hcju/isabelleps huggingface dataset:
inf_sup_aci_5: fixes inf :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and less_eq :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and less :: "'x \<Rightarrow> 'x \<Rightarrow> bool"   and sup :: "'x \<Rightarrow> 'x \<Rightarrow> 'x"   and x :: "'x"   and y :: "'x" assumes "class.lattice inf less_eq less sup" shows "sup x y = sup y x"
atlas_is_atlas: fixes a1 :: "('a, 'e) chart"   and a2 :: "('a, 'e) chart" assumes "a1 \<in> src.atlas"   and "a2 \<in> src.atlas" shows "k-smooth_compat a1 a2"
abc_twice_def: shows "abc_twice = (let (aprog, ary, fp) = rec_ci rec_twice in aprog [+] dummy_abc (Suc 0))"
finsert_is_funion: fixes a :: "'x"   and A :: "'x fset" shows "finsert a A = {|a|} |\<union>| A"
get_child_nodes_impl: shows "get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M = CD.a_get_child_nodes"
not_ide_fromArr: fixes x :: "bool" shows "\<not> ide (fromArr x)"
terminalE: fixes b :: "'a \<times> 'a"   and a :: "'a \<times> 'a" assumes "CC.terminal b"   and "CC.ide a" obtains f :: "'a \<times> 'a" where "CC.in_hom f a b" and "\<And>f'. CC.in_hom f' a b \<Longrightarrow> f' = f"
scan_length: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'c list"   and a :: "'b" shows "length (scan f xs a) = length xs"
sup_absorb1: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<squnion> y = x"
edges_restrict_nodes: fixes g :: "'b ugraph"   and V :: "'b set" shows "edges (restrict_nodes g V) = Restr (edges g) V"
sq_norm_vec_as_cscalar_prod: fixes v :: "'a vec" shows "\<parallel>v\<parallel>\<^sup>2 = v \<bullet>c v"
seqr_convr: fixes p :: "('c, 'd) urel"   and q :: "('d, 'b) urel" shows "(p ;; q)\<^sup>- = q\<^sup>- ;; p\<^sup>-"
trcl_uncons_cases: fixes c :: "'a"   and e :: "'b"   and w :: "'b list"   and c' :: "'a"   and S :: "('a \<times> 'b \<times> 'a) set" assumes "(c, e # w, c') \<in> trcl S" obtains ch :: "'a" where "(c, e, ch) \<in> S" and "(ch, w, c') \<in> trcl S"
projection_orthogonal: fixes M :: "'b set"   and m :: "'b"   and h :: "'b" assumes "closed_csubspace M"   and "m \<in> M" shows "is_orthogonal (h - projection M h) m"
one_N_def: shows "1 \<equiv> Abs_N (0::'b)"
content_dvd_coeffs: fixes c :: "'b"   and p :: "'b poly" assumes "c \<in> set (coeffs p)" shows "content p dvd c"
wp_def: fixes pr :: "bool \<Rightarrow> ('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "wp pr \<equiv> pr True"
tendsto_powr2: fixes f :: "'a \<Rightarrow> real"   and a :: "real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F"   and "\<forall>\<^sub>F x in F. 0 \<le> f x"   and "0 < b" shows "((\<lambda>x. f x powr g x) \<longlongrightarrow> a powr b) F"
consistent_imp_is_solution: fixes A :: "(('b, 'c) vec, 'd) vec"   and b :: "('b, 'd) vec" assumes "consistent A b" shows "is_solution (solve_consistent_rref (Gauss_Jordan A) (P_Gauss_Jordan A *v b)) A b"
fns1: fixes fa :: "'a \<Rightarrow> 'a"   and fr :: "'b \<Rightarrow> 'b" assumes "Rep \<circ> fa = fr \<circ> Rep \<or> fa \<circ> Abs = Abs \<circ> fr" shows "Abs \<circ> fr \<circ> Rep = fa"
simps: fixes fs :: "('a \<Rightarrow>\<^sub>0 'c) list"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and r :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.trd_aux fs p r = (if p = 0 then r else case punit.find_adds fs (punit.lt p) of None \<Rightarrow> punit.trd_aux fs (punit.tail p) (r + monomial (punit.lc p) (punit.lt p)) | Some f \<Rightarrow> punit.trd_aux fs (punit.tail p - local.punit.monom_mult (punit.lc p / punit.lc f) (punit.lp p - punit.lp f) (punit.tail f)) r)"
simps_2: fixes U :: "('a, 'g) aezfun set"   and Us :: "('a, 'g) aezfun set list" shows "ActingGroup.remisodups (U # Us) = (if \<exists>W\<in>set Us. \<exists>T. FGModuleIso G (*) U (*) T W then ActingGroup.remisodups Us else U # ActingGroup.remisodups Us)"
ODord_le_ref: fixes X :: "'c Order set"   and Y :: "'c Order set" assumes "X \<in> ODnums"   and "Y \<in> ODnums"   and "X \<sqsubseteq> Y"   and "Y \<sqsubseteq> X" shows "X = Y"
hyps_1: shows "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>"
delta: fixes S :: "'a set"   and a :: "'a"   and b :: "'a \<Rightarrow> 'b" assumes "finite S" shows "(\<Sum>k\<in>S. if k = a then b k else (0::'b)) = (if a \<in> S then b a else (0::'b))"
dghm_comp_is_iso_dghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<GG> :: "V"   and \<AA> :: "V"   and \<FF> :: "V" assumes "\<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<CC>"
atrm_pls: fixes t :: "'trm"   and t' :: "'trm" assumes "t \<in> atrm"   and "t' \<in> atrm" shows "pls t t' \<in> atrm"
dyn_data_dependence_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and n :: "'b"   and V :: "'d"   and n' :: "'b"   and as :: "'a list" assumes "CFG_wf sourcenode targetnode kind valid_edge Entry Def Use state_val" shows "CFG_wf.dyn_data_dependence sourcenode targetnode valid_edge Def Use n V n' as \<equiv> V \<in> Def n \<and> V \<in> Use n' \<and> CFG.path sourcenode targetnode valid_edge n as n' \<and> (\<exists>a' as'. as = a' # as' \<and> (\<forall>n''\<in>set (CFG.sourcenodes sourcenode as'). V \<notin> Def n''))"
assms_5: shows "\<sigma> \<in> space (state_measure V \<Gamma>)"
right_neutral: fixes a :: "'a::bounded_semilattice_inf_top" shows "(a::'a::bounded_semilattice_inf_top) \<sqinter> \<top> = a"
nth'_def: fixes a :: "'a array" shows "Array.nth' a = Array.nth a \<circ> nat_of_integer"
length_map: fixes f :: "'d \<Rightarrow> 'c"   and xs :: "'d list" shows "length (list.map f xs) = length xs"
finite_iff_cardOf_nat: fixes A :: "'d set" shows "finite A = (|A| <o |UNIV|)"
dgrad_set_le_def: fixes d :: "'a \<Rightarrow> nat"   and S :: "'a set"   and T :: "'a set" shows "dgrad_set_le d S T = (\<forall>s\<in>S. \<exists>t\<in>T. d s \<le> d t)"
injh: shows "inj_on index_f (UNIV // eq_app_right language)"
E_def: shows "E = {(11 / 10, - (9 / 10\<^sup>2))..(12 / 10, - (8 / 10\<^sup>2))}"
DERIV_continuous: fixes f :: "'a \<Rightarrow> 'a"   and D :: "'a"   and x :: "'a"   and s :: "'a set" assumes "(f has_field_derivative D) (at x within s)" shows "continuous (at x within s) f"
c_of_initial_no_mon: fixes csp :: "'a list multiset" assumes "\<And>s. s \<in># csp \<Longrightarrow> \<exists>p. s = [entry fg p] \<and> initialproc fg p" shows "mon_c fg csp = {}"
path_singleton: fixes weights :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and s :: "'b"   and l1 :: "'a"   and l2 :: "'a" assumes "graph weights"   and "s \<in>\<^sub>A weights l1 l2" shows "graph.path weights l1 l2 [(l1, s, l2)]"
simps_1: fixes ops :: "'b arith_ops_record"   and x :: "'b"   and xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "coeffs_minus_i ops (x # xs) (y # ys) = arith_ops_record.minus ops x y # coeffs_minus_i ops xs ys"
assms_1: shows "enat p + INV t \<le> INV s"
ParamUses_call_source_length: fixes a :: "'a"   and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool"   and r :: "'e"   and p :: "'f"   and fs :: "(('c \<Rightarrow> 'd option) \<Rightarrow> 'd option) list"   and ins :: "'c list"   and outs :: "'c list" assumes "valid_edge a"   and "kind a = Q:r\<hookrightarrow>\<^bsub>p\<^esub>fs"   and "(p, ins, outs) \<in> set procs" shows "length (ParamUses (sourcenode a)) = length ins"
pc': shows "poly_cond (abs_int_poly (poly_uminus p))"
simps_6: fixes M :: "('b, 'c) kripke"   and s :: "'c"   and i :: "'b"   and p :: "'b fm" shows "(M, s \<Turnstile> K i p) = (\<forall>t\<in>\<K> M i s. M, t \<Turnstile> p)"
1: shows "splits_wrt (ps1, qs1) (cone (monomial (1::'a) (monomial 1 x + t), U)) F"
divide_eq_imp: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and c :: "'c"   and b :: "'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus"   and "c \<noteq> zero"   and "b = times a c" shows "divide b c = a"
ra_ia_rb_ib_rc_ic_def_4: shows "ib = Im b"
edges_graph: fixes V :: "'b set"   and E :: "('b \<times> 'b) set" assumes "finite V"   and "finite E" shows "edges (graph V E) = E \<union> E\<inverse> - Id"
assms_3: shows "P,E \<turnstile> \<langle>es,s\<^sub>1\<rangle> [\<rightarrow>]* \<langle>map Val vs,(h\<^sub>2, l\<^sub>2)\<rangle>"
density_distr: fixes f :: "'a \<Rightarrow> ennreal"   and N :: "'a measure"   and X :: "'b \<Rightarrow> 'a"   and M :: "'b measure" assumes "f \<in> borel_measurable N"   and "X \<in> M \<rightarrow>\<^sub>M N" shows "density (distr M N X) f = distr (density M (\<lambda>x. f (X x))) N X"
assms_1: shows "A, n \<turnstile> (ps, a) # branch"
eventually_at_ball: fixes d :: "real"   and z :: "'b"   and A :: "'b set" assumes "0 < d" shows "\<forall>\<^sub>F t in at z within A. t \<in> ball z d \<and> t \<in> A"
the_elem_def: fixes X :: "'b set" shows "the_elem X = (THE x. X = {x})"
distrib_left: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semiring plus times" shows "times a (plus b c) = plus (times a b) (times a c)"
assms_3: shows "\<forall>x y. f x \<le> y \<longrightarrow> (\<exists>z\<ge>x. y = f z)"
trivial_group_subset: shows "trivial_group (reduced_homology_group (int p) (nsphere p)) = (carrier (reduced_homology_group (int p) (nsphere p)) \<subseteq> {\<one>\<^bsub>reduced_homology_group (int p) (nsphere p)\<^esub>})"
simps_2: fixes x2 :: "'a"   and y2 :: "'a" shows "(Neg x2 = Neg y2) = (x2 = y2)"
Sup_empty: 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 bot :: "'a"   and top :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup {} = bot"
comm_monoid_axioms: shows "comm_monoid (\<oplus>) \<zero>"
poly_roots_linear: fixes x :: "'a" shows "poly_roots [:- x, 1::'a:] = {#x#}"
span_insert_idI: fixes r :: "'b"   and B :: "'b set" assumes "r \<in> span B" shows "span (insert r B) = span B"
distinct_entries: fixes t :: "('a, 'c) rbt" assumes "rbt_sorted t" shows "distinct (map fst (entries t))"
degree_pdevs_of_list_scaleR: fixes c :: "real"   and xs :: "'a list" shows "degree (pdevs_of_list (map ((*\<^sub>R) c) xs)) = (if c \<noteq> 0 then degree (pdevs_of_list xs) else 0)"
key_sets_msg_def: fixes X :: "msg"   and Y :: "msg" shows "key_sets_msg X Y \<equiv> key_sets X {Y}"
weight_le_stable: fixes s :: "('f, 'c) Term.term"   and t :: "('f, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('f, 'd) Term.term" assumes "weight_le s t" shows "weight_le (s \<cdot> \<sigma>) (t \<cdot> \<sigma>)"
\<Psi>_sel_2: shows "capacity \<Psi> = cap"
cone_at_ide_is_cone: fixes x :: "('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and \<chi> :: "'j \<Rightarrow> ('a \<Rightarrow> 'b, 'a \<Rightarrow> 'b) concrete_category.arr"   and a :: "'a" assumes "local.cone x \<chi>"   and "A.ide a" shows "Limit.cone (\<cdot>\<^sub>J) (\<cdot>\<^sub>B) (at a D) (A_B.Map x a) (at a \<chi>)"
exec_step_ind_Getfield: fixes v :: "val"   and stk :: "val list"   and D :: "char list"   and fs :: "Objects.fields"   and h :: "heap"   and D' :: "char list"   and b :: "staticb"   and t :: "ty"   and P :: "jvm_prog"   and C :: "char list"   and F :: "char list"   and loc :: "val list"   and C\<^sub>0 :: "char list"   and M\<^sub>0 :: "char list"   and pc :: "nat"   and ics :: "init_call_status"   and frs :: "frame list"   and sh :: "sheap" assumes "v = hd stk"   and "(D, fs) = the (h (the_Addr v))"   and "(D', b, t) = field P C F"   and "v \<noteq> Null"   and "P \<turnstile> D has F,NonStatic:t in C" shows "exec_step_ind (StepI (Getfield F C)) P h stk loc C\<^sub>0 M\<^sub>0 pc ics frs sh (None, h, (the (fs (F, C)) # tl stk, loc, C\<^sub>0, M\<^sub>0, pc + 1, ics) # frs, sh)"
cases: fixes x :: "'b \<times> 'b" obtains l :: "'b" and u :: "'b" where "x = (l, u)"
subring_is_ring: fixes H :: "'c set" assumes "subring H S" shows "ring (S\<lparr>carrier := H\<rparr>)"
that: fixes P' :: "'i \<Rightarrow> ('a \<times> 'a) set"   and k :: "'i" assumes "profile A Is P'"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P' i)\<^esub>\<prec> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<prec> y"   and "\<And>i. \<lbrakk>i \<in> Is; x \<^bsub>(P' i)\<^esub>\<approx> y\<rbrakk> \<Longrightarrow> x \<^bsub>(P i)\<^esub>\<preceq> y"   and "k \<in> Is \<and> x \<^bsub>(P' k)\<^esub>\<approx> y \<and> x \<^bsub>(P k)\<^esub>\<prec> y"   and "card {i \<in> Is. x \<^bsub>(P' i)\<^esub>\<prec> y} = card {i \<in> Is. y \<^bsub>(P' i)\<^esub>\<prec> x}" shows "thesis"
univ_poly_is_ring: fixes R :: "('c, 'd) ring_scheme"   and K :: "'c set" assumes "domain R"   and "subring K R" shows "ring (K [X]\<^bsub>R\<^esub>)"
card_residue_primroots: fixes m :: "nat" assumes "\<exists>g. residue_primroot m g" shows "card {g \<in> totatives m. residue_primroot m g} = totient (totient m)"
add_left_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c \<squnion> a \<le> c \<squnion> b"
nth_append_length: fixes xs :: "'a list"   and x :: "'a"   and ys :: "'a list" shows "(xs @ x # ys) ! length xs = x"
div_imp_le_less: fixes n :: "nat"   and m :: "nat"   and k :: "nat" assumes "n div m = k"   and "0 < m" shows "k * m \<le> n \<and> n < Suc k * m"
cond_spmf_fst_map_prod_inj: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'd \<Rightarrow> 'c"   and p :: "('a \<times> 'd) spmf"   and x :: "'a" assumes "inj f" shows "cond_spmf_fst (map_spmf (\<lambda>(x, y). (f x, g x y)) p) (f x) = map_spmf (g x) (cond_spmf_fst p x)"
WTrtFAssNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'addr expr"   and e2 :: "'addr expr"   and T2 :: "ty"   and F :: "String.literal"   and D :: "String.literal" assumes "P,E,h \<turnstile> e1 : NT"   and "P,E,h \<turnstile> e2 : T2" shows "P,E,h \<turnstile> e1\<bullet>F{D} := e2 : Void"
assms_3: fixes g :: "recf"   and xs :: "nat list"   and rest :: "(recf \<times> nat list \<times> nat list) list" assumes "g \<in> set gs"   and "valid ((g, xs, []) # rest)" shows "correct ((g, xs, []) # rest, None)"
stutter_extend_simps_1: fixes G :: "('a, 'b) graph_rec_scheme" shows "g_V (stutter_extend G) = g_V G"
swap_cols_rows_def: fixes k :: "nat"   and l :: "nat"   and A :: "'a mat" shows "swap_cols_rows k l A = swaprows k l (swapcols k l A)"
rsplit0: fixes bs :: "real list"   and t :: "Ferrack.num" shows "Inum bs (case rsplit0 t of (x, xa) \<Rightarrow> CN 0 x xa) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
def_coinduct_set: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and a :: "'a"   and X :: "'a set" assumes "A \<equiv> gfp f"   and "mono f"   and "a \<in> X"   and "X \<subseteq> f (X \<union> A)" shows "a \<in> A"
trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "R x y"   and "R y z" shows "R x z"
vector_space_pair_ow_def: fixes U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 :: "'d \<Rightarrow> 'd"   and scale\<^sub>1 :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f set"   and plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f"   and minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f \<Rightarrow> 'f"   and uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 :: "'f \<Rightarrow> 'f"   and scale\<^sub>2 :: "'e \<Rightarrow> 'f \<Rightarrow> 'f" shows "vector_space_pair_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2 \<equiv> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>1 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>1 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>1 scale\<^sub>1 \<and> vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 plus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 zero\<^sub>V\<^sub>S\<^sub>_\<^sub>2 minus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 scale\<^sub>2"
FunctorId3Dom: fixes f :: "'c" assumes "f \<in> mor\<^bsub>CatDom F\<^esub>" shows "F ## id\<^bsub>CatDom F\<^esub> (dom\<^bsub>CatDom F\<^esub> f) = id\<^bsub>CatCod F\<^esub> (dom\<^bsub>CatCod F\<^esub> (F ## f))"
R_def: shows "R \<equiv> Regions.region \<X> I r"
D: shows "disjoint_family_on (\<lambda>y. {p \<in> space (PAR x). select_first x p y}) (I x)"
eqvt: fixes x :: "'a" shows "\<pi> \<bullet> f x = f (\<pi> \<bullet> x)"
simps_6: fixes v :: "'b"   and va :: "('a, 'b) Digit"   and vb :: "('a, 'b) FingerTreeStruc"   and vc :: "('a, 'b) Digit"   and xs :: "('a, 'b) Node list"   and x :: "('a, 'b) Node" shows "FingerTreeStruc.app3 (Deep v va vb vc) xs (Single x) = FingerTreeStruc.nrcons (FingerTreeStruc.rconsNlist (Deep v va vb vc) xs) x"
that: fixes P :: "'a mat"   and Q :: "'a mat" assumes "P \<in> carrier_mat n n"   and "Q \<in> carrier_mat n n"   and "char_poly A = (\<Prod>(na, a)\<leftarrow>n_as. [:- a, 1::'a:] ^ na)"   and "\<And>k. A ^\<^sub>m k = P * jordan_matrix n_as ^\<^sub>m k * Q" shows "thesis"
1: shows "wordinterval_CIDR_split1 r = (Some s, u)"
map_commutes_a: fixes Q :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c" shows "concat (map (\<lambda>n. if Q (f n) then [f n] else []) []) = filter Q (map f [])"
fls_deriv_eq_0_iff: fixes f :: "'a fls" shows "(fls_deriv f = 0) = (f = fls_const (fls_nth f 0))"
symm_weak_reduction_simulation_is_bisimulation: fixes Rel :: "('a \<times> 'a) set"   and Cal :: "'a processCalculus" assumes "sym Rel"   and "weak_reduction_simulation Rel Cal" shows "weak_reduction_bisimulation Rel Cal"
f: fixes M :: "'a measure" assumes "P M" shows "f \<in> M' \<rightarrow>\<^sub>M M"
chain_map_cmul: fixes p :: "nat"   and g :: "'d \<Rightarrow> 'c"   and a :: "int"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'd) \<Rightarrow>\<^sub>0 int" shows "chain_map p g (frag_cmul a c) = frag_cmul a (chain_map p g c)"
init_state_def: fixes t :: "(nat \<times> linear_poly) list" shows "init_state t = State t Mapping.empty Mapping.empty (Mapping.tabulate (Simplex.vars_list t) (\<lambda>v. 0::'b)) False None"
path_mono: fixes E :: "('b \<times> 'b) set"   and E' :: "('b \<times> 'b) set"   and u :: "'b"   and p :: "'b list"   and v :: "'b" assumes "E \<subseteq> E'"   and "path E u p v" shows "path E' u p v"
more_temp_simps3_3: fixes F :: "'a \<Rightarrow> bool" shows "\<diamond>\<diamond>F \<equiv> \<diamond>F"
ap_vec_def: fixes f :: "('e \<Rightarrow> 'c, 'd) vec"   and x :: "('e, 'd) vec" shows "Applicative.ap f x = (\<chi>i. (f $ i) (x $ i))"
eq_iff: fixes a :: "('a, 'b) Term.term"   and b :: "('a, 'b) Term.term" shows "(a = b) = (b \<sqsubseteq> a \<and> a \<sqsubseteq> b)"
cring_simprules_14: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e"   and y :: "'e" assumes "cring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> y = y \<otimes>\<^bsub>R\<^esub> x"
noFaultStuckn_execD: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, bool) com option"   and c :: "('e, 'd, bool) com"   and s :: "('e, bool) xstate"   and n :: "nat"   and t :: "('e, bool) xstate" assumes "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow>\<notin>{Fault True, Fault False, Stuck}"   and "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t" shows "t \<notin> {Fault True, Fault False, Stuck}"
singleton: fixes f :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and x :: "'g" assumes "semilattice_set f" shows "semilattice_set.F f {x} = x"
poincare_mapsto_congI2: fixes A :: "'a set"   and B :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set"   and C :: "'a set"   and D :: "'a set"   and E :: "('a \<times> 'd \<Rightarrow>\<^sub>L 'a) set" assumes "c1_on_open_euclidean.poincare_mapsto g Y A B C D E"   and "closed A" shows "poincare_mapsto A B C D E"
Fract_coprime: fixes a :: "int"   and b :: "int" shows "Fract (a div gcd a b) (b div gcd a b) = Fract a b"
map_add_subsumed1: fixes f :: "'a \<Rightarrow> 'b option"   and g :: "'a \<Rightarrow> 'b option" assumes "f \<subseteq>\<^sub>m g" shows "f ++ g = g"
deriv_subc: fixes z :: "('c, 'd) form list"   and p :: "('c, 'd) form"   and c :: "'c"   and s :: "'c FOL_Fitting.term" assumes "infinite UNIV"   and "z \<turnstile> p" shows "subcs c s z \<turnstile> subc c s p"
continuous_at_enn2ereal: fixes x :: "ennreal"   and A :: "ennreal set" shows "continuous (at x within A) enn2ereal"
invrules_2: assumes "L \<noteq> {}" shows "\<forall>B\<in>P\<^sub>1 \<union> wrap B\<^sub>1. B \<inter> {u \<in> U. real c / 2 < w u} \<noteq> {}"
right_option_is_option: fixes x :: "game"   and g :: "game" assumes "zin x (right_options g)" shows "zin x (options g)"
incl: fixes a :: "'a"   and H :: "'a set" assumes "a \<in> H" shows "a \<in> generate H"
dist_decreases_closed_segment: fixes x :: "'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "x \<in> closed_segment a b" shows "dist c x \<le> dist c a \<or> dist c x \<le> dist c b"
all_edges_subset_Pow: fixes A :: "nat set" shows "all_edges A \<subseteq> Pow A"
prob_sums: fixes s :: "'s"   and P :: "nat \<Rightarrow> 's stream \<Rightarrow> bool"   and Q :: "'s stream \<Rightarrow> bool" assumes "\<And>n. {x \<in> space (T s). P n x} \<in> T.events s"   and "{x \<in> space (T s). Q x} \<in> T.events s"   and "AE x in T s. (\<forall>n. P n x \<longrightarrow> Q x) \<and> (Q x \<longrightarrow> (\<exists>!n. P n x))" shows "(\<lambda>n. T.prob s {x \<in> space (T s). P n x}) sums T.prob s {x \<in> space (T s). Q x}"
unreachable_set_bounded: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and Q :: "'b set"   and b :: "'b"   and Qx :: "'b"   and Qy :: "'b" assumes "MinkowskiUnreachable \<E> \<P> betw"   and "Q \<in> \<P>"   and "b \<notin> Q"   and "b \<in> \<E>"   and "Qx \<in> Q - MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b"   and "Qy \<in> MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b" shows "\<exists>Qz\<in>Q - MinkowskiPrimitive.unreachable_subset \<E> \<P> Q b. betw Qx Qy Qz \<and> Qx \<noteq> Qz"
distinct_3: fixes x31 :: "ty_Rep"   and x32 :: "ty_Rep" shows "TBase_Rep \<noteq> Arrow_Rep x31 x32"
continuous_ge_on_closure: fixes s :: "'b set"   and f :: "'b \<Rightarrow> real"   and x :: "'b"   and a :: "real" assumes "continuous_on (closure s) f"   and "x \<in> closure s"   and "\<And>x. x \<in> s \<Longrightarrow> a \<le> f x" shows "a \<le> f x"
delete_edges_simp2: fixes G :: "'a graph"   and E :: "('a \<times> 'a) set" shows "FiniteGraph.delete_edges G E = \<lparr>nodes = nodes G, edges = edges G - E\<rparr>"
suminf_add: fixes f :: "nat \<Rightarrow> 'c"   and g :: "nat \<Rightarrow> 'c" assumes "summable f"   and "summable g" shows "suminf f + suminf g = (\<Sum>n. f n + g n)"