hcju/isabelleps huggingface dataset:
lbI: fixes A :: "nat \<Rightarrow> nat set"   and j :: "nat"   and T :: "nat \<Rightarrow> nat" assumes "\<forall>x\<in>{1..m}. \<forall>y\<in>{1..m}. x \<noteq> y \<longrightarrow> A x \<inter> A y = {}"   and "\<Union> (A ` {1..m}) = {1..j}"   and "\<forall>x\<in>{1..m}. sum t (A x) = T x" shows "lb T A j"
four_block_mat_preserves_column: fixes A :: "'b mat"   and n :: "nat"   and m :: "nat"   and B :: "'b mat"   and C :: "'b mat"   and i :: "nat" assumes "A \<in> carrier_mat n m"   and "B = four_block_mat (1\<^sub>m 1) (0\<^sub>m 1 (m - 1)) (0\<^sub>m (m - 1) 1) C"   and "C \<in> carrier_mat (m - 1) (m - 1)"   and "i < n"   and "0 < m" shows "(A * B) $$ (i, 0) = A $$ (i, 0)"
assms_2: fixes x :: "'a" assumes "x \<in> U"   and "P x" shows "Q x"
less_eq_list_antisym: fixes xs :: "'a list"   and ys :: "'a list" assumes "local.less_eq_list xs ys"   and "local.less_eq_list ys xs" shows "xs = ys"
field_cont_on_typeII_region_cont_on_edges: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real"   and F :: "real \<times> real \<Rightarrow> 'a"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "typeII_twoCube twoC"   and "continuous_on (cubeImage twoC) F"   and "(k, \<gamma>) \<in> boundary twoC" shows "continuous_on (\<gamma> ` {0..1}) F"
meval_MPred: fixes n :: "nat"   and t :: "nat"   and db :: "(char list \<times> 'b list) set"   and e :: "char list"   and ts :: "'b MFOTL.trm list" shows "meval n t db (MPred e ts) = ([Option.these ((map_option (\<lambda>f. tabulate f 0 n) \<circ> match ts) ` (\<Union>(e', x)\<in>db. if e = e' then {x} else {}))], MPred e ts)"
fst_UNIV: fixes A :: "('d \<times> 'e) set" shows "A \<subseteq> fst ` A \<times> UNIV"
get_dom_component_def: shows "Shadow_DOM_DOM_Components.get_dom_component \<equiv> l_get_dom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_dom_component Shadow_DOM.get_root_node Shadow_DOM.to_tree_order"
deriv: fixes x :: "(real, 'n) vec" assumes "x \<in> S" shows "(f has_derivative f' x) (at x within S)"
Norm_def: fixes "is" :: "floatarith list" shows "Norm is = Sqrt (Sum\<^sub>e (\<lambda>i. i * i) is)"
order_uniform_limitD1: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> real"   and T :: "'c set"   and l :: "'c \<Rightarrow> real"   and x :: "'b"   and b :: "real" assumes "uniform_limit T f l (at x)"   and "continuous_on T l"   and "compact T"   and "\<And>t. t \<in> T \<Longrightarrow> l t < b" shows "\<forall>\<^sub>F y in at x. \<forall>t\<in>T. f y t < b"
no_olen_add_nat: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> x + y) = (unat x + unat y < 2 ^ LENGTH('b))"
arr_tensor: fixes f :: "'a"   and g :: "'a" assumes "arr f"   and "arr g" shows "arr (tensor f g)"
row_def: fixes A :: "'b mat"   and i :: "nat" shows "row A i = vec (dim_col A) (\<lambda>j. A $$ (i, j))"
replicating_portfolio_def: fixes M :: "'c measure"   and F :: "nat \<Rightarrow> 'c measure"   and Mkt :: "('c, 'd) discrete_market"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'c \<Rightarrow> real"   and der :: "'c \<Rightarrow> real"   and matur :: "nat" assumes "disc_equity_market M F" shows "disc_equity_market.replicating_portfolio M F Mkt pf der matur = (stock_portfolio Mkt pf \<and> disc_filtr_prob_space.trading_strategy F pf \<and> self_financing Mkt pf \<and> AEeq M (cls_val_process Mkt pf matur) der)"
label_stab_map_surj_on_simplices: fixes \<phi> :: "'a \<Rightarrow> 'a" assumes "label_wrt C0 \<phi>"   and "fixespointwise \<phi> C0" shows "(\<lambda>v. {w \<in> \<langle>\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}\<rangle>. w \<rightarrow> \<phi> v = v}) \<turnstile> X = CoxeterComplex.TheComplex (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g})"
leg1_simps_6: shows "cod s\<^sub>1 = s\<^sub>1"
L3: fixes k :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "take k (map f xs) = map f (take k xs)"
simps_1: fixes eff :: "'d \<Rightarrow> 'e \<Rightarrow> 'e fset \<Rightarrow> bool"   and rules :: "'d stream"   and "structure" :: "'f set"   and sat :: "'f \<Rightarrow> 'e \<Rightarrow> bool"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> ('g \<times> bool \<times> 'g) set"   and Ord :: "('h \<times> 'h) set"   and \<sigma> :: "'g \<times> 'f \<Rightarrow> 'h"   and inv :: "'e \<times> 'i \<Rightarrow> bool"   and pred :: "'e \<times> 'i \<Rightarrow> 'd \<Rightarrow> 'e \<times> 'i \<Rightarrow> bool"   and t :: "('e \<times> 'd) tree"   and a :: "'i" assumes "Infinite_Soundness eff rules structure sat \<delta> Ord \<sigma>" shows "shd (Infinite_Soundness.konigDtree inv pred t a) = (tree.root t, a)"
matching_Goto: assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[r], s\<rangle> \<Rightarrow> t"   and "\<not> no_matching_Goto \<gamma> p [r]" shows "thesis"
terminating_path_root_end_aux: fixes x :: "'a" assumes "terminating_path x" shows "\<exists>r e. terminating_path_root_end r x e"
nsMulDistrL: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r  n \<times>\<^bsub>R\<^esub> y =  n \<times>\<^bsub>R\<^esub> (x \<cdot>\<^sub>r y)"
redlove_def: fixes p :: "fm" shows "redlove p \<equiv> let (q, B, d) = chooset p; mq = simpfm (minusinf q); md = evaldjf (\<lambda>j. simpfm (subst0 (C j) mq)) [1..d] in if md = T then T else let qd = evaldjf (stage q d) B in decr (MIR.disj md qd)"
assms: shows "know P (sKS m) \<or> know Q (sKS m)"
R: shows "stabilizer G \<phi> x #> inv g \<in> rcosets stabilizer G \<phi> x"
that: shows "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"
externals_def: fixes asig :: "'b signature" shows "externals asig \<equiv> inputs asig \<union> outputs asig"
sel: fixes x2 :: "'f" shows "the (Some x2) = x2"
equals_rsp: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Abs :: "'b \<Rightarrow> 'c"   and Rep :: "'c \<Rightarrow> 'b"   and xa :: "'b"   and xb :: "'b"   and ya :: "'b"   and yb :: "'b" assumes "Quotient3 R Abs Rep"   and "R xa xb"   and "R ya yb" shows "R xa ya = R xb yb"
iteratei_abs_rule_insert_P: assumes "set_iterator_abs_genord \<alpha> invar it S0 (\<lambda>_ _. True)"   and "I {} \<sigma>0"   and "\<And>S \<sigma> x. \<lbrakk>c \<sigma>; invar x; \<alpha> x \<in> S0 - S; I S \<sigma>; S \<subseteq> S0; \<forall>y\<in>S0 - S - {\<alpha> x}. True; \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> I (insert (\<alpha> x) S) (f x \<sigma>)"   and "\<And>\<sigma>. I S0 \<sigma> \<Longrightarrow> P \<sigma>"   and "\<And>\<sigma> S. \<lbrakk>S \<subseteq> S0; S \<noteq> S0; \<not> c \<sigma>; I S \<sigma>; \<forall>x\<in>S0 - S. \<forall>y\<in>S. True\<rbrakk> \<Longrightarrow> P \<sigma>" shows "P (it c f \<sigma>0)"
rel_inject_2: fixes R :: "'l \<Rightarrow> 'm \<Rightarrow> bool"   and x2 :: "'l"   and y2 :: "'m" shows "rel_option R (Some x2) (Some y2) = R x2 y2"
map_upds_cut_irrelevant: fixes tab :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and vn :: "'b"   and el :: "'a"   and tab' :: "'b \<Rightarrow> 'a option" assumes "(tab(xs [\<mapsto>] ys)) vn = Some el"   and "(tab'(xs [\<mapsto>] ys)) vn = None" shows "tab vn = Some el"
assms_1: fixes x :: "ZF" assumes "x \<in> A" shows "f x \<in> range explode"
f_Exec_Cons: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and x :: "'e"   and xs :: "'e list"   and c :: "'d" shows "f_Exec_Comp trans_fun (x # xs) c = f_Exec_Comp trans_fun xs (trans_fun x c)"
mat_uniqueness: fixes nr1 :: "nat"   and nc :: "nat"   and M :: "'b list list"   and nr2 :: "nat"   and z :: "'b list" assumes "mat nr1 nc M"   and "mat nr2 nc M"   and "z = hd M"   and "M \<noteq> []" shows "\<forall>x\<in>set M. nr1 = nr2"
Gromov_completion_rel_def: fixes u :: "nat \<Rightarrow> 'a"   and v :: "nat \<Rightarrow> 'a" shows "Gromov_completion_rel u v = (Gromov_converging_at_boundary u \<and> Gromov_converging_at_boundary v \<and> (\<forall>a. (\<lambda>x. ereal (Gromov_product_at a (u x) (v x))) \<longlonglongrightarrow> \<infinity>) \<or> (\<forall>n m. u n = v m \<and> u n = u m \<and> v n = v m))"
LeftDerivation_impossible: fixes a :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list"   and i :: "nat" assumes "LeftDerivation a D b"   and "i < length a"   and "is_nonterminal (a ! i)"   and "derivation_ge D (Suc i)" shows "D = []"
swap_set_in: fixes a :: "atom"   and S :: "atom set"   and b :: "atom" assumes "a \<in> S"   and "b \<notin> S"   and "sort_of a = sort_of b" shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
has_integral_integrable_integral: fixes f :: "'b \<Rightarrow> 'c"   and i :: "'c"   and s :: "'b set" shows "(f has_integral i) s = (f integrable_on s \<and> integral s f = i)"
r_into_rtranclp: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g" assumes "r x y" shows "r\<^sup>*\<^sup>* x y"
simps_2: fixes f :: "'f"   and ts :: "('f, 'a) Term.term list" shows "weight (Fun f ts) = (let n = length ts; scff = scf (f, n) in w (f, n) + sum_list (map2 (\<lambda>ti i. weight ti * scff i) ts [0..<n]))"
edka5_def: fixes am :: "nat \<Rightarrow> nat list" shows "edka5 am \<equiv> edka5_tabulate am \<bind> (\<lambda>(x, y). edka5_run x y)"
fv_subterms_subst_eq: fixes t :: "('b, 'c) Term.term"   and \<theta> :: "'c \<Rightarrow> ('b, 'a) Term.term" shows "fv\<^sub>s\<^sub>e\<^sub>t (subterms (t \<cdot> \<theta>)) = fv\<^sub>s\<^sub>e\<^sub>t (subterms t \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>)"
degree_det: fixes A :: "('c poly, 'b) sq_matrix" assumes "\<And>j. \<lbrakk>j permutes UNIV; j \<noteq> id\<rbrakk> \<Longrightarrow> degree (\<Prod>i\<in>UNIV. to_fun A i (j i)) < degree (\<Prod>i\<in>UNIV. to_fun A i i)" shows "degree (Square_Matrix.det A) = degree (\<Prod>i\<in>UNIV. to_fun A i i)"
is_natural_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (G f) (\<tau> (partial_magma.dom A f)) = \<tau> f"
bivar_fun_mem: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'d set"   and C :: "'e set"   and a :: "'c"   and b :: "'d" assumes "f \<in> A \<rightarrow> B \<rightarrow> C"   and "a \<in> A"   and "b \<in> B" shows "f a b \<in> C"
keys_empty_iff: fixes p :: "'b \<Rightarrow>\<^sub>0 'c" shows "(keys p = {}) = (p = 0)"
is_valid_operator_sas_plus_then_3: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and op :: "('c, 'd) sas_plus_operator" assumes "is_valid_operator_sas_plus \<Psi> op" shows "\<forall>(v, a)\<in>set (effect_of op). v \<in> set (\<Psi>\<^sub>\<V>\<^sub>+)"
p: shows "map snd cms\<^sub>1' = map snd cms\<^sub>2' \<and> (add_initial_modes cmds, mem\<^sub>2) \<rightarrow>\<^bsub>k\<^esub> (cms\<^sub>2', mem\<^sub>2') \<and> makes_compatible (cms\<^sub>1', mem\<^sub>1') (cms\<^sub>2', mem\<^sub>2') mems'"
S'Q': shows "(S', Q') = Smith_1xn_aux (Suc i) A (S, Q)"
iadd_Suc: fixes m :: "enat"   and n :: "enat" shows "eSuc m + n = eSuc (m + n)"
bounded_cbilinear_cblinfun_apply: shows "bounded_cbilinear cblinfun_apply"
matching_def: fixes sigma :: "Runs.var \<Rightarrow> msg option"   and sigma' :: "Runs.var \<Rightarrow> msg option" shows "matching sigma sigma' \<equiv> \<forall>x. x \<in> dom sigma \<inter> dom sigma' \<longrightarrow> sigma x = sigma' x"
rel_Grp: fixes A1 :: "'i set"   and f1 :: "'i \<Rightarrow> 'k"   and A2 :: "'j set"   and f2 :: "'j \<Rightarrow> 'l" shows "rel_sum (BNF_Def.Grp A1 f1) (BNF_Def.Grp A2 f2) = BNF_Def.Grp {x. Basic_BNFs.setl x \<subseteq> A1 \<and> Basic_BNFs.setr x \<subseteq> A2} (map_sum f1 f2)"
jointfun_def: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'e"   and m :: "nat"   and g :: "nat \<Rightarrow> 'e" shows "jointfun n f m g = (\<lambda>i. if i \<le> n then f i else g (sliden (Suc n) i))"
digraph_dg_Rel: shows "digraph \<alpha> (dg_Rel \<alpha>)"
P_LCons: shows "P = LCons v0 (ltl P)"
regions_closed: fixes \<R> :: "('d \<Rightarrow> real) set set"   and X :: "'d set"   and k :: "'d \<Rightarrow> nat"   and R :: "('d \<Rightarrow> real) set"   and v :: "'d \<Rightarrow> real"   and t :: "real" assumes "\<R> = {region X I r |I r. valid_region X k I r}"   and "R \<in> \<R>"   and "v \<in> R"   and "0 \<le> t" shows "[v \<oplus> t]\<^sub>\<R> \<in> \<R>"
aux: fixes b :: "'d option" shows "(\<bottom> \<noteq> b) = (\<exists>y. b = \<lfloor>y\<rfloor>)"
nn_integral_return: fixes x :: "'h"   and M :: "'h measure"   and g :: "'h \<Rightarrow> ennreal" assumes "x \<in> space M"   and "g \<in> borel_measurable M" shows "integral\<^sup>N (return M x) g = g x"
verts_mem: fixes i :: "nat"   and G :: "graph" shows "(i \<in> verts G) = (i < Dijkstra.size G)"
SUP_Fx: fixes x :: "'a" assumes "x \<in> space M" shows "(\<Squnion>n. F n x) = f x"
if_pred_eq: fixes b :: "bool" shows "Predicate.if_pred b = (if b then Predicate.single () else bot)"
tt_uminus: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "tt (- p) = tt p"
ord_minimal: fixes m :: "nat"   and n :: "nat"   and a :: "nat" assumes "0 < m"   and "m < ord n a" shows "[a ^ m \<noteq> 1] (mod n)"
lunit'_simps_2: fixes f :: "'a" assumes "ide f" shows "src \<l>\<^sup>-\<^sup>1[f] = src f"
c0: shows "(a1 = pCons (coeff (pCons b c2) (degree q) div lead_coeff q) c1 \<and> a2 = pCons b c2 - smult (coeff (pCons b c2) (degree q) div lead_coeff q) q) = (Some (a1, a2) = div_int_poly_step q b (Some (c1, c2)))"
abs_le_self_iff: fixes abs :: "'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 a :: "'b" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (abs a) a = less_eq zero a"
in_obs_intra_valid_2: fixes n' :: "'node"   and n :: "'node"   and S :: "'node set" assumes "n' \<in> obs_intra n S" shows "valid_node n'"
NewHeapElem: fixes P :: "'a prog"   and a :: "'b"   and hT :: "htype" shows "saction P (NormalAction (NewHeapElem a hT))"
simps: fixes s :: "('e, 'f) locks \<times> (('f \<Rightarrow> ('x \<times> 'e \<Rightarrow>f nat) option) \<times> 'g) \<times> ('f \<Rightarrow> 'h wait_set_status option) \<times> 'f set"   and t :: "'f"   and a1 :: "'e + 'f + 'f"   and a2 :: "'f set" shows "must_wait s t a1 a2 = ((\<exists>l t' Ts. a1 = Inl l \<and> a2 = Ts \<and> has_lock (locks s $ l) t' \<and> t' \<noteq> t \<and> t' \<in> Ts) \<or> (\<exists>t' Ts. a1 = Inr (Inl t') \<and> a2 = Ts \<and> not_final_thread s t' \<and> t' \<in> Ts) \<or> (\<exists>Ts t'. a1 = Inr (Inr t') \<and> a2 = Ts \<and> all_final_except s Ts \<and> t' \<notin> interrupts s))"
Edges_compl: fixes vs :: "nat list"   and x :: "nat"   and y :: "nat" assumes "distinct vs"   and "x \<in> set vs"   and "y \<in> set vs"   and "x \<noteq> y" shows "Edges (x # between vs x y @ [y]) \<inter> Edges (y # between vs y x @ [x]) = {}"
\<pi>_at_top: shows "filterlim \<pi> at_top at_top"
IH': fixes \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec" shows "dterm_sem I (Tsubst \<theta> \<sigma>) \<nu> = dterm_sem (local.adjoint I \<sigma> \<nu>) \<theta> \<nu>"
nth_map: fixes n :: "nat"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'c" assumes "n < length xs" shows "map f xs ! n = f (xs ! n)"
assms_8: shows "poincare_distance z u = poincare_distance z v"
l_one: fixes x :: "nat \<times> nat" assumes "x \<in> carrier S" shows "\<one>\<^bsub>S\<^esub> \<otimes>\<^bsub>S\<^esub> x = x"
h6: shows "get_matrix_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))) = M_mat_R (get_signs_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2)))) (get_subsets_R (snd (combine_systems_R p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
less_level: fixes l1 :: "level"   and l2 :: "level" shows "l1 < l2 \<equiv> l1 = Lo \<and> l2 = Hi"
row_add_iterate_eq_Gauss_Jordan_in_ij: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c"   and j :: "'b" shows "row_add_iterate (mult_row (interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n)) i ((1::'a) / interchange_rows A i (LEAST n. A $ n $ j \<noteq> (0::'a) \<and> i \<le> n) $ i $ j)) (nrows A - 1) i j = Gauss_Jordan_in_ij A i j"
assms_3: shows "\<lbrakk>M; map (\<lambda>d. \<langle>ac: pair (t, s) \<doteq> pair (snd d)\<rangle>\<^sub>s\<^sub>t) Di\<rbrakk>\<^sub>d \<I>"
exhaust: fixes y :: "'a \<times> 'b" obtains x1 :: "'a" and x2 :: "'b" where "y = (x1, x2)"
when_False: fixes X :: "'d \<Rightarrow> (unit \<times> 'd) \<times> bool" shows "when1 False X = return ()"
hom_restrict: fixes h :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c"   and H :: "('c, 'd) monoid_scheme"   and t :: "('a \<Rightarrow>\<^sub>0 int) \<Rightarrow> 'c" assumes "h \<in> hom (free_Abelian_group S) H"   and "\<And>g. g \<in> carrier (free_Abelian_group S) \<Longrightarrow> h g = t g" shows "t \<in> hom (free_Abelian_group S) H"
is_TNil_def: fixes tllist :: "('f, 'g) tllist" shows "is_TNil tllist = (\<exists>x1. tllist = TNil x1)"
elims_3: fixes wtFsym :: "'b \<Rightarrow> bool"   and wtPsym :: "'c \<Rightarrow> bool"   and arOf :: "'b \<Rightarrow> 'a list"   and resOf :: "'b \<Rightarrow> 'a"   and x :: "'b trm" assumes "Signature TYPE('a) wtFsym wtPsym"   and "\<not> Signature.wt wtFsym arOf resOf x" obtains f :: "'b" and Tl :: "'b trm list" where "x = Fn f Tl" and "\<not> (wtFsym f \<and> list_all (Signature.wt wtFsym arOf resOf) Tl \<and> arOf f = map (Signature.tpOf resOf) Tl)"
monoidal_category_axioms: shows "monoidal_category (\<cdot>) T \<alpha> \<iota>"
rbt_max_opt_induct: fixes P :: "('b, unit) RBT_Impl.rbt \<Rightarrow> bool"   and t :: "('b, unit) RBT_Impl.rbt" assumes "P rbt.Empty"   and "\<And>color t1 a b t2. \<lbrakk>P t1; P t2; t2 = rbt.Empty\<rbrakk> \<Longrightarrow> P (Branch color t1 a b t2)"   and "\<And>color t1 a b t2. \<lbrakk>P t1; P t2; t2 \<noteq> rbt.Empty\<rbrakk> \<Longrightarrow> P (Branch color t1 a b t2)" shows "P t"
c_pos: shows "\<forall>\<^sub>F n in sequentially. 0 < c n"
imonad1: fixes u :: "unit"   and f :: "unit \<Rightarrow> 'b nres" shows "RETURN u \<bind> f = f u"
add_cancel_left_right: fixes a :: "'a"   and b :: "'a" shows "(a + b = a) = (b = (0::'a))"
two_ordering: fixes ord :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "\<exists>f. TernaryOrdering.ordering f ord {a, b}"
assms_2: shows "sigma_finite_measure (distr M borel (\<lambda>x n. f ((T ^^ n) x)))"
assoc: fixes a :: "'k"   and b :: "'k"   and c :: "'k" shows "a * b * c = a * (b * c)"
guessed_frame_Resp_xgny: fixes R :: "fid_t" assumes "role (guessed_runs R) = Resp" shows "sklvl1.guessed_frame R xgny = Some (Exp Gen (NonceF (R $ ny)))"
list_all2_reorder_left_invariance: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'d list"   and ys :: "'e list"   and xs' :: "'d list" assumes "list_all2 R xs ys"   and "mset xs' = mset xs" shows "\<exists>ys'. list_all2 R xs' ys' \<and> mset ys' = mset ys"
list_order_monotonic: fixes A :: "('a \<times> 'a option) list"   and B :: "('a \<times> 'a option) list"   and x :: "'a"   and y :: "'a" assumes "insert_ops A"   and "insert_ops B"   and "set A \<subseteq> set B"   and "list_order A x y" shows "list_order B x y"
bisim1_inline_call_Throw: fixes P :: "'a prog"   and h :: "'heap"   and e :: "'addr expr1"   and e' :: "'addr expr1"   and xs :: "'addr val list"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and a :: "'addr"   and M :: "String.literal"   and vs :: "'addr val list"   and n0 :: "nat"   and A :: "'addr" assumes "P,e,h \<turnstile> (e', xs) \<leftrightarrow> (stk, loc, pc, None)"   and "call1 e' = \<lfloor>(a, M, vs)\<rfloor>"   and "compE2 e ! pc = Invoke M n0"   and "pc < length (compE2 e)" shows "n0 = length vs \<and> P,e,h \<turnstile> (inline_call (Throw A) e', xs) \<leftrightarrow> (stk, loc, pc, \<lfloor>A\<rfloor>)"
infsetsum_Im: fixes f :: "'c \<Rightarrow> complex"   and M :: "'c set" assumes "f abs_summable_on M" shows "(\<Sum>\<^sub>ax\<in>M. Im (f x)) = Im (infsetsum f M)"
emp_shift_rev: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat"   and I :: "nat set" shows "({} = local.shift cl n I) = (I = {})"
rel_predI: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'c set"   and B :: "'d set" assumes "(R ===> (=)) (\<lambda>x. x \<in> A) (\<lambda>y. y \<in> B)" shows "rel_pred R A B"
m1_inv4_rfreshI: fixes x :: "'a m1_state_scheme" assumes "\<And>Rb1 Rb2 A1 A2 B1 B2 Kab Ts1 Ts2 Ta1 Ta2. \<lbrakk>runs x Rb1 = Some (Resp, [A1, B1], [aKey Kab, aNum Ts1, aNum Ta1]); runs x Rb2 = Some (Resp, [A2, B2], [aKey Kab, aNum Ts2, aNum Ta2]); B1 \<notin> bad; A1 \<notin> bad; Kab \<notin> Domain (m1_state.leak x)\<rbrakk> \<Longrightarrow> Rb1 = Rb2" shows "x \<in> m1_inv4_rfresh"
det_perm_rows: fixes p :: "'c \<Rightarrow> 'c"   and A :: "('d, 'c) sq_matrix" assumes "p permutes UNIV" shows "Square_Matrix.det (perm_rows A p) = of_int (sign p) * Square_Matrix.det A"
new_msg_in_set_implies_occurrence: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and c' :: "('a, 'b, 'c) configuration"   and m :: "'c message"   and i :: "nat"   and p :: "'a"   and q :: "'a" assumes "distributed_system channel"   and "distributed_system.next channel trans send recv c ev c'"   and "m \<notin> set (msgs c i)"   and "m \<in> set (msgs c' i)"   and "channel i = Some (p, q)" shows "occurs_on ev = p"
super_orientation_irreflexive_inf: fixes x :: "'a"   and y :: "'a" assumes "super_orientation x y" shows "irreflexive_inf x"
map_graph_returns_restricted: fixes G :: "('f, 'e) labeled_graph"   and f :: "('e \<times> 'g) set" assumes "vertices G = Domain f" shows "graph (map_graph f G)"
lossless_response: fixes r :: "nat"   and w :: "nat"   and e :: "nat" shows "lossless_spmf (response r w e)"
Sup_upper: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "\<Sqinter> A \<le> x"
hyps_3: assumes "\<forall>P\<in>S - {Q}. uhnf P \<and> valid P" shows "\<exists>P'. uhnf P' \<and> valid P' \<and> P \<equiv>\<^sub>e P' \<and> depth P' \<le> depth P"
is_hom_ord_def: fixes x :: "'x" shows "is_hom_ord x = (\<forall>s t. deg_pm s = deg_pm t \<longrightarrow> (s \<preceq> t) = (except s {x} \<preceq> except t {x}))"
finite_quote_all: fixes V :: "name set"   and p :: "perm" assumes "finite V" shows "finite (quote_all p V)"
EndPhase0_HInv3_nL: fixes s :: "state"   and s' :: "state"   and t :: "Proc"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase0 s s' t"   and "\<not> HInv3_L s p q d"   and "t \<noteq> p"   and "t \<noteq> q" shows "\<not> HInv3_L s' p q d"
abs_eq: fixes xa :: "'b"   and x :: "('b \<times> real) nonzero" shows "chart_basis_domainP xa (Proj x) = chart_basis_domain_nonzeroP xa x"
assms_3: fixes x :: "'b" assumes "x \<in> space M" shows "0 \<le> f x"
ordertype_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "ordertype A r \<equiv> set (ordermap A r ` A)"
assoc: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and m :: "'a \<Rightarrow> 'a \<Rightarrow> 'a option"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "merkle_interface h bo m" shows "m a b \<bind> m c = m b c \<bind> m a"
eq: shows "MPoly_Type.monom (monomial (Suc 0) x') (1::'a) = Var x'"
DIM_precond_real: shows "DIM_precond TYPE(real) 1"
write_reg_def: fixes w :: "word32"   and win :: "'b word"   and ur :: "word5" shows "write_reg w win ur \<equiv> do modify (user_reg_mod w win ur); return () od"
LLL_inv_wD_1: fixes fs :: "int vec list" assumes "LLL_invariant_weak fs" shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs)"