hcju/isabelleps huggingface dataset:
even_power: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and n :: "nat" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (power.power one times a n) = (dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a \<and> 0 < n)"
inv_QP: shows "invertible (Q ** mult_row (mat (1::'a)) i (ass (A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))) div A $ i $ (LEAST n. A $ i $ n \<noteq> (0::'a))))"
ipresIGWls_def: fixes h :: "'z \<Rightarrow> 'aa"   and MOD :: "('ab, 'ac, 'ad, 'ae, 'af, 'ag, 'z, 'ah, 'ai) model_scheme"   and MOD' :: "('aj, 'ak, 'al, 'ae, 'am, 'an, 'aa, 'ao, 'ap) model_scheme" shows "ipresIGWls h MOD MOD' \<equiv> \<forall>s X. igWls MOD s X \<longrightarrow> igWls MOD' s (h X)"
assms: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', (\<alpha>\<cdot>P) \<oplus>\<^sub>\<top> \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q, \<alpha>\<cdot>(\<tau>.P) \<oplus>\<^sub>\<top> Q) \<in> Rel"
three: fixes x :: "'a" shows "3 \<times> x = x \<squnion> x \<squnion> x"
cases: fixes a :: "'a llist" assumes "valid_path a" obtains "a = LNil"   | v :: "'a" where "a = LCons v__ LNil" and "v__ \<in> V"   | v :: "'a" and w :: "'a" and Ps :: "'a llist" where "a = LCons v Ps" and "v \<in> V" and "w \<in> V" and "v \<rightarrow> w" and "valid_path Ps" and "\<not> lnull Ps" and "lhd Ps = w"
1: fixes i :: "'a" assumes "i \<in> K'" shows "0 < r i \<and> r i \<le> 1"
hd_init: fixes f :: "nat \<Rightarrow> nat option"   and n :: "nat" shows "e_hd (f \<triangleright> n) = the (f 0)"
power_Suc2: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc n) = times (power.power one times a n) a"
Map_cod: fixes f :: "('a, 'a set) arr" assumes "arr f" shows "Map (cod f) = \<lbrakk>local.Cod f\<rbrakk>\<^sub>B"
foldr_append: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and xs :: "'d list"   and ys :: "'d list"   and a :: "'c" shows "foldr f (xs @ ys) a = foldr f xs (foldr f ys a)"
pred_sexp_CONS_I1: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "(M, CONS M N) \<in> pred_sexp\<^sup>+"
ideals_def: shows "ord.ideals (\<lambda>x y. y \<le> x) = {X. X \<noteq> {} \<and> X \<in> ord.downsets (\<lambda>x y. y \<le> x) \<and> ord.directed (\<lambda>x y. y \<le> x) X}"
is_prefix_empty: fixes a :: "'b list" shows "is_prefix [] a"
Equiv2Cl: fixes T :: "(ZF, ZF) Axioms"   and A :: "ZF"   and e :: "(ZF, ZF) Expression"   and d :: "(ZF, ZF) Expression"   and B :: "ZF" assumes "Axioms T"   and "Vx : A \<turnstile> e\<equiv>d : B \<in> Axioms.Theory T" shows "[A,e,B]\<^bsub>T\<^esub> = [A,d,B]\<^bsub>T\<^esub>"
S_V: shows "S \<subseteq> V\<^bsub>remove_vertex v\<^esub>"
take_Suc_conv_app_nth: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "take (Suc i) xs = take i xs @ [xs ! i]"
Entry_path: 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 n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit"   and "CFG.valid_node sourcenode targetnode valid_edge n" shows "\<exists>as. CFG.valid_path' sourcenode targetnode kind valid_edge get_return_edges Entry as n"
drop_all_iff: fixes n :: "nat"   and xs :: "'b list" shows "(drop n xs = \<epsilon>) = (\<^bold>|xs\<^bold>| \<le> n)"
elims: fixes x :: "('b literal multiset \<times> nat) multiset \<times> ('b literal multiset \<times> nat) multiset \<times> ('b literal multiset \<times> nat) multiset \<times> nat"   and y :: "'b literal multiset set \<times> 'b literal multiset set \<times> 'b literal multiset set" assumes "state_of_wstate x = y" obtains N :: "('b literal multiset \<times> nat) multiset" and P :: "('b literal multiset \<times> nat) multiset" and Q :: "('b literal multiset \<times> nat) multiset" and n :: "nat" where "x = (N, P, Q, n)" and "y = (set_mset (image_mset fst N), set_mset (image_mset fst P), set_mset (image_mset fst Q))"
bound_on_all_plans_bounds_MPLS_charles: fixes P :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> bool"   and f :: "(('c, bool) fmap \<times> ('c, bool) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> finite PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB x. P PROB \<and> finite PROB \<longrightarrow> x \<in> MPLS_charles PROB \<longrightarrow> x < f PROB"
changes_increasing: fixes xs :: "'a list" assumes "\<And>i. i < length xs - 1 \<Longrightarrow> xs ! i \<le> xs ! (i + 1)"   and "1 < length xs"   and "hd xs < (0::'a)"   and "(0::'a) < last xs" shows "changes xs = 1"
convex_translation_eq: fixes a :: "'a"   and s :: "'a set" shows "convex ((+) a ` s) = convex s"
comm_monoid_mult_ow_axioms: shows "comm_monoid_mult_ow U (*\<^sub>o\<^sub>w) 1\<^sub>o\<^sub>w"
higherPowers_lowerPowers: fixes n :: "nat"   and i :: "nat"   and x :: "nat \<Rightarrow>\<^sub>0 'a" shows "higherPowers n i (lowerPowers n i x) = x"
hyps: shows "\<exists>rs'. s \<star> rs = s \<star> rs' \<and> set rs' \<subseteq> set rs \<and> (\<forall>r\<in>set rs'. \<not> contains s r)"
ring_simprules_16: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "ring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
simps: fixes l :: "name"   and s :: "tm"   and k :: "tm"   and v :: "tm"   and u :: "tm"   and sl :: "name"   and sl' :: "name"   and m :: "name"   and n :: "name"   and sm :: "name"   and sm' :: "name"   and sn :: "name"   and sn' :: "name"   and x :: "tm"   and x' :: "tm" assumes "atom l \<sharp> (s, k, v, u, sl, sl', m, n, sm, sm', sn, sn')"   and "atom sl \<sharp> (s, v, u, sl', m, n, sm, sm', sn, sn')"   and "atom sl' \<sharp> (s, v, u, m, n, sm, sm', sn, sn')"   and "atom m \<sharp> (s, n, sm, sm', sn, sn')"   and "atom n \<sharp> (s, sm, sm', sn, sn')"   and "atom sm \<sharp> (s, sm', sn, sn')"   and "atom sm' \<sharp> (s, sn, sn')"   and "atom sn \<sharp> (s, sn')"   and "atom sn' \<sharp> s" shows "SeqSubstFormP v u x x' s k = LstSeqP s k (HPair x x') AND All2 l (SUCC k) (SyntaxN.Ex sl (SyntaxN.Ex sl' (HPair (Var l) (HPair (Var sl) (Var sl')) IN s AND (SubstAtomicP v u (Var sl) (Var sl') OR SyntaxN.Ex m (SyntaxN.Ex n (SyntaxN.Ex sm (SyntaxN.Ex sm' (SyntaxN.Ex sn (SyntaxN.Ex sn' (Var m IN Var l AND Var n IN Var l AND HPair (Var m) (HPair (Var sm) (Var sm')) IN s AND HPair (Var n) (HPair (Var sn) (Var sn')) IN s AND (Var sl EQ Q_Disj (Var sm) (Var sn) AND Var sl' EQ Q_Disj (Var sm') (Var sn') OR Var sl EQ Q_Neg (Var sm) AND Var sl' EQ Q_Neg (Var sm') OR Var sl EQ Q_Ex (Var sm) AND Var sl' EQ Q_Ex (Var sm'))))))))))))"
neqE: fixes x :: "'a"   and y :: "'a" assumes "x \<noteq> y"   and "x \<in> carrier"   and "y \<in> carrier" obtains u :: "'a set" and v :: "'a set" where "u \<in> nhds x" and "v \<in> nhds y" and "u \<inter> v = {}"
countableI: fixes f :: "'b \<Rightarrow> nat"   and S :: "'b set" assumes "inj_on f S" shows "countable S"
SUP_cong: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'd"   and D :: "'c \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd" assumes "A = B"   and "\<And>x. x \<in> B \<Longrightarrow> C x = D x" shows "Sup (C ` A) = Sup (D ` B)"
chine_composite_in_hom_1: shows "\<guillemotleft>\<mu>\<nu>_\<pi>.chine : Chn ((local.dom \<mu> \<star> local.dom \<nu>) \<star> local.dom \<pi>) \<rightarrow>\<^sub>C Chn ((cod \<mu> \<star> cod \<nu>) \<star> cod \<pi>)\<guillemotright>"
validI: fixes Q :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool"   and ds :: "'b set"   and P :: "'b set \<Rightarrow> 'a set \<Rightarrow> bool" assumes "Q ds {}"   and "\<And>fp. \<lbrakk>P ds fp; Q ds fp\<rbrakk> \<Longrightarrow> Q ds (cop_F ds fp)" shows "valid ds P Q"
prems_2: shows "abc_steps_l (0, lm) ap (Suc stp) = (as, am)"
set_subseqs_eq: fixes ys :: "'b list" shows "set (subseqs ys) = {xs. subseq xs ys}"
SN_on_Image_normalizable: fixes r :: "('b \<times> 'b) set"   and A :: "'b set" assumes "SN_on r A" shows "\<forall>a\<in>A. \<exists>b. b \<in> r\<^sup>! `` A"
bis_imp_2: fixes c1 :: "('test, 'atom) com"   and c2 :: "('test, 'atom) com" assumes "c1 \<approx>01T c2" shows "c1 \<approx>01 c2"
hyps: assumes "z + y \<cdot> x \<le> y" shows "z \<cdot> x ^ n_ \<le> y"
funion_upper2: fixes B :: "'a fset"   and A :: "'a fset" shows "B |\<subseteq>| A |\<union>| B"
of_nat_ge_zero: fixes n :: "nat" shows "(0::'b) \<le> of_nat n"
simps_2: fixes g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat"   and y :: "nat"   and x :: "nat" shows "PrimRecOp g h (Suc y) x = h y (PrimRecOp g h y x) x"
one_extra_update_subsumes: fixes t1 :: "transition"   and t2 :: "transition"   and r :: "nat"   and u :: "vname aexp"   and c :: "nat \<Rightarrow>f value option" assumes "Label t1 = Label t2"   and "Arity t1 = Arity t2"   and "set (Guards t1) \<subseteq> set (Guards t2)"   and "Outputs t1 = Outputs t2"   and "Updates t1 = (r, u) # Updates t2"   and "not_updated r t2"   and "c $ r = None" shows "subsumes t1 c t2"
tensor_mat_trace: fixes m1 :: "'a mat"   and m2 :: "'a mat" assumes "m1 \<in> carrier_mat d1 d1"   and "m2 \<in> carrier_mat d2 d2" shows "trace (tensor_mat m1 m2) = trace m1 * trace m2"
reachable_refl: fixes v :: "'a" assumes "v \<in> verts G" shows "v \<rightarrow>\<^sup>* v"
atm_of_Max_lit: fixes C :: "'b literal multiset" assumes "C \<noteq> {#}" shows "atm_of (Max_mset C) = Max (atms_of C)"
lookup_tensors_from_net: fixes m :: "real mat convnet"   and "is" :: "nat list"   and j :: "nat" assumes "valid_net' m"   and "is \<lhd> input_sizes m"   and "j < output_size' m" shows "Tensor.lookup (tensors_from_net m $ j) is = evaluate_net m (base_input m is) $ j"
array_ensure_def: fixes a :: "'b array"   and s :: "nat"   and x :: "'b" shows "array_ensure a s x \<equiv> Array.len a \<bind> (\<lambda>l. if s \<le> l then return a else Array.new s x \<bind> (\<lambda>a'. blit a 0 a' 0 l \<bind> (\<lambda>_. return a')))"
iff: fixes x :: "'a" assumes "x \<in> topspace X" shows "(\<forall>T. x \<in> T \<and> openin X T \<longrightarrow> (\<exists>y. y \<noteq> x \<and> y \<in> S \<and> y \<in> T)) = (\<forall>T\<subseteq>topspace Y. f x \<in> T \<longrightarrow> openin Y T \<longrightarrow> (\<exists>y. y \<noteq> f x \<and> y \<in> f ` S \<and> y \<in> T))"
harry_step_def: fixes xs :: "bool list" shows "harry_step xs = flip xs (heads xs - 1)"
rem_closure: fixes Cn :: "'b set \<Rightarrow> 'b set"   and K :: "'b set"   and A :: "'b set"   and B :: "'b set"   and \<phi> :: "'b" assumes "Tarskian_logic Cn"   and "K = Cn A"   and "B \<in> Tarskian_logic.remainder Cn K \<phi>" shows "B = Cn B"
arity_var\<^sub>h_lt_\<omega>_if_\<delta>\<^sub>h_gt_0: fixes x :: "'v" assumes "0 < \<delta>\<^sub>h" shows "arity_var\<^sub>h x < \<omega>"
assms_2: shows "\<forall>i\<in>S. p i \<le> i"
SPEC_rule_conjunct1: fixes A :: "'b nres"   and P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "A \<le> SPEC (\<lambda>v. P v \<and> Q v)" shows "A \<le> SPEC P"
eqpoll_imp_lepoll: fixes A :: "'d set"   and B :: "'e set" assumes "A \<approx> B" shows "A \<lesssim> B"
NOT_eq: fixes x :: "'b word" shows "NOT x = - x - 1"
remdups_filter: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "remdups (filter P xs) = filter P (remdups xs)"
some_case: fixes r :: "'a"   and \<sigma> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and \<tau> :: "'b \<Rightarrow> ('a, 'b, 'c) val option"   and e :: "('a, 'b, 'c) expr" assumes "s r = Some (\<sigma>, \<tau>, e)" shows "s' (\<alpha> r) = Some (\<R>\<^sub>L \<alpha> \<beta> (\<sigma>, \<tau>, e))"
tendsto_Complex: fixes f :: "'b \<Rightarrow> real"   and a :: "real"   and F :: "'b filter"   and g :: "'b \<Rightarrow> real"   and b :: "real" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. Complex (f x) (g x)) \<longlongrightarrow> Complex a b) F"
adds_antisym: 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.ulcs_powerprod plus minus zero lcs"   and "plus.adds plus s t"   and "plus.adds plus t s" shows "s = t"
NS_staticSecret_parts_agent_weak: fixes m :: "msg"   and C :: "agent"   and evs :: "event list"   and A :: "agent" assumes "m \<in> parts (knows C evs)"   and "m \<in> staticSecret A"   and "evs \<in> ns_public" shows "A = C \<or> (\<exists>D E X. Says D E X \<in> set evs \<and> m \<in> parts {X})"
Uniq_D: fixes P :: "'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "Uniq P"   and "P a"   and "P b" shows "a = b"
Meet_equality: fixes A :: "'b set"   and inf :: "'b" assumes "is_Inf A inf" shows "\<Sqinter>A = inf"
typeI_twoCube_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "typeI_twoCube twoC \<equiv> \<exists>a b g1 g2. a < b \<and> (\<forall>x\<in>{a..b}. g2 x \<le> g1 x) \<and> twoC = (\<lambda>(x, y). ((1 - x) * a + x * b, (1 - y) * g2 ((1 - x) * a + x * b) + y * g1 ((1 - x) * a + x * b))) \<and> g1 piecewise_C1_differentiable_on {a..b} \<and> g2 piecewise_C1_differentiable_on {a..b}"
IH_1: fixes t2 :: "('a, 'b) rbt" assumes "rbt_sorted t2" shows "rbt_lookup (RBT_Impl.fold rbt_insert l t2) = rbt_lookup t2 ++ map_of (rev (RBT_Impl.entries l))"
hpair_in_nonrestrict_iff: fixes a :: "hf"   and b :: "hf"   and s :: "hf"   and X :: "hf" shows "(\<langle>a, b\<rangle> \<^bold>\<in> nonrestrict s X) = (\<langle>a, b\<rangle> \<^bold>\<in> s \<and> a \<^bold>\<notin> X)"
image_map_upd: fixes x :: "'b"   and A :: "'b set"   and m :: "'b \<Rightarrow> 'c option"   and y :: "'c" assumes "x \<notin> A" shows "m(x \<mapsto> y) ` A = m ` A"
indep_var_restrict: fixes M' :: "'b \<Rightarrow> 'c measure"   and X :: "'b \<Rightarrow> 'a \<Rightarrow> 'c"   and I :: "'b set"   and A :: "'b set"   and B :: "'b set" assumes "indep_vars M' X I"   and "A \<inter> B = {}"   and "A \<subseteq> I"   and "B \<subseteq> I" shows "indep_var (Pi\<^sub>M A M') (\<lambda>\<omega>. \<lambda>i\<in>A. X i \<omega>) (Pi\<^sub>M B M') (\<lambda>\<omega>. \<lambda>i\<in>B. X i \<omega>)"
pred_rel: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b literal" shows "pred_literal P x = rel_literal (eq_onp P) x x"
simps_42: fixes \<phi> :: "formula"   and \<psi> :: "formula" shows "safe_formula (And \<phi> \<psi>) = (safe_formula \<phi> \<and> (safe_assignment (fv \<phi>) \<psi> \<or> safe_formula \<psi> \<or> fv \<psi> \<subseteq> fv \<phi> \<and> (is_constraint \<psi> \<or> (case \<psi> of Neg x \<Rightarrow> safe_formula x | _ \<Rightarrow> False))))"
ProcModifyReturn: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'d"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and ModifAbr :: "'e \<Rightarrow> 'e set" assumes "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return' c Q,A"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>\<sigma>. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/UNIV \<^esub>{\<sigma>} Call p (Modif \<sigma>),(ModifAbr \<sigma>)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P call init p return c Q,A"
second: assumes "rs\<^sub>1 = []"   and "rs\<^sub>2 = [x]" shows "thesis"
eventually_filtercomap: fixes P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and F :: "'d filter" shows "eventually P (filtercomap f F) = (\<exists>Q. eventually Q F \<and> (\<forall>x. Q (f x) \<longrightarrow> P x))"
div_div_div_same: 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 d :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times d b"   and "dvd.dvd times b a" shows "divide (divide a d) (divide b d) = divide a b"
Suc: shows "of_real_hom.mat_hom A ^\<^sub>m k * P = P * J ^\<^sub>m k"
coboundedI1: fixes a :: "'d"   and c :: "'d"   and b :: "'d" assumes "a \<le> c" shows "a \<sqinter> b \<le> c"
exec: shows "\<Gamma>\<turnstile> \<langle>c,s\<rangle> =n\<Rightarrow> t"
fps_XD_def: shows "fps_XD = (*) fps_X \<circ> fps_deriv"
r: shows "r = p - q - monom_mult (lookup (p - q) (t \<oplus> lt f) / lc f) t f"
A_to_prodag_mem: fixes A :: "('d, 'e) aGroup_scheme"   and I :: "'f set"   and B :: "'f \<Rightarrow> ('g, 'h) aGroup_scheme"   and S :: "'f \<Rightarrow> 'd \<Rightarrow> 'g"   and x :: "'d" assumes "aGroup A"   and "\<forall>k\<in>I. aGroup (B k)"   and "\<forall>k\<in>I. S k \<in> aHom A (B k)"   and "x \<in> carrier A" shows "A_to_prodag A I S B x \<in> carr_prodag I B"
prod_mset_multiplicity: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and M :: "'b multiset" assumes "class.comm_monoid_mult times one" shows "comm_monoid_mult.prod_mset times one M = comm_monoid_mult.prod times one (\<lambda>x. power.power one times x (count M x)) (set_mset M)"
set_pmf_of_set: fixes S :: "'a set" assumes "S \<noteq> {}"   and "finite S" shows "set_pmf (pmf_of_set S) = S"
reps_def: shows "reps \<equiv> \<lambda>(l, R). if R \<in> \<R> then (l, SOME u. u \<in> R) else (l, \<lambda>_. - 1)"
a: shows " Var x \<longrightarrow>\<^sub>1 M"
PUpdate_select: fixes U :: "'b pupdate" shows "PUpdate (Rep_pupdate U)"
simps: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and a1 :: "'c + 'e"   and a2 :: "'d + 'f" shows "rel_sum R1 R2 a1 a2 = ((\<exists>a c. a1 = Inl a \<and> a2 = Inl c \<and> R1 a c) \<or> (\<exists>b d. a1 = Inr b \<and> a2 = Inr d \<and> R2 b d))"
that: fixes d :: "real" assumes "0 < d"   and "sgn (poly p (x - d)) \<noteq> sgn (poly p (x + d))"   and "sgn (poly p (x - d)) \<noteq> 0"   and "0 \<noteq> sgn (poly p (x + d))"   and "\<forall>d'>0. d' \<le> d \<longrightarrow> sgn (poly p (x + d')) = sgn (poly p (x + d)) \<and> sgn (poly p (x - d')) = sgn (poly p (x - d))" shows "thesis"
complete_Int_closed: fixes S :: "'b set"   and t :: "'b set" assumes "complete S"   and "closed t" shows "complete (S \<inter> t)"
Overriding: fixes new :: "qtname \<times> methd" assumes "methd G C sig = Some new"   and "G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old" shows "thesis"
rev_upt_upt: fixes i :: "nat"   and j :: "nat" shows "rev [i..<j] = [j>..i]"
landau_flip_2: fixes f :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and g :: "'a \<Rightarrow> 'b" shows "(f \<in> o[F](g)) = (g \<in> \<omega>[F](f))"
disc_7: fixes x3 :: "nat" shows "is_HOLD (HOLD x3)"
assms_1: fixes i :: "'a" assumes "i \<in> I" shows "Group.group (G i)"
last_seg_impl_refine: fixes s :: "'a list \<times> nat list \<times> ('a \<Rightarrow> node_state option) \<times> (nat \<times> 'a set) list"   and p :: "'a set list"   and D :: "'a set"   and pE :: "('a \<times> 'a) set" assumes "(s, p, D, pE) \<in> GS_rel"   and "p \<noteq> []" shows "last_seg_impl s \<le> \<Down> Id (RETURN (last p))"
Rep_bit: fixes x :: "bit" shows "Rep_bit x \<in> UNIV"
ccSup_le_iff: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and A :: "'b set"   and b :: "'b" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top"   and "countable A" shows "less_eq (Sup A) b = (\<forall>a\<in>A. less_eq a b)"
Q3_a_def: shows "Q3_a \<equiv> [: X, mrk \<leadsto> X', mrk' . \<exists>x\<in>X. \<exists>y. (x, y) \<in> next \<and> y \<notin> mrk \<and> X' = X \<union> {y} \<and> mrk' = mrk \<union> {y} :]"
absorb_iff2: fixes a :: "'c"   and b :: "'c" shows "(a \<le> b) = (max a b = b)"
borel_measurable_indicator: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> sets M" shows "indicator A \<in> borel_measurable M"
const_on_def: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and x :: "'d" shows "const_on f S x = (\<forall>y\<in>S. f y = x)"
zero_poly_mapping_def: shows "0 \<equiv> Abs_poly_mapping (\<lambda>k. 0::'c)"
assoc'_naturality: fixes \<mu> :: "'d"   and \<nu> :: "'d"   and \<tau> :: "'d" assumes "D.arr \<mu>"   and "D.arr \<nu>"   and "D.arr \<tau>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>"   and "src\<^sub>D \<nu> = trg\<^sub>D \<tau>" shows "\<a>\<^sub>D\<^sup>-\<^sup>1[D.cod \<mu>, D.cod \<nu>, D.cod \<tau>] \<cdot>\<^sub>D (\<mu> \<star>\<^sub>D \<nu> \<star>\<^sub>D \<tau>) = ((\<mu> \<star>\<^sub>D \<nu>) \<star>\<^sub>D \<tau>) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[D.dom \<mu>, D.dom \<nu>, D.dom \<tau>]"
bibd_block_number: shows "\<b> = \<Lambda> * \<v> * (\<v> - 1) div (\<k> * (\<k> - 1))"
associatedE2: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c"   and thesis :: "bool" assumes "monoid_cancel G"   and "a \<sim>\<^bsub>G\<^esub> b"   and "\<And>u. \<lbrakk>a = b \<otimes>\<^bsub>G\<^esub> u; u \<in> Units G\<rbrakk> \<Longrightarrow> thesis"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "thesis"
ord_le_eq_trans: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "less_eq a b"   and "b = c" shows "less_eq a c"
q_def: fixes x :: "nat"   and y :: "nat"   and u :: "nat" shows "q x y u = p y x u"
compat2: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<succ> y"   and "z \<le> y" shows "x \<succ> z"
Noetherian_iff_PreLoebian: fixes x :: "'a" shows "Noetherian x = PreLoebian x"
mergesort_by_rel_merge_simps_3: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and ys :: "'a list" shows "mergesort_by_rel_merge R [] ys = ys"
assoc'_simps_2: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "ide a"   and "ide b"   and "ide c" shows "local.dom (assoc' a b c) = local.prod a (local.prod b c)"
rep_eq: fixes x :: "'b \<Rightarrow> 'a"   and xa :: "'b fset" shows "fset (x |`| xa) = x ` fset xa"
Cons_2: shows "set (b # p) \<subseteq> carrier R"
insertAssertionWeaken: fixes \<Psi> :: "'b"   and F :: "'b frame" shows "\<langle>\<epsilon>, \<Psi>\<rangle> \<hookrightarrow>\<^sub>F insertAssertion F \<Psi>"
pr_int_mapk_analz: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and n :: "nat"   and run :: "nat"   and b :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "IntMapK (S (Card n, n, run)) = Some b" shows "Pri_AgrK b \<notin> analz (A \<union> spies evs)"
start_point_iff1: fixes x :: "'a" assumes "path x" shows "is_point (start_points x) = (\<not> no_start_points x)"
Rep: fixes x :: "'a Word.word" shows "sint x \<in> sints LENGTH('a)"
starfun_norm: fixes f :: "'a \<Rightarrow> 'b" shows "*f* (\<lambda>x. norm (f x)) = (\<lambda>x. hnorm ((*f* f) x))"