y = OpAddNum"
eq_inf_symp: fixes w\<^sub>1 :: "'item stream"   and w\<^sub>2 :: "'item stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2" shows "w\<^sub>2 =\<^sub>I w\<^sub>1"
thetaWhileW0_RetrT: shows "thetaWhileW0 \<subseteq> RetrT (thetaWhileW0 \<union> BisT)"
Diff_insert_absorb: fixes x :: "'b"   and A :: "'b set" assumes "x \<notin> A" shows "(x \<triangleright> A) - {x} = A"
that: fixes p\<^sub>A :: "'a \<Rightarrow> 'a"   and p\<^sub>B :: "'b \<Rightarrow> 'b" assumes "p\<^sub>A permutes A"   and "p\<^sub>B permutes B"   and "\<forall>x\<in>A. f x = p\<^sub>B (f' (p\<^sub>A x))" shows "thesis"
intersects_x_axis_positive_mk_circline: fixes A :: "complex"   and B :: "complex" assumes "is_real A"   and "A \<noteq> 0 \<or> B \<noteq> 0" shows "intersects_x_axis_positive (mk_circline A B (cnj B) A) = (Re B / Re A < - 1)"
fset_cong: fixes S :: "'b fset"   and T :: "'b fset" shows "(fset S = fset T) = (S = T)"
assms: fixes e :: "real" assumes "0 < e" shows "x \<le> y + ereal e"
max_fun_diff: fixes r :: "('d \<times> 'd) set"   and s :: "('e \<times> 'e) set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'e \<Rightarrow> 'd" assumes "wo_rel2 r s"   and "f \<noteq> g"   and "f \<in> FinFunc r s"   and "g \<in> FinFunc r s" shows "\<exists>a b. a \<noteq> b \<and> a \<in> Field r \<and> b \<in> Field r \<and> f (wo_rel.max_fun_diff s f g) = a \<and> g (wo_rel.max_fun_diff s f g) = b"
tbx: shows "tighten_poly_bounds_for_x p (rat_of_int \<lfloor>r'\<rfloor>) l' r' sr' = (l'', r'', sr'')"
f1: fixes R' :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and gpv :: "('a, 'c, 'd) gpv"   and gpv' :: "('b, 'f, 'e) gpv"   and A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and A' :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "('a, 'c, ('a, 'c, 'd) rpv) generat"   and y :: "('b, 'f, ('b, 'f, 'e) rpv) generat"   and a :: "('a, 'c, 'd) rpv"   and b :: "('b, 'f, 'e) rpv"   and a' :: "'d"   and \<alpha> :: "'e"   and \<beta> :: "'d"   and b' :: "'e" assumes "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv gpv gpv' A A'"   and "x \<in> set_spmf (the_gpv gpv)"   and "y \<in> set_spmf (the_gpv gpv')"   and "a \<in> generat_conts x"   and "b \<in> generat_conts y"   and "R' a' \<alpha>"   and "R' \<beta> b'" shows "??.Generative_Probabilistic_Value.foo R' results_gpv results_gpv (a a') (b b') A A'"
arg_cong_ffilter: fixes f :: "'b fset"   and p :: "'b \<Rightarrow> bool"   and p' :: "'b \<Rightarrow> bool" assumes "\<forall>e|\<in>|f. p e = p' e" shows "ffilter p f = ffilter p' f"
induct: fixes P :: "formula \<Rightarrow> bool"   and a0 :: "formula" assumes "\<And>l. P (Ands l)"   and "\<And>v va. P (Pred v va)"   and "\<And>v va vb. P (formula.Let v va vb)"   and "\<And>v va. P (formula.Eq v va)"   and "\<And>v va. P (Less v va)"   and "\<And>v va. P (LessEq v va)"   and "\<And>v. P (Neg v)"   and "\<And>v va. P (Or v va)"   and "\<And>v va. P (And v va)"   and "\<And>v. P (Exists v)"   and "\<And>v va vb vc vd. P (Agg v va vb vc vd)"   and "\<And>v va. P (Prev v va)"   and "\<And>v va. P (Next v va)"   and "\<And>v va vb. P (Since v va vb)"   and "\<And>v va vb. P (Until v va vb)"   and "\<And>v va. P (MatchF v va)"   and "\<And>v va. P (MatchP v va)" shows "P a0"
obj_simps_1: fixes a :: "('a S.E.term, 'a) S.arr" assumes "S.obj a" shows "S.arr a"
prod_list_def: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "class.monoid_mult one times" shows "monoid_mult.prod_list one times \<equiv> monoid_list.F times one"
not_Cons_self2: fixes x :: "'a"   and xs :: "'a list" shows "Cons x xs \<noteq> xs"
finprod_one_eqI: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'a vec" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = 0\<^sub>v n" shows "finsum V f A = 0\<^sub>v n"
simps: fixes a1 :: "'a"   and a2 :: "'b list"   and a3 :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" shows "((a1, a2, a3) \<in> trcl t) = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c a c' w c''. a1 = c \<and> a2 = a # w \<and> a3 = c'' \<and> (c, a, c') \<in> t \<and> (c', w, c'') \<in> trcl t))"
digraph_isomorphism_def: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" shows "H.digraph_isomorphism hom \<equiv> wf_digraph (app_iso hom G) \<and> inj_on (iso_verts hom) (verts (app_iso hom G)) \<and> inj_on (iso_arcs hom) (arcs (app_iso hom G)) \<and> (\<forall>a\<in>arcs (app_iso hom G). iso_verts hom (tail (app_iso hom G) a) = iso_tail hom (iso_arcs hom a) \<and> iso_verts hom (head (app_iso hom G) a) = iso_head hom (iso_arcs hom a))"
final_edge_Low: fixes c :: "'com"   and n :: "'node" assumes "final c"   and "n \<triangleq> c" shows "\<exists>a. valid_edge a \<and> sourcenode a = n \<and> targetnode a = (_Low_) \<and> kind a = \<Up>id"
P: fixes x :: "'a \<times> 'b" assumes "\<And>y. y < x \<Longrightarrow> P y" shows "P x"
echelon_form_of_column_k_det_def: fixes bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and A' :: "'c \<times> (('c, 'd) vec, 'e) vec \<times> nat"   and k :: "nat" shows "echelon_form_of_column_k_det bezout A' k = (let (det_P, A, i) = A'; from_nat_i = mod_type_class.from_nat i; from_nat_k = mod_type_class.from_nat k in if i \<noteq> nrows A \<and> A $ from_nat_i $ from_nat_k = (0::'c) \<and> (\<exists>m>mod_type_class.from_nat i. A $ m $ mod_type_class.from_nat k \<noteq> (0::'c)) then (- (1::'c) * det_P, echelon_form_of_column_k bezout (A, i) k) else (det_P, echelon_form_of_column_k bezout (A, i) k))"
ul_g: fixes K :: "complex set" assumes "compact K"   and "K \<subseteq> S" shows "uniform_limit K \<F> g sequentially"
dom: shows "\<D>\<^sub>\<circ> (fflip (fflip f)) = \<D>\<^sub>\<circ> f"
elims_2: fixes x :: "'b \<times> 'c"   and xa :: "'b \<times> 'c" assumes "alpha_residual x xa" obtains \<alpha>1 :: "'b" and P1 :: "'c" and \<alpha>2 :: "'b" and P2 :: "'c" where "x = (\<alpha>1, P1)" and "xa = (\<alpha>2, P2)" and "[bn \<alpha>1]set. (\<alpha>1, P1) = [bn \<alpha>2]set. (\<alpha>2, P2)"
span_mono: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'b set"   and B :: "'b set" assumes "module scale"   and "A \<subseteq> B" shows "module.span scale A \<subseteq> module.span scale B"
gcs_adds: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and lcs :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and s :: "'c"   and t :: "'c" assumes "class.lcs_powerprod plus minus zero lcs" shows "plus.adds plus (lcs_powerprod.gcs plus minus lcs s t) s"
p2c_mono_c2p_strict: fixes c :: "'id"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and n' :: "nat" assumes "\<langle>c \<and> t\<rangle> \<le> n"   and "n < \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n'" shows "\<^bsub>c\<^esub>\<down>\<^bsub>t\<^esub>n < n'"
IH: assumes "\<forall>a\<in>{(i, t, s)} \<union> set D. case a of (j, p) \<Rightarrow> \<forall>a\<in>{(i, t, s)} \<union> set D. case a of (k, q) \<Rightarrow> (\<exists>\<delta>. Unifier \<delta> (pair p) (pair q)) \<longrightarrow> j = k"   and "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) D\<rbrakk>\<^sub>d \<I>" shows "list_all (\<lambda>d. fst d = i) D"
top_edge_explicit: shows "(\<lambda>x. twoC (x, 1)) = (\<lambda>x. (g2 b + x *\<^sub>R (g1 b - g2 b), b))"
crt1: shows "continuous (at x0 within S) (return_time p1)"
coprime_Suc_0_left: fixes n :: "nat" shows "coprime (Suc 0) n"
hequiv_trans: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b"   and k :: "'b"   and j :: "'b" assumes "Hintikka A H"   and "i \<in> A"   and "k \<in> A"   and "hequiv H i j"   and "hequiv H j k" shows "hequiv H i k"
iso_imp_ordertype_eq_ordertype: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "iso r r' f"   and "wf r"   and "Total r"   and "small (Field r)" shows "ordertype (Field r) r = ordertype (Field r') r'"
unitary_gen_inv: fixes M :: "complex \<times> complex \<times> complex \<times> complex" assumes "unitary_gen M" shows "unitary_gen (mat_inv M)"
leD: fixes y :: "('a, 'b) Term.term"   and x :: "('a, 'b) Term.term" assumes "y \<sqsubseteq> x" shows "\<not> x \<sqsubset> y"
execn_blockStuck: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and bdy :: "('e, 'd, 'f) com"   and init :: "'e \<Rightarrow> 'e"   and s :: "'e"   and n :: "nat"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com" assumes "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Stuck" shows "\<Gamma>\<turnstile> \<langle>block init bdy return c,Normal s\<rangle> =n\<Rightarrow> Stuck"
dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t_Cons_4: fixes l :: "'d strand_label"   and t :: "('e, 'f) Term.term"   and s :: "('e, 'f) Term.term"   and A :: "('d strand_label \<times> ('e, 'f) stateful_strand_step) list" shows "dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t ((l, insert\<langle>t,s\<rangle>) # A) = (l, insert\<langle>t,s\<rangle>) # dual\<^sub>l\<^sub>s\<^sub>s\<^sub>t A"
index_i: shows "index (insort a (x # xs)) a \<le> i"
attractor_set_VVpstar: fixes v :: "'a"   and p :: "Player"   and W :: "'a set" assumes "\<not> deadend v"   and "\<And>w. v \<rightarrow> w \<Longrightarrow> w \<in> attractor p W" shows "v \<in> attractor p W"
rhs: shows "Hom(A,_) \<^bsub>\<a>\<^esub> g \<odot> Hom(A,_) \<^bsub>\<a>\<^esub> f = \<lparr>set_dom = Hom A (Dom f), set_func = restrict ((\<bullet>) (g \<bullet> f)) (Hom A (Dom f)), set_cod = Hom A (Cod g)\<rparr>"
cInf_eq_minimum: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and z :: "'a"   and X :: "'a set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "z \<in> X"   and "\<And>x. x \<in> X \<Longrightarrow> less_eq z x" shows "Inf X = z"
card_of_less: fixes n :: "nat" shows "|{0..<n}| =o natLeq_on n"
ocU_def: fixes p :: "'a" shows "local.ocU p = local.general_U (return_pmf p)"
assms_4: shows "Y = Im (2 * z / (1 + z * cnj z))"
iT_Minus_cut_le2: fixes t :: "nat"   and k :: "nat"   and I :: "nat set" assumes "t \<le> k" shows "(k \<ominus> I) \<down>\<le> (k - t) = k \<ominus> I \<down>\<ge> t"
eucl_of_list_prod_if: fixes xs :: "real list" assumes "length xs = DIM('a) + DIM('b)" shows "eucl_of_list xs = (eucl_of_list (take DIM('a) xs), eucl_of_list (drop DIM('a) xs))"
in_gamma_update: fixes num' :: "int \<Rightarrow> 'b"   and plus' :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<gamma> :: "'b \<Rightarrow> int set"   and s :: "char list \<Rightarrow> int"   and S :: "'b st"   and i :: "int"   and a :: "'b"   and x :: "char list" assumes "Gamma_semilattice num' plus' \<gamma>"   and "s \<in> \<gamma>_st \<gamma> S"   and "i \<in> \<gamma> a" shows "s(x := i) \<in> \<gamma>_st \<gamma> (update S x a)"
real_sq_exp: fixes a :: "real"   and b :: "real" shows "(a * b)\<^sup>2 = a\<^sup>2 * b\<^sup>2"
carrier_quotient_ring_iff: fixes X :: "('a \<times> 'a) set" shows "(X \<in> local.carrier_quotient_ring) = local.valid_frac X"
awo: fixes t :: "real" assumes "0 < t" shows "at t within {0<..} = at t"
atom_neg: fixes x :: "'a" assumes "atom x" shows "x \<noteq> \<bottom> \<and> (\<forall>y z. x \<le> y \<or> x \<le> - y)"
RangeI: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
take_drop_update_second: fixes j :: "nat"   and ds :: "'c list"   and cs :: "'c list" assumes "j < length ds"   and "length cs = length ds" shows "(take j ds @ drop j cs)[j := cs ! j] = take j ds @ drop j cs"
scale_left_diff_distrib: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'a"   and b :: "'a"   and x :: "'b" assumes "module scale" shows "scale (a - b) x = scale a x - scale b x"
insert: fixes A :: "'b set"   and x :: "'b"   and g :: "'b \<Rightarrow> 'a measure" assumes "finite A"   and "x \<notin> A" shows "sup_measure.F g (insert x A) = sup (g x) (sup_measure.F g A)"
a_2: shows "A \<inter> {x. P x} \<inter> (A \<inter> - {x. P x}) = {}"