hcju/isabelleps huggingface dataset:
CFGExit_wf_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list" shows "CFGExit_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses \<equiv> CFGExit sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit \<and> CFG_wf sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Def Use ParamDefs ParamUses \<and> CFGExit_wf_axioms Exit Def Use"
map_def: shows "\<rr>'.map = transformation_by_components.map (\<cdot>) (\<cdot>) local.map (\<lambda>a. local.inv (\<rr> a))"
or_list_semantic: fixes w :: "nat \<Rightarrow> 'a set"   and xs :: "'a ltln list" shows "w \<Turnstile>\<^sub>n or_list xs = (\<exists>x\<in>set xs. w \<Turnstile>\<^sub>n x)"
fv_ik_subset_fv_sst': fixes S :: "('d, 'c) stateful_strand_step list" shows "fv\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>s\<^sub>s\<^sub>t S) \<subseteq> fv\<^sub>s\<^sub>s\<^sub>t S"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "\<omega> \<Rightarrow> 'a"   and f2 :: "\<sigma> \<Rightarrow> 'a"   and \<upsilon> :: "\<upsilon>" shows "P (case \<upsilon> of \<omega>\<upsilon> x \<Rightarrow> f1 x | \<sigma>\<upsilon> x \<Rightarrow> f2 x) = ((\<forall>x1. \<upsilon> = \<omega>\<upsilon> x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x2. \<upsilon> = \<sigma>\<upsilon> x2 \<longrightarrow> P (f2 x2)))"
closure_Int_ball_not_empty: fixes S :: "'b set"   and T :: "'b set"   and x :: "'b"   and r :: "real" assumes "S \<subseteq> closure T"   and "x \<in> S"   and "0 < r" shows "T \<inter> ball x r \<noteq> {}"
set_get_eq: fixes l :: "('c \<times> 'd) list"   and k :: "'c" assumes "unique l" shows "((k, the (table_of l k)) \<in> set l) = (table_of l k \<noteq> None)"
cindex_polyE_0_1: fixes a :: "real"   and b :: "real"   and p :: "real poly" shows "cindex_polyE a b 0 p = 0"
ground_fatom_of_hatom: fixes a :: "char list \<times> hterm list" shows "ground\<^sub>t\<^sub>s (snd (fatom_of_hatom a))"
ct_1: fixes sa :: "'s" assumes "sa \<in> S" shows "ct sa \<in> K sa"
linear_surjective_right_inverse: fixes f :: "'b \<Rightarrow> 'b" assumes "linear (*s) (*s) f"   and "surj f" shows "\<exists>g. linear (*s) (*s) g \<and> f \<circ> g = id"
sums_snoc: fixes xs :: "'a list"   and x :: "'a" shows "sums (xs @ [x]) = sums xs @ [sum_list (xs @ [x])]"
tensor_prod_of_id_2_x: shows "Quantum.Id 2 \<Otimes> X = M3"
b: shows "b = 3 * p\<^sup>2 * q - 3 * q ^ 3 \<or> b = - (3 * p\<^sup>2 * q) + 3 * q ^ 3"
isSubobj_Subobjs: 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 C :: "char list"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "is_subobj P (C, Cs)"   and "wf_prog wf_md P" shows "Subobjs P C Cs"
sum_list_prod_fun_apply: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b \<Rightarrow> 'a"   and xys :: "('c \<times> 'd) list"   and z :: "'b" shows "(\<Sum>(x, y)\<leftarrow>xys. f x y) z = (\<Sum>(x, y)\<leftarrow>xys. f x y z)"
cot_altdef: fixes x :: "'a" shows "cot x = inverse (tan x)"
IH: assumes "k \<le> m" shows "\<forall>\<^sub>F z in at_infinity. B \<le> norm (\<Sum>i\<le>m. c i * z ^ i)"
pos_le_divideC_eq: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<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 less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and c :: "complex"   and a :: "'b"   and b :: "'b" assumes "class.ordered_complex_vector scaleR scaleC plus zero minus uminus less_eq less"   and "0 < c" shows "less_eq a (scaleC (inverse c) b) = less_eq (scaleC c a) b"
assms: shows "wffp (sl1 @ [Idle st1] @ [Idle st2] @ sl2)"
is_strongly_dom_component_safe_def: shows "Core_DOM_DOM_Components.is_strongly_dom_component_safe \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_is_strongly_dom_component_safe Core_DOM_Functions.get_root_node Core_DOM_Functions.to_tree_order"
cases: fixes F :: "'c set"   and r :: "('c \<times> 'c) set"   and a1 :: "nat"   and a2 :: "'c"   and a3 :: "'c" assumes "ntrancl_onp F r a1 a2 a3" obtains a :: "'c" and b :: "'c" where "a1 = 0" and "a2 = a__" and "a3 = b__" and "a__ = b__" and "a__ \<in> F"   | a :: "'c" and b :: "'c" and n :: "nat" and c :: "'c" where "a1 = Suc n" and "a2 = a" and "a3 = c" and "(a, b) \<in> r" and "ntrancl_onp F r n b c" and "a \<in> F"
topology_eq: fixes T1 :: "'c topology"   and T2 :: "'c topology" shows "(T1 = T2) = (\<forall>S. openin T1 S = openin T2 S)"
simps_2: fixes x :: "'f"   and xs :: "'f list" shows "rev (x # xs) = rev xs @ [x]"
list_distinct_conv_distinct: shows "equal_base.list_distinct (=) = distinct"
root_nat_ceiling: fixes p :: "nat"   and x :: "nat" shows "root_nat_ceiling p x = \<lceil>root p (real x)\<rceil>"
le_the_leftI: fixes i :: "nat"   and j :: "nat"   and xs :: "int list" assumes "i \<le> j"   and "xs ! i < xs ! j" shows "i \<le> the_left xs j"
hv_vector: fixes x :: "real"   and y :: "real"   and z :: "real"   and u :: "real"   and v :: "real"   and q :: "real"   and g :: "real" shows "Hv (vector [x, y, z, u, v, q, g]) = Octo 0 x y z u v q g"
Diff_idemp: fixes A :: "'b set"   and B :: "'b set" shows "A - B - B = A - B"
satisfies_properties_sys: shows "satisfies_properties_R p qs subsets signs matrix"
small_refl_iff: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "(f \<in> o[F](f)) = (\<forall>\<^sub>F x in F. f x = (0::'c))"
make_tr_len: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and i :: "nat" shows "length (make_tr f n i) = n"
forall_intros: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and B :: "Core.term"   and frees :: "(variable \<times> typ) list" assumes "wf_theory \<Theta>"   and "\<Theta>,\<Gamma> \<turnstile> B"   and "\<forall>(x, \<tau>)\<in>set frees. (x, \<tau>) \<notin> FV \<Gamma> \<and> typ_ok \<Theta> \<tau>" shows "\<Theta>,\<Gamma> \<turnstile> mk_all_list frees B"
lucas_lehmer_add_cong_1: fixes m :: "nat"   and x :: "nat \<times> nat"   and y :: "nat \<times> nat" shows "[fst (lucas_lehmer_add m x y) = fst x + fst y] (mod m)"
rbt_insert_with_key_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and k :: "'b"   and v :: "'c"   and t :: "('b, 'c) rbt" shows "ord.rbt_insert_with_key less f k v t = paint color.B (ord.rbt_ins less f k v t)"
bottom_in_unsynlr: fixes a :: "'b"   and R :: "('a, 'b) synlr" shows "(\<bottom>, a) \<in> unsynlr R"
rel_fun_def_butlast: fixes R :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and S :: "'i \<Rightarrow> 'k \<Rightarrow> bool"   and T :: "'j \<Rightarrow> 'l \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'i \<Rightarrow> 'j"   and g :: "'h \<Rightarrow> 'k \<Rightarrow> 'l" shows "(R ===> S ===> T) f g = (\<forall>x y. R x y \<longrightarrow> (S ===> T) (f x) (g y))"
e_var_intro: fixes \<rho> :: "(nat \<times> val) list"   and x :: "nat"   and v' :: "val"   and v :: "val" assumes "lookup \<rho> x = Some v'"   and "v \<sqsubseteq> v'" shows "v \<in> E (EVar x) \<rho>"
find_indices_distinct: fixes ss :: "'b list"   and i :: "nat" assumes "distinct ss"   and "i < length ss" shows "find_indices (ss ! i) ss = [i]"
wf_mkTree: fixes s :: "'state"   and rs :: "'rule stream" assumes "s \<in> S"   and "fair rs" shows "local.wf (mkTree rs s)"
unfold_mapStep: fixes s :: "'b"   and f :: "'d \<rightarrow> 'c"   and h :: "'b \<rightarrow> ('d, 'b) Step" assumes "s \<noteq> \<bottom>" shows "unfold\<cdot>(mapStep\<cdot>f\<cdot>h)\<cdot>s = mapL\<cdot>f\<cdot>(unfold\<cdot>h\<cdot>s)"
f_def: shows "f = sum_upto (\<lambda>d. mangoldt d / real d)"
length_replicate: fixes n :: "nat"   and x :: "'a" shows "length x\<^bsup>n\<^esup> = n"
linear_split_sm: fixes xs :: "('b \<times> 'c) list"   and p :: "'c"   and ys :: "('b \<times> 'c) list"   and ls :: "('b \<times> 'c) list"   and rs :: "('b \<times> 'c) list" assumes "linear_split_help xs p ys = (ls, rs)"   and "sorted_less (separators (ys @ xs))"   and "\<forall>sep\<in>set (separators ys). sep < p" shows "\<forall>sep\<in>set (separators ls). sep < p"
not_Interp_imp_not_INTERP: fixes D' :: "'b literal multiset"   and D :: "'b literal multiset"   and N :: "'b literal multiset set" assumes "D' \<le> D"   and "D' \<in> N \<and> {#} = {#} \<or> Max (atms_of D') < Max (atms_of D)"   and "\<not> Interp N D \<Turnstile> D'" shows "\<not> INTERP N \<Turnstile> D'"
converse_rtranclp_induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and P :: "'b \<Rightarrow> bool" assumes "r\<^sup>*\<^sup>* a b"   and "P b"   and "\<And>y z. \<lbrakk>r y z; r\<^sup>*\<^sup>* z b; P z\<rbrakk> \<Longrightarrow> P y" shows "P a"
False: shows "\<not> (x = (0::'a) \<or> p = (0::'a) \<or> is_unit p)"
hseqI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a"   and \<nu> :: "'a"   and c :: "'a" assumes "\<guillemotleft>\<mu> : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>\<nu> : b \<rightarrow> c\<guillemotright>" shows "hseq \<nu> \<mu>"
assms_2: shows "\<forall>b'\<in>contours_in_ve ve. Q b'"
Rep_perm: fixes x :: "perm" shows "Rep_perm x \<in> perm"
ordLeq_countable_subsets: fixes A :: "'f set" shows "|A| \<le>o |{X. X \<subseteq> A \<and> countable X}|"
dgrad_p_set_le_def: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and G :: "('a \<Rightarrow>\<^sub>0 'd) set" shows "punit.dgrad_p_set_le d F G = dgrad_set_le d (local.punit.pp_of_term ` Keys F) (local.punit.pp_of_term ` Keys G)"
Number_Pair: fixes X :: "nat"   and X' :: "msg"   and Y' :: "msg" shows "Number X \<noteq> \<langle>X', Y'\<rangle>"
minimal_unsat_core_tabl_atoms_def: fixes I :: "'c set"   and t :: "(nat \<times> linear_poly) list"   and as :: "('c \<times> 'd atom) set" shows "minimal_unsat_core_tabl_atoms I t as = (I \<subseteq> fst ` as \<and> (\<nexists>v. v \<Turnstile>\<^sub>t t \<and> (I, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>s as) \<and> (distinct_indices_atoms as \<longrightarrow> (\<forall>J\<subset>I. \<exists>v. v \<Turnstile>\<^sub>t t \<and> (J, v) \<Turnstile>\<^sub>i\<^sub>a\<^sub>e\<^sub>s as)))"
HFinite_hypreal_sqrt_iff: fixes x :: "real star" assumes "0 \<le> x" shows "((*f* sqrt) x \<in> HFinite) = (x \<in> HFinite)"
MakeFtorObj: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and X :: "'d" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>" shows "MakeFtor F @@ X = F @@ X"
assms_2: assumes "\<exists>n\<in>elts \<omega>. n \<approx>\<^sub>\<circ> A" shows "P"
lemma_3_5_i_2: fixes a :: "'a"   and b :: "'a" shows "(d2 a b = (1::'a)) = (a = b)"
m_lcomm: fixes x :: "'a vec"   and y :: "'a vec"   and z :: "'a vec" assumes "x \<in> carrier_vec n"   and "y \<in> carrier_vec n"   and "z \<in> carrier_vec n" shows "x + (y + z) = y + (x + z)"
wlt_constrains_wlt: fixes F :: "'c program"   and B :: "'c set" shows "F \<in> wlt F B - B co wlt F B"
inv_swap_id: fixes a :: "'a"   and b :: "'a" shows "inv (Fun.swap a b id) = Fun.swap a b id"
Ana_vars_2: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and t :: "('a, 'b) Term.term"   and K :: "('a, 'b) Term.term list"   and M :: "('a, 'b) Term.term list" assumes "intruder_model arity Ana"   and "Ana t = (K, M)" shows "fv\<^sub>s\<^sub>e\<^sub>t (set M) \<subseteq> fv t"
L_transform_Tree\<^sub>\<alpha>_Not\<^sub>\<alpha>: fixes t\<^sub>\<alpha> :: "('a, 'b, 'c, 'd) FL_Formula.Tree\<^sub>\<alpha>" shows "L_transform_Tree\<^sub>\<alpha> (FL_Formula.Not\<^sub>\<alpha> t\<^sub>\<alpha>) = Formula.Not\<^sub>\<alpha> (L_transform_Tree\<^sub>\<alpha> t\<^sub>\<alpha>)"
ennreal_cases: fixes x :: "ennreal" obtains r :: "real" where "0 \<le> r" and "x = ennreal r"   | "x = top"
fold_Mapping_filter_Some_None: fixes ts :: "('b, 'a) mapping"   and as :: "'b"   and t :: "'a"   and X :: "'b set"   and ds :: "('a \<times> 'b set) list" assumes "Mapping.lookup ts as = Some t"   and "as \<in> X"   and "(t, X) \<in> set ds" shows "Mapping.lookup (fold (\<lambda>(t, X) ts. Mapping.filter (filter_cond X ts t) ts) ds ts) as = None"
int_transE_nv2T: fixes T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "wt T"   and "I.wtE \<xi>"   and "infTp (tpOf T) \<or> (\<forall>x\<in>nv2T T. tpOfV x \<noteq> tpOf T)" shows "Ik.int (transE \<xi>) T = I.int \<xi> T"
rec_left_def: shows "rec_left = Cn 1 rec_lo [recf.id 1 0, constn (Pi 0)]"
exact_Nil_iff: fixes qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and k :: "nat" assumes "valid_decomp X qs"   and "standard_decomp k qs" shows "(exact k qs = []) = (qs = [])"
REC_unfold: fixes B :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> 'd" assumes "trimono B" shows "REC B = B (REC B)"
inteq_reflection: fixes x :: "'a \<Rightarrow> 'b"   and y :: "'a \<Rightarrow> 'b" assumes "\<turnstile> x = y" shows "x \<equiv> y"
higher_differentiable_on_imp_continuous_nth_derivative: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and n :: "nat"   and v :: "'b" assumes "higher_differentiable_on S f n" shows "continuous_on S (\<lambda>x. nth_derivative n f x v)"
cases: fixes a1 :: "'tp"   and a2 :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" assumes "mcalc2 a1 a2" obtains \<sigma> :: "'tp" and c :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" where "a1 = \<sigma>'__" and "a2 = c__" and "infTp \<sigma>'__"   | c :: "(('fsym, 'tp) efsym, ('psym, 'tp) epsym) lit list" and \<sigma> :: "'tp" where "a1 = \<sigma>" and "a2 = c" and "\<And>l x. \<lbrakk>l \<in> set c; x \<in> nv2L l; GE.tpOfV x = \<sigma>\<rbrakk> \<Longrightarrow> isGuard x (grdOf c l x)"
sset_shift: fixes xs :: "'b list"   and s :: "'b stream" shows "sset (xs @- s) = set xs \<union> sset s"
morphism_ringed_spaces_axioms_def: fixes X :: "'d set"   and \<O>\<^sub>X :: "'d set \<Rightarrow> 'e set"   and \<rho>\<^sub>X :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str\<^sub>X :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and one_str\<^sub>X :: "'d set \<Rightarrow> 'e"   and Y :: "'f set"   and is_open\<^sub>Y :: "'f set \<Rightarrow> bool"   and \<O>\<^sub>Y :: "'f set \<Rightarrow> 'g set"   and \<rho>\<^sub>Y :: "'f set \<Rightarrow> 'f set \<Rightarrow> 'g \<Rightarrow> 'g"   and d :: "'g"   and add_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and mult_str\<^sub>Y :: "'f set \<Rightarrow> 'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and one_str\<^sub>Y :: "'f set \<Rightarrow> 'g"   and f :: "'d \<Rightarrow> 'f"   and \<phi>\<^sub>f :: "'f set \<Rightarrow> 'g \<Rightarrow> 'e" shows "morphism_ringed_spaces_axioms X \<O>\<^sub>X \<rho>\<^sub>X b add_str\<^sub>X mult_str\<^sub>X zero_str\<^sub>X one_str\<^sub>X Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y f \<phi>\<^sub>f \<equiv> morphism_sheaves_of_rings Y is_open\<^sub>Y \<O>\<^sub>Y \<rho>\<^sub>Y d add_str\<^sub>Y mult_str\<^sub>Y zero_str\<^sub>Y one_str\<^sub>Y (im_sheaf.im_sheaf X \<O>\<^sub>X f) (im_sheaf.im_sheaf_morphisms X \<rho>\<^sub>X f) b (im_sheaf.add_im_sheaf X add_str\<^sub>X f) (im_sheaf.mult_im_sheaf X mult_str\<^sub>X f) (im_sheaf.zero_im_sheaf X zero_str\<^sub>X f) (im_sheaf.one_im_sheaf X one_str\<^sub>X f) \<phi>\<^sub>f"
inject: fixes x11 :: "name"   and x12 :: "name"   and y11 :: "name"   and y12 :: "name" shows "(OutputR_Rep x11 x12 = OutputR_Rep y11 y12) = (x11 = y11 \<and> x12 = y12)"
ProcUniv: shows "FLPSys.Proc = {p0, p1}"
unit_factor_mult_unit_left: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semidom_divide_unit_factor divide plus minus zero times one unit_factor"   and "dvd.dvd times a one" shows "unit_factor (times a b) = times a (unit_factor b)"
prevVertex_nextVertex: fixes f :: "face"   and v :: "nat" assumes "distinct (vertices f)"   and "v \<in> \<V> f" shows "f\<^bsup>-1\<^esup> \<bullet> (f \<bullet> v) = v"
new_thread_memory: fixes t :: "'t"   and s :: "'x1 \<times> 'm1"   and ta :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action"   and s' :: "'x1 \<times> 'm1"   and t' :: "'t"   and x :: "'x1"   and m :: "'m1" assumes "t \<turnstile> s -1-ta\<rightarrow> s'"   and "NewThread t' x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "m = snd s'"
map_real_pair: fixes f :: "real \<Rightarrow> real \<Rightarrow> 'a"   and g :: "val \<Rightarrow> 'a"   and i :: "real"   and j :: "real" shows "map_real_pair f g <|RealVal i, RealVal j|> = f i j"
F_one: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "sum_0 g A + (0::'a) = sum_0 g A"
width_bound_V_le: fixes G :: "('c, 'd) Graph_scheme"   and T :: "('e, 'f) Graph_scheme"   and bag :: "'e \<Rightarrow> 'c set" assumes "TreeDecomposition G T bag" shows "TreeDecomposition.width T bag \<le> card V\<^bsub>G\<^esub> - 1"
idempotent_bot_closed: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c" assumes "class.non_associative_left_semiring one times sup less_eq less bot" shows "times bot bot = bot"
mem_family_sub_Un: fixes A :: "'c set"   and C :: "'c set set" assumes "A \<in> C" shows "A \<subseteq> \<Union> C"
for_comb_for_rec: fixes f :: "'b \<Rightarrow> nat \<Rightarrow> 'b nres"   and a :: "'b"   and n :: "nat" shows "for_comb f a n = for_rec f a n"
assms_3: shows "ik\<^sub>s\<^sub>t (proj_unl n B) \<cdot>\<^sub>s\<^sub>e\<^sub>t \<I> \<turnstile> t \<cdot> \<I>"
emp_UNpart1: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and dl :: "('d, 'e, 'f) cmd list" assumes "\<And>n. n < length cl \<Longrightarrow> {} \<notin> P n" shows "{} \<notin> UNpart1 cl dl P"
submonoid_pr_ideal: fixes R :: "'c set"   and I :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" assumes "pr_ideal R I addition multiplication zero unit" shows "submonoid (R\<setminus>I) R multiplication unit"
int_mod_lem: fixes n :: "int"   and b :: "int" assumes "0 < n" shows "(0 \<le> b \<and> b < n) = (b mod n = b)"
assms_3: shows "a +\<^sub>o\<^sub>w b = 0\<^sub>o\<^sub>w"
preserves_ide: fixes a :: "'c \<times> 'c" assumes "C'.VV.ide a" shows "C.ide (fst a \<star>\<^sub>C\<^sup>o\<^sup>p snd a)"
swapped_skel: fixes X :: "('c, 'd, 'varSort, 'var, 'e) qTerm set"   and Y :: "('c, 'd, 'varSort, 'var, 'e) qTerm set" assumes "good X"   and "(X, Y) \<in> swapped" shows "skel Y = skel X"
Domain_set_zip: fixes A :: "'b list"   and B :: "'c list" assumes "length A = length B" shows "Domain (set (zip A B)) = set A"
simps_3: fixes r :: "real"   and x :: "octo" shows "Im2 (r *\<^sub>R x) = r * Im2 x"
abs_eq_iff: fixes x :: "'d group"   and y :: "'d group" shows "(Abs_group_iso x = Abs_group_iso y) = group_iso_rel x y"
simps: fixes k :: "'a"   and u :: "'a Vector" shows "k ** u = \<lparr>tdir = k * tdir u, xdir = k * xdir u, ydir = k * ydir u, zdir = k * zdir u\<rparr>"
restricted_deduct_if_restricted_ik: fixes M :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and m :: "('fun, 'var) Term.term" assumes "\<langle>M;P\<rangle> \<turnstile>\<^sub>r m"   and "\<forall>m\<in>M. P m"   and "\<forall>t t'. P t \<longrightarrow> t' \<sqsubseteq> t \<longrightarrow> P t'" shows "P m"
preserves_cod: fixes f :: "'c" assumes "C.arr f" shows "C.cod (src\<^sub>C f) = src\<^sub>C (C.cod f)"
D: shows "X = (\<Union>n\<le>length xs. \<Union>W\<in>R n. W) \<and> (\<forall>W\<in>R 0. xs \<notin> sentences P \<and> None \<notin> set xs \<and> (xs, W) \<in> failures P \<or> xs \<in> sentences P \<and> (\<exists>U V. (xs, U) \<in> failures P \<and> ([], V) \<in> failures Q \<and> W = insert None U \<inter> V)) \<and> (\<forall>n\<in>{0<..length xs}. \<forall>W\<in>R n. take (length xs - n) xs \<in> sentences P \<and> (drop (length xs - n) xs, W) \<in> failures Q)"
jump_poly_sgn: fixes p :: "real poly"   and x :: "real"   and q :: "real poly" assumes "p \<noteq> 0"   and "poly p x = 0" shows "jump_poly (pderiv p * q) p x = sign (poly q x)"
bind_gpv_sel': fixes r :: "('k, 'i, 'j) gpv"   and f :: "'k \<Rightarrow> ('h, 'i, 'j) gpv" shows "the_gpv (r \<bind> f) = the_gpv r \<bind> (\<lambda>generat. if is_Pure generat then the_gpv (f (generat.result generat)) else return_spmf (IO (generat.output generat) (\<lambda>input. continuation generat input \<bind> f)))"
fold_simps_2: fixes f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'b"   and xs :: "'b list"   and s :: "'a" shows "fold f (x # xs) s = fold f xs (f x s)"
syzygy_module_closed_map_scale: fixes s :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and c :: "'a \<Rightarrow>\<^sub>0 'c" assumes "s \<in> pmdl.syzygy_module B" shows "c \<cdot> s \<in> pmdl.syzygy_module B"
dagger_unfoldr_eq: fixes x :: "('a \<times> 'a) set" shows "Id \<union> x ; x\<^sup>* = x\<^sup>*"
bdia_whilei: fixes p :: "('b \<times> 'b) set"   and i :: "('b \<times> 'b) set"   and t :: "('b \<times> 'b) set"   and q :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" assumes "rel_antirange_kleene_algebra.ars_r p \<subseteq> rel_antirange_kleene_algebra.ars_r i"   and "rel_antirange_kleene_algebra.ars_r i ; rel_ad t \<subseteq> rel_antirange_kleene_algebra.ars_r q"   and "rel_antirange_kleene_algebra.bdia x (rdom t ; rel_antirange_kleene_algebra.ars_r i) \<subseteq> rel_antirange_kleene_algebra.ars_r i" shows "rel_antirange_kleene_algebra.bdia (rel_antidomain_kleene_algebra.whilei t i x) p \<subseteq> rel_antirange_kleene_algebra.ars_r q"
eq_onp_to_Domainp: fixes P :: "'e \<Rightarrow> bool"   and Abs :: "'e \<Rightarrow> 'f"   and Rep :: "'f \<Rightarrow> 'e"   and T :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "Quotient (eq_onp P) Abs Rep T" shows "Domainp T = P"
interval_cases: fixes S :: "'c set" assumes "\<And>a b x. \<lbrakk>a \<in> S; b \<in> S; a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> x \<in> S" shows "\<exists>a b. S = {} \<or> S = UNIV \<or> S = {..<b} \<or> S = {..b} \<or> S = {a<..} \<or> S = {a..} \<or> S = {a<..<b} \<or> S = {a<..b} \<or> S = {a..<b} \<or> S = {a..b}"
transpose_vector: fixes x :: "('a, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" shows "x v* A = Finite_Cartesian_Product.transpose A *v x"