hcju/isabelleps huggingface dataset:
Int_mono: fixes A :: "'g set"   and C :: "'g set"   and B :: "'g set"   and D :: "'g set" assumes "A \<subseteq> C"   and "B \<subseteq> D" shows "A \<inter> B \<subseteq> C \<inter> D"
pivot_tableau_valuated: fixes s :: "('i, 'a) state"   and x\<^sub>i :: "nat"   and x\<^sub>j :: "nat" assumes "\<triangle> (\<T> s)"   and "x\<^sub>i \<in> lvars (\<T> s)"   and "x\<^sub>j \<in> rvars_eq (eq_for_lvar (\<T> s) x\<^sub>i)"   and "\<nabla> s" shows "\<nabla> (pivot x\<^sub>i x\<^sub>j s)"
pfinite_sim: shows "pfinite_converter \<I>_middle \<I>_inner sim_inner"
vlrestriction_vsubset_vrange: fixes A :: "V" assumes "\<D>\<^sub>\<circ> r \<subseteq>\<^sub>\<circ> A" shows "r \<restriction>\<^sup>l\<^sub>\<circ> A = r"
solves_odeD_2: fixes y :: "real \<Rightarrow> 'a"   and f :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and T :: "real set"   and X :: "'a set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
apfst_conv: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and y :: "'b" shows "apfst f (x, y) = (f x, y)"
beta_reduction_fvs: fixes M :: "'a trm"   and M' :: "'a trm" assumes "M \<rightarrow>\<beta> M'" shows "fvs M' \<subseteq> fvs M"
r_cancel_one: fixes x :: "'c"   and a :: "'c" assumes "x \<in> carrier S"   and "a \<in> carrier S" shows "(a \<oplus>\<^bsub>S\<^esub> x = x) = (a = \<zero>\<^bsub>S\<^esub>)"
operative_integrableI: fixes e :: "'a"   and f :: "'b \<Rightarrow> 'c" assumes "(0::'a) \<le> e" shows "operative (\<and>) True ((integrable_on) f)"
base: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b" shows "rtrancl_path r x [] x"
Block_lem: fixes l :: "'a \<Rightarrow> 'b option"   and Vs :: "'a list"   and ls :: "'b list"   and l' :: "'a \<Rightarrow> 'b option"   and ls' :: "'b list"   and V :: "'a"   and v :: "'b" assumes "l \<subseteq>\<^sub>m [Vs [\<mapsto>] ls]"   and "l' \<subseteq>\<^sub>m [Vs [\<mapsto>] ls', V \<mapsto> v]"   and "V \<in> set Vs \<Longrightarrow> ls ! last_index Vs V = ls' ! last_index Vs V"   and "length ls = length ls'"   and "length Vs < length ls'" shows "l'(V := l V) \<subseteq>\<^sub>m [Vs [\<mapsto>] ls']"
step: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and ys :: "'a list"   and z :: "'a" assumes "r x y"   and "rtrancl_path r y ys z" shows "rtrancl_path r x (y # ys) z"
wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t_def: fixes S :: "('d, 'c) stateful_strand_step list" shows "wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t S \<equiv> \<Union> (set (map wfrestrictedvars\<^sub>s\<^sub>s\<^sub>t\<^sub>p S))"
t: shows "t\<^sub>p [x, y] x (fst (snd (rTS h0) ([x, y], h) x)) = 0"
AtomicP_graph_aux_def: shows "AtomicP_graph_aux \<equiv> lfp (\<lambda>p x1 x2. \<exists>AtomicP_aux t u y. x1 = y \<and> x2 = SyntaxN.Ex t (SyntaxN.Ex u (TermP (Var t) AND TermP (Var u) AND (y EQ Q_Eq (Var t) (Var u) OR y EQ Q_Mem (Var t) (Var u)))) \<and> atom t \<sharp> (u, y) \<and> atom u \<sharp> y)"
chine_is_identity: shows "C.ide \<mu>\<nu>\<pi>.chine"
seqE: fixes g :: "'a"   and f :: "'a" assumes "L'.C.seq g f" obtains "L'.C.arr f" and "L'.C.arr g" and "L'.C.dom g = L'.C.cod f"
deg0: fixes g :: "'a fls" assumes "fls_subdegree g = 0" shows "fls_regpart (f / g) = fls_regpart f / fls_regpart g"
a0n_agreementI: fixes x :: "a0n_state" assumes "\<And>A B M. 0 < signals x (Commit A B M) \<Longrightarrow> 0 < signals x (Running A B M)" shows "x \<in> a0n_agreement"
mono_gamma_o: fixes sa :: "'av Abs_State.st option"   and sa' :: "'av Abs_State.st option" assumes "sa \<sqsubseteq> sa'" shows "local.\<gamma>\<^sub>o sa \<subseteq> local.\<gamma>\<^sub>o sa'"
assms_2: shows "A \<subseteq>\<^sub>\<circ> C"
functor_axioms: shows "functor (\<cdot>\<^sub>A\<^sub>1\<^sub>x\<^sub>A\<^sub>2) (\<cdot>\<^sub>B) F"
subst_bvs1'_Abs: fixes args :: "Core.term list"   and T :: "typ"   and t :: "Core.term"   and lev :: "nat" assumes "\<forall>x\<in>set args. is_closed x" shows "subst_bvs1' (Abs T t) lev args = Abs T (subst_bvs1' t (Suc lev) args)"
unbounded: shows "\<not> bdd_above {n. c < root n (norm (f n))}"
simps_43: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt"   and x51 :: "'c"   and x52 :: "('a, 'b, 'c) Stmt"   and x53 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stmt.If x51 x52 x53"
simps_2: fixes x :: "'a"   and xs :: "'a list" shows "distinct (x # xs) = (x \<notin> set xs \<and> distinct xs)"
mat_distr: fixes f :: "'b \<Rightarrow> 'e \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'e \<Rightarrow> 'd"   and h :: "'e \<Rightarrow> 'c \<Rightarrow> 'd" shows "f \<oplus> g \<otimes> h = (f \<otimes> h) \<oplus> (g \<otimes> h)"
csv_length_same_as_qlist: fixes p :: "real poly"   and x :: "real"   and sign :: "rat list"   and qs :: "real poly list" assumes "p \<noteq> 0"   and "x \<in> {x. poly p x = 0}"   and "sign = consistent_sign_vec qs x" shows "length sign = length qs"
simps_1: fixes I :: "'c \<Rightarrow> 'd option"   and P :: "'d \<Rightarrow> 'c \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool" shows "upd_invs I P [] = I"
FactGroup_iso: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('e, 'f) monoid_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "group_hom G H h"   and "h ` carrier G = carrier H" shows "G Mod kernel G H h \<cong> H"
vector_derivative_linepath_at: fixes a :: "'a"   and b :: "'a"   and x :: "real" shows "vector_derivative (linepath a b) (at x) = b - a"
reach_snoc: fixes tr :: "'a \<Rightarrow> 'b \<Rightarrow> 'a"   and p :: "'a"   and bs :: "'b list"   and q :: "'a"   and b :: "'b" assumes "reach tr p bs q" shows "reach tr p (bs @ [b]) (tr q b)"
gen_cong_reflp: fixes x :: "'a"   and y :: "'a"   and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "x = y" shows "gen_cong R x y"
c_in: shows "C \<in> set CAs \<union> {DA}"
plus_Collect_helper: fixes x :: "'b word"   and P :: "'b word \<Rightarrow> bool" shows "(+) x ` {xa. P xa} = {xa. P (xa - x)}"
Even_lemma: shows "{{Z=Arg+1}. BODY Odd .{Res_ok}}|-{Z=Arg+0}. evn .{Res_ok}"
Sup_upper: fixes x :: "'d \<times> 'e"   and A :: "('d \<times> 'e) set" assumes "x \<in> A" shows "less_eq_prod x (Sup_prod A)"
frombd_F23_tobd_F23: fixes b :: "'d"   and x :: "('e, 'f, 'd) F2"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" assumes "b \<in> F2set3 x" shows "fromCard (F2set3 x) (dir_image (r1 +c bd_F2) Abs_bd_type_F) (toCard (F2set3 x) (dir_image (r1 +c bd_F2) Abs_bd_type_F) b) = b"
pair_imageI: fixes a :: "'a"   and b :: "'b"   and A :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'd" assumes "(a, b) \<in> A" shows "f a b \<in> (\<lambda>(a, b). f a b) ` A"
LAssThrow: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e' :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and V :: "char list" assumes "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>V:=e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
pell_valuation_def: shows "pell_valuation = (\<lambda>x. case case x of (x, y) \<Rightarrow> (real_of_int x, real_of_int y) of (a, b) \<Rightarrow> a + b * sqrt (real D))"
case2: assumes "\<not> x \<le> z" shows "l_sort_aux (x # xs, ys @ [z], zs) \<in> l_sort_set (x # xs, ys @ [z], zs)"
uint32_div_def: fixes x :: "uint32"   and y :: "uint32" shows "uint32_div x y = fst (uint32_divmod x y)"
trace_eq_simcl_bind2: fixes q :: "'a spmf"   and S :: "'b spmf \<Rightarrow> 'c spmf \<Rightarrow> bool"   and p :: "'b spmf"   and f :: "'a \<Rightarrow> 'c spmf" assumes "\<forall>x\<in>set_spmf q. trace_eq_simcl S p (f x)"   and "lossless_spmf q" shows "trace_eq_simcl S p (q \<bind> f)"
join: fixes a :: "'b\<^sub>m"   and b :: "'b\<^sub>m" assumes "rhb a = rhb b"   and "a \<in> B" shows "\<exists>ab. mb a b = Some ab \<and> bob a ab \<and> bob b ab \<and> (\<forall>u. bob a u \<longrightarrow> bob b u \<longrightarrow> bob ab u)"
tobd_F23_inj: fixes x :: "'d"   and xa :: "('e, 'f, 'd) F2"   and y :: "'d"   and r1 :: "(bd_type_F1 \<times> bd_type_F1) set" assumes "x \<in> F2set3 xa"   and "y \<in> F2set3 xa" shows "(toCard (F2set3 xa) (dir_image (r1 +c bd_F2) Abs_bd_type_F) x = toCard (F2set3 xa) (dir_image (r1 +c bd_F2) Abs_bd_type_F) y) = (x = y)"
prime_elem_imp_irreducible: fixes p :: "'a" assumes "prime_elem p" shows "irreducible p"
set_id_hom: fixes A :: "'b set"   and U :: "'b set" assumes "A \<in> Ob\<^bsub>set_cat U\<^esub>" shows "Id\<^bsub>set_cat U\<^esub> A \<in> Hom\<^bsub>set_cat U\<^esub> A A"
int_of_integer_shift: fixes n :: "integer"   and k :: "nat" shows "int_of_integer (n >> k) = int_of_integer n div 2 ^ k"
Max_wrt_subset: fixes carrier :: "'c set"   and le :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "Order_Predicates.preorder_on carrier le" shows "Max_wrt le \<subseteq> carrier"
gim: shows "g ` (sphere (0::'a) 1 \<inter> S) \<subseteq> T"
simps_4: fixes T :: "typ"   and t :: "Core.term"   and U :: "typ"   and u :: "Core.term"   and subs :: "((variable \<times> String.literal set) \<times> typ) list" shows "assoc_match_term (Abs T t) (Abs U u) subs = Option.bind (assoc_match T U subs) (assoc_match_term t u)"
abs_RED: fixes \<tau> :: "ty"   and x :: "name"   and t :: "lam"   and \<sigma> :: "ty" assumes "\<forall>s\<in>RED \<tau>. t[x::=s] \<in> RED \<sigma>" shows "Lam [x].t \<in> RED (\<tau> \<rightarrow> \<sigma>)"
assms_7: fixes s :: "('i, 'a) state" shows "P s = P' (<) \<B>\<^sub>i\<^sub>u \<B>\<^sub>i\<^sub>l \<B>\<^sub>u \<B>\<^sub>l \<B>\<^sub>i\<^sub>u_update \<I>\<^sub>u \<I>\<^sub>l Leq Geq s"
f: shows "prime (9 * 2 ^ (2 * 4 - 1) - 1)"
\<V>_def: fixes state :: "('b, 'a) state" shows "\<V> state = (case state of State x1 x2 x3 x4 x5 x6 \<Rightarrow> x4)"
isGlbI1: fixes R :: "'a set"   and S :: "'a set"   and x :: "'a" assumes "greatestP (isLb R S) x" shows "isGlb R S x"
that: fixes z :: "'a"   and zs :: "'a list" assumes "ys = xs @ z # zs" shows "thesis"
feq: fixes x :: "'a" assumes "x \<in> topspace X" shows "f x = x"
unsigned_numeral: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and n :: "num" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (numeral n) = semiring_1.of_nat one plus zero (take_bit LENGTH('c) (numeral n))"
ereal_divide_less_pos: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "0 < x"   and "x \<noteq> \<infinity>" shows "(y / x < z) = (y < x * z)"
I: fixes t :: "real" assumes "t \<in> {a..b}" shows "D - K \<le> infdist (f t) H"
brnL_def: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n :: "nat" shows "brnL cl n \<equiv> lsum brn (take n cl)"
IH1: assumes "p1 \<in> subterms (\<Gamma> pn)" shows "sterms \<Gamma> p1 \<subseteq> cterms \<Gamma>"
Yb: shows "[[a\<^sub>1 a\<^sub>n b]]"
n1: shows "(q, czero) \<notin> ordIso \<and> Card_order q"
rComm2: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and P :: "('a, 'b, 'c) psi"   and M :: "'a"   and xvec :: "name list"   and N :: "'a"   and P' :: "('a, 'b, 'c) psi"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and Q :: "('a, 'b, 'c) psi"   and K :: "'a"   and Q' :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list"   and C :: "'d" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>Q) P (M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle>) P' A\<^sub>P \<Psi>\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<And>C. Prop C (\<Psi> \<otimes> \<Psi>\<^sub>P) Q (K\<lparr>N\<rparr>) Q' A\<^sub>Q \<Psi>\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "distinct xvec"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C" shows "Prop C \<Psi> (P \<parallel> Q) (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q') (A\<^sub>P @ A\<^sub>Q) (\<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q)"
sorted_wrt_insort: fixes A :: "'b set"   and R :: "('b \<times> 'b) set"   and xs :: "'b list" assumes "linorder_on A R"   and "set xs \<subseteq> A" shows "Linorder_Relations.sorted_wrt R (insort_wrt R xs)"
lead_coeff_poly_add: fixes p :: "'a poly"   and q :: "'a poly" assumes "monic p"   and "monic q"   and "0 < degree p"   and "0 < degree q" shows "lead_coeff (poly_add p q) = poly_add_sign (degree p) (degree q)"
simps: fixes T :: "Arraytype"   and a :: "nat" shows "arr_len (arrV T a) = arrLenLoc T a"
that_3: fixes n :: "nat"   and Y' :: "nat \<Rightarrow> Value \<rightarrow> Value" assumes "Y = (\<lambda>m. if m < n then \<bottom> else Fn\<cdot>(Y' (m - n)))"   and "chain Y'" shows "thesis"
rf2': shows "f \<in> BaseFGMod.indspace G FH V"
simps: fixes a1 :: "boolean_binop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'a OCL_Types.type"   and a4 :: "'a OCL_Types.type" shows "boolean_binop_type a1 a2 a3 a4 = ((\<exists>\<tau> \<sigma> \<rho>. a1 = AndOp \<and> a2 = \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?]) \<or> (\<exists>\<tau> \<sigma> \<rho>. a1 = OrOp \<and> a2 = \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?]) \<or> (\<exists>\<tau> \<sigma> \<rho>. a1 = XorOp \<and> a2 = \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?]) \<or> (\<exists>\<tau> \<sigma> \<rho>. a1 = ImpliesOp \<and> a2 = \<tau> \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> \<tau> \<squnion> \<sigma> = \<rho> \<and> \<rho> \<le> Boolean[?]))"
2: fixes z :: "'a" assumes "z \<in> I +> x \<otimes> y" shows "\<exists>a\<in>I +> x. \<exists>b\<in>I +> y. z \<in> I +> a \<otimes> b"
smult_1: fixes A :: "'b tensor" shows "A = (1::'b) \<cdot> A"
at_within_nhd: fixes "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and S :: "'a set"   and T :: "'a set"   and U :: "'a set" assumes "class.topological_space open"   and "x \<in> S"   and "open S"   and "T \<inter> S - {x} = U \<inter> S - {x}" shows "topological_space.at_within open x T = topological_space.at_within open x U"
simps_1: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "vc SKIP Q = True"
dvd_0_right: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and a :: "'a" assumes "class.comm_semiring_1 times one plus zero" shows "dvd.dvd times a zero"
nth_length_takeWhile: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list" assumes "length (takeWhile P xs) < length xs" shows "\<not> P (xs ! length (takeWhile P xs))"
is_const_simps_1: fixes size :: "'c \<Rightarrow> nat"   and frees :: "'c \<Rightarrow> name fset"   and subst :: "'c \<Rightarrow> (name, 'c) fmap \<Rightarrow> 'c"   and "consts" :: "'c \<Rightarrow> name fset"   and app :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unapp :: "'c \<Rightarrow> ('c \<times> 'c) option"   and const :: "name \<Rightarrow> 'c"   and unconst :: "'c \<Rightarrow> name option"   and free :: "name \<Rightarrow> 'c"   and unfree :: "'c \<Rightarrow> name option"   and name :: "name" assumes "class.pre_term size frees subst consts app unapp const unconst free unfree" shows "pre_term.is_const unconst (const name)"
tm3': fixes x :: "'a"   and z :: "'a"   and y :: "'a" shows "!x \<cdot> !(t z \<cdot> t y) = !(!(!x \<cdot> !y) \<cdot> !(!x \<cdot> !z))"
smult_1: fixes d :: "'a" shows "smult d 1 = [:d:]"
mat_comp_allI: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and B :: "'b mat"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "A \<in> carrier_mat nr nc"   and "B \<in> carrier_mat nr nc"   and "\<And>i j. \<lbrakk>i < nr; j < nc\<rbrakk> \<Longrightarrow> r (A $$ (i, j)) (B $$ (i, j))" shows "mat_comp_all r A B"
uniform_groups_alt: shows "\<forall>G\<in>\<G>. card G = \<m>"
wellformed_transaction_send_receive_trm_cases_2: fixes T :: "('e, 'f, 'g, 'h) prot_transaction"   and t :: "(('e, 'f, 'g) prot_fun, (('e, 'f, 'g) prot_fun, 'f prot_atom) Term.term \<times> nat) Term.term" assumes "wellformed_transaction T"   and "t \<in> trms\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T)" shows "send\<langle>t\<rangle> \<in> set (unlabel (transaction_send T))"
op_set_enumerate_def: fixes S :: "'e set" shows "op_set_enumerate S \<equiv> SPEC (\<lambda>f. dom f = S \<and> inj_on f S)"
prems: shows "\<not> one_call_in_path x (x' # xs)"
rel_option_reflI: fixes y :: "'c option"   and P :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>x. x \<in> set_option y \<Longrightarrow> P x x" shows "rel_option P y y"
proper_part_transitivity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "M4"   and "PP x y"   and "PP y z" shows "PP x z"
is_const_list_vs_to_es_list: fixes vs :: "v list" shows "const_list ($$* vs)"
eval_lift_ms: fixes C :: "'a" shows "eval (lift_ms C) = eval C"
sim: shows "weak_reduction_simulation (indRelLTPO TRel) (STCal Source Target)"
subtype_implies_ok: fixes \<Gamma> :: "binding list"   and S :: "ty"   and T :: "ty" assumes "\<Gamma>\<turnstile>S<:T" shows "\<turnstile> \<Gamma> ok"
dom_simp: fixes f :: "(bool, unit) concrete_category.arr" assumes "partial_magma.arr (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f" shows "partial_magma.dom (concrete_category.COMP UNIV (\<lambda>a b. if a = b then {()} else {}) (\<lambda>_ _ _ _ _. ())) f = f"
closed_Inter: fixes "open" :: "'a set \<Rightarrow> bool"   and K :: "'a set set" assumes "class.topological_space open"   and "\<forall>S\<in>K. topological_space.closed open S" shows "topological_space.closed open (\<Inter> K)"
segment_convex_hull: fixes a :: "'b"   and b :: "'b" shows "closed_segment a b = convex hull {a, b}"
eSuc_eq_infinity_iff: fixes n :: "enat" shows "(eSuc n = \<infinity>) = (n = \<infinity>)"
SUP_sup_const1: 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"   and I :: "'b set"   and c :: "'a"   and f :: "'b \<Rightarrow> 'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}" shows "Sup ((\<lambda>i. sup c (f i)) ` I) = sup c (Sup (f ` I))"
1: shows "e \<le> - - v * - v\<^sup>T"
d: shows "map_of (effect_to_assignments op) = map_of as\<^sub>2 ++ map_of as\<^sub>1"
simps: fixes f :: "'c \<rightarrow> 'd \<rightarrow> 'b"   and x :: "'d"   and y :: "'c" shows "flip\<cdot>f\<cdot>x\<cdot>y = f\<cdot>y\<cdot>x"
t': shows "t \<rightarrow>\<^sub>\<beta>\<^sup>* t'"
step: shows "\<Gamma>\<turnstile> (cs, css, s) \<rightarrow> (cs', css', t)"
ord_le_eq_trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "subseq a b"   and "b = c" shows "subseq a c"
scale: fixes r :: "'a"   and b :: "'b" shows "f (r *s b) = scaleC r (f b)"
sconst_eq: fixes x :: "'d"   and y :: "'d" shows "(sconst x = sconst y) = (x = y)"
prv_lcnj_imp: fixes \<chi> :: "'fmla"   and \<phi>s :: "'fmla list"   and \<phi> :: "'fmla" assumes "\<chi> \<in> fmla"   and "set \<phi>s \<subseteq> fmla"   and "\<phi> \<in> set \<phi>s"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (lcnj \<phi>s) \<chi>)"
generate_valid_stateful_policy_IFSACS_wf_stateful_policy: fixes G :: "'b graph"   and edgesList :: "('b \<times> 'b) list"   and M :: "'b SecurityInvariant_configured list" assumes "wf_graph G"   and "set edgesList = edges G" shows "wf_stateful_policy (generate_valid_stateful_policy_IFSACS G M edgesList)"
simps_24: fixes \<C> :: "t_context"   and i :: "nat"   and ts :: "checker_type" shows "check_single \<C> (Get_local i) ts = (if i < length (local \<C>) then type_update ts [] (Type [local \<C> ! i]) else Bot)"
comm_ring_1_ow: shows "class.comm_ring_1 = comm_ring_1_ow UNIV"
monom_mult_dist_right: fixes c :: "'c"   and t :: "'b"   and p :: "'b \<Rightarrow>\<^sub>0 'c"   and q :: "'b \<Rightarrow>\<^sub>0 'c" shows "MPoly_Type_Class.punit.monom_mult c t (p + q) = MPoly_Type_Class.punit.monom_mult c t p + MPoly_Type_Class.punit.monom_mult c t q"
assms_2: fixes u :: "'t" assumes "u \<in> keys p" shows "v \<preceq>\<^sub>t u"
step_n: shows "lnth Ns n \<leadsto>GC lnth Ns (Suc n)"
cs_select_def: fixes \<pi> :: "nat \<Rightarrow> 'n"   and xs :: "'n list" shows "(\<pi> \<exclamdown> xs) = (THE k. cs\<^bsup>\<pi>\<^esup> k = xs)"
steps_Fault_prop: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and c' :: "('a, 'b, 'c) com"   and s' :: "('a, 'c) xstate"   and f :: "'c" assumes "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* (c', s')"   and "s = Fault f" shows "s' = Fault f"
cinner_mult_right': fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and a :: "'b"   and b :: "'b"   and m :: "complex" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner a (b * of_complex m) = cinner a b * m"
compseriesTr0: fixes G :: "('e, 'f) Group_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e set"   and i :: "nat" assumes "Group G"   and "compseries G n f"   and "i \<le> n" shows "G \<guillemotright> f i "
ennreal_add_diff_cancel_right: fixes y :: "ennreal"   and x :: "ennreal" assumes "y \<noteq> top" shows "x + y - y = x"
change_respecting_subst: fixes dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and eval :: "((('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) \<times> ('c \<times> (Mode \<Rightarrow> 'a set)) \<times> ('a \<Rightarrow> 'b)) set"   and cms :: "'c \<times> (Mode \<Rightarrow> 'a set)"   and mem :: "'a \<Rightarrow> 'b"   and cms' :: "'c \<times> (Mode \<Rightarrow> 'a set)"   and mem' :: "'a \<Rightarrow> 'b"   and X :: "'a set" assumes "sifum_security_init dma \<C>_vars eval"   and "sifum_security_init.change_respecting \<C>_vars eval (cms, mem) (cms', mem') X" shows "\<forall>\<sigma>. dom \<sigma> = X \<longrightarrow> ((cms, subst \<sigma> mem), cms', subst \<sigma> mem') \<in> eval"
assms_2: shows "fst ` X0 \<inter> {x. c \<le> x \<bullet> n} = {}"
assms: shows "R \<cdot> 1\<^sub>\<pi> \<parallel> Multirelations.U = R"