hcju/isabelleps huggingface dataset:
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 abs_pred :: "('c \<Rightarrow> bool) \<Rightarrow> 'c \<Rightarrow> bool"   and t\<^sub>1 :: "term"   and t\<^sub>2 :: "term"   and u :: "'c" assumes "class.term size frees subst consts app unapp const unconst free unfree abs_pred" shows "term.match unapp unconst (t\<^sub>1 $ t\<^sub>2) u = unapp u \<bind> (\<lambda>(u\<^sub>1, u\<^sub>2). term.match unapp unconst t\<^sub>1 u\<^sub>1 \<bind> (\<lambda>env\<^sub>1. term.match unapp unconst t\<^sub>2 u\<^sub>2 \<bind> (\<lambda>env\<^sub>2. Some (env\<^sub>1 ++\<^sub>f env\<^sub>2))))"
divide_nonpos_nonpos: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs"   and "less_eq x zero"   and "less_eq y zero" shows "less_eq zero (divide x y)"
pderiv_closed: fixes p :: "nat \<Rightarrow> 'a" assumes "p \<in> carrier P" shows "pderiv p \<in> carrier P"
ccBindsExtra_simp: fixes ccExp :: "exp \<Rightarrow> Arity \<rightarrow> CoCalls"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and i :: "(Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls" shows "CoCallAnalysis.ccBindsExtra ccExp \<Gamma>\<cdot>i = snd i \<squnion> CoCallAnalysis.ccBinds ccExp \<Gamma>\<cdot>i \<squnion> (\<Squnion>x\<mapsto>e\<in>map_of \<Gamma>. fv e G\<times> ccNeighbors x (snd i))"
integral_dominated_convergence: fixes f :: "'c \<Rightarrow> 'd"   and M :: "'c measure"   and s :: "nat \<Rightarrow> 'c \<Rightarrow> 'd"   and w :: "'c \<Rightarrow> real" assumes "f \<in> borel_measurable M"   and "\<And>i. s i \<in> borel_measurable M"   and "integrable M w"   and "AE x in M. (\<lambda>i. s i x) \<longlonglongrightarrow> f x"   and "\<And>i. AE x in M. norm (s i x) \<le> w x" shows "(\<lambda>i. integral\<^sup>L M (s i)) \<longlonglongrightarrow> integral\<^sup>L M f"
preserves_reflects_arr: fixes f :: "'d \<times> 'c" shows "S.arr (\<Phi> f) = DopxC.arr f"
pathfinish_def: fixes g :: "real \<Rightarrow> 'c" shows "pathfinish g = g 1"
ag_r_zero: fixes A :: "('h, 'i) aGroup_scheme"   and x :: "'h" assumes "aGroup A"   and "x \<in> carrier A" shows "x \<plusminus>\<^bsub>A\<^esub> \<zero>\<^bsub>A\<^esub> = x"
fun_upd_restrict: fixes m :: "'c \<Rightarrow> 'd option"   and D :: "'c set"   and x :: "'c"   and y :: "'d option" shows "(m |` D)(x := y) = (m |` (D - {x}))(x := y)"
eSuc_minus_eSuc: fixes n :: "eint"   and m :: "eint" shows "n + 1 - (m + 1) = n - m"
Gramian_determinant_2: fixes k :: "nat" assumes "k \<le> length fs" shows "(0::'a) < Gramian_determinant fs k"
Composition: fixes SCompose :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and SImp :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and SBottom :: "'d"   and SChanEq :: "'f \<Rightarrow> 'f \<Rightarrow> 'e"   and \<Psi> :: "'d"   and \<Psi>' :: "'d"   and \<Psi>'' :: "'d" assumes "assertion SCompose SImp SBottom SChanEq"   and "assertionAux.AssertionStatEq SImp \<Psi> \<Psi>'" shows "assertionAux.AssertionStatEq SImp (SCompose \<Psi> \<Psi>'') (SCompose \<Psi>' \<Psi>'')"
open_Compl: fixes S :: "'a set" assumes "closed S" shows "open (- S)"
STL3: fixes F :: "'a \<Rightarrow> bool" shows "\<turnstile> (\<box>\<box>F) = (\<box>F)"
prems: fixes x :: "'a" assumes "x \<in> insert x__ A" shows "f x \<noteq> 0"
rel_compp: fixes R :: "'c \<Rightarrow> 'e \<Rightarrow> bool"   and S :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_pmf (R OO S) = rel_pmf R OO rel_pmf S"
lincomb_same: fixes S :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "S \<subseteq> mat_kernel A" shows "lincomb a S = NC.lincomb a S"
upd_obj_def: fixes a :: "loc"   and f :: "fname"   and v :: "val"   and s :: "state" shows "upd_obj a f v s \<equiv> let (C, fs) = the (heap s a) in hupd(a\<mapsto>(C, fs(f \<mapsto> v))) s"
sup_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z \<squnion> x \<le> z \<squnion> y"
mkProg_inverse: fixes y :: "db" assumes "y \<in> {P. closed P}" shows "unProg (mkProg y) = y"
order_monic_linear: fixes x :: "'a"   and y :: "'a" shows "order x [:y, 1::'a:] = (if y + x = (0::'a) then 1 else 0)"
ipcidr_to_interval_def: fixes cidr :: "'b word \<times> nat" shows "ipcidr_to_interval cidr \<equiv> (ipcidr_to_interval_start cidr, ipcidr_to_interval_end cidr)"
conductor_le_modulus: shows "\<Phi>.principal.conductor \<le> c"
times_poly_def: fixes p :: "'a poly"   and q :: "'a poly" shows "p * q = fold_coeffs (\<lambda>a p. smult a q + pCons (0::'a) p) p 0"
axioms_2: fixes S :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e" assumes "sheaf_of_rings S is_open \<FF> \<rho> b add_str mult_str zero_str one_str" shows "sheaf_of_rings_axioms S is_open \<FF> \<rho> zero_str"
nhds_metric: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b" assumes "class.metric_space dist uniformity open" shows "topological_space.nhds open x = (INF e\<in>{0<..}. principal {y. dist y x < e})"
compactin_standard_simplex: fixes p :: "nat" shows "compactin (powertop_real UNIV) (standard_simplex p)"
quasi_isometry_onI: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and lambda :: "real"   and C :: "real" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> dist (f x) (f y) \<le> lambda * dist x y + C"   and "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> 1 / lambda * dist x y - C \<le> dist (f x) (f y)"   and "1 \<le> lambda"   and "0 \<le> C" shows "lambda C -quasi_isometry_on X f"
alpha_qFresh_qSwap_compose: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and x :: "'var" assumes "qGood X"   and "qFresh zs y X"   and "qFresh zs z X" shows "X #[[y \<and> x]]_zs #[[z \<and> y]]_zs #= X #[[z \<and> x]]_zs"
inj_image_ord_of_nat: shows "inj ((`) ord_of_nat)"
nn_integral_cmult_indicator: fixes A :: "'c set"   and M :: "'c measure"   and c :: "ennreal" assumes "A \<in> sets M" shows "\<integral>\<^sup>+x\<in>A. c\<partial>M = c * emeasure M A"
mismatch: shows "s ! (i - 1) \<noteq> s ! (j - 1)"
iso_free_Abelian_group_sum: fixes S :: "'b \<Rightarrow> 'c set"   and I :: "'b set" assumes "pairwise (\<lambda>i j. disjnt (S i) (S j)) I" shows "(\<lambda>f. sum' f I) \<in> Group.iso (sum_group I (\<lambda>i. free_Abelian_group (S i))) (free_Abelian_group (\<Union> (S ` I)))"
update_until_def: fixes I :: "\<I>"   and pos :: "bool"   and rel1 :: "'b option list set"   and rel2 :: "'b option list set"   and nt :: "nat"   and aux :: "(nat \<times> 'b option list set \<times> 'b option list set) list" shows "update_until I pos rel1 rel2 nt aux = map (\<lambda>x. case x of (t, a1, a2) \<Rightarrow> (t, if pos then join a1 True rel1 else a1 \<union> rel1, if mem (nt - t) I then a2 \<union> join rel2 pos a1 else a2)) aux @ [(nt, rel1, if left I = 0 then rel2 else empty_table)]"
transition_relations_equivalent: fixes s :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option"   and r :: "'r"   and s' :: "'r \<Rightarrow> (('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('l \<Rightarrow> ('r, 'l, 'v) val option) \<times> ('r, 'l, 'v) expr) option" assumes "reachable s" shows "revision_step r s s' = revision_step_relaxed r s s'"
ltl_FG_logical_characterization: fixes w :: "nat \<Rightarrow> 'b set"   and \<phi> :: "'b ltl" shows "w \<Turnstile> F G \<phi> = (\<exists>\<G>\<subseteq>\<^bold>G (F G \<phi>). G \<phi> \<in> \<G> \<and> closed \<G> w)"
containsNotTerminal': fixes gamma :: "(signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and f :: "nat \<Rightarrow> (signs \<times> predicate \<times> vbl list) list \<times> (nat \<times> formula) list"   and iA :: "nat \<times> formula"   and n :: "nat" assumes "branch subs gamma f"   and "\<forall>n. \<not> proofTree (tree subs (f n))"   and "contains f iA n" shows "\<not> terminal subs (f n)"
predicate2D_conj: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R :: "bool"   and x :: "'c"   and y :: "'d" assumes "P \<le> Q \<and> R" shows "R \<and> (P x y \<longrightarrow> Q x y)"
rank_Some_time: fixes x :: "nat"   and n :: "nat"   and j :: "nat" assumes "\<UU>.rank x n = Some j" shows "x \<le> n"
ordinal_exp_left_cancel: fixes w :: "ordinal"   and x :: "ordinal"   and y :: "ordinal" assumes "oSuc 0 < w" shows "(w ** x = w ** y) = (x = y)"
distinct_15: fixes x2 :: "'c" shows "Notify x2 \<noteq> Notified"
mult_right_mono: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "a * c \<le> b * c"
weakTransitiveCoinduct2: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and X :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "(\<Psi>, P, Q) \<in> X"   and "eqvt X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<lessapprox><X> Q"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> \<Psi> \<rhd> P \<leadsto><{(\<Psi>, P, Q) |\<Psi> P Q. \<exists>P' Q'. \<Psi> \<rhd> P \<approx> P' \<and> (\<Psi>, P', Q') \<in> X \<and> \<Psi> \<rhd> Q' \<sim> Q}> Q"   and "\<And>\<Psi> P Q \<Psi>'. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi> \<otimes> \<Psi>', P, Q) \<in> X"   and "\<And>\<Psi> P Q. (\<Psi>, P, Q) \<in> X \<Longrightarrow> (\<Psi>, Q, P) \<in> X" shows "\<Psi> \<rhd> P \<approx> Q"
simps_11: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x21 :: "'e"   and x22 :: "'e list"   and y21 :: "'f"   and y22 :: "'f list" shows "list_all2 R (x21 # x22) (y21 # y22) = (R x21 y21 \<and> list_all2 R x22 y22)"
map_def: fixes f :: "('a, 'a) CC.arr" shows "CC.map f = (if CC.arr f then f else CC.null)"
that: fixes f :: "'a \<Rightarrow> 'b" assumes "linear f"   and "f ` S = T"   and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) = norm x" shows "thesis"
Assign': fixes a :: "aexp"   and x :: "char list"   and l :: "nat" assumes "sec a \<le> sec x"   and "l \<le> sec x" shows "l \<turnstile>' x ::= a"
differentiable_sum_fun: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and a :: "'d" assumes "\<And>i. i \<in> I \<Longrightarrow> f i differentiable at a" shows "sum f I differentiable at a"
induct_gen_abs: fixes lst :: "'a list"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and h :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "\<And>a r. a \<in> set lst \<Longrightarrow> P (h a * r) (g a * r)"   and "\<And>x y z. \<lbrakk>P x y; P y z\<rbrakk> \<Longrightarrow> P x z"   and "P (prod_list (map g lst)) (prod_list (map g lst))" shows "P (prod_list (map h lst)) (prod_list (map g lst))"
t': shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t'"
mkArr_restrict_eq: fixes S :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and img :: "'b \<Rightarrow> 'b"   and \<AA> :: "('c \<times> 'c) set"   and A :: "'b set"   and B :: "'b set"   and F :: "'b \<Rightarrow> 'b" assumes "set_category_given_img S img \<AA>"   and "partial_magma.arr S (set_category_given_img.mkArr S img \<AA> A B F)" shows "set_category_given_img.mkArr S img \<AA> A B (restrict F A) = set_category_given_img.mkArr S img \<AA> A B F"
paste_1: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "\<nu> \<star> \<mu> = (cod \<nu> \<star> \<mu>) \<cdot> (\<nu> \<star> local.dom \<mu>)"
cpx_mat_cnj_cnj: fixes M :: "complex mat" shows "M\<^sup>\<star>\<^sup>\<star> = M"
norm_triangle_mono: fixes a :: "'a"   and r :: "real"   and b :: "'a"   and s :: "real" assumes "norm a \<le> r"   and "norm b \<le> s" shows "norm (a + b) \<le> r + s"
simps_1: fixes n :: "nat" shows "ack 0 n = Suc n"
IH: shows "CF_pure f \<leftrightarrow> unlift f"
A: shows "P x @ P' x' @ P'' \<Turnstile> cfs\<box>"
vector_7_5: fixes x1 :: "'b"   and x2 :: "'b"   and x3 :: "'b"   and x4 :: "'b"   and x5 :: "'b"   and x6 :: "'b"   and x7 :: "'b" shows "vector [x1, x2, x3, x4, x5, x6, x7] $ 5 = x5"
finprod_closed: fixes f :: "'d \<Rightarrow> 'a"   and A :: "'d set" assumes "f \<in> A \<rightarrow> carrier R" shows "finsum R f A \<in> carrier R"
uw\<theta>\<omega>: shows "ide w \<and> \<guillemotleft>\<theta> : f \<star> w \<Rightarrow> u\<guillemotright> \<and> \<guillemotleft>\<omega> : local.dom \<omega> \<Rightarrow> r \<star> u\<guillemotright>"
dom_inter: fixes p :: "'b \<Rightarrow> 'c option"   and q :: "'b \<Rightarrow> 'd option"   and x :: "'b"   and y :: "'c" assumes "dom p \<inter> dom q = {}"   and "p x = \<lfloor>y\<rfloor>" shows "q x = \<bottom>"
orthogonal_mult: fixes a :: "'a"   and b :: "'a"   and x :: "real"   and y :: "real" assumes "orthogonal a b" shows "orthogonal (x *\<^sub>R a) (y *\<^sub>R b)"
simps_3: fixes thy :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and vs :: "(variable \<times> typ) list"   and ns :: "variable set"   and Hs :: "Core.term list"   and T :: "typ"   and p :: "proofterm" shows "replay' thy vs ns Hs (Abst T p) = (if typ_ok thy T then let (s', ns') = variant_variable (Free STR ''default'') ns in map_option (mk_all s' T) (replay' thy ((s', T) # vs) ns' Hs p) else None)"
cases: fixes eff :: "'a \<Rightarrow> 'b \<Rightarrow> 'b fset \<Rightarrow> bool"   and rules :: "'a stream"   and a :: "('b \<times> 'a) tree" assumes "RuleSystem_Defs.wf eff rules a" obtains t :: "('b \<times> 'a) tree" where "a = t" and "snd (tree.root t) \<in> i.R rules" and "eff (snd (tree.root t)) (fst (tree.root t)) ((fst \<circ> tree.root) |`| cont t)" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> RuleSystem_Defs.wf eff rules t'"
inject_1: fixes x1 :: "ct list"   and y1 :: "ct list" shows "(TopType x1 = TopType y1) = (x1 = y1)"
compat_def: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" shows "compat r r' f \<equiv> \<forall>a b. (a, b) \<in> r \<longrightarrow> (f a, f b) \<in> r'"
xb: shows "x * b mod q = (s2 + (q - r)) mod q"
mono_afilter: fixes r :: "'a"   and r' :: "'a"   and S :: "'a Abs_State.st option"   and S' :: "'a Abs_State.st option"   and e :: "aexp" assumes "r \<sqsubseteq> r'"   and "S \<sqsubseteq> S'" shows "afilter e r S \<sqsubseteq> afilter e r' S'"
assms_1: fixes f :: "three \<Rightarrow> three \<Rightarrow> three"   and xs :: "three list" assumes "associative f"   and "xs \<noteq> []" shows "candidate f xs = scanl1 f xs"
is_UP_monom_def: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "UP_ring.is_UP_monom R = (\<lambda>f. f \<in> carrier (UP R) \<and> f = monom (UP R) (f (deg R f)) (deg R f))"
PHI'_4: fixes x0 :: "'a"   and h'' :: "real"   and phi :: "real \<Rightarrow> 'a" assumes "x0 \<in> X0"   and "0 \<le> h''"   and "h'' \<le> h"   and "phi \<in> Abstract_Rigorous_Numerics.cfuncset 0 h'' PHI'" shows "x0 + integral {0..h''} (\<lambda>t. ode (phi t)) \<in> PHI''"
callee_invariant_on_axioms: shows "callee_invariant_on callee2 I2 \<I>"
distinct_11: fixes x31 :: "'b tree234"   and x32 :: "'b"   and x33 :: "'b tree234"   and x34 :: "'b"   and x35 :: "'b tree234"   and x41 :: "'b tree234"   and x42 :: "'b"   and x43 :: "'b tree234"   and x44 :: "'b"   and x45 :: "'b tree234"   and x46 :: "'b"   and x47 :: "'b tree234" shows "\<langle>x31, x32, x33, x34, x35\<rangle> \<noteq> \<langle>x41, x42, x43, x44, x45, x46, x47\<rangle>"
cbsub_3: shows "cbox (- c) c \<subseteq> cbox (- (b + One)) (b + One)"
simps_2: fixes x :: "'b"   and xs :: "'b list" shows "remove_nth 0 (x # xs) = xs"
next_suc_suffix: fixes n :: "nat"   and s :: "nat \<Rightarrow> 'b" shows "next (Suc n) s = nextsuffix (next n s)"
mtf20: fixes x :: "'c"   and xs :: "'c list" shows "mtf2 0 x xs = xs"
graph_homomorphism_Id: fixes a :: "('a, 'b) labeled_graph" shows "graph_homomorphism (restrict a) (restrict a) (Id_on (vertices a))"
mm_equiv_sym: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and c\<^sub>1 :: "'a"   and mds\<^sub>1 :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and c\<^sub>2 :: "'a"   and mds\<^sub>2 :: "Mode \<Rightarrow> 'b set"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and dma :: "'b \<Rightarrow> Sec" assumes "sifum_security stop eval some_val some_val'"   and "(((c\<^sub>1, mds\<^sub>1), mem\<^sub>1), (c\<^sub>2, mds\<^sub>2), mem\<^sub>2) \<in> sifum_security.mm_equiv dma eval" shows "(((c\<^sub>2, mds\<^sub>2), mem\<^sub>2), (c\<^sub>1, mds\<^sub>1), mem\<^sub>1) \<in> sifum_security.mm_equiv dma eval"
machine_type_preservation: fixes t :: "lam"   and t' :: "lam"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes "<t,[]> \<mapsto>* <t',[]>"   and "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> t' : T"
subS: shows "{z. norm z \<le> m \<and> (\<exists>t\<in>T -` P. norm (z - t) \<le> e)} \<subseteq> S"
poly_deg_prod_le: fixes f :: "'e \<Rightarrow> ('c \<Rightarrow>\<^sub>0 'b) \<Rightarrow>\<^sub>0 'd"   and A :: "'e set" shows "poly_deg (prod f A) \<le> (\<Sum>a\<in>A. poly_deg (f a))"
cases: fixes G :: "('c, 'd) Graph_scheme"   and a :: "'c llist" assumes "Digraph G"   and "Digraph.maximal_path G a" obtains "a = LNil"   | v :: "'c" where "a = LCons v__ LNil" and "\<not> (\<exists>w\<in>V\<^bsub>G\<^esub>. v__ \<rightarrow>\<^bsub>G\<^esub> w)"   | Ps :: "'c llist" and v :: "'c" where "a = LCons v Ps" and "\<not> lnull Ps" and "Digraph.maximal_path G Ps"
open_Int: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set"   and T :: "'a set" assumes "class.topological_space open"   and "open S"   and "open T" shows "open (S \<inter> T)"
assms_3: shows "a \<rightarrow>\<^sup>* b2"
polypoly'_def: fixes bs :: "'b list"   and p :: "poly" shows "polypoly' bs p = map (Ipoly bs \<circ> decrpoly) (coefficients p)"
inf_mono_left: fixes a :: "'a::type"   and b :: "'a::type"   and c :: "'a::type" assumes "(a::'a::type) \<sqsubseteq> (b::'a::type)" shows "(a::'a::type) \<sqinter> (c::'a::type) \<sqsubseteq> (b::'a::type) \<sqinter> c"
simps_2: fixes uv :: "'c \<Rightarrow> 'b dres" shows "dSUCCEED \<bind> uv = dSUCCEED"
seqI': fixes f :: "'j"   and a :: "'j"   and b :: "'j"   and g :: "'j"   and c :: "'j" assumes "J.in_hom f a b"   and "J.in_hom g b c" shows "J.seq g f"
reals_Archimedean2: fixes x :: "'b" shows "\<exists>n. x < of_nat n"
SUP_least: 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 :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and u :: "'b" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq (f i) u" shows "less_eq (Sup (f ` A)) u"
norm_diff_triangle_less: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and e1 :: "real"   and z :: "'a"   and e2 :: "real" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open"   and "norm (minus x y) < e1"   and "norm (minus y z) < e2" shows "norm (minus x z) < e1 + e2"
expand_map_def: fixes f :: "'c \<Rightarrow> 'd set option" shows "expand_map f \<equiv> expand (case_option {None} ((`) Some) \<circ> f)"
conversion_sym: fixes A :: "('b \<times> 'b) set" shows "sym (A\<^sup>\<leftrightarrow>\<^sup>*)"
frame_rem1: fixes P :: "assn" shows "P \<Longrightarrow>\<^sub>t P"
filter_less': fixes n1 :: "int"   and n2 :: "int"   and a1 :: "'av"   and a2 :: "'av"   and b1 :: "'av"   and b2 :: "'av" assumes "filter_less' (n1 < n2) a1 a2 = (b1, b2)"   and "n1 \<in> \<gamma> a1"   and "n2 \<in> \<gamma> a2" shows "n1 \<in> \<gamma> b1 \<and> n2 \<in> \<gamma> b2"
fresh_delete: fixes a :: "atom"   and \<Gamma> :: "('d \<times> 'e) list"   and v :: "'d" assumes "a \<sharp> \<Gamma>" shows "a \<sharp> delete v \<Gamma>"
fds_eq_iff: fixes f :: "'b fds"   and g :: "'b fds" shows "(f = g) = (\<forall>n>0. f $ n = g $ n)"
simps_1: shows "ket_zero_k 0 = ket_zero"
N_mu: fixes i :: "nat"   and j :: "nat" assumes "i < length fs"   and "j \<le> i" shows "(\<mu> i j)\<^sup>2 \<le> N ^ Suc j"
nat_pow_mult: fixes x :: "'c"   and n :: "nat"   and m :: "nat" assumes "x \<in> carrier S" shows "x [^]\<^bsub>S\<^esub> n \<otimes>\<^bsub>S\<^esub> x [^]\<^bsub>S\<^esub> m = x [^]\<^bsub>S\<^esub> (n + m)"
rev_upt_code: fixes i :: "nat"   and j :: "nat" shows "[i>..j] = rev_upt_aux i j []"
that: fixes s :: "'a" assumes "s \<le> t \<and> 0 < zcount (zmset_of (mset_set (set_antichain (frontier M)))) s" shows "thesis"
bmod_fdia_comp: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'f \<Rightarrow> 'd"   and g :: "'g \<Rightarrow> 'd"   and x :: "'e" shows "\<otimes> R f g x =  |{(x, y, z) |x y z. R x y z}\<rangle> (\<lambda>(x, y). f x \<cdot> g y) x"
append_prefixD: fixes xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" assumes "xs \<cdot> ys \<le>p zs" shows "xs \<le>p zs"
from_list_correct_aux_2: fixes l :: "('b \<times> 'b \<times> nat) list"   and g :: "'a" assumes "valid_graph_rep l"   and "edges_undef l (\<alpha> g)"   and "invar g" shows "\<forall>u v d. (\<alpha> (fold (\<lambda>(u, v, d). add_edge (u, v) d) l g) (u, v) = enat d) = (\<alpha> g (u, v) = enat d \<or> (u, v, d) \<in> set l)"
union_def: fixes i :: "nat_int"   and j :: "nat_int" shows "i \<squnion> j = Abs_nat_int (Rep_nat_int i \<union> Rep_nat_int j)"
hom_mult: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme"   and h :: "'a \<Rightarrow> 'c"   and x :: "'a"   and y :: "'a" assumes "group_hom G H h"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
reduced_word_for_lists: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and A :: "'b set"   and a :: "'b"   and as :: "'b list" assumes "class.monoid_add plus zero"   and "monoid_add.reduced_word_for plus zero A a as" shows "as \<in> lists A"
freshChain: fixes P :: "pi"   and P' :: "pi"   and x :: "name" assumes "P \<Longrightarrow>\<^sub>\<tau> P'"   and "x \<sharp> P" shows "x \<sharp> P'"